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Nanofluids: Mathematical, Numerical, and Experimental Analysis
 0081029330, 9780081029336

Table of contents :
Cover
Nanofluids: Mathematical, Numerical, and
Experimental Analysis
Copyright
Dedication
Contents
Preface
1 Introduction to nanofluids
1.1 History of nanofluids
1.1.1 Preparing nanofluids
1.1.2 Synthesis of nanofluids
1.2 Structures and different types
1.2.1 Case 1: Single phase: different shapes of nanoparticles in a wavy-wall square cavity filled with power-law non-Newton...
1.2.2 Case 2: Two-phase nanofluid thermal analysis over a stretching infinite solar plate
1.3 Nanofluid properties
1.3.1 Case 1: A modified multisphere Brownian model to predict the thermal conductivity of colloid suspension of wide volum...
1.3.1.1 Models for the thermal conductivity of colloidal suspensions
1.3.1.2 Multisphere Brownian model
1.3.1.3 Modification of the multisphere Brownian model
1.3.1.4 Establishment of database with various experimental results
1.3.1.5 Thermal conductivity measured in our experiments
1.3.1.6 Determining the parameters for the modified model
1.3.1.7 Dependency of the n values on the volume fractions
1.4 Benefits and applications
1.5 Other forces on nanoparticles in base fluid
1.5.1 Case 1: Natural convection heat transfer in an NF-filled cavity with double sinusoidal wavy walls
1.5.2 Case 2: The effects of nanoparticle aggregation on convection heat transfer investigated using a combined NDDM and DP...
1.5.2.1 Nanoparticle diameter distribution model
1.5.2.1.1 Discrete phase model
1.5.2.2 Problem description in second case and its solution
References
2 Mathematical analysis of nanofluids
2.1 Mathematical modeling of nanofluids properties
2.2 Weighted residual method for nanofluid modeling
2.2.1 Case 1: Heat transfer and nanofluid flow through circular concentric heat pipes
2.2.2 Case 2.A: Condensation of nanofluids
2.2.3 Case 2.B: Magnetohydrodynamic flow over porous medium
2.2.3.1 Solution of condensation of nanofluids
2.2.3.2 Solution of magnetohydrodynamic flow between parallel plates
2.2.4 Case 3: Peristaltic nanofluid flow in a divergent asymmetric wavy-wall channel
2.3 Differential transformation method for nanofluid modeling
2.3.1 Case 1: Thermal boundary-layer analysis of nanofluid flow over a stretching flat plate
2.3.2 Case 2: Peristaltic flow of nanofluids in a sinusoidal wall channel
2.3.3 Case 3: Inclined rotating disk
2.4 Other analytical/mathematical modeling
2.4.1 Case 1: Nanofluids over a cylindrical tube under the magnetic field effect
2.4.2 Case 2: Nanofluid passing over a porous moving semiinfinite flat plate
2.4.3 Case 3: Two-phase nanofluid flow over a stretching infinite solar plate
References
3 Numerical analysis of nanofluids
3.1 Finite element method in nanofluid
3.1.1 Case 1: Hot tubes in a wavy porous channel and nanofluid under variable magnetic field
3.1.2 Case 2: Circular-wavy cavity filled by nanofluid
3.1.3 Case 3: Wavy porous cavity filled with nanofluid in the presence of solar radiation
3.2 Finite volume method in nanofluid
3.2.1 Case 1: Nanofluid natural convection for an F-shaped cavity under magnetic field effects
3.2.2 Case 2: Different nanofluid flow through venturi
3.3 Lattice-Boltzmann method in nanofluid
3.3.1 Two-phase lattice Boltzmann method
3.3.2 Population balance equation
3.3.3 Coupling population balance equations and the lattice Boltzmann method method
3.3.4 Case 1: Dynamic nanoparticle aggregation for a flowing colloidal suspension
3.4 Finite difference method in nanofluid
3.4.1 Case 1: Alumina–water nanofluid in an inclined direct absorption solar collector
3.5 Runge–Kutta–Fehlberg numerical method
3.5.1 Case 1: Nanofluid analysis in a porous medium under magnetohydrodynamic effect
3.5.2 Case 2: Ferrofluid flow influenced by rotating disk
3.5.3 Case 3: Solar radiation effect on the magnetohydrodynamic nanofluid flow over a stretching sheet
References
4 Experimental analysis of nanofluids
4.1 Brownian motion of nonspherical particle
4.1.1 Model validation
4.1.2 Experimental thermal conductivity of nanocube nanofluid
4.2 Different properties of nanofluids
4.2.1 Viscosity
4.2.2 Thermal conductivity
4.2.3 Optical properties
4.2.3.1 Experimental properties attained over aggregation radii
4.2.4 Surface tension
4.3 Optical properties of nanofluids
4.3.1 Aggregation of nanofluid
4.3.1.1 Monte Carlo simulation
4.3.1.2 Population balance equation
4.3.1.3 Brownian dynamic simulation
4.3.2 Optical properties of nanofluids
4.3.2.1 Rayleigh scattering theory
4.3.2.2 Mie scattering theory
4.3.2.3 Maxwell–Garnett effective medium theory
4.3.3 Optical models considering aggregation
4.3.3.1 Generalized multiparticle Mie solution method
4.3.3.2 Finite difference time domain method
4.3.4 Experimental research on optical properties
4.4 Experimental correlations of nanofluid properties
4.4.1 Preparation of alumina nanofluids with controlled particle aggregation properties
4.4.2 Measurement of the absorption coefficient of the nanofluid
4.4.3 Measurement of particle size distribution
4.4.4 Experimental result of absorption coefficients
4.4.5 Experimental result of particle size distributions
4.4.6 Predicted results by theoretical model
4.5 Experimental application of nanofluids
4.5.1 Magnetic electrolyte nanofluids for a hybrid photovoltaic/thermal solar collector application
4.5.1.1 Application of electrolyte nanofluid in photovoltaic/thermal system
4.5.2 Highly dispersed nanofluid in a concentrating photovoltaic/thermal system
4.5.2.1 Optical properties and thermal conductivity of the nanofluids
4.5.2.2 Application of optimized nanofluids in a model photovoltaic/thermal system
References
Further reading
5 Nanofluid analysis in different media
5.1 Nanofluids in porous media
5.1.1 Case 1: Lid-driven T-shaped porous cavity
5.1.2 Case 2: Nanofluid in porous-filled absorber tube of solar collector
5.1.3 Case 3: Porous half-annulus enclosure filled by Cu–water nanofluid under the uniform magnetic field
5.2 Nanofluids in magnetic field (magneto hydrodynamics–ferrofluid)
5.2.1 Case 1: Variable magnetic field effect on a half-annulus cavity filled by nanofluid
5.2.2 Case 2: Nanofluid flow over a porous plate under the variable magnetic field effect
5.2.3 Case 3: Ferrofluids under external magnetic field
5.2.3.1 Measurement setup
5.3 Nanofluids under thermal radiation
5.3.1 Case 1: Polydisperse colloidal particles in the presence of thermal gradient
5.3.2 Case 2: Carbon nanotube-water analysis between rotating disks under the thermal radiation conditions
5.3.3 Case 3: Ethylene glycol (C2H6O2) carbon nanotubes in rotating stretching channel with nonlinear thermal radiation
References
Further reading
6 Nanofluid analysis in different applications
6.1 Nanofluids for cooling and heating
6.1.1 Case 1: Design of microchannel heat sink with wavy and straight wall
6.1.2 Case 2: Nanofluids in the heating process of an heating, ventilation, and air conditioning system model
6.2 Nanofluids in nuclear engineering
6.2.1 Case 1: Turbulent nanofluids flow in pressured water reactor
6.2.2 Case 2: Nanoparticles around the heated cylinder in a wavy-wall enclosure
6.3 Nanofluids in renewable energies
6.3.1 Case 1: Nanofluid-based Concentrating Parabolic Solar Collector
6.3.2 Case 2: Wavy direct absorption solar collector filled with different nanofluids
6.4 Nanofluid in industry
6.4.1 Case 1: Condensation of nanofluids
6.4.2 Case 2: Heat transfer of nanofluids between parallel plates
6.5 Nanofluid analysis in other applications
6.5.1 Case 1: Hybrid nanoparticles (Cu–Al2O3) over a porous medium
6.5.1.1 Flow analysis
6.5.1.2 Heat transfer analysis
6.5.2 Case 2: Engine radiator heat recovery using different nanofluids
References
Index
Back Cover

Citation preview

NANOFLUIDS

NANOFLUIDS Mathematical, Numerical, and Experimental Analysis

MOHAMMAD HATAMI

Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

DENGWEI JING

Xi’an Jiaotong University, Xi’an, P.R. China

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/ permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-102933-6 For Information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisition Editor: Brian Guerin Editorial Project Manager: Rafael G. Trombaco Production Project Manager: Nirmala Arumugam Cover Designer: Victoria Pearson Typeset by MPS Limited, Chennai, India

Dedicated to: • My doctoral supervisor, Professor Liejin Guo, who always supported me for learning new knowledge and doing scientific researches (Dengwei Jing) • My dear wife, who consumed her time and helped me during the book preparing (Mohammad Hatami)

Contents Preface

ix

1. Introduction to nanofluids

1

1.1 History of nanofluids 1.2 Structures and different types 1.3 Nanofluid properties 1.4 Benefits and applications 1.5 Other forces on nanoparticles in base fluid References

1 3 18 25 33 48

2. Mathematical analysis of nanofluids

51

2.1 Mathematical modeling of nanofluids properties 2.2 Weighted residual method for nanofluid modeling 2.3 Differential transformation method for nanofluid modeling 2.4 Other analytical/mathematical modeling References

51 52 70 91 111

3. Numerical analysis of nanofluids

113

3.1 Finite element method in nanofluid 3.2 Finite volume method in nanofluid 3.3 Lattice-Boltzmann method in nanofluid 3.4 Finite difference method in nanofluid 3.5 Runge Kutta Fehlberg numerical method References

113 129 138 148 153 169

4. Experimental analysis of nanofluids

171

4.1 Brownian motion of nonspherical particle 4.2 Different properties of nanofluids 4.3 Optical properties of nanofluids 4.4 Experimental correlations of nanofluid properties 4.5 Experimental application of nanofluids References Further reading

vii

171 180 190 199 206 219 221

viii

Contents

5. Nanofluid analysis in different media 5.1 Nanofluids in porous media 5.2 Nanofluids in magnetic field (magneto hydrodynamics ferrofluid) 5.3 Nanofluids under thermal radiation References Further reading

6. Nanofluid analysis in different applications 6.1 Nanofluids for cooling and heating 6.2 Nanofluids in nuclear engineering 6.3 Nanofluids in renewable energies 6.4 Nanofluid in industry 6.5 Nanofluid analysis in other applications References

Index

223 223 240 258 279 281

283 283 295 306 317 329 346

349

Preface Learn the science and knowledge, which at the Shah (God)’s presence Anyone who has not politeness (knowledge), is not deserved for conversation! Hafez, Persian Poet (1315 90)

Nanofluids are formed by dispersing metal or metallic oxide nanoparticles into selected base fluids. Since nanofluids have two different phases (solid and liquid phases), their thermophysical properties are completely different from the each separated phases. The useful properties of nanofluids have motivated a large number of researchers to study them in a variety of environments such as in industrial engineering processes, oil and gas industries, automobile manufacturing processes, nuclear engineering models, chemical engineering processes, renewable energies such as solar collectors, etc. While there are several books available on the field of nanofluids some of them are numerical studies and some are just experimental research. With the aim of providing a comprehensive text that includes mathematical, numerical, and experimental studies in the field of nanofluids and to fill the gap in the available texts this book came to be. This reference is based on my experience in mathematical and numerical modeling of nanofluids, along with the valuable experience of Professor Dengwei Jing in experimental studies in this area. Of benefit to engineers, researchers, and graduate students who want to develop their knowledge of basic and applicable phenomena of nanofluids, this resource is a tool that can be used by professionals at all levels. In Chapters 1 3, the mathematical and numerical modeling of nanofluids along with their preparation and properties are covered. In Chapters 4 6, application of nanofluids in different case studies is presented and several examples of recently published papers from highquality journals are introduced to illuminate the subject. We welcome reader comments and suggestions on the contents of this book and hope it will be beneficial to them. Finally, we would like to express our sincere thanks to the staff at Elsevier for their helpful support.

Mohammad Hatami

Department of Mechanical Engineering, Esfarayen University of Technology (EUT), Esfarayen, Iran

ix

C H A P T E R

1 Introduction to nanofluids

1.1 History of nanofluids The idea of nanofluids (NFs) was first proposed by Choi [1] after performing experimental studies on various nanoparticle suspensions. NFs can be formed by dispersing metal or metallic oxide nanoparticles into a selected base fluid, such as water, oils, and ethylene glycol. The thermal conductivity, viscosity, specific heat, and density of the base fluid can be changed after adding nanoparticles. Many NFs have been investigated, with results showing a high degree of heat exchange with some pressure drop. The convective heat transfer rate and flow performance of Cuwater NFs in a straight tube were investigated by Xuan and Li [2], and they found that the Nusselt number could be increased by more than 39% for an NF with 2.0% of Cu nanoparticle volume fraction. Azmi et al. [3] considered the heat transfer coefficient of TiO2 NF in a circular tube under turbulent flow. They reported a maximum enhancement of 22.8% at 50 C for Nusselt number at 1.5% particle concentration. Experimental studies by Zhang et al. [4] showed that a maximum heat transfer enhancement of 10.6% could be obtained using Al2O3water NFs through a circular microchannel. However, there are many problems that must be solved before an NF-based heat transfer system can be commercially available. This analysis can be obtained by numerical simulation, which can significantly reduce calculations, increase the range of experimental investigations, and provide theoretical guidance for system optimization. In this book a number of approaches or models are proposed to simulate NF heat transfer. The discrete phase model (DPM) is a two-phase model used to simulate the motion of particles through a base flow with a forcebalance

Nanofluids DOI: https://doi.org/10.1016/B978-0-08-102933-6.00001-9

1

© 2020 Elsevier Ltd. All rights reserved.

2

1. Introduction to nanofluids

equation. In the DPM, the nanoparticles are tracked using the Lagrangian approach, while the governing equations for the base fluid are solved using the Eulerian approach. The biggest advantage of the LagrangianEulerian approach (DPM) is that the thermophysical properties of the base fluid and the nanoparticles can be given separately, unlike the single-phase model in which the thermophysical properties need to be specified either through experimental data or various approximation model [5]. In fact, heat transfer with NFs in laminar flow using the DPM has proved the applicability of the DPM for engineering estimation of heat transfer performance [6,7]. Unfortunately, agglomeration of nanoparticles, one of the main problems NFs have in base fluids, occurs often due to van der Walls forces, electrical double-layer interaction forces, etc., which substantially decrease system performance as often reported. For instance, agglomeration of carbon nanotubes (CNTs) in an NF-based solar thermal collector will result in fouling, clogging, and a considerable reduction in the absorbance of incident solar rays [8]. NFs that are colloidal suspensions containing a kind of dispersed nanoparticles smaller than 100 nm (,100 nm) in a base fluid which show thermal properties superior to traditional fluid media. NFs have also been proposed as next-generation heat transfer fluids for numerous heat transfer applications [9]. In recent years, the thermal conductivity of NFs (the essential heat transfer property) has been studied extensively by researchers. Improvement of the thermal conductivity of various NFs has been widely reported and several proposed models have been developed based on various mechanisms as presented in the literature [1014]. However, in many of the studies, the obtained thermal conductivities did not agree well with each other due to various measurement conditions and experimental deviations. Thus further theoretical investigation on the thermal conductivity of NFs is still required for a deeper understanding of the heat transfer mechanisms in suspensions of nanoparticles.

1.1.1 Preparing nanofluids To show the NF preparation processes, three kinds of aqueous NFs were prepared using TiO2, CNT, and SiO2 nanoparticles [15]. TiO2 (85 nm), SiO2 (12 nm), and CNTs (6.2 nm diameter and 15 μm length) nanoparticles were purchased from Sigma-Aldrich. Solutions with the chosen volume concentration of nanoparticles were obtained by mixing the appropriate amounts of distilled water and nanoparticles. The mixtures were stirred for 15 min for stable dispersion of the nanoparticles, and the solutions were sonicated for approximately 3 h. To obtain

Nanofluids

3

1.2 Structures and different types

TABLE 1.1

Thermophysical properties of some nanoparticles and base fluids.

Material/ properties

ρ (kg m23)

Cp (J kg21 K21)

k (W m21 K21)

μ (kg m21 s21)

β (K21) 3 1026

Water, H2O

997.1

4179

0.613

0.0010003

210

Ethylene glycol, (CH2OH)2

1076

2664

0.261

0.003036

CuO

6320

531.8

76.5



18.0

Al2O3

3970

765

40



8.5

TiO2

4250

686.2

8.9538



9.0

Fe3O4

5200

670

6



Cu

8933

385

401



16.7

Ag

10,500

235

429



18.9

characterization of the NFs, the morphology and microstructure of the samples were studied using scanning electron microscopy. As shown in Ref. [15] the average particle sizes were 85 and 12 nm (nanometer) for SiO2 and TiO2, respectively, and 6.2 nm diameter and 15 μm length for CNTs. Table 1.1 shows the physical properties of these nanoparticles. The effective density ρnf , the effective heat capacity ρCp nf , and the thermal expansion ðρβ Þnf of the NFs are explained in the next chapter. Note that all experiments were performed for 0.001 g nanoparticles in 3 L water [15].

1.1.2 Synthesis of nanofluids A complete discussion on the different approaches to nanoparticle synthesis such as solgel and analysis methods like transmission electron microscopy (TEM) is not provided here but can be found in Refs. [1620].

1.2 Structures and different types Comparison of single- and two-phase NF modeling can be found in the literature. For instance, a comparison of the results of single-phase and two-phase numerical methods for NFs in a circular tube was reported by Haghshenas Fard et al. [21]. They reported that the average relative error between the experimental data and computational fluid dynamics (CFD) results was 16% for a Cuwater single-phase model

Nanofluids

4

1. Introduction to nanofluids

while it was 8% for a two-phase model. In another numerical study, Go¨ktepe et al. [22] compared the two models for NF convection at the entrance of a uniformly heated tube and found the same results, further confirming that the accuracy of two-phase modeling is greater than single-phase modeling. In the following cases single-phase and twophase modelings are presented.

1.2.1 Case 1: Single phase: different shapes of nanoparticles in a wavy-wall square cavity filled with power-law non-Newtonian nanofluid Fig. 1.1 shows the geometry of the studied wavy-wall square cavity [23]. The left wavy wall is hold in cold temperature (TL) and the right flat wall is kept in a constant hot temperature (TH). Two insulated bottom and upper flat walls were fixed in their locations. The domain was filled with Fe3O4/non-Newtonian shear-thinning NF in the presence of a magnetic field. The thermophysical properties used are given in Table 1.1. A magnetic field with three different angles (γ 5 0, 30, and 60 degrees) was applied on the cavity. The flow was incompressible, steady, and laminar. The density variation was approximated by the standard Boussinesq model. A nondimensional cosine function with Am as the wave amplitude was used to simulate the wavy wall as shown in Fig. 1.1 [23]. In this single-phase case, the temperature and velocity fields in wavy square cavity of Fig. 1.1 were obtained by solving the continuity, NavierStokes, and energy equations. Therefore the governing equations are simplified by using the following assumptions:

FIGURE 1.1 Schematic of the wavy-wall square cavity and generated mesh [23].

Nanofluids

1.2 Structures and different types

5

• The flow is incompressible and NF is power-law non-Newtonian. • Relative movement between fluid and Fe3O4 particles is zero and thermal equilibrium exists between them. • The temperature and velocity fields are laminar, steady state, and 2D. • The effects of radiation and viscous dissipation are neglected. The governing equations (continuity, momentum, and energy equations) and definitions of dimensionless variables can be introduced as follows based on the above assumptions [23,24]: @v @u 1 50 (1.1) @y @x    2      @u @u 1 @P @ u @2 u 2 2 1u 1 μnf v 2 1 2 1σnf B v sin γ cos γ 2 u sin γ 5 @y @x ρnf @x @x2 @y "    2  @v @v 1 @P @ v @2 v 1 μnf 1 v 1u 5 1 gρnf β nf ðT 2 Tc Þ 2 @y @x ρnf @y @x2 @y2 #   2 2 1 σnf B u sin γ cos γ 2 v cos γ 

ρCp



 nf

  2  @T @T @ T @2 T 1u v 1 2 5 knf @y @x @x2 @y

(1.2)

(1.3)

(1.4)

To solve the governing equations for single-phase modeling, the NF effective properties are required, and can be calculated as a single phase by the equations introduced in the next section (Eqs. 1.271.32). (    2   )ðn21Þ 2 @u 2 @v @v @u 2 1 12 1 μf 5 N 2 @x @x @y @x

(1.5)

In this case, the Hamilton equation is applied to calculate the thermal conductivity of the NF:   knf kp 1 ðm 1 1Þkf 2 ðm 1 1Þϕ kf 2 kp   5 (1.6) kf kp 1 ðm 1 1Þkf 1 ϕ kf 2 kp It can be shown that for spherical nanoparticles m 5 3 and for other nanoparticle shapes, m can be calculated by using m 5 3=ψ and Table 1.2. In Eq. (1.5), N is the consistency coefficient and n is the power-law index. Therefore the deviation of n from 1.0 specifies the degree of deviation from Newtonian behavior. For n 6¼ 1, the constitute Eq. (1.5)

Nanofluids

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1. Introduction to nanofluids

TABLE 1.2 Constants of Eq. (1.6). Nanoparticle shape

ψ

Spherical

1

Platelet

0.52

Cylindrical

0.62

Brick

0.81

represents a pseudoplastic fluid (n , 1) and for (n . 1) it represents a dilatant fluid. The boundary condition of the problem is defined as: 8 @T > > 50 Insulated walls: u 5 v 5 0; > < @y (1.7) > Wavy wall: u 5 v 5 0; T 5 TC > > : Right wall: u 5 v 5 0; T 5 TH The following dimensionless variables are considered in the solution: Τ5

T 2 TC uL vL y x pL2 ; U5 ; V5 ; Y 5 ; Χ 5 ; P 5  2 0:5 0:5 αf Ra αf Ra L L TH 2 TC ρf αf Ra (1.8)

Using Eq. (1.8), Eqs. (1.1)(1.4) can be written in dimensionless form as: @V @U 1 50 (1.9) @Y @X 2 0 1 0 0 113 ρ μ μ @U @U @P Pr f 1 42 @ @ f @U A1 @ @ f @@U 1@V AA5 1U 52 1 pffiffiffiffiffiffi V @Y @X @X @Y N @Y @X Ra ρnf ð12ϕÞ2:5 @X N @X 2 3   Ha2 Pr σnf =σf 5 1 1 pffiffiffiffiffiffi 4 V sin γ cos γ 2U sin2 γ 2:5 Ra ρnf =ρf ð12ϕÞ (1.10) 0 0 113 μf @V μf @U @V @V @V @P Pr ρf 1 @ @ 42 @ A1 @ @ 1 AA5 52 1 pffiffiffiffiffiffi V 1U @Y @X @Y @X N @Y @X Ra ρnf ð12ϕÞ2:5 @Y N @Y 2 3 ðρβ Þnf   Ha2 Pr σnf =σf 5 1 1 pffiffiffiffiffiffi 4 U sin γ cos γ 2V cos2 γ 1Pr T 2:5 ρnf β f Ra ρnf =ρf ð12ϕÞ 2

0

1

(1.11)

Nanofluids

7

1.2 Structures and different types

" #  knf =kf @Τ @Τ 1 @2 Τ @2 Τ    1V 5 pffiffiffiffiffiffi  U 1 @X @Y Ra ρCp nf = ρCp f @X2 @Y2 With dimensionless parameters: ρf β f gL3 ðTH 2 TC Þ

Raf 5 ; αf μf

υf Pr 5 ; αf

sffiffiffiffiffi σf Ha 5 LB μf

8 > > Insulated walls: >
Wavy wall: > > : Right wall:

U 5 V 5 0; U 5 V 5 0;

T50 T51

(1.12)

(1.13)

(1.14)

To numerically solve these dimensionless governing equations, the finite element method was applied using FlexPDE commercial software. In addition, the summation of residual convergence criterion was considered to be less than 1024. To validate the independence of the grids, the Num of the wavy wall was considered as the criterion, and three different geometries of wavy walls with amplitude of Am 5 0.1, 0.2, and 0.3 were considered. In this case, the main goal is to model the NF to find its best heat transfer performance with the least possible amount of entropy generation. Table 1.1 lists the thermophysical properties of Fe3O4 nanoparticles. The following dimensionless parameters were considered in this study: • • • • •

Nanoparticles volume fraction (ϕ): changes from 2% to 6%. The wavy-wall amplitude (Am): varies from 0.1 to 0.3. Hartmann number (Ha): changes between 20 and 100. Magnetic field angle (γ): varies from 0 to 60. Nanoparticles shapes: platelet, cylindrical, spherical, and brick shapes.

Fig. 1.2 shows the streamline and isotherm contours in the three different geometries of the cavity. The contours were drawn for the Rayleigh number of 10,000, Ha 5 20, n 5 1, γ 5 0, and spherical nanoparticles with ϕ 5 2%. It can be observed that in all temperature contours the left wavy side is cooler than the right wall due to boundary conditions. By increasing the Am, the peak of the wavy shape will be more near to hot wall, which makes a separation in the natural flow pattern in the streamline. As seen in Fig. 1.2, for Am 5 0.3 the separation of streamlines completely occurred and two different areas were created. Also, in the temperature

Nanofluids

8

1. Introduction to nanofluids

Temp.

Streamline

A m = 0. 1

A m = 0. 2

A m = 0. 3

FIGURE 1.2 Effect of Am on temperature and streamlines [23].

contours there are lower temperatures in the cavity center due to the cold temperature of the wavy wall. Fig. 1.3 shows the effects of nanoparticle volume fraction on the contour results, and based on these results, Fig. 1.4 shows the effect of Am and ϕ on the Nusselt number. Increasing the ϕ caused an increase in the Nusselt number due to more heat transfer from greater nanoparticle volume fraction.

Nanofluids

1.2 Structures and different types

Temp.

9

Streamline

φ = 0.0 2

φ = 0.0 4

FIGURE 1.3 Effect of phi on temperature and streamlines [23].

The effects of the four described nanoparticle shapes (i.e., platelet, cylindrical, spherical, and brick) are depicted in Figs. 1.5 and 1.6. It can be seen that the maximum average Nusselt number occurred for the brick-shaped nanoparticles, while the spherical-shaped nanoparticles had the minimum Nusselt number.

1.2.2 Case 2: Two-phase nanofluid thermal analysis over a stretching infinite solar plate In this case, the aim is to show two-phase NF flow modeling [25]. An NF flow over a solar plate was considered by Khan et al. [26] as shown in Fig. 1.7. As in the last case, the flow was incompressible and induced due to the plate being stretched in two directions by nonlinear functions. The plate was kept at constant temperature and the nanoparticle mass flux at the wall was expected to be zero. The three-dimensional governing equations are [25,26]:

Nanofluids

10

1. Introduction to nanofluids

2

Average Nusselt

1.8

1.6

1.4

1.2 0.1

0.15

0.2

0.25

0.3

Wave amplitude (Am) 1.24

1.22

Average Nusselt

1.2

1.18

1.16

1.14

1.12

1.1

0.02

0.03 0.04 0.05 Nanoparticles volume fraction (ϕ)

FIGURE 1.4 Nusselt numbers in different Am and ϕ [23].

Nanofluids

0.06

1.2 Structures and different types

Temp.

11

Streamline

Platelet

Brick

Spherical

FIGURE 1.5 Effect of ψ on temperature and streamlines [23].

@u @v @w 1 1 50 @x @y @z

(1.15)

u

@u @u @u @2 u 1v 1w 5 υf 2 @x @y @z @z

(1.16)

u

@v @v @v @2 v 1 v 1 w 5 υf 2 @x @y @z @z

(1.17)

Nanofluids

12

1. Introduction to nanofluids

FIGURE 1.6 Nusselt numbers in different nanoparticle shapes [23].

FIGURE 1.7 Schematic of the nanofluid flow over a stretched solar plate [25].

"  # @T @T @T @2 T @C @T DT @T 2 1v 1w 5 α 2 1 τ DB 1 u @x @y @z @z @z @z TN @z u

  2 @C @C @C @2 C DT @ T 1v 1w 5 DB 2 1 @x @y @z @z TN @z2

(1.18)

(1.19)

where u and v are the velocities in the x and y directions, respectively; T is the temperature, C is the concentration; DB is the Brownian diffusion

Nanofluids

13

1.2 Structures and different types

coefficient of the diffusing species; and DT is the thermophoretic diffusion coefficient. By this assumption that the plate is infinite and stretched in two directions by nonlinear functions, the boundary conditions will be:  n  n u 5 uw 5 a x1y ; v 5 vw 5 b x1y ; w 5 0; T 5 Tw ;

DB

@C DT @T 1 50 @z TN @z

u-0; v-0; T-TN ; C-CN

at z 5 0

as z-0

By introducing these parameters:  n  n u 5 a x1y f 0 ðηÞ; v 5 a x1y g0 ðηÞ 0 1 n21       2 pffiffiffiffiffiffiffi @ n 1 1 f 1 g 1 n 2 1 η f 0 1 g0 A w 5 2 aυf x1y 2 2 T 2 TN C 2 CN θðηÞ 5 ; φðηÞ 5 ;η5 Tw 2 TN CN

(1.20)

(1.21)

n21 sffiffiffiffiffi  2 a x1y z υf

and substituting above variables into Eqs. (1.15)(1.19), we have:    n11 f 1 g fv 2 n f 0 1 g0 f 0 5 0 2    n11 f 1 g gv 2 n f 0 1 g0 g0 5 0 gw 1 2  1 n11 θv 1 f 1 g θ0 1 Nb φ0 θ0 1 Nt θ02 5 0 Pr 2 n11 Nt Scðf 1 gÞφ0 1 θv 5 0 φv 1 2 Nb fw 1

(1.22) (1.23) (1.24) (1.25)

These systems of nonlinear equations can be solved by a powerful numerical or analytical method. In this study, the optimal collocation method (which will be introduced in Chapter 2: Mathematical analysis of nanofluids) was applied with these boundary conditions: f ð0Þ 5 0; f 0 ð0Þ 5 1; gð0Þ 5 0; g0 ð0Þ 5 λ; θð0Þ 5 1; Nb φ0 ð0Þ 1 Nt θ0 ð0Þ 5 0 f 0 ðNÞ-0; g0 ðNÞ-0; θðNÞ-0; φðNÞ-0 (1.26) where Pr (Prandtl number), Sc (Schmidt number), Nb (Brownian motion parameter), Nt (thermophoresis parameter), and λ 5 b/a (ratio of the stretching rate along y to x directions) are the defined parameters found in Ref. [26].

Nanofluids

14

1. Introduction to nanofluids

Here, the KellerBox method was used to solve the problem in Maple 15.0 software. The KellerBox scheme, as described in Chapter 3, Numerical analysis of nanofluids, is a face-based method for solving partial differential equations (PDEs) that has several great mathematical and physical properties. These properties are due to the fact that the scheme exactly discretizes partial derivatives and only makes estimations in the algebraic constitutive relations given in the PDE. The discrete calculus associated with the KellerBox scheme is different from all other simulated numerical procedures. Essentially, KellerBox is a variation of the finite volume method (FVM) in which unknown coefficients are stored at control volume faces rather than at the traditional cell centers. It is due to the fact that in spacetime equations, the unknowns sit at the corners of the spacetime control volume, which is a box in one space dimension on a fixed mesh. The original method [27] was distributed with parabolic initial value problems such as the unsteady heat transfer equation. First of all, the infinite range must be clear to solve the governing equations. This means that η can be considered as the infinite number. To discover this value three different numbers (4, 7, and 10) were studied, the results of which are presented in Fig. 1.8. As seen for all the depicted graphs (velocity, temperature, and nanoparticle concentrations) the two last values (i.e., 7 and 10) have the same profiles, so increasing this value has no significant effect on the results (7 was chosen in the solution procedure). The effect of the power-law index (n) on the x- and y-components of dimensionless velocities (f0 (η) and g0 (η) functions), temperature, and nanoparticle volume fraction boundary layer profiles (θ(η) and ϕ(η)) are depicted in Fig. 1.9. As can be seen, by increasing the power-law index, n, dimensionless velocity will increase in both the x- and y-directions. Also, it can be observed that the reductions of velocity in the x- and y-directions are roughly equal. The results show that both velocity profiles are decreasing functions of the power index (n). It is also obvious that the thermal boundary layer will be thinner for larger n values while the nanoparticle concentration profile will become thicker, which increases the rate of heat transfer from the sheet. Furthermore, it can also be understood that the greater the n value the faster the decline of θ. It has been suggested that increasing n can enhance the convective properties of the fluid since it will increase the deformation by the shear stress from the wall to the fluid. Fig. 1.10 demonstrates the effect of n and λ on the shear stress at the surface (i.e., fvð0Þ and gvð0Þ) and reduced Nusselt number (Nur). This means that increasing both the n and λ parameters reduces the shear stress, but increases the Nusselt number.

Nanofluids

1.2 Structures and different types

15

FIGURE 1.8 Effect of η on profiles when n 5 1; Nb 5 0:7; Nt 5 0:4; Sc 5 3; Pr 5 1; λ 5 0:5: [25].

Nanofluids

16

1. Introduction to nanofluids

FIGURE 1.9 Effect of power index (n) when ηN 5 7; Nb 5 0:7; Nt 5 0:4; Sc 5 3; Pr 5 1;

λ 5 0:5 [25].

Nanofluids

1.2 Structures and different types

17

FIGURE 1.10 Effect of n and λ on Nur, fvð0Þ, and gvð0Þ when ηN 5 7; Nb 5 0:7; Nt 5 0:4; Sc 5 3; Pr 5 1 [25].

Nanofluids

18

1. Introduction to nanofluids

1.3 Nanofluid properties The following are some of the most applicable formulas for NF properties. These demonstrations of NF properties are obtained based on the nanoparticle volume fraction as follows: 

ρeff 5 ρs φ 1 ρf ð1 2 φÞ     ρCp eff 5 ρCp s φ 1 ρCp f ð1 2 φÞ

(1.27)

β eff 5 β f ð1 2 φÞ 1 β s φ

(1.29)



μeff 5

μf

ð12φÞ2:5   2 2φ kf 2 ks 1 2kf 1 ks   keff 5 kf φ kf 2 ks 1 2kf 1 ks

3 σσsf 2 1 φ σnf



511 σs σs σf 1 2 2 2 1 φ σf σf

(1.28)

(1.30) (1.31)

(1.32)

As mentioned above, these equations are traditionally used to model NF properties. But many studies have been performed to present novel or more accurate formulas for these properties. In the following some of these equations are reviewed and an experimental study is presented.

1.3.1 Case 1: A modified multisphere Brownian model to predict the thermal conductivity of colloid suspension of wide volume fraction ranges As mentioned above, traditional thermal conductivity models based on classical effective medium theory, such as the Maxwell [28] model, fail to describe the improvement of thermal conductivity in NFs. This is due to the fact that these models only consider the effects of nanoparticle concentration. However, it is known that the enhancement of NF thermal conductivity is dependent on many factors, such as the nanoparticle material, volume fraction, particle size and shape, base fluid, temperature, chemical additives, etc. Improved theoretical models that consider the effects of various heat conduction mechanisms (i.e., Brownian motion of nanoparticle, nanolayer, clustering, and the nature of heat transport in nanoparticles) have been suggested [29]. Nanoconvection is one of the commonly accepted mechanisms for enhanced heat transfer of NFs, which is caused by the nanoparticle Brownian motion. A novel model considering the effect of this

Nanofluids

1.3 Nanofluid properties

19

micromixing due to the Brownian motion with the conventional static contribution as determined by the Maxwell model was established by Kumar et al. [30]. They derived a moving particle model from the StokesEinstein formula to explain the temperature effect. Prasher et al. [31] determined that the local convection caused by the Brownian motion of the nanoparticles is primarily responsible for the irregular enhancement of NF thermal conductivity through an order-ofmagnitude analysis of various possible mechanisms. They also noted that the thermal conductivity for colloidal suspensions with large particle sizes can be explained by traditional conduction-based theories such as the MaxwellGarnett model. Consequently, they suggested a method combining the MaxwellGarnett conduction model and the Brownian motioninduced convection from multiple nanoparticles called the multisphere Brownian model (MSBM). 1.3.1.1 Models for the thermal conductivity of colloidal suspensions By collecting the volume fraction data from the available theoretical models in the literature for the thermal conductivity of NFs, it can be seen that the validity of most existing models has to be limited to NFs with volume fractions larger than 0.1%. Summaries of some of the models for thermal conductivity of NFs and their volume fraction ranges are listed in Table 1.3 [29]. 1.3.1.2 Multisphere Brownian model The MSBM is one of the most reliable and commonly used models for predicting the thermal conductivity of NFs [34,35]. After investigating several available heat conduction mechanisms and related models, Prasher et al. [31] determined that the local convection heat transfer in the liquid due to the Brownian motion of the particles is the main contributor to the enhancement of the thermal conductivity in NFs. They suggested the effective thermal conductivity of the semiinfinite area based on the BrownianReynolds number as a part of the modified MaxwellGarnett model. In addition, this model considers the effect of interfacial thermal resistance between nanoparticles and different base fluids. Considering all the abovementioned factors at the same time, the multisphere Brownian model can be written as: !  kp ð1 1 2αÞ 1 2km 1 2ϕ kp ð1 2 αÞ 2 km kn  m 0:333 5 1 1 ARe Pr φ (1.38) kf kp ð1 1 2αÞ 1 2km 2 ϕ kp ð1 2 αÞ 2 km where Re 5 1/ν f(18kbT/πρpdp)0.5, ν f is the kinematic viscosity of the liquid, α 5 2Rbkm/dp is the nanoparticle Biot number, Rb is the selective

Nanofluids

TABLE 1.3

Typical models for thermal conductivity of nanofluids proposed in the literature [29].

References Prasher et al. [31]

Models 0 1  kp ð1 1 2αÞ 1 2km 1 2ϕ kp ð1 2 αÞ 2 km kn  m 0:333 @ A 5 1 1 ARe Pr ϕ kf kp ð1 1 2αÞ 1 2km 2 ϕ kp ð1 2 αÞ 2 km vffiffiffiffiffiffiffiffiffiffiffiffi u   2Rb km 1 u18kb T ; km 5 kf 1 1 1=4 Re Pr ; Re 5 t α5 ν f πρp dp dp

Volume fraction

Eq.

ϕ . 1%

(1.33)

ϕ 5 1% and 4%

(1.34)

0:1% # ϕ # 3%

(1.35)

Al2 O3 : 1% # ϕ # 10% ZnO: 1% # ϕ # 7% CuO: 1% # ϕ # 6%

(1.36)

0:2% # ϕ # 9%

(1.37)



Chon et al. [32]

kn 5 1 1 64:7ϕ0:7460 kf Pr 5

Patel et al. [33] Vajjha and Das [34]

μf ρf αf

; Re 5

!0:3690 df dp

!0:7476 kp kf

Pr0:9955 Re1:2321

ρf kb T 3πμf 2 lf

 0:273  0:547  0:234 kp kn T 100 5 1 1 0:135 ϕ0:467 20 dp kf kf 2 # " vffiffiffiffiffiffiffiffiffi  3 uk T kp 1 2kf 2 2ϕ kf 2 kp u b 4 4 5   f ðT; ϕÞ kf 1 5 3 10 βϕρf cpf t kn 5 ρp dp kp 1 2kf 1 ϕ kf 2 kp 1 T f ðT; ϕÞ 5 2:8217 3 10 ϕ 1 3:917 3 10 @ A T0   1 23:0669 3 1022 ϕ 2 3:91123 3 1023 

22

23



0

β 5 8:4407ð100ϕÞ21:07304 for Al2 O3 β 5 8:4407ð100ϕÞ21:07304 for ZnO Corcione [35]

β 5 9:881ð100ϕÞ20:9446 for CuO !10 !0:03 kn kp 0:4 0:66 T 5 1 1 4:4Re Pr ϕ0:66 kf Tfr kf Re 5

2ρf kb T πμ2f dp

1.3 Nanofluid properties

21

FIGURE 1.11 Schematic of the interaction among nanoparticles in colloid suspension of low volume fraction (A) and high volume fraction (B) [29].

interfacial thermal resistance, km is the matrix thermal conductivity, and A and m are empirical constants determined by experimental data. The coefficient A is independent of the fluid type and equals 40,000, while the value of the exponent m depends on the type of the base fluid. For water-based NFs and assuming Rb to be 0.77 3 1028 K m2 W21, m will have a value of 2.5% 6 15% by comparison with the experimental values. 1.3.1.3 Modification of the multisphere Brownian model In a stable suspension, Brownian nanoparticles move accidentally and thus will convey the large surrounding liquid to near places. As reported by Prasher et al. [31] the thickness of the thermal boundary layer will exceed the interparticle distance when the volume fraction is larger than 0.0055% at room temperature. This means that even at very small nanoparticle concentrations, convection currents due to multiple particles will interact with each other. The existence of φ in Eq. (1.38) indicates strong interaction between the convection currents from various spheres [31]. However, as shown in Fig. 1.11, the intensity of this interaction will be completely different when the volume fraction of the NFs changes. Obviously, the interaction among nanoparticles will be much weaker in colloid suspension of low volume fraction in which the average interparticle distance increases. Also, there are exponential coefficients for the volume fraction term in the mathematical expressions of some suggested models as shown in Table 1.3. As can be seen, nanoparticle volume fraction is considered to be a crucial factor in these models. Furthermore, it is tried to modify the exponential part of the item for volume fraction φ in MSBM. Assuming A has the same value of 40,000 as the original model, a modified multisphere Brownian model is proposed by: !  kp ð1 1 2αÞ 1 2km 1 2ϕ kp ð1 2 αÞ 2 km kn  m 0:333 n 5 1 1 40; 000Re Pr φ kf kp ð1 1 2αÞ 1 2km 2 ϕ kp ð1 2 αÞ 2 km (1.39)

Nanofluids

22

1. Introduction to nanofluids

where the m and n parameters are modifiable empirical constants determined by experimental data. The next crucial step for utilization of the modified MSBM is the suitable determination of m and n parameters and their valid application ranges. Prasher et al. [31] concluded that m 5 2.5% 6 15% by comparing the MSBM with experimental data for water-based NFs. For the modified MSBM, it is also expected that the deviation in m is centered on m  2.5, so the major emphasis is on the determination of the n values. Validation of this assumption can be found in the next section. 1.3.1.4 Establishment of database with various experimental results To find a widely applicable model, typical data for water-based NFs containing alumina, titania, and copper oxide nanoparticles of volume fractions ranging from 0.01% to 10% were considered for our experiment: that is, Al2O3 (0.01%0.03%) from Lee et al. [36], TiO2 (0.2%1%) from Reddy and Rao [37], CuO (1%3.5%) from Kazemibeydokhti et al. [38], and CuO (5%7%) and Al2O3 (5%10%) from Mintsa et al. [39]. Due to the lack of data available on the thermal conductivity of NFs at low particle volume fractions (,0.1%), we obtained the necessary data by experiment for two typical NFs [i.e., Al2O3 (0.001%0.005%) and TiO2 (0.008%0.1%)]. The thermal conductivity of NFs was measured using the laser flashbased thermal analyzer LFA467 HyperFlash from NETZSCH instruments. 1.3.1.5 Thermal conductivity measured in our experiments In Fig. 1.12, the measured results of the thermal conductivities of water-based Al2O3 (30 nm) and TiO2 (21 nm) NFs are presented with the low volume fractions. The effects of volume fraction and temperature on the thermal conductivity of NFs were also investigated simultaneously. Clearly, the thermal conductivity will increase with volume fraction increment. Fig. 1.12A shows a nearly linear increase of the thermal conductivity with an increasing volume fraction of nanoparticles. Likewise, the thermal conductivity also increases with the temperature. 1.3.1.6 Determining the parameters for the modified model Table 1.4 gives the best-fit values of m and n, from which m is also found to be 2.5% 6 15%, while n is dependent on the volume fraction. Fig. 1.13 represents a comparison between the modified MSBM and experimental outcomes from the literature. It indicates good agreement between the thermal conductivity ratio predicted by the modified model and the available experimental data. To find the validity of our proposed modified model, it was applied to the measured data obtained for kn under different temperatures. Fig. 1.14 (for TiO2water NFs)

Nanofluids

23

1.3 Nanofluid properties

(A)

Thermal conductivity (W m–1K–1)

0.650 Al2O3 /H 2O nanofluid

0.648

0.646

0.644

0.642

0.640 1

2

3

4

5

3

Volume fraction ×10 (%)

Thermal conductivity (W m–1K–1)

(B) 0.67 TiO2 /H2O nanofluid

0.66 0.65 0.64 0.63 0.62 0.61 0.60 20

30

40

50

60

Temperature (ºC)

FIGURE 1.12 Thermal conductivities of (A) Al2O3/H2O nanofluid against volume fraction at temperature 50 C and (B) TiO2/H2O NF against temperature with volume fraction of 0.008% [29].

shows that the modified model matches very well with the data by assuming m 5 2.5 and n 5 0.725 when φ 5 0.008%, whereas the original MSBM significantly underpredicts the thermal conductivity of NFs when its particle volume fraction is quite low (i.e., φ 5 0.008%).

Nanofluids

24

1. Introduction to nanofluids

TABLE 1.4 Best-fit values of m and n for various reference data [29]. Type of particles

References

Volume fraction

Temperature 

m

n

Present work

Al2O3—30 nm

0.001% 0.005%

50 C

2.45

0.7

Present work

TiO2—21 nm

0.008% 0.1%

20 C60 C

2.5

0.7250.86

Lee et al. [36]

Al2O3—30 nm

0.01% 0.3%

21 C

2.5

0.750.89

Reddy and Rao [37]

TiO2—21 nm

0.2%1%

30 C

2.5

0.891

Chon et al. [32]

Al2O3—47 nm

1%

20 C60 C

2.415

1

Kazemibeydokhti et al. [38]

CuO—23 nm

1%3.5%

27 C

2.43

1

Mintsa et al. [39]

CuO—29 nm

5%7%

21 C23 C

2.65

1

2.85

1

Mintsa et al. [39]

Al2O3—47 nm

5%10%





21 C23 C

Lee et al. [36] Al 2 O3 , 30 nm Reddy and Rao [37] TiO2 , 21 nm

Thermal conductivity ratio from database (kn/k f )

1.25

Chon et al. [32] Al 2 O3 , 47 nm Kazemibeydokhti et al. [38] CuO , 23 nm Mintsa et al. [39] Al 2 O3 , 47 nm Mintsa et al. CuO , 29 nm Present work , Al 2 O3 , 30 nm

1.20

Present work , TiO2 , 21 nm ——

Line of slope = 1

1.15

1.10

1.05

1.00 1.00

1.05

1.10

1.15

1.20

1.25

Thermal conductivity ratio predicted by our model (kn /k f ) FIGURE 1.13 Comparison of our modified model with the experimental results; the corresponding values of m and n are given in Table 1.4 [29].

Nanofluids

25

1.4 Benefits and applications

1.020

Thermal conductivity ratio (kn /k f )

φ = 0.008% TiO2 /water (data) φ = 0.008% TiO2 /water (modified MSBM) φ = 0.008% TiO2 /water (original MSBM)

1.015

1.010

1.005

1.000 20

30

40

50

60

Temperature (ºC)

FIGURE 1.14 Comparison of predicted data by original and modified MSBM with the experimental data over TiO2water suspensions [29]. MSBM, Multisphere Brownian model.

1.3.1.7 Dependency of the n values on the volume fractions Fig. 1.15 shows the dependence of the n values on the volume fractions of NFs. The appropriate values of n were determined under different volume fractions by comparison of the modified MSBM with various experimental results. It was concluded that n has a value of unity when the volume fraction is larger than 1%. This means that the modified MSBM can be changed to the original model for high nanoparticle concentrations. Furthermore, n approximately equals 0.7 when the volume fractions of NFs are lower than 0.005%. Between these two volume fractions, n is observed to have a nearly linear relation with the logarithm of volume fraction. This dependence relationship can be described by a brief mathematical formula as follows: 8 for φ . 1% > < 1:0 n 5 0:13 3 logðφÞ 1 1:26 for φ 5 0:005%  1% (1.40) > : 0:7 for φ , 0:005%

1.4 Benefits and applications Most of the applications of NFs are covered in Chapter 6: Nanofluids analysis in different applications. In this section, we look at the

Nanofluids

26

1. Introduction to nanofluids

Lee et al. [36] Al 2 O3 , 30 nm Reddy and Rao [37] TiO2 , 21 nm

1.1

Chon et al. [32] Al 2 O3 , 47 nm Kazemibeydokhti et al. [38] CuO , 23 nm Mintsa et al. [39] Al 2 O3 , 47 nm Mintsa et al. CuO , 29 nm Present work , Al 2 O3 , 30 nm

1.0

Present work , TiO 2 , 21 nm

n

0.9

0.8

0.7

0.6 1E–3

0.01

0.1

1

10

Volume fraction (%) FIGURE 1.15

Dependence of the n values on the volume fractions of various nano-

fluids [29].

characteristic oscillation phenomenon after head-on collision of two NF droplets to show the benefits and applications of NFs [40]. These experiments aimed to study the oscillation characteristic of NF droplets by looking at one drop fall from height and impact with low velocity onto another drop on a fixed platform. Since drops undergo spreading and recoiling after impact, the images of the drops corresponding to different shape changes were recorded and then analyzed with digitization. The experimental setup used to study the drop impact consists of three subsystems, as shown in Fig. 1.16: 1. drops generation subsystem (syringe pump, injector, and needles with different sizes); 2. supporting syringe pump and platform subsystem (adjustable height for the drop falling); and 3. high-speed photographic optical subsystem. A single drop is shaped from a needle attached to a syringe pump. After growth, it separates from the needle due to its gravity. The impact velocity of the drop depends on the height of the platform. Then the drop falls vertically onto a sessile droplet on a horizontal platform.

Nanofluids

27

1.4 Benefits and applications

(A)

(B) D H

v Vertical direction

Horizontal direction

FIGURE 1.16 (A) Experimental setup for studying the oscillation characteristic when two droplet collide head-on. (B) Illustration of the oscillation process [40].

The platform is made of polytef in order to guarantee the high contact angle of the drop on the platform. By measuring with a TR200 roughness meter, the Ra value (absolute surface roughness) was determined to be 0.253 μm. To avoid affecting vibration, the whole apparatus was placed on a vibration isolation table. As mentioned above, in the third subsystem the high-speed video camera for continuously capturing images of the drop as it impacts, spreads, recoils, and rebounds on the substrate is essential equipment for this research. The camera system can record at a framing rate up to 20,000 full images per second, but we chose 250 images per second, which was enough to meet our research needs. Images of drops subsequent to oscillation were analyzed using the MATLAB software to obtain the shapes of the droplets quantitatively. After analyzing, we obtained the drop size change, impact velocity, and oscillation frequency. As presented in Fig. 1.16B, D is the spread radius, H is the height of the drop rebound, and t refers to time measured from the instant the drop impacts. TiO2water NF with different nanoparticle mass fraction was used for the droplets, which was prepared with distilled water as the base fluid. In order to keep the NF suspension stable, sodium dodecyl benzene sulfonate surfactant was also added to the fluid. Also, to obtain homogeneous nanoparticle dispersion, ultrasonic irradiation was applied to it. The particle size and morphology of the samples were observed on TEM images from a FEI Tecnai G2 F30 transmission electron microscope at an accelerating voltage of 300 kV. Particle size distribution (PSD) evolution of TiO2 in the suspension was studied by dynamic light scattering using a Malvern laser particle analyzer (Spraytec 300, Malvern, United Kingdom). As mentioned above, three nanoparticle concentrations, 0.001%, 0.01%, and 0.1%, were used in our study and the images for the prepared NFs are shown in Fig. 1.17. As can be seen, the color of the fluid

Nanofluids

28

1. Introduction to nanofluids

0%

0.001% TiO2–water

0.01% TiO2–water

0.1% TiO2–water

FIGURE 1.17 Images for various TiO2water nanofluids employed in this research [40].

Surface tensioin σ (mN m–1)

50

3.5

40 3.0 30

2.5

20

2.0

10

1.5

0%

0.001% 0.01% 0.1% Nanoparticle mass fraction

0.5%

Dynamic viscosity μ (mPa s–1)

4.0 Surface tension Dynamic viscosity

1.0

FIGURE 1.18 Surface tension and viscosity of TiO2water nanofluid with various nanoparticle mass fraction [40].

turns from almost transparent to completely opaque or milky when the loading amount of TiO2 increases from 0% to 0.1%. Fig. 1.18 shows the experimentally measured surface tension σ and dynamic viscosity μ of NFs for different mass fractions. It can be observed that the very low amount of 0.001% TiO2 leads to a minor

Nanofluids

29

1.4 Benefits and applications

(A)

(B) 16 0.001% 0.01% 0.1%

14

Percent (%)

12 10 8 6 4 2 0 –2

0

200

400

600

800

Particle diameter (nm)

FIGURE 1.19 TEM image and nanoparticle diameter distribution of test NF. (A) TEM image of TiO2 nanoparticles; (B) nanoparticles diameter distribution in three different mass fraction NF [40]. NF, Nanofluid.

decrease of both surface tension and dynamic viscosity compared with pure water. Subsequently, with the increase in nanoparticle concentrations, the σ and μ will also increase. It must be noted that the increasing trend is not strictly linear. Abnormal fluctuation appears when the nanoparticle mass fraction is around 0.1% (see Fig. 1.18). However, it is interesting that the changing trends for both surface tension and dynamic viscosity are identical. As is known, the surface tension of a surfactant solution is determined by the physicochemical nature of the surfactant molecules, which both have hydrophilic and hydrophobic groups and consequently have a tendency to stay at the liquid/air interface. Actually, the surfactant molecules form the monomolecular film in the liquid surface and then decrease or increase the interfacial surface tension. It was found that the dynamic viscosity indicates the internal friction between the liquid molecules. In our case, the concentration of surfactant already guaranteed the formation of micelles and the layer. Therefore it is assumed that TiO2 addition could also change the interactions at the liquid surface and lead to fluctuation of the surface tension and viscosity. However, more experiments are required to confirm these assumptions. The initial kinetic energy (inertial force) and the surface energy (surface tension force) play the main roles in finding the dynamics of drop impact and spreading on a substrate in low-speed spreading experiments. The initial kinetic energy is mainly obtained by the impact velocity U and the size, or drop radius Dd of the falling drop, while the surface energy is related to the surface tension σ. A TEM image of a TiO2 nanoparticle which can be considered as an indication of nanoparticle aggregation when they are dispersed in water is shown in Fig. 1.19A. For additional validation of the occurrence of nanoparticle aggregation, the dynamic light scattering method was

Nanofluids

30

1. Introduction to nanofluids

t = 0 ms

t = 4 ms

t = 8 ms

t = 16 ms

t = 20 ms

t = 24 ms

t = 12 ms

t = 28 ms

FIGURE 1.20

The shape change of drop as a function of time (nanofluid mass fraction is 0.1%, U 5 0.6 m s21, initial droplet diameter is 2.4 mm) [40].

employed as depicted in Fig. 1.19B. As can be seen, the maximum nanoparticle diameter will reach 800 nm, with an initial size of 21 nm before being dispersed in water. Fig. 1.20 shows the evolution process of droplet shape after one drop impacts a sessile drop. As can be seen, after head-on collision and merging of the two drops, a new drop forms that then experiences spreading, recoiling, and rebounding. As the impacting velocity in our study was only 0.51.5 m s21, droplets did not break up during impact because their kinetic energy was too low to overcome surface tension. A droplet at maximum velocity U 5 1.5 m s21, a droplet diameter of 2.8 mm, and nanoparticle mass fraction of 0.001% were chosen to calculate the kinetic energy, the energy of surface tension, and the viscidity of the dissipated work, respectively. Fig. 1.21 shows the history of droplet shape change after one drop impacts another sessile drop. In this case, the diameter of the droplet is 2.4 mm and the dropping velocity for one of them is U 5 0.6 m s21, and for the other one is U 5 1.5 m s21. Fig. 1.21 shows the deformation, expansion, and contraction of the drop upon impact in dimensionless terms as normalized drop radius, D/Dd, and normalized height, H/Dd, as a function of time measured from the instant of impact. Right after two droplets coalesce, the initial kinetic energy of the falling drop will be transferred to the sessile drop, then a newly merged droplet will be formed and oscillating with time in both the horizontal and vertical directions simultaneously. The drop will undergo damped oscillations on the platform. By comparing Fig. 1.21A, C, E, and G, and B, D, F, and H, it can be observed that the vibration amplitude of the drop in the

Nanofluids

(B) 1.4

(A)

4.5

φ = 0%

φ = 0%

U = 0.6 m s–1 U = 1.5 m s–1

1.2

U = 1.5 m s–1 U = 0.6 m s–1

4.0

1.0

H/Dd

D/Dd

3.5 3.0 2.5

0.8 0.6 0.4

2.0

0.2 0

40

80

120

160

200

240

280

0

40

80

Time (ms)

120

160

200

240

280

Time (ms)

(D)

(C)

1.4

4.5

φ = 0.001%

φ = 0.001% U = 1.5 m s–1 U = 0.6 m s–1

D/Dd

3.5

U = 0.6 m s–1 U = 1.5 m s–1

1.2 1.0

H/Dd

4.0

0.8 0.6

3.0

0.4 0.2

2.5 0

40

80

120

160

200

240

280

0

40

80

Time (ms)

(E)4.5

120

160

200

240

280

Time (ms)

(F) φ = 0.01% U = 1.5 m s–1 U = 0.6 m s–1

4.0

φ = 0.01%

2.4

U = 0.6 m s–1 U = 1.5 m s–1

2.0

3.5

D/Dd

1.6

H/Dd

3.0

1.2

2.5 0.8

2.0 1.5

0.4

0

40

80

120

160

200

240

(G)4.5

0.0

280

Time (ms)

40

80

120

160

200

240

280

Time (ms) φ = 0.1%

0.8

φ = 0.1% U = 1.5 m s–1 U = 0.6 m s–1

4.0

0

(H)

U = 0.6 m s–1 U = 1.5 m s–1

0.7 0.6

H/Dd

D/Dd

3.5 3.0 2.5

0.5 0.4 0.3 0.2

2.0 0

40

80

120

160

Time (ms)

200

240

280

0.1

0

40

80

120

160

200

240

280

Time (ms)

FIGURE 1.21 Time courses for the shape evolution of TiO2water nanofluid drops of various nanoparticle loadings impacting and spreading on a PTEF surface, Dd 5 2.4 mm, U 5 0.6, and 1.5 m s21, 0.05% SDBS. Dimensionless radius of spreading drops, D/Dd; dimensionless height of spreading drops, H/Dd [40]. PTFE, Polytetrafluoroethylene; SDBS, sodium dodecyl benzene sulfonate surfactant.

