Handbook of Multiphase Flow Science and Technology 9789814585866

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Basic Theory and Conceptual Framework of Multiphase Flows Guan Heng Yeoh and Jiyuan Tu

Abstract

The fundamentals of computational multiphase fluid dynamics are presented using discrete and continuum frameworks. Depending on the number, type, and size of phases and their interaction between phases within the flow system, a multiscale consideration of the multiphase flow physics allows the adoption of a number of possible approaches. The Lagrangian formulation can be utilized to track the motion of discrete constituents of identifiable portion of particular phases occupying the flow system. This represents the most comprehensive investigation that can be performed to analyze the multiphase flow physics. Because of the complexity of the microscopic motions and thermal characteristics of each discrete constituent which can be prohibitive at the (macro) device scale, the Eulerian formulation which characterizes the flow of discrete constituents as a fluid can be adopted for practical analysis of the flow system. This results in the development of a multifluid approach which solves for the conservation equations of continuous and dispersed phases. In order to better resolve the microphysics of the discrete constituents at the mesoscale, population balance allows the synthesization of the behavior and dynamic evolution of the population of the discrete constituents occupying the flow system. Such an approach allows the consideration of the spatial and temporal evolution of the geometrical structures as a result of formation and destruction of agglomerates or clusters through G.H. Yeoh (*) School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW, Australia Australian Nuclear Science and Technology Organisation (ANSTO), Kirrawee DC, NSW, Australia e-mail: [email protected] J. Tu School of Aerospace, Mechanical and Manufacturing Engineering, Platform Technologies Research Institute (PTRI), RMIT University, Melbourne, VIC, Australia e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_1-1

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interactions between the discrete constituents and, more importantly, the collisions with turbulent eddies. Keywords

Multiphase flows • Lagrangian framework • Eulerian framework • Population balance

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Solid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Solid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Liquid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Phase Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Basic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-Scale Consideration of Multiphase Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models for Particulate-Particulate Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Particulate-Particulate Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Fluid-Particulate Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eulerian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpenetrating Media Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivation of Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Population Balance Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction Multiphase flows in the context of fluid mechanics can be perceived as a flow system that consists of two or more distinct phases flowing in a fluid mixture where the level of separation between the phases is at a scale well above the molecular level. Principally, multiphase flows can be categorized by the number, type, and size of phases and their respective interaction between each phase in the fluid mixture. A multiphase flow system can thus be classified according to different types of flows depending on the combination of different phases within the fluid mixture. The physical understanding of multiphase flows especially when dealing with more than one phase in the fluid mixture offers problems of complexity that are immeasurably far greater than in single-phase flows. These phases generally do not uniformly mix, and the prevalence of small-scale interactions occurring between the phases can have profound effects on the macroscopic properties of the flow system. This clearly reflects the ubiquitous challenges that persist when modeling the inherently complex nature of the multiphase flow physics.

Basic Theory and Conceptual Framework of Multiphase Flows

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Gas-Solid Flows Gas-solid or gas-particle flows concern the motion of suspended solid particles in the gas phase. When the particle number density is relatively small, the gas flow influences the flow of solid particles. Such flows are governed predominantly by the surface and body forces acting on the solid particles. These types of flows are generally known as dilute gas-particle flows. When the particle number density is very small, these solid particles are mere tracers in the gas phase. These types of flows are commonly known as very dilute gas-particle flows. When the particle number density is sufficiently large, particle-particle interactions now govern the motion of solid particles. Collisions that exist between solid particles will significantly alter the migration of these particles within the gas phase. These types of flow are referred to as dense gas-particle or granular flows. For gas-particle flows in conduits, the motion of solid particles following impact on the boundary walls is affected by the surface characteristics and material properties, which is different when compared to the free flight of solid particles in the gas phase. In gas-particle flows, the solid particles constitute the dispersed or discrete phase and the gas is the continuous phase.

Liquid-Solid Flows Liquid-solid flows comprise the transport of solid particles in the liquid phase. Such flows are driven largely by the presence of pressure gradients since the density ratio between the two phases is generally low and the drag between the phases constitutes the dominant effect in such flows. These types of flows are generally known as liquid-particle flows or slurry flows. In liquid-particle flows, the solid particles constitute the dispersed phase and the liquid represents the continuous phase. The major concern in such flows is the characterization of the sedimentation behavior in the liquid-particle mixture which is governed by the range of size of solid particles traveling in the liquid phase.

Gas-Liquid Flows Gas-liquid flows can exist in a number of different forms – the motion of gas bubbles in the liquid phase or the motion of liquid droplets in the gas phase. For the former, the liquid is taken as the continuous phase and the gas bubbles are considered as discrete constituents of the gas phase or the dispersed phase. For the latter, the gas is regarded as the continuous phase and the droplets are taken as the finite fluid particles of the liquid phase or the dispersed phase. Since gas bubbles or liquid droplets are allowed to deform within the continuous phase, several different geometrical shapes are possible which include spherical, elliptical, distorted, toroidal, cap, and other complex configurations. Gas-liquid flows undergo a spectrum of flow transition regimes. Change of interfacial structures between the phases are due

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to bubble-bubble interactions via coalescence and breakup of bubbles and any phase change process from gas-to-liquid or liquid-to-gas. For the special case of separated flows such as liquid film in gas phase or gas film in liquid phase and liquid core and gas film or gas core and liquid film in conduits, such flows possess well-defined interfaces, and they belong to the specific consideration of immiscible flows. Each phase is treated as a continuous fluid co-flowing simultaneously with each other.

Three-Phase Flows Three-phase gas-liquid-solid flows are encountered in a number of engineering applications of technical relevance. Principally, this particular class of multiphase flows considers the solid particles and gas bubbles as being the discrete constituents of the dispersed phase co-flowing with the continuous liquid phase. The coexistence of the three phases considerably complicates the computational modeling due to the requirement of understanding the phenomena associated with particle-particle, bubble-bubble, particle-bubble, particle-fluid, and bubble-fluid interactions modifying the multiphase flow physics.

Definitions of Basic Terms Some basic definitions that are fundamental to the description of multiphase flows are introduced herein. For convenience, the notation that will be adopted corresponds to the Cartesian tensor format. The lowercase subscripts (ijk) are employed in the conventional manner to denote vector or tensor components. The single uppercase subscript (n) refers to the property of a specific phase. In general, the generic subscripts n = c (continuous liquid), n = d (dispersed or discrete phase), n = l (liquid), n = g (gas), and n = s (solid) are employed for clarity in depicting the different classes of multiphase flows. Specific properties frequently encountered are as follows. The volume fraction of the continuous phase can be defined as αc ¼ lim

δV!δV

o

δV c δV

(1)

where δVc is the volume of the continuous phase in the volume of δV. The volume δVo represents the limiting volume whereby a stationary averaging is performed. Equivalently, the volume fraction of the dispersed or discrete phase can be written as αd ¼ lim

δV!δV

o

δV d δV

(2)

where δVd is the volume of the dispersed or discrete phase in the volume of δV. This volume fraction is also referred to as the void fraction which describes the portion of

Basic Theory and Conceptual Framework of Multiphase Flows

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the channel or pipe occupied by the dispersed gas phase at any instant in space and time. In the chemical engineering terminology, it is known as holdup. For the case of two-phase flow, it follows that αd = 1  αc. Hence, the sum of the volume fractions P of different constituents in a multiphase mixture must be equal to unity, i.e., n αn ¼ 1. The mixture density can be evaluated in accordance with ρ¼

X

α ρ n n n

(3)

The bulk density of the dispersed phase is related to the material density by ρd ¼ αd ρd

(4)

of which the material density, in terms of a limit, is defined as ρd ¼ limδV!δV o

δMd δV

(5)

with δMd being the mass of the dispersed phase. The bulk and material densities of the continuous phase are analogously defined in accordance with the definition of the bulk and material densities of the dispersed phase. Conversely, the specific enthalpy, h, and specific entropy, s, in terms of per unit mass are weighted similar to the mixture density to the following: ρh ¼ ρs ¼

X

α ρh n n n n

(6)

α ρs n n n n

(7)

X

It should be noted that properties such as mixture viscosity or thermal conductivity may not be obtained through such simple weighted averaging. Other means of evaluating these properties are required. The true velocities (or actual local velocities) of the different phases are the velocities the phases actually travel, which is the local instantaneous velocities of the fluids. Defining uc and ud to be the local instantaneous or phase velocities of continuous and dispersed phases, the superficial velocities and the phase velocities are related by the volume fraction according to U c ¼ αc uc

(8)

Ud ¼ αd ud

(9)

For the case of two-phase flow, the total superficial velocity is U = Uc + Ud. Hence, the totalPsuperficial velocity for a multiphase mixture can be analogously written as U ¼ n αn un .

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Of more specific definitions, the quality of a mixture comprising liquid and vapor is defined by X¼

ρd ρ

(10)

while the dispersed phase mass concentration, which is the ratio of the mass of the dispersed phase to that of the continuous phase in a mixture, is given as C¼

ρd ρc

(11)

Multi-Scale Consideration of Multiphase Flows Within the conceptual framework of computational multiphase fluid dynamics, the Eulerian or Lagrangian formulation of multiphase flows requires the multiscale consideration of the multiphase flow physics. The different physical characteristics at different length scales are illustrated in Fig. 1. For the microscale physics, it is paramount that interaction of the gas bubbles, liquid drops, and solid particles with the continuum phase is properly understood through tracking the motion of the individual discrete constituents in space and time. For the mesoscale physics, significant interaction between the discrete constituents may result in local structural changes due to agglomeration/coalescence and breakage/breakup processes of gas bubbles, liquid drops, and solid particles. For the macroscale physics, the hydrodynamic behavior of the background fluid on the clusters of gas bubbles, liquid drops, and solid particles may yield large scale flow structures influencing the different individual phases co-flowing with the continuous phase within the flow system. Computational approaches can be utilized to reveal details of particular multiphase flow physics that otherwise could not be visualized through experiments or clarify specific accentuating mechanisms that are consistently being manifested. Techniques that are being adopted based on the utilization of advanced numerical methods and models usually contain very detailed information, producing an accurate realization of the fluid flow.

Lagrangian Formulation The concept of Lagrangian tracking entails following the motion of individual identities of the identifiable portion of a particular phase occupying the flow system. Such consideration therefore includes molecular dynamics, Brownian dynamics, and discrete element method, which is represented by an illustration of a plot depicting the characteristic time scale versus length scale in Fig. 2. In general, these identities being considered represent a wide range of discrete elements including atoms, molecules, nuclei, cells, aerosol or colloidal particles, and granules.

Basic Theory and Conceptual Framework of Multiphase Flows

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Micro-Scale

Meso-Scale

Macro-Scale

Interactions of Discrete Elements with Continuum Fluid

Formation of Clusters of Bubbles, Drops and Particles

Hydrodynamic Behaviour of Phases at Device Scale

Motion of Discrete Elements

Local Structural Changes

Large Scale Flow Structures

Length Scale Fig. 1 Multiscale consideration of multiphase flows (Yeoh et al. 2014)

Fig. 2 Illustration of time and length scales for Lagrangian simulations: molecular dynamics, Brownian dynamics, and discrete element method (Yeoh et al. 2014)

Molecular dynamics, Brownian dynamics, and discrete element method share the common characteristics whereby the discrete elements are allowed to interact for a period of time under prescribed interaction laws, and the motion of each individual element is resolved by solving the linear momentum and angular momentum equations, subject to forces and torques arising both from particle interaction with each other and those imposed on the particles by the surrounding fluid (Li et al. 2011). Consideration of which Lagrangian tracking approach should be adopted

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stems primarily through the formulation of appropriate particle interaction laws as well as by the imposition of random forcing to mimic collisions or interaction with molecules of the surrounding fluid. By carefully identifying these particle interaction laws, the physical behavior of the discrete element at a certain range of time and length scales can be efficiently captured. Considering the discrete elements which could be solid particles, liquid droplets, or gas bubbles, the instantaneous velocity Ud and rotation rate Ωd for the discrete particulate (particle, droplet, or bubble) through the time-driven discrete element method can be obtained through solution of the linear and angular momentum equations (Newton’s second law): DUd X ¼ Fd Dt DΩd X Id ¼ Md Dt md

(12) (13)

where md is the mass of particulate, and Id is the moment of inertia of particulate. On the right hand side, the time derivatives are essentially the material derivatives of the particulate velocity and particulate rotation rate. On the left hand side, the source terms represent the sum or cumulative of forces and moments acting on the particulate. For multiphase flow associated with heat exchange and phase change between particulate and surrounding fluid, these heat and mass transfer processes are concurrently tracked along the discrete particulate trajectories and solved by the particulate conservation equations of mass and energy: Dmd ¼ Smd Dt

(14)

DT d ¼ ST d Dt

(15)

where Td is the temperature of the particulate. The source term Smd represents the interphase mass transfer between the particulate and surrounding fluid, while the source term ST d is governed primarily by the interphase convection heat transfer, latent heat transfer associated with mass transfer, net radiative power absorbed by the particulate, and particulate-particulate interaction due to conduction heat transfer. Note that the product of the particulate mass, specific heat of constant pressure, and material derivative of the particulate temperature denote the sensible heating term of the particulate energy equation.

Basic Theory and Conceptual Framework of Multiphase Flows

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Models for Particulate-Particulate Interaction Through specific consideration of using discrete element method for large particulate assemblies, the hard-sphere or soft-sphere model can be effectively applied. Soft contact forces are subsequently derived from a point on the bodies for the hardsphere model or the overlap of bodies for soft-sphere model.

Hard-Sphere Model Main assumptions that are concerned with the particulate shape, deformation history during collision, and nature of collisions are: • • • • • •

Particulates are generally taken to be spherical and quasirigid Shape of particulates is retained after impact Dynamics of idealized binary collision Collisions between particulates are instantaneous Contact of particulates during collision occurs at a point Interaction forces are taken to be impulsive and all other finite forces are negligible during collisions

These assumptions are believed to be sufficiently realistic for collisions of relatively coarse particulates (>100 μm). One characteristic feature of this model is the ability to process a sequence of collisions one at a time. Also, Lagrangian tracking of particulates can be readily performed with realistic values of the restitution and friction coefficients. During the impact of two particles such as described in Fig. 3, the motion is governed by the linear and angular impulse momentum laws for a binary collision of two spheres:   md,k Ud,k  U0d,k ¼ J

(16)

  md,l Ud,l  U0d,l ¼ J

(17)

 I d,k  Ωd,k  Ω0d,k ¼ J  n rk  I d,l  Ωd,l  Ω0d,l ¼ J  n r

(18) (19)

l

where superscript 0 denotes conditions just before collision, md,k and md,l are the mass, rk and rl are the radii, Ωd,k and Ωd,l are the angular velocities, Id,k and Id,l are the moment of inertia of particulate k and particulate l. Velocities prior to collisions are the velocities determined at the last time step just before collision. Note that the corresponding time difference should not be larger than 104 s. In Fig. 3, the normal and tangential unit vectors that define the collision coordinate system are:

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G.H. Yeoh and J. Tu

y

rl

rk

l

k

Gd,kl x

Ud,l Ud,k

Wd,k

n

Wd,l

Rk k

Rl l t

J

z Fig. 3 Contact between two particles for hard-sphere model

xk  xl jxk  xl j   Ud,kl  G0d,kl ∙n n    t ¼   Ud,kl  G0d,kl ∙n n n¼

(20)

(21)

By definition, I ¼ mr 2gyration where rgyration is the radius of gyration of the particulate and J is the impulse vector. Adopting (J  n)  n = J  (J ∙ n)n, the relative velocity at the contact point between two particulates with velocities Ud,k and Ud,l is derived as   Ud,kl  U0d,kl ¼ Bd,k J  Bd,k  Bd,l ðJ∙nÞn

(22)

  Ud,kl ¼ Gd,kl  r k Ωd,k  r l Ωd,l  n

(23)

where

The relative velocity of particulate centroids Gd , kl is given by Gd,kl ¼ Ud,k  Ud,l

(24)

Some parameters are established to relate the velocities before and after collisions. The first collision parameter is the coefficient of normal restitution, en:   Ud,kl ∙n ¼ en U0d,kl ∙n The normal component of the impulse vector can thus be written as:

(25)

Basic Theory and Conceptual Framework of Multiphase Flows

 J n ¼ ð1þen Þ

U0d,kl ∙n

11

 (26)

B1

where B1 = (md,k + md,l)/md,kmd,l. The second and third collision parameters consist of the coefficient of tangential restitution, et, and the coefficient of friction, μf. These two parameters concern the two kinds of collisions – particulate sticking and sliding in the tangential impact process. For the case where the tangential component of the impact velocity is sufficiently high or the friction coefficient is small by comparison, 

0 ð1þet Þ Ud,kl ∙t μf < B2 Jn

 (27)

where B2 ¼

1 1 r2 r2 þ þ k þ l md , k md , l I d , k I d , l

Here, sliding occurs throughout the whole duration of the contact. Applying Coulomb’s law, the tangential component of the impulse is then given by J t, sliding ¼ μf J n

(28)

On the other hand, if the friction coefficient is sufficiently high, 

0 ð1þet Þ Ud, kl ∙t μf  B2 Jn

 (29)

in which sticking collisions occur after an initial sliding phase – the relative tangential velocity between two colliding particles becomes zero – the tangential impulse for this case is:  J t,sliding ¼ ð1þet Þ

U0d,kl ∙t



B2

(30)

where the coefficient of tangential restitution, et, is defined as:   Ud,kl ∙t ¼ et U0d,kl ∙t

(31)

Once all the impulse vectors are known, the postcollision velocities can be appropriately determined. More details on a two-step approach in determining the hard-sphere particle dynamics can be found in Hoomans et al. (1996).

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G.H. Yeoh and J. Tu

y

x Ud,j

rj

ri

Ud,i

j

i

Wd,i

x

n

Wd,j Rj

Ri

j i

t

z Fig. 4 Contact between two particles for soft-sphere model

Soft-Sphere Model This model requires that particulate collisions are of finite durations. The duration of contact is related to the nonfinite particulate stiffness which is specified as the particulate property. Thus, the force at contact is continuously changing as the particulates begin to deform, which can be represented by the assumption of a small overlap distance. Forces at all contacts are determined at one instant and Newton’s equations of motion are then solved to obtain new particulate locations and velocities. For dense flows, this model is considered to be much more efficient than the hard-sphere model. More importantly, it can be applied to any configurations, including static and dynamic situations. As illustrated in Fig. 4, at any instant during the collision of two particulates, the resultant force acting between the two particulates, assuming spheres for the purpose of illustration, can be expressed by their respective normal and tangential components. Note that the formation of normal and tangential stresses during impact is described as a decoupled problem. The normal and tangential unit vectors can be expressed as n ¼ 

xi  xj  xi  xj 

(32)

Utd  Utd 

(33)

t ¼ 

From above, Utd is the slip velocity of the point of contact or tangential velocity which can be obtained from   Utd ¼ Urd  Urd ∙n n with the relative velocity of each particulate given by

(34)

Basic Theory and Conceptual Framework of Multiphase Flows

  Urd ¼ Ud,i  Ud,j  r i Ωd,i  r j Ωd,j  n

13

(35)

The particulate deformation during contact can be characterized by the presence of a normal overlap or displacement ξ of the two particulates, viz.,     ξ ¼  Ri þ Rj þ xi  xj ∙n

(36)

while the normal displacement rate ξ_ can be evaluated based upon the relative translational velocities projected in the direction of the normal unit vector:   ξ_ ¼  Ud, i  Ud, j ∙n

(37)

The total collision force and torque fields on particle i can be written as FCd ¼ Fnd þ Ftd

(38)

t MC d ¼ ri n  F

(39)

where Fnd and Ftd are the normal and tangential contact forces.

Classification of Particulate-Particulate Forces For the soft sphere model, the fundamentals of particulate-particulate interactions can be represented using a combination of particulate and continuum mechanics. Figure 5 illustrates the different levels of classification for the dynamics of random packing in particulate beds, clusters, or agglomerates: continuum mechanics, micromechanics, and molecular dynamics. Using this model with stiff particulates and soft contacts, the influence of elastic-plastic repulsion in a representative particulate contact can be demonstrated via contact force equilibrium. Also, a sphere-sphere adhesion model without any contact deformation can be combined with a plate-plate adhesion model for nanocontact flattening. Through this model, various contact deformation paths can be realized. With the increasing flattening, normal and tangential contact stiffness, rolling and twisting resistance, energy absorption, and friction work increase. The level of continuum mechanics constitutes the formulation of threedimensional continuum models as described by tensor equations of representative elemental finite volumes. Balances of mass, momentum, moment, and energy are considered for three translational and three rotational degrees of freedom. The level of micromechanics constitutes the microtransition to a geometrical equivalent of two-dimensional plane with a finite number of discrete subelements, i.e., particulates. At this level, all particulate-particulate interactions within the random packing structure are described by contact forces in the normal and tangential directions in conjunction with fundamental laws for elastic force-displacement, inelastic deformations or plastic dislocations, solid friction, and viscous damping.

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Fig. 5 Classification of different levels of particulate dynamics: continuum mechanics, micromechanics, and molecular dynamics

These microscopic simulations of a small sample can be utilized to derive macroscopic constitutive relationships as required to describe the behavior within the framework of the macroscopic continuum theory. The level of molecular dynamics constitutes the dominant molecular interactions at particulate contact surfaces. In general, fundamental material properties such as elasticity; bonding strength; viscoelasticity; electromagnetic, thermal, and wave propagation characteristics; or phase conversion enthalpies can be physically explained by the molecular interaction energies and potential forces.

Contact Forces at the Level of Micromechanics Four deformation effects can be considered in accordance with particulate-surface contacts and their force-response (stress-strain) behaviors: • Elastic contact deformation which is reversible, independent of deformation rate and consolidation time effects and such deformation is valid for all particulate solids

Basic Theory and Conceptual Framework of Multiphase Flows

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Table 1 Literatures on elastic, plastic, viscoelastic, and viscoplastic contact deformation Elastic contact Hertz (1882) Huber (1904)

Mindlin (1949)

Plastic contact Derjaguin (1934) Krupp and Sperling (1965) Greenwood and Williamson (1966) Schubert et al. (1976) Molerus (1975, 1978) Maguis and Pollock (1984) Walton and Braun (1986) Fleck et al. (1992)

Deresiewicz (1954)

Greenwood (1997)

Lurje (1963) Krupp (1967)

Thornton (1997) Thornton and Ning (1998) Tomas (2000, 2001)

Derjaguin (1934) Bradley (1936) Fromm (1927) Cattaneo (1938) Foppl (1947)

Greenwood and Williamson (1966) Johnson et al. (1971) Dahneke (1972) Maw et al. (1976) Cundall and Strack (1979) Tsai et al (1991) Thornton and Yin (1991), Thornton (1991) Sadd et al. (1993) Tavares and King (2002) Vu-Quoc and Zhang (1999) Di Renzo and Di Maio (2004)

Viscoelastic contact Pao (1955) Lee and Radok (1960) Hunter (1960) Yang (1966) Krupp (1967) Rumpf et al. (1976) Walton (1993) Sadd et al. (1993) Leszczynski (2003) Brilliantov and Poschel (2005)

Viscoplastic contact Rumpf et al. (1976) Kuhn and McMeeking (1992) Bouvard and McMeeking (1996) Stroakers et al. (1997) Stroakers et al. (1999) Parhami and McMeeking (1998) Parhami et al. (1999) Redanz and Fleck (2001) Heylinger and McMeeking (2001) Tomas (2004a, b) Luding et al. (2005)

Vu-Quoc and Zhang (1999) Vu-Quoc et al. (2000) Mesarovic and Johnson (2000) Luding and Herrmann (2001) Luding et al. (2001) Delenne et al. (2004)

• Plastic contact deformation which is irreversible, deformation rate and consolidation time being independent • Viscoelastic contact deformation which is reversible and dependent on deformation rate and consolidation time

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G.H. Yeoh and J. Tu

Normal Impact

Shearing

Twisting

Rolling

Fig. 6 Different modes of particulate-particulate interactions

• Viscoplastic contact deformation which is irreversible and dependent on deformation rate and consolidation time More details on the respective models accounting for the above deformation effects can be found in the literatures listed in Table 1. In Fig. 6, the forces and torques acting on the particulates can be described according to the contact forces, moments, and degrees of freedom: • Along the line normal to the particulate centers • Resistance from shearing or sliding, twisting, and rolling of one particulate over another

For the simulation particulate dynamics by the discrete element method, the linear spring-dashpot model (Cundall and Strack 1979) is normally employed. This model can be combined to include viscous damping (Saluena et al. 1999), plasticity (Luding and Herrmann 2001; Luding et al. 2001), and liquid bridge bonds (Lian et al. 1993). The basic nonlinear elastic behavior derived by Hertz (1882), also

Basic Theory and Conceptual Framework of Multiphase Flows

17

known as the nonlinear spring-dashpot model, can also be combined to include viscoelasticity (Pao 1955, Lee and Radok 1960; Hunter 1960; Yang 1966) and constant adhesion (Johnson et al. 1971). Linear plastic behavior has been described by Walton and Braun (1986). The increase of adhesion due to plastic contact information has nonetheless been introduced by Molerus (1975) and Schubert et al. (1976). Nonlinear plastic, displacement-driven contact has been investigated by Vu-Quoc and Zhang (1999) while contact softening was considered by Tomas (2001). Considering the above theories and constitutive models, a general contact model for load, time, and rate dependent elastic, viscoelastic, plastic, viscoplastic, adhesion, and dissipative behaviors can be developed to characterize the particulate-particulate interactions for the specific problem of interest.

Contact Forces at the Level of Molecular Dynamics The phenomenon of adhesion of particulates for fine ( :

if ðx, y, zÞ is in kth phase at time t otherwise

(59)

Drew and Passman (1999) have demonstrated that the topological equation reflecting the material derivatives of φk for each kth phase following the interface velocity Uint vanish as   @φk þ Uint ∙∇ φk ¼ 0 @t

(60)

Based on Eq. 60, it can be demonstrated that both partial derivatives of φk in space and time vanish away from the interface. The phase indicator function φk on the interface can be regarded as the jump condition which remains constant so that the material derivatives following the interface vanish. If mass transfer exists across the interface from one fluid to the other, the interface moves not only by convection but also by the amount of mass being transferred between the fields. It should thus be noted that the interface velocity is not equivalent to the neighboring velocities of each phase.

Basic Theory and Conceptual Framework of Multiphase Flows

25

Using Eq. 60 and multiplying φk to both sides of Eq. 58, the following equation can be derived:       @ ρk φ k þ ∇∙ ρk Uk φk ¼ ρk Uk  Uint ∙∇φk @t

(61)

Applying averaging to Eq. 61 and using the Reynolds, Leibnitz, and Gauss rules, the instantaneous averaged equation for the mass conservation is given by   @ρk φk þ ∇∙ρk Uk φk ¼ ρk Uk  Uint ∙∇φk @t |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(62)

Γk

where Γk denotes the interfacial mass transfer.

Momentum Conservation The instantaneous equation for the kth phase momentum conservation can be derived through the consideration of Newton’s second law of motion which states that: Rate increase of momentum of the elemental volume ¼ Sum of forces acting on the elemental volume

(63)

On the left hand side of Eq. 63, given that the mass of any fluid mk is ρkdV, the rate of increase of momentum within the elemental volume is given by mk ak ¼ ρk ak dV

(64)

Since the acceleration ak in Eq. 64 is simply the material derivative of Uk, Eq. 64 can be rewritten as ρk

DUk dV Dt

(65)

On the right hand side of Eq. 63, the sum of forces acting on the elemental volume comprises of two sources: surface forces and body forces. Defining Tk to be the Cauchy stress tensor representing the respective stresses along the directions of x, y, and z such as depicted in Fig. 5: σ kxx , σ kyy , σ kzz , τkxy , τkxz , τkyz and the body forces to be Fk,bodyforces, these forces are normally incorporated as additional source or sink terms in Eq. 63 as ∇ ∙ Tk dV þ

X

F k,bodyforces dV

Combining Eqs. 65 and 66 and dividing by the volume dV yields:

(66)

26

G.H. Yeoh and J. Tu

 k  X @U k k þ U ∙ ∇U ¼ ∇ ∙ Tk þ ρ Fk,bodyforces @t k

(67)

It is customary to represent Tk in terms of pressure pk and extra stress tensor τk. This entails expressing the normal stresses σ kxx , σ kyy , σ kzz as the combination of pressure and normal viscous stresses: σ kxx ¼ pk þ τkxx , σ kyy ¼ pk þ τkyy , σ kzz ¼ pk þ τkzz : Therefore, Eq. 67 can be alternatively expressed as ρk



@Uk þ Uk ∙ ∇Uk @t



¼ ∇pk þ ∇ ∙ τk þ

X

Fk, bodyforces

(68)

Similar to the derivation of the averaged equation governing the mass conservation, the averaged equation governing the momentum conservation can be derived by multiplying Eq. 68 with the phase indicator function. After some rearrangement,         @ ρk φk Uk þ ∇ ∙ ρk φk Uk Uk ¼ ∇ φk pk þ ∇ ∙ φk τk þ pk ∇φk @t   þ ρk Uk Uk  Uint ∙∇φk  τk ∙∇φk X Fk, bodyforces þ φk

(69)

Applying the Reynolds, Leibnitz, and Gauss rules, the averaged equation governing the momentum conservation is given by X @〈ρk φk Uk 〉 þ ∇∙〈ρk φk Uk Uk 〉 ¼ ∇〈φk pk 〉 þ ∇∙〈φk τk 〉 þ 〈φk 〉〈 Fk, bodyforces 〉 @t   þ〈ρk Uk Uk  Uint ∙∇φk 〉 þ 〈pk 〉〈∇φk 〉  〈τk ∙∇φk 〉 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Ωk

(70) where Ω denotes the interfacial momentum transfer. k

Energy Conservation Based on the first law of thermodynamics, the instantaneous equation for the kth phase energy conservation can be derived assuming no additional or removal of heat due to external heat sources as Rate increase of energy of the elemental volume ¼ Net rate of heat added to the elemental volume þ Net rate of work done on the elemental volume

(71)

On the left hand side of Eq. 71, analogous to the consideration of the conservation of momentum, the time rate of change of energy for the elemental volume of the kth

Basic Theory and Conceptual Framework of Multiphase Flows

27

phase is simply the product of the density ρk and material derivative of energy Ek. The rate of increase of energy is thus given by ρk

DEk dV Dt

(72)

On the right hand side of Eq. 71, with reference to Fig. 5, the rate of work done on the elemental volume in the x direction is the product between the surface forces (caused by the normal stress σ kxx and tangential stresses τkyx and τkzx ) and the velocity component in the x direction. Work done due to surface stress components along the y direction and z direction is derived accordingly. In addition to the work done by surface forces on the fluid element, possible work done due to body forces is also considered. Hence, the net rate work done can be written as X

X   W_ ¼ ∇∙ Uk ∙Tk dV þ Uk ∙ Fk, bodyforces dV

(73)

Also, the net rate of heat transfer to the fluid due to the heat flow is given by the difference between the heat entering and leaving the elemental volume. The total rate of heat added is expressed by X

Q_ ¼ ∇∙qk dV

(74)

Combining Eqs. 72, 73, and 74 and dividing by the volume dV yields:  k  X   @E k k þ U ∙∇E ¼ ∇∙qk þ ∇∙ Uk ∙Tk þ Uk ∙ Fk, bodyforces ρ @t k

(75)

Equation 75 can also be alternatively expressed in terms of combination of pressure, normal viscous stresses, and extra stress tensor as ρk



@Ek þ Uk ∙∇Ek @t

þ Uk ∙

X



    ¼ ∇∙qk  ∇∙ Uk pk þ ∇∙ Uk ∙τk

Fk, bodyforces

(76)

Similar to the derivation of the averaged equation governing the mass and momentum conservation, the averaged equation governing the energy conservation can be derived by multiplying Eq. 76 with the phase indicator function. After some manipulation,

28

G.H. Yeoh and J. Tu

        @ ρ k E k φk þ ∇∙ ρk Uk Ek φk ¼ ∇∙ φk qk  ∇∙ φk Uk pk þ qk ∙∇φk @t   þ Uk pk ∙∇φk þ ρk Ek Uk Uk  Uint ∙∇φk   þ ∇∙ φk Uk ∙τk  Uk ∙τk ∙∇φk X þ φk Uk ∙ Fk, bodyforces

(77)

Applying the Reynolds, Leibnitz, and Gauss rules, the averaged equation governing the energy conservation is given by @〈ρk Ek φk 〉 þ ∇∙〈ρk Uk Ek φk 〉 ¼ ∇∙〈φk qk 〉  ∇∙〈φk Uk pk 〉 þ ∇∙〈φk Uk ∙τk 〉þ @t P 〈φk〉〈Uk ∙ Fk, bodyforces 〉þ  〈ρk Uk Uk  Uint ∙∇φk 〉 þ 〈qk ∙∇φk 〉 þ 〈Uk pk 〉∙〈∇φk 〉  〈Uk ∙τk ∙∇φk 〉

(78)

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Πk

where Πk denotes the interfacial energy transfer. It should be noted that the specific energy Ek of a fluid can often be defined as the sum of the specific internal energy and kinetic energy of the kth phase: Ek ¼

ek

þ

|{z} specific internal energy for the kth phase

1  k k U ∙U 2 |fflfflfflfflfflffl{zfflfflfflfflfflffl}

(79)

kinetic energy for the kth phase

Conservation equations for specific internal energy and kinetic energy may be derived by substituting Eq. 79 into Eq. 78. Based on the definitions of the sensible enthalpy hks and the total enthalpy Hk of a fluid given as hks ¼ ek þ

pk ρk

H k ¼ hks þ

1  k k U U 2

(80)

and combining these two definitions yields H k ¼ ek þ

pk 1  k k  pk þ U ∙U ¼ Ek þ k k 2 ρ ρ

(81)

Using Eq. 81, the equation for the total enthalpy can be formulated which may be written in the averaged form as:

Basic Theory and Conceptual Framework of Multiphase Flows

@〈ρk H k φk 〉 @〈pk φk 〉 þ ∇∙〈ρk Uk H k φk 〉 ¼   ∇∙〈φk qk 〉 þ ∇∙〈φk Uk ∙τk 〉þ @t @t P 〈φk 〉〈Uk ∙ Fk, bodyforces 〉þ   @φk 〉  〈Uk ∙τk ∙∇φk 〉 〈ρk Uk Uk  Uint ∙∇φk 〉 þ 〈qk ∙∇φk 〉 þ 〈pk @t Πk |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

29

(82)

The term ∇ ∙ 〈φkqk〉 at the right hand side in Eq. 82 can normally be formulated by applying the Fourier’s law of heat conduction that relates the heat flux to the local temperature gradient. In other words, qk = ∇(λkTk) where λk is the thermal conductivity for the kth phase. It is also worth noting that term @φk/@t in Eq. 82 represents the local acceleration term of the material derivative of the phase indicator function as stipulated by Eq. 60. It may be rewritten as @φk/@t = Uint ∙ ∇φk

Conservation Equations for Turbulent Multiphase Flow In most practical flows of interest, turbulence is associated with the existence of random fluctuations in the fluid. The presence of the random nature of the fluid flow generally precludes computations based on the equations that describe the fluid motion to be carried out to the desired accuracy. It is therefore preferable that some means of practically resolving the random transient distribution of the instantaneous flow field with time or space is realized for practical computations. Reynolds-Averaged Closure One feasible approach to handle turbulent multiphase flow in the context of interpenetrating media framework is to consider any instantaneous field ϕ to be decomposed into a steady mean motion ϕ and its fluctuating component ϕ0 : ϕ ¼ ϕ þ ϕ0

(83)

This particular flow decomposition presents an attractive way of characterizing a turbulent flow by the mean values of flow properties with its corresponding statistical fluctuating property. The time averaged of the fluctuating component ϕ0 is zero. In other words, 0

ϕ ¼ limT!1

1 T

ð

ϕ0 ¼ 0

(84)

By applying volume-averaging or ensemble-averaging, the instantaneous averaged field can be written as 〈ϕ〉 ¼ 〈ϕ〉 þ ϕ00 In accordance with Eq. 84, the time averaged of the fluctuating component ϕ also, by definition, zero.

(85) 0 0

is

30

G.H. Yeoh and J. Tu

In multiphase flow analysis, preference is normally given to the Favre-averaging approach in order to alleviate the complication of modeling additional correlation terms containing averages of fluctuating quantities. Two types of averaged variables are employed. The phase-weighted average for the field ζ can be defined by 〈ζ〉 ¼

〈ζφk 〉 〈φk 〉

(86)

and the mass-weighted average of the field ψ can also be defined in accordance with 〈ψ〉 ¼

〈ψρk 〉 〈ρk 〉

(87)

It follows that ζ 00 〈φk 〉 and ψ 00 〈ρk 〉 are zeros. The local volume fraction (volumetric concentration or relative residence time) represents an important parameter in multiphase flow investigations. It can usually be defined as the fraction of time in which the continuous or dispersed phase occupies a particular given point in space. In principal, the local volume fraction αk can be regarded as the ratio of the fractional volume ϑk of the kth phase in an arbitrary small region over the total volume V of the region within the multiphase flow. It also corresponds to the volume-averaged of the phase indictor function, i.e., αk= ϑk /ϑ =〈φk〉. In the event where the transport equations governing the conservation of mass, momentum, and energy are volume-averaged and subsequently timeaveraged, suitable forms of these equations via the phase-weighted and massweighted averages for the multifluid model can be formulated. Dropping the bars and parentheses which they denote be default the Favreaveraging and volume-averaging processes being performed on the transport equations, the so-called Reynolds-averaged form of the governing equations written in terms of the local volume fraction and products of averages can be derived as Mass conservation:   @  k k ρ α þ ∇∙ ρk Uk αk ¼ Γ0k @t

(88)

Momentum conservation:         @  k k k ρ α U þ ∇∙ ρk αk Uk Uk ¼ ∇ pk αk þ ∇∙ αk τk  ∇∙ αk τk00 @t X ak Fk, bodyf orces þ Ω0k Energy conservation:

(89)

Basic Theory and Conceptual Framework of Multiphase Flows

31

      @  k k k @  k k 00 ρ α H þ ∇∙ ρk αk Uk Hk ¼  p α  ∇∙ αk λk T k  ∇∙ αk qk @t @t     00 þ ∇∙ αk Uk ∙τk  ∇∙ αk Uk ∙τk þ αk Uk ∙

X

Fk, bodyforces þ Π0k

(90)

It is worth noting that if ensemble averaging is performed on the governing equations, and fluctuating quantities are subsequently introduced into the equations, the final forms of the governing equations are no different from those of the volumeaveraged and time-averaged conservation equations. In Eqs. 88, 89, and 90, the interfacial terms Γ0 k, Ω0 k, and Π0k represent the Favreaveraging that is subsequently performed on top of the volume-averaged terms: Γk, Ωk, and Πk. These interfacial exchange terms can be modeled in accordance with Γ0k ¼ Ω0k ¼

XN  l¼1

XN

ðm_ lk  m_ kl Þ

(91)

 k, drag k, nondrag þ FD m_ lk Ul  m_ kl Uk þ pkint ∇αk þ FD

(92)

l¼1

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} FkD

Π0k ¼

N  X

m_ lk H l  m_ kl H k Þ þ Qint

(93)

l¼1

where m_ lk and m_ kl characterizes the mass transfer from the lth phase to kth phase and from the kth phase to lth phase, respectively. From mass conservation, m_ kk ¼ m_ ll ¼ 0: The interfacial force FkD in Eq. 92 is usually decomposed in terms k, drag k, nondrag and nondrag forces in FD . The interfacial drag force of the drag force FD k, drag FD and interfacial heat source Qint can usually be expressed as k, drag FD 

N X

  Bkl Ul  Uk

(94)

l¼1

Qint 

N X

  Ckl T l  T k

(95)

l¼1

where Bkl and Ckl are the interphase drag and heat transfer terms. Through appropriate modeling considerations, closure to the interfacial exchange terms can be attained through prescribed algebraic functions of the governing flow parameters.

32

G.H. Yeoh and J. Tu

In most multiphase flow problems, the fluid is normally assumed to be Newtonian. Therefore, the normal and shear stress τk can be taken to be proportional to the time rate of strain, which is the velocity gradient. The normal and shear viscous stress components for the kth phase according to the Newton’s law of viscosity are h  T i 2 k   τk ¼ μk ∇Uk þ ∇Uk  μ ∇∙Uk I 3

(96)

where μk is the dynamic viscosity for the kth phase. In Eqs. 89 and 90, τk 00 and qk00 are turbulent fluxes which can be estimated from the eddy viscosity and eddy diffusivity hypotheses. According to Lopez de Bertodano et al. (1994a, b), this concept suggests that the Reynolds stress τk 00 can be correlated with the mean rates of deformation for different phases in analogous to the Newton’s law of viscosity. The Reynolds stress τk 00 can thus be expressed as h  T i 2 k   2 τk ¼ μkt ∇Uk þ ∇Uk  μt ∇∙Uk I  ρk kk I 3 3

(97)

where kk is the turbulent kinetic energy and μkt is the so-called turbulent or eddy viscosity which is taken to be a function of the flow rather than of the fluid and it is required to be prescribed. Similar to the eddy viscosity hypothesis, the eddy diffusivity hypothesis for the Reynolds flux qk00 can be taken to be proportional to the gradient of the transported quantity. For the total enthalpy, the Reynolds flux term can be modeled as 00

qk ¼ Γkt ∇H k

(98)

The term Γkt in the above expression is the eddy diffusivity for total enthalpy of the fluid. Since the turbulent transport of momentum and heat can be attributed through the same mechanisms – eddy mixing – the value of the eddy diffusivity in Eq. 98 can be taken to be close to that of the eddy viscosity μkt . By definition, the turbulent Prandtl number, which is the ratio between momentum diffusivity (viscosity) and thermal diffusivity, can thus be written as Pr kt ¼

μkt Γkt

(99)

To satisfy dimensional requirements, at least two scaling parameters are required to relate the Reynolds stress to the rate of deformation. One feasible choice is the use of the turbulent kinetic energy kk and the rate of dissipation of turbulent energy ek. The local turbulent viscosity in Eq. 99 can be obtained either from dimensional analysis or from analogy to the laminar viscosity as μkt / ρk vt l . Based on the

Basic Theory and Conceptual Framework of Multiphase Flows

33

pffiffiffiffiffi characteristic velocity vt defined as kk and the characteristic length l as (kk)3/2/ek, the turbulent viscosity can be calculated according to μkt

 k 2 k ¼ Cμ ρ ek k

(100)

where Cμ is an empirical constant and local values of kk and ek are obtained in time and space through solutions of appropriate two-equation turbulence models. Following proposals by Lopez de Bertodano et al. (1994a) and Lahey and Drew (2001b), the system of turbulent scalar equations at high Reynolds numbers for the continuum equations in the interpenetrating media framework is straightforward generalizations of the single-phase counterpart of the two-equation k  e model developed by Tennekes and Lumley (1976) and Versteeg and Malalasekera (1995). Other turbulence models such as the Reynolds stress and shear stress transport (SST) could also be applied in place of the two-equation k  e model. The Reynolds stress model has a greater potential to account for flows having strongly anisotropic turbulence. SST which combines the two-equation k  e model and two-equation k  ω model where ω is the turbulent frequency can be employed to better handle nonequilibrium boundary layer regions.

Spatial Filtering In contrast to Reynolds-averaged closure, another approach to handling multiphase flows within the interpenetrating media framework is through the consideration of large eddy simulation (LES). The basic idea behind LES is that the large scale motions are solved directly and the small scale motions are represented in terms of subgrid scale models. Significant advances in computational resources are looking towards LES as the preferred methodology to be adopted for many turbulence investigations of fundamental fluid dynamics problems. Since all real-world flows are inherently unsteady, LES provides the means of obtaining such solutions and is gradually replacing the use of two-equation turbulence models. The Favre-averaging approach, as utilized in Reynolds-averaged closure, is adopted in LES in order to alleviate the complication of modeling additional correlation terms containing fluctuating quantities. Here, the phase-weighted average for the field ζ can also be defined as ζ ðx0 , tÞφk ðx0 , tÞ ζ ðg x0 , tÞ ¼ ¼ φk ðx0 , tÞ

Ð

Δ ζ ðx

0

, tÞφk ðx0 , tÞGðx  x0 Þdx0 Ð k 0 0 Δ φ ðx , tÞcdx

(101)

and the mass-weighted average of the field ψ can also be defined in accordance with ψ ðx0 , tÞρk ðx0 , tÞ ¼ ψ ðg x0 , t Þ ¼ ρk ð x0 , t Þ

Ð

, tÞρk ðx0 , tÞGðx  x0 Þdx0 k 0 0 0 Δ ρ ðx , tÞGðx  x Þdx

Δ ψÐðx

The instantaneous variables of ζ and ψ are:

0

(102)

34

G.H. Yeoh and J. Tu

ζ ðx0 , tÞ ¼ ζ ðg x0 , tÞ þ ζ 00 ðx0 , tÞ

(103)

x0 , tÞ þ ψ 00 ðx0 , tÞ ψ ðx0 , tÞ ¼ ψ ðg

(104)

x0 , tÞ represent the filtered or resolvable components (essentially where ζ ðg x0 , tÞ and ψ ðg local averages of the complete field) and ζ 00 (x0 , t) and ψ 00 (x0 , t) are the subgrid scale components that account for the unresolved spatial variations at a length smaller than the filter width Δ. In Eqs. 101 and 102, G(x  x0 ) represents an appropriate spatial filter function to be applied for the problem in question. The most common localized filter functions can be represented by Top hat: 8 < 1 for jx  x0 j < Δ 0 2 Gðx  x Þ ¼ Δ : 0 otherwise

(105)

Gaussian: ! rffiffiffiffiffiffiffiffi 0 2 6 6 ð x  x Þ G ð x  x0 Þ ¼ exp πΔ2 Δ2

(106)

Spectral cutoff: G ð x  x0 Þ ¼

sin ðke ðx  x0 ÞÞ , π ðx  x0 Þ

ke ¼

π Δ

(107)

Within the interpenetrating media framework, it is also customary in the context of LES that the volume fraction can be expressed as αk ðx0 , tÞ ¼ ζ ðx0 , tÞ ¼

ð Δ

φk ðx0 , tÞGðx  x0 Þdx0

(108)

Dropping the bars which by default denote Favre-averaging, the filtered conservation equations are: Mass conservation:   @  k k 00 ρ α þ ∇∙ ρk Uk αk ¼ Γ k @t Momentum conservation:

(109)

Basic Theory and Conceptual Framework of Multiphase Flows

      @  k k k ρ α U þ ∇∙ ρk αk Uk ⨂Uk ¼ ∇ pk αk þ ∇∙ αk τk @t   X k, bodyf orces F  ∇∙ αk τk00 αk þ Ω00k

35

(110)

Energy conservation:         @  k k k ρ α U þ ∇∙ ρk αk Uk ⨂Uk ¼ ∇ pk αk þ ∇∙ αk τk  ∇∙ αk τk00 @t

(111)

where Γ 0 0 k, Ω0 0k, and Π0 0 k are the filtered interfacial mass, momentum, and energy balance source terms. Note that τk has been defined in Eq. 96 for Newtonian fluid. The unresolved subgrid stress tensor τk 0 0 in Eqs. 89 and 90 can be modeled in analogous with the Reynolds-averaged closure via the Boussinesq hypothesis: 00 fk ¼ 2μk S~ k þ 1 τk 00 I fk ⨂U τk ¼ ρk Ukg ⨂Uk  ρk U T SGS 3 1 h k  k T i 1 k ~ S ¼ ∇U þ ∇U  ∇∙Uk I 2 3

(112)

where S~ is the strain rate of the large scale or resolved field and μkT SGS is the subgrid scale eddy viscosity for the kth phase are determined through solutions of appropriate subgrid scale models. Also, the unresolved subgrid enthalpy flux qk00in Eq. 111 is modeled in a manner similar to the subgrid turbulence stresses by the standard gradient diffusion hypothesis as μk 00 g fk H k ~ ¼  T SGS ∇H H  ρk U qk ¼ ρk U Pr kT SGS

(113)

where Pr kT SGS is the subgrid turbulent Prandtl number for the kth phase. Since the smallest turbulence eddies are almost isotropic, the Boussinesq hypothesis provides a good description of the unresolved eddies (Smagorinsky 1963). Taking the length scale to be the filter width, the velocity scale can be expressed as the product of the length scale and the average strain rate of the resolved flow. This thus brings about the formulation of the Smagorinsky-Lilly model which assumes that the subgrid scale eddy viscosity can be described in terms of a length and a velocity scale. Nevertheless, the Smagorinsky-Lilly model has been designed for flow which is highly turbulent, fully developed, and isotropic but does not accommodate any eventual departure of the flow from these assumptions. In order to attain an automatic adaptation of the model for inhomogeneous flows, simulations of multiphase flows have been performed on the dynamic formulation of the model (Germano et al. 1991).

36

G.H. Yeoh and J. Tu

Population Balance Approach Presence of particulates regardless whether they are inherently present within or deliberately introduced into the flow system significantly affects the behavior of the multiphase flow. Because of mounting interests in determining the influence of particulates within the flow system, population balance has the capacity of resolving the microphysics which occurs at the mesoscale level. This in turn allows the behavior and dynamic evolution of the population of discrete particulates to be better synthesized. Much consideration has been concentrated towards describing the spatial and temporal evolution of the geometrical structures as a result of the formation and destruction of agglomerates or clusters through interactions among and between discrete particulates and collisions with turbulent eddies. The primary issue concerning the use of macroscopic formulation based on averaging of the transport equations is the determination of the interfacial rates of the interphase interaction terms. These terms provide the appropriate closure to the equations which can be realized through the consideration of suitable mechanistic models accounting for the physical interaction between the different phases. One such approach to determine the local size distribution in space and time is population balance modeling. In essence, the population balance of any system is a record for the number of particulates whose presence or occurrence governs the overall behavior of the flow system under consideration. Record of these discreet elements is dynamically dependent on the birth and death processes that create new discrete and terminate existing particulates within a finite or defined space. Since a multiphase flow generally contains millions or billions of discrete particulates that are simultaneously varying in space and time, the feasibility of direct numerical simulation in resolving such flow is still far beyond the capacity of existing computational resources. Population balance, which records the number of these particulates as an averaged function through the population balance equation, has shown to be extremely promising in handling the flow complexity because of its comparatively lower computational requirements.

Definition of Density Function and Continuous Phase Vector In this framework, the population of particulates can be treated of not only being distributed in the physical space but also in an abstract property space. Dependent variables of these entities can be taken to exist in two different coordinates: internal and external coordinates. Mathematically, the internal coordinates are the property coordinates while the external coordinates are the spatial coordinates. Figure 7 illustrates the internal and external coordinates involved in the population balance for a three-phase flow. The joint space comprising the internal and external coordinates is referred to as the space of particulate phase. In this space, the quantity of basic interest which characterizes the distinct particulates is the consideration of the density function.

Basic Theory and Conceptual Framework of Multiphase Flows

37

Liquid flow Internal Coordinates

External Coordinates

Number

Fractional volume occupy by gas particles at (r2,t2) Size

Changes of internal properties caused by the “Birth” or “Death” processes Number

Size

Gas Particles Solid particles Changes of external variables resulted from the Fluid Motions Fractional volume occupy by gas particles at (r1, t1)

Fig. 8 Population balance for gas-liquid-particle flow (Yeoh et al. 2014)

Based on Ramkrishna (2000), the density function f1(x, r, t) can be defined as the number of particles per unit volume of the particle phase at time t at the internal coordinates x  (x1, x2, . . . , xn) where n represents the number of different quantities associated with the particle and at the external coordinates r  (r1, r2, r3) which may be employed to indicate the position vector of the particulate. This number density f1(x, r, t)is usually taken to be smooth so that it can be differentiated with respect to any of its arguments. The particle state vector accounts for both internal and external coordinates where the domain of internal coordinate shall be taken to be Vx while the domain of external coordinates to be Vr that represents a set of points in the physical space in which particulates are present. The behavior of each particulate being affected by the continuous phase may also be collated into a finite dimensional vector field. The continuous phase vector which can be defined by Y = (r, t) is a function of only the external coordinates r and time t, and it is calculated from the governing conservation equations associated with the particular problem. It should be noted that a continuous phase may not be necessarily considered for some applications where interaction between the population of particulates and the continuous phase does not result in a substantial change in the continuous phase. Analysis on the population thereby reduces to only the consideration of population balance (Fig. 8).

Population Balance Equation The development of the population balance equation can be traced back to as early as the end of eighteenth century via the Boltzmann-type equation, proposed by Ludwig

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G.H. Yeoh and J. Tu

Boltzmann. Such an equation could be regarded as the first population balance equation which could be expressed in terms of a statistical distribution of molecules or particles in a state space. The fundamental variable is the particle distribution function along with an appropriate choice of internal coordinates pertaining to a particular problem being solved. Defining p(x, r, c, t)dxdrdc to be the particle number density distribution function which is assumed to be continuous and specifies the probable number density of particles with internal coordinates about x in the range of dx, about position r in the range of dr, and about velocity c in the range dc, at about time t, an equation for the particle number density distribution function can be written as pðx þ dx, r þ dr, c þ dc, t þ dtÞdxdrdc  pðx, r, c, tÞdxdrdc ¼ Sp dxdrdcdt

(114)

Where dr = cdt, dc = Fdt in which F is the force per unit mass acting on a particle and Sp consists of the rates of change of p(x, r, c, t) due to death and birth of particulates as well as other sources or sinks due to particulate interactions (for example, rebounding of particulates). By assuming that the change of particulate velocity within the time interval t to t + dt is negligible, Eq. 113 reduces to @f þ ∇r ∙ðvr f Þ þ ∇x ∙ðvx f Þ ¼ Sf @t

(115)

The above equation is analogous to the Boltzmann-type equation in describing the temporal and spatial rate of change of the distribution function: f ðx, r, tÞ ¼

ð þ1 pdc

(116)

1

which simply denotes the probable number density of particulates with internal coordinates about x in the range of dx, about position r in the range of dr, and at about time t. Incidentally, that this function is by definition the same as the density function f1(x, r, t). The source/sink term at the left hand side of Eq. 116 is Sf ¼

ð þ1 1

Sp dc

(117)

In Eq. 115, integrating the force F over the whole velocity space results in no net contribution due to this force and thus it does not explicitly appear in the population balance equation since the distribution vanishes as the velocity approaches to infinity (1). Also, the source/sink term consists of only the net generation rate of particles due to death and birth processes while other sources or sinks due to particle interactions vanish as the number of particulates is conserved especially for the case during the rebounding processes in the system.

Basic Theory and Conceptual Framework of Multiphase Flows

39

Integrated Forms of Population Balance Equation For a nonreactive and isothermal multiphase flow system, it is customary to assume that all relevant internal variables can be calculated from the consideration of the particulate volume or diameter. If the internal coordinate is taken to be the particulate volume Vp or diameter dp in describing incompressible particulate dispersions. For a multiphase flow system where compressibility effect becomes important, the use of particulate mass mp as internal coordinate may prove to be more advantageous. This requirement is particularly important in the gas phase because this quantity is required to be conserved under pressure changes. The population balance equation can now be written in accordance with the particulate volume, diameter, and mass as         @  @  f 1 V p , r, t þ ∇r ∙ vr V p , r, Y, t f 1 V p , r, t þ ̇ V p f 1 V p , r, t @t @V p   ¼ Sf 1 V p , r, Y, t

(118)

        @  @  f dp , r, t þ ∇r ∙ vr dp , r, Y, t f V p , r, t þ ̇ dp f 1 d p , r, t @t @V p   ¼ Sf 1 dp , r, Y, t

(119)

        @  @  f 1 mp , r, t þ ∇r ∙ vr mp , r, Y, t f 1 mp , r, t þ ̇ m p f 1 V p , r, t @t @V p   ¼ Sf 1 mp , r, Y, t

(120)

where ̇ V p denotes the time rate of change of particle volume, d p_ represents the time rate of change of particle diameter, and ̇ m p represents the time rate of change of particle mass. To close the population balance problem, closure is required for the growth of particulates as well as the death and birth kernels through appropriate models in the source/sink term Sf 1 . Note that these kernels are required to be consistent with the internal coordinate used. In practice, the population balance of particulates is solved through the consideration of moment transformation. If the particle volume Vp is adopted as the internal coordinate, the mth moments of the number density function in terms of volume can be defined as mk ðr, tÞ ¼

1 ð

  f 1 V p , r, t V kp dV p

(121)

0 th

Various m moments in equation (3–18) contribute to physically important moments of the number density function. In retrospective, particle number density, particle mass density, interfacial area concentration, and local volume fraction

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essentially correspond to zero, first, second, and third order moments. If the volume Vp is treated as the independent internal coordinate, the particle number density N, average particle volume V , interfacial area concentration Ai, and local volume fraction αp are: N ðr, tÞ ¼ m0 ðr, tÞ ¼

1 ð

  f 1 V p , r, t dV p

(122)

0

01 1 ð   V ðr, tÞ ¼ m1 ðr, tÞ ¼ @ f 1 V p , r, t V p dV p A=N ðr, tÞ

(123)

0

Ai ðr, tÞ ¼ m2 ðr, tÞ ¼

1 ð

  f 1 V p , r, t πD2e dV p

(124)

 π f 1 V p , r, t D3e dV p 6

(125)

0

αp ðr, tÞ ¼ m3 ðr, tÞ ¼

1 ð

0

In Eqs. 124 and 125, it is noted that the particulate surface area and volume are expressed in terms of equivalent diameters.

Source/Sink Term of Population Balance Equation The death and birth processes in the source/sink term for the population balance equation are taken to occur simultaneously. Thus, Sf 1 ðx, r, Y, tÞ can be obtained via Sf 1 ðx, r, Y, tÞ ¼

ð

νðx0 , r, Y, tÞbðx0 , r, Y, tÞPðx, rj x0 , r, Y, tÞf 1 ðx, r, tÞdV x

Vx

bðx, r, Y, tÞf 1 ðx, r, tÞ þ 

ð

ð Vx

1 @ ðx~, rÞ aðx~,~ r ; x0 , r, Y, tÞf 2 ðx~,~r ; x0 , r, tÞ dV x δ @ ðx, rÞ

aðx0 , r0 ; x, r, Y, tÞf 2 ðx0 , r0 ; x, r, tÞdV x

(126)

Vx

The terms ν(x0 , r, Y, t) and P(x, r| x 0 , r, Y, t) denote the average number of particulates, and probability density function for particulates from the breakage of a single particle of state (x0 , r0 ) in an environment of Y at time t. The probability density function P(x, r| x 0 , r, Y, t) is generally taken as a continuously distributed

Basic Theory and Conceptual Framework of Multiphase Flows

41

fraction over particle state space and is commonly associated with the daughter particle size distribution function denoting the size distribution of daughter particles produced upon the breakage of a parent particulate. This quantity needs to be determined through either detailed modeling of the breakage process or from experimental observation. The term b(x, r, Y, t) represents the specific breakage rate of particulates or more commonly known as the breakage frequency. In Eq. 126, δ represents the number of times identical pairs have been considered in the interval of integration where 1δ is introduced to correct for the redundancy. The aggregation frequency aðx~, r; x0 , r, Y, tÞ represents the fraction of pairs of particles of states ðx~,~ r Þ and (x0, r0) that will aggregate. A coarse approximation of the pair density function is assumed for f 2 ðx~,~ r ; x0 , r, tÞ f~1 ðx~,~ r , tÞf 01 ðx0 , r0 , tÞ, which implies the absence of any statistical correlation between particles of state spaces (x, r) and ~ (x0 , r0 ) at any instant of time t. The term @@ ððxx,, rrÞÞ represents the Jacobian determinant. In practice, the transport of particle number density is solved within the framework of computational multiphase fluid dynamics. By taking the volume Vp as the independent internal coordinate. Eq. 126 becomes 1 ð

Sf 1 ðx, r, Y, tÞ ¼

          ν V 0p , r, Y b V 0p , r, Y P V p , rj V 0p , r, Y, t f 1 V 0p , r, t dV 0p  b V p , r, Y

Vp

V ðp      1 ~ f 1 V p , r, t þ a V p ,~ r ; V 0p , r0 , Y f~1 V~ p ,~ r , t f 01 V 0p , r0 , t dV 0p 2





0 1 ð



    a V 0p , r0 ; V p , r, Y f 2 V 0p , r0 ; V p , r, t dV 0p

(127)

0

Owing to the complexity associated with both the particle growth term as well as birth and death processes in the source/sink term, appropriate constitutive relationships are inevitably required to close the population balance equation. In most cases, the detailed functionality of closure relationships and even the physical insights of the birth and death processes are generally unknown or unresolved. Also, formulation of the growth terms that may be dependent on the choice of internal coordinates or particle properties leads to additional complexity of suitably characterizing the particle phase in the population balance framework. Therefore, this parameterization process represents the weakest component which still demands the greatest attention.

Practical Considerations of Population Balance Equation For practical solutions, various numerical approaches have been developed to solve the population balance equation. The three common methods are the Monte Carlo methods, method of moments, and class methods. Monte Carlo methods solve the population balance equation based on the statistical ensemble approach (Domilovskii et al. 1979; Liffman 1992; Maisels et al.

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2004). The main advantage is the flexibility and accuracy to track particulate changes in a multidimensional system. Nevertheless, the accuracy of Monte Carlo methods is greatly dependent on the number of simulation particles; extensive computational time is thus required to track a multitude of particulates. Monte Carlo methods are generally not easily coupled with the conceptual framework of computational multiphase fluid dynamics. The basic principle behind method of moments centers in the transformation of the problem into lower-order of moments of the particulate size distribution. The primary advantage is its numerical economy that condenses the problem substantially by only tracking the evolution of a small number of moments (Frenklach 2002). More importantly, method of moments do not suffer from truncation errors in the approximation of the particulate size distribution. Mathematically, the transformation from the space of the particulate size distribution to the space of moments is extremely rigorous. Fraction moments, representing mean diameter or surface area, pose serious closure problem. To overcome this, Frenklach and Wang (1991, 1994) have proposed an interpolative scheme to aptly determine the fraction moment from integer moments, namely, the method of moments with interpolative closure. Another different approach to computing the moment is through the use of numerical quadrature scheme such as the quadrature method of moments as suggested by McGraw (1997). With the aim to solve multidimensional problems, Marchisio and Fox (2005) extended the method by developing the direct quadrature method of moments where the quadrature abscissas and weights are subsequently formulated as transport equations. The main idea of direct quadrature method of moments is to track the primitive variables appearing in the quadrature approximation, instead of moments of the. In general, the method of moments represent a sound mathematical approach and an elegant tool for solving the population balance equation with limited computational burden. Instead of inferring the particulate size distribution to derivative variables such as moments, class methods directly simulate the main characteristics using primitive variables. One approach is the adoption of an averaged quantity to represent the overall changes of the particle population. Ishii and coworkers have formulated the interfacial area concentration transport equation to simulate different flow regimes and conditions (Hibiki and Ishii 2002; Sun et al. 2004a, b). In order to be consistent with the form of conservation equations in the multifluid approach, the transport equation of the averaged bubble number density can be adopted (Cheung et al. 2007a, b). A more sophisticated model namely the homogeneous multiple-sizegroup model that was developed by Lo (1996) can provide the feasibility of accounting different shapes and velocities. The inhomogeneous multiple-sizegroup model developed by Krepper et al. (2005) consisted of further subdividing the dispersed phase into N number of velocity fields.

Basic Theory and Conceptual Framework of Multiphase Flows

43

Summary Owing to the increasing reliance on computational investigations of multiphase flows of natural and technological significance, the purpose of this chapter is to present the appropriate conceptual frameworks that can be applied to aptly resolve and attain a fundamental understanding of many different classifications of multiphase flows. For small scale flow system, it is becoming ever more possible to solve directly the transport equations governing the conservation of mass, momentum, and energy for each phase and compute every detail of the multiphase flow, the motion of all the fluid around every particulate, and the position of every interface via the Lagrangian formulation. For large scale flow system, such comprehensive treatment remains prohibitive which is only restricted to turbulent flows of low Reynolds number and the dynamics of a limited amount of individual particulates. The use of multiphase flow continuum via the Eulerian formulation within the interpenetrating media framework provides the effective way of predicting the gross features of the multiphase flows. Because of mounting interests in determining the influence of particulates affecting the flow system, the need of population balance to resolve the microphysics which occurs at mesoscale is required in order to better synthesize the behavior and dynamic evolution of particulates occupying the flow system. The overarching issue especially using the macroscale formulation based on averaging the transport equations governing the conservation of mas, momentum, and energy is the determination of the interfacial transfer terms in providing the realistic physical interaction between the different phases. One such approach to determine the local size distribution of the particulates in space and time is through the population balance approach. The population balance of any flow system is a record for the number of particulates whose presence or occurrence governs the overall behavior of the flow system under consideration. Record of these particulates is dynamically dependent on the birth and death processes that terminate existing particulates and create new particulates within defined space and time.

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"https://pure.tudelft.nl/portal/en/persons/s-luding(b29c6ff7-35b1-4123-8724-e2ee249b6bb3). https://pure.tudelft.nl/portal/en/publications/micromacro-transition-for-cohesive-granularmedia(b254117d-f6f6-47e4-8e2e-2eb53e1017b6).html"Micro-Macro Transition for Cohesive Granular Media. in S Diebels (ed.), Zur Beschreibung komplexen Materalverhaltens, Institut für Mechanik.. pp. 121-134. S. Luding, K. Manetsberger, J. Müller, A discrete model for long time sintering. J. Mech. Phys. Solids 53, 455–491 (2005) A.J. Lurje, Räumliche Probleme der Elastizitätstheorie (Akademie-Verlag, Berlin, 1963) A. Maisels, F.E. Kruis, H. Fissan, Direct simulation Monte Carlo for simulation nucleation, coagulation and surface growth in dispersed systems. Chem. Eng. Sci. 59, 2231–2239 (2004) D.L. Marchisio, R.O. Fox, Solution of population balance equations using the direct quadrature method of moments. J. Aerosol Sci. 36, 43–73 (2005) D. Maugis, H.M. Pollock, Surface forces, deformation and adherence at metal microcontacts. Acta Metall. 32, 1323–1334 (1984) N. Maw, J.R. Barber, J.N. Fawcett, The oblique impact of elastic spheres. Wear 38, 101–114 (1976) R. McGraw, Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. Technol. 27, 255–265 (1997) S.D. Mesarovic, K.L. Johnson, Adhesive contact of elastic-plastic spheres. J. Mech. Phys. Solids 48, 2009–2033 (2000) R.D. Mindlin, Compliance of elastic bodies in contact. Trans. ASME J. Appl. Mech. 16, 259–267 (1949) O. Molerus, Theory of yield of cohesive powders. Powder Technol. 12, 259–275 (1975) O. Molerus, Effect of interparticle cohesive forces on the flow behaviour of powders. Powder Technol. 20, 161–175 (1978) R.J. Panton, Flow properties for the continuum viewpoint of a non-equilibrium gas particle mixture. J. Fluid Mech. 31, 273–304 (1968) Y.H. Pao, Extension of the hertz theory of impact to the viscoelastic case. J. Appl. Phys. 26, 1083–1088 (1955) F. Parhami, R.M. McMeeking, A network model for initial stage sintering. Mech. Mater. 27, 111–124 (1998) F. Parhami, R.M. McMeeking, A.C.F. Cocks, Z. Suo, A model for the sintering and coarsening of rows of spherical particles. Mech. Mater. 31, 43–61 (1999) D. Ramkrishna, Population Balances. Theory and Applications to Particulate Systems in Engineering (Academic Press, San Diego, 2000) P. Redanz, N.A. Fleck, The compaction of a random distribution of metal cylinders by the discrete element method. Acta Mater. 49, 4325–4335 (2001) H. Rumpf, K. Sommer, K. Steier, Mechanismen der Haftkraftverstärkung bei der Partikelhaftung durch plastisches Verformen, Sintern und viskoelastisches Fließen. Chem. Ing. Tech. 48, 300–307 (1976) M.H. Sadd, Q. Tai, A. Shukla, Contact law effects on wave propagation in particulate materials using distinct element modelling. Int. J. Non-Linear Mechanics 28, 251–265 (1993) P.G. Saffman, The lift on small sphere in slow sphere flow. J. Fluid Mech. 22, 385–400 (1965) C. Saluena, T. Pöschel, S.E. Esipov, Dissipative properties of vibrated granular materials. Phys. Rev. E 59, 4422–4425 (1999) H. Schubert, K. Sommer, H. Rumpf, Plastisches Verformen des Kontaktbereiches bei der Partikelhaftung. Chem. Ing. Tech. 48, 716 (1976) J.S. Shirolkar, C.F.M. Coimbra, M.Q. McQuay, Fundamental Aspects of Modeling Turbulent Particle Dispersion in Dilute Flows. Prog. Energy Combust. Sci. 22, 363–399 (1996) J. Smagorinsky, General circulation experiment with the primitive equations: part I. The basic experiment. Mon. Weather Rev. 91, 99–164 (1963) B. Storakers, S. Biwa, P.L. Larsson, Similarity analysis of inelastic contact. Int. J. Solids Struct. 34, 3061–3083 (1997)

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B. Storakers, N.A. Fleck, R.M. McMeeking, The viscoplastic compaction of composite powders. J. Mech. Phys. Solids 47, 785–815 (1999) X. Sun, S. Kim, M. Ishii, S.G. Beus, Modeling of bubble coalescence and disintegration in confined upward two-phase flow. Nucl. Eng. Des. 230, 3–26 (2004a) X. Sun, S. Kim, M. Ishii, S.G. Beus, Model evaluation of two-group interfacial area transport equation for confined upward flow. Nuc. Eng. Des. 230, 27–47 (2004b) L.M. Tavares, R.P. King, Modeling of particle fracture by repeated impacts using continuum damage mechanics. Powder Technol. 123, 138–146 (2002) H. Tennekes, J.L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, MA, 1976) C. Thornton, Interparticle sliding in the presence of adhesion. J. Phys. D. Appl. Phys. 24, 1942–1946 (1991) C. Thornton, Coefficient of restitution for collinear collisions of elastic–perfectly plastic spheres. Trans. ASME J. Appl. Mech. 64, 383–386 (1997) C. Thornton, Z. Ning, A theoretical model for stick/bounce behaviour of adhesive elastic-plastic spheres. Powder Technol. 99, 154–162 (1998) C. Thornton, K.K. Yin, Impact of elastic spheres with and without adhesion. Powder Technol. 65, 153–166 (1991) J. Tomas, Particle adhesion fundamentals and bulk powder consolidation. KONA Powder Part 18, 157–169 (2000) J. Tomas, Assessment of mechanical properties of cohesive particulate solids – part 1: particle contact constitutive model. Part. Sci. Technol. 19, 95–110 (2001) J. Tomas, Fundamentals of cohesive powder consolidation and flow. Granul. Matter 6, 75–86 (2004a) J. Tomas, Product design of cohesive powders – mechanical properties, compression and flow behavior. Chem. Eng. Technol. 27, 605–618 (2004b) C. Tsai, D. Pui, B. Liu, Elastic flattening and particle adhesion. Aerosol Sci. Technol. 13, 239–255 (1991) P. Vernier, J.M. Delhaye, General two-phase flow equation applied to the thermohydrodynamics of boiling nuclear reactors. Acta Tech. Belg. Energie Primaire 4, 3–43 (1968) H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics – The Finite Volume Method (Prentice Hall, Pearson Education Ltd., England, 1995) L. Vu-Quoc, X. Zhang, An accurate and efficient tangential force–displacement model for elastic frictional contact in particle-flow simulations. Mech. Mater. 31, 235–269 (1999) L. Vu-Quoc, X. Zhang, O.R. Walton, A 3-D discrete-element method for dry granular flows of ellipsoidal particles. Comput. Methods Appl. Mech. Eng. 187, 483–528 (2000) O.R. Walton, R.L. Braun, Viscosity, Granular Temperature and Stress Calculations for Shearing Assemblies of Inelastic, Frictional Disks. J. Rheology. 30, 949–980 (1986) O.R. Walton, Numerical Simulation of Inelastic, Frictional Particle–Particle Interactions, Particulate Two-Phase Flow (Ed. M. C. Roco), Butterworth–Heinemann, chap. 25, pp. 884–911 (1993) G. Yadigaroglu, R.T. Lahey Jr., On the various forms of the conservation equations in two-phase flow. Int. J. Multiphase Flow 2, 477–494 (1976) W.H. Yang, The contact problem for viscoelastic bodies. Trans. ASME J. Appl. Mech. 33, 395–401 (1966) G.H. Yeoh, C.P. Cheung, J.Y. Tu, Multiphase Flow Analysis Using Population Balance Modelling (Butterworth-Heinemann, Elsevier, 2014)

Recent Advances in Modeling Gas-Liquid Flows Sherman C. P. Cheung, Lilunnahar Deju, and Sara Vahaji

Abstract

Gas-liquid flows are commonly encountered in industrial flow systems. In order to adequately capture the distribution and its effect on the local hydrodynamics in vertical gas-liquid flow, this chapter presents a numerical assessment on seven combinations of six widely adopted bubble coalescence and bubble breakage kernels. Three different coalescence kernels by Coulaloglou and Tavlarides (Chem Eng Sci 32:1289–1297, 1977), Prince and Blanch (AIChE J 36:1485–1499, 1990), and Lehr et al. (AIChE J 48:2426–2443, 2002) have been selected and combined with three different breakage kernels where each kernel considers a different shape of the daughter size distribution of the bubbles such as the U-shape proposed by Luo and Svendsen (AIChE J 42:1225–1233, 1996), the bell-shape proposed by Maritnez-Bazan (J Fluid Mech 401:157–182, 1999a; J Fluid Mech 401:157–182, 1999b), and the M-shape proposed by Wang et al. (Chem Eng Sci 58:4629–4637, 2003). Numerical predictions of the void fraction, bubble size distribution, and interfacial area concentration are compared against the TOPFLOW experimental data (Lucas et al. 2010). Numerical results reveal that the predicted two-phase flow structure is very sensitive to the choice of coalescence and breakage kernels. Bases on the results, the model of Wang et al. (Chem Eng Sci 58:4629–4637, 2003) is found to have a tendency to predict higher breakage rate than the other two kernels. Moreover, although similar order of magnitude of breakage rates are given by Luo and Svendsen (AIChE J 42:1225–1233, 1996) and Martinez-Bazan (J Fluid Mech 401:157–182, 1999a; J Fluid Mech 401:157–182, 1999b), the bell-shape daughter size distribution by Martinez-Bazan (J Fluid Mech 401:157–182, 1999a; J Fluid Mech 401:157–182, 1999b) is found to be favorable for equal breakage of bubbles, leading to overprediction of larger bubbles. S.C.P. Cheung (*) • L. Deju • S. Vahaji School of Engineering, RMIT University, Melbourne, VIC, Australia e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_3-1

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S.C.P. Cheung et al.

Keywords

Gas-liquid flows • Bubble coalescence • Bubble breakage

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Population Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coalescence Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Breakage Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact of the Kernels on the Void Fraction Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact on the Bubble Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation of Interfacial Area Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consideration of Collision Frequency and Coalescence Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . Consideration of Breakage Rate and Daughter Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Greek Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Super/Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 4 8 8 10 13 13 15 15 17 19 19 23 24 25 26 27 27

Introduction Two-phase gas-liquid flows appear in many industrial applications, including chemical, civil, nuclear, mineral, energy, food, pharmaceutical, and metallurgy. Depending on the flow conditions, the two-phase flow pattern could evolve dynamically and transit to different flow regimes. Such transition of flow regimes unfortunately poses significant impact on the system performance or even incurs safety issues in the operation. Aiming to improve the performance and assess the safety of a particular system, it is therefore essential to grasp the phenomenological understanding of bubble size or interfacial area and its dispersion behavior in the complex two-phase flow structures. Subject to local flow conditions, previous experimental studies have shown that bubbles within the bulk liquid flow could undergo significant coalescence and break-up processes leading to a wide spectrum of bubble size distribution. Especially in the transition from the bubbly to slug flow regime, rigorous bubble coalescence and breakup processes gradually transform the bubble size distribution from a single-peaked to a bimodal profile. The bimodel profile signifies the coexistence of the two types of bubbles within the system: spherical and cap/distorted bubbles. Obviously, with different geometrical shape, interfacial area concentration and its corresponding heat and mass transfer processes exhibit substantial difference between the two types of bubbles. More importantly, the shape of bubbles also affects its transversal motion within the system. Based on the previous studies, it has been widely accepted that small bubbles are subject to positive lift

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3

forces and have tendency traveling toward the wall region (Bothe et al. 2006; Tomiyama 1998; Lucas et al. 2007). Cap and distorted bubbles generally travel in opposite direction migrating towards the pipe center, which could eventually turn Taylor bubbles via additional coalescence. Accurate knowledge of these hydrodynamic variables throughout the entire system is thus paramount to the successful design and operation of the system. Developing a robust and reliable mathematical framework for modeling the aforementioned complex flow structure remains as a great challenge at the moment. Theoretically speaking, the evolution of bubble size distribution can be modeled by the population balance equation (PBE) which is expressed in integro-differential form with corresponding coalescence and breakup kernels. The development of population balance model stems from the Boltzmann equation back in the eighteenth century which governs the statistical distribution of particles in a state space. Nevertheless, the generic population balance concept was first introduced in the middle of the nineteenth century. Adopting the statistical mechanics framework, Hulburt and Katz (1964) presented the population balance concept to solve particle size variation due to nucleation, growth, and agglomeration processes. A series of research developments were thereafter presented by Fredrickson et al. (1967), Ramkrishna and Borwanker (1973), and Ramkrishna (1979, 1985) where the treatment of population balance equations were successfully generalized with various internal coordinates. The detailed descriptions of the mathematical and the generic issues of population balance have been documented in the textbook by Ramakrishna (2000). Although the concept of population balance has been formulated over many decades, implementation of population balance modeling was only realized until very recent years. The breakthrough was facilitated by the rapid development of computational fluid dynamics (CFD) and in situ experimental measuring techniques. The flourish of commercial CFD packages in the past decades has made a reliable framework for solving the PBE. The field information obtained from the CFD framework enabled solution algorithms to be developed within the internal coordinates. The advancement in measuring bubble size or population balance variables from experiment has also provided vital information for model calibrations and validations. Among all existing methods, the class method (CM) is widely adopted, in which the internal coordinate (e.g., particle length or volume) is discretized into a finite series of bins. Encouraging results using CM in the form of MUSIG model for bubbly flow simulations can be found in literatures (Frank et al. 2004; Olmos et al. 2001; Pohorecki et al. 2001; Bordel et al. 2006; Krepper et al. 2005, 2007; Cheung et al. 2007, 2013). Although encouraging results have been obtained, all of the aforementioned studies at best have demonstrated the feasibility and performance of various numerical approaches in solving the population balance of bubbles within the two-phase flow domain. Limited emphasis has been devoted towards the understanding and modeling of the mechanisms of bubble coalescence and bubble breakage under different flow conditions. Chen et al. (2005) presented a numerical assessment on the performance of several bubble coalescence and breakage kernels. Predictions by the coalescence models of Chesters and Hoffman (1982), Prince and Blanch (1990), as well as breakage kernels

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of Luo and Svendsen (1996) and Martinez-Bazan et al. (1999a, b) were validated against three different sets of experimental data. Numerical results show that the choice of bubble coalescence and bubble breakage kernels did not pose significant impact on the final predictions. Nonetheless, model validations were mainly focused on the comparison of velocity and turbulent kinetic energy. Limited comparison of the gas phase distribution was presented. Sensitivity of the coalescence and breakage kernels on the bubble size distribution and phase distribution was rather inconclusive. In view of current developments of the state of the art, this chapter aims to further exploit the recent advancements of population balance modeling for multiphase flows from two aspects. First, it attempts to implement some of the widely adopted bubble coalescence and breakage kernels and assess their applicability in predicting the local hydrodynamic variables. Second, it seeks to investigate and understand the physical mechanisms of each kernel and the effect of the interfacial forces affecting the two-phase flow structure with the two-fluid model. A total of six coalescence and breakage kernels are considered. The widely adopted breakage kernels by Luo and Svendsen (1996), Martinez-Bazan et al. (1999a, b), and Wang et al. (2003) are selected. For the coalescence kernels, the models by Coulaloglou and Tavlarides (1977), Prince and Blanch (1990), and Lehr and Mewes (2001) are chosen. Numerical results are assessed against the experimental data by Lucas et al. (2010) for air-water flow in a tall vertical pipe with an inner diameter of 195.3 mm.

Mathematical Models With reference to the formulation of the PBE, one should notice that the left-hand side of the equation denotes the time and spatial variations of the PSD, which depends on the external variables. By incorporating the PBE within CFD solver, external variables can be obtained. In this section, governing equations of the two fluid model and its associated model for handling interfacial momentum and mass transfer are introduced.

Two-Fluid Model The three-dimensional two-fluid model solves the ensemble-averaged of mass, momentum, and energy transport equations governing each phase. Denoting the liquid as the continuum phase (αl) and the vapor (i.e., bubbles) as disperse phase (αg), these equations can be written as: Continuity equation of liquid phase

Recent Advances in Modeling Gas-Liquid Flows


@ρl αl ⇀ þ ∇  ρl αl u l ¼ Γlg @t

5

(1)

Continuity equation of vapor phase   @ρg αg f i ⇀ þ ∇  ρg αg f i u g ¼ Si  f i Γlg @t

(2)

Momentum equation of liquid phase h 

 i
 ⇀ @ρl αl u l þ ∇  ρl αl u⇀l u⇀l ¼ αl ∇P þ αl ρl g⇀ þ∇ αl μe ∇u⇀l þ ∇u⇀l T þ Γlg u⇀g  Γgl u⇀l þ Flg l @t

(3) Momentum equation of vapor phase   h  ⇀
 i
 @ρg αg u g þ ∇  ρ αg u⇀g u⇀g ¼ αg ∇P þ αg ρ g⇀ þ∇ αg μe ∇u⇀g þ ∇u⇀g T þ Γgl u⇀l  Γlg u⇀g þ Fgl g g g @t

(4) Energy equation of liquid phase
   
@ρl αl H l ⇀ þ ∇  ρl αl u l Hl ¼ ∇ αl λel ð∇T l Þ þ Γgl H l  Γlg H g @t

(5)

Energy equation of vapor phase   h
i
 @ρg αg Hg ⇀ þ ∇  ρg αg u g H g ¼ ∇ αg λeg ∇T g þ Γgl H l  Γlg H g @t

(6)

On the right-hand side of Eq. 2, Si represents the additional source terms due to coalescence and breakage. For isothermal bubbly turbulent pipe flows, it should be noted that the mass transfer rate Γlg and Γgl are essentially zero. The total interfacial force Flg appearing in Eq. 3 is formulated according to appropriate consideration of different subforces affecting the interface between each phase. For the liquid phase, the total interfacial force is given by: lift lubrication þ Fdispersion Flg ¼ Fdrag lg þ Flg þ Flg lg

(7)

The subforces appearing on the right hand side of Eq. 7 are: drag force, lift force, wall lubrication force, and turbulent dispersion force. More detailed descriptions of these subforces can be found in Anglart and Nylund (1996). Note that for the gas phase, Fgl =  Flg.

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The lift coefficient CL has been correlated as a function of the Eotvos number, Eo (59); it allows the lift coefficient to be positive or negative depending on the bubble size. CL can be expressed as: 8 Eo < 0:4 < min½0:288tanhð0:121 Reb Þ, f ðEod Þ f ðEod Þ ¼ 0:00105Eo3d  0:0159Eo2d  0:0204Eod þ 0:474 4  Eo  10 CL ¼ : 0:29 Eo > 10

(8) where the modified Eotvos number, Eod ¼

gðρl ρg ÞD2H σ

; DH corresponds to maximum
1=3 horizontal dimension given by Wellek et al. (1966) as DH ¼ Ds 1 þ 0:163 Eo0:757 .

Turbulence Modeling for Two-Fluid Model In handling bubble induced turbulent flow, unlike single phase fluid flow problem, no standard turbulence model is tailored for multiphase flow. For simplicity, the standard k-e model has been employed with encouraging results in early studies (Davidson 1990; Schwarz and Turner 1988). Nonetheless, based on our previous study (Cheung et al. 2012), the Menter’s (1994) k-ω based Shear Stress Transport (SST) model were found superior to the standard k-e model. Similar observations have been also reported by Frank et al. (2004). Based on their bubbly flow validation study, they discovered that standard k-e model predicted an unrealistically high gas void fraction peak close to wall. Interestingly, they also found that the two turbulence models behaved very similar by reducing the inlet gas void fraction to a negligible value. This could be attributed to a more realistic prediction of turbulent dissipation close to wall provided by the k-ω formulation. It revealed that further development should be focused on multiphase flow turbulence modeling in order to better understand or improve the existing models. The SST model is a hybrid version of the k-e and k-ω models with a specific blending function. Instead of using empirical wall function to bridge the wall and the far-away turbulent flow, it solves the two turbulence scalars (i.e., k and ω) explicitly down to the wall boundary. The ensemble-averaged transport equations of the SST model are given as:  
μ t, l @ρl αl kl ⇀  þ ∇  ρl αl u l kl ¼ ∇  αl μl þ ∇kl þ αl Pk, l  ρl β0 kl ωl @t σ k3  
μ t, l @ρl αl ωl ⇀  þ ∇  ρl αl u l ωl ¼ ∇  αl μl þ ∇ωl @t σ ω3 1 @kl @ωl ωl  2ρ1 αl ð1  F1 Þ þ α l γ 3 P k , l  ρ l β 3 ωl 2 σ ω2 ωl @xj @xj kl (9) where σ k3, σ ω3, γ 3, and β3 are the model constants which are evaluated based on the blending function F1. The shear induced turbulent viscosity μts , l is given by:

Recent Advances in Modeling Gas-Liquid Flows

μts, l ¼

pffiffiffiffiffiffiffiffiffiffiffiffi ρa1 kl , S ¼ 2Sij Sij , maxða1 ωl , SF2 Þ

7

(10)

The success of SST model hinges on the use of blending functions of F1 and F2 which govern the crossover point between the k-ω and k-e models. The blending functions are given by: "
4 F1 ¼ tanh Φ1 , Φ1 ¼ min max

! # pffiffiffiffi kl 500μl 4ρl kl , , 2 0:09ωl dn ρl ωl d2n Dþ ω σ ω2 d n ! pffiffiffiffi
 kl 500μl F2 ¼ tanh Φ22 , Φ2 ¼ max , , 0:09ωl dn ρl ωl d2n

(11)

Here, default values of model constants were adopted. More detailed descriptions of these model constants can be found in Menter (1994). In addition, to account the effect of bubbles on liquid turbulence, the Sato’s bubble-induced turbulent viscosity model was also employed (Sato et al. 1981). The turbulent viscosity of liquid phase is therefore given by: μt, l ¼ μts, l þ μtd, l

(12)

and the particle induced turbulence can be expressed as: ⇀  ⇀  μtd, l ¼ Cμp ρl αg DS U g  U l 

(13)

For the gas phase, dispersed phase zero equation model was adopted, and the turbulent viscosity of gas phase can be obtained as: μt, g ¼

ρg μt, l ρl σ g

(14)

where σ g is the turbulent Prandtl number of the gas phase.

Population Balance Equation The foundation development of the PBE stems from the consideration of the Boltzman equation. Such equation is generally expressed in an integrodifferential form describing the particle size distribution (PSD) as follow:

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 @f ðξ, r, tÞ þ ∇  uðξ, r, tÞ f ξ, r, t @tð ð1 1 ξ 0 0 0 0 0 ¼ aðξ  ξ , ξ Þf ðξ  ξ , tÞf ðξ , tÞdξ  f ðξ, tÞ aðξ  ξ0 , ξ0 Þf ðξ0 , tÞdξ0 0 Ð21 0 0 þ ξ γ ðξ Þbðξ0 Þpðξ=ξ0 Þf ðξ0 , tÞds  bðξÞf ðξ, tÞ

(15)

where f(ξ, r, t) is the particle size distribution dependent on the internal space vector ξ, whose components could be characteristics dimensions, surface area, volume, and so on. r and t are the external variables representing the spatial position vector and physical time in external coordinate, respectively. u(ξ, r, t) is velocity vector in external space. On the RHS, the first and second terms denote birth and death rate of particle of space vector ξ due to merging processes, such as: coalescence processes; the third and fourth terms account for the birth and death rate caused by the breakage processes, respectively. a(ξ, ξ0) is the coalescence rate between bubbles of size ξ and ξ0. Conversely, b(ξ) is the breakage rate of bubbles of size ξ. γ(ξ0) is the number of fragments/daughter bubbles generated from the breakage. Two-phase gas-liquid flows of a bubble of size ξ0 and p(ξ/ξ0) represents the probability density function for a bubble of size ξ to be generated by breakage of a bubble of size ξ0. Driven by practical interest, numerical approaches have been developed to solve the PBEs. The most common methods are Monte Carlo methods, method of moments, and class methods. Because of their relevance in CFD applications, the widely adopted MUltiple-Size-Group (MUSIG) model (Lo 1996) has been used as the numerical technique for solving PBE in the present study.

Coalescence Kernels For coalescence between two colliding bubbles of i and j group, the coalescence efficiency a(Mi, Mj) could be calculated as a product of collision frequency, h(Mi, Mj), and coalescence efficiency λ(Mi, Mj).




 Mi , Mj ¼ h Mi , Mj λ Mi , Mj

(16)

In Liao and Lucas (2010) paper, they have listed a variety of mechanisms for the collision frequency among the bubbles in the turbulent flow: • • • • •

Random motion induced collision due to fluctuating turbulent eddies. Collision due to velocity gradient. Collision due to capture in turbulent eddies. Collision due to buoyancy. Collision due to wake interaction.

Recent Advances in Modeling Gas-Liquid Flows

9

Coulaloglou and Tavlarides Coalescence Kernel Coulaloglou and Tavlarides (1997) developed their model based on the consideration of turbulent random motion induced collisions as primary source of bubble coalescence and, the model is formulated according to the film drainage model for deformable particle with immobile surface. The total coalescence rate is given by: "  # 1=2 1
2  2=3 di dj 4 μl ρl ϵ 2= =3 3 a Mi , Mj ¼ C2 d i þ dj di þ dj e exp CC&T  2 σ di þ dj




(17) For the current set of experimental data, the coalescence efficiency parameter (CC & T) was selected as 0:183  1010 cm2 .

Prince and Blanch Coalescence Kernel Turbulent random collision is considered for the bubble coalescence by Prince and Blanch (1990). Based on their model, coalescence process in turbulent flows has been described in three steps. Firstly, the bubbles trap small amount of liquid between them. Then the liquid drains out until the liquid film thickness equals to the critical thickness. The two bubbles are then finally ruptured and merged into one bigger bubble. Similar to particle kinetic theory, the coalescence rate of bubbles can be related to the collision frequency of bubbles and the probability of successful coalescence. The derivation of the kernel can be found out from the paper by Prince and Blanch (1990). The total coalescence rate by Prince and Blanch is calculated as following:  1=2 1

2  2= tij 2= = a Mi , Mj ¼ C3 di þ d j di 3 þ dj 3 e 3 exp  τij

(18)

Here τij is the contact time for two bubbles, and tij is the time required for two bubbles to coalesce having diameters di and dj.

Lehr et al. Coalescence Kernel Lehr et al. (2002) proposed the coalescence frequency based on the critical approach velocity model. An experimental investigation has been conducted to determine the criterion of collision between two bubbles resulting in coalescence or bouncing. They have found that the colliding bubbles might result in coalescence or bounce back depending on the relative approach velocity perpendicular to the surface of contact. They found that the critical approach velocity (ucritical) for distilled water and air is of 0.08 m/s. They have also defined the critical velocity as the maximum velocity of bubbles resulting in coalescence, which has no dependency on the size of the bubbles. Coalescence will only occur when the relative approach velocity of bubbles perpendicular to contact surface is lower than the critical approach velocity. The total coalescence rate is given as following:

10

S.C.P. Cheung et al.

2 !2 3 1=   1=2 1 3 2  2=3 α 2= = 5min ucritical , 1 a Mi , Mj ¼ C4 di þ d j di þ dj 3 e 3 exp4 max  1 u0 α1=3







(19)

Breakage Kernels For breakage of bubble from j group to i group, the partial breakage frequency γ(Mi, Mj) is a function of total breakage frequency, γ(Mi), and the daughter size distribution, β(Mi, Mj).
 γ Mi , Mj β Mi , Mj ¼ γ ðM i Þ




(20)

According to Liao and Lucas (2009) paper, the variety of mechanisms for the breakage of particles in the turbulent flow could be categorized as: • • • •

Turbulent fluctuation and collision Viscous shear force Interfacial instability Shearing off process

Luo and Svendsen Breakage Kernel The bubble breakup rate by Luo and Svendsen (1996) is developed with the assumption of binary breakup under isotropic turbulence influence. Breakage event is determined by the energy level of arriving eddy with smaller or equal length scale compared to the bubble diameter to induce the oscillation. The daughter size distribution is accounted using a stochastic breakup volume fraction fBV. Denoting 2=3 the increase coefficient of surface area as cf = [f BV +(1fBV)2/3–1], the breakage rate in terms of mass can be obtained as: !  1=3 ð 1 12cf σ e ð 1 þ ξ Þ2 γ Mi , Mj ¼ 0:923 1  αg n exp  dξ dj βρe2=3 d j 5=3 ξ11=3 ξ11=3 ξmin









(21)

where ξ = λ/dj is the size ratio between an eddy and a bubble in the inertial subrange and consequently ξmin = λmin/dj and β = 2.0 are defined based on the consideration of bubbles breakup in turbulent dispersion systems. For binary breakage, the value of the dimensionless variable describing breakage volume fraction should be between 0 and 1 (0 < CL ¼ f ðEo⊥ Þ > : 0:27

Eo⊥ < 4 for

4 < Eo⊥ < 10 10 < Eo⊥

(30)

with f ðEo⊥ Þ ¼ 0:00105Eo3⊥  0:0159Eo2⊥  0:0204Eo⊥ þ 0:474: This coefficient depends on the modified Eötvös number given by Eo⊥ ¼

gðρL  ρG Þd2⊥ , σ

(31)

14

R. Rzehak

where d⊥ is the maximum horizontal dimension of the bubble. It is calculated using an empirical correlation for the aspect ratio by Wellek et al. (1966) with the following equation: p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 d⊥ ¼ dB 1 þ 0:163Eo0:757 , (32) where Eo is the usual Eötvös number. The experimental conditions on which Eq. 30 is based were limited to the range  5.5 log10 Mo 2.8, 1.39 Eo 5.74, and values of the dimensionless shear rate 0 Sr ¼ d B ̇ γ=uterm 2. The water-air system at normal conditions has a Morton number Mo = 2.63∙1011 which is quite different, but good results have nevertheless been reported for this case (Lucas and Tomiyama 2011).

Wall Force A bubble translating next to a wall in an otherwise quiescent liquid also experiences a lift force. This wall lift force, often simply referred to as wall force, has the general form Fwall ¼

2 CW ρL αG juG  uL j2 ^y , dB

(33)

where ^ y is the unit normal perpendicular to the wall pointing into the fluid. The dimensionless wall force coefficient CW depends on the distance to the wall y and is expected to be positive so the bubble is driven away from the wall. Based on the observation of single bubble trajectories in simple shear flow of a glycerol water solution, Tomiyama et al. (1995) and later Hosokawa et al. (2002) concluded a functional dependence CW ðyÞ ¼ f ðEoÞ


2 dB : 2y

(34)

In the limit of small Morton number, the correlation f ðEoÞ ¼ 0:0217Eo

(35)

can be derived from the data of Hosokawa et al. (2002). The experimental conditions on which Eq. 35 is based are 2.2 Eo 22 and 6.0 log10 Mo 2.5 which is still different from the water-air system with Mo = 2.63∙1011, but good predictions have been obtained also for air bubbles in water (Rzehak et al. 2012).

Turbulent Dispersion Force The turbulent dispersion force describes the effect of the turbulent fluctuations of liquid velocity on the bubbles. Burns et al. (2004) derived an explicit expression by Favre averaging the drag force as

Euler-Euler Modeling of Poly-Dispersed Bubbly Flows

F

disp


 3 αG μturb 1 1 L ¼  CD juG  uL j þ gradαG : 4 dB σ TD αL αG

15

(36)

In analogy to molecular diffusion, σ TD is referred to as a Schmidt number. In principle it should be possible to obtain its value from single bubble experiments also for this force by evaluating the statistics of bubble trajectories in wellcharacterized turbulent flows, but to our knowledge this has not been done yet. A value of σ TD = 0.9 is typically used.

Virtual Mass Force When a bubble is accelerated, a certain amount of liquid has to be set into motion as well. This may be expressed as a force acting on the bubble as FVM ¼ CVM ρL αG


 DG uG DL uL  , Dt Dt

(37)

where DG / Dt and DL / Dt denote material derivatives with respect to the velocity of the indicated phase. For the virtual mass coefficient, a value of CVM = 0.5 has been derived for isolated spherical bubbles in inviscid and creeping flows by Auton et al. (1988) and Maxey and Riley (1983), respectively. Results of direct simulations of a single bubble by Magnaudet et al. (1995) suggest that this value also holds for intermediate values of Re.

Turbulence Modeling Due to the low density and small spatial scales of the dispersed gas, it suffices for bubbly flows to consider turbulence in the continuous liquid phase. Two contributions to the turbulent fluctuations have to be taken into account, which are referred to as shear-induced and bubble-induced turbulence. Despite some discussion in the literature (e.g., Rensen et al. 2005, Riboux et al. 2010), we here treat both contributions as indistinguishable and describe them by means of a single total turbulent kinetic energy. To this end a standard two-equation model is used, which is known to work well for single phase flows where only shear-induced turbulence exists. Details are given in section “Basic Turbulence Model”. To account for the bubble-induced contribution, this model is augmented with suitable source terms which are described in section “Source Terms for Bubble-Induced Turbulence”. To avoid the need to resolve the viscous sublayer near solid walls, a turbulent wall function is applied. Lacking definite results for two-phase flows this is presently taken the same as for single phase flow. Details are given in section “Turbulent Wall Function”.

16

R. Rzehak

Basic Turbulence Model Turbulence in bubbly flows is here described by an SST model (Menter et al. 2003, Menter 20091) with additional source terms for the bubble-induced contribution. The SST model is a wall-distance dependent blend of k-ε and k-ω models that combines the respective advantages of both. Usually k and ω are employed as independent variables by noting that the ε-equation may be transformed to an equivalent equation for ω that contains a cross-diffusion term, which is not present in the usual ω-equation. Hence the equations for the turbulent kinetic energy kL and the turbulent frequency ωL to be solved are     @ 1 turb ðαL ρL kL Þ þ ∇  ðαL ρL uL kL Þ ¼ ∇  αL μmol ∇kL L þ σ k μL @t   þαL Pk  Cμ ρL ωL kL þ SkL     @ 1 turb ∇ω ðαL ρL ωL Þ þ ∇  ðαL ρL uL ωL Þ ¼ ∇  αL μmol L þ σ ω μL @t
 ρ Pk ∇kL  ∇ωL þαL CωP Lturb  CωD ρL ω2L þ αL 2 σ 1 þ SωL : ω2 ρL ð1  F1 Þ ωL μL

(38)

(39)

Here F1 denotes the blending function, which assumes a value of one for the k-ω model and zero for the k-ε model. It is defined as
pffiffiffiffiffi  114 3 kL 500μmol L 6B CC 7 B max Cμ ωL y , ρ ωL y2 , 6B L CC 7 B 6B CC 7 B F1 ¼ tanh6BminB 1 CC 7: 4σ ω2 ρL kL 6@ @   AA 7 5 4 10 2 L y2 max σω2 ρL ∇kLω∇ω , 1:0  10 L 20

0

(40) 1 The model constants Cμ, CωP, CωD, σ 1 k , and σ ω are also interpolated between the corresponding values of the k-ω model (index “1”) and the k-ε model (index “2”) using the blending function F1 as

χ ¼ F1 χ 1 þ ð1  F1 Þχ 2 :

(41)

All turbulence model parameters take their usual single phase values as summarized in Table 1. Note that these values deviate slightly from those commonly used for the k-ω and k-ε models alone (NASA 2014). In terms of the strain rate tensor

1 Note that the ANSYS CFX User Guide (ANSYS 2012) quotes (Menter 1994) but describes the later modifications as implemented in the code.

Euler-Euler Modeling of Poly-Dispersed Bubbly Flows

17

Table 1 Parameter values for k-o and k-E models k-ω model (index “1”) k-ε model (index “2”)

Cμ 0.09 0.09

CωP 0.5532 0.4463

CωD 0.075 0.0828

σ1 k

σ1 ω

0.85034 1.0

2.0 0.85616

  S ¼ 1=2 ∇uL þ ð∇uL ÞT ,

(42)

the production term is Pek ¼ 2μturb L S : uL but a limiter is introduced to prevent the build-up of turbulent kinetic energy in stagnation zones so that   Pk ¼ min Pek , 10Cμ ρL ωL kL :

(43)

Since bubble-induced effects are included in k and ω due to the respective source terms, the turbulent viscosity is evaluated from the standard relation of the SST pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi model which includes a limiter based on the generalized shear rate γ ¼ 2 S : S μturb ¼ L

ρ k  L L , max ωL , Cγ F2 γ

(44)

where F2 is a second blending function defined as "
F2 ¼ tanh


pffiffiffiffiffi 2 # 2 kL 500μmol L , max Cμ ωL y ρL ωL y2

(45)

and Cγ = 1 / 0.31 is a further model constant. The effective viscosity is simply μLeff = μLmol þ μLtrub. Boundary conditions on k and ω are taken the same as for the single phase case, which is consistent with the view that the full wall shear stress is exerted by the liquid phase, as it contacts the complete wall area.

Source Terms for Bubble-Induced Turbulence Concerning the source term SkL describing bubble effects in the k-equation, there is large agreement in the literature (e.g., Kataoka et al. 1992, Troshko and Hassan 2001). A plausible approximation is provided by the assumption that all energy lost by the bubble due to drag is converted to turbulent kinetic energy in the wake of the bubbles. Hence, the k-source becomes SkL ¼ Fdrag  ðuG  uL Þ: L

(46)

The source term SLe that describes the effect of bubble-induced turbulence in the ε-equation of k-ε models is derived using similar heuristics as for the single phase case, namely, the k-source is divided by some time scale τ so that

18

R. Rzehak

SeL ¼ CeB

SkL : τ

(47)

Further modeling then focuses on the time scale τ proceeding largely based on dimensional analysis. This follows the same line as the standard modeling of shearinduced turbulence in single phase flows (Wilcox 1993), where production terms in the ε-equation are obtained by multiplying corresponding terms in the k-equation by an appropriate time scale that represents the lifetime of a turbulent eddy before it breaks up into smaller structures. In single phase turbulence the relevant variables are obviously k and ε from which only a single time scale τ = kL/eL can be formed. For the bubble-induced turbulence in two-phase flows, the situation is more complex and several plausible expressions for the time scale are conceivable. In the absence of theoretical arguments to decide which of these is the most relevant one, a comparison of all four alternatives has shown the best performance for the choice τ = dB /√ kL (Rzehak and Krepper 2013a, b). For the coefficient CeB a value of 1.0 was found to give reasonable results. For use with the SST model, the ε-source is transformed to an equivalent ω-source which gives SωL ¼

1 e ωL k S  S : Cμ k L L k L L

(48)

This ω-source is used independently of the blending functions in the SST model since it should be effective throughout the fluid domain.

Turbulent Wall Function To avoid the need to resolve the viscous sublayer, a single phase turbulent wall function assuming a smooth wall is applied, which consists of a blend between inertial and viscous sublayers (ANSYS 20122). This treatment is facilitated by the availability of analytical solutions for both sublayers in the k-ω model (Wilcox 1993, Sect. 4.6.3) that cover the near wall region in the SST model. It allows mesh refinement near the wall to a degree that the viscous sublayer becomes resolved. Since turbulence is considered in the liquid phase only, all variables in the following presentation refer to this phase, but the index “L” has been dropped for notational convenience. In terms of variables nondimensionalized with the friction velocity and corresponding viscous length, denoted by subscript “+”, the analytical solutions for the viscous and inertial sublayers are uviscous ¼ yþ , þ

2

(49)

This approach is termed automatic near-wall treatment in the ANSYS CFX User Guide. No reference is quoted there and only partial accounts could be found in the literature (e.g., Vieser et al. 2002, Esch et al. 2003).

Euler-Euler Modeling of Poly-Dispersed Bubbly Flows

 1  uinertial ¼ ln Eyþ , þ κ

19

(50)

where κ = 0.41 and E = 9.8 for smooth walls. A simple interpolation between both regimes is furnished by ucompound ¼ þ

h

uviscous þ

4

 4 i1=4 þ uinertial : þ

(51)

If u+ is known, the friction velocity can be obtained by straight forward inversion of Eq. 51 or equivalently from ucompound ¼ τ

h

uviscous τ

4

 4 i1=4 þ uinertial , τ

(52)

where uτviscous and uτinertial are found from inverting Eqs. 49 and 50, respectively. In case u+ vanishes, the friction velocity and the viscous length scale become ill-defined. This may happen at points where the boundary layer separates. To alleviate this issue, following Launder and Spalding (1974) the alternative velocity scale 1=2 uinertial ¼ C1=4 k μ k

(53)

is introduced for the inertial sublayer. Variables nondimensionalized with this alternative velocity and corresponding viscous length will be denoted by subscript “*”. With this modification, explicit expressions for the two terms in Eq. 52 are uviscous τ

sffiffiffiffiffiffiffiffiffi μ j uj ¼ ρy

¼ uinertial τ

κ juj : lnðE y Þ

(54)

(55)

In addition, the overall alternative velocity scale is defined by ucompound ¼ k

h

uviscous τ

4

 4 i1=4 þ uinertial : k

(56)

Now for u and k the known values at the wall adjacent grid cell are taken, uτ and uk are calculated, and from these the wall shear stress τW ¼ ρuτ uk ,

(57)

which provides the flux of momentum into this cell through the wall. Similarly to the above, the turbulent frequency ω is computed from the solutions in the viscous and inertial sublayer

20

R. Rzehak

ωviscous ¼

6μ , CωD1 ρ y2

1 uk ωinertial ¼ pffiffiffiffiffiffi  , Cμ  κ y

(58) (59)

however, with a different blending, namely ωcompound ¼

h 2  2 i1=2 ωviscous þ ωinertial :

(60)

The value from the analytical solution is specified in the wall adjacent grid cell for ω. The boundary condition for k is a vanishing normal derivative at the wall.

Other Model Aspects This section collects some aspects which do not strictly belong to closure modeling in the sense that they would describe bubble-scale phenomena that have been averaged out in the two-fluid model. However, they are necessary ingredients for any computation and will certainly have an impact on the results obtained. Hence, validation of the closures as presented in the previous sections also depends on these aspects of the overall model. These miscellanies comprise the calculation of the wall distance (section “Wall Distance Calculation”) and inlet and outlet conditions (section “Inlet and Outlet Conditions”).

Wall Distance Calculation An often overlooked and somewhat fuzzy aspect of bubbly flow modeling in complex geometries as they frequently occur in engineering applications is the calculation of the normal wall distance, which is required for both the blending functions for the SST turbulence model (see section “Basic Turbulence Model”) and the wall force (see section “Wall Force”). For a simple geometry, one might simply take the closest distance to the nearest wall, but for complex geometries involving curved boundaries or sharp corners, a more global measure is needed. Such a global measure is obtained by solving a partial differential equation for a distance function ϕ and computing the normal wall distance y from ϕ (see e.g., Tucker 2003). An admittedly crude but widely used approach attributed by Liu et al. (2010) to a contribution by Spalding (1994), which however apparently never appeared in print, is as follows. The distance function is obtained as solution to the Poisson equation ∇2 φ ¼ 1:

(61)

Euler-Euler Modeling of Poly-Dispersed Bubbly Flows

21

On walls a Dirichlet boundary condition φ = 0 is specified and on other types of boundaries a homogeneous Neumann boundary condition @@ φn ¼ 0 is used. The wall distance y is then computed from y ¼ j∇φj þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j∇φj2 þ 2φ,

(62)

which gives the exact result for the one-dimensional case.

Inlet and Outlet Conditions While conditions at solid walls or symmetry planes are determined from fundamental physical principles as discussed in section “Initial and Boundary Conditions”, conditions at inlets and outlets are specific to each case. For model validation these conditions are best taken from measurements to eliminate any error. For predictive simulations, however, this is not possible and additional modeling is necessary. Commonly used prescriptions are described in the following. Typically volumetric fluxes JL and JG of liquid and gas through a reference area Aref are given as part of the problem specification. Care must be taken correctly to consider the reference area. Inlet conditions for the liquid velocity and turbulence are frequently set to fully developed single phase flow profiles corresponding to the required liquid volume flux. In general these profiles have to be calculated numerically. This calculation may be combined with the simulation at hand by introducing a sufficiently long flow development zone. For a round pipe with radius R, a useful approximation of the normal velocity u is given by the empirical power law profile (Schlichting 1979, p. 599) u umax

 r 1=7 ¼ 1 , R

(63)

where the peak value umax is calculated such that the desired volume flux JL results. The turbulent kinetic energy k is often computed from the liquid velocity and a specified value for the turbulence intensity Iturb as 3 k ¼ ðI turb juL jÞ2 : 2

(64)

A common value used for the turbulence intensity is Iturb = 5%. The turbulent frequency ω is computed from the turbulent kinetic energy k and the integral turbulent length scale Lturb as ω ¼ C1 μ

k1=2 : Lturb

(65)

For pipe flows, the integral turbulent length scale is often estimated as Lturb = 0.2 R.

22

R. Rzehak

Concerning the gas two options are commonly used. For the first, the gas is assumed to be distributed over the full cross section at the inlet. In this case the inlet areas for gas and liquid coincide and a value for the gas fraction must be specified. The gas velocity may be set to a constant value or a profile that corresponds to the liquid velocity shifted by the relative velocity of the bubbles. The constant value or the relative velocity, respectively, is determined to match the desired gas flux JG. The gas is taken to be uniformly distributed over the inlet so that the gas fraction is constant. Assuming that the bubbles enter with zero relative velocity to the liquid, its value is obtained in terms of the given volume fluxes JL and JG as αG ¼

JG , JG þ JL

(66)

whereas for a nonvanishing relative velocity
 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1 JG þ JL 1 JG þ JL 2 JG 1þ 1þ :  α¼  2 4 urel urel urel

(67)

For the second option, the true area through which gas enters is taken as the gas inlet. Since only gas enters, the gas fraction is specified as 1. The liquid inlet then must exclude this area and a gas fraction of 0 is specified there. The gas velocity is typically set to a constant value over the gas inlet. If the gas injector works in the bubbling regime, a suitable value is obtained as uG ¼ J G

4Aref : πd2B

(68)

Precise conditions at the inlet do not matter as long as the axial distance to the measurement location is large enough for fully developed conditions to be attained. That this is indeed the case may be checked by looking at the axially resolved fields. At an outlet, typically a constant pressure is prescribed while the normal derivatives of the tangential velocity components of gas and liquid are set to zero. This still allows the normal derivative of the normal components as well as the values of all velocity components of both liquid and gas velocities to adjust in the simulation. For other variables, a vanishing normal derivative is prescribed. For flow in pipes or ducts, this corresponds with the assumption of fully developed flow. For cases like bubble columns, where the liquid cannot leave the domain, a common alternative is the so-called degassing condition. There the liquid normal velocity is set to zero which means that now the pressure becomes part of the solution. Its distribution over the outlet area can be considered to describe the effect that while the fluid surface is taken fixed in the simulation, it may deform in reality.

Euler-Euler Modeling of Poly-Dispersed Bubbly Flows

23

If the outlet is sufficiently far away from the measurement location, the outlet condition should have no influence on the observed results. If necessary an artificial flow abatement zone may be added to the physical domain to ensure this. A length of ~10% of the main flow section is a reasonable practical estimate for the required length.

Model Validation The closure models defined in the previous sections have been implemented in ANSYS-CFX and also in OpenFOAM (Rzehak and Kriebitzsch 2015). The basic equations which can be augmented by user-defined closures are available also in other closed and open software packages. Several applications to bubbly pipe flow and flow in bubble columns will be presented in the following to illustrate the state of the art. An overview of the pertinent experimental conditions is given in Table 2, where D is the pipe or column diameter or width, JL and JG are liquid and gas volume fluxes, 〈dB〉 is the average bubble size, and 〈αG〉 the average gas fraction. The results are taken from previous publications (Rzehak and Krepper 2013a, b; Rzehak et al. 2014, 2015; Rzehak and Krepper 2015; Ziegenhein et al. 2015). All of these applications are limited to upward vertical flows in simple geometries and air bubbles in water (see Zidouni et al. 2015; Liao et al. 2016; Rzehak et al. 2017a, 2017b, for more complex applications). The bubble size or its distribution is taken from the experiment rather than being modeled reflecting the state of model development. While it may be reasonably assumed that other materials are also well described due the use of nondimensional numbers in the correlations, other geometries and boundary conditions as well as largely different parameter values require additional validation. Many works on such cases using different models may be found in the literature as well, a review of which, however, is beyond the scope of this chapter. Depending on the test under investigation, different setups were used. The calculations were made either in stationary mode imposing plane/axisymmetric conditions by considering only a thin slice/sector of the domain together with symmetry conditions or in transient mode with subsequent averaging of the results and fully 3D on the same domain as the experiments. The reason to choose the stationary or quasi-2D approximation whenever applicable is that it drastically reduces the computation time. For the transient simulations, the reported quantities are averages over the statistically steady state. At the inlet a uniform distribution of gas throughout the cross section was assumed or the injection nozzles or needles were modeled as individual surface patches. For the liquid, fully developed single phase velocity and turbulence profiles were assumed in the pipe flow cases. At the top, a pressure boundary condition was set for the pipe flow cases while the degassing condition was employed for the bubble column cases.

24 Table 2 Main experimental conditions for the selected test cases

R. Rzehak

Name

D JL JG m s1 mm m s1 Liu (1998): round pipe L21B 57.2 1.0 0.14 L21C 57.2 1.0 0.13 L22A 57.2 1.0 0.22 L11A 57.2 0.5 0.12 MTLoop: round pipe MT039 51.2 0.4050 0.0111 MT041 51.2 1.0167 0.0115 MT061 51.2 0.4050 0.0309 MT063 51.2 1.0167 0.0316 Mudde et al. (2009): round bubble column M1 150 – 0.015 M2 150 – 0.017 M3 150 – 0.025 M4 150 – 0.032 M5 150 – 0.039 M6 150 – 0.049 TOPFLOW: round pipe TL12–041 195.3 1.017 0.0096 bin Mohd Akbar et al. (2012): flat bubble column A1 240 – 0.003 A2 240 – 0.013

〈dB〉 mm

〈αG〉 %

3.03 4.22 3.89 2.94

10.6 9.6 15.7 15.2

4.50 4.50 4.50 4.50

1.89 1.00 5.03 2.64

4.02 4.06 4.25 4.47 4.53 4.44

6.1 7.6 11 16 20 25

4.99

1.1

4.3 5.5

1.4 6.2

Concerning bubble size, a monodisperse approximation was used whenever the bubbles are smaller than the diameter of ~6 mm where the lift force changes its sign for the water-air system (cf. section “Lift Force”). In the other cases, two size and velocity groups corresponding to bubbles smaller and larger than 6 mm were used. If there is a significant variation of pressure within the domain, the gas density will change according to the ideal gas law and consequently the bubble size changes, since mass is conserved. Yet the flow of both gas and liquid remains incompressible to a good approximation. It is possible to keep the computational advantage of treating both gas and liquid as incompressible fluids with constant material properties if a fully developed flow is considered by adjusting the gas flux at the inlet to the value obtained by evaluating D=2 ð

J G ¼ 8=D

αG ðr ÞuG ðr Þrdr

2

(69)

0

using the data at the measurement location (Rzehak et al. 2012). In cases in which only uL but not uG has been measured, an estimate of the latter may be obtained from

Euler-Euler Modeling of Poly-Dispersed Bubbly Flows

25

the former and αG based on the assumption of fully developed stationary flow. Where this procedure has been applied, the adjusted values are given in Table 2. Turbulence data frequently give the axial intensity of turbulent fluctuations while in the simulations based on two-equation models only the turbulent kinetic energy is available. For a comparison, it has to be considered that wall-bounded turbulence is anisotropic with the axial component of fluctuating velocity being larger than those in radial and azimuthal directions. Data on axial and radial components for typical turbulent two-phase flow conditions are given, e.g., in Michyoshi and Serizawa (1986), Wang et al. (1987), or Shawkat et al. (2008), while data on the azimuthal component could not be found in the literature. Concerning turbulent kinetic energy k, an estimate of the ratio √ k / u’ may be obtained from the quoted works by assuming that azimuthal and radial components are equal as √ k / u’  0.8 . . .1.2. The lower bound is a bit larger than the value √(1/2)  0.71 corresponding to the unidirectional limiting case while the upper bound is almost the same as √(3/2)  1.22 obtained for isotropic turbulence. Taking u’ as an estimate for √ k thus provides an estimate that is accurate to within ~20%.

Tests of Liu (1998) The system studied by Liu (1998) is vertical upflow of water and air in a round pipe with inner diameter D = 57.2 mm at a temperature of 26 C and presumably atmospheric pressure. The total length of the flow section was H = 3.43 m. Gas was delivered through a tube of 9.7 mm inner diameter located at the center of the main pipe. A special gas injector was used that allowed to adjust the bubble size independently of liquid and gas superficial velocities. A variety of combinations of these three parameters are available. Radial profiles of gas fraction, mean bubble-size, axial liquid velocity, and axial liquid turbulence intensity were measured at an axial position H / D = 60 corresponding to fully developed conditions. In Rzehak and Krepper (2013a, b) stationary axisymmetric simulations were done assuming a uniform distribution of gas throughout the pipe cross-section at the inlet. Since the pressure effect is significant for the 3.43 m long pipe, gas volume fluxes were adjusted to allow treating the gas as incompressible. A selection of test cases was made for which a monodisperse bubble size distribution is sensible. The bubble size was set equal to the average of the measured profiles. The comparison of calculated and measured profiles in Fig. 2 shows reasonable agreement for the gas fraction αG and the axial liquid velocity uL. Notable deviations occur in the region close to the wall where the simulations predict the peak in the gas fraction too high and the gradient of the liquid velocity too steep. For the turbulent kinetic energy kL, the agreement between simulation and measurement is not as good, but possibly to some extent due to the isotropic approximation.

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Fig. 2 Fully developed gas fraction αG, axial liquid velocity uL, and square root of turbulent kinetic energy √ kL for the tests of Liu (1998). Lines: simulation, symbols: experiment

MTLoop Tests The MTLoop facility described in detail in Prasser et al. (2003) and Lucas et al. (2005) was used to study upward vertical flow of air and water at a slightly elevated temperature of 30 C and atmospheric pressure. The test section consists of a circular pipe with inner diameter D = 51.2 mm. Gas was injected through up to 19 needles depending on the required gas volume flux. The distance between the gas injector and the measurement location was varied between H = 0.03 m and H = 3.03 m which corresponds to a ratio H / D  60. Thus at the highest level fully developed flow may be expected. Radial profiles of gas fraction and gas velocity as well as distributions of bubble size were measured by a wire-mesh sensor. A large number of combinations of liquid and gas volumetric fluxes JG and JL were investigated.

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Simulations of four selected tests in the bubbly flow regime have been performed with a setup that aims at calculating the fully developed flow at the end of the test section (Rzehak et al. 2015). To this end, the gas volumetric flux was adjusted to capture the gas expansion due to the drop of hydrostatic pressure over the height of the pipe and the bubble size distribution measured at this location was imposed. To capture the different direction of motion for large and small bubbles according to the lift-force, two velocity groups were used for bubbles larger and smaller than 6 mm in size. Values of bubble size dj and relative amount fj of both classes for all tests are given in Table 3. At the inlet, a uniform distribution of gas was imposed and axisymmetric flow was assumed so that only a narrow sector of the pipe needs to be considered as the computational domain. Measured and calculated radial profiles of gas fraction αG and gas velocity uG are compared in Fig. 3. For the gas fraction the overall agreement with the measured values is quite reasonable. Here the peak near the wall is predicted quite well, but deviations are seen in the pipe where the predicted gas fraction is sometimes too high and sometimes too low. For the gas velocity the agreement between simulation and experiment is very good. Table 3 MUSIG group sizes dj and relative amounts fj for MTLoop simulations

Case MT039 MT041 MT061 MT063

d1 [mm] 4.85 4.79 4.89 5.00

d2 [mm] 6.52 6.51 6.69 6.54

f1 0.9364 0.9621 0.7964 0.8783

f2 0.0636 0.0379 0.2036 0.1217

Fig. 3 Fully developed gas fraction αG and gas velocity uG, for the MTLoop tests. Lines: simulation, symbols: experiment

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Table 4 Length of test section for different levels of gas injection for the TOPFLOW tests Level H H/D

[mm] [-]

A 221 1.1

C 335 1.7

F 608 3.1

I 1552 7.9

L 2595 13.3

O 4531 23.2

R 7802 39.9

TOPFLOW Tests The TOPFLOW facility has been specifically designed to obtain high quality data for the validation of CFD models. The tests used here (Lucas et al. 2010) have been run for cocurrent vertical upward flow of air and water in a round pipe with an inner diameter of D = 195.3 mm. Measurements were made by a wire mesh sensor at the top end of the pipe while gas injection occurs at different levels below. The operating conditions were set to a temperature of 30 C and a pressure of 0.25 MPa at the location of the active gas injection. In this way the flow development can be studied as it would be observed for gas injection at a fixed position and measurements taken at different levels above. Distances H between the injection devices and the sensor are given in Table 4 for the different levels. The values of mean bubble size and average gas fraction given in Table 2 correspond to the highest measurement level R. Instrumentation with a wire mesh sensor allows collection of data on radial profiles of gas - fraction and gas - velocity as well as distributions of bubble size. A large range of liquid and gas superficial velocities was investigated in which all flow regimes from bubbly to annular occur. In the detailed report (Beyer et al. 2008) it has been noted that for bubbly flows the gas volume fluxes calculated from the measured profiles by integrating the product of gas fraction and velocity were systematically larger than those measured by the flow meter controlling the inlet. This deviation is likely to be caused by the distance between the sending and receiving wire planes, which leads to an increased value of gas fraction, but a detailed explanation is not available yet. The ratio of the values calculated from the profiles (cf. Eq. 37) to the values measured directly at the inlet has an approximately constant value of 1.2 over the bubbly flow regime (Beyer et al. 2008, Fig. 1–19). In the simulations the values measured by the flow meter will be used to set the inlet boundary condition. To get the same integral value of this conserved quantity for each cross-sectional plane, all measured void gas fractions are divided by 1.2 throughout this section. For this data set, the investigation of developing flow is possible. To this end tests have been selected by (Rzehak and Krepper 2015) in which a significant polydispersity is present but the distribution of bubble masses does not change appreciably and a treatment by fixed polydispersity is possible. The change in bubble size due to the decrease in hydrostatic pressure with height is included in the simulations. Stationary axisymmetric simulations were done with the gas inlet modeled as individual nozzles. Two MUSIG groups were used with sizes of 4.4 mm and 6.6 mm at the inlet and relative amounts of 94.4% and 5.6%. As already mentioned, the sizes increase with height due to the calculation based on mass discretization and the pressure effect on gas density.

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Fig. 4 Developing gas fraction αG and gas velocity uG, for the TOPFLOW test case TL12–041. Top: experimental data corrected as described in the text, bottom: simulation results

Results for the development of gas fraction αG and gas velocity uG for one of the cases from Rzehak and Krepper (2015) are shown in Fig. 4. It can be seen that near the inlet the wall peak in the gas fraction is underestimated by the simulation but at the higher levels it is overpredicted. Likewise the initial width of the peak comes out too broad in the simulations but the shoulder that develops subsequently has a narrower range than in the experiments. In particular at the highest level, R, the gas has reached the pipe center in the experiment but not yet in the simulation. For the gas velocity, experimental values exist only where a certain minimum amount of gas is present. The simulated profiles are in reasonable accord with the data up to level O. A large deviation is seen at the highest level, R, which probably corresponds with the gas not being distributed over the full pipe radius.

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Tests of Mudde et al. (2009) The setup of Mudde et al. (2009) consists of a round bubble column with diameter D = 150 mm again operated with air and water at ambient conditions. The ungassed fill height was H = 1.3 m. Measurements of gas fraction and axial liquid velocity profiles were taken at different levels of which the one at 0.6 m above the inlet has been chosen for the comparison here. The sparger was designed specifically to provide highly uniform inlet conditions. Several values of the gas superficial velocity are available reaching rather large values of gas fraction. The mean bubble size and variation around it have been measured at two locations close to the inlet and close to the top water level. Since a slight increase is observed, the average value of both measurements corresponding to the middle level has been used in the simulations. Transient 3D simulations of these tests were performed by Rzehak et al. (2014) assuming a uniform distribution of gas at the inlet. The height of the computational domain was obtained by adding the average gas volume from the measurements to the volume of liquid. A monodisperse bubble size distribution corresponding to the measured values was used. For the column height of 1.3 m, the pressure effect is still small enough to be neglected. Measured and calculated values are compared in Fig. 5. Clearly the gas fractions αG are predicted within the experimental errors. The calculated liquid velocity profiles uL do not depend on the total gas hold up. Since the measured profiles do not show any systematic trend as a function of this variable, their variation is most likely an indication of the measurement errors.

Fig. 5 Gas fraction αG, axial liquid velocity uL, for the tests of Mudde et al. (2009). Lines: simulation results, symbols: measured values. Only half of the column is shown

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Tests of bin Mohd Akbar et al. (2012) The experiments of bin Mohd Akbar et al. (2012) were conducted in a flat bubble column of width D = 240 mm using air bubbles in water at ambient conditions. Without gas supply the water level was at H = 0.7 m. Profiles of gas fraction, axial liquid velocity, and axial turbulence intensity as well as the bubble size distribution were measured at a plane 0.5 m above the inlet. The bubble size distribution in addition was measured also near the inlet. Two values of superficial gas velocities JG are available. From the measured bubble size distributions, it can be seen that no significant change occurs over the column height. Hence a treatment by fixed polydispersity is possible. Due to the small column height the pressure effect on material properties is negligible. In particular the gas density remains constant to a good approximation so that there is no expansion of the bubbles. The simulations were run in transient (URANS) mode on the full 3D domain (Ziegenhein et al. 2015). The resulting data were averaged over a sufficiently long period to obtain stationary averages. For the evaluation of turbulent kinetic energy, the axial component of the resolved transient fluctuations has been added to the unresolved part obtained from the turbulence model. The change in the water level due to the gas is small enough not to affect the flow at the measurement level and was hence neglected. Individual needles of the sparger were represented by area elements roughly corresponding to the size of the bubbles. For the lower value of gas volume flux JG, a monodisperse bubble size distribution was imposed. For the higher value, two MUSIG size and velocity groups were used with diameters of 5.3 mm and 6.3 mm and relative amounts of 63% and 37%, respectively. This treatment allows to capture the effect of the sign change of the lift coefficient (see section “Lift Force”), which for air bubbles in water takes place at a bubble size of dB  6 mm. A comparison between simulation results and measured data is shown in Fig. 6. As may be seen, the agreement between both is quite good for gas fraction αG and axial liquid velocity uL. Slight differences are that the predicted gas fraction profile is a little bit too peaked near the wall and there is a small dip in the predicted liquid velocity in the center of the column. The turbulent kinetic energy in the column center is somewhat underpredicted by the simulations, and the peak in kL near the wall is not reproduced by the simulations.

Discussion and Conclusions As shown by the examples discussed in section “Model Validation,” in its present state the Euler-Euler simulation of polydispersed bubbly flows may be used for case studies, the design of better experiments, to enhance understanding in combination with experiments, and within a limited range for prediction. In the future, full predictive capability and use for optimization are envisaged, but further model development and validation are needed to reach this goal.

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Fig. 6 Gas fraction αG, axial liquid velocity uL, and square root of turbulent kinetic energy √kL for the tests of bin Mohd Akbar et al. (2012). Lines: simulation results, symbols: measured values. Only half of the column is shown

Concerning the basic framework presented in section “Euler-Euler Framework of Interpenetrating Continua,” the most important desideratum would be a stronger link between terms derived by averaging and closure models. At present this replacement is treated heuristically. Possible approaches that allow a more fundamental correspondence to be exploited are discussed, e.g., in Prosperetti (1998) and Buffo and Marchisio (2014). In this respect it would also be desirable to relax the point particle approximation. A possible way to achieve this is discussed in Tomiyama et al. (2006, Sect. 4.2ff) together with the expected benefits. For the closure models discussed in section “Baseline Closure Relations” concerning bubble forces, the biggest issue in view of applications to industrial problems is probably an extension of the range of applicability towards higher gas fractions by the inclusion of so-called swarm effects. Suggestions in this direction have been made, e.g., in Roghair et al. (2011) showing that direct numerical simulations provide a viable tool to improve closures on bubble forces. This could also be used to extend the range of applicability towards larger bubbles. Another

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aspect requiring further developments is a general lift force combining shear-, wall-, and other effects. Concerning turbulence including the turbulent dispersion force, there is obviously a great need for model improvement until a similar state as in single phase flows is reached. This comprises better source term models for the k-ε/ω models as well as two-phase wall functions and also the nature of the turbulent dispersion process. Extension of two-equation to full Reynolds stress models with source terms is needed for more complex geometries. Direct numerical simulations hold a great promise to make progress particularly in this area. The modeling of coalescence and breakup processes is a cornerstone in a full model of polydisperse bubbly flows since all other submodels strongly depend on the bubble size. An attempt to include these processes has been made, e.g., in Liao et al. (2015). However, the nature of these processes is not yet sufficiently understood especially for turbulent flows. In addition, physicochemical properties of the involved materials have significant influence on the rates of these processes. For the validation of models, more comprehensive CFD-grade data sets are needed. This means in particular that local values are needed, e.g., cross-sectional profiles at different locations along the main flow direction. Full sets of observables including in addition to phase fractions and phasic velocities also the bubble size distribution and turbulent fluctuations are highly desirable. This serves both to cover all aspects of the various submodels as well to facilitate the analysis of possible deviations between the model predictions and the validation data. Parametric variations of all independent variables – gas and liquid fluxes, bubble size, pipe/column diameter and height, material properties, etc. – are necessary to obtain a range of applicability that is large enough to make the models useful for practical applications. Finally, accurate measurements in particular with small systematic errors are needed. Ideally this criterion would be fulfilled by any good measurement technique, but multiphase flows pose significant challenges in this respect. Concerning all of these aspects, there is a dynamic ongoing development which may be expected to lead to fully predictive models in the future.

Nomenclature Notation CD CL CTD CVM CW Cμ dB d⊥ D

Unit – – – – – – m m m

Denomination Drag coefficient Lift coefficient Turbulent dispersion coefficient Virtual mass force coefficient Wall force coefficient Shear-induced turbulence coefficient (k-ε model) Bulk bubble diameter Bubble diameter perpendicular to main motion Pipe / column diameter or width (continued)

34

Notation Eo F g H Iturb J k Lturb Mo p r R Re t T u uτ U u’ x y z α δ e μ ν ρ σ τW ω

R. Rzehak

Unit – N m3 m s2 m – m s1 m2 s2 m – Pa m m – s N m2 m s1 m s1 m s1 m s1 m m m – m m2 s3 kg m1 s1 m2 s1 kg m3 N m1 N m2 s1

Denomination Eötvös Number Force Acceleration of gravity Pipe length / column height Turbulence intensity Superficial velocity = volumetric flux Turbulent kinetic energy Characteristic eddy size Integral turbulent length scale Morton Number Pressure Radial coordinate Pipe / column radius or half-width Reynolds number Time Stress tensor Velocity Friction velocity Velocity scale Fluctuation velocity Axial coordinate Wall normal coordinate Spanwise coordinate Phase fraction Viscous length scale Turbulent dissipation rate Dynamic viscosity Kinematic viscosity Density Surface tension Wall shear stress Turbulent frequency

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Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between Different Flow Morphologies Thomas Höhne

Abstract

Stratified two-phase flows are relevant in many industrial applications, e.g., pipelines, horizontal heat exchangers, and storage tanks. Special flow characteristics as flow rate, pressure drop, and flow regimes have always been of engineering interest. The numerical simulation of free surface flows can be performed using phase-averaged multi-fluid models, like the homogeneous and the two-fluid approaches, or non-phase-averaged variants. The approach shown in this chapter within the two-fluid framework is the algebraic interfacial area density (AIAD) model. It allows the macroscopic blending between different models for the calculation of the interfacial area density and improved models for momentum transfer in dependence on local morphology. An approach for the drag force at the free surface was introduced. The model improves the physics of the existing two-fluid approaches and is already applicable for a wide range of industrial two phase flows. A further step of improvement of modeling the turbulence was the consideration of sub-grid wave turbulence (SWT) that means waves created by Kelvin-Helmholtz instabilities that are smaller than the grid size. The new approach was verified and validated against horizontal two-phase slug flow data from the HAWAC channel and smooth stratified flow experiments of a different rectangular channel. The results approve the ability of the AIAD model to predict key flow features like liquid holdup and free surface waviness. Furthermore, an evaluation of the velocity and turbulence fields predicted by the AIAD model against experimental data was done. The results are promising and show potential for further model improvement. Keywords

CFD • Horizontal flow • AIAD • Two-phase flow • HAWAC • HZDR T. Höhne (*) Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Dresden, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_5-1

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T. Höhne

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Free Surface Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The CFD Approaches Applicable to Free Surface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Interfacial Area Density Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling the Free Surface Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sub-grid Wave Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal Wavy and Stratified Flow: Fabre Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CFD Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal Slug Flow: HAWAC Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction In the last decade, applications of computational fluid dynamic (CFD) methods for industrial applications received more and more attention, as they proved to be a valuable complementary tool for design and optimization. The main interest toward CFD consists in fact in the possibility of obtaining detailed 3D complete flow-field information on relevant physical phenomena at lower cost than experiments. Stratified two-phase flows are relevant in many industrial applications, e.g., pipelines, horizontal heat exchangers, and storage tanks. Special flow characteristics as flow rate, pressure drop, and flow regimes have always been of engineering interest. Wallis and Dobson (1973) analyzed the onset of slugging in horizontal and near-horizontal gas-liquid flows. Flow maps which predict transitions between horizontal flow regimes in pipes were introduced, e.g., by Taitel and Dukler (1976) and Mandhane et al. (1974). The most important flow regimes are smooth stratified flow, wavy flow, slug flow, and elongated bubble flow. Taitel and Dukler (1976) explained the formation of slug flow by the Kelvin-Helmholtz instability. They also proposed a model for the frequency of slug initiation (Taitel and Dukler 1977). The viscous Kelvin-Helmholtz analysis proposed by Lin and Hanratty (1986) generally gives better predictions for the onset of slug flow. Typically, free surfaces manifest as stratified, wavy, or slug flows in horizontal flow domain where gas and liquid are separated by gravity. The simulation of slug formation is a sensitive test case for the model setup regarding the quality of the models for interfacial friction, respectively, momentum transfer. A general overview on the phenomenological modeling of slug flow was given by Hewitt (2003) and Valluri et al. (2008). Various multidimensional numerical models were developed to simulate stratified flows: marker and cell (Harlow and Welch 1965), Lagrangian grid methods (Hirt et al. 1974), volume of fluid (VOF) method (Hirt and Nichols 1981), and level set method (Osher and Sethian 1988). These methods can in principle capture accurately most of the physics of the stratified flows. However, they cannot capture all the morphological formations such as small bubbles and droplets, if the

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

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grid is not sufficiently small. One of the first attempts to simulate mixed flows was ^ presented by Cerne et al. (2001) who coupled the VOF method with a two-fluid model in order to bring together the advantages of both formulations. Mouza et al. (2001) were numerically investigating the characteristics of horizontal wavy stratified flow in circular pipes and rectangular channels. They used the CFD code CFX for a simulation of the gas and liquid flow in separate domains, setting the time-averaged values of interfacial velocity and shear as boundary condition at the free surface. They used the data set by Fabre et al. (1987) as test case for rectangular channel flows. In a validation study for a preliminary version of the NEPTUNE_CFD code, Yao et al. (2003) conducted 2D simulations of the experiments by Fabre et al. (1987) as one of three test cases. They report a qualitatively good agreement of the calculated profiles of velocity, turbulence kinetic energy, and turbulent shear stresses for the cases with zero and medium gas velocity. But some quantitative deviations occur. In the case with high gas velocity (case 400), the code fails in predicting the turbulence parameters, which they account to the inability of the 2D model to capture the transverse flow reported for that experiment. Terzuoli et al. (2008) used the data set as test case for a cross-code comparison of three different CFD codes and to validate the free surface flow models of the respective codes. They compared the scientific code NEPTUNE_CFD and the commercial codes ANSYS CFX and FLUENT. By comparing 2D and 3D simulations with the experiments, they found that three-dimensional effects should not be neglected. Furthermore, they pointed out the fundamental role of the drag modeling at the free surface. A series of five different experiments were used by Coste et al. (2012) for the validation of the NEPTUNE_CFD code, the experimental cases 250 and 400 from Fabre et al. (1987) being among them. They were able to achieve a good agreement of their numerical data with the experimental data for velocity and turbulence in case 250. For case 400, they found a significant deviation between their simulations and the experiment, which they account to the inability of the NEPTUNE_CFD code to predict the transverse flows occurring in case 400. In general, CFD simulations for free surface flows require the modeling of the non-resolved scales. For modeling of interfacial transfers, it is necessary to select the adequate interfacial transfer models and to determine the interfacial area. The numerical solution can resolve the statistically averaged motion of the free surface (including waves) which may not be too small relatively to the channel height and to the characteristic length of the spatial discretization. However, the detailed structure of interacting boundary layers of the separated continuous phases and surface ripples cannot be resolved. Instead, its influence on the average flow must be modeled. Non-resolved small-scale structures of the interface have influence on mass, momentum, and heat transfer between the phases. The type of required models depends on the general modeling approach used. To model the momentum transfer, e.g., in the frame of the two-fluid model, the correlations for the interfacial drag are used. In the past, due to the lack of appropriate models, often drag correlations valid for bubbly flows or correlations developed for 1D codes were used to simulate the

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interfacial momentum transfer at the free surface. Such approaches do not properly reflect the physics of the phenomena. From this point of view, in the framework of the two-field model, it is interesting to consider, close to the interface, an anisotropic momentum exchange between liquid and gas. This is done for the algebraic interfacial area density (AIAD) model (Höhne and Vallée 2010; Höhne et al. 2011) which allows using different models to calculate the drag force coefficient and the interfacial area density for the free surface and for bubbles or droplets. A further step of improvement of modeling the turbulence is the consideration of non-predicted free surface waves or so-called sub-grid waves that means waves created by Kelvin-Helmholtz instabilities that are smaller than the grid size. So far in the present code versions, they are neglected. However, the influence on the turbulence kinetic energy of the liquid side can be significantly large. A region of marginal breaking is defined according to Brocchini and Peregrine (2001). In addition, turbulence damping functions should cover all the free surface flow regimes, from weak to strong turbulence.

Modeling Free Surface Flows The CFD Approaches Applicable to Free Surface Flow The three main types of two-phase CFD, namely, the RANS approach, the spacefiltered approaches (such as LES methods), and the pseudo-DNS approaches, are in principle applicable to free surface flow (see Bestion 2010a, b). Table 1 shows the main characteristics of these methods. If only two continuous fields (continuous liquid and continuous gas) exist in the flow without any bubble below the free surface and without any droplet in the gas flow, a one-fluid approach (homogeneous model) is applicable together with an interface tracking method (ITM) to predict the free surface. Since there may be some bubble entrainment below the free surface, a two-fluid approach was also used to be able to deal with various types of interface configurations including both large interfaces (free surface) and interface of dispersed fields (bubbles, droplets). Detailed derivation of the two-fluid model can be found in the book of Ishii and Hibiki (2006). On both sides of the free surface, shear layers are expected which require a specific attention since complex phenomena with turbulent transfers coupled to possible interfacial waves take place. It was found necessary to be able to track the interface position in order to treat this zone in a similar way as a wall boundary layer using wall functions. When trying to use a two-fluid approach, the development of an interface recognition method was found necessary. The AIAD method belongs to the third time and space filtering type. Because the model can be directly applied for industrial cases, it is classified as a macroscale model (Fox 2013). A different approach in this group, for instance, is done in the NEPTUNE code (see Coste et al. 2007; Coste and Laviéville 2009).

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . . Table 1 Flow rates and resulting mean liquid level and superficial velocities of the experimental case 250 (Fabre et al. 1987)

Case

5 250 3.0

Liquid flow rate V_ L in l/s Gas flow rate V_ G in l/s

45.4

Mean liquid level hmean in mm Liquid superficial velocity Ws,L in m/s Gas superficial velocity Ws,G in m/s

38 0.395 3.66

Fig. 1 Different morphologies under slug flow conditions

Algebraic Interfacial Area Density Model Figure 1 shows different morphologies that occur under slug flow conditions. Separate models are necessary for droplets or bubbles and separated continuous phases (interfacial drag, etc.). The basic idea of the AIAD model is: • The interfacial area density allows the detection of the morphological form and the corresponding switching of each correlation from one object pair to another. • It provides a law for the interfacial area density and the drag coefficient for a full range of phase volume fractions from no liquid to no gas (Fig. 2). • The model improves the physical modeling in the asymptotic limits of bubbly and droplet flows. • The interfacial area density in the intermediate range is set to the interfacial area density for free surface (Fig. 2).

The approach used in the AIAD model is to define blending functions depending on the volume fraction that enable switching between the morphologies of dispersed droplets, dispersed bubbles, and the free surface. Based on these blending functions, different equations for the interfacial area density and the drag coefficient can be applied according to the local morphology. The blending functions are defined as Eqs. 1 and 2 for droplet and bubble region, respectively, and Eq. 3 for the free surface: h i1 f D 1 þ eaD ðαL αD, limit Þ

(1)

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Fig. 2 Air volume fraction and corresponding morphologies/models

h i1 f B ¼ 1 þ eaB ðαG αB, limit Þ

(2)

f FS ¼ 1  f D  f B

(3)

with aD and aB being the blending coefficients for droplets and bubbles, respectively, and αD,limit and αB,limit the volume fraction limiters. In the simulations presented here, values of aD = aB = 70 and αD,limit = αB,limit = 0.3 were used. For all model coefficients, same values were used as in previous studies (Höhne and Vallée 2010; Höhne 2013). They were chosen independent of the actual geometry and flow regime and no tuning of the AIAD model was done for the work presented here. The threshold value αB,limit = 0.3 is a critical volume fraction before the coalescence rate increases sharply and is verified by experiments in both vertical and horizontal flows. Published data agree that bubbly flow rarely exceeds a gaseous volume fraction of about 0.25–0.35 when the transition to resolved structures occurs (Griffith and Wallis 1961; Taitel et al. 1980; Murzyn and Chanson 2009). Parameter studies also indicated that the model is not very sensitive toward a change of the blending function parameters. In Fig. 3, the blending function for bubbles is plotted against the gas volume fraction for different values of aB and αB,limit. For simplicity, bubbles and droplets are for now assumed to be of spherical shape, with a constant diameter of dB and dD, respectively. Non-drag forces in the regions of dispersed flow are neglected. The resulting formulation for the interfacial area density for droplets, AD, is given by AD ¼

6αL dD

(4)

in which αL is the volume fraction of the liquid phase. The IAD for bubbles, AB, is formulated analogous. The IAD of the free surface, AFS, is defined as the magnitude of the gradient of the liquid volume fraction αL, as given in Eq. 5, with n being the normal vector of the free surface:

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

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Fig. 3 Blending function fB plotted for different values of aB and αB,limit

AFS ¼ j∇αL j ¼

@αL @n

(5)

The local interfacial area density A is then calculated as the sum of Aj, weighted by the blending functions fj: A¼

X

A, j fj j

j ¼ FS, B, D

(6)

Modeling the Free Surface Drag In the general case of a two-phase flow, there is a velocity difference between the fluids, which is commonly called slip velocity. In contradiction to most volume of fluid (VOF) methods where only one velocity field is present, in the multi-fluid framework, each phase has its own velocity and turbulence model. Thereby, a drag force is induced at the phase boundary that is acting on both phases. The drag force can be correlated with the slip velocity U, the fluid density ρ, the surface area a, and the dimensionless drag coefficient CD. For geometry-independent modeling, the drag force is expressed as the volumetric force density FD and a is then replaced by the area density. The magnitude of the drag force density is given by Eq. 7: 1 jFD j ¼ CD A ρjUj2 2

(7)

For a dispersed flow, the density of the continuous phase is used in Eq. 7. In case of a free surface, the phase-averaged density is used, i.e., ρ ¼ α G ρG þ α L ρL

(8)

with ρG and ρL being the density of the gas and the liquid, respectively. In simulations of free surface flows, Eq. 7 does not represent a realistic physical model. It is reasonable to expect that the velocities of both fluids in the vicinity of the interface are rather similar. In Höhne et al. (2011), it is assumed that the shear stress near the surface behaves like a wall shear stress on both sides of the interface in order to reduce the velocity differences of both phases. It is supposed that the morphology region “free surface” is acting like a wall, and a wall-like shear stress is introduced at

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the free surface which influences the loss of gas velocity. The components of the ! normal vector n at the free surface are taken from the gradients of the void fraction. To use these directions of the normal vectors, the gradients of gas/liquid velocities, which are used to calculate the wall shear stress onto the free surface, are weighted with the components of the normal vector. From theory, shear stress is a ! symmetric tensor τ; if we have a surface normal vector n , then the wall-like free surface shear stress vector τW is the product of the viscous stress tensor and the surface normal vector: 2

3 2 3 τxz nx τyz 5  4 ny 5 τzz nz

(9)

τW, x ¼ τxx nx þ τxy ny þ τxz nz τW, y ¼ τyx nx þ τyy ny þ τyz nz τ W, z p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τzx nx þ τzy ny þ τzz nz τW ¼ τW, x 2 þ τW, y 2 þ τW, z 2

(10)

τxx τW ¼ 4 τyx τzx

τxy τyy τzy

This results in the following equations:

with nx , ny , nz being the components of the normal vector, u,v,w the components of the velocity vector, and μ the dynamic viscosity of the fluid; it can also be written:          @u @u @v @u @w þ þ τW, x ¼ nx  μ  2 þ ny  μ  þ nz  μ  @x @y @x @z @x          @u @v @v @v @w þ þ τW, y ¼ nx  μ  þ ny  μ  2 þ nz  μ  @y @x @y @z @y 

        @w @u @w @v @w τW, z ¼ nx  μ  þ þ þ ny  μ  þ nz  μ  2 @x @z @y @z @z τW ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τW, x 2 þ τW, y 2 þ τW, z 2 (11)

It is assumed that the drag force Eq. 7 is equal to the wall shear stress force acting at the free surface in the vicinity of the free surface: FW ¼ τ i A ¼ FD

(12)

As a result, the modified drag coefficient is dependent on the viscosities of both phases, the wall-like shear stresses (local gradients of gas/liquid velocities normal to the free surface), the mixture density, and the slip velocity between the phases:

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

  2 αL τW , L þ αG τW , G

CD, FS ¼

ρjUj2

9

(13)

The AIAD model uses the following three different drag coefficients: CD,B = 0.44 (Newton) for bubbles, CD,D = 0.44 for the droplets, and CD,FS (Höhne and Vallée 2010) according to Eq. 13 for the free surface. The advantage of this definition is that the resulting drag force is not depending on U anymore, since |U|2 is being eliminated, but instead the local velocity gradients and the viscosities of both phases are included in the calculation. This is a more physical definition for the interfacial drag force in a shear-driven flow. The total drag coefficient for a unit volume is calculated analogous to the interfacial area density as the weighted sum of the drag coefficients for all morphologies: CD ¼

X j

f j CD, j , j ¼ FS, B, D

(14)

Sub-grid Wave Turbulence Small waves (Fig. 4) created by Kelvin-Helmholtz instabilities that are smaller than the grid size are neglected in traditional two-phase flow CFD simulations, but the influence on the turbulence kinetic energy of the liquid side can be significantly large. Brocchini and Peregrine (2001) try to quantify this in the L-q diagram (Fig. 5) pffiffiffiffiffiffi which predicts the free surface shape as a function of the liquid turbulence q ¼ 2 k and a length scale L. They supposed that both gravity and surface tension act at a liquid surface so the surface behavior depends on both the turbulent Froude number 1=

Fr ¼ q=ð2gLÞ 2 and Weber number We = q2Lρ/2σ (σ is the interfacial tension coefficient). Thus, the value of both parameters must be considered. Their effect is discussed by seeking to delineate a critical region of parameter space between quiescent surfaces and surfaces that break up completely. The shaded area in Fig. 5 represents the region of marginal breaking and has been obtained by using the two estimated values for both the critical Weber number and the critical Froude number. So far we assume that the local length scale L can be obtained by local surface morphology created by the larger interface structures, which are resolved. The shaded area in the diagram also indicates the range of variations between surface that is no longer smooth or possibly broken because of turbulent flow. Depending on the values of the Froude and Weber numbers, the following four regimes can be classified: – Weak turbulence Fr  1, We  1: The turbulence is not strong enough to cause significant surface disturbances.

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Fig. 4 Surface instabilities at the free surface (HAWAC)

Fig. 5 Diagram of the (L,q)-plane for water (Brocchini and Peregrine 2001)

– “Knobbly” flow Fr  1, We  1: The turbulence is strong enough to deform the surface against gravity, but its turbulent length scale is small. Surface tension causes the surface shape to be very smooth and rounded. – Turbulence is dominated by gravity Fr  1, We  1: Surface distortions are primarily counters by gravity, resulting in a nearly flat free surface. The turbulent energy is sufficient to disturb the surface at relatively small scales, leading to small regions of waves, vortex dimples, and scars. – Strong turbulence Fr  1, We  1: The turbulence is strong enough to counter gravity and surface tension which is no longer sufficient to prevent the surface from breaking up into droplet and bubbles. Since turbulence can have different length scales at the same time, many of these regions can occur close to each other. For an upper critical value, Brocchini and Peregrine (2001) compared the turbulent kinetic energy density per unit volume of a blob that can disturb the surface with the energy of a surface disturbance per unit surface area (Fig. 6). It is assumed that a blob is just able to generate a spherical drop that touches the surface when it has lost

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

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Fig. 6 Example for the upper and lower bound of the transition zone

any overall motion. This leads to an estimate for the upper bound of the transition zone of qu 2 

!

π πσ g Lþ 24 n 2LρL

(15)

!

with gn ¼ g  n FS . The gravity vector normally stabilizes the free surface in horizontal flows. In cases of slug flows or vertical flows, the vector can even destabilize the flow. Therefore, the normal vector of the free surface is also taken into account. If gn turns into negative values, the term is destabilizing the surface. The interface normal vector is formulated using the volume fraction gradient of the liquid phase: !

n¼

∇αl j∇αl j

(16)

With this formulation, the interface normal vector is always pointing into the liquid. The product becomes zero in the case of an interface located parallel to the gravity vector

!

!

n perpendicular to g n . Additionally, it becomes negative if the

inward-directed interface normal vector is pointing into the opposite direction ! compared to g , which is the case occurring with Rayleigh-Taylor type of instabilities. The model surface for the lower bound is derived at the molecular level from the continuum surface model from Shikhmurzaev (1997). Surface diffusion studies show that the first indication of turbulence “breaking” the surface can be represented by the creation of fresh surface by the breaking of the surface skin (Fig. 6). It is considered as a linear down welling feature bounded by two, convex-upward quarter-circles. The result of the lower bound of the transition region is then

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Fig. 7 Diagram of the (L,k)plane for water

  5 π gn L ðπ  2Þσ  þ ql  3 2 125 5LρL 2

(17)

We assume that we must add the potion of turbulent kinetic energy created by the small waves at the liquid side which is so far missing in the simulation to the overall turbulent kinetic energy (see Fig. 5 shaded area):   kSWT ¼ 0:5 q2u  q2l

(18)

The result is depended on the local length scale L shown in Fig. 7. The consequence of the specific turbulent kinetic energy k is prescribed in the following source term: Pk, SWT ¼ f FS

2 @Ui ρ k 3 @xi L

(19)

where the gradients of the local velocities and the liquid density are present and which are added to the total turbulent kinetic energy (k-ω Model, Wilcox 1994): @ ðρkÞ þ ∇  ðρUkÞ ¼ ∇  @t

 μþ

  μt ∇k þ Pk  β0 ρkω þ Pk, SWT σk

(20)

The term Pk, SWT in Eq. 20 is blended only in the vicinity of the free surface.

Turbulence Damping Damping of turbulence as the interface is approached – on both sides – was found vital for the modeling of interface deformations under strong gas-side shear (Reboux

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

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et al. 2006). The phenomenon is similar to wall-turbulence decay, where it is known and rigorously derived that the eddy viscosity scales with the cube of the distance to the wall. The same has been found via DNS for interfacial flows, on the gas side which obviously perceives the interface as a solid wall (Fulgosi et al. 2003); it was also found that the liquid-side eddy viscosity scales with the square of the interface distance. In fact, it was found that when no damping is introduced, a spurious amount of eddy viscosity is generated at the interface, which tends to smear highfrequency surface instabilities, like wrinkling and fingering, and introduces strong errors in estimating the interfacial shear, which is the most important ingredient for mass transfer modeling, as explained in the corresponding section. Furthermore, without any special treatment of the free surface, the high velocity gradients at the free surface, especially in the gaseous phase, generate levels of turbulence that are too high throughout the two-phase flow when using differential eddy viscosity models like the k-ε or the k-ω model. Therefore, a certain amount of damping of turbulence is necessary in the region of the interface, because the mesh is too coarse to resolve the velocity gradient in the gas phase at the interface. A few empirical models have been suggested, which address the turbulence anisotropy at the free surface; see among others Celik and Rodi (1984). However, no model is applicable for a wide range of flow conditions, and all of them are nonlocal: they require, for example, explicit specification of the liquid layer thickness, of the amplitude and period of surface waves, etc. Direct and large eddy simulations of turbulent multi-material flow have been applied to model surface waves. Specifically, Reboux et al. (2006) used DNS to quantify the damping of turbulence approaching the interface and incorporated this knowledge into the damping of LES turbulence models (Liovic and Lakehal 2007a, b). Nourgaliev et al. (2008), Boeck and Zaleski (2005), and Coward et al. (1997) reported representations of surface instabilities that were obtained by DNS. For the two-fluid formulation, Egorov (2004) proposed a symmetric damping procedure. This procedure provides a solid wall-like damping of turbulence in both gas and liquid phases. It is based on the standard ω-equation, formulated by Wilcox (1994) as follows: @ α i  ρ i  ωi ðαi  ρi  ωi Þ þ ∇  ðαi ρi  Ui  ωi Þ ¼ α   τt, i  S_ i  αi  β  ρi  ω2i @t ki    þ ∇ αi μi þ σ ω  μt, i  ∇ωi þ SD, i i ¼ g, l (21) where α = 0.52 and β = 0.075 are the k-ω model closure coefficients of the generation and the destruction terms in the ω-equation, σω = 0.5 is the inverse of the turbulent Prandtl number for ω, τt is the Reynolds stress tensor, and S_ is the strain-rate tensor. Asymptotic analysis in the viscous sublayer near the wall shows that the k-ω model properly describes the turbulence damping in the internal part of the boundary layer, if the following Dirichlet boundary condition is specified for ω on the wall (subscript W):

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T. Höhne

ωw , i ¼ B 

6μi βρi Δn2

(22)

Here Δn is the near-wall grid cell height, and B is a coefficient. In order to mimic the effect of this boundary condition near the free surface, the following source terms have been introduced in the right-hand side of the gas and liquid phase ω-equations:  SD, i ¼ a  Δy β ρi B 

6μi βρi Δn2

2 ,

i ¼ g, l

(23)

Here A is the interface area density, Δy the local characteristic lengths scale, Δn is the typical grid cell size across the interface, and ρi and μi are density and viscosity of the corresponding phase i. The factor A activates this source term only at the free it cancels the standard ω-destruction term of the ω-equation surface, where  αi βρi ω2i and enforces the required high value of ωi and thus the turbulence damping. As a consequence, there is a sink term and a source term in the k equation and probably the effect might cancel it out. Nevertheless, the physical effects of sub-grid turbulence and turbulence damping due to high velocity gradients are not the same and both should be considered.

Verification and Validation Horizontal Wavy and Stratified Flow: Fabre Channel The first validation case of the AIAD model is an experimental data set from Fabre et al. (1987). In a quasi-horizontal channel with rectangular cross section, experiments with air and water were conducted. The flow regime investigated was smooth cocurrent stratified flow. The channel consists of six segments made of Plexiglas, with a total length of lc = 12.92 m and an inner cross section of height and width of hc = 0.1 m and wc = 0.2 m, respectively. Water flows into the channel out of a tranquilization tank. Air and water inlet are separated by a floating Plexiglas sheet. At the channel outlet, the water is discharged into a tank. Air and water are recirculated in separate loops. In Fabre et al. (1987), measurement data along the vertical axis at z = 9.1 m from the inlet are reported. The instantaneous interface height was measured with vertically mounted capacitance wires. Velocity in gas was measured using hot-wire anemometry and velocity in liquid was measured with laser Doppler anemometry. A more detailed description of the experimental setup and measurement techniques can be found in Fabre et al. (1987). The experiment which has been investigated in this work is case 250. The liquid flow rate was V_ L ¼ 3:0 l=s. The gas flow rate was adjusted at V_ L ¼ 45:4 l=s for case 250. The resulting mean liquid levels and corresponding superficial velocities are

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

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Table 2 Number of nodes and cell width in each spatial direction of the meshes used in the grid study Mesh Mesh 1 Mesh 2 Mesh 3

No. of nodes 237,446 473,946 1,174,845

Δx [mm] 4.76 4.76 3.03

Δy [mm] 4.77 4.77 2.94

Δz [mm] 10.00 5.00 5.00

summarized in Table 2. The specific case was chosen, since case 250 was reported to be smooth stratified flow.

CFD Setup An unstructured mesh with hexahedral cells was used. The AIAD model was implemented into the commercial CFD code ANSYS CFX 14.5 by CFX Command Language (CCL). The data for the comparison with the experiments were taken at a vertical center line 2.0 m from the inlet. To generate appropriate inlet boundary conditions, velocity profiles were taken from preliminary 2D simulations of the fulllength channel at 7.1 m from the inlet. The water level measured in the experiments was specified at the inlet and the boundary velocity profiles of air and water were scaled accordingly. The same water level was specified as initial condition for the whole domain, together with the corresponding gas and liquid superficial velocities and a hydrostatic pressure. As outlet boundary condition, a constant pressure outlet was defined, with the volume fraction function and the hydrostatic pressure also used for the initial conditions. Figure 8 shows an illustration of the domain at the initial state. The channel inclination was neglected in this study, since it is very low. All simulations were run in transient mode. In a grid study, three meshes were compared. The number of nodes and the cell widths in all three spatial directions is given in Table 3. For the coarsest mesh 1, convergence was not achieved and no results are presented. For mesh 2 and mesh 3, the profiles of the velocity w in main flow direction for air and water are shown in Figs. 9 and 10, respectively. There is no qualitative difference between the profiles and the quantitative deviation is sufficiently small to stop the grid study. For further calculations, the finest mesh 3 was used, since it showed better convergence behavior and allowed wider time steps.

Results and Discussion The focus of the analysis is put on the profiles of the velocity w in flow direction and the profiles of the turbulence kinetic energy k, because in previous validation studies of the AIAD model, these parameters were not available. The time course of the water level was also evaluated. Calculations with different turbulence models were done, which confirm the necessity of turbulence damping in free surface flows.

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T. Höhne

Fig. 8 Illustration of the initial volume fraction and the inlet velocity profile in case 400 (detail view)

Fig. 9 Vertical profiles of the streamwise velocity w in air

Smooth Stratified Flow The importance of turbulence modeling at the free surface was shown, e.g., by Fulgosi et al. (2003), so it is interesting and important to know how the use of different turbulence models influences the solution and how the AIAD turbulence modeling performs in comparison to standard turbulence models. To get an insight into this, an assessment of turbulence models was done. The AIAD k-ω model with turbulence damping and small wave turbulence is compared to three reference models. The models used as reference were k-ε (Launder and Sharma 1974), unmodified k-ω (Wilcox 1998), and shear stress transport (SST) (Menter 1994), all of which can be regarded as established standard models. All other parameters of the setup and the AIAD model were left unchanged. The time-dependent water levels at z = 2 m are shown in Fig. 11. Since no flow history data is reported in Fabre et al. (1987), the simulation results are compared to the time-averaged water level reported for the experiment. For all setups, an initial

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Fig. 10 Vertical profiles of the streamwise velocity w in water

Fig. 11 Development of the water level from the simulations of case 250 for different turbulence models compared to the experimental mean water level from Fabre et al. (1987)

decrease of the water level occurs before a quasi-steady state is reached. There is hardly any deviation in the course of the water level between k-ε and k-ω, which both show the poorest performance among the models used. Also the SST model predicts water levels that are much lower than in the experiment. For the k-ω model with turbulence damping, as used in the AIAD model, good agreement with the experiment is achieved in the quasi-steady state, but minor deviations leave room for

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T. Höhne

Fig. 12 Scaled vertical profiles of the streamwise velocity w in air

further improvement. The results clearly show the necessity of turbulence damping at the free surface in order to correctly predict the liquid holdup. The ability of the AIAD model to qualitatively predict the key features of free surface flows as well as liquid holdup was already demonstrated in the previous work. But in those studies, measurement data on the kinematic structure of the flows was not available from the experiments. Therefore, the analysis of the velocity and turbulence data was the most important part of the work presented here. Figures 12 and 13 show the vertical profiles of the streamwise velocity w at the measurement location for the different turbulence models in comparison to the experimental data. To restore comparability of the velocity profiles despite the different water levels, the velocity values are scaled with the superficial velocityws, i ¼ V_ i =Aave, i of the phase i, where Aave,i is the average cross-sectional area occupied by phase i. The height is scaled with the average water level hmean in a way that for water hL ¼ hL =hmean and for air hG ¼ ðhG  hmean Þ=ðhc  hmean Þ. In Fig. 12, especially the non-damped models expose a flattened profile in comparison to the experimental data, with the velocity maximum shifted to the upper part of the channel close to the wall. This trend can also be observed for the AIAD k-ω model in a weaker form, but in that case, the qualitative agreement with the experiment is better in the upper part of the channel. Nonetheless, the trend in the free flow region is similar to the other turbulence models, but a steeper gradient is observed close to the free surface. In water, a satisfying agreement of the scaled velocity profiles is achieved between the SST model and the experiment, as can be seen in Fig. 13. The non-damped k-ω and k-ε models also deliver good results in the free flow region,

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Fig. 13 Scaled vertical profiles of the streamwise velocity w in water

but predict too low velocities in the vicinity of the free surface. When using the AIAD turbulence modeling, the velocity close to the free surface is calculated too high. This inevitably distorts the profile also in the lower part of the channel. In order to further investigate this problem, the turbulence parameters are analyzed in the following. The numerical coupling between the turbulence model and the conservation equations in RANS-based eddy viscosity turbulence models is done via the turbulence viscosity or eddy viscosity μt, which is a function of turbulence kinetic energy and turbulence dissipation rate. For the k-ω model and the k-ε model, the eddy viscosity is expressed by Eqs. 24 and 25, respectively, with Cμ = 0.09 being a constant: μt, kω ¼ ρ

k ω

μt, ke ¼ Cμ ρ

k2 e

(24) (25)

In the SST model, the eddy viscosity is expressed by Eq. 26: μt, SST ¼ ρ 

a1 k maxða1 ω, ΩF2 Þ

(26)

with a1 = 5/9 being a SST model coefficient, Ω the absolute value of the vorticity, and F2 a blending function of the SST model:

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T. Höhne

  F2 ¼ tan h arg22  pffiffiffi  2 k 500v arg2 ¼ max 0 , 2 β ωy y ω

(27) (28)

with y being the wall distance, v the kinematic viscosity of the fluid, and β0 = 0.09 an SST model coefficient. Eddy viscosity and molecular viscosity of the fluid μ are added up to the effective viscosity μeff = μ + μt. Since the eddy viscosity is a numerical construct, no experimental data is available to compare with the simulation results. But an analysis of the eddy viscosity can give an insight on the influence of the turbulence modeling on the velocity field and the free surface position. The effective viscosity acts as proportionality factor between the viscous force Fμ and the velocity gradient normal to the flow direction @w/@h, as expressed in the standard shear stress in Eq. 29, with τeff = Fμ/A: τeff ¼ μeff

@w @h

(29)

Therefore, if the flow is incompressible and driven by a constant force, an increase of the effective viscosity has the effect of a decrease in the velocity gradient and vice versa. Since the molecular viscosity is assumed to be constant and is orders of magnitude smaller than the turbulent viscosity in the case examined here, the latter one is the governing parameter of the effective viscosity. Figures 14 and 15 show the vertical profiles of the calculated eddy viscosity in air and water, respectively. For better comparability, μt is also correlated with h*. At the walls, all models are more or less in agreement, but the deviations between the models grow drastically toward the free surface. There the unmodified k-ω and k-ε model reach the maximum μt, while with SST and AIAD k-ω the maximum of μt is located in the free flow region and toward the free surface μt decreases again. These trends are observed in both phases. From these observations, it can be explained why there are such strong deviations between the velocity profiles and free surface positions predicted by the different turbulence models. According to Eq. 29, an increase in turbulence viscosity has to result in a decrease of the velocity gradient. The effect of this is a smoothing of the velocity profile, as can be observed in Figs. 12 and 13. The velocity gradients also strongly affect the calculation of the free surface drag, since the free surface drag force resulting from the combination of Eqs. 7 and 13 is only depending on the area density and the free surface shear stresses. The latter ones again are depending on the velocity gradients at the free surface according to Eq. 11. The result of the problems discussed above is therefore a reduction in the calculated drag force, which forces a reduction in the velocity difference between the phases. The only remaining degree of freedom is the interface position, with the result of a shifting of the free surface toward the slower phase, which is the water in this case.

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

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Fig. 14 Vertical profiles of the turbulent viscosity ηt in air

Fig. 15 Vertical profiles of the turbulent viscosity ηt in water

The above considerations in combination with the analysis of the eddy viscosity explain why the prediction of the holdup is poor when using the non-damped turbulence models. With the turbulence damping at the free surface used in the

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T. Höhne

Fig. 16 Profiles of the turbulence kinetic energy k in air

AIAD model, steeper velocity gradients at the free surface are allowed, and this effect is weakened. This explains the better performance of the AIAD model in predicting the liquid holdup. To get a deeper insight into the turbulence structure of the flow, also the turbulence kinetic energy is analyzed, for which measurement data is available. The vertical profiles of k in the gas phase are plotted in Fig. 16. For comparability, k is scaled with ws2 and correlated with h*. It is obvious that with all non-damped turbulence models, k is strongly overestimated, up to one order of magnitude close to the free surface in case of the unmodified k-ω. According to Egorov (2004), this is a well-known problem and was the initial motivation for the development of turbulence damping functions. The k-ω model with damping term predicts a turbulence kinetic energy profile much closer to the experimental one. But still there are deviations, especially in the region below h* = 0.5. For the liquid phase, the vertical profiles of the turbulence kinetic energy are shown in Fig. 17. The situation is comparable to the gas phase; the standard turbulence models fail in predicting k, while the k-ω with damping is much closer. Noteworthy is the strong increase of k close to the gas-liquid interface when using damped k-ω and SST. Unfortunately, there is no measurement data available above h* = 0.9, so it cannot be determined whether this is an artifact of the turbulence modeling or a realistic effect. From the examination of the turbulence parameters, it can be deduced that the free surface turbulence damping has an effect not only in the small region where the damping functions are blended in but also influences a major part of the core flow. This effect is beneficial, because it leads to a more realistic calculation of the turbulence kinetic energy. But the problem remains that the performance of the k-ω model away from the walls is not optimal, which is a wellknown drawback (Menter 1994). Future development of the AIAD model should therefore focus on the turbulence modeling. It has to be determined more in detail, how the free surface turbulence modeling takes effect on the core flow and on the free surface drag. An asymmetric turbulence damping should be considered to overcome the weaker performance of the model in the liquid phase, which was up to now not done for reasons of numerical stability. Also a detailed assessment of the influence of the small wave

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

23

Fig. 17 Profiles of the turbulence kinetic energy k in water

turbulence is necessary. One prerequisite for these tasks are measurement data on the velocity and turbulence fields close to the free surface, which are not available in Fabre et al. (1987), as in most experimental studies on stratified flow, due to limitations in the measurement techniques. A solution might be the analysis of DNS data as complement to experimental studies. As an alternative approach, the damping functions should be implemented with the SST model, since the k-ω model is known to create problems in the free flow region (Menter 1994).

Horizontal Slug Flow: HAWAC Channel The horizontal air/water channel (HAWAC) (Fig. 18) is devoted to cocurrent flow experiments. A special inlet device provides defined inlet boundary conditions by separate injection of water and air into the test section. A blade separating the phases can be moved up and down to control the free inlet cross section for each phase, which influences the evolution of the two-phase flow regime. The cross section of this channel is smaller than the channel used in an earlier study described by Vallée et al. (2008): its dimensions are 100  30 mm2 (height  width). The test section is about 8 m long, and therefore the length-to-height ratio L/h is 80. In terms of the hydraulic diameter, the dimensionless length of the channel is L/Dh = 173. The inlet device (Fig. 19) is designed for separate injection of water and air into the channel. The air flows through the upper part and the water through the lower part of this device. As the inlet geometry introduces many perturbations into the flow (bends, transition from pipes to rectangular cross section), four wire mesh filters are mounted in each part of the inlet device. The filters are made of stainless steel wires

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T. Höhne

Fig. 18 Schematic view of the horizontal channel with inlet device for a separate injection of water and air into the test section

Fig. 19 The HAWAC inlet device

with a diameter of 0.63 mm and have a mesh size of 1.06 mm. The wire mesh filters are used to provide homogenous velocity profiles at the test section inlet. Moreover, the filters produce a pressure drop that attenuates the effect of the pressure surge created by slug flow on the fluid supply systems. Air and water come in contact at the final edge of a 500 mm long blade that divides both phases downstream of the filter segment. The free inlet cross section for each phase can be controlled by inclining this blade up and down. In this way, the perturbation caused by the first contact between gas and liquid can be either minimized or, if required, a perturbation can be introduced (e.g., hydraulic jump). Both, filters and the inclinable blade, provide well-defined inlet boundary conditions for the CFD model and therefore offer very good validation possibilities. Optical measurements were performed with a high-speed video camera.

CFD Setup The new approach was implemented via the command language CCL into ANSYS CFX-14 (ANSYS 2012). An Euler-Euler multiphase model using fluid-dependent RANS k-ω turbulence models was applied. The high-resolution discretization scheme was used for convection terms in the equations. For time integration, the fully implicit second-order backward Euler method was applied with a constant time step of dt = 0.001 s and a maximum of 30 coefficient loops per time step.

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

25

Convergence was defined in terms of the RMS values of the residuals, which was less than 104 most of the time. An initial water level of half of the channel was assumed for the entire model length. The inlet conditions are given by both superficial velocities; a pressure distribution is set at the channel outlet. The grid consists of 1.2  106 hexahedral elements. A slug flow experiment at a water velocity of 2.0 m/s and an air velocity of 10.0 m/s was chosen for the CFD calculations. These velocities correspond to the following correlation proposed by Mishima and Ishii (1980) for the evaluation of the onset of slugging in a horizontal pipe: cG  cL ¼ 0:487

rffiffiffiffiffiffiffiffiffiffiffi gy0 ρL ρG

(30)

where y0 = 50 mm is the height of the gas flow part of the channel (the distance of the interface to the top) and ci the critical velocity of phase i. In the experiment, the inlet blade was in the horizontal position. Accordingly, the model inlet was divided into two parts: in the lower half of the inlet cross section, water was injected with air injected in the upper half. To allow for upstream flow development, the inlet blade was modeled as shown in Fig. 20a. An initial water level of y0 = 50 mm was assumed (Fig. 20b). In the simulation, both phases have been treated as isothermal and incompressible, at 25 C and at a reference pressure of 1 bar. A hydrostatic pressure was assumed for the liquid phase. At the inlet, the turbulence properties were set using the “medium intensity and eddy viscosity ratio” option of the flow solver. This is equivalent to a turbulence intensity of 5% in both phases. The inner surface of the channel walls are defined as hydraulically smooth with a nonslip boundary condition applied to both gaseous and liquid phases. The channel outlet was modeled with a pressure-controlled outlet boundary condition. The parallel transient calculation of 15.0 s of simulation time on eight processors took 21 CPU days.

Verification To first verify the damping procedures, this slug flow experiment of the HAWAC channel was used. The horizontal two-phase flow was modeled with and without the damping procedures. In addition, a single-phase flow of the gaseous phase was modeled using the upper part (50%) of the channel, and a moving wall with constant velocity of the liquid phase of 2 m/s at the lower wall boundary was utilized to mimic the free surface. Figure 21 shows the verification of the k-ω turbulence model damping procedures using a HAWAC experiment. On the left side, a case is shown with gas velocity 10 m/s and liquid velocity 2 m/s with and without damping functions. On the right side, a single-phase case with only air flowing over a moving wall with 2 m/s is displayed. Figure 22 shows the results of the gas velocity field over the channel height. The simulations used the k-ω turbulence model with and without damping procedures inside the AIAD model framework in comparison to single gas-phase flow. The damping functions have a strong effect on the gas velocity field.

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Fig. 20 Model and initial conditions of the volume fractions

Moving Wall 2m/s Single phase case with only air flowing over moving wall

Case with gas velocity 10 m/s and liquid velocity 2 m/s with and without damping functions

Fig. 21 Verification of the k-ω turbulence model damping procedures 16.00 14.00

Air Velocity [m/s]

12.00

K omega no Damping Single Phase

10.00

k omega Damping

8.00 6.00 4.00 2.00 0.00 0

0.02

0.04 0.06 Channel height [m]

0.08

0.1

Fig. 22 Results of the gas velocities over the channel height, k-ω turbulence model with and without damping procedures using the AIAD model in comparison with single gas-phase flow

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27

Fig. 23 Measured picture sequence at JL = 1.0 m/s and JG = 5.0 m/s with Δt = 50 ms (depicted part of the channel: 0–3.2 m after the inlet)

AreaDensVar

[m^-1]

0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 1.0 00 00 00 00 00 00 00 00 00 00 0 0e 0e 0e 0e 0e 0e 0e 0e 0e 0e 0e +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 00 01 01 01 01 01 01 01 01 01 02

Fig. 24 Calculated picture sequence at JL = 1.0 m/s and JG = 5.0 m/s with Δt = 50 ms (depicted part of the channel: 0.4–3.6 m after the inlet)

It is obvious that without modification of the turbulence at the free surface, the velocity fields of a horizontal two-phase flow are not predicted correctly.

Results and Discussion Picture sequences in Figs. 23 and 24 compare predictions of the phase distribution from the CFD calculation and comparable camera frames of slugging behavior observed in the experiment. In both cases, a slug is generated. The sequences show that the qualitative behavior of the creation and propagation of the slug is similar in both the experiment and the CFD calculation.

28

T. Höhne Air at 25 C.Volume Fraction

Air at 25 C.Volume Fraction 1.000e+000

1.000e+000

7.500e-001

7.500e-001

5.000e-001

5.000e-001

2.500e-001

2.500e-001

1.000e-015

1.000e-015

Y

Y X

X time = 9.21 [s]

time = 8.97 [s]

8.97 s after start of simulation

9.21 s after start of simulation

Air at 25 C.Volume Fraction uboot Default 1.000e+000

Air at 25 C.Volume Fraction uboot Default 1.000e+000

7.500e-001

7.500e-001

5.000e-001

5.000e-001

2.500e-001

2.500e-001

1.000e-015

1.000e-015

Y

Y

X

X time = 9.52 [s]

9.52 s after start of simulation

time = 9.91 [s]

9.91 s after start of simulation

Fig. 25 Calculated picture sequences at JL = 1.0 m/s and JG = 5.0 m/s with isosurface at αL = 0.5, development of waves and slug formations in the channel.

In addition, Fig. 25 shows the development of the slug. These slugs are induced only by instabilities generated in the simulation, and the single effects leading to slug formation that can be simulated in this model are: • Instabilities and small waves randomly generated by the interfacial momentum transfer. As a result, bigger waves are generated. • The waves can have different velocities and can merge together. • Bigger waves roll over smaller waves and can close the channel cross section.

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

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Fig. 26 Sub-grid wave turbulence [W m3], development of a slug formation in the channel

The slug formation can also be induced by a perturbation appearing downstream at the inlet blade which separates the water and the air flow. In the simulation as well as in the experiment, this perturbation is induced from the blade lip. Figure 26 shows the influence of the sub-grid wave turbulence during a slug development in the channel. In order to extract quantitative information, a local position was chosen 1.5 m after the blade position in the middle of the channel over the channel height. The sub-grid wave turbulence exists only in the area of the free surface and follows the slug formations. At the wavy front and back of the slugs, the value of the sub-grid wave turbulence is the highest in the channel. The turbulence intensity with and without sub-grid wave turbulence is displayed in Fig. 27. The turbulence intensity Tu is calculated according to Eq. 31: pffiffiffiffiffiffiffiffiffiffiffi 2=3ki Tu ¼ ui

with i ¼ g, l

(31)

with the additional source term of the sub-grid wave turbulence in the water turbulence kinetic energy the value is slightly higher in the vicinity of the interface. The slug frequency analysis was done using fast Fourier transform (FFT). The power spectral density (PSD), which describes how the power of a signal or time series is distributed over different frequencies, is used. The position of 2.5 m away from inlet blade in the middle of the channel (0.015 m, 0.05 m), where waves are generated, was utilized to describe the change of water level in the channel. A characteristic slug frequency of around 2.0 Hz is seen Fig. 28, which corresponds roughly to the experimental value of approximately 2.4 Hz.

Summary and Conclusions Stratified two-phase flows are relevant in many industrial applications, e.g., pipelines, horizontal heat exchangers, and storage tanks. Special flow characteristics as flow rate, pressure drop, and flow regimes have always been of engineering interest. The numerical simulation of free surface flows can be performed using phaseaveraged multi-fluid models, like the homogeneous and the two-fluid approaches,

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T. Höhne 0.10 0.09

Air SWT Water SWT Air Water

channel hight [m]

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 1.0E-03

1.0E-02

1.0E-01

1.0E+00

turbulence intensity [-]

Fig. 27 Comparison of air/water turbulence intensity [] with and without sub-grid wave turbulence 0,01

Power Spectral Density [(ˆ2)/Hz]

0,0001

1e-06

1e-08

1e-10

1e-12

1e-14 0,1

1

10

100

Frequency [Hz]

Fig. 28 Power spectral density (PSD) of water level at the middle of the channel, CFD simulation

Euler-Euler Modeling of Segregated Flows and Flows with Transitions Between. . .

31

or non-phase-averaged variants. The approach shown in this paper within the two-fluid framework is the algebraic interfacial area density (AIAD) model. It allows the macroscopic blending between different models for the calculation of the interfacial area density and improved models for momentum transfer in dependence on local morphology. An approach for the drag force at the free surface was introduced. The model improves the physics of the existing two-fluid approaches and is already applicable for a wide range of industrial two-phase flows. A further step of improvement of modeling the turbulence was the consideration of sub-grid wave turbulence that means waves created by Kelvin-Helmholtz instabilities that are smaller than the grid size. A first CFD validation of the approach was done for an adiabatic case of the HAWAC channel. In addition, an experimental test case of the Fabre channel was simulated by using the AIAD model as closure for stratified gas-liquid flows. For the first time, the AIAD model was validated with a detailed set of velocity and turbulence data, and its performance was compared with three standard turbulence models. It was confirmed that with the AIAD model, it is possible to predict key flow features of stratified flow like waviness of the free surface and liquid holdup with good accuracy, where the established turbulence models fail. In predicting the streamwise velocity and turbulence kinetic energy in the gas phase, the AIAD model was also superior to the standard turbulence models, but there are still existing deviations to the experimental data. In calculating the liquid velocity, the SST turbulence model is showing the best performance, although it again fails in predicting the turbulence kinetic energy, as well as the other standard turbulence models. The results show that a sophisticated modeling of drag and turbulence at the free surface is necessary in order to correctly model stratified flow. The AIAD model is fulfilling these requirements from a qualitative point of view. But further research is necessary on the turbulence damping and its effect on the free surface drag. More verification and validation of the approach is necessary – more CFD-grade experimental data are required for the validation.

References ANSYS CFX, User Manual. (Ansys, 2012) D. Bestion, Extension of CFD code application to two-phase flow safety problems. Nucl. Eng. Technol 42, 365–376 (2010a) D. Bestion, Applicability of Two-Phase CFD to Nuclear Reactor Thermal Hydraulics and Elaboration of Best Practice Guidelines (CFD4NRS-3, Washington, DC, 2010b) Sept 2010, to be published in a special issue of Nuc. Eng. Des. T. Boeck, S. Zaleski, Viscous versus inviscid instability of two-phase mixing layers with continuous velocity profile. Phys. Fluids 17, 032106-1–032106-11 (2005) M. Brocchini, D.H. Peregrine, The dynamics of strong turbulence at free surfaces. Part1. Description, J. Fluid Mech. 449, 225–254 (2001) I. Celik, W. Rodi, A Deposition-Entrainment Model for Suspended Sediment Transport. Report SFB 210/T/6, Strömungstechnische Bemessungsgrundlagen für Bauwerke. University of Karlsruhe, Karlsruhe (1984) G. Cerne, S. Petelin, I. Tiselj, Coupling of the interface tracking and the two-fluid models for the simulation of incompressible two-phase flow. J. Comput. Phys. 171(2), 776–804 (2001)

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A. Mouza, S. Paras, A. Karabelas, CFD code application to wavy stratified gas-liquid flow. Trans. IChemE 79(Part A), 561–568 (2001) F. Murzyn, H. Chanson, Experimental investigation of bubbly flow and turbulence in hydraulic jumps. Environ. Fluid Mech. 9, 143–159 (2009) R. Nourgaliev, M.-S. Liou, T.G. Theofanous, Numerical prediction of interfacial instability: Sharp Interface Method (SIM). J. Comput. Phys. 227, 3940–3970 (2008) S. Osher, J.A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988) S. Reboux, P. Sagaut, D. Lakehal, Large-eddy simulation of sheared interfacial flow, Phys. Fluids 18, 105 (2006) Y.D. Shikhmurzaev, Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 334, 211–249 (1997) Y. Taitel, A.E. Dukler, A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AICHE J. 22, 47–55 (1976) Y. Taitel, A.E. Dukler, A model for slug frequency during gas-liquid flow in horizontal and near horizontal pipes. Int. J. Multiphase Flow 3, 585 (1977) Y. Taitel, D. Bornea, A.E. Dukler, Modelling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AlChE J. 26, 345–354 (1980) F. Terzuoli, M. Galassi, D. Mazzini, F. D’Auria, CFD code validation against stratified air-water flow experimental data. Sci. Technol. Nucl. Installations 2008, 596 (2008) C. Vallée, T. Höhne, H.M. Prasser, T. Sühnel, Experimental investigation and CFD simulation of horizontal stratified two-phase flow phenomena. Nuc. Eng. Des. 238(3), 637–646 (2008) P. Valluri, P.D.M. Spelt, C.J. Lawrence, G.F. Hewitt, Numerical simulation of the onset of slug initiation in laminar horizontal channel flow. Int. J. Multiphase Flow 34, 206–225 (2008) G.D. Wallis, J.E. Dobson, Onset of slugging in horizontal stratified air-water flow. Int. J. Multiphase Flow 1, 173–193 (1973) D.C. Wilcox, Turbulence Modelling for CFD (DCW Industries Inc., La Cañada, 1994) W. Yao, P. Coste, D. Bestion, M. Boucker, Two-Phase Pressurized Thermal Shock Investigations Using a 3D Two-Fluid Modeling of Stratified Flow with Condensation. Proceedings of 10th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-10), pp. 5–9 (2003)

Recent Advances in Modeling Gas-Particle Flows Luca Mazzei

Abstract

This chapter is concerned with the mathematical modeling of dense fluidized suspensions and focuses on the so-called Eulerian or multifluid approach. It introduces newcomers to some of the techniques adopted to model fluidized beds and to the challenges and long-standing problems that these techniques present. After introducing the principal approaches for modeling fluid-solid systems, we focus on the multifluid, overviewing the main averaging techniques that consent to describe granular media as continua. We then derive the Eulerian equations of motion for fluidized powders of a finite number of monodisperse particle classes, employing volume averages. We present the closure problem and overview constitutive relations for modeling the granular stress and the interaction forces between the phases. To conclude, we introduce the population balance modeling approach, which permits handling suspensions of particles continuously distributed over the size and any other property of interest. Keywords

Multiphase flows • Fluidization • Fluidized bed modeling • Averaging methods • Multifluid modeling • Computational fluid dynamics • Closure problem • Population balance modeling

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 An Overview of Fluidized Bed Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 An Overview of Averaging Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Statistical Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Volume Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

L. Mazzei (*) Department of Chemical Engineering, University College London, London, UK e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_8-1

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Time Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Final Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Averaged Equations of Motion for Fluid-Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Problem of Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid-Particle Interaction Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-Particle Interaction Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Population Balance Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid-Phase Volume Average of Point Variable Spatial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . Fluid-Phase Volume Average of Point Variable Time Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-Phase Volume Average of Point Variable Time Derivatives . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 14 14 14 18 23 23 27 33 34 39 39 39 40 41 41

Introduction Fluidization is a well-established technology used in several industrial processes such as coal combustion, biomass gasification, waste disposal, and food processing. To design fluidized beds, engineers have resorted for many years to experimental correlations and pilot plants. This practical approach to fluidization is well reflected in the first textbooks on the subject (see, for instance, Leva 1959). These correlations, however, lack general validity and can help design only standard units: they cannot tell us how changes in vessel geometry, introduction of internals (like heat exchanger tubes), or feed repartition over various entry points affect fluid dynamics and performance. To answer these and similar questions, and improve the design of conventional units, one needs a theory for predicting how dense fluidized powders behave; pilot plants are not a convenient alternative, being expensive, timeconsuming, and not always leading to adequate scale up. When fluidized beds were first employed in the 1920s–1940s, engineers did not appreciate this problem, probably because at the time, the required plant performance was either not critical (like in FCC plants) or easily achievable (like in roasting and drying). Nevertheless, when later the problem revealed itself in other and more demanding applications, with some plants falling far short of the expected conversions previously achieved in pilot units, it became clear that this matter had to be addressed thoroughly. Researchers hence endeavored to find more reliable methods to predict the dynamics of fluidized suspensions. In the 1960s, scientists began to adopt the conservation laws of mass, momentum, and energy to analyze nearly any physical and chemical problem. This innovative approach, most probably fostered by the release of the influential textbook Transport Phenomena (Bird et al. 1960), led to significant theoretical headway, bolstered the hope to explain theoretically the behavior of fluidized powders and prompted the first trials to develop fluid dynamic models based on transport equations.

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Anderson and Jackson (1967) were among the first to model fluidized systems; starting from the continuity and dynamical equations for single-phase, incompressible fluids and the Newtonian equations for rigid-body motion, they derived averaged equations of conservation for the fluid and solid phases by applying a formal mathematical process of volume averaging. Afterward, several researchers did the same, refer, for instance, to Whitaker (1969), Drew (1971), and Drew and Segel (1971). Initially, they used these models to understand better the complex behavior of multiphase systems, but without regarding them as a viable way to design real industrial units. Nonetheless, when faster computer processors and advanced numerical methods to integrate partial differential equations became available, they realized that a mathematical theory of multiphase flows could indeed become a useful design tool. With the further development of new and more rigorous formulations of multiphase equations of motion (Buyevich 1971; Hinch 1977; Nigmatulin 1979; Drew 1983; Jenkins and Savage 1983), the late 1970s and early 1980s witnessed the first endeavors to simulate numerically granular flowing systems (Pritchett et al. 1978; Gidaspow and Ettehadieh 1983; Gidaspow et al. 1986). The promising results of these few pioneering studies generated an increasing interest in computational fluid dynamics (CFD) and multiphase flows, which rapidly started to turn into research areas in their own right. Nowadays, CFD has become an almost indispensable tool to solve problems of academic and industrial interest. In the field of fluidization, CFD has assisted to understand fluid-solid interactions and has enabled to predict numerous macroscopic phenomena encountered in particulate systems. Similarly, multiphase flows and fluidization dynamics have become topics of interest not only for the scientific community but also for the industrial world. Over the years, many researchers have developed mathematical models to predict the dynamics of dense fluidized suspensions, proposing several approaches and mathematical schemes; we now briefly overview some of them, highlighting their advantages and limitations.

An Overview of Fluidized Bed Modeling Fluidized beds can be modeled at various levels of detail. At the most fundamental, the motion of the system is determined by the Newtonian equations for translation and rotation of each particle and the Navier-Stokes transport equations to be satisfied at every point occupied by the interstitial fluid. In this approach, referred to as Eulerian-Lagrangian, the flow field of the fluid is modeled at a length scale far smaller than the particle size; one, therefore, is able to determine the pressure and velocity gradient over the surface of each particle and from there the interaction force between the fluid and the particles (see, for instance, Pan et al. 2002). No closure problem hence arises for this force. Furthermore, since the particles are considered individually, the solid phase is not modeled as a continuum, retaining its granular nature; the equations describing its motion, consequently, feature no granular stress.

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Also the closure problem for this term, which arises in macroscopic models, is therefore absent. This modeling strategy is appealing: it is conceptually quite simple – being probably the most natural for describing the dynamics of particulate systems – and it is the least affected by closure problems. It presents, nonetheless, a few disadvantages. First of all, it is extremely demanding computationally: simulations of this kind have been performed only for systems containing a relatively small number of particles; extending these calculations to dense suspensions in large domains, like those found in industrial fluidized beds, is presently inconceivable. Moreover, even if this were feasible, the information provided by the solution would be much too detailed, and one would have to filter the results to render them useful. Note that such results are of little direct interest to most end users: the simulations yield the position and velocity of each particle at any given time, but what are the spatial distributions of the observables, such as the granular temperature and pressure, which are of real interest in applications? With the output of Eulerian-Lagrangian simulations, one can obtain these distributions (if one knows how the observables of interest are related to the fluid and particle dynamics at the microscopic length scale, a knowledge that is acquired when one derives the averaged equations which characterize continuum models), but a complex calculation is required. These observations suggest that it might be convenient to formulate equations of change governing the evolution of these observables directly. In this approach, we renounce to capture the details described above, satisfying ourselves with a far reduced description of the flow. Although there exists no guarantee that these simplified equations can be really obtained – in closed form, that is – several studies have been conducted in such a pursuit (Anderson and Jackson 1967; Whitaker 1969; Drew 1971, 1983; Drew and Segel 1971; Drew and Lahey 1993; Gidaspow 1994; Zhang and Prosperetti 1994; Enwald et al. 1996; Jackson 1997, 1998, 2000). Owing to the complexity of the problem, one does not aim to derive the general exact averaged multiphase equations of motion; the intent is merely to formulate models which may describe satisfactorily phenomena of interest for industrial applications. Various mathematical techniques yield such equations, and several claims have been advanced as to the superiority of each form of averaging versus the others. However, the resulting transport equations are very similar and present many common features. Two are the most significant. First, they are all written in terms of mean variables defined over the entire physical domain, so, they resemble those that one would write for a set of imaginary fluids which interpenetrate one another and occupy simultaneously the same physical volume. The model, known as Eulerian-Eulerian or multifluid, thus takes the form of coupled differential equations subjected to initial and/or boundary conditions assigned only on the boundaries of the domain containing the mixture, and no longer on the surface of the particles, as in Eulerian-Lagrangian models. Second, the process of averaging generates a number of indeterminate terms unrelated to the averaged variables but associated with details of the motion at the particle (that is, microscopic) length scale. These are key terms, represented by the fluid and solid stress tensors and by the interaction forces exchanged by the phases. A closure problem hence arises, which one cannot usually

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solve analytically; in fact, there is no guarantee that a solution exists. So, one has to resort to empirical relations, this being the main shortcoming of the method. Besides these two approaches, there is a third that can be regarded as a hybrid between them. Averaged equations of motion are used for the fluid phase, but rigidbody Newtonian equations are solved for every particle of the system. These do not interact with the fluid through its microscopic velocity field – as is the case in Eulerian-Lagrangian models – but with the averaged value of the latter. For instance, the overall force exerted by the fluid on each particle is not computed by integrating over the particle surface the local traction arising from the fluid velocity gradients: the force is instead calculated in terms of slip velocity between the average fluid velocity and the velocity of the particle center of mass and by resorting to empirical relations. This strategy, called discrete particle modeling (DPM), is significantly less demanding computationally than the Eulerian-Lagrangian and has met with resounding success (Tsuji et al. 1993; Hoomans et al. 1996; Xu and Yu 1997; Ouyang and Li 1999; Kafui et al. 2002; Lu et al. 2005; Zhu et al. 2008; Di Renzo et al. 2011; He et al. 2012; Wang et al. 2013; Deen et al. 2014). To describe particle collisions, modelers use two approaches: hard and soft sphere. In the first, particles interact via binary, instantaneous, pointwise collisions. Their velocities after an encounter are computed by requiring that linear and angular momentum are conserved in the collision. This approach was pioneered by Allen and Tildesley (1990). Since their publication many authors have found it useful to model the collision dynamics in granular systems. Hoomans et al. (1996) used it in their model for gas-fluidized beds; it was the first time that the technique had been applied to a dense system. Many authors have since published papers with this strategy (see, for instance, Ouyang and Li 1999). The soft-sphere model for fluidized beds was instead pioneered by Tsuji et al. (1993), who developed their approach on the basis of earlier work done by Cundall and Strack (1979). Here, during an encounter, particles overlap slightly, and the contact forces are calculated from the deformation history of the solids using a linear spring-dashpot model. This has been employed by Xu and Yu (1997), Pandit et al. (2005), Ye et al. (2005), and several other researchers. Among the three modeling approaches discussed, the second is often preferred for its valuable feature of being computationally less demanding. Due to the number of particles present in industrial plants, Eulerian (continuum) modeling is unlikely to be replaced by its Lagrangian (discrete) counterparts in the near future; furthermore, Eulerian models appeal more to end users, because they provide information of direct interest. The role of discrete modeling is yet paramount. The method, to be considered more as an effective research tool than a practical design instrument, by yielding information about the dynamics of multiphase systems at the microscopic length scale, may significantly help to develop and improve continuous models through the derivation of accurate closure relations. Eulerian-Lagrangian and DPM simulations are to multiphase flows what direct numerical simulations are to turbulent flows. This multiscale modeling strategy is represented in Fig. 1. The goal of the strategy is clear, but how to link the models and extract from each the information needed by those higher up in the hierarchy is an open challenge.

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Fig. 1 Multiscale modeling strategy

An Overview of Averaging Theory In the present section, we focus on three techniques that one can adopt to derive averages of point variables: statistical, volume, and time averaging.

Statistical Averaging As previously mentioned, predicting the dynamics of fluidized suspensions may appear conceptually simple: one has to solve the Newtonian equations of motion for each particle and the Navier-Stokes equations for the fluid. In practice, doing so is extremely demanding, because the number of particles is quite large; however, one could argue that this is only a practical and temporary issue, which future generations of computers will certainly overcome. There is, nevertheless, a more fundamental problem: to integrate the equations, one has to know the initial positions and velocities of all the particles. For large particle numbers, this information is impossible to obtain. The problem, therefore, cannot be addressed deterministically: a statistical approach is necessary. To clarify this concept, let us be more definite. Consider a fluidized suspension of ν identical, spherical, smooth particles, and let xs(t) and vs(t) be the position vector and linear velocity of the sth particle center, respectively. Initially, the sth particle is located in the point xs with velocity vs. If Fs(t) denotes the unit mass force acting on the particle, as time advances the latter moves obeying the equations: x_ s ðtÞ ¼ vs ðtÞ;

v_s ðtÞ ¼ Fs ðtÞ;

xs ð0Þ ¼ xs ;

vs ð0Þ ¼ vs (1)

Consequently, if we know the initial conditions for each particle and the functional expression of the overall force acting on each particle, by integrating the above

Recent Advances in Modeling Gas-Particle Flows

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differential equations, we may predict, with certainty, the particle positions and velocities at any future time. For a realistic number of particles, knowing the initial positions and velocities of the particles appears to be impossible even in principle. For systems comprising a great number of particles, therefore, we cannot know the initial state of every particle; in consequence, we cannot assign the initial conditions deterministically, as we did in Eq. 1. What we usually know are solely macroscopic – and therefore measurable – properties of the system, such as its local density, temperature, or mean velocity. But there are infinite system configurations yielding the same macroscopic properties, each with a certain probability of occurrence. So, we must replace the deterministic initial conditions above with probabilistic initial conditions. To this end, we introduce a probability density function (PDF) defined so that: π ν ðx1 , . . . , xν , v1 , . . . , vν ; tÞdx1 . . . dxν dv1 . . . dvν

(2)

gives the joint probability that at time t the first particle has position and velocity in the ranges dx1 and dv1 about the real-space point x1 and the velocity-space point v1, the second particle has position and velocity in the ranges dx2 and dv2 about the realspace point x2 and the velocity-space point v2, and so on up to the last particle forming the particulate system. If we let: r  ð x1 , . . . , x ν , v 1 , . . . , v ν Þ

(3)

we can regard r as the position point identifying the state of the entire particulate system in an abstract phase space of 6ν dimensions. Then π ν(r; t)dr is the probability that at time t the configuration of the particulate system lies in the range dr about the phase-space point r. We refer to the function π ν(r; t) as the ν-particle joint PDF or master joint PDF. At any given time, we do not know the exact configuration of the system, but the master joint PDF states how probable each configuration is. π ν(r; 0) dr, in particular, is the probability that the initial configuration lies in the range dr about r. No determinism is present: the system can be in any configuration, but for each one, the PDF tells us the probability of occurrence. Knowing the master joint PDF means having complete statistical knowledge of the population of particles. Another useful function is the one-particle marginal PDF, which gives the probability of finding a single particle in a differential neighborhood of a given state – independently of the states of all the other particles. For a system in which the particles are identical, and consequently indistinguishable, the ν-particle joint PDF is symmetrical with respect to the particle state variables, and the one-particle marginal PDF is equal for all the particles. Let π 1(x1, v1; t) denote the latter (for the real-space and velocity-space variables, the subscript is unimportant; the one reported has been selected only for convenience). By definition, π 1(x1, v1; t)dx1dv1 is the probability of finding at time t a particle – any particle of the population, not just the first particle – in the ranges dx1 and dv1 about the points x1 and v1. The one-particle marginal PDF contains significantly less information about the particulate system than the master

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joint PDF; however, as we shall see, in most cases of real interest, this is all the information that is truly required. We reduce the ν-particle joint PDF to the one-particle marginal PDF by integrating out the state variables of every particle but the first one: ð ð

ð π 1 ðx1 , v1 ; tÞ 

Ωx

...

ð

Ωx Ωv

...

ν

Ωv

π ν ðr; tÞ ∏ dxs dvs

(4)

s¼2

where Ωx and Ωv represent the ranges of variation of the particle positions and velocities, respectively; the former coincides with the region of physical space enclosed by the vessel containing the suspension, whereas the latter is unbounded and coincides with ℝ3. Knowing the master joint PDF allows calculating any average associated with the population of particles. Take a function b(r) that associates a scalar value with the state of the particulate system. This is referred to as dynamical function (Balescu 1975). If the system changes configuration (that is, the value of r changes), the value of the function changes. The observable 〈b〉s associated with the function b is the average value of the latter over all the system configurations; therefore, it is: ð hbis ðtÞ 

Ωr

bðrÞπ ν ðr; tÞdr

(5)

where Ωr represents the range of variation of r. In the integral above, we are summing all the values that the dynamical function can take, each one weighted by the probability of occurrence of the system configuration to which that value refers. Being macroscopic variables, observables do not depend on the microscopic state of the system but can be functions of the time, real-space, and velocity-space coordinates; in the expression above, 〈b〉s is only a function of time, for b does not depend on real-space and velocity-space coordinates; in general, however, this dependence will be present. Equation 5 suggests that to calculate any kind of macroscopic property of the system, one needs to know the master joint PDF (that is, one has to have complete knowledge of the system). In general this is true, but fortunately this is not always the case. A class of dynamical functions of particular theoretical importance is given by functions which take the following mathematical form: bðrÞ ¼

ν X

b1 ðxs , vs Þ

(6)

s¼1

where b1 is an arbitrary function of the phase-space state of one particle. This dynamical function depends on the state of the entire system, but the state of each particle is taken one at a time. An example is given by the total kinetic energy of the system, in which case b1  m(vs  vs)/2, where m is the particle mass. For this class of dynamical functions, we can write:

Recent Advances in Modeling Gas-Particle Flows

h bi s ð t Þ ¼ ð ð ¼v

ν ð X s¼1

Ωx Ωv

9

 Ωr

b1 ðxs , vs Þπ ν ðr; tÞdr ð ð

b1 ðx1 , v1 Þπ 1 ðx1 , v1 ; tÞdx1 dv1 ¼

Ωx Ωv

b1 ðx1 , v1 Þf 1 ðx1 , v1 , tÞdx1 dv1 (7)

where it is: f 1 ðx1 , v1 , tÞ  νπ 1 ðx1 , v1 ; tÞ

(8)

Here we have exploited the symmetry properties of the ν-particle PDF, which hold insofar as all the particles are identical. Thus, for this class of dynamical functions, to calculate the observables, one needs to know only the one-particle marginal PDF or equivalently the scalar function f1(x1, v1, t). Known as number density function (NDF), the latter arises naturally from the passages shown above but has as well an important physical interpretation: f1(x, v, t) dxdv represents the average number of particles present at time t in the range (or infinitesimal volume) dx around the real-space point x with velocity in the range dv around the velocity-space point v (we have removed the subscript from the arguments of the NDF for convenience). f1(x, v, t), in other words, is an observable representing the mean particle number density in the six-dimensional phase space formed by the union of the real space Ωx and velocity space Ωv. To prove this, we must show that the NDF is the mean value of the number density of particles present in the real-space point x with velocity v. This density has this expression: φ1 ðr; x, vÞ 

ν X

δðxs  x, vs  vÞ

(9)

s¼1

This is because if no particle is located in x with velocity v the density is zero, while if a particle is therein located, the density diverges (assuming that the volume of the particles is negligibly small compared with the macroscopic volumes of interest). Notice that φ1(r; x, v) belongs to the special class of dynamical functions defined by Eq. 6. Thus, we have: ð ð δðx1  x, v1  vÞπ 1 ðx1 , v1 ; tÞdx1 dv1 ¼ νπ 1 ðx, v; tÞ (10) hφ1 is ðx, v, tÞ ¼ ν Ωx Ωv

This differs from the NDF defined in Eq. 8 merely in notation. Because of the important physical meaning that the NDF possesses, one usually favors the latter over the one-particle marginal PDF; knowledge of either function, however, permits calculating observables associated with dynamical functions of the class defined by Eq. 6. Of course, to determine observables of this kind, one needs to know how the

10

L. Mazzei

NDF evolves in each phase-space point; this, as we shall see at the end of this chapter, is extremely challenging. To conclude this section, we present three examples of observables that are particularly significant: mass, linear momentum, and energy density. The dynamical functions with which these macroscopic quantities are associated take the following expressions: ν X bM ðr; xÞ  m δðxs  xÞ;

bL ðr; xÞ  m

ν X vs δðxs  xÞ

s¼1

s¼1

ν X bE ðr; xÞ  ðm=2Þ ðvs vs Þδðxs  xÞ

(11)

s¼1

Eq. 7 then gives: hbM is ðx, tÞ ¼ mnðx, tÞ;

hbL is ðx, tÞ ¼ mnðx, tÞhvis ðx, tÞ

hbE is ðx, tÞ ¼ ðm=2Þnðx, tÞhvvis ðx, tÞ

(12)

where n represents the expected number of particles per unit real-space volume, or equivalently the expected number density of particles in real space, 〈v〉s the expected particle velocity, and 〈v  v〉s twice the expected kinetic energy per particle unit mass. Mathematically, their expressions are: nðx, tÞ 

Ð

Ωv f 1 ðx,

1 hvis ðx, tÞ  nðx, tÞ

v, tÞdv; ð 1 ðvvÞf 1 ðx, v, tÞdv hvvis ðx, tÞ  nðx, tÞ Ωv

ð Ωv

v f 1 ðx, v, tÞdv (13)

In this section, to simplify the treatment, we have assumed that the state of each particle is identified only by position in real space and velocity. Additional coordinates can be introduced, such as the particle size, but the concepts presented do not change. For a more general treatment of this subject, we refer to the literature, in particular to the textbook by Marchisio and Fox (2013).

Volume Averaging Another method of deriving observables relies on volume averages; these are computed over spatial domains that are large enough to contain a statistically significant number of particles, but which are small compared with the length scale of variation of the observables. There are two kinds of volume averages: hard and soft. In the former, a volume Vx bounded by a surface Sx is attached to every spatial point x; within this volume, one averages the property of interest by using the mean value theorem of integral calculus. The values of the property within Vx are accounted for and ascribed the

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same weight in the average, while those outside Vx are ignored. Soft averages are based on an alternative technique, more elegant and convenient from a mathematical viewpoint, that uses radial weighting functions. These are continuous, monotone, decreasing functions of the radial distance from the spatial point in which the average is evaluated. This mathematical device ascribes a weight to the property values within the whole physical domain; however, the length scale over which the weighting function decays significantly (referred to as weighting function radius) identifies a spherical volume around the point of average x outside which the property values affect the average negligibly. The two averaging schemes, accordingly, are not as different as they may appear. For mathematical convenience, in what follows we favor soft averages. Volume-averaged variables might appear to depend on the specific choice of volume Vx or of weighting function (in particular on its radius). The larger the ratio between the smallest length scale of variation of the observables and the particle size, the more such a dependence dwindles provided that the weighting function radius is properly chosen. If this radius is denoted by r2, the particle radius by r1, and the macroscopic length scale by r3, the local average is expected not to depend on the particular form of weighting function provided that the condition r1  r2  r3 is satisfied. In such a case, there is said to exist separation of scales between the macroscopic fluid dynamic problem and the detailed motion at the scale of a single particle. Only in this instance the volume-averaged variables have an unambiguous physical meaning. In multiphase systems, made up of one continuous phase (the fluid) and one or more discrete phases (the particles), one can employ volume averages to obtain mean properties for each phase. We now first introduce formally the weighting functions and then report how such averages are defined.

Weighting Functions Weighting functions are characterized by the following mathematical properties: 1. The weighting function ψ is a scalar function of r defined for r > 0, where r denotes the distance of a point z from a point x in Euclidean space: ψ ¼ ψ ðr Þ,

r  jx  zj

(14)

2. ψ(r) is positive for any value of r, decreases monotonically with r and possesses continuous derivatives of any order. In other words, it is a function of class C1. 3. ψ(r) is normalized, so that, if Ωx denotes the spatial domain occupied by the system of interest (assumed here to stretch out to infinity), it is: ð Ωx

ψ ðjx  zjÞdz ¼ 4π

ð1 0

ψ ðr Þr 2 dr ¼ 1

(15)

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L. Mazzei

In the integral on the left-hand side, z is the spatial variable of integration, while x is the spatial position in which the volume average is computed. The radius of the weighting function is defined as the scalar r2 that satisfies the following equation: ð r2 4π

ψ ðr Þr 2 dr ¼ 4π

0

ð1

ψ ðr Þr 2 dr ¼

r2

1 2

(16)

The weighting function radius is thus a measure of the linear size of the spherical neighborhood of x in which the spatial points have appreciable weight in the averaging process.

Fluid-Phase Volume Averages The void fraction, or fraction of space occupied by the fluid, and the fluid-phase volume average of a generic point variable ζ(x, t) calculated in x at time t are so defined: ð eðx, tÞ 

ψ ðjx  zjÞ dz;

Λe

1 hζ ie ðx, tÞ  eðx, tÞ

ð Λe

ζ ðz, tÞψ ðjx  zjÞdz

(17)

In the equations above, Λe represents the domain occupied by the fluid phase at time t (we have left out the explicit dependence on t to simplify the notation).

Solid-Phase Volume Averages In a system with ν solid phases, the volume fraction of the rth phase Sr and the solidphase volume average of a generic point variable ζ(x, t) calculated in x at time t are so defined: ϕr ðx, tÞ 

Xð Sr

Λr

ψ ðjx  zjÞ dz;

hζ irs ðx, tÞ 

Xð 1 ζ ðz, tÞψ ðjx  zjÞ dz ϕr ðx, tÞ S Λr r

(18) where Λr is the region of physical space occupied by a generic particle of phase S r at time t. The summation is over all the particles of phase Sr. In hζ irs the subscript s indicates that this is a solid-phase volume average, while the superscript r indicates that the average refers to solid phase Sr. This average, used by several researchers (Enwald et al. 1996; Drew and Passman 1998), operates on the microscopic properties of the particle material, considering point fields ζ(x, t) that vary within the particles. It is an average which exactly parallels the one given for the fluid. Another approach, advanced by Anderson and Jackson (1967), is based on properties ζ r(t) of each particle as a whole.

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Particle-Phase Volume Averages Since the particles are rigid, their motion is determined by the translation of their centers of mass and by the rotation of their bodies about instantaneous axes of rotation. Thus, the resultant forces and torques acting on the particles suffice to establish their motion. We can then introduce a different kind of volume average that depends only on properties of the particles as a whole. We define the number density of particles of class Sr calculated in x at time t as follows: nr ðx, tÞ 

X

ψ ðjx  zr ðtÞjÞ

(19)

Sr

zr(t) being the position occupied at time t by the center of mass of a generic particle of solid phase Sr. The volume fraction ϕr(x, t) is related to the number density nr(x, t) as follows: ϕr ðx, tÞ  nr ðx, tÞV r

(20)

in which Vr is the volume of a particle of solid phase Sr. As indicated, this equation is approximate, but it is accurate if the separation-of-scale requirement is met and the weighting function radius is selected correctly; this, as said, must be far larger than the particle radius. Generalizing the averaging scheme of Jackson (1997), we define the particlephase volume average for a particle property ζ r(t) of solid phase Sr calculated in x at time t as: hζ irp ðx, tÞ 

X 1 ½ζ r ðtÞψ ðjx  zr ðtÞjÞ nr ðx, tÞ S

(21)

r

In hζ irp the subscript p indicates that this is a particle-phase volume average, while the superscript r indicates that the average refers to solid phase Sr.

Time Averaging The third averaging method available is time averaging. Let us consider a field ζ(x, t); for any fixed spatial position x, ζ(x, t) is a function of time that fluctuates irregularly. We denote the time scale that represents these fluctuations as τ1. In x, we can obtain a mean value of ζ(x, t) by time averaging over a large number of fluctuations, considering a time interval τ2 much larger than the time scale of the fluctuations. Again, we resort to the mean value theorem, this time writing: hζ it ðx, tÞ 

1 τ2

ð tþα tα

ζ ðx, τÞdτ,

α  τ2 =2

(22)

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L. Mazzei

where 〈ζ〉t(x, t) denotes the time average and τ is a dummy integration variable. Also now, the mean value is expected to be insensitive to the averaging time scale provided that τ1  τ2  τ3, where τ3 represents the time scale of the mean flow variations. That is, there has to be separation of scale (now in the time domain) between the macroscopic motion of the fluid-solid mixture and the microscopic motion of the particles; only in this case the time-averaged variables have an unambiguous physical meaning.

A Final Remark Before concluding this section on averaging, we would like to point out that different averaging schemes can lead to different average values. If the values are equal, the system is said to be ergodic, but not all systems present this feature. For details, refer, for instance, to Jackson (2000).

Averaged Equations of Motion for Fluid-Particle Systems We now derive the averaged equations of motion for a generic fluid-particle system of ν solid phases by using volume averages. Similar equations can be obtained with statistical and time averages; refer, for instance, to Gidaspow (1994), Drew and Passman (1998), and Brilliantov and Poschel (2004). Our treatment is an extension of the work of Jackson (2000) and Owoyemi et al. (2007). Below, we adopt Einstein’s convention: repeated indices are summed over the values one to three, with the exception of r and s, used as phase indices, and of e and p, used to specify the volume average type.

Fluid Phase Let us first derive the volume-averaged continuity equation. The starting point is the microscopic continuity equation for the fluid. If we assume that the latter is incompressible, this reads: @ a ua ¼ 0

(23)

where @ a  @/@xa and ua is the ath component of the fluid velocity vector u(x, t) with respect to a generic orthonormal vector basis. Let us multiply both sides by ψ(|x  z|) and integrate over Λe with respect to z; doing so yields this averaged equation: ð eh@a ua ie ¼

Λe

½@a ua ðz, tÞψ ðjx  zjÞdz ¼ 0

(24)

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15

In this form the equation is not useful, because it is written in terms of averaged derivatives of point variables instead of derivatives of averaged point variables. We may, however, manipulate the equation by using these mathematical relations, whose proof is given in the appendix: 



eh@a ζ ie ¼ @a ehζ ie 

ν Xð X r¼1

ν Xð   X eh@t ζ ie ¼ @t ehζ ie þ r¼1

Sr

ζ ðz, tÞka ðz, tÞψ ðjx  zjÞ dσ z

(25)

ζ ðz, tÞuðz, tÞkðz, tÞψ ðjx  zjÞ dσ z

(26)

@Λr

Sr

@Λr

Here @ t  @/@t, k(x, t) is the outward unit normal to the surface @Λr bounding Λr, while ka(x, t) is the ath component of the unit vector k(x, t). Setting ζ  ua and ζ  1 in Eqs. 25 and 26, respectively, and adding the results yields the averaged continuity equation in the form that we sought:   @t e þ @a ehua ie ¼ 0

(27)

In this equation, as we should have expected, the fluid volume fraction takes on the role that the fluid density has for single-phase compressible fluids. Let us go on to derive the volume-averaged linear momentum balance equation for the fluid. The starting point is the corresponding microscopic balance equation: ρe ½@t ua þ @b ðua ub Þ ¼ @b T ab þ ρe ga

(28)

where ρe is the (constant) fluid density and Tab(x, t) is the abth component of the point fluid stress tensor, while ga is the ath component of the gravitational field. Multiply both sides by ψ(|x  z|) and integrate over Λe with respect to z. To treat the left-hand side of the averaged equation obtained, write Eqs. 25 and 26 with ζ  ua ub and ζ  ua, respectively, while to treat the right-hand side, use Eq. 25 with ζ  Tab. With these relations, the averaged equation becomes:      ρe @t ehua ie þ @b ehua ub ie ν X X   ¼ @b ehT ab ie þ eρe ga  r¼1 S r

ð @Λr

T ab ðz, tÞkb ðz, tÞψ ðjx  zjÞdσ z

(29)

The last term on the right-hand side is the sum over all particle classes of the mean resultant traction forces exerted by the fluid on the particles of each class. The force: Xð Sr

@Λr

T ab ðz, tÞkb ðz, tÞψ ðjx  zjÞdσ z

(30)

is the sum of the average resultant forces exerted by the fluid on the rth phase particles. To compute this force for each particle, we first weight the differential

16

L. Mazzei

traction forces acting on each infinitesimal region dσ z of the particle surface using the value of ψ(|x  z|) corresponding to each region, and then we sum the (infinite number of) contributions. The fluid-solid interaction force, defined by Eq. 30, couples the linear momentum balance equation of the fluid to that of each particle class. For reasons that will be clear later (when we deal with the solid phases), it is convenient to express Eq. 30 differently. To do so, we expand the weighting function in a Taylor series about the center zr(t) of a generic particle of phase Sr, writing: 8z  @Λr : ψ ðjx  zjÞ  ψ ðjx  zr jÞ (31)

 2   ½@b ψ ðjx  zr jÞr r kb ðzÞ þ ð1=2Þ @bc ψ ðjx  zr jÞ r 2r kb ðzÞkc ðzÞ

where rr is the radius of the particles of phase Sr. As the particle radius is far smaller than the radius of the weighting function, we may truncate the Taylor series at the second-order term with acceptably small error. Using this relation, we approximate the force in Eq. 30 as:   2 nr hf a irp  @b nr hAab irp þ ð1=2Þ@bc nr hBabc irp

(32)

where it is: nr ðx, tÞhf a irp ðx, tÞ

ð X  ψ ð j x  zr j Þ

@Λr

Sr

nr ðx, tÞhAab irp ðx, tÞ 

ð X ψ ðjx  zr jÞr r

nr ðx, tÞhBabc irp ðx, tÞ



X Sr

T ad ðz, tÞkd ðz, tÞdσ z

(33) 

@Λr

Sr



T ad ðz, tÞkd ðz, tÞkb ðz, tÞdσ z

(34) 

ð ψ ðj x 

zr jÞr 2r

@Λr

T ad ðz, tÞkd ðz, tÞkb ðz, tÞkc ðz, tÞdσ z (35)

The quantities defined above are the components of a vector, a second-order tensor, and a third-order tensor, respectively. The force in Eq. 30 is obtained by first weighting the differential traction forces exerted on the infinitesimal surface elements of the fluid-particle interface, using the values of the weighting function at the locations of the elements, and then by summing such contributions. The force in Eq. 33, on the other hand, is obtained by first calculating the forces acting on the entire surface of each particle, then by weighting them using the values of the weighting function at the particle centers, and finally by summing such contributions. This second average interprets better the fluid-particle interaction force and fulfills the principle of action and reaction, as we will see in section “Solid Phases”; this is why we prefer to operate in terms of this average force and of the additional contributions appearing in Eq. 32.

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The convective term in Eq. 29 features the average of the product of point velocity components. We find it convenient to decompose it into the sum of a product of average velocity components and of an average of velocity fluctuations; thus, we write: hua ub ie  hua ie hub ie þ h^u a ^u b ie

(36)

Here hatted variables denote the deviations of point variables from their respective mean values. The relation above is not exact, holding only when there is separation of scale between the microscopic and macroscopic descriptions of the flow. Introducing Eqs. 32 and 36 into Eq. 29 yields: ν  X      ρe @t ehua ie þ @b ehua ie hub ie ¼ @b hSab ie  nr hf a irp þ eρe ga

(37)

r¼1

where it is: hSab ie  ehT ab ie þ

ν h  i X nr hAab irp  ð1=2Þ@c nr hBabc irp  eρe h^u a ^u b ie

(38)

r¼1

This term is the fluid-phase effective stress tensor. Finding an analytical closure for it is extremely complex, but Jackson (1997) did so for the limiting case of diluted, Stokesian, monodisperse suspensions fluidized by Newtonian fluids. We will address the problem of closure, for all the terms featuring on the right-hand side of Eq. 37 and of the ν solid-phase-averaged dynamical equations, later on in section “The Problem of Closure.”

Solid Phases The volume-averaged continuity equation for the generic solid phase S r can be derived quite easily by using this mathematical relation, whose proof is given in the appendix:  

r nr ζ_ p ¼ @t nr hζ irp þ @a nr hζva irp

(39)

where the dot denotes a total time derivative and hζva irp ðx, tÞ is the average of the product of ζ r(t) and of the ath component of the velocity vr(t) of the particle center. Setting ζ r  1 gives:  @t nr þ @a nr hva irp ¼ 0

(40)

which is the equation sought. Here the particle number density, or equivalently the volume fraction, takes on the (compressible) fluid density role.

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L. Mazzei

To derive the volume-averaged equation of motion for the generic solid phase S r, we adopt the equation governing the motion of the generic particle of such phase: ð ρr V r v_r, a ðtÞ ¼

@Λr

ν X X

T ab ðz, tÞkb ðz, tÞdσ z þ

s¼1

Ss

f rs, a ðtÞ þ ρr V r ga

(41)

where ρr denotes the density of the particles of phase S r, v_r, a ðtÞ the ath component of the acceleration of the particle center of mass while frs,a(t) the ath component of the force exerted on the r particle by the generic s particle of phase S r when a collision takes place. This force does not vanish only if particles r and s are in direct contact (it is zero for most s particles). Notice also that frs,a(t) vanishes when r and s refer to the same particle, because particles r and s need, of course, to be different. The surface integral on the right-hand side of the equation is the overall force exerted by the fluid on the particle. To average Eq. 41, we multiply both sides by ψ(|x  zr|) and sum over all the particles belonging to phase S r. Doing so gives: ρr V r

X

 ð  X ψ ðjx  zr ðtÞjÞv_r, a ðtÞ ¼ ψ ðjx  zr ðtÞjÞ

Sr

þ

X

Sr

" ψ ðjx  zr ðtÞjÞ

Sr

ν X X s¼1 S s

#

f rs, a ðtÞ þ ρr V r ga

X

 @Λr

T ab ðz, tÞkb ðz, tÞdσ z

(42) ψ ðjx  zr ðtÞjÞ

Sr

We now employ Eqs. 19, 21, 33, and 39, with ζ r  v_r, a in the second and ζ r  vr, a in the fourth, and the following relation, whose proof is left to the reader: X

" ψ ðjx  zr ðtÞjÞ

Sr

ν X X s¼1

Ss

# f rs, a ðtÞ ¼

ν X X

" ψ ðjx  zr ðtÞjÞ

s¼1 S r

X Ss

# f rs, a ðtÞ

(43)

to obtain: h   i ρr V r @t nr hva irp þ @b nr hva vb irp " # ν X X X r ¼ nr hf a ip þ nr ρr V r ga þ ψ ðjx  zr ðtÞjÞ f rs, a ðtÞ s¼1 S r

(44)

Ss

The first term on the right-hand side is the fluid-particle interaction force – which also features, with opposite sign, in Eq. 37. This force satisfies the action-andreaction principle, as it should. The final term combines the resultant forces arising from the particle-particle contacts among particles that belong to the same phase (s = r) and to different phases (s 6¼ r). These contributions are conceptually different, insofar as the former is a self-interaction term that represents the stress internal to the phase under examination, while the latter is a contact force acting between the

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Eulerian solid phases. To let the collisional solid stress tensor associated with phase S r appear explicitly in Eq. 44, we need to manipulate the equation further. We first consider the following double sum over the particles r and s of the rth phase: X X Sr

Sr

ψ ðjx  zrs ðtÞjÞf rs, a ðtÞ



(45)

in which zrs(t) denotes the position vector of the point of mutual contact between the rigid particles r and s. This double sum vanishes, inasmuch as zrs = zsr and, for the action-and-reaction principle, frs,a = fsr,a. If we then expand ψ(|x  zrs|) in a Taylor series around zr, letting krs denote the unit vector of the vector zrs  zr, we obtain from the equation above that: X

" ψ ðjx  zr ðtÞjÞ

Sr

X Sr

#

h  i f rs, a ðtÞ  @b nr hMab irp  ð1=2Þ@c nr hN abc irp

(46)

where it is: nr ðx, tÞhMab irp ðx, tÞ



X

( ψ ðjx  zr ðtÞjÞr r

Sr

nr ðx, tÞhN abc irp ðx, tÞ



X

X Sr

( ψ ðjx 

zr ðtÞjÞr 2r

Sr

 f rs, a ðtÞkrs, b ðtÞ

)

X  f rs, a ðtÞkrs, b ðtÞkrs, c ðtÞ

(47) ) (48)

Sr

The second-order tensor so defined:  hCab irp  nr hMab irp  ð1=2Þ@c nr hN abc irp

(49)

is the collisional stress tensor of the rth particle phase which accounts for the transfer of linear momentum at collisions between alike particles over the distance 2rr separating their centers. This physical phenomenon is important in dense fluidized suspensions, where the total volume occupied by the particles is not negligible compared with the volume of the vessel containing them. For rarefied granular gases, which one can model adopting the Boltzmann-Grad limit, defined as: r r ! 0;

νr ! 1;

νr r 2r bounded

(50)

in which νr represents the overall number of particles belonging to solid phase S r, because zr ! zrs ! zs, the collisional stress vanishes (this result is well known in kinetic theory of gases; see, for instance, Chapman and Cowling 1970 or Gidaspow 1994). This is consequence of the principle of action and reaction, insofar as in the Boltzmann-Grad limit it is:

20

L. Mazzei

X

" ψ ðjx  zr ðtÞjÞ

Sr

X Sr

# f rs, a ðtÞ !

X X  ψ ðjx  zrs ðtÞjÞf rs, a ðtÞ ¼ 0 Sr

(51)

Sr

Consider now the other contribution to the overall particle-particle contact force appearing on the right-hand side of Eq. 44. This term, which represents the contact forces acting between r particles of phase S r and s particles of phase S s, can be expressed as: X

" ψ ðjx  zr ðtÞjÞ

Sr

X Ss

# f rs, a ðtÞ

(52)

where s 6¼ r and with particles r and s belonging to phases S r and S s, respectively. Given its definition, this force should fulfill the principle of action and reaction, so that: X

" ψ ðjx  zr ðtÞjÞ

X

Sr

Ss

# f rs, a ðtÞ ¼ 

X

" ψ ðjx  zs ðtÞjÞ

Ss

X Sr

# f sr, a ðtÞ

(53)

Clearly, this condition is not satisfied, since even if frs,a = fsr,a, it is ψ(|x  zr|) 6¼ ψ(|x  zs|). Only when rr ! 0 and rs ! 0, this equation holds. We conclude that the force in Eq. 52 cannot be regarded as the interaction force between phases S r and S s but must include an additional contribution which does not satisfy the actionand-reaction principle. To find this force, we expand ψ(|x  zrs|) in a Taylor series about the point zr. Doing so gives: X

" ψ ðjx  zr ðtÞjÞ

Sr

X Ss

# f rs, a ðtÞ  nr hf a irs p h  i rs þ @b nr hPab irs (54) p  ð1=2Þ@c nr hQabc ip

where it is: nr ðx, tÞhf a irs p ðx, tÞ



X

( ψ ðjx  zrs ðtÞjÞ

Sr

nr ðx, tÞhPab irs p ðx, tÞ



X Sr

nr ðx, tÞhQabc irs p ðx, tÞ 

X Sr

(

X Ss

) f rs, a ðtÞ

X  ψ ðjx  zr ðtÞjÞr r f rs, a ðtÞkrs, b ðtÞ

(55) ) (56)

Ss

( ψ ðjx  zr ðtÞjÞr 2r

X Ss

f rs, a ðtÞkrs, b ðtÞkrs, c ðtÞ

) 

(57)

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The second-order tensor so defined:  rs rs hDab irs p  nr hPab ip  ð1=2Þ@c nr hQabc ip

(58)

is the collisional stress tensor related to the momentum transferred at collisions between phases S r and S s. We find it natural now to introduce the following tensor: hSab irp  hCab irp þ

ν X

v a^v b irp hDab irs p  nr ρr V r h^

(59)

s6¼r¼1

This is the effective stress tensor of phase S r. The first two contributions, taken together, represent the (total) collisional stress tensor, while the last, which arises from the Reynolds decomposition of the convection term in the averaged dynamical equation, represents the kinetic stress tensor. In light of the results obtained above, we can express the averaged linear momentum balance equation for phase S r as follows: h   i ρr V r @t nr hva irp þ @b nr hva irp hvb irp ¼ @b hSab irp þ nr hf a irp þ

ν X

nr hf a irs p þ nr ρr V r ga

(60)

s6¼r¼1

In this equation, the interaction forces between the phases (represented by the second and third terms on the right-hand side) satisfy the action-and-reaction principle. Table 1 reports, in absolute notation, the multifluid equations of motion just derived.

The Problem of Closure The averaged equations of motion for the fluid and solid phases just derived are mathematically unclosed, for they feature terms related to point (i.e., microscopic) variables. In their current form, therefore, the equations cannot be solved. An example of such terms is given by the interaction force between the fluid phase and the generic solid phase S r. This force, as seen, is equal to: h f irp ðx, tÞ

 ð X 1  ψ ð j x  zr j Þ Tðz, tÞkðz, tÞdσ z nr ðx, tÞ S @Λr

(61)

r

To calculate it, one needs to know the point fluid stress distribution T(x, t) over the surface of each particle as well as the position of each particle. This distribution is related to the point velocity field of the fluid, not to its volume average; because in a macroscopic description of the flow this field and the particle positions are unknown, Eq. 61 has no practical use (in a macroscopic modeling context).

22

L. Mazzei

Table 1 Eulerian-Eulerian averaged equations of motion for a system of n-particle classes Continuity equation – fluid phase @ te + @ x  (ehuie) = 0 Continuity equation – particle phase r  @ t ϕr þ @ x  ϕr hvirp ¼ 0 Dynamical equation – fluid phase ν      P ρe @ t ehuie þ @ x  ehuie huie ¼ @ x hSie  nr hfirp þ eρe g r¼1

Dynamical equation – particle phase r h   i ν P ρr @ t ϕr hvirp þ @ x  ϕr hvirp hvirp ¼ @ x hSirp þ nr hfirp þ nr hfirs p þ ϕr ρr g s6¼r¼1

We manipulated the averaged equations of motion in such a way that the closure problem is confined to a small number of well-defined terms. These are the effective stress tensor for each phase and the interaction forces between the fluid and each solid phase and between each pair of solid phases. Overcoming the closure problem means deriving expressions for them in terms of averaged variables only. Analytical closures based on purely theoretical arguments are prohibitively difficult to obtain; there is no guarantee that such equations even exist. Usually, the goal is far less ambitious and is finding equations that consent to analyze the systems of interest with the desired accuracy; such equations should be the simplest able to capture enough physics to describe the fluid dynamics of the suspension satisfactorily. In what follows, we first present some strategies for modeling the effective fluid and solid stress tensors; we then analyze the mean fluid-particle interaction force, laying emphasis on the buoyancy and drag forces, and the mean particle-particle interaction force.

Effective Stress Owing to the many contributions, yielded by the averaging process, that make up the effective stress tensors, these are complex to model. Closing these quantities is further complicated by the absence of experimental measurements having a direct bearing on them. Notwithstanding, researchers usually suppose that both fluid and solid phases behave as Newtonian fluids, writing: 

 2 Se ¼  pe  κe  μe trDe I þ 2μe De ; 3 

 2 Sr ¼  pr  κr  μr trDr I þ 2μr Dr 3

(62)

where pe, pr, κe, κr, μe, and μr are the averaged pressures, dilatational viscosities and shear viscosities of the fluid, and rth solid phase, respectively; furthermore, I is the

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23

identity tensor, while De and Dr. are the rate of deformation (or strain) tensors, defined as: De 

 1 @x ue þ @x uTe ; 2

Dr 

 1 @x vr þ @x vTr 2

(63)

From now on, as done in these last expressions, we simplify the notation by leaving out the angular brackets that imply averaging. Experimental evidence has shown that in several fluidization regimes, the assumption of Newtonian behavior is satisfactory, especially for powders far from maximum packing. If Eq. 62 holds, the closure problem reduces to finding suitable constitutive expressions for the pressure, dilatational viscosity, and shear viscosity of each phase. As done in section “Averaged Equations of Motion for Fluid-Particle Systems,” often one assumes that the fluid is incompressible, so that no constitutive equation is required for pe, that κe is zero and that μe is proportional to e, the proportionality constant being the shear viscosity of the pure fluid. For the solid phases, constitutive expressions for these quantities have been derived from granular kinetic theory (Gidaspow 1994; Brilliantov and Poschel 2004), a generalization of the mathematical theory of dense nonuniform gases (Chapman and Cowling 1970). The idea is that, because dense granular gases resemble in many ways dense molecular gases, the constitutive equations that govern the two should be derivable, at least in part, from the same theoretical framework. Similarly to a molecular gas, particle pressure and viscosities are functions of a granular temperature, which is governed by a balance equation for a pseudointernal energy related to the particle peculiar velocity. For solid phase S r, the balance equation is: ρr ½@t ðϕr U r Þ þ @x ðϕr U r vr Þ ¼ @x qr þ Sr : @x vr þ Gd, r  Sv, r  Sc, r

(64)

where Ur  3/2Θr is the pseudointernal energy per unit mass, Θr being the granular temperature, and qr. is the pseudothermal heat flux. The equation differs from the classical internal energy balance equation (Bird et al. 1960) because of a sink term Sc,r(x, t) representing energy degradation caused by inelastic collisions, a source term Gd,r(x, t) representing generation of particle velocity fluctuations by fluctuating fluid-particle forces, and a sink term Sv,r(x, t) representing their dampening by viscous resistance to particle motion. qr. is usually modeled using Fourier’s law, writing: qr ¼ kr @x Θr

(65)

where kr is the granular thermal conductivity of the rth solid phase. Different closures have been developed for this parameter; Gidaspow et al. (1992), for instance, proposed:

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L. Mazzei

kr ¼

150ρr sr ðπΘr Þ1=2 ½1 þ ð6=5Þϕr αr ð1 þ er Þ2 384αr ð1 þ er Þ þ 2ϕ2r ρr sr αr ð1 þ er ÞðΘr =π Þ1=2

(66)

where sr denotes the particle diameter, er the coefficient of restitution for particle collisions, and αr the radial distribution function for the rth solid phase. Various expressions are available for this function; for instance, that advanced by Iddir and Arastoopour (2005) reads: αr ¼ ½1  ðϕ=ϕmax Þ1 þ ð3sr =2Þ

ν X

ðϕs =ss Þ

(67)

s¼1

where ϕ is the overall solid volume fraction and ϕmax is the maximum solid compaction (i.e., the maximum value which ϕ can take). Here αr diverges positively when ϕ approaches ϕmax. An expression in which αr is bounded is that of Lebowitz (1964), where ϕmax does not feature: " αr ¼ ð1=eÞ 1 þ ð3sr =2eÞ

ν X

# ðϕs =ss Þ

(68)

s¼1

To close Eq. 64, one needs constitutive equations also for the terms Gd,r, Sv,r, and Sc,r. For briefness, we do not report them; the interested reader may refer, for instance, to Gidaspow (1994), Syamlal et al. (1993), Fan and Zhu (1998) and Jackson (2000). Various closures for the solid pressure are available in the literature, all derived from the granular kinetic theory. As an example, we report the expression advanced by Lun et al. (1984), suitably extended to cater for polydisperse suspensions: " pr ¼ 1 þ 2

ν X

# 3

ðsrs =sr Þ ϕs αrs ð1 þ ers Þ ϕr ρr Θr

(69)

s¼1

where sr and ss are the particle diameters for phases r and s, respectively, ers is the coefficient of restitution for collisions between particles of phases r and s, while: srs  ðsr þ ss Þ=2;

αrs  ðsr αs þ ss αr Þ=ðsr þ ss Þ

(70)

As an example of constitutive equations for the solid-phase dilatational and shear viscosities, we report those given in Gidaspow (1994), even if several are available in the literature: κr ¼ ð4=3Þϕ2r ρr sr αr ð1 þ er ÞðΘr =π Þ1=2

(71)

Recent Advances in Modeling Gas-Particle Flows

μr ¼

25

10ρr sr ðπΘr Þ1=2 ½1 þ ð4=5Þϕr αr ð1 þ er Þ2 96αr ð1 þ er Þ þ ð4=5Þϕ2r ρr sr αr ð1 þ er ÞðΘr =π Þ1=2

(72)

These expressions are those originally developed for monodisperse suspensions and do not directly account for the presence of the other solid phases. The expressions given above, as said, are based on the kinetic theory of granular flows. This assumes that particles are smooth and spherical, that collisions are binary and instantaneous, and that the suspension is far from the frictional packing limit, which marks the transition from the viscous to the frictional flow regime. In the first regime, particles undergo transient contacts, momentum transfer is translational and collisional, and the granular kinetic theory holds; in the second, particles undergo enduring contacts, and momentum transfer is mainly frictional. Granular kinetic theory does not account for these interactions, and thus in the frictional flow regime, the closures reported above are inadequate. In regions of high solid volume fraction, particles interact with multiple neighbors and the mechanism for stress generation is not merely due to kinetic and (particularly) collisional contributions but also to sustained contacts among particles. Such contacts make particles dissipate considerable energy, letting them form very dense regions in the bed. This increases the ability of the granular assembly to resist shearing, for tangential frictional forces at contact points are now present. Hence, the solid viscosity is larger than that predicted by the granular kinetic theory. To describe the frictional stress, other models, empirical, phenomenological, or based on the theory of soil mechanics, are needed. Usually, one assumes that it is: ⋆ ⋆ S⋆ r ¼ pr I þ 2μr Dr

(73)

where the star indicates that the quantity refers to the frictional flow regime. Syamlal et al. (1993) proposed this equation for the frictional pressure: ⋆ p⋆ r ¼ ϕr p ,

 B p⋆  10A ϕ  ϕf

(74)

where ϕf denotes the frictional solid packing (the solid volume fraction threshold value at which the powder enters the frictional flow regime). The coefficients A and B are very high, with typical values of 25 and 10, respectively. For other constitutive equations, the reader is referred to the literature. An expression often used for the frictional shear viscosity is that of Schaeffer (1987), which reads: μ⋆ r

ν X p⋆ r sin ϑr ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p ϕs 2 I 2 ðDr Þ s¼1

!1 ,

I 2 ðD r Þ 

i 1h ðtrDr Þ2  trD2r 2

(75)

where ϑr is the angle of internal friction of the rth granular material, while I2(Dr) is the second invariant of the rate of deformation tensor. Other expressions are

26

L. Mazzei

available in the literature. In the frictional flow regime, one usually accounts also for the kinetic and collisional contributions to the solid stress; the easiest way to do this is adding the viscous stress tensor to the frictional one.

Fluid-Particle Interaction Force There are five main contributors to the fluid-particle interaction force. The first is the buoyancy force, whose definition in the context of multiphase flows is not unique and needs to be discussed. The second acts in the direction of the fluid-particle slip velocity – that is, the fluid velocity relative to an observer moving with the same local mean velocity as the particles. The third is normal to the slip velocity, the fourth is parallel to the relative acceleration between the phases, and the fifth is proportional to the local mean acceleration of the fluid. The last four terms are commonly referred to as drag force, lift force, virtual mass force, and local fluid acceleration force, respectively. As we shall see, the local fluid acceleration force is not always present but features only when one definition of buoyancy force is used – in particular, the classical definition presented later on. Among these five terms, often the buoyancy and drag forces are dominant.

Buoyancy Force A first definition sets this force equal to the weight of the fluid displaced by the solid; accordingly, if we refer the force to the unit volume of suspension, it is: nr fB⋆, r  ϕr ρe g

(76)

Since it is consistent with the Archimedes’ principle original formulation, we call this classical definition. For a given value of ϕr, this force is constant, being unrelated to the flow. The second definition relates the force to the effective fluid stress tensor, as reported by Jackson (2000); per unit volume of suspension, it is: nr fB•, r  ϕr @x Se

(77)

Another definition often encountered in the literature considers solely the isotropic part of the effective stress tensor of the fluid; the closure therefore takes the form: nr fB∘, r  ϕr @x pe

(78)

These definitions lead to different values of the buoyancy force. There is nothing wrong with this, for we are free to define this force as we like: what is crucial is that the total fluid-particle interaction force nr fr, which has an objective physical meaning, be correctly calculated. Thus, modelers who adopt different definitions of buoyancy force will also need to employ different expressions for the complementary force that makes up the total fluid-particle interaction force. The value of the

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27

latter must be the same in all models. So, for instance, if one opts to use Eq. 78, the contribution of the deviatoric part of Se must be included in the complementary force: it will still be present, the only difference being that it is not regarded as part of the buoyancy force, but as part of the complementary force. To better understand the meaning of these definitions, consider a monodisperse suspension of motionless particles equally distributed in space (i.e., an ideal homogeneous bed). The second and third definitions here coincide, because De vanishes and Se is therefore isotropic: Se ¼ pe I ) nfB• ¼ nfB∘

(79)

where n is the particle number density. We can derive an expression for this force using the mean dynamical equations reported in Table 1; these reduce to: nf ¼ @x pe þ eρe g;

nf ¼ ϕρs g

(80)

where ϕ denotes the solid volume fraction, ρs the particle density, and n f the fluidparticle interaction force. Subtracting the two equations and using Eq. 78 gives: @x pe ¼ ðeρe þ ϕρs Þg;

nfB∘ ¼ nfB∘ ¼ ϕðeρe þ ϕρs Þg

(81)

Thus, for ideal uniform fluidized beds, the difference between the first and the other two definitions reduces to the density choice in the force expression: the first requires the fluid density while the second the suspension bulk density. For further details about this topic, we refer to Jackson (2000).

Local Fluid Acceleration Force If the classical definition of buoyancy force is employed, the complementary force to the total fluid-particle interaction force must include a term known as local fluid acceleration force (this term is absent otherwise). Per unit volume of suspension, this force is given by: nr fA⋆, r  ϕr ρe Det ue ,

Det ue  @t ue þ ue  @x ue

(82)

where the derivative on the right-hand side is a material derivative relative to a Lagrangian observer moving with the locally averaged velocity of the fluid. If the fluid acceleration is far less than the gravitational acceleration, the local fluid acceleration force is far less than the buoyancy force, and so its contribution is negligible. This force, nonetheless, is conceptually important, as the following thought experiment reveals. Consider a uniform assembly of particles at rest in a body of fluid. The fluid is also at rest in a vertical container placed on a horizontal plane. The system resides in a uniform gravitational field. If the plane supporting the container and the constraints keeping the particles at rest are suddenly removed, the entire system falls freely with an acceleration equal to g. Since the mean velocity fields of both phases are uniform and no pressure gradients are present, the effective

28

L. Mazzei

stress tensors of both phases vanish, and the dynamical equations in Table 1 reduce to: eρe Det ue ¼ nf þ eρe g;

ϕρs Dst vs ¼ nf þ ϕρs g

(83)

For convenience, we have used the nonconservative formulation of the equations; to obtain them, one must combine the dynamical and continuity equations (see, for instance, Bird et al. 1960). The material derivative for the solid phase is defined similarly to that for the fluid phase. In the case at hand, both material derivatives are equal to the gravitational acceleration, and Eq. 83 leads to the same result: the fluidparticle interaction force must vanish. This condition can be met only if the local fluid acceleration force is accounted for. As the two phases move identically, no slip velocity and acceleration are present between them; consequently, the drag, virtual mass, and lift forces are all zero (see the sections below dedicated to these forces). Conversely, if its classical definition is adopted, the buoyancy force is nonzero. So, the total fluid-particle interaction force can vanish only if the local fluid acceleration force is considered: nf ¼ nfB⋆ þ nfA⋆ ¼ ϕρe g þ ϕρe g ¼ 0

(84)

If the other definitions of buoyancy force are adopted, the local fluid acceleration force must not be included, because in both cases, being the effective fluid stress tensor zero, the buoyancy force vanishes. Note that this is not only true for solid suspensions but also for single bodies moving in pure fluids.

A Consideration on the Complementary Force As stated, all models need to agree on the value ascribed to the total fluid-particle interaction force, but they can use different repartitions for such force. A model may use the classical definition for the buoyancy force and include the local fluid acceleration force, while another may adopt one of the buoyancy force definitions given in Eqs. 77 and 78 without including the local fluid acceleration force. Both choices are acceptable, but the models will have to adopt different expressions for the complementary force to the total fluid-particle interaction force. Let us write:   nr fr ¼ ϕr ρe Det ue  g þ nr fr⋆ ;

nr fr ¼ ϕr @x Se þ nr fr•

(85)

We wonder how the complementary forces nr fr⋆ and nr fr• are related. To answer this question, we consider the following relation, obtained by combining the equations above:   nr fr• ¼ nr fr⋆ þ ϕr ρe Det ue  ρe g  @x Se

(86)

Then, using the dynamical equation for the fluid phase reported in Table 1, with a few mathematical passages not reported for briefness, one can prove that:

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29

ν X   ϕr ρe Det ue  ρe g  @x Se ¼ ðϕr =eÞ ns fs•

(87)

s¼1

We now use this relation in Eq. 86 and then sum both sides of the resulting equation over the phase index r. Doing so gives: ν  X  nr fr⋆  ð1=eÞnr fr• ¼ 0

(88)

r¼1

This is the condition that needs to be satisfied to render the two models consistent. This condition can be met by imposing the following restriction:   nr fr• ¼ e nr fr⋆

(89)

If we denote as τ e the deviatoric part of the effective stress tensor of the fluid, then, using the equation above, we immediately obtain:   nr fr∘ ¼ e nr fr⋆ þ ϕr @x τ e

(90)

where nr fr∘ represents the complementary force to the total fluid-particle interaction force that arises when Eq. 78 is employed for defining the buoyancy force. The main constituents of the complementary forces defined above are the drag force, the lift force, and the virtual mass force. We will now discuss how these forces are expressed constitutively.

Drag Force By definition, the drag force is parallel to the fluid-particle slip velocity (a vector that fulfills the principle of material frame-indifference); hence, it is: nr f D , r  β r ð ue  v r Þ

(91)

where βr denotes the drag coefficient for the rth particle phase. Finding a closure for the drag force amounts to finding a constitutive expression for βr. We now report some of these expressions, written in a way that is consistent with the classical definition of buoyancy force. Ergun and Orning (1949) developed an empirical correlation for assessing the unrecoverable pressure drop through packed beds. Extending its range of validity to homogeneous fluidized suspensions, one obtains the following constitutive equation: βr ¼ 150

μ e ϕr ð 1  e Þ ðesr Þ

2

þ 1:75

ρe ϕr j ue  v r j esr

(92)

Gidaspow (1994) recommends using this closure for values of the void fraction up to 0.80, even if the Ergun equation was developed (and has been extensively verified)

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L. Mazzei

for fixed beds in which the void fraction is small, with values close to 0.40. For void fraction values larger than 0.80, Gidaspow (1994) recommends using the expression of Wen and Yu (1966), which is one of the most popular correlations for the calculation of the drag coefficient in dense fluidized suspensions: 3 ρ ϕ jue  vr j 2:70 βr ¼ CD ðRer Þ e r e sr 4

(93)

where: ρ ejue  vr jsr Rer  e ; CD ðRer Þ ¼ μe



  ð24=Rer Þ 1 þ 0:15Re0:687 r 0:44

for Rer < 1000 for Rer  1000 (94)

The expression above has been proposed by Schiller and Naumann (1935). In Eq. 93, as we see, the exponent in the voidage function eα is constant and equal to 2.70. Di Felice (1994) suggested that the exponent α should be a function of the particle Reynolds number; the expression that he proposed is: h i αðRer Þ ¼ 2:70  0:65exp ð1=2Þð1:50  log10 Rer Þ2

(95)

The value of the exponent reduces to that employed in the expression of Wen and Yu (1966) for very small and very large values of the Reynolds number. In the intermediate region, nevertheless, the deviation from 2.70 is significant, the exponent reaching a minimum value of 2.05 when Rer  32. The closures reported above are extensively used; nevertheless, they are not consistent with the empirical equation developed by Richardson and Zaki (1954) to describe the expansion of homogeneous fluidized beds of non-cohesive particles. Since this equation is very accurate, the inconsistency is a shortcoming of the drag force closures. To overcome this limitation, Mazzei and Lettieri (2007) derived an expression that is consistent with the Richardson and Zaki correlation over the entire range of fluid dynamic regimes and for any value of the suspension void fraction. It has the following formulation: 9 8h i2 > > = < 0:63 þ 4:80ðRer =eγ Þ1=2 2ð1γ Þ αðe, Rer Þ ¼ ð1=lneÞln e   2 > > ; : 0:63 þ 4:80Re1=2 r

(96)

where: γ ðe, Rer Þ ¼

4:80 þ 2:40  0:175ðRer =eγ Þ3=4 1 þ 0:175ðRer =eγ Þ3=4

(97)

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31

To calculate γ one needs to solve a nonlinear equation. Since γ has a very narrow range of variation, finding the solution requires few iterations. For a detailed discussion on how this closure was derived and on how it compares with the other expressions reported above, we refer to Mazzei and Lettieri (2007).

Virtual Mass and Lift Forces If a body immersed in a fluid accelerates, some of the surrounding medium must also accelerate; this results in a force, named virtual mass force, equal to:   nr fV , r  ϕr ρe CV ðϕr Þ Det ue  Drt vr

(98)

where Det ðÞ and Drt ðÞ are the material derivatives associated with the fluid and rth solid phase, respectively. The virtual mass coefficient, denoted as CV(ϕr), depends on the particle shape and on the volume fraction of the solid phase considered. For very dilute mixtures of spherical particles, CV(ϕr) is taken to be 1/2, since this is the calculated value for a single sphere in an infinite fluid (Maxey and Riley 1983). The same result was found by Zhang and Prosperetti (1994) for an inviscid fluid and low particle concentration. For larger values of the solid volume fraction, the coefficient is expected to increase. Using lattice-Boltzmann simulations (but for bubbly suspensions), Sankaranarayanan et al. (2002) showed that CV(ϕr) is nearly linear; at moderate values of the solid volume fraction, Zuber (1964) suggested that: CV ðϕr Þ ¼ ð1 þ 3ϕr Þ=2

(99)

The virtual mass force is important when the density of the fluid is higher than that of the disperse phase; so, in fluidized beds, especially when the fluidization medium is a gas, this force usually plays a secondary role. In bubble columns, conversely, it strongly affects the dynamics of the system. If an object moves in a fluid which is in shearing flow, it experiences a force transverse to the direction of relative motion. This lift force is equal to: nr fL, r  ϕr ρe CL ðϕr Þð@x ue Þ ðue  vr Þ

(100)

The lift coefficient, denoted as CL(ϕr), depends on the particle shape and on the volume fraction of the solid phase considered. For very dilute mixtures of spherical particles, CL(ϕr) is also taken to be 1/2. One reason for this is that Eqs. 98 and 100 are not frame independent when taken separately, but their sum satisfies the principle of material objectivity if the coefficients of the two forces are equal. Hence, to satisfy this principle, one should set CL(ϕr) = CV(ϕr). In fluid-solid systems, the lift force is often (slightly) more important than the virtual mass force, but both forces are outweighed by the drag force. For more details about these forces, we refer to Marchisio and Fox (2013) and the references therein provided.

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Other Forces Other contributions to the fluid-particle force could be considered. A comprehensive overview can be found in Drew and Passman (1998). Here we cite only the Faxen force and a history-dependent term analogous to the Basset force for the motion of isolated particles (Basset 1888). For the latter, we can reasonably believe that, for fluidized suspensions, the averaging of history-dependent forces should result in a vanishing contribution, since averaging would most probably erase any historical effects of the motion of the particles on the fluid in their immediate neighborhood. We thus expect this force to be negligible.

Particle-Particle Interaction Force In fluidized mixtures of several monodisperse particle classes, each class exchanges linear momentum with all the others; this momentum transfer arises from particle collisions and results into a particle-particle drag force. Soo (1967) was among the first to quantify it, deriving a theoretical expression for the force acting on a single particle of species r in a cloud of particles of species s. Nakamura and Capes (1976) and Arastoopour et al. (1982) made similar efforts. Many authors have since then put forward other correlations, most of them being variations of earlier works. The force is expressed as the product of a drag coefficient by the velocity of slip between the particle classes: nr frs  ζ rs ðvs  vr Þ

(101)

where nrfrs is the force exerted by phase s on phase r per unit volume of suspension (see Table 1) and ζ rs is the particle-particle drag coefficient for the two-particle classes involved. The closure problem reduces to finding a constitutive expression for ζ rs. Gidaspow et al. (1985) advanced the relation: "

# ϕr ϕs ρr ρs ðsr þ ss Þ2 ζ rs ¼ Crs ð1 þ ers Þ j vs  vr j ρr s3r þ ρs s3s

(102)

where ρr, ρs, sr, and ss are the densities and diameters of the particles of classes r and s, respectively, ers is their coefficient of restitution, and Crs is given by: 3Φ1=3 þ ðϕr þ ϕs Þ1=3 i Crs  h rs 1=3 4 Φ1=3 rs  ðϕr þ ϕs Þ where:

(103)

Recent Advances in Modeling Gas-Particle Flows

Φrs  ð1  srs Þ½Φr þ ð1  Φr ÞΦs ð1  Xrs Þ þ Φr for Xrs 

33

Φr Φr þ ð1  Φr ÞΦs

Φrs  ½ðΦr  Φs Þ þ ð1  srs Þð1  Φr ÞΦs ½Φr þ ð1  Φr ÞΦs 

Xrs þ Φs otherwise Φr (104)

In the relations above, Φr and Φs are the particle volume fractions at maximum packing for phases r and s, respectively; moreover, it is: ϕr Xrs  ; ϕ r þ ϕs

1=2 1=2 ss sr srs  if sr  ss and srs  otherwise sr ss

(105)

Another popular closure is that of Syamlal (1987), which reads: " #  3 π ϕr ϕs ρr ρs grs ðsr þ ss Þ2 ζ rs ¼ ð1 þ ers Þ 1 þ Frs j vs  vr j 4 4 ρr s3r þ ρs s3s

(106)

in which Frs denotes a coefficient of friction for phases r and s while grs the radial distribution function of Lebowitz (1964), given by Eq. 68. Gera et al. (2004) suggested that the equations above should include an additional term that is necessary to prevent the particle phases from segregating when they are fully packed. Without it, Eqs. 102 and 106 permit packed particles of different size to segregate, a phenomenon which is not observed experimentally. To prevent this, they recommended adding to the coefficient ζ rs the term Ψp⋆, where p⋆ is given by Eq. 74 and Ψ is a coefficient that must be adjusted to match the actual segregation rate of the powder considered. The value that Gera et al. (2004) used was 0.30, but they stressed that this is not of general validity. Ψp⋆ is added so that when the powder approaches maximum packing, the particle-particle drag increases sufficiently to make the solid phases r and s move together as if they were one phase, thereby hindering segregation. This additional term is included only for ϕ > ϕf.

Population Balance Modeling In the previous sections, we have presented the Eulerian equations of motion for dense fluidized suspensions constituted of ν-particle classes, the rth class being characterized by a density ρr and a diameter sr. A serious limitation of this modeling approach is that changes in particle size are not permitted: particles can segregate or mix, and so the particle size distribution (PSD) in every real-space point can change in time, but the size of the particles for each class is fixed. In general, nevertheless, particles can grow, shrink, aggregate, and break, and new particles, of vanishing small size, may nucleate; these size changes reflect the physical and chemical processes taking place in the system and strongly affect the evolution of the PSD.

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Predicting this evolution is essential for a realistic description of the system behavior. We now introduce a modeling approach, referred to as population balance modeling, which has this capability. Population balance modeling is statistical in nature; it can be regarded as a generalization of the statistical modeling approach presented in section “Statistical Averaging.” There, the state of a particle was identified by two coordinates: position in real space x and velocity v; this number can be increased, if additional properties are required to fully characterize the particle state. Here we will add only the particle size s. The complete description (in a statistical sense) of the system is given by the master joint PDF, which we introduced in section “Statistical Averaging”; now, this function depends on 7ν internal coordinates plus the time coordinate (the phase space of the entire particulate system has 7ν dimensions). To calculate many of the macroscopic properties of practical interest, however, the one-particle marginal PDF, or equivalently the number density function, suffices; this is because many of the microscopic functions of interest take the following form: bð r Þ ¼

ν X s¼1

b1 ðxs , vs , ss Þ ¼

ν X

b1 ð r s Þ

(107)

s¼1

where b1 is an arbitrary function of the phase-space state of one particle, and where r and rs are the position points of the entire particulate system and of particle s in their phase spaces, respectively. For a function of this kind, one can prove that: ð ð ð hbis ðtÞ ¼

Ωx Ωv Ωs

b1 ðx1 , v1 , s1 Þf 1 ðx1 , v1 , s1 , tÞ dx1 dv1 ds1

(108)

where Ωs is the range of variation of s, while f1 is the NDF. By definition, f1(x, v, s, t) dxdvds represents the expected number of particles located at time t in the volume dx around the point x with velocity in the range dv around the velocity v with size in the range ds around the size s. f1(x, v, s, t)dxdvds, therefore, is an observable representing the mean particle number density in the seven-dimensional phase space made up by the union of the real space Ωx, velocity space Ωv, and size space Ωs. Knowing the NDF is equivalent to knowing the particle size and velocity distributions in any real-space point at any time. In Eq. 108, 〈b〉s is only a function of time, since b does not depend on real-space, velocity-space, and size-space coordinates. So, knowing the NDF permits calculating observables associated with microscopic functions of the class defined by Eq. 107. To calculate observables of this kind, one has to know how the NDF evolves in the phase space of one particle. An evolution equation for it is hence necessary. This is called (generalized) population balance equation (PBE). One may derive this equation rigorously starting from the microscopic description of the particulate system, given by the transport equation of the master joint PDF, which is a generalization of the Liouville equation (Marchisio and Fox 2013). Here we follow a less rigorous, and therefore easier, method that

Recent Advances in Modeling Gas-Particle Flows

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regards the population balance equation as a simple continuity statement written in terms of the number density function in the phase space of one particle (this space in our case has seven dimensions: three in real space, three in velocity space, and one in size space). This is the most popular derivation method used in the literature on polydisperse fluid-particle systems. Consider an arbitrary, fixed control volume Λr  Λx[Λv[Λs in the phase space of a single particle. The number of particles that accumulate in it per unit time is: ð ACC ¼ @t

ð

Λr

f 1 dx ¼

Λr

@t f 1 dr

(109)

where, to simplify the notation, we have denoted as r the position point of the particle in its phase space. The operations of time differentiation and space integration can be interchanged insofar as the control volume is not time dependent. The net number of particles entering Λr per unit time is: ð IN  OUT ¼ 

@Λr

ð _ r dσ r ¼  f 1 rn

Λr

_ @r f 1 rdr

(110)

where r_ and @ r are the particle velocity and the nabla operator in phase space, respectively, and where nr is the unit vector normal to the hypersurface bounding Λr directed outward. To turn the surface integral into a volume integral, we have used the Gauss theorem. The difference between the two terms above has to balance the net number of particles generated per unit time within Λr. Particle generation is caused by collisions, breakage, aggregation, and similar instantaneous processes (no process, of course, is instantaneous; however, these processes have characteristic times that are so smaller than those characterizing the evolution of the NDF that we can regard them as instantaneous). For instance, if two particles located outside Λv collide, their velocities vary abruptly, and after the collision, one particle (or even both) might be located within Λv, having thus entered Λr without crossing its boundaries. If we denote as Gr the net number of particles generated per unit volume of phase space and unit time owing to instantaneous phenomena, it is: ð GEN ¼

Λr

Gr dr

(111)

If we equate the accumulation term to the sum of the convection and generation terms, we obtain, after a few minor rearrangements, the following integral equation: ð Λr

ð@t f 1 þ @r f 1 r_  Gr Þdr ¼ 0

(112)

Because the integration volume Λr is arbitrary, and the integrand is (assumed to be) continuous, we conclude that the integrand must vanish:

36

L. Mazzei

@t f 1 ¼ @r f 1 r_ þ Gr

(113)

This is the PBE. We find it convenient to rewrite it in terms of the velocities which the particles possess in the real, velocity, and size spaces. Letting @ v and @ s represent the nabla operators in the velocity and size spaces, respectively, and v_ and ṡ be the particle velocities in the velocity and size spaces, respectively, ( v_ represents the particle acceleration in real space and s_ the particle growth rate), we can write: _ þ Gr @t f 1 ¼ @x f 1 v  @v f 1 v_  @s ðf 1 sÞ

(114)

_ þ Gr @t f 1 ¼ v@x f 1  @v f 1 v_  @s ðf 1 sÞ

(115)

or equivalently:

The two expressions are equivalent insofar as the real-space particle velocity v is an independent coordinate and not a function of x; the same is not true for v_ and ṡ, which in general may depend on the coordinates v and s, respectively. For instance, since v_ is the real-space particle acceleration, and since this is equal to the total force per unit mass acting on the particle, if the latter depends on the particle velocity, also v_ will. This is surely the case in fluidized beds, where a component of the force is the drag. The PBE, as said, is the transport equation of the NDF. Solving it allows determining the NDF evolution. The equation, nevertheless, can be solved only if it is closed. Here by closed we mean that all the terms in the equation can be computed from knowledge of the number density function (of course, these functionals need to be known). This is not the case for the generation term, because, as we know from statistical mechanics, it involves correlations between two particles (involving therefore the two-particle marginal PDF). A closure, consequently, will have to be introduced to express Gr in terms of the NDF. This is a significant challenge, because these closures are in general complex to derive (Balescu 1975). Once this has been done, the PBE is closed, but its solution will be extremely difficult to obtain. This is because in general, the PBE results to be a nonlinear, integral, partial differential, functional equation in a seven-dimensional space. As a consequence, one does not usually attempt to solve it, using the equation to extract solely the information about the system behavior that is of interest in the application at hand. The topics of how to close the PBE and how to solve it are vast. We thus refer to the specialized literature (we strongly recommend Chapman and Cowling 1970; Gidaspow 1994; Ramkrishna 2000 and Marchisio and Fox 2013). Here we just briefly mention a powerful solution method that allows tuning the PBE into a set of four-dimensional equations that can be solved with normal computational fluid dynamics numerical codes. This is called quadrature-based moment method (several variants exist, but all of them are based on the same idea, which we will now present). Often engineers are only interested in few integral properties of the NDF. Called moments, these may be important because they control the product quality or

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because they are simple to measure and monitor. The idea behind the method of moments is to derive transport equations for the moments of interest by integrating out the coordinates v and s from the PBE. For any given function φ(v, s), we can write: ð ð @t

Ωv Ωs

ð ð φðv, sÞf 1 dvds ¼ 

ð ð

Ωv Ωs

_ φðv, sÞ@r f 1 rdvds þ

Ωv Ω s

φðv, sÞGr dvds (116)

The integral on the left-hand side is the moment of f1 associated with the function φ(v, s) and depends only on x and t. As a consequence, the equation above, which governs the evolution in time and real space of the moment of the NDF associated with φ(v, s), can be solved by any CFD numerical code. We have therefore overcome the dimensionality issue. The problem with the equation above is that it is usually unclosed, since for any set of moments which the modeler wishes to track, obtained with a finite set of functions φ1, . . ., φn, the equations involve also moments external to the set. To overcome the closure problem, we can operate as follows. As mentioned, the moment method aims to solve the dimensionality issue by turning a problem involving one higher-dimensional differential equation into a problem involving a set of four-dimensional differential equations solvable by a CFD code. To capture all the information contained in the PBE, one would have to consider an infinite set of equations. But since we neither want nor can solve an infinite number of equations, the idea behind the method of moments is to satisfy only a finite number of them. This leaves the NDF largely undetermined, because only the infinite set would yield the correct NDF. This means that we can choose – to a certain extent – the NDF arbitrarily and then let the moment equations determine the details which we have left unspecified. Moment methods differ in the choice of the function φ and in the arbitrary input for the NDF. Their common feature is to choose the latter so that f1 is a given function of v and s containing 3α undetermined parameters (two scalars and one vector) depending on x and t. So, if we take 5α scalar moment transport equations, we obtain 5α differential equations for the unknown parameters. One hopes that, for α sufficiently large, the result is accurate enough and independent of the form chosen for the NDF. The quadrature methods of moments are examples of this approach; they overcome the closure problem by assuming that the NDF has the following functional expression: f 1 ðx, v, s, tÞ ¼

α X

nr ðx, tÞδ½v  vr ðx, tÞδ½s  sr ðx, tÞ

(117)

r¼1

This is a quadrature formula, in which α is the number of nodes, vr(x, t) and sr(x, t) are the rth quadrature nodes, and nr(x, t) is the rth quadrature weight. This formula represents the particle population by means of α solid phases, the rth having number density nr(x, t) and being made up of particles with velocity vr(x, t) and size sr(x, t).

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The difference between this representation and that used in section “Averaged Equations of Motion for Fluid-Particle Systems” is that here the size of each particle class is not fixed but evolves in time and space. Here the 3α parameters which one must obtain via the moment transport equations are nr(x, t), vr(x, t), and sr(x, t). For details about how this is done, we refer the reader to the literature previously cited.

Conclusions We presented three strategies for modeling fluidized beds: Eulerian-Lagrangian modeling, discrete particle modeling, and Eulerian-Eulerian modeling. Tracking the motion of each particle, the first two give a detailed description of the system dynamics; these methods, however, are too expensive computationally to be of any use for describing systems of industrial interest. We thus focused on EulerianEulerian modeling, describing the averaging techniques that turn granular systems into continuous media and deriving the volume-averaged equations of mass and linear momentum balance for fluidized suspensions made up of ν-particle classes. We then addressed the closure problem, describing the main constitutive equations used by modelers to express the fluid-particle and particle-particle interaction forces and the effective fluid dynamic stress. We concluded the chapter by introducing the population balance modeling, which permits describing systems in which the particles are continuously distributed over the size and in which the size is free to vary owing to continuous and discontinuous processes, such as chemical reaction, growth, aggregation, and breakage.

Appendix Fluid-Phase Volume Average of Point Variable Spatial Derivatives We intend to derive an expression for the fluid-phase volume average of point variable spatial derivatives; to this end, we start by considering the derivative:   @a eðx, tÞhζ ie ðx, tÞ

(118)

Now, using the definition of fluid-phase volume average given in Eq. 17 and the derivation chain rule, we write the quantity above as: @xa ¼

Ð

Ð

Ð

Ð

Λe ζ ðz,tÞψ ðjx  zjÞdz ¼ Λe ζ ðz,tÞ@xa ψ ðjx  zjÞdz ¼  Λe ζ ðz,tÞ@za ψ ðjx  zjÞdz

Λe ½@za ζ ðz,tÞψ ðjx  zjÞdz 

Ð

Λe @za ½ζ ðz,tÞψ ðjx  zjÞdz

For the first integral, we can write:

(119)

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ð Λe

½@za ζ ðz, tÞ ψ ðjx  zjÞdz ¼ eðx, tÞh@a ζ ie ðx, tÞ

(120)

For the second, the Gauss theorem allows writing: Ð

Λe @za ½ζ ðz, tÞψ ðjx

¼

Ð

 zjÞdz

@Λx ζ ðz, tÞna ðz, tÞψ ðjx

 zjÞdσ z 

ν Xð X r¼1 S r

@Λr

ζ ðz, tÞka ðz, tÞψ ðjx  zjÞdσ z (121)

where @Λx is the surface bounding the domain containing the mixture and na(x, t) is the ath component of the unit vector normal to @Λx pointing away from the mixture. If the shortest distance from the generic point x  @Λx is considerably larger than the weighting function radius, the first term of the right-hand side of the equation above is much smaller than the second. Neglecting it, we obtain Eq. 25.

Fluid-Phase Volume Average of Point Variable Time Derivatives Similarly, to derive an expression for the fluid-phase volume average of point variable time derivatives, we start by considering the derivative:   @t eðx, tÞhξie ðx, tÞ

(122)

Using the definition of fluid-phase volume average given in Eq. 17 and then applying the Leibnitz theorem allows writing this as: @t

Ð



Λe ζ ðz, tÞψ ðjx ν Xð X r¼1 S r

þ

Ð

@Λr

 zjÞdz ¼

Ð

Λe ½@t ζ ðz, tÞψ ðjx

 zjÞdz

ζ ðz, tÞuðz, tÞkðz, tÞψ ðjx  zjÞdσ z

@Λx ζ ðz, tÞuðz, tÞnðz, tÞψ ðjx

(123)

 zjÞdσ z

The integral on @Λx can be neglected for the same reasons given in Appendix “FluidPhase Volume Average of Point Variable Spatial Derivatives.” Now, using the definition of fluid-phase volume average given in Eq. 17, we have: ð Λe

½@t ζ ðz, tÞψ ðjx  zjÞdz ¼ eðx, tÞh@t ζ ie ðx, tÞ

(124)

Obtaining Eq. 26 is then immediate. Note that if Λx is time independent, u(z, t) = 0 on @Λx, and therefore the last integral on the right-hand size of Eq. 123 rigorously vanishes.

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Particle-Phase Volume Average of Point Variable Time Derivatives We intend to derive an expression for the particle-phase volume average of point variable time derivatives; in this case, we consider the derivative: h i @t nr ðx, tÞhζ irp ðx, tÞ

(125)

Now, employing the definition of particle-phase volume average given in Eq. 21, we can express the partial derivative above as: X

@t ½ζ r ðtÞψ ðjx  zr ðtÞjÞ ¼

Sr

X

 ζ_r ðtÞψ ðjx  zr ðtÞjÞ

Sr

þ

X

½ζ r ðtÞ@t ψ ðjx  zr ðtÞjÞ

(126)

Sr

From the definition of particle-phase volume average, it is: X 

r ζ_r ðtÞψ ðjx  zr ðtÞjÞ ¼ nr ðx, tÞ ζ_ p ðx, tÞ

(127)

Sr

Applying the derivation chain rule yields: X

X  ζ r ðtÞvr, a ðtÞ@xa ψ ðjx  zr ðtÞjÞ Sr Sr h i X  ζ r ðtÞvr, a ðtÞψ ðjx  zr ðtÞjÞ ¼ @xa nr ðx, tÞhζva irp ðx, tÞ ¼ @xa ½ζ r ðtÞ@t ψ ðjx  zr ðtÞjÞ ¼ 

(128)

Sr

having used again the partial derivatives commutative property and the definition of particle-phase average. Replacing these last two results in Eq. 126 yields Eq. 39.

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Numerical Modelling of Pulverised Coal Combustion Zhao F. Tian, Peter J. Witt, Mark P. Schwarz, and William Yang

Abstract

Many thermal power generation plants rely on combustion of pulverised coal carried out in large furnaces. Design and improvement of these furnaces can be effectively assisted by using numerical modelling with Computational Fluid Dynamics (CFD) techniques to develop a detailed picture of the conditions within the furnace, and the effect of operating conditions, coal type, and furnace design on those conditions. The equations governing CFD models of pulverised coal combustion are described, with a focus on sub-models needed for devolatilisation, combustion and heat transfer. The use of the models is discussed with reference to examples of CFD modelling of brown coal fired furnaces in the Latrobe Valley in Australia and black coal fired furnaces described in the literature. Extensions to the CFD models that are required to tackle specific industrial and environmental issues are also described. These issues include control of NOx and SOx emissions and the effect of slagging and fouling on furnace and boiler operation. Keywords

CFD • Coal combustion • Tangentially fired • Drying model • Devolatilisation model • Char Combustion model • NOx • Soot model • Turbulence model

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Phase Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Phase Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 4

Z.F. Tian (*) School of Mechanical Engineering, The University of Adelaide, Adelaide, SA, Australia e-mail: [email protected] P.J. Witt • M.P. Schwarz • W. Yang CSIRO, Mineral Resources, Clayton, VIC, Australia # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_9-1

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Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Phase Combustion: Gas Reaction Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models for Particle Phase Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eulerian-Eulerian Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eulerian-Lagrangian Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emissions Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 10 15 15 15 24 26 31 33

Introduction Coal-fired electricity generation is still dominant in the power industries of many countries, despite the rapid increase of renewable electricity generation in recent years. In 2010, coal-fired electricity generation accounted for 40% of the electricity generation worldwide (EIA 2013). In Australia, coal including black coal and brown coal generated about 64% of electricity in 2012–2013 (BREE 2014). It is particularly noteworthy that in the state of Victoria, brown coal from the Latrobe Valley region produces over 85% of the state’s electricity supply (Allardice 2000). Pulverised coal (PC) combustion is one of the major technologies for the conversion of chemical energy in coal into electricity. In the brown coal fired power plants in the Latrobe Valley, all existing boilers use PC combustion technologies. With advances in computing power and modelling techniques, computational fluid dynamics (CFD) has evolved into a feasible tool for scientists and engineers who can apply it to better understand PC combustion in furnaces (Tian et al. 2009) and therefore optimise the design and operation of PC boilers. Nevertheless, PC combustion in furnaces is one of the most difficult processes to model mathematically, since it generally involves the simultaneous coupled processes of threedimensional gas-particle fluid dynamics, turbulent mixing, heat transfer, and complex homogeneous and heterogeneous chemical reaction kinetics (Viskanta and Mengüç 1987). In modelling real industrial installations, additional complications arise from slagging and fouling of heat transfer surfaces, variability in feed characteristics and inevitable uncertainties in actual structural geometry due, for example, to occasional damage or maintenance issues. Figure 1 shows the major physical and chemical processes that occur during the burning of pulverised coal particles in a tangentially-fired PC furnace. Fine coal particles, pulverised in mills, are blown into the furnace through a number of burners. Once the particles enter the furnace, they are heated by hot furnace gases and radiation from the flame, they start to dry when their temperature reaches about 100–110
C (Wu 2005). When the particles are heated further to a certain critical temperature (depending on the coal type and size), devolatilisation starts and volatiles are released from the particles. The products of devolatilisation include non-condensable volatiles (light gases), condensable volatiles (tars) and remaining solid particles that normally comprise char and mineral matter. The volatiles react with oxygen from the combustion air and other oxidants in the furnace. Finally char

Numerical Modelling of Pulverised Coal Combustion Fig. 1 A schematic drawing of PC coal fired furnace and some typical CFD sub-models required to model these processes. Submodels: (1) particle phase model, (2) evaporation/drying model, (3) devolatilisation model, (4) char combustion model, (5) turbulence model, (6) turbulence-reaction interaction model, (7) radiation model, (8) soot model, (9) NOx model, (10) SOx model, (11) slagging model

3 Ash, flue gas, and emissions (including NOx, SOx, etc. )

Convective parts (Superheaters, Reheaters, Economizer)

Furnace gas

PC particles (1,2,3,4)

Slag on furnace wall(11)

Turbulent flame zone (5-10)

Raw Coal

Burners

Mill

Combustion air

particles react with gases in the furnace, leaving mineral matter and probably a small fraction of unburnt char in the solid particles. These particles (ash) and the furnace gas flow through convective heat transfer sections such as superheaters, reheaters and economisers, exchanging heat with the working fluid (water/steam) in the convective devices. Typical exhaust gases comprise CO2, N2, H2O, O2, and small amounts of NOx, SOx, CO and particulate matters (PM). After leaving the convective passes exhaust gases will normally go through various air pollution control equipment before being discharged through the stack. In the CFD approach, mathematical descriptions of each of these processes in the furnaces are called “sub-models” because they can be developed and updated in the same way that modules in circuit boards can be replaced (Niksa 1996). As shown in Fig. 1, a CFD code for modelling coal combustion probably needs the following sub-models: (1) model for particle phase motion, (2) evaporation/drying model, (3) devolatilisation model, (4) char combustion model, (5) turbulence model, (6) turbulence-reaction interaction model (gas phase reaction models), (7) radiation model, (8) soot model, (9) NOx model, (10) SOx model, (11) slagging model. Figure 2, adapted from Tian et al. (2010a) shows some sub-models available in the commercial CFD code ANSYS/CFX 14. This book chapter briefly reviews and describes the mathematical equations of some of these sub-models. The authors have implemented some of these sub-models into a CFD model of a tangentially-fired PC furnace at the TRUenergy Yallourn power plant, Latrobe Valley, Australia (Tian et al. 2010a). This CFD model was developed based on the commercial CFD code, ANSYS/CFX. The model has been

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CFX Coal combustion solver

Radiation models

Emissions models

Turbulence models

Particle phase models

Gas combustion models

Standard k-ε model

Lagrangian model

EDM model

P-1 model

Soot

FRC

DT model

NOx

RSM model Drying model k-ω model

SST model

Devolatilisatio n model

Combined model

Monte Carlo model

Char oxidation

model

Fig. 2 The structure of CFD coal combustion solver (Adapted from Tian et al. 2010a). RSM: Reynolds Stress model; SST: Shear stress transport model; EDM: Eddy dissipation model; FRC: Finite rate chemistry model; Combined model: Combined EDM/FRC model; DT: Discrete transfer model

validated against plant measurements and applied to investigate the effects of several operating conditions at full load, such as different out-of-service firing groups and different combustion air distributions on the coal flames (Tian et al. 2010b). The CFD furnace model was then used to assess the combustion of pre-dried brown coal in the furnace that was designed for raw or non-pre-dried brown coal (Tian et al. 2012). In this book chapter, additional results of the CFD furnace model are reported as examples of the sub-model applications.

Gas Phase Model Gas Phase Governing Equations In CFD models of pulverized coal combustion, gases in the furnace are normally considered to be a mixture consisting of the gaseous components, O2, H2O, CO2, CO, N2, NO, and volatiles (Tian et al. 2010a). Volatiles can be taken as one single gas component, a mixture of light gas and tar, or a mixture of individual gases such as CH4, C2H2, etc. To reduce the computing time, these components are normally assumed to be mixed at the molecular level, hence having the same mean velocity, pressure, temperature and turbulence fields (Tian et al. 2010a). The Navier-Stokes equations are used to solve the continuity and momentum equations of the gas mixture. The gas phase equations solved in ANSYS/CFX are given below as an example of the steady state governing equations for CFD models. Equations 1 and 2

Numerical Modelling of Pulverised Coal Combustion

5

are the continuity equation and momentum equation of the mean steady state after Reynolds averaging:
 ∇  ρg U g ¼ 0

(1)


 n h o  T i  ρg U g U g þ SM , ∇  ρg U g U g ¼ ∇pg þ ∇  μg ∇U g þ ∇Ug

(2)

here Ug is the gas phase mean velocity vector; Pg is the gas phase mean pressure; SM is the external momentum source such as gravity and forces from the coal particle phase; and ρg is the gas mixture density defined as: ρg ¼

Nc X

Y I ρI ,

(3)

I¼1

where ρI is the mass density of the component I. Nc is the number of modelled species in the gas mixture, and YI is the mass fraction of the species I, solved by the following equation:
 ∇  ρg U g Y I ¼ ∇  ðΓI:eff ∇Y I Þ þ SI

(4)

In this equation, SI is the source term of the species related to generation or destruction of the species by reaction. Other properties of the gas mixture, such as the gas mixture molecular viscosity μg and the gas mixture specific heat capacity at constant pressure Cp,g, are calculated in the same manner as Eq. (3): αg ¼

Nc X

Y I αI ,

(5)

I¼1

where αg in the gas mixture property being considered. The effective diffusion coefficient of species I, ΓI.eff, in Eq. (4) is defined as: ΓI:eff ¼ ΓI þ

μt , Sct

(6)

Where ΓI is the molecular diffusion coefficient of species, ΓI = ρIDI, here DI is the kinematic diffusivity of the species I. Sct is the turbulent Schmidt number and μt is turbulent viscosity. The source term SI in Eq. (4) is due to chemical reaction involving the species. There is one transport equation of each gas component except the constraint gas N2. The mass fraction of N2 is calculated by using the following equation: Nc X I¼1

YI ¼ 1

(7)

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The Reynolds averaged energy equation for the gas mixture can be:


∇  ρg U g hg



  μt ∇hg þ SE , ¼ ∇  λg ∇T g þ Prt

(8)

where hg is the gas mixture enthalpy. Prt is the turbulent Prandtl number. The energy source term SE includes thermal energy from chemical reactions and thermal radiative heat transfer. In most CFD coal combustion models, the Reynolds averaged Navier Stokes (RANS) modelling approach is used to handle turbulence. In RANS models, the Reynolds stresses terms ρg Ug Ug in the momentum Eq. 2 are modeled based on the Boussinesq hypothesis: h   T i 2
ρg U g Ug ¼ μt ∇U g þ ∇U g  δij ρg kg þ μt ∇  Ug , 3

(9)

where μt is the turbulence viscosity that can be calculated by applying turbulence models that will be discussed later. Δij is the Kronecker delta that is 1 when i = j and 0 when i 6¼ j.

Turbulence Models Turbulent mixing is one of the major factors controlling the local proportions of fuel and oxygen throughout the primary flame zones and thereby exerting a predominant effect on heat release rates, heat fluxes onto steam tubes, carbon burnout times and pollutant formation rates (Niksa 1996). In CFD techniques, turbulence models are used to close the equations for the fluctuating quantities and thereby include the effects of eddies on the time averaged flow. The Reynolds stresses in Eq. (9) can be also directly calculated from six transport equations and this is called the Reynolds Stress model (RSM). The RSM is a second order RANS model but has been used to predict coal combustion by only a few researchers such as Weber et al. (1995), Zhang and Nieh (1997). The RSM has been shown to perform better than k-e models in predicting of isothermal swirl jets (German and Mahmud 2005; Weber et al. 1990) due to two equation models not being able to reliably resolve flows with strong streamline curvatures. This limitation of two equations models can be partly overcome by a curvature correction term to the turbulence production term. However work at International Flame Research Foundation (IFRF) (Weber et al. 1995) demonstrated that this advantage of the RSM over k-e models was not found in the burning jet applications (Niksa 1996). Backreedy et al. (2006) found that the performance of RSM in modelling a swirl coal flame in a pilot-scale furnace is not better than that of k-e models. This is confirmed by a recent modelling project that compared the performance of several RANS models in modelling swirling flow in a vortex flow reactor (Tian et al. 2015). The BSL RSM can predict the anisotropic Reynolds stresses that the SST model and the

Numerical Modelling of Pulverised Coal Combustion

7

standard k-e model cannot predict, but this does not make it more accurate in predicting the mean flow-field than the other two models. Application of the RSM requires more computational resources than is required by the standard k-e and similar first order models, partially due to the need to solve additional transport equations of the Reynolds Stresses and probably due to the poor convergence characteristics and stability of RSM. The standard k-e model is commonly applied in studies of coal combustion in furnaces. For the standard k-e model the turbulent viscosity, μt, in Eq. (8) is computed from: μt ¼ Cμ ρg k2g =eg

(10)

where kg is the turbulence kinetic energy of the gas mixture and eg is the kinetic energy dissipation rate of the gas mixture. In the standard k-e model, the turbulence kinetic energy and the kinetic energy dissipation rate of the gas mixture are calculated by solving two transport equations. The turbulence kinetic energy equation for the standard k-e model is:  
 μ ∇  ρg U g kg ¼ ∇  μg þ t ∇kg þ Pk  ρg eg σk

(11)

Here, the rate of production of turbulence kinetic energy Pk is modeled by:

  T  2 Pk ¼ μt ∇U g  ∇Ug þ ∇U g  ∇  U g μt ∇  U g þ ρg kg 3

(12)

The kinetic energy dissipation rate equation is:  
  eg
μ Ce1 Pk  Ce2 ρg eg ∇  ρg U g eg ¼ ∇  μg þ t ∇eg þ σe kg

(13)

The values of the constants are Cμ = 0.09, σ k = 1.0, σ e = 1.3, C1e = 1.44, C2e = 1.92 (Launder and Spalding 1974). Compared to the RSM, the standard k-e model is less computationally intensive while providing a similar level of predictive accuracy to the RSM for most coal combustion applications not involving strong swirling flow fields. The standard k-e model has been used to simulate coal flames in pilot-scale furnaces (Tian et al. 2010a), e.g., Lockwood and Salooja (1983), Truelove and Holcombe (1991), Zhou et al. (2002) and many others, and full scale furnaces by workers such as Belosevic et al. (2006), Xu et al. (2001). One of the major shortcomings of the standard k-e model is that it cannot predict the adverse pressure gradient properly; the standard k-e model significantly overpredicts shear stress levels and thereby delays separation (Menter 1992). Another shortcoming relates to the numerical stiffness of the equations when integrated through the viscous sublayer (Menter 1992). Many modified k-e models have been

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derived from the standard k-e model in order to overcome these shortcomings, one of which is the Re-Normalization Group (RNG) k-e model that has been used in coal combustion modelling. The transport equations for gas phase kg and eg in the RNG k-e model are given as follows: 
 ∇  ρg Ug kg ¼ ∇  μg þ

μt



∇kg þ Pk  ρg eg

(14) σ k, RNG  
  eg
μt ∇  ρg U g e g ¼ ∇  μ g þ Ce1, RNG Pk  Ce2, RNG ρg eg (15) ∇eg þ σ e, RNG kg Ce1,RNG is calculated as: Ce1, RNG ¼ 1:42  f η ,

(16)

where
η  η 1 4:38 fη ¼ ð1 þ βRNG η3 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pk η¼ ρg Cμ, RNG eg

(17)

(18)

and where βRNG is 0.012. The values of the other constants are Cμ,RNG = 0.0845, σ k,RNG = 0.7179, σ e,RNG = 0.7179, Ce2,RNG = 1.68 (Versteeg and Malalasekera 2007). The Re-Normalisation Group (RNG) k-e model has been used in some coal combustion modelling projects. Fan et al. (2001) compared modeling results from the RNG k-e model and the standard k-e model in a tangentially fired furnace against experimental data, and found the RNG k-e model gave better results for swirling flow and sharper flow gradients within calculated regions than the standard k-e model. Backreedy et al. (2006) used the RNG k-e model in a coal test furnace model, as the RNG k-e model is believed to have advantages over the standard k-e model in swirling flows. Nevertheless, these advantages of the RNG k-e model over the standard k-e model in swirling flows have been a matter of some controversy (Saqr 2011). The Wilcox k-ω model (Wilcox 1988), Menter k-ω model that also called Baseline (BSL) model, and Shear-stress transport (SST) model (Menter 1994) are another class of two-equation RANS models. The SST model is a hybrid approach between the standard k-e model and the k-ω model. In the SST model, in the region near walls, the k-ω model is used as it performs well for near wall flows and can avoid the use of wall functions also it allows the accurate specification of ω values on the wall surface hence avoiding issues of defining e near wall for fine grids. In the region far away from wall, the standard k-e model is used as it is robust in the free

Numerical Modelling of Pulverised Coal Combustion

9

stream while k-ω model is sensitive to the free stream value of ω (Versteeg and Malalasekera 2007). The transport equations of kg and ωg in the SST model are shown below, with ωg = eg/kg. The transport equations of kg and ωg of the SST model are:  
 μ ∇  ρg U g kg ¼ ∇  μg þ t ∇kg þ Pk  β0 ρg kg ωg σ k3   
 ωg μt ∇  ρg Ug ωg ¼ ∇  μg þ ∇ωg þ α3 Pk σ ω3 kg þ ð1  F1 Þ2ρg

1 ∇ωg ∇kg  β3 ρg ω2g σ ω , 2 ωg

(19)

(20)

The turbulent viscosity, μt, is calculated as: μt ¼

ρ kg α1 g , max α1 ωg , SF2

pffiffiffiffiffiffiffiffiffiffiffiffi where S ¼ 2Sij Sij . The blending function F1 in Eq. calculated as:  20is p  ffiffiffiffi  4 kg 500μg F1 ¼ tanh arg1 , arg1 ¼ min max 0:09ωg y , ρ y2 ωg , g

(21)

4ρg kg 2 σ ω, 2 Dþ ωy



 1 1 10 Dþ ¼ max 2ρ ∇k ∇ω , 10 g g g ω σ ω, 2 ωg

(22)

where y is the distane to the nearest wall. The blending function F2 in Eq. 21 is given as: 

 F2 ¼ tanh arg22 , arg2 ¼ max

! pffiffiffiffiffi 2 kg 500μg , : 0:09ωg y ρg y2 ωg

(23)

The values of the constants employed in the SST model are β0=0.09, α1 = 5/9, α3 = 0.44, β3 = 0.0828, σ ω,2 = 1/0.856 (Versteeg and Malalasekera 2007). Only a few studies of coal combustion using the k-ω or the SST model can be found in the literature. In a CFD modelling study (Tian et al. 2009), six first order RANS models , namely, the standard k-e model, a modified k-e model, RNG k-e model, Wilcox k-ω model, BSL k-ω model and SST models were used to simulate a non-swirling coal flame in a pilot-scale furnace of IFRF. The standard k-e model, RNG k-e model, BSL and SST models were found to be generally in good agreement with the experimental data. Predictions using the SST model and BSL k-ω model were almost identical, and results of the standard k-e model and the RNG k-e model were similar (Tian et al. 2010b). The SST model and the standard k-e model were further tested in modelling an isothermal gas-particle flow in three inclined

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rectangular jets in crossflow (Tian et al. 2011). The flow configuration and flow conditions were scaled based on typical flow conditions experienced in the Victorian brown coal furnace burners (Tian et al. 2010a). Gas and particle flows predicted by both models were found to be in reasonable agreement with the detailed experimental data, although the SST model showed a slightly better agreement with the measurements than the standard k-e model (Tian et al. 2010a). The SST and the standard k-e model were then employed in a CFD model of a 375 MW tangentially fired furnace (Tian et al. 2010a) burning high-moisture brown coal. Both turbulence models provide similar predictions that were in good agreement with the plant data (Tian et al. 2010a). The standard k-e model has been found to perform particularly well in confined flows where Reynolds shear stresses are most important (Versteeg and Malalasekera 2007). In tangentially fired furnaces, the strong vortex at the center formed by the impinging jets from corners or walls greatly increases the turbulence transport in the furnaces (Basu et al. 1999), therefore the flows in the furnaces can be taken as turbulence transport dominated flows. The major advantage of the SST model over the standard k-e model is the inclusion of the shear rate magnitude, S, in Eq. 21, which ensures the ratio of turbulence production to turbulence dissipation larger than one in adverse pressure gradient flows (Menter 1992). However, this supposed advantage does not appear to result in a clearly better prediction forthe tangentially fired flames, though the separation of flows are indeed found in the furnace as shown in Fig. 3. Figure 3a-c show the predicted flow vectors on a horizontal plane at the exit of the upper main burner when firing units 3&6, 5&6 and 2&6 are out of service, respectively. Flow separations can be found in Fig. 3 between jets. For example, as shown in Fig. 3a, primary gas flows in the furnace as jets and these high speed jets entrain furnace gas and secondary air. This entrainment helps to form recirculations and flow separations. The details of the CFD model and boundary conditions for the cases shown in Fig. 3 can be found in previous papers (Tian et al. 2010a; Tian et al. 2010b).

Gas Phase Combustion: Gas Reaction Kinetics The gas phase reactions in coal fired furnaces are very complex. Not all the species and reaction chemistry can be included in the CFD models, partially due to the large computing time required to transport all the species and the stiffness of the large number of intermediate reactions. This problem is further complicated by the heterogeneous nature of coal and the devolatilisation process making knowing the detailed chemical composition of volatiles and subsequent reactions extraordinarily difficult. As discussed in Yeoh and Yuen (2009), CFD techniques for partial differential equations require computing time roughly proportional to Ns2 (Ns is the number of species). If all the reaction species found in the PC furnace are included in the CFD model, the computing time will be excessive. Furthermore, except for some simpler alkane hydrocarbon fuels such as CH4 and C2H2, comprehensive reaction chemistry for complex fuel is still not well determined (Yeoh and Yuen

Numerical Modelling of Pulverised Coal Combustion

11

Fig. 3 Predicted flow patterns for cases (a) Firing units (FU) 3&6 out-of-service (b) Firing units (FU) 5&6 out-of-service, (c) Firing units (FU) 6&7 out-of-service

2009). Therefore global reaction schemes are normally used in CFD modelling of the gas phase volatile combustion of coal. As noted earlier and described in detail below coal combustion consists of a number of stages with a critical stage being the devolatilisation of the solid particle to produce gas phase volatiles: Coal ! Volatiles þ CðcharÞ

(24)

Volatile combustion can be modeled by the global single step reaction: VolatilesðHCÞ þ O2 ! CO2 þ H2 O

(25)

Tian et al. (2010b) notes that the concentration of CO cannot be calculated by the above single step reaction. If understanding the CO concentration is important for

12

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Z.F. Tian et al.

b

CO.Mass Fraction CO concentration

0.030 0.025 0.020 0.015 0.010 0.005 0.000

Fig. 4 Predicted CO concentration for cases (a) Firing units 2&6 out-of-service (b) Firing units 5&6 out-of-service. CFD model details and boundary conditions can be found in Tian et al. (2010a, b)

the coal combustion modeling work being undertaken, the following reaction scheme was proposed in Tian et al. (2010b): Coal ! Volatiles þ CðcharÞ

(26)

VolatilesðHCÞ þ O2 ! CO þ H2 O

(27)

1 CðcharÞ þ O2 ! CO 2

(28)

1 CO þ O2 ¼ CO2 2

(29)

Figure 4 shows the predicted CO concentration in the furnace of Tian et al. (2010b) based on the combustion scheme shown above. More species and reactions can be added in the CFD model, however, as mentioned above, the computing time will increase and details of the chemistry are required. When predition of NOx and/or SOx emissions is required, the NOx species and SOx species can be added to the gas mixture. NOx and SOx models are briefly reviewed later. Several approaches can be used to calculate behaviours of gas species specified by the chemistry. The most straight forward one is a species transport approach. In this approach, transport equations for each species (or each species except a

Numerical Modelling of Pulverised Coal Combustion

13

constraint species) are solved. To close the transport equations (such as Eq. (4)), the source term, SI, needs to be calculated. Usually the gas phase reactions in coal fired boilers can be taken as a fast reaction system in respect to their modelling. A fast reaction system means the chemical reaction rates are much faster than the mixing processes in the system, in other words, reaction rates in the system are controlled by the mixing process. Another characteristic of flames in coal-fired boilers is that they generally can be classified as non-premixed combustion. The source term SI can be computed as the sum of the reaction sources of reactions involving species I, SI ¼ W I

XNKI   v00kI  v0kI RkI , k¼1

(30)

where WI is the molar mass of species I. RkI is the reaction rate of species I in the reaction k, which can be calculated by using either a finite rate approach or the eddy dissipation model. v0kI is the stoichiometric coefficient of species I in the reaction k as a reactant and v00kI is the stoichiometric coefficient of species I in the reaction k as a product. NkI is the number of reactions that component I involves in. For finite rate chemistry model, the reaction rate of reaction k, Rk, is calculated as: 0

00

rkI r kI Nc Rk ¼ Fk ∏Nc I¼A, B, :::: ½I   Bk ∏I¼A, B, :::: ½I 

(31)

here [I] is the molar concentration of species I. Nc is the number of species in the reaction k. The forward rate constant Fk can be calculated by the Arrhenius rate:   Ek Fk ¼ Ak T βk exp  RT

(32)

where Ak is the pre-exponential factor; T is temperature; βk is the temperature exponent; Ek is the activation energy; and R is the universal gas constant, 8.314 J/ molK. If applicable, the backward rate constant Bk can be calculated as: Bk ¼ Ab T

βbk



Ebk exp  RT

 (33)

The finite rate chemistry model is applicable to laminar flames as the effects of turbulence on the reactions are not included. This approach can be used for combustion with relatively slow chemistry and small turbulent fluctuations (Yeoh and Yuen 2009). In coal-fired flames, the eddy dissipation model (Magnussen and Hjertager 1977) can be used to model the turbulence-chemistry interaction. In the eddy dissipation model for pre-mixed flames, Rk is directly related to the time required to mix reactants at the molecular level, i.e., a mixing time defined by the turbulent kinetic energy of gas mixture, kg, and dissipation rate, eg:

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Z.F. Tian et al.

  eg ½I  RkI ¼ A min 0 vkIR kg

(34)

where v0kIR is the stoichiometric coefficient for reactant I in reaction k; and A is a constant with a value of 4. The advantage of the eddy dissipation model is that it is simple and takes accounts effects of turbulence on chemistry. However, the eddy dissipation model, as shown in Eq. (34), does not account for the effects of temperature on reaction rates and it can only be used for one-step or two-step reactions without giving detailed chemistry effects. When more detailed reaction kinetics are required in the CFD model, the generalized eddy dissipation concept model can be used. In the generalised eddy dissipation concept model, the mean reaction rate of component I, RI, is assumed to occur in small turbulence structures over a mean residence time τ(Magnussen and Hjertager 1981). The fine turbulence structures in a computational cell are characterised a mean length fraction, ξ. The mean reaction rate of I, RI, is calculated as: RI ¼

 W ρðξ Þ2   h I i YY  YI 3  τ 1  ðξ Þ

(35)

Here Y Y is the species mass faction in the fine structures after reacting over the time scaleτ; Y Y can be determined through a laminar finite rate model (Yeoh and Yuen 2009). The mean residence time τis calculated as:  1=2 vg τ ¼ Cτ , eg 

(36)

here Cτ is a constant with a default value of 0.4082. The mean length fraction, ξ is calculated as: 

ξ ¼ Cξ

v g eg kg 2

!1=4 ,

(37)

here Cξ is a constant with a default value of 2.1377. More advanced turbulence-chemical interaction models such as the joint probability density function (PDF) and laminar flamelet models can be applied to coal fired combustion. Their advantages over the EDC model and eddy dissipation models have been found to be less pronounced in coal flames than the gas flames, partially due to the fact that gaseous phase reactions are just a part of the reaction sequence involved in coal combustion (Vascellari and Cau 2012): the heterogeneous reaction of char is also important in the coal flame.

Numerical Modelling of Pulverised Coal Combustion

15

Models for Particle Phase Motion In the most CFD studies of PC combustion in furnaces, two categories of approaches are typically used for prediction of the coal particle phase motion: the EulerianEulerian model or the Eulerian-Lagrangian model.

Eulerian-Eulerian Model The Eulerian-Eulerian model calculates the particle flow using Eulerian or fluid-like equations, e.g. modified Navier-Stokes Eqs. (1) and (2). These equations can be implemented efficiently in the existing solvers resulting in relatively less computational time being required to calculate mean parameters, such as velocity and volume fraction for the particle flow. Some simplified Eulerian-Eulerian models, which assumed a mechanical and thermal equilibrium between the two phases, have been developed and used to model coal combustion, e.g., Benim et al. (2005), Fiveland and Wessel (1988), Zhou et al. (2002) developed a two-fluid-trajectory model and simulated coal combustion in a tangentially fired boiler. This two-fluid-trajectory model uses Eulerian gas-phase equations, Eulerian particle-phase continuity and momentum equations, two-phase turbulence models, and Lagrangian ordinary differential equations of particle temperature and mass change to take into consideration of the history effects (Zhou et al. 2002). The Eulerian-Eulerian coal combustion model has been incorporated in the commercial CFD code PHEONICS. Nevertheless, there are some inherent problems in the use of Eulerian models for gas-particle flows as reviewed by Tian et al. (2005), propably making this approach less attractive in modeling PC combustion.

Eulerian-Lagrangian Model Most commercial and research CFD codes make use of the Eulerian-Lagrangian approach to model pulverised coal particle combustion, for examples, ANSYS/CFX and FLUENT. The Lagrangian model tracks the individual particle motion and therefore overcomes some difficulties associated with the Eulerian model for the particles (Tian et al. 2005). The equations of particle motion are: dxp ¼ up dt mp

dup ¼ FD þ Fg þ Fother , dt

(38) (39)

16

Z.F. Tian et al.

here FD is the drag force, Fg is the gravity force and Fother includes other forces such as buoyancy force, virtual mass force, pressure gradient force, etc. The drag force FD is calculated from (Tian et al. 2010a): FD ¼ m p

ug, instant  up , τr

(40)

here ug , instant is the instantaneous gas velocity. The discrete random walk (DRW) model is widely used to model the effects of turbulence on particle trajectories. In the DRW model, ug, instant = ug + u0 when particle dispersion is taken into consideration (u0 is an approximation to the eddy fluctuation velocity determined using a random walk approach); ug, instant = ug when the particle dispersion is off. The particle relaxation time τr is given by: τr ¼

ρp d2p 18μg f D

(41)

and the drag coefficient fD for a sphere can be calculated based on empirical equations, e.g., (Tian et al. 2010a):  fD ¼

1 þ 0:15Re0:687 , Rep  1000 p 0:01833Rep , Rep > 1000

(42)

24 f Rep D

(43)

CD ¼

One major concern of the Eulerian-Lagrangian approach is the expensive computing time that may be experienced when tracking a substantial number of the particles to obtain good statistical information of the particle phase (Tian et al. 2005). With the progress in computer speeds, multi-core processors and parallelisation techniques, the time expense of Eulerian-Largrangian models has been significantly reduced and it has become a popular tool for coal combustion simulations. The Eulerian-Lagrangian model is extended in coal combustion models to take into account the particle combustion processes occurring in the furnace. The most widely used coal drying models, devolatilisation models and char oxidation models that have been implemented the Eulerian-Largrangian models are reviewed in the next section.

Coal Devolatilisation and Char Oxidation Models When pulverised coal particles enter the furnace through the burners, they rapidly mix with hot intermediates and combustion products. The particles undergo the following four well-defined steps during combustion in the furnace (Wu 2005) shown in Fig. 5:

Numerical Modelling of Pulverised Coal Combustion

17

Volatile Combustion

Original Coal Particle

1.

2.

Heating & Drying Process

Devolatilisation

4. Char Combustion

Fig. 5 Coal particle combustion processes (After Wu 2005)

• Coal particle heating and drying; • Devolatilisation of the coal particle to produce non-condensable volatiles (light gases), condensable volatiles (tars), and a carbonaceous char; • Gas phase volatile combustion; • Char combustion.

To model these processes in the Lagrangian particle tracking model, the coal particles are normally treated as spheres that do not interact with other particles. Each particle is able to undergo internal reactions as well as being fully coupled through the transfer of mass, momentum and energy with the gas phase, which enables heat transfer and chemical reactions to occur between the particle and gas phase. Several models are required to model the combustion of the coal particles shown in Fig. 5, namely, a drying model for the raw coal particles, a devolatilisation model, and a char combustion model. Coal normally contains moisture that can be divided into surface moisture and inherent (or bound) moisture. Surface moisture is moisture on the coal surface including inter particle voids and contact points of particles; while inherent moisture exists in the coal internal pore structure (Wu 2005). Old coal such as bituminous coals has 1–12.2% moisture as received and subbituminous coals have moisture as received in the range of 14.1–31% (Tillman 1991; Wu 2005). Young coals such as brown coal and lignite normally contain higher moisture content, e.g., Victorian brown coal typically has 66–70% moisture by weight (Tian et al. 2010b). In pulverised coal furnaces, a fraction of water in coal particles is released from coal during the pulverising process and also in the pre-drying process if there is any. The content of the water in particles entering the furnace depends on the coal type and the boiler type.

18

Z.F. Tian et al.

Sometimes the modelled drying process of coal particles in PC furnaces can be incorporated in the devolatilisation process, or it can be modelled in a separate drying model. When the water content of the particles is small, e.g., coal particles after a pre-drying process, evaporation of water in the pre-dried particle can be modelled as a species of volatile gas. This will slightly reduce the complexity of the coal modelling process by eliminating the need for a separate drying model. However, it is more accurate to model the evaporation of water separately from the devolatilisation process, because in real furnace combustion, most water in the particles evaporates before the start of devolatilisation process. Typical equations used to model the water evaporation in coal particles assuming mass transfer control are given in Bhambare et al. (2010). The change of mass of coal particles during the drying process can then be calculated as below:     dmp ¼ kc Cvapour, s  Cvapour, 1 Ap Mvapour dt drying

(44)

here Ap is particle surface area. Mvapour is the molar mass of water vapour. kc is mass transfer coefficient. Cvapour,s is vapour concentration at the coal particle surface: Cvapour, s ¼

psat ðTP Þ , RT p

(45)

here psat(TP) is the saturated vapour pressure at the particle temperature, Tp. Cvapour,1 in Eq. (44) is the vapour concentration in the bulk gas: Cvapour, 1 ¼ ½H2 O

pop , RT 1

(46)

here, [H2O] is the mole fraction of H2O vapour and pop is the operating pressure. The mass transfer coefficient, kc, in Eq. (44) is calculated using the Nusselt number: Nu ¼

kc d P 1=2 ¼ 2:0 þ 0:6Red Sc1=3 Dvapour, m

(47)

where Dvapour,m is the diffusion coefficient of vapour, Sc is the Schmidt number. In the CFD model, the reduction of particle mass can be accounted for either by reducing the particle density without changing of the particle diameter or reducing the particle diameter with a constant density.

Devolatilisation Models After drying in the burner exit region, coal particles are heated to higher temperatures rapidly, and start to decompose to produce non-condensible volatiles and tars. Non-condensible volatiles are non-condensable gases that consist mainly of a

Numerical Modelling of Pulverised Coal Combustion

19

mixture of CO2, H2O vapour and combustible gases including CO, H2 and hydrocarbons such as CH4, C2H4, C2H6, etc. (Field et al. 1967). The tar is a heavy hydrocarbon-like substance that is condensable, with an atomic ratio of H/C > 1.0 (Tillman 1991; Wu 2005). Again, the exact products of the devolatilisation process are determined by the coal types and decomposition condition that can be either rapid or slow. Pulverised coal combustion always involves a high rate of heating (104 K/s or greater) that is classified as rapid decomposition (Field et al. 1967). Evolution of volatile matter under the influence of heat and the subsequent combustion of the vapours evolved is an integral part of the combustion of coal, including brown coal (Mulcahy et al. 1991). In fact, about 65% of the heat released by combustion of Yallourn coal, one kind of Latrobe Valley coal, is derived from combustion of the volatiles (Jones and Stacy 1986). Two groups of devolatilisation models have been developed and used for PC combuston models: simple global kinetic models and more comprehensive computer-based network models. In CFD models, the products of devolatilisation process are assumed to be the gas(es) and that remaining coal particles that comprise char and ash only. Volatile gas(es) in the model can be a single species or several major volatile components such as CH4, C2H4, C2H6, etc. Two simple global kinetic models are widely used in PC coal combustion modelling. These are the single firstorder reaction (SFOR) model and the competing reaction model. In SFOR model the devolatilisation of coal particles is assumed to be independent of the particle size, porosity, specific surface area and surface/mass ratio, and other coal characteristics (Tillman 1991). The rate of devolatilisation is assumed to be firstorder dependent on the amount of volatiles remaining in the particle: 

dmp dt



 

¼ kv mp  1  f v, 0 mp, 0

(48)

devo

where mp is the instantaneous particle mass, mp,0 is the initial particle mass after drying process if there is a separate drying model. fv,0 is the initial mass fraction of volatiles in the particle before devolotiliation and kv is the kinetic rate: 

Tv kv ¼ Av exp  Tp

 (49)

The pre-exponential factor Av and the activation temperature Tv are constants determined experimentally for the particular coal. Some experiments have found that the yield of volatiles from PC particles can be greater by as much as a factor of two than the proximate value in PC furnaces (ANSYS/CFX 2015). The competing reaction model takes this into consideration by assuming that two devolatilisation process undergo simultaneously, one reaction dominants at low temperatures and the other at high temperatures (ANSYS/CFX 2015). Therefore, Eq. (48) can be written as:

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Z.F. Tian et al.

     

dmp ¼  kv, 1 α1 þ kv, 2 α2 mp  1  f v, 0 mp, 0 , dt devo

(50)

where α1is near the proximate volatile fraction where α2 is higher, close to unity, reflecting the characteristics of devolatilisation at high temperature (Wu 2005). kv , 1 and kv , 2 are the kinetic rate at low and high temperature, respectively. A chemical percolation model for devolatilisation (CPD) model has been developed in Grant et al.(1989) by applying the lattice statistics. In contrast to the above devolatilisation models based on empirical rate relations, the CPD model characterises the devolatilisation behaviour of rapidly heated coal based on the physical and chemical transformations of the coal structure (ANSYS/FLUENT 2015). The CPD has been implemented into several CFD codes such as ANSYS/ FLUENT and has been used for some coal combustion modelling, e.g., in Jovanovic et al.(2012). In this chapter, the CPD model is not discussed in details due to the space limitation. Interested readers can read more details in Grant et al.(1989), Wu (2005) . Devolatilisation of Latrobe Valley brown coal under fast heating rates (normally larger than 104 K/s), which is experienced in pulverised brown coal combustion, has been investigated in several studies using different methods such as a vertical laminarflow furnace (Roberts and Loveridge 1969), a plug-flow reactor (Duong 1985), and a pressurised drop-tube furnace (Yeasmin et al. 1999), and corresponding kinetic parameters for the SFOR model have been calculated based on the experimental measurements. Duong (1987) conducted measurements of pulverised brown coal combustion in a plug-flow reactor under different inlet conditions. It is found that the fuel/air mass ratio is seen to be the only factor affecting both the rates and mechanism of the volatile release. However, this cannot explain the large difference between parameters developed in Roberts and Loveridge (1969), Yeasmin et al. (1999), given that both experiments were carried out in an inert atmosphere of nitrogen. The kinetics parameters from Yeasmin et al. (1999), Duong (1987) have been tested in a drop tube furnace (Ouyang et al. 1998) and it was found the parameters from run 3 and run 5 of Duong (1987) give better agreement than other parameters for the measured particle mass loss along the axis line in the drop tube furnace. It has been found that some bituminous coals swell considerably during heating. A swelling coefficient can be used in CFD codes to take into account such swelling effects during devolatilisation. The value of the swelling coefficient is determined by coal types and combustion conditions. Experiments have shown that Latrobe Valley coals do not undergo swelling but develop an internal bubble structure when devolatilised in nitrogen (Sainsbury and Hawksley 1969). The non-swelling characteristic is confirmed by the observations of several combustion experiments such as in Street (1979). Therefore, the particle swelling can be neglected when simulating the Latrobe Valley coal combustion using CFD.

Char Combustion Models Char remaining in the coal particle after devolatilisation contains fixed carbon and subsequently undergoes a heterogeneous reaction with gaseous species at elevated

Numerical Modelling of Pulverised Coal Combustion

21

Fig. 6 Steps in heterogeneous reactions (After Williams et al. 2000)

Oxidising reactants 1

5 4

3

2

temperatures (Wu 2005). Combustion of the residual char is relatively slow due to the small reaction surface. The heterogeneous reactions in a coal particle include five steps shown in Fig. 6 (Williams et al. 2000; Wu 2005): Step 1. Diffusion of oxidants through the gas boundary layer surrounding the particle (external diffusion) and through the pores of the particle (internal diffusion) to the particle surface Step 2 adsorption of reactants onto the particle surface Step 3 surface reactions to form solid products Step 4 desorption of the solid products into the gas phase Step 5 diffusion of gas phase oxidisation products through the pores of the particle and through the ambient gas phase to the gas stream (Williams et al. 2000; Wu 2005).

In pulverised coal combustion, the main heterogeneous reactions include: C þ O2 ! CO2

(51)

1 C þ O2 ! CO 2

(52)

Reaction 51 dominates at lower temperature and reaction 52 is dominant with increasing temperature. Furthermore, residual char may also react as follows: C þ CO2 ! 2CO

(53)

C þ H2 O ! CO þ H2

(54)

CO and H2 produced in the above heterogeneous reactions diffuse away from the char particle into the ambient gas stream and react as follows:

22

Z.F. Tian et al.

CO þ 1=2 O2 ! CO2

(55)

H2 þ 1=2 O2 ! H2 O

(56)

CO þ H2 O ! CO2 þ H2

(57)

Combustion of a char particle is controlled by the rate of oxygen diffusion to the particle or the chemical reaction rate, or a combination of the two factors (Wu 2005). In the low temperature zone ( pv

(62)

(63)

where Cc and Ce are condensate and evaporate parameters and need to be determined empirically. Dynamic Cavitation Model (DCM) It may be noticed that the expression of ℜb in FCM is independent of the local pressure, which is not a good assumption for unsteady cavitation. Therefore, there is

Cavitation Flow of Cryogenic Fluids

23

still room to improve the FCM, regardless of its success in modeling the quasi-steady sheet cavitation (Singhal et al. 2002; Zhang et al. 2008a), by considering the effects of pressure on the bubble radius. In general, the liquid and vapor may not be in equilibrium locally, and the bubble will essentially experience a nonequilibrium thermodynamic path and metastable state during the phase-change process (Zein et al. 2010; Gavrilyuk and Saurel 2002). However, as indicated in section “Thermodynamic State of Sheet Cavitation,” the local thermodynamic equilibrium during the phase-change process is a valid assumption. At thermodynamic equilibrium, the temperature and chemical potentials between the liquid phase and vapor phase are equal. Integrating the Gibbs-Duhem equation (Carey 1992) from the initial pressure P = Psat to an arbitrary pressure P for the liquid and vapor phases, respectively, and using the ideal gas equation, a general relation between the pressure inside and outside of the bubble is obtained. These two pressures should satisfy the Young-Laplace equation. Combining the two equations finally leads to the following expressions relating the local pressure and the bubble radius. Details regarding the derivation can be found elsewhere (Zhang et al. 2013; Carey 1992). For the evaporation process: ℜb , e ¼

2σ  Psat ðT l Þexp ½Pl  Psat ðT l Þ=ðρl RTÞl  Pl

(64)

and for the condensation process: ℜb , e ¼

2σ ρl RTv ln½Pv =Psat ðT v Þ  Pv þ Psat ðT v Þ

(65)

where the subscript sat denotes the saturation state. Pl and Pv in above equations represent the local pressure P during the evaporation or condensation process, while Tl and Tv represent the local temperature Tref. Substituting Eqs. 60, 64, and 65 into 60 and considering the fact that the per unit volume phase-change rates should be proportional to the volume fractions of the donor phase, leads to the expressions: When Psat > P (evaporation process) m_ re ¼ Ce

   Psat ðT ref Þexpf½P  Psat ðT ref Þ=ðρl RTref Þg  P 2 Psat  P 1=2   ρv 1  f v  f gas σ 3 ρl

(67) When Psat P (condensation process)   ρl RTref ln½P=Psat ðT ref Þ þ Psat ðT ref Þ  P 2 P  Psat 1=2  ρl fv m_ rc ¼ Cc σ 3 ρl

(68)

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X. Zhang and Z. Jiakai

In the above equations, the relation αv = fρ/ρv is used to calculate the vapor volume fraction. Ce and Cc are empirical constants, while fgas is the non-condensable gas mass fraction that is set to 108 simulations to avoid its influence. No assumption has been made that the bubble pressure is equal to the saturation pressure in the derivation of the bubble radius expression. However, those two pressures are practically very close from the thermodynamic viewpoint, which means that Eqs. 67 and 68 can be further simplified. Notice also that during the phase change the bubble radius is a function of the local pressure and the saturation pressure. For isothermal cavitation in water, both the saturation pressure and the surface tension are constant; thus the bubble radius only varies with the local pressure. For the evaporation process, the evaporation rate is larger if the local pressure P is lower; for the condensation process, the condensation rate is larger if the local pressure P is higher. The effects of turbulence on cavitation are considered in the same fashion as in the FCM (Singhal et al. 2002) by simply raising the phase-change threshold pressure value as: Pcav ¼ ½Psat ðT ref Þ þ Pt =2

(69)

where Pt = 0.39ρκ is the turbulent pressure. Thus, Eqs. 67 and 68 can be rewritten as When Pcav > P (evaporation process)    Pcav ðT ref Þexpf½P  Pcav =ðρl RTref Þg  P 2 Pcav  P 1=2   ρv m_ re ¼ Ce 1  f v  f gas σ 3 ρl

(70) When Pcav P (condensation process)   ρl RTref lnðP=Pcav Þ þ Pcav  P 2 P  Pcav 1=2  ρl fv m_ rc ¼ Cc σ 3 ρl

(71)

Turbulence Model For Quasi-Steady Cavitating Flow For quasi-steady cavitating flow, the realizable turbulence model is utilized (Theory Guide 2012): 
   @ ð ρm κ Þ @ ρm uj κ @ μt @κ þ ¼ μþ þ G k  ρm e @xj @t @xj σ κ @xj 
   @ ð ρm e Þ @ ρm uj e @ μt @e e2 pffiffiffiffiffi þ ¼ μþ þ ρm C1 Se  C2 ρm @xj @t @xj σ e @xj κ þ νe

(72)

(73)

Cavitation Flow of Cryogenic Fluids

25




 pffiffiffiffiffiffiffiffiffiffiffiffi @uj η κ C1 ¼ max 0:43, , η ¼ S , S ¼ 2Sij Sij , Gκ ¼ ρu0i u0j ηþ5 e @ui In these equations, Gκ represents the generation of turbulence kinetic energy due to the mean velocity gradients. σ κ and σ e are the turbulent Prandtl numbers for k and e. k is the turbulent kinetic energy and e is turbulent dissipation rate. The eddy viscosity is computed from μt ¼ ρm Cμ

κ2 e

(74)

The default constants for this turbulence model are given as follows: C2 ¼ 1:9, σ κ ¼ 1:0, σ e ¼ 1:2, Prt ¼ 0:85

(75)

For Unsteady Cavitating Flow For unsteady cavitating flow, the large eddy simulation model is used. The LES equations are given below (Theory Guide 2012):  @ ðρm Þ @ ρm uj þ ¼0 @t @xj  @ ð ρm ui Þ @ ρm ui uj @p @  @τij þ ¼ þ σ ij  @t @xi @xj @xj @xj   @ui @uj 2 @uk σ ij ¼ μm þ  δij @xj @xi 3 @xk

(76)

(77)

where the over-bar denotes the filtered quantities. τij is the sub-grid scale (SGS) stress and defined as:  τij ¼ ρm ui uj  ui uj

(78)

It is assumed that the SGS stress is proportional to the modulus of the strain rate tensor, Sij , of the filtered large-scale flow: 1 τij  τkk δij ¼ 2μt Sij 3

(79)

where μt is the sub-grid scale turbulent viscosity. In the LES Wall-Adapting Local Eddy-Viscosity (WALE) model (Nicoud and Ducros 1999), μt and Sij are modeled by

26

X. Zhang and Z. Jiakai

 μt ¼ ρL2s 

Sij Sij

5=2

3=2

 5=4 þ Sdij Sdij

(80)

  1 @ui @uj þ 2 @xj @xi

(81)

 1 1 2 @ui gij þ g2ji  δij g2kk , gij ¼ , Ls ¼ minðκd, Cw ΔÞ 2 3 @xj

(82)

Sij ¼ Sdij ¼

Sdij Sdij

where Ls is the mixing length for the SGS, κ is von Karman’s constant, d is the distance to the closest wall, Δ represents the local grid scale based on the cell volume, and the WALE constant Cw is 0.325 (Theory Guide 2012).

Pressure Correlation Equation and Update of Thermodynamic Properties Thermodynamic properties (such as saturation vapor pressure, densities, specific heat, thermal conductivity, viscosity, etc.) are specified as the function of temperature from the data generated from Prop v7.0 (Lemmon et al. 2007). The key step in the implicit pressure-based algorithm is to establish the pressure correlation equation for the cavitating flows. To satisfy the continuity constraint and ensure numerical stability, the pressure correlation equation is built based on the total volume continuity (Li and Vasquez 2012):    1 @ρl αl 1 @ρv αv _ þ þ ∇  ðρl αl uÞ  mg þ ∇  ðρv αv uÞ  ðm_ Þ ¼ 0 ρl ρv @t @t

(83)

where m_ ¼ R_ e þ R_ c and αl + αv = 1.

Steady Flow It is noted that for steady flow computations, the pressure-density coupling scheme only affects the convergence path, and the final solution is independent of the choice because of the nature of pressure correlation (Senocak and Shyy 2004a, b; Li and Vasquez 2012). The conclusions are verified that the calculations for quasi-steady cavitation based on the incompressible fluids for both liquid and gas phases obtained the quite accordant results with the experimental data (Senocak and Shyy 2004a; Zhang and Khoo 2013) and the compressible calculations (Zhang and Khoo 2014). Therefore, compressibility of either liquid or gas is neglected. Equation 83 is easily converted to  ∇u¼

 1 1  m_ ρl ρv

(84)

Cavitation Flow of Cryogenic Fluids

27

The set of the pressure-based continuity equation and momentum equation are simultaneously solved in the following sequences (Theory Guide 2012): 1. The momentum equations are discretized using the finite volume method to obtain the linear algebraic equations, Au = t  ∇p, which is then used to obtain the relation, u = AH/AD  (1/AD)  ∇p and u0 = (1/AD)  ∇p0, where A is the coefficient matrix of the algebraic equations for velocity vectors (u); AD = diag(A), AH = t  ANu, and AN = A  diag(A); u* and p* are the estimation velocities and pressure; u0 and p0 are the correction velocities and pressure, respectively. 2. Substituting p = p + p0 and u = u + u0, together with the relation u = AH/ AD  (1/AD)  ∇p, into Eq. 17, yields the implicit governing equation for the correction pressure p0 , which can be easily solved. It is emphasized that the mass transfer term m_ also contains p* and p0 . 3. Pressure and velocities are updated by p = p + p0 and u = u + u0. Scalars (T, k, e, αv) are solved by Eqs. 50, 76, 77, and 51, respectively, and the fluid properties including the mixture density are then updated.

Unsteady Flow For unsteady flow computations, the speed of sound of vapor and liquid will influence the final results. Thus, it is important to establish a pressure correlation equation to account for the compressibility. Integrating Eq. 83 over a control volume and assuming that ρl ¼ ρl þ ρ0l ; ρv ¼ ρv þ ρ0v ; αl ¼ αl þ ρ0l ; αv ¼ αv þ ρ0v ; ρ0l ¼ en,  , 0 ; V en ¼ V en,  þ V en, 0 ; α0  0; α0  0; V en þ V en l ¼ Vl v v v l v l

@ρl 0 0 @ρ p ; ρv ¼ v p0 ; @p @p

The discretized pressure correction equation can be obtained (Li and Vasquez 2012): )   αl Vol @ρl 0 X 1 @ρl e,  0  m Vol þ F p p þ Δt @p ρle,  @p e l e e ( )   X 1 @ρ  1 X e,  , 0 þ m0 Vol þ αv Vol @ρv p0 þ e,  0 v αq ρv Aen V en F p þ q ρv Δt @p ρev,  @p e v e e ( ) ( ) 1 αl ρl Vol  α0l ρ0l Vol0 X e,  1 αv ρv Vol  α0v ρ0v Vol0 X e,  þ þ Fl  m0 Vol  Fv þ m0 Vol ¼ ρl ρv Δt Δt e e

1 ρl

( X

,0 αeq,  ρl Aen V en q

0

(85) Here, the superscript * and 0 , respectively, represent old values and corrections. Vol is the cell volume, Δt is the time step, A is the area at face “e,” and F is the phase mass flux. It should be noted that the extra terms related to @ρ/@p are the compressible effect in the pressure correction equation. @ ρl = @ p and @ ρv = @ p are the

28

X. Zhang and Z. Jiakai

speed of sound in liquid and vapor, respectively, which are given as the function of saturation temperature and coupled into the codes of fluent before simulations. In the coupled algorithm (Theory Guide 2012), it solves the pressure correction equation and momentum equation simultaneously to get the updated pressure and velocities: p = p + p0, u = u + u0. Then scalars T and αv are solved by Eqs. 50 and 51, respectively, with the updated fluid properties.

Validation and Recalibration of the Cavitation Model The Hord group from NASA has performed subscale tests of cavitation in cryogenic fluids (liquid nitrogen and liquid hydrogen) in a transparent plastic blowdown tunnel. The hydrofoil and ogive were placed in the center of the tunnels, respectively. And several pressure transducers and thermal couples were mounted along the wall of them. The accuracy for pressure and temperature was 6,900 Pa and 0.2 K, respectively. Details of the experiments can be found in Hord (1972, 1973a, b). The experimental data are usually used as bench mark to examine the cavitation modeling framework. Full Cavitation Model The full cavitation model (FCM) was used for quasi-steady cryogenic cavitation modeling by Zhang et al. (2008a, b). The realizable turbulence model is used to investigate the effects of turbulent mixing. In the near wall treatment, the standard wall function and the enhanced wall treatment are compared. Both the standard wall function and the enhanced wall treatment are applicable for modeling cavitating turbulent flows in liquid hydrogen, and comparable results with Hord’s experimental data are obtained. However, the enhanced wall treatment provides a better temperature and pressure distribution than the standard wall function, because it assumes a local equilibrium that exists between the production of kinetic energy and its dissipation rate at the wall-adjacent cells. As illustrated in Fig. 6 in liquid hydrogen cavitating flow over the hydrofoil 229C and ogive 390B, the variation of the vaporization and condensation rate parameters has a great effect on the simulation results. As these parameters become larger, the temperature and pressure depressions also increase. The reason for this behavior is that although the evaporation and condensation rate change proportionally, the change in the absolute magnitude of the evaporation rate is greater than that of condensation rate. Also, the locations of the lowest temperature and pressure move away from the leading edge when the parameters decrease. Apparently, the simulated results with the default model parameters (Ce = 0.02, Cc = 0.01) correlate better with the experimental data than those with significantly modified model parameters. As introduced in section “Parameters Developed to Estimate Thermal Effects on Cavitation”, the basic assumptions in the B-factor theory is that liquid flows over the cavity as if it were a solid body and no convection of liquid mass flows across the vapor-liquid surface. Therefore, the primary factor governing the vaporization is the viscous diffusion of energy across vapor-liquid interface, which sustains the continuous vaporization of the liquid. In the simulated liquid nitrogen cavitating flow (Zhang et al. 2008b), the particle traces of the fluid streamlines are plotted near the

Cavitation Flow of Cryogenic Fluids

29 4x105

20.4 20.0

Ce=0.01, Cc=0.005 Ce=0.02, Cc=0.01 Ce=0.04, Cc=0.02 Hord, 1973

19.6 19.2

Pressure (Pa)

Temperature (K)

20.8

Ce=0.01,Cc=0.005 Ce=0.02,Cc=0.01 Ce=0.04,Cc=0.02 Hord,1973

3x105 2x105 1x105

18.8 0.13

0.14

0.15

0.16

0.17

0.13

0.14

0.15

0.16

0.17

x (m)

x (m)

Hydrofoil - 229C 4x105

21.6 21.2 20.8

Ce=0.01,Cc=0.005 Ce=0.02,Cc=0.01 Ce=0.04,Cc=0.02 Hord,1973

20.4 20.0 19.6 0.05

0.06

0.07 0.08 x (m)

0.09

Pressure (Pa)

Temperature (K)

22.0 Ce=0.01,Cc=0.005 Ce=0.02,Cc=0.01 Ce=0.04,Cc=0.02 Hord,1973

3x105 2x105 1x105

0.10

0.05

0.06

0.07 0.08 x (m)

0.09

0.10

Ogive - 390B

Fig. 6 Wall temperature and pressure distribution with different cavitating model parameters (Zhang et al. 2008a) Fig. 7 Closeup particle traces of the fluid streamlines near the hydrofoil wall of NASA hydrofoil 283C (Zhang et al. 2008b)

hydrofoil wall in Fig. 7. It is noticed that only the outskirt of the cavity, where the vapor fraction is relatively small, has the liquid flow through it; however, the cavity core near the wall, where it has the largest vapor fraction, has no liquid flow through it. Then, it is deduced that the sustaining mechanism of the cryogenic cavity maybe contains two parts: for the outskirt of the cavity, it is mainly sustained by the convection of liquid flowing across the vapor-liquid surface, and for the cavity core, it is still mainly controlled by the viscous diffusion of energy from the surrounding mixture, other than from the vapor-liquid surface contrasted to the B-factor theory. The distribution of pressure and temperature along the wall of hydrofoil 293A are plotted together in Fig. 8 to present a much clearer understanding of the process of cavitation in liquid nitrogen. First, as the cross area of passage shrinks, the static pressure decreases. When it becomes 104,700 Pa (the saturation pressure

30

X. Zhang and Z. Jiakai

78.0

temperature depression starting point

6x105

dp

Temperature (k)

77.7 77.4

5x105

cavitation region

4x105 77.1 3x105

76.8

pressure recovery

computed T computed P saturation P of computed T

76.5 76.2

Pressure (pa)

Fig. 8 Pressure and temperature distribution in liquid nitrogen cavitation over the wall of hydrofoil 293A (Cao 2011)

2x105 1x105

0.13

0.14

0.15 X (m)

0.16 –0

corresponding to the temperature), theoretically cavitation will happen. However, the turbulence increases the cavitation pressure so that cavitation occurs in advance. The evaporation of liquid dominates at the front of the cavity with a temperature depression. At the same time, the static pressure varies together with saturation pressure corresponding to the local temperature. When the temperature decreases to the lowest value, the intensity of cavitation is the strongest. At that time, the pressure recovers as the cross area becomes bigger, which suppresses the development of cavity. And the condensation dominates this stage and the released heat heated the surrounding liquid. Due to the low conductivity of liquid nitrogen, the recovery of temperature experiences a long distance along the hydrofoil wall. The cavity closes earlier than the recovery of temperature.

Dynamic Cavitation Model The dynamic cavitation model (DCM) was used for quasi-steady cryogenic cavitation modeling by Zhang et al. (2013). As shown in Fig. 9, the results are in good agreement with the full cavitation model of the experimental data. The dynamic cavitation model can perform as precisely as the full cavitation model in computing temperature and pressure fields of quasi-steady cryogenic cavitation. This ability of dynamic cavitation model in modeling steady and unsteady cavitation in water is also conducted in Zhang et al. (2014, 2015). For quasi-steady cavitating flow, the CFD code with the dynamic cavitation model and the full cavitation model is applied to a number of validation and demonstration problems, which include cavitating flow over submerged cylindrical bodies with different forehead geometries and the NACA66 (MOD) hydrofoil at different boundary conditions. The predictions of the pressure distribution from the dynamic cavitation model are found to agree well with the experimental data and the results from the full cavitation model. The computed cavitation region using the dynamic cavitation model spreads out over a smaller space than the results from the

Cavitation Flow of Cryogenic Fluids Fig. 9 Temperature and pressure distributions along the ogive wall for liquid nitrogen cavitating flow (Zhang et al. 2013)

31

a 84.0

Temperature (K)

83.8

Ogive-419A, LN2

83.6 83.4 FCM

83.2

DCM 83.0

Experiment

82.8 0.05

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5 b 3.4x10

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FCM

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DCM Experiment

3.0x105 2.8x105

Ogive-419A, LN2

2.6x105 2.4x105 2.2x105 2.0x105 1.8x105 0.05

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full cavitation model, because the former model incorporates a higher sensitivity to pressure than the latter. Unsteady cavitating flows through a NACA66 hydrofoil were modeled. Figure 10 gives the contours of the volume fraction α at different times, which shows the transient evolutions of the cavity’s developing, shedding, and collapsing. The corresponding experimental observations (Leroux et al. 2004) at the same process are also given for comparisons. Qualitatively, they are well accordant. A long and narrow cavity is formed in the leading edge at t = 1.26 s and then it grows larger along the wall of the hydrofoil. When the length is more than about 0.5c, the trailing of the cavity becomes so unstable that the cavity breaks off and sheds from the wall. At the same time, the length of the cavity that adheres to the wall becomes smaller in Fig. 10f–g.

Sauer-Schnerr Model Zhu et al. extended the Sauer-Schnerr model for quasi-steady cryogenic cavitation (2015). The realizable turbulence model is used based on the recommendation from FCM modeling results stated above. The bubble number density per unit liquid (n) is

X. Zhang and Z. Jiakai DCM (a) t=1.26 s

Experiments

9.00e-01

9.89e-01

32

(c) t=1.34 s

6.33e-01

7.22e-01

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(b) t=1.30 s

(e) t=1.42s

3.67e-01

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(d) t=1.38 s

(g) t=1.50s

1.00e-01

1.89e-01

2.78e-01

(f) t=1.46 s

Fig. 10 Contours of the water volume fraction of DCM at different times. (Fluids flow from right to left) (Zhang et al. 2014)

calibrated, as shown in Fig. 11; when n exceeds 1010, the simulation results become unstable and can’t get to convergence. When n equals to 1010, the pressure distribution is obviously away from the experiment. When n is fixed at 105, 108, and 109, the pressure and temperature distributions along the hydrofoil wall are all consistent well with the experimental data. Thus, 108 is finally chosen.

9x10

5

8x10

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5x10

5

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33 84.0

Hord,1973;P n=1x105 n=1x108 n=1x109 n=1x1010

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n=1x105 n=1x108 n=1x109 n=1x1010

81.5 81.0 80.5

5

1x10 0.12

290C

83.5

290C T/K

P / Pa

Cavitation Flow of Cryogenic Fluids

0.14

0.16

0.18

0.20

0.22

80.0 0.12

x/m

0.14

0.16

0.18

x/m

0.20

0.22

Fig. 11 Computed temperature and pressure depression in liquid nitrogen compared with experimental data of hydrofoil 290C with variable bubble number density (Zhu et al. 2015)

Together with the large eddy simulation model (described in section “Turbulence Model”-b), the calibrated Sauer-Schnerr model is further utilized to model unsteady cryogenic cavitation (Zhu et al. 2016). The pressure correction equation described in section “Pressure correlation equation and update of thermodynamic properties”-b is adopted to take account the compressibility of liquid and vapor. The modeling results are analyzed to reveal the interactions of vortices, thermal effects, and cavitation in liquid hydrogen cavitating flows. General Observations of Unsteady Cavitation in LH2

To illustrate the dynamic evolution of cavitation, 15 numerical snapshots with an interval of 0.3 ms are shown in Fig. 12 (Zhu et al. 2016). The whole cavity is vaporous with less than 60% vapor in most regions and periodically detaches from the wall with a frequency of 275 Hz. Two completely different cavitation shedding phenomena can be observed in a complete period from the figures. From (a) to (f), the shape of the primary cavity almost remains unchanged, while the size shrinks during this time. Simultaneously, the small cavitation clouds are found to flow out of the primary cavity along the ogive surface. This stage is defined as the partially shedding mode (PSM). From (j) to (m), different from the PSM, the primary cavity becomes unstable and fully shed off from the lead edge of the ogive, which stage is defined as the fully shedding mode (FSM). The other two stages represent the transition between the PSM and FSM.

Detailed Observations of PSM

In the PSM, the primary cavity is quasi-steady, while, it is interesting to see that there are small cavitation clouds intermittently flowing out of the rear of the primary cavity and collapsing downstream (Zhu et al. 2016) . In order to better understand the occurrence mechanism of the small clouds, the vorticity transport equation in a variable density flow is employed as follows:

y (m)

a

0.03

0.03

0.03

Fig. 12 (continued)

0 0.025

0.005

0.01

0.015

g

0 0.025

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0.01

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d

0 0.025

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0.01

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0.035

0.035

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0.04

X (m)

0.04

X (m)

0.04

0.045

0.045

0.045

0.05

0.05

0.05

0.055

0.055

0.055

y (m) y (m) y (m)

b

0 0.025

0.005

0.01

0.015

h

0 0.025

0.005

0.01

0.015

e

0 0.025

0.005

0.01

0.015

0.03

0.03

0.03

0.035

0.035

0.035

X (m)

0.04

X (m)

0.04

X (m)

0.04

0.045

0.045

0.045

0.05

0.05

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0.055

0.055

0.055

y (m)

c

i

0 0.025

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0.01

0.015

0.025

0

0.005

0.01

0.015

f

0 0.025

0.005

0.01

0.015

y (m) y (m)

y (m)

y (m)

0.03

0.03

0.03

0.035

0.035

0.035

0.045

0.045

0.045

X (m)

0.04

X (m)

0.04

X (m)

0.04

0.05

0.05

0.05

0.055

0.055

0.055

34 X. Zhang and Z. Jiakai

m

0 0.025

0.005

0.01

0 0.025

0.005

0.01

0.015

y (m)

0.015

0.03

0.03

0.045

0.05

0.035

X (m)

0.04

0.045

0.05

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

X (m)

0.04

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

0.035

0.055

0.055

y (m) y (m)

0.03

0 0.025 0.03

0.005

0.01

0.015

n

0 0.025

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0.01

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k

0.045

0.05

0.035

X (m)

0.04

0.045

0.05

0.05 0.15 0.25 0.35 0.45 0.550.65 0.750.85

X (m)

0.04

0.05 0.15 0.25 0.35 0.45 0.550.65 0.75 0.85

0.035

0.055

0.055

0 0.025

0.005

0.01

0.015

o

0 0.025

0.005

0.01

0.015

l

X (m)

0.04

0.045

0.035

0.04

X (m)

0.045

0.05

0.05

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

0.035

0.05 0.150.25 0.35 0.45 0.55 0.65 0.750.85

0.03

0.03

Fig. 12 Instantaneous vapor volume fraction contours for unsteady LES calculations shown at intervals of 0.3 ms (Zhu et al. 2016)

y (m)

y (m) y (m)

j

0.055

Cavitation Flow of Cryogenic Fluids 35

36

X. Zhang and Z. Jiakai ! D ω  !  ! !  ! ∇ρm  ∇p ! ¼ ω  ∇ V  ω ∇ V þ þ ðνm þ νt Þ∇2 ω ρ2m Dt

(86)

!

In this equation, the term Dω =Dt is the material derivative of the vorticity vector ! ω , which describes the rate of change of vorticity of the fluid particle. The first term on the right-hand side represents the vortex stretching term, which is zero for a 2D case. The second term is the vortex dilatation term due to volumetric expansion/ contraction, describing the effects of fluid compressibility. If the vapor and liquid !

density are regarded as constants (Ji et al. 2013; Bensow and Bark 2010), ∇ V ¼ ! _ then the vortex dilatation term becomes ω ð1=ρv  1=ρl Þm, _ indið1=ρv  1=ρl Þm, cating the link between the mass transfer rate and the vorticity generation. The third term means the baroclinic torque resulting from the misaligned pressure and density gradients. The last term is viscous diffusion of vorticity and can be ignored because of the much smaller effect on the vorticity transport in high Reynolds number flow (Ji et al. 2014; Huang et al. 2014b). Figure 13 presents the contours of the calculated vapor volume fraction, pressure, ! temperature, streamline, and distribution of Dω=Dt near the ogive wall in the PSM at decreased intervals of 0.1 ms. In Fig. 13a, small cavitation clouds are found to shed at an interval of about 0.4 ms, which corresponds to a frequency of 2,500 Hz. The small clouds are generated near the leading edge inside the cavity. Its center has the elevated vapor fraction and decreased pressure and temperature values compared with the rest of the primary cavity (Fig. 13b, c). As the small cloud travels beyond the cavity, the vapor in the cloud begins to condensate because of the high surrounding pressure field, and even a higher temperature than that of inlet liquid occurs around the cloud, indicating the warming effect. In Fig. 13d for streamline, it is found that the liquid can flow into the cavity, and the streamlines are nearly straight apart from near the ogive wall, where several small discrete vortexes appear with the clockwise rotation direction. The positions of the vortexes are the same as the small cavitation clouds, implying that the formation of the small cloud within the primary one is primarily due to the vorticity structures. Correspondingly, because the vortex interior has the lowest pressure, there triggers the most violent cavitation phase change, resulting in the highest vapor phase fraction and the lowest temperature. The ! vorticity structure is further verified quantitatively by the contours of the term Dω=Dt in Fig. 13e. Since the vortex has the largest mass transfer rate due to cavitation and baroclinic torque term, respectively, corresponding to the calculations of the right ! second term and third one of Eq. 86, it is not surprising that the Dω=Dt is intensive near the ogive surface, but negligible in the part of cavity away from the wall. These results demonstrate that there are strong vortex-cavitation interactions near the ogive wall and in the shedding small vapor cloud. The pressure fluctuations and the corresponding power spectrum density (PSD) at position x = 0.04 m on the ogive surface are plotted in Fig. 14. The primary frequency of pressure fluctuations is 275 Hz, being equivalent to the primary cavitation shedding events. It is also noted that pressure fluctuation at the higher

0.02

x (m)

0.04

0.045

x (m)

0.04

0.045

0.05

0.05

0.045

0.03

x (m)

0.04

0.045 0.05

0.035

0.04

0.045

0.05

100000 160000 220000 280000 340000 400000 460000

0.035

0.05

0.015

0.02

Fig. 13 (continued)

0 0.025

x (m)

0 0.025

x (m)

0.03

0.03

0 0.025

0.03

0.04 x (m)

0.045

0.04 x (m)

0.045

0.035

x (m)

0.04

0.045

0.05

100000 160000 220000 280000 340000 400000 460000

0.035

0.05

0.05

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

0.035

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

Pressure (Pa)

0 0.025

0.005

0.01

0.015

0.02

0 0.025

0.005

0.045

x (m)

0.04

0.005

0.04

0.035

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

Pressure (Pa)

0.03

0.03

0.005

0.01

0.015

0.005

0.015

0.02

0 0.025

0.005

0.01

0.015

0.02

0 0.025

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

0.01

0.035

100000 160000 220000 280000 340000 400000 460000

0.035

0.05

0.005

0.01

0.015

0.01

0.03

Pressure (Pa)

0.03

0.035

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

0.03

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

0.01

0.015

0.02

0 0.025

0.005

0.01

0 0.025

0.005

0.01

0.015

0.015

b

y (m)

y (m)

y (m)

y (m) y (m) y (m)

y (m) y (m) y (m)

a

Cavitation Flow of Cryogenic Fluids 37

c

y (m)

y (m)

0.03

0.035

0.03

0.035

Temperature (K)

0 0.025

0.005

0.01

0.015

0.02

0.035

Temperature (K)

0 0.025

0.005

0.01

0.015

0.02

0.03

0.05

0.05

0.04 0.045 x (m)

0.05

20 20.2 20.4 20.6 20.8 21 21.2 21.4 21.6

0.04 0.045 x (m)

20 20.2 20.4 20.6 20.8 21 21.2 21.4 21.6

0.04 0.045 x (m)

100000 160000 220000 280000 340000 400000 460000

Pressure (Pa)

0 0.025

0.005

0.01

0.015

0.02

Fig. 13 (continued)

y (m)

y (m) 0.02

0.035

0.035

Temperature (K)

0 0.025 0.03

0.005

0.01

0.035

Temperature (K)

0 0.025 0.03

0.005

0.01

0.015

0.02

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0.05

0.04 0.045 x (m)

0.05

20 20.2 20.4 20.6 20.8 21 21.2 21.4 21.6

0.04 0.045 x (m)

0.05

21 21.2 21.4 21.6

0.045

20 20.2 20.4 20.6 20.8

0.04 x (m)

100000 160000 220000 280000 340000 400000 460000

Pressure (Pa)

0 0.025

0.005

0.01

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0.02

0.015

y (m) y (m)

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Temperature (K)

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Temperature (K)

0.03

20.2 20.4 20.6 20.8

21

0.05

21.2 21.4 21.6

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0.04 x (m)

0.045

0.05

20 20.2 20.4 20.6 20.8 21 21.2 21.4 21.6

0.04 0.045 x (m)

20

0.04 0.045 x (m)

100000 160000 220000 280000 340000 400000 460000

Pressure (Pa)

0 0.025

0.005

0.01

0.015

0.02

0.015

y (m) y (m) y (m)

38 X. Zhang and Z. Jiakai

y (m)

e

Fig. 13 (continued)

0 0.025

0.005

0.01

0.015

0.02

0 0.025

0.005

0.01

0.015

0.02

0 0.025

0.005

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0.04 x (m)

0.045

0.04 x (m)

0.035 0.04 x (m)

0.045

2E+08

0.045

–1E+09 –5E+08 –2E+08

0.035

0.05

5E+08

1E+09

0.05

0.05

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

0.035

: S–2

0.03

Dt

→

Dw

0.03

0.03

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

y (m)

0 0.025

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y (m) y (m)

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y (m)

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0.035

0.04 x (m)

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2E+08

0.045

–1E+09 –5E+08 –2E+08

0.035

0.05

5E+08

1E+09

0.05

0.05

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

0.035

: S–2

0.03

Dt

→

Dw

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0.03

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85

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d

→

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2E+08

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–1E+09 –5E+08 –2E+08

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: S–2

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Dw

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5E+08

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1E+09

Cavitation Flow of Cryogenic Fluids 39

0 0.025

0.005

0.01

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Dt

:

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–1E+09 –5E+08 –2E+08

0.035 0.04 x (m)

S–2

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0.05

5E+08

1E+09

!

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: S–2

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→

Dw

2E+08

0.04 0.045 x (m)

–1E+09 –5E+08 –2E+08

0.05

5E+08

1E+09

Fig. 13 The predicted vapor volume fraction, pressure, temperature, streamline, and distribution of Dω =Dt in the partially shedding mode are shown at intervals ! of 0.1 ms (Zhu et al. 2016). (a) Vapor content, (b) pressure, (c) temperature, (d) streamlines, (e) distribution of Dω =Dt

y (m)

→

y (m)

Dw

y (m)

0.02

40 X. Zhang and Z. Jiakai

Cavitation Flow of Cryogenic Fluids 3.5x105

3.5x107

x=0.04 m

3.0x105 2.5x105

3.0x107 Power

Pressure (Pa)

41

2.0x105 1.5x105

0.00

2.0x107 1.5x107 1.0x107

1.0x105 5.0x104

2.5x107

x=0.04 m f =275 Hz

5.0x106 0.01

0.02 Time (s)

0.03

0.04

100

1000 Frequency (Hz)

Fig. 14 Pressure fluctuations and power spectrum density (PSD) at the position x = 0.04 m (Zhu et al. 2016)

frequency at 600 Hz and beyond is not ignorable, especially a signal peak exists around 2,000 Hz. These higher frequencies are closely related to the faster shedding process of the vortex-induced small cavitation clouds. Thermal Effects on Vortex Formation in PSM

To explore the thermal effects on the cavity developments, the simulations of cavitation in LH2 in isothermal condition are also carried out, of which the energy equation (Eq. 50) is not solved (Zhu et al. 2016). The results are compared in Fig. 15 together. It is found that most regions in the isothermal cavity contain more than 85% vapor, much larger than the value in the nonthermal cavity. The shedding phenomena of the small cavitation cloud in the non-isothermal cases are not observed; in contrast, an obvious reentrant jet exists in the rear of the cavity, which is considered to be the reason of the shedding of the attached cavity (Stutz and Reboud 1997a, b; Le et al. 1993). The incoming liquid cannot enter the interior of the cavity and has to flow around it, which indicates that the pressure in cavitation zone is maintained mainly through the evaporation of liquid at the gas-liquid interface. Therefore, the dilatation term, due to the close relationship with the mass transfer, is found to be dominant along the liquid-vapor interface but negligible inside the cavity region. In addition, due to the larger vapor fraction in the cavity zone compared to the non-isothermal cases, there is a larger density gradient near the cavity interface for the isothermal cavity. As a result, the highest levels of the vortex baroclinic torque term exist near the closure of the cavity. The combination of these two terms in ! Eq. 86 predominates the vorticity transport; thus the highest levels of Dω =Dt appear at the interface and closure of the cavity. However, for the non-isothermal cavity, both the mixture density and temperature, as well as the saturated pressure, experience a gradual variation across the cavity except near the ogive surface, as shown in Fig. 13. These smooth transitions at the cavity interface lead to the negligible dilatation term and baroclinic torque term, as shown in Fig. 15b. While the gradients of vapor content and temperature, so as the saturated pressure, are the largest immediately near the ogive surface inside the

0.02

0 0.025

0.005

0.01

0.015

b

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0.055

0.06

0.05 0.055

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0.045

–1E+09 –5E+08 –2E+08 2E+08

0.045 x (m)

0.05

5E+08 1E+09

0.06

→

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→

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→

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0.055

0.05 0.055

–1E+09 –5E+08 –2E+08 2E+08

0.03 0.035 0.04 0.045 x (m)

w (∇.V ) :[s–2]

0.05

–1E+09 –5E+08 –2E+08 2E+08

0.04 0.045 x (m)

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

w (∇.V ) :[s–2]

→

0 0.025

0.005

0.01

0.015

0.02

0 0.025

0.005

0.01

0.015

0.02

0 0.025

0.005

0.01

0.015

0.02

0 0.025

0.005

0.01

0.015

0.02

0.03

r2m

0.03

r2m

0.05

0.055

0.04

0.045 x (m)

0.05

x (m)

0.04

0.045

0.05

5E+08

0.06

1E+09

1E+09

5E+08

0.06

0.055

–1E+09 –5E+08 –2E+08 2E+08

0.045 x (m)

–1E+09 –5E+08 –2E+08 2E+08

0.035

:[s–2]

0.035

:[s–2]

0.04

100000 160000 220000 280000 340000 400000 460000

0.035

∇rm x∇r

∇rm x∇r

0 0.025

0.005

0.01

0.015

0.02

0.03

Pressure (Pa)

0 0.025

0.005

0.01

0.015

0.02

Fig. 15 Comparison of the features in isothermal and non-isothermal cavity in FSM (Zhu et al. 2016). (a) Isothermal, (b) non-isothermal (PSM)

0.03

→

Dw :As–2E Dt

0.045 x (m)

–1E+09 –SE+07 –2E+07 –2E+08 SE+08 1E+09

0.04

0.03 0.035 0.04

0 0.025

0.005

0.01

0.015

0.02

0 0.025

0.005

0.01

0.035

Dw :As–2E Dt

→

0.03

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

y (m)

0.02

0.015

y (m)

y (m)

y (m)

y (m) y (m)

y (m) y (m) y (m)

a

42 X. Zhang and Z. Jiakai

Cavitation Flow of Cryogenic Fluids Fig. 16 Mechanisms of the LH2 unsteady cavities with strong thermal effects (a) and without thermal effects (b) (Zhu et al. 2016)

a

43

Liquid S tre a m line

Vapor

Ogive

b

Liquid

Vapor

Ogive

cavity. Therefore, there are high levels of both the dilatation term and baroclinic torque term. The different vortex-occurring characteristics of the isothermal and non- isothermal fluids mean the different cavitation dynamics. The unique phenomenon is illustrated by the interaction of vortex and the cavity shown in Fig. 16. For the non-isothermal cavitation, the vortex is small compared with the primary cavity, and the interactions between the small vortex and cavitation only happen in the boundary layer along the wall, resulting in the phenomenon of the coexistence of the quasisteady primary cavity and the unsteady shedding of small cavitation cloud. However, for the isothermal cavitation, the rear vortex travels upstream to the leading edge and can develop to the comparable size of the primary cavity, leading to the shedding off of the whole cavity. The experimental observations seem accordant with the phenomena. Niiyama et al. (2012) conducted experiments on the NACA16-012 hydrofoil using LN2 with an angle of attack 8 . As the cavitation number decreases, the primary cavity just gradually expands to the trailing edge. In this process, no largescale cavitation clouds sheds off and the observed cavity consists of lots of bubbles smaller than those in water. Similar phenomenon was also found in other thermosensitive fluids (Cervone et al. 2006; Gustavsson et al. 2008). However, around the NACA16-012 hydrofoil in water, periodically oscillating cavity caused by the reentrant jet was observed when the cavity length developed over chords of 0.5–0.7  C hydrofoil (Franc and Michel 1985).

Conclusion In this chapter, historical theoretical models about thermal effects estimation in cryogenic cavitation are classified by their hypothesis. Those models are limited to quasi-steady cavitation. Appearances, thermodynamic state and features of cryogenic cavitation are summarized from past research reports. The thermal effects inhibit the development of cryogenic cavitation bubbles and make the cavity porous

44

X. Zhang and Z. Jiakai

and cloudy. The numerical simulation of cryogenic cavitation shows that the liquid can penetrate the cavity to evaporate instead of along the vapor liquid interface in isothermal cavitation situations. What’s more, the vorticity mainly transfers along the surfaces of objects within the cavity resulting in a unique partially shedding mode for cryogenic cavitation. The details of a robust numerical framework are presented and three cryogenic cavitation models are validated again the experimental data.

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I. Senocak, W. Shyy, Interfacial dynamics-based modeling of turbulent cavitating flows, part-1: model development and steady-state computations. Int. J. Numer. Meth. Fl. 44, 975–995 (2004b) A.K. Singhal, M.M. Athavale, H.Y. Li, et al., Mathematical basis and validation of the full cavitation model. J. Fluids Eng. 124(3), 617–624 (2002) E. Sinibaldi, F. Beux, M.V. Salvetti, A numerical method for 3D barotropic flows in turbomachinery. Flow Turbul. Combust. 76, 371–381 (2006) H. Soyama, D. Macodiyo, Improvement of fatigue strength on stainless steel by cavitating jet in air. Fifth International Symposium on Cavitation, Osaka, 2003 W.A. Spraker, The effects of fluid properties on cavitation in centrifugal pumps. J. Eng. Power 87, 309–318 (1965) H.A. Stahl, A.J. Stephanoff, Thermodynamic aspects of cavitation in centrifugal pumps. ASME J. Basic Eng. 78, 1691–1693 (1956) B. Stutz, J. Reboud, Two-phase flow structure of sheet cavitation. Phys. Fluids 9, 3678–3686 (1997a) B. Stutz, J.L. Reboud, Experiments on unsteady cavitation. Exp. Fluids 22, 191–198 (1997b) M. Tanguay, T. Colonius, Progress in modeling and simulation of shock wave lithotripsy (SWL). Fifth International Symposium on Cavitation, Osaka, 2003 Theory Guide, ANSYS, FLUENT 14.5 Documentation, 2012 S.S. Thipse, Cryogenics (Alpha Science International, Oxford, UK, 2013) Y. Utturkar, J. Wu, G. Wang, W. Shyy, Recent progress in modeling of cryogenic cavitation for liquid rocket propulsion. Prog. Aerosp. Sci. 41, 558–608 (2005) M. Watanabe, L. Nagaura, S. Hasegawa, et al., Direct visualization for cavitating inducer in cryogenic flow (The 3rd report: visual observations of cavitation in liquid nitrogen), (in Japanese) JAXA Research and Development Memorandum, JAXA-RM-09-010, 2010 Y. Yoshida, Y. Sasao, M. Watanabe, et al., Thermodynamic effect on rotating cavitation in an inducer. ASME J. Fluids Eng. 131, 091302-1–091302-7 (2009) Y. Yoshida, H. Nanri, K. Kikuta, et al., Thermodynamic effect on subsynchronous rotating cavitation and surge mode oscillation in a space inducer. ASME J. Fluids Eng. 133, 0613011–061301-7 (2011) I. Yutaka, N. Tsukasa, N. Takao, Periodical shedding of cloud cavitation from a single hydrofoil in high-speed cryogenic channel flow. J. Therm. Sci. 18, 58–64 (2009a) I. Yutaka, N. Tsukasa, N. Takao, Cavitation patterns on a plano-convex hydrofoil in a high-speed cryogenic cavitation tunnel. Proceedings of the 7th International Symposium on Cavitation, Ann Arbor, 17–22 Aug 2009b A. Zein, M. Hantke, G. Warnecke, Modeling phase transition for compressible two-phase flows applied to metastable liquids. J. Comput. Phys. 229, 2964–2998 (2010) L.X. Zhang, B.C. Khoo, Computations of partial and super cavitating flows using implicit pressurebased algorithm (IPA). Comput. Fluids 73, 1–9 (2013) L.X. Zhang, B.C. Khoo, Dynamics of unsteady cavitating flow in compressible two-phase fluid. Ocean Eng. 87, 174–184 (2014) X.B. Zhang, L.M. Qiu, H. Qi, X.J. Zhang, Z.H. Gan, Modeling liquid hydrogen cavitating flow with the full cavitation model. Int. J. Hydrog. Energy 33, 7197–7206 (2008a) X.B. Zhang, L.M. Qiu, Y. Gao, X.J. Zhang, Computational fluid dynamic study on cavitation in liquid nitrogen. Cryogenics 48, 432–438 (2008b) X. Zhang, Z. Wu, S. Xiang, L. Qiu, Modeling cavitation flow of cryogenic fluids with thermodynamic phase-change theory. Chin. Sci. Bull. 58, 567–574 (2013) X.B. Zhang, W. Zhang, J.Y. Chen, et al., Validation of dynamic cavitation model for unsteady cavitating flow on NACA66. Sci. China Technol. Sci. 57, 819–827 (2014) X.B. Zhang, J.K. Zhu, L.M. Qiu, et al., Calculation and verification of dynamical cavitation model for quasi-steady cavitating flow. Int. J. Heat Mass Transf. 86, 294–301 (2015) D.F. Zhao, J.K. Zhu, L. Xu, et al., Visualization experiment of cavitating flow of cryogenic fluid in venturi tube. (In chinese). Cryog. Eng. (submitted)

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Experiments on Gas-Liquid Flow in Vertical Pipes D. Lucas, M. Beyer, and L. Szalinski

Abstract

A comprehensive database on upward two-phase flows in vertical pipes was obtained using the wire-mesh sensor technologies for gas-liquid flows in vertical pipes. The investigations were done for different pipe diameter as well as for flows with and without phase transfer. Wire-mesh sensors provide detailed information on the structure of the gas-liquid interphase. Basic characteristics of gas-liquid flows can be observed in such experiments and are discussed in this chapter. The quantitative results obtained in the measurements as radial volume fraction profiles, radial gas velocity profiles, bubble size distributions, distributions of interfacial area density, and others are valuable data for the development and validation of Computational fluid dynamics (CFD) codes for multiphase flows. Keywords

Gas-liquid flow • Pipe flow • Bubble size distribution • Phase transfer • Experiment • Database Nomenclature

CFD DN50 Dbub Dorifice Dpipe HZDR IA IAD ID

Computational fluid dynamics Test section with the 52.3 mm (ID) pipe Bubble diameter Diameter of the gas injection orifices for the variable gas injection Pipe diameter Helmholtz-Zentrum Dresden – Rossendorf Interfacial area Interfacial area density Inner diameter

D. Lucas (*) • M. Beyer • L. Szalinski Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_15-1

1

2

D. Lucas et al.

K16 L/D L12 L20 PR17 r TOPFLOW UG ai db JG JL ε

Experimental test series for condensing steam-water flow Length-to-diameter ratio Experimental test series for air-water flow in 195.3 mm (ID) pipe Experimental test series for air-water flow in 52.3 mm (ID) pipe Experimental test series for evaporating flow in case of pressure relief Radius [m] Transient two-phase FLOW test facility Gas velocity [m/s] Interfacial area density [1/m] Bubble diameter [m] Gas superficial velocity [m/s] Liquid superficial velocity [m/s] Gas volume fraction [%]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Setup and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The TOPFLOW Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wire-Mesh Sensors and Data Evaluation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Sections, Measuring Procedures, and Test Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separation of Small and Large Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the Air-Water Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interfacial Area Density (IAD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steam Bubble Condensation in Sub-cooled Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient Data from Pressure Relief Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 3 4 7 15 16 19 22 27 30 34 42 43

Introduction Gas-liquid flows in vertical pipes are well suited to study basic phenomena and general characteristics of two-phase flows with deformable interfaces. Here, the flow develops under clear boundary conditions and can be investigated for a relatively long distance. Different flow pattern such as bubbly flow, slug flow, churn-turbulent flow, and annular flow as well as transitions between them may occur depending on the flow rates and size of the pipe. For many practical applications as pipelines, steam generator tubes, and many others, pipe flow is of interest on its own, but more generally the findings obtained for two-phase flows in pipes can be transferred to more complex flow situations using computational fluid dynamics (CFD). For medium- and large-scale applications, the two- or multi-fluid approach is frequently applied. Here closure models are essential to reflect the non-resolved phenomena on local scale. The establishment of CFD setups with sets of closures which can be applied to a wide range of flow conditions without

Experiments on Gas-Liquid Flow in Vertical Pipes

3

Table 1 Test series

Test L20

Pipe diameter [mm] 52.3

L12

195.3

K16

195.3

PR17

195.3

Fluids Airwater Airwater Steamwater Steamwater

Flow type Adiabatic

Range of liquid flow rates [m/s] 0.04–4

Range of gas flow rates [m/s] 0.0025–19

2.5

Adiabatic

0.04–1.6

0.0025–3.2

10, 20, 40, 65 10, 20, 40, 65

Condensing steam

0.4–1.0

0.09–0.53

Evaporation by pressure relive

0 or 1.0

–

Pressure [bar(a)] 1.1–2.1

any tuning is an important requirement as discussed in chapter “▶ Euler-Euler-Modelling of Poly-dispersed Bubbly Flows”. High-quality data on vertical pipe flow are one important basis to develop, adapt, and validate such closure models. Afterward the validated model setup can also be used for more complex flow situations. The aim of the experiments discussed in this chapter is to achieve a better understanding of two-phase flows and to provide a comprehensive high-quality database suitable for the CFD model development and validation. The latter issue requires data with high resolution in space and time as they are provided by innovative measuring techniques as presented in chapter “▶ Imaging Measuring Techniques”. The database includes data for adiabatic air-water flows and steam-water flows with phase transfer for different pressure levels up to 6.5 MPa. An overview is given in Table 1. For all these cases flow rates and other parameter were varied over a wide range.

Experimental Setup and Boundary Conditions The TOPFLOW Facility TOPFLOW is an acronym standing for Transient twO-Phase FLOW test facility. It is designed for the generic and applied study of transient two-phase flow phenomena in the power and process industries. By applying innovative measuring techniques, TOPFLOW provides data suitable for CFD code development and qualification. TOPFLOW allows to perform steam-water or air-water experiments. The facility is described in detail by Schaffrath et al. (2001) and Prasser et al. (2006). The TOPFLOW facility can be operated at pressures up to 7 MPa and the corresponding saturation temperature of 286
C. The maximum steam mass flow is about 1.4 kg/s, produced by a 4 MW electrical heater. The maximum saturated water mass flow rate through the vertical test section is 50 kg/s. Different test sections can be operated between the heat source (steam generator) and the heat sink (cooling systems).

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Fig. 1 The integration of the Variable Gas Injection test section in the TOPFLOW facility in case of the K16 experiments

Figure 1 shows a scheme of the integration of the vertical pipe test section loop in the TOPFLOW facility for the example of the K16 experiments. Water is circulated in loop entering the vertical pipe test sections from below. Air or steam is supplied from gas injection systems which are different for the different tests and which are described below. The TOPFLOW steam drum is used as separator for all experiments.

Wire-Mesh Sensors and Data Evaluation Procedures The challenges to measure the gas-liquid phase distribution are discussed in detail in chapter “▶ Imaging Measuring Techniques”. The aim of the experiments discussed here was to get data on the evolution of gas-liquid flows in a pipe for a wide range of flow rates with high resolution in space and time. Such data can be provided using the so-called wire-mesh sensor techniques, originally developed by Prasser et al. (1998). Improvements of this technology are presented by Prasser et al. (2001) and Pietruske and Prasser (2007).

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A wire-mesh sensor consists of two grids of parallel wires, which span over the measurement cross section (see Fig. 1 in chapter “▶ Imaging Measuring Techniques”). The wires of both planes cross under an angle of 90
but do not touch. Instead there is a vertical distance between the wires at the crossing points. At these points, the conductivity is measured. According to the different conductivity of gas and water, the phase present in the moment of the measurement at the crossing point can be determined. Many different types of wire-mesh sensors were built and successfully used by an international community during the last 15 years. Sensors with 16  16 wires (52.3 mm pipe) and 64  64 wires (195.3 mm pipe) were used in the presented experiments. For both sensor sizes, measurements were done with a frequency of 2,500 Hz, i.e., each crossing point was measured 2,500 time per second. This allows to measure all gas structures larger than about 3 mm sphere equivalent diameter. The measuring time was 10 s, i.e., the result of one single measurement is a threedimensional matrix of 16*16*25.000 or 64*64*25.000 values of the instantaneous local conductivity, respectively. By a calibration procedure, a matrix of the instantaneous local volume void fraction with the same dimensions is calculated. The void fraction values can be visualized to get detailed insights on the flow characteristics. However, more important is the derivation of quantitative data by using averaging procedures (Prasser et al. 2005b). Most important is the time averaging, which, e.g., leads to time-averaged two-dimensional gas volume fraction distributions in the pipe cross section. Due to the radial symmetry of the data, the statistical error can be further lowered by an azimuthally averaging. To do this, the cross section is subdivided into ring-shaped domains with equal radial width. The contribution of each mesh cell is calculated by weight coefficients obtained from a geometrical assignment of the fractions of a mesh belonging to these rings. In the result, radial gas volume fraction profiles are obtained. For the measurements, two sensors were used which measurement planes have a distance in the range of few centimeters. This allows to cross-correlate the gas volume fraction values of the two planes for all mesh points which are located above each other. From the maxima of the cross-correlation functions, the typical time-shift of the local void fraction fluctuations can be determined. Since the distance between the measuring planes is known, the local time-averaged gas velocity can be calculated. The point-to-point two-dimensional gas velocity distributions in the pipe cross section are obtained in the results of this procedure. Again an azimuthally averaging is applied to obtain the radial profiles of the gas velocity. This procedure has clear limits for the transition region to annular flow with a gas core in the pipe center. Here the gas velocity can be much larger than the velocity of the interfaces determined here. Please consider that the second sensor is only used for the determination of the gas velocities. Due to the perturbing effect of the first sensor, other data, especially bubble size distributions obtained from the second sensor, would be distorted. The next step of the data evaluation procedure is the identification of single bubbles. A bubble is defined as a region of connected gas-containing elements in void fraction matrix which is completely surrounded by elements containing the

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liquid phase. A complex procedure, introduced by Prasser et al. (2001), applies a filling algorithm combined with sophisticated stop criteria to avoid artificial coalescence as well as artificial fragmentation of bubbles. In the result, the same identification number is assigned to all volume elements which belong to the same bubble. Different bubbles receive different identification numbers. These numbers are stored in the elements of a second array. This array has the same dimension as the void fraction array. Combining the information from the void faction and bubble number, arrays together with the radial profiles of the gas velocity characteristic data of the single bubbles as bubble volume, sphere equivalent bubble diameter, maximum circle equivalent bubble diameter in the horizontal plane, coordinates of the bubble center of mass, moments characterizing asymmetries, and others are obtained. Based on these data, cross section and time-averaged bubble size distributions and radial gas volume fraction profiles decomposed according to the bubble size are calculated. The bubble size distributions are defined volume fraction related, i.e., they present the volume fraction per width of a bubble diameter class (equivalent diameter of a sphere with the measured bubble volume Vb is considered). Besides these evaluation procedures, the wire-mesh sensor data comprise also information on the interfacial area density (IAD). To obtain it, first the interfacial area (IA) has to be determined. This is done by the evaluation of the surface area of each single bubble measured. Due to the spatial resolution of the wire-mesh sensor of usually 3  3 mm2 in the plane of the pipe cross section, the IA cannot be determined directly from the measurement data. If a bubble moves through the wire mesh, its surface area information is stored as averaged void fraction values. As originally proposed by Prasser 2007, an iterative approach is used for the determination of the bubble surface. Based on an initial threshold for void fraction which is characteristic for each single bubble, the surface area and the enclosed volume are calculated. Since the bubble volume is already known from the bubble identification algorithm, the void fraction threshold is modified iteratively until both volumes are equalized. Thereby the whole bubble surface area is assembled from single area parts from each volume element of the wire-mesh sensor data related to the current bubble surface. These volume elements are cuboids of 3*3* gas velocity/measurement frequency mm3, on whose corner points void fraction data are available. Resulting from the interpolation between these void fraction values, the algorithm calculates geometrical points which are elements of the bubble surface area under consideration of the void fraction threshold at the bubble surface. Using these points, a reconstruction of the bubble surface is possible. Thereby the IA algorithm considers different geometrical arrangements of the area parts inside the volume elements, whereas these area parts may again consist of some more little subarea parts. Based on the single area assembling, the algorithm slightly overestimates the IA in dependence on the bubble size. As smaller the bubble, as more this effect influences the results. For this reason, the calculated surface area of small bubbles is corrected using the known bubble volume and the maximal cross section area to obtain a surface area of an equipotential ellipsoid. Depending on the bubble size and

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related to the wire pitch of the mesh sensor, a weighting function determines the final bubble surface. Due to the determination of the surface area of each single bubble, the algorithm also provides detailed distributions. Thus, time-dependent cross section averaged time sequences of the IA and IAD as well as azimuthal and time-averaged radial distributions are calculated. Furthermore the algorithm provides distributions of both parameters related to bubble size classes including quotients of the calculated IA and the surface area of a volume equivalent sphere. Additionally for each measurement, a matrix of 64  64 time-averaged IAD values is calculated that can be used for a quick view over the flow data. For further investigations of the flow structure, extensive files are available that contain detailed parameter for each detected single bubble of the measured flow.

Test Sections, Measuring Procedures, and Test Matrices As discussed in the previous section, wire-mesh sensors can provide quite detailed information on the gas-liquid phase distribution in pipe flows. In addition it can also be applied for oil-water flows as shown, e.g., by Szalinski et al. (2010) who presents measurements on air-water flows in comparison with experiments done for oil-water flows. However, the wire-mesh sensor is an intrusive measuring technique, which is a clear disadvantage of this technology. Several investigations on this effect were done; see, e.g., Prasser et al. (2001, 2005a), Sharaf et al. (2011), and Nuryadin et al. (2015). Summarizing these investigations, it can be said that the sensor signal well reproduces the bubble sizes and shapes as long as there is a minimum liquid flow rate equivalent to a superficial velocity JL (volume flow rate divided by the cross section area of the pipe) of about 0.2 m/s. For lower flow rates, a deceleration of the bubbles may occur. Unfortunately the flow behind the sensor is clearly disturbed. Most important is here the effect of a bubble fragmentation by the sensor. As mentioned before, the aim of the experiments is to investigate the evolution of the flow along the pipe. Because of the disturbed flow behind the sensor, it is however not meaningful to place several sensors in the pipe to get the information of the gas distribution at different inlet lengths. Instead two different methods were used. For the 52.3 mm pipe (test series L20), pipe sections of different lengths were used to vary the distance between the fixed position of the gas injection and the measuring plane. For the 195.3 mm pipe, such a procedure, which requires to rebuild the facility for measurements at different height positions, would be not feasible. For this reason, the so-called Variable Gas Injection test section was constructed. Here the measuring plane is located at the top of the pipe for all measurements. To vary the inlet length, gas injection devices are placed on several distances below the measuring plane. They allow a gas injection through holes in the pipe wall to minimize the disturbance of the flow by the non-active gas injections.

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Fig. 2 Vertical test section DN50 (a), pipe section with gas injection module (b), and gas injection device (c)

Vertical Test Section DN50 The TOPFLOW test facility is equipped with two vertical pipes for the investigation of vertical two-phase flows. Both pipes are connected with a circulation pump and a separation tank (steam drum) to a test section circuitry (Fig. 1). Both test sections can operate separately using isolating valves. The DN50 test section has an inner diameter of 52.3 mm and a length of about 9 m. It is assembled from several sections with various lengths, to allow different inlet length between the gas injection and the measurement plane. The water from the steam drum flows through the circulation pump and is injected into the vertical test section from the bottom. The gas was mixed into the water flow also from the bottom, using a central injection device. It is equipped with eight orifices of 4 mm diameter (Fig. 2). Leaving the vertical pipe, the two-phase mixture flows via a horizontal pipe section into the steam drum, where it separates. The gas is released into the blow-off tank and finally to the atmosphere. The water is pumped again in the test section.

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9

Fig. 3 General test matrix of HZDR for vertical pipe flow with the measured points for the DN50 air-water upward flow series L20 marked in gray

Applying this test section, comprehensive experiments were done including a wide range of combinations of flow rates (represented by the superficial velocities JL and JG). Four different inlet lengths were investigated, 100 (A), 1600 (C), 3100 (D), and 7910 (F/F2) mm, so that the evolution of the flow along the vertical pipe can be analyzed. Thereby the gas injection always was installed at the bottom of the pipe, while one wire-mesh sensor is shifted to the different positions. The corresponding length to diameter ratios L/D are 1.9 (A), 30.6 (C), 59.3 (D), and 151.2 (F/F2), respectively. Figure 3 shows the test matrix for upward air-water flows. For the matrix points with red numbers, the measurements for the largest inlet length were measured twice to allow reproducibility checks. The first measurement is denoted with F, the second with F2. After detailed consistency checks, measurements for some height positions were omitted to include only reliable data into the database. Due to the different position of the wire-mesh sensor, the pressure at sensor position is slightly different. It varies between 0.11 and 0.21 MPa (absolute pressure). During the measurements, the temperature changed in a range from 16
C to 45
C.

The Variable Gas Injection Test Section This test section (Fig. 4) consists of a vertical steel pipe with an inner diameter of 195.3 mm and a length of about 8 m. The measuring plane is located at the upper end of the test section. The device is equipped with six (L12 test series) or seven (K16 test series) gas injection units which allow to inject air or steam via orifices in the pipe wall. The gas injection via wall orifices offers the advantage that the two-phase flow can rise smoothly to the measurement plane, without being influenced by the feeder within the tube at other height positions. The injection devices are arranged almost logarithmically over the pipe length since the flow structure varies quite fast close to the gas injection mainly caused by the radial redistribution of the gas. Six of the gas injection modules consist of three injection chambers each. Two of the three chambers (the uppermost and the lowest) have 72  1 mm orifices. The middle chamber has 32  4 mm orifices, which is used to vary the initial bubble size

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Fig. 4 The test section “Variable Gas Injection” for the L12 experiments and a gas injection module

distribution. For rotation-symmetric gas injection, all orifices per chambers are equally distributed over the circumference of the pipe. For the K16 experiments, an additional injection chamber with 1 mm orifices was installed as close to the measuring plane as possible (40 mm between gas injection and measurement plane of the first wire-mesh sensor in flow direction; L/D ~ 0.2). This was done to provide

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more detailed information on the injected steam bubbles. Only one injection chamber is activated for a single measurement. L/D is increased by using the gas injection chambers from “A” (L12) or “@” (K16) through “R” and “B” through “Q” for 1 and 4 mm injection, respectively. Caused by the implementation of the additional injection chamber, the injection lengths are 83 mm larger in the K16 experiments compared to the data given in Fig. 4 for the L12 experiments.

Procedure and Test Matrix for the L12 Test Series During each experiment, water is fed from the steam drum via the test section pump through the vertical test section (see Fig. 1). For the experiments, water mass flow rates between 1.2 and 48 kg/s were applied. The water temperature was kept constant at 30
C  1 K during the experiments by a special procedure explained in detail by Beyer et al. (2008). More information on the L12 experiments and their results can be found in Lucas et al. (2010a, b). Because of the hydrostatic pressure and friction pressure loss, the pressure changes along the pipe. To reflect the same situation as having a fixed position for the gas injection and shifting the measuring plane, the system pressure was adjusted to guarantee the nominal value of 0.25 MPa exactly at the position of the respectively activated injection device. In contrast to the K16 experiments discussed below, no measurement for the pressure difference between the top of the pipe and the single injection devices was available during the L12 test series. For this reason, the pressure differences were estimated by a procedure described by Beyer et al. (2008). The measured combinations of air and water superficial velocities are shown in Fig. 5. Two test series with constant liquid superficial liquid velocities JL of about 0.4 and 1.0 m/s and two with constant gas superficial velocities JG about 0.01 m/s and 0.2 m/s were done. To investigate the evolution of the flow along the pipe, all levels (A–R) were measured for any points smaller than 149. The maximum possible gas

Fig. 5 General experimental test matrix for vertical pipe flows with marked points measured for the L12 test series

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flow rate, which can be injected through the injection chambers with a diameter of 1 mm, is limited. For this reason, for the points 149, 151, 160, and 162, both injection chambers with 1 mm orifices were operated parallel. For the measurement points 171, 173, 182, and 184, only the 4 mm injections were used. The investigated combinations of flow rates lead to different flow pattern starting from bubbly flow, via churn-turbulent flow up to wispy-annular flow.

Procedure in the K16 Test Series In this experimental series, condensing pipe flows were investigated. To do that, some extensions of the test section “Variable Gas Injection” were done compared to the L12 experiments. The experiments base of experiences obtained in a previous test presented by Lucas and Prasser (2007). The test section pump (see Fig. 1) circulates the saturated water from the steam drum to the lower end of the variable gas injection. In addition cold water is injected through a mixing device at the lower end of the test section. This allows to obtain a sub-cooling of the water of several Kelvin depending on the flow rates. Several temperature measurements were added to check and control the temperature. As in the L12 experiments, the nominal pressure is set at the position of the respectively activated injection chamber. Thus switching between different positions of the injection provides the same conditions like in case of a fixed location of the injection and shifting the measuring plane. This is especially important for the condensation experiments since the saturation temperature and by that also the sub-cooling depends on pressure. To adjust the pressure, the absolute value is measured at the upper end of the test section. In addition the differential pressure between this measurement position and the position of the single gas injection is determined using a newly installed measuring system. After adjusting the pressure for the selected flow rates and for the position of the activated steam injection, the aspired sub-cooling has to be set up. The water temperature is measured by thermocouples, mounted in the saturated and the cold water pipes as well as in the Variable Gas Injection pipe below the injection levels R and O for the mixing temperature. The total water mass flow rate (saturated water from the loop and the additional cold water injection) and the water temperature are adjusted together finally to reach the aspired values. Measurements were done for four different pressure levels in the range from 1 up to 6.5 MPa. Furthermore steam and water flow rates as well as the sub-cooling of the water were varied as indicated in Table 2. PR17 Test Series The “Variable Gas Injection” test section was also used for experiments on evaporating pipe flow. The evaporation was induced by pressure relief, i.e., the gas injection devices itself were not used in these experiments. Two different experimental procedures were applied as illustrated in Fig. 6. In case of the first procedure (left-hand side of Fig. 6), water was circulated with a superficial velocity of about 1 m/s and flows upward through the test section. Before starting the pressure release saturation, conditions are obtained in the steam drum. Since the circulation pump is located at much lower elevation, the pressure exceeds

Experiments on Gas-Liquid Flow in Vertical Pipes Table 2 Test matrix for the K16 experiments

Point

JL [m/s]

118 138 140

1.017 0.405 1.017

118 138 140

1.017 0.405 1.017

118 138 140

1.017 0.405 1.017

096 116 118 138 140

1.017 0.405 1.017 0.405 1.017

13 JG [m/s] 1 MPa 0.219 0.534 0.534 2 MPa 0.219 0.534 0.534 4 MPa 0.219 0.534 0.534 6.5 MPa 0.0898 0.219 0.219 0.534 0.534

ΔT [K] 3.9, 5.0 4.7, 5.3, 6.3, 6.6, 7.2 3.7, 4.8, 5.0, 6.0 3.7, 4.9, 6.0 4.8, 6.6, 8.7 3.2, 5.0, 6.8 2.7, 5.0, 7.2 2.6, 6.6, 12.6 2.6, 5.0, 7.6 2.4, 5.0, 7.6 2.9, 6.0, 9.0 2.5, 5.0, 7.4 2.5, 9.8, 17.2 2.3, 5.0, 8.7

Fig. 6 Schemas of the two experimental procedures for the PR14 experiments (left procedure 1, right procedure 2) (From Mikuz et al. (2015))

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Test 1 MPa

R

time t1

Test 2

Pressure R t1 t2 t3 R t1 t2 t3

Opening level of the blow-off valve

t2

t1

t3

% s

s

s % s

s

s

1

60 21 30 30 50 18 34 30

2

50 18 34 30 40 14 42 30

4

30 11 48 30 25 9 52 30

6,5

25 9 52 30 20 7 56 30

Fig. 7 Test matrix for procedure 1 in the PR17 test series, R is the relative degree of opening of the blow-off valve (From Liao et al. (2013))

the saturation pressure at this position. It is important to maintain sub-cooled conditions at the position of the pump also during the pressure relief to avoid cavitation. This condition limits the speed of the depressurization which can be used in the experiments. Therefore, the blow-off valve which is located at the steam drum was only partially opened. According to the rather small pressure gradients, also the maximum void fraction generated in test section by evaporation is limited. The blow-off valve used in the procedure has relative long opening and closing times. For this reason, the valve was opened and closed according to the ramp shape shown at the left-hand side of Fig. 6. The relative degree of opening at the plateau and the corresponding durations are shown in the test matrix, Fig. 7. In total there are eight tests. Each test was repeated using the same conditions to check the reproducibility. For the second procedure, the facility was equipped with an additional blow-off line which was mounted at the upper end of the test section (right-hand side of Fig. 6). After heating up and before the initiation of the depressurization, the test section was separated from the loop by valves, i.e., the experiments run from stagnant liquid. The new blow-off line is equipped with a fast opening valve allowing an opening ramp as shown at the left-hand side of Fig. 8. An orifice with a diameter of 20 mm was implemented in the blow-off line to limit the speed of depressurization. Much steeper pressure gradients resulting in much larger void fraction are obtained by this procedure. The test matrix is presented in Fig. 8. As in case of procedure 1, the pressure relief was start from 4 different pressure values. Opening times of 10 and 20 s were used. Again each of the eight runs was repeated once again. Both procedures have advantages and disadvantages. The advantage of the first procedure is the relatively large velocity of the fluid at the measuring plane. Previous investigations on the intrusive effect of the wire-mesh sensor have shown that the uncertainties of the measurements increase for small water velocities. Bubbles may be considerably decelerated due to the interaction with the wires for water superficial velocities below 0.2 m/s. For a water superficial velocity of 1 m/s as applied in this procedure, this undesired effect is rather negligible, and reliable data are obtained. For the second procedure, it is expected that the bubbles are also pushed through the

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Test 1

Test 2

Pressure t1 t2 t3 t1 t2 t3 Opening level of the fast-acting blow-off valve

MPa

open

time

closed t1

t2

t3

s

s

s

s

s

s

1

5 10 35 5 20 55

2

5 10 85 5 20 75

4

5 10 85 5 20 75

6,5

5 10 85 5 20 75

Fig. 8 Test matrix for procedure 2 in the PR17 test series (From Mikuz et al. (2015))

sensor due to the boiling up, but it is rather difficult to quantify a possible interaction between the sensor wires and the bubbles. Another disadvantage of the second procedure is caused by the fact the valves which separate the test section from the loop (Fig. 6, right) have relatively long closing times. Due to heat losses during the waiting time before the pressure relief is started, a slight sub-cooling will be obtained in the test section. Nevertheless the second procedure has the advantage that stepper pressure gradients can be realized leading to higher void fractions.

Results For all the experiments described above, well-documented data on boundary conditions and on information obtained from the wire-mesh sensor measurements are available. It is a comprehensive database containing detailed information on the phase distributions in vertical pipe flows under various conditions. Examples for derived quantitative data are radial gas volume fraction, interfacial area concentration, and gas velocity profiles as well as bubble size distributions. This database is frequently used for CFD model development and validation (e.g., Kaji et al. 2009; Duan et al. 2011; Qi et al. 2012). The single documentations on the experiments also discuss uncertainties and include plausibility checks of the measured data. Some considerations on uncertainties specifically for the L12 experiments are discussed by Beyer et al. (2010). In frame this handbook these findings will not be repeated. Instead some more basic observations from the experiments are discussed in this section.

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Fig. 9 Examples for different flow pattern observed in the L20 experiments at L/D = 151 (fully developed flow)

121

040

085

118

129

140

215

Table 3 Assignment of flow pattern in Fig. 9 Test 121 040 085 118 129 140 215

JL [m/s] 4.047 0.641 1.017 1.017 1.017 1.017 0.405

JG [m/s] 0.2190 0.0096 0.0574 0.2190 0.3420 0.5340 12.14

Flow pattern Finely dispersed Bubbly flow with wall peak Bubbly flow with wall and center peak Bubbly flow with center peak Bubbly to slug flow transition Slug flow Annular flow

Flow Structure Flow pattern maps for vertical pipe flow were introduced long time ago, e.g., by Taitel et al. (1980). Other authors published similar maps using different criteria or introducing sub-pattern. The wire-mesh sensor data can be visualized, e.g., by plotting the gas distribution in a plane along the pipe diameter or by a virtual side view generated by a special ray tracing algorithm developed by Manera et al. (2005). This allows to distinguish flow pattern subjectively. Figure 9 shows an example of such views, and Table 3 presents the assignment to different flow pattern for the 52.3 mm pipe. Figure 9 demonstrates the dependency of the flow pattern on the flow rates, but in general there is also a dependency on the pipe size and on the inlet length. Figure 10 compares flow structures for the 52.3 mm pipe and the 195.3 mm pipe for the same flow rates.

Experiments on Gas-Liquid Flow in Vertical Pipes

0.037 0.057 0.090 0.14

0.22

0.34

0.53

0.84

1.3

17

0.037 0.057 0.090 0.14

0.22

0.34

0.53

J air [m/s]

J air [m/s]

Dpipe = 52.3 mm

Dpipe = 195.3 mm

0.83

1.3

Fig. 10 Dependency of air-water flow structures for JL = 1.017 m/s and increasing air flow rates on the diameter of the pipe

The main difference is the shape of the large gas structures. While in the 52.3 mm pipe slightly distorted Taylor bubbles are observed, rather irregular large gas structures can be found in the 195.3 mm pipe. This is in accordance with literature. Because of the decreasing confining effect of the pipe with increasing pipe diameter, Taylor bubbles become more and more unregularly finally forming highly distorted gas structures which characterize the churn-turbulent flow regime. However for both pipe sizes, bimodal bubble size distributions can be found for the same combination of flow rates as shown in Fig. 11. The peak representing the large bubbles is shifted to larger bubble sizes and is also broader for the larger pipe. This effect was discussed in detail by Prasser et al. (2005b, 2007). The wire-mesh sensor technology allows to extract single large gas structures from a churn-turbulent flow as shown in the same paper. Such extracted gas structures can be understood as a strongly distorted Taylor bubble. For this reason, there is some similarity between slug and churn-turbulent flows with the most important difference of more regular shapes in smaller pipes and more distorted shapes in larger ones. Also, it can be concluded from these experiments and measurements done for other pipe diameters that there is a smooth transition between slug flow and churn-turbulent flow in dependence on the pipe diameter.

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Fig. 11 Bubble size distributions for the same flow rates (JL = 1.017 m/s and JG = 0.53 m/s) but different pipe diameter

The quantitative data obtained by the wire-mesh sensors allow to define more objective criteria to define flow pattern as introduced, e.g., by Lucas et al. (2005) for experiments done in a 51.6 mm pipe. Here the transition between bubbly flow and slug flow was defined basing on the sphere equivalent diameter of the largest bubbles. If it exceeds the pipe diameter, the flow was assigned to slug flow, otherwise to bubbly flow. This criterion may hold for pipes with a diameter in the range up to 50 mm. In case of larger pipes, the flow characteristics are typical closer to churn turbulent if bubbles larger 50 mm sphere equivalent diameter occur. For this reason, a more generalized criterion for the transition between bubbly and slug or churnturbulent flow can be formulated. The flow can be assigned to bubbly flow if db < min (Dpipe, 50 mm). For annular flow, only small liquid drops – which cannot be registered by the wire-mesh sensor – should occur in the pipe center. For the L12 experiments, some water bridges are still observed for the largest gas superficial velocity of 3.185 m/s. These measuring points still lie in the transition between churn-turbulent and annular flow. Sometimes this region is also called wispy-annular flow. In this case, some liquid wisps are overserved to detach from the liquid film as shown in Fig. 12. Here also some larger drops are visible in the core region of the pipe. For the bubbly flow region, in addition a distinction between flows with wall and center peaks of the radial gas volume fraction profiles can be used to characterize the flow structure. The nature of these profiles is discussed in detail in section “Separation of Small and Large Bibbles.” Also for some flow rates, bubbly flows in the transition between these two situations occur, showing a wall peak as well as a center peak of these profiles.

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Fig. 12 Instantaneous liquid distribution in the cross section of the 195.3 mm pipe, JL = 0.405, JG = 3.185, L/D = 39.9

Using the criteria introduced above, flow pattern maps can be established for the L20 and L12 experiments. Such maps are shown in Fig. 13. While the general tendencies are rather similar for both pipe configurations, some differences can be seen for the positions of the transition lines. Also, for some conditions, transitions from bubbly flow to churn-turbulent flow caused by bubble coalescence can be found for the L12 experiments. Another interesting aspect is the influence of the initial bubble size distribution on the flow characteristics. As explained before, the L12 measurements were done for the same combinations of flow rates but injecting the gas through holes with different diameter. This results in different initial bubble size distributions. The influence on bubble size distributions and correspondingly to other characteristics like radial gas volume fraction and gas velocity profiles is still visible for the largest L/D measured in case of low gas volume rates. Starting from point 072 (JG = 0.0368 m/s) for JL = 0.405 m/s and point 107 (JG = 0.14 m/s) for JL = 1.017 m/s, the distributions and profiles are the same under consideration of the experimental uncertainties.

Separation of Small and Large Bubbles The distribution of the bubbles over the pipe cross section can be characterized by radial gas volume fraction profiles because of the symmetry of the flow, which was checked for all experiments. For bubbly flows, the maximum of this distribution can be located in the near wall region or in the center, while it is always in the pipe center for all other flow regimes (Fig. 14). These profiles are established by the

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Fig. 13 Flow pattern maps for the L20 experiments (upper figure) and L12 experiments (lower figure)

action of the so-called non-drag forces (Lucas et al. 2001, 2007, chapter “▶ EulerEuler-Modelling of Poly-dispersed Bubbly Flows”), which act perpendicular to the main flow direction. The lateral lift force acts in an upward vertical pipe flow to push the small bubbles toward the pipe wall. However it strongly depends on the bubble size as found experimentally by Tomiyama et al. (2002) and in Direct Numerical Simulation (DNS), e.g., by Ervin and Tryggvason (1997) and Bothe et al. (2007). In case of deformed bubbles, the lateral lift force even changes its sign. According to the correlation obtained by Tomiyama et al. (2002), this change of sign of the lift force occurs for air-water flows at ambient conditions for a sphere equivalent bubble diameter of about 5.8 mm. Since the bubble deformation depends on the relation between buoyancy and surface tension reflected by the Eötvös number, it also depends on the material properties and decreases for steam-water flows at

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Fig. 14 Examples for radial gas volume fraction distributions for developed flow; left, L20 (inner diameter 52.3 mm); right, L12 (inner diameter 195.3); (a) bubbly flow with wall peak, (b) bubbly flow with transition between wall peak and center peak, (c) slug or churn-turbulent flow, (d) transition to annular flow

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6.5–3.5 mm sphere equivalent diameter. Due to the change of sign, larger bubbles can be found in upward flows preferred in the pipe center, i.e., in general there is a clear separation of small and large bubbles in pipe flow. This separation can be clearly shown by the wire-mesh sensor data as discussed by Prasser et al. (2002). Since the bubble sizes are known, the radial volume fraction profiles can be decomposed according to the bubble size. Figure 15 shows examples for such decomposition. The separation of small and large bubbles can clearly be seen. The correlation from Tomiyama et al. (2002) was obtained for single bubbles in a highly viscous linear laminar shear flow. Surprisingly the correlation fits at least regarding the change of the sign of the lift force very well with the observed separation of small and large bubbles in the TOPFLOW experiments as discussed in detail by Lucas and Tomiyama (2011). Here the flow is clearly turbulent, there are cases with high void fraction, and the material parameters are clearly different. The agreements hold not only for air-water flows in the L20 and L12 experiments but also in the experiments for steam-water flows K16 under different pressure levels as shown in Fig. 15. It also holds for the refrigerant dichlorodifluoromethane (R12) as shown by Krepper et al. (2013). For the CFD modeling of polydisperse bubbly flows, it is essential to reflect this separation of small and large bubbles as soon as the bubble sizes are in the range of the change of the sign of the lift force since the local distribution may have a strong influence on the evolution of the flow as discussed by Krepper et al. (2005, 2008), Frank et al. (2008), and Lucas et al. (2011). Accordingly the knowledge of bubble size distributions is important. The wire-mesh sensor technology allows the measurement of such bubble size distributions as discussed above. For fully developed air-water flows, a mono-modal distribution is observed for flow with a maximum gas volume fraction of about 10%. For larger gas volume fractions, the bubble size distribution is usually bi-model with one peak in the range of 6–10 mm volume equivalent bubble diameter which sharply decreases (especially for the smaller diameter pipe) and a second peak which represents bubbles stabilized by the stabilizing effect of the pipe as shown in Fig. 11. More examples for bubble size distributions will be given in the next sections.

Evolution of the Air-Water Flows Beside the fully developed flow, the experiments discussed here provide valuable data on the evolution of the flow. In the L12 experiments, gas is injected via orifices in the pipe wall. For this reason close to the injection (i.e., at small L/D), the radial gas volume fraction profile always shows a wall peak. With increasing distance from the injection, there is a redistribution of the bubbles depending on flow rates and bubble sizes as shown in Fig. 16. The letters in the legend refer to the different distances between gas injection and measuring plane according to Fig. 4. In both cases presented in this figure, the bubbles start to distribute over the cross section of the pipe. For the case with the lower gas flow, a wall peak is maintained for all

Fig. 15 Decomposed radial volume fraction profiles: upper line left, L20 experiments, JL = 1.611 m/s; JG = 0.219 m/s, L/D = 151.2; right, L12, JL = 1.017 m/s; JG = 0.219 m/s, L/D = 39.9, injection via 1 mm orifices; lower line left, K16, 2 MPa, JL = 1.017 m/s; JG = 0.219 m/s, L/D = 39.9, injection via 1 mm orifices; right, K16, 6.5 MPa, JL = 1.017 m/s; JG = 0.0898 m/s, L/D = 39.7, injection via 4 mm orifices

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Fig. 16 Evolution of the radial gas volume fraction profile for the L12 experiments DOrifice = 1 mm; top, JL = 1.017 m/s; JG = 0.0062 m/s; bottom, JL = 1.017 m/s; JG = 0.219 m/s

distances, while a core peak of the gas volume fraction is established for the larger gas flow rate. This again is caused by the action of the lateral lift force, and the effects fit well with the evolution of the corresponding bubble size distributions shown in Fig. 17. For the low gas flow rate, there is a slight increase of the bubble sizes along the pipe which is mainly caused by the expansion caused by the decreasing pressure with height position. However, most bubbles remain below 5.8 mm sphere equivalent diameter for all positions resulting in the wall peak. Since the fraction of larger

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Fig. 17 Evolution of the bubble size distributions for the L12 experiments DOrifice = 1 mm; top, JL = 1.017 m/s; JG = 0.0062 m/s; bottom, JL = 1.017 m/s; JG = 0.219 m/s

bubbles increases, the wall peak becomes smaller, and the gas volume fraction in the core region increases with the distance from the gas injection. For the high gas flow rate, the bigger part of the gas is transported by bubbles larger than 5.8 mm, and thus most of these bubbles migrate toward the pipe center. The evolution of the bubble size distributions shows an increase of the small bubble peak caused by breakup processes, while the large bubbles further increase by coalescence. The data are used for the development and validation of improved models for bubble coalescence and breakup; see, e.g., Liao et al. (2011, 2015).

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Fig. 18 Evolution of the radial gas velocity profile for the L12 experiments DOrifice = 1 mm; top, JL = 1.017 m/s; JG = 0.0062 m/s; bottom, JL = 1.017 m/s; JG = 0.219 m/s

The profiles of the gas velocity shown in Fig. 18 are mainly influenced by the gas volume fraction profiles. The wall peak leads to a flattening of the liquid velocity profile and in the result also of the gas velocity profile. Accordingly with the decreasing height of the wall peak, the velocity profiles become more and more pronounced along the pipe for the case with lower gas flow rate. Because of the high gas volume fraction in the case with higher gas flow rate, the characteristics of the gas velocity profile follow the gas volume fraction profile. In the region above the injection, the gas is still close to the wall that also leads to a maximum gas (and

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liquid) velocity in the wall region. Along the pipe, there is a transition to a core peaked velocity profile in accordance with the gas volume fraction profiles. It is worth to mention that all the measured data are highly consistent with each other. Even for the small distances between the injection chambers with 1 mm orifices from one injection device (see Fig. 4), the evolution of the flow always show a clear trend in all the measurements (solid and dashed curves in the figures above).

Interfacial Area Density (IAD) The IAD is an important parameter in two-phase flows with mass transfer. Condensation and evaporation rates in dynamic flows or heterogeneous chemical reactions are proportional to this parameter. As mentioned before, the interfacial area is determined by a special algorithm for each single bubble. In the following, only some general trends will be discussed for the example of the L12 experiments. To get an overview on the dependency of the IAD ai on flow rates, data for developed flow, i.e., for largest length-to-diameter ratio (L/D) which is about 40, are presented first in Fig. 19. The left-hand side shows the results for the two lines of the experimental matrix (see Fig. 5) with constant liquid superficial velocities JL of about 0.4 m/s and 1.0 m/s and increasing gas superficial velocity JG, while the righthand side shows the two lines with constant JG of about 0.01 m/s and 0.22 m/s and increasing JL. It can be seen at the upper left picture that the IAD first increases with JG (in the regions of bubbly flow and the transition to churn-turbulent flow) then reaches a maximum in the churn-turbulent flow regime and starts to decrease as soon as the transition to wispy-annular and annular flow regimes plays a role. For JG = 0.01 m/s, the IAD deceases with increasing JL but remains almost constant for JG = 0.22 m/s which is mainly in the churn-turbulent flow regime. This general behavior corresponds to the expectations since in the bubbly flow regime, more and more bubbles occur with increasing JG, while for churn-turbulent flow, more and more large gas structures are formed which contribute only little to the gas-liquid interfacial area. Finally in an annular flow, the gas-liquid interface is small. For bubbly flow with monodispersed spherical bubbles, the IAD is related to the gas volume fraction ε and the bubble diameter db according to ai ð d b Þ ¼

6e , db

(1)

i.e., for such a flow, the IAD should be proportional to the void fraction. In case of monodisperse spherical bubbles with 5 mm diameter, the IAD to a ratio would be 12 (%*m)1. In the middle row of Fig. 19, the IAD related to the void fraction is presented. The graphs clearly shows two different parts – an almost constant part with values between 11 and 14 (%*m)1 which is characteristic for bubbly flows and a decreasing part for churn-turbulent flows. For the churn-turbulent flows, the increase of IAD is less than that of ε.

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Fig. 19 IAD (top), IAD related to void fraction (middle), and relation between real IAD and the values obtained from the bubble size distribution assuming spherical bubbles (bottom). Left, experiments with constant JL; right, experiments with constant JG. Dashed lines (middle pictures) present the IAD related to the void fraction under the assumption of spherical bubbles

The graphs for the two different injection orifices collapse to one graph starting from about JG = 0.04 m/s for JL = 1 m/s and from about JG = 0.01 m/s for JL = 0.4 m/s. Below these gas flow rates, there is still a dependency of the IAD-to-ε ratio on the injection. The smaller bubbles in case of the 1 mm injection lead to higher values of this ratio compared to the 4 mm injection. For constant JG shown at the right picture in the middle row of Fig. 19, the trends that the ratio is in the range between 11 and 14 for the bubbly flow region but clearly smaller for the churn-turbulent flow regime are confirmed.

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Fig. 20 IAD (top), IAD related to void fraction (middle), and relation between real IAD and the values obtained from the bubble size distribution assuming spherical bubbles (bottom). Selected matrix points. Left, JL = 0.4 m/s; right, JL = 1.0 m/s

The pictures in the middle row also show the IAD-to-ε ratio obtained from the bubble size distributions assuming spherical bubbles (dashed lines). They show quite similar trends but are smaller for all cases. This has to be expected since spheres represent the smallest possible surface. To quantify the deviation from the assumption of spherical bubbles, the ratio between IAD obtained with the new algorithm and the one obtained from the bubble size distribution assuming spherical bubbles is shown at the bottom of Fig. 19. For bubbly flow, this ratio lies between

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A(304 mm) D (577 mm) G (1521 mm) J (2564 mm) M (4500mm) B (361 mm) E (634 mm) H (1578 mm) K (2621 mm) N (4557 mm)

Fig. 21 Virtual side projects (left columns) and central cuts (right columns) for different distances from the steam injection. JL = 1.017 m/s, JG = 0.534 m/s, 1 MPa, sub-cooling 3.7 K; left, injection trough 1 mm orifices; right, injection trough 4 mm orifices

1 and 1.5, while it becomes much larger for churn-turbulent and (wispy-)annular flow. The evolution of the IAD along the pipe is presented in Fig. 20 for the same parameter as discussed above for the matrix points 019, 074, 107, 129, and 184 (left column) for JL = 1 m/s and the corresponding points (017, 072, 105, 127, and 182) for the JL = 0.4 m/s line. Only for the case with the lowest JG shown in these figures, a decrease of the IAD-to-ε ratio is observed in the injection region. This corresponds to the measured bubble size distribution which only shows for this case a dominance of bubble coalescence. For the other cases, because of the injection of relatively large bubbles, the flow behind the gas injection is dominated by bubble breakup related to an increase of the IAD-to-ε ratio. As the lower pictures in Fig. 20 demonstrate, the bubbles are highly deformed in the vicinity of the injection. This deformation decreases along the pipe due to bubble breakup. Also the radial profiles of the IAD show very similar shapes like the gas volume fraction profiles. The IAD-to-ε ratio always shows a wall peak for all cases of developed flow (L/D = 40). This is due to the decreasing averaged bubble size from the pipe center toward the wall.

Steam Bubble Condensation in Sub-cooled Liquid A detailed discussion on the results of the K16 experiments is given by Lucas et al. (2010c, 2013). Beside the variations listed in the test matrix, Table 2, the 1 mm and 4 mm orifices were used for the injection. Here again the 4 mm orifices lead to larger bubbles, i.e., to a lower interfacial area density comparing both injections for the same flow rates, sub-cooling, and pressure. As an example, Fig. 21 compares the evolution of the

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Fig. 22 Evolution of the cross section averaged bubble size distributions. Case 118 (JL = 1,017 m/s, JG = 0,219 m/s), 2 MPa, sub-cooling 3.7 K, top 1 mm orifices, bottom 4 mm orifices

flow along the pipe for case 140 at 1 MPa pressure and a sub-cooling of 3.7 K qualitatively.

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Fig. 23 Evolution of the time and cross section averaged gas volume fraction along the pipe. Case 118 (JL = 1,017 m/s, JG = 0,219 m/s), 4 MPa

For both cases, the experimental conditions are exactly the same, but the size and number of injection orifices are different. This is also true for the quantitative results shown in Fig. 22. Clearly the bubbles are much larger in case of the 4 mm orifices. Figure 23 shows the evolution of the time and cross section averaged gas volume fraction for three different values of the initial sub-cooling. Of course the condensation rate increases with the sub-cooling leading to a faster decrease of the steam volume fraction. In addition, the abovementioned effect of the lower IAD in case of the 4 mm injection clearly leads to lower condensation rates resulting in a slower decrease compared to the 1 mm injection for the same boundary conditions. For the case with the lowest sub-cooling saturation, conditions are achieved at about one half of the pipe length. For this reason, condensation stops, and a slight reevaporation can be observed. This effect is more pronounced in the case shown in Fig. 24. On the one hand, the condensation of the injected steam heats up the water, and on the other hand, the pressure and accordingly also the saturation temperature decrease along the pipe. Again the condensation rate is higher in case of the 1 mm injection for low L/D. In both cases, a minimum gas volume fraction of about 15% is reached. After that there is an increase of the gas volume fraction caused by reevaporation. The temperature measurement obtained by one lance of thermocouples which was mounted directly above the wire-mesh sensor and which spans over the whole pipe diameter confirms these phenomena, in principle. The saturation temperature was calculated from the pressure measured at the sensor position. First the water temperature increases, but starting from about L/D = 12, the temperature again decreases in accordance to the saturation temperature. The interfacial area has an important influence as long a clear sub-cooling occurs. It is important to mention that the water temperature should be slightly larger than the saturation temperature. That fact that this is not true for the example shown in Fig. 24 can be attributed to the uncertainties of the temperature measurement. As soon as the saturation temperature is reached, the fluid is in thermal

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Fig. 24 Evolution of the time and cross section averaged void fraction (top) and averaged water temperature (bottom) along the pipe. Case 140 (JL = 1.017 m/s, JG = 0.534 m/s), 2 MPa and 3.2 K sub-cooling

equilibrium, and the interfacial area density should only determine the amount of small overheat mentioned before. Figure 25 demonstrates the influence of pressure on the condensation rate. Because of the increase of the steam density with pressure, condensation along all the pipe only occurs at 1.0 MPa, while reevaporation occurs for higher pressure. An example for the evolution of the radial gas volume fraction profiles and gas velocity profiles is shown in Fig. 26. For small L/D, the gas velocity (and

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Fig. 25 Evolution of the time and cross section averaged void fraction (left) and averaged water temperature (right) along the pipe. Case 140 (JL = 1.017 m/s, JG = 0.534 m/s), 5.0 K sub-cooling

accordingly the liquid velocity) is larger in the near wall region compared to the pipe center. As already discussed for the L12 experiments, this is caused by the increase of the vertical component of the liquid velocity caused by the injected gas. With increasing L/D, the bubbles start to migrate toward the pipe center but are condensing at the same time. The velocity profiles develop a center-peaked profile. The data were used for the validation of CFD models for condensing polydisperse bubble flows; see, e.g., Krepper et al. (2011) and Liao et al. (2014).

Transient Data from Pressure Relief Experiments The time-plots of the pressure clearly differ for the two procedures used in the pressure relief experiments (see section “PR17 Test Series”). Due to the slow opening of the blow-off valve in case of procedure 1, the pressure transients are rather smooth (Fig. 27, left). For procedure 2 (Fig. 27, right), a sharp decrease of pressure occurs immediately after the opening of the valve. This leads to a considerable sub-cooling of the liquid followed by a strong evaporation process. During a short period, the volume of steam generated by evaporation exceeds to discharge volume leading to an increase of pressure. After closing the valve, an increase of pressure is observed which is more pronounced at the 1 MPa experiment compared to the 6.5 MPa experiment. It is caused by the heat input from the walls. In the following, the experiment for procedure 1 at 6.5 MPa and 25% opening of the blow-off valve is discussed more in detail. The opening of the blow-off valve starts at 2 s, i.e., according to the numbers given in Fig. 7, the valve is opened to the

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Fig. 26 Evolution of the radial gas volume fraction profile (top) and the radial gas velocity profile (bottom). Case 140 (JL = 1.017 m/s, JG = 0.534 m/s), 4 MPa, 7.6 K sub-cooling, injection via 4 mm orifices

desired value (25%) at 11 s and closing starts at 63 s. The valve is closed completely at 72 s. Figure 28 shows the cross section averaged void fraction obtained from the wiremesh sensor measurement as function of time. The delay of the increase and decrease of the void fraction compared to the opening ramp of the valve results from the delay of the evaporation process but mainly from the fact that the measuring plane is

Fig. 27 Time-dependent pressure for the two procedures; left, procedure 1; right, procedure 2; top, 1 MPa; bottom, 6.5 MPa. The numbers on brackets indicate the first and second realization of the test

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Fig. 28 Cross section averaged void fraction for procedure 1, 6.5 MPa and 25% opening of the blow-off valve

located at the upper end of the pipe. The steam which is produced along the pipe needs some time to travel to the sensor. This fact is also reflected in the evolution of the bubble size distribution with time as shown in Fig. 29. With the increase of the averaged void fraction, also the bubble sizes increase (Fig. 29, top, left). During the period in which a plateau of the averaged void fraction is observed, also the bubble size distributions remain almost unchanged (Fig. 29, top, right). The bubble sizes decrease with the averaged void fraction after closing the blow-off valve (Fig. 29, bottom). The boiling up during the pressure relief process is also reflected in the radial profiles of the gas velocity which are shown in Fig. 30. Because the first bubbles are generated at the pipe wall (see Fig. 31, top, left) in the first seconds after the start of the blowdown, the velocity increases first only in the near wall region (up to 18 s in Fig. 30, left). Later on the maximum of the gas volume fraction shifts away from the pipe wall and forms intermediate peaks (Fig. 31, top line – 23–69 s) followed by a center peak from 71 to 83 s. The velocity profiles have their maxima in the pipe center for t > 29 s. They are again flattened with the decrease of the boiling process after closing of the blow-off valve. Starting from about 90 s, bubbles are observed only in the near wall region (Fig. 31, bottom, right). Compared to steady-state experiments for air-water flows and condensing steam-water flow as discussed in the previous sections in which the radial profiles and bubble size distributions were obtained from an averaging over the whole measuring time (10 s), here the statistics of the data are not so good. The averaging over period of only 2 s leads to some fluctuations of the profiles as it can be seen in Fig. 31. The fluctuations are more pronounced in the core region of the pipe compared to the wall region due to the

Fig. 29 Bubble size distributions during the transient, procedure 1, 6.5 MPa, 25% opening of the blow-off valve

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Fig. 30 Radial profiles of the gas velocity during the transient, procedure 1, 6.5 MPa, 25% opening of the blow-off valve

occurrence of few large bubbles and the lower statistics in central positions compared to larger radii caused by the equal width of the radial rings (see section “WireMesh Sensors and Data Evaluation Procedures”). During the period of the plateau of the averaged void fraction in Fig. 28, almost stationary conditions are observed in respect to bubble size distributions (Fig. 29, top, right), radial profiles of gas velocity (Fig. 30, right), and radial gas volume fraction profiles (Fig. 31, top, right). For this reason, it should be justified to do a

Fig. 31 Radial profiles of the gas volume fraction during the transient, procedure 1, 6.5 MPa, 25% opening of the blow-off valve

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Fig. 32 Radial volume fraction profiles decomposed according to the bubble size and averaged over the gas volume fraction plateau (31–61 s) , procedure 1, 6.5 MPa, 25% opening of the blow-off valve

Fig. 33 Volume averaged void fraction for different pipe sections for procedure 1, 6.5 MPa and 25% opening of the blow-off valve. The sections are characterized by length to diameter ratio (L/D) measured from the axial position of the wire-mesh sensor

time averaging over this period in order to improve the statistics. This is especially important for the radial volume fraction profiles decomposed according to the bubbles size. Such profiles are presented in Fig. 32. Obviously the peak at half of the pipe radius in the total gas volume fraction profiles shown in Fig. 31, top right, is

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caused mainly by bubbles larger than 10 mm sphere equivalent bubble diameter. Smaller bubbles are more or less equally distributed over the pipe cross section. The measured data for the pressure difference recorded for several height positions were used to obtain some information on the axial void distribution along the pipe. Results are shown in Fig. 33. Considerable uncertainties arise for these measurements from the fluctuations in signals of the measured pressure differences and from the correlations used for two-phase pressure drop due to friction and acceleration. Compared to the averaged void fraction measured by the wire-mesh sensor (see Fig. 28), some lower values are obtained for the topmost section. In contrast to experiments done at 1 MPa in case of the 6.5 MPa experiment, steam is also observed at the lowest section between the measurement positions for the pressure difference, as shown in Fig. 33. This observation agrees with the temperature measurement at the lower end of the test section. Before the opening of the valve, the water temperature is slightly below the saturation temperature which corresponds to the pressure measured at this position. After the start of the blow-off, the measured temperature and saturation temperature agree quite well. The data that were from this experimental series were used for the validation of one-dimensional system codes (Mikuz et al. 2015) as well as for the validation of CFD codes (Liao et al. 2013).

Conclusion This chapter presents experimental setups on gas-liquid vertical pipe flow, explains which quantitative data can be obtained using the wire-mesh sensor technology, and discusses observations regarding two-phase flow characteristics obtained in these experiments. Flows without phase transfer and in condensing and evaporating flows are considered. Results from two different pipe sizes are discussed, and the experimental boundary conditions were varied over a wide range to obtain different flow pattern. The latter is not only dependent on the gas and liquid volume flow rates but also on pipe diameter and often initial bubble size distribution which depends on the gas injection device. The reason lies in the fact that the evolution of bubbly flows is strongly influenced by the bubble size distribution. It is essential to consider this fact in a proper simulation of such flows. There is a complex interaction between local gas-liquid phase fractions and the local bubble size distribution which changes caused by bubble coalescence and break. The situation becomes even more complex for flows with phase transfer. The interfacial area density was shown to have an important influence on the dynamics of the condensation of steam bubbles injected into sub-cooled water. The evolution of such flows along the pipe also depends on flow rates, pressure, and sub-cooling. Evaporation was initiated in the presented experiments by pressure relief. Transient boiling processes were investigated by limited valve opening times. In case of flows with phase transfer, the bubble size distribution changes because of phase transfer in addition to the complex coalescence and breakup processes.

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Beside qualitative insights in gas-liquid flows, a comprehensive database for CFD code development and validation is obtained. In this book chapter, wire-mesh sensor measurements were considered. It was shown that it is an innovative technology which is able to provide valuable insights into two-phase flows. However it is an intrusive measuring technique, i.e., it influences the flow. For this reason, the range of applicability is limited. For the cases discussed here, always a basic liquid flow with a superficial velocity of at least 0.2 m/s was present. This limits the influence of the interaction between the wires and bubbles on the results of the measurements because the bubbles are pushed through the sensor by some momentum. Detailed discussion on accuracy for such flows can be found in the cited references giving more details on the experiments. For noninvasive measurements, radiation-based methods can be applied. Recently the ultrafast electron-beam X-ray tomography was developed at HZDR (see chapter “▶ Imaging Measuring Techniques”). Presently is used to obtain similar data as discussed in this chapter for a wider range of flow conditions including counter-current air-water vertical pipe flow and cocurrent downward air-water and steam-water pipe flows (Banowski et al. 2015). Since there is no interaction between the measurement and the flow, bubbles measured in the upstream measuring plane can be found with a high probability also in the downstream measuring plane. This allows to determine velocities of single bubbles. There is a high potential for getting new insights into the flow characteristics and getting better quantitative data from these experiments. Acknowledgments This work is carried out in the frame of a research project funded by the German Federal Ministry of Economic Affairs and Energy, project numbers 150 1329 and 150 1411. The authors like to thank especially Prof. Dr. Horst Michael Prasser who planned the experiments discussed here and developed the wire-mesh sensor technology as well as all members of the TOPFLOW team who contributed to the successful performance of these experiments.

References M. Banowski, D. Lucas, L. Szalinski, A new algorithm for segmentation of ultrafast X-ray tomographed gas-liquid flows. Int. J. Therm. Sci. 90, 311–322 (2015) M. Beyer, D. Lucas, J. Kussin, P. Schütz, Air-water experiments in a vertical DN200-pipe. Report Forschungszentrum Dresden-Rossendorf (2008), http://www.hzdr.de/publications/PublDoc5374.pdf M. Beyer, D. Lucas, J. Kussin, Quality check of wire-mesh sensor measurements in a vertical air/water flow. Flow Meas. Instrum. 21, 511–520 (2010) D. Bothe, M. Schmidtke, H.-J. Warnecke, Direct numerical computation of the lift force acting on single bubbles. 6th International Conference on Multiphase Flow, ICMF 2007, 09.-13.07.2007, Leipzig, Germany (2007) X.Y. Duan, S.C.P. Cheung, G.H. Yeoh, J.Y. Tu, E. Krepper, D. Lucas, Gas–liquid flows in medium and large vertical pipes. Chem. Eng. Sci. 66, 872–883 (2011) E.A. Ervin, G. Tryggvason, The rise of bubbles in a vertical shear flow. J. Fluids Eng. 119, 443–449 (1997) T. Frank, P. Zwart, E. Krepper, H.-M. Prasser, D. Lucas, Validation of CFD models for mono- and polydisperse air-water two-phase flows in pipes. Nucl. Eng. Design 238, 647–659 (2008)

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Multiphase Flows in Biomedical Applications Jingliang Dong, Kiao Inthavong, and Jiyuan Tu

Abstract

A sound understanding of the physics of multiphase flow is important for studying biofluid dynamics in human, and the air flow in the respiratory system and blood flow in the cardiovascular system remain two of the most important fields. This chapter presents case studies covering 3D model reconstruction, gasparticle modelling in the human respiratory system and liquid-particle flow modelling in the human arterial capillary by reviewing the current multiphase modelling techniques and its challenges. The potential benefits of using computational fluid dynamics (CFD) in human biofluids modelling are demonstrated. Keywords

Modelling • Gas particle flow • Blood flow • Nasal cavity • Capillary • Biofluid • Inhalation exposure • Drug delivery • Red blood cells

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiphase Flow in Human Respiratory System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiphase Flow in Human Cardiovascular System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Study: Gas-Particle Flow in Human Nasal Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nasal Cavity Reconstruction and Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inhalation and Particle Deposition Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inhalation Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Deposition in the Human Nasal Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Study: Nasal Spray Efficiency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inhalation Airflow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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J. Dong (*) • K. Inthavong • J. Tu School of Engineering, RMIT University, Bundoora, VIC, Australia e-mail: [email protected]; [email protected]; [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_16-1

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Spray Trajectory Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Deposition Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Study: Modelling of Blood Flow in Capillary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CFD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Field and Mean Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction Airflow in the respiratory system and blood flow in the cardiovascular system remain two of the most important fields within biomedical engineering. As these flows constitute biofluids either in liquid or gaseous phase, fluid dynamics principles can solve the biofluid mechanics to support the diagnosis and decision-making for treatment of clinical pathologies. Exploring the relationship between flow phenomena and pathophysiological observations is enhanced by continual advances in imaging modalities, measurement techniques, and computational modelling capabilities (Siebes and Ventikos 2010). In particular, computational fluid dynamics (CFD) is a compelling, nonintrusive, virtual modelling technique with powerful visualization capabilities that enables engineers to solve complex airflow in realistic airways (Inthavong et al. 2006, 2008; Wen et al. 2008; Shang et al. 2015) or blood flow in arterial vessels (Antiga et al. 2008; Dong et al. 2013, 2015). This chapter presents case studies covering gas-particle modelling in the human respiratory system and liquid-particle flow modelling in the human arterial capillary by reviewing the current multiphase modelling techniques and its challenges. This chapter can serve as an illustration of the necessary modelling framework in numerical applications. Combined with experimental measurements and clinical observations, multiphase CFD modelling can address unmet clinical needs, predominantly in the direction of enhanced diagnosis, as well as assessment and prediction of treatment outcomes.

Multiphase Flow in Human Respiratory System Based on function, the respiratory system can be divided into the conducting zone (nose to bronchioles) and respiratory zone (alveolar duct to alveoli) shown in Fig. 1. Alternatively, it can be divided based on anatomy leading to the upper (all structures before larynx) and lower respiratory tract. As the first passageway for air entering the body, the nasal cavity acts a critical role in the respiratory system, such as air conditioning, hazardous airborne particles filtration, and the sense of olfaction. The multiphase flow in this field is classified as gas-particle flows occurring during inhalation. Dust particles, pollen, and spray droplets that exist in the micron size range can be treated as discrete particles, which are generally visible and tend to deposit on respiratory surfaces by inertial impaction (Kelly et al. 2005; Frank et al.

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Fig. 1 Schematic of the respiratory system divided by anatomy into the upper and lower respiratory tract

2012). Finer particles such as fumes, gases, and smoke that exist in the submicron range are less visible exhibiting low Stokes numbers, and its deposition is considered to diffusion dominant (Cheng et al. 1990; Kreyling et al. 2006). Occupational hazards associated with exposure to micron and submicron particles are widely recognized in work environments (Yokel and MacPhail 2011). For example, workers exposed to manganese-containing welding fumes may face hazardous risks associated with translocation of ultrafine manganese oxide particles (5 μm), particle inertia dominates its motion. As a result, particles hit the nasal vestibule wall immediately after being released from the spray device. In contrast, when particle size reduces to 5 μm, the influence of particle inertia turns to be minor, and spray particles can follow the airflow streamlines well and travel further downstream of the human up airway.

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Fig. 15 Particle deposition comparison

Particle Deposition Comparison Sprayed particle deposition between 15 and 5 μm is compared in Fig. 15. Despite a greater total deposition fraction for 15 μm particles, the 5 μm particle deposition is more dispersed. For 15 μm particles, although the particle deposition achieves 11% deposition in the middle region of the nasal cavity, the majority of them are deposited in the vestibule region of the nasal cavity with a deposition of 75%. In the middle region, there is comparable deposition between the two particle sizes. However, due to the reduced particle inertia, a larger number of 5 μm particles escaped through the pharynx. This will lead to deposition later downstream in the respiratory tract and may reach the lungs, which can cause adverse health responses.

Summary Nasal drug delivery offers an attractive alternative to invasive drug delivery for small and large molecular weight drugs. The major advantages are the straightforward and needle-free application mode and the permeable application site in the nasal cavity that allow a rapid onset of local and systemic drug actions. However, the limited area in the nasal cavity with an optimal drug absorption capability requires a precise drug delivery to the targeted area during the nasal spray process. This case presents an illustration of how to numerically assess the drug delivery efficiency of commercial nasal spray devices in detail. Due to high inertia effects, large-sized particles are immediately impacted on the anterior vestibule region, indicating an inefficient drug delivery performance. While for fine particles (5 μm), the sprayed drug particles can

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Table 2 RBCs case studies for different volume fractions Pressure drop (Pa) 48 48 24

VF % 5 15 30

Tube diameter (D, μm) 150 150 150

Tube length 2D 2D 1D

No. of RBCs 3,400 10,200 10,200

travel further downstream the nasal cavity, with an increased likelihood to reach the targeted epithelium area.

Case Study: Modelling of Blood Flow in Capillary As a physiological fluid, blood is a complex suspension of polydisperse, flexible, chemically and electrostatically active cells, which are suspended in an electrolytic fluid consisting of numerous active proteins and organic substances (Baskurt and Meiselman 2003). Macroscopic vessels represent only a small fraction of circulatory system, although the largest veins contain 50% of blood. The vascular tissue is made of microscopic capillary channels. There are about 1010 blood vessels whose diameters are comparable with the dimensions of the red blood cells (RBCs), i.e., 5–10 μm (Eggleton and Popel 1998). Therefore, the majority of defects in circulatory blood system occur in capillary vessels where blood flows less vigorously than in larger macroscopic vessels. When analyzing disease development in arteries, it is important to understand the local variations in blood rheology. Blood flow in large arteries is often assumed to behave as a homogeneous fluid, an assumption that is not entirely correct (Chien et al. 1970; Gijsen et al. 1999). The local viscosity changes with the local concentration, and the rate of shear strongly influences the wall shear stress (WSS) and its gradients, physiological parameters important in the study of atherosclerosis and aneurysm (Evju et al. 2013; Xiang et al. 2012). Moreover, the flow behavior of RBCs is influenced by the geometric structure of the flow environment, while rheological properties across a tube cross section are difficult to measure if noninvasive techniques are to be used (Fullana et al. 2007; Boynard et al. 2007). In this section, a numerical study employing the Lattice Boltzmann method to model the blood as a particle suspension of RBCs was reviewed (van Wyk et al. 2014), in which a tube flow with the RBC volume fractions (VF) of 5%, 15%, and 30% was studied.

CFD Models The geometry is a simplified cylindrical arterial vessel and meshed into O-grid cells to ensure near orthogonality close to the walls of the tubes. Cells are progressively finer toward the walls to ensure resolution of the boundary layer. Common boundary conditions are applied, such as the no-slip wall condition and pressure outlets. The

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Fig. 16 Instantaneous RBC distributions of the tube flow cases described in Table 2. (a) VF 5%, (b) VF 15%, (c) VF 30% (Figure adapted with permission from van Wyk et al. (2013))

zero gradient condition is set at the walls for the scalar transport equation since the RBCs never diffuse across arterial walls. Other boundary conditions are listed in Table 2.

Multiphase Flows in Biomedical Applications Fig. 17 Mean velocity distributions for (a) VF 5%, (b) VF 15%, and (c) VF 30% (Figure adapted with permission from van Wyk et al. (2013))

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0.8 VF 5% VF 15% VF 30%

0.7 0.6

UX / UN

0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

r/R

Flow Field and Mean Velocity Distribution An isometric view of the RBC distributions at an instant in time for each of the cases is shown in Fig. 16. The image on the right for each flow case plots all particles in semitransparent form to compare the particle density. The results demonstrate that RBCs get forced closer to the walls as the bulk VF is increased. This is partially due to the decrease in available volume in the center and therefore an increase in the viscosity. The mean radial velocity distributions of the studied tube flow cases are displayed in Fig. 17. The mean radial velocity profile of the 5% case is still in a parabolic shape, whereas the 15% and 30% cases develop a more pluglike flow shape in the center of the tube. The error bars show that the velocity profiles of the 5% and 15% cases are within an acceptable range of approximately 2%. The lack of data for the 30% case is partly responsible for the larger variation near the center of the tube. However the concentration of RBCs near the center is greater leading to larger variations in the flow field.

Summary The application of multiphase flow modelling on Poiseuille flow of blood suspensions in small capillaries was reviewed. The modelled suspension of monodispersed RBCs exhibits distinct flow behavior compared with homogeneous plane-shear flow, and the variation of the RBC volume fraction significantly alters the flow field. In conjunction with physiological realistic arterial vessel and flow conditions, this modelling approach can offer more fluid dynamic insights to the study of arterial disease, such as atherosclerosis.

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Conclusion This chapter outlines the main procedure of solving biofluid dynamics problems, and provides practical modelling solutions through case study demonstrations. Multiphase modelling of inhalation exposure, nasal drug delivery and red blood cells in capillary were presented in detail. This chapter can serve as an illustration of the necessary modelling framework in numerical applications. Combined with experimental measurements and clinical observations, multiphase CFD modelling can address unmet clinical needs, predominantly in the direction of enhanced diagnosis, as well as assessment and prediction of treatment outcomes.

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D.O. Frank, J.S. Kimbell, S. Pawar, J.S. Rhee, Effects of anatomy and particle size on nasal sprays and nebulizers. Otolaryngol. Head Neck Surg. 146, 313–319 (2012) J.M. Fullana, N. Dispot, P. Flaud, M. Rossi, An inverse method for non-invasive viscosity measurements. Eur. Phys. J.-Appl. Phys. 38, 79–92 (2007) M.C. Fung, K. Inthavong, W. Yang, P. Lappas, J. Tu, External characteristics of unsteady spray atomization from a nasal spray device. J. Pharm. Sci. 102, 1024–1035 (2013) Q. Ge, X. Li, K. Inthavong, J. Tu, Numerical study of the effects of human body heat on particle transport and inhalation in indoor environment. Build. Environ. 59, 1–9 (2013) F.J.H. Gijsen, F.N. Van de vosse, J.D. Janssen, The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model. J. Biomech. 32, 601–608 (1999) H.L. Goldsmith, J.C. Marlow, Flow behavior of erythrocytes. 2. Particle motions in concentrated suspensions of ghost cells. J. Colloid Interface Sci. 71, 383–407 (1979) E.A. Gross, J.A. Swenberg, S. Fields, J.A. Popp, Comparative morphometry of the nasal cavity in rats and mice. J. Anat. 135, 83–88 (1982) Y. Hoi, H. Meng, S.H. Woodward, B.R. Bendok, R.A. Hanel, L.R. Guterman, L.N. Hopkins, Effects of arterial geometry on aneurysm growth: three-dimensional computational fluid dynamics study. J. Neurosurg. 101, 676–681 (2004) K. Inthavong, Z.F. Tian, H.F. Li, J.Y. Tu, W. Yang, C.L. Xue, C.G. Li, A numerical study of spray particle deposition in a human nasal cavity. Aerosol Sci. Technol. 40, 1034–10U3 (2006) K. Inthavong, Z.F. Tian, J.Y. Tu, W. Yang, C. Xue, Optimising nasal spray parameters for efficient drug delivery using computational fluid dynamics. Comput. Biol. Med. 38, 713–726 (2008) K. Inthavong, Q.J. Ge, X.D. Li, J.Y. Tu, Detailed predictions of particle aspiration affected by respiratory inhalation and airflow. Atmos. Environ. 62, 107–117 (2012) K. Inthavong, Q.J. Ge, A. Li, J.Y. Tu, Source and trajectories of inhaled particles from a surrounding environment and its deposition in the respiratory airway. Inhal. Toxicol. 25, 280–291 (2013) K. Inthavong, M.C. Fung, X.W. Tong, W. Yang, J.Y. Tu, High resolution visualization and analysis of nasal spray drug delivery. Pharm. Res. 31, 1930–1937 (2014a) K. Inthavong, Y.D. Shang, J.Y. Tu, Surface mapping for visualization of wall stresses during inhalation in a human nasal cavity. Respir. Physiol. Neurobiol. 190, 54–61 (2014b) J.A. Johnson, K.C. Bloch, B.N. Dang, Varicella reinfection in a seropositive physician following occupational exposure to localized zoster. Clin. Infect. Dis. 52, 907–909 (2011) J.T. Kelly, B. Asgharian, J.S. Kimbell, B.A. Wong, Particle deposition in human nasal airway replicas manufactured by different methods. Part I: inertial regime particles. Aerosol Sci. Technol. 38, 1063–1071 (2004) J.T. Kelly, B. Asgharian, B.A. Wong, Inertial particle deposition in a monkey nasal mold compared with that in human nasal replicas. Inhal. Toxicol. 17, 823–830 (2005) J.S. Kimbell, J.D. Schroeter, B. Asgharian, B.A. Wong, R.A. Segal, C.J. Dickens, J.P. Southall, F.J. Miller, Optimization of nasal delivery devices using computational models. Respir. Drug Deliv. IX 1, 233–238 (2004) C.M. King SE, K. Inthavong, J. Tu, Inhalability of micron particles through the nose and mouth. Inhal. Toxicol. 22, 287–300 (2010) R. Krams, J.J. Wentzel, J.A.F. Oomen, R. Vinke, J.C.H. Schuurbiers, P.J. Defeyter, P.W. Serruys, C.J. Slager, Evaluation of endothelial shear stress and 3D geometry as factors determining the development of atherosclerosis and remodeling in human coronary arteries in vivo – combining 3D reconstruction from angiography and IVUS (ANGUS) with computational fluid dynamics. Arterioscler. Thromb. Vasc. Biol. 17, 2061–2065 (1997) W.G. Kreyling, M. Semmler-behnke, W. Moller, Ultrafine particle-lung interactions: Does size matter? J. Aerosol Med.-Depos. Clearance Eff. Lung 19, 74–83 (2006) D.N. Ku, Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434 (1997) M.G. Menache, F.J. Miller, O.G. Raabe, Particle inhalability curves for humans and small laboratory-animals. Ann. Occup. Hyg. 39, 317–328 (1995)

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Flow Boiling Enhancement via Cross-Sectional Expansion Patrick Phelan and Mark Miner

Abstract

Heat transfer enhancements available from expanding the cross section of a microchannel or microchannel system in flow boiling are presented, including recommendations appropriate for design and selection of expanding channel heat sinks. The principal relevant operating parameters of a boiling-channel heat sink are the attainable critical heat flux (CHF), which limits the practical heat flux permissible and the pressure drop across the channel, which may impose substantial pumping costs on the loop and is coupled to stability of flow in the channels. Keywords

Microchannel • Liquid cooling • High heat flux • Flow boiling • Pressure drop • Two-phase

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microchannels: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluating Predictive Criteria for Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observed Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 6 9 15 18 19 20

P. Phelan (*) School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, USA e-mail: [email protected] M. Miner School of Earth and Space Exploration, Arizona State University, Tempe, AZ, USA e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_17-1

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A perturbation of the channel diameter may be employed to examine CHF and pressure drop relationships from the literature with the aim of identifying those adequately general and suitable for use in a scenario with an expanding channel. Optimum rates of expansion which maximize the critical heat flux may be extracted from some criteria, and optima in expansion have been observed by investigators. The boiling number is considered, and it is seen that expansion typically produces an increase in the boiling number in the region explored, though no optima are observed. Pressure drop relationships admit improvement with expansion according to the same perturbation analysis, and no optimum appears. The relevant phenomena surrounding flow boiling pressure drop are considered, along with a handful of dimensionless numbers as qualitative selection aids. Decrease in the pressure drop across the evaporator is observed with expanding channels, though low-frequency oscillations are not necessarily damped out. Expansion is seen to improve stability of the flow by reducing the dependence of the pressure drop on heat flux.

Introduction The thermal management engineer may be justly likened to the undertaker; it is best not to need, but the need is inevitable. Heat exchange apparatuses are required in every thermodynamic cycle, no matter how simple. If the engineer wishes to accomplish most any task consuming or generating useful work, heat exchange will be required. How a cycle is accomplished directly affects its utility; wasted work becomes entropy, and the exergy of the system is reduced. Heat transfer across a finite temperature difference generates entropy; therefore, devices with low resistance to heat flow allow any given process to capture more of the desirable output with lower overhead. Looking beyond efficiency, device effectiveness remains stubbornly tied to heat exchange. Low-power design and thoughtful component layout can mitigate the need to pump away unwanted heat, but performance demands inevitably outstrip efficient architecture, and the waste heat generated by ever-smaller, higher-power, and higher-frequency devices must be dissipated. The ability to employ processes which rapidly convert energy into beneficial formats is increasingly limited by the ability to reject substantial amounts of heat energy in space- or weight-limited devices. Many engineering examples offer themselves, such as turbine blades, high-concentration photovoltaics, portable catalytic reforming, and advanced batteries, forcing the conclusion that thermal limitations are a ubiquitous challenge across engineering disciplines. Changing the thermodynamic phase of a substance is not strictly a mode of heat transfer, as the substance undergoing the change must accept or reject heat via one of the usual modes, but phase change does represent an opportunity for significant energy arbitrage. Since the temperature of a liquid-vapor mixture is coupled to the pressure of a system, the boiler or evaporator designer has a measure of predictability. The coupling of convection with phase change to obtain improved cooling

Flow Boiling Enhancement via Cross-Sectional Expansion

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performance presents potent possibilities, where the cooling device scale enforces proximity of coolant molecules to the heated walls. A useful tool in this application is the microchannel. This chapter presents an advance in flow boiling heat-exchanger design, analytically and experimentally treating the possibility of optimizing the streamwise shape of a channel or array of channels to enhance heat rejection and reduce associated system pumping cost. Reducing the required work input directly increases first-law efficiency metrics of a given cycle, and reducing the resistance to heat transfer of either the accepting or rejecting heat exchanger improves second-law efficiency, reducing entropy generation.

Microchannels: An Overview Microchannels promise a compact, versatile, tunable solution to a substantial set of thermal management problems. Enhancing convection by shrinking the scale of the channel is an extremely attractive option at small device scales. Conventional convective relationships for predicting the heat transfer coefficient look like h¼

Nu  k d

(1)

where h is the convective heat transfer coefficient (in mW2 K ); Nu the dimensionless Nusselt number which represents the ratio of convection to conduction in the fluid and is typically
W  obtained via experimental correlation, k, the thermal conductivity of ; and d the appropriate length scale of the flow, which for internal flow the fluid mK is the hydraulic diameter of the tube carrying the fluid. The Nusselt number is an experimentally derived value, with a variety of correlations tailored to specific flow regimes. In laminar flow, which is typical for single-phase microchannels, the Nusselt number carries a subunity dependence on the length scale, meaning that the limit of Eq. 1 tends to infinity as d goes to zero. This behavior clearly extrapolates from an experimental-physical correlation into nonphysical values, but for some region, as the diameter of a tube decreases, the resistance of the flow to convective heat transfer also decreases. Single-phase cooling in microchannels is widely employed and presents many interesting challenges, but it is always limited by Poiseuille’s law and often hampered by the necessarily nonuniform fluid temperatures obtaining in sensible heating. The latent heat of vaporization of a liquid coolant holds out promise for moving beyond some of the limitations of single-phase flow. The principal drawback, on the surface, is the exchange of a quasi-static system that operates at a predictable and repeatable steady state for the inherently dynamic boiling process, where flow is dramatically less predictable in time and in space. For all that, flow boiling remains a tantalizing tool to employ in the pursuit of higher heat-density cooling applications, and the prospects of direct boiling for steam generation could be attractive due to reduced capital requirements. Phase change promises obvious advantages in large

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latent heats, reduced coolant inventories, and stable, pressure-selectable sink temperatures. Engineering challenges, though, are to be found right the way down. An excellent survey of the field is available (Kandlikar 2012), which, with some overlap into single-phase flow, weighs well the state of the art and the challenges impeding the widespread application of microscale flow boiling. The common themes of pressure drop oscillation and critical heat flux (CHF) are sounded time and again, and the reader is directed to Kandlikar’s extensive bibliography for further exploration. The challenge of harnessing microscale flow boiling is not the theory, but difficulties which appear in the practical application of the phenomenon. Many investigators (Zhang et al. 2009; Chang and Pan 2007; Qu and Mudawar 2003; Kandlikar 2004) have observed and investigated the unstable nature of multiphase microchannel flow, where local stochastic processes like bubble nucleation cascade up into global unpredictability of the entire heat sink. Much research has subsequently targeted the reduction or elimination of these thermohydraulic instabilities, which tend to limit the achievable heat flux. Thermal instabilities become pressure instabilities, as local temperature irregularities cause the flow to boil in a more inhomogeneous fashion than usual, and the subsequent changes in the flow field feed back into thermal behavior because of local dryout or flow reversal. Because thermal loads will often be somewhat variable, due to the locality of heating and time-varying loads in circuitry or other applications, the principal emphasis of instability suppression has been to break oscillatory cycles by manipulating the flow field through modification of the pressure profile. Elimination of flow reversal in microchannels can be readily accomplished by imposition of a steep pressure gradient at or near the inlet of the channels. This clearly approaches the limiting case of an expansion valve in a vapor-compression refrigerant loop, which is a known stable configuration. By using less aggressive chokes in the flow path, called inlet restrictors, many investigators (Odom et al. 2012; Revellin and Thome 2007a; Koşar et al. 2006; Kandlikar et al. 2006; Basu et al. 2011) have successfully reduced oscillatory instabilities, but at the cost of a significant pressure penalty. While a single restriction is beneficial, some investigators (Mukherjee and Kandlikar 2009) further posited a greater beneficial effect from a series of finite constrictions. An expanding vapor bubble in such a channel was modeled and the output was promising. Channel expansion allowed the bubble growth to proceed predominately downstream, somewhat ameliorating the flow reversal that can be introduced by bubble expansion in a straight channel. As a series of finite constrictions could be a challenge to manufacture, this work suggested that continuous expansion would serve better than none. A separate approach of reducing instability is to attempt to homogenize the flow field by distributing nucleation sites in desired areas, and some investigators (Kuo and Peles 2008) fabricated reentrant cavities and laser-etched nucleation sites into microchannel walls. This reduces the required wall superheat for bubble nucleation, but may not directly affect pressure behavior. However, Pan’s group continued these researches and obtained significant improvement in heat sink behavior by combining artificial nucleation sites with the idea of an expanding channel cross section (Lu and

Flow Boiling Enhancement via Cross-Sectional Expansion

5

Pan 2009). This concept was first introduced to the thermal literature in an examination of the pressure effects of varying cross-sectional channels on ethanol-CO2 flows, and benefits were noted from expansion (Hwang et al. 2005). Pan and colleagues continued to pursue a number of related experiments, using continuously expanding etched silicon microchannels; sometimes as a single channel (Lee and Pan 2007), but also in parallel arrays (Lu and Pan 2008, 2009, 2011). More recently, investigation of these arrays of expanding microchannels extended to other flow patterns with a mind toward engineering applications (Liu et al. 2012). Balasubramanian et al. have also inquired into expanding microchannels, using copper with wire-EDM-cut channels; discrete cross-sectional area changes are made by chopping a channel wall at a given point along the flow direction, thus merging two neighboring channels and adding the wall space in between (Balasubramanian et al. 2011). Results from all of these groups have been uniformly positive: expanding the cross section of a microchannel reduces instabilities and improves both CHF and pressure drop characteristics. Miner et al. numerically, experimentally, and analytically pursued the optimization of expanding microchannel geometry (Miner et al. 2013a, b, 2014), seeking to maximize the critical heat flux (CHF) attainable within a fixed footprint. These investigations pointed to a maximum CHF at a relatively small expansion angle and realized this in laboratory experiments. These same experiments demonstrated that expanding microchannels offered a substantial reduction in pumping costs and a partial decoupling of the thermal and pressure effects (Miner and Phelan 2013). Another avenue of research has examined microchannels of constant cross section with an open manifold above, a configuration that allows vapor to leave the channel orthogonal to the flow direction of refrigerant (Kandlikar et al. 2013; Kalani and Kandlikar 2014, 2015a, b). These studies considered both a tapered and a constant cross-sectional manifold and noted improvement with the taper. This method facilitates rewetting of the heated surface, which is a substantial benefit, and effectively acts as a modified form of pool boiling, where the superficial velocity of the coolant is not a driver of performance. The approach eliminates confinement effects at the scale of the channel dimension and may allow for reduced manufacturing costs, as the tapered manifold may be simpler to produce than a tapered channel array. Excellent visualization of the flow is presented by these investigators, and flow pattern mapping is pursued. Finally, a beautiful set of investigations with high-speed photography (Tamanna and Lee 2015a, b) enabled further flow pattern mapping in an expanding microgap (effectively a single, wide channel), and the fine-grained apparatus employed also demonstrated conclusively that expansion has a substantial stabilizing effect on wall temperatures, which is a major motivation for electronics-oriented researchers. These investigators also reported an optimum expansion rate, also at a relatively small expansion angle. While reporting reduced pressure drop, they also observed large-scale slow pressure oscillations in the system, which seemed to intensify as the expansion rate increased. The research group of Odom et al. observed similar large-scale pressure oscillation behavior early in their investigations, but determined it to be caused by periodic

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cycling in the chiller. The chiller was taken out of service, and an ice bath was employed to stabilize the heat sink temperature. Higher heat rejection with expansion increases chiller loading, increasing the amplitude of these cycles, which are the result of systematic pressure reduction in the liquid-vapor system as heat is removed by the chiller. It was also noted that chillers advertising PID control fail to note that compressors are binary and effectively pursue a pulse-width modulation cooling scheme, adding an unwanted oscillation to the loop. Researchers employing chillers may wish to take additional care to reduce these effects. Pressure drop in small channels has been addressed by a number of researchers, beginning with Poiseuille, but modern consideration of boiling flow was dominated by macroscale applications, such as boilers for steam generation and, later, evaporators in refrigeration applications. The results of these researchers were carried downward in channel size for some time, and departure from the venerable Lockhart and Martinelli correlation in mini- and microchannels began to be contemplated by Lazarek and Black (1982), who modified it for their frictional pressure drop in short tubes with fluorocarbon refrigerants. Kandlikar provided an excellent review of the literature up to the year 2001, and it is notable that only 10 of the 28 investigations he surveyed took up pressure drop directly. The reader is directed to the extensive bibliography in Kandlikar’s paper (Kandlikar 2002) for further investigation. Since that survey, much more attention has been paid to understanding the mechanisms of the boiling pressure effects, especially in light of their tight coupling to heat transfer phenomena. Subsequent investigators took up the questions of classification and prediction of pressure drop through microchannels, and as usual, new correlations work well for the data sets from which they are derived, but a lack of generality persists. Mudawar surveyed the challenges encountered in his career investigating boiling flow and noted in particular the multiplicity of factors contributing to pressure drop across microchannel evaporators and the ongoing disagreement over their significance and prediction (Mudawar 2011). However, his extensive paper is also a paean to a variety of employments found for microchannels in emerging and established technology and leaves the researcher encouraged that the applications engineers may be catching on and catching up to the state of the art.

Evaluating Predictive Criteria for Expansion The body of data considering microchannels is growing rapidly, and many investigators have published correlations predicting both critical heat flux events and pressure drop across a microchannel array. These correlations are generally derived from limited data sets and invariably produce acceptable outcomes for the researchers deriving them. The correlations typically extend only to very similar circumstances and configurations, and extrapolation of any correlation is a danger. In the interest of clarifying which relationships may be of use to the microchannel engineer who wishes to improve flow boiling, a variety of correlations available

Flow Boiling Enhancement via Cross-Sectional Expansion

7

from the literature are examined to obtain a verdict on the effect of channel crosssectional expansion. A simple test may be posed to some relationship predicting a flow parameter, such as critical heat flux correlation q_ CHF ¼ F , or pressure drop dp dz ¼ G , purporting generality: if the diameter terms are construed as d = di + ez, where di is the inlet diameter, e the rate of channel expansion, and z the station downstream from the inlet, what is the sign on the expression resulting from the partial with respect to e, e.g., @F/@e? The results will plainly be best for small e, but if the effect on the prediction corresponds to observed physical reality, the correlation may contain adequate information and prove generally useful. If, in a reasonable parameter space, @F/@e > 0, the prediction behaves realistically and predicts enhanced critical heat flux from expansion. This is known to be the case from experiment, and the predictive criterion may be useful. Similarly, if @G/@e 645 > 376

(5)

d 0:280Lh WeF

(13)

Lh d Lh d

Qi et al. (2007) Koşar and Peles (2007) Revellin and Thome (2007b)

Lh d Lh d

> :003

> 376 > 83:33Coþ50:97 0:765Co h 1 @P >  23:12 @e

17:281P 2 :0934P :34P 1:3104

i



 17  37  ðρL ρV Þgd2 1 > :075 VVVL 7 4σ L

Lh d

 37

@ V

V @e

VV



@V L @e



VL

cannot be negative, the result may be presented as an inequality. If there is a competition between effects permitting the sign to change under certain conditions, the expression has captured competing physics, which is encouraging. Additionally, 2 the optimum situation where @@e2 ¼ 0 may be explored analytically if the calculation is tractable, or it may be considered numerically across desired parameter spaces. Revellin and Thome’s criterion is displayed as in their work (Revellin and Thome 2007b) and not in the guise of a Bond or confinement number, since the authors wished to call attention to the most dangerous wavelength in the Kelvin-Helmholtz instability. Investigators who desire to employ the CHF enhancements available from manipulating channel geometry must choose wisely among predictive criteria, some of which may not function well for even simple enhancements such as an expanding cross section. Relationships which contemplate the flow variables directly influencing CHF respond better to this interrogation by partials, but require measurement, approximation, or modeling of the flow variables required. Some of these criteria depend on geometry and flow variables easily obtainable early in the design phase of a system, rendering them nearly as useful as the purely upstreamdependent relations. In general, the more physical phenomena which are reflected in the criterion, the more accurate it will be (Table 1). A similar study was performed across a variety of pressure drop predictions in microchannels (Miner et al. 2014) as an adjunct to an experimental study. In all cases, expansion was predicted to reduce pumping cost, and it seems that pressure relationships may be more generally physically grounded than the CHF relationships noted above. Thus, the discussion of pressure drop correlations is somewhat more limited in scope than that of critical heat flux prediction. While many investigators have discussed the issue, their concern is typically with refining certain parameters within an unanimous framework or characterizing and including pressure drops associated with other elements of the apparatus, such as inlet contraction and outlet expansion. Consistent across all investigators considered is the distinction between

Flow Boiling Enhancement via Cross-Sectional Expansion

9

frictional and acceleration pressure drop, the former due to bubble-induced tortuosity and surface effects and the latter due to the thrust developed from evaporating liquid turning to faster-moving vapor. The physical forces affecting these terms will be taken up below.

Basic Phenomenology The physical scientist will wish to know the nature of the forces acting inside a boiling microchannel. The engineer will wish to know which of these forces matter and what may be done to obtain a given objective. Kandlikar provides an excellent discussion of these effects and settles on the inertia, surface tension, shear, buoyancy, and evaporation momentum forces as the candidates for significance (Kandlikar 2010). The formulations here employed are those which consider force over a unit area and are cast into variables more appropriate to a system designer. The inertial force scale Fi considers momentum flux through a given area as: Fi 

_ G2 mV m_ 2  4 A ρ ρd

(2)

where m_ is the mass flow rate, V is the average velocity, A is the local flow crosssectional area, ρ is the fluid density (as appropriate to the quality at a given station), and G is the mass flux. The final formulation, involving the constant mass flux and quality-dependent density, will substantially increase in magnitude as the density of the flow drops during boiling. The surface tension force scale Fσ follows from the Young-Laplace equation applied to the bulk channel dimension as for a circular channel cross section: Fσ 

σ d

(3)

where σ is the surface tension of the liquid phase of the fluid and d is the relevant channel dimension; either radius or diameter may suffice, as the order of magnitude is unchanged by the factor of 2. Plainly, this term is significant only in proportion to the liquid fraction in the channel and will disappear for a pure vapor flow. The shear force scale Fτ will be treated as Newtonian, with shear stress proportional to the velocity gradient at the wall. When the channel is full of liquid, flow is likely to be laminar and have a velocity gradient proportional to the velocity. As vapor is generated in what is here assumed to be an annular core, the liquid film against the walls is subjected to a zero-velocity no-slip condition at the wall and a matching-velocity no-slip condition at the vapor. When the vapor density is significantly less than the liquid density, as is common in refrigerants, the vapor velocity will be large. Regardless of the particular shape of the velocity distribution in the liquid, the velocity gradient in the increasingly thin liquid film may be reasonably

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assumed to vary linearly with the vapor core velocity, as this situation approaches a type of Couette flow. Thus, the force scale may be simply represented as: Fτ 

μV vG m_ μ 3 d d ρd

(4)

where μ is the liquid viscosity, V is the velocity of whatever phase is appropriate, ρ is again the quality-weighted bulk fluid density, and ν is the momentum diffusivity, which is generally higher in the liquid phase than in the vapor phase, making this term reduce in magnitude during boiling. Gravity may affect the flow, and the characteristic buoyancy force Fg is a function of the density difference as: Fg  ðρL  ρV Þgd sinðθÞ

(5)

where g is the acceleration due to gravity and θ the deviation of the channel from horizontal. The fluid boiling in the channel advances in the vapor phase, with a higher velocity than the liquid phase. Thus, as fluid is boiled, the conservation of mass requires that it is accelerated forward, developing a thrust. This evaporation momentum force FM scales as:   _ m_ _ gen ðV V  V L Þ mV 1 1 qL   FM  d2 hLV d 3 ρV ρL

(6)

_ gen is the rate of vapor mass generation (intimately related to the heat input) where mV and L is the channel length. The heat flux may be considered locally, as when writing a spatially resolved solver, or globally, as when designing a channel ab initio. These are the most significant fundamental forces at work within a channel, and it will be instructive to consider how they may react to alterations in the channel configuration. Until flow reversal occurs, the inertial forces will tend downstream. The liquidvapor interface in annular separated flow is distended from the minimum area shape sought by the surface tension, which will tend to pull liquid upstream; this follows from consideration of a channel as a capillary tube. Shear forces will resist motion, so until flow reversal occurs, these forces will be directed upstream. Gravity will be directed according to the channel orientation. Evaporation momentum thrust will always tend to drive the interface upstream. The system driving pressure, typically from a pump or compressor, will define the downstream direction. Notably, only driving pressure, evaporation momentum, and capillary forces have orientations independent of the instantaneous flow behavior. This physically grounds the observation that rewetting dry channels or overcoming flow reversals is a difficult task: the inertia force has reversed direction and must now be overcome. Additionally, the evaporation momentum thrust tends to increase during dryout events, as thin liquid

Flow Boiling Enhancement via Cross-Sectional Expansion

11

Table 2 Sensitivity of forces to channel expansion Inertia

Surf. tens.

Shear

m_ 2 ρd4

σ d

μm_ ρd3

@F @e

4m_ 2 z ρd 5

σz d2

_ 3μmz ρd 4

Fig. 1 Representative force scales in 10 mm long microchannels with 3 gs R-134a at 350 kPa, x = 0.5, W q_ ¼ 500 cm 2

Force Magnitude

Force Expression

Buoyancy (ρL  ρV)gd (ρL  ρV)gz

Evaporation thrust   _ m_ qL 1  ρ1 hLV d 3 ρV L   _ m_ 3qL 1  ρ1 h d4 ρ V

LV

L

100 10–6

10–5

10–4

10–3

–2

10

Diameter, m Inertia dF/dε Magnitude

Evap. Thrust Shear Surf. Tens Buoyancy 100 10–6

10–5

10–4 Diameter, m

10–3

10–2

films evaporate extremely well, temporarily increasing the vapor mass generation rate. The sensitivity of each of these force scales to the channel expansion is a quantity of interest. Table 2 presents the results of the sensitivity analysis, in which the diameter is defined as d = d0 + ez and the resulting expression is differentiated with respect to the small parameter e. A visualization of these force scales, following in a familiar format (Kandlikar 2010), is presented in Fig. 1, assuming a representative 3 gs flow of R-134a at 350 kPa in a 10 mm long channel with an average quality W of 0.5 and a heat flux of 500 500 cm 2. All of the forces diminish with channel expansion, with the exception of buoyancy, and all force scale effects increase linearly with the length of the channel. The more significant quantity may be the strength of the relationship between a force and the channel diameter d and by extension the expansion parameter e. The inertia force is the most sensitive, dropping off as d5. This force scale tends to have the largest magnitude in small channels and may be directed downstream in typical flow patterns or upstream during flow reversal. Reduction of the inertial force scale allows the conclusion that an expanding channel will be easier to rewet after a flow reversal.

12

P. Phelan and M. Miner

Shear and evaporation momentum forces fall off as d4, a steep reduction, but not so steep as inertia. The evaporation thrust force is always working against the desired flow direction, so its reduction with expansion is seen as a significant benefit. The reduction in shear force comes from the easing of the velocity gradient with an expanding channel; this is seen as a benefit, as it contributes to the diode-like properties of the channel, easing flow in the direction of expansion. The surface tension force diminishes as well, but not nearly so rapidly. The d2 dependence means that this force will increase in magnitude with respect to inertia, shear, and evaporation thrust; the surface tension is helpful in rewetting walls, which are drier in the downstream direction, though it continues to pull liquid toward the smaller end of the channel. Buoyancy alone becomes more significant as the channel expands. This effect would be most pronounced in vertical orientations of the evaporator, but since Lazarek and Black (1982) did not report any significant directionality to their results for upflow and downflow, directionality is not expected to be a first-order influence on the flow boiling behavior. If the critical parameter for predicting pressure drop in a microchannel array is driven solely by evaporation thrust and inertia, we may require that FM < Fi:   _ m_ 1 1 m_ 2 qL  < 4 3 ρ ρL ρd hLV d V

(7)

which may be rearranged to yield the dimensionless term that predicts this behavior as:   _ 1 1 qLdρ  > 1, L/d >> 1, We >> 1, and Eo  1, leaving the boiling number and Euler number as the variables most readily manipulated. Given a certain refrigerant and a target heat flux, minimization of the sum on the left-hand side of Eq. 12 would illuminate the desirable parameter space in D and L, which are likely to be the design variables in evaporator or boiler design. The insertion of the expanding channel diameter d = d0 + ez would add e to the parameter space and allow the applications engineer to select a desirable geometry and estimate the attendant pressure drop penalty.

Observed Improvements Observed improvements from a variety of investigators are presented in Table 3 to allow an overview of the gains obtainable from expanding microchannels. Where the investigators observed an optimum, it is noted. Table 3 demonstrates an unanimity in the results obtained for the various expanding flow passage configurations: stability is enhanced, wall temperatures are moderated, and pressure oscillations lessen. Not all of the investigators pursued experiments up to critical heat flux, but those that did observed increased CHF available from expansion. The results for overall pressure drop are mixed, with the very small channels tending toward higher pressure drops, and the larger channels, manifolds, and gaps observing reduced pumping costs associated with expansion. The dominant forces at very small (verging on nanochannel) scales are not the same as those at the microchannel and nearly minichannel scales, and this behavior is not surprising. Example data were provided by Miner et al., and some of the key features of expansion are illustrated in the figures that follow. The typical increase of critical heat flux with mass flux is observed in Fig. 2 (Miner et al. 2013b). These data trend monotonically upward with mass flux, and expanding channel cross-sectional shifts the curves toward higher heat fluxes for a

16

P. Phelan and M. Miner

Fig. 2 CHF versus mass flow rate Critical Heat Flux (W/cm2)

500 450 400 350 300

0° 250

0.5°

200

1° 2°

150 400

600

800

1000

1200

1400

1600

Mass Flux (kg / (m2∗s))

given mass flux. Because the mass flux varies through the channel, what is presented here is taken at the channel average cross-sectional area. A very surprising feature was noted in a straight microchannel apparatus after experimentation was concluded: damage appeared in the copper, looking very much like early-stage cavitation scarring. Figure 3 shows the outlet face of the straight channel turret, and bubble-like damage patterns are circled on the end faces of some of the fins. This damage did not appear on the other channel arrays in that investigation. The pressure drop occurring when liquid refrigerant exits the channels to the outlet plenum can result in flashing, and the lowest pressure regions would be the outlet faces of the fins. These sites may therefore be prone to cavitation-type nucleation-induced damage. A reduction in the outlet pressure accompanies expansion of the channel cross section, and this may ameliorate the likelihood of damage and account for the lack of observed damage in the other turrets. The largest observed damage site was approximately 60 μm diameter, and sites are observed down to 15 μm in diameter and below, though it becomes increasingly difficult to distinguish between possible damage sites and surface roughness features. A brief examination of the critical bubble radius indicates that these site sizes lie within the bounds of the minimum and maximum critical bubble radii. This damage mechanism in an evaporator may be negligible at larger scales, but could pose a threat to the longevity of a microchannel evaporator. Suitable engineering of the plenum could ameliorate this; however, the open-manifold arrays employed by Kalani et al. and the gap configuration of Tamanna and Lee would be much less susceptible to this phenomenon. Extensive data on the pressure drop and CHF coupling are presented in Fig. 4 (Miner et al. 2014). The figure shows results averaged over periods of constant heater input power for every experiment performed in the study. Heat flux into the channel array lies on the abscissa, and the ordinate plots the pressure drop over mass flux. The trend shown in Fig. 4 is clear: cross-sectional expansion of the microchannel reduces the deleterious effect of heat input on the pressure drop across the

Flow Boiling Enhancement via Cross-Sectional Expansion

17

Fig. 3 Damage at straight channel outlet

Fig. 4 Heat flux effect on pressure drop

0.8

ΔP/G (kPa∗s∗mm2 / g)

0° 0.5°

0.6

1° 2°

0.4

0.2

0

0

1

3 2 Heat Flux (W / mm2)

4

5

array. This augurs well for the employment of expanding microchannels, as the feedback effect from higher heat rates increasing the required pressure drop is ameliorated. It bears repeating that pressure drop and heat input remain strongly coupled in the expanding channels; the degree of this coupling decreases for larger channel expansions. Additionally, the decoupling seems to stop at the optimum 1 degree expansion observed in that investigation, with the larger 2 degree angle exhibiting effectively identical coupling between thermal and pressure effects. Improved stability is also evident from an examination of the outlet qualities attained by the expanding channels. A more stable channel will tolerate a higher outlet quality and continue to function. The general trend is for expanding channels

18

P. Phelan and M. Miner

Fig. 5 Outlet vapor quality

0.9 0° Outlet Quality (dim)

0.5° 0.8

1° 2°

0.7

0.6

0.5 1.5

2

2.5 3 3.5 Mass Flow Rate (g/s)

4

4.5

to allow increased outlet quality, indicating a stability improvement. The enhanced outlet quality is not adequately explained by greater inlet subcooling of the expanding channels, as the additional heat input required to raise the quality is about four times greater than the heat input required to make up for the largest subcooling shown in Fig. 5, and the most dramatic outlet quality difference occurs for lower mass flow rates, where there was no appreciable difference in the subcooling between the expansions. This leaves expansion as the most logical explanation for the improvement in boiling efficiency.

Applications Expanding microchannels in all flavors allow enhanced cooling in high heat flux applications. Chip cooling is especially amenable to these devices, as the typical chip geometry requires a constrained footprint operating at high-power dissipations. Expanding channels and expanding manifolds have both shown promise, using a wide variety of coolants, including air. Chip cooling is expected to remain a driver of research in this field, and the application is not limited to processors. A key advantage of the expanding microchannel array is the improved uniformity in wall temperature. This ameliorates hotspots on chips, which are almost invariably the precursors to failure. Steam generation in small formats could profitably employ the expansion advantage, and steam generation in large format boilers could be substantially aided by the concomitant pressure drop reduction, though it is not as clear what the CHF effects would be in larger channels where different boiling modes (and traditional models) hold sway. However, the conventional scale models are ripe for examination by the methods discussed herein. Reforming of fuels in fuel-cell applications demands high heat transfers and the accommodation of multiphase and reacting flows, all of which are temperature and

Flow Boiling Enhancement via Cross-Sectional Expansion

19

pressure sensitive to some degree. The enhanced control offered by expanding microchannels or tapered manifolds can make a beneficial difference in the optimal operation of these reactors, with wall temperature stability being directly linked to catalytic performance. The heat buildup in modern lithium-ion batteries may be especially amenable to expanding channel cooling, as the only possible coolants for use in batteries must be dielectric fluids, a category in which air and common refrigerants readily fall, and fluids for which expanding microchannel cooling has proven effective. Developers of solar receivers could consider expanding microchannels as either steam generation units for a true solar-thermal plant or for photovoltaic module cooling under highly concentrated light. The amazing advances in manufacturing technology developed in the last generation enable the fabrication of microchannels with greater ease than ever before. Micromachining, electron discharge techniques, and selective etching allow almost any desired channel profile to be placed on a suitable substrate. The demonstrable improvements available from enhanced microchannels are well within reach of the system designer.

Conclusions This chapter presents geometric expansion as an eminently usable enhancement to microchannel evaporators and boilers. The effects of channel cross-sectional expansion or manifold expansion have been demonstrated to increase the critical heat flux attainable by an evaporator while reducing the pump work required and improving the stability of the cooling system. Experimental and analytical investigations provide much support for the idea of optimizable expansion for critical heat flux and significant reduction of pressure drop across microchannel systems. Experiment bears out the optimization of expansion of the cross section of a microchannel in the flow direction. The location of the optimum tends to higher expansions when mass flow rates in the channel are higher. Boiling numbers are also increased by cross-sectional expansion, indicating improved consumption of latent heat in expanding channel evaporators and reflecting the improved stability of these devices. Expansion of the microchannel cross section in the flow direction is readily done with conventional fabrication techniques, and the optimum improvement does not require dramatic expansion of the channel, leaving conduction heat paths minimally affected. The effects of microchannel cross-sectional expansion appear to be uniformly beneficial: reduced associated pumping cost in the loop, reduced or eliminated low-frequency oscillatory instabilities in the system, and notable decoupling of the thermal and mechanical behaviors of the system. Applying this method to conventional channels requires a greater understanding of the scaling of the relevant phenomena, but besides fabrication costs, there are few perceived disadvantages. Cross-sectional expansion may prove to be an expedient method of making multiphase flow more tractable to the applications engineer by enhancing the

20

P. Phelan and M. Miner

predictability of evaporators or boilers subject to large heat fluxes. The reduction in cyclic stresses from oscillatory instabilities may allow longer and more reliable service life to a boiling system. Examination of force scales which are likely to be relevant to the flow allows insight to the effects of cross-sectional expansion on the fundamental forces in a boiling channel. These considerations are phenomenologically specific with respect to major fluid properties and apparatus geometry and are expected to yield qualitative guidance to the engineer wishing to design and employ channel expansion as a design tool. Care must be taken when selecting and balancing dominant mechanisms in the optimization, as these will scale with the channel dimensions. Channel cross-sectional expansion has a similar principle of effect as a flow restriction, namely, the insertion of a diode into the flow circuit. Making the heat source itself act as the enforcer of unidirectional flow reduces the part count of the system and means that the entire system will behave better. The unpleasant side effects of flow boiling are not merely contained, but are abated; the heat fluxes available to the apparatus are increased; the system becomes more stable and more capable. Well-tuned cross-sectional expansion in a boiler will provide the applications engineer with a system that is more capable, durable, and predictable at high heat fluxes. As exergetic efficiency becomes a stronger driver of design in the face of resource and lifetime cost constraints, it is hoped that this method will find increasing use in the engineering community at large. Acknowledgments The authors gratefully acknowledge the Office of Naval Research for funding and support. This work was partially supported by the Office of Naval Research as a MURI award (prime award number N00014-07-1-0723).

References K. Balasubramanian, P.C. Lee, L. Jin, S. Chou, C. Teo, S. Gao, Experimental investigations of flow boiling heat transfer and pressure drop in straight and expanding microchannels. A comparative study. Int. J. Therm. Sci. 50(12), 2413–2421 (2011) S. Basu, S. Ndao, G.J. Michna, Y. Peles, M.K. Jensen, Flow boiling of R134a in circular microtubes part II: study of critical heat flux condition. ASME J. Heat Transf. 133(5), 051503 (2011) K. Chang, C. Pan, Two-phase flow instability for boiling in a microchannel heat sink. Int. J. Heat Mass Transf. 50(11), 2078–2088 (2007) J. Hwang, F. Tseng, C. Pan, EthanolCO2 two-phase flow in diverging and converging microchannels. Int. J. Multiphase Flow 31(5), 548–570 (2005) A. Kalani, S.G. Kandlikar, Evaluation of pressure drop performance during enhanced flow boiling in open microchannels with tapered manifolds. ASME J. Heat Transf. 136(5), 051502 (2014) A. Kalani, S.G. Kandlikar, Effect of taper on pressure recovery during flow boiling in open microchannels with manifold using homogeneous flow model. Int. J. Heat Mass Transf. 83, 109–117 (2015a) A. Kalani, S.G. Kandlikar, Flow patterns and heat transfer mechanisms during flow boiling over open microchannels in tapered manifold (OMM). Int. J. Heat Mass Transf. 89, 494–504 (2015b) S.G. Kandlikar, Fundamental issues related to flow boiling in minichannels and microchannels. Exp. Thermal Fluid Sci. 26(2), 389–407 (2002)

Flow Boiling Enhancement via Cross-Sectional Expansion

21

S. Kandlikar, Heat transfer mechanisms during flow boiling in microchannels. ASME J. Heat Transf. 126(1), 8–16 (2004) S.G. Kandlikar, Scale effects on flow boiling heat transfer in microchannels: a fundamental perspective. Int. J. Therm. Sci. 49(7), 1073–1085 (2010) S.G. Kandlikar, History, advances, and challenges in liquid flow and flow boiling heat transfer in microchannels: a critical review. ASME J. Heat Transf. 134(3), 034001–1–15 (2012) S.G. Kandlikar, W.K. Kuan, D.A. Willistein, J. Borrelli, Stabilization of flow boiling in microchannels using pressure drop elements and fabricated nucleation sites. ASME J. Heat Transf. 128(4), 389–397 (2006) S.G. Kandlikar, T. Widger, A. Kalani, V. Mejia, Enhanced flow boiling over open microchannels with uniform and tapered gap manifolds. ASME J. Heat Transf. 135(6), 061401 (2013) Y. Katto, H. Ohno, An improved version of the generalized correlation of critical heat flux for the forced convective boiling in uniformly heated vertical tubes. Int. J. Heat Mass Transf. 27(9), 1641–1648 (1984) A. Koşar, Y. Peles, Critical heat flux of R-123 in silicon-based microchannels. ASME J. Heat Transf. 129(7), 844–851 (2007) A. Koşar, C.-J. Kuo, Y. Peles, Suppression of boiling flow oscillations in parallel microchannels by inlet restrictors. ASME J. Heat Transf. 128(3), 251–260 (2006) C.-J. Kuo, Y. Peles, Flow boiling instabilities in microchannels and means for mitigation by reentrant cavities. ASME J. Heat Transf. 130(7), 72402 (2008) G.M. Lazarek, S.H. Black, Evaporative heat transfer, pressure drop and critical heat flux in a small vertical tube with R-113. Int. J. Heat Mass Transf. 25(7), 945–960 (1982) P.C. Lee, C. Pan, Boiling heat transfer and two-phase flow of water in a single shallow microchannel with a uniform or diverging cross section. J. Micromech. Microeng. 18(2), 025005–1–13 (2007) T.-L. Liu, B.-R. Fu, C. Pan, Boiling two-phase flow and efficiency of co and counter-current microchannel heat exchangers with gas heating. Int. J. Heat Mass Transf. 55(21–22), 6130–6141 (2012) C.T. Lu, C. Pan, Stabilization of flow boiling in microchannel heat sinks with a diverging crosssection design. J. Micromech. Microeng. 18(7), 075035–1–13 (2008) C.T. Lu, C. Pan, A highly stable microchannel heat sink for convective boiling. J. Micromech. Microeng. 19(5), 055013–1–13 (2009) C.T. Lu, C. Pan, Convective boiling in a parallel microchannel heat sink with a diverging cross section and artificial nucleation sites. Exp. Thermal Fluid Sci. 35(5), 810–815 (2011) M.J. Miner, P.E. Phelan, Effect of cross-sectional perturbation on critical heat flux criteria in microchannels. ASME J. Heat Transf. 135(10), 101009 (2013) M.J. Miner, B.A. Odom, C.A. Ortiz, J. Sherbeck, R. Prasher, P.E. Phelan, Optimized expanding microchannel geometry for flow boiling. ASME J. Heat Transf. 135(4), 042901 (2013a) M.J. Miner, P.E. Phelan, C.A. Ortiz, B.A. Odom, Experimental measurements of critical heat flux in expanding microchannel arrays. ASME J. Heat Transf. 135(10), 101501 (2013b) M.J. Miner, P.E. Phelan, C.A. Ortiz, B.A. Odom, Experimental measurements of pressure drop in expanding microchannel arrays. ASME J. Heat Transf. 136(3), 031502 (2014) I. Mudawar, Two-phase microchannel heat sinks: theory, applications, and limitations. ASME J. Electron. Packag. 133(4), 041002 (2011) A. Mukherjee, S.G. Kandlikar, The effect of inlet constriction on bubble growth during flow boiling in microchannels. Int. J. Heat Mass Transf. 52(21), 5204–5212 (2009) B.A. Odom, M.J. Miner, C.A. Ortiz, J. Sherbeck, R. Prasher, P.E. Phelan, Microchannel two-phase flow oscillation control with an adjustable inlet orifice. ASME J. Heat Transf. 134(12), 122901 (2012) S.L. Qi, P. Zhang, R.Z. Wang, L.X. Xu, Flow boiling of liquid nitrogen in micro-tubes: part II heat transfer characteristics and critical heat flux. Int. J. Heat Mass Transf. 50(25), 5017–5030 (2007) W. Qu, I. Mudawar, Measurement and correlation of critical heat flux in two-phase micro-channel heat sinks. Int. J. Heat Mass Transf. 47(10), 2045–2059 (2003)

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R. Revellin, J.R. Thome, Adiabatic two-phase frictional pressure drops in microchannels. Exp. Thermal Fluid Sci. 31(7), 673–685 (2007a) R. Revellin, J.R. Thome, A theoretical model for the prediction of the critical heat flux in heated microchannels. Int. J. Heat Mass Transf. 51(5), 1216–1225 (2007b) A. Tamanna, P.S. Lee, Flow boiling heat transfer and pressure drop characteristics in expanding silicon microgap heat sink. Int. J. Heat Mass Transf. 82, 1–15 (2015a) A. Tamanna, P.S. Lee, Flow boiling instability characteristics in expanding silicon microgap heat sink. Int. J. Heat Mass Transf. 89, 390–405 (2015b) T. Zhang, T. Tong, J.-Y. Chang, Y. Peles, R. Prasher, M.K. Jensen, J.T. Wen, P.E. Phelan, Ledinegg instability in microchannels. Int. J. Heat Mass Transf. 52(25), 5661–5674 (2009)

Nanoparticle-Laden Flow for Solar Absorption Vikrant Khullar, Sanjeev Soni, and Himanshu Tyagi

Abstract

Nanoparticle-laden fluid (or more popularly “nanofluid”) could be engineered to efficiently absorb as well as transport solar energy. This flow involves suspension of nano-sized particles (particle size 5 cm E: Experimental results are in good agreement with the model predictions M: 1. Tailoring extinction profile as a function of depth improves thermal efficiency 2. Efficiency improvement of 6% as compared to conventional surface absorption-based systems E: 1. The shear viscosity increases with increasing volume fraction and decreases with increase in

Major findings E: Experimental

(continued)

(Han et al. 2011)

(Otanicar et al. 2011)

References (Lenert and Wang 2012)

Nanoparticle-Laden Flow for Solar Absorption 21

NCT

NCT

Concentrating type (CT)/nonconcentrating type (NCT)

Cylinder

Cylinder

D’ = 0.035 m

H = 0.150 m

D’ = 0.026 m

H = 0.150 m

Receiver geometry Shape Dimensions

Solar thermal system

Table 3 (continued)

Solar simulator (Newport Co. Oriel Xenon Arc lamp)

Sun

Irradiance source

Gold nanoparticles (GNP)/water

CNTs/ ethylene glycol

Nanoparticle dispersion employed

E: 20–34
C

E: 24–40
C

Operating temperatures (
C) temperature at the same shear rate 2. Thermal conductivity of carbon black nanofluids increases with increase in volume fraction and temperature 3. Carbon black nanofluids have good solar absorption efficiency E: 1. CNTs with oxidation treatment exhibit good dispersing performance 2. At room temperature, 18% enhancement was found in the photothermal conversion efficiency of the 0.5% mass fraction CNTs glycol nanofluids in comparison to the basefluids E: 1. GNPs have good photo-thermal conversion efficiency, at particle concentration of 0.15 ppm,

M: Modeling

Major findings E: Experimental

(Zhang et al. 2014)

(Meng et al. 2012)

References

22 V. Khullar et al.

CT, parabolic dish collector

NCT, flat plate collector

NCT, flat plate collector

Cuboid

Cuboid

Cuboid

H = 0.001 m

L’ = 0.02 m W = 0.02 m

A’ = 1.51 m2

A’ = 1.51 m2

H = 0.003 m

Sun

Sun

Sun

Graphite/ Therminol VP-1

Al2O3/Water

MWCNT/ water

E: 270
C

E: 30–40
C

E: 30–40
C

GNP increases the photothermal conversion efficiency of basefluid by 20% and reaches a specific absorption rate of 10 kWg1 2. Photothermal conversion efficiency increases with increasing volume fraction E: Efficiency of solar collector increases as the pH is increased or decreased with respect to the isoelectric point E: Using 0.2 Wt% Al2O3 nanofluid as absorbing medium in a flat plate solar water heater increases the efficiency by 28.3% M: 1. In case of nanofluidbased receivers, efficiency enhancement of 10% can be achieved relative to surface absorption-based receivers when concentration ratios are in the range of 100–1000 2. Graphite nanofluids with volume fraction on the order of 0.001% are suitable for 10–100 MWe power plants E: Experiments on laboratory scale nanofluid(continued)

(Taylor et al. 2011)

(Yousefi et al. 2012b)

(Yousefi et al. 2012a)

Nanoparticle-Laden Flow for Solar Absorption 23

CT

Concentrating type (CT)/nonconcentrating type (NCT)

Cuboid

H= 0.001 m–0.01 m

Receiver geometry Shape Dimensions

Solar thermal system

Table 3 (continued)

Sun

Irradiance source

Graphite/ Therminol VP-1

Nanoparticle dispersion employed

–

Operating temperatures (
C) based dish receiver suggest that 10% increase in efficiency can be achieved relative to surface absorption collector if operating conditions are carefully chosen M: 1. An analytical model was formulated to investigate the effect of heat loss, particle loading, solar concentration, and channel height on the receiver efficiency 2. Model predicts an optimum total efficiency of 0.35 for a volumetric receiver (dimensionless receiver length of 0.86) employing graphite nanoparticles dispersed in Therminol VP-1

M: Modeling

Major findings E: Experimental

Veeraragavan et al. 2012

References

24 V. Khullar et al.

CT, potential application in central receivers

CT

NCT

Cylinder

Cuboid

Cuboid

D’ = 0.008 m, 0.035 m H = 0.008 m, 0.010 m

H= 0.000150 m

A’ = 0.0015 m2

H = 0.0012 m

Artificial light source with optical fiber, color temperature 3400 K

SuperPAR64 lamp (color temperature of 3158 K)

Sun

Amorphous carbon/ ethylene glycol MWCNT/ distilled water

Graphite, CNT, silver/ water

Aluminum/ Water

E: 95
C

E: 25–45
C

M: 34–35
C M: 1. The presence of nanoparticles increases the absorption of incident radiation by more than nine times over that of pure water 2. Under similar operating conditions, the efficiency of a direct absorption collector using nanofluid as the working fluid is found to be up to 10% higher than of conventional flat plate collector E: 1. Efficiency improvements of up to 5% in solar thermal collectors by utilizing nanofluids as the absorption mechanism 2. Experimental results demonstrate an initial rapid increase in efficiency with volume fraction, followed by leveling off in efficiency as volume fraction continues to increase E: Under similar operating conditions, higher average stagnation temperatures are achievable in the case of nanofluid-based volumetric absorption systems as compared to solar selective surface absorption-based system (continued)

(Khullar et al. 2014)

(Otanicar et al. 2010a)

(Tyagi et al. 2009)

Nanoparticle-Laden Flow for Solar Absorption 25

CT, Fresnel lens

Concentrating type (CT)/nonconcentrating type (NCT)

H = 0.002 m

Receiver geometry Shape Dimensions L’ = 0.1 v m W = 0.05 m Cuboid

Solar thermal system

Table 3 (continued)

Irradiance source Sun

CuSO4 solution

Nanoparticle dispersion employed Graphite/ water

M: 70
C

Operating temperatures (
C) E: 70
C M: Modeling M: Increase in concentration ratio of solar radiation significantly enhances the system efficiency E: Compared to the system employing basefluid the absorbing liquid based system has 28% higher thermal efficiency

Major findings E: Experimental

References (Kaluri et al. 2015)

26 V. Khullar et al.

Nanoparticle-Laden Flow for Solar Absorption

27

Conclusions and Future Scope Extensive research has being going on in recent years into the theoretical as well as experimental aspects of applications of nanoparticle-laden flows in direct absorption solar volumetric collectors. In principle, it can be concluded from these investigations that increase in efficiency can definitely be achieved relative to surface absorption-based solar collector if operating conditions are carefully chosen. Continued research into these systems is warranted in order to realize such nanofluidbased collectors on a commercial scale. As for as the theoretical modeling pertinent to nanofluid-based volumetric collectors is concerned, investigators have developed various numerical (Tyagi et al. 2009; Otanicar et al. 2010a; Lenert and Wang 2012) and analytical (Veeraragavan et al. 2012) heat transfer models for predicting the efficiency of the collectors employing nanoparticle dispersions. These attempts have been quite successful in understanding the radiative and conductive heat transfer mechanisms in direct absorption nanofluid-based receivers. On the experimental front, the investigations have been carried out on both miniature collectors having stagnant nanofluid layers as well as nanofluid-based micro-channel flow collectors under laboratory conditions. Majority of these studies are limited to low operating temperatures (maximum operating temperatures being approx. 270
C). However, the real benefit is expected to be demonstrated at higher operating temperatures (typically 400–500
C) and under high solar concentration ratios owing to the fact these systems shall offer better conversion efficiency when large amount of solar irradiance needs to be transferred to the working fluid. Solar power towers seem to be the most promising solar thermal systems where these nanofluids could be used as the working fluid to volumetrically absorb the solar thermal energy. Furthermore, heat mirror-based covers could be engineered into these systems to further improve upon the efficiency of nanoparticle-laden fluid-based solar thermal systems. In terms of the choice of nanoparticle material is concerned, recent improvements in the stability of MWCNT-based nanofluids and the fact that they have broad absorption spectra lend carbon-based nanoparticle to be the most appropriate and inexpensive proposition for volumetric absorption of the solar thermal energy. Currently, there is real impetus to fabricate and test nanofluid-based solar volumetric collectors under outdoor conditions. Furthermore, this exercise shall play a seminal role in identification of the key parameters governing the overall performance of such volumetric solar collectors.

References M. Abdelrahman, P. Fumeaux, P. Suter, Study of solid-gas-suspensions used for direct absorption of concentrated solar radiation. Sol. Energy 22(1), 45–48 (1979) N. Arai, Y. Itaya, M. Hasatani, Development of a “volume heat-trap” type solar collector using a fine-particle semitransparent liquid suspension (FPSS) as a heat vehicle and heat storage medium Unsteady, one-dimensional heat transfer in a horizontal FPSS layer heated by thermal radiation. Sol. Energy 32(1), 49–56 (1984).

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Review of Subcooled Boiling Flow Models Eckhard Krepper and Wei Ding

Abstract

In this chapter, the present capabilities of CFD modelling for wall boiling in industrial applications are described. The basis is the Eulerian two-fluid framework of interpenetrating continua. From the first attempts, heat flux partitioning algorithms were used to describe boiling at a heated wall. Based on a mechanistic model representation of the microscopic processes, the framework is described by empirical correlations. The developments of the main correlations for the bubble size at detachment and for the nucleation site density are described. Different approaches for the bubble size in the bulk are presented. Further the extension of the conventional heat partitioning model toward the high heat flux will be also stated. Finally an outlook on further model improvement is given. Keywords

Boiling • Bubble detachment frequency • Bubble diameter at detachment • CFD • Euler-Eulerian approach • Nucleation site density • Wall boiling

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wall Boiling in the Framework of Euler/Euler Two-Phase Flow CFD Modelling . . . . . . . . . . . . . Modelling of Boiling at a Heated Wall by Heat Flux Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 4 4 4

E. Krepper (*) Institute of Fluid-Dynamics, Helmholtz-Zentrum Dresden – Rossendorf, Dresden, Germany Computational Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany e-mail: [email protected] W. Ding Computational Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_20-1

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Bubble Detachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleation Site Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Influence Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaporation and Condensation in the Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bubble Size in the Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Extensions Toward Higher Heat Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook of Further Model Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Subprocess Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 9 11 13 14 18 19 19 21 22 22 24

Introduction Subcooled flow boiling becomes an essential phenomenon in many industrial applications when large heat transfer coefficients are required. For lots of flow conditions, heat transfer from a wall into a fluid can be much more effective than for single-phase convection. The design or the safety assessment of such applications can be supported by simulation of these processes. The computational fluid dynamic (CFD) modelling describes the phenomena dependent only on local quantities. Therefore, it is suited especially for design and optimization of the flow geometry. For engineering calculations, currently the most widely used CFD approach to model two-phase flows with significant volume fractions of both phases is the Eulerian two-fluid framework of interpenetrating continua (see, e.g., Ishii 1975; Drew and Passman 1998; Yeoh and Tu 2010). In this approach, balance equations for mass, momentum, and energy are written for each phase, i.e., gas and liquid, separately and weighted by the so-called volume fraction which represents the ensemble averaged probability of occurrence for each phase at a certain point in time and space. Microscopic phenomena are averaged and considered by empirical correlations. Exchange terms between the phases appear as source/sink terms in the balance equations. These exchange terms express the interfacial forces, as well as heat and mass fluxes, as functions of the average flow parameters. Since most of these correlations are highly problem specific, their range of validity has to be carefully considered. Up to medium values of the gas volume fraction, the two-phase flow is in the bubbly flow regime. In this instance, the mass and momentum exchange between the phases is conveniently parametrized by the bubble size. To account for deformations the equivalent spherical diameter is used as a measure of bubble size. At high Reynolds numbers common to engineering flows, the bubbles strongly affect the turbulence intensity and structure in the bulk and near walls. These questions have been investigated extensively for adiabatic flows (see, e.g., Lucas et al. 2007), but not all issues have been resolved satisfactorily yet.

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3

For the case of boiling flows, where heat is transferred into the fluid from a heated wall at such high rates that vapour is generated, additional source terms describing the physics of these processes at the heated wall have to be included. In the past, several CFD wall boiling model approaches following the lines of Kurul and Podowski (1990, 1991) were calibrated and validated by several authors against different experimental results. In this approach according to a microscopic model concept, a given overall heat flux is divided into various components. An application of continuing high interest is the thermal hydraulic flow in the core of a nuclear reactor. Accordingly in most of these tests, subcooled flow boiling of water at high pressure flowing upward in a vertical pipe heated from the outside was investigated, and measurements of the axial development of gas volume fraction, wall temperature, and cross-sectionally averaged liquid temperature were provided. However, typical flow conditions encountered in this application do not particularly lend themselves to experimental investigation. High pressure, high temperature, narrow channels, and small expected sizes of steam bubbles represent significant challenges for measurements. The use of refrigerants can greatly relieve this burden. Advantages are that this allows a choice of test parameters that is more convenient for the measurement compared to the water/steam system at high pressure. The same vapour/liquid density ratio can be achieved at a much lower system pressure, and the same Reynolds number can be achieved at a larger diameter of the heated pipe. This enables a measurement of radial profiles for gas volume fraction, temperature, liquid, and gas velocities and of bubble sizes which allows a stringent validation of CFD models.

Critical Heat Flux For lots of flow conditions, heat transfer from a wall into a fluid can be much more effective than for single-phase convection. However, the efficient heat transfer mechanism provided by vapour generation is limited at a point where liquid is expelled from the surface over a significant area. This occurs at the critical heat flux where the heat transfer coefficient begins to decrease with increasing temperature leading to an unstable situation. In this event, a rapid heater temperature excursion occurs which potentially leads to heater melting and destruction. For a given working fluid, the critical heat flux depends on the flow parameters as well as the geometry of the flow domain. The verification of design improvements and their influence on the critical heat flux requires expensive experiments. Therefore, the supplementation or even the replacement of experiments by numerical analyses is of high interest in industrial applications. In the past, many different empirical correlations for critical heat flux were developed and fitted to data obtained from experimental tests. These have been implemented mainly in purpose-specific 1D codes and applied for engineering design calculations. However, these correlations are valid only in the limited region of fluid properties, working conditions, and geometry corresponding to the tests to which they were fitted. Using large lookup tables based on a great number of

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experiments, a significant range of fluid properties and working conditions can be covered. But this method is still limited to only that specific geometry for which they were developed. Independence of the geometry can only be achieved by the application of computational fluid dynamic (CFD) methods. Existing CFD models, however, are not yet able to describe critical heat flux reliably. A precondition would be the complete understanding and simulation of boiling as a preliminary state toward critical heat flux.

Wall Boiling in the Framework of Euler/Euler Two-Phase Flow CFD Modelling The general equations for diabatic two-phase flow in the Euler/Euler framework of interpenetrating continua have been reviewed in many places before (e.g., Ishii 1975; Drew and Passman 1998; Yeoh and Tu 2009 and the corresponding chapter of the actual handbook). Therefore, here the focus is directed only on those issues that are particularly relevant for the simulation of boiling. The first major block is the wall boiling model describing vapour generation at the wall and transfer of sensible heat to the liquid. Here, most of the CFD model implementations closely follow the heat flux partitioning approach. A second issue are the phase change phenomena occurring in the bulk fluid. Here, the relative amount of liquid and vapour is allowed to change by condensation/ evaporation depending on the net transport of heat to the liquid-vapour interface. To simplify matters, the vapour bubbles are assumed to be at saturation temperature everywhere which is a rather good approximation except close to the critical heat flux. In the first model approaches, a monodispersed size distribution was assumed with a size dependent on the liquid temperature. More developed models consider the bubble size distribution by momentum methods respective by population balance models. Turbulent fluctuations are modelled by a shear stress turbulence (SST) model according to Menter (1994, 2009) applied to the liquid phase. This corresponds to a k-ω model near the walls and a k-ε model far from walls. The enhancement of liquid turbulence caused by the bubbles is considered following Sato et al. (1981) by an additive contribution to the effective viscosity. In addition, a wall function for boiling flows suggested by Ramstorfer et al. (2005) and subsequently verified by Koncar and Krepper (2008) and Koncar and Mavkov (2010) was tested. For momentum exchange between the phases, finally, lift and turbulent dispersion wall forces are included in the model in addition to the ubiquitous drag force. The consideration of these forces has been found in good agreement with data for adiabatic air water bubbly flow (e.g., Lucas et al. 2007).

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5

Modelling of Boiling at a Heated Wall by Heat Flux Partitioning General Model Structure Modelling of wall boiling is based on the microscopic phenomena observed in the near wall region and transformed either by direct simulation or by empirical correlations to the Euler/Euler framework. Real wall surfaces have irregularities such as pits or scratches. The initial nucleation of a vapour bubble typical occurs at such imperfections of the surface. The bubble grows during a time tg and finally leaves the wall. During this time, heat QE is consumed by the vapour generation. After the release of the initial bubble, cooler bulk liquid comes in contact to the surface where the bubble was previously attached. This new liquid adjacent to the wall leads to cooling QQ. This mechanism, which is obviously not present in single-phase flows, is termed quenching. After a waiting time tw, the thermal layer near the wall is reformed, and a new bubble occurs at the same place. This so-called ebullition cycle extends over a time τ = tg + tw . Consequently, the departure frequency f is calculated as f = 1/τ. On parts of the wall, where no bubbles reside, heat QC flows directly to the subcooled liquid in the same way as in single-phase flow. The complete cycle is presented in Fig. 1. Judd and Hwang (1976) first proposed the heat flux partitioning concept which was further developed by Kurul and Podowski (1990, 1991). Detailed informations on heat flux partitioning are also given by Yeoh et al. (2008). A given overall het flux Qtot is divided into various components according to a microscopic model concept (see Fig. 1). Accordingly, the given external heat flux Qtot, applied to the heated wall, is written as a sum of three parts: Qtot ¼ QC þ QQ þ QE

Fig. 1 Microscopic concept of the heat flux partitioning model

(1)

6 Fig. 2 Principal algorithm of the heat flux partition model approach

E. Krepper and W. Ding

TL and U from flow solution guess for TW (nested intervals) calculation of evaporation heat flux convective heat flux quenching heat flux

π 3 d BW fNh1g 6 QC = ( 1 − AW )hC (TW − TL )

QE =

QQ = AW hQ (TW − TL )

overall heat balance already fulfilled? No

?

QW = QE + QF + QQ

Yes where QC, QQ, and QE denote the heat flux components due to single-phase turbulent convection, quenching, and evaporation, respectively. The individual components in this heat flux partitioning are then modelled as functions of the wall temperature and other local flow parameters. Once this is accomplished, Eq. 1 can be solved iteratively for the local wall temperature TW, which satisfies the wall heat flux balance (see Fig. 2). Denoting the fraction of area influenced by the bubbles as AW, the heat flux components are expressed as discussed in the following. The turbulent convection heat flux is calculated in the CFX model version in much the same way as for a pure liquid flow without boiling, but multiplied by the fraction of area unaffected by the bubbles, i.e., QC ¼ AconvL hCL ðT W  T L Þ

(2)

Here AconvL = 1  AW, with AW as the area fraction occupied by bubbles. In some implementations, TL is taken from the first grid cell adjacent the heated wall. To avoid a grid size dependency, Wintterle (2004) proposed to consider here the turbulent wall function. Here hCL is the heat transfer coefficient which is written using the temperature wall function T+(y+) known from Kader (1981) as hCL ¼

ρ C P uτ Tþ

(3)

where nondimensional variables (indicated by superscript “+”) and the friction velocity uτ are defined as usual. QQ is represented in terms of the quenching heat transfer coefficient hQ: QQ ¼ AW hQ ðT W  T L Þ

(4)

Review of Subcooled Boiling Flow Models

7

A grid independent solution for QQ is obtained by evaluating the nondimensional temperature profile at a fixed value of y+. The evaporation heat flux QE is obtained via the evaporation mass flux at the wall: QE ¼ m_ W H LG

(5)

where the generated vapour mass m_ W is expressed in terms of the bubble diameter at detachment dW, bubble generation frequency f, and nucleation site density N as π m_ W ¼ ρG d3W f N 6

(6)

Correlations for the yet undetermined quantities used in the heat flux partitioning wall boiling model are discussed in the following.

Bubble Detachment The bubble size at detachment depends on the liquid subcooling. Also the liquid properties, the system pressure, wall material and surface properties, and the heat flux have an influence. Finally the mechanical attraction of the surrounding flow, as indicated by the fluid velocity, determines the detachment of a growing bubble. For CFD modelling of boiling, different correlations for bubble detachment were developed derived from measurements or analytical considerations.

Tolubinsky and Kostanchuk An investigation of the bubble size at detachment was performed by Tolubinsky and Kostanchuk (1970) for water at different pressures and subcoolings. The observed dependence on the liquid subcooling at atmospheric pressure can be fitted to a correlation T

sat  ΔT

d W ¼ dref e

T L refd

(7)

To match the tests of Bartholomej and Chanturiya (1967) which were conducted at much higher pressures relevant under typical nuclear energy applications, the values of dref and DTrefdwere adjusted to dref = 0.6 mmand ΔTrefd = 45K(e.g., Krepper et al. 2007).

Fritz An attempt to determine the bubble diameter at detachment at atmospheric pressure was performed by Fritz (1935). He integrated the surface of the arising bubble and analyzed a force balance.
 σ 0:5 dw ¼ CΘ gΔρ

(8)

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Here Θ is the wall contact angle and was estimated for water/steel to π/4. σ is the surface tension and Δρ the density difference between liquid and gas.

Kocamustafaogullari To extend the range of validity, the correlation of Eq. 8 was extended by Kocamustafaogullari (1983): dw ¼ 2:64  10

5




σ gΔρ

0:5

Δρ ρg

!0:9 (9)

The correlation was supported by experiments with water in the pressure range of 0.01–14 MPa. For atmospheric pressure, Eq. 10 reduces to Eq. 9.

Unal A very detailed correlation for the bubble diameter at detachment was developed by Unal (1976). He tried to consider both the thermal wall material properties of the heater and the fluid field of the surrounding flow: dw ¼

2:42  105 P0:709 a

ðr bΦffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ0:5 ffi ΔT sup ρS CPS λS a¼ π 2ρg H lv ΔT sub ! b¼ ρl 2 1 ρ
g  v Φ ¼ max , 1½m=s v0

(10)

Bubble Detachment Based on Force Balance Klausner et al. (1993) proposed a mechanistic force balance model for the prediction of both bubble departure and lift-off sizes in the nucleate boiling condition. They applied the model successfully in the various flow conditions in both horizontal and vertical channels under pool and flow boiling. Later, many investigators tried to improve the model to achieve a general applicability for flow direction and fluids. Zeng et al. (1993a, b) applied the model for both horizontal and vertical channels under pool and flow boiling, whereas Situ et al. (2005) and Yeoh and Tu (2005) extended its application to steam-water boiling flow condition. The concept of Klausner et al. (1993) is based on a force balance at the arising bubble (see Fig. 3). Based on the conservation of momentum, in the direction parallel (tangential t – direction) and perpendicular (vertical n- direction) to the heating wall, the forces acting on the bubble can be obtained as

Review of Subcooled Boiling Flow Models

9

Fig. 3 Schematic diagram of a growing bubble at a heated surface

Fn ¼ Fgroth, n þ Fdrag, n þ Fcpress, n þ Fsl, n þ Fb, n þ Fsurf , n

(11)

Ft ¼ Fgrowth, t þ Fdrag, t þ Fb, t þ Fsurf , t

(12)

Fgrowth is the bubble growth force, Fdrag the quasi-steady drag force due to the viscous fluid flowing around the bubble, Fcpress a contact pressure due to the effect of the wall, Fsl a force resulted from the flow distribution, Fb the buoyancy force, and Fsurf the surface tension. When the bubble is fixed at the heating surface and growing out of an nucleation site, the sum of all forces has to be zero:Fn = Ft = 0. In horizontal pool boiling, the detachment or lift off occurs when the Fn becomes positive. In flow boiling or non-horizontal pool boiling, detachment occurs when Fn or Ft becomes larger than 0. If Ft becomes greater than 0, firstly the bubble departs from the nucleation site and starts sliding along the wall until it lift off when the force in perpendicular direction becomes positive. Conversely, the bubble directly lifts off from the nucleation site without sliding when Fn becomes positive firstly. The phenomenon of sliding bubbles was observed also in experiments (e.g., Prodanovic et al. 2002). This model concept was further developed by different authors formulating more precise the single components of the forces in Eqs. 11 and 12 (e.g., Chen et al. 2012).

Nucleation Site Density If the nucleation temperature of the liquid is reached, a vapor bubble is formed. The nucleation temperature slightly exceeds the saturation temperature of the fluid. The

10

E. Krepper and W. Ding

initial nucleation of vapor typically occurs at a cavity or crevice on the heated surface.

Lemmert and Chawla Lemmert and Chawla (1977) proposed a quite simple correlation for the calculation of the nucleation site density dependent on the wall superheating temperature:
N ¼ N ref

TW  TL ΔT refN

p (13)

More Detailed Models Kocamustafaogullari and Ishii (1983) developed a nucleation site density correlation based on measurements. They considered both surface conditions and fluid properties. N¼

2:157  107 ρ3:2 ð1 þ 0:0049ρ Þ4:13 RC 4:4  Δρ ,ρ ¼ ρg dw 2

(14)

with the critical cavity size RC. Additional in this analysis, they derived a correlation for the bubble size at detachment (see section “Kocamustafaogullari”, Eq. 9). Yang and Kim (1988) considered the probability distribution of the cavity size. They investigated the activation capability of a cone dependent on the cone angle. Wang and Dhir (1993) proposed an empirical correlation at pool boiling based on measurements including the effect of contact angle. During the tests, they influenced the wettability of the heated surface and measured the effect. For pool boiling, they derived N ¼ 7:81  1029 ð1  cos θÞRC 6:0

(15)

with RC as the critical cavity radius. Benjamin and Balakrishnan (1997) investigated the effect of the heat flux on the nucleation site density. They considered different heater materials (stainless steel and aluminum) with different surface treatment by polishing. Furthermore, different fluids were investigated. Their correlation for pool boiling is N ¼ 218:8Pr

1:63


 1 0:4 Ω ΔT W 3 γ

(16)

with the Prandtl number Pr, a surface-liquid interaction parameter γ, and a dimensionless surface roughness parameter Ω. Also Basu et al. (2002) analyzed experiments and tried to consider the contact angle. During the tests, they controlled the surface wettability following a welldefined surface treatment.

Review of Subcooled Boiling Flow Models

11

Hibiki and Ishii (2003) performed a very detailed investigation of the different dependencies of the nucleation site density. They tried to collect the experiences of the previous described correlations. The dependency on surface conditions and fluid properties was expressed as dependencies on pressure and on the minimum cavity size.  N ¼ N ref


     0 θ2 þ λ 1  exp  2 exp f ðρ Þ 1 8μ Rc

(17)

with the parameters m = 0.722, L0 = 2.5.106 m, the contact angle Θ, the critical cavity radius Rc n

o 2σ 1 þ ρg =ρf =Pf

   Rc ¼ exp ifg T g  T sat = RTg T sat  1

(18)

and the function f ðρþ Þ ¼ 0:01064 þ 0:48246ρþ  0:22712ρþ2 þ 0:05468ρþ3 , ρþ ¼ logðρ Þ (19) Later, it was proposed to replace the function f(ρ+) by measurements. Kandlikar and Steinke (2002) performed corresponding experiments.

General Considerations on Nucleation Site Density Most of the theoretical and experimental investigations show a clear dependency of the nucleation site density on the wall superheating temperature. This is expressed in the correlation of Lemmert and Chawla (1977); see Eq. 7. However, a recent compilation (Kolev 2006) shows that vastly different parameter values are required to match different data sets. A likely reason for this fact is that nucleation site density is highly dependent on the microscale topography of the boiling surface, which in turn depends strongly on the processes that were used to finish the surface. These processes are very diverse, in most boiling experiments not specifically controlled, and in most cases an unknown boundary condition for the simulation. This is the reason for the doubts of the authors, to find any reasonable predictive model for nucleation site density. Krepper et al. (2013) proposed the following solution strategy: Dealing with the model framework shows that the nucleation site density has almost no influence on the liquid temperature and a small influence on the gas volume fraction, but a strong influence on the wall superheatTW  Tsat. In the most common case of missing information on the nucleation site density, this can be compensated by the measured wall temperature which is in most boiling experiments easily accessible. It might be reasonable to apply a quite simple relation between nucleation site density and wall temperature as in Eq. 13 and determine Nref in a way that the calculated wall temperature agrees to the measured value.

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Other Influence Factors Bubble Influence Area/Factor In terms of bubble detachment diameter dW and nucleation site density N, the wall area fraction AW, influenced by vapor bubbles, is given by


dW AW ¼ π a 2

2 N

(20)

Here, a is the so-called bubble influence factor, for which a value of 2 is commonly used (Kurul and Podowski 1990, 1991). Direct experimental evidence concerning this quantity is rather scarce. Probably the most relevant source is Han and Griffith (1965) who in some “rough” experiments determined the hydrodynamic disturbance caused by lifting a spherical particle from a horizontal surface and found that it has a range of twice the size of the particle. A similar size has been claimed by Cieslinski (2005) from PIV measurement of the flow field around departing bubbles, although the quality of the images presented is rather poor. AW = 1 corresponds to the case where the whole surface is under the influence of bubbles. The parameters dW, a, and N should be such that the calculated therefrom AW by Eq. 20 results in smaller values. Moreover, it should be kept in mind that already as AW approaches 1, the assumptions of the model are not really satisfied anymore.

Bubble Detachment Frequency The bubble detachment frequency f is given according to Cole (1960) as a function of the detachment size dW. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4gðρL  ρG Þ f ¼ 3CD dW ρL

(21)

Relations of this type have been reviewed critically by Ivey (1967) and CeumernLindenstjerna (1977).

Quenching Heat Transfer Coefficient The quenching heat transfer coefficient is calculated using the analytical solution for one-dimensional transient conduction, as suggested by Mikic and Rohsenow (1969); Mikic et al. (1970): 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hQ ¼ pffiffiffi f twait kL ρL CPL π

(22)

where twait is waiting time between the bubble departure and the appearance of a next bubble at the same nucleation site. A simple assumption by Kurul and Podowski

Review of Subcooled Boiling Flow Models

13

(1990, 1991) is adopted also here, where the waiting time takes a fixed fraction of the bubble departure period: twait ¼

0:8 f

(23)

Data supporting this simple estimate come from the work of del Valle and Kenning (1985) which however is limited to the regime where the heat flux is 75% of the critical heat flux and larger.

Turbulent Wall Functions for Boiling Flow The first wall boiling model originally was developed for the application in a 1D system code. Then the liquid temperature can be taken from the cell center. In the first CFD implementations of the model, the temperature was taken from the first cell adjacent to the heated wall, resulting in a grid dependency. Ramstorfer et al. (2005) proposed and subsequently Koncar and Krepper (2008) and Koncar and Mavkov (2010) verified a turbulent wall function for boiling flow where bubbles grow on the heated wall by analogy with flow over a rough wall. The presence of the bubbles forces the liquid into a similar flow pattern as that observed in single-phase turbulent flow with wall roughness. The latter is described by a modified law of the wall (White 1991; Pope 2000). u 1 y

1 y

¼ ln þ B ¼ ln þ B uτ κ δ κ s

(24)

s / Nd 3W

(25)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where uτ ¼ ðτW =ρÞis the friction velocity, δ = μ/(ρuτ) is the viscous length scale, s is the hydrodynamic roughness length scale, and k  0.41 is the von Karman constant. For flow over smooth walls, B  const  5.5, while for flow over rough walls, B* is a function of s/δ which in the limit of large s, i.e., for the so-called fully rough walls, tends to a constant value of 8.5. An essentially new concept used here is to directly relate the hydrodynamic roughness s directly to the bubble size and nucleation site density as

The constant of proportionality is not known from theoretical considerations at present, so its value is determined by matching the data. As will be shown by the comparison in section 7.1, the representation of radial velocity profiles is greatly enhanced by employing this two-phase boiling wall function over the often used single-phase wall function. Moreover, a single value for the constant applies to a range of flow situations.

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E. Krepper and W. Ding

Evaporation and Condensation in the Bulk Vapor is assumed to be at saturation condition. Where the liquid is subcooled, i.e., TL < Tsatvapor is condensing with the mass transfer rate per unit volume: m_ ¼ max


 hLG ðT sat  T L ÞAI ,0 H LG

(26)

while in superheated liquid, fluid is evaporating at the rate
 hLG ðT L  T sat ÞAI m_ ¼ max ,0 H LG

(27)

Here, hLG is the interfacial heat transfer coefficient, calculated according to Ranz and Marshall (1952) as hLG ¼

kL kL Nu ¼ 2 þ 0:6Re1=2 Pr1=3 dB dB

(28)

and AI is the interfacial area, which is expressed in terms of gas volume fraction and equivalent spherical bubble diameter as AI ¼

6 αG dB

(29)

Bubble Size in the Bulk RPI Approach To close the phase transition model in the bulk bubble flow with a mean bubble diameter dB, Kurul and Podowski (1990, 1991) and also Anglart et al. (1997) proposed to calculate the bubble diameter dB locally as a linear function of liquid subcooling Tsub:   dB, 1 T sub  T sub, 2 þ dB, 2 T sub  T sub, 1 dB ¼ T sub, 2  T sub, 1

(30)

Reference subcooling conditions for typical nuclear energy applications have been given as dB,1 = 0.1 mm at Tsub , 1 =  13.5 Kand dB,2 = 2 mm at Tsub,2 = 5 K in Anglart et al. (1997). Near a heated wall with increasing wall distance, the subcooling temperature decreases. Calculating dB according Eq. 30 also dB would decrease with increasing wall distance. In boiling tests however in many cases, an increase in dB with increase

Review of Subcooled Boiling Flow Models

15

in wall distance is observed, which might be caused by bubble coalescence. This effect cannot be described by Eq. 30.

Zeitoun and Shoukri Zeitoun and Shoukri (1996) performed experiments in a vertical channel under low pressure and mass flux conditions. High-speed video techniques combined with digital image processing were applied to measure the bubble sizes of the generated vapor. For the bubble size in the bulk db, the following correlation was derived:

1:326 0:0683 ρ =ρ l g db pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0

1:326 1 σ=gΔρ 149:2 ρl =ρg B C Re0:324 @Ja þ A Bo0:487 Re1:6

(31)

iMUSIG Approach A more detailed calculation of the interfacial area density is possible by an additional transport equation, by a momentum method, or most comprehensively by a bubble class approach. Here the latter method is described short for the application to boiling processes. A detailed description can be found on another place of the current handbook. To describe polydispersed flows within a purely Eulerian approach, a number of different (MUltiple) bubble size groups i = 1 . . . M are considered, each representing bubbles of typical size di. The fraction of gas volume contained in each bubble size group is denoted as αi so that the total gas volume fraction is given by αG ¼

M X

αi

(32)

i¼1

From a theoretical point of view, there would be no problem to define each size group i as additional phase with the complete setup of Navier-Stokes equations. Practical reasons of computational effort and numerical stability are limiting factors of such a procedure. Instead, the different size groups are considered only in the continuous equation. Between the size groups, the exchange both by bubble coalescence and fragmentation and by condensation and evaporation is organized. The advantage is that a large number of bubble size groups can be considered while keeping the computational effort within reasonable bounds. On the other hand, profound effects of bubble size are missed entirely like, for example, the change in the sign of the lift force as discussed in other places of this handbook. To capture such phenomena, provision has to be made that bubbles of different size may move with different velocities. To overcome these limitations, the inhomogeneous MUSIG model (Frank et al. 2008; Krepper et al. 2008) was developed and applied to boiling

16

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Inhomgeneous MUSIG model Velocity groups J=1..N

V2

...

condensation

evaporation

V1

Size fractions K=1..∑MJ d1

dM1 dM1+1

bubble break-up

dM1+M2

VN

d∑MJ

bubble coalescence

Fig. 4 Principle schema of the inhomogeneous MUSIG model including phase transfer

phenomena in Krepper et al. (2013) and Krepper and Rzehak (2014). The schema is shown in Fig. 4. It is then useful to define occupation numbersfi = ai/aG giving the contribution of each size group to the total gas volume fraction. Obviously we then have Σi fi = 1. For each size group, the equation of mass conservation assumes the form @ ðαi ρG Þ þ ∇ðαi ρG uG Þ ¼ Γ topo þ Γ phase i i @t

(33)

where the right-hand side gives the net source of mass for group i which results from topological changes due to coalescence and breakup as well as phase change due to condensation and evaporation. For the homogeneous MUSIG model, only one momentum and energy equation for the total amount of vapor is considered as well as the conservation equations of the liquid of course. In these equations, the total gas volume fraction αG is calculated according to Eq. 32. In addition, also the bubble size dB appears which is taken in the Sauter sense representing the interfacial area AI = 6αG/dB. In order to preserve this interpretation, dB is calculated from the occupation number and bubble size for each group as dB ¼

M X fi d i¼1 i

!1 (34)

Bubble Coalescence and Breakup The net mass source for size group i due to bubble coalescence and breakup can be expressed as the sum of bubble birth rates due to the breakup of larger bubbles from groups j > i to group i and coalescence of smaller bubbles from size groups j < i to group i as well as bubble death rates due to breakup of bubbles from size group i to smaller bubbles in groups j < i and the coalescence of bubbles from size group i with bubbles from any other group to even larger ones which belong to groups j > i. That is,

Review of Subcooled Boiling Flow Models

Γtopo ¼ Bbreak  Dbreak þ Bcoal  Dcoal i i i i i

17

(35)

The birth and death rates in turn are commonly expressed in terms of the coalescence and breakup kernels such that X  mi b mj , mi f j Δmi mj j>i X  1 Di break ¼ ρG αG b mi , mj f i Δmj 2 j 0, i:e: evaporation Γ Γ condensation mi  mi1 miþ1  mi

(38) where mi = r pd3/6 is the mass of each bubble in size group i. Basing the calculation on bubble mass rather than size for compressible flows has the advantage that since

18

E. Krepper and W. Ding

mass is conserved, no extra terms arise in the equations. Conversion to the corresponding bubble size which depends on the local density can be done straight forwardly as needed. For incompressible flows, no differences between mass- and size-based groups arise. The liquid side heat transfer coefficient finally is calculated according to Ranz and Marshall (1952) as in Eq. 28. In addition to the source terms for the continuity equations for the bubble size groups, there is also a mass source for the liquid phase continuity equation which is given by ΓL ¼ 

X

ei Γ

(39)

i

Moreover, corresponding secondary sources appear in the momentum and energy equations. To include the generation of vapor bubbles at the heated wall, an additional source term, Srpi, is included in Eq. 40 (Lifante et al. 2011). This source term applies only to the equation corresponding to the size group whose diameter is the closest to the bubble detachment diameter dW. It is given by the evaporation mass flux computed in the wall heat partitioning distributed evenly throughout the grid cells adjacent to the heated wall, i.e., Srpi ¼ m_ W

S V

(40)

where m_ W is given by Eq. 6 and S and V are wall surface area and volume of the corresponding grid cell, respectively.

Model Extensions Toward Higher Heat Fluxes Newer implementations of the heat flux partitioning model (e.g., Lifante 2013; Mimouni et al. 2016) include model developments toward the simulation of higher heat fluxes. The original heat flux partitioning model considers steam only at saturation conditions. For lower vapor void fractions the energy equation has to be solved only for the liquid phase. Gas will occur only at saturated conditions. At higher heat fluxes, larger vapor void fractions will occur and this simplification has to be extended by solving also the energy equation for gas and superheated steam can be simulated. The heat flux partitioning algorithm has to be extended by an additional component, the convective heat flux into gas QG. Equation 1 reads now as Qtot ¼ QC þ QQ þ QE þ QG

(41)

Review of Subcooled Boiling Flow Models

19

The additional degree of freedom has to be closed by a function describing the relation of heat transferred to liquid and to gas. For the CFX implementation, f(a)is proposed (Lifante et al. 2013): 9 8 1 ½20ðααcrit Þ > > > e 1  α α = < crit > 2
20αcrit f ðαÞ ¼ 1 α > > > α  αcrit > ; : 2 αcrit

(42)

Equation 2 reads now as QC ¼ ð1  AW Þf ðαL ÞhCL ðT W  T L Þ and similarly QG ¼ ð1  AW Þf ðαG ÞhCG ðT W  T G Þ

(43)

In the same way as Eq. 3, hCG ¼

ρG CPG uτ Tþ G

(44)

Figure 5 shows the proposed distribution functions for aLcrit = 0.2 respective αGcrit = 0.8. With good adjusted values, a typical behavior for film boiling can be described (Lifante et al. 2013). The weakness of the above referred implementation is the arbitrariness of the applied distribution function independent on the microscopic phenomena. These can be achieved by a detailed consideration of the microscopic processes in the near bubble region.

Outlook of Further Model Improvements Overview As is introduced, the Euler/Euler two-phase flow CFD-Model has been already well developed to model the global boiling process. Nevertheless, this approach is far away from a predictive tool due to the correlations described by bubble dynamics. A critical review of the detailed correlations shows that some of the parameters are not suited for a broad usage for different fluids or different pressure levels but have to be carefully recalibrated for the intended applications. In order to improve the CFD model, it is necessary to investigate the individual subprocess models that are able to present the bubble dynamics on the wall correctly. The difficulties here are due to the small scales of bubbles which have a non-negligible impact on the heat transfer; the high dynamics of boiling; the

E. Krepper and W. Ding 1.0

1.0

0.8

0.8

F(αG) [-]

F(αL) [-]

20

0.6 0.4 0.2

0.6 0.4 0.2

0.0

0.0 0.0

0.2

0.4

0.6 αL [-]

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

αG [-]

Fig. 5 Distribution functions F(αL)and F(αG) for αLcrit = 0.2 resp. αGcrit = 0.8

complexity of the transient heat transfer between solid, liquid, and vapor; and largescale heated surface. In the recent decades, several different CFD models were developed to investigate the bubble dynamics based on single-bubble simulation. Lee and Nydahl (1989) simulate the single-bubble growth in nucleation boiling based on moving mesh with the generalized arbitrary Lagrangian-Eulerian (ALE) approach. However, in their models, the assumption of a constant wall temperature and lack of suitable modeling of the detachment lead to only partial accurate prediction of the bubble dynamics. Later in order to avoid the complex mesh generation process, other interface tracking methods (ITM) were applied in the CFD simulations, such as the level set method, the volume of fluid method (VOF), the front tracking method (FTM), the color/ density function, the phase-field method (PFM), and the lattice Boltzmann approach (LB). Son et al. (1999) succeeded in modeling the bubble dynamics using level set method to track the liquid/vapor interface. The computed bubble dynamics were in good agreement with experimental measurement. However, the constant temperature assumption for the heat transfer surface makes the model be unable to predict the bubble waiting time and frequency which is very important in the global boiling process. Further, Stephan and Busse (1992) tried to simulate the single bubble dynamics based on the volume of fluid method, In this model, the temperature on the wall is allowed to vary. Due to lack of validation, the accuracy of the model is still uncertain. Recently, Sato and Niceno (2013) developed a new model based on color/density function. In this model, they could simulate the dry spot underneath the bubble; they could even show the bubble growth rate and the temperature distribution over the heat transfer surface which were in good agreement with experimental data. However, because of the large computational effort, their model is only limited in few nucleation sites and only available for pool nucleation boiling. Another big disadvantage is that the simulation domain in the models above is strictly limited in the millimeter to centimeter size. Though computational science was well developed, it is still impossible to directly simulate both small scale of bubbles (~mm, ~cm) and large scale (~meter)

Review of Subcooled Boiling Flow Models

21

of the global boiling process in one model. Nevertheless, the direct numerical simulations could yield essential information for the derivation of correlation valid in a wide range of parameters. For industrial applications, the simplification of the subprocess models based on microphysics is still required in the numerical simulation.

Individual Subprocess Models Microlayer Model Cooper and LIoyd (1969) considered the shape of the bubble is hemispherical at the first growth phase of a bubble due to the hindering by the liquid surrounding the bubble. Then there should be a “microlayer” between the vapor hemisphere and heated surface. This “microlayer” was proposed by Snyder (1956) and verified by Judd and Hwang (1976). They accounted for up to 30% of the total heat transfer for the pool boiling of dichloromethane experimentally. In the last decades, based on the microlayer theory, Zhao et al. (2002a, b, c) have established a dynamic microlayer model that focuses on individual bubbles and can be used to explain pool boiling mechanism. Their model can predict heat flux in the nucleate boiling region at high heat flux and in the transition boiling region. They concluded that in the nucleate boiling and the low-super heat region of transition boiling, the heat flux is mainly contributed by the evaporation of the microlayer. In the high-super heat region of transition boiling, the heat flux is contributed by both microlayer and macrolayer. Bubble Growth The bubble growth takes place as inertia controlled growth and thermal diffusion controlled growth. The inertia controlled growth follows the rule of Rayleigh solution established by Mikic et al. (1970). In this period, a microlayer is formed under the bubble. Later the evaporation of the microlayer and the evaporation around cap controlled by thermal diffusion contribute to the bubble growth. The thermal diffusion controlled growth follows the rule of Labuntsov (1975) solution. During the bubble growth, the new microlayer is formed due to the expansion of gas liquid interface on the wall. The evaporation of microlayer causes the dry out on the boiling wall. Bubble Departure As is introduced, Klausner et al. (1993) developed a model based on the balance of the forces acting on the bubble to predict the departure and lift off of the bubble (see also section “Bubble Detachment Based on Force Balance”). He obtained a satisfactory predictive accuracy against own data of flow boiling with refrigerant R113. In the model, the base diameter (contact diameter dw) of bubble was recommended to

22

E. Krepper and W. Ding

be 0.09 mm and advancing/receding contact angle (βad, βre) of π/4 and π/5, respectively. Later, modified versions of the Klausner model have been applied by many different authors to predict their own experimental data such as Thorncroft et al. (2001), Situ et al. (2005), Sugrue (2012), and Chen (2012) but with their own calibration of dw, βαd, and βre. Klausner applied the Mikic et al. (1970) model to simulate the bubble growth, while Situ and most of latter authors employed the Zuber (1961) formulation. Zuber included in his formulation a parameter b to account for bubble sphericity. This parameter has been used by Sugrue (2012) with different value from 0.24 to 24 as calibration to fit the models with their experiment data. With above well-developed subprocess model, the single bubble life cycle could be modelled without complex vapor/liquid interface tracking. The complexity of this sub-model should be suitable to be implemented in the modelling of the global boiling process.

Summary For the CFD modelling of boiling at a heated wall by an Euler/Euler description of two-phase flow in the past, a heat flux partitioning model describing the microscopic phenomena at the wall by empirical correlations adapted to experimental data has gained some success. Such an approach was used and adjusted to boiling experiments with water at a pressure of several MPa and also to experiments applying different refrigerants. Nevertheless, this approach is far away from a predictive tool. A critical review of the detailed correlations shows that some of the parameters are not suited for a broad usage for different fluids or different pressure levels but have to be carefully recalibrated for the intended applications. Then some deviations can be calculated with good success. An improvement of the models can only be achieved by a more detailed microphysics. This particularly is necessary when critical heat flux should be simulated. Then besides the physical micro phenomena, the morphology of the fluid has to be considered. In many cases, the models can process more information than is available from the experiment. One example is the nucleation site density; their importance on boiling modelling was shown. A model of the nucleation site density on a heated wall is very difficult, if not impossible. Even when the heat properties of the material are known commonly, no information exists on surface treatment. As a compromise the gap can be closed by matching the measured and calculated temperature of the heated wall which has a sensitive dependence on the nucleation site distribution. A suitable model must be starting from the question to be solved considering a good weight between necessary information and calculated parameters.

Review of Subcooled Boiling Flow Models

23

Nomenclature Notation a AconvL AW AI CB CD CL CP CTD CVM CW Cμ dB d⊥ dW D Eo f FD FL FTD FVM FW g hCL hCG hLG hQ H HLG Ja kL k L m_ W

Unit – – – – – – – J K1 kg3 – – – – m m m m – Hz N m3 N m3 N m3 N m3 N m3 m s2 W m2K1 W m2K1 W m2K1 W m21 J kg3 J kg3 – W m1 K1 m2s2 m kg s1

Denomination Bubble influence factor Area fraction for heat convection to liquid Area fraction occupied by bubbles (AW + AconvL = 1) Interfacial area density Bubble-induced turbulence coefficient (Sato et al. 1981) model) Drag coefficient Lift coefficient Specific heat capacity at constant pressure Turbulent dispersion coefficient Virtual mass force coefficient Wall force coefficient Shear-induced turbulence coefficient (k-ε model) Bulk bubble diameter Bubble diameter perpendicular to main motion Bubble detachment diameter Pipe diameter Eötvös Number Bubble detachment frequency Drag force Lift force Turbulent dispersion force Virtual mass force Wall force Acceleration of gravity Heat transfer coefficient for single-phase convection to liquid Heat transfer coefficient for single-phase convection to gas Heat transfer coefficient for bulk evaporation / condensation Heat transfer coefficient for quenching Specific enthalpy Specific evaporation enthalpy Jakob number Thermal conductivity Turbulent kinetic energy Length scale Vapour mass flux

Mo N p Pr Qtot QC

– m3 Pa – W m2 W m2

Morton number Nucleation site density Pressure Prandtl number Total wall heat flux Heat flux due to single-phase convection (continued)

24

Notation QQ QE r Re s t twait T TL Tsat Tsub Tsup TW u uτ U V x y α δ ε ΔT μ ν ρ σ τW Θ

E. Krepper and W. Ding

Unit W m2 W m2 m – m s s K K K K K K m s1 m s1 m s1 m3 m m – m m2s3 K kg m1s1 m2s1 kg m3 N m1 N m2 grd

Denomination Heat flux due to quenching Heat flux due to evaporation Radial coordinate Reynolds number Hydrodynamic wall roughness Time Waiting time Temperature Liquid temperature Saturation temperature Liquid subcooling Wall superheat Wall temperature Velocity Friction velocity Velocity scale Volume Axial coordinate Distance to the wall Volume fraction Viscous length scale Turbulent dissipation rate Temperature scale Dynamic viscosity Kinematic viscosity Density Surface tension Wall shear stress Wall contact angle

References H. Anglart, O. Nylund, N. Kurul, M.Z. Podowski, CFD prediction of flow and phase distribution in fuel assemblies with spacers, NURETH-7, 1995 Saratoga Springs, New York. Nucl. Eng. Des. 177, 215–228 (1997) G.G. Bartolomej, V.M. Chanturiya, Experimental study of true void fraction when boiling subcooled water in vertical tubes. Therm. Eng. 14, 123–128 (1967.) translated from Teploenergetika 14(2), 80–83 N. Basu, G.R. Warrier, V.K. Dhir, Onset of nucleate boiling and active nucleation site density during subcooled flow boiling. ASME J. Heat Transf. 124, 717–728 (2002) R.J. Benjamin, A.R. Balakrishnan, Nucleation site density in pool boiling of saturated pure liquids: effect of surface microroughness and surface and liquid physical properties. Exp. Thermal Fluid Sci. 15, 32–42 (1997)

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W.C.B. Ceumern-Lindenstjerna, in Heat Transfer in Boiling, ed. by E. Hahne, U. Grigull. Bubble departure diameter and release frequencies during nucleate pool boiling of water and aqueous NaCl solutions (Academic Press and Hemisphere, 1977) D. Chen, L. Pan, S. Ren, Prediction of bubble detachment diameter in flow boiling based on force analysis. Nucl. Eng. Des. 243, 263–271 (2012) J.T. Cieslinski, J. Polewski, J.A. Szymczyk, Flow field around growing and rising vapour bubble by PIV measurement. J. Vis. 8, 209 (2005) R. Cole, A photographic study of pool boiling in the region of the critical heat flux. AIChE J. 6, 533–542 (1960) M.G. Cooper, A.J.P. Lloyd, The microlayer in nucleate pool boiling. Int. J. Heat Mass Transf. 12, 895–913 (1969) M. del Valle, D. Kenning, Subcooled flow boiling at high heat flux. Int. J. Heat Mass Transf. 28, 1907 (1985) D.A. Drew, S.L. Passman, Theory of Multicomponent Fluids (Springer, New York, 1998) W. Fritz, Phys. Z. 36, 379 (1935) T. Frank, P. Zwart, E. Krepper, H.-M. Prasser, D. Lucas, Validation of CFD models for mono- and polydisperse air-water two-phase flows in pipes. Nucl. Eng. Des. 238, 647–659 (2008) J. Garnier, E. Manon, G. Cubizolles, Local measurements on flow boiling of refrigerant R12 in a vertical tube. Multiph. Sci. Technol. 13, 1–111 (2001) C.-Y. Han, P. Griffith, The mechanism of heat transfer in nucleate pool boiling. Int. J. Heat Mass Transf. 8, 887 (1965) T. Hibiki, M. Ishii, Active nucleation site density in boiling systems. Int. J. Heat Mass Transf. 46, 2587–2601 (2003) Ishii, M., Thermo-Fluid Dynamic Theory of Two-Phase Flow (Paris, Eyrolles, 1975) H.J. Ivey, Relationships between bubble frequency, departure diameter, and rise velocity in nucleate boiling. Int. J. Heat Mass Transf. 10, 1023 (1967) R.L. Judd, K.S. Hwang, A comprehensive model for nucleate pool boiling heat transfer including microlayer evaporation. J. Heat Transf. 98, 623–629 (1976) B.A. Kader, Temperature and concentration profiles in fully turbulent boundary layers. Int. J. Heat Mass Transf. 24, 1541–1544 (1981) S.G. Kandlikar, M.E. Steinke, Contact angles and interface behavior during rapid evaporation of liquid on a heated surface. Int. J. Heat Mass Transf. 45, 3771–3780 (2002) J. Klausner, R. Mei, D. Bernhard, L. Zeng, Vapor bubble departure in forced convection boiling. Int. J. Heat Mass Transf. 36, 651–662 (1993) G. Kocamustafaogullari, Pressure dependence of bubble diameter for water. Int. Commun. Heat Mass Transf. 10, 501–509 (1983) G. Kocamustafaogullari, M. Ishii, Interfacial area and nucleation site density in boiling systems. Int. J. Heat Mass Transf. 26, 1377 (1983) N.I. Kolev, Uniqueness of the elementary physics driving heterogeneous nucleate boiling and flashing. Nucl. Eng. Technol. 38, 175 (2006) B. Koncar, E. Krepper, CFD simulation of convective flow boiling of refrigerant in a vertical annulus. Nucl. Eng. Des. 238, 693 (2008) B. Koncar, B. Mavko, Simulation of boiling flow experiments close to CHF with the neptune CFD code. Sci. Technol. Nucl. Installations 2008, 732158 (2008) B. Koncar, B. Mavko, Wall function approach for boiling two-phase flows. Nucl. Eng. Des.. 2010 240, 3910 (2010) E. Krepper, R. Rzehak, CFD for subcooled flow boiling: analysis of DEBORA tests. J. Comput. Multiphase Flows 6, 329–359 (2014) E. Krepper, B. Koncar, Y. Egorov, Modelling of subcooled boiling – concept, validation and application to fuel assembly design. Nucl. Eng. Des. 237, 716–731 (2007) E. Krepper, D. Lucas, T. Frank, H.-M. Prasser, P. Zwart, The inhomogeneous MUSIG model for the simulation of polydispersed flows, Nucl.Eng. Des. 238, 1690–1702 (2008)

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E. Krepper, R. Rzehak, C. Lifante, T. Frank, CFD for subcooled flow boiling: coupling wall boiling and population balance models. Nucl. Eng. Des. 255, 330–346 (2013) N. Kurul, M.Z. Podowski, Multidimensional effects in forced convection subcooled boiling, in Proc. 9th Int. Heat Transfer Conf., Jerusalem, Israel (1990) N. Kurul, M. Podowski, On the modeling of multidimensional effects in boiling channels, in ANS Proceedings of 27th National Heat Transfer Conference, Minneapolis (1991) D. Labuntsov, Current theories of nucleate boiling of liquids. Heat Transf. – Sov. Res. 7, 1–15 (1975) R.C. Lee, J.E. Nydahl, Numerical calculation of bubble growth in nucleate boiling from inception through departure. J. Heat Trans. 111, 474–479 (1989) M. Lemmert, J.M. Chawla, in Heat Transfer in Boiling, ed. by E. Hahne, U. Grigull, Influence of Flow Velocity on Surface Boiling Heat Transfer Coefficient (Academic Press and Hemisphere, 1977), ISBN: 0-12-314450-7, 237–247 C. Lifante, F. Reiterer, T. Frank, A. Burns, Coupling of wall boiling with discrete population balance model, The 14th International Topical Meeting on Nuclear Reactor Thermalhydraulics, NURETH-14, Toronto, Ontario, Canada, September 25–30, (2011), NURETH14-087 C. Lifante, T. Frank, A. Burns, RPI wall boiling extension towards CHF 2013, presentation (2013) D. Lucas, E. Krepper, CFD Models for Polydispersed Bubbly Flows (Forschungszentrum FZD, Dresden, 2007), p. 486 D. Lucas, E. Krepper, H.-M. Prasser, Use of models for lift, wall and turbulent dispersion forces acting on bubbles for poly-disperse flows. Chem. Eng. Sci. 62, 4146–4157 (2007) D.M. Lucas, T. Frank, C. Lifante, P. Zwart, A. Burns, Extension of the inhomogeneous MUSIG model for bubble condensation. Nucl. Eng. Des. 241, 4359–4367 (2011) F. Menter, Two-equation eddy-viscosity turbulence models for engineering applications. AIAA-J. 32(8), 1598–1605 (1994) F.R. Menter, Review of the shear-stress transport turbulence model experience from an industrial perspective. Int. J. Comput. Fluid Dyn. 23, 305–316 (2009) B.B. Mikic, W.M. Rohsenow, A new correlation of pool-boiling data including the fact of heating surface characteristics. ASME J. Heat Transf. 91, 245–250 (1969) B. Mikic, W. Rohsenow, P. Griffith, On bubble growth rates. Int. J. Heat Mass Transf. 13, 657 (1970) S. Mimouni, C. Baudry, M. Guingo, J. Lavieville, N. Merigoux, N. Mechitoua, Computational multi-fluid dynamics predictions of critical heat flux in boiling flow. Nucl. Eng. Des. 299, 28–36 (2016) S.B. Pope, Turbulent flow, ISBN 0521 59125 2 (2000) V. Prodanovic, D. Fraser, M. Salcudean, Bubble behaviour in subcooled flow boiling of water at low pressures and low flow rates. Int. J. Multiphase Flow 28, 1–19 (2002) F. Ramstorfer, B. Breitschädel, H. Steiner, G. Brenn, Modelling of the near-wall liquid velocity field in subcooled boiling flow, in Proceedings of HT2005, ASME Summer Heat Transfer Conference 2005, San Francisco (2005) W.E. Ranz, W.R. Marshall, Evaporation from drops. Chem. Eng. Prog. 48(3), 141–146 (1952) Y. Sato, B. Niceno, A sharp-interface phase change model for a mass-conservative interface tracking method. J. Comput. Phys. 249, 127–161 (2013) Y. Sato, M. Sadatomi, K. Sekoguchi, Momentum and heat transfer in two-phase bubble flow-I. Int. J. Multiphase Flow 7, 167–177 (1981) R. Situ, T. Hibiki, M. Ishii, M. Mori, Bubble lift-off size in forced convective subcooled boiling flow. Int. J. Heat Mass Transf. 48, 5536–5548 (2005) N.W. Snyder, Summary of conference on bubble dynamics and boiling heat transfer held at the jet propulsion laboratory, JPL Memo No. 20–137 (Jet Propulsion Laboratory, California Institute of Technology, 1956) G. Son, V.K. Dhir, N. Ramanujapu, Dynamics and heat transfer associated with a single bubble during nucleate boiling on a horizontal surface. J. Heat Trans. 121, 623–631 (1999)

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P.C. Stephan, C.A. Busse, Analysis of the heat transfer coefficient of grooved heat pipe evaporator walls. Int. J. Heat Mass Transf. 35, 383–391 (1992) R.M. Sugrue, The effects of orientation angle, subcooling, heat flux, mass flux, and pressure on bubble growth and detachment in subcooled flow boiling, Master Thesis in Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA (2012) G.E. Thorncroft, J.F. Klausner, R. Mei, Bubble forces and detachment models. Multiph. Sci. Technol. 13, 35–76 (2001) V.I. Tolubinsky, D.M. Kostanchuk, Vapour bubbles growth rate and heat transfer intensity at subcooled water boiling; Heat Transfer 1970, Preprints of papers presented at the 4th International Heat Transfer Conference, Paris, 5, Paper No. B-2.8 (1970) H.C. Ünal, Maximum bubble diameter, maximum bubble growth time and bubble growth rate. Int. J. Heat Mass Transf. 19, 643–649 (1976) C.H. Wang, V.K. Dhir, Effect of surface wettability on active nucleation site density during pool boiling of saturated water. J. Heat Transf. 115, 659–669 (1993) F.M. White, Viscous Fluid Flow (McGraw-Hill, 1991), ISBN 0-07-069712-4 T. Wintterle, Development of a numerical boundary condition for the simulation of nucleate boiling at heated walls, Diploma Thesis University Stuttgart, IKE - 8- D- 014 (2004) S. Yang, R. Kim, A mathematical model of the pool boiling nucleation site density in terms of the surface characteristics. Int. J. Heat Mass Transf. 31, 1127–1135 (1988) G.H. Yeoh, J.Y. Tu, A unified model considering force balances for departing vapour bubbles and population balance in subcooled boiling flow. Nucl. Eng. Des. 235, 1251–1265 (2005) G.H. Yeoh, J.Y. Tu, Modelling Subcooled Boiling Flows, (Nova Science Publishers, Inc., 2009) ISBN 978-1-60456-943-8 G.H. Yeoh, J.Y. Tu, Computational Techniques for Multiphase Flows – Basics and Applications (Butterworth-Heinemann, Elsevier Science and Technology, 2010) G.H. Yeoh, S.C.P. Cheung, J.Y. Tu, M.K.M. Ho, Fundamental consideration of wall heat partition of vertical subcooled boiling flows. Int. J. Heat Mass Transf. 51, 3840–3853 (2008) O. Zeitoun, M. Shoukri, Bubble behaviour and mean diameter in subcooled flow boiling. ASME J. Heat Transf. 118, 110–116 (1996) L. Zeng, J. Klausner, D. Bernhard, R. Mei, A unified model for the prediction of bubble detachment diameters in boiling systems – I. Pool boiling. Int. J. Heat Mass Transf. 36, 2261–2270 (1993a) L. Zeng, J.F. Klausner, D.M. Bernhard, R. Mei, A unified model for the prediction of bubble detachment diameters in boiling systems – II Flow boiling. Int. J. Heat Mass Transf. 36, 2271–2279 (1993b) Y.-H. Zhao, T. Masuoka, T. Tsuruta, Unified theoretical prediction of fully developed nucleate boiling and critical heat flux based on a dynamic microlayer model. Int. J. Heat Mass Transf. 45, 3189–3197 (2002a) Y.-H. Zhao, T. Tsuruta, T. Masuoka, Prediction of bubble-behavior in subcooled pool boiling based on microlayer model. JSME Int. J. B: Fluids Therm. Eng. 2002(45), 346–354 (2002b) Y.-H. Zhao, T. Tsuruta, T. Masuoka, Critical heat flux prediction of subcooled pool boiling based on the microlayer model. JSME Int. J. B: Fluids Therm. Eng. 2002(45), 712–718 (2002c) N. Zuber, The dynamics of vapor bubbles in nonuniform temperature fields. Int. J. Heat Mass Transf. 2, 83–98 (1961)

Multiphase Flows in Pharmaceutical Applications Z.B. Tong, R.Y. Yang, H.K. Chan, and A.B. Yu

Abstract

Dry powder inhalers (DPI) have been widely used for drug delivery to the respiratory tract to treat disease such as asthma and cystic fibrosis. It is therefore important to improve their efficiency through better understanding of dispersion mechanisms of powders in devices. In this aspect, numerical modeling has been playing an increasingly important role. This chapter reviews the recent progress using computational fluid dynamics (CFD) and the discrete element method (DEM) to investigate the airflow and de-agglomeration in DPIs. In particular, with the coupled CFD and DEM approach, the models are able to generate detailed information at the particle scale, leading to improved and novel designs of next generations of DPI systems. Keywords

Aerosol drug delivery • Dry powder inhaler • Particle-scale modeling • Discrete element method • Computational fluid dynamics • Powder dispersion

Z.B. Tong (*) Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, China e-mail: [email protected] R.Y. Yang School of Materials Science and Engineering, University of New South Wales, Sydney, NSW, Australia e-mail: [email protected] H.K. Chan Faculty of Pharmacy, University of Sydney, Camperdown, NSW, Australia e-mail: [email protected] A.B. Yu Department of Chemical Engineering, Monash University, Clayton, VIC, Australia e-mail: [email protected] # Springer Nature Singapore Pte Ltd. 2016 G.H. Yeoh (ed.), Handbook of Multiphase Flow Science and Technology, DOI 10.1007/978-981-4585-86-6_22-1

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Developments of Numerical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Fluid Dynamics (CFD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Element Method (DEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CFD-DEM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of CFD to DPIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DEM Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CFD-DEM Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 3 4 4 5 5 12 15 22 23

Introduction Dry powder inhalation is a popular aerosol drug delivery for airway disease. Since developed 45 years ago (Bell et al. 1971), the technique has been used to treat asthma, cystic fibrosis (CF), and chronic obstructive pulmonary disease (COPD) (Gonda 1992; Frijlink and De Boer 2004, Zhou et al. 2014). Attempts have also been made to use the inhalation route to deliver many drugs such as proteins and genes for systemic effects (e.g., using the lung to deliver insulin to the bloodstream for diabetics) (Patton 1998). Dry powder inhalation has several advantages comparing to other aerosol drug delivery methods, such as portable, chemically stable, delivery of drug using the patient’s respiration and obvious environmental benefits. Currently, various commercial dry powder inhalers (DPIs) are already on the market, and more are under development. The current pharmaceutical DPIs, however, have rather poor aerosol performance with less than 30% of the dose being able to reach the lungs. Particles with aerodynamic size range of 1–5 μm are required for the delivery to the more distal parts of the respiratory tract. Such fine powders are often very cohesive and difficult to disperse. To improve dispersion performance, development has been focused on reducing cohesion between particles, e.g., using porous particles, micronized carrier powders, or corrugated particles (Chan 2006a). On the other hand, powder dispersion performance is more efficient by increasing the dispersion/de-agglomeration forces in devices, which can be achieved by changing inhaler design, environmental conditions, and inhalation flow rate. The dispersion process generally involves multiple physical factors due to inhaler-particle interactions as well as particleparticle and particle-fluid interactions, as shown in Fig. 1 (Finlay 2001). But how they exactly occur and what is their relative importance are still not clear. The consequent lack of basic understanding of the dispersion process is a major hindrance to the advance of pharmaceutical powder aerosol technology. Detailed information of the fluid flow and particle behavior will help understand dispersion mechanisms of DPIs. However, obtaining such information from experiments is extremely difficult (Zhu et al. 2007), if not impossible because of the small particle size (less than 5 μm) and extremely short duration of impacts. So far

Multiphase Flows in Pharmaceutical Applications

Drug Agglomerates 1 Particle – air interaction Eddies

3

2 Particle – device impaction

3 Particle – particle impaction

Fig. 1. De-agglomeration mechanisms (Finlay 2001)

experiments have been largely limited to post-dispersion analysis. On the other hand, numerical modeling has been playing an increasingly important role in this area. Computation fluid dynamic (CFD) has been used to understand the airflow inside DPIs and to investigate the effect of design on dispersion performance (Wong et al. 2012; Ruzycki et al. 2013). The studies have shown that aerosol performance depends on the inhaler characteristics and prevailing flow conditions. They also highlighted the important role of computational modeling in inhaler design for pharmaceutical aerosol generation. While CFD is unable to model the dynamics and interaction of discrete particles, such limitation can be overcome by the discrete element method (DEM)-based models (Calvert et al. 2009). The DEM models explicitly calculate the interparticle forces (Cundall and Strack 1979) and provide dynamic information of individual particles. By combining CFD with DEM, the approach can generate information in two important aspects: flow structure and force structure at individual particle scale, thus allowing the dispersion of powders to be accurately described. This chapter will review the development in this area and discuss a perspective on the future work.

Developments of Numerical Models Computational Fluid Dynamics (CFD) CFD is a tool widely used to simulate and analyze fluid flow. It is the numerical technique to solve the Navier-Stokes equations which describe the behavior of fluid flow. While direct numerical simulation (DNS) and large eddy simulation (LES) are more accurate to model turbulent flows, their applications in this area are rare due to their extremely high computational cost. The most commonly used approach for drug aerosol modeling is to solve the Reynolds averaged Navier-Stokes (RANS) equations, given by (Anderson and Jackson 1967): ∇
ðuÞ ¼ 0 ρ


@ ðuÞ ρ∇
ðuuÞ ¼ ∇P þ ∇
ðτÞ þ ρg þ ρ∇
ρ  u0 u0 @t

(1) (2)

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where ū, u0, ρ, P, and τ are fluid mean velocity, turbulent velocity fluctuation, fluid density, pressure, and fluid viscous stress tensor, respectively. The Reynolds averaging of the transport equations introduces additional Reynolds stress term ρ  u0 u0 whose closure is typically via a two-equation eddy viscosity model (Walters and Leylek 2004; Genç et al. 2009) or a transported turbulent shear stress model (Launder et al. 1975; Wang et al. 2006).

Discrete Element Method (DEM) The macroscopic behavior of particle flow is governed by the interactions between individual particles as well as interactions of particles with surrounding fluid and wall. The DEM was originally developed by Cundall and Strack (1979) to study granular dynamics and since has been extensively used in the study of various phenomena, such as particle packing and compaction, particle flow, and particlefluid flow (Zhu et al. 2008). In the DEM approach, the governing equations for the translational and rotational motion of particle i with mass dvi ¼ Fi dt

(3)

dωi ¼ Mi dt

(4)

mi Ii

where vi and ωi are the translational and angular velocities of particle i, respectively, Fi and Mi are the total force and torque acting on particle i. The total force may include mechanical contact forces, noncontact forces such as van der Waals force, capillary force and electrostatic force, particle-fluid force (e.g., the drag force, Saffman and Magnus lift forces) as well as gravity (Zhu et al. 2008). Various models have been proposed to calculate these forces and torques (Tong et al. 2015). Once the forces and torques are known, Eqs. (3) and (4) can be solved numerically. Thus, the trajectories, velocities, and the transient forces of all particles in a system considered can be determined.

CFD-DEM Model The CFD-DEM approach was firstly proposed by Tsuji et al. (1992), and then followed by many others (Xu and Yu 1997). As mentioned above, with this approach, the motion of discrete particles is described by the DEM based on the Newton’s second law of motion and the flow of continuum fluid by CFD based on the local averaged Navier-Stokes equations. The CFD-DEM coupling is achieved by exchanging the information obtained from the DEM and the CFD at each step, as shown in Fig. 2. At every time step with a given fluid flow condition, the DEM

Multiphase Flows in Pharmaceutical Applications

Discrete Model

5

vi , (x, y, z )i ,ε i

kc

fpf, i

ui , (u ‒ v )i , Re p,i

Fpf =

∑ fpf,i i=1

ΔV

Continuum Model

Fig. 2. Coupling and information exchange between continuum (CFD) and discrete (DEM) models (Xu et al. 2001)

determines particle-related information such as the positions, velocities, and forces of individual particles. The porosity and volumetric particle-fluid interaction force in the individual computational cells are then calculated and passed to the CFD which uses these data to determine the air flow field and determines the interaction from the fluid to individual particles. The resulting forces are then passed to the DEM to determine the state of particles at the next time step, and the process continues (Chu et al. 2009). In some situations particles have a minimum effect on fluid flow (e.g., dispersion of single agglomerate in a uniform flow), a one-way coupling method in which only the interaction of fluid on particles is considered is preferred to reduce simulation time.

Model Applications Applications of CFD to DPIs CFD simulations have been performed to investigate the flow field inside the DPIs (Matida et al. 2003; Coates et al. 2004; Matida et al. 2004; Coates et al. 2005). The studies by Coates et al. showed that the change to DPI designs, such as grid size (Coates et al. 2004), mouthpiece geometry (Coates et al. 2007), inlet air dimension (Coates et al. 2006), the presence of a capsule (Coates et al. 2005), and air flow rate, can affect aerosol performance significantly. Figure 3 shows the effect of the structure of the inhaler grid on flow velocity. However the modified mouthpiece

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Fig. 3. Different designs of the grid and their effects on velocity profile (Coates et al. 2004)

designs of Aerolizer ® (Fig. 4) are able to reduce throat deposition but have no effect on the overall dispersion performance characterized in terms of fine particle fraction (FPF) (Coates et al. 2007). The change of the inhaler mouthpiece length was also found to play a less significant role. On the other hand, air inlets control the levels of turbulence and particle impaction velocities generated in the device, as well as the flow development rate and device emptying times. Modification of their dimensions (Coates et al. 2006) has a varying effect on the inhaler performance depending on flow rates (Fig. 5). De Boer et al. (2012) used CFD to investigate the performance of a commercial high-dose disposable DPI named Twincer™ as shown in Fig. 6. Their results indicated that the flow across the inhaler was independent of the pressure drop. The dispersion efficiency or deposition was not affected by the flow rate. By

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Fig. 4. Schematic of the different modified mouthpiece designs (Coates et al. 2007)

Fig. 5. Schematic of the device designs of the air inlet size on inhaler performance (Coates et al. 2006)

reducing the resistance of the classifier bypass channels, the classifier symmetry can be improved. Donovan et al. (2012) simulated two commercial DPIs, Handihaler ® and Aerolizer ® (Fig. 7), to investigate the effects of carrier particle size and shape on inhaler performance. The results indicated the number of particle-wall collisions increased with carrier particle size in Aerolizer ® but was independent of carrier particle size in Handihaler ®. Milenkovic (Milenkovic et al. 2013; Milenkovic and Alexopoulos 2014), on the other hand, investigated the airflow and particle

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

100 μM; 4 kPa

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15

3 μM; 4 kPa

Fig. 6. Particle trajectories in Twincer™ colored by residence time for particles of different sizes (de Boer et al. 2012)

10 μM; 4 kPa

0.000

1 μM; 4 kPa

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Multiphase Flows in Pharmaceutical Applications Fig. 7. Handihaler ® (left) and Aerolizer ® (right) a contour of velocity magnitude; b carrier particle trajectory (Donovan et al. 2012)

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a

Speed (m/s) 90 80 70 60 50 40 30 20 10 0

b

deposition in the Turbuhaler ®, as shown in Fig. 8. Their simulation results for particle deposition and fine particle fraction agreed well with available experimental data. Following the studies proposed by Coates et al. (2006), the effect of device design of the Aerolizer ® on the aerosolization of a carrier-based formulation has been investigated by Zhou et al. (2013). The results showed the air inlet size and grid

10 Fig. 8. a Contour of velocity magnitude (Milenkovic et al. 2013); b deposited particles for steady flow in the Turbuhaler ® (Milenkovic and Alexopoulos 2014)

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a

32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

b 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1

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11 Mouthpiece

Grid

Inhaler chamber

Air inlet Original

1/3 mouthpiece Iength

1/3 inlet size

Cross grid

Fig. 9. Diagram of the original and modified Aerolizer ® devices (Zhou et al. 2013)

Fig. 10. Agglomerate-based formulation (left) and Carrier-based formulation (right)

structure of the Aerolizer ® have significant effect on the aerosolization performance (Fig. 9). The work mentioned above improved our understanding of the performance of DPIs which can be extended to other dry powder inhaler systems to provide critical information for optimal inhaler design. On the other hand, the work also showed the limitations of simulating the powder dispersion process (Iimura et al. 1998; Higashitani et al. 2001; Iimura and Higashitani 2005). The limitation of CFD model can be overcome by explicit calculation of the particle contact mechanics using the DEM.

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DEM Applications The DEM has been applied to investigate the mechanical impaction between agglomerates and devices. There are two kinds of drug powder formulations used in DPIs, e.g., agglomerate-based and carrier-based formulation, as shown in Fig. 10. The applications of DEM modeling in the dispersion of two formulations due to impaction are discussed separately in the following sections. (i) Dispersion of agglomerate-based formulation due to impaction In the early DEM research, Thornton and his co-workers (Thornton et al. 1996, 1999; Ning et al. 1997) have reported impact simulations on lactose agglomerates (Fig. 11) and observed a minimum velocity below which no significant damage occurs, and the agglomerate behaves like a large single particle. Similar conclusions were also obtained by Subero et al. (1999) who showed that the extent of breakage increased with impact velocity but eventually reached a limit beyond which the breakage approached an asymptotic value. As shown in Fig. 12, a modified Weber 2

oÞ D number We0 ¼ ρðUU was linked to the damaged ratio. Γ Moreno et al. (2003, 2006) studied the effect of impact angle and surface energy on the breakage of agglomerates and found that the normal component of impact velocity was the dominant factor. Comparing the previous work, the wider range of surface energies have been tested by Moreno et al., and a new dimensionless group, Δ ¼ WeI2=3 e , was able to provide a much better unification of data compared to the modified Weber number We0 proposed by Subero et al. (1999), as shown in Fig. 13. In DPIs, powders are loosely bonded in order to have easy dispersion when in use (Chan 2006b). Comparing with hard, strong agglomerates, soft agglomerates behave quite differently on impacts and often disintegrate into many small pieces instead of breaking into several large chunks (Ning et al. 1997; Boerefijn et al. 1998). Yang et al. (2008) investigated the agglomeration of fine particles down to 5 μm in size under an assumed centripetal force. Their results showed that the strength of an agglomerate can be predicted by the modified Rumpf model (Rumpf 1962). Recently, Tong et al. (2009) simulated the loose and weak agglomerates impacted with a wall at different velocities and angles. The simulation results showed that the agglomerates experienced significant plastic deformations before breaking into many small pieces/fragments. While the normal (90 ) impaction generated large forces inside the agglomerate, the deformation area was much larger with the 45 impaction. This suggested that the shear force played a very important role in the breakage of the agglomerates. This is different from the breakage of hard agglomerates of which the normal impact is the dominant breakage factor (Moreno et al. 2003). This was further confirmed by analyzing the size distribution of fragments after impaction. As shown in Fig. 14a, at a given impact velocity, the 45 impaction always generated the smallest fragment size characterized by the 80 % passing size (P80). To quantify the breakage efficiency, the total impact energy (work) from the wall to the agglomerates was analyzed. The results showed that the mechanical

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breakage efficiency was governed by the total impact energy which includes energy in both the normal and shear directions, as shown in Fig. 14b. (ii) Dispersion of carrier-based formulation The carrier-based formulation is more popular in the DPIs than the agglomeratebased formulation. Recently, the DEM method (Yang et al. 2013) has been used to investigate the dispersion process of the carrier-based agglomerate due to wall impaction. The effects of impact velocity, impact angle, and work of adhesion on the dispersion performance were analyzed. As shown in Fig. 15, the results showed that the impact-induced dispersion performance for carrier-based DPI formulations can be well approximated using a cumulative Weibull distribution function that is governed by the ratio of overall impact energy and adhesion energy.

Fig. 11. Space lattice at a t = 9 μs and b t = 12 μs of an agglomerate upon impaction (Thornton et al. 1999)

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Yang et al. (2013; 2015) also used the DEM method to investigate the mixing process for carrier-based dry powder inhaler formulations (Fig. 16) and found that amplitude and frequency of the vibration velocity can be controlled to maximize the mixing of small particles with the carrier. The results also showed that the electrostatic force can result in a different mixing behavior comparting to the van der Waals force.

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CFD-DEM Applications This section discusses the main applications of CFD-DEM models on the dispersion of agglomerate-based and carrier-based formulation with the airflow. (i) Dispersion of agglomerate-based formulation with airflow

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Fig. 15. The variation of dispersion ratio with the energy ratio from the perspective of the whole agglomerate (Yang et al. 2013)

Fig. 16. Snapshots of drug and carrier particles mixing at different time instances

Fig. 17. Dispersion behavior of an agglomerate in a uniform flow field (Calvert et al. 2011)

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Fig. 18. The change in the agglomerate after impact (Nguyen et al. 2014)

The dispersion of agglomerates by airflow has been investigated by Iimura et al. (2009a, b) and Calvert (Calvert et al. 2011; Calvert et al. 2013). As shown in Fig. 17, the results showed beyond a threshold of particle-fluid velocity, dispersion occurs quickly and approaches a completely dispersed state asymptotically. Agglomerate dispersions due to impaction with a target particle or cylindrical obstacle with airflow were simulated by Iimura et al. (2009a, b) and Nguyen et al. (2014). Figure 18 showed the fine fragments were attached to the target due to the restructuring mechanism that occurred during impact (Nguyen et al. 2014). Recently, the effect of multiple impactions on the agglomerate dispersion (Fig. 19) has been investigated by Tong et al. (2011). The results indicated that the generation of fine particles were mainly caused by the second impaction. After the first impaction, the agglomerate broke into large-sized fragments which are

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Fig. 19. De-agglomeration of powders in different throat designs (Tong et al. 2011)

subsequently broken into smaller pieces upon the second impaction. While increasing flow rate and the number of impactions increased breakage, they also resulted in larger powder deposition on the wall. To have an optimal dispersion, both throat design and flow rate should be well considered. While some studies (French et al. 1996; Li et al. 1996; Dunbar et al. 1998) suggested that the drag force generated by turbulent flow is a major factor of

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Fig. 20. Flow field about the particle cluster and resulting total forces on the fine particles for three Reynolds numbers (Cui et al. 2014)

de-agglomeration, others (Finlay 2001; Voss and Finlay 2002) indicated that mechanical impaction is more important for powder dispersion. The aerodynamic dispersions of a loose aggregate (Calvert et al. 2011) and a carrier-based DPI formulation (Yang et al. 2014) in a uniform fluid flow have been investigated. The results showed that unless the relative velocity was above a threshold limit, the loose aggregate or carrier-based DPI formulation would not deform and disperse but accelerate as a single entity. As the flow velocity in DPIs is normally smaller than the threshold limit, the dispersion directly induced by air flow only plays a minor role in powder dispersion in DPIs. This was also confirmed by the study of a cyclonic flow model (Tong et al. 2010).

20 Fig. 21. a Schematic view of the inhaler; b size distribution of powders forming the agglomerate. The inset shows the formed agglomerate (colors represent particle diameters)

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(ii) Dispersion of carrier-based formulation with airflow Work has also been conducted to investigate the aerosolization of carrier-based formulation caused by airflow (Cui et al. 2014; Yang et al. 2014). For the detachment of the drug particles from the carrier through the fluid stresses (normal and tangential force), the adhesion force (van der Waals) and the friction force have to be overcome. Figure 20 shows the detachment probability by the three mechanisms as a function of carrier particle Reynolds number. These results can be used for deriving the drug powder detachment model. (iii) Powder dispersion in Aerolizer ® Powder dispersion in a commercial Aerolizer ® inhaler model was studied. Initially, the agglomerate-based formulation was simulated. Figure 21 shows the inhaler

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Fig. 22. Impaction of single carrier particle with a wall in a uniform shear flow (Tong et al. 2015)

device and agglomerates used in the study. The inhaler consists of a chamber with two inlets, a barrel, and a grid in between. A capsule is inside the chamber to store agglomerated powders (Tong et al. 2013). The simulations showed that the dominant mechanism was agglomerate-device impaction. The turbulent flow was not strong enough to break the agglomerates, and the agglomerate-agglomerate impactions occurred only at the very early stage when the agglomerates were spun out from the capsule. The performance of the inhaler was very sensitive to the flow condition. While better dispersion performance was observed with increasing airflow velocity, larger flow rates also increased the amount of powder deposition and exceedingly high flow rate actually reduced the performance of the inhaler. Thus the flow rate should be controlled for optimum dispersion efficiency. The carrier-based DPI formulations enable the fine drug particles being detached from the carrier upon inhalation. However, simulating such a system has a very computational cost due to the large number of particles involved but also the large carrier-particle size ratio. Therefore, a multi-scale approach was proposed (Tong et al. 2015). This approach was based on an assumption that the powder dispersion in an inhaler depends on the detachment of fine drug powders from carriers and energy distribution inside the inhaler. Figure 22 shows the detachment of fine powders from the carrier impaction. The detachment behavior of particles depended on the characteristics and material properties of carrier and drug particles. By linking such information with the collision energy distribution inside the inhaler (Fig. 23), the overall aerosolization performance of the inhaler was predicted. The results showed

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Fig. 23. Spatial distribution of total particle-wall (left) and particle-particle (right) collision energy (the colors represent magnitude (mJ) of collision energy) (Tong et al. 2015)

that the dispersion performance of carrier-based formulation decreased with the carrier particle size and increased with the air flow velocity.

Conclusion and Perspectives The numerical studies of DPI have demonstrated that the approach is an effective tool in analyzing and predicting the performance of various inhalation devices and formulations. The numerical results can generate information leading to a better understanding of the internal flow structure of inhaler systems by analyzing the flow patterns of particle phase at particle scale and fluid phase at computational cell scale and particle-fluid, particle-particle, and particle-wall interaction forces. The results can be used to improve the performance of existing DPIs or designing new inhalers with desirable dispersion. It also has a wide applicability to a variety of inhalers and operational conditions.

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Future effort should be made in the following three aspects: (i) Model development: Theoretical development is the foundation and guarantees the rationality of the model. One major area is to develop a more comprehensive model (e.g., LES) to study the complicated situation. (ii) Computational performance: the accuracy, robustness, versatility, and speed (e.g., GPU) (Stone et al. 2010), which are normally the key factors to judge numerical techniques, can be improved in the future. (iii) Application: The findings on the role and relative importance of particledevice, particle-particle, and particle-air interactions can be applied to developing better design of inhalers for improved de-agglomeration. The new designs will be evaluated by measuring the powder dispersion performance and compared with the model prediction. Acknowledgments Authors are grateful to the Australian Research Council (ARC) for the financial support.

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