Mass Transfer Processes with a Solid Phase Participation

Учебное пособие предназначено для обучения магистров по направлению «Химическая технология» и его содержание соответству

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Mass Transfer Processes with a Solid Phase Participation

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Russian Federation Ministry of Education and Science State Educational Establishment Of Higher Professional Education "Kazan National Research Technological University"

A.I. Razinov, P.P. Sukhanov



Kazan KNRTU 2012

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УДК 66.015.23 Процессы массопереноса с участием твердой фазы: Учебное пособие /А.И. Разинов, П.П. Суханов; Казан. нац. исслед. технол. ун-т. – Казань: КНИТУ, 2012. – 96 с. Учебное пособие предназначено для обучения магистров по направлению «Химическая технология» и его содержание соответствует ФГОС 3-го поколения для дисциплины «Процессы массопереноса в системах с участием твердой фазы». Изложенный в учебном пособии материал позволяет студентам восполнить и систематизировать знания по теории массообменных процессов, знакомит со спецификой массопереноса в системах с участием твердой фазы, а также с такими процессами, как адсорбция, ионный обмен, кристаллизация, растворение, мембранное разделение, конструкциями соответствующих аппаратов и методами их расчетов. Подготовлено на кафедре «Процессы и аппараты химической технологии». Печатается по решению редакционно-издательского совета Казанского национального исследовательского технологического университета. Tutorial is intended for the course «Mass Transfer Processes with a Solid Phase Participation» of Masters Degree Program in «Chemical Technology». Contents of the tutorial introduces students to the fundamentals of mass transfer processes theory and peculiarities of the methods of calculation of adsorption, ion exchange, crystallization, dissolution and membrane separation processes.

Рецензенты: Зав. кафедрой технологии воды и топлива Казанского государственного энергетического университета, д-р техн. наук, проф. А.Г. Лаптев Д-р техн. наук, проф. кафедры энергосберегающих технологий и энергообеспечения предприятий Казанского государственного энергетического университета А.Я. Мутрисков


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3 5

CHAPTER 1. FUNDAMENTALS OF MASS TRANSFER PROCESSES 1.1. Mass Transfer Mechanisms and Equations 1.2. The Law of Mass Conservation 1.3. Interfacial Mass Transfer 1.4. Phase Equilibrium 1.5. Various Modifications of Mass Delivery and Mass Transfer Equations 1.6. Peculiarity of Mass Transfer in Systems with a Solid Phase Participation CHAPTER 2. ADSORPTION AND ION EXCHANGE 2.1. General Information 2.2. Types of Adsorbents and Their Characteristics 2.3. Equilibrium in Adsorption 2.4. Kinetics of Periodic Adsorption 2.5. Continuous Adsorption 2.6. Desorption 2.7. Device and Operation Principles of Adsorption Machines 2.8. Calculation of Adsorbers 2.9. Ion Exchange CHAPTER 3. CRYSTALLIZATION AND DISSOLUTION 3.1. General Information 3.2. Equilibrium in Crystal-Solution System 3.3. Kinetics of Crystallization Processes 3.4. Techniques of Crystallization 3.5. Design of Crystallizers 3.6. Calculation of crystallizers 3.7. Dissolution


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4.1. General Information. Classification of Membrane Separation Techniques 4.2. Types of Membranes. Theory of Membrane Separation 4.3. Design of Membrane Devices 4.4. Calculation of Membrane Devices

80 84 86 89

APPENDIX 1. Some Information from the Field of Mathematics 2. Algorithms of Diffusion Coefficients Calculation

90 90 93




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ABBREVIATIONS a – active; adsorption; adsorbent b – boundary bd – bulk density c - convective; concentration d – diffusion; dynamic dr - drying e – equivalent; effective; electric f – fluid; phase (surface) fin – final; output in – initial; input ion - ionite id - ideal L - liquid lam – laminar

^ - laboratory (frame of reference, etc.) lw - lower m – medium; mass; mean n – nucleus s – solid; saturated skс – skipping concentration skt – skipping time ss - supersaturated T – turbulent; thermal; temperature skti – skipping time (ideal) up – upper v - volume


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CHAPTER 1. FUNDAMENTALS OF MASS TRANSFER PROCESSES 1.1. Mass Transfer Mechanisms and Equations We can distinguish three mechanisms of transfer: molecular, convective and turbulent (eddy). Molecular mechanism of transfer is due to the thermal motion of molecules. Interaction between molecules can be imagined roughly as "hard" repulsion at short distances between their centers ( l ) and "soft" attraction at large distances. A typical view of the potential energy of intermolecular interaction ϕ( l ) is shown on fig.1.1. ϕ( )


Fig. 1.1. A typical view of an intermolecular interaction potential: l - distance between the centers of molecules The force of interaction F = dϕ dl is negative at distances

l < l 0 (repulsion) and is positive if l > l0 (attraction). The nature of molecular motion can be different, depending on the phase state of a substance. Molecules in gases move chaotically. A larger share of the time of molecular` movement here constitutes the so called «free path» (collisions free) rundown, i.e. due to the low density of the system the movements of molecules are virtual - without any interaction with each other. Lowering temperature of system leads to the decrease of kinetic energy of molecules. They loose the ability to overcome the intermolecular forces of attraction, and a system condenses – transforms


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from the gaseous state into the liquid state. The chaotic nature of molecular motion in this case mainly remains. However, the role of intermolecular interaction becomes much more substantial because of a considerable increase of density, and a large share of the volume of a system is occupied by molecules. As the result, exit of a molecule out of its nearest environment becomes hampered. Majority of molecular systems transforms into a crystalline state in case of a further lowering of temperature, when the kinetic energy of a molecule becomes not enough to exit a cell, formed by surrounding molecules. From the energy point of view, it leads to the formation of the most advantageous structure of the crystal lattice. Movement of molecules from one node to another is possible only with the help of violations of structure regularity - dislocations, "holes". Thermal motion of molecules within the cell becomes prevailing. Molecule, moving from one point of space to another, transports all three types of substance parameters (or «substances») - mass, momentum and energy. Visible macroscopic transport of substances is not observed in the conditions of equilibrium, because the transfer of molecules in any direction is equally probable, when system rests and concentrations of components and temperature in all points are of the same value. In the absence of equilibrium, a predominant probability of molecular transfer of masses appears in the direction from a larger values of concentration to a lower, of momentum - from a large values of velocities to a smaller, of energy - from a higher temperature to a lower. This leads to the observed macroscopic phenomena of transfer. Convective mechanism of substance transfer is caused by the motion of macroscopic volumes of a medium (fluid) as a whole. The characteristic scale of engineering tasks allows you to operate on macroscopic values, that can be specified in each point of space by averaging the microscopic values. The whole set of quantities of physical values, clearly defined at each point of some part of space, is called a field of this value (field of density, concentration, pressure, velocity, temperature, etc.). The motion of macroscopic volumes of a fluid leads to transfer of →

ρ- density or mass ρ, momentum ρ W and energy ρE' of a single volume (ρ →

mass of a unit volume, ρ W  - momentum of a single volume, ρE' energy of a single volume).


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We distinguish free and forced convection according to the causes, which create convective flows. Free convection is due to natural causes and occurs under the action of force of gravity, but forced convection arises as the result of artificial (external) affect, what can be realized with the usage of pumps, compressor machines, by mixing, etc. Turbulent mechanism of transfer occupies an intermediate place between molecular and convective mechanisms from the point of view of the space-time scale. It is necessary to perform maximum number of conditions for creation of a turbulent motion. So, molecular thermal motion occurs in any system (including the equilibrium systems), which temperature differs from the absolute zero, i.e., almost always. The convective motion is observed only in non-equilibrium systems in case of displacement of macroscopic volumes of the medium. Turbulent motion arises only under the certain conditions of the convective motion: a sufficient distance from a phase boundary and heterogeneity of convective velocity field. Macroscopic layers move regularly, parallel to each other, in case of a small convective velocity of a medium (gas or liquid) motion with respect to a phase boundary. This kind of movement is called laminar. Incidental or artificial small perturbations, arising in real conditions, impact the regularity of movement (roughness of a surface, limiting the flow, etc.) and do not increase over time, but, on the contrary, dampen. However, the sustainability of the motion with respect to small perturbations is violated if heterogeneity of velocity and distance from a boundary of the phase exceed a certain value. Therefore begins the development of irregular chaotic movement of individual volumes of the medium (vortices). This movement is called turbulent. Here are introduced the values, characterizing turbulent motion, which are analogous to the characteristics of molecular movement, where are used mean square speed of molecules and their size. The notion of the scale of turbulence, which determines the size of vortices, is used too. In contrast, for example, of molecules, vortices are not sustainable, clearly limited in space formations. They arise, break down into smaller vortices and vanish up with the transition of energy to heat (dissipation of energy). Therefore, the scale of turbulence - is an averaged statistical value. And turbulent vortex (vortex flux) carries out all kinds of transfer - mass, momentum and energy - similarly to the random motion of molecules.


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Equations of transfer. Let us consider the local equations of mass transfer, i.e., equations that are true for each point of space. The examination is conducted at macroscopic level - all used values are macroscopic. The task is to obtain a mathematical expression for mass flux (quantity of mass, transferred per unit of time through a unit surface area) under the action of various transfer mechanisms. Convective mechanism. Flow of mass (m) for the account of convective mechanism in any point of a system in the laboratory (^) frame →

of reference ^ jm is connected with convective speed: →

2 ^ jm = ρ^ W , kg/m s.


In case of multi-component environment we can consider the flow of mass of each component: →

, ^ jm i = ρi ^ W


where i is a number of components, ρi - density of component i. Often it is more convenient to use the flow of substance, and not of mass: → ^ ji = ^ jm i →

2 m i = c i ^ W , kmol/m s,


where m i - mole mass of component i, kg/kmol; c i - mole concentration, kmol/m3. Note that convective velocity and flow are analyzed in the laboratory system of reference, i.e. relative to the reference frame, associated with a device. Convective velocity relative to a machine in the conditions of hydro-mechanical equilibrium is not only a constant value, but the constant value, equal to zero. Molecular mechanism. Actually molecular mechanism of mass transfer can be observed in thermodynamically equilibrium system, where are present only the gradients of concentration of marked particles of the sort i (i' - isotopes of molecules of the sort i): →

^ ji ' = − D i ∇ c i '


The sign «minus» testifies to the opposite direction of vectors of substance flow and gradient of concentration. Gradient of concentration


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(see Appendix 1.3) is directed towards the maximum increase of concentration, but the gradient of substance flow - in the direction of reducing it (alignment of heterogeneity). It was shown by Einstein, that coefficients of proportionality in this ratio characterize mean square displacement of molecules per unit of time due to the random thermal motion:

Di = ∆l 2i 6∆t .


These values are called Einsteinian coefficients of diffusion. They are experimentally determined with the help of methods of tagged atoms or nuclear magnetic resonance, as well as on the basis of numerical experiment by the method of molecular dynamics (simulation of motion of an ensemble of particles on computer). D i depend on dynamic characteristics of molecules (mass, potential of interaction), as well as on pressure and temperature of the system (see Appendix 2). Since D i characterize the mobility of molecules, they greatly depend on the phase state of the system. In gases D i are of the order of 10-6 m2/s, in liquids 10-9 m2/s, in solids - 10-12 m2/s, and they increase with an increase in temperature and decrease of pressure. It is assumed in accordance with the concept of independent diffusion, that actually diffusion fluxes under non-equilibrium conditions can be described by Einsteinian coefficients of diffusion. Then a flow of a component i in isothermal system in the absence of turbulence can be written as the sum of diffusion (d) and convective (c) flows: ∧

j i = − Di

→ → → ci → ∇ µ i + ci ∧ W = ∧ jdi + ∧ jci , RT

i = 1,n ,


where µ i is chemical potential of a component i , n is a number of components in system. It should be borne in mind, that convective velocity under non-equilibrium conditions can arise only at the expense of diffusion too. An example may clarify this situation. Let us consider a device, in one part of which (compartment) is situated a component 1 and in the other - component 2, which are separated from each other with the help of a partition. Pressure and temperature in both parts of the device are the same. If you remove a partition, then due to molecular diffusion there will appear the oppositely directed flows of components. However, magnitudes of the flows will vary in accordance to the differences of


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dynamic characteristics of molecules of this components and, therefore, of Einsteinian coefficients of diffusion D i . We shall suppose, that D1 > D 2 . Then the diffusion flux of the first component will be greater than the →

second one, because from Gibbs-Dougem ratio c1 ∇ µ1 = −c 2 ∇ µ 2 . Molecular mechanism will initiate the resulting transfer of a substance from the first part of the system to the second, which in the closed device (system) will lead to a rise of gradient of the density of the number of particles and, respectively, pressure (p II > p I ) . And this will cause the oppositely directed convective flow, equalizing the pressure gradient (Fig. 1.2). ∧ jg

cI , pI ∧jg


cII , pII



Fig. 1.2. Diffusion flow (g) and convective speed, caused by diffusion in the closed device Thus, it is difficult under the non-equilibrium conditions observe and study in pure form the molecular transfer of mass, because it requires artificial maintenance of the constancy of a pressure in a system. The difficulty exists in experimental determination of the values of D i and →

convective velocity

^w .

Even if you have measured the flows of all

components ^ ji and the field of concentrations ci in laboratory system of reference, you can't solve a system of n equations (1.6), since it contains →

n+1 of unknown values (Di, w ). So, usually the diffusion flows are determined in the reference system, whose velocity relatively to laboratory system can be set quite easily, and, as a rule, in this case are used the frames of reference of medium-mass or medium-volume type. Reference system is given by the condition of equality to zero of the total flow of corresponding feature (let us denote it zi) in this frame of reference (z):


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∑ ji z i = 0 z


i =1

In the medium-mass reference system zi = mi (molar mass of a component), and in the medium-volume reference system zi = Vi (partial molar volume of a component Vi, m3/kmol):  ∂V   Vi =   ∂N i  p , T

Then the mole flow of component i in laboratory reference system can be presented in the form of →

^ ji = z ji + c i

z −∧


(1.8) →



W =

∑ ∧ ji z i , i =1



∑ ciz i i =1

where z −∧ W is the velocity of appropriate reference system relative to laboratory reference system, which can be found, if experimental values of r flows Λ ji are measured. Flows in the reference system z, in accordance with the concept of independent diffusion, have the form → ci →  →  (1.10) ∇ µ i + ci  ∧ W− z − ∧ W  , RT   → → → where  ∧ W−z−∧ W =z W - convective velocity in the reference system z.   z


ji = − Di

W can be expressed, using the optional equation (1.7), through D i , µ i ,

c i , i = 1, n , and the flows can be submitted in the form of → ∧ n cj → . z ji = − ∑ z Dij ∇ µj RT j=1



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It is more convenient to use in practice the diffusion coefficients, which are linking the flows not with gradients of chemical potentials, but with gradients of concentration. So, we can express chemical potentials through mole concentrations and use the ratio of n

∑ Vi ∇ c i = 0 , i =1

which allows to reduce the number of independent variables on one. Then it is possible to write: → → n −1 (1.12) z j = − zD ∇c .





Thus, macroscopic flow of each component in the reference system z depends on gradients of concentrations of all components. Coefficients of proportionality in (1.12) is called the matrix of multicomponent diffusion coefficients, and they are defined by the properties of components of a medium and the choice of reference system. Experimental finding of diffusion coefficients is performed, as a rule, in closed device. In these conditions, total flow of the volume is equal to zero, i.e. the laboratory system of reference coincides with the medium volume. Therefore, the experimental data on coefficients of diffusion is usually cited for the medium-volume (v) frame of reference. In the special case of a twocomponent composition the matrix of ∨ Dij is degenerated in the only factor of binary (*) (or mutual) diffusion D*ij (D*ij = D*ji ): →

^ ji = - D*ij · ∇ c i ,



This ratio is called the first Fick`s law. Using concept of the independent diffusion, it's possible to express coefficients of mutual diffusion through the Einsteinian coefficients: D*ij = ( D x + D x ) ∂ ln γ i x i . (1.14) i j j i ∂ ln x i where γ i , х i is the coefficient of activity and mole fraction of component respectively. You can do this and for multi-component mixtures, linking elements of matrix zDij with Di (see Appendix 2.1). It should be


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remembered about the dependence of coefficients of diffusion (in addition to Einsteinian) on the choice of reference frame. So, for example, coefficients of binary diffusion in medium-mass system of reference (m) do not have the property of symmetry (mD*ij ≠ mD*ji) and can be expressed through vDij: m D*ij = vDij · mj / (ρVj) (1.15) The flows of substance, in case of interfacial transfer, are often written on the phase boundary. Thus, as a rule, one or several components are not migrated through the phase interface (inert components). In a binary mixture, whose first component crosses the phase boundary (b) and another component does not, flows in one phase can be written in the frame of reference, where the flow of the second component is equal to zero : 2


j 1 = − 2 D12 ∇ c1 = − D1

j 2 = − 2 D21 ∇ c2 = − D2

→ c1 →  →  ∇ µ1 + c1  ∧W −b−∧ W  , RT  


→ c2 →  →  ∇ µ2 + c2  ∧ W −b −∧ W  = 0 . (1.17) RT  

→ → → Expressing b W =  ∧ W −b−∧ W  of (1.17) and substituting in (1.16), you can   obtain relations for coefficients of binary diffusion in the frame of reference, corresponding to the zero flow of the second component,


 cV  D12 = 1 + 1 1  ∨ D12 ;  c2V2 


When c 2 → 0 then 2 D12 → ∞ , that clearly demonstrates the infinitely large range of changes of coefficients of binary diffusion depending on the choice of reference frame. They can take both positive and negative values according to the choice of reference frame and elements of z Dij . Only the Einsteinian diffusion coefficients of Di (1.5) do not depend on the choice of the frame of reference, always positive, and have a clear physical meaning.


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Turbulent mechanism. Turbulent transfer of mass can be considered by the analogy with molecular and can be respectively represented as a result of the chaotic movement of vortices. Then can be introduced coefficient of turbulent diffusion D т , which depends on the properties of the medium, of heterogeneity of velocity and distance from interfacial surface. Total turbulent substance flux, concerning laboratory system of reference, can be written as →

^ j i = − Di

→ → ci → ∧ ∇ µ i − Dт ∇ ci + ci W RT


or n −1 n −1 → → → → → r ^ j i = −∑ z Dij ∇ c j − Dт ∑ zα ij ∇ c j + ci z −∧ W = z jid + ci z −∧ W , (1.20) j =1

j =1


  z α ij = δ ij − c i  z j − n Vj  Vn  


∑ ск z к . к =1


For the frame of reference, in which the flow of component n is equal to zero, n (1.22) α ij = δ ij − c i Vj cn Vn ,

δ ij = 1, i = j ,

δ ij = 0, i ≠ j;


where δ ij is Kronecker symbol. You can enter coefficients of turbulent diffusion in two-component mixture in the corresponding reference system: z

D тi = D т z α ij .


For medium-volume system of reference ∨ Dт1 = ∨ D т 2 = D т , and for medium-mass system: m D т1 = m 2 D т (V2ρ )≠ m D т 2 = m 1 D т (V1ρ ). Because volumes of medium, which participate in the turbulent pulsations, significantly exceed molecular dimensions, the intensity of the turbulent mass transfer may be significantly higher than of molecular mass


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transfer. The ratio of coefficients of turbulent and molecular diffusion in the wall (boundary) region reaches D т D i ~ 10 2 − 10 5 . 1.2. The Law of Mass Conservation In the analysis of technological processes and calculation of devices the laws of conservation of mass, momentum and energy are used. It should be recalled, that fundamental laws are formulated on the basis of a vast experimental material and do not involve any theoretical justification. Relativistic effects of the relationship of mass and energy in the chemical technology, as a rule, are negligible. Conservation laws can be recorded with regard to the entire system or its parts (integral form), as well as to individual points of space (local form), in addition, can be used for the environment in general (medium) or individual components. Essence of the law of conservation of mass is that mass (weight) cannot disappear or emerge, i.e. the total number of mass in a closed system is constant (closed system does not exchange by mass with the environment), therefore, ∆M = 0 or dM/dt = 0. Let us consider mass conservation law for open systems. Integral form of the law of mass conservation (material balance). Change in mass in a fixed volume V is caused by the difference between input and output of mass inside a selected volume:

∆M = V ∆ρ = M in − M fin ,


where ∆ρ - change of density. For description of continuous processes often more convenient to use the notion of mass consumption (flow) G, which corresponds to the amount of mass, that passed in a unit time. Let us the values, included in equation (1.25), rewrite for infinitely small time intervals:

dM dρ (1.26) =V = Gin − G fin . dt dt If the density of a substance does not change (the environment is incompressible) or the process takes place in stationary conditions (steadystate), then material balance can be simplified:


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dM dρ =V = 0 → Gin = Gfin . dt dt


You can write the equation of material balance for each component:

∆M i = V ∆ρi = M i ,in − M i , fin .


This equation is not universal and is valid only in the absence of chemical reactions in a system, as in the latter case some components can transform into the other. In general case, equation of material balance for each component will have the form

∆M i = V ∆ρi = M i ,in − M i , fin + rm,iVt ,


where r m , i is the mass of a component i, arising in unit volume per unit time (source of mass). Summing up equation (1.29) for all components, we must obtain equation (1.25) for the whole mass in general. Hence arise the natural condition for the sources of mass of individual components (negative sources of mass sometimes are called sinks): n

∑ rm , i = 0



You can rewrite equation (1.29) in terms of flows: dM i d ρi =V = Gi ,in − Gi , fin + rm ,iV . dt dt


Local form of the law of mass conservation (continuity equation). Law of mass conservation in the local form may be formulated similarly to material balance. The difference consists only in the fact, that in this case we analyze not the finite volume V, but an infinitely small volume dV:


∂ρ = Gin − G fin . ∂t



 ∂ ∧ jm ∂ ∧ jm  ∂ρ ∂ ∧ jm y x z . = − + +  ∂t ∂y ∂z   ∂x



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This formula can be represented, using differential operator ∇ , as well as the expression for mass flux, in the form of → → → → → ∂ρ = − ∇⋅∧ jm = − div ∧ jm = − ∇⋅ (ρm −∧ W) . ∂t


This equation, that expresses the law of mass conservation in the local form, bears the name of continuity equation. Left part of this equation describes the change in time of the density of a moving medium in a fixed point of space (elementary volume dV), which is motionless in respect to laboratory system of reference. The movement of a medium as a whole has been analyzed earlier. It is easy to get the law of conservation of mass for multi-component systems in the local form for each component: → → ∂ρi = − ∇⋅∧ jmi + rm,i ∂t


In general, the law of mass conservation in relation to a single volume can be formulated as follows:

rate  resulting rate source        of accumulation = of reception  + of  of masses  of masses  masses       Equation of the type (1.35), when considering multiphase systems, can be written for each phase separately. If there occurs a transfer of mass component from one phase to another, then it may be taken into account in the source of mass rm,i . It was said earlier, that for multi-component medium are usually used flows not of a mass, but substance, and respectively, instead of densities of the components are used their mole concentrations. Dividing equation (1.35) on mole mass of the component mi, we get → → r ∂c i m ,i . = − ∇⋅∧ ji + ∂t mi


Let us consider the simplest case of mass transfer in twocomponent medium in the absence of chemical reactions. Using expression


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for the flow of a component in medium-mass frame of reference, you can write the equation of unsteady-state convective diffusion: → → → ∂ci = − ∇⋅ (ci m −∧ W − ( m Dij + m Dтi ) ∇ ci ) . ∂t


Using the assumption of the constancy of ρ and taking into account →

that in this case ∇⋅m − ∧ W = 0 , we obtain → → → Dci → m = ∇⋅ (( Dij + m Dтi ) ∇ ci ) = − ∇⋅m jid , Dt


Dci - full (total, substantial) derivative, which characterize the Dt change in concentration over time to an observer, moving together with a where

medium, with the velocity m − ∧ W . In case, when the medium-mass velocity in laboratory frame of →

reference is equal to zero ( m − ∧ W = 0 ⇒ Dт = 0 , Dci Dt = ∂c i ∂t ) m and assuming that Dij = const , we obtain the equation, called the second

Fick`s law:

∂ ci m = Dij ∇ 2 ci . ∂t


If we assume the stationarity (steady state) of the process, then it can be more simplified: (1.40) ∇ 2c i = 0 . Thus, here on the basis of the law of conservation and equations of mass transfer are received differential equations, by solving which we can define the fields of concentrations and mass flow components in any machine. Integration of differential equations gives a common solution for a whole class of processes. For obtaining a specific private decision the equation should be supplemented with the single-valuedness conditions. 1.3. Interfacial Mass Transfer Conducting of basic processes of chemical technology is accompanied by transfer of substances from the nucleus of one phase through the phase boundary to another phase. We can identify the transfer


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of mass, heat and momentum according to the type of transferred parameter of a substance. Process of interfacial mass transfer can be divided into three stages: transfer of mass from nucleus of the first phase towards the phase boundary, transfer directly through the phase boundary, and transfer from the phase boundary towards nucleus of the second phase. Transfer of mass from the boundary to the nucleus of a phase or from the nucleus to the boundary is called mass delivery (internal mass transfer). Transfer of mass directly through the phase boundary is called mass transfer (external mass transfer). Decision of the engineering tasks often does not require knowledge of magnitudes of the fields of velocity, pressure, temperature, concentrations, as well as of the flows of substances in the whole volume of a device. For such purposes would be enough to obtain the magnitudes of these variables, averaged over cross-section of each of the phases in the output from machine, especially when the input magnitudes of these variables are known. This can be done with the help of laws of conservation of mass, momentum and energy in the integral form, knowing the quantity of substance and (or) the values of its parameters, transferred from one phase to another. For determination of the latter the equations of interfacial transfer of a substance are used. Thus, in engineering practice, solution of a system of differential equations with partial derivatives is replaced by solution of a system of linear algebraic equations of the balance of substances and interfacial transfer, which is the significant simplification. However, theoretical definition of coefficients in equations of interfacial transfer of substances, derivation of equations and their proper application, can be made only on the basis of transfer equations. Mass delivery equation. Local form of equation. Imagine an elementary, infinitely small section of an interfacial surface dF, and the rectangular reference system, which has been oriented so, that the plane X - Z coincides with the surface of the phase, and the X-axis - with the direction of the movement of the marked phase (because the interfacial surface area is selected infinitely small, then the interfacial surface can be considered flat). Flow of mass through the boundary of a phase will occur along the normal to the boundary, i.e. along the Y axis. Let us consider a flow of mass, which exists at the expense of molecular and turbulent transfer


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→ d

mechanisms. We can name it diffusion flow of mass ji . In contrast to the total flow, this value does not take into account convective mechanism of mass transfer, in spite of the fact that the latter can be present. Projection of the flux of mass by means of diffusion can be recorded on the Y-axis (when considering the change of this magnitude only in the direction of the у) in the form of

jdiy = −( b D*ij + Dт )

∂сi ∂y

= −( b D*ij + Dт ) x,z = const

dсi . dy i


Let us introduce the value ji *(y), which characterizes the ratio between the magnitudes of the flow at a distance y from the interfacial boundary and on the phase boundary:

jdiy (y)

ji * (y) = ji * =

jdiy (0)


jdiy jdb iy



Equation (1.41) can be rewritten, using the value ji *(y), then it can be solved in respect to dс i and integrated, using the model of diffusion boundary layer (area near the phase boundary, in which occur 99% of changes in concentration), from the phase boundary up to the outer boundary of the layer (δ δd). Respectively, concentration will change from the magnitude on the interfacial (boundary (b)) surface (с i b) up to the magnitude in the nucleus (n) of the phase (с i n). It should be borne in mind, that coefficient of turbulent diffusion is a function of the distance from the phase boundary Dт(y) and may not be taken out from under the integral ci n

∫ dc

ci b

δd i

db iy

= −j

∫( 0


ji * dy . D*ij + Dт )


We can obtain the equation of mass delivery by the way of solving the equation (1.43) in respect to jdb . The expression in its right side, which iy stands before the difference of concentrations, is called coefficient of mass delivery:


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 δd ji * dy  b n j =∫ b *  (сi − сi ) .   0 D ij + Dт  db iy


This allows you to determine coefficient of mass delivery (1.46): ∧

jiydb = β i ( µib − µin ) = β i (cib − cin ) , βi =

1 δq

j*idy ∫0 (b D*ij + Dт )


(1.45) (1.46)

The difference of the numbers of chemical potentials or concentrations at a phase boundary and in the nucleus of a phase is called moving (driving) force of mass delivery. Difference of the moving force value from the zero is the necessary condition of the process. The meaning of coefficients in equation (1.45) can be easy understood, if we will solve the equation for obtaining this coefficients: ∧

βi =

βi =

jdb iy (µbi − µni ) jdb iy (cbi − cni )



dM i b , mol2/J·m2s; dFdt(µbi − µni )

dM bi , m/s; dFdt(cbi − cni )



Coefficient of mass delivery – amount of substance of component i, transferred from the boundary of the phase into the nucleus of the phase or in the opposite direction per unit time, through the unit of the interfacial ∧

surface per unit of the moving force. Coefficients β i and β I are defined by equations (1.47) and (1.48), depending on the use as a moving force of a process, respectively, the difference of chemical potentials or concentrations. In future we will use mainly coefficients β i as the most frequently used in practice. Equations (1.41) - (1.48) are valid for twocomponent media. In case of multicomponent systems should be used matrixes of coefficients of mass delivery.


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Integral form of mass delivery equation. Equations of interfacial transfer of substances in the integral form are more often used in practice in case, when they are obtained by means of averaging of local equations for a part or for overall interfacial surface F: F



& b = dMi = jdbdF = β (cb − c n )dF = β F(cb − c n ) , M i i i i ∫0 iy ∫0 i i i dt


In general, such a record is conditional in case of a simultaneous variation of coefficient of mass delivery and of moving force over the interfacial surface, because it is impossible to divide the procedure of averaging of coefficient and of moving force (integral of a product is not equal to a product of integrals). In extreme cases, we can conduct an independent averaging of one value, but then the averaged value of the second variable will depend on the nature of variation and on the method of averaging of the first value. As a rule, moving force undergoes a much more change, than coefficient of mass delivery. Therefore, kinetic coefficient can be considered constant and then the moving force can be averaged as one variable. Determination of coefficients of mass delivery, similarity of the corresponding processes. Mass delivery coefficients cannot be found theoretically from (1.46 ) for most practically important cases, therefore, they are usually determined by using method of physical modeling. It is based on a generalization of experimental data with the use of the theory of similarity in the form of criteria equations. In case of mass delivery, the determinate will be diffusion Nusselt criterion (number) Nu d , also called Sherwood criterion (number) Sh, and the determinative criteria - Reynolds number Re, diffusion Prandtl criterion Prd , also called Schmidt number Sc, and diffusion criterion (number) of Fourier Fod :

Re =

Wl 0 ν


Fod =


D* ijt l 20



ν βil 0 (1.52), (1.53) Prd ≡ Sc = b * * D ij D ij Equations of mass transfer. Local form of equations. In this section transfer of mass component i from the phase I through the phase Nu d ≡ Sh =



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barrier surface into the phase II by means of molecular and turbulent mechanisms will be discussed. Let us suppose, that interfacial surface resistance towards the transfer of substances can be neglected. This is equivalent to the assumption about the establishment of equilibrium at a phase boundary. Then you can write the equality of chemical potentials of component i from both sides of interfacial surface

µiIb = µiIIb


Let us consider the derivation of mass transfer equation. If we denote by index I the first phase, out of which the transfer of substance occurs, then µiI > µiII . Then we shall direct Y-axis from phase I to II. In view of the above, we can write equations of mass delivery of each of the phases (1.45), dividing them on the corresponding coefficients:

jdb iy ∧

= µ niI − µ biI ,


= µ biII − µniII .


β iI

jdb iy ∧

β iII We can add up these two equations, bearing in mind the assumption (1.54), and solve them in respect to the substance flow of component i, which passes through the interfacial surface:    1 + 1  = µ n − µn , (1.57) jdb iy iI iII ∧ ∧   β iI β iII  −1

   1 + 1  (µ n − µ n ) . (1.58) jdb = iy iI iII ∧ ∧   β iI β iII  Equation (1.58) is called equation of mass transfer, and the value before the difference between chemical potentials in its right-hand part is called coefficient of mass transfer. Thus, ∧

n n , jdb iу = K i (µ iI − µ iII )



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Ki =



β iI



1 ∧

β iII

where K i is coefficient of mass transfer. Its meaning differs from the coefficient of mass delivery (1.47) only by the fact, that it characterizes the transfer of substances from one phase to another, but not inside phases. The moving (driving) force in this case is the difference between chemical potentials of a component in the nuclei of two phases. Thus, equation (1.59) is of extremely easy content, which testifies to the proportionality of interfacial flow of mass to the deviation of system from equilibrium state. But it would be a mistake to substitute in (1.59) concentration сi instead of µi, as the equality of concentrations of component in phases is not a condition of equilibrium. There can be a process of mass transfer in case of equality of ciIb = ciIIb, and, on the contrary, equilibrium, i.e. the lack of interface transfer, when ciIb ≠ ciIIb. There is a possibility of presentation of the moving force of mass flow (transfer) through the difference in concentrations, however, it will be the difference of working and the equilibrium concentration of a component in one of phases. The ratio (1.60) can be rewritten differently:

1 ∧



1 ∧

β iI


1 ∧



β iII

The values, which are inverse to considered kinetic coefficients, are named

resistances: 1/ Ki , - resistance of mass transfer (interfacial ∧

resistance), and 1/ β i , - resistance of mass delivery (phase resistance). It is easy to see that the ratio (1.61) expresses the additivity of phase resistances. Integral form of mass transfer equation. You can get the integral form of equation by means of averaging of the local equation of mass transfer on the surface:


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F ∧

& b = jdbdF = K i (µ b − µ b )dF = K i F(µ b − µ b ) , M i iI iII iI iII ∫ iy ∫ 0



In general, like for equation (1.49), such a record is conditional in case of a simultaneous change of kinetic coefficient and moving force on the interfacial surface, because it is impossible to divide the procedure of the averaging of kinetic coefficient and driving force. Similarly to mass delivery, it is possible to average one of the values independently, but then the magnitude of the second value will depend on the nature of variation of the first value. 1.4. Phase Equilibrium Main purpose of this section is to establish the nature of relationship between concentrations of components in phases in equilibrium. Mass-exchange processes in chemical technology are taking place in systems, consisting of two or more components, so here will be considered binary and multi-component mixtures. The material is presented taking into account a detailed study of this issue in the courses of physics, thermodynamics and physical chemistry. Conditions of equilibrium in a hetero phase system, which is not under the influence of external forces, are the equality of pressure, temperature and chemical potentials of components in all phases. The chemical potential of component i in each of phases is defined as the partial derivative of Gibbs energy with respect to the number of moles of the i-th component at the conditions of fixed pressure, temperature and quantity of matter of other components:

 ∂G  p, T, N j = const ; µ i =  N ∂  i

G = G(p, T, N1 , N2 ...Nn ) .

µiI = µiII , µiI = µiIII ,..., µiI = µiF , i = 1, n .



Since the chemical potential is a function of temperature, pressure and composition - µ i = µ i (P, T, x1, x2, ... xn-1) - as it follows from (1.63), then equation (1.64) establish a connection between these values, reducing the number of independent variables on the number of equations.


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With this in mind, a number of degrees of freedom C (a number of independent variables) in the equilibrium hetero phase (F) system will be equal to C = F ( n − 1) + 2 − ( F − 1) n = n − F + 2 (1.65) Thus we have received the ratio, known as Gibbs phase rule. Now we proceed to establish the connection between concentrations of components in the phases in equilibrium. Formal relations, which replace (1.64), can be written very simply, they are called equations of equilibrium: I

, c i = m Ii ,,cII c II i

c Ii = m iI,,cIII c III i ,...


The proportionality coefficients between concentrations in different phases bear the name of coefficients of distribution. The problem is to find them. Using Gibbs phase rule (1.65), you can determine the number of independent variables, from which will be dependent coefficients of distribution. Specific values can be found from (1.66), using experimental data about equilibrium concentrations, or from (1.64), if the nature of the concentration dependence of chemical potential is already set. It should be noted, that the number of distribution coefficient depends on the way of expressing of concentrations in (1.66). So, if instead of the volumetric mole concentration сi use the molar share xi or relative mass concentration X i , the distribution coefficients will be different: m i , c ≠ m i , x ≠ m . Because later we will consider only two-phase i,x

systems, the upper indices of distribution coefficients can be omitted. If in each of the phases is not more than two components, then usually the corresponding them values are not marked by the lower index i, which shows the number of components. The implication is that this values refer to the distributed component. If interfacial boundary comes through the both components, then this values refer to the easy-flowing component. Concentration of the component in phases can be indicated by different letters, what allows not to mark a number of the phase. So, for the designation of mole fraction in a gaseous phase is usually used letter y, and in a liquid phase - x. In this case, equilibrium equations can be written in the form yi =

c Ii c Ii

+ c Ij

, =y

xi =

c II i c II i

+ c IIj

= x,


y = mx • x ,


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Yi =

m′i c Ii , =Y m′j c Ij

Хi =

m′i c II , i =X II ′ mj cj

Y = mx • Х ,


where m′i - mole mass of component i. The connection between distribution coefficients, which correspond to the different ways of expressing the concentration, is given in [9]. Finding coefficients of distribution from the experimental data on equilibrium concentrations with the usage of relations of the type (1.66)(1.68) is extremely simple. However, it should be borne in mind, that even for two-component two-phase system, where in accordance with (1.65) С=2, coefficient of distribution will depend on two variables: mx=mx(T,x)=mx(р,x)=mx(T,р). Therefore, it is necessary to have a set of experimental data on equilibrium over the whole area of application of both parameters. Situation becomes even more complex in case of increasing in the number of components in the system. In the absence of the necessary volume of experimental data, and also if you wish to have an analytical dependence of coefficients of distribution from state parameters, you can use the other way – try to establish a connection of concentrations with chemical potentials, for which the condition of equilibrium has a simple form (1.64). Let us try to install this link. Lewis proposed to write the chemical potential in the form of

µ i = µ i+ + RTln(f i ) ,



where µ i is some function which depends only on temperature, and fi volatility or fugitiveness of component i. More convenient is to use not the absolute value of volatility, but the relative аi, assigning it to some standard volatility fi0. The relative volatility аi is called the activity of component i:

µ i = µ i0 + RTln(ai ) ,

µ 0 i = µ i+ + RTl n(f i0 )


In actual gases (vapors) volatility of a component does not coincide with its partial pressure. At this case volatility can be expressed through activity аi or coefficient of volatility V*i, using as a standard status the pure ideal gas of i kind (id) at a temperature and pressure of a real mixture:


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о ,b i


id ,b i yi →1


= р,


ai =

fi b = V *i yi f i о ,b

fi b = ai fi о ,b = ai р = V *i yi р .

(1.72) (1.73)

For liquid mixtures are usually used two different standard statuses. If temperature and pressure of a mixture correspond to the liquid state of a pure component i, then just it must be selected as the standard status. If a pure component exists in the gaseous or solid state at the same temperature and pressure of a mixture, then the status of infinitely diluted system must be chosen for him as the standard: fi 0, L = fi L ` = fi L , xi → 1 ,


fi L = ai′ fi L ` = γ i′xi fi L` ,


fiL,о = fiL" = fiL , x i → 0 , (1.78), fiL = a′′ifiL" = γ ′′i xifiL" ,

fi L = γ i′xi fi L `


xi → 1, γ′i → 1, a′i → xi


ai′ =


a′′i =

fiL = γ ′′i ⋅ x i fiL"


x i → 0, γ ′i′ → 1, a′i′ → x i (1.81)

Here and hereinafter, the values, characterizing pure component i (xi → 1), are indicated by one stroke, in the state of infinite dilution (xi → 0) - by two strokes, except the activities аi and coefficients of activity γi , for which a stroke indicates only the choice between the first or the second standard status. With the help of the notion of volatility, the condition of phase equilibrium can be formulated in another form, than in (1.64). Taking into account the equality of temperatures of phases, and, consequently, µi+, we obtain from (1.64) and (1.69): (1.82) f i I = f i II = ... = f i F , i = 1, n . Now we shall find the coefficients of distribution mi for different cases of equilibrium in two-phase systems. Balance in a liquid - liquid system. Some liquids have limited solubility, which leads to the disintegration of a system into phases, for example, oil - water. Only strongly non-ideal mixtures, in which the energy of attraction of heterogeneous molecules is much less than


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homogeneous, can exfoliate. Exfoliation can be sometimes observed only within a certain range of temperature and concentration values, and outside of this field the system will be a single phase. Pressure has little effect on the equilibrium in a liquid - liquid system. Condition for equilibrium in two-phase liquid system has the form

f i LI = fi LII


II x Ii γ Ii = x II i γi ,

mi,x =

x Ii x II i


γ II i .

i = 1, n ,

(1.83) (1.84)

γ Ii

Thus, finding of distribution coefficient is reduced to the calculation of activity coefficients of component i in different phases. Determination of coefficients of activity is a complex problem. In practice the model equations, describing the deviation of a liquid mixture from the ideal, are often used. They are based on various models of determination of Gibbs energy, excessive in relation to the ideal mixture, what allows you to calculate coefficients of activity, using several parameters for each pair of components, which compose the mixture. These parameters are extracted from experimental data on phase equilibrium of binary systems and are listed in the reference literature. The most universal can be considered the uniform chemical equation UNIQUAC. In addition, for this purpose are also widely used more simple equations of Wilson and NRTL. For mixtures of substances, for which there are no experimental equilibrium data, it is possible to forecast the calculation of coefficients of activity with the help of equations UNIFAC. It is based on the equation of UNIQUAC and the method of group components, the essence of which consists in the determination of the substance properties by the way of analysis of its structural groups (methyl, carbonyl, etc.). Equilibrium in a solid - liquid system. Let us consider the simplest case, when a pure component i is a solid (s). The condition of its equilibrium with a mixture of liquid components will have the form or (1.85) fi s = fi L fis` = γ ′′i x if iL" The use of distribution coefficient in this case does not have any sense, as the mole fraction of component i in the solid phase is known (is equal to unity). The value, which have to be determined, is the equilibrium


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concentration of component i in the liquid phase, called the concentration of saturation, or solubility: f s` (1.86) x*i = i L" . γ ′′i fi In the particular case of two-component two-phase system, the number of degrees of freedom is equal to two. Taking into consideration a small dependence of properties of condensed media from the pressure, temperature can be considered as the only parameter, that defines solubility of solid substances in a mono-component solvent. Solubility of solid substances in various solvents in certain temperatures are given in the reference literature. 1.5. Various Modifications of Mass Delivery and Mass Transfer Equations The equation of mass transfer (1.59), obtained in the section 1.3, contains the moving (driving) force, which is determined as the difference of chemical potentials of components in the nuclei of different phases. Calculation of chemical potential is a rather complex task, as evidenced by the content of the previous section. That’s why in practice are usually used equations of mass delivery and mass transfer, containing the difference of concentrations of a component as the moving force of a process. Often a big problem of usage of the integral form of equations of mass delivery and mass transfer ((1.49), (1.62)) consists in the definition of the surface of phases (interfacial) contact in a real machine, which can be formed of the surfaces of sprays, drops, bubbles, foam. In this case are applied the modified equations, which do not contain the amount of interfacial surface. Before proceeding to the derivation of modified equations of mass delivery and mass transfer, we shall get some necessary for it ratios. In addition, further we will consider two-component mixtures. Equations of material balance, workers and equilibrium lines of mass-transfer processes. Imagine, that two phases I and II with the flows G and L respectively, move counter-currently to each other in a typical cylindrical vertical machine for mass-transfer processes. Let us denote concentration of the distributable component in them as y and x. Let us assume, that concentration may be changed only with the height of apparatus and being permanent or averaged for each cross-section, i.e. we simplify the task to one-dimensional. Units of measurement of flows is


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better to choose in such a way, that G and L does not change with the height of apparatus (kg/s of inert component in case of absorption, kmol/s of a mixture in case of rectification, etc.).

Fig. 1.3. Scheme of mass exchange process in a vertical counter-current machine: L, G – flows of phases; x, y - the concentration of distributable component in phases; indices «in» and «fin» - initial and final state; I, II – number of phases; A – A is an arbitrary cross-section of an apparatus In stationary conditions, the law of conservation of mass (matter) for the whole apparatus (Fig. 1.3) can be recorded in accordance with (1.27) in the form of the equation of material balance: total input of mass (matter) should be equal to its total output (consumption):

Gin + Lin = G fin + L fin .


You can write the material balance for distributable component in the absence of chemical reactions:

Gin yin + Lin xin = G fin y fin + L fin x fin . 32


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In case of constancy of flows (G, L = const), equation (1.88) can be simplified: (1.89) G ( yin − y fin ) = L( x fin − xin ) , or for an elementary section of the apparatus (1.90)

− Gdy = Ldx

The sign «minus» testifies to the opposite change of concentration of distributable component in phases: if in one of phases the concentration increases, then in the other decreases. Equation of the working lines can be obtained from the equation of material balance. Let us write down the equation of material balance for a part of an apparatus from the lower section up to a certain current section A - A (Fig. 1.3) and resolve this equation in respect to the concentration of distributed component in one of the phases:

Gin y in + Lx = Gy + L fin x fin , y=


L fin G L x + in yin − x fin G G G


This equation is called equation of the working line of the countercurrent mass exchange process. It describes a set of the working concentrations values of a distributed component in the phases for arbitrary cross-section of an apparatus. Under the working concentrations is understood the concentrations of nucleus of the phase, whose values are averaged upon the section, or being constant across the cross-section of an apparatus. In case of the constancy of flows, the equation of the working line is simplified:

L L x + yin − x fin ; G, L = const G G This equation of a straight line can be represented in the form of y=

y = Ax + B ,

where A = L , G


B = yin −

L x fin G



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Similarly may be obtained the equation of the working lines for the forward-current movement of phases:

Gin y in + Lin xin = Gy + Lx , y=−


y = − Ax + B′ ,


G L L x + in yin + in xin , G G G

L L x + yin + xin ; G G

where A = L . , G


G, L = const


L xin G


B′ = yin +

Let us write down equation of the equilibrium line, which connects the working concentration of the distributed component in one of the phases with its equilibrium concentration in the other phase. Under equilibrium concentration in a random cross-section of an apparatus is understood the concentration of the component in the phase, which is in equilibrium with the other phase, whose composition is determined by the working concentration. The equation of the equilibrium line can be written in accordance with (1.67): (1.99) y* = mx , where y* is the equilibrium concentration in the phase I, x - the working concentration in the phase II, m – distribution coefficient. Methods of determination of coefficient of distribution are considered in the previous section. The value of m can be constant (for dilute solutions), then the equilibrium line will be direct, or can depend on x, then the equilibrium line will be curved.


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y B’ yA В

1 А


y*(x) y*A





Fig. 1.4. Working (1 and 2) and the equilibrium line on the x-y plot: 1 – counter-current and 2 – forward-current movement of phases In Fig. 1.4 are shown the working line and the equilibrium line in case, when the working concentration in the phase I exceeds the equilibrium concentration. With the aspiration of the system for a state of equilibrium the working concentration in each of the phases is moving closer to equilibrium. If the concentration of distributable component in the phase is above equilibrium, then this component will go away from this phase to another, where its concentration is below the equilibrium. In this case, the distributed component will move from phase I to phase II, as y А > y *А , x А < x *А . In case, when the concentration of a component is equal to equilibrium, the interfacial transfer of a substance is absent. Thus, by mutual orientation of the working and equilibrium lines we can make a conclusion about the existence or absence of mass transfer process, as well as about its direction. We can also assume that the value of the interfacial flow of a component will be proportional to the deviation of the system from the equilibrium state, i.e., the difference of the working and equilibrium concentrations. Let us confirm this assumption. Mass transfer equation in the local form. Let us write down the equations of mass delivery for the two phases I and II and mark them by indices y and x, respectively. We shall use the difference in concentrations as the moving (driving) forces. In order to simplify the record of equations, we shall omit the upper index «d» and lower «y» in the designation of interfacial flow, the upper index «I» when referring to concentration, the


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lower index «i», which corresponds to the number of component. Let us suppose, that the distributed component moves from phase I to phase II:

j b = β x ( x b − x)

j b = β y ( y − y b ) , (1.100),


where x, y – working concentrations of the distributable component in the phases. Let us use the assumption (1.54) about the absence of resistance of interfacial surface towards the transfer of a substance, or about equilibrium at the phase boundary, expressing it in the form of

µiIb = µiIIb

y b = m( xb ) xb


(1.102) Further derivation of the mass transfer equation is similar to (1.55) - (1.58). We can express xb from (1.102) and substitute it in (1.101). We can solve equations (1.100) and (1.101) relative to the difference of concentrations, summarize both equations and then find from the received equation the expression for the flow:  1 m(xb )  j = +  β β x   y b



 1 m(xb )  y − m(x )x =  +   β x   βy b



 m(xb )  1.103) y * y− m(x)  

It is easy to see that, if the distribution coefficient does not depend on the composition of the phase, m(xb) = m(x) = m, then the equation (1.103) can be simplified:

j b = K y ( y − y*) ,


1 1 m = + K y βy βx


In general, equation (1.103) can be transformed to a traditional record of the equation of mass transfer with the help of (1.100) - (1.102). However, coefficient of mass transfer Ky, in case of dependence of m on the composition, will be determined as follows:


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1 1 m1y , = + K y βy βx

m 1y =

m( x г ) x г − m( x ) x xг − x


y г* − y *



xг − x

For finding m1y we need to know the limit value of concentration x , which is determined from the solution of the system of equations (1.100) - (1.102). If the equilibrium line for the cross-section of an apparatus А-А in the area from xA до xA* can be approximated by a straight line, then for finding m1y there is no need to solve the system of equations and determine xAb, because in this case b

m1 y ≈

dy * dx x= xA + x∗A


m1 y ≈


y − y * y A − m( x A ) x A = x * − x y A m( y A ) − x A


If the value yb of (1.102) substitute into (1.100) and repeat the above transformations, then the equation of mass transfer will take the form of

jb = K x ( x * − x ) , 1 1 1 = + K x β x m 1x β y

1 1 1 (1.110), = + K x β x mβ y

m1x =

y − yb y − yb = y m( x) − y b m( xb ) x * − xb∗

(1.109) (1.111)


In case of approximation of the line of equilibrium by a straight line in the area from yA* to yA, the value m1x = m1y = m1 and it is determined by the ratio (1.108). Thus, we have obtained the equations of mass transfer, where as the moving forces are used the differences of the working and equilibrium concentrations of a component in one of the phases. The usage of one of two mass-transfer coefficients Ky or Kx depends on the choice of the phase,


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with the help of whose concentrations the moving (driving) force has been recorded. During the calculation and usage of coefficients of mass delivery and mass transfer we must monitor their accordance with the dimensionalities of flows, moving forces, coefficients of distribution, mass delivery and mass transfer. If the moving force is expressed in mole fractions, and the flow of a matter - in kmol/(m2s), coefficients of mass delivery and mass transfer will have the dimension of kmol /(m2s · mole fraction). In this case distribution coefficient must also connect the equilibrium concentrations of a component, expressed in mole fractions. From the equations (1.104) and (1.109) it is easy to establish the link between these coefficients: K x y − y * y − m( x )x (1.113) = = K y x * − x y m( y ) − x In particular cases, ratio (1.113) can be simplified. So, when m = const, it can be reduced to the ratio Kx (1.114) =m Ky In case of approximation of the line of equilibrium by a straight line in the area from x to x*, we can get from (1.107) the ratio Kx (1.115) = m1 Ky Integral form of mass transfer equation. We can obtain equations of mass transfer in the integral form by the way of integrating the equations (1.104), (1.109) over the whole interfacial surface of the apparatus or of its part: F



M& b = ∫ j b dF = ∫ K y ( y − y*)dF = ∫ K x ( x * − x)dF 0




These equations acquire the practical meaning, only if the values of coefficients of mass transfer can be considered constant (Ky, Kx= const) at the considered area of integration. Then they can be withdrawn out from under integral and rewritten as


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M& b = K y ∫ ( y − y*)dF = K y F ∆ym ,




1 ∆ym = ∫ ( y − y*)dF , F0



M& b = K x ∫ ( x * − x) dF = K x F ∆xm ,




∆xm =

1 ( x * − x)dF . F ∫0


Equations (1.117) and (1.119) bear the name of basic equations of mass transfer. We can define the average moving forces of mass transfer process for the model of ideal displacement and constancy of the flows along the height of an apparatus (G, L = const). The quantity of distributable component, which moves out of the phase I (y) into the phase II (x) per unit of time dМ& b across an elementary part of the interfacial surface dF, can be expressed either from the equation of material balance (1.90), or from the equation of mass transfer (1.104):

dM& b = j b dF = −Gdy = K y ( y − y*)dF


Let us divide the variables and then integrate them over the surface of interfacial contact in the analyzed apparatus (part of apparatus): Yfin

−∫ Yin

F K KF dy = ∫ y dF = y y − y* 0 G G


Using the integral form of equations of material balance, you can write:

M& b = G( yin − y fin )


We can define G from (1.123) and substitute it in (1.122), then change the limits of integration in the left part of this equation to get rid of the sign «minus», and solve the equation for М& b :


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yin − y fin M& b = K y F Yin dy ∫Y y − y * fin


Comparing (1.124) and (1.117), we find:

∆ym =

yin − y fin Yin

Y fin


dy y − y*

Similarly, you can obtain the expression:

∆xm =

x fin − xн X fin

∫ X in

dх x * −x


The calculation of the average moving (driving) forces of mass transfer provides for a finding of a definite integral. In particular case, when the distribution coefficient m = const within the limits of integration, or the line of equilibrium can be approximated by a straight line, the average moving (driving) force is determined by the average logarithmic value. This can be seen, substituting the corresponding dependence in (1.125) or (1.126):

∆ym =

∆yup − ∆ylw  ∆y  ln  up   ∆ylw 



where ∆yup and ∆ylw - the moving force of mass transfer in the upper (up) and lower (lw) sections of an apparatus (part of an apparatus). A similar ratio is true for ∆xm.


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Fig. 1.5. Determination of the average moving force of mass transfer in case of approaching of equilibrium line towards the form of a straight line: ∆yup and ∆ylw - moving force of mass transfer in the upper and lower sections of the apparatus respectively Volumetric coefficients of mass delivery and mass transfer. As it has been already mentioned, it is often difficult to determine the surface of the interfacial contact in a real machine, because this surface can be composed of the surfaces of sprays, bubbles, drops, etc. This makes difficult to use fundamental equation of mass transfer, which contains the value of F. One of the ways of solving this problem is the usage of modified equations of mass delivery and mass transfer, in which interfacial surface F is not included. We shall use the notion of specific surface area of contact of phases a as the contact surface, formed in the unit of the working volume of an apparatus: 2 3 (1.128) a = F V , m /m . Expressing F = a /V, we can rewrite the equations of mass delivery and mass transfer in the form of


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M& b = β y aV ( y − y b ) = β yV V ( y − y b ) ,


M& b = β x aV ( x b − x) = β xV V ( x b − x) ,


M& b = K y aV ∆ym = K yV V ∆ym ,


M& b = K x aV ∆xm = K xV V ∆xm .


Coefficients of mass transfer and mass delivery with the index “v” are called volumetric. They are connected with the usual coefficients, which are referred to the surface of interfacial contact, with the help of simple relations, as it follows from (1.129) - (1.132):

β yV = β y a,

β xV = β x a,

K yV = K y a ,

K xV = K xa


Expressions of volumetric coefficients of mass transfer through the volumetric coefficients of mass delivery are similar to (1.105), (1.106), (1.110), (1.111). If the values of volumetric coefficients of mass delivery or mass-transfer are already known, then, depending on the formulation of the task, we can extremely easy determine the working volume of the apparatus V or the quantity of the substance, which moves from one phase to the other in a unit of time M& b , with the help of equations (1.129)(1.132). At this case it is not necessary to solve a difficult task of defining the specific surface of the interfacial contact а. However, volumetric coefficients cannot be easily defined theoretically. It is difficult to obtain for them generalized equations with the help of the method of physical modeling too. You should pay attention to the dimensionality of volumetric coefficients of mass delivery and mass transfer. In the SI system they are measured in [s-1]. For calculating of apparatus with a stepped contact of phases, coefficients of mass delivery and mass transfer is more convenient to refer not to the volume of an apparatus, but to the area of the working section of a contact device f, for example, to the square of the working section of a plate. After defining the specific surface of contact of phases as af (interfacial surface, formed at the given contact device and referred to the


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working section), we can write the equation of mass transfer in the following way: (1.135) M& b = K y a f f ∆ym = K yf f ∆ym af = F f , (1.134) Similarly, you can rewrite and other equation (1.129)-(1.132) by using the factors of mass transfer and mass delivery, referred to the area of the working section of a contact device: β yf = βy af , β xf = β xa f , K yf = K y af , K xf = K xaf (1.136) Number and height of transfer units. Main geometrical parameter of the most common type of mass-transfer apparatus (cylindrical vertical columns), which depends on mass transfer rate, is the height of a device H. If the cross-sectional area of the machine S is constant, then its volume can be written as (1.137) V = SH

& from (1.123) in equation Substituting V from (1.137) and М (1.131), and then solving this equation on H, we obtain b


G yin − y fin = hoy noy K yV S ∆ym


This is another modification of the equation of mass transfer, also does not containing the interfacial surface value. Similarly, you can transform and the other equations (1.129) - (1.132):





x fin − xin

K xV ⋅ S



yin − y fin

β yV S ( y − y b ) L

x fin − xin

β xV S ( xb − x)


= hoх noх ,

= hy ny ,

= hx nx




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The first cofactors of these equations is called the Height of Transfer Unit (HTU), and the second - the Number of Transfer Unit (NTU). Moreover, before a name of the value, characterizing the process of mass transfer (hoy, hox, noy, nox), is added the definition «total» («common»). The values, characterizing the process of mass delivery (hy, hx, ny, nx), are called «partial», or «phasic». Number of units of transfer is the change of the working concentration of the phase in the limits of a part of an apparatus, relatively to the moving (driving) force of the process, averaged for this part. Then one unit of transfer corresponds to the part of the apparatus, for which the change of working concentration of the phase is equal to the average moving force in this division. Using expressions (1.125), (1.126) or (1.128) - (1.141), you can use many ways to provide total numbers of transfer units:

noy =

nox =

yin − y fin ∆ym x fin − xin ∆xm



Y fin

FK y VK yV fK yf dy , = = = y − y* G G G

X fin


∫ X in

fK xf FK x VK xV dx . = = = x * −x L L L



Similar relations can be obtained for partial numbers of units of transfer: yin − y fin Yin dy F β y V β yV f β yf , (1.144) ny = = ∫ = = = b G G G ( y − y b ) Y fin y − y

nx =

x fin − x’ ( xb − x )

X fin


∫ X in

f β xf F β x V β xV dx . = = = x −x L L L b


In case of possibility of calculation of the averaged moving (driving) force as an averaged logarithmic value, NTU can be easily obtained analytically. As coefficients of mass transfer are expressed through coefficients of mass delivery, total numbers of units of transfer could be expressed through the number of partial units of transfer. So, when m = const


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1 1 1 = + noy ny Am nx

1 1 A = + m nox nx ny



where Am = L/(mG) - factor of the mass transfer process. The height of the units of transfer corresponds to the height of the part of the apparatus, equivalent to one unit of transfer. As it follows from the equations (1.138) - (1.141), HTU are inversely proportional to the coefficients of mass delivery and mass transfer. The larger these coefficients, the less HTU and the lower height H will have a device, which secures the required separation of substances. Thus, the aim must be to design devices with smaller HTU, providing their lower steel capacity, of course, taking into account all other cost items too. Total heights of the units of transfer can be also expressed in partial heights:

hoy = hy +

hx , Am

hox = hx + Am hy



As volumetric coefficients of mass delivery and mass transfer, the heights of the units of transfer are usually found from the equations, obtained by generalization of experimental data. 1.6. Pecularity of Mass Transfer in Systems with a Solid Phase Participation Peculiarity lies in the transfer of distributable component inside the porous solids, which are used, as a rule, in the processes of extraction, adsorption, ion exchange, drying, membrane separation. The main mechanism of transfer is molecular, but in large pores in the presence of pressure gradient and capillary forces it can be supplemented by convective. Molecular diffusion in narrow pores acquires the specifics of limited or Knudsen`s diffusion, when the molecules of distributable component to a greater extent interact with molecules of the solid frame, than with similar molecules (see Appendix 2.2.). In addition, diffusion can be carried out and in a matrix of a porous body, as well as on the surface of pores. Theoretical description of all these effects causes certain difficulties, therefore, in practice are often used empirical coefficients of mass


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conductivity Ki or effective diffusion coefficients Die , using them instead of coefficient of molecular diffusion in the equations of the first (1.13) and the second (1.39) Fick's laws. Mass-carry with the solid porous bodies participation is described similarly to the heat exchange. For the case of mass-carry in twocomponent mixtures differential equations of unsteady-state convective diffusion (1.37) and of thermal conductivity are identical. This allows to use the ratios, describing heat exchange, for the description of mass exchange in case of hydrodynamic similarity of the flows and identity of the initial and boundary conditions of heat and mass movements. In this case most convenient is to use the dimensionless ratios, in which the criteria of thermal (T) similarity Nuт, Prт are simply replaced with the diffusion criteria Nud , Prd . Usage of equations of mass conductivity allows you, having replaced the thermal conductivity coefficient of λ on Ki, to applicate the thermal analogy to the description of mass transfer with a solid porous body. So, unsteady-state mass exchange with the boundary conditions of the third kind can be represented by analogy with the heat exchange in the form of

Bi d =

βi δ Ki

ci (y, t) − c*i = f (Fo d , Bi d , y / δ ) , coi − c*i Kt (1.151), Fod = 2i δ

(1.150) (1.152)

where ci is the concentration of the distributable components in the solid phase: ci (y, t) - in point y at time t, cio - in the initial moment of time; ci* - equilibrium with the concentration of component i in the nucleus of the flow; Вi d , Fod - diffusion criteria of Bio and Fourier; δ - a characteristic linear size. Dependences, presented in the reference literature for the heat exchange, can be also used for the mass exchange too. Let's consider one more typical case of a mass exchange between a solid phase (gas or liquid) and distributed in it elements of disperse phase (solid particles). For dissolving solid particles, all resistance of mass flow is concentrated in the solid phase. The solution for spherical particles of small diameter, slowly moving relative to the solid phase (Red