32

1. Introduction to nanofluids

FIGURE 1.22 The image of nanoparticle moving to the triple line obtained by a chatelier-type microscope [40].

vertical direction is stronger and clearer than that in the horizontal direction. Here, we assume that the locking effect will dominate at the interface between the substrate and drop. This effect is expected to restrict the motion of the drop on the radius direction [41]. In our experiments, it was also detected that after the droplet reached its maximum radius, the contact line remain fixed throughout the period of oscillation, while the contact angle changed in a range, as shown in Fig. 1.20. Fig. 1.21B, D, F, and H also confirms that the impact velocity has a substantial influence on droplet height vibration. Hong et al. [42] found that water-based Fe3O4 ferrofluids have non-Newtonian behavior. Clearly, the rheological behavior is also important in droplet oscillation. We examined the rheological properties of NFs and nanoparticle-free fluids in these cases. Basaran and DePaoli [43] noted that the damping frequency and rate of drop oscillation is a significant function of drop size, so that the frequency of oscillation rises as drop size falls. Fig. 1.22 shows the TiO2water NFs droplet image with 0.001% mass fraction obtained by a chatelier-type microscope. The triple-phase contact line, which is formed on a solid surface by two mutually dissoluble fluids, is important for studying droplet dynamics. As can be seen from Fig. 1.22, the nanoparticle will move to the triple-phase contact line from the indicated circle after droplet contact with the solid surface. Moreover, it can be observed that nanoparticles cluster at the edge of the droplet, and from the rim to center, the nanoparticles become sparse [41]. With low-impact velocity, the surfactant has sufficient time to restructure on the surface, resulting in uniform spreading and with reduced uniform surface tension. Thus in accordance with that at lowimpact velocities, the lower surface tension causes the larger maximum spreading radius of the drop. With the impact velocity increase, we found that two main effects will influence the drop maximum spreading radius. First, based on the GibbsMarangoni effect, the surface tension

Nanofluids

1.5 Other forces on nanoparticles in base fluid

33

FIGURE 1.23 Schematic illustration of the characteristic oscillation phenomenon after head-on collision of two nanofluid droplets (A) before impacting, (B) process of spreading, and (C) process of recoiling [40].

of fluid with more surfactant molecules should be less than that with fewer surfactant molecules. In these cases, it is considered that there also exists a Marangoni effect. As shown before, after droplet impact, the surfactant molecules will move to the vicinity of the droplet contact line due to inertness, and consequently the surfactant concentration at the droplet edge will be higher than in the droplet center. Accordingly, the surface tension near the center of the drop will be larger than that near the contact line. Therefore this gradient of surface tension will inhibit the spreading of the drop. Second, the effect of particle inertia is enhanced with high velocity and particle concentration. This described effect will opposes the inhibition of surface tension gradient. Thus there may be a velocity value that gives rise to a stronger effect on particle inertia than the surface tension gradient. As a result, the maximum drop spreading radius (at high particle concentration) will be greater than low concentration under high velocity. Fig. 1.23 shows a schematic for this mechanism, illustrating nanoparticle drop shape evolution when there is a surfactant addition [41].

1.5 Other forces on nanoparticles in base fluid There are other forces applied on nanoparticles in base fluid. Drag force (Fd), gravity force (Fg), Brownian force (Fb), force due to

Nanofluids

34

1. Introduction to nanofluids

FIGURE 1.24 Physical model for the wavy-wavy cavity (A) and the quadrilateral mesh for the calculated domain when γ 5 1.77 rad and A 5 0.11 (B) [44].

thermophoresis effect (FT), force due to lift motion (Fv), force due to virtual mass (Fl), and force due to pressure gradient (Fp) per unit mass are the main forces on nanoparticles, which are discussed in the following two cases.

1.5.1 Case 1: Natural convection heat transfer in an NF-filled cavity with double sinusoidal wavy walls This case is based on the earlier study of Sheikholeslami et al. [45]. As schematically presented in Fig. 1.24, a 2D quarter circular cavity with two wavy walls was modeled to study the natural convection heat transfer process using four different kinds of NFs. The quadrilateral dominant meshes used in the FVM program are presented in Fig. 1.24. The inner and outer walls were maintained at constant temperatures Th and Tc, respectively. The other two boundaries were considered to be adiabatic. The wavy shape of the inner and outer walls were assumed to be the following cosine functions [44,45]: r 5 rin 1 A cos ðN ðζ ÞÞ

(1.41)

r 5 rout 1 A cos ðN ðζ ÞÞ

(1.42)

where rin and rout are the circle radii of the inner and outer walls; A and N are the amplitude and number of undulations, respectively; and ζ is the rotation angle. In this case, for simplification, it is assumed that A and N of the inner wall are 0.3 and 9, respectively, based on our previous optimization study [46]. The number of boundary layers is five and the maximum cell size is 0.01. There are nine meshes created by

Nanofluids

1.5 Other forces on nanoparticles in base fluid

35

automethod and it is assumed that the mesh quality is beyond 0.5, which means they meet the needs of our calculations. The wavy-wall cavity in this case is based on and transformed from a quarter of a cyclic annular cavity with a 9 μm inner radius and a 20 μm outer radius. The nanoparticle was tracked using the Lagrangian model and the governing equations are calculated under the Eulerian frame. The volume fraction parameters were calculated under an implicit scheme. Liquid water was used as the base fluid and various kinds of nanoparticles (Ag, CuO, Al2O3, and TiO2 with thermophysical properties in Table 1.1) with averaged diameter of 20 nm were added to it. The pressurevelocity coupling equation used the phase-coupled semiimplicit method for pressure-linked equations (SIMPLE) scheme. The solution procedure showed that all the cases converge within 1000 steps approximately. Thermophoretic and Brownian forces are not considered here, but the equations for laminar natural convection under the Eulerian model are: Continuity equation: @ρ 1 r ðρVÞ 5 0 @t



(1.43)

Momentum equation: @ ðρ vl Þ 1 r ðρl vl vl Þ 5 2 rp 1 r τ l 1 ρl g 2 Sp @t l





where Sp can be obtained by: Sp 5 Energy equation:

X

Fmp Δt

(1.45)

  @τ 1 vl rτ 5 r ðkrτ Þ ρl c @t



(1.44)



(1.46)

where τ 1 can be obtained by: 2 τ l 5 μl rvl 1 rvTl 2 μl r vl I 3



(1.47)

Eqs. (1.43)(1.47) describe the movement of continuous water phase. ρl is the fluid density, vl is the velocity, t is the time, p is the pressure, g is the gravitational acceleration, c is the heat capacity, k is the thermal conductivity, and T is the fluid temperature, Here, τ is the stress tensor, F is the total force acting on a particle, I is the unit vector, and mp is the mass of the nanoparticle.

Nanofluids

36

1. Introduction to nanofluids

In the Lagrangian frame of reference, the equation of motion of a nanoparticle is the following equation based on the total force applied on the nanoparticles: dvp 5F dt

(1.48)

F 5 FD 1 FG 1 FL 1 FP 1 FV

(1.49)

where FD is the hydrodynamic drag force from the fluid, which can be calculated by applying the Stokes’ law [47]: FD 5 6πμl ri ðui 2 up Þ

(1.50)

FG is the force due to gravity: FG 5

gðρp 2 ρl Þ ρp

(1.51)

The Staffman’s lift force, FL, was derived by Staffman as [48]: FL 5

2KS v1=2 ρl dij ρp dp ðdlk dkl Þ1=4

ðvl 2 vp Þ

(1.52)

where KS 5 2.594 is a constant, dij is the deformation tensor defined as dij 5 1/2(vli,j 1 vlj,i), and FP is the force due to the gravity gradient: ! ρl (1.53) FP 5 vp rvl ρp



FV is the virtual mass force: FV 5

1 ρl d ðvl 2 vp Þ 2 ρp dt

(1.54)

where h is the surface heat transfer coefficient calculated by AnsysFluent 15.0. The following assumptions were used to simplify the problem. 1. The working fluid is considered as incompressible laminar flow across a cavity and maintains the condition of single phase. 2. The thermophysical properties of the nanoparticles are constant. 3. The two straight surfaces of the cavity are well insulated. 4. The diameter of each nanoparticle is uniform. After this simplification, commercial CFD AnsysFluent 15.0 software based on the FVM was used to obtain the numerical solution. The

Nanofluids

1.5 Other forces on nanoparticles in base fluid

37

computational domain and created mesh were produced using the commercial preprocessing code ICEM CFD 15.0. As mentioned before, a LagrangianEulerian approach was used here. The nanoparticle was tracked using the Lagrangian approach while the governing equations for the base fluid were solved using the Eulerian approach. It is worth noting here that Bianco et al. [49] compared the accuracy of the single-phase model and LagrangianEulerian approach in a laminar flow with waterAl2O3. They reported that the results from both models were similar. In another study, Kumar and Puranik [50] studied the convective heat transfer of NFs in turbulent flow using a LagrangianEulerian approach. They compared the single-phase model with the LagrangianEulerian model and determined that the LagrangianEulerian approach was a more accurate model for simulating forced convection heat transfer when the NF has a particle volume fraction less than 0.5%. Since the nanoparticle concentrations in all of our examined cases are under 0.5%, the LagrangianEulerian approach is used here to model the heat transfer process. The pressure terms in the governing equations were discretized using the second-order implicit term, whereas the momentum and energy equations were discretized using the second-order upwind scheme. The pressurevelocity coupling was implemented following the SIMPLE algorithm. Moreover, the absolute convergence criterion was set to be 1025 for the continuity equation and velocity equation while it was 1026 for the energy equation. The Boussinesq approximation was used for both water and nanoparticles to simulate natural convection. Pure water was added to the cavity first where it was under laminar natural convection and then the nanoparticles were added to the water and the Eulerian model was used to simulate the two-phase flow. The evolution of the streamline when the volume fraction of Ag rises from 0% to 0.9% is depicted in Fig. 1.25. This figure demonstrates that by increasing the volume fraction, the high mass flow rate region of the NF decreases as well as the low mass flow rate region. In the interior region, the low mass flow region (which is blue in Fig. 1.25) becomes smaller and the underpart of this blue region gradually disappears. The velocity tends to become uniform because when the Ag volume fraction rises from 0% to 0.9%, the viscosity of NFs increases, so the velocity wave conveys slower. The surface heat transfer coefficient (h) of Ag NF when the volume fraction increases from 0% to 0.9% is shown in Table 1.5. It is obvious that it increases quickly with increasing volume fraction and the increased speed is higher with increased volume fraction. Fig. 1.26 shows an obvious increase of the outer wall Nusselt number compared to a circular wavy enclosure, thus indicating that considering the circular outer wall to be sinusoidal is meaningful from a heat transfer viewpoint.

Nanofluids

38

1. Introduction to nanofluids

FIGURE 1.25 Streamline evolution of water phase with Ag nanofluid s at A 5 0.1, γ 5 0 for (A) ϕ 5 0%, (B) ϕ 5 0.1%, (C) ϕ 5 0.3%, (D) ϕ 5 0.5%, (E) ϕ 5 0.7%, and (F) ϕ 5 0.9% [44]. TABLE 1.5 Absolute surface heat transfer coefficient of outer wall of Ag nanofluids at A 5 0.1, γ 5 0 for ϕ 5 0%, ϕ 5 0.1%, ϕ 5 0.3%, ϕ 5 0.5%, ϕ 5 0.7%, and ϕ 5 0.9%. 22

have (W m

21

K )

ϕ50

ϕ 5 0.1%

ϕ 5 0.3%

ϕ 5 0.5%

ϕ 5 0.7%

ϕ 5 0.9%

38,811.5

41,101.11

44,588.64

50,133.55

55,852.2

64,291.73

1.5.2 Case 2: The effects of nanoparticle aggregation on convection heat transfer investigated using a combined NDDM and DPM method In this case the effect of nanoparticle heat transfer (including the forces applied on the nanoparticles) was investigated using a combined nanoparticle diameter distribution model (NDDM) and DPM method. First, we defined the NDDM and DPM methods [51]. 1.5.2.1 Nanoparticle diameter distribution model Based on our previous study [52], the process of nanoparticle aggregation includes two main steps, coagulation and fragmentation, and this process can be defined by the following population-balance equation:

Nanofluids

39

1.5 Other forces on nanoparticles in base fluid

22

CWE WWE

20

Local Nusselt number

18 16 14 12 10 8 6 4 0

10

20

30

40

50

60

70

80

90

ζ

FIGURE 1.26

Comparison of CWE with WWE at different angles [44]. CWE, Circular

wavy enclosure. i22 i21 X dNi X 1 2 5 2j2i11 β i21;j Ni21 Nj 1 β i21;i21 Ni21 2 Ni 2j2i β i;j Nj 2 dt j51 j51

2 Ni

imax 21 X

imax X β i;j Nj 2 Si Ni 1 Γ i;j Sj Nj

j51

(1.55)

j51

where Ni is the number concentration of flocs with volume Vi. The β i,j is the collision rate of particles. It is assumed that the collision is mainly produced by the shear deformation of the fluid as well as the Brownian motion, so β i,j can be considered as: rffiffiffiffiffiffiffiffiffiffiffiffiffi  1 1 3 8πkB T 2 Df Df σ β i;j 5 0:31Gvp xi 1xj 1 (1.56) m where G is the shear rate, vp is the volume of primary particle, Df is the fractal dimension, xi is the number of primary particles in an aggregate in section i, m is the effective mass of particles, kB is the Boltzmann constant, and σ is the equivalent diameter. The rate of particle breakage is called the “fragmentation kernel Si,” which is dependent on the shear rate and aggregate size as follows:    q  3=Df 1 η φtot G dc;i Si 5 kb v3p (1.57) τ dp

Nanofluids

40

1. Introduction to nanofluids

where η is a variety of correlations between the particle volume fraction and suspension viscosity and can be calculated by: η0 ηðΦtot Þ 5  (1.58) 2 12Φtot =φm where η0 is the viscosity of basic fluid of dilute solution and Φtot is the effective of total volume concentration of the aggregates: imax X

Φtot 5

Ni vc;i

(1.59)

i51

where vc,j is the collision volume of aggregate. Also, the fragment distribution function Гi,j in Eq. (1.55) has been assumed to be [51,52]:

2 j5i11 (1.60) Γi;j 5 0 else

1.5.2.1.1 Discrete phase model

As mentioned, in the DPM, the Lagrangian approach was used for nanoparticle tracking, while the Eulerian approach was applied on the governing equations for the base fluid solution. The coupling between the two approaches was reached by adding the interphase momentum and energy exchanges as source terms in the suitable governing equations. The governing equations are as follows: Continuity equation:   @ρb 1 r ρb Vb 5 0 @t

(1.61)

  @ρb vb 1 r ρb vb vb 5 2 rp 1 r Tb 2 Sp 1 ρb g @t

(1.62)



Momentum equation:



Energy equation:



  @T 1 vb rT 5 r ½krT  ρb c @t





(1.63)

In Eqs. (1.61)(1.63), ρb is the base fluid density, vb is the base fluid velocity, t is the time, p is the pressure, g is the acceleration of gravity, c is the heat capacity, k is the thermal conductivity, and T is the fluid temperature. αb in Eq. (1.64) is the stress tensor, which can be determined as:

Nanofluids

1.5 Other forces on nanoparticles in base fluid

41

2 αb 5 μb rvb 1 rvTb 2 μb rvb I 3

(1.64)

where μb is the shear viscosity of the fluid phase and I is the unit vector. Sp in Eq. (1.65) is the source term in order to signify the momentum transfer between the base fluid and particle phases. Sp can be determined by computing the particle momentum variation as they pass through the control volume for the fluid phase, as shown by [51]: X Fmp rt (1.65) Sp 5 In Eq. (1.65), F is the total force acting on a particle in the base fluid and is determined to be the summation of drag force (Fd), gravity force (Fg), Brownian force (Fb), force due to thermophoresis effect (FT), force due to lift motion (Fv), force due to virtual mass (Fl), and force due to pressure gradient (Fp) per unit mass. Thus the relation can be written as [51]: F 5 Fd 1 Fg 1 Fb 1 FT 1 Fv 1 Fp 1 Fl

(1.66)

Using the Lagrangian frame of reference, the equation of motion for a nanoparticle is: dvp 5F dt

(1.67)

where vp is the particle velocity. The StokesCunningham drag law is offered in AnsysFluent 14.5 for calculation of drag force per unit mass. It is given by: Fd 5

18μb d2p ρp Cc

(1.68)

where Cc is called the Cunningham correction factor and is defined as:

2λ 1:257 1 0:4e2ð1:1dp =2λÞ Cc 5 1 1 (1.69) dp where dp is the particle diameter and λ is the molecular mean-free path of fluid. The gravity force is also a significant force and can be obtained by:   g ρd 2 ρb Fg 5 (1.70) ρp Because some researchers pointed out that the convective heat transfer coefficient in laminar flow is affected more by the thermal

Nanofluids

42

1. Introduction to nanofluids

conductivity than by the viscosity, the lift force plays a limited role in this process and can be neglected in the process as done in this case. The Brownian force Fb can be expressed as: rffiffiffiffiffiffiffiffi πS0 Fbi 5 ζ i (1.71) Δt where ζ i is the unit varianceindependent Gaussian random number with zero mean and S0 is defined by: S0 5

216vkB T ρ 2 π2 ρb d5p ρp Cc

(1.72)

b

where kB is the Boltzmann constant and vb is the kinetic viscosity of fluid. The thermophoresis force, which acts on the small particles suspended in a fluid under temperature gradient, is called the thermophoretic force and can be given by: FT 5 2

6πdp μ2b Cs ðK 1 Ct KnÞ 1 r Τ ρb ð1 1 3Cm KnÞð1 1 2K 1 2Ct KnÞ mp T



(1.73)

where Cm 5 1.146, Cs 5 1.147, and Ct 5 2.18 are the momentum exchange coefficient, the thermal slip coefficient, and the temperature jump coefficient, respectively. Also, Kn is the Knudsen number. The pressure gradient produces another force acting on nanoparticles in the fluid and can be expressed as: ! ρb dvb (1.74) Fp 5 up ρp dx The heat exchange between the “discrete phase” and the “base fluid phase” can be calculated using the expression: mp cp

  dTp 5 hAp Tb 2 Tp dt

(1.75)

where Ap is the particle superficial area, Tb is the base fluid temperature, and Tp is the particle temperature. The last equation assumes that there is no heat transfer through the radiation process and the internal thermal resistance within the nanoparticle is insignificant. Finally, the heat transfer coefficient h can be considered from the Ranz and Marshall expression [51]: Nu 5

hdp ð1=3Þ 5 2 1 0:6Re0:5 p Pr Kb

Nanofluids

(1.76)

1.5 Other forces on nanoparticles in base fluid

43

Constant surface heat flux at wall

Pressure outlet

Velocity inlet

Nanofluids

Constant surface heat flux at wall

FIGURE 1.27 Three-dimensional physical model for the horizontal microchannel circular tube [51].

1.5.2.2 Problem description in second case and its solution A schematic of the computational setup is shown in Fig. 1.27. A tube with an inner diameter of 0.4 mm and length of 0.12 m was considered for the numerical simulation and 3D DPM were used for the solution. The defined boundary condition for this tube was as follows: the wall of the tube has a constant surface heat flux of 69.9 kw m22, the inlet condition is velocity inlet, and the outlet type is considered to be pressure outlet. All the nanoparticles were assumed to be spherical and their initial sizes were 60 nm. The inlet temperature of the fluid was set to 291 K and details for the DPM settings were as follows. For the nanoparticle injection “surface injection” was selected and nanoparticles were injected from the inlet surface. The mass flow rate of the particle was determined using the nanoparticle density, NF velocity, and volume fraction. The boundary condition for the inlet and outlet surfaces were all set as “Escape” for particles, while the tube wall was defined as “Reflect.” The nanoparticle temperature and velocity at the inlet were considered to be the same as those for the base fluid. AnsysFluent 15.0 based on the FVM was used to model and analyze this case. The preprocessing meshing software ICEM was applied for generating the grid. The SIMPLE algorithm was used to solve the governing equations. During the solution, the DPM was enabled when the base fluid medium achieved a converged solution. The option for interaction with the continuous phase was enabled along with the option for updating the continuous phase after each iteration in the DPM phase. The maximum number of steps was not constant for all runs, and was actually based on the particle velocity for particle tracking which causes that all the injected particles exit from the outlet, completely. The numerical solution was validated based on the available experimental outcomes of Zhang et al. [53] where Al2O3 NF was considered as the working fluid. The average values of the heat transfer coefficient number were used under the two different Re numbers as the evaluated index.

Nanofluids

44 (A)

1. Introduction to nanofluids (B)

0.35

φ = 0.5%

Re = 250 Re = 750

0.25 0.20 0.15 0.10 0.05

φ = 1%

0.35

Particle number fraction (Ni /Ntot )

Particle number fraction (Ni /Ntot )

0.30

Re = 500 Re = 1000

0.00

Re = 500 Re = 1000

0.30 0.25 0.20 0.15 0.10 0.05 0.00

1

10

100

1

10

Dimensionless f loc diameter (d i /dp )

100

Dimensionless f loc diameter (d i /dp )

(C) 0.35

(D) 0.30

φ = 2%

Re = 250 Re = 750

0.30

Re = 500 Re = 1000

Re = 500 0.25

Particle number fraction (Ni /Ntot )

Particle number fraction (Ni /Ntot )

Re = 250 Re = 750

0.25 0.20 0.15 0.10 0.05

ϕ = 0.5%

ϕ2

0.20

0.15

ϕ1 ϕ3

0.10

0.05

0.00

0.00 1

10

100

1

10

100

Dimensionless f loc diameter (d i /dp )

Dimensionless f loc diameter (d i /dp )

FIGURE 1.28 Particles diameter distribution for various Re numbers [51].

The particle size evolution and distribution were essentially the combined results of Brownian motion and shear deformation. As mentioned in the literature, Brownian motion plays a dominant role in particle coagulation procession in the early stage (i.e., when the particles are very small (below micrometer)). After the aggregated particle increases to a certain size as introduced by Eq. (1.56), shear deformation will dominate the evolution of the size distribution and the agglomeration rate will decrease. The PSD at the dynamic equilibrium state was demonstrated in Fig. 1.28. It can be seen that the particle distribution profile is like the normal distribution curve and thus it can be concluded that the decrease of shear rate will lead to particle size increase. In order to simplify the follow-up simulation of particle-laden flow, the particle number fraction was generally divided into three parts when dynamic balance was achieved. As the curve shape of the steady state particle diameter distribution was approximately a normal distribution, three sections (small, middle, and large particle diameter values) were chosen and the average particle diameter of each part was considered for calculation, as shown in Fig. 1.28D. A summary of the results is given in Table 1.6.

Nanofluids

45

1.5 Other forces on nanoparticles in base fluid

TABLE 1.6

Results of particle diameter distribution after agglomeration.

ϕ1

Average particle diameter (nm)

ϕ2

ϕ3

Average particle diameter (nm)

0.05%

179.54

0.43%

803.25

0.02%

2931.48

500

0.02%

100.41

0.14%

318.77

0.01%

2112.51

750

0.01%

76.95

0.48%

230.84

0.01%

1964.10

1000

0.03%

67.80

0.47%

159.39

0.01%

1714.50

250

0.01%

179.54

0.92%

923.36

0.06%

3212.99

500

0.06%

115.42

0.91%

401.62

0.03%

2280.04

750

0.03%

87.71

0.93%

253.01

0.04%

1964.10

1000

0.01%

67.80

0.98%

183.22

0.01%

1964.10

250

0.12%

133.22

1.82%

5065.01

0.06%

2469.80

500

0.14%

133.22

1.77%

506.01

0.09%

2469.80

750

0.09%

100.41

1.84%

318.77

0.07%

2112.51

1000

0.05%

79.65

1.87%

200.81

0.08%

1832.17

Initial particle (nm)

Volume fraction

Re number

60

0.50%

250

1%

2%

Average particle diameter (nm)

As is known, Reynolds numbers have a very significant effect on the heat transfer process. Fig. 1.29 shows comparisons of the heat transfer coefficients of Al2O3water NFs with different nanoparticle volume fraction. As depicted in Fig. 1.29, it is clear that by adding Al2O3 nanoparticles to the water, its heat transfer coefficient can be improved and the enhancement is more significant with the increasing nanoparticle volume fractions. Furthermore, the heat transfer rate is improved as the Reynolds number increases. The main reason for the heat transfer improvements is due to the much higher thermal conductivity of the solid particles than the base liquid. As described before, after adding the nanoparticles, the properties of the base fluid will also change, resulting in enhanced heat transfer in the compound fluid interior. Furthermore, Brownian motion force or other forces acting on the nanoparticles in the fluid is another reason for heat transfer improvement, which could result in Brownian diffusion and heat diffusion. It is correct to note that in the current study, the h values of the base fluid (water) increased relatively slowly and had a tendency to level off at high Re numbers, but raise quickly for NFs. For example, from Re 250 to 1000, the heat transfer coefficient of 2% Al2O3water NFs increases by 112.44%, which is larger than the increment of 96.23% for water. This may be due to the degree of microscale disturbance from the

Nanofluids

46

1. Introduction to nanofluids

11000 10000

Water 0.5% Al2 O3 1%

Al2 O3

2%

Al2 O3

h (w m–2 k–1)

9000 8000 7000 6000 5000 4000 3000 200

400

600

800

1000

Re FIGURE 1.29 Variations of the heat transfer coefficient with Reynolds for the Al2O3water nanofluids with different nanoparticle volume fractions [51].

nanoparticles, which increases as Re increases and would be more significant when the Re number is higher [51]. Fig. 1.30 shows the nondimensional cross-sectional temperature distribution of the base fluid and NFs for various volume fractions at the outlet under Re 5 250. Comparisons of nondimensional temperature distribution before and after particle agglomeration are also presented in Fig. 1.30. In this case, with constant heat flux on the wall, nondimensional temperature can be considered as: T 5

T 2 Tin qw R=λ0



(1.77)

where R is the inner radius of the tube, qw is the heat flux, Tin is the inlet temperature of coolant, and λ0 is the NF thermal conductivity. A smooth parabolic profile of temperature distribution for base fluid can be found in Fig. 1.30, and temperature profiles after adding nanoparticles were similar to that of the base fluid. Also, it can be seen that by increasing the nanoparticle volume fractions, the nondimensional temperature numbers near the wall decreased. For instance, when the nanoparticle volume fraction increased from 1% to 2%, the maximum dimensionless temperature decreased from 0.85 to 0.65 before nanoparticle agglomeration. In the literature, Xuan and Li [54] also reported that the temperature gradient between the fluid and the wall increased and

Nanofluids

1.5 Other forces on nanoparticles in base fluid

47

FIGURE 1.30 Nondimensional temperature distribution of the base fluid and Al2O3 nanofluids at Re 5 250 (A) before agglomeration and (B) after agglomeration [51].

the heat transfer rate near the wall boundary can improved after adding nanoparticles to the base fluid. It is assumed that the nanoparticles with the same temperature as the liquid of the central region of the tube move to the area near the solid wall and become the heat sinking points by absorbing heat, and then quickly move back to the tube central region after transferring the heat to the solid wall. It is assumed that thermal equilibrium between the nanoparticle and the liquid near the center of the tube cross section is achieved rapidly due to the large specific surface area of the nanoparticles, which contributes to the heat exchange area between the two phases. Thus nanoparticles are an efficient medium of heat transfer resulting in high-temperature gradients

Nanofluids

48

1. Introduction to nanofluids

near the wall. It is also meaningful to note that by the volume fraction increasing, the heat transfer rate is also improved, and consequently the nondimensional temperature near the wall is reduced. Additionally, it can be observed from Fig. 1.30 that after nanoparticle clustering, the nondimensional temperature near the wall is greater. Therefore the temperature difference between the fluid and the wall is decreased, leading to a declined heat transfer rate. It is important to mention that the degree of Brownian motion will be weakened after nanoparticle clustering, which causes a significant reduction of the heat transfer performance of NFs.

References [1] S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME Publ. Fed. 231 (1995) 99106. [2] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, ASME J. Heat Transf., 125, 2003, pp. 151155. [3] W.H. Azmi, K. Abdul Hamid, R. Mamat, K.V. Sharma, Effects of working temperature on thermo-physical properties and forced convection heat transfer of TiO2, nanofluids in waterethylene glycol mixture, Appl. Therm. Eng. 106 (2016) 11901199. [4] H. Zhang, S. Shao, H. Xu, C. Tian, Heat transfer and flow features of Al2O3water nanofluids flowing through a circular microchannel  experimental results and correlations, Appl. Therm. Eng. 61 (2013) 8692. [5] N. Kumar, B.P. Puranik, Numerical study of convective heat transfer with nanofluids in turbulent flow using a Lagrangian-Eulerian approach, Appl. Therm. Eng. 111 (2016) 16741681. [6] M. Mahdavi, M. Sharifpur, J.P. Meyer, CFD modelling of heat transfer and pressure drops for nanofluids through vertical tubes in laminar flow by Lagrangian and Eulerian approaches, Int. J. Heat Fluid Flow 88 (2015) 803813. [7] S. Sonawane, U. Bhandarkar, B. Puranik, Modeling forced convection nanofluid heat transfer using an Eulerian-Lagrangian approach, J. Therm. Sci. Eng. Appl. 8 (2016) 031001031008. [8] J. Zhou, X. Wang, D. Song, D. Jing, The effects of nanoparticle aggregation on the convection heat transfer investigated by a combined NDDM and DPM method, Numer. Heat Transf. Part A Appl. 71 (7) (2017) 754768. [9] S. Krishnamurthy, P. Bhattacharya, P.E. Phelan, R.S. Prasher, Enhanced mass transport in nanofluids, Nano Lett. 6 (2006) 419423. [10] W. Yu, D.M. France, Review and comparison of nanofluid thermal conductivity and heat transfer enhancements, Heat Transf. Eng. 29 (2008) 432460. [11] J.H. Lee, S.H. Lee, C. Choi, S. Jang, S. Choi, A review of thermal conductivity data, mechanisms and models for nanofluids, Int. J. Micronano Scale Transp. 1 (2015) 269322. [12] C. Pang, J. Lee, H. Hong, Y. Kang, Heat conduction mechanism in nanofluids, J. Mech. Sci. Technol. 28 (2014) 29252936. [13] H. Aybar, M. Sharifpur, M.R. Azizian, M. Mehrabi, J.P. Meyer, A review of thermal conductivity models for nanofluids, Heat Transf. Eng. 36 (2015) 10851110. [14] G. Tertsinidou, M.J. Assael, W.A. Wakeham, The apparent thermal conductivity of liquids containing solid particles of nanometer dimensions: a critique, Int. J. Thermophys. 36 (2015) 13671395.

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[15] M. Hatami, G. Domairry, S.N. Mirzababaei, Experimental investigation of preparing and using the H2O based nanofluids in the heating process of HVAC system model, Int. J. Hydrog. Energy 42 (12) (2017) 78207825. [16] S.K. Das, S.U. Choi, W. Yu, T. Pradeep, Nanofluids Science and Technology, Wiley, 2008. [17] A.V. Rane, K. Kanny, V.K. Abitha, S. Thomas, Methods for synthesis of nanoparticles and fabrication of nanocomposites, in: Synthesis of Inorganic Nanomaterials, Elsevier, 2018, pp. 121139. [18] S. Roy, C.K. Ghosh, C.K. Sarkar (Eds.), Nanotechnology: Synthesis to Applications., CRC Press, 2017. [19] M. Aliofkhazraei (Ed.), Handbook of Nanoparticles, Springer International Publishing, Cham, 2016. [20] M. Hosokawa, K. Nogi, M. Naito, T. Yokoyama (Eds.), Nanoparticle Technology Handbook., Elsevier, 2012. [21] M. Haghshenas Fard, M. Nasr Esfahany, M.R. Talaie, Numerical study of convective heat transfer of nanofluids in a circular tube two-phase model versus single-phase model, Int. Commun. Heat Mass Transf. 37 (2010) 9197. [22] S. Go¨ktepe, K. Atalık, H. Ertu¨rk, Comparison of single and two-phase models for nanofluid convection at the entrance of a uniformly heated tube, Int. J. Therm. Sci. 80 (2014) 8392. [23] M. Hatami, Different shapes of Fe3O4 nanoparticles on the free convection and entropy generation in a wavy-wall square cavity filled by power-law non-Newtonian nanofluid, Int. J. Heat Technol. 36 (2) (2018) 509524. [24] G.H.R. Kefayati, Simulation of heat transfer and entropy generation of MHD natural convection of non-Newtonian nanofluid in an enclosure, Int. J. Heat Mass Transf. 92 (2016) 10661089. [25] S. Mosayebidorcheh, M. Vatani, M. Hatami, D.D. Ganji, Two-phase nanofluid thermal analysis over a stretching infinite solar plate using Keller Box Method (KBM), Iran. J. Chem. Chem. Eng. 37 (2018) 247256. [26] J.A. Khan, M. Mustafa, T. Hayat, A. Alsaedi, Three-dimensional flow of nanofluid over a non-linearly stretching sheet: an application to solar energy, Int. J. Heat Mass Transf. 86 (2015) 158164. [27] H.B. Keller, A new difference scheme for parabolic problems, in: B. Hubbard (Ed.), Numerical Solutions of Partial Differential Equations, Part II, Academic Press, New York, 1971, pp. 327350. [28] J.C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon, Oxford, 1892. [29] J. Jin, M. Hatami, D. Jing, Experimental investigation and prediction of the thermal conductivity of water-based oxide nanofluids with low volume fractions, J. Therm. Anal. Calorim. 135 (1) (2019) 257269. [30] D.H. Kumar, H.E. Patel, V.R.R. Kumar, T. Sundararajan, T. Pradeep, S.K. Das, Model for heat conduction in nanofluids, Phys. Rev. Lett. 93 (2004) 144301. [31] R. Prasher, P. Bhattacharya, P.E. Phelan, R. Prasher, P. Bhattacharya, Brownianmotion-based convective-conductive model for the effective thermal conductivity of nanofluids, J. Heat Transf. 128 (2006) 588595. [32] C.H. Chon, K.D. Kihm, S.P. Lee, S.U.S. Choi, Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3), thermal conductivity enhancement, Appl. Phys. Lett. 87 (2005) 395403. [33] H.E. Patel, T. Sundararajan, S.K. Das, An experimental investigation into the thermal conductivity enhancement in oxide and metallic nanofluids, J. Nanopart. Res. 12 (2010) 10151031. [34] R.S. Vajjha, D.K. Das, Experimental determination of thermal conductivity of three nanofluids and development of new correlations, Int. J. Heat Mass Transf. 52 (2009) 46754682.

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1. Introduction to nanofluids

[35] M. Corcione, Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, Energy. Convers. Manag. 52 (2011) 789793. [36] J.H. Lee, K.S. Hwang, S.P. Jang, B.H. Lee, J.H. Kim, S.U.S. Choi, et al., Effective viscosities and thermal conductivities of aqueous nanofluids containing low volume concentrations of Al2O3 nanoparticles, Int. J. Heat Mass Transf. 51 (2008) 26512656. [37] M.C.S. Reddy, V.V. Rao, Experimental studies on thermal conductivity of blends of ethylene glycol-water-based TiO2 nanofluids, Int. Commun. Heat Mass Transf. 46 (2013) 3136. [38] A. Kazemibeydokhti, S.Z. Heris, N. Moghadam, M. Shariatiniasar, A.A. Hamidi, Experimental investigation of parameters affecting nanofluid effective thermal conductivity, Chem. Eng. Commun. 201 (2013) 14431444. [39] H.A. Mintsa, G. Roy, T.N. Cong, D. Doucet, New temperature dependent thermal conductivity data for water-based nanofluids, Int. J. Therm. Sci. 48 (2009) 363371. [40] J. Zhou, Y. Wang, J. Geng, D. Jing, Characteristic oscillation phenomenon after headon collision of two nanofluid droplets, Phys. Fluids 30 (7) (2018) 072107. [41] M. Pasandideh-Fard, Y.M. Qiao, S. Chandra, J. Mostaghimi, Capillary effects during droplet impact on a solid surface, Phys. Fluids 8 (3) (1996) 650659. [42] R.Y. Hong, Z.Q. Ren, Y.P. Han, H.Z. Li, Y. Zheng, J. Ding, Rheological properties of water-based Fe3O4 ferrofluids, Chem. Eng. Sci. 62 (21) (2007) 59125924. [43] O.A. Basaran, D.W. DePaoli, Nonlinear oscillations of pendant drops, Phys. Fluids 6 (9) (1994) 29232943. [44] W. Tang, M. Hatami, J. Zhou, D. Jing, Natural convection heat transfer in a nanofluid-filled cavity with double sinusoidal wavy walls of various phase deviations, Int. J. Heat Mass Transf. 115 (2017) 430440. [45] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, MHD natural convection in a nanofluid filled inclined enclosure with sinusoidal wall using CVFVM, Neural Comput. Appl. 24 (3) (2014) 873882. [46] M. Hatami, D. Song, D. Jing, Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition, Int. J. Heat Mass Transf. 98 (2016) 758767. [47] K. Apostolou, A.N. Hrymak, Discrete element simulation of liquid-particle flows, Comput. Chem. Eng. 32 (2008) 841856. [48] P.G. Staffman, The lift on a small sphere in a slow shear flow, J. Fluid Mech. 22 (1965) 385. [49] V. Bianco, F. Chiacchio, O. Manca, S. Nardini, Numerical investigation of nanofluids forced convection in circular tubes, Appl. Therm. Eng. 29 (2009) 36323642. [50] N. Kumar, B.P. Puranik, Numerical study of convective heat transfer with nanofluid in turbulent flow using a Lagrangian-Eulerian approach, Appl. Therm. Eng. 111 (2017) 16741681. [51] J. Zhou, X. Wang, D. Song, D. Jing, The effects of nanoparticle aggregation on the convection heat transfer investigated by a combined NDDM and DPM method, Numer. Heat Transf. Part A Appl. 71 (7) (2017) 754768. [52] D. Song, D. Jing, J. Geng, Y. Ren, A modified aggregation based model for the accurate prediction of particle distribution and viscosity in magnetic nanofluids, Powder Technol. 283 (2015) 561569. [53] H. Zhang, S. Shao, H. Xu, C. Tian, Heat transfer and flow features of Al2O3water nanofluids flowing through a circular microchannel  experimental results and correlations, Appl. Therm. Eng. 61 (2013) 8692. [54] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, ASME J. Heat Transf., 125, 2003, pp. 151155.

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C H A P T E R

2 Mathematical analysis of nanofluids

2.1 Mathematical modeling of nanofluids properties In our last book [1], analytical/mathematical methods for engineering problems were introduced including the adomian decomposition method, variational iteration method, differential transformation method (DTM), homotopy perturbation method, homotopy analysis method (HAM), and weighted residual method (WRM). In this chapter those methods that are more reliable and accurate for the nanofluids (NFs) modeling are presented including example cases to be valuable for readers. From the mathematical viewpoint, the NFs can be modeled in two main manners; single- and two-phase methods. As noted in Chapter 1, Introduction to nanofluids, for single-phase model, the nanoparticles and base fluid are considered as a single phase, so an equivalent properties can be considered for it. However, in the twophase model the governing equations must be solved for the both nanoparticle and fluid phases at the same time. To find the best model, comparison of the single- and two-phase modeling has been considered by the researchers. For instance, Haghshenas Fard et al. [2] compared the outcomes of the single-phase and two-phase numerical methods for NFs in a circular tube and found that the two-phase model is more efficient. The single-phase model is also a simple and suitable model for NF modeling studies. In this model, the Maxwell and Brinkman correlations for NF thermal conductivity and viscosity are usually used in the literature, which determine the NF properties based on the nanoparticle volume fraction. Also, it is possible to use other suggested experimental

Nanofluids DOI: https://doi.org/10.1016/B978-0-08-102933-6.00002-0

51

© 2020 Elsevier Ltd. All rights reserved.

52

2. Mathematical analysis of nanofluids

TABLE 2.1 Water and nanoparticle properties. ρ (kg m23) H2O

Cp (J kg 21 K21)

997.1

K (W m21 K21)

4179

0.613 8.9538

β (K21) 3 1026 210

TiO2

4250

686.2

9.0

Al2O3

3970

765

40

CuO

6320

531.8

76.5

Cu

8933

385

401

16.7

Ag

10,500

235

429

18.9

8.5 18.0

formulas for thermal conductivity and viscosity as previously presented in Chapter 1, Introduction to nanofluids: 

ρeff 5 ρs φ 1 ρf ð1 2 φÞ     ρCp eff 5 ρCp s φ 1 ρCp f ð1 2 φÞ

(2.1)

β eff 5 β f ð1 2 φÞ 1 β s φ

(2.3)



μeff 5

μf

ð12φÞ2:5   2 2φ kf 2 ks 1 2kf 1 ks   keff 5 kf φ kf 2 ks 1 2kf 1 ks

(2.2)

(2.4) (2.5)

where φ is the nanoparticle volume fraction; μf is the dynamic viscosity of the basic fluid; ρf and ρs are the densities of the base fluid and nanoparticles, respectively; (ρCp)f and (ρCp)s are the specific heat parameters of the base fluid and nanoparticles; and kf and ks are the thermal conductivities of the base fluid and nanoparticles, respectively. Using Eqs. (2.1)(2.5) and nanoparticle/base fluid properties such as those in Table 2.1 (or Table 1.1), the NF properties for a single-phase model can be obtained.

2.2 Weighted residual method for nanofluid modeling To better understand the least squares method (LSM), suppose that a differential operator D is acted on a function u to produce a function p [3]: DðuðxÞÞ 5 pðxÞ

Nanofluids

(2.6)

2.2 Weighted residual method for nanofluid modeling

53

~ a linear combination of basic where u is estimated by a function u, functions chosen from a linearly independent set, which means uDu~ 5

n X

ci ϕi

(2.7)

i51

Now, by substituting into the differential operator, D, an error or residual will occur, so the result of the operations generally is not p(x). The residual function can be calculated by: ~ RðxÞ 5 DðuðxÞÞ 2 pðxÞ 6¼ 0

(2.8)

In WRMs it is important to force the residual to zero in some average sense over the domain. We have: ð RðxÞWi ðxÞ 5 0 i 5 1; 2; . . .; n (2.9) X

where Wi, the number of weight functions, is accurately equal to the ~ If the continuous summation of number of unknown constants ci in u. all the squared residuals be minimized, the LSM as a WRM is applied. In other words, a minimum of ð ð S 5 RðxÞRðxÞdx 5 R2 ðxÞdx (2.10) X

X

In the next step, to achieve a minimum of this scalar function, the derivatives of S (with respect to all the unknown parameters) must be zero, that is, ð @S @R 5 2 RðxÞ dx 5 0 (2.11) @ci @ci X Comparing Eq. (2.11) with Eq. (2.9), the weight functions are: Wi 5 2

@R @ci

(2.12)

Since the “2” coefficient can be eliminated, it cancels out from the equation. Thus the weight functions for the LSM are just the derivatives of the residual with respect to the unknown constants, that is [1], Wi 5

@R @ci

(2.13)

In the following, three cases of this method for NF analysis are introduced.

Nanofluids

54

2. Mathematical analysis of nanofluids

FIGURE 2.1 Schematic of the problem [4].

2.2.1 Case 1: Heat transfer and nanofluid flow through circular concentric heat pipes As shown in Fig. 2.1, a circular concentric heat pipe that conveys waterbased NFs is considered. The flow is hydrodynamically and thermally fully developed, the r-axis is considered normal to the pipe walls, and the x-axis is aligned horizontally. ri and ro are the inner and outer pipe radii, respectively, and the walls of the pipe are permitted to carry different and opposite slip velocities. The assumed boundary conditions for the walls are a constant heat flux, q, for the outer wall with the inner wall insulated, or the outer wall at adiabatic wall condition with the inner wall considered as a constant wall heat flux, q. A constant pressure gradient with respect to the x-coordinate is the main cause of motion in this case. It is assumed that all nanoparticles (Ag, Cu, Cuo, Al2O3, and TiO2) are dispersed equally and the NF layer is developed in accordance with a single-phase model. The governing equations of single-phase modeling are [4]:   1d du rμnf 5 px r dr dr     @T 1d @T 5 ρcp nf u rknf @x r dr @r du du at r 5 ri ; u 5 2 λ2 at r 5 ro dr dr   @T @T 5 0 or knf 5 q at r 5 ri @r @r   @T @T 5 q or 5 0 at r 5 r knf @r @r u 5 λ1

Nanofluids

(2.14)

55

2.2 Weighted residual method for nanofluid modeling

where u is the velocity in the r direction, T is the local temperature, and p is the pressure. λ1 and λ2 are the slip velocity factors, where the subscript nf stands for the NF property. The NF properties appearing in Eq. (2.14) are selected as [4]: μnf 5 ð12ϕÞ22:5 μf ρnf 5 ð1 2 ϕÞρf 1 ϕρs   ks 1 2kf 2 2ϕ kf 2 ks   knf 5 kf ks 1 2kf 1 ϕ kf 2 ks

(2.15)

where ϕ is the solid volume fraction or concentration; ρf and ρs are the densities of the pure fluid and nanoparticle; and kf and ks are the thermal conductivities of the base fluid and nanoparticle, respectively. It should be noted that the effective thermal conductivity of the NF knf approximated above is based on the MaxwellGarnett model introduced in Chapter 1, Introduction to nanofluids. Furthermore, the velocity slip given in Eq. (2.14) is due to both the base Ð r fluid and the nanoparticle. By defining the mean velocity as (um 5 rio rudr) and integrating the energy equation (Eq. 2.14) a constant mean temperature gradient will appear [5] and by presenting the following transformation: u5

 1  2px r2o U; μnf

T 5 Tw 2

r0 q θ; knf

η5

r ; ro

R5

ri ; ro

λ1 5

λ1 ; ro

λ2 5

λ2 ro

(2.16) Eq. (2.14) can be changed to   1 d dU dUðRÞ dUð1Þ ; Uð1Þ 5 2 λ2 η 5 2 1; UðRÞ 5 λ1 η dη dη dη dη   1 d dθ U dθðβÞ 521 ; θðβÞ 5 0; η 5α η dη dη Um dη

(2.17)

It should be noted that the energy equation in Eq. (2.17) is combined into a single form so that for the constant outer heat flux α 5 2β 5 21 and for the constant inner heat flux α 5 β 5 R, respectively. The local Nusselt number, Nu, for both thermal conditions is calculated by: Nu 5

knf 2ð1 2 RÞ 2ð1 2 RÞ 5 kf ðTw 2 Tb Þ θb kf

(2.18)

where Tb is the bulk mean temperature with the following dimensionless form: Ð1 ηUθdη θb 5 ÐR1 (2.19) R ηUdη Nanofluids

56

2. Mathematical analysis of nanofluids

To obtain the relative effects of velocity slip λ on the rate of heat transfer, Nur is introduced:  knf θb λ50 Nur 5 (2.20) kf θ b which is normalized by the Nusselt number (Eq. 2.18) at the no slip conditions and λ 5 0. Additionally, the obtained velocity and temperature fields depend on the nanoparticle concentration, so u* is defined as follows for analyzing the results: u 5

ð12ϕÞ5=2 U 2Um

(2.21)

Regarding boundary conditions (BCs) and Eq. (2.17), the following expressions for the velocity and temperature trial functions when λ1 5 λ2 5 0 are assumed: U ðηÞ5 a0 ðη2 RÞðη 2 1Þ1a1 ðη2RÞ2 ðη2 1Þ1a2 ðη 2RÞðη21Þ2 1a3 ðη2RÞ2 ðη21Þ2 1a4 ðη2RÞ3 ðη21Þ2 1a5 ðη2RÞ2 ðη21Þ3 (2.22) θðηÞ52 ðη 2β Þ 1a6 ηðη2β Þ2 1 a7 ηðη2β Þ3 1a8 ηðη2β Þ4 1 a9 ηðη2β Þ5 1a10 ηðη2β Þ6

(2.23)

By computing the values of ai, i 5 0, 1, . . ., 10, the approximate LSM solution of the problem can be obtained. For example, when λ1 5 λ2 5 0; R 5 0:5; α 5 2 1; β 5 1 the unknown coefficients can be found and Eq. (2.22) will be: U ðηÞ520:30209ðη2RÞðη21Þ20:31379ðη2RÞ2 ðη21Þ10:55978ðη2RÞðη21Þ2 20:90590ðη2RÞ2 ðη21Þ2 11:43708ðη2RÞ3 ðη21Þ2 21:18040ðη2RÞ2 ðη21Þ3 (2.24) These analytical results were compared with the numerical solution of the problem obtained by Maple 15.0 as presented in Table 2.2. As seen here, the LSM results are excellent with low percentage error (averagely 0.03%).

2.2.2 Case 2.A: Condensation of nanofluids Consider a disk rotating in its own plane with angular velocity Ω as shown in Fig. 2.2A. The angle between the horizontal axis and disk is β. An NF film with h thickness is created by spraying, with the W velocity on the disk. It is assumed that the disk radius is large enough to ignore

Nanofluids

57

2.2 Weighted residual method for nanofluid modeling

TABLE 2.2 Comparison of LSM results with the numerical solution for λ1 5 λ2 5 0; R 5 0:5; α 5 2 1; β 5 1 [4]. U 0 ðηÞ

U ðηÞ

η Numerical

LSM (Eq. 2.24)

Error (%)

Numerical

LSM

Error (%)

0.50

0.0

0.0

0.0

0.291010

0.290994

0.00549

0.55

0.012656

0.012664

0.06321

0.216827

0.216810

0.00784

0.60

0.021818

0.021831

0.05958

0.150842

0.150822

0.01325

0.65

0.027845

0.027853

0.02873

0.091126

0.091103

0.02524

0.70

0.031075

0.031015

0.19308

0.036457

0.036412

0.12343

0.75

0.031555

0.031548

0.02218

2 0.014365

2 0.014351

0.09745

0.80

0.029638

0.029632

0.02024

2 0.061864

2 0.061832

0.05172

0.85

0.025412

0.025411

0.00393

2 0.106751

2 0.106717

0.03185

0.90

0.018999

0.019002

0.01579

2 0.149437

2 0.149401

0.02409

0.95

0.010499

0.010503

0.03810

2 0.190257

2 0.190210

0.02470

1

0.0

0.0

0.0

2 0.229458

2 0.229409

0.02135

LSM, Least squares method.

FIGURE 2.2 (A) Schematic of nanofluid condensation and (B) schematic of the problem [6].

the end effects. Also, vapor shear occurs at the interface of vapor and fluid are usually unimportant. The gravitational acceleration, g, acts in the downward direction on the fluid layer. The boundary conditions are the temperature on the disk, Tw; the temperature on the film surface, T0; and the ambient pressure on the film surface, p0 . It can be considered that the pressure is only a function of z. The NF is a two-component mixture with the following assumptions: Incompressible, no-chemical reaction, negligible radiative heat transfer, nanosolid-particles, and the base fluid are in thermal

Nanofluids

58

2. Mathematical analysis of nanofluids

equilibrium and no slip occurs between them. The thermo physical properties of the base fluid and nanoparticles can be found in Table 2.1 or Table 1.1. Neglecting viscous dissipation, the continuity, momentum, and energy equations for steady state condition are assumed as follows [6]: @u @v @w 1 1 50 @x @y @z   @u @u @u μnf @2 u @2 u @2 u 1v 1w 5 u 1 1 1 g sinβ @x @y @z @y2 @z2 ρnf @x2   @v @v @v μnf @2 v @2 v @2 v 1 1 u 1v 1w 5 @x @y @z @y2 @z2 ρnf @x2   @w @w @w μnf @2 w @2 w @2 w pz 1v 1w 5 1 2 1 2 2 g cosβ 2 u 2 @x @y @z @y @z ρnf @x ρnf  2  knf @T @T @T @ T @2 T @2 T  1v 1w 5 1 1 u @x @y @z @y2 @z2 ρCp nf @x2

(2.25) (2.26)

(2.27)

(2.28)

(2.29)

where u, v, and w indicate the velocity components in the x, y, and z directions, respectively:   The effective density ðρnf Þ, the effective heat  capacity ρCp nf of the NF, and the effective heat capacity ρCp nf of the NF are defined as: ρnf 5 ð1 2 φÞ ρf 1 φ ρs ; μnf 5

μf ð12φÞ2:5

; ðρ Cp Þnf 5 ð1 2 φÞ ðρ Cp Þf 1 φ ðρ Cp Þs (2.30)

In this case, as well as the last case, the effective thermal conductivity of the NF is approximated by the MaxwellGarnetts model as: knf ks 1 2kf 2 2φðkf 2 ks Þ 5 ks 1 2kf 1 φðkf 2 ks Þ kf

(2.31)

The boundary conditions with these assumptions (zero slip on the disk and zero shear stress on the film surface) are: u 5 2 Ωy; v 5 Ωx; w 5 0; T 5 Tw at z 5 0 uz 5 0; vz 5 0; w 5 2 W; T 5 T0 ; p 5 p0 ; at z 5 h

Nanofluids

(2.32)

2.2 Weighted residual method for nanofluid modeling

59

For this problem, Wang introduced the following transform [7]: u 5 2 Ω y gðηÞ 1 Ω x f 0 ðηÞ 1 g kðηÞ sin v 5 Ω x gðηÞ 1 Ω y f 0 ðηÞ 1 g sðηÞ sin pffiffiffiffiffiffiffiffiffiffiffi w 5 2 2 Ω υnf fðηÞ T 5 ðT0 2 Tw Þ θ ðηÞ 1 Tw

β Ω0

β Ω0

(2.33)

and η was introduced as follows: sffiffiffiffiffiffi Ω η5z υnf

(2.34)

Continuity (Eq. 2.25) is automatically satisfied, and Eqs. (2.26) and (2.27) can be written as follows: fw 2 ðf 0 Þ2 1 g2 1 2f fv 5 0 0

0

(2.35)

gv 2 2g f 1 2f g 5 0

(2.36)

kv 2 kf 0 1 sg 1 2f k0 1 1 5 0

(2.37)

sv 2 kg 2 sf 0 1 2f s0 5 0

(2.38)

If the temperature is only a function of the z distance, Eq. (2.29) becomes θv 1 2 Pr

A2 A3 0 f θ 50 A1 A4

  ρCp nf knf  ; A4 5 A1 5 ; A2 5 ; A3 5  ρf μf kf ρCp f ρnf

μnf

(2.39)

υ

where Pr 5 αff is the Prandtl number of the base fluid. The boundary conditions for Eqs. (2.35)(2.39) are: fð0Þ 5 0; gð0Þ 5 1; kð0Þ 5 0; sð0Þ 5 0; θð0Þ 5 0;

f 0 ð0Þ 5 0;

fvðδÞ 5 0

0

g ðδÞ 5 0; k0 ðδÞ 5 0; 0

s ðδÞ 5 0; θðδÞ 5 1;

Nanofluids

(2.40)

60

2. Mathematical analysis of nanofluids

and δ is the constant normalized thickness as: sffiffiffiffiffiffi Ω δ5h υnf

(2.41)

which is known through the condensation or spraying velocity by: W fðδÞ 5 pffiffiffiffiffiffiffiffiffiffiffi 5 α 2 Ω υnf

(2.42)

In this case the nondimensional Nusselt number is obtained as:   knf @T @z w 5 A4 δ θ0 ð0Þ (2.43) Nu 5 k f T0 2 Tw

2.2.3 Case 2.B: Magnetohydrodynamic flow over porous medium In this case, consider the three-dimensional (3D), incompressible, and steady flow of an electrically conducting viscous liquid between two parallel plates at y 5 6 h (see Fig. 2.2B). Both the plates and the fluid rotate uniformly with a constant angular velocity in direction y. The upper plate at y 51 h is fixed and the flow is produced due to shrinking the lower porous plate at y 5 2 h. The governing equations of the problem are [6]: @u @v 1 50 @x @y  2 @u @u 1 @p @ u @2 u σB20 1v 1 2Ωw 5 2 1ν u u 1 2 @x @y ρ @x @x2 @y2 ρ  2 @u 1 @p @ v @2 v 52 1ν u 1 @y ρ @y @x2 @y2  2 @w @w @ w @2 w σB2 1v 2 2Ωu 5 ν 1 2 0w u 2 2 @x @y @x @y ρ

(2.44) (2.45) (2.46) (2.47)

Subject to the boundary conditions: u 5 2ax; u 5 0;

v 5 2V; v 5 0;

w50

w50

at y 5 2h

at y 51 h

(2.48)

where u, v, and w are the velocity components in the x, y, and z directions, respectively; ρ is the density; σ is the electrical conductivity; ν is

Nanofluids

61

2.2 Weighted residual method for nanofluid modeling

the kinematic viscosity; p* is the pressure term; B0 is the magnetic induction; and V is the suction velocity. By introducing the following parameters: y η5 ; h

u 5 2axf 0 ðηÞ;

v 5 ahf ðηÞ;

w 5 axgðηÞ

(2.49)

The conservation law is satisfied automatically. Eqs. (2.44)(2.47) can be written as follows:   (2.50) f IV 2 M2 fv 2 2Kp2 g0 2 Re f 0 fv 2 ffw 5 0   gv 2 M2 g 1 2Kp2 f 0 2 Re f 0 g 2 fg0 5 0 (2.51) where primes express the differentiation with respect to η and the following parameters are used for simplifying: λ52

V ; ah

Re 5

ah2 ; ν

M2 5

σB20 h2 ; ρν

K2 5

Ωh2 ν

(2.52)

The transformed boundary conditions are: f ð 21Þ 5 λ;

f 0 ð 21Þ 5 2 1;

f ð1Þ 5 0;

0

f ð1Þ 5 0;

gð 21Þ 5 0 gð 1Þ 5 0

(2.53)

2.2.3.1 Solution of condensation of nanofluids Now we want to apply LSM as a WRM to this problem. Because trial functions must satisfy the boundary conditions in Eq. (2.40), they can be considered as: 8 ! 3 2 > η δη > > 2 1? > fðηÞ 5 c1 η2 ðη2δÞ3 1 c2 > > 6 2 > > > > ! > > 2 > η > 2 > 2 δη 1 ? gðηÞ 5 1 1 c3 ηðη2δÞ 1 c4 > > > 2 > > > > ! > < η2 2 (2.54) kðηÞ 5 c5 ηðη2δÞ 1 c6 2 δη 1 ? > 2 > > > > > ! > > > > η2 2 > > 2 δη 1 ? sðηÞ 5 c7 ηðη2δÞ 1 c8 > > 2 > > > > > > η 2 > > > θðηÞ 5 1 c9 ηðη 2 δÞ 1 c10 ηðη2δÞ 1 ? : δ

Nanofluids

62

2. Mathematical analysis of nanofluids

It must be noted that every trial function that satisfies the boundary condition of the problem can be examined and its accuracy can be improved by increasing the number of its statements. In this case, five coupled equations (Eqs. 2.352.39) are presented, so five residual functions and ten unknown coefficients (c1c10) are found. By substituting the residual functions, R1(c1c10, η), R2(c1c10, η), R3(c1c10, η), R4(c1c10, η), and R5(c1c10, η), into Eq. (2.11), a set of equations (including 10 equations) are found and c1c10 coefficients can be determined by solving this system of equations. For example, using LSM for a Cuwater NF with φ 5 0.04, Pr 5 6.2, and δ 5 0.5 the following equations are obtained: 8 ! > η3 η2 > 3 2 > > 2 fðηÞ 5 2 0:008811728449η ðη20:5Þ 2 0:9607426021 > > 6 4 > > > > ! > > > η2 η > 2 > > 2 gðηÞ 5 1 1 0:07794439347ηðη20:5Þ 1 0:1969431465 > > 2 2 > > > < ! (2.55) η2 η > 2 kðηÞ 5 0:008808870176ηðη20:5Þ2 2 0:9838801251 > > 2 2 > > > > > ! > > > 2 > η η > > sðηÞ 5 0:03931075727ηðη20:5Þ2 1 0:09985540140 2 > > > 2 2 > > > > : θðηÞ 5 2η 2 0:1982408923ηðη 2 0:5Þ 2 0:2492463454ηðη20:5Þ2 In the same manner for CuWater NF when δ 5 1 the results will be [6]: 8 ! 3 2 > η η > > > fðηÞ 5 2 0:01914194896η2 ðη21Þ3 2 0:6564549829 2 > > 6 2 > > > > ! > > 2 > η > 2 > > 2η gðηÞ 5 1 1 0:08162945698ηðη2δÞ 1 0:4634992471 > > 2 > > > < ! (2.56) η2 2 > kðηÞ 5 0:03926545013ηðη21Þ 2 0:8513453189 2η > > 2 > > > > > ! > > > 2 > η > 2 > sðηÞ 5 0:04388591688ηðη21Þ 1 0:2695506788 2η > > > 2 > > > > : θðηÞ 5 η 2 0:4379485922ηðη 2 1Þ 2 0:2403810627ηðη21Þ2

Nanofluids

2.2 Weighted residual method for nanofluid modeling

63

FIGURE 2.3 Comparison of LSM and NUM for Cuwater nanofluid with φ 5 0.04 and δ 5 0.5, 0.75, 1 [6]. LSM, Least squares method.

Fig. 2.3 shows a comparison between the obtained results and the numerical results. This excellent accuracy gives high confidence to users about the validity of this method and is an excellent method for NF

Nanofluids

64

2. Mathematical analysis of nanofluids

treatment modeling. The effect of normalized thickness on the velocity and temperature profiles is shown in Fig. 2.3. Increasing normalized thickness leads to an increase in f; f 0 and a decrease in g; θ. The effect of normalized thickness on k and s has a similar effect on f0 and g, respectively. 2.2.3.2 Solution of magnetohydrodynamic flow between parallel plates To apply the LSM in this case, trial functions are considered as follows because we must satisfy the boundary conditions [6]:       λ 1 λ  2 λ21  2 2 f ðxÞ 5 λ 2 ðx 1 1Þ 1 x 21 1 x 2 1 ð x 1 1Þ 2 2 4 4  2  2  3 1 c1 x2 21 1 c2 x2 21 ðx 1 1Þ 1 c3 x2 21 









gðxÞ 5 x2 2 1 1 c4 x2 2 1 ðx 1 1Þ 1 c5 x2 21

2



2

(2.57)

1 c6 x2 21 ðx 1 1Þ (2.58)

Once applying these trial functions to Eqs. (2.50) and (2.51) to obtain the residual functions, the unknown parameters of the above equations can be determined. For example, by considering the following parameters: M 5 0:5; Kp 5 0:5; Re 5 0:2; λ 5 1

(2.59)

The LSM solution finds:  2  2  3 1 1 1 f ðxÞ5 2 x1 x2 10:0024 x221 10:0054 x221 ðx 11Þ2 0:0009 x221 4 2 4 (2.60)       2 2 gðxÞ 5x2 2 12 0:2694 x2 21 ðx1 1Þ1 0:0298 x221 2 0:0052 x221 ðx 11Þ (2.61) Fig. 2.4 shows a comparison of the results obtained with the HAM solution to show the efficiency and accuracy of this method.

2.2.4 Case 3: Peristaltic nanofluid flow in a divergent asymmetric wavy-wall channel In this case, the peristaltic and incompressible NF flow through a 2D asymmetric wavy divergent channel as shown in Fig. 2.5 is

Nanofluids

65

2.2 Weighted residual method for nanofluid modeling

0.4 Present result HAM

f ′ (η)

0

–0.4

λ = 0, 0.5, 1, 1.5

–0.8

–1

–0.8

–0.6

–0.4

–0.2

0

η

0.2

0.4

0.6

0.8

1

FIGURE 2.4

Comparison of the results of the proposed hybrid method and homotopy analysis method when M 5 0:5; Kp 5 0:5, and Re 5 0.2 [6].

investigated. The peristaltic waves move along the x-axis with speed c, while the y-axis is perpendicular to it. A uniform magnetic field in the y-direction, B 5 (0, B0, 0), is applied to the channel. The lower and upper wavy walls are designed with the following functions [9]:  2π ðX 2 ctÞ 1 φ H 1 ðX; tÞ 5 2 d 2 k0 X 2 b1 sin λ   (2.62) 2π ðX 2 ctÞ H 2 ðX; tÞ 5 d 1 k0 X 1 b2 sin λ where b1 and b2 are the wave amplitude moving along H 1 ðX; tÞ and H 2 ðX; tÞ, respectively; φ is the phase difference between the upper and lower walls, which varies between 0 # φ # π; d is the half-width of channel; and k0 ({1) is the nonuniform parameter of the tapered asymmetric channel. The following equation is considered between defined parameters: b21 1 b22 1 2b1 b2 d cos φ # ð2dÞ2

(2.63)

By considering an extra stress tensor in pseudoplastic fluid as presented in Ref. [8], the governing equations are as follows: @U @V 50 1 @X @Y

Nanofluids

(2.64)

66

2. Mathematical analysis of nanofluids

FIGURE 2.5 Schematic of the sinusoidal wall channel [8].



   @ @ @ @P @  @  1U ρf U52 SX X 1 SX Y 2 σ0 B20 U (2.65) 1V 1 @t @X @X @X @Y @Y     @ @ @ @P @  @  ρf 1U 1 V52 SY X 1 SY Y 1V (2.66) @t @X @Y @Y @X @Y      2 @ @ @ @ T @2 T @C @T @C @T 1U T5k 1 D 1V 1 1 ð ρc Þ ðρCÞf p B 2 2 @t @X @X @X @Y @Y @Y @X @Y !  2  2  DT @T @T DB KT @2 C @2 C 1 ðρcÞp 1 1 1 2 2 Tm cs @X @Y @X @Y (2.67)  2      @ @ @ @C @2 C DB KT DT @ 2 T @2 T 1U C 5 DB 1 1 1 1V 1 2 2 2 Tm Tm @X2 @t @X @Y @X @Y @Y   2 k1 C 2 C 0 (2.68) where SX X ; SX Y ; SY X ; and SY Y denote the components of the extra stress tensor; σ0 is the fluid electrical conductivity; C is the nanoparticle concentration; f and p represent the fluid and nanoparticles properties, respectively; DB is the Brownian diffusion coefficient; DT is the

Nanofluids

67

2.2 Weighted residual method for nanofluid modeling

themophoretic diffusion coefficient; cs is the concentration susceptibility; KT is the thermal diffusion ratio; and k1 is the chemical reaction parameter. Additionally, the extra stress components must satisfy some related partial differential equations as presented in Ref. [8]. In this case, the boundary conditions are: (

U 5 0; T 5 T0 ; C 5 C0 ;

at Y 5 H 1

U 5 0; T 5 T1 ; C 5 C1 ;

at Y 5 H 2

(2.69)

Hayat et al. [8] showed that by defining the transformation in the shape of: x 5 X 2 ct;

y 5 Y;

uðx; yÞ 5 UðX; Y; tÞ 2 c

vðx; yÞ 5 VðX; Y; tÞ;

(2.70)

pðx; yÞ 5 PðX; Y; tÞ

And considering the following nondimensional parameters: x5

x ; λ

h2 5 Pr 5

y5

H2 ; d μc0f

;

y ; d1

a5

Sc 5

υ ; DB

b1 ; d

σ5

k sffiffiffiffi0 σ M5 dB0 ; μ

u5

t5

ct λ

b5

u5 b2 ; d

u ; c p5

v v5 ; c d2 p ; cλμ

δ5

θ5

Nb 5

τDB ðC1 2 C0 Þ ; υ

k 1 d2 ; υ

Sr 5

DB KT ðT1 2 T0 Þ ; υTm ðC1 2 C0 Þ

@ψ ; @y

v52δ

h1 5

T 2 T0 ; T1 2 T0

C 2 C0 ; C1 2 C0 γ5

d ; λ

Nt 5

H1 ; d

Re 5

ρf cd μ

τDT ðT1 2 T0 Þ Tm υ

Du 5

DB KT ðC1 2 C0 Þ μcs c0f ðT1 2 T0 Þ

@ψ @x (2.71)

where δ is the wave number; Re the Reynolds number; Pr is the Prandtl number; M is the Hartman number; Nb is the Brownian motion parameter; Nt is the thermophoresis parameter; Du is the Dufour number; and Sr and Sc are the Soret and Schmidt numbers, respectively. It must be pointed out that some assumptions such as low Reynolds number and eliminating the pressure term by extra stress components are considered for this case and the governing equations can be transferred to the final set of the following equations [8]:

Nanofluids

68

2. Mathematical analysis of nanofluids 2 @2   2@ ψ s 50 2 M xy @y2 @y2

or

@2 @y2

ψyy

!

1 1 ζψ2yy

2 M2

@2 ψ 50 @y2

   2 @2 θ @σ @θ @θ @2 σ 1 N Pr Pr 1 Pr Da 50 1 N t b @y2 @y @y @y @y2   @2 σ Nt @2 θ 1 Sc Sr 1 2 γScσ 5 0 @y2 Nb @y2 where sxy is defined based on the stream function: h i21 sxy 5 ψyy 11ζψ2yy Thus Eq. (2.72) will be: ψyy @2 2 @y 1 1 ζψ2yy

! 2 M2

@2 ψ 50 @y2

(2.72)

(2.73) (2.74)

(2.75)

(2.76)

With the boundary condition of: 8 F @ψ > > 5 0; θ 5 0; σ 5 0 at y 5 h1 ψ52 ; > > 2 @y < F @ψ > > > > : ψ 5 2 ; @y 5 0; θ 5 1; σ 5 1 at y 5 h2

(2.77)

where ζ 5 λ21 1 μ21 F 5 Θ 1 a sin½2πx 1 φ 1 b sin½2πx

(2.78)

In the first step of the LSM solution, the trial functions must satisfy the boundary conditions and are considered as follows:       F F  F ψ y 52 1 y 2 h1 1 y 2 h 1 y 2 h2 2 2 h2 2 h1 ðh2 2h1 Þ  2    2  2 2F y2h1 y 2 h2 1 c1 y2h1 y2h2 2 3 ðh2 2h1 Þ  3  2  3  3 1 c2 y2h1 y2h2 1 c3 y2h1 y2h2   θ y 5

     2   1  y 2 h1 1 c4 y 2 h1 y 2 h2 1 c5 y2h1 y 2 h2 h2 2 h1  2  2  3  2 1 c6 y2h1 y2h2 1 c7 y2h1 y2h2

Nanofluids

(2.79)

(2.80)

69

2.2 Weighted residual method for nanofluid modeling

  σ y 5

     2   1  y 2 h1 1 c8 y 2 h1 y 2 h2 1 c9 y2h1 y 2 h2 h2 2 h1  2  2  3  2 1 c10 y2h1 y2h2 1 c11 y2h1 y2h2

(2.81)

By minimization of the residuals, the unknown parameters ci can be calculated. In this case: when a 5 0:5; b 5 0:7; φ 5 π=2; Θ 5 1:5; Pr 5 2; γ 5 Sr 5 Du 5 0:2; Nt 5 0:1; M 5 1; x 5 0:65, the following residual functions can be obtained: R1 50:9519y2204:68c3 149:71c2 133:49c1 1148:95c2 y1445:44c3 y2 211:39c3 y 212:c1 y2 10:307c1 y220c2 y3 217:72c2 y2 230c3 y4 11:535c3 y3 20:012 (2.82) R2 51:99c4 12:53c5 19:47c6 213:13c7 10:04210:0001c24 10:024c4 c5 20:0012c4 c6 20:059c4 c7 20:02c24 y11:13c25 y2 22:9c25 y10:002c26 10:28c6 c7 10:77c26 y3 118:02c26 y2 20:46c26 y16:77c27119:84c27 y10:8c24 y2 11:8c25 y4 13:6c25 y313:2c26 y6 20:24c26 y5 215:18c26 y4 15c27 y8111:81c27 y7 221:97c27 y6 262:73c27 y5 20:02c11 c4 22:80c11 c5 10:14c11 c6 16:77c11 c7 20:0006c10 c4 20:05c10 c5 10:002c10 c6 10:14c10 c710:012c9 c4 11:16c9 c5 20:058c9 c5 22:80c9 c710:0001c8 1? (2.83) R3 52:5c4 20:084y13:78c5 211:96c6 217:97c7 20:26c8 y2 20:26c9 y2 20:26c10 y4 20:26c11 y520:38c11 y4121:25c11 y3119:57c11 y2230:46c11 y17:56c5 y115:12c6 y2 20:38c6 y125:2c7 y3 122:33c7 y2 236:48c7 y10:006c8 y20:39c9 y2 16:62c9 y 10:013c10 y3113:23c10 y220:33c10 y216:5c11210:96c1013:94c912:61c820:128 (2.84) By solving the values of ci , the following solution can be found:      ψ y 5 2 0:0096 1 0:7533y 1 0:2444 y 1 1:5280 y 2 1:5536   2  2  2  2 0:1587 y11:5280 y 2 1:5536 1 0:0121 y11:5280 y21:5536  3  2 2 0:0078 y11:5280 y21:5536 (2.85)      θ y 5 0:4958 1 0:3245y 2 0:0621 y 1 1:5280 y 2 1:5536   2  2  2  2 0:0066 y11:5280 y 2 1:5536 2 0:0015 y11:5280 y21:5536  3  2 2 0:00004 y11:5280 y21:5536 (2.86) Nanofluids

70

2. Mathematical analysis of nanofluids

     σ y 5 0:4958 1 0:3245y 1 0:0977 y 1 1:5280 y 2 1:5536  2    2  2 1 0:0215 y11:5280 y 2 1:5536 1 0:0040 y11:5280 y21:5536  3  2 1 0:0003 y11:5280 y21:5536 (2.87) Table 2.3 compares the LSM results with the numerical solution of the problem. As can be seen, the accuracy of the approximate solution is acceptable, so the LSM can be used for problems like these to obtain accurate solutions. Also, as seen in Table 2.3, the LSM calculated errors are in 10E 2 4 and 10E 2 5 order. The effect of the geometry parameter (a) using the LSM on the results is presented in Fig. 2.6.

2.3 Differential transformation method for nanofluid modeling DTM is another efficient technique for the mathematical analysis of NFs. To better understand this method, suppose that x(t) is an analytic function in domain D, and t 5 ti represents any point in the domain [1]. The function x(t) is then represented by one power series whose center is located at ti. The Taylor series expansion function of x(t) is in the form of:  N X ðt2ti Þk dk xðtÞ xðtÞ 5 ’tAD (2.88) dtk t5ti k! k50 The Maclaurin series of x(t) can be found by taking ti 5 0 in Eq. (2.88) as: xðtÞ 5

N k k X t d xðtÞ k50

k!

dtk

’tAD

(2.89)

t50

So, the differential transformation of the function x(t) is defined as follows:  N X H k dk xðtÞ XðkÞ 5 (2.90) k! dtk t50 k50 where x(t) is the original function and X(k) represents the transformed function. The differential spectrum of X(k) is restricted within the interval tA½0; H. H really depends on the time steps, which for classical DTM is considered unity, but for multistep DTM or MsDTM it is equal to the time step value. When H-N, common DTM has not

Nanofluids

TABLE 2.3 Comparison of LSM results with numerical solution of problem [9].   ψ y

y

  θ y

  σ y

LSM

Num

Error

LSM

Num

Error

LSM

Num

Error

21.528

2 1.16084

2 1.16084

0

0

0

0

0

0

0

21.2

2 1.07975

2 1.08022

4.7E 2 4

0.16322

0.16322

3.0E 2 6

0.01519

0.01522

3.2E 2 5

20.9

2 0.89384

2 0.89520

1.36E 2 3

0.30217

0.30217

4.2E 2 7

0.04249

0.04254

4.8E 2 5

20.6

2 0.63678

2 0.63854

1.7E 2 3

0.43122

0.43121

1.2E 2 5

0.08333

0.08336

2.8E 2 5

20.3

2 0.3359

2 0.33714

1.2E 2 3

0.55009

0.55006

2.8E 2 5

0.13901

0.13900

2.9E 2 6

0

2 0.01389

2 0.01394

5.0E 2 5

0.65819

0.65815

3.9E 2 5

0.21151

0.21150

1.4E 2 5

0.3

0.308965

0.310121

1.1E223

0.75464

0.75460

4.0E 2 5

0.30362

0.30362

4.8E 2 6

0.6

0.612478

0.61423

1.7E 2 3

0.83829

0.83827

2.4E 2 5

0.41899

0.41902

3.6E 2 5

0.9

0.874265

0.87569

1.4E 2 3

0.90764

0.90764

3.8E 2 6

0.56224

0.56229

5.3E 2 5

1.2

1.067464

1.06803

5.6E 2 4

0.96088

0.96089

7.0E 2 6

0.73906

0.73909

3.3E 2 5

1.553

1.160836

1.15839

0

1

1

0

1

1

0

LSM, Least squares method.

72

2. Mathematical analysis of nanofluids

FIGURE 2.6 Effect of geometry parameter a on the velocity, temperature, and nanoparticle fraction profiles for b 5 0:7; φ 5 π=2; Θ 5 1:5; Pr 5 0:6; γ 5 Du 5 0:2; Sr 5 0:2; Nt 5 0:1; M 5 1 [8].

correct results and Pade approximation should be applied on the final results. The differential inverse transform of X(k) is defined as: N  k X t XðkÞ (2.91) xðtÞ 5 H k50 It is obvious that the concept of DTM is based on the Taylor series expansion. The values of X(k) function at values of argument k are referred to as discrete, that is, X(0) is known as the zero discrete, X(1) as the first discrete, and so on. The more discrete the more precise, and it is possible to restore the unknown function. The x(t) function consists of the T-function X(k), and its value is given by the sum of the T-function with (t/H)k as its coefficient. In actual applications, with the right choice of constant H, the larger values of argument k make the discrete of the

Nanofluids

2.3 Differential transformation method for nanofluid modeling

TABLE 2.4 [10].

The operations for the one-dimensional differential transform method

Original function   wðxÞ 5 uðx 6 vðx

Transformed function   W ðkÞ 5 Uðk 6 Vðk

 wðxÞ 5 λuðx

W ðkÞ 5 λU ðkÞ; λ is constant

wðxÞ 5

duðxÞ dx

dr uðxÞ dxr   wðxÞ 5 uðx vðx wðxÞ 5

wðxÞ 5

73

duðxÞ dvðxÞ dx dx

  W ðkÞ 5 ðk 1 1 Uðk    W ðkÞ 5 ðk 1 1Þðk 1 2 ?ðk 1 r Uðk 1 r W ðkÞ 5 W ðkÞ 5

wðxÞ 5 uðxÞ

dvðxÞ dx

W ðkÞ 5

wðxÞ 5 uðxÞ

d2 vðxÞ dx2

W ðkÞ 5

Pk r50

UðrÞVðk 2 rÞ

Pk

r50 ðr 1 1Þðk 2 r 1 1ÞUðr 1 1ÞVðk 2 r 1 1Þ

Pk

r50 ðk 2 r 1 1ÞUðrÞVðk 2 r 1 1Þ

Pk

r50 ðk 2 r 1 2Þðk 2 r 1 1ÞUðrÞVðk 2 r 1 2Þ

spectrum reduce rapidly. The function x(t) is expressed by a finite series and Eq. (2.91) can be written as: n  k X t XðkÞ (2.92) xðtÞ 5 H k50 where n is the number of statements of the DTM. Generally, by increasing the n, the DTM accuracy can be increased, but this depends on H and the time step’s length. Some important mathematical operations performed by the DTM are given in Table 2.4. The following three examples of NF problems are analyzed using DTM for more clarification.

2.3.1 Case 1: Thermal boundary-layer analysis of nanofluid flow over a stretching flat plate In this case, we investigate the incompressible and laminar boundary-layer flow of an NF past a permeable stretching sheet. Two equal and opposite forces along the x-axis induce flow generation as a result of nonlinear stretching of the sheet. The sheet is stretched with a powerlaw velocity of uw 5 axn in which the origin location is fixed. Furthermore, lateral mass flux (i.e., wall transpiration) is considered in this case. A schematic of the problem is depicted in Fig. 2.7. It should be

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FIGURE 2.7 Nanofluid stretching sheet-flow physical regime [10].

noted that the constant temperature and concentration of the stretching surface (Tw and Cw ) are assumed to be larger than the ambient temperature and concentration (TN and CN ). Also, the isothermal condition is used for the walls. Neglecting buoyancy forces, edge effects, and pressure gradient presence, the governing equations (i.e., mass, momentum, energy, and nanoparticle species conservation) will be [10]: @u @v 1 50 @x @y @u @u @2 u 1v 5ν 2 @x @y @y "   # @T @T @C @T DT @T 2 2 1v 5 αm r T 1 τ D B 1 u @x @y @y @y TN @y u



u

@C @C @2 C DT @2 T 1v 5 DB 2 1 @x @y @y TN @y2

(2.93) (2.94)

(2.95)

(2.96)

where u and v are the velocity components along the x- and y-axis; ν is the NF kinematic viscosity; αm 5 km =ðρcÞf is the NF thermal diffusivity; τ 5 ðρcÞp =ðρcÞf is the ratio between the effective heat capacity of the nanoparticles and heat capacity of the base fluid; DB is the Brownian diffusion coefficient; DT is the thermophoretic diffusion coefficient; ρp and ρf are the density of the nanoparticles and base fluid, respectively; and c is the volumetric volume expansion coefficient. The following boundary conditions are considered:

Nanofluids

2.3 Differential transformation method for nanofluid modeling

at y 5 0: v 5 vwðxÞ ; uw 5 axn ; T 5 Tw ; C 5 Cw at y-N: u 5 v 5 0; T 5 TN ; C 5 CN

75 (2.97)

In Eq. (2.29), vwðxÞ is the variable velocity component in the vertical direction at the stretching surface, and vwðxÞ , 0 represents the suction cases and vwðxÞ . 0 represents the injection. By introducing similarity transformations: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n21 a ð n 1 1Þ 2 0 x ; u 5 axn f ðηÞ η5y 2ν 0 1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n21 0 2 aν ðn 1 1Þ n 2 1A 0 A x @f 1 @ ηf v52 2 n11 θðηÞ 5

(2.98)

T 2 TN C 2 CN ; φðηÞ 5 Tw 2 TN Cw 2 CN

By applying these transformations into the governing Eqs. (2.93) (2.96), the reduced form of the conservation equations for momentum, energy (heat), and species (nanoparticle) concentration can be found [10]:   2n 0 0 (2.99) f v 1 ffv 2 f 2 50 n11 1 2 θv 1 fθ0 1 Nb θ0 φ0 1 Nt ðθ0 Þ 5 0 Pr 1 Nt θv 5 0 φv 1 Lefφ0 1 2 Nb

(2.100) (2.101)

and the transformed boundary conditions will be: η 5 0: f 5 fw ; f 0 5 1; θ 5 1; φ 5 1 0

η-N: f 5 1; θ 5 0; φ 5 0

(2.102) (2.103)

where Pr 5 ν=α is the Prandtl number; Nb 5 ðρcÞp 3 DB ðCw 2 CN Þ=ðρcÞf ν is the Brownian motion parameter; Nt 5 ðρcÞp DT ðTw 2 TN Þ=ðρcÞf νTN is the thermophoresis parameter;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n21 Le 5 ν=DB is the Lewis number; and fw 5 2 vw ðxÞ= aν ðn 1 1Þ=2x 2 is the wall transpiration parameter (suction/injection). Now we try to find the solution of the governing Eqs. (2.99)(2.103) using DTMPade´. By applying the one-dimensional DTM from Table 2.4 to each term of Eqs. (2.99)(2.101), the following transformed equations can be obtained:

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2. Mathematical analysis of nanofluids

f 0 v-ðk 1 3Þðk 1 2Þðk 1 1ÞFðk 1 3Þ k X

0

ff -

(2.104)

ðk 1 2 2 rÞðk 1 1 2 rÞFðrÞFðk 1 2 2 rÞ

(2.105)

ðk 1 1 2 rÞðr 1 1ÞFðr 1 1ÞFðk 1 1 2 rÞ

(2.106)

θv -ðk 1 2Þðk 1 1ÞΘðk 1 2Þ

(2.107)

r50 0

f 2-

k X r50

0

fθ -

k X

ðk 1 1 2 rÞFðrÞΘðk 1 1 2 rÞ

(2.108)

ðk 1 1 2 rÞðr 1 1ÞΘðr 1 1ÞΘðk 1 1 2 rÞ

(2.109)

r50 0

θ 2-

k X r50

0

0

θφ-

k X

ðk 1 1 2 rÞðr 1 1ÞΘðr 1 1ÞΦðk 1 1 2 rÞ

(2.110)

φv -ðk 1 2Þðk 1 1ÞΦðk 1 2Þ

(2.111)

r50

0

fθ -

k X ðk 1 1 2 rÞFðrÞΦðk 1 1 2 rÞ

(2.112)

r50

where FðkÞ; ΘðkÞ; and ΦðkÞ are the transformed ðkÞ; θðkÞ; and φðkÞ, respectively, which are given by: f ðηÞ 5

N X

functions

of

FðkÞηk

(2.113)

ΘðkÞηk

(2.114)

ΦðkÞηk

(2.115)

k50

θðηÞ 5

N X k50

φðηÞ 5

N X k50

Substituting Eqs. (2.104)(2.113) into Eqs. (2.99)(2.101) and using boundary conditions (2.102) and (2.103), we have: ðk 1 1Þðk 1 2Þðk 1 3ÞFðk 1 3Þ 1

k X

FðrÞðk 2 r 1 2Þðk 2 r 1 1ÞFðk 2 r 1 2Þ

r50

2

k 2n X ðr 1 1ÞFðr 1 1Þðk 2 r 1 1ÞFðk 2 r 1 1Þ 5 0 n 1 1 r50

(2.116)

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2.3 Differential transformation method for nanofluid modeling

k X 1 ðk 1 1Þðk 1 2ÞΘðk 1 2Þ 1 FðrÞðk 2 r 1 1ÞΘðk 2 r 1 1Þ Pr r50

1 Nb

k X

ðr 1 1ÞΘðr 1 1Þðk 2 r 1 1ÞΦðk 2 r 1 1Þ

(2.117)

r50

1 Nt

k X

ðr 1 1ÞΘðr 1 1Þðk 2 r 1 1ÞΘðk 2 r 1 1Þ 5 0

r50 k 1 X ðk 1 1Þðk 1 2ÞΦðk 1 2Þ 1 Le FðrÞðk 2 r 1 1ÞΦðk 2 r 1 1Þ 2 r50

(2.118)

Nt ðk 1 1Þðk 1 2ÞΘðk 1 2Þ 5 0 1 Nb Fð0Þ 5 fw ; Fð1Þ 5 1; Fð2Þ 5 α

(2.119)

Θð0Þ 5 1; Θð1Þ 5 β

(2.120)

Φð0Þ 5 1; Φð1Þ 5 γ

(2.121)

Moreover, by substituting Eqs. (2.119)(2.121) into Eqs. (2.116) (2.118) and by a recursive method the values of FðkÞ; ΘðkÞ; and ΦðkÞ can be calculated. Therefore substituting all FðkÞ; ΘðkÞ; and ΦðkÞ into Eqs. (2.113)(2.115), the series solutions will be:   1 2n 2 1 2fw α η3 f ðηÞDfw 1 η 2 αη 1 6 11n    (2.122) 1 8nα 2n 4 . . . 2α 2 2 fw 1 2fw α η 1 1 24 11n 11n   1 θðηÞD1 2 βη 1 Prη2 2 Nt β 2 2 Nb βγ 1 βfw 0 2   β 1 2Nt Prβ 2Nt β 2 2 Nb βγ 1 βfw B   B B 2 Prfw 2Nt β 2 2 Nb βγ 1 βfw B " B B 1 2 B 1 Prη3 B 2 Nb 2 Prγð2 Nt β 2 Nb βγ 1 βfw Þ B 6 B 0 B   1# B 2 B N Pr 2N β 2 N βγ 1 βf Leγf t t w A b w @ 2 β@ 2 Nb 2

1 C C C C C C C ... C1 C C C C C A (2.123)

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2. Mathematical analysis of nanofluids

 ! 1 2 Leγfw Nt Pr 2Nt β 2 2 Nb βγ 1 βfw 2 φðηÞD1 2 γη 1 η 2 Nb 2    1   0 1 Leγfw Nt Pr 2Nt β 2 2 Nb βγ 1 βfw 2 C B 2 2 Le 2γ 1 fw Nb 2 C B C B C B   C B 1 C B 2 Nt Prβ 1 2Nt Prβ 2 Nt β 2 2 Nb βγ 1 βfw C 1 ... B Nb C B C 1 3B   C B 1 ηB 2 C 2 N β 2 N βγ 1 βf 2 Prf w t w 6 B b C C B C B 2 N ð 2 Prγð 2 Nt β 2 2 N βγ 1 βfw Þ b b C B C B 0 1!! C B 2 C B A @ 2 β @Leγfw 2 Nt Prð 2 Nt β 2 Nb βγ 1 βfw A 2 Nb (2.124) As mentioned before, when the boundary conditions are infinite, the analytical solution obtained by the DTM cannot satisfy boundary conditions, so Pade´ approximation must be applied to the final results [1]. Fig. 2.8 shows the accuracy of the DTM compared to the numerical method. As seen in this figure as well as Figs. 2.92.11, which depict f ðηÞ, f 0 ðηÞ, θðηÞ; and φðηÞ, the DTM results cannot be converged in infinite and reach infinite or zero for different profiles, so it is not a very reliable method for these kinds of problems. For accurate results, Pade´ [10,10] approximation is applied as shown in Fig. 2.12. Tables 2.5 and 2.6 compare the results of DTMPade´ with the previous numerical methods presented in the literature. As seen, there is excellent agreement between this approximation and the numerical method.

2.3.2 Case 2: Peristaltic flow of nanofluids in a sinusoidal wall channel As seen in Fig. 2.13, a peristaltic fluid flow through a sinusoidal wavy channel is considered. The geometry of the wavy wall is considered using the sinusoidal function with the parameters defined in Fig. 2.13 [11]: ~ ξ; ~ tÞ ~ ~ 5 a 1 b sin 2π ðξ~ 2 ctÞ hð λ

(2.125)

~ and ξ~ represent time, transverse vibration of the wall, and ~ h, where t, axial coordinate respectively; and a, b, c, and λ also existent the half

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2.3 Differential transformation method for nanofluid modeling

1 DTM−Padé [10,10] DTM Numerical

0.9 0.8 0.7

f(η)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4 η

5

6

7

8

FIGURE 2.8 f ðηÞ by the DTM and the DTMPade´ [10,10] compared to the numerical solution [10]. DTM, Differential transformation method.

1 DTM−Padé [10,10] DTM Numerical

0.9 0.8 0.7

f’(η)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4 η

5

6

7

8

FIGURE 2.9 f 0 ðηÞ by the DTM and the DTMPade´ [10,10] compared to the numerical solution [10]. DTM, Differential transformation method.

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2. Mathematical analysis of nanofluids

1 DTM−Padé [10,10] DTM Numerical

0.9 0.8 0.7

θ (η)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4 η

5

6

7

8

FIGURE 2.10 θðηÞ by the DTM and the DTMPade´ [10,10] compared to the numerical solution [10]. DTM, Differential transformation method.

1 DTM−Padé [10,10] DTM Numerical

0.9 0.8 0.7

φ(η)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4 η

5

6

7

8

FIGURE 2.11 φðηÞ by the DTM and the DTMPade´ [10,10] compared to the numerical solution [10]. DTM, Differential transformation method.

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2.3 Differential transformation method for nanofluid modeling

1 f(η) θ(η) φ(η) df/dη Numerical

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

4 η

3

5

6

7

8

Comparison of f ðηÞ, f 0 ðηÞ, θðηÞ, and φðηÞ obtained by the DTMPade´ [10,10] and numerical solution [10].

FIGURE 2.12

TABLE 2.5 Comparison of the results of the numerical and DTMPade´ solutions, when fw 5 0; Le 5 Pr 5 n 5 2 and Nb 5 Nt 5 0:5 [10]. 0

f ðηÞ η

Numerical solution

DTMPade´

0:0

1:000000

1:000000

1:0

0:348380

0:348380

2:0

0:128123

0:128123

3:0

0:048153

0:048152

4:0

0:018249

0:018248 θðηÞ

0:0

1:000000

1:000000

1:0

0:4716978

0:4754076

2:0

0:1384595

0:1399197

3:0

0:0289441

0:0287842

4:0

0:0049499

0:0049464

DTM, Differential transform method.

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2. Mathematical analysis of nanofluids

TABLE 2.6 Comparison of the results of the DTMPade´ [10,10] and numerical solutions, when fw 5 0; Nt 5 Nb 5 0:5; Pr 5 1; n 5 1:5; and Le 5 2 [10]. 0

f ðηÞ

θðηÞ

φðηÞ

η

DTMPade´

Numerical

DTMPade´

Numerical

DTMPade´

Numerical

0:0

1:000000

1:000000

1:000000

1:000000

1:000000

1:000000

0:5

0:592838

0:592838

0:798988

0:798988

0:843667

0:843667

1:0

0:355768

0:355769

0:600981

0:600981

0:710694

0:710694

1:5

0:215166

0:215169

0:430405

0:430405

0:588966

0:588966

2:0

0:130746

0:130749

0:296400

0:296400

0:475138

0:475138

2:5

0:079650

0:079655

0:197860

0:197860

0:371684

0:371683

3:0

0:048566

0:048572

0:128858

0:128859

0:281979

0:281976

3:5

0:029595

0:029602

0:082282

0:082281

0:207844

0:207828

4:0

0:017992

0:017999

0:051697

0:051687

0:149150

0:149089

DTM, Differential transform method.

FIGURE 2.13 Schematic of the sinusoidal wall channel [11].

width of the channel, amplitude of the wave, wave velocity and wave length, respectively. Using the OberbeckBoussinesq approximation, the governing equation of total mass, momentum, thermal energy, and nanoparticle fraction can be given as follows [11]:



r v50 (2.126)   h n oi @v 1 v rv 5 2 rp 1 μr2 v 1 Fρp 1 ð1 2 FÞ ρf ð1 2 βðT 2 T0 ÞÞ g ρf @t (2.127)



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2.3 Differential transformation method for nanofluid modeling

ðρcÞf

     @T DT 1 v rT 5 kr2 T 1 ðρcÞp DB rφ rT 1 rT rT (2.128) @t T0   @F DT 1 v rF 5 DB r2 φ 1 (2.129) r2 T @t T0









The boundary conditions are: T 5 T0 ; F 5 F0 T 5 T1 ; F 5 F1

at η 5 0 at η 5 h

(2.130)

where the subscripts 0 and 1 represent the centerline and the channel wall properties, respectively. Considering the nanoparticle concentration to be dilute, and selecting a suitable choice for the reference pressure, the momentum equation can be rewritten as:   h i @v 1 v rv 5 2 rp 1 μr2 v 1 ðρp 2 ρf0 ÞðF 2 F0 Þ 1 ð1 2 F0 Þρf0 βðT 2 T0 Þ g ρf @t (2.131)



Now by utilizing standard boundary-layer approximation, the governing equations can be considered as follows [11]: @u~ @v~ 50 1 ~ @~η @ξ

!  @ @ @ @p~ @2 u~ @2 u~ ρf u~ 5 2 1 2 1 u~ 1 v~ 1μ 2 @~η @t~ @~η @ξ~ @ξ~ @ξ~ h i 1 g ð1 2 F0 Þρf0 βðT 2 T0 Þ 2 ðρ 2 ρf0 ÞðF 2 F0 Þ !   @ @ @ @p~ @2 v~ @2 v~ ρf 1 2 1 u~ 1 v~ v~ 5 2 1μ 2 @~η @t~ @~η @ξ~ @ξ~ @ξ~ h i 1 g ð1 2 F0 Þρf0 βðT 2 T0 Þ 2 ðρ 2 ρf0 ÞðF 2 F0 Þ !   @ @ @ @2 T @2 T ðρcÞf 1 2 1 u~ 1 v~ T5k 2 @~η @t~ @~η @ξ~ @ξ~ (    2  2 !) @F @T @F @T DT @T @T 1 ðρcÞp DB 1 1 1 ~ ~ ~ @~η @~η @~η T0 @ξ @ξ @ξ ! !   @ @ @ @2 F @2 F DT @2 T @2 T 1 v~ 1 2 1 1 2 1 u~ F 5 DB 2 @~η T0 @ξ~ 2 @t~ @~η @~η @ξ~ @ξ~

(2.132)



Nanofluids

(2.133)

(2.134)

(2.135)

(2.136)

84

2. Mathematical analysis of nanofluids

Introducing the following nondimensional quantities: ξ5

~ 2 u~ v~ pa ct~ η~ ξ~ ; η5 ; t5 ; u5 ; v5 ; p5 ; c cδ λ a λ μcλ

h5

h~ a b μ T 2 T0 F 2 F0 5 1 1 φ sinð2πξÞ; δ 5 ; φ 5 ; v 5 ; θ5 ; Φ5 ; a λ a ρf0 T1 2 T 0 F1 2 F0

Re 5 Pr 5

ρf ca μ

; GrT 5

vðρcÞf k

; Nb 5

ga3 ðρ 2 ρf0 ÞðF1 2 F0 Þ βga3 ð1 2 F0 ÞðT1 2 T0 Þ ; Gr 5 ; F v2 ρf0 v2 ðρcÞp DB ðF1 2 F0 Þ k

; Nt 5

ðρcÞp DT ðT1 2 T0 Þ kT0 (2.137)

Using the low Reynolds number and large wavelength approximations, the governing Eqs. (2.130)(2.134) lead to the following nondimensional equations: @u @v 1 50 @ξ @η

(2.138)

@p @2 u 5 2 1 GrT θ 2 GrF Φ @ξ @η

(2.139)

@p 50 @η

(2.140)

 2 @2 θ @θ @Φ @θ 1 Nt 1 Nb 50 @η2 @η @η @η

(2.141)

@2 Φ N t @2 θ 1 50 @η2 Nb @η2

(2.142)

where ξ is the nondimensional axial coordinate; η is the nondimensional transverse coordinate; t is the dimensionless time; u and v are the nondimensional axial and transverse velocity components, respectively; h is transverse vibration of the wall; δ is the wave number, p is the dimensionless pressure; ϕ is the amplitude ratio; υ is the nanofluid kinematic viscosity; Re is the Reynolds number; θ is the dimensionless temperature; Φ is the rescaled nanoparticle volume fraction; GrT is the thermal Grashof number; GrF is the basic-density Grashof number; Pr is the Prandtl number; Nb and Nt are the Brownian motion and thermophoresis parameters, respectively; and k is the nanofluid thermal conductivity. The boundary conditions are: θ50;

Φ50;

@u @η 5 0

Nanofluids

at η 5 0

(2.143)

85

2.3 Differential transformation method for nanofluid modeling

θ51;

Φ51; u 50

at η 5 h

(2.144)

Eqs. (2.143) and (2.144) are the velocity, temperature, and nanoparticle fraction boundary conditions in the bottom and top boundaries of the wavy channel, respectively. Here, the DTM is applied to the governing equations (Eqs. 2.1362.140) and boundary conditions (Eq. 2.141). As we know, DTM usually can be applied on the initial value problems and is not suitable for the boundary-value problems. Thus one of the boundary conditions is replaced with an unknown initial condition (ai, i 5 1,2,3), that is: θð0Þ 5 0; θ0 ð0Þ 5 a1 φð0Þ 5 0; φ0 ð0Þ 5 a2 uð0Þ 5 a3 ; u0 ð0Þ 5 0

(2.145)

Finally, the unknown parameters (ai, i 5 1,2,3) can be determined from other boundary conditions at η 5 h. By applying the DTM on Eqs. (2.136)(2.140) at η 5 0, the following equations are found: ðk 1 1Þðk 1 2ÞU ðk 1 2Þ 1 GrT ΘðkÞ 2 GrT ΦðkÞ 2 ðk 1 1Þðk 1 2ÞΘðk 1 2Þ 1 Nb

@p δ ðkÞ 5 0 @ξ

k X ðr 1 1ÞΘðr 1 1Þðk 2 r 1 1ÞΦðk 2 r 1 1Þ r50

k X 1 Nt ðr 1 1ÞΘðr 1 1Þðk 2 r 1 1ÞΘðk 2 r 1 1Þ 5 0 r50

ðk 1 1Þðk 1 2ÞΦðk 1 2Þ 1

Nt ðk 1 1Þðk 1 2ÞΘðk 1 2Þ 5 0 Nb (2.146)

Concerning the initial conditions at η 5 0, we have: Θð0Þ 5 0; Θð1Þ 5 a1 Φð0Þ 5 0; Φð1Þ 5 a2 U ð0Þ 5 a3 ; U ð1Þ 5 0

(2.147)

Now, the Taylor series coefficients can be achieved using the previous equations. The values of the unknown parameters can also be calculated from the boundary conditions at η 5 h. For example, the DTM solution of the problem for ζ 5 1; Φ 5 0:5; Nb 5 1; Nt 5 0:5; @p @ξ 5 2; GrT 5 2; GrF 5 1 will be [11]:

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2. Mathematical analysis of nanofluids

θðxÞ 5 1:9308x 2 1:4481x2 1 0:7240x3 2 0:2715x4 1 0:0814x5 2 0:0203x6 1 0:0043x7 2 0:0008x8 1 0:0001x9 φðxÞ 5 0:5345x 1 0:7240x2 2 0:3620x3 1 0:1357x4 2 0:0407x5 1 0:0101x6 2 0:0022x7 1 0:0004x8 2 0:0001x9 uðxÞ 5 2 0:6752 1 x2 2 05545x3 1 0:3017x4 2 0:0905x5 1 0:0226x6 2 0:0048x7 1 0:0009x8 2 0:0001x9 (2.148) The governing Eqs. (2.136)(2.140) were solved and the results of ζ 5 1; Φ 5 0:5; Nb 5 1; Nt 5 0:5; @p @ξ 5 2; GrT 5 2; GrF 5 1 are compared in Tables 2.72.9 for nanoparticle concentration, temperature, and velocity profiles, respectively. It can be concluded that the DTM is more accurate than the LSM due to lower errors.

2.3.3 Case 3: Inclined rotating disk This problem was described in Section 2.2 for case 2.A (see Fig. 2.2A). Here, DTM will be applied to solve the governing equation. TABLE 2.7 Comparison of results of LSM, DTM, and exact solution for φðηÞ [11]. φðηÞ

x Exact

LSM

DTM

Error (LSM)

Error (DTM)

0

0

0

0

0

0

0.1

0.06035

0.06025

0.06035

0.0001

2 1.6E 2 10

0.2

0.133188

0.132993

0.133188

0.000195

2 2E 2 10

0.3

0.216774

0.216573

0.216774

0.000202

4E 2 10

0.4

0.309611

0.309493

0.309611

0.000118

1.1E 2 09

0.5

0.410411

0.410423

0.410411

2 1.2E 2 05

9E 2 09

0.6

0.518063

0.518193

0.518063

2 0.00013

5.72E 2 08

0.7

0.631614

0.631798

0.631614

2 0.00018

2.64E 2 07

0.8

0.750243

0.750394

0.750242

2 0.00015

9.89E 2 07

0.9

0.873241

0.8733

0.873238

2 5.9E 2 05

3.17E 2 06

1

1

1

1

0

0

DTM, Differential transform method; LSM, least squares method.

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2.3 Differential transformation method for nanofluid modeling

TABLE 2.8

Comparison of results for LSM, DTM, and exact solution for φðηÞ [11]. φðηÞ

x Exact

LSM

DTM

Error (LSM)

Error (DTM)

0

0

0

0

0

0

0.1

0.179299

0.179422

0.179299

0.000123

3E 2 10

0.2

0.333623

0.33392

0.333623

0.000296

5E 2 10

0.3

0.466451

0.466782

0.466451

0.00033

2 3E 2 10

0.4

0.580777

0.580975

0.580777

0.000198

2 2.2E 2 09

0.5

0.679179

0.679148

0.679179

2 3E 2 05

2 1.8E 2 08

0.6

0.763874

0.76363

0.763874

2 0.00024

2 1.1E 2 07

0.7

0.836771

0.836427

0.836772

2 0.00034

2 5.3E 2 07

0.8

0.899515

0.899228

0.899517

2 0.00029

2 2E 2 06

0.9

0.953518

0.953403

0.953525

2 0.00012

2 6.3E 2 06

1

1

1

1

0

0

DTM, Differential transform method; LSM, least squares method.

TABLE 2.9

Comparison of results for LSM, DTM, and exact solution for uðηÞ [11].

x

uðηÞ Exact

LSM

DTM

Error (LSM)

Error (DTM)

0

2 0.67523

2 0.67512

2 0.67523

0.000114

0

0.1

2 0.66575

2 0.6658

2 0.66575

2 4.4E 2 05

4E 2 10

0.2

2 0.63921

2 0.63946

2 0.63921

2 0.00025

3E 2 10

0.3

2 0.59796

2 0.59826

2 0.59796

2 0.0003

5E 2 10

0.4

2 0.54384

2 0.544

2 0.54384

2 0.00017

3E 2 09

0.5

2 0.4782

2 0.47812

2 0.4782

7.87E 2 05

2.06E 2 08

0.6

2 0.40201

2 0.4017

2 0.40201

0.000307

1.26E 2 07

0.7

2 0.31589

2 0.31548

2 0.31589

0.000408

5.85E 2 07

0.8

2 0.22017

2 0.21984

2 0.22018

0.000334

2.2E 2 06

0.9

2 0.11493

2 0.11479

2 0.11493

0.000135

7.05E 2 06

1

0

0

2 2E 2 05

0

2E 2 05

DTM, Differential transform method; LSM, least squares method.

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2. Mathematical analysis of nanofluids

To transform Eqs. (2.35)(2.38), a one-dimensional transform is used and the following transforms are obtained [12]: fv0 -ði 1 3Þði 1 2Þði 1 1ÞFði 1 3Þ ffv-

i X

(2.149)

ði 1 2 2 rÞði 1 1 2 rÞFðrÞFði 1 2 2 rÞ

(2.150)

ði 1 1 2 rÞðr 1 1ÞFðr 1 1ÞFði 1 1 2 rÞ

(2.151)

gv-ði 1 2Þði 1 1ÞGði 1 2Þ

(2.152)

kv-ði 1 2Þði 1 1ÞKði 1 2Þ

(2.153)

sv-ði 1 2Þði 1 1ÞSði 1 2Þ

(2.154)

θv-ði 1 2Þði 1 1ÞΘði 1 2Þ

(2.155)

r50 0

f 2-

i X r50

g2 -

i X

GðrÞGði 2 rÞ

(2.156)

r50 0

fg -

i X ði 1 1 2 rÞFðrÞGði 1 1 2 rÞ

(2.157)

r50 0

gf -

i X

ði 1 1 2 rÞGðrÞFði 1 1 2 rÞ

(2.158)

ði 1 1 2 rÞFðrÞKði 1 1 2 rÞ

(2.159)

ði 1 1 2 rÞKðrÞFði 1 1 2 rÞ

(2.160)

ði 1 1 2 rÞFðrÞSði 1 1 2 rÞ

(2.161)

ði 1 1 2 rÞSðrÞFði 1 1 2 rÞ

(2.162)

ði 1 1 2 rÞFðrÞΦði 1 1 2 rÞ

(2.163)

r50 0

fk -

i X r50

0

kf -

i X r50

0

fs -

i X r50

0

sf -

i X r50

0

fθ -

i X r50

Substituting this equation into Eqs. (2.35)(2.38) and according to the boundary conditions, we have:

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89

2.3 Differential transformation method for nanofluid modeling

1 i X ði 1 1 2 rÞðr 1 1Þ  Fðr 1 1ÞFði 1 1 2 rÞ C B2 C B r50 C B i C B X C B ði 1 3Þði 1 2Þði 1 1ÞFði 1 3Þ 5 2 B 1 GðrÞGði 2 rÞ C C B r50 C B C B i X A @ 1 2 ði 1 1 2 rÞði 1 2 2 rÞFði 1 2 2 rÞFðrÞ 0

r50

1

0

i X B 2 2 ði 1 1 2 rÞGðrÞFði 1 1 2 rÞ C C B r50 C ði 1 2Þði 1 1ÞGði 1 2Þ 5 2 B C B i X A @ 1 2 ði 1 1 2 rÞFðrÞGði 1 1 2 rÞ

(2.164)

(2.165)

r50

1 i X B 2 2 ði 1 1 2 rÞKðr 1 1ÞFði 1 1 2 rÞ C C B r50 C B i C B X C B ði 1 2Þði 1 1ÞKði 1 2Þ 5 2 B 1 SðrÞGði 2 rÞ C C B r50 C B C B i X A @ 1 2 ði 1 1 2 rÞFðrÞKði 1 1 2 rÞ 1 δD ðiÞ 0

r50

0

i X

1

2 GðrÞKði 2 rÞ C B C B r50 C B i C B X C B ði 1 2Þði 1 1ÞSði 1 2Þ 5 B 2 ði 1 1 2 rÞSðrÞFði 1 1 2 rÞ C C B r50 C B C B i X A @ 1 2 ði 1 1 2 rÞFðrÞSði 1 1 2 rÞ

(2.166)

(2.167)

r50

ði 1 2Þði 1 1ÞΦði 1 2Þ 5 2 2

i A2 A3 X Pr ði 1 1 2 rÞFðr 1 1ÞΦði 1 1 2 rÞ (2.168) A1 A4 r50

Fð0Þ 5 0; Fð1Þ 5 0; Fð2Þ 5 a1 Gð0Þ 5 1; Gð1Þ 5 a2 Kð0Þ 5 0; Kð1Þ 5 a3 Sð0Þ 5 0; Sð1Þ 5 a4 Φð0Þ 5 0; Φð1Þ 5 a5

Nanofluids

(2.169)

90

2. Mathematical analysis of nanofluids

where FðiÞ; GðiÞ; KðiÞ; SðkÞ; and ΦðkÞ are the transformation functions of f ðiÞ; gðiÞ; kðiÞ; sðiÞ; and θðkÞ, respectively, and are defined by: f ðηÞ 5

N X

FðiÞηi

(2.170)

GðiÞηi

(2.171)

KðiÞηi

(2.172)

SðiÞηi

(2.173)

ΦðkÞηk

(2.174)

i50

gð ηÞ 5

N X i50

k ð ηÞ 5

N X i50

sðηÞ 5

N X i50

θ ð ηÞ 5

N X k50

After using iteration i 5 0; 1; . . ., and using Eqs. (2.170)(2.174), when Pr 5 1; ϕ 5 0:4; and δ 5 1; we have: η3 η4 a 2 1 5 2 ... η a2 1 2 2 (2.175) 60 6 12    2η3 a1 η5 a 2 1 4 1 η 22a1 a2 1 2 2 1 2a1 a2 1 ηa2 2 1 gðηÞ 5 1 1 1 ... 12 2 3 15 (2.176)  

η2 1 5 a3 a3 1 a2 a3 η 21 2 2 2a1 2 2 2 12 k ð ηÞ 5 a 3 η 2 1 20 4 2 6 6 3 (2.177)  1 1 1 a1 ð 21 1 4a1 a3 2 a4 Þ 1 η3 ð 21 1 4a1 a3 2 a4 Þ 1 . . . 3 6     η3 a3 1 4 1 1 5 a2 1 a4 η 2 1 a2 a3 1 η 2 1 ð 21 1 4a1 a3 2 a4 Þ 2 1 s ð ηÞ 5 12 2 20 6 6 2 6 . . . 1 ηa4 1 f ð ηÞ 5 a 1 η2 2

(2.178) η a5 1 1

a2 a5 . . . 2 η3 a1 a5 1 η5 a21 a5 1 θðηÞ 5 a5 η 1 1 3 10 36 12 4

(2.179)

Finally, by applying the boundary conditions the solution and unknown coefficients can be obtained as depicted in Fig. 2.14.

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2.4 Other analytical/mathematical modeling

91

FIGURE 2.14 Comparison of DTM and shooting method for Cuwater NF with ϕ 5 0.04 and δ 5 0.5, 0.75, 1 [12]. DTM, Differential transformation method.

2.4 Other analytical/mathematical modeling As in our last book [1], a complete list of analytical/mathematical methods usable for engineering problems is presented. But in this chapter, based on our experience, the three most efficient techniques (based on the boundary condition types) are presented for NF modeling. The optimal collocation method (OCM) is the last method introduced in this section. Actually, the collocation method is one of the WRMs presented in Section 2.2. The OCM is also an optimized version of the CM. To understand the main idea behind this method, suppose a differential operator D is acted on a function u to produce a function p [1]:

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92

2. Mathematical analysis of nanofluids

DðuðxÞÞ 5 pðxÞ

(2.180)

~ which is a linear combiWe want to approximate u by a function u, nation of basic functions chosen from a linearly independent set, that is: u 5 u~ 5

n X

ci ϕi

(2.181)

i51

Now, when substituted into the differential operator, D, the result of the operations is not pðxÞ, so an error or residual will exist as: ~ 2 pðxÞÞ 6¼ 0 EðxÞ 5 RðxÞ 5 DðuðxÞ

(2.182)

Like the LSM, the main idea in the collocation is to force the residual to zero in some average sense over the domain, that is, ð RðxÞ Wi ðxÞ 5 0; i 5 1; 2; . . .; n (2.183) x

where n is the number of weight functions and Wi is exactly equal to ~ The result is a set of n algethe number of unknown constants ci in u. braic equations for the unknown constants ci . For the collocation method, the weighting functions are different from the LSM and are taken from the family of Dirac δ functions in the domain (i.e., wi ðxÞ 5 δðx 2 xi Þ). The Dirac δ function is defined as: 1 if x 5 xi δðx 2 xi Þ 5 (2.184) 0 otherwise The residual function in Eq. (2.182) must be forced to be zero at specific points. In the OCM as the optimized CM, a suitable finite value of η (- ~ ), say ηN is chosen. Thus the equations and boundary conditions should be transformed to another set of equations. Finally, the extra asymptotic condition can be applied to obtain the ηN . Considering a very small value for ηN would not assure a uniformly valid convergent solution. Also, selecting too large a value for ηN results either in asymptotic divergent series or in slow convergence of the series. Therefore a method must be introduced to logically estimate the value of ηN . In the CM (as in other analytical methods), the user must define infinity with a number, but in the OCM the mathematical concept of infinity is directly used and at the end the numerical value of infinity can be found. This is one of the advantages of the OCM compared to other analytical methods. This method removes the restrictions of the CM while maintaining all its advantages [7]. To illustrate the application of this method, following are three examples of analyzing NF using the OCM [7].

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2.4 Other analytical/mathematical modeling

FIGURE 2.15

Schematic of the problem [7].

2.4.1 Case 1: Nanofluids over a cylindrical tube under the magnetic field effect As shown in Fig. 2.15, consider the steady laminar flow of an incompressible electrically conducting fluid over a stretching tube of radius a in the axial direction. The z-axis is considered along the axis of the tube and the r-axis is measured in the radial direction. For the boundary condition, the surface of the tube is considered at constant temperature Tw and the fluid temperature is T1, where Tw . T1. Furthermore, the uniform magnetic field of intensity B0 acts in the radial direction. The viscous dissipation, Ohmic heating, and Hall effects are neglected in this case. The base fluid is water and three different types of nanoparticles (Cu, Al2O3, and TiO2) were added to it. Like the other cases, it is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. After the above assumptions, the boundary-layer equations can be written in dimensional form as [7]: @ðruÞ @ðrwÞ 1 50 @x @y   @w @w μnf @2 w 1 @w σβ 20 u 1w 5 1 w 2 @r @z r @r ρnf @r2 ρnf   @u @u 1 @p μnf @2 u 1 @u u 1w 52 1 2 2 1 u @r @z ρnf @r r @r r ρnf @r2    2    @T @T @ T 1 @T 1u 1 5 knf ρCp nf w @z @r @r2 r @r

Nanofluids

(2.185) (2.186)

(2.187)

(2.188)

94

2. Mathematical analysis of nanofluids

Subject to the following boundary conditions: u 5 0 ; W 5 Ww ; W-0 ; T-TN

T 5 Tw at r 5 a as r-N

(2.189)

where u and w are the velocity components along the r- and z-axes, respectively; and Ww 5 2cz, where c is a positive constant and a is a constant. Further, v; ρ; T, and α are the kinematic viscosity, fluid density, fluid temperature, and thermal diffusivity, respectively. The effective density ρnf, the effective dynamic viscosity μnf, the heat capacitance (ρCp)nf, and the thermal conductivity knf of the NF are given as Eqs. (2.1)(2.5). By introducing the similarity transformation as

r 2 fðηÞ T 2 TN u 5 2 ca pffiffiffi ; W 5 2cf 0 ðηÞz; η 5 ; θ5 (2.190) η a Tw 2 TN where prime refers to differentiation with respect to η. Substituting Eq. (2.190) into Eqs. (2.186) and (2.188), the following ordinary differential equations can be obtained [7]: 1 1 2:5 ð1 2 ϕÞ 1 ð12ϕÞ

ρs ρf

 ϕ

 fwη 1 fv 2

M ð1 2 ϕÞ 1

0

ρs ρf

ϕ

f 0 2 Re f 2 1 Re ffv 5 0 (2.191)

ðθv η 1 θ0 Þ 1 fθ0 Re Pr

ð1 2 ϕÞ 1

ðρCp Þs ϕ ðρCp Þf

ks 1 2kf 2 2ϕðkf 2 ks Þ ks 1 2kf 1 ϕðkf 2 ks Þ

50

(2.192)

where Re 5 ca2 =ð2υnf Þ is the Reynolds number and M 5 σB20 a2 =ð4υnf ρnf Þ is the magnetic parameter. The boundary conditions (Eq. 2.189) become: fð1Þ 5 0; f 0 ð1Þ 5 1; θð1Þ 5 1 fðNÞ-0; θðNÞ-0

(2.193)

The pressure can be assumed from Eq. (2.187) in the following form: p 2 pN Re 5 2 f 2 ðηÞ 2 2f 0 ðηÞ η ρcv

(2.194)

The interested physical quantities are the skin friction coefficient and the Nusselt number, which are defined as follows: Cf 5

τw aqw ; Nu 5 ρWw =2 kðTw 2 TN Þ

(2.195)

Moreover, τ w and qw are the skin friction and the heat transfer from the surface of the tube, respectively, and are known as:

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95

2.4 Other analytical/mathematical modeling

  @W τw 5 μ ; @r r5a



@T qw 5 2 k @r

 (2.196) r5a

where k is the thermal conductivity. Considering variables of Eq. (2.190), we get:   Re z (2.197) 5 fvð1Þ; Nu 5 22θ0 ð1Þ Cf a



As noted earlier, the OCM was first introduced by Khazayinejad et al. [13] and Hatami et al. [14] for the solution of boundary-layer problems as a modification of the CM. To explain the method, the physical region η 5 ½1; NÞ is changed to the region η 5 ½1; ηN , where ηN is sufficiently large, and the maximum values of η at the edge of the boundary layer are used. It is clear that the values of ηN change when the variable parameters (such as nanoparticle volume fraction, suction parameter, density, magnetic parameter, thermal conductivities, heat capacitance, and Reynolds and Prandtl numbers) in the problem change. Thus ηN is a function of ϕ, S, ρ, Mn, K, Cp , Re, and Pr, which should be determined as a part of the solution. By introducing the following nondimensional h i variable λ 5 η=ηN , the problem transforms into the interval of

1 ηN

;1

instead of ½1; ηN Þ. Introducing the above transformations, the governing equations (Eqs. 2.191 and 2.192) can be transformed into the following shape:   3 d d2 d gðλÞ λ 1 2 gðλÞ gðλÞ 3 2  dλ dλ dλ d    2 ηN Re gðλÞ 2 ηN M  ρ ρ dλ 12ϕ1ϕ s 1 2 ϕ 1 ϕ s ð12ϕÞ2:5 ρf ρf 1 ηN Re gðλÞ



d2 gðλÞ 5 0 dλ2

ðρc Þ 1 2 ϕ 1 ϕ ðρcpp Þs

(2.198)

 d2 d d f hðλÞ 1 ηN Re Pr gðλÞ hðλÞ Ks 1 2K 2 2ϕðK 2 Ks Þ 5 0 hðλÞ λ1 f f dλ dλ dλ2 Ks 1 2 Kf 1 ϕðKf 2 Ks Þ

(2.199) hðλÞ 5 θðηÞ ηN , and the “prime” represents the derivatives h i 1 with respect to λA η ; 1 . Also, the boundary conditions can be transwhere gðλÞ 5

fðηÞ ηN ,

N

formed into:

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96

2. Mathematical analysis of nanofluids

1 S 1 .g 5 ; g0 5 1; h 5 ηN ηN ηN

λ5

λ 5 1.g0 5 0; h 5 0

(2.200) (2.201)

In the OCM, the last boundary condition is achieved using the asymptotic condition: η-N.f 00 5 0; θ0 5 0

(2.202)

The extra boundary conditions (Eq. 2.202) can be substituted by the conditions: λ 5 1.g00 5 0;

h0 5 0

(2.203)

The asymptotic condition (Eq. 2.203) is imposed for computing ηN . It is expected to obtain an approximate solution for this problem in the interval of 1 , λ , ð1=ηN Þ. To make a trial solution, the following functions are determined: gðλÞ 5 c1 1 c2 λ 1 c3 λ2 1 ? 1 c12 λ11

(2.204)

hðλÞ 5 c13 1 c14 λ 1 c15 λ2 1 ? 1 c24 λ11

(2.205)

The accuracy of the solution can be enhanced by increasing the number of trial function terms. Since the trial solution must satisfy the boundary conditions of Eqs. (2.200) and (2.201) for all values of “c” constants: c1 1

c2 c3 c12 S 1 1?1 5 ηN ηN ηN 2 ηN 11

(2.206)

c2 1

2 c3 3 c4 11 c12 1 1?1 51 2 ηN ηN ηN 10

(2.207)

c2 1 2 c3 1 3 c4 1 ? 1 11 c12 5 0 c13 1

c14 c15 c24 1 1 1?1 5 ηN ηN ηN 2 ηN 11

c13 1 c14 1 c15 1 ? 1 c24 5 0

(2.208) (2.209) (2.210)

ηN in Eqs. (2.206)(2.210) can be determined by using the extra boundary conditions given in Eq. (2.203). This means: 2 c3 1 6 c4 1 12 c5 1 ? 1 110 c12 5 0

(2.211)

c14 1 2 c15 1 3 c16 1 ? 1 11 c24 5 0

(2.212)

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97

2.4 Other analytical/mathematical modeling

By introducing gðλÞ and hðλÞ to the governing equations, the residual functions will be:   R1 c1 ;c2 ; c3 ;. ..;c12 ;ηN ;λ 52 2 ηN Re c23 λ2 23 ηN Re c24 λ4 1? 2 c3 12 c4 λ 1 1 10 10 ρ ρ @1 2ϕ 1 ϕ s Að12ϕÞ2:5 @1 2ϕ 1 ϕ s Að12ϕÞ2:5 ρf ρf (2.213) 1? 2

10 ηN M c11 λ9 11 ηN M c12 λ10 ρ 2 ρ 1 2ϕ 1 ϕ s 1 2ϕ 1 ϕ s ρf ρf

R2 ðc1 ;c2 ;c3 ;...;c24 ;ηN ;λÞ5c14 14 c15 λ19 c16 λ2 1?2 2

4 Re Pr ηN ϕ c1 c17 λ3 Kf Ks 12 Kf 22 ϕ Kf 12 ϕ Ks

11 Re Pr ηN ϕ2 ðρcp Þs c12 λ21 c24 Ks 10 Re Pr ηN ϕ c1 c18 λ4 Ks 1?2 Ks 12 Kf 22 ϕ Kf 12 ϕ Ks ðKs 12 Kf 22 ϕ Kf 12 ϕ Ks Þ ðρcp Þf (2.214)

In the OCM, the numbers of weight functions, Wi , are: nwi 5 nci 2 nb

(2.215)

where nci are the numbers of unknown constants and ci and nb are the number of equations that satisfy the boundary conditions. Additionally, the residual function must h be iclose to zero. For this purpose, specific points in the domain λA η1 ; 1 should be considered. These points are: N       9 1 ηN 8 1 2ηN 1 1 9ηN R1 5 0; R1 5 0; . . .; R1 5 0 (2.216) 10 ηN 10 ηN 10 ηN       9 1 ηN 8 1 2ηN 1 1 9ηN R2 5 0; R2 5 0; . . .; R2 5 0 (2.217) 10 ηN 10 ηN 10 ηN In this problem, 25 algebraic equations appeared: five equations (Eqs. 2.2062.210) that satisfy the boundary conditions; two equations (Eqs. 2.211 and 2.212) that satisfy the extra boundary conditions; and 18 equations (Eqs. 2.216 and 2.217) that force the residual function to zero. By solving this system of equations, unknown coefficients ci , ηN can be determined. Finally, after specifying these unknown parameters, the velocity and temperature distribution can be determined. For example, using the OCM for a Cu 2 H2 O NF with ϕ 5 0:05,S 5 1, M 5 0:1, Re 5 5, Ks 5 401, Kf 5 0:613, ðcp Þs 5 0:1, ðcp Þf 5 0:1, ρs 5 8933, ρf 5 997:1, and Pr 5 7, fðηÞ and θ ðηÞ are as follows:

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2. Mathematical analysis of nanofluids

FIGURE 2.16 (A) Comparison of OCM and numerical results for stream function for various values of S [7]. (B). Comparison of OCM and numerical results for stream function for various values of Re [7]. (C) Comparison of OCM and numerical results for stream function for various values of M [7]. (D) Comparison of OCM and numerical results for stream function for various values of ϕ [7]. OCM, Optimal collocation method.

fðηÞ 5 2 214:324914 1 1365:510780η 2 3995:967289η2 1 7094:956363η3 1 ? 1 0:751988η11 θ 5 ðηÞ 5 124; 291 2 864; 328:0η 1 272; 1530η2 2 512; 1740η3 1 ? 1 648:549608η11 (2.218) Fig. 2.16AD and Table 2.10 demonstrate the accuracy of this method for different values of constant numbers, S, Re, M, and ϕ parameters. For all variables this method gives a high accuracy solution for fðηÞ compared to the numerical solution.

Nanofluids

99

2.4 Other analytical/mathematical modeling

TABLE 2.10 Comparison of the OCM and numerical results when ϕ 5 0:1,S 5 1, M 5 0:01, and Re 5 1 [7]. η

θðηÞ

fðηÞ Numerical

OCM

Error

Numerical

OCM

Error

1

1.00000

1.00000

0.00000

1.00000

1.00000

0.00000

1.2

1.16329

1.16384

0.00055

0.32345

0.34944

0.02598

1.4

1.27450

1.27559

0.00109

0.11430

0.12273

0.00842

1.6

1.35373

1.35510

0.00136

0.04405

0.04636

0.00231

1.8

1.41210

1.41359

0.00149

0.01837

0.01939

0.00102

2

1.45618

1.45771

0.00153

0.00821

0.00864

0.00043

2.2

1.49010

1.49161

0.00150

0.00390

0.00395

0.00005

2.4

1.51657

1.51800

0.00143

0.00195

0.00187

0.00009

2.6

1.53742

1.53874

0.00131

0.00102

0.00092

0.00011

2.8

1.55394

1.55511

0.00117

0.00056

0.00041

0.00015

3

1.56705

1.56807

0.00101

0.00032

0.00015

0.00017

3.2

1.57745

1.57828

0.00083

0.00018

0.00002

0.00017

3.4

1.58563

1.58627

0.00064

0.00011

0.00009

0.00002

3.6

1.59201

1.59244

0.00043

0.00006

0.00013

0.00007

3.8

1.59687

1.59709

0.00022

0.00004

0.00023

0.00019

4

1.60048

1.60046

0.00001

0.00002

0.00025

0.00023

4.2

1.60301

1.60276

0.00025

0.00001

0.00013

0.00012

4.4

1.60463

1.60413

0.00049

0.00001

0.00032

0.00032

4.6

1.60546

1.60472

0.00074

0.00000

0.00050

0.00050

OCM, Optimal collocation method.

2.4.2 Case 2: Nanofluid passing over a porous moving semiinfinite flat plate A two-dimensional steady and laminar boundary-layer flow of Cuwater NF over a porous moving semiinfinite flat plate is considered as shown in Fig. 2.17A and B. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. The plate is expected to move with constant velocity Uw in the same direction as the free stream UN . Using boundary-layer approximation, under zero pressure gradient, the governing equations can be written as [13]:

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100

2. Mathematical analysis of nanofluids

FIGURE 2.17 (A) Geometry of problem when 0 , ε , 0:5ðUw , UN Þ; (B) geometry of problem when 0:5 , ε , 1ðUw . UN Þ [13].

@u @v 1 50 @x @y  @u @u @2 u 1v ρnf u 5 μnf 2 @x @y @y

(2.219) (2.220)

The boundary conditions for the above equations are: y 5 0.u 5 Uw ; v 5 Vw

(2.221)

y-N.u-UN

(2.222)

where u and v are the velocity components in the x- and y-directions, respectively. For simplicity of the equations, the following transformations are applied: y η 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(2.223)

  2 ψ x; y f 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Uw 1 UN Þυf x

(2.224)

υf x ð Uw 1 UN Þ

Nanofluids

2.4 Other analytical/mathematical modeling

101

where the stream function ψ is defined as u 5 2 @ψ=@y and v 5 @ψ=@x and f is the similarity function dependent on η. To ensure the existence of similarity solutions of Eqs. (2.219) and (2.220), we take qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Uw 1 UN Þ υf Vw 5 2 fð0Þ . Applying the above assumptions leads to a 2 x reduction of the basic equations as follows: " #! 1 ρs 2:5 ð12ϕÞ ð1 2 ϕÞ 1 ϕ fwðηÞ 1 fðηÞfvðηÞ 5 0 (2.225) 2 ρf



and the boundary conditions: η 5 0.f 5 s

(2.226)

η 5 0.f 0 5 ε

(2.227)

0

η-N.f 5 1 2 ε

(2.228)

where fð0Þ 5 s is the suction/injection quantity, which is a nondimensional parameter, with s , 0 for injection, s . 0 for suction, and s 5 0 for impermeable surface and ε 5 Uw =ðUw 1 UN Þ is the velocity ratio parameter. In this example, only the case of 0 # ε # 1 is considered. It should be noted that when ε 5 0 and ε 5 1 correspond to the flow over a stationary surface caused by the free stream velocity and moving plate in fluid, respectively. Moreover, the case 0 , ε , 1 says the plate and fluid are moving in the same direction. For 0 , ε , 0:5ðUw , UN Þ or 0:5 , ε , 1ðUw . UN Þ there are two different physical problems. The mechanical characteristic for the flow is represented by the shear stress at the surface (fvð0Þ). Like the previous case, to apply the OCM to the problem, the physical region ½0; NÞ is transformed into the region [0,ηN ] and by introducing the following variable z 5 η=ηN , the problem is transformed in the interval of [0,1] instead of [0,ηN ]. Using the above transformation, the governing equation (Eq. 2.225) can be converted into the following form: " #! ηN 2 ρs 2:5 ð12ϕÞ ð1 2 ϕÞ 1 ϕ gwðzÞ 1 gðzÞgvðzÞ 5 0 (2.229) 2 ρf



where gðzÞ 5 fðηÞ=ηN and the “prime” denotes the derivatives with respect to zA ½0; 1. Similarly, the three boundary conditions can be changed to: z 5 0.g 5

s ηN

z 5 0.g0 5 ε

Nanofluids

(2.230) (2.231)

102

2. Mathematical analysis of nanofluids

z 5 1.g0 5 1 2 ε

(2.232)

In the OCM, the last boundary condition is achieved by using the asymptotic condition: η-N.f 00 5 0

(2.233)

The extra boundary condition Eq. (2.233) can be replaced by the condition: z 5 1.f 00 5 0

(2.234)

The asymptotic condition Eq. (2.234) is used to compute ηN . An approximate solution for this problem in the interval of 0 , z , 1 is expected. The trial function is considered as follows, which contains seven unknown coefficients “c”: g ðzÞ 5 c0 1 c1 z 1 c2 z2 1 c3 z3 1 c4 z4 1 c5 z5 1 c6 z6 1 c7 z7

(2.235)

Because the trial solution must satisfy the boundary conditions of Eqs. (2.230)(2.232), we have the following: c0 5

s ηN

(2.236)

c1 5 ε

(2.237)

c1 1 2c2 1 3c3 1 4c4 1 ? 1 7c7 5 1 2 ε

(2.238)

The “ηN ” in Eq. (2.229) can be determined by using the extra boundary condition presented in Eq. (2.234). This yields: 2c2 1 6c3 1 12c4 1 ?42c7 5 0

(2.239)

By introducing g ðzÞ to the differential equations (Eq. 2.229), the residual function will be:   η 2 ð12ϕÞ2:5 ϕ ρs sc2 R c1 ;c2 ;c3 ;...;c7 ;ηN ;z 56c3 1 N 2ηN 2 ð12ϕÞ2:5 ϕ s c2 ρf 1ηN 2 ð12ϕÞ2:5 s c2 0 1 2:5 2 3 η ð 12ϕ Þ ϕ ρ s c s 3 1@ N 1?13ηN 2 ð12ϕÞ2:5 sc3 1ηN 2 ð12ϕÞ2:5 εc2 Ax ρf 0 1 2:5 2 2 21η ð 12ϕ Þ ϕ ρ c N s 7 121ηN 2 ð12ϕÞ2:5 c27 Ax1250 1?1@221ηN 2 ð12ϕÞ2:5 ϕ c271 ρf (2.240)

Nanofluids

2.4 Other analytical/mathematical modeling

103

In the OCM, the numbers of weight functions Wi are: nwi 5 nci 2 nb

(2.241)

where nci are the numbers of unknown constants and ci and nb are the number of equations that satisfy the boundary conditions. Also, the residual function must be close to zero. To reach this aim, five specific points in the domain zA ½0; 1 are chosen as: ! ! ! ! ! 1 2 3 4 5 R 5 0; R 5 0; R 5 0; R 5 0; R 50 6 6 6 6 6 (2.242) In this problem, we have a system of equations including nine algebraic equations: Three equations (Eqs. 2.2362.238) that satisfy the boundary conditions; one equation (Eq. 2.239) that satisfies the extra boundary condition; and five equations (Eq. 2.242) that force the residual function to zero. By solving this system of equations, unknown coefficients ci and ηN can be determined. For example, using the OCM for a Cu 2 H2 O NF with ϕ 5 0:2, s 5 1, ε 5 0:4, ρs 5 8933, and ρf 5 997:1, fðηÞ is as follows: fðηÞ 5 1 1 0:4 η 1 0:1106302 η2 2 0:0339097 η3 1 0:0061957 η4 2 0:0006729 η5 1 0:0000401 η6 2 0:0000010 η7

(2.243)

The range of the nanoparticle fraction ϕ is considered as 0 # ϕ # 0:2. In order to examine the accuracy of the present results, the outcomes corresponding to the stream function ð fðηÞÞ and velocity profiles ð f 0 ðηÞÞ for ϕ 5 0, s 5 0, and ε 5 0 (i.e., pure water, impermeable surface, and stationary surface) are compared with the available published results and are presented in Fig. 2.18. Furthermore, the approximate solution of the f 0 ðηÞ obtained by the OCM is compared to the obtained results by the numerical method listed in Table 2.11.

2.4.3 Case 3: Two-phase nanofluid flow over a stretching infinite solar plate In the last case of this chapter, an NF flow over a stretching infinite plate under the solar radiation is considered for the mathematical analysis. It is assumed that the flow is incompressible and induced due to a plate stretched in two directions by nonlinear functions. Also, the plate is maintained at constant temperature and the mass flux of the nanoparticles at the wall is supposed to be zero. The 3D governing equations are [15]:

Nanofluids

104

2. Mathematical analysis of nanofluids

FIGURE 2.18 Comparison of the OCM and numerical results for stream function and velocity profiles when ϕ 5 0, S 5 0, and ε 5 0 [13]. OCM, Optimal collocation method.

@u @v @w 1 1 50 @x @y @z u

@u @u @u @2 u 1v 1w 5 υf 2 @x @y @z @z

@v @v @v @2 v 1 v 1 w 5 υf 2 @x @y @z @z "  # @T @T @T @2 T @C @T DT @T 2 1v 1w 5 α 2 1 τ DB 1 u @x @y @z @z @z @z TN @z u

  2 @C @C @C @2 C DT @ T 1v 1w 5 DB 2 1 u @x @y @z @z TN @z2

(2.244) (2.245) (2.246)

(2.247)

(2.248)

where u and v are the velocities in the x- and y-directions, respectively; T is the temperature; C is the concentration; and DB and DT are the Brownian and thermophoretic diffusion coefficients, respectively.

Nanofluids

2.4 Other analytical/mathematical modeling

TABLE 2.11 Results obtained in this study compared with the numerical method for f 0 ðηÞ [13]. η

fðηÞ Numerical

OCM

Error OCM

0.0

0.0000

0.0000

0.0000

0.2

0.0664

0.0692

0.0028

0.4

0.1328

0.1371

0.0043

0.6

0.1989

0.2041

0.0052

0.8

0.2647

0.2705

0.0058

1.0

0.3298

0.3361

0.0063

1.2

0.3938

0.4005

0.0068

1.4

0.4563

0.4635

0.0073

1.6

0.5168

0.5245

0.0078

1.8

0.5748

0.5830

0.0082

2.0

0.6298

0.6383

0.0086

2.2

0.6813

0.6901

0.0088

2.4

0.7290

0.7380

0.0090

2.6

0.7725

0.7815

0.0091

2.8

0.8115

0.8206

0.0090

3.0

0.8460

0.8550

0.0090

3.2

0.8761

0.8850

0.0089

3.4

0.9018

0.9105

0.0087

3.6

0.9233

0.9319

0.0085

3.8

0.9411

0.9495

0.0083

4.0

0.9555

0.9637

0.0082

4.2

0.9670

0.9750

0.0080

4.4

0.9759

0.9837

0.0079

4.6

0.9827

0.9904

0.0077

4.8

0.9878

0.9953

0.0075

5.0

0.9915

0.9985

0.0069

OCM, Optimal collocation method.

Nanofluids

105

106

2. Mathematical analysis of nanofluids

Since the plate is infinite and is stretched in two directions by nonlinear functions, the relevant boundary conditions are:  n  n u 5 uw 5 a x1y ; v 5 vw 5 b x1y @C DT @T 1 5 0 at z 5 0 w 5 0; T 5 Tw ; DB (2.249) @z TN @z u-0; v-0; T-TN ; C-CN as z-0 By introducing these parameters:  n  n u 5 a x1y f 0 ðηÞ; v 5 a x1y g0 ðηÞ pffiffiffiffiffiffiffi w 5 2 aυf ðx1yÞ

n21 2

 n21  0  n11 f 1g 1 η f 1 g0 2 2

! (2.250)

n21 sffiffiffiffiffi  2 T 2 TN C 2 CN a x1y θðηÞ 5 ; φðηÞ 5 ; η5 z υf Tw 2 TN CN

and substituting the above variables into Eqs. (2.244)(2.248), we have:    n11 f 1 g fv 2 n f 0 1 g0 f 0 5 0 2    n11 f 1 g gv 2 n f 0 1 g0 g0 5 0 gw 1 2  1 n11 0 θv 1 f 1 g θ0 1 Nb φ0 θ0 1 Nt θ 2 5 0 Pr 2 n11 Nt Scðf 1 gÞφ0 1 θv 5 0 φv 1 2 Nb fw 1

(2.251)

Finally, the OCM is applied to solve this nonlinear system of equations with these boundary conditions: f ð0Þ 5 0; f 0 ð0Þ 5 1; gð0Þ 5 0; g0 ð0Þ 5 λ; θð0Þ 5 1; Nb φ0 ð0Þ 1 Nt θ0 ð0Þ 5 0 f 0 ðNÞ-0; g0 ðNÞ-0; θðNÞ-0; φðNÞ-0 (2.252) where the Pr (Prandtl number), Sc (Schmidt number); Nb (Brownian motion parameter), Nt (thermophoresis parameter), and λ 5 b/a (ratio of the stretching rate along the y- to x-directions) are as defined in Ref. [4]. To apply this method, the physical region η 5 ½0; NÞ is transformed into the region η1 5 ½0; ηN1  and η2 5 ½0; ηN2  along the x- and y-directions for the hydrodynamic boundary layer, and η3 5 ½0; ηN3  and η4 5 ½0; ηN4  for the thermal and nanoparticle volume fraction boundary

Nanofluids

107

2.4 Other analytical/mathematical modeling

layer, respectively. Note that, ηN is sufficiently large and a function of Sc,Pr,Nb ,Nt ,λ. Furthermore, by introducing the following variables z1 5 η1/ηN1, z2 5 η2/ηN2, z3 5 η3/ηN3, and z4 5 η4/ηN4, the problem transforms into the interval ½0; 1 instead of [0,ηN]. Applying the above transformations, the governing equations can be changed into the following new shape:     d2 d3 H ðz1 Þ ðn 1 1Þ ηN1 H ðz1 Þ 1 ηN2 Kðz2 Þ H ðz1 Þ dz1 3 dz1 2 1 ηN1 2 2 ηN1 (2.253) 0 10 1 d d d 2n @ H ðz1 Þ 1 K ð z 2 Þ A@ H ðz1 ÞA 5 0 dz1 dz2 dz1 3

d K ðz2 Þ dz2 3 ηN2 2 0

1

0 1   d2 ðn 1 1Þ ηN1 H ðz1 Þ 1 ηN2 Kðz2 Þ @ 2 Kðz2 ÞA dz2

2ηN2 10 1 d d d 2 n@ H ðz1 Þ 1 K ð z 2 Þ A@ K ðz2 ÞA 5 0 dz1 dz2 dz2

d2 M ðz3 Þ dz3 2

(2.254)

0

1 0 1   d n 1 1 A ηN1 H ðz1 Þ 1 ηN2 Kðz2 Þ @ 1@ Mðz3 ÞA 1 2 dz3 ηN3 Pr 0 10 1  2 d d d @ A @ A N ðz4 Þ Mðz3 Þ 1 Nt dz3 Mðz3 Þ 5 0 Nb dz4 dz3      d n11 1 N ðz4 Þ Sc ηN1 H ðz1 Þ 1 ηN2 Kðz2 Þ 2 dz4 ηN 4

2 Nt dzd3 2 Mðz3 Þ 1 50 Nb ηN3

d2 dz4 2

N ðz4 Þ

(2.255)



(2.256)

kðz2 Þ 5 gðη2 Þ=ηN 2 , Mðz3 Þ 5 θðη3 Þ=ηN3 , where hðz1 Þ 5 fðη1 Þ=ηN1 , Nðz4 Þ 5 φðη4 Þ=ηN 4 , and the “prime” refers to the derivatives with respect to zA[0,1]. Also, the boundary conditions can be transformed into: z1 5z2 5z3 5z4 50.H 50; K50; H 0 51; K0 5λ; M5

Nanofluids





1 ;Nb M0 1Nt N 0 50 ηN3 (2.257)

108

2. Mathematical analysis of nanofluids

z1 5z2 5z3 5z4 51.h0 50; k0 50; M50; N 50

(2.258)

In this method, the last boundary conditions are obtained by using the asymptotic conditions: η-N.f 00 5 0; g00 5 0; θ0 5 0; φ0 5 0

(2.259)

and the extra boundary conditions can be replaced by the following conditions: z1 5 z2 5 z3 5 z4 5 1.H00 5 0; K00 5 0; M0 5 0; N 0 5 0

(2.260)

The asymptotic condition, Eq. (2.260), is imposed for computing ηN 1 , ηN 2 , ηN 3 , and ηN 4 . We hope to obtain an approximate solution for this problem in the interval 0 , z , 1. The trial functions are considered to be: H ðz1 Þ: 5 z1 1 c1 z21 1 c2 z31 1 c3 z41 1 ? 1 c6 z71

(2.261)

Kðz2 Þ 5 λz2 1 c7 z22 1 c8 z32 1 c9 z42 1 ? 1 c12 z72

(2.262)

M ðz3 Þ 5

1

1 c13 z3 1 c14 z23 1 c15 z33 1 ? 1 c18 z63

(2.263)

Nt c13 z4 1 c21 z24 1 c22 z34 1 ? 1 c25 z64 Nb

(2.264)

ηN 3

N ðz4 Þ 5 c19 2

whereas the trial solution must satisfy the boundary conditions of Eqs. (2.259) and (2.260) for all values of “c,” thus: 1 1 2c1 1 3c2 1 ? 1 7c6 5 0

(2.265)

λ 1 2c7 1 3c8 1 ? 1 7c12 5 1

(2.266)

1 ηN3

1 c13 1 c14 1 ? 1 c18 5 0

(2.267)

Nt c13 1 c21 ? 1 c25 5 1 Nb

(2.268)

c19 2

ηN 1 , ηN 3 , ηN 4 , and ηN 4 in Eqs. (2.253)(2.256) can be calculated using the extra boundary conditions given in Eq. (2.260). This yields: 2c1 1 6c2 1 12c3 ? 1 42c6 5 0

(2.269)

2c7 1 6c8 1 12c9 1 ? 1 42c12 5 0

(2.270)

c13 1 2c14 1 3c15 1 ? 1 6c18 5 0

(2.271)

Nt c13 1 2c21 1 ? 1 6c25 5 0 Nb

(2.272)

2

Nanofluids

2.4 Other analytical/mathematical modeling

109

By introducing gðz1 Þ, hðz2 Þ, Mðz3 Þ, and Nðz  4 Þ into Eqs. (2.253)(2.256),  the residual functions can be found: R 1 c1 ; c2 ; . . .; c12 ; ηN1 ; ηN2 ; z1 ; z2 ,  R2 c1 ; c2 ; . . .; c12  ; ηN1 ; ηN2 ; z1 ; z2 , R3 c1 ; c2 ; . . .; c25 ; ηN1 ; ηN2; ηN3 ; ηN4 ; z1 ; z2 ; z3 ; z4 Þ and R4 c1 ; c2 ; . . .; c25 ; ηN1 ; ηN2 ; ηN3 ; ηN4 ; z1 ; z2 ; z3 ; z4 . In the OCM, the number of weight functions Wi are: nwi 5 nci 2 nb

(2.273)

where nci are the numbers of unknown constants and ci and nb are the number of equations that satisfy the boundary conditions. Also, the residual function must be close to zero. To reach this aim, specific points in the domain z A ½0; 1 should be chosen. These points are:       1 2 5 R1 5 0; R1 5 0; . . .; R1 50 (2.274) 6 6 6       1 2 5 R2 5 0; R2 5 0; . . .; R2 50 (2.275) 6 6 6       1 2 5 R3 5 0; R3 5 0; . . .; R3 50 (2.276) 6 6 6       1 2 5 R4 5 0; R4 5 0; . . .; R4 50 (2.277) 6 6 6 Here there is a set of nine algebraic equations: four equations (Eqs. 2.2652.268) that satisfy the boundary conditions; four equations (Eqs. 2.2692.272) that satisfy the extra boundary conditions; and 20 equations (Eqs. 2.2742.277) that kept the residual function close to zero. By solving this system of equations, unknown coefficients ci , ηN1 , ηN3 , ηN4 , ηN4 and consequently the velocity and temperature distributions can be determined. For example, using the OCM for an NF with Sc 5 1, Pr 5 25, Nb 5 0:1, Nt 5 0:1, λ 5 0:5, and n 5 3 fðηÞ, gðηÞ, θðηÞ, and ϕ(η) are as shown here: f ðηÞ51:0000000000η20:9682222966η2 10:6478596724η31?10:0018627118η7 gðηÞ50:5000000003η20:4841111523η210:3239298503η31?10:0009313565η7 θðηÞ51:0000000000219:3130049200η1154:9379597000η2 1? 1987:7535926000η6 φðηÞ520:9615385160119:3130049300η2161:1354819000η2 1? 21201:7536980000η6 (2.278)

Nanofluids

TABLE 2.12 Comparison of the OCM and numerical shooting in predicting fv(0) and gv(0) values [15]. n

λ

2f0 (0) shooting

OCM

OCM

2gv(0) shooting

OCM

Error

1

0

1

0.992650

0.007350

0.000000

0.000000

0.000000

1

0.5

1.224745

1.215743

0.009002

0.612372

0.607871

0.004501

1

1

1.414214

1.403819

0.010395

1.414214

1.403819

0.010395

3

0

1.624356

1.606584

0.017772

0.000000

0.000000

0.000000

3

0.5

1.989422

1.936444

0.052978

0.994711

0.968222

0.026489

3

1

2.297186

2.236013

0.061173

2.297186

2.236013

0.061173

OCM, Optimal collocation method.

TABLE 2.13

Comparison of the OCM and numerical results when λ 5 0:5, n 5 3 [15].

η

Numerical

f0 (η) OCM

Error

Numerical

g0 (η) OCM

Error

0

1.000000

1.000000

0.000000

0.500000

0.500000

0.000000

0.1

0.821284

0.824577

0.003293

0.410642

0.412288

0.001646

0.2

0.677489

0.681129

0.003640

0.338745

0.340565

0.001820

0.3

0.560912

0.563781

0.002869

0.280456

0.281890

0.001434

0.4

0.465764

0.467639

0.001875

0.232882

0.233819

0.000937

0.5

0.387641

0.388668

0.001026

0.193821

0.194334

0.000513

0.6

0.323159

0.323571

0.000412

0.161579

0.161786

0.000206

0.7

0.269685

0.269681

0.000004

0.134842

0.134840

0.000002

0.8

0.225157

0.224858

0.000299

0.112578

0.112429

0.000149

0.9

0.187941

0.187400

0.000541

0.093970

0.093700

0.000271

1

0.156735

0.155962

0.000774

0.078368

0.077981

0.000387

1.1

0.130494

0.129483

0.001011

0.065247

0.064741

0.000506

1.2

0.108371

0.107124

0.001247

0.054185

0.053562

0.000623

1.3

0.089677

0.088215

0.001462

0.044839

0.044107

0.000731

1.4

0.073850

0.072210

0.001640

0.036925

0.036105

0.000820

1.5

0.060426

0.058654

0.001771

0.030213

0.029327

0.000886

1.6

0.049022

0.047157

0.001865

0.024511

0.023579

0.000932

1.7

0.039321

0.037379

0.001942

0.019661

0.018690

0.000971

1.8

0.031059

0.029023

0.002037

0.015530

0.014511

0.001018

1.9

0.024016

0.021837

0.002178

0.012008

0.010919

0.001089

2

0.018005

0.015632

0.002373

0.009003

0.007816

0.001187

2.1

0.012873

0.010296

0.002577

0.006436

0.005148

0.001288

2.2

0.008487

0.005830

0.002657

0.004244

0.002915

0.001329

2.3

0.004738

0.002390

0.002349

0.002369

0.001195

0.001174

2.4

0.001533

0.000331

0.001201

0.000766

0.000166

0.000601

OCM, Optimal collocation method.

References

111

In order to ensure the accuracy of the present results, the obtained results corresponding to the shear stress at the surface (i.e., f 00 ð0Þ and g00 ð0Þ) for various values of n and λ are compared with the available published results presented in Table 2.12. This table confirms that excellent agreement exists between the presented analytical method and previous shooting method. For better understanding, sample data and absolute error are presented for other parameter values and velocity profiles in the x- and y-directions in Table 2.13. This table also confirms the high accuracy of this method.

References [1] M. Hatami, Weighted Residual Methods; Principles, Modifications and Applications, first ed., Academic Press, Elsevier, 2017. [2] M. Haghshenas Fard, M. Nasr Esfahany, M.R. Talaie, Numerical study of convective heat transfer of nanofluids in a circular tube two-phase model versus single-phase model, International Communications in Heat and Mass Transfer 37 (2010) 9197. [3] M.N. Ozisik, Heat Conduction, second ed., John Wiley & Sons Inc., 1993. [4] M. Hatami, S. Mosayebidorcheh, J. Geng, D. Jing, Heat transfer and nanofluids flow through the circular concentric heat pipes: a comparative study using least square method (LSM), J. Math. Comput. Sci 17 (2017) 235245. [5] M. Turkyilmazoglu, Anomalous heat transfer enhancement by slip due to nanofluids in circular concentric pipes, Int. J. Heat Mass Transf. 85 (2015) 609614. [6] M. Hatami, S. Mosayebidorcheh, D. Jing, Two-phase nanofluid condensation and heat transfer modeling using least square method (LSM) for industrial applications, Heat Mass Transf. 53 (6) (2017) 20612072. [7] S.S. Nourazar, M. Hatami, D.D. Ganji, M. Khazayinejad, Thermal-flow boundary layer analysis of nanofluid over a porous stretching cylinder under the magnetic field effect, Powder Technol. 317 (2017) 310319. [8] T. Hayat, R. Iqbal, A. Tanveer, A. Alsaedi, Soret and Dufour effects in MHD peristalsis of pseudoplastic nanofluid with chemical reaction, J. Mol. Liquids 220 (2016) 693706. [9] S. Mosayebidorcheh, M. Hatami, Analytical investigation of peristaltic nanofluid flow and heat transfer in an asymmetric wavy wall channel (Part II: Divergent channel), Int. J. Heat Mass Transf. 126 (2018) 800808. [10] M.A. Yousif, M. Hatami, B.A. Mahmood, M. Mehdi Rashidi, Thermal boundary layer analysis of nanofluid flow past over a stretching flat plate in different transpiration conditions by using DTM-Pade´ method, J. Math. Comput. Sci. (JMCS) 17 (1) (2017) 8495. [11] M. Hatami, S. Mosayebidorcheh, D. Jing, Peristaltic flow and heat transfer of nanofluids in a sinusoidal wall channel: two-phase analytical study, J. Anal. 27 (3) (2018) 117. [12] M. Hatami, D. Jing, M.A. Yousif, Three-dimensional analysis of condensation nanofluid film on an inclined rotating disk by efficient analytical methods, Arab J. Basic Appl. Sci. 25 (1) (2018) 2837. [13] M. Khazayinejad, M. Hatami, D. Jing, M. Khaki, G. Domairry, Boundary layer flow analysis of a nanofluid past a porous moving semi-infinite flat plate by optimal collocation method, Powder Technol. 301 (2016) 3443. [14] M. Hatami, M. Khazayinejad, D. Jing, Forced convection of Al2O3water nanofluid flow over a porous plate under the variable magnetic field effect, Int. J. Heat Mass Transf. 102 (2016) 622630. [15] Z. Jiandong, J. Dengwei, M. Hatami, M. Khazayinejad, Three-dimensional and twophase nanofluid flow and heat transfer analysis over a stretching infinite solar plate, Therm. Sci. 22 (2) (2018) 871884.

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C H A P T E R

3 Numerical analysis of nanofluids

3.1 Finite element method in nanofluid In this chapter, some of the useful numerical software as well as a short theory of the methods are presented for the numerical nanofluid (NF) modeling. A finite element method (FEM) is considered by a variational formulation, a discretization strategy, one or more solution algorithms, and postprocessing techniques. This method involves two main steps: (1) dividing the domain of the problem into a collection of subdomains, and (2) recombining all sets of element equations into a global system of equations for the final calculation, systematically. In the FEM, the solution region is considered as built up of many small, interconnected subregions called finite elements. The solution of a general continuum problem by the FEM always follows an orderly step-by-step process [1]. FlexPDE and COMSOL are two open source software packages that enable users to model and solve problems including nonlinear and partial coupled differential equations using FEM. The main advantages of FlexPDE commercial software are: • It does not need discretization by users and solves the governing equations directly. • Grid number changes until the defined convergence condition (dynamic mesh number) is reached. • Possibility to add any source term in the governing equations, • Faster analyzing time and more CPU processing time savings. The following three NF cases are analyzed using the FlexPDE software.

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© 2020 Elsevier Ltd. All rights reserved.

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3. Numerical analysis of nanofluids

3.1.1 Case 1: Hot tubes in a wavy porous channel and nanofluid under variable magnetic field In this example, as shown in Fig. 3.1, a two wavy-side channel including heated tubes is considered under the variable magnetic field effect. This problem has applications such as in the cooling process of fluid-conveying tubes in different industries. The magnetic source is employed by placing a magnetic wire on the center of heated tubes in the xy-plane at the point ða; bÞ, vertically. ðH x ; H y Þ are the components of the magnetic field intensity with the magnetic field strength H, which are considered in the following forms [2]: h  2 i21 γ   y2b HX 5 ðx2a Þ2 1 y2b (3.1) 2π h  2 i21 γ Hy 5 2 ðx2a Þ2 1 y2b ðx 2 a Þ (3.2) 2π

FIGURE 3.1 Schematic of the problem and a sample generated mesh [1].

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H5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 HX 1 Hy

(3.3)

where γ is the magnetic field strength at the source. The boundary conditions are as follows: the two side walls are insulation, wavy walls have cold temperature (Tc), and hot tubes have high temperature (Th). Fe3O4water NF is used in this example due to its magnetic properties and it is assumed to be Newtonian, incompressible, and laminar flow. Using the Boussinesq approximation the governing equations for a steady incompressible two-dimensional laminar NF flow are [1,2]: @v @u 1 50 @y @x    2  @u @u @P @ u @2 u 1u 1 μnf ρnf v 1 2 52 @y @x @x @x2 @y 2 μ20 σnf Hy2 u 1 μ20 σnf Hx Hy v 2

(3.4)

μnf

(3.5)

u K   @v @v @P ρnf v 1 u 1 ρnf β nf gðT 2 Tc Þ 52 @y @x @y  2  μnf @ v @2 v 1 μnf v 1 1 μ20 σnf Hx Hy u 2 μ20 σnf Hx2 v 2 2 2 @x @y K 

ρCp



 nf

(3.6)  2 2   2 @T @T @ T @ T 1u v 1 2 1 σnf μ20 uHy 2vHx 5 knf 2 @y @x @x @y (    2   ) (3.7) @u 2 @v @v @u 2 1 1 μnf 2 12 1 @x @x @y @x 

where ρnf, (ρCp)nf αnf, β nf, μnf, knf, and σnf are defined as presented in Chapters 1, Introduction to nanofluids, and Chapter 2, Mathematical analysis of nanofluids. The following dimensionless variables are presented:     Hy ; Hx ; H p P 5  2 ; Hy ; Hx ; H 5 H0 α ρf Lf T 2 Tc uL vL a b y x ; U 5 ; V5 ; a5 ; β 5 ; Y5 ; Χ 5 αf αf L L L L Th 2 Tc     H; H x ; H y pR2 γ P 5  2 ; H; Hx; Hy 5 ; H0 5 H ða; 0Þ 5 2πjbj H0 ρf αf

Θ5

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(3.8)

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3. Numerical analysis of nanofluids

Using Eq. (3.8), Eqs. (3.4)(3.7) can be written in dimensionless form as: @V @U 1 50 (3.9) @Y @X " #  μnf =μf @2 U @U @U @2 U V 1U 51 Pr 1 @Y @X @X2 @Y2 ρnf =ρf " # " #  @P σnf =σf  2 Pr μnf =μf 2 2 2 Ha Pr Hy U 2 Hx Hy V 2 U @X Da ρnf =ρf ρnf =ρf 2

V

30

(3.10)

1

2 2 μnf =μf @V @V 5@@ V 1 @ V A 1U 5 Pr4 @Y @X @X2 @Y2 ρnf =ρf 2 3   σ =σ @P nf f 5 H 2 V 2 Hx Hy U 2 Ha2 Pr4 2 x @Y ρnf =ρf 2 3 2 3 μnf =μf β nf Pr 4 5V 1 RaPr4 5Θ 2 Da ρnf =ρf βf

(3.11)

2

3 2 3 σnf  2  7 @ Θ @2 Θ 6 σ f 7

@Θ @Θ 6 7 7 2UHy 1VHx 2 1V 56 U 1 1 Ha2 Ec6 4 5 4 5 2 2 ðρCp Þnf ðρCp Þnf @X @Y @X @Y ðρCp Þf ðρCp Þf 2 3 (   μnf  2  2 ) 2 6 μf 7 @U @V @U @V 7 1 Ec6 4ðρCp Þnf 5 2 @X 1 2 @Y 1 @Y 1 @X ðρCp Þf knf kf

(3.12) With dimensionless parameters: sffiffiffiffiffi gβ f L3 ðΔTÞ υf σf  ; Pr 5 ; Ha 5 Lμ0 H0 Raf 5  ; αf μf αf υf   αf μ f kf K i ; Da 5 2 ; αf 5   ; Ec 5 h  L ρC 2 p f ρCp f ðΔTÞL

υf 5

μf

(3.13)

ρf

The thermophysical properties of Fe3O4 and water were presented in Chapter 1, Introduction to nanofluids. The stream function, vorticity, and local and average Nusselt numbers are defined as:

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FIGURE 3.2 Validation of the code for streamlines (right) and temperatures (left). (A) Results by Sheikholeslami and Ganji [2]; (B) present study for ϕ 5 0.04, Ha 5 10, and Ra 5 104 [1].

Ω5

ωL2 ; αf

Ψ5

ψ ; αf

ν52

@ψ ; @x

u5 (

@ψ ; @y

ω52

@u @v 1 @y @x

u 5 v 5 0.Ψ 5 0:0 Τ 5 Τ c .Θ 5 0:0 8 < u 5 v 5 0.Ψ 5 0:0 @Θ On all the insulated walls: : @ν 5 0:0 ( u 5 v 5 0.Ψ 5 0:0 On the heated tubes walls: Τ 5 Τ h .Θ 5 1:0

(3.14)

On the cold wavy walls:

Also, the local and average Nusselt numbers are:   knf @Θ Nuloc 5 kf @r ð 1 ‘ Nuave 5 Nuloc ðζÞdζ ‘ 0

(3.15)

(3.16) (3.17)

where r is the tube radius and ‘ is the length of the curve that local and average Nusselt numbers wants to be calculated over them. Firstly, the validation of FEM is confirmed by the model using a square geometry from Sheikholeslami and Vajravelu [3]. For the FlexPDE code, it is necessary to replace the pressure gradient in the governing equation with the equivalent velocity gradient from the literature. Fig. 3.2 compares the results of streamlines and temperature contours for ϕ 5 0.04,

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FIGURE 3.3 Nanoparticle volume fraction effect on temperature and streamlines when a 5 0.5, Da 5 10, Ha 5 20, and Ra 5 104 [1].

Ha 5 10, and Ra 5 104. As seen, the value of temperatures and streamlines are an acceptable agreement with Sheikholeslami and Vajravelu [3]. Also, as seen in Eqs. (3.1) and (3.2), the intensity of the magnetic field varies radially, so near the source has strength magnetic force while the outer walls are under small magnetic effect. The effect of nanoparticle volume fraction is presented in Figs. 3.3 and 3.4 as contours and curves, respectively. Fig. 3.3 shows that the larger nanoparticle volume fraction makes wider vortexes around the tubes, which leads to better heat transfer, so temperature contours and consequently Nusselt numbers will increase in these locations. Fig. 3.4

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3.1 Finite element method in nanofluid

Local Nusselt number

(A) 35

ϕ = 0.02 ϕ = 0.04 ϕ = 0.06

30

25

20

0

0.05

0.1

0.15

0.2

0.25

0.3

Central circle length (B) 20

ϕ = 0.02 ϕ = 0.04 ϕ = 0.06

Local Nusselt number

15

10

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Bottom wavy wall length

FIGURE 3.4 Local Nusselt number for (A) center circle and (B) wavy bottom wall for different nanoparticle volume fractions [1].

confirms that ϕ has a significant and clear effect on the Nusselt numbers, so it is more suitable to consider this parameter as the most changeable parameter to reach the required heat transfer. Increasing ϕ leads to an increase in heat transfer as well as the average Nusselt number as presented in Fig. 3.4.

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FIGURE 3.5 Schematic of the cavity with wavy inner wall function: r 5 rin 1 A cos(N (ξ)) [4].

3.1.2 Case 2: Circular-wavy cavity filled by nanofluid In the second case of FEM analysis, a circular-wavy cavity as shown in Fig. 3.5 is modeled that has an inner wavy wall with the function of: r 5 rin 1 A cosðNðζÞÞ

(3.18)

where rin is the base circle radius; r is the true inner wall; A and N are amplitude and number of undulations, respectively; and ζ is the rotation angle. More parameter and boundary conditions are presented in Fig. 3.5. Since the condition of this study is natural convection of NFs in the cavity, the density of the NFs is considered as presented in Chapter 2, Mathematical analysis of nanofluids [4]: n o (3.19) ρ 5 φρp 1 ð1 2 φÞρp Dφρp 1 ð1 2 φÞ ρf0 ð1 2 β ðT 2 Tc ÞÞ where ρ, ρp, and ρf are the densities of the NF, nanoparticles, and base fluid, respectively; Tc is the reference temperature; ρf0 is the base fluid’s density at the reference temperature; and β is the volumetric coefficient of expansion [5].

(3.20) ρDφρp 1 ð1 2 φÞ ρ0 ð1 2 β ðT 2 Tc ÞÞ @u @v 1 50 (3.21) @x @y

 2  @u @u @p @ u @2 u 1v 1μ ρf u 1 52 (3.22) @x @y @x @x2 @y2

 2      @v @v @p @ v @2 v  1 2ρ 52 1μ 2 φ2φ g1 12φc ρf0 ðT2Tc Þg ρ ρf u 1v c p f 0 @x @y @y @x2 @y2 (3.23)

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3.1 Finite element method in nanofluid

"

 2  ðρcÞp @T @T @ T @2 T @φ @T @φ @T 1 u 1v 5α 1 1 D B @x @y @x2 @y2 @x @x @y @y ðρcÞf  ( 2  2 )# DT @T @T 1 1 @x @x Tc





2

  2

@φ @φ @ φ @2 φ DT @ T @2 T 1 1 1 u 1v 5DB @x @y @x2 @y2 @x2 @y2 Tc

(3.24)

(3.25)

where u and v represent the velocities in the x and y directions; T is temperature; p is pressure; and ϕ gives the nanoparticle concentration. Continuity, momentum under Boussinesq approximation, and energy equations for the laminar and steady state natural convection in a twodimensional form are presented in Eqs. (3.22)(3.25). Boundary conditions are considered constant values for temperature and nanoparticle concentration for inner and outer walls while their normal gradient is assumed zero for other boundaries due to insulation, that is: T 5 Th ; φ 5 φh on the inner sinusoidal boundary T 5 Th ; φ 5 φc on the outer circular boundary @T @φ 5 5 0 on two other insulation boundaries @n @n

(3.26)

ψ 5 0 on all the solid boundaries The stream function and vorticity are considered as follows: u5

@ψ @ψ @v @u ; v52 ; w5 2 @y @x @x @y

(3.27)

By introducing the following nondimensional variables: X5

x y wL2 ψ T 2 Tc φ 2 φc ; Y5 ; Ω5 ; Ψ 5 ; Θ5 ; Φ5 L L α α Th 2 Tc φh 2 φc

(3.28)

and using these dimensionless parameters, the governing equations will be [5]:      2  @Ψ @Ω @Ψ @Ω @Ω @2 Ω @Θ @Θ 2 2 Nr 1 5 Pr (3.29) 1 Pr Ra @Y @X @X @Y @X2 @Y2 @X @X      2  2 ! @Ψ @Θ @Ψ @Θ @ 2 Θ @2 Θ @Φ @Θ @Φ @Θ @Θ @Θ 2 2 1 5 2 1 2 1Nb 1Nt @Y @X @X @Y @X @Y @X @X @Y @Y @X @Y (3.30)

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3. Numerical analysis of nanofluids



    2  @Ψ @Φ @Ψ @Φ 1 @2 Φ @2 Φ Nt @ Θ @2 Θ 2 1 1 5 1 @Y @X @X @Y Le @X2 @Y2 Nb Le @X2 @Y2



@2 Ψ @2 Ψ 1 52Ω @X2 @Y2

(3.31) (3.32)

where the thermal Rayleigh number, the buoyancy ratio number, the Prandtl number, the Brownian motion parameter, the thermophoretic parameter, and the Lewis number of the NF are defined as: Ra 5 (1 2 φc) ρf0gβL3(Th 2 Tc)/(μα), Nr 5 (ρp 2 ρ0)(φh 2 φc)/[(1 2 φc)ρf0gβL(Th 2 Tc)], Pr 5 μ/ρfα, Nb 5 (ρc)pDB(φh 2 φc)/((ρc)fα), Nt 5 (ρc)pDT(Th 2 Tc)/[(ρc)fαTc], and Le 5 α/DB, respectively. The boundary conditions of these dimensionless parameters are as follows: Θ 5 1; Φ 5 1 on the inner sinusoidal boundary Θ 5 1; Φ 5 1 on the outer circular boundary @Θ=@n 5 @Φ=@n 5 0 on two other insulation boundaries Ψ 5 0 on all solid boundaries

(3.33)

The local Nusselt number on the cold circular wall can be introduced as [5]: Nuloc 5 2

@Θ @n

(3.34)

where n is the normal direction to the outer cylinder surface. The average number on the defined wall is defined as: ð 1 0:5π Nuave 5 Nuloc ðξÞdξ (3.35) 0:5π 0 Fig. 3.6 compares the present results with previous works for different Rayleigh numbers when Pr 5 0.7. This figure confirms that our FEM code is accurate enough. Figs. 3.73.9 are achieved from the numerical methods of nine different cases using geometries that show the isotherm, streamlines, and nanoparticle volume fractions, respectively.

3.1.3 Case 3: Wavy porous cavity filled with nanofluid in the presence of solar radiation For the third and last FEM case, a two-dimensional schematic diagram of a porous wavy cavity is considered as shown in Fig. 3.10A. Like the pervious examples, the NF flow in the cavity is assumed to be incompressible and laminar. Furthermore, water and nanoparticles are

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3.1 Finite element method in nanofluid

123

FIGURE 3.6 Comparison of the present results with previous works for different Rayleigh numbers when Pr 5 0.7 [4].

FIGURE 3.7 Temperature contours for nine different cases proposed by central composite design (CCD) [4].

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3. Numerical analysis of nanofluids

FIGURE 3.8 Streamlines contours for nine different cases proposed by central composite design (CCD) [4].

in thermal equilibrium and no slip occurs between them. The top wall of the cavity has a wavy shape due to the function mentioned on Fig. 3.10A and in the presence of solar radiation and the bottom wall is completely insulated. Other walls are in constant and cold temperature. The porous area of the cavity is considered to be isotropic, homogeneous, and saturated with a single-phase. For modeling the NF, first its thermal properties must be calculated as the formula presented in Chapter 2, Mathematical analysis of nanofluids. It is suitable to solve the governing equations in nondimensional forms, so the following variables are used to obtain the nondimensional governing equations [6]:

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3.1 Finite element method in nanofluid

125

FIGURE 3.9 Contours of nanoparticle volume fraction for nine different cases proposed by central composite design (CCD) [4].

x L y Y5 L uL U5 αnf vL V5 αnf X5

P5

pL2 ρnf α2nf

θ5

ðT 2 TN Þk qvL

Nanofluids

(3.36)

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3. Numerical analysis of nanofluids

The nondimensional governing parameters defined in the present simulation are as follows: vnf αnf K Da 5 2 L

Pr 5

  gβ nf L3 qvL =k Ra 5 vnf αnf

(3.37)

where L is the length of the cavity’s wall. Also, K in the Darcy equation for a packed bed as a close model of a spherical bed with diameter d can be defined as: 2 2 1 ε dp K5 150 ð12εÞ2

(3.38)

Using the abovementioned assumptions, the nondimensional forms of the governing equations for steady, laminar, natural convection inside the porous cavity in the form of the NavierStokes and energy equations are [7]: @U @V 1 50 @X @Y  2  @U @U @P @ U @2 U Pr U 1V 52 1 Pr U 1 2 2 2 @X @Y @X @X @Y Da  2  @V @V @P @V @2 V Pr 1V 52 1 Pr V 1 RaPrθ U 1 2 @X @Y @Y @X2 @Y2 Da U

@θ @θ @2 θ @2 θ 1V 5 1 @X @Y @X2 @Y2

(3.39) (3.40) (3.41) (3.42)

And the local Nusselt number on the wavy wall can be expressed as: Nuloc 5

1 θs

(3.43)

where s is the wavy surface. The boundary condition is as follows: all solid boundaries U 5 V 5 0. For the top plate, a constant heat flux is considered due to solar radiation and the bottom wall is insulated and the side walls are in constant temperature (θ 5 0). The equations that define the movement of nanoparticles and forces acting on them are based on the Lagrangian frame of reference as introduced in Section 1.5:

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3.1 Finite element method in nanofluid

127

dvp 5F dt

(3.44)

F 5 FD 1 FG 1 FL 1 FP 1 FV

(3.45)

where FD was the hydrodynamic drag force from the fluid and was defined as: FD 5 6πμl ri ðui 2 up Þ

(3.46)

And FG was the force due to gravity: FG 5

gðρp 2 ρl Þ

(3.47)

ρp

The Staffman’s lift force, FL, was defined by: FL 5

2KS v1=2 ρl dij

ðvl 2 vp Þ

ρp dp ðdlk dkl Þ1=4

(3.48)

where Ks 5 2.594 was a constant, dij was the deformation tensor defined as dij 5 1/2(vli,j 1 vlj,i), and FP was the force due to the gravity gradient: ! ρl (3.49) FP 5 vp rvl ρp



FV was the virtual mass force: FV 5

1 ρl d ðvl 2 vp Þ 2 ρp dt

(3.50)

In Ref. [6] by applying BoxBehnken Design (BBD), 25 different geometry cases (details are in Tables 3.1 and 3.2) were modeled for optimization. Here, the results of the natural convection heat transfer for the first six cases are presented in the wavy cavity shown in Fig. 3.10. Furthermore, the temperature contours of these six cases obtained by FEM are presented in Fig. 3.11 and their streamlines are also shown in Fig. 3.12. TABLE 3.1 Parameters under study and their levels.

Heat flux

21 Level

0 Level

11 Level

1

3

5

25

23.30

Darcy

10

10

1023

Rayleigh

103

104.7

105

Am

0.1

0.2

0.3

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3. Numerical analysis of nanofluids

TABLE 3.2 Tests proposed by BoxBehnken Design (BBD) (25 different case3 central points) [6]. StdOrder

Am

Rayleigh number

Darcy number

Solar heat flux

1

0

0

21

21

2

0

0

21

1

3

0

0

1

21

4

0

0

1

1

5

21

21

0

0

6

21

1

0

0

7

1

21

0

0

8

1

1

0

0

9

21

0

0

21

10

21

0

0

1

11

1

0

0

21

12

1

0

0

1

13

0

21

21

0

14

0

21

1

0

15

0

1

21

0

16

0

1

1

0

17

0

21

0

21

18

0

21

0

1

19

0

1

0

21

20

0

1

0

1

21

21

0

21

0

22

21

0

1

0

23

1

0

21

0

24

1

0

1

0

25

0

0

0

0

26

0

0

0

0

27

0

0

0

0

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3.2 Finite volume method in nanofluid

129

FIGURE 3.10 (A) Schematic of the porous wavy cavity and (B) sample mesh generated [6].

FIGURE 3.11

Temperature contours for all six cases [6].

3.2 Finite volume method in nanofluid The second presented numerical method is the finite volume method (FVM) using some well-known numerical software such as ANSYS-FLUENT based on this method. More details about the

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FIGURE 3.12 Streamlines contours for all six cases [6].

numerical FVM and its application can be found in Ref. [8]. The convergence criteria in this method must be defined as [9,10]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X δn11 2δn 2 # 1028 (3.51) n11 i;j δ j i where n is number of iterations and δ shows the general variable of (U, V, θ). To verify the accuracy of the results, a comparison between the obtained results and other available results must be done as in previous methods. The following two examples (using FORTRAN code and ANSYS-FLUENT) are investigated for better understanding of this method application.

3.2.1 Case 1: Nanofluid natural convection for an F-shaped cavity under magnetic field effects In the first case of FVM (using FORTRAN code) based on Fig. 3.13, a two-dimensional closed F-shape cavity filled by silver-water NF with length and width 5 L is modeled. A constant magnetic field is affected on the cavity. The left vertical wall is at constant temperature Th, the middle horizontal walls are at Tc, and the other walls are insulated [10].

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3.2 Finite volume method in nanofluid

It is assumed that the NF is a Newtonian and incompressible fluid, the flow is laminar, steady, and without any energy generation or absorption, and the viscosity dissipation is neglected. The dimensionless governing equations can be written as [11]: @U @V 1 50 @X @Y

(3.52)

  μnf @2 U @U @U @P @2 U U 1V 52 1 1 (3.53) @X @Y @X @Y2 ρnf ν f @X2   μnf @2 V @2 V ðρβ Þnf ρf σnf 2 @V @V @P 1V 52 1 U 1 Ra Pr θ2 Ha Pr V 1 2 2 @X @Y @Y ρnf ν f @X @Y ρnf β f ρnf σf

 



U

@θ @θ αnf @2 θ @2 θ 1V 5 1 @X @Y αf @X2 @Y2



 

(3.54) (3.55)

The dimensionless variables are as follows: X5

x y uL vL h l0 pL2 T 2 TC ; 5 ; U5 ; V5 ; H 5 ; L0 5 ; P 5 ; θ5 L L αf αf L L ρnf αf 2 Th 2 T C (3.56)

The dimensionless Reyleigh, Prandtl, and Hartman numbers are defined as: sffiffiffiffiffiffiffiffiffi νf σf gβL3 ðTh 2 TC Þ Ra 5 (3.57) ; Pr 5 ; Ha 5 B0 L ν f αf αf ρf ν f and Gr 5 gβ f ðTh 2 TC ÞL3 =ϑf 2 is Grashof number. Due to boundary conditions, because there is no suction or injection, U,V 5 0. Moreover, the thermal boundary condition for the heat source surface is θ 5 1, the right wall with cold temperature is θ 5 0, and the temperature gradient is zero for the insulated walls. The Nusselt number and local Nusselt number on the hot walls are defined as Eqs. (3.58) and (3.59), respectively:   knf @θ Nu 5 2 (3.58) kf @X X50 ð  1 1 Nu dY (3.59) Num 5 L 0 The NF property relations are chosen as presented in Chapter 1, Introduction to nanofluids, and a FORTRAN program is used for

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3. Numerical analysis of nanofluids

FIGURE 3.13 F-shaped cavity under the magnetic field [10].

FIGURE 3.14

Comparison between finite volume method (FVM) and the reported

data [10].

numerical simulation of the problem. The governing equations are solved using the FVM with the SIMPLE algorithm. In Ref. [10], the FVM results of average Nusselt number inside a Cshaped cavity with three inner walls in low temperature and three outer walls in high temperature are compared to available results in the literature [9]. Various nanoparticle ratios with Ra 5 105 for dimensional ratio of (AR 5 L/H) have been studied. According to Fig. 3.14,

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3.2 Finite volume method in nanofluid

TABLE 3.3

Network point effect on average Nusselt number. 40 3 40

Grid Num

60 3 60

80 3 80

100 3 100

120 3 120

Ra 5 10

3.253

3.219

3.205

3.200

3.200

Ra 5 106

17.880

17.821

17.789

17.786

17.774

4

FIGURE 3.15

Cavity meshing [10].

it can be concluded that the obtained results are accurate. For the grid study, it must be shown that the results are independent of the grid numbers. The results of the grid study are shown in Table 3.3, which shows that the uniform mesh 100 3 100 has enough accuracy for computation (see Fig. 3.15). Finally, sample results are presented in Fig. 3.16.

3.2.2 Case 2: Different nanofluid flow through venturi In the second case, the effect of convergent/divergent channels on the heat transfer of NFs in a venturi is investigated using FVM based on ANSYS-FLUENT software [12]. In this example, the k-ε turbulence model with enhanced wall function using ANSYS software is used for solving the problem. In this case, three different nanoparticles (SiO2, Al2O3, and CuO) in water base fluid are considered (Fig. 3.17). The following assumptions are considered for obtaining the governing equations:

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3. Numerical analysis of nanofluids

Ra = 104

Ra = 105

Ra = 106

Isotherm

Streamline

Ra = 103

FIGURE 3.16 Flow and temperature lines with different Reyleigh numbers in Ha 5 30 for pure fluid (—) and nanofluid of ϕ 5 0.04 (- - -) [10].

FIGURE 3.17 Venturi tube and boundary conditions [12].

• The flow is steady and noncompressible. • All the thermal properties are independent of temperature. • Venture is under constant heat flux. The governing equations (continuity, momentum, and energy) for the NF flow in a venturi are as follows. The continuity equation will be [12]:  @  ρnf ui 5 0 (3.60) @xi where ρnf is the NF density and ui is the axial velocity in the tube. The momentum equation is:      @uj @  @P @ @ui @  0 0 ρnf ui uj 5 2 1 μnf 1 2ρnf ui uj 1 (3.61) @xj @xi @xj @xj @xj @xi

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3.2 Finite volume method in nanofluid

135

where μnf is the viscosity of the NF, u0 is the fluctuating velocity, and uj is the axial velocity components. The term of ρnf ui 0 uj 0 is turbulent shear stress or Reynolds stress. The energy equation will be:    @  @ @T ρnf ui T 5 ð Γ 1 Γt Þ (3.62) @xi @xj @xj where Γ and Γt are: Γ5

μnf Pr

and Γt 5

μt;nf Prt

(3.63)

As mentioned before, the standard k 2 ε turbulence model is used for the modeling:     @uj @ui 0 0 1 2ρnf ui uj 5 μt;nf (3.64) @xj @xi where eddy or turbulent viscosity will be: μt;nf 5 ρnf Cμ

k2 ε

(3.65)

The turbulence kinetic energy and dissipative rate equations are:    i μt;nf @k @ h @ ρ kui 5 μnf 1 (3.66) 1 Gk 2 ρnf ε @xi nf @xj σk @xj    i μt;nf @ε @ h @ ε ε2 (3.67) ρnf εui 5 μnf 1 1 C1ε Gk 1 C2ε ρnf @xi @xj k σε @xj k where Gk represents the generation of turbulence kinetic energy and ερ is its dissipation rate. The following constants are used based on experimental procedures: C1ε 5 1.44, C2ε 5 1.92, Cμ 5 0.09, σk 5 1.0, and σε 5 1.3. The thermophysical properties of the NFs including density, heat capacity, thermal conductive coefficient, and viscosity (by considering the Brownian motion for the nanoparticles) will be [12]: ρnf 5 ð1 2 ϕÞρf 1 ϕρnp

(3.68)

ðρCp Þnf 5 ð1 2 ϕÞðρCp Þf 1 ϕðρCp Þnp  10  0:03 knf knp 0:4 0:66 T 5 1 1 4:4Renp Pr ϕ0:66 Tfr kf kf

(3.69)

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(3.70)

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3. Numerical analysis of nanofluids

In Eq. (3.70), Re is the Reynolds number for the nanoparticles, Pr is the Prandtl for the base fluid, T represents the NF temperature, Tfr is the freezing temperature of the base fluid, knp is thermal conductivity for the nanoparticles, and ϕ is the nanoparticle volume fraction. The thermal diffusivity is: αnf 5

knf ðρCp Þnf

(3.71)

And the nondimensional Nusselt number is defined as: Nu 5

2q_ Ts 2 Tbulk

kf

Dh

(3.72)

where q_ is the heat flux in the lower wall, Tbulk is the average temperature of the inlet and outlet, and Dh is the hydraulic diameter for the tube. The boundary conditions and assumptions must be defined in the software as: inlet velocity 0.005 m s21, inlet temperature 283 K, heat flux on the wall 40 kW m22, and convection coefficient 30 W m22 K21. The grids are tetrahedrons with 78,419 nodes and 167,640 elements, and boundary layer meshing is used from the inflation item in ANSYSFLUENT 17.0 with 8 faces and 1.2 growth rate. Fig. 3.18 shows the ICEM meshing as a sample case and Figs. 3.19 and 3.20 are depicted from the outcomes of the NF modeling. Fig. 3.19 depicts Al2O3water NF and shows that by increasing the nanoparticle volume fraction, the temperature in the centerline will increase due to greater thermal conductivity (Fig. 3.19A), while the velocity in the centerline (Fig. 3.19B) will decrease due to denser fluid with greater volume fraction. The surface Nusselt number on the venturi section is depicted

FIGURE 3.18 Generated mesh for venturi [12].

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FIGURE 3.19 (A ) Central temperature, (B) velocity magnitude, (C) Nusselt number, and (D) temperature contour for Al2O3water nanofluid with different nanoparticle volume fraction [12].

FIGURE 3.20 (A) Central temperature, (B) velocity magnitude, (C) Nusselt number, and (D) velocity contour for different nanoparticles [12].

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3. Numerical analysis of nanofluids

in Fig. 3.19C and confirms that more heat transfer from the constant wall heat flux to the NF in the lower volume fractions; a schematic of the temperature solution is presented in Fig. 3.19D. Fig. 3.20 compares the three different effects on heat transfer and confirms that the SiO2water NF has the highest centerline temperature among the NFs, while CuO has the most velocity in the downstream of the venturi. Also, SiO2 has the highest surface Nusselt number among the tested cases. A sample velocity contour for the Al2O3water NF is depicted in Fig. 3.20D where the maximum velocity occurred in the throat section due to the continuity principle.

3.3 Lattice-Boltzmann method in nanofluid In this section, a comprehensive model is introduced by coupling a two-phase lattice Boltzmann method (LBM) and population balance equations (PBEs) to describe the dynamic particle aggregations in flowing and heated colloidal suspensions [13]. Our theoretical model will be presented in three parts: (1) two-phase LBM for flow and heat transfer, (2) population balance equations for particle size distribution (PSD), and the (3) coupling of these two methods.

3.3.1 Two-phase lattice Boltzmann method LBM is a mesoscopic method that is widely applied for studying fluid flow and transport processes. Both single-phase and two-phase lattice Boltzmann models have been employed to explore the flow and heat transfer of suspensions in previous studies [13,14]. Since the singlephase model cannot consider the interphase forces (such as drag force, Brownian force, etc.) and particle migration is not very accurate, the two-phase model is presented here to achieve a better accuracy. The density evolution equation of a two-phase LB model can be written as [13,14]: fασ ðr 1 eα δt ; t 1 δt Þ 2 fασ ðr; tÞ 52

 2τ σf 2 1 Fσα δt eα 1 σ 0 σeq f ð r; t Þ 2 f ð r; t Þ 1 1δt Fσα : α α σ σ 2 τf 2τ f Bα c (3.73)

where σ 5 1 and 2 expresses the liquid and solid phases; eα and α are the lattice velocity vector and direction; r is the lattice position; τ σf is σeq the dimensionless collisionrelaxation time of the flow field; fα ðr; tÞ is the local equilibrium distribution function of fασ ðr; tÞ; t and δt are time and time step, respectively; c 5 δx/δt is the reference lattice velocity; Bα is the weight coefficient to distribute the total interparticle interaction forces

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3.3 Lattice-Boltzmann method in nanofluid 0

to the lattice; Fσα and Fσα are interphase force and external force, respectively, where Fσα includes buoyancy force, drag force, interaction potential σ 0 σeq force, Brownian force, etc.; Fσα 5 G ðeα 2p u Þ fα ðr; tÞ, G 5 2 β ðT 2 T0 Þg is the effect of buoyancy lift induced by density change; uσ is the macroscopic velocity of the lattice; β is the thermal expansion coefficient; and T and T0 are local and reference temperature, respectively. For a D2Q9 model, nine discrete velocities eα exist, which are given at different direction α as follows: 8 0 2 α50 > > 0 3 2 31 > > > > > > @ 4ðα 2 1Þπ5; sin4ðα 2 1Þπ5A > α 5 1; 2; 3; 4 < cos 2 2 : eα 5 8 2 3 2 39 > > > < = > > > pffiffi2ffi cos4ðα 2 5Þπ 1 π5; sin4ðα 2 5Þπ 1 π5 > α 5 5; 6; 7; 8 > > : 2 4 2 4 ; :



(3.74) The density equilibrium distribution function can be introduced as [15]:    eα uσ ðeα uσ Þ2 uσ uσ fασeq 5 ωα ρσ 1 ρσ0 1 2 ; (3.75) c2s 2c4s 2c2s pffiffiffi P where cs 5 c= 3 refers to velocity of the lattice; ρσ0 5 8α50 fασ ðr; tÞ is the P density; ρσ  ρσ0 for low velocity flow (Ma , 0.3); uσ 5 8α50 eα fασ ðr; tÞ the macroscopic velocity; and ωα is equal to 4/9 (α 5 0), 1/9 (α 5 1, 2, 3, 4), and 1/36 (α 5 5, 6, 7, 8). Based on Boussinesq theory, the density change with temperature is assumed only due to the volume force in the momentum equation and other parameters such as pressure are supposed to be constant, the temperature evolution equation is reported as [16]:



Tασ ðr 1 eα δt ; t 1 δt Þ 2 Tασ ðr; tÞ 5 2





 1  σ Tα ðr; tÞ 2 Tασeq ðr; tÞ ; σ τT

(3.76)

where Tασ ðr; tÞ is the temperature distribution function and τ σT is the dimensionless collisionrelaxation time of temperature. The temperature equilibrium distribution function is:   eα u σ 1 ðeα uσ Þ2 uσ uσ σeq σ Tα 5 ω α T 1 1 1 2 ; (3.77) 2 2c4s c2s 2c2s P where Tσ 5 8α50 Tασ ðr; tÞ is the temperature of the given lattice.



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140

3. Numerical analysis of nanofluids

TABLE 3.4 The interaction force between particles and liquid [13]. Force

Expression

Buoyancy force FH

3 FH 5 2 4πa 3

Drag force FD

FD 5 2 6πμaΔu qffiffiffi C FB 5 Gi dt

Brownian force FB Interaction potential force FA

FA 5

P8 i51

Action target 0

Particles

gΔρ

Liquid and particles Liquid and particles

A ni @V @r i

Particles

The controlled criterion numbers are Rayleigh number (Ra), Mach number (Ma), and Prandtl number (Pr) and can be written as: Ra 5

gβΔTH 3 Pr νσ uσc ; Pr 5 ; Ma 5 ; χσ cs ν σ2

(3.78)

where g is the acceleration of gravity; χσ is the thermal diffusion coefficient; H is the characteristic height of a cavity; and ν σ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uσc 5 gβΔTH are the kinematic viscosity and the characteristic velocity of natural convection, respectively. The dimensionless collisionrelaxation time and two parameters ν σ and χσ can be obtained by: pffiffiffiffiffiffiffiffi     MaH 3Pr σ 3ν σ 1 2 σ 1 1 2 σ 1 σ σ σ pffiffiffiffiffiffi ;τ T 50:51 τ f 50:51 ;ν 5 c τ f 2 δt ;χ 5 c τ T 2 δt : 3 2 3 2 Prc2 δt c2 δt Ra (3.79) The forces between particles and liquid that previously were pre0 sented in Section 1.5 are listed in Table 3.4. In those equations, Δρ and Δu are the density and velocity difference between the liquid and solid phase; Gi is the Gaussian random number  with zero mean;

C 5 2 3 ð6πμaÞkB T; VA 5 2 16 A

2a2 L2cc 2 4a2

1

2a2 L2cc

1

L2cc 2 4a2 L2cc

is the interaction

potential where A is Hamaker constant; Lcc is the center-to-center distance between nanoparticles; and ni is the particle number in the adjacent lattice i. The total per-unit volume force on a lattice for particles Fp and liquid Fw can be written as [13]: Fp 5

nðFH 1 FD 1 FB 1 FA Þ nðFD 1 FB Þ ; Fw 5 2 ; V V

(3.80)

where n is the number of particles in the given lattice and V is the volume of lattice. Since the external and internal forces can affect the macroscopic velocities of both solid and liquid phases, they are introduced as [13]:

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3.3 Lattice-Boltzmann method in nanofluid

upnew 5 up 1

F p Δt F w Δt ; uwnew 5 uw 1 : p 2ρw 2ρ

141 (3.81)

In the same way, the temperatures of the two phases are modified as: σ Tnew 5 Tσ 1 δtτ T

dT 5 Tσ 1 δtτ T Φαβ ; dt

(3.82)

where hαβ is the convective heat transfer coefficient of the colloid sus  pension and Φαβ 5 hαβ Tβ ðx; t 2 δtÞ 2 Tα ðx; t 2 δtÞ =ρα cpα aα is the energy exchange between the nanoparticles and base fluid. The definition of the Nusselt number is: Nu 5

hH ; knf

(3.83)

where h 5 qw =ðTH 2 TL Þ and knf 5 2 qw =ð@T=@xÞ are the heat transfer coefficient and thermal conductivity, respectively. By introducing h and knf, the Nusselt number will be: ð1   @T H ; Nuavg 5 Nu y dy: (3.84) Nu 5 2 @x TH 2 TL 0



3.3.2 Population balance equation The Brownian motion and shear rate can lead to drastic collisions between nanoparticles, which may further induce particle aggregation. Because the aggregation processes can dramatically affect the properties of the colloid suspension, such as its thermal conductivity and viscosity, the particle aggregation and size evolution should be considered in natural convection. The PBE can be applied to predict the evolution of PSD, which can be written as [17]: @ @ nðx; tÞ 1 ½Gðx; tÞnðx; tÞ 5 Qnuc 1 Qagg 1 Qbreak ; @t @t

(3.85)

where nðx; tÞ is the PSD of solid phase; Gðx; tÞ is the growth rate of the particle with volume x; and Qnuc , Qagg , and Qbreak represent the rate of nucleation, aggregation, and breaking, respectively, which can be written as:

Qagg 5

1 2

ðx

Qnuc 5 B0 ðtÞδðx0 Þ; nðx 2 x0 ; tÞnðx0 ; tÞaðx 2 x0 ; x0 Þdx0 2

0

ðN

(3.86) nðx; tÞnðx0 ; tÞaðx; x0 Þdx0 ;

0

(3.87)

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3. Numerical analysis of nanofluids

Qbreak 5

ðN

bðx; x0 ÞΓðx0 Þnðx0 ; tÞdx0 2 ΓðxÞnðx; tÞ;

(3.88)

x

where B0 is the nucleation rate; δ is the Dirac delta function; and x0 is the volume of nuclei. aðx; x0 Þ indicates the aggregation frequency for two particles with volume x and x0 to form a new particle with volume x 1 x0 . Similarly, bðx; x0 Þ is the breaking frequency for a x0 particle to split two particles with volume x and x0 2 x, and ΓðxÞ is the breakage rate. For a colloid suspension, the growth rate Gðx; tÞ and nucleation rate Qnuc are zero. As hyperbolic partial differential equations, the analytical solution of PBEs is difficult to obtain in most cases. Spicer and Pratsinis [18] proposed a sectional model to solve this equation where particle volumes satisfy xi 5 2i21 x0 , i 5 1, 2, 3. . . and x0 is the volume of the primary nanoparticle. According to the discrete volume values, the PBE can be simplified to [18]: i22 i21 X X dNi 1 2 5 2j2i11 β i21;j Ni21 Nj 1 β i21;i21 Ni21 2 Ni 2j2i β i;j Nj 2 dt j51 j51

2 Ni

imax 21 X j51

β i;j Nj 2 Si Ni 1

imax X

(3.89)

Γ i;j Sj Nj ;

j5i

where Ni is the number concentration of aggregates with volume xi ; β i;j and Si are collision kernel and fragmentation kernel, respectively; and Γ i;j is the fragment distribution function. The values of β i;j and Si are related to particle size, flow field, viscosity, and aggregate structure, as shown by the following equations [34]:  1 3 rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 8πkB T 2 Df Df σ ; β i;j 5 0:31Gx0 wi 1wj 1 (3.90) m    q  3=Df μ φtot G 1=3 dc;i x ; (3.91) Si 5 kb 0 τ dp where G represents the local shear rate of flow field; wi 5 xi =x0 is the number of primary nanoparticles in the xi aggregate; Df is the fracture   pffiffiffiffiffi pffiffiffiffiffiffi dimension; m 5 mi mj = mi 1 mj and σ 5 dp ð 3 wi 1 3 wj Þ=2 are the effective mass and effective collision diameter, respectively; mi 5 xi ρp is the mass of the xi aggregate; kb 5 1cm21 s21 is a coefficient to balance the dimension and the colloidal sus is a constant regardless of the kind of ð1=D Þ pensions; μ φtot is the viscosity of the NF; dc;i 5 dp wi f is the collision diameter of xi aggregate; and τ  is introduced to nondimensionalize the

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3.3 Lattice-Boltzmann method in nanofluid

shear stress term as an index indicating the strength of the aggregate. It must me mentioned that τ  is directly related to the interparticle forces, such as the interaction potential force. Thus τ  can be obtained based on the interparticle forces and aggregate structure that calculates particle number and distribution in the aggregates. Therefore, in our study, a typical value of τ  from the literature was used to simplify the model. The complete fragment distribution function Γi;j is very complex. Here only aggregate splitting will be considered [13]: 2 j5i11 : (3.92) Γi;j 5 0 else

3.3.3 Coupling population balance equations and the lattice Boltzmann method method To obtain the PSD by solving the PBEs, the parameters of the flow field (such as local shear rate and concentration) should be identified first. The PSD can also affect the forces of liquid and particles as shown in Table 3.4 and further change the flow field. Thus the calculations of flow field (LBM method) and PSD (PBEs) are joined closely with each other with respect to physical consideration. To relate them in mathematics, three steps (i.e., unifying unit, calculating parameter, and feedback) will be employed. Firstly, in PBEs all the parameters have their own physical units, which is incompatible with LBM units. Thus the first step is to transform the lattice unit to a physical unit using the following relations [19]:   ρ hLi ht i hcs i ; ur 5 ; ρr 5 : (3.93) Lr 5 ; tr 5 ρ L t cs where Lr, ρr, ur, and tr are reference length, density, velocity, and time, respectively. Similarly, hLi, hρi, hcsi, hti, and L, ρ, cs, t are the physical and lattice parameters. Generally, L, ρ, cs, t, and viscosity ν are known in a specific simulation, and hρi, hcsi, hνi are the physical property parameters of the material and should be preset. The reference parameters ρr and ur can thus be known. Additional relations should be considered to obtain the Lr, tr as follows: Lr ur 5

hν i Lr ; tr 5 ; ν ur

(3.94)

where hνi and ν are the physical and lattice kinetic viscosity, respectively.

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3. Numerical analysis of nanofluids

Secondly, when solving PBEs, shear rate G, dynamic viscosity μ(φ), and particle number n in a lattice are required. Based on those reference values, the required parameters can be calculated by the following equations: 8   #2 912  2 " u y 

> > > > >

> >^ > > > : xN ðtÞ; tA½tN ; tN11  where ti 5 ði 2 1ÞH. The multistep DTM for every subdomain defined as: Nanofluids

3.4 Finite difference method in nanofluid

X i ðkÞ 5

  H k dk xi ð t Þ dtk t5ti k!

149 (3.104)

The inverse multistep DTM is: xi ð t Þ 5

  t2ti k X i ðkÞ H k50

N X

(3.105)

Some of the properties of the DTM and the multistep DTM are presented in Chapter 2, Mathematical analysis of nanofluids. To solve the partial equation (including x and y such as in the next example) in the domain xA½0; 1 and yA½0; 1 using hybrid multistep differential transformation method (MDTM) and FDM, the finite difference approximate is applied in the y direction and the MDTM in the x direction. The following finite difference scheme is used based on the uniform mesh. The length in the direction of y is divided into Ny equal intervals.   The y coordinates of the grid points can be obtained by yj 5 j Δy , j 5 0:Ny , where Δy is the mesh size. The x domain is divided into Nx sections. It is assumed that the x subdomains are equal; therefore the length of each subdomain will be H 5 1=Nx . Thus a separate function for every subdomain is defined as: 8   T1 x; j ; xA½x1 ; x2 ; 0 # j # Ny > > > > > >^   <   (3.106) Ti x; y 5 Ti x; j ; xA½xi ; xi11 ; 0 # j # Ny > > > ^ > > >     : TNx x; j ; xA xN ; xNx 11 ; 0 # j # Ny where xi 5 ði 2 1ÞH: An example of applying FDMDTM on NF flow is presented here.

3.4.1 Case 1: Aluminawater nanofluid in an inclined direct absorption solar collector Consider a flat channel of length L and thickness H and an angle θ with the horizontal plane as shown in Fig. 3.25 as a direct solar absorption collector filled by water/alumina NF. NF is allowed to directly absorb the incident solar flux passing through an optically transparent upper-surface panel. It is assumed that the bottom panel is coated with a heat-transferring material or it is presumed to be totally isothermal with a prescribed bottom surface temperature to reduce the heat losses. Other assumptions can be found in Refs. [24,25]. The energy equation when considering a heat generation q* due to solar radiation is [23]:

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3. Numerical analysis of nanofluids

FIGURE 3.25 Schematic of the nanofluid-based direct absorption solar collector [23].

ρnf Cpnf U

@T  @2 T  5 k 1 q nf @x @y2

(3.107)

where 0 , y* , H and 0 , x , L. From the momentum equation the average velocity of the NF will be: U5

ρnf gH2 sin θ

(3.108)

12μnf

By fixing the θ 5 π=9 and approximating the heat source by an appropriate function obtained from solar radiation [24]: q 5

Gs β 0 β 1



H 11β 0 y  H21

β1 11

(3.109)

where Gs is equal to 1000 and the betas function describes the NF properties and optical values of the collector (their values are presented in Refs. [24,25]). The following boundary conditions are assumed: • Convection heat transfer is considered for the upper surface (y* 5 0) with two conditions for the bottom surface: • An insulated base; by definition, a zero convection heat transfer coefficient (hb 5 0). • Isothermal heat transfer with a constant surface temperature. By the above definitions and the following scaled variables: x 5 Lx;

y 5 Hy;



T 5 ΔT T 1 Ti ;

Nanofluids

ΔT 5

Gs Lβ 0 β 1 Hρnf Cpnf U

(3.110)

151

3.4 Finite difference method in nanofluid

the energy equation will be: @T @2 T 1 5γ 2 1  β 11 @x @y 11β 0 y 1

(3.111)

and the boundary conditions are: T x50 5 0   @T 5 Nu T y50 2 Th @y y50

A:



@T @y y51

5

 Nu  hb T y51 2 Th hs

(3.112)

B: T y51 5 Te

where Th and Te are temperature difference and bottom panel temperature, respectively. Gamma represents the thermal diffusion parameter and Nu is the Nusselt number defined by: Th 5

Tamb 2 Ti ; ΔT

Te 5

Te 2 Ti ; ΔT

γ5

knf L ; ρnf Cpnf UH 2

Nu 5

Hhs knf

(3.113)

where Te and Ti denote the panel temperature and the inlet temperature at x 5 0, respectively. The solution of the system of Eq. (3.111) and the nonlinear boundary conditions of Eq. (3.112) can be assumed as: for 1 # i # Nx ; 0 # j # Ny m   k   X i Ti x; j 5 T i k; j x2x H 

xA½xi ; xi11 

(3.114)

k50

   where Ti k; j is the differential transform of T x; y . After taking the second-order central finite difference approximation with respect to y and applying MDTM on Eq. (3.114) for the x domain, the following equations can be obtained: for 1 # i # Nx ;

1# 8 j # Ny 2 1 9 =        H <  1 Ti k 1 1; j 5 Ti k; j 1 1 2 2Ti k; j 1 Ti k; j 2 1 1  β1 11 ; k11: 11β 0 yj (3.115) Applying MDTM on the initial condition in the x direction:   for 0 # j # Ny ; T 1 0; j 5 0

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(3.116)

152

3. Numerical analysis of nanofluids

The boundary conditions in Eq. (3.112) can be transformed as follows:   T i ðk; 1Þ 2 T i ðk; 0Þ 5 Nu Ti ðk; 0Þ 2 Th Δy       T i k; Ny 2 Ti k; Ny 2 1 Nu   hb T i k; Ny 2 Th 5 hs Δy   T i 0; Ny 5 Te   T i k; Ny 5 0 for k $ 1

(3.117) (3.118) (3.119)

To solve the problem in total time subdomains, the continuity condition in each x subdomain must be applied by substituting x 5 xi11 into Eq. (3.114): for 1 # j # Ny 2 1; 2 # i # Nx m   X   Ti 0; j 5 T i21 k; j

(3.120)

k50

Fig. 3.26 compares the results of the hybrid method with the RungeKutta numerical method. Also, for an exact comparison, Table 3.6 gives the computational errors. As seen in this table and Fig. 3.26, the introduced method has excellent agreement with the numerical method and is faster and simpler. Furthermore, the effect of the nanoparticle volume fraction for the insulated base is presented in Fig. 3.27, which illustrates that by increasing the nanoparticle volume fraction, NF temperature is increased due to higher solar radiation absorption by the alumina nanoparticles.

FIGURE 3.26 Accuracy of applied method compared to numerical method when Te 5 0:01; φ 5 0:003 [23].

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153

3.5 RungeKuttaFehlberg numerical method

TABLE 3.6 Comparison of the present results and numerical solution at different locations [23].

y

Hybrid differential transformation method finite difference method (DTM FDM)

Numerical

Error

0

35.7449

35.7473

2.4E 2 3

0.1

35.7174

35.7196

2.2E 2 3

0.2

35.6474

35.6502

2.8E 2 3

0.3

35.5717

35.57621

4.5E 2 3

0.4

35.5004

35.5051

4.7E 2 3

0.5

35.4379

35.4415

3.6E 2 3

0.6

35.3857

35.3898

4.1E 2 3

0.7

35.3449

35.3465

2.6E 2 3

0.8

5.3159

35.3173

1.4E 2 3

0.9

35.2988

35.2999

1.1E 2 3

1.0

35.2939

35.2945

6.0E 2 4

FIGURE 3.27 Effect of the nanoparticle volume fraction on the nanofluid temperature when hb 5 0; x 5 1 [23].

3.5 RungeKuttaFehlberg numerical method The Maple package uses a fourth-order RungeKuttaFehlberg technique for solving nonlinear boundary value (BV) problems. The algorithm was verified to be exact and accurate for solving a wide range of mathematical and engineering problems such as NF analysis. Actually, the available submethods in Maple 15.0 are a combination of base

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3. Numerical analysis of nanofluids

FIGURE 3.28 Geometry of the problem [27].

schemes; trapezoid or midpoint method. There are two major considerations when choosing a submethod for a problem. The trapezoid technique is usually efficient for typical problems, but the midpoint method is capable of handling harmless endpoint singularities that the trapezoid method cannot. The midpoint method, also known as the fourth-order RungeKuttaFehlberg method, further develops the Euler method by adding a midpoint in the step, which increases the accuracy by one order. Thus the midpoint method is used as a suitable numerical technique for NF modeling [26]. The following three cases are analyzed using this technique.

3.5.1 Case 1: Nanofluid analysis in a porous medium under magnetohydrodynamic effect Suppose a steady, two-dimensional magnetohydrodynamic (MHD) laminar flow and heat transfer of a dusty micropolar fluid suspended with Fe3O4 nanoparticles on a permeable stretching sheet with velocity Uw ðxÞ 5 bx ðb . 0Þ along the x-axis direction as shown in Fig. 3.28. The sheet is in the presence of a uniform magnetic field B0 along the y-axis direction. Also, the effects of thermal radiation, Lorentz force, and Joule

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3.5 RungeKuttaFehlberg numerical method

155

heating are considered. The nonlinear differential equations are obtained as follows [27]: The momentum boundary layer equations are:

@u @u k 1v 5 ν nf 1 u @x @y ρnf

!

@u @v 1 50 @x @y

(3.121)

ν nf @2 u KS k @N σB0 2 2 1 ðu 2 uÞ 1 u2 0 u p 2 @y ρnf ρnf @y ρnf k

  @N @N γ @2 N k @u 1v 5 2N 1 2 u @x @y ρnf j @y2 ρnf j @y @up @vp 1 50 @x @y   @up @up 1 vp ρp u p 5 KSðu 2 up Þ @x @y

(3.122) (3.123) (3.124) (3.125)

where (u, v) and (up , vp ) are the velocity components of the NF and dust phases in the directions of x and y, respectively; μf is the dynamic viscosity of the fluid; K 5 6πμf a refers to the Stokes drag constant; γ 5 ðμnf 1 ðk=2ÞÞ j 5 μnf ð1 1 ðR=2ÞÞ is the spin-gradient viscosity; j 5 ν nf =a denotes the reference length; R 5 k=μnf is the material parameter; and ρp 5 rS is the density of the dust particles. Also, S is the density number of the dust particles. The NF properties can be calculated from the equations presented in Chapter 1, Introduction to nanofluids. Furthermore, the suitable boundary conditions for the above equations are presented according to the assumptions as follows: @u at y 5 0 @y (3.126) u 5 v 5 0; up -0; vp -v; N-0 as y-N pffiffiffiffiffiffiffi where Vw ðxÞ 5 2 f0 bν f is the suction velocity. The following variations have been defined to simplify Eqs. (3.121)(3.125):  1  1 1 Uw ðxÞ 2 Uw ðxÞ 2 η5 y; ψ 5 ðν f xUw ðxÞÞ2 fðηÞ; N 5 Uw ðxÞ hðηÞ (3.127) νf x νf x u 5 Uw ðxÞ; v 5 Vw ðxÞ; N 5 2 n

It should be mentioned that the stream function for both Eqs. (3.121) and (3.124) have the same treatment and the continuity Eqs. (3.121) and (3.124) are similar to the stream function ψ. Also, it can be applied for up and vp :

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u5

@ψ @ψ ; v52 ; @y @x

(3.128)

Thus the nonlinear governing equations of the problem (Eqs. 3.122, 3.123, and 3.125) after changing PDEs to ordinary differential equations (ODEs) by Eq. (3.127) are derived as follows:     ρs 2:5 ð1 1 RÞfw 1 ð12φÞ ð1 2 φÞ 1 φ ðffv 2 ðf 0 Þ2 Þ 1 ð12φÞ2:5 lβðF0 2 f 0 Þ 2 Haf 0 ρf 1 Rh0 2 kp f 0 5 0 " #   R ρ 11 hv 1 ð12φÞ2:5 ð1 2 φÞ 1 φ s ðfh0 2 f 0 hÞ 2 Rð2h 1 fvÞ 5 0 2 ρf FFv 2 ðF0 Þ2 1 βðf 0 2 F0 Þ 5 0;

(3.129) (3.130) (3.131)

where l 5 rS=ρf is the mass concentration; β 5 1=bτ v is the fluid particle interaction parameter for velocity; τ v 5 r=K is the relaxation time of the particle phase; Ha 5 σB20 =ρf b is the Hartman number (magnetic field parameter); and Kp 5 ν f =bk0 is the permeability parameter. Introducing Eq. (3.127) into Eq. (3.126), the boundary conditions of Eqs. (3.129)(3.131) will be: f 0 ðηÞ 5 1; fðηÞ 5 f0 ; hðηÞ 5 2 m at η 5 0 f 0 ðηÞ 5 0; F0 ðηÞ 5 0; FðηÞ 5 1; hðηÞ 5 0 as η-N

(3.132)

and the thermal boundary layer equations for dusty NFs are:    2 ρp cpf ρp @T @T @2 T @u 1v ðTp 2 TÞ 1 ðup 2uÞ2 1 μnf ðρcp Þnf u 5 Knf 2 1 @x @y @y @y τT τv 1  ðρp cmf Þ up

σB0 2 2 1 @qr u 2 ðρcp Þnf @y ðρcp Þf 

(3.133) ρp cpf

@Tp @Tp 1 vp ðTp 2 TÞ 52 @x @y τT

(3.134)

where the thermal conductivity (knf ) and the heat capacitance of the NF. ðρCp Þnf are defined as presented in Chapter 2, Mathematical analysis of nanofluids: knf ks 1 2kf 2 2φðkf 2 ks Þ ; 5 ks 1 2kf 1 2φðkf 2 ks Þ kf

ðρcp Þnf 5 ð1 2 φÞðρcp Þf 1 φðρcp Þs

Nanofluids

(3.135)

3.5 RungeKuttaFehlberg numerical method

157

Also, in Eq. (3.133), qr is the radiative heat flux and using the Rosseland approximation for radiation, the thermal flux is defined as follows: qr 5 2

4σ @T4 ; 3k @y

(3.136)

where σ is the StefanBoltzmann constant and k is the mean absorption coefficient. Since the temperature difference in the flow is very low, the Taylor series approximation for T4 according to TN is defined as: 3 3 2 3TN T4  4TTN

(3.137)

After substituting Eqs. (3.136) and (3.137) into Eq. (3.133), the final equation can be obtained as:     ρp cpf @T @T 16σ TN 3 @2 T ρ 1v ðρcp Þnf u 1 ðTp 2 TÞ 1 P ðup 2uÞ2 5 Knf  2 @x @y @y 3k knf τT τv  2 2 σB0 2 @u 1 u 1 μnf ; @y ðρcp Þf (3.138) The suitable boundary conditions for Eqs. (3.133) and (3.134) in the prescribe heat flux (PHF case) and the prescribe surface temperature (PST) cases are as follows: T 5 Tw 5 TN 1 A

 x 2 l

ðPST caseÞ; 2 K

 2 @T 5 qw 5 D xl ðPHF caseÞ at y 5 0 @y

T-TN ; Tp -TN as y-N (3.139) pffiffiffiffiffi where l 5 ν f =b is the characteristic length. The dimensionless temperature functions for the liquid (θðηÞ) and dust (θp ðηÞ) phases are defined as follows: θðηÞ 5

Tp 2 TN T 2 TN ; θp ðηÞ 5 Tw 2 TN Tw 2 TN

(3.140)

pffiffiffiffiffi where T2TN 5Aðx=lÞ2 (PST case), Tw 2TN 5 ðD=tÞðx=lÞ2 ð ν f =bÞ (PHF case). Finally, by introducing Eqs. (3.127), (3.135), and (3.140) into Eqs. (3.134) and (3.128), the ODEs will be: 2 3    knf ðρc Þ 4 p s 5ðfθ0 2 2f 0 θÞ 1 Pr lβ T ðθp 2 θÞ 1 1 Nr θv 1 Pr4ð1 2 φÞ 1 φ 3 kf ðρcp Þf 0

1 lEcβðF0 2f 0 Þ2 1 Ecf v2 1 Ec Ha2 f 2  5 0 (3.141)

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3. Numerical analysis of nanofluids

0

Fθp 2 2F0 θp 1 γβ T ðθ 2 θp Þ 5 0;

(3.142)

where Nr 5 4σ TN 3 =k knf shows the radiation parameter; β T 5 1=bτ T is the fluid particle interaction parameter for temperature; Pr 5 ðμcp Þf =kf is the Prandtl number; Ec 5 b2 l2 =Acpf (PST case) and Ec 5 pffiffiffi 2 2 ðb l t=Dcpf Þð b=ν f Þ (PHF case) are the Eckert numbers; and γ 5 cpf =cmf is the ratio of specific heat. Finally, the boundary conditions of Eqs. (3.141) and (3.142) are as follows: θðηÞ 5 1 ðPST caseÞ; θ0 ðηÞ 5 2 1 ðPHF caseÞ at η 5 0 θðηÞ-0; θp ðηÞ-0 as η-N

(3.143)

Now, the nonlinear ordinary differential Eqs. (3.129)(3.131) and (3.141) and (3.142) with the boundary conditions (3.132) and (3.143) are solved numerically using the RungeKuttaFehlberg fourth-fifth method. In this technique the boundary value problem must be transformed into the initial value problem. Furthermore, a finite value of ηN must be selected. Approximation of fourth and fifth orders to the solution are as follows:   25 1408 2197 1 k0 1 k2 1 k3 2 k4 ym11 5 ym 1 h (3.144) 216 2565 4109 5   16 6656 28561 9 2 k0 1 k2 1 k3 2 k4 1 k5 ym11 5 ym 1 h (3.145) 135 12825 56430 50 55 Also, in this method for the exact solution of the problem, step size should be properly selected. Each step consists of six steps as follows: 8   > k0 5 f xm 1 ym > >   > > > h hk0 > > k1 5 f xm 1 ; ym 1 > > > 4 4 > > >    > >  3 > 3 9 > > k0 1 k1 h k2 5 f xm 1 h; ym 1 > > > 8 32 32 > > <     12 1932 7200 7296 h; ym 1 k0 2 k1 1 k2 h k3 5 f xm 1 > > > 13 2197 2197 2197 > > > >   >   439 > 3860 845 > > k0 2 8k1 1 k2 2 k3 h k4 5 f xm 1 h; ym 1 > > > 216 513 4104 > > > >     > > h 8 3544 1859 11 > > > > k5 5 f xm 1 2 ; ym 1 2 27 k0 1 2k1 2 2565 k2 1 4104 k3 2 40 k4 h : (3.146)

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3.5 RungeKuttaFehlberg numerical method

159

It must be noted that for solving the differential equations using this method, the nonlinear boundary value problem must first be converted to the system of first-order differential equations. Thus the defined variables for the functions are as follows: f 5 q1 ; f 0 5 q2 ; fv 5 q3 ; h 5 q4 ; h0 5 q5 ; F 5 q6 ; F0 5 q7 ; θ 5 q8 ; θ0 5 q9 ; θp 5 q10 (3.147) After substituting Eq. (3.147) into Eqs. (3.129)(3.131) and (3.141) and (3.142), the new ODE system will appear as:     ρs 0 2:5 ð1 1 RÞq3 1 ð12φÞ ð1 2 φÞ 1 φ ðq1 q3 2 q22 Þ 1 ð12φÞ2:5 lβðq7 2 q2 Þ 2 Ha q2 ρf 1 Rq5 2 Kpq2 5 0 (3.148)     R 0 ρ 11 q 1 ð12φÞ2:5 ð1 2 φÞ 1 φ s ðq1 q5 2 q1 q4 Þ 2 Rð2q4 1 q3 Þ 5 0 2 5 ρf (3.149) 0

q6 q7 2 q27 1 βðq2 2 q7 Þ 5 0 (3.150) 2 3  knf  ðρcp Þs 5 4  0 ðq1 q9 2 2q2 q8 Þ 1 Pr lβ T ðq10 2 q8 Þ 1 1 Nr q9 1 Pr4ð1 2 φÞ 1 φ 3 kf ðρcp Þf  1 ‘Ecβðq7 2q2 Þ2 1 Ecq3 2 1 EcHa2 q2 2 5 0 (3.151) 0

q6 q10 2 2q7 q10 1 γβ T ðq8 2 q10 Þ 5 0

(3.152)

Also, the boundary conditions (3.133) and (3.144) become as follows: q1 ð0Þ 5 f0 ; q2 ð0Þ 5 1; q3 ð0Þ 5 s1 ; q4 ð0Þ 5 2 m; q5 ð0Þ 5 s2 ; q6 ð0Þ 5 s3 q7 ð0Þ 5 s4 (3.153) q8 ð0Þ 5 1; q9 ð0Þ 5 s5 ; q10 ð0Þ 5 s6 ; ðPST caseÞ

q8 ð0Þ 5 s5 ; q9 ð0Þ 5 2 1;

q10 ð0Þ 5 s6 ; ðPHF caseÞ (3.154) where unknown initial conditions s1 ; s2 ; s3 ; s4 ; s5 and s6 will be obtained using an iterative method called the shooting method until the boundary conditions f 0 ðηÞ-0; hðηÞ-0; FðηÞ-0; F0 ðηÞ-0; θðηÞ-0; θp ðηÞ-0 as η-N are satisfied. The above procedure will be repeated since the nonlinear solution converges with a defined convergence criterion of 1026 . Furthermore, the step size is chosen as Δη 5 0:001:

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FIGURE 3.29 (A) Influence of φ on velocity profile; (B) influence of φ on temperature

profile for PST case; (C) influence of φ on temperature profile for PHF case; and (D) influence of φ on temperature profile for PHF case [27]. PHF, Prescribe heat flux; PST, prescribe surface temperature.

The effects of increase of nanoparticle volume fraction parameter (φ) on velocity and temperature functions are presented in Fig. 3.29. This figure reveals a velocity reduction for both liquid and dust phases by increasing the φ, but the obtained results for temperature functions are completely different. Also, Fig. 3.29 demonstrate the effects of φ on temperature distribution in two PST and PHF cases. As seen, in both PST and PHF cases, φ has a reduction effect on the temperature distribution of both fluid and dust phases [27] (Table 3.7).

3.5.2 Case 2: Ferrofluid flow influenced by rotating disk Consider the laminar flow produced by a disk rotating with angular velocity Ω 5 (0, 0, Ω) along the vertical z-axis in an infinite ferrofluid Nanofluids

161

3.5 RungeKuttaFehlberg numerical method

TABLE 3.7

Thermophysical properties of ethylene glycol and iron (II; III) oxide [27]. Physical properties ρ ðkg m Þ

cp ðJ kg21 K21 Þ

k ðW m21 K21 Þ

C2 H6 O2

1115

2430

0.253

Fe3 O4

5180

670

9.7

23

FIGURE 3.30

Coordinate system for rotating disk system in ferrofluid [28].

media as shown in Fig. 3.30. A cylindrical coordinate system (r, θ, z) is considered such that θ is oriented in the direction of rotation with vr, vθ, and vz representing the radial, tangential, and axial components of the velocity vector, respectively. The basic assumptions are: 1. 2. 3. 4. 5. 6.

The flow is steady ð@=@t 5 0Þ and axisymmetric ð@=@θ 5 0Þ. The fluid density ρ is constant. The fluid and disk are electrically nonconducting. The magnetic field affects only viscosity and no other properties. The fluid and the ferrous particles have the same velocity. The flow is completely described by the continuity and conservation of the momentum equations.

By considering the above assumptions and for the effect of rotation, the momentum equation is modeled as [28]:   @V ρ 1 ðV rÞV 5 2 rp 1 μ0 ðM rÞH 1 μf r2 V 1 2ρðΩ 3 V Þ 1 rjΩ 3 rj2 ρ @t 2 (3.155)





Applying the general assumptions, the transport equations for mass conservation and conservation of momentum can be simplified as follows [28]: Continuity equation @vr vr @vz 1 1 50 @r r @z Momentum equations Nanofluids

(3.156)

162 

3. Numerical analysis of nanofluids

In the radial direction:  2   @vr @vz v2θ 1 @p μ0 @ @ vr @  vr  @ 2 vr 1 jMj jHj1υ vr 1vz 2 1 52 1 2 12Ωvθ ρ @r ρ @r @r @z r @r2 @r r @z (3.157) In the tangential direction:  2    @vθ @vθ vr vθ @ vθ @  vθ  @ 2 vθ vr 1 vz 1 1 2 2Ωvr 5υ 1 @r r @r @z r @r2 @z2

(3.158)

In the axial direction:  2    @vz @vz 1 @p μ0 @ @ vz 1 @vz @2 v z jMj jH j 1 υ vr 1 1 vz 1 2 1 52 ρ @r @r r @r @r @z ρ @r2 @z (3.159) The terms 2Ωvθ and r 2Ωv are the projections of the Coriolis force onto the axes r and θ, respectively, while the projection of centrifugal force onto the axis r adjusts with the pressure term. The boundary conditions for the velocity components at the surface and far away from the disk are given by: z 5 0; vr 5 0; vθ 5 rω; vz 5 0 (3.160) z 5 N; vr 5 0; vθ 5 0 The value of vz disappears near the surface of the disk, because there is no penetration. However, the value of vz as z-N is not definite; it adjusts to a negative value, which delivers enough fluid. In contrast to the axial velocity, both the radial and tangential velocities reach zero at large axial distances from the disk. By introducing the appropriate nondimensional transformation variable such as: rffiffiffiffi ω η5z (3.161) υ And the following set of dimensionless velocity and pressure components, that is: EðηÞ 5

vr vθ vz p ; fðηÞ 5 ; gðηÞ 5 pffiffiffiffiffiffi ; PðηÞ 5 ρωυ rω rω υω

(3.162)

This similarity transformation indicates that all three dimensionless velocity components and pressure are related only to the distance from the disk, α. The boundary conditions (Eq. 3.160) are changed into the α coordinate as follows [29]:

Nanofluids

3.5 RungeKuttaFehlberg numerical method

Eð0Þ 5 gð0Þ 5 0; fð0Þ 5 1; Pð0Þ 5 P0 EðNÞ 5 fðNÞ 5 0

163 (3.163)

In the similarity solution, the governing partial differential equation set will be transformed into a set of ODEs. Thus Eqs. (3.156)(3.159) will be converted to: Continuity equation g0 1 2E 5 0

(3.164)

Momentum equations in the radial direction: kEv 2 gE0 2 E2 1 f 2 2 1 5 0

(3.165)

In the tangential direction: Fv 2 GF0 2 2EF 5 0

(3.166)

P0 2 kGv 1 GG0 5 0

(3.167)

In the axial direction:

where prime denotes differentiation with respect to α. Fig. 3.31 shows the numerical solution of the problem (applied as in the previous case) in different ηinf to find the best value for infinite in this case study.

3.5.3 Case 3: Solar radiation effect on the magnetohydrodynamic nanofluid flow over a stretching sheet Consider a two-dimensional convectively heated sheet stretched through two equal and opposite forces along the x-axis by keeping the origin fixed with the velocity Uw 5 ax and an incompressible boundary layer of NF that flows over it. Let UN(x) 5 bx be the fluid’s velocity outside the boundary layer. A uniform magnetic field of strength H0 is applied perpendicular to the direction of the flow. It is assumed that the external electrical field is zero and the electric field due to the polarization of charges is negligible. Heat transfer analysis is carried out in the presence of thermal radiation, Joule heating, and viscous dissipation effects. The combined effects of Brownian motion and thermophoresis due to the presence of nanoparticles are considered. Tf denotes the convective surface temperature while TN is the ambient fluid’s temperature. The steady boundary layer equations governing the twodimensional incompressible stagnation-point flow of the NF can be written as [30,31]:

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FIGURE 3.31 Variation of f ðηÞ, gðηÞ, and EðηÞ for different values of ηInf [28].

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3.5 RungeKuttaFehlberg numerical method

@u @v 1 50 @x @y u

@u @u duN @2 u σe H02 1v 5 uN 1 vf 2 2 ðu 2 uN Þ @x @y @y @x ρf

165 (3.168)

(3.169)

where x and y are the coordinates along and normal to the sheet, respectively; vf and σe are the kinematic viscosity and electrical conductivity of the fluid, respectively; H0 is the uniform magnetic field along the y-axis; and u and v are the velocity components along the x and y directions, respectively. The boundary conditions for this problem are: u 5 Uw ðxÞ 5 ax; v 5 0 at y 5 0 u-uN ðxÞ 5 bx at y-N Using the dimensionless variables [31]: rffiffiffiffi pffiffiffiffiffiffiffi a y; u 5 axf 0 ðηÞ; v 5 2 avf fðηÞ η5 vf

(3.170)

(3.171)

Eq. (3.168) is identically satisfied and Eqs. (3.169) and (3.170) take the forms: fw 1 ffv 2 f 02 1 λ2 1 Mðλ 2 f 0 Þ 5 0

(3.172)

fð0Þ 5 0; f 0 ð0Þ 5 1; f 0 ð 1 NÞ-λ

(3.173)

where M 5 σH02 =ρf is the magnetic parameter and λ 5 b=a is the ratio of the rates of free stream velocity to the velocity of the stretching sheet. For M 5 0 and l 5 1, Eq. (3.172) reduces to the classical problem first formulated by Hiemenz. Furthermore, l 5 0 the exact solution of pffiffiffiffiffiffiffiffi when pffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (3.172) is given by f 5 ð1 2 e2 11Mη Þ= 1 1 M. u

vf @u2 @T @T @2 T 1  @qr  σe H02 1v 5α 2 1 2 ðuN 2uÞ2 1 @x @y @y ðρCÞf @y Cf @y ðρCÞf 2 3 (3.174) @T2 @T @C D T 4 5 1 1 τDB @y @y TN @y u

@C @C @2 C D T @2 T 1v 5 DB 2 1 @x @y @y TN @y2

(3.175)

where T is the temperature; C is the nanoparticle concentration; a is the thermal diffusivity; Cf is the specific heat of the fluid; DB and DT are the Brownian motion and the thermophoretic diffusion coefficients, respectively; τ 5 ðρCÞp =ðρCÞf is the ratio of the effective heat capacity of the

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3. Numerical analysis of nanofluids

nanoparticle material to the heat capacity of the fluid; and qr is the radiative heat flux. The radiative heat flux is given by Brewster [32] by applying the Rosseland approximation for thermal radiation to an optically thick media as: qr 5 2

4σ @T4 16σ 3 @T 5 2 T @y 3k @y 3k

(3.176)

where σ and k are the StefanBoltzman constant and the mean absorption coefficient, respectively. Now Eq. (3.174) can be expressed as: 20 1 3 !2 @T @T @ 4@ 16σ T3 A @T5 vf @u 1v 5 α1 1 u @x @y @y 3ðρCÞf k @y Cf @y 2 (3.177) !2 3 σe H02 @T @C D @T T 5 1 1 ðuN 2uÞ2 1 τDB 4 @y @y ðρCÞf TN @y where the nonlinear Rosseland approximation for radiative heat flux is considered. The relevant boundary conditions for convective heat transfer can be written as: 2k

@T 5 hðT 2 Tf Þ; C 5 CW @y

T-TN ; C-CN

at y 5 0

(3.178)

as y-N

By defining the nondimensional temperature θðηÞ 5 T 2 TN =Tf 2 TN with T 5 TN ð1 1 ðθw 2 1ÞθÞ and θw 5 Tf =TN (temperature parameter), the first term  on the right-hand side of Eq. (3.177) can be written as αð@=@yÞ ð@T=@yÞð1 1 Rd ð1 1 ðθw 2 1Þθ3 Þ where Rd 5 16σ TN 3 =3kk represents the radiation parameter, and Rd 5 0 provides no thermal radiation effect. The last expression can be further reduced to: αðTf 2 TN Þ  ð1 1 Rd ð1 1 ðθw 2 1Þθ3 Þθ0 0 Pr

(3.179)

where Pr 5 vf =α is the Prandtl number. Eqs. (3.175) and (3.177) take the following forms: 1  0 ð1 1 Rd ð1 1 ðθw 2 1Þθ3 Þθ0 0 1 fθ0 1 Nb θ0 φ0 1 Nt θ 2 1 Ec f v2 1 MEc ðλ2f 0 Þ2 5 0 Pr (3.180) φv 1 Lefφ0 1

Nt θv 5 0 Nb

Nanofluids

(3.181)

3.5 RungeKuttaFehlberg numerical method

167

with the boundary conditions: (3.182) θ0 ð0Þ 5 γ ½1 2 θð0Þ; φð0Þ 5 1; θð 1 NÞ-0; φð 1 NÞ-0 pffiffiffiffiffiffiffiffiffiffi where γ 5 h=k vf =α is the Biot number, Nb 5 τDB ðCw 2 CN Þ=vf is the Brownian motion parameter, Nt 5 τDT ðTw 2 TN Þ=TN vf is the thermo2 =Cp ðTw 2 TN Þ is the local Eckert numphoresis parameter, and EC  5 UW ber. It should be noted that the x-coordinate could not be eliminated from the energy equation. Thus the availability of local similarity solutions must be examined. The surface heat and mass fluxes are defined by the following equations: 0 1 sffiffiffiffi    @T A a @ qw 5 2 k 1 1 Rd θ3w θ0 ð0Þ 1 qr w 5 2 kðTf 2 TN Þ @y vf y50 0 1 (3.183) sffiffiffiffi @C a jw 5 2DB @ A 5 2DB ðCW 2 CN Þ φ0 ð0Þ @y vf y50

and with the help of the local Nusselt number Nux 5 xqw =kðTf 2 TN Þ and local Sherwood number Sh 5 xjw =DB ðCW 2 CN Þ we obtain:   Nux Sh pffiffiffiffiffiffiffi 5 2 1 1 Rd θ3w θ0 ð0Þ 5 Nur; pffiffiffiffiffiffiffi 5 2 φ0 ð0Þ 5 Shr: Rex Rex

(3.184)

where Rex 5 UW ðxÞ=v is the local Reynolds number. In this case, an efficient numerical method is used for solving the problem using Maple 15.0 software. The boundary layer equations were nondimensional through appropriate similarity transformations and the resulting differential system has been solved for the numerical solutions. Also, heat transfer analysis is performed in the presence of thermal radiation, and Joule heating and viscous dissipation are considered by using the Rosseland approximation for thermal radiation. After solution, the obtained results are compared with the fourth-fifth order RungeKutta method presented by Mushtaq et al. [31]. Fig. 3.32 display this comparison for both temperature and nanoparticle concentration profiles for different radiation, Brownian motion, and thermophoresis parameters. As seen, the RungeKutta numerical method has excellent agreement with the KellerBox method in a wide range of constant parameters. Tables 3.8 and 3.9 confirm this agreement in a special case for temperature and nanoparticle values, respectively.

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3. Numerical analysis of nanofluids

FIGURE 3.32 Comparison of RungeKutta (Mushtaq et al. [31]) and numerical method for (A) nanoparticle concentration and (B) temperature profile [30]. TABLE 3.8 Compared results for (η) with those of Mushtaq et al. [31] when Pr 5 7, M 5 0.5, λ 5 0.5, Ec 5 0.2, Nt 5 0.1, Le 5 1, γ 5 0.5, θW 5 1.5, and Rd 5 0 [30].

η

Present work (Nb 5 0.3)

Mushtaq et al. (Nb 5 0.3)

Present work (Nb 5 0.5)

Mushtaq et al. (Nb 5 0.5)

Present work (Nb 5 0.7)

Mushtaq et al. (Nb 5 0.7)

0

0.433131

0.433378

0.559314

0.559820

0.713093

0.713220

0.5

0.192426

0.192910

0.302830

0.302897

0.459378

0.459814

1

0.031337

0.031472

0.061149

0.061267

0.116772

0.116978

1.5

0.002061

0.002183

0.004317

0.004429

0.009590

0.009621

2

0.000090

0.000096

0.000144

0.000148

0.000284

0.000290

2.5

0.000005

0.000005

0.000006

0.000006

0.000007

0.000007

3

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

TABLE 3.9 Compared results for (η) with those of Mushtaq et al. [31] when Pr 5 7, M 5 0.5, λ 5 0.5, Ec 5 0.2, Nb 5 0.1, Le 5 1, γ 5 0.5, θW 5 1.5, and Rd 5 0 [30].

H

Present work (Nt 5 0.3)

Mushtaq et al. (Nt 5 0.3)

Present work (Nt 5 0.5)

Mushtaq et al. (Nt 5 0.5)

Present work (Nt 5 0.7)

Mushtaq et al. (Nt 5 0.7)

0

1.000000

1.000000

1.000000

1.000000

1.000000

1.000000

1

0.887265

0.887329

1.269700

1.270348

1.715403

1.715880

2

0.247857

0.247973

0.367775

0.367910

0.517634

0.517911

3

0.040873

0.040981

0.060668

0.607173

0.085411

0.085503

4

0.004035

0.004057

0.005989

0.006101

0.008432

0.008601

5

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

Nanofluids

References

169

References [1] M. Hatami, Cross-sectional heat transfer of hot tubes in a wavy porous channel filled by Fe3O4water nanofluid under a variable magnetic field, Eur. Phys. J. Plus 133 (9) (2018) 374. [2] M. Sheikholeslami, D.D. Ganji, Numerical analysis of nanofluid transportation in porous media under the influence of external magnetic source, J. Mol. Liq. 233 (2017) 499507. [3] M. Sheikholeslami, K. Vajravelu, Nanofluid flow and heat transfer in a cavity with variable magnetic field, Appl. Math. Comput. 298 (2017) 272282. [4] M. Hatami, D. Song, D. Jing, Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition, Int. J. Heat Mass Transf. 98 (2016) 758767. [5] M. Sheikholeslami, M. Gorji-Bandpy, S. Soleimani, Two phase simulation of nanofluid flow and heat transfer using heatline analysis, Int. Commun. Heat Mass Transf. 47 (2013) 7381. [6] A. Ghorbanian, M. Tahari, M. Hatami, Physical optimization of a wavy porous cavity filled by nanofluids in the presence of solar radiations using the BoxBehnken design (BBD), Eur. Phys. J. Plus 132 (6) (2017) 278. [7] K. Milani Shirvan, M. Mamourian, S. Mirzakhanlari, R. Ellahi, K. Vafai, Numerical investigation and sensitivity analysis of effective parameters on combined heat transfer performance in a porous solar cavity receiver by response surface methodology, Int. J. Heat Mass Transf. 105 (2017) 811825. [8] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington DC, 1980, pp. 113137. [9] M. Mahmoodi, S.M. Hashemi, Numerical study of natural convection of a nanofluid in C-shaped enclosures, Int. J. Therm. Sci. 55 (2012) 7689. [10] A. Yadollahi, A. Khalesidoost, A. Kasaeipoor, M. Hatami, D. Jing, Physical investigation on silver-water nanofluid natural convection for an F-shaped cavity under the magnetic field effects, Eur. Phys. J. Plus 132 (8) (2017) 372. [11] A. Kasaeipoor, B. Ghasemi, S.M. Aminossadati, Convection of Cuwater nanofluid in a vented T-shaped cavity in the presence of magnetic field, Int. J. Therm. Sci. 94 (2015) 5060. [12] M. Hatami, A. Kheirkhah, H. Ghanbari-Rad, D. Jing, Numerical heat transfer enhancement using different nanofluids flow through venturi and wavy tubes, Case Stud. Therm. Eng. 13 (2019) 100368. [13] D. Song, M. Hatami, J. Zhou, D. Jing, Dynamic nanoparticle aggregation for a flowing colloidal suspension with nonuniform temperature field studied by a coupled LBM and PBE method, Ind. Eng. Chem. Res. 56 (38) (2017) 1088610899. [14] C. Qi, Y. He, S. Yan, F. Tian, Y. Hu, Numerical simulation of natural convection in a square enclosure filled with nanofluid using the two-phase Lattice Boltzmann method, Nanoscale Res. Lett. 8 (2013) 56. [15] X. He, L.S. Luo, Lattice Boltzmann model for the incompressible NavierStokes equation, J. Stat. Phys. 88 (1997) 927944. [16] Z. Guo, B. Shi, C. Zheng, A coupled lattice BGK model for the Boussinesq equations., Int. J. Numer. Methods Fluids 39 (2002) 325342. [17] A. Majumder, V. Kariwala, S. Ansumali, A. Rajendran, Lattice Boltzmann method for population balance equations with simultaneous growth, nucleation, aggregation and breakage, Chem. Eng. Sci. 69 (2012) 316328. [18] P.T. Spicer, S.E. Pratsinis, Coagulation and fragmentation: Universal steady-state particle-size distribution, AIChE J. 42 (1996) 16121620. [19] L. He, Y. Wang, Q. Li, Lattice Boltzmann Method: Theory and Applications, Science Press, Beijing, 2009.

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[20] K. Kahveci, Buoyancy driven heat transfer of nanofluids in a tilted enclosure, J. Heat Transf 132 (2010) 062501. [21] S. Mosayebidorcheh, M. Hatami, D.D. Ganji, T. Mosayebidorcheh, Investigation of transient MHD Couette flow and heat transfer of dusty fluid with temperaturedependent properties, J. Appl. Fluid Mech. 8 (4) (2010) 921929. [22] M. Mahmoodi, S.M. Hashemi, Numerical study of natural convection of a nanofluid in C-shaped enclosures, Int. J. Therm. Sci. 55 (2012) 7689. [23] M. Hatami, S. Mosayebidorcheh, D. Jing, Thermal performance evaluation of alumina-water nanofluid in an inclined direct absorption solar collector (IDASC) using numerical method, J. Mol. Liq. 231 (2017) 632639. [24] V. Cregan, T.G. Myers, Modelling the efficiency of a nanofluid direct absorption solar collector, Int. J. Heat Mass Transf. 90 (2015) 505514. [25] M. Turkyilmazoglu, Performance of direct absorption solar collector with nanofluid mixture, Energy Convers. Manag. 114 (2016) 110. [26] A. Aziz, Heat Conduction With Maple, R.T. Edwards, Philadelphia, PA, 2006. [27] S.S. Ghadikolaei, Kh Hosseinzadeh, D.D. Ganji, M. Hatami, Fe3O4(CH2OH) 2 nanofluid analysis in a porous medium under MHD radiative boundary layer and dusty fluid, J. Mol. Liq. 258 (2018) 172185. [28] M. Hatami, S. Mosayebidorcheh, D. Jing, Three-dimensional analysis of Ferrofluid flow influenced by rotating disk using numerical approach, J. Heat Mass Transf. 14 (4) (2017) 445460. [29] P. Ram, A. Bhandari, K. Sharma, Effect of magnetic field-dependent viscosity on revolving ferrofluid, J. Magn. Magn. Mater. 322 (2010) 34763480. [30] S.E. Ghasemi, M. Hatami, D. Jing, D.D. Ganji, Nanoparticles effects on MHD fluid flow over a stretching sheet with solar radiation: a numerical study, J. Mol. Liq. 219 (2016) 890896. [31] A. Mushtaq, M. Mustafa, T. Hayat, A. Alsaedi, Nonlinear radiative heat transfer in the flow of nanofluid due to solar energy: a numerical study, J. Taiwan Inst. Chem. Eng. 45 (2014) 11761183. [32] M.Q. Brewster, Thermal Radiative Transfer Properties, John Wiley and Sons, New York, 1972.

Nanofluids

C H A P T E R

4 Experimental analysis of nanofluids

4.1 Brownian motion of nonspherical particle The Brownian motion of nanoparticles in nanofluid (NF) is a complicated process that can be affected by size, shape, and density of particles as well as the base fluid properties [1]. Actually, most thermal conductivity models assume that particles are spherical for Brownian motion [2,3]. For example, Shukla et al. [2] suggested a thermal conductivity model for spherical nanoparticles considering both Brownian motion and microconvection effects. They extended the suggested model to nonspherical nanoparticles by using a simplified volumeequivalent diameter dep 5 (6V/π)1/3. Although it has been confirmed that the rotating Brownian motion of nonspherical particles can significantly affect the particle diffusion and consequently the properties of suspension, few studies have considered its effect on thermal conductivity. In this case study, cubic shape will be considered for nanoparticles and the contribution of rotating Brownian motion of cubic particle to the thermal conductivity improvement is examined experimentally and theoretically [1].

4.1.1 Model validation The particle motion in a fluid media can be explained by the NavierStokes equation: @u 1 1 ðu rÞu 5 2 rp 1 νr2 u 1 f @t ρ



Nanofluids DOI: https://doi.org/10.1016/B978-0-08-102933-6.00004-4

171

(4.1)

© 2020 Elsevier Ltd. All rights reserved.

172

4. Experimental analysis of nanofluids

where u is velocity (m s21); t is time (s); ρ and p are density of the fluid (kg m23) and pressure (Pa), respectively; ν is the kinematic viscosity of fluid (m2 s21); and f is volume force (m s22). From the left-hand side of Eq. (4.1), the first two statements define the inertia effects produced by the unsteady and uneven flow field, called the“unsteady” term and “convective” term, respectively. Furthermore, the second term on the right-hand side is called the “dissipative” term. Here, we present the characteristic time t0, length L, and velocity U of flow field [1]: @u @t νr2 u



5

ðu rÞu 5 νr2 u

U @u t0 @t νU 2  r u L2

U2 L

~

L2 5 Ns νt0



ðu r Þu UL 5 Re ~ νU 2  ν r u L2

(4.2)

(4.3)

From Eqs. (4.2) and (4.3), it can be concluded that the Stokes number (Ns) and Reynolds number (Re) are the indicators of the relative values of “unsteady” and “convective” statements compared to the “dissipative” term. Because length L of the nanoparticles is very small, Re and Ns should be much smaller than 1. Moreover, by neglecting the volume force, the NavierStokes equation can be shortened to: r2 u 5

1 rp μ

(4.4)

where μ is the dynamic viscosity, μ 5 νρ, (Pa s21). Eq. (4.4) is called the Stokes equation, which is usable when Re{1. The continuity equation is:



r u50

(4.5)

As a typical result of the Stokes equation, the resistance force of moving spherical particles in fluid is: F r 5 6πμUr 5 Rc

s

 U;

(4.6)

where Fr is the resistance force (N), Rc_s indicates the resistance coefficient of the spherical particle (N s21 m21), and r denotes the radius of the particle (m). Assuming a stable cubic nanoparticle (with side length of 2a, m) moves with velocity U in a uniform flow fluid, the boundary conditions can be considered as: x; y or z-N:u 5 U; v 5 w 5 0 x 5 6 a; and y; zAð 2a; aÞ: u 5 v 5 w 5 0 y 5 6 a; and x; zAð 2a; aÞ: u 5 v 5 w 5 0 z 5 6 a; and x; yAð 2a; aÞ: u 5 v 5 w 5 0

Nanofluids

(4.7)

4.1 Brownian motion of nonspherical particle

173

Regrettably, except for some regular rotators, the analytical solution cannot be found by mathematical techniques for most of the shapes using the boundary condition of Eq. (4.7). Using the basic conclusion of Stokes flow, the resistance of a cubic particle can be defined as: Fr 5 Rc

c

 U;Rc

c

5 Rc c ðμ; 2aÞ;

(4.8)

where Rc_c is the resistance coefficient of cube particle (N s m21). From Section 1.5, the forces on a cubic nanoparticle in fluid can be categorized into three main forces: surface force (resistance), volume force (gravity, or electromagnetic force), and random force (Brownian force). Thus the motion equation of a cubic nanoparticle can be considered as: m

d2 r 5 2 Rc dt2

c

dr 1 F Bc 1 f dt

  dLo 5 Mxy 1 Myz 1 Mzx 1 MðF r Þ 1 M f dt

(4.9) (4.10)

where, r is the position vector with three components (x,y,z), m; Rc_c denotes the resistance coefficient of the cube; LO indicates the momentum moment in the centroid of cube O; and kg m2 s21, and M(Fr), M(f) are force moments of resistance Fr and volume force f, N m, respectively. The FB_c, Mxy, Myz, and Mzx are components of random Brownian force FB as displayed in Fig. 4.1 and presented in detail in Ref. [1]. Also, the momentum moment can be related to a kinematic parameter, angular velocity ω, rad s21, which is calculated in Ref. [1]. Based on the above assumptions, we have: qffiffiffiffipffiffiffiffiffiffi     8 2 Rc c 6kB T C1 3π ω Rc c 2 C1 2 3kB T qffiffiffiffiffiffiffiffi ;that is;ω2 51:6405ð2aÞ2 ω2 51:2808a Rm c πm C2 Rm c C2 π2 m 8kB T πm

(4.11)

Actually, C1 and C2 constants have the same physical origin assuming that the direction of Brownian force affects the variations of motion parameters. These constants can be variable with changes to particle mass, shape, density, etc. To simplify the current case, it is assumed that factor C1 could equal C2 for the same particle. As seen in Chapter 1, Introduction to nanofluids, and Chapter 2, Mathematical analysis of nanofluids, the thermal conductivity of a NF has a close relationship with the particle concentration, the material, size and shape of nanoparticles, etc. For NFs with high concentrations, the nanoparticles may aggregate into larger secondary particles, so thermal conductivity of base fluid will vary. Also, in Chapter 1, Introduction to nanofluids, the traditional thermal conductivity mode

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4. Experimental analysis of nanofluids

Brownian motion and micro-convection

Component moments of force Mxy, Mxz

Brownian force FB

Thermal conductivity enhancement

L2

z Component force FB_c

L1

y x

Cubic nanofluid

FIGURE 4.1 Illustration of random Brownian force FB on a cubic nanoparticle (FB can be divided into a force through the center of the cube FB_c, and two moments of force perpendicular to FB). The relations between these parameters are |FB| 5 |FB_c|, |Mxy| 5 | FB_c|L1, and |Mxz| 5 |FB_c|L2 [1].

MaxwellGarnett (MG) model [4] was introduced, which considers both concentration of nanoparticles and interfacial thermal resistance as:     kp ð1 1 2αÞ 1 2kf 1 2Φ kp ð1 2 αÞ 2 kf kn    5  (4.12) kf kp ð1 1 2αÞ 1 2kf 2 Φ kp ð1 2 αÞ 2 kf where α 5 2Rbkf/d, Φ is the volume fraction of solid phase; kn, kf, and kp are the thermal conductivity of the suspension, base fluid, and nanoparticle, respectively; W (m K)21, d indicates the diameter (equivalent diameter) of the nanoparticle; and Rb (about 0.77 3 1028 K m2 W21 for water-based NFs) denotes the interfacial thermal resistance between the liquid and solid phases, K m2 W21. One of the shortcomings of the MG model is that it neglects the Brownian motion of nanoparticles. To consider the effect of motion of nanoparticles, consider a sphere inserted in a semiinfinite medium of thermal conductivity km, where the Nu based on the radium of particle r is 1 (i.e., h 5 km/r [5]). In the Stokes regime, h can be shown as h 5 kf/r(1 1 1/ 4Re 3 Pr) [6], thus the effective k of the fluid affected by the movement of a single sphere is km 5 kf (1 1 1/4Re 3 Pr). Consequently, Eq. (4.12) will be [1]:      kp ð1 1 2αÞ 1 2kf 1 2Φ kp ð1 2 αÞ 2 kf kn 1    5 1 1 Re 3 Pr  (4.13) 4 kf kp ð1 1 2αÞ 1 2kf 2 Φ kp ð1 2 αÞ 2 kf

Nanofluids

4.1 Brownian motion of nonspherical particle

175

In Eq. (4.13) (called the single-sphere Brownian model) only one particle is considered. Thus the interactions between particles are ignored in this case, Also for a colloid suspension, the hydrodynamic boundary layer of the particle is much larger than its original size. Prasher et al. [7] proposed a general correlation for heat transfer coefficient by considering the Nu correlation for particle-to-fluid heat transfer in fluidized beds, its h 5 kf/r(1 1 ARemPr0.333Φ) for Brownian motion-induced convection. Here, m is dependent on the fluid type, while A is not. In fact, m 5 2.5 and A 5 40,000 have been reported for water-based NFs [7]. By similarly with the deduction of Eq. (4.13), the modified MG model considering multiple nanoparticles Brownian motion can be introduced as [1]:      kp ð1 1 2αÞ 1 2kf 1 2Φ kp ð1 2 αÞ 2 kf kn  m 0:333     5 1 1 ARe Pr Φ (4.14) kf kp ð1 1 2αÞ 1 2kf 2 Φ kp ð1 2 αÞ 2 kf where Re 5 UL/ν and Pr 5 cpμ/k are the Reynolds number and Prandtl number, respectively. For nonspherical nanoparticles, both the Brownian motions of translation and rotation occur, so the maximum relative velocity in the phase interface of base fluid and nanoparticle is qffiffiffiffiffiffiffiffi u 5 ut 1 ur, where ut and ur are the root-mean-square velocity ( v2 ) qffiffiffiffiffiffiffiffiffi and the rotating linear velocity (deq ω2 =2), respectively. In Ref. [1], the maximum relative velocity u is selected as the characteristic velocity of Reynolds number, so Re can be shown as: qffiffiffiffiffiffiffiffiffi 0 1 rffiffiffi qffiffiffiffiffiffiffiffi ρdeq @ 2 deq ω2 A 3 6 ; for cube deq 5 v 1 Re 5 Ret 1 Rer 5 ð2aÞ (4.15) 2 π μ where deq shows the volume-equivalent diameter. If only the translation Brownian motion exists, the Reynolds number will be Ret 5 q ffiffiffiffiffiffiffiffi ðρdeq =μÞ v2 ; and if only rotation exists, the Reynolds number will be qffiffiffiffiffiffiffiffiffi Rer 5 ðρdeq =μÞðdeq ω2 =2Þ. Thus Ret and Rer are the donations of translation and rotation Brownian motion, respectively, to the Reynolds number. As seen in Eq. (4.11), the angular velocity of rotating Brownian motion of cubic particles is related to Rc_c and Rm_c. Thus to find the effect of rotating Brownian motion, these two coefficients must be clear first. Fig. 4.2A displays numerically the variation of the resistance on moving spherical particles Fr against the particle size r and viscosity of fluid μ. Here, the fluid velocity v is set to be changed from 1025 to 1027 m s21 and Fr/v is always a constant with the changing v. It is obvious that a good lineal fit for Fr against r can be found at a certain μ. Also, the slopes of the three fitted lines are in proportion with μ (i.e.,

Nanofluids

176

4. Experimental analysis of nanofluids

(A)

(B)

–8

1.2 × 10

k/μ ≈ 13.5

–9

7.0 × 10

Fr/v (Ns m−1)

Fr/v (Ns m−1)

–9

8.0 × 10

k/μ ≈ 6π

–8

1.0 × 10

–9

9.0 × 10

μ = 1μ

–9

8.0 × 10

water

μ = 2μ

water

μ = 3μ

–9

6.0 × 10

water

6.0 × 10–9

μ = 1μ

water

μ = 2μ water

–9

5.0 × 10

μ = 3μ water

–9

4.0 × 10

–9

3.0 × 10

–9

4.0 × 10

2.0 × 10–9 –9

2.0 × 10

–9

1.0 × 10

0.0 0.0

5.0 × 10–8 1.0 × 10–7 1.5 × 10–7 2.0 × 10–7

5.0 × 10

r (m)

–8

1.0 × 10

–7

1.5 × 10

–7

2.0 × 10

–7

2a (m)

(C)

(D) ω

k/μ ≈ 1.64

Pressure

X

Y

Z

X

0.01 0.008 0.006 0.004 0.002 0 –0.002 –0.004 –0.006 –0.008 –0.01

M(Fr)/ω(Ns m−1)

Z

10–23

μ = 1μ

water

μ = 2μ

10–24

water

μ = 3μ

water

10–25

10–26 10

–23

10–22

10–21

10–20

(2a)3 (m3)

Y

FIGURE 4.2 Translation resistance coefficients of (A) spherical and (B) cubic particles, (C) the pressure distribution on rotating cube surfaces, and (D) rotating resistance coefficient of cubic particle obtained by finite volume method (FVM), where μwater  0.001 Pa s21 and k is the slope of the fitted line [1].

Fr/v ~ μr). Moreover, from the values of k/μ, it can be found that Fr/ v  6πμr [1]. The results of the changes of resistance imposed on the moving cubic particle with various side lengths 2a and fluid viscosity μ are depicted in Fig. 4.2B. Similar treatments can be observed in Fig. 4.2B compared to Fig. 4.2A (i.e., Fr/v ~ 2a μ). Therefore the resistance on the cubic particle can be introduced as:



F r 5 Rc

c

 U;Rc

c

 13:5μð2aÞ

Nanofluids

(4.16)

4.1 Brownian motion of nonspherical particle

177

The rotating resistance coefficient of the cubic particle is also determined and presented in Fig. 4.2C and D. As seen in Fig. 4.2C, the pressure field on the surfaces of the rotating cube is irregular from the left to the right of the cube, which gives rise to a moment along the z axis. Also, the pressure distribution is symmetrical from top to bottom, so there is no force moment along the x- and y-axes. The momentum of resistance M(Fr) can be obtained by integrating the products of pressure on the cubic surface and the corresponding moment arm. The variation of force moments of resistance on the cubic particle against some relevant parameters are demonstrated in Fig. 4.2D. As seen, the angular velocity ω is altered from 6.28 to 628 rad s21 and M (Fr)/ω stays unchanged. It can be shown that M(Fr)/ω increases in proportion to the increases of (2a)3 and μ. A specific expression of Rm_c is obtained by analyzing the slopes of lines in Fig. 4.2D: MðF r Þ 5 2 Rm c ω; Rm

c

5 1:64μð2aÞ3

(4.17)

Note that the expressions of Rc_c and Rm_c are developed by fitting several values in different cases, so their application range is not as wide as Eq. (4.8).

4.1.2 Experimental thermal conductivity of nanocube nanofluid Here, Ag cubic nanoparticles (AgNCs) were dispersed in ethylene glycol solvent (EG) using a method presented by Sun and Xia [8]. Also, polyvinyl pyrrolidone (PVP) (molecular weight of B55,000 g) was dissolved in 10 mL of EG as a stabilizer. First, 70 mL of EG was added to a 100 mL round bottom flask. The obtained solution was stirred and heated at 140 C for 1 h in an oil bath. After that, 800 mg PVP (in 10 mL EG) was added to it. Then, the temperature was increased slowly to 155 C. Second, 5 min after adding PVP, 0.8 mL of 3 mM sodium sulfide was added to the EG. Finally, 5 mL of AgNO3 (in the EG) was added to the mixture and the reaction was kept for 45 min. The thermal conductivity of NFs was measured by a laser flash thermal analyzer, LFA 467 HyperFlash, Netzsch. Measurements were done at least three times for a single sample and their average was used to certify the reproducibility of the experimental results. Actually, this instrument can measure the thermal diffusion coefficient a, m2 s21, and the thermal conductivity of the NF, knf, W (m K)21, is calculated by multiplying the thermal diffusion coefficient with the specific heat capacity, cp, J (kg K)21, and density, ρ, kg m23 (i.e., kc 5 a 3 cp 3 ρ). Here a comparison of the experimental results with the theoretical prediction is given. As seen in Fig. 4.3, a scanning electrical microscope (SEM) was used for the characterization of the synthesized cubic

Nanofluids

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4. Experimental analysis of nanofluids

(A)

(B)

(C)

(D) Ferquency (%)

25

30 nm

50 nm

60 nm

20 15 10 5 0

20 25 30 35 40 40 45 50 55 60 50 55 60 65 70

Edge length (nm)

FIGURE 4.3 Scanning electrical microscope (SEM) (JSM-7800f, JEOL) images (AC) and particle size distributions (PSDs) (D) of cubic Ag nanoparticles (AgNCs); insets in the top-right corners of the images (AC) give an enlarged view to show the particle sizes [1].

AgNCs. Fig. 4.3AC shows the images of AgNCs in different sizes, while Fig. 4.3D shows the statistic AgNCs particle size distributions (PSDs) obtained from more than 10 SEM images. From Fig. 4.3AC, we can see that the side lengths for three cubic AgNCs samples are around 30, 50, and 60 nm, respectively. From Fig. 4.3D, it can be concluded that the AgNCs have good monodispersity in PSD, and the nominal sizes of 30, 50, and 60 nm are dominant and reasonable for the three examined cases. Fig. 4.4 compares the thermal conductivity enhancements of cubic Ag NFs versus solid volume concentration Φ and cubic length predicted by theoretical Maxwell model, Hamilton et al.’s model, Prasher et al.’s model [6], and the present model. As described before, the former model does not consider the size and shape of particles and the latter model considers only the effects of particles with spherical shape. In Fig. 4.4, experimental results are also presented for comparison. In this figure, the term “without considering ω” shows when the rotating Brownian motion (i.e., ω 5 0 and Re 5 Ret) is ignored so Eq. (4.14) can be reduced to Prasher et al.’s model [6].

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179

4.1 Brownian motion of nonspherical particle

(A)

(B) 2.5 Maxwell model

2.0

Hamilton et al.

100 × (kn/kf –1)

100 × (kn/kf –1)

Experiment

Considering ω, Present model

1.5

Without considering ω, Prasher et al.

2a = 30 nm

1.0 0.5 0.0

0.0000

0.0005

0.0010

0.0015

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

0.0020

Maxwell model Hamilton et al. Considering ω, Present model Without considering ω, Prasher et al.

2a = 50 nm

0.0000

0.0005

0.0010

0.0015

0.0020

Volume fraction

Volume fraction (C)

(D)

1.4

2.5

Experiment

1.2

100 × (kn/kf –1)

Maxwell model

100 × (kn/kf –1)

Experiment

Hamilton et al. Considering ω, Present model

1.0

Without considering ω, Prasher et al.

0.8 0.6

2a = 60 nm

0.4

Experiment Maxwell model

2.0

Hamilton et al. Considering ω, Present model Without considering ω, Prasher et al.

1.5

Φ = 0.002 1.0

0.2 0.5

0.0 0.0000

0.0005

0.0010

0.0015

0.0020

Volume fraction

10–7

10–6

2a (m)

FIGURE 4.4 Experimental results and theoretical prediction of thermal conductivity values of various cubic Ag nanofluid (NF) where particle lengths are (A) 30 nm, (B) 50 nm, and (C) 60 nm, and (D) is the relations between the thermal conductivity values and cubic lengths 2a changing from 20 to 1000 nm [7].

Fig. 4.4A depicts for the thermal conductivities of 30 nm cubic Ag NFs against volume fractions. It can be seen that the thermal conductivity values increase linearly with the particle volume fraction for all models. However, the thermal conductivity of cubic Ag NFs without considering ω models (i.e., the Maxwell’s, Hamilton et al.’s, and Prasher et al. models) are significantly smaller than that those “considering ω” case and the experiment results. Since Maxwell’s and Hamilton et al.’s models have not considered the effect of Brownian motion and interfacial thermal resistance, they cannot predict the thermal conductivity of the NF accurately. Although the Prasher et al.’s model considered the translation Brownian motion, the important rotating Brownian motion was ignored and thus a large error compared to the experimental results was also observed in their model. Fig. 4.4B and C illustrates the experimental and theoretical predicted thermal conductivity values for the 50 and 60 nm cubic Ag NFs in different Φ. The same treatment of Fig. 4.4A can also be observed here. The experimental results again reveal the reliability of the

Nanofluids

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4. Experimental analysis of nanofluids

proposed model. It can be found that by increasing the particle size, the thermal conductivity for the cubic Ag NFs decreases. This also confirms that the reverse relation of thermal conductivity with the particle size will not be significantly altered due to the particle shape. Moreover, by comparing the three figures, it can be concluded that the differences between the predicted thermal conductivity values for “considering ω” and “without considering ω” become smaller by increasing the cubic size for the same particle concentration. Fig. 4.4D illustrates this fact. As Fig. 4.4D shows, the results of the thermal conductivity values “considering ω” and “without considering ω” decrease and increase with the increase of particle size, respectively, while the Maxwell’s and Hamilton et al.’s models are constant with the change of cubic length. This treatment of two described models is due to the fact that models do not consider the effect of particle size. It is interesting to note that when the cubic length is close to 1000 nm, the difference between the “considering ω” and “without considering ω” cases mostly disappears. Also, the changes in the thermal conductivity become very small and even constant with the increasing sizes after certain values for both cases. For example, when the cubic length changes from 20 to 100 nm, the variations of the two cases are 1.7 and 0.15, respectively; while when the cubic length increases from 100 to 1000 nm, the variations are only 0.19 and 0.09. Clearly, the changes of velocity decrease significantly. This fact comes from changes to two main factors on thermal conductivity: the interfacial thermal resistance and the Re obtained by Brownian motion.

4.2 Different properties of nanofluids In this section, different thermophysical properties of NFs are presented based on the particle equivalent radius. Usually, three kinds of equivalent diameters are used to define the aggregate size: hydrodynamic radius (Rh), gyration radius (Rg), and smallest sphere enclosing radius (Rs) [9]. The hydrodynamic radius (Rh) refers to the size of the equivalent sphere that moves with the same velocity as the aggregate in a fluid, so Rh depends on the geometrical arrangement of particles. The complete equations of these radiis are presented in Ref. [9]. Here, viscosity, thermal conductivity, and optical properties of NFs are the most important thermophysical properties predicted based on equivalent radius. Finally, the predicted values obtained by different equivalent radii will be compared with experimental results to find which equivalent is more suitable in these models.

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4.2 Different properties of nanofluids

181

4.2.1 Viscosity Many models have been developed to describe the viscosity of NF. Nwosu et al. [10] introduced a simple method to show which viscosity models can be used in different cases. Earlier we reported that the semiempirical relation proposed by Krieger and Dougherty [11] would be suitable to describe the shear viscosity of NFs when considering particle aggregation [12]. This model includes a relationship between intrinsic viscosity and average particle radius [11], which represents the effect of aggregate size on viscosity. The model can be introduced as:   ½ Φ η Φ 2η m 5 12 (4.18) η0 Φm where η and η0 indicate the viscosity of base fluid and NF, respectively; and Φ and Φm are the nanoparticle volume fraction and maximum nanoparticle volume fraction, respectively. The value range of Φm is defined between 0.58 and 0.69 [13] and [η] is the intrinsic viscosity, which can be calculated by [14]:   1 R 3 ½η 5 2:5 (4.19) N r where R is an equivalent radius of aggregate. In Ref. [9], R values are chosen from Rh, Rg, or Rs, respectively, and results are compared with experimental outcomes.

4.2.2 Thermal conductivity In this section, a thermal conductivity model representing the relationship between aggregate shape and thermal conductivity of NF has been selected as suggested by Hamilton and Crosser [15]. It is:   kp 1 ðn 2 1Þkf 2 ðn 2 1ÞΦ kf 2 kp kn   5 (4.20) kf kp 1 ðn 2 1Þkf 1 Φ kf 2 kp where kn, kf, and kp indicate the thermal conductivities of the NF, base fluid, and nanoparticles, respectively; Ф is the nanoparticle volume fraction; and n is the shape factor where for spherical particles the value of n is 2, then Eq. (4.20) changes to Maxwell model [16]. For nonspherical particles, n can be obtained by n 5 3/ψ2, where ψ is the sphericity defined as ψ 5 Ss/Sa and Ss and Sa are the surface areas of the volumeequal sphere and aggregate, respectively. For a nanoparticle in an aggregate, due to the high specific surface area and the effect of surfactant, the observing layer of base fluid and surfactant has a thickness of

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4. Experimental analysis of nanofluids

Δ where ΔBRp. Thus the surface areas of the volume-equal sphere will be Ss 5 4π(3Vs/4/π)2/3, where Vs is the volume of the equal sphere and is calculated by Vs 5 4π/3N(Rp 1 Δ)3. To calculate the surface areas of aggregate Sa, three equivalent radii were chosen to find the most suitable value, Sa 5 4πR2 (R is equal to Rh, Rg, or Rs).

4.2.3 Optical properties There are three important theories for the explanation of the optical properties: Rayleigh scattering theory, MG effective medium theory, and Mie and Gans theory [9]. A complete discussion on these theories is given in Section 4.3 due to its importance. Among these theories, Rayleigh scattering theory is used in most studies due to its simplicity and accuracy. However, the Rayleigh scattering theory is most suitable for small particles and has significant errors for large particles. In Ref. [17], it is noted that the Rayleigh scattering theory (for d , nλ) and the Fraunhofer diffraction theory (for d . nλ) are suitable for determining the absorption coefficients of NFs where n is equal to 4 and satisfying results were obtained. This model can be presented as: 3 Qaλ 4 R 3 Qsλ ksλ 5 4 R

kaλ 5

(4.21a) (4.21b)

where kaλ and ksλ represent the absorption and scattering coefficients, respectively; R denotes the equivalent diameter of an aggregate; and Qaλ and Qsλ can be calculated from the following equations when d , nλ: 2

  m 21 α2 m2 2 1 m4 1 27m2 1 38 Qaλ 5 4αIm 1 1 (4.22) m2 1 1 2m2 1 3 15 m2 1 2

 2  2 8 m 21 (4.23) Qsλ 5 α4 Re 3 m2 12 where α 5 2πR/λ is called the size parameter, which depends on the aggregate radius and the incident wavelength λ; m is the relative complex refractive index defined as m 5 mparticle/mbasefluid; and Re and Im are the real and imaginary parts of a complex number, respectively. When d . nλ, Qaλ and Qsλ satisfy the Fraunhofer diffraction theory as: Qaλ 5 Qsλ 5 1

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(4.24)

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4.2 Different properties of nanofluids

By comparison of the Eqs. (4.21a) and (4.12b), it can be found that the factor “3/(4R)” is equivalent to the projected area dividing the volume of a sphere [i.e., 3/(4R) 5 πR2/(4/3πR3)]. However, most aggregates cannot be considered as spherical shapes when considering their volume. Thus a parameter Rva is defined. If an aggregate is a sphere, the Rva is equal to 3/(4R). If not, it is Rva 5 Sa/Va, where Va and Sa are the volume and projected area, respectively; and Va is calculated by 4π/ 3NRp3. Because it is difficult to obtain the particle number N, a relation for the mass-equivalent radius Rm is presented as Rm 5 Rg(Df/3) [9], and Va can be stated as Va 5 4/3πRm3. Moreover, the projected area of aggregate Sa is an important parameter to represent the effects of equivalent radii on optical properties. Here, it is assumed that Sa 5 πR2, where R indicates one of the equivalent radii (i.e., R 5 Rh, Rg, or Rs). Substituting the new relations of Va and Sa into Eqs. (4.21a) and (4.12b), we have: kaλ 5 ksλ 5

πR2 Df 4 3 πRg

πR2 Df 4 3 πRg

Qaλ 5 Qsλ 5

3R2 D

4Rg f 3R2 D

4Rg f

Qaλ

(4.25)

Qsλ

(4.26)

It must be noted that we need to choose the proper equivalent radius to obtain α. In Ref. [17], α was assumed equal to 2πRm/λ and reasonable results of absorption coefficient were found. 4.2.3.1 Experimental properties attained over aggregation radii In this case, first the hydrodynamic radius of aggregates in TiO2/ water NFs was measured experimentally with different nanoparticle volume fractions of 0.01%, 0.04%, and 0.1% using the dynamic light scattering (DLS) method. Cetyltrimethylammonium bromide (CTAB) surfactant as antiagglomeration material was added to obtain a more stable suspension. Also, NFs were preserved with magnetic stirring (600 rpm min21) for more than 1 h and ultrasonic fragmentation (40 kHz) for more than 0.5 h; the DLS measurements started immediately after the ultrasonic process. Fig. 4.5 shows the hydrodynamic radius curves for NFs with different concentrations. Although DLS can only give the hydrodynamic radius of aggregates and their volume friction, in Ref. [9], by an approximation method, the gyration radius and smallest sphere enclosing radius were calculated and are presented in Fig. 4.5. Fig. 4.5AC shows that for all three equivalent radii, two peaks coexist over the curve, where higher peak relates to the small particle size and the lower peak relates to the large particle size. This interesting property suggests a dynamic equilibrium process of coagulation and

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4. Experimental analysis of nanofluids

(A)

(B) 9

9 2Rh 2Rg 2Rs

Percentage (%)

7 6 5 4 3

7 6 5 4 3

2

2

1

1

0 0.1

1

2Rh 2Rg 2Rs

8

Percentage (%)

8

0 0.1

10

1

Particle size (μm)

(D)

9 2Rh 2Rg 2Rs

8

Percentage (%)

7 6 5 4 3 2 1 0 0.1

1

Particle size (μm)

10

Average particle size (μm)

(C)

10

Particle size (μm) 2Rh 2Rg 2Rs

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0.01%

0.04%

0.1%

Concentration

FIGURE 4.5 Experimental result of hydrodynamic, gyration, and smallest sphere enclosing radii distributions of aggregates in nanofluid (NFs), where solid volume fraction (A) Ф 5 0.01%, (B) 0.04%, and (C) 0.1%, primary particle radius Rp 5 10.5 nm. (D) the average particle size of aggregates [9].

fragmentation of aggregates. By considering different particle concentrations, it can be concluded that the percentage of the higher peak decreases as the concentration increases. This fact means that some small aggregates will further aggregate into larger ones leading to the decrease of the volume fractions of larger aggregates. Fig. 4.5D shows the increase of general particle size in NFs with higher concentration. The average hydrodynamic diameters (two times the radius) reported by the DLS method for Ф 5 0.01%, 0.04%, and 0.1% are 0.3347, 0.3695 and 0.4058, respectively. Furthermore, the average gyration and smallest sphere enclosing diameters are 0.4282, 0.4727, and 0.5192; 0.5941, 0.6559, and 0.7203, respectively. Based on the presented data, an order of Rh , Rg , Rs can be concluded for all three particle concentrations. In the next section, average particle size will be employed for calculations of viscosity, thermal conductivity, and optical properties due to the requirement of the mathematic models. Viscosities of NFs with different concentrations (Ф 5 0.01%, 0.04%, and 0.1%) are shown in Fig. 4.6. Because these NFs are diluted, they can

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4.2 Different properties of nanofluids

1.14

Relative viscosity

1.12 1.10

Experiment Rh Rg Rs

1.08 1.06 1.04 1.02 1.00 0.01%

0.04%

0.1%

Concentration FIGURE 4.6 Relative viscosity of nanofluids (NFs) with Ф 5 0.01%, 0.04%, and 0.1% where the experimental result was obtained using an Ubbelohde viscometer at a velocity very close to zero. Relative viscosity is defined as the ratio of viscosities of NF to base fluid (deionized water 1 a little surfactant) [9].

be considered as Newtonian fluids, so their viscosities are constant with a changing shear velocity. For the measurements, an Ubbelohde viscometer with an inner diameter of 0.6 mm was used to obtain the true values of the viscosities. Details of using method with a Ubbelohde viscometer are presented in Ref. [9]. Here, both experimental and calculation results obtained by Rh, Rg, and Rs are given in Fig. 4.6. The experimental measurements of relative viscosities are 1.0059, 1.0293, and 1.0772 for Ф 5 0.01%, 0.04%, and 0.1%, respectively, showing a general trend of increase with the increasing particle concentrations. As noted above, due to the larger value of Rs, the calculation results of viscosity achieved by Rs are significantly larger than Rh and Rg. The result obtained by Rh shows better agreement with the experimental result while the results obtained over Rg and Rs obviously fail to match the true value. Thus hydrodynamic radius is the most appropriate equivalent radius to describe the viscosity of NFs. Fig. 4.7 compares both the experimental and numerical results of thermal conductivities for NFs with different concentrations. It must be noted that the experimental data of relative thermal conductivities, which is the ratio between thermal conductivities of NFs and base fluids, were achieved by the transient hot-wire method with a conductometer (TC3000, Xi’an Xiatech Electronics; see details in Ref. [9]). The experimental measurements show 1.0011, 1.0048, and 1.0113 for NFs with Ф 5 0.01%, 0.04%, and 0.1%, respectively, which indicates that higher particle concentration leads to larger thermal conductivity. With the same treatment, the numerical results for Rh, Rg, and Rs show an

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4. Experimental analysis of nanofluids

Relative thermal conductivity

1.020 1.017 1.014

Experiment Rh Rh Rs

1.011 1.008 1.005 1.002 0.999 0.996 0.01%

0.04%

0.1%

Concentration FIGURE 4.7 Numerical and experimental thermal conductivities of nanofluids (NFs) with Ф 5 0.01%, 0.04%, and 0.1% [9].

increasing trend, possibly due to the larger radius of aggregates, which induces a larger n in Eqs. (4.21a) and (4.12b) and further increases the relative thermal conductivity. Here, the values for the thermal conductivity of the base fluid (deionized water 1 surfactant) and TiO2 nanoparticle of 0.6047 and 11.9 W (m K)21 were used to solve Eqs. (4.21a) and (4.12b). By comparing the experimental results and the three numerical results, it can be concluded that the results obtained by the gyration radius Rg, 1.0009, 1.0042, and 1.0122, are closest to the experimental data. The absorption coefficients of NFs with Ф 5 0.01%, 0.04%, and 0.1% are depicted in Fig. 4.8, which includes both numerical results for Rh, Rg, and Rs and experimental results measured by a ultravioletvisible (UVVIS) spectrophotometer, the Hitachi U-4100 (see details on this measurement technique in Ref. [9]). From the figure, it can be seen that the absorption coefficient generally increases with the increase of particle concentration, while the increase is not proportional to the variation of particle concentration. The average increase of absorption coefficient is about 1.2 cm21 from when Ф increased from 0.01% to 0.04%, while it is less than 0.5 cm21 for Ф increasing from 0.04% to 0.1%. Also, three curves of the predicted values over Rh, Rg, and Rs show an increase in order while the values obtained by radii are in the order of Rsc, RgcRh. Although the predicted results for all equivalent radii, Rh, Rg, and Rs, displayed a deviation, the predicted results of Rs show a more suitable match with the experimental results.

4.2.4 Surface tension Nanoparticlesurfactant suspensions, as complicated fluid systems, are colloidal dispersions of nanoparticles and surface active agents in

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4.2 Different properties of nanofluids

(B)

5 Experimet Rh Rg Rs

Abs (cm–1)

4 3 2

8 7 6

Abs (cm–1)

(A)

5

Experimet Rh Rg Rs

4 3 2

1

1 0 300 400 500 600 700 800 900 1000 1100 1200 Wavelength (nm)

Abs (cm–1)

(C)

0 300 400 500 600 700 800 900 1000 11001200 Wavelength (nm)

10 9 Experimet Rh 8 Rg 7 Rs 6 5 4 3 2 1 0 300 400 500 600 700 800 900 1000 1100 1200 Wavelength (nm)

FIGURE 4.8 Absorption coefficient of nanofluids (NFs) with (A) Ф 5 0.01%, (B) 0.04%, and (C) 0.1% where the experimental result was obtained using the Hitachi U-4100 ultravioletvisible (UVVIS) spectrophotometer [9].

suitable fluid such as water, organic solvent, etc. [18]. In this section, the effect of interactions between hydrophilic nanoparticles and ionic surfactants on the liquidair interface tension is investigated in a wide range of nanoparticle and surfactant concentrations, namely 02 wt.%, and 0.25, 0.5, and 1 CMC (critical micelle concentration), respectively. For this aim, two types of commercial hydrophilic nanoparticles, namely SiO2 (740 nm, Aladdin Chemistry Co. Ltd., China, specific surface area, B380 m2 g21) and Degussa P25 TiO2 (21 nm, 80% anatase, 20% rutile, specific surface area, B50 m2 g21), were used and dodecyl sodium sulfate (SDS) was used as the anionic surfactant and hexadecyl trimethylammonium bromide (CTAB) was the cationic surfactant. Furthermore, deionized water was chosen as the base fluid. More details on the preparation of the NF and experiment conditions can be found in Ref. [18]. The surface tension was measured using a contact angle meter (SL200B, Shanghai Zhongchen Digital Technology Apparatus Co. Ltd., China) with the principle of drop shape tensiometry and the results were compared with those measured by a JK99C1 tensiometer (Shanghai Zhongchen Digital Technology Apparatus Co. Ltd., China) based on the Wilhelmy plate method [18].

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4. Experimental analysis of nanofluids

FIGURE 4.9 Scanning electron microscope (SEM) images of nanoparticles for (A) SiO2 and (B) TiO2 and transmission electron microscope (TEM) images of nanoparticle suspensions for (C) SiO2 and (D) TiO2 [18]. (A)

(B)

16 SiO2 SiO2-SDS SiO2-CTAB

14 12

14

TiO2

12

TiO2-CTAB

TiO2-SDS

10 Intensity (%)

Intensity (%)

10

16

8 6

8 6

4

4

2

2

0

0 10

100

1000

10,000

10

Size (nm)

100

1000

10,000

Size (nm)

FIGURE 4.10 Particle size distributions of nanoparticle suspensions with or without surfactants determined by DLS for (A) SiO2 and (B) TiO2. Both the concentrations of dodecyl sodium sulfate (SDS) and CTAB for DLS measurement are 1 CMC (critical micelle concentration) [18]. DLS, dynamic light scattering; CTAB, cetyltrimethylammonium bromide.

Fig. 4.9AD shows the SEM and TEM images of SiO2 and TiO2 nanoparticles in the suspensions and reveals that the nanoparticles are approximately spherical. As presented in Fig. 4.10A and B, in the absence of surfactant, the average particle sizes of the nanoparticles

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4.2 Different properties of nanofluids

TABLE 4.1 Comparison of measured surface tension values of the pendant drop and Wilhelmy plate methods [18]. Suspensions

Pendant drop method (mN m21)

Wilhelmy plate method (mN m21)

Difference (%)

0.25 CMC SDS

51.22 6 0.42

50.26 6 0.45

1.91

0.25 CMC SDS 1 1 wt. % SiO2

45.58 6 0.53

44.79 6 0.36

1.76

0.5 CMC SDS

38.60 6 0.37

38.21 6 0.24

1.02

0.5 CMC SDS 1 1 wt.% SiO2

36.56 6 0.26

36.11 6 0.28

1.25

CMC SDS, Critical micelle concentration dodecyl sodium sulfate.

TABLE 4.2 Surface tension of water at liquidair interface with and without nanoparticles [18]. Nanoparticle (wt.%)

SiO2 suspensions (mN m21)

TiO2 suspensions (mN m21)

0

72.19 6 0.40

72.19 6 0.40

0.1

72.40 6 0.42

72.06 6 0.13

0.5

72.04 6 0.35

72.25 6 0.71

1

72.19 6 0.28

72.11 6 0.63

were determined to be 168.1 and 180.8 nm for SiO2 and TiO2 suspensions, respectively. Furthermore, Fig. 4.10 shows that by adding the surfactant a small change will occur in the size distributions of nanoparticles and aggregation because of the ionic concentration change and potential energy between nanoparticles. For accuracy, the obtained surface tension values from the pendant drop method are compared with the Wilhelmy plate method data. As seen in Table 4.1, the surface tension values measured by the two methods are in good agreement, with a difference of less than 2% [18]. The results of surface tension of nanoparticle suspension in the absence of surfactants as a function of nanoparticle concentration are summarized in Table 4.2. As can be seen, both SiO2 and TiO2 nanoparticles are not surface active and the addition of nanoparticles has an insignificant effect on the surface tension of the base fluid. Fig. 4.11 shows the surface tension as a function of surfactant concentration for systems containing surfactant only and surfactant with additional nanoparticles. It can be found that the surface tension of surfactant solution drops more suddenly in the presence of nanoparticles at low surfactant

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190 (A)

4. Experimental analysis of nanofluids

(B)

80

80 In the absence of TiO2

In the absence of SiO2

70

In the presence of 1 wt.% TiO2

−1

Surface tension (mN m )

In the presence of 1 wt.% SiO2

−1

Surface tension (mN m )

70 60 50 40

60 50 40

30 30 20

0

4

8

12

16

SDS concentration (mM)

0.0

0.5

1.0

1.5

2.0

CTAB concentration (mM)

FIGURE 4.11 Variation of surface tension of aqueous surfactant solutions with increase in the concentration of surfactants in the absence and presence of nanoparticles with like charge for (A) dodecyl sodium sulfate (SDS) and (B) cetyltrimethylammonium bromide (CTAB) [18].

concentrations. While at high surfactant concentrations, no obvious surface tension reduction is observed. The combined effects of nanoparticle and surfactant on the surface tension of colloid suspensions are studied in detail using Fig. 4.12, which summarizes the SiO2 suspension surface tension results based on the pendant drop method with and without SDS at different concentrations of nanoparticles and surfactants. The results reveal that the surface tension of surfactant solutions will further decrease with the inclusion of nanoparticles.

4.3 Optical properties of nanofluids In Section 4.2 a short study on the optical properties was presented. Here a complete study on the described theories is given. Before this, some terms need to be defined.

4.3.1 Aggregation of nanofluid Aggregation is an important phenomenon in NF due to its significant effect on the stabilization and thermodynamic properties of NF. Many factors are introduced as influence factors on the aggregation process, such as particle size, shape, method of preparation, and solid concentration of NF. To control aggregation, some common methods are performed such as chemical treatment on nanoparticles, addition of surfactant, changing pH value, etc. Commonly, the aggregation processes in NFs can be

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191

4.3 Optical properties of nanofluids

DI water

0.1 wt.% SiO2

IFT = 72.19 mN m−1

IFT = 72.40 mN m−1

IFT = 72.04 mN m−1

IFT = 72.19 mN m−1

IFT = 51.22 mN m−1

IFT = 49.71 mN m−1

IFT = 47.19 mN m−1

IFT = 45.58 mN m−1

IFT = 37.92 mN m−1

IFT = 37.19 mN m−1

IFT = 36.56 mN m−1

0.5 wt.% SiO2

1 wt.% SiO2

0%

0.25 CMC SDS

IFT = 38.60 mN m−1

0.5 CMC SDS

FIGURE 4.12 Illustration of the variation in the surface tension of SiO2 suspensions, SDS solutions, and SiO2SDS suspensions at different nanoparticle and surfactant concentrations with respect to deionized water from the pendant drop shape analysis [18]. SDS, Dodecyl sodium sulfate.

defined by diffusion limited cluster aggregation (DLCA) theory, which says if collision happens, aggregates will form [19]. Monte Carlo simulation [20], the population balance equation [13], and Brownian dynamic simulation [21] are the three main theoretical methods used to describe the aggregation process of nanoparticles in NF. Actually, in the population balance equation, only equivalent mass radius can be acquired, while Monte Carlo simulation can give more information on aggregation than the population balance equation, such as structure, kinds of equivalent radii, etc. On the other hand, when applying Monte Carlo simulation, some essential properties of nanoparticles and base fluid (e.g., density, viscosity, and temperature) have to be neglected. Thus the population balance equation can be considered as a more accurate method than Monte Carlo simulation. The third theoretical method (i.e., Brownian dynamic simulation) is similar to Monte Carlo simulation, and is probably the more physically sound method [19].

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4. Experimental analysis of nanofluids

4.3.1.1 Monte Carlo simulation Monte Carlo simulation works by randomly putting the initial N nanoparticles with radius Rp in a cubic box of size L, according to the chosen initial particle volume fraction Ф [20]: Ф5

4πR3p N 3L3

(4.27)

After that, a nanoparticle or aggregate is selected randomly to move over a distance equal to a nanoparticle diameter along a random direction. The criterion of choice is based on the StokesEinstein equation [19]. In this case (based on DLCA theory) the fragmentation mechanism has not been introduced, so the algorithm continues until all the initial primary particles finally aggregate into a single one. Since Monte Carlo simulation considers every collision, including the location and time, it is easy to obtain the structural properties of the aggregate, such as Rg, Rs, the scattering structure factor S(q), etc. The particleparticle correlation function g(r) is selected to obtain the structural properties [22]. g(r) can be expressed by: ðN i 2 1 5 4π gðrÞr2 dr (4.28) 0

This means that particle number i in an aggregate is equal to the infinite integral of g(r)4πr2dr. Lattuada et al. [21], using Monte Carlo simulation, derived the expression of g(r) as: 8   Nnn > > for r 5 2Rp > >4πð2R Þ2 δ r 2 2Rp > > p > > > > < a b for 2Rp , r , 4Rp (4.29) gðrÞ 5 Rpb13 r > > >     > > c Df 23 r > > > r exp 2 γ for r . 4Rp > Df > ξ : Rp where Nnn indicates the average number of nearest neighboring particles; δ(x) shows the Dirac delta function; a and b are two parameters describing the structure of the second coordination shell, respectively; and c, Df, ξ, and γ are the empirical constant factor, the fractal dimension, the cut-off length, and the cut-off exponent, respectively. 4.3.1.2 Population balance equation The population balance equation evaluates PSD based on primary particle number conservation, and the interactions between

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4.3 Optical properties of nanofluids

193

nanoparticles, nanoparticle and aggregate, and particles and base fluid. The population balance equation was first proposed by Blatz and Tobolsky [23] and Spicer and Pratsinis [24] introduced the following sectional model to solve its equation: i22 i21 X dNi X 1 2 5 2j2i11 β i21;j Ni21 Nj 1 β i21;i21 Ni21 2 Ni 2j2i β i;j Nj 2 dt j51 j51

2 Ni

imax 21 X j51

β i;j Nj 2 Si Ni 1

imax X

Γi;j Sj Nj

(4.30)

j5i

where Ni indicates the aggregate number in the ith section; the ith section denotes the set of aggregates containing 2(i21) primary particles; t is time; β i,j and Si are the collision (coagulation) and fragmentation kernel of aggregates in the ith and jth sections; and Γi,j shows the fragment distribution function. More details about this model can be found in Ref. [19]. 4.3.1.3 Brownian dynamic simulation As mentioned earlier, Brownian dynamic simulation is similar to Monte Carlo simulation in the first few steps of modeling and for the posttreatment of results. Since Brownian dynamic simulation is based on the Langevin equation, its collision rule is more realistic. In this method, the forces and motion of particles must be analyzed. Jiang et al. [25] explored possible forces on nanoparticles and aggregates as forces were introduced in Section 1.5. The forces on the nanoparticles include Fatt (attractive force between nanoparticles); Frep (electric double-layer repulsive force between nanoparticles); Fre (fluid resistance force); Fbrown (Brownian force), Fbuo (buoyancy force); and Fg (gravity force), respectively. The resultant force can be presented as: ~att 1 F ~rep 1 F ~brown 1 F ~buo 1 F ~g ~5 F ~re 1 F F

(4.31)

Fre can be calculated by the Stokes approximation, Fatt and Frep can be calculated by the equations presented in Ref. [19], and all other forces are shown below [25]: ~re 5 2 3πμ dv F f 0 1 2 4 4 m m X X Ad R 2 0:5d p 1 p C B ~att 5 ~att;i 5 F F @ 2nc0 nci  2 A 3 i51 i51 R3i R3i 2d2p

Nanofluids

(4.32)

(4.33)

194

4. Experimental analysis of nanofluids

~rep 5 F

m X i51

~rep;i 5 F

m X

   ! exp 2κ Ri 2 dp    1 1 exp 2κ Ri 2 dp sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6πμf dkB T

nc0 nci πεf κdp ζ 2p

i51

~brown 5 G F

Δt

~buo 5 nc0 F ~g 5 nc0 F

πd3p ρf g 6

πd3p ρp g 6

(4.34)

(4.35) (4.36) (4.37)

where μf and εf represent the dynamic viscosity and electric permittivity of the base fluid, respectively; κ is the inverse of double-layer thickness; ζ p is the zeta potential of the nanoparticles; G is the random number satisfying normal distribution; kB is the Boltzmann constant; T is the temperature; Δt is the time step; and ρf and ρp are the density of the base fluid and nanoparticles, respectively. Based on Newton’s second law, the nanoparticles and aggregates satisfy: ð Δt F a 5 ; v 5 v0 1 adt (4.38) m 0 where a is acceleration and v is velocity.

4.3.2 Optical properties of nanofluids The optical properties of particles depend on the particle size and shape [19]. With aggregation in NFs, the size and shape of particles change, so the optical properties also change. Therefore there are two ways to calculate the optical properties of NFs where aggregations occur: (1) assuming aggregates are sphere and homogeneous and then applying theories to work out optical properties and (2) obtaining the microstructures of aggregates by simulation and applying a multiple scattering theory to describe this scattering process. As described before, some of the general optical theories used include Rayleigh scattering theory, MG effective medium theory, and Mie and Gans theory described in the following. 4.3.2.1 Rayleigh scattering theory The Rayleigh scattering model was first proposed by Tyagi et al. [26] to obtain the optical properties of NFs. A basic supposition of the Rayleigh scattering theory is that particle size (around 1050 nm) is

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4.3 Optical properties of nanofluids

195

very small compared with the incident wavelength (greater than 300 nm). Thus this model cannot be used when aggregation occurs in NFs due to an increase in nanoparticle sizes by aggregation. Here, the contribution of the particles to the optical properties of NFs is defined as in Ref. [26]: 3 Qaλ 2 D 3 Qsλ ksλ 5 2 D

kaλ 5

(4.39) (4.40)

where kaλ and ksλ indicate the absorption and scattering coefficients, respectively; D is the equivalent diameter of an aggregate; and absorption efficiency Qaλ and scattering efficiency Qsλ are given by the following relations: 2

  m 21 α2 m2 2 1 m4 1 27m2 1 38 Qaλ 5 4αIm 11 (4.41) m2 1 1 2m2 1 3 15 m2 1 2

 2  2 8 4 m 21 (4.42) Qsλ 5 α Re 3 m2 12 where α 5 πD/λ is the size parameter, which depends on the aggregate radius and the incident wavelength, λ; m 5 mparticle/mbasefluid is the relative complex refractive index; and Re and Im are the real and imaginary parts of a complex number, respectively. 4.3.2.2 Mie scattering theory The Mie scattering theory describes the scattering process of large particles (d . λ). This theory also requires that scatters are homogeneous spheres. Since an aggregate can hardly meet the condition of Mie scattering, reasonable assumptions are needed to obtain satisfactory results [19]. In this theory, the optical efficiencies Qaλ and Qsλ in Eqs. (4.39) and (4.40) can be calculated by: N 2 X ð2n 1 1ÞRe ðan 1 bn Þ α2 n51

(4.43)

N   2 X 2 2 j j j j ð 2n 1 1 Þ a 1 b n n α2 n51

(4.44)

Qeλ 5 Qsλ 5

Qaλ 5 Qeλ 2 Qsλ where an and bn, called the Mie scattering coefficients, satisfy:

Nanofluids

(4.45)

196

4. Experimental analysis of nanofluids 0

0

an 5

ψn ðαÞψn ðmαÞ 2 mψn ðαÞψn ðmαÞ 0 0 ξ n ðαÞψn ðmαÞ 2 mξn ðαÞψn ðmαÞ

bn 5

mψn ðαÞψn ðmαÞ 2 ψn ðαÞψn ðmαÞ 0 0 mξn ðαÞψn ðmαÞ 2 ξn ðαÞψn ðmαÞ

0

(4.46)

0

(4.47)

where ψn(z) and ξn(z) are RicattiBessel functions, and can be determined by the semiinteger order Bessel function and Hamkel function as: πz1=2 ψn ðzÞ 5 Jn11=2 ðzÞ (4.48) 2 πz1=2 ð2Þ Hn11=2 ðzÞ (4.49) ξn ðzÞ 5 2 Thus to solve the Mie scattering theory, the RicattiBessel functions must be solved first. Usually, the recursive method is used to obtain the values of the functions [19]. According to the literature, n 5 1, 2, 3. . .nstop, and nstop is equal to α 1 cα1/3 1 b, where c 5 44.05 and b 5 12. Thus ψn(z) and ξn(z) satisfy the following repetition equations: 2n 2 1 ψn21 ðzÞ 2 ψn22 ðzÞ z n 0 ψn ðzÞ 5 2 ψn ðzÞ 1 ψn21 ðzÞ z 2n 2 1 ξn21 ðzÞ 2 ξn22 ðzÞ ξ n ðzÞ 5 z 0 n ξn ðzÞ 5 2 ξ n ðzÞ 1 ξn21 ðzÞ z

ψ n ðzÞ 5

(4.50) (4.51) (4.52) (4.53)

with initial conditions: ψ21 ðzÞ 5 cosðzÞ

(4.54)

ψ0 ðzÞ 5 sinðzÞ

(4.55)

ξ21 ðzÞ 5 expð2izÞ

(4.56)

ξ0 ðzÞ 5 iexpð2izÞ

(4.57)

where n 5 1, 2, 3. . ., nstop, i is an imaginary unit, and z can be α or mα. 4.3.2.3 MaxwellGarnett effective medium theory The MG effective medium theory, in contrast to the two previous theories that consider NF as multiphase, assumes NF as homogeneous with a compositive dielectric function, that is [19]:

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4.3 Optical properties of nanofluids

    # 3φ εp 2 εf = εp 1 2εf     εeff 5 εf 1 1 1 2 φ εp 2 2εf = εp 1 2εf

197

"

(4.58)

where εeff, εf, and εp are the dielectric functions of the NF, base fluid, and nanoparticles, respectively. By relating to the dielectric function, the refractive index can be presented by: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi uqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ε0 2 1 εv2 1 ε0 t eff eff eff neff 5 (4.59) 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ε0 2 1 εv2 2 ε0 t eff eff eff κeff 5 (4.60) 2 where neff, κeff represent the real and imaginary parts of the refractive 0 index and εeff and εveff indicate the real and imaginary parts of the dielectric function of the NF.

4.3.3 Optical models considering aggregation Basic optical theories require very strict conditions to be applicable. For instance, when aggregation occurs in the NF, Rayleigh scattering theory cannot be used to describe the scattering process due to the increase of particle size. Thus based on the above explanations, some optical models that consider aggregation are introduced in the following. 4.3.3.1 Generalized multiparticle Mie solution method The generalized multiparticle Mie solution (GMM) method, which is an extension of the Mie theory, is a practical approach to calculating the optical properties of aggregates of spheres as proposed by Xu [27]. This method considers both the incident radiation and the scattered radiation of other particles in the simulation area [19]. Thus this model can solve all optical problems of aggregates with different sizes and structures. It must be mentioned that in this theory the particles in aggregate must be homogeneous spheres because the Mie scattering theory is applied to them. In this method, the incident electromagnetic waves and scattering waves of all nanoparticles in the aggregate are extended in terms of vector spherical wave functions [20], that is: Eiinc 5

N X n X 2 X

ð1 Þ pimnp Nmnp ðkri Þ

n51 m52n p51

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(4.61)

198

4. Experimental analysis of nanofluids

Eisca 5

N X n X 2 X

ð3 Þ aimnp Nmnp ðkri Þ

(4.62)

n51 m52n p51 ð1Þ where k is the wave number; ri is the position vector; and Nmnp and ð3Þ Nmnp are the vector spherical wave functions (VSWF) for incoming and outgoing waves [28], respectively. Scattering coefficients aimnp can be calculated from pimnp using the Mie scattering theory:

aimnp 5 a imn pimnp

(4.63)

where a imn refers to the Mie coefficient of single nanoparticle i and pimnp is the expansion coefficients of the total incident electromagnetic field Eiinc for particle i. Eiinc includes two parts, the original incident wave Eio and the scattering wave from other particles Eis , which can be described as: Eiinc 5 Eio 1

N X j6¼i

 Eis j

(4.64)

The first step in the GMM solution is to make clear the structural information of the aggregate, because this method considers the relative positions of all primary particles in an aggregate. Then, the incident electromagnetic field and scattering field should be expanded in terms of vector spherical wave functions. Finally, the incident electromagnetic field on a single particle can become a first-kind vector spherical harmonics expansion [20]. The detailed solution of GMM can be found in Ref. [27]. 4.3.3.2 Finite difference time domain method The finite difference time domain (FDTD) method is an explicit timemarching algorithm (based on Maxwell’s equations) used to solve Maxwell’s curl equations on a discretized spatial grid [29]. This method can be used to explore both the near- and far-field electromagnetic responses of irregular and heterogeneous systems. The dielectric function of the scatterer is the only parameter required for this method. The electric flux-dependent form of Maxwell’s equations is [29]: ~ @D ~ 5r3H @t

(4.65)

DðωÞ 5 εðωÞEðωÞ

(4.66)

~ @D 1 ~ 52 r3E @t μ0

(4.67)

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4.4 Experimental correlations of nanofluid properties

199

where D and H represent the electric displacement vector and magnetic field intensity, respectively; and E indicates the electric field intensity and μ0 the permeability of vacuum. Based on the Yee algorithm [30], both D and H are offset in both space and time. For a one-dimensional system, the discrete form is:     Dn11=2 ðkÞ 2 Dn21=2 ðkÞ H n k 1 1=2 2 H n k 2 1=2 5 (4.68) Δt Δx where k and n are spatial and time coordinate. The time step must be small enough (no very small which increase computation time) to ensure that the information is not traveling faster than the speed of light in that medium [29]. This is called the “Courant condition,” that is: Δt 5 SC

Δx pffiffiffi c0 d

(4.69)

where d is the dimensionality and SC lies between 0 and 1 (usually between 0.5 and 0.9 [29]) and is called the Courant number. Eq. (4.69) indicates that time step (Δt) is related to mesh size (Δx), so mesh size can meaningfully influence the computation time and accuracy. A detailed application of the process can be found in Ref. [29].

4.3.4 Experimental research on optical properties The optical properties of NFs including absorption, scattering, and extinction coefficient (the sum of absorption and scattering coefficients). Usually, experiment optical coefficients are obtained based on a spectrophotometer (UV, VIS, and near infrared) [19] or self-designed apparatus based on LambertBeer law [31]. According to the following two main defects, the gained experimental absorption coefficient includes the contribution of some scattering light. First, because NF samples must be dispersed in a glass or plastic container, the transmission and reflection of container walls will be affected on the optical properties of NFs. Second, NF is an inhomogeneous two-phase mixture and light transporting in an inhomogeneous medium may accompany scattering, which leads to diffusion in all directions, not just the incident direction [19].

4.4 Experimental correlations of nanofluid properties In this section, as an application of last section, we investigate the relationship between the aggregated particle sizes and optical properties of NFs. For this aim, the absorption coefficients of NFs, PSDs (DLS method), and the absorption coefficient (using Rayleigh scattering and

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4. Experimental analysis of nanofluids

Mie scattering theories) are predicted and by comparison of experimental and numerical results, a high accuracy prediction model for absorption coefficients in NFs is presented [32].

4.4.1 Preparation of alumina nanofluids with controlled particle aggregation properties Here, nanoparticles were chemically functionalized by polar or nonpolar functional groups such as carboxylic and alkyl groups and were dispersed in organic or inorganic liquids such as ethyl alcohol and water to obtain NFs of different particle aggregation properties. Hydrophilic and lipophilic alumina nanoparticles with diameter of 30 nm, purchased from Aladdin (99.9% purity), were used. Deionized water and pure ethyl alcohol were employed as the organic and inorganic base fluids, respectively [32]. As in the previous procedure, nanoparticles were dispersed, stirred, and ultrasonic treatment applied to them. As presented in Fig. 4.13AD, four kinds of NFs were prepared: (A) AluminaP/water, (B) AluminaP/ethyl alcohol, (C) AluminaN/water, and (D) AluminaN/ethyl alcohol NFs, where

(A)

(B)

(C)

(D)

(G)

(H) 30 nm 30 nm

AluminaP/water

AluminaP/EA

AluminaN/water

AluminaN/EA

(I)

(J) 3 μm

(E)

4.5 μm

Base fluid (F)

(L) 6μ

m

(K)

2.5 μm Functional group

Alumina Hydrophobicity nanoparticle region

FIGURE 4.13 Prepared nanofluids (NFs): (A) AluminaP/water, (B) AluminaP/ ethyl alcohol, (C) AluminaN/water, and (D) AluminaN/ethyl alcohol NFs. Schematic description of the effects of interactions between functional groups and base fluids on the aggregation processes of NFs with (E) similar polarity and (F) opposite polarity between particle and base fluid; scanning electrical microscope (SEM) images of (G) AluminaP, (H) AluminaN, (I) aggregates in NFs of AluminaP/water, (J) AluminaP/ethyl alcohol, (K) in AluminaN/water, and (L) AluminaN/ethyl alcohol [32].

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4.4 Experimental correlations of nanofluid properties

201

P and N indicate nanoparticles with polar or nonpolar functional groups, respectively. Fig. 4.13E and F shows schematically the interaction effect between the functional groups and base fluids, which may affect the aggregation processes. In the case of (E), the base fluid and functional group have similar polarity and thus the surfaces of nanoparticles are covered with thick base fluid layers. This layer caused a difficult aggregate due to the absorbability between the nanoparticle and base fluid. In the case of (F), due to the different polarities between the base fluid and nanoparticle, there are hydrophobic regions around the nanoparticles, so probability of coalescence increases. As presented in the SEM images of Fig. 4.13GL, the shapes of nanoparticles are close to spheres and the particle diameters of both kinds of Al2O3 powders are about 30 nm. Fig. 4.13IL shows the aggregate sizes in the four different NFs corresponding to Fig. 4.13AD. As described, aggregate sizes in AluminaP/water, and AluminaN/ethyl alcohol NFs are smaller than those for AluminaP/ethyl alcohol and AluminaN/water NFs.

4.4.2 Measurement of the absorption coefficient of the nanofluid Generally, for NF with particles smaller than 20 nm, the scattering coefficient can be overlooked, but when particle sizes are larger than 20 nm, the influence of light scattering cannot be ignored [32]. To obtain the precise absorption coefficients of NFs, a similar experimental procedure by Cabrera et al. [33] was employed and a simple radiation transport model was used to split the experimentally obtained extinction into absorption and scattering coefficients, respectively. For the optical measurement of the NFs, a UVVIS spectrophotometer equipped with a lab-sphere diffuse reflectance accessory (U-4100, Hitachi) was used. A schematic diagram of the experimental measurement and data analysis are given in Fig. 4.14. The main three steps (AC) are shown in Fig. 4.14 and detailed in Ref. [32]. As seen in this figure, for each step a coefficient will be determined. Note that the obtained “absorption coefficient” is larger than the true absorption coefficient and smaller than the extinction coefficient. Thus, as shown in Fig. 4.14C, in order to find the pure absorption coefficient, the scattering spatial distribution function (the phase function) and radiative transport equation are introduced to process the experimental data. The radiative transport equation (RTE) can be used to exactly define the radiation field light distributions inside the reactor. To solve this equation, at least two parameters (the absorption and scattering

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4. Experimental analysis of nanofluids

Forward scattering

Back (A) scattering

Detector

Transmission light

Incident light

Integrating sphere

Sample

Slip (D) 12 mm

(B)

Slit

Back scattering

5 mm

(C)

μ=0 θ0

θ μ = cos(θ)

μ0

Sheet iron μ=1

μ=0

FIGURE 4.14 Schematic illustration of the procedure to obtain the accurate absorption coefficient of nanofluids (NFs); the measurements of (A) extinction coefficient, (B) absorption and forward scattering coefficient, (C) working out accurate absorption coefficient, and (D) size and material of slip [32].

coefficients) and one scattering spatial distribution function (the phase function) are required. The RTE can be presented as [33]:  ð β λ 2 κλ 1 @Iλ ðx; μÞ 1 β λ Iλ ðx; μÞ 5 μ Iλ ðx; μ0 Þpðμ; u0 Þdu0 (4.70) @x 2 μ0 521   (4.71) Iλ ð0; μÞ 5 I0 δ μ 2 μ0 μ . 0 Iλ ðL; μÞ 5 0 μ , 0

(4.72)

where μ is equal to cos(θ) as shown in Fig. 4.14C and μ0 is the cosine of the angle between the incident and the scattered rays (from 21 to 1), dimensionless, I refers to radiation intensity, unit einsteins m22 s21 sr21, β, κ, and b are volumetric extinction, scattering and absorption coefficients (unit m21), respectively (see Fig. 4.14). Also, p is the phase function and can be calculated by [34]:    pffiffiffiffiffiffiffiffiffiffiffiffi 8 0 1 2 μ0 2 μ0 cos21 μ0 pð μ Þ 5 (4.73) 3π For a given μ0 , the value of p can be determined. Actually, to apply the diffuse reflection phase function model, a significant assumption must be considered as the surface is an ideal diffuser with 100% diffuse reflection.

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4.4 Experimental correlations of nanofluid properties

203

4.4.3 Measurement of particle size distribution DLS is employed here for PSD. DLS uses the interferences between different scattering light by particles or aggregates to acquire the information of aggregate diffusion and Brownian motion. The scattering light signal of particles determined by Brownian motion is easily fluctuant in an observation time scale, so an autocorrelation function would be correct to describe this process [32]:    Iðq; 0ÞIðq; tÞ T ð2Þ  gT q; t 5 (4.74)   2 I q T where q 5 (4πn/λ0)sin(θ/2) calculates the scattering wave vector, n is the refractive index of the NF, λ0 is the wavelength of the incident light, θ is the scattering angle, and gT(2) indicates the average autocorrelation function of I. The following two main assumptions are considered for the simplification: (1) all particles and aggregates are independent of Brownian motion at the observation scale and (2) the number of particles and aggregates in the scattering region is large. Thus the Siegert relation can be used for the time averaged autocorrelation function, and the dynamic structure factor f(q,t) is determined [35]:     2 gð2Þ (4.75) T q; t 5 1 1 c f q; t where c, called the coherence factor, is the instrument constant. The autocorrelation function is investigated by fitting the following thirdorder polynomial equation to the experimental value [36]:      1 ln f q; t 5 1 2 Γ1 t 1 0:5Γ2 t2 2 (4.76) Γ3 t 3 6 where Г1 is called the first cumulant, which is related to the diffusion coefficient of an aggregate or particle through Г1/q2 5 D. Furthermore, in a mathematical sense, the hydrodynamic radius satisfies: Г1 kB T 5D5 2 6πηRh q

(4.77)

where the StokesEinstein relation has been used; D and kB are the diffusion coefficient of the aggregate or nanoparticle and Boltzmann constant, respectively; T indicates thermodynamic temperature; and η is the dynamic viscosity of the base fluid. To compare the aggregate sizes for different cases, two average diameters are considered as: P ND3 D32 5 P (4.78) ND2

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4. Experimental analysis of nanofluids

P ND4 D43 5 P ND3

(4.79)

where N shows the number of aggregates or nanoparticles whose diameters are D. Here, D32 and D43 are the Sauter average diameter (indicating the translational motion property of aggregates) and the biquadrate moment average diameter (indicating the rotation motion property of aggregates diameter), respectively.

4.4.4 Experimental result of absorption coefficients The absorption properties of nanoparticles, base fluids, and NFs were measured and the results are depicted in Fig. 4.15. In the infrared region, the absorption coefficient of the base fluid is quite large and the light absorption improvement caused by the nanoparticles will not be as significant as in the visible light region. As predicted by Fig. 4.15A, the values are around 0.06 in the wavelength range from 300 to 850 nm, and there is a small difference (less than 1%) of absorbance for (A)

(B)

0.14

Alumina-P Alumina-N

Water EA

0.35

Abs (cm–1)

Abs (a.u.)

0.12

0.40

0.10 0.08 0.06

0.30 0.25 0.20

0.04 0.15

0.02 0.00

300

400

500

600

700

800

0.10

300

400

Wavelength (nm)

(C)

600

700

800

2.0 Al2O3-P/water Al2O3-P/EG Al2O3-N/water Al2O3-N/EG

1.8 1.6

Abs (cm–1)

500

Wavelength (nm)

1.4 1.2 1.0 0.8 300

400

500

600

700

800

Wavelength (nm)

FIGURE 4.15 Absorbances of (A) two alumina powders, absorption coefficients of (B) base fluids, and (C) four Al2O3 nanofluids (NFs) obtained by ultravioletvisible (UVVIS) spectrophotometer (U-4100, Hitachi) [32].

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205

4.4 Experimental correlations of nanofluid properties

nanoparticles with different functional groups. Also, as depicted in Fig. 4.15B, the optical properties of the two base fluids, water and ethyl alcohol, also show no significant difference. In Fig. 4.15C, two characters of absorption coefficients of NFs can be found. First, with the change of wavelength, the absorption coefficients of all NFs have a regular decrease. Second, the values of absorption coefficient for the four kinds of NFs are in the order of AluminaN/ethyl alcohol . AluminaP/ water . AluminaP/ethyl alcohol . AluminaN/water. Studying the interactions between functional groups and base fluids, we can see that the PSDs of NFs are different, which may be a significant factor leading to variation in the optical properties of the four alumina NFs.

4.4.5 Experimental result of particle size distributions The equilibrium PSDs of the four NFs were measured by a Malvern Laser Particle Analyzer [32]. As seen in Fig. 4.16A, different NFs have different PSDs. The variation curves of AluminaP/water and AluminaN/ethyl alcohol NFs are the same and most of the particle are small (from 1 to 5 μm). The main particle size in AluminaP/ethyl alcohol NF is a little larger than the first two, but smaller than AluminaN/water. An comparison of the particle sizes is presented Pin Fig.P4.16B. Two kinds of average diameters, D32 and D43, satisfy Dab 5 NDa/ NDb, where N is the number of particles with the same diameter D. The average diameter values from left to right are 2.336, 2.765, 3.797, 2.316 μm for D32, and 4.627, 6.272, 12.56, 2.762 μm for D43, respectively. As an interesting result, two average diameters show similar trend to absorption coefficient, that is, AluminaN/ethyl alcohol . AluminaP/water . AluminaP/ethyl alcohol . AluminaN/water. Therefore there is a close relationship between the PSD and the absorption coefficients.

(B) 9 8 7 6 5 4 3 2 1 0 –1 0.1

Alumina-P/water Alumina-P/EA Alumina-N/water Alumina-N/EA

1

10 100 Particle size (μm)

Particle size (μm)

Percentage (%)

(A)

1000

13 12 11 10 9 8 7 6 5 4 3 2 1 0

Alumina-P/water Alumina-P/EA Alumina-N/water Alumina-N/EA

D43

D32

Average diameter

FIGURE 4.16 (A) Particle size distribution of the four nanofluids (NFs) and (B) two kinds of average diameters, D32 and D43, of the four kinds of NFs [32].

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206

4. Experimental analysis of nanofluids 4 × 104 3 × 104

(A)

Al2O3-P/water Al2O3-P/EA Al2O3-N/water Al2O3-N/EA

2 × 104 1 × 104

Abs(cm–1)

0 0.6

(B)

0.5 0.4 0.3 1.8

(C)

1.6 1.4 1.2 1.0 300

400

500

600

700

800

900

Wavelength (nm) FIGURE 4.17 Absorption coefficients of the four different Al2O3 nanofluids (NFs) obtained by (A) Rayleigh scattering theory and (B, C) Mie scattering theory. In (B) and (C), the projected areas of particles are acquired by the hydrodynamic diameter and the smallest sphere enclosing diameter, respectively [32].

4.4.6 Predicted results by theoretical model The Rayleigh scattering theory is first applied by introducing PSDs from Fig. 4.16 into Eqs. (4.78) and (4.79), and the absorption coefficient results are presented in Fig. 4.17A. However, due to aggregation and experimental condition λ{D the results of this theory do have good agreement with the experimental data in Fig. 4.15C [32]. Therefore the Mie scattering theory is applied as a second step as shown in Fig. 4.17B. These values have similar orders of absorption as the experimental values for the four kinds of NFs. Also, like the experiment, all absorption curves decrease with increase of wavelength. Thus the corrected absorption coefficients are calculated based on the smallest sphere enclosing diameter (details are described in Ref. [32] and the results presented in Fig. 4.17C). As seen, by introducing the projected areas of the smallest sphere enclosing diameter, an enhanced degree of matching between the theoretically and experimentally obtained optical absorption is reported.

4.5 Experimental application of nanofluids In this section, two applications of experimental NF analysis are presented for a hybrid photovoltaic/thermal (PV/T) system. Nanofluids

4.5 Experimental application of nanofluids

207

FIGURE 4.18 The UVvisible absorption spectral and physical images of the selected component materials of the electrolyte nanofluids (ENFs) [37].

4.5.1 Magnetic electrolyte nanofluids for a hybrid photovoltaic/ thermal solar collector application Common PV modules just convert 4%17% of the incident solar irradiation into electrical energy, depending on the type of solar cells and the working conditions, and more than 50% of the incoming solar energy is converted to heat, not electricity. One proposed solution to this problem is to combine a PV cell with a solar thermal collector, often referred to as hybrid PV/T system [37]. Also, electrolyte nanofluid (ENF), colloidal suspensions of nanoparticles in electrolyte solution, is used as a facile design of ENF as optical filters for the hybrid PV/T solar collector. Here, two stable ENFs [i.e., Fe3O4/methylene blue (MB) and Fe3O4/copper sulfate (CS)] are prepared by dispersing Fe3O4 nanoparticles in MB solution and copper sulfate (CS) solution. Fig. 4.18 shows the absorption spectral and physical images of the selected component materials of the ENFs [37]. All the materials listed in Table 4.3 were of analytical reagent grade without further purification. Details of the preparation process are given in Ref. [37]. The thermal conductivity of the ENFs was measured by the laser flash technique using a LFA 467 HyperFlash thermal analyzer from NETZSCH Instruments.

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4. Experimental analysis of nanofluids

TABLE 4.3 Data on purchased materials used in the study [37]. Materials

Manufacturers

Remarks

Fe3O4 nanoparticles

Aladdin, Shanghai, China

Particle size: 20 nm

Methylene blue

BSAF, Tianjin, China

C16H18ClN3S 3H2O

Copper sulfate

Sinopharm, Shanghai, China

CuSO4 5H2O

PEG-4000

Sinopharm, Shanghai, China



Ethylene glycol

Sinopharm, Shanghai, China







CS, Copper sulfate; EG, ethylene glycol; MB, methylene blue.

For independent scattering, the contribution of volume fraction φ to the extinction coefficient of particles, σnp, can be derived as [38]: σnp 5

3 Qext φ 2 D

(4.80)

where Qext indicates the extinction efficiency and D is the particle diameter. The absorbance, Aes, of the electrolyte solution can also be calculated from the BeerLambert law:   I0 Aes 5 log10 5 kcL (4.81) I where k is the molar absorptivity; and c and L are the molar concentration and length along the incident direction, respectively. The relation between the extinction coefficient σes and the absorbance Aes is [33]: σes 5

2:303Aes L

(4.82)

Combining Eqs. (4.81) and (4.82), σes can be obtained as the function of molar concentration: σes 5 2:303kc

(4.83)

By introducing Eqs. (4.80) and (4.83) to the main equation [37]: σtotal 5

3 Qext φ 1 2:303kc 2 D

(4.84)

This means the total extinct coefficient of the ENF is directly related to φ and c. Thus the volume fraction of the Fe3O4 nanoparticles and the molar concentration of each type of electrolyte are the two key parameters for experimental optimization procedures. To determine the

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209

4.5 Experimental application of nanofluids

TABLE 4.4 Comparison of optimized electrolyte nanofluid (ENF) filters with the reported water-based core/shell nanoparticle filters. Water-based core/shell nanoparticle nanofluid filters Design options

Optimized ENF filters

Particle 1 (φ)

Particle 2 (φ)

RMSE

Nanoparticle (φ)

Electrolyte (c)

RMSE

Si

2 nm Au, 50 nm SiO2 (7.3 3 1027)

30 nm pure Ag (2.5 3 1025)

25.82% [20 mm]

Fe3O4 (1.0 3 1025)

MB , 1.1 3 1024 .

16.91% [10 mm]

InGaP

4 nm Au, 30 nm SiO2 (2.1 3 1028)

4 nm Au, 40 nm SiO2 (6.8 3 1027)

27.05% [18.5 mm]

Fe3O4 (5.0 3 1026)

CS , 8.0 3 1022 .

19.16% [10 mm]

Values in brackets [] represent filter thickness, values in parentheses () represent particle volume fraction, and values in angle brackets ,. represent electrolyte molar concentration (mol L21). CS, Copper sulfate; MB, methylene blue; RMSE, root-mean-square error.

optical efficiency of an ENF-based optical filter the following integral form for root-mean-square error (RMSE) is used as the objective function: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uÐ longλ u ðT2Tideal Þ2 dλ 3 100% (4.85) ζ 5 t shortλ Ð longλ shortλ dλ where Tideal is the transmittance of ideal filters. The smaller value of ζ, the better designed filter. Details on choosing the appropriate NFs for two typical PV cells, Si and InGaP, can be found in Ref. [37]. Table 4.4 provides a comparison of optimized ENF filters with NF filters based on core/shell nanoparticles. One of the advantages of NFs is that they can easily be pumped in and out of a system or controlled by magnetic/electric fields. For solar energy-collecting applications, this is especially advantageous because an NF-based filter can also be applied as the heat transfer and thermal storage medium. Fig. 4.19 shows the TEM images of the optimized ENF optical filters. The TEM images confirm that the shape of the Fe3O4 nanoparticles are approximately spherical and seem to be uniformly dispersed. After careful examination, the average primary particle sizes for the optimized ENF filters were determined to be 15 6 2 nm. The particle sizes for the optimized NF filters were also considered by the laser light-scattering technique. The technique is based on the principle that particles passing through a laser beam will scatter light at an angle that is directly related to their size. DLS is a common method for the determination of PSD in nanoparticle suspensions. As presented in Fig. 4.20, the averaged particle sizes of the secondary particles were found to be 178 and 205 nm for ENF-1 and ENF-2, respectively [37].

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4. Experimental analysis of nanofluids

FIGURE 4.19 Low-magnification (A) and high-magnification (B) transmission electron microscope (TEM) images of electrolyte nanofluid 1 (ENF-1) [37]. (B)

(A)

30

20

25 Volume (%)

Volume (%)

15

10

20 15 10

5 5 0

0 10

100 Size (nm)

1000

10

100 Size (nm)

1000

FIGURE 4.20 Particle size distributions determined by dynamic light scattering (DLS) for (A) electrolyte nanofluid 1 (ENF-1) and (B) electrolyte nanofluid 2 (ENF-2) [37].

Thermal conductivity is another key parameter in the investigation of NF application in a PV/T system. Here, the thermal conductivity of ENFs are measured using an LFA 467 HyperFlash thermal analyzer from NETZSCH Instruments. It can be seen from Fig. 4.21A that there is good agreement (less than 1% deviation) between the measured results and the reported experimental outcomes. Fig. 4.21B shows the variation of the thermal conductivity of the two optimized NF filters under various temperatures. Clearly, the thermal conductivity increases with temperature increments. Also, it can be observed that both the optimized ENF filters have higher thermal conductivities than the base fluid (50% water/50% EG). 4.5.1.1 Application of electrolyte nanofluid in photovoltaic/thermal system A reported theoretical model [39] was applied to obtain the performance of the NF-based hybrid PV/T collector. Thus the collector was

Nanofluids

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4.5 Experimental application of nanofluids

(A)

(B) 0.43

Present work Sengers and Watson (1986)

0.66

Thermal conductivity W (m K)−1

Thermal conductivity W (m K)−1

0.67

0.65 0.64 0.63 0.62 0.61 0.60

50% water/50% EG ENF-1 ENF-2

0.42 0.41 0.40 0.39 0.38

0.59

0.37 10

20

30

40

50

60

20

70

30

Temperature (ºC)

40

50

60

Temperature (ºC)

FIGURE 4.21 (A) Thermal conductivity for pure water under various temperatures measured in the present work in comparison with reference data; (B) thermal conductivities for optimized electrolyte nanofluid (ENF) filters and base fluid [50% water/50% ethylene glycol (EG)] under various temperatures [37]. qr,amb

qc,amb Tamb

Tnf,in

Tg2a Tg2b

qglass 2

Glass 2

qnf,out qr,1-2

Nanofluids qnf,in qglass 1

Glass 1 Vaccum

Tnf,out

Tg1a Tg1b

qr,PV-1 TPV

PV cell Insulation

qnf

qins

FIGURE 4.22 Schematic of the hybrid photovoltaic/thermal (PV/T) collector using an nanofluid (NF)-based spectral filter [37].

designed as shown in Fig. 4.22, where the total size of the PV cell is 1 m 3 1 m with thickness of 0.2 mm. Furthermore, the height of the NF channel is 10 mm. The radiation is assumed to be due to the incoming solar radiation at an air mass AM 5 1.5. The related values of spectral irradiance were taken from ASTM G173-03 [37]. The thermal energy balance for each surface can be considered as: ð 2400nm    τ glass;1 ðλÞ τ glass;2 ðλÞ 1 2 αnf ðλÞ 1 2 ηPV ðλÞ GðλÞ dλ 5 qins 1 qr;PV-1 300nm

(4.86)

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4. Experimental analysis of nanofluids

qnf;in 1 qr;1-2 1

ð 2400nm

qr:PV-1 5 qglass;1

(4.87)

qglass;1 5 qnf;in 1 qr;1-2

(4.88)

τ glass;2 ðλÞαnf ðλÞGðλÞ dλ 5 qnf;out 1 qnf

(4.89)

300nm

qnf;out 5 qglass;2   4 1 qc;amb qglass;2 5 εσ T4 2 Tamb

(4.90) (4.91)

where τ glass(λ) represents the transmittance of the glasses at wavelength λ, αnf(λ) and G(λ) are the absorptance of NFs (taken from experiments) and the solar irradiance, respectively; ηPV(λ) is the photoelectric conversion efficiency for the selected PV cells (Si: 24.2%, InGaP: 42.3%) [40]; and q denotes the heat transfer per unit area. Here, the subscripts r and c are radiation and convection, respectively. Moreover, ε is the emittance of the glass; σ denotes the radiation constant; and T and Tamb are the temperatures of the glass and ambience, respectively. Now, six equations for the energy balance and six unknown temperatures exist. Additional parameters required by the model can be found in the literature [37]. The PV cell output power for the simple system (without an optical filter) and PV/T system (with an optical filter) can be computed by Eqs. (4.92) and (4.93), respectively: ð 2400nm τ glass ðλÞηPV ðλÞGðλÞ dλ (4.92) PPV;s-a 5 PPV;PVT 5

ð 2400nm 300nm

300nm

  τ glass;1 ðλÞτ glass;2 ðλÞ 1 2 αnf ðλÞ ηPV ðλÞGðλÞ dλ

(4.93)

The thermal output of the hybrid PV/T system can be calculated by:   _ p Tnf;out 2 Tnf;in Pth;PVT 5 mC (4.94) where Tnf,out and Tnf,in are the NF outlet temperature and inlet temperature, respectively; m˙ the mass flow rate; and cp is the specific NF heat. A weight factor, w, is assumed here for electrical power that accounts for its intrinsically higher economic value compared to heat and is set as a value of 3 [37]. The merit function (MF) is used to determine the performance of different working fluids. For the two kinds of PV cells, MF is the sum of the PV cell and thermal power in the hybrid PV/T system, divided by the PV output in the simple system with the same cell. Note that both PV outputs are multiplied by the weight factor:

Nanofluids

4.5 Experimental application of nanofluids

213

The value of merit function (MF)

1.8

Core/shell nanoparticle filter ENF filter 1.6

1.524

1.553 1.428

1.462

1.4

1.2

1.0

InGaP

Si

FIGURE 4.23 Comparison of the value of merit function (MF) between electrolyte nanofluid (ENF) and core/shell nanoparticle nanofluid (NF) filters at mass flow rate 0.01 kg s21 [37].

MF 5

wPPV;PV=T 1 Pth;PV=T wPPV;s-a

(4.95)

Finally, the MF values for the hybrid PV/T system with Si and InGaP PV cells at mass flow rate 0.01 kg s21 were calculated and are shown in Fig. 4.23. It confirms that the MF values for both systems are larger than 1, and also that the PV/T systems have better performance than simple systems without optical filters.

4.5.2 Highly dispersed nanofluid in a concentrating photovoltaic/thermal system In this case, a facile one-step preparation of highly dispersed SiO2/ H2O NF with nanoparticles of narrow size distributions was prepared for a PV/T system. The particles sizes were determined to be 5, 10, 25, and 50, respectively. The transmittance and thermal conductivity of the SiO2 NFs of various particle sizes were measured. In this twodimensional model, the NFs were circulated both above and below the PV panel to filter the infrared part of the incident light and to remove the heat generated in the photoelectric conversion process, respectively. This design is useful to reduce the operation temperature of PV cells and is thus to increase PV photoelectric efficiency. For this aim, silica particles in the size range of 550 nm are synthesized by hydrolysis and condensation of tetraethoxysilane (TEOS) in alcohol solvent in the presence of ammonium hydroxide. First, a certain amount of deionized H2O, NH3, and alcohol were mixed together for 10 min. Second, a known volume of TEOS was added into

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4. Experimental analysis of nanofluids

Transmittance (%)

140

DI water 1.0%–5 nm SiO2

120

1.0%–10 nm SiO2 1.0%–25 nm SiO2 1.0%–50 nm SiO2

100 80 60 40 20

(B) 140

0.5%–5 nm SiO2 1.0%–5 nm SiO2 2.0%–5 nm SiO2

120 Transmittance (%)

(A)

100 80 60 40 20

0 –20

–20 0

500 1000 1500 2000 2500 λ (nm)

0

500 1000 1500 2000 2500 λ (nm)

FIGURE 4.24 Transmittance of (A) silica/water nanofluids (NFs) with different particle sizes; (B) 5 nm SiO2/water NFs at different volume fractions [41].

the above solution at certain temperatures under stirring. After that, the solution was constantly stirred for 4 h. Four kinds of NFs with different particle sizes of 5, 10, 25, and 50 nm were prepared and labeled as NF-1, NF-2, NF-3, and NF-4, respectively. The size and morphology of the resulting particles were evaluated on a field emission SEM JSM7800F and a TEM FEI Tecnai G2F30 S-Twin. More details are presented in Ref. [41]. 4.5.2.1 Optical properties and thermal conductivity of the nanofluids Fig. 4.24A demonstrates that the transmittance of the silica/ water NFs is lower than that of deionized water. Furthermore the transmittance of silica/water NFs decreases with increased particle sizes. Fig. 4.24B confirms that the transmittance of 5 nm SiO2/ water NFs is very near to pure water even for the 2% volume fraction of NF, and the transmittance is also observed as high as 97% in this case. Fig. 4.25 shows the changes in the thermal conductivity of the SiO2/ water NFs as a function of temperature. As in the last example, the thermal conductivity increases with temperature. Fig. 4.25 also shows that the thermal conductivity increases as the volume fraction of the NF increases. Comparison of Fig. 4.25A and B shows the nanoparticle size effect and reveals that the thermal conductivity of the SiO2/water NFs with smaller particles is higher than that of the SiO2/water NFs with larger particle sizes. Considering both optical properties and thermal conductivity of the NFs, 5 nm SiO2/water NFs with 2% volume fraction is chosen to be used in the PV/T system.

Nanofluids

4.5 Experimental application of nanofluids

(A)

(B) 0.80

0.80 DI water 0.5% 1.0% 2.0%

0.75 k (W m–2 K–1)

k (W m–2 K–1)

0.75 0.70 0.65 0.60

215

5 nm

DI water 0.5% 1.0% 2.0%

0.70 0.65 10 nm

0.60

15 20 25 30 35 40 45 50 55 T (°C)

15 20 25 30 35 40 45 50 55 T (°C)

FIGURE 4.25 Effect of volume fraction and temperature on thermal conductivities of nanofluids (NFs) with particle size of (A) 5 nm and (B) 10 nm [41].

FIGURE 4.26 Two-dimensional (2D) sketch of the decoupled photovoltaic/thermal (PV/T) system [41].

4.5.2.2 Application of optimized nanofluids in a model photovoltaic/thermal system As presented in Fig. 4.26, NF is flowing below and above a PV cell, so it is applicable to decouple the PV and thermal systems so that each can operate at an optimum temperature and the temperature of the PV cell can be controlled appropriately. It is assumed that the PV cell operates at its optimal working condition and can generate electricity with enhanced efficiency [41]. The monocrystalline silicon solar cell was employed in the simulation. However, this model is also considered to be applicable to other solar cells working at relatively higher temperatures when the light concentration is high (i.e., 3 200). The size of the PV/T system is

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4. Experimental analysis of nanofluids

156 mm 3 15 mm with thickness of 0.2 mm. The optimal working temperature is considered to be 25 C with temperature coefficient of 0.4%. This PV module consists of 100 cells in total with a size of 1.56 m 3 1.56 m. The height of the channel for NF flow is considered to be 10 mm. Here, the discrete ordinates (DO) radiation model in ANSYSFLUENT 14.0 was used for simulation. Also, by using a gray-band model, the software allows for modeling of nongray radiation. Considering the irradiation of infrared light and for simplification, the gray DO model was used in this simulation. The DO model uses the radiative transfer equation (RTE) in the direction ~ s as a field equation and can be introduced as: ð σs 4π  0   0 0  0 2 ~ ~ r ðIλ ðr ;~ s dΩ Iλ ~ s Þ~ s Þ 1 ðαλ 1 σs ÞIλ ðr ;~ s Þ 5 αλ n Ibλ 1 r ;~ s Φ ~ s ~ 4π 0 (4.96)





where Iλ ð~ r ;~ s Þ indicates the spectral intensity; and αλ and Ibλ are the spectral absorption coefficient and the black 2 body intensity ðgiven by the Planck functionÞ, respectively. In this model, the scattering coefficient, σs , the refractive index n, and the scattering phase function are assumed independent of wavelength. In the modeling process, the transmittance of NF was determined by experimental measurements in Fig. 4.24. The heat absorbed by the upper NF can be obtained by Eq. (4.97), which was set as the inner heat source in the simulation procedure. Hup 5 Esun ðλÞTglass ðλÞ½1 2 Tup ðλÞ

(4.97)

where Hup shows the heat absorbed by the upper fluid; Esun ðλÞ is the solar radiation at wavelength λ; and Tglass ðλÞ and Tup ðλÞ are the transmittance of glasses and the transmittance of NF at wavelength λ, respectively. It can be shown that the photoelectric conversion efficiency of the PV cell commonly decreases with increased temperature. Due to assumptions, the temperature coefficient is 0.4%. Thus the photoelectric conversion efficiency can be obtained by the solar cell temperature as: ηel ðλÞ 5 Tglass ðλÞTup ðλÞηcell ðλÞ½1 2 ðT 2 298Þ 3 0:4%

(4.98)

where ηel and ηcell ðλÞ are the photoelectric conversion efficiency and the photoelectric conversion efficiency with respect to wavelength for the selected Si PV cell, respectively. The wavelength interval for the interpolation calculation is 0.2 nm, and T is its working temperature. The heat generated within the PV panel, Hcell , can be calculated by:

Nanofluids

4.5 Experimental application of nanofluids

217

Hcell 5 Esun ðλÞTglass ðλÞTup ðλÞ½1 2 Esun ðλÞηel 

(4.99)

By considering the temperature of the inlet NF as 25 C, the thermal output efficiency of the PV/T system, ηth , can be obtained by: ηth 5

Cp ρAνðTout 2 298Þ Esun

(4.100)

where Cp is the heat capacity of the fluid; ρ is the density of the fluid; A is the cross-section area of the flow channel; υ is the flow rate; andTout is the outlet temperature of the NF. A small wind of 0.1 m s21 on the PV module surface is assumed to simulate the real natural conditions. The radiative heat exchange between the upper glass surface and the outer space is introduced by: φrad 5 εAσðT 4 2 Tsky 4 Þ

(4.101)

where ε indicates emissivity and A is the surface area of the glass; σ refers to the radiation constant; and T and Tsky are the temperatures of the glass and sky, respectively. Also, the exergy efficiency for incident light to thermal output for the PV/T system can be obtained by:   Ten Exth 5 ηth 1 2 (4.102) Tout where Ten is the temperature of the environment. The exergy efficiency incident light to electricity output can be obtained by: Exel 5 ηel

(4.103)

The overall exergetic efficiency of the PV/T system can be obtained by [41]:   en ηth 12 TTout 1Exel Exth 1Exel Exoverall 5 5 (4.104) Esun Esun Details on the numerical simulation are given in Ref. [41]. Based on the above model, Fig. 4.27 is depicted for the exergetic efficiency of the PV/T system with various light concentrations against the velocities of the NF flow. Fig. 4.27A and B are plotted under no concentration, and as can be seen the change of exergetic efficiency for the PV/T system using NF is close to that of deionized (DI) water. Also, the exergetic efficiency of the system is quickly increased at first and then decreases a little with the increase of fluid velocity. For the concentration of 40, the system efficiency decreases continuously as the flow velocity increases. While the exergetic efficiency of the NF system is usually higher than that of the water system, it can be

Nanofluids

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4. Experimental analysis of nanofluids

(A)

(B) 0.185 DI water

0.180

Ex overall

Ex overall

Nanofluid

0.24

0.175 0.170 0.165

0.23 0.22

0.160

0.21

0.155

0.20 0.4

(C)

DI water

0.25

Nanofluid

0.8 1.2 1.6 V  10–3 (m s–1)

2.0

(D)

0.171

0.165

0.168

0.162

0.015

0.020

0.025 0.030 V (m s–1)

0.03

0.165 Ex overall

Ex overall

0.159 0.162 0.159

0.156

0.156

0.153

0.153

0.150 0.10

0.15 0.20 0.25 V (m s–1)

0.30

0.30 0.35 0.40 0.45 0.50 0.55 0.60 V (m s–1)

FIGURE 4.27 Exergetic efficiency of the photovoltaic/thermal (PV/T) system with various light concentrations versus the velocities of the nanofluid (NF) flow: (A) 3 1; (B) 3 40; (C) 3 100; (D) 3 150 [41].

concluded that the maximum exergetic efficiencies occurred at the flow velocity of 0.015 m s21 for both working fluids. Fig. 4.27C and D shows that for high concentration conditions, the exergetic efficiency of the NF system is clearly greater than that of the water system, especially at low flow velocities. A comparison of the exergetic efficiency of PV/T system working at optimized concentrations using NF and DI water is presented in Fig. 4.28. As can be seen, the exergetic efficiency of the PV/T system using NF as the working fluid is always greater than that with water as working fluid. The maximum exergetic efficiency and an enhancement of 7.0% exergetic efficiency was observed when the concentration is 40 and at the flow velocity of 0.015 m s21. Also, when the concentration is 100 and at a flow velocity of 0.1 m s21, an exergetic efficiency enhancement of 9.5% can be obtained. Thus there is an obvious advantage to using NF as working fluid in a PV/T system especially at certain concentrations and low flow velocity [41].

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References

0.275 DI water Nanofluid

0.015 m s–1 0.250

Ex overall

0.225

0.200

0.175

0.1 m s–1

0.0003 m s–1

0.3 m s–1

0.150 1

40 100 Concentration

150

FIGURE 4.28

Comparison of exergetic efficiency of photovoltaic/thermal (PV/T) system working at optimized concentrations and nanofluid (NF) flow velocity using deionized (DI) water [41].

References [1] D. Song, Y. Yang, D. Jing, Insight into the contribution of rotating Brownian motion of nonspherical particle to the thermal conductivity enhancement of nanofluid, Int. J. Heat Mass Transf. 112 (2017) 6171. [2] K.N. Shukla, T.M. Koller, M.H. Rausch, A.P. Fro¨ba, Effective thermal conductivity of nanofluids  a new model taking into consideration Brownian motion, Int. J. Heat Mass Tranf. 99 (2016) 532540. [3] D.H. Kumar, H.E. Patel, V.R. Rajeev Kumar, T. Sundararajan, T. Pradeep, S.K. Das, Model for heat conduction in nanofluids, Phys. Rev. Lett. 93 (2004) 144301. [4] J. Zhou, M. Hatami, D. Song, D. Jing, Design of microchannel heat sink with wavy channel and its time-efficient optimization with combined RSM and FVM methods, Int. J. Heat Mass Transf. 103 (2016) 715724. [5] F.P. Incropera, D.P. Dewitt, Fundamentals of Heat and Mass Transfer, Wiley, New York, 1996. [6] A. Acrivos, T.D. Taylor, Heat and mass transfer from single spheres in Stokes flow, Phys. Fluids 5 (1962) 387394. [7] R. Prasher, P. Bhattacharya, P.E. Phelan, Thermal conductivity of nanoscale colloidal solutions (nanofluids), Phys. Rev. Lett. 94 (2005) 025901. [8] Y. Sun, Y. Xia, Shape-controlled synthesis of gold and silver nanoparticles, Science 298 (2002) 21762179. [9] D. Song, J. Zhou, Y. Wang, D. Jing, Choice of appropriate aggregation radius for the descriptions of different properties of the nanofluids, Appl. Therm. Eng. 103 (2016) 92101. [10] P.N. Nwosu, J. Meyer, M. Sharifpur, Nanofluid viscosity: a simple model selection algorithm and parametric evaluation, Comput. Fluids 101 (2014) 241249. [11] I.M. Krieger, T.J. Dougherty, A mechanism for non-Newtonian flow in suspensions of rigid spheres, Trans. Soc. Rheol. (1957-1977) 3 (1959) 137152.

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4. Experimental analysis of nanofluids

[12] D.X. Song, D.W. Jing, J.F. Geng, Y.X. Ren, A modified aggregation based model for the accurate prediction of particle distribution and viscosity in magnetic nanofluids, Powder Technol. 283 (2015) 561569. [13] G. Barthelmes, S.E. Pratsinis, H. Buggisch, Particle size distributions and viscosity of suspensions undergoing shear-induced coagulation and fragmentation, Chem. Eng. Sci. 58 (2003) 28932902. [14] S. Tang, Y. Ma, C. Shi, Modelling the mechanical strength of fractal aggregates, Colloids Surf. A 180 (2001) 716. [15] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous two-component systems, Mem.of Sagami Inst. Technol. 11 (1962) 2740. [16] J.C. Maxwell, Treatise on Electricity and Magnetism, (Clarendon Press, Oxford, 1873. [17] D.X. Song, M. Hatami, Y.C. Wang, D.W. Jing, Y. Yang, Prediction of hydrodynamic and optical properties of TiO2/water suspension considering particle size distribution, Int. J. Heat Mass Transf. 92 (2016) 864876. [18] J. Jin, X. Li, J. Geng, D. Jing, Insights into the complex interaction between hydrophilic nanoparticles and ionic surfactants at the liquid/air interface, Phys. Chem. Chem. Phys. 20 (22) (2018) 1522315235. [19] D. Jing, D. Song, Optical properties of nanofluids considering particle size distribution: experimental and theoretical investigations, Renew. Sustain. Energy Rev. 78 (2017) 452465. [20] M. Du, G. Tang, Optical property of nanofluids with particle agglomeration, Sol. Energy 122 (2015) 864872. [21] M. Lattuada, H. Wu, M. Morbidelli, A simple model for the structure of fractal aggregates, J. Colloid Interface Sci. 268 (2003) 106120. [22] C. Johnson, Laser Light Scattering, Dover, New York, 1990. [23] P. Blatz, A. Tobolsky, Note on the kinetics of systems manifesting simultaneous polymerization-depolymerisation phenomena, J. Phys. Chem. 49 (1945) 7780. [24] P. Spicer, S. Pratsinis, Coagulation and fragmentation: universal steady-state particle size distribution, AIChE J. 42 (1996) 16121620. [25] W. Jiang, G. Ding, H. Peng, H. Hu, Modeling of nanoparticles’ aggregation and sedimentation in nanofluid, Curr. Appl. Phys. 10 (2010) 934941. [26] H. Tyagi, P. Phelan, R. Prasher, Predicted efficiency of a low-temperature nanofluidbased direct absorption solar collector, ASME J. Sol. Energy Eng. 131 (4) (2009) 041004. [27] Y. Xu, Electromagnetic scattering by an aggregate of spheres, Appl. Opt. 34 (21) (1995) 45734588. [28] Y. Xu, Radiative scattering properties of an ensemble of variously shaped small particles, Phys. Rev. E 67 (2003) 046620. [29] C. Oubre, P. Nordlander, Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method, J. Phys. Chem. B 108 (2004) 1774017747. [30] K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 13 (1966) 302307. [31] S. Lee, S.P. Jang, Extinction coefficient of aqueous nanofluids containing multi-walled carbon nanotubes, Int. J. Heat Mass Transf. 67 (2013) 930935. [32] D. Song, Y. Wang, D. Jing, J. Geng, Investigation and prediction of optical properties of alumina nanofluids with different aggregation properties, Int. J. Heat Mass Transf. 96 (2016) 430437. [33] M.I. Cabrera, O.M. Alfano, A.E. Cassano, Absorption and scattering coefficients of titanium dioxide particulate suspensions in water, J. Phys. Chem. 100 (1996) 2004320050.

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Further reading

221

[34] R. Siegel, J. Howell, Thermal Radiation Heat Transfer, third ed., Hemisphere, Bristol, PA, 1992. [35] P.N. Pusey, M.W. Van, Dynamic light scattering by non-ergodic media, Physica A Stat. Mech. Appl. 157 (1989) 705741. [36] P. Sandkqhler, M. Lattuada, H. Wu, J. Sefcik, M. Morbidelli, Further insights into the universality of colloidal aggregation, Adv. Collioid Interface Sci. 113 (2005) 6583. [37] J. Jin, D. Jing, A novel liquid optical filter based on magnetic electrolyte nanofluids for hybrid photovoltaic/thermal solar collector application, Sol. Energy 155 (2017) 5161. [38] R. Xu, Particle Characterization: Light Scattering Methods, Kluwer Academic Publishing, Dordrecht, 2000. [39] T.P. Otanicar, I. Chowdhury, R. Prasher, P.E. Phelan, Band-gap tuned direct absorption for hybrid concentrating solar photovoltaic/thermal system, J. Sol. Energy Eng. 133 (2011) T20046-1T20046-8. [40] M.A. Green, K. Emery, Y. Hishikawa, W. Warta, Solar cell efficiency tables (version 37), Prog. Photovolt. Res. Appl. 19 (2011) 8492. [41] D. Jing, Y. Hu, M. Liu, J. Wei, L. Guo, Preparation of highly dispersed nanofluid and CFD study of its utilization in a concentrating PV/T system, Sol. Energy 112 (2015) 3040.

Further reading S. Ahmad, U. Riaz, A. Kaushik, J. Alam, Soft template synthesis of superparamagnetic Fe3O4 nanoparticles a novel technique, J. Inorg. Organomet. Polym. 19 (2009) 355360.

Nanofluids

C H A P T E R

5 Nanofluid analysis in different media

5.1 Nanofluids in porous media In this chapter, three main media (porous media, magnetic field, and thermal radiation area) are introduced and nanofluid treatments are studied both experimentally and numerically. For each area three example cases are introduced and results are discussed. In the first media (i.e., the porous media) the penetration of nanoparticles from the porosities is an most important parameter, and it depends on the nanofluid aggregation, porosity, permeability, nanoparticle size, etc. Each of these parameters were introduced in earlier chapters, so in this section, three example cases are introduced to show their effects.

5.1.1 Case 1: Lid-driven T-shaped porous cavity Consider a T-shaped porous cavity with four defined dimensions (a, b, c, and d) as presented in Table 5.1 and Fig. 5.1. Boundary conditions are considered as follows: The top wall of the cavity is moving at a constant velocity u0 and low temperature Tc; all other walls are fixed and isolated except the bottom wall, which is heated and kept at high temperature, Th. The cavity is filled by various nanofluids (NFs) with the thermal properties shown in Chapter 2, Mathematical analysis of nanofluids. For this problem, the flow is supposed to be steady and laminar. NF density is considered to be variable by Boussinesq approximation. As described in Chapter 1, Introduction to nanofluids, single-phase modeling is applied in this case. Also, no-slip condition is considered for the fluid boundary conditions on the wall sides. The governing equations are as follows [1]:

Nanofluids DOI: https://doi.org/10.1016/B978-0-08-102933-6.00005-6

223

© 2020 Elsevier Ltd. All rights reserved.

224

5. Nanofluid analysis in different media

TABLE 5.1 Variation of average Nusselt numbers in different central composite design (CCD) proposed cases for Al2O3water nanofluid when Re 5 50, ϕ 5 0.05, Ri 5 0.1, and Da 5 0.001 [1]. Case number

a

b

c

d

Nusselt

1

0.2

0.6

0.6

0.6

0.3097

2

0.2

0.2

0.2

0.2

0.8874

3

0.6

0.2

0.6

0.6

2.1915

4

0.6

1.0

0.6

0.6

0.5324

5

1.0

0.2

1.0

1.0

2.2821

6

0.6

0.6

0.6

0.2

1.3324

u=u0, v=0, T=Tc d

g Adiabatic

C b

y

u=v=0, T=Th

a

x

FIGURE 5.1 Schematic of the porous T-shaped cavity [1].

The continuity equation: ρnf

  @u @v 1 50 @x @y

(5.1)

The momentum equation for porous cavity:    2  μnf @u @u @p @ u @2 u 1v 1 μnf ρnf u u (5.2) 1 52 2 2 2 @x @y @x @x @y K    2  μnf @v @v @p @ v @2 v 1 μnf v 1 gðρβ Þnf ðT 2 TC Þ ρnf u 1 v 1 2 2 52 @x @y @y @x2 @y K (5.3)

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5.1 Nanofluids in porous media

The last term on the right-hand side of Eq. (5.2) demonstrates the porosity effect as well as the same term in Eq. (5.3). In these terms, K denotes the permeability of the porous medium (m2). The energy equation is:  2  @T @T @T @2 T 1v 5 αnf u 1 2 (5.4) @x @y @x2 @y The density of the NF, ρnf, depends on the solid volume fraction, and it can be expressed as described in Chapter 2, Mathematical analysis of nanofluids. To solve the governing equations in nondimensional forms, the following nondimensional variables are applied to obtain the nondimensional governing equations: x y Y5 L L u U5 u0 v p V5 ; P5 ; u0 ρnf u20 X5

(5.5) Θ5

T 2 TC ; Th 2 Tc

Da 5

K L2

where L is equal to a 1 2c and Da indicates the Darcy number for the porous medium. The nondimensional governing parameters introduced in the present case are:   μCp f ρf u0 L Re 5 ; Pr 5 kf μf (5.6) 2 3 gρf β f ðTh 2 Tc ÞL Gr Gr 5 ; Ri 5 2 Re μ2f After applying these parameters, the nondimensional forms of the continuity, momentum, and energy equations will be: @U @V 1 50 @X @Y   μnf 1 @2 U μnf @U @U @P @2 U 1 U 1V 52 1 U 1 2 2 2 @X @Y @X @Y ρnf vf Re @X ρnf vf Re Da

(5.7) (5.8)

  μnf 1 @2 V μnf ðρβ Þnf @V @V @P @2 V 1 U 1V 52 1 V1 1 Riθ 2 @X @Y @Y ρnf vf Re @X2 @Y2 ρnf vf Re Da ρnf β f (5.9)

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5. Nanofluid analysis in different media

 2  @Θ @Θ αnf 1 @ Θ @2 Θ 1V 5 U 1 @X @Y @Y2 αf Re Pr @X2

(5.10)

where U, V are the velocities in the x and y directions, respectively; P refers to the pressure; and vf indicates the kinematic viscosity of base fluid. The subscripts nf and f are related to the NF properties and base fluid, respectively. The average fluid temperature inside the cavity can be computed using the following equation: Θav 5

ð 1 ΘdA A

(5.11)

The heat transfer phenomena is evaluated using local and average Nusselt numbers of the heated bottom wall, which are expressed by the following relations:  @Θ dX @Y 0 ð   knf L a=L @Θ ha 52 dX Average: Nu 5 kf @Y Y50 kf a 0

Local: Nu 5 2

knf kf

ða 

(5.12)

where h is the convection coefficient heat transfer. As presented in the boundary conditions, a T-shaped lid-driven porous cavity is considered under the mixed convection for different NFs. In this function, geometry parameters (a, b, c, and d) are considered as the understudy parameters for this cavity. Since Satyajit et al. [2] supposed constant numbers for these dimensions for a simple (nonporous) cavity and Al2O3water NFs, here, the equations are derived for a porous cavity and we try to find optimal constants for these parameters to reach a higher Nusselt number with different nanoparticles (Al2O3, Cu, and TiO2). For this aim, the central composite design (CCD) technique is applied for three levels of the factors as presented in Table 5.1. The first six different geometries (obtained from CCD) with various dimensions have been chosen as shown in Table 5.1. By using the FEM, the average Nusselt number for all these cases were calculated when Re 5 50, ϕ 5 0.05, Ri 5 0.1, and Da 5 0.001. Table 5.1 shows that the Nusselt number for cases 3 and 5 are significantly greater than other geometries. Figs. 5.2 and 5.3 are obtained from the numerical outcomes of the initial six cases, which show the isotherm and streamlines, respectively.

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5.1 Nanofluids in porous media

227

FIGURE 5.2 Temperature contour of central composite design (CCD) proposed cases for Al2O3water nanofluid when Re 5 50, ϕ 5 0.05, Ri 5 0.1, and Da 5 0.001 [1].

FIGURE 5.3 Streamline contour of central composite design (CCD) proposed cases for Al2O3water nanofluid when Re 5 50, ϕ 5 0.05, Ri 5 0.1, and Da 5 0.001 [1].

Because here we want to show the effect of porous media and NF flow, the effect of Darcy (Da) and nanoparticle volume fraction (ϕ) on isotherm lines are presented in Fig. 5.4. This figure confirms that the Da effect is more visible on the isotherm lines, while Fig. 5.5 confirms that Nanofluids

228

5. Nanofluid analysis in different media

FIGURE 5.4 Effect of Da and ϕ on the temperature contours for optimized case when Re 5 50, Ri 5 0.1 [1].

(A)

(B) ϕ=0.1 ϕ=0.0

5

Da=0.0001 Da=0.001 Da=0.01

8

Local Nusselt

Local Nusselt

7 4 3 2

6 5 4 3 2

1

0

0.2

0.4 Wall length

0.6

0

(C) 5

0.2

0.4 Wall length

0.6

ϕ=0.0 ϕ=0.1

4.5 Average Nusselt

4 3.5 3 2.5 2 1.5 1

0

0.002

0.004

0.006 Da

0.008

0.01

FIGURE 5.5 Effect of Da an ϕ on local and average Nusselt numbers for Al2O3water nanofluid in porous cavity [1]. (A) Local Nusselt number for different nanoparticle volume fraction along wall length; (B) Local Nusselt number for different Darcy numbers along wall length; (C) Average Nusselt number for different nanoparticle volume fraction and Darcy numbers.

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5.1 Nanofluids in porous media

229

both Da and ϕ have a valuable effect on Nusselt number and consequently heat transfer. Increasing the ϕ and Da lead to an increase in Nusselt number due to severely dependent of Nu to ϕ via NF thermal conductivity. Also, the effect of different nanoparticles (TiO2, Al2O3, and Cu) on isotherm, streamlines, and local Nusselt number is depicted in Fig. 5.6. As can be observed, the maximum local Nusselt number occurs for Cu nanoparticles while its minimum values are reported for TiO2 due to greater thermal conductivity of Cu nanoparticles compared to TiO2 [1].

FIGURE 5.6 Effect of different nanoparticles on the isotherm, streamlines, and local Nusselt number for optimized case when Da 5 0.01, ϕ 5 0.1 [1].

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5. Nanofluid analysis in different media

5.1.2 Case 2: Nanofluid in porous-filled absorber tube of solar collector In this case, a two-dimensional NF-based concentrating parabolic solar collector is modeled under the constant heat flux as depicted in Fig. 5.7. The absorber tube is filled by porous material and four types of nanoparticles are added in the absorber based on the properties in Chapter 1, Introduction to nanofluids [3]. The boundary condition is constant solar heat flux at the outer wall and constant low temperature Tc for the inner wall; the two remaining walls are considered insulation due to symmetric condition. The NF used in this analysis is water based and supposed to be Newtonian, incompressible, laminar, steady, and incompressible flow. By considering the above approximations, the governing equations are [4]: @v @u 1 50 (5.13) @y @x    2  μnf @u @u @P @ u @2 u 1u 1 μnf ρnf v u (5.14) 1 52 2 2 2 @y @x @x @x @y K    2  μnf @v @v @P @ v @2 v 1 ρnf β nf gðT 2 Tc Þ 1 μnf ν ρnf v 1 u 1 2 2 52 @y @x @y @x2 @y K (5.15)    2    @T @T @ T @2 T 1u 1 2 ρCp nf v 5 knf (5.16) 2 @y @x @x @y where ρnf, (ρCp)nf αnf, β nf, μnf, knf, and σnf were defined in Chapter 1, Introduction to nanofluids, based on nanoparticle concentration. The stream function and vorticity are presented as follows to remove pressure source terms: @ψ @ψ @v @u ; v52 ; w5 2 (5.17) u5 @y @x @x @y

FIGURE 5.7 Geometry and boundary conditions and generated mesh considered in this case [3].

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231

5.1 Nanofluids in porous media

Also, the following dimensionless variables are defined: T 2 TC ; Θ5  qvR=kf P5

U5

pR2  2 ; ρnf αnf

uR ; αnf Ω5

V5 wR2 ; αnf

vR ; αnf Ψ5

Y5

y ; R

Χ5

x ; R (5.18)

ψ αnf

Using Eq. (5.18), Eqs. (5.13)(5.16) can be introduced in dimensionless form as [4]: @2 Ψ @2 Ψ 1 52Ω (5.19) @X2 @Y2 2 3 2 32 1    30 2 2 ρCp nf = ρCp f μnf =μf @Ψ @Ω @Ψ @Ω @ Ω @ Ω 4 5 5 Pr4 54 5@ A 2 1 @Y @X @X @Y @X2 @Y2 ρnf =ρf knf =kf 2 3 2 32   3     ρCp nf = ρCp f ðρβ Þnf =ðρβ Þf ðρCp Þnf =ðρCp Þf 2 @Θ μnf =μf Pr 5 4 54 5Ω 2 1 Pr Ra4 knf =kf @X Da ρnf =ρf ρnf =ρf knf =kf 

(5.20)

  2  @Ψ @Θ @Ψ @Θ @ Θ @2 Θ 2 1 5 @Y @X @X @Y @X2 @Y2

With dimensionless parameters:   gβ f R3 qvR=kf   Ra 5 ; αf υf

Pr 5

υf ; αf

Da 5

(5.21)

K R2

(5.22)

Furthermore, boundary conditions will be: On the cold side walls On all walls

Θ 5 0:0

ψ 5 0:0

On all the insulated walls

@Θ=@n 5 0:0

@Θ=@n 5 2 1:0   knf 1 Nuloc 5 j kf Θ Ro ð 1 π Nuave 5 Nulox rdθ πRo 0

(5.23)

On the heat flux

(5.24) (5.25)

The FEM applied by FlexPDE software is the solution instrument. In this case, different nanoparticles (Fe3O4, Al2O3, TiO2, and Cu) are chosen from Chapter 2, Mathematical analysis of nanofluids, and the results of temperature contour as well as streamline are illustrated in Fig. 5.8.

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5. Nanofluid analysis in different media

FIGURE 5.8 Effect of nanoparticle type on temperature and streamline contour when Ra 5 10e4, ϕ 5 0.08, and Da 5 50 [3].

Based on this figure, Fig. 5.9 is plotted to show the local and average Nusselt number treatment. As can be seen, Cu and Al2O3 are the most suitable nanoparticles due to greater thermal conductivities from a heat transfer viewpoint. Thus Cu is selected for the next analysis to find the effect of porosity on the heat transfer of NF. Fig. 5.10 is presented for different Cunanoparticle volume fraction which confirms that the greater ϕ, moves down the streamline loop center and based on Fig. 5.11, it can be concluded that larger ϕ has greater Nusselt number and enhances the heat transfer. The effect of Darcy number on porous effect on the Nusselt numbers is investigated in Fig. 5.12. It can be concluded that increasing the Da number caused an increase in streamline

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233

5.1 Nanofluids in porous media 6

5

Average Nusselt number

Local Nusselt number

5

Cu Al2O3 Fe3O4 TiO2

5.5

4.5 4 3.5 3

4.5

4

3.5

2.5 2

0

0.2

0.4

0.6 0.8 1 1.2 Outer wall length

1.4

1.6

3

Cu

Al2O3

Fe3O4

TiO2

FIGURE 5.9 Local and average Nusselt number for different nanoparticles [3].

FIGURE 5.10 Effect of Cunanoparticle volume fraction on temperature and streamline contour when Ra 5 10e4 and Da 5 50 [3].

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234

5. Nanofluid analysis in different media

Local Nusselt number

5.5

5

ϕ=0.08 ϕ=0.06 ϕ=0.04 ϕ=0.02

Average Nusselt number

6

5 4.5 4 3.5 3

4.5

4

3.5

2.5 2 0

3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Outer wall length

FIGURE 5.11

0.02 0.04 0.06 0.08 Nanoparticle volume fraction

Local and average Nusselt number for different Cunanoparticle vol-

ume fraction [3].

FIGURE 5.12 Effect of Darcy number on the temperatures and streamline contours of Cuwater when Ra 5 10e4 and ϕ 5 0.08 [3].

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235

5.1 Nanofluids in porous media

Local Nusselt number

5.5 5

5

Da=0.005 Da=0.5 Da=50 Da=500

Average Nusselt number

6

4.5 4 3.5 3 2.5 2 0

4.5

4

3.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Outer wall length

3

0.005

0.5 50 Darcy number

500

FIGURE 5.13 Local and average Nusselt number for different Darcy numbers of Cuwater nanofluid [3].

values and a decrease in temperature value contours, thus Nusselt will be increased. Consequently increasing the Da leads to an increase in both local and average Nusselt number as reported in Fig. 5.13.

5.1.3 Case 3: Porous half-annulus enclosure filled by Cuwater nanofluid under the uniform magnetic field In this case, as presented in Fig. 5.14, a porous halfannulus cavity under a uniform magnetic field (UMF) considering heat generation is modeled. The magnetic force is applied on the geometry with angle γ and internal and external curve walls are in hot and cold temperatures, respectively. The other walls are insulated. Cuwater NF is used in the analysis and it is assumed to be Newtonian, incompressible, and laminar flow [5,6]. Based on Rashad et al. [7] study, Boussinesq approximation is used for NF density approximation and the Tiwari and Das model is applied for obtaining the temperature distribution. The properties of porous medium are supposed to be homogenous and isotropic whereas the effects of viscous dissipation, Joule heating, and radiation are ignored. With these assumptions, the governing equations are [5,7]:





r q50

(5.26)

K q 5 2 rp 1 pg 1 I 3 B μnf

(5.27)

 qv0 q r T 5 αnf r2 T 1  0 pCp nf

(5.28)





r I50

Nanofluids

(5.29)

236

5. Nanofluid analysis in different media

FIGURE 5.14

Geometry, boundary conditions, and generated mesh considered in this

case [5].

  I 5 σnf 2rΦ 1 q 3 B

(5.30)

,

Here, q 5 ðu; vÞ denotes the Darcy velocity vector; K refers to the permeability of the porous medium; g indicates the gravitational acceleration vector; μnf and ρnf are the NF dynamic viscosity and density, respectively; p is the pressure; I is the electric current; B is the external magnetic field; T0 is the temperature of the upper wall; Φ shows the electric potential; αnf , σnf and β nf are thermal diffusivity, electric conductivity,and  coefficient of thermal expansion of the NF, respectively; and also Cp nf is the heat capacitance of the NF. Eqs. (5.29) and (5.30) can be reduced to [7]: r2 Φ 5 0

Nanofluids

(5.31)

237

5.1 Nanofluids in porous media

As there is continuously an electrically insulating boundary around the cavity, Eq. (5.31) has the following single solution: rΦ 5 0

(5.32)

Removing the pressure term in Eq. (5.27) and using Eqs. (5.30) and (5.32), the governing equations can be changed to: @u @v 1 50 (5.33) @x @y   gβ nf K @T σnf KB20 @u @v @u @v @v 2 52 1 2 sin2 γ 1 2 sin γ cos γ 1 cos2 γ @y @x @y @y @x υnf @x μnf (5.34)

 2  @T @T @T @2 T qv0 1v 5 αnf u 1 1  0 2 2 @x @y @x @y ρCp nf

(5.35)

The defined boundary conditions are: u 5 0; v 5 0;

T 5 T0 at outer wall ðR in Fig: 5:14Þ @T 5 0 at flat side walls in y 5 0 @y

By defining the following dimensionless variables: X5

x ; R

Y5

y ; R

U5

R u; αf

V5

R v; αf

θ5

T 2 T0 ; ΔT

ΔT 5

(5.36)

  qw0 R2 kf (5.37)

The dimensionless stream function ψ can be introduced as: U5

@Ψ ; @Y

V52

@Ψ @X

(5.38)

when Eqs. (5.37) and (5.38) are substituted into Eqs. (5.34) and (5.35), and the pressure terms can be eliminated and the results are: 2 3    2 2 ρ β @ Ψ @ Ψ p p 5 @θ 1 A2 2 5 2 Rað12φÞ2:5 4ð1 2 φÞ 1 φ @X2 @Y @X ρf βf 2 3 3ðζ 2 1Þφ 5 2 Ha2 ð12φÞ2:5 41 1 (5.39) ðζ 1 2Þ 2 ðξ 2 1Þφ 0 1 2 2 2 @ Ψ @ Ψ @ Ψ 2 2 sin γ cos γ 1 3 @A2 2 sin γ 1 2A cos γ A @y @X@Y @X2

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238  A

5. Nanofluid analysis in different media

  2   2  knf @ θ @Ψ @θ @Ψ @θ 1 2 @ θ   1 A 1 1 2 5    kf @X2 @Y @X @X @Y @Y2 1 2 φ 1 φ ρCp p = ρCp f

(5.40) In the above equations, due to the circular shape of the nanoparticles A 5 1 and ζ 5 σp =σf : The dimensionless forms of the boundary conditions will be Ψ 5 0;

θ 5 0 at outer wall ðR in Fig: 5:14Þ

Ψ 5 0;

θ 5 1 at inner wall ðhot wallÞ (5.41) @Ψ @θ Ψ 5 0; 5 0; 5 0 at flat side walls in Y 5 0 @Y @Y  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Ra Ra 5 gKβ f ΔTR=υf αf and Ha 5 B0 σf K=μf are the Rayleigh and Hartmann numbers for the porous media, respectively. It should be pointed out that γ 5 0 and γ 5 π=2 correspond to the horizontal and vertical magnetic fields, respectively. In order to analyze the heat transfer improvement, the local Nusselt number (Nu) and the average Nusselt number (Num) were defined on the outer wall as follows [7]: knf @θ Nu 5 2 (5.42) kf @n r5R ð 1 πR Num 5 Nuðζ Þ dζ (5.43) πR 0 where ζ is the rotation angle [8]. In first step, to confirm the results of the FEM, Fig. 5.15 is depicted for the contours (temperature and streamlines) in the porous halfannulus under the magnetic field of different angles (γ 5 0, π/6, and π/3 or 0, 30, and 60 degrees) and other conditions mentioned in the figure. As demonstrated by the three contours on the left-hand side of this figure, the temperature values change a little with the variation of the magnetic field angles. From the streamlines presented on the right-hand side of Fig. 5.15, the effect of inclination angle of magnetic field can be clearly observed. When γ 5 0 degree, two symmetric counter-rotating vortexes appeared in the streamlines. As γ increases, this symmetrical shape will be disturbed and the positive vortex on the left half of the cavity will be dominant; it keeps expanding in the direction of the applied magnetic field. As an important outcome, the maximum local Nusselt number will not occur at the center of the outer wall as presented in Fig. 5.16. Actually, the maximum local heat Nusselt number

Nanofluids

5.1 Nanofluids in porous media

239

FIGURE 5.15 Effect of magnetic field angle on temperature (left) and streamlines (right) when ϕ 5 0.01, Ha 5 20, and Ra 5 103 [5].

FIGURE 5.16 Effect of magnetic field angle on the local Nusselt number of outer wall for conditions in Fig. 5.15 [5].

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240

5. Nanofluid analysis in different media

FIGURE 5.17 Effect of nanoparticles volume fraction on temperature (left) and streamlines (right), when γ 5 0, Ha 5 20, and Ra 5 103 [5].

will shift to one side of the outer wall depending on the angle of the imposed magnetic field. The effect of copper nanoparticle volume fraction on the heat transfer performance is shown in Figs. 5.17 and 5.18. It is obvious that Cu particles improve the heat transfer due to their greater thermal conductivity, so the cooling process is enhanced and temperature values decrease by increasing the Cu volume fractions. Thus higher ϕ leads to a larger Nusselt number.

5.2 Nanofluids in magnetic field (magneto hydrodynamicsferrofluid) Some of NFs such as Fe3O4 have magnetic properties and need to be treated differently. In this section the effect of magnetic field (constant and variable) on the NF treatment is investigated both experimentally and numerically by the following three cases.

5.2.1 Case 1: Variable magnetic field effect on a half-annulus cavity filled by nanofluid In this first case, as depicted in Fig. 5.19, a variable magnetic field (VMF) is affected on a half-annulus two-dimensional cavity under

Nanofluids

FIGURE 5.18 Effect of nanoparticle volume fraction on the local Nusselt number of outer wall for conditions given in Fig. 5.17 [5].

FIGURE 5.19 study [9].

Geometry, boundary conditions, and generated mesh considered in this

242

5. Nanofluid analysis in different media

constant heat flux. The magnetic source is located by inserting a magnetic wire on to the xy-plane at the point ða; bÞ, vertically. The components of the magnetic field intensity ðH x ; H y Þ and the magnetic field strength H are calculated from the following equations [9]: Hx 5

  γ 1 y2b   2 2π ðx2a Þ2 1 y2b

Hy 5 2

H5

γ 1 ðx 2 a Þ 2π ðx2a Þ2 1 y2b 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ 1 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hx 1 Hy 5  2ffi 2π ðx2a Þ2 1 y2b

(5.44)

(5.45)

(5.46)

where γ is the magnetic field strength at the source (the wire). The boundary conditions are the outer wall: insulation; inner wall: constant heat flux; and two remaining walls are at constant temperature (Tc). Fe3O4water NF due to its magnetic properties is used here and assumed to be Newtonian, incompressible, and laminar flow. The governing equations for the steady, two-dimensional, and laminar NF flow are [10]: @v @u 1 50 @y @x

(5.47)

   2  @u @u @P @ u @2 u ρnf v 1u 1 μnf 1 52 2 σnf B2y u 1 σnf Bx By v (5.48) @y @x @x @x2 @y2    2  @v @v @P @ v @2 v 52 1 ρnf β nf γ ðT 2 Tc Þ 1 μnf ρnf v 1 u 1 1 σnf Bx By u 2 σnf B2x v @y @x @y @x2 @y2

(5.49)    2     2 @T @T @ T @2 T 1u ρCp nf v 1 2 1 σnf uBy 2vBx 5 knf @y @x @x2 @y 8 9  2  2 = < @u2 @v @v @u 1 1 μnf 2 12 1 : @x @x @y @x ;

(5.50)

B y 5 μ0 Hy where ρnf, (ρCp)nf αnf, β nf, μnf, knf, and σnf are as defined in Chapter 2, Mathematical analysis of nanofluids.

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243

5.2 Nanofluids in magnetic field (magneto hydrodynamicsferrofluid)

T 2 TC ; Θ5  qvR=kf pR2 P 5  2 ; ρ f αf

U5 

uR ; αf

V5 

H; Hx ; Hy 5



vR ; αf

a ; R

a5

H; H x ; H y



H0

;

β5

b ; R

Y5

H 0 5 H ða; 0Þ 5

y ; R

Χ5

x ; R

γ 2πjbj (5.51)

Using Eq. (5.51), Eqs, (5.47)(5.50) can be written in nondimensional form as @V @U 1 50 @Y @X 2 3   2 μnf =μf @U @U @ U @2 U 4 5 1U 51 Pr V 1 @Y @X @X2 @Y2 ρnf =ρf 2 3 σnf =σf  2 5 H U 2 Hx Hy V 2 @P 2 Ha2 Pr4 y @X ρnf =ρf   2  μnf =μf @V @V @V @2 V 1U 5 Pr V 1 @Y @X @X2 @Y2 ρnf =ρf     β nf σnf =σf @P 2 Ha2 Pr 2 Hy2 V 2 Hx Hy U 1 RaPr Θ @Y ρnf =ρf βf 2 3   2 knf =kf @Θ @Θ 4 @ Θ @2 Θ 5     1V 5 U 1 @X @Y @X2 @Y2 ρCp nf = ρCp f 2 3

σnf =σf    5 UHy 1VHx 2 1 Ha2 Ec4 ρCp nf = ρCp f 9 2 38  2  2 = < @U 2 μnf =μf @V @U @V   5 2 1 1 Ec4 12 1 @Y @X @Y ; ρCp nf = ρCp f : @X With dimensionless parameters:   gβ f R3 qvR=kf υf   Raf 5 ; Pr 5 ; α αf υf f  αf μ f Ec 5 h   i ρCp f qvR=kf R2

Nanofluids

(5.52)

(5.53)

(5.54)

(5.55)

sffiffiffiffiffi σf Ha 5 Rμ0 H0 μf (5.56)

244

5. Nanofluid analysis in different media

As mentioned above, the thermophysical properties of water Fe3O4 are presented in Chapter 1, Introduction to nanofluids. The stream function, vorticity, local, and average Nusselt numbers are considered as: Ω5

ωR2 ; αf

ψ5

ψ ; αf

ν52

@ψ ; @x

u5

On the cold walls On all walls

@ψ ; @y

ω52

(5.57)

Θ 5 0:0

ψ 5 0:0

On insulated walls

@u @v 1 @y @x

@Θ=@n 5 0:0

On the heat flux @Θ=@n 5 2 1:0   knf 1 Nuloc 5 j k f Θ Ri ð 1 π Nuave 5 Nulox rdθ πRi 0

(5.58)

(5.59) (5.60)

The half-annulus generated mesh using FEM-based FlexPDE software is presented in Fig. 5.19B, and the solutions achieved show the effect of different parameters on the heat transfer treatment. Fig. 5.20 displays the variable magnetic strength in different directions. Actually, the intensity of the magnetic field is different radially, and near the source has greater force while the outer walls are under smaller magnetic effect. The effect of the Hartmann number on the results is presented in Fig. 5.21, which reveals that by increasing the Ha, the

FIGURE 5.20 Variable magnetic field applied on the problem. (A) Hx, (B) Hy, and (C) H [9].

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5.2 Nanofluids in magnetic field (magneto hydrodynamicsferrofluid)

245

FIGURE 5.21 Effect of Hartmann on temperature (left) and streamlines (right) when ϕ 5 0.02, Ec 5 0.00001, and Ra 5 103 [9].

streamlines values reduced significantly and the center of the vortexes relocated upward due to lower magnetic effect near the above situations. This reduction may be due to Lorentz force resulting from the presence of the magnetic field that slows down the fluid motion and thus reduces the heat transfer too as Fig. 5.22 confirms [9]. In the highlighted area in Fig. 5.22, it can be observed that the maximum local Nusselt number occurs when Ha 5 0 and by increasing the Ha, Nu decreased as presented in Fig. 5.22. Figs. 5.23 and 5.24 show that increasing the Fe3O4 nanoparticle volume fraction improves the heat transfer due to their higher thermal conductivity, improving the cooling process and decreasing temperature contours. In the zoomed area of Fig. 5.24, it can be seen that for greater ϕ, there is larger Nusselt number.

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246

5. Nanofluid analysis in different media

FIGURE 5.22 Effect of Hartmann number on the local Nusselt number of outer wall [9].

5.2.2 Case 2: Nanofluid flow over a porous plate under the variable magnetic field effect As seen in Fig. 5.25, consider variable magneto hydrodynamics (MHD) affected on the two dimensional-forced convection boundary layer of an incompressible NF flow over a horizontal surface. By considering boundary layer approximation as well as the viscous dissipation term, the simplified two-dimensional governing equations of the flow in the boundary layer of a steady, laminar, and incompressible flow are [11,12]: @u @v 1 50 @x @y   @u @u 1 @2 u 1v 5 u μnf 2 2 σBðxÞ2 u @x @y ρnf @y  2 μnf @T @T @2 T @u  1v 5 αnf 2 1  u @x @y @y ρCp nf @y

Nanofluids

(5.61) (5.62)

(5.63)

5.2 Nanofluids in magnetic field (magneto hydrodynamicsferrofluid)

247

FIGURE 5.23 Effect of nanoparticle volume fraction on temperature (left) and streamlines (right) when Ha 5 20, Ec 5 0.00001, and Ra 5 103 [9].

where u and v are the xand ycomponents of velocity, respectively; σ indicates the electrical conductivity; BðxÞis the VMF acting in the perpendicular direction to the horizontal flat plate; μnf and ρnf are  the  viscosity and the density of the NF, respectively; and αnf and ρCp nf are the thermal diffusivity and the heat capacitance of the NF, respectively. The physical boundary conditions are defined as: u 5 uw ;

v 5 vw ;

u-0;

T 5 Tw

T-TN

at y 5 0

as y-N

(5.64) (5.65)

It is assumed that the surface is stretched in its plane with velocity uw ðxÞ 5 bxm , where b is constant and m is the nonlinear stretching parameter. Moreover, vw shows the velocity of the mass transfer, and T, Tw , and TN are the temperatures across the thermal boundary layer, constant temperatures of the wall, and ambient fluid, respectively.

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248

5. Nanofluid analysis in different media

FIGURE 5.24 Effect of nanoparticle volume fraction on the local Nusselt number of outer wall [9].

FIGURE 5.25 Schematic of the nanofluid boundary layer on a horizontal plate in the presence of variable magnetic field [11].

Actually, when y-N a sufficiently large value of y has been reached so that u and T reach free stream velocity and temperature, respectively. The VMF is defined as [13,14]: BðxÞ 5 B0 ðxðm21Þ=2 Þ

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(5.66)

5.2 Nanofluids in magnetic field (magneto hydrodynamicsferrofluid)

249

where B0 and m are constant. The following nondimensional similarity variable is applied to transform the partial governing equations into ordinary differential equations (ODEs): y η 5 Rex 1=2 (5.67) x ρf uw ðxÞ Rex 5 x (5.68) μf The dimensionless stream function and dimensionless temperature are: ψðx; yÞðRex Þ1=2 uw ðxÞ T 2 TN θðηÞ 5 Tw 2 TN

fðηÞ 5

(5.69) (5.70)

where the stream function ψðx; yÞ is defined as: u5

@ψ ; @y

v52

@ψ @x

(5.71)

By adding the similarity transformation parameters, the momentum equation (Eq. 5.62) and the energy equation (Eq. 5.63) can be changed to new shapes:     2 ρs 2:5 m11 fw 1 ð1 2 ϕÞ 1 ϕ ffv ð12ϕÞ 2 ρf     (5.72) ρ 2 12ϕ 1ϕ s ð12ϕÞ2:5 ðmÞf 02 2 ðð12ϕÞ2:5 MnÞf 0 5 0 ρf  ! ρCp s Ec Pr  Pr fθ0 1 θv 1 ð1 2 ϕÞ 1 ϕ  fv2 5 0 (5.73) ρCp f ð12ϕÞ2:5 And also, the transformed boundary conditions will be: fð0Þ 5 S;

f 0 ð0Þ 5 1;

θð0Þ 5 1;

θðNÞ 5 0

f 0 ðNÞ 5 0

(5.74)

The dimensionless parameters of S,Mn,Pr,Ec, and Rex are the suction parameter, magnetic parameter, Prandtl, Eckert, and Reynolds numbers, respectively, defined as:   ρCp f ρf uw ðxÞ σB20 uw ðxÞ2 ; Pr 5 ; Rex 5 ν f ; Ec 5 x Mn 5 ρf b kef Cp ΔT μf (5.75) vw ; S 5 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bνðm 1 1Þ=2 xðm21Þ=2

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5. Nanofluid analysis in different media

Eqs. (5.72) and (5.73) are rewritten as: fw 1 Affv 2 Bf 02 2 Cf 0 5 0

(5.76)

θv 1 Dfθ0 1 Efv2 5 0

(5.77)

where coefficients A, B, C, D, and E are written as:     2 ρs 2:5 m11 A 5 ð1 2 ϕÞ 1 ϕ ð12ϕÞ 2 ρf    ρ B 5 ð1 2 ϕÞ 1 ϕ s ð12ϕÞ2:5 ðmÞ ρf C 5 ðð12ϕÞ2:5 MnÞ  ! ρCp s  Pr D 5 ð1 2 ϕÞ 1 ϕ  ρCp f

(5.78) (5.79) (5.80) (5.81)

To solve this method by optimal collocation method (OCM), the physical region η 5 ½0; NÞ is transformed into the new region of η1 5 ½0; ηN 1  and η2 5 ½0; ηN 2  for the hydrodynamic and thermal boundary layer, respectively. Note that ηN is a function of ϕ, S, ρf , ρs , Mn, m, Pr, and Ec, which should be determined as a part of the solution. Thus by considering the change of variable asz1 5 η1 =ηN1 and z2 5 η2 =ηN2 , the problem will transform in the interval ½0; 1 instead of ½0; ηN . Finally, the governing equations (Eqs. 5.72 and 5.73) can be transformed into the following forms: 0 12    2    ρs ð m11 Þ d 2:5 @ A ð12ϕÞ 12ϕ 1ϕ gðz 1 Þ gðz 1 Þ ρf 2 dz21 0 12           ρs d d 2:5 ð12ϕÞ 50 g z1 m @ gðz1 ÞA 2 ð12ϕÞ2:5 Mn 2 ð12ϕÞ1ϕ ρf dz1 dz1

ðd3 =dz1 3 Þgðz1 Þ 1 ηN1 2





ðρCp Þs ðd2 =dz22 Þhðz2 Þ 1 ð12ϕÞ1ϕ ðρCp Þf ηN2

!

(5.82)  2 2  ðd =dz21 Þgðz1 Þ d Ec Pr Pr ηN1 gðz1 Þ hðz2 Þ 2  50 ηN1 2 dz2 ð12ϕÞ2:5 

(5.83) where gðz1 Þ 5 fðη1 Þ=ηN1 , hðz2 Þ 5 θðη2 Þ=ηN 2 , and the “prime” indicates the derivatives with respect to zA ½0; 1. Also, the boundary conditions can be changed to: z1 5 z2 5 0.g 5

S ; ηN1

g0 5 1;

Nanofluids

h5

1 ηN2

(5.84)

251

5.2 Nanofluids in magnetic field (magneto hydrodynamicsferrofluid)

z1 5 z2 5 1.g0 5 0;

h50

(5.85)

In OCM, as introduced in Chapter 2, Mathematical analysis of nanofluids, the last boundary condition will be obtained by considering the asymptotic condition: η-N.fv 5 0;

θ0 5 0

(5.86)

The extra boundary conditions (Eq. 5.86) can be replaced by: z1 5 z2 5 1.gv 5 0;

h0 5 0

(5.87)

To construct a trial solution in the interval 0 , z , 1, the following basic functions are chosen with undetermined or unknown coefficients “c”: gðz1 Þ: 5

S ηN1

        1 1 1 1 1 c1 z1 2 z21 1 c2 z1 2 z31 1 c3 z1 2 z41 1 ? 1 c8 z1 2 z91 2 3 4 9

(5.88) hð z 2 Þ 5

1 1 c9 z2 1 c10 z2 2 1 c11 z2 3 1 ? 1 c17 z2 9 ηN2

(5.89)

The accuracy of the solution can be enhanced by increasing the number of statements. Because the trial solution must satisfy the boundary conditions of Eqs. (5.84) and (5.85) for all values of “c,” thus:

1 ηN2

c1 1 c2 1 c3 1 ? 1 c8 5 1

(5.90)

1 c9 1 c10 1 c11 1 ? 1 c17 5 1

(5.91)

ηN 1 and ηN 2 in Eqs. (5.82) and (5.83) can be obtained by using the extra boundary conditions introduced in Eq. (5.87). This yields: 2c1 2 2c2 2 3c3 2 ? 2 8c8 5 0

(5.92)

c9 1 2c10 1 3c11 1 ? 1 9c17 5 0

(5.93)

By introducing gðz1 Þ and hðz2 Þ to differential Eqs. (5.82) and (5.83), residual functions will be found:   1 1 3 R1 c1 ; c2 ; c3 ; . . .; c8 ; ηN1 ; z1 5 2 ð12ϕÞ2:5 c21 z1 2 ð12ϕÞ2:5 c22 z21 2 ð12ϕÞ2:5 c23 z31 4 2 4 2

ð12ϕÞ2:5 ϕ ρs m2 c8 z41 c4 2 ð12ϕÞ2:5 ϕ ρs S c8 z1 7 1 ð12ϕÞ2:5 Mn c8 z81 1 ? 2 ρf ρf ηN1

(5.94)

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5. Nanofluid analysis in different media

  6c11 z2 12c12 z22 20c13 z32 4Ec Pr c22 z21 R2 c9 ; c10 ; . . .; c17 ; ηN1 ; ηN2 ;z1 ;z2 5 1 1 1?1 ηN2 ηN2 ηN2 ð12ϕÞ2:5 η2N1 1

8Pr ϕ ðρ Cp Þs S c16 z72 9Pr ϕðρ Cp Þs S c17 z82 9Ec Pr c23 z41 1?1 1 2:5 2 ðρ Cp Þf ðρ Cp Þf ð12ϕÞ ηN1

(5.95) In the OCM, the numbers of weight functions Wi are: nwi 5 nci 2 nb

(5.96)

where nci denote the numbers of unknown constants and ci and nb indicate the number of equations that satisfy the boundary conditions. To reach a zero residual function, specific points in the domain zA ½0; 1 must be chosen as:   1 5 0; 8   1 R2 5 0; 9 R1

    2 7 5 0; . . .; R1 50 8 8     2 8 R2 5 0; . . .; R2 50 9 9 R1

(5.97)

(5.98)

Finally, a set of nine algebraic equations appear: 2 equations (Eqs. 5.90 and 5.91) satisfy the boundary conditions; 2 equations (Eqs. 5.92 and 5.93) satisfy the extra boundary conditions; and 15 equations (Eqs. (5.97) and (5.98)) force the residual function to zero. By solving this system of equations, unknown coefficients ci , ηN1 , and ηN2 can be determined. After specifying these unknown parameters, the velocity and temperature profile equation will be known. For example, using OCM for a AL2 O3 2 H2 O NF with ϕ 5 0:05,S 5 1,Mn 5 2,m 5 1,Ec 5 0:1,Pr 5 6:2,ρs 5 3890, and ρf 5 997:1,fðηÞ, and θðηÞ are as follows: f ðηÞ 5 0:9999999999 1 0:9999999900 η 2 1:1177446490 η2 1 ? 1 0:0000822028 η9

(5.99) θðηÞ 5 0:9999999999 2 6:8602041110 η 1 21:6807722800 η2 1 ? 2 0:1680955649 η9

(5.100) Figs. 5.26 and 5.27 demonstrate the accuracy of this method for different values of constant numbers, S, Mn, ϕ, and m parameters. For all variables the method gives a high accuracy solution for fðηÞ (stream function) compared to the numerical solution. Also, the effect of magnetic number and nanoparticle volume fraction is observed in these figures.

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253

FIGURE 5.26 Comparison of numerical and optimal collocation method (OCM) solutions for different S and Mn numbers for stream function [11]. (A) Different S parameters; (B) Different Mn parameters.

FIGURE 5.27 Comparison of numerical and optimal collocation method (OCM) solutions for different ϕ and m numbers for stream function [11]. (A) Different ϕ numbers; (B) Different m numbers.

5.2.3 Case 3: Ferrofluids under external magnetic field Ferrofluids, also called smart fluids, are colloidal suspensions of magnetic nanoparticles (Fe3O4) dispersed in a suitable conveyer liquid. Upon applying suitable magnetic fields, the anisotropic particle motion, optical properties, heat transfer, etc., from ferrofluids make them perfect model systems from both a fundamental and application viewpoint Nanofluids

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5. Nanofluid analysis in different media

FIGURE 5.28 Schematic of the experimental setup to measure the time evolution of transmitted light intensity in a ferrofluid sample under an external magnetic field [15].

[15]. Here, both the time- and wavelength-dependent light scattering in water-based ferrofluids containing Fe3O4 nanoparticles under an external magnetic field are investigated [15]. In the experimental setup, as seen in Fig. 5.28, a stable water-based colloidal suspension of Fe3O4 nanoparticles with a diameter of around 15 nm is considered, and coated with sodium oleate (SO) as the primary layer and polyethylene glycol 4000 (PEG-4000) as the secondary layer. Fe3O4 powders with desired amount were dispersed in deionized water, and the mixture was kept under ultrasonic vibration at a frequency and amplitude of 25 kHz and 60% for 1 h. More details on the NF preparation are presented in Ref. [15]. 5.2.3.1 Measurement setup A schematic of the experimental setup used for the light scattering measurement is presented in Fig. 5.28. The ferrofluid sample in a standard quartz cuvette (10 mm) is located at the center of a Helmholtz coil, where the magnetic field intensity can be variable by changing its current using a DC power supply. As seen, the direction of incident light is parallel to the applied magnetic field. The double-beam UVvis spectrophotometer (Hitachi U-4100), equipped with a 50 W WI lamp and an integrating sphere detector, was applied to measure the time-dependent intensities of the light scattered in the forward direction through the sample [15]. In the present case, to explore the field-induced response behavior of the nanoparticles inside the suspension, the thermal conductivity of the prepared samples parallel (k||) and perpendicular (k\) to the field direction was measured and compared with our numerical outcomes [16]. Table 5.2 summarizes all the parameters used in this model. The experimental and numerical results of relative thermal conductivity k/kf

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5.2 Nanofluids in magnetic field (magneto hydrodynamicsferrofluid)

TABLE 5.2

Values of parameters employed in our numerical model [16].

Parameters

Values

Boltzmann constant, kB

1.381 3 10223 J

The saturation magnetization of the particles, MS

39 Am2 kg21

The compression parameter, κ

10

The maximum particle volume fraction, ϕm

6.38

Temperature, T

300K

The vacuum magnetic permeability, μ0

4π 3 1027 Tm A21

(B)

1.09 1.08

Experimental result k ⎢⎢

1.07

Experimental result k ⊥

1.06

Numerical result k ⎢⎢

Relative thermal conductivity (k/kf )

Relative thermal conductivity (k/kf )

(A)

Numerical result k ⊥

1.05 1.04 1.03 1.02 1.01

1.12

0.4%

0.1%

1.10

0.5%

1.08 1.06 1.04 1.02 1.00

1.00

0.98

0.99 0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0

300

600

900

1200

1500

1800

Time (s)

Volume fraction (%)

FIGURE 5.29 (A) Comparison of experimental and numerical results of relative thermal conductivity k/kf parallel and perpendicular to the field direction and (B) variation of relative thermal conductivity k/kf parallel to the field direction with time for Fe3O4water ferrofluids of various volume fractions under constant 100 Gs magnetic field [15].

are depicted in Fig. 5.29A. As seen in this figure, the experimental values of thermal conductivity are in excellent agreement with the numerical results, indicating the reasonable response behavior of the particles under magnetic field. The variation of k|| versus time at different volume fractions under constant 100 Gs magnetic field is investigated in Fig. 5.29B. It can be observed that k|| values stay approximately constant. Similar outcomes have also been reported by Vinod and Philip [17] under a weak magnetic field condition (B50 Gs). However, they reported that the thermal conductivity can be dependent on time as the field intensity increases due to the growth of particle chains both in length and width. The measured transmitted intensity I under the applied magnetic field is normalized with respect to the absolute value of intensity I0 in the absence of any magnetic field. Fig. 5.30A shows the time progress of

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256

5. Nanofluid analysis in different media

(A)

(B)

2.2 2.2

100

1.6 60

1.05 1.00

1.4

0.95 0.90

1.2

I

II

III

40

0.85

τ0

0.80

1.0

0.75

20

0.70

0.8

0.65 –50

0

50

100

150

200

250

0

300

Normalized transmitted intensity (I/I0 )

80

0.6

Cycle 3

Cycle 2

Cycle 1

1.8

Magnetic field intensity (Gs)

Normalized transmitted intensity (I/I0 )

2.0 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

0

600

1200

1800

2400

3000

3600

4200

0

Time (s)

2000

4000

6000

8000

10,000

12,000

Time (s)

FIGURE 5.30 (A) Normalized transmitted intensity (left) and magnetic field intensity (right) as a function of time for ferrofluid sample (volume fraction ϕ 5 0.005%) under incident light with wavelength of 600 nm [15]. (B) Normalized transmitted intensity in three time cycles.

normalized transmitted intensity and its dependence on the external magnetic field for the described conditions. From Fig. 5.30A, the curve of transmitted intensity can be divided into three characteristic regions. The first area (zone I) relates to a sharp decrease in the light intensity directly after the field is turned on. When a uniform magnetic field H is applied to the suspension, the field makes a magnetic dipole moment in the particles m. The ratio between the maximum magnitude interaction energy and the thermal energy as a significant dimensionless parameter is [15]: λ5

μ0 m2 2πd3 kB T

(5.101)

where μ0 indicates the vacuum magnetic permeability, d represents the diameter of the nanoparticles, and kB is the Boltzmann constant. For the 100 Gs field intensity in this case, the value of λ is calculated to be 0.17. Thus the magnetic energy of a single nanoparticle is weaker than thermal energy, and flexible chains will form affected by thermal fluctuations [18]. The second region (zone II) demonstrates the time interval between the initial state and the time τ 0 when the minimum intensity occurs. Under the effect of the external magnetic field, the sizes of the scatterers will grow with time through the structural evolution of the chains (i.e., linear aggregation and lateral coarsening) [15]. At time τ 0, the number of scatterers that satisfy the resonance is considered to reach the maximum leading to the minimum light intensity. On the other hand, the aggregation process also makes a smaller cross section for the scattering of incident light, and thus more light will be transmitted through the sample [15]. This should be the reason for the inversed increase after the minimum. In the third region (zone III), the intensity grows gradually until it gets saturated at a particular time.

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257

Interestingly, the transmitted intensity returns fully to the initial value right after the magnetic field is switched off. Fig. 5.30B explains the effect of frequent magnetic field cycle on the time-dependent light scattering. Once the external magnetic field is turned off, the magnetization of the particles can decrease by two mechanisms. Firstly, the relaxation happens by particle rotation in the liquid known as the Brownian relaxation with a rotational diffusion time is obtained by [19]: τB 5

3ηVh kB T

(5.102)

where η indicates the viscosity of the base fluid, and Vh shows the hydrodynamic volume of the particle. Secondly, the relaxation is due to rotation of magnetic vector within the particle, recognized as the Ne´el relaxation mechanism with a characteristic time defined by [20]:   KVm τ N 5 τ 0 exp (5.103) kB T where τ 0B1029 s, K 5 1.5 3 104 J m23 refers to the anisotropy constant here and Vm is the magnetic volume of the particle. T 5 300 K and η 5 1.01 3 1023 Pa s for water. The Brownian and Ne´el relaxation time are obtained from the equations as 3.49 3 1025 s and 6.03 3 1027 s, respectively. Also, the values of τ B and τ N will be 1.03 3 1025 s and 6.7 3 1027 s for a particle of 10 nm size, respectively. Fig. 5.31 shows the TEM images of the field-induced columnar structures in the ferrofluid sample. The interaction between the adjacent particles is considered by

FIGURE 5.31 TEM images of the aggregated structures of nanoparticles in ferrofluid sample under the magnetic field. The inset shows a zoomed image of the field-induced columnar structure along the field direction [15].

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5. Nanofluid analysis in different media

TABLE 5.3 Characteristic times for relaxation processes and experimental measurements [15]. Characteristic times

Values (s)

Brownian relaxation, τ B

3.49 3 1025

Ne´el relaxation, τ N

6.03 3 1027

Brownian motion, τ b

1.29 3 1026

Experimental time scale

0.1

the Van der Waals’ attractive force. The VdW interaction free energy, Evdw, between two spheres of radius r at surface separation s, can be derived by Eq. (5.104) [21]:   A r Evdw 5 2 (5.104) 12 s where A shows the Hamaker constant. The characteristic time scale to cover a distance equal to the diameter of the particle due to the Brownian motion is given by: τb 5

πηd3 2kB T

(5.105)

τ b is calculated to be 1.29 3 1026 s for the ferrofluid sample at room temperature. Because these time scales (τ B, τ N, and τ b) are very small compared to those of the experimental measurements as presented in Table 5.3, the aggregated nanoparticles can disperse accidentally again right after the field is turned off as explained by the reversible light transmission presented in Fig. 5.30.

5.3 Nanofluids under thermal radiation In this section, three different examples of NFs including both experimental and numerical studies are presented for thermal radiation. Since most NF applications include the presence of thermal radiation such as a solar collector’s application, this section plays an important role in NF studies.

5.3.1 Case 1: Polydisperse colloidal particles in the presence of thermal gradient From an experimental viewpoint, applying a thermal gradient to colloids causes the migration of charges and particles. The particle current in a dilute colloidal suspension can be obtained by [22]: J 5 2 Drn 2 nDT rT

Nanofluids

(5.106)

5.3 Nanofluids under thermal radiation

θ

n

t

Δpe

259

GT Op Oe

FIGURE 5.32 Misalignment Δpe of the centers and of particle Op and diffusion layer ions Oe under the thermal gradient GT (the center of the Stern layer ions overlaps with Op). The n and t are the normal and tangent vectors of the particle, and θ is the included angle of n and GT [22].

where the 2Drn refers to Fick’s law with the Einstein diffusion coefficient D, and the 2nDT rT shows the thermally induced particle flow with thermophoretic coefficient DT . This equation proposes that the thermal gradient can carry out an inhomogeneous distribution of particles. The charged particles also sense the electrostatic force from the ions around them under the thermal gradient. As depicted in Fig. 5.32, the thickness of electrical double layers is uneven pffiffiffiffi due to the nonuniform temperature based on the relation 1=κ ~ T, where 1=κ denotes the thickness of the electrical double layers [23]. The ions in the Stern layer and in the diffusion layer have the reverse sign, and an additional force on the particle will occur if their centers do not overlap. This force can be calculated from the Maxwell tensor:   3 X 1 Fi 5 t Γij dS; Γ ij 5 ε Ei Ej 2 E2 δij (5.107) 2 j51 where Fi is the component force on the charges in surface S and E 5 2 rψ is the electrical field. The electrical potential ψ of a flat surface for the case of a thin diffusion layer can be calculated from DebyeHuckel’s equation r2 ψ 5 κ2 ψ as: ψ 5 ψ0 e2κx

(5.108)

where x is the distance to the surface. Really, the derivation of Eq. (5.108) needs a constant temperature field or assuming the change of temperature in a region of colloidal particle size is very small. In this case, the small difference between the thermal gradient in a colloidal particle and the externally applied thermal gradient is ignored, so the temperature field near a particle will not be affected by the particle. The influence of electrostatic forces from the diffusion layer on particle motion ΔrEi can be obtained by [22]: ΔrEi 5

1 kT

 DTii  FΔtB

Nanofluids

(5.109)

260

5. Nanofluid analysis in different media

TABLE 5.4 The set values of parameters in the Brownian dynamic simulation [22]. Parameter

Value

EA

0.52

ER Lκ

520 pffiffiffiffiffiffiffiffiffiffiffi 10 T0 =T

η0

1025

Δt0

1025

Φ

3.35%13.7%

ai

a0 3a0

l0

0.25   0:5 1 z=50a0 kT0

kTðzÞ

Some parameters from the governing force can be collected to form dimensionless parameters that are also carefully related to the particle properties and solvent. By considering a0 5 minfa1 ; a2 ; . . . ; aN g, m0 5 minfm1 ; m2 ; . . . ; mN g, kT0 as the unit length, mass, and energy, respectively, the nondimensional parameters will be:

 εa0 ψ20i 1 ψ20j A η Δt EA 5 ; ER 5 Lκ 5 κa0 ; η0 5 ; Δt0 5  1=2 21=2 22 kT0 kT0 ðm0 kT0 Þ a0 a0 m0 =kT0

(5.110) where m1 ; m2 ; . . . ; mN represent the mass of particles; EA and ER are the dimensionless potential energy of interaction between the particles and their double layers, respectively; and Lκ , η0 , and Δt0 indicate the DebyeHuckel length, viscosity, and time step, respectively. Actually, EA and ER are the ratio of electrostatic force from other particles to the random Brownian force. Table 5.4 shows these parameter values in the Brownian dynamic simulation. A cubic enclosure with size 50a0 3 50a0 3 50a0 3D is used for the Brownian dynamic model. The enclosure is located in the first quadrant and one of its vertexes is on the origin of coordinates O as shown in Fig. 5.33. The temperatures of the top and bottom faces are constant as TH and TL where a relation TH . TL is considered to generate a thermal gradient. Assuming only the heat conductive process and constant thermal conductivity, the thermal gradient @T=@z is constant in the whole computational region. Particles added in the cube could be any size as long as the phase separation does not occur while the particles are

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5.3 Nanofluids under thermal radiation

261

TH G

V(r)

0.372 0.23

50

0.084

z

O O

1 1.5 2 2.5 3

/

y

x TL

FIGURE 5.33

Geometry of cubic simulation region of 50a0 3 50a0 3 50a0 [22].

polydisperse with a Gaussian distribution. As presented in Fig. 5.33, the average radius of the polydisperse particles is 2a0 , and five particle radii a0 , 1:5a0 , 2a0 , 2:5a0 , and 3a0 ; with volume fractions 0.084, 0.23, 0.372, 0.23, and 0.084, respectively, are chosen to meet the Gaussian distribution. More details of this case can be found in Ref. [22]. The main aim of this study is divided into two steps: (1) calculating the forces and (2) updating the particle positions under the thermal gradient. For the boundary conditions, the top and bottom sides are considered as the reflecting wall and the remaining four sides are the periodic conditions. Thus the particle positions see the following improvement: ( xi ; yi 5 xi ; yi 6 50a0 xi ; yi , 0orxi ; yi . 50a0 ri 5 (5.111) zi 5 0or100a0 2 zi zi , 0orzi . 50a0 where xi , yi, and zi indicate the position components of particle i with ri ðt 1 ΔtÞ 5 xi ; yi ; zi . To avoid particle overlap, it is assumed that collisions between particles are entirely inelastic where particle positions are tangent when they overlap. The relation can be presented as: 8 aj > > > ri 5 ri 2 r rij < ij (5.112) a i > > rj 5 ri 1 rij > : rij By the above adjustment, the impractical positions in the simulation can be avoided. To quantitatively define the aggregate structure, aggregation degree, the position, and migration of aggregates and particles, five statistical parameters, namely aggregation ratio (RA ), porosity and fractal dimension of largest aggregate (PBA and Df , respectively),

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5. Nanofluid analysis in different media

aggregate and concentration distributions in space domain (DISA and DISC , respectively), are introduced here. The definitions of all these parameters are summarized from Ref. [22]. RA is given by: RA 5

BA BM

(5.113)

where BA indicates the number of bonds between particles, and BM shows the maximal number of bonds that might be attained in the system. For monodisperse particles with radius 2a0 , the number of bonds is 12Na0 =2. For the polydisperse particles, of bonds of particle  the number 2 i is related to its surface area (i.e.; 12 ai =2a0 ). Thus the BM of a given particle component is: BM 5

X 12n2 Nna n

2

0

(5.114)

The porosity PBA indicates the structure of the aggregate and can be presented as: P VBA 2 Vi BV PBA 5 (5.115) VBA where Vi BV is the volume of particle i in the largest aggregate and VBA indicates the volume of the aggregate. The VBA can be written as: VBA 5

4π 3 a 3 G

(5.116)

Here, aG shows the gyration radius of the aggregate, which stays equal to the square root of the ratio between rotational inertia and mass. The porosity from 0 to 1 describes the density of aggregates from great to little. The fractal dimension Df , as another parameter to describe the structure of aggregates, can be defined as:   ln NE =kf ; Df 5  (5.117) ln aG =a0 where NE shows the number of particles in the aggregate 3 P  equivalent obtained by NE 5 ai =a0 , and kf is a fractal prefactor equal to 1.117 in this case [24]. Assuming the entirely symmetrical conditions in the x and y axes directions, only the distributions of aggregate and concentration in the z-axis direction are discussed. Thus the DISC can be written as: ZZ DISC ðzÞ 5 ΦðrÞdxdy (5.118)

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263

5.3 Nanofluids under thermal radiation

where ΦðrÞ is the concentration (i.e., solid volume fraction) of particles. The calculation of DISA is based on the consideration that the larger number of bonds in a region, the larger and more aggregates there will be. Thus the DISA is: ZZ CPEAL ðrÞdxdy DISA ðzÞ 5 (5.119) CPEA where ÐÐÐCPEAL ðrÞ refers to the local number of bonds with CPEA 5 CPEAL ðrÞdxdydz. Be aware that microscopically ΦðrÞ and CPEAL ðrÞ are nondifferentiable functions, so integration for this case is a summation where dx, dy are changed with Δx, Δy, respectively. Figs. 5.34 and 5.35 demonstrate the concentration distribution DISC and aggregation distribution DISA beside the z-axis at t 5 tB for the monodisperse and polydisperse particle systems, respectively. As shown, the DISC and DISA of the monodisperse particle systems are approximately symmetrical with line z 5 25 for all Φ and EA 1 ER . Also, the initial distributions are constant for the whole region. Moreover, the average z-coordinate values of various concentrations are 25.4, 25.6, and 25.7 for the DISC , and 25.4, 25.9, and 26.1 for the DISA with EA 1 ER equal to 5.5, 11, and 22, respectively, presenting a minor migration along the thermal gradient GT [22]. Fig. 5.35 approximates the DISC and DISA for the polydisperse particle systems where the particle migration against thermophoresis effect can also be considered. As seen, the average z-coordinate values are 25.5, 26.9, and 25.8 for DISC and 26.5, 29.3, and 27.5 for DISA . The z-coordinate values larger than the values in Fig. 5.34 show that the effect of unsynchronized aggregation is stronger for polydisperse

(A)

0.05

DISC

0.00 0.10

(B)

(D)

25.4

0.05

EA+ER=11

0.00 0.10

25.6

0.05 0.00 0.10

0.10 EA+ER=5.5

DISA

0.10

(C)

Φ=3.35% Φ=5.79% Φ=9.20% Φ=13.7%

EA+ER=22

25.7

0.05 0.00 0

10

20

30

40

(E)

EA+ER=11

25.9

0.05 0.00 0.10

EA+ER=5.5

25.4

(F)

Φ=3.35% Φ=5.79% Φ=9.20% Φ=13.7%

EA+ER=22

26.1

0.05 0.00

50

z (a0)

0

10

20

30

40

50

z (a0)

FIGURE 5.34 (AC) The concentration distribution DISC and (DF) aggregation distribution DISA along the z-axis at t 5 tB with various concentration Φ and potential energy EA 1 ER for monodisperse particle systems, where the average z-coordinate value of each figure is marked [22].

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5. Nanofluid analysis in different media

0.10

(A)

DISC

(B)

Φ=5.79%

Φ=3.35% Φ=9.20%

25.5

0.05 0.00 0.10

EA+ER=5.5

Φ=13.7%

EA+ER=11

DISA

0.15

26.9

0.05 0.00 0.10

0.20 0.15 0.10 0.05 0.00 0.10

(D)

26.5 (E)

Φ=5.79% Φ Φ Φ=13.7%

Φ=3.35% Φ=9.20%

EA+ER=3

EA+ER=6

29.3

0.05 0.00

(C)

EA+ER=22

0.10

25.8

0.05

(F)

27.5

0.05

0.00

EA+ER=12

0.00

0

10

20

30

40

50

0

z (a0)

10

20 0

30

40

50

z (a0)

FIGURE 5.35 (AC) The concentration distribution DISC and (DF) aggregation distribution DISA along the z-axis at t 5 tB with various concentration Φ and potential energy EA 1 ER for polydisperse particle systems, where the average z-coordinate value of each figure is marked [22].

particles. In addition, the average z-coordinate values of particles are very near 25, which means that the migration of particles under the thermal gradient is not noticeable at t 5 tB. To find the effect of thermophoresis on the particle migration and aggregation in a long action time, DISC , and DISA are calculated as: t 5 tB 1 5000Δt0 ;

tB 1 10; 000Δt0 ;

and

tB 1 20; 000Δt0

(5.120)

As presented in Figs. 5.36 and 5.37 for the monodisperse and polydisperse particle systems, respectively, when Φ 5 1:76% and EA 1 ER 5 6. It can be observed that by increasing the time, the particles migrate clearly at the inverse direction of the thermal gradient. For example, in Fig. 5.36AC, the prediction of particles clearly moves along the z direction and the quantitative migration is presented in the concentration distribution in Fig. 5.37DF where the average z-coordinate values are 23.1, 19.7, and 15.8 at t 5 tB 1 5000Δt0, tB 1 10,000Δt0, and tB 1 20,000Δt0, respectively. Compared to the monodisperse particles, the aggregation structures of the polydisperse particles are more incompact as shown before. Moreover, the average z-coordinate values are 20.8, 14.1, and 11.2 for the DISC and 12.5, 8.1, and 5.8 for the DISA in the polydisperse particle system in Fig. 5.37, significantly smaller than those in Fig. 5.36DI. The aggregates of monodisperse particles shown in Fig. 5.36AC are very compact and include approximately all particles, while there are many small aggregates including several primary particles in the polydisperse particle system as shown in Fig. 5.37AC [22]. By studying the results of Figs. 5.36 and 5.37, it can be concluded that the average z-coordinate values of DISC and DISA are suitable indicators of the equilibrium state of thermophoresis like RA as

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5.3 Nanofluids under thermal radiation

tB+5000 Δt0

tB+10,000Δt0

(B)

(D)

(E)

(F)

0

23.1

tB+ 5000

19.7

tB+ 10,000

tB+ 20,000

15.8

10

20

30

4 40

50

DISA

DISC

(A) 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.10 0.08 0.06 0.04 0.02 0.00 0.10 0.08 0.06 0.04 0.02 0.00

tB+20,000Δt0

(C) 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.10 0.08 0.06 0.04 0.02 0.00 0.10 0.08 0.06 0.04 0.02 0.00

(G)

(H)

23.2

tB+5000

19.5

tB+10,000

15.6

(I)

0

10

tB+20,000

20

30

40

50

z (a0)

z (a0)

FIGURE 5.36 (AC) The aggregate structure, (DF) concentration distribution DISC , and (GI) aggregation distribution DISA at t 5 tB 1 5000Δt0, tB 1 10,000Δt0, and tB 1 20,000Δt0, respectively, of a monodisperse particle system with Φ 5 3:35% and EA 1 ER 5 22 [22].

an indicator of the particle aggregation. Also, the results show that the particle aggregation is faster than the particle migration under a certain thermal gradient. Furthermore, if the thermal gradient is too large, the thermophoresis of particles could be accelerated and consequently the particle migration under the thermal gradient might be faster than aggregation.

5.3.2 Case 2: Carbon nanotube-water analysis between rotating disks under the thermal radiation conditions In this case, two parallel infinite disks are considered as shown in Fig. 5.38 which between them is filled by incompressible NF. Single and multiwalled carbon nanotubes (SWCNTs and MWCNTs) are added to the base fluid and their thermophysical properties are presented in Table 5.5. Disks are rotating with different angular velocities Ω1 and Ω2. Also, disks are stretching in the radial direction with different rates of a1 and a2. The thermal boundary conditions are: the lower disk is heated

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5. Nanofluid analysis in different media

tB+5000Δt0

tB+10,000Δt0

0.08 0.06 0.04 0.02 0.00 0.08 0.06 0.04 0.02 0.00 0.10 0.08 0.06 0.04 0.02 0.00

(B)

(D)

20.8

(E)

0

tB+10,000

14.1

(F)

11.2

10

20

tB+5000

DISA

DISC

(A)

tB+20,000Δt0

tB+20,000 30

40

50

(C) 0.10 0.08 0.06 0.04 0.02 0.00 0.10 0.08 0.06 0.04 0.02 0.00 0.12 0.09 0.06 0.03 0.00

z (a0 )

(G)

(H)

(I)

12.5

tB+5000

tB+10,000

8.1

tB+20,000

5.8

0

10

20

30

40

50

z (a0 )

FIGURE 5.37 (AC) The aggregate structure, (DF) concentration distribution DISC , and (GI) aggregation distribution DISA at t 5 tB 1 5000Δt0, tB 1 10,000Δt0, and tB 1 20,000Δt0, respectively, of a polydisperse particle system with Φ 5 3:35% and EA 1 ER 5 22 [22].

FIGURE 5.38 Schematic of the problem [25].

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5.3 Nanofluids under thermal radiation

TABLE 5.5

Thermal properties of base fluid (water) and nanoparticles [26].

Properties

Unit 21

21

Water

SWCNT

MWCNT

Heat capacitance

J kg

4179

425

796

Density

kg m23

997.1

2600

1600

Thermal conductivity

W m21 K21

0.613

6600

3000

K

MWCNT, Multiwalled carbon nanotubes; SWCNT, single carbon nanotubes.

in T0 and the upper disk is in T1 temperatures. Thus the governing equations in a cylindrical coordinate system (r, θ, z) are [26]: @u u @w 1 1 50 @r r @z  2  @u @u υ2 1 @p @ u 1 @u @2 u u u 1w 2 1 ν nf 1 2 2 2 52 1 @r @z ρnf @r @r2 r @r @z r r  2  @υ @υ uυ @ υ 1 @υ @2 υ υ 1w 2 5 ν nf 1 1 2 u @r @z r @r2 r @r @z2 r2  2  @w @w 1 @p @ w 1 @w @2 w 1w 52 1 ν nf 1 2 1 u @r @z ρnf @z @r2 r @r @z 

ρcp

 nf

 u

(5.121) (5.122)

(5.123) (5.124)

  2    @T @T @T 1 @T @2 T 16σ T13 @2 T 1 @T @2 T 5 knf 1 1w 1 1 1 1 @r @z @r2 r @r @z2 @r2 r @r @z2 3k

(5.125) where p and T are the pressure and temperature, respectively; σ indicates the Stefan Boltzmann constant; and k is the mean absorption coefficient. For this case, the boundary conditions are: u 5 ra1 ;

υ 5 rΩ1 ;

w 5 0;

u 5 ra2 ;

υ 5 rΩ2 ;

w 5 0;

@T 5 2 h1 ðT0 2 T Þ at z 5 0 @z @T 5 2 h2 ðT 2 T1 Þ at z 5 h knf @z knf

(5.126)

Here, based on Imtiaz et al. [25] and Chapter 1, Introduction to nanofluids, the following equations are applied to obtain the CNTwater NF properties: μf μnf 5 (5.127) ð12ϕÞ2:5 ρnf 5 ρf ð1 2 ϕÞ 1 ρCNT ϕ

Nanofluids

(5.128)

268

5. Nanofluid analysis in different media

ðρcp Þnf 5 ðρcp Þf ð1 2 ϕÞ 1 ðρcp ÞCNT ϕ     knf ð1 2 ϕÞ 1 2ϕ kCNT =ðkCNT 2 kf Þ ln ðkCNT 1 kf Þ=2kf     5 kf ð1 2 ϕÞ 1 2ϕ kf =ðkCNT 2 kf Þ ln ðkCNT 1 kf Þ=2kf

(5.129) (5.130)

By considering the following transformation function: u 5 rΩ1 f 0 ðηÞ; υ 5 rΩ1 gðηÞ;  w 5 2 2hΩ1 f ð ηÞ T 2 T1 1 r2 ; p 5 ρf Ων f PðηÞ 1 ε ; θðηÞ 5 2 h2 T0 2 T1

η5

z h

(5.131)

Eqs. (5.122)(5.126) will be transformed to: ð12ϕÞ

2:5



1 1 2 ϕ 1 ðρCNT =ρf Þϕ

  fv0 1 Re 2ffv 2 f 02 1 g2 2

ε 50 1 2 ϕ 1 ðρCNT =ρf Þϕ

(5.132) 



1  gv 1 Re 2fg0 2 2f 0 g 5 0 1 2 ϕ 1 ðρCNT =ρf Þϕ ð12ϕÞ 2:5

(5.133)

1 2  P0 5 2 4Re ff 0 2  fv 2:5 1 2 ϕ 1 ðρCNT =ρf Þϕ 1 2 ϕ 1 ðρCNT =ρf Þϕ ð12ϕÞ !     ρcp CNT 1 knf 1 Rd θv 1 2Re 1 2 ϕ 1   ϕ fθ0 5 0 Pr kf ρcp f

(5.134) (5.135)

And the boundary conditions: fð0Þ 5 0; θ0 ð0Þ 5 2

fð1Þ 5 0;

f 0 ð0Þ 5 A1 ;

kf γ ½1 2 θð0Þ; knf 1

f 0 ð1Þ 5 A2 ;

θ0 ð1Þ 5 2

kf γ θð1Þ; knf 2

gð0Þ 5 1;

gð1Þ 5 Ω

Pð0Þ 5 0 (5.136)

where the Reynolds (Re), Prandtl (Pr), scaled stretching parameters (A1, A2), rotation parameter (Ω), radiation parameter (Rd), and thermal Biot numbers (γ 1, γ 2) are:   ρcp f ν f Ω 1 h2 a1 a2 Re 5 ; Pr 5 ; A1 5 ; A2 5 νf kf Ω1 Ω2 (5.137)  3 Ω2 16σ T1 h1 h h2 h ; γ1 5 Ω5 ; Rd 5 ; γ2 5 kf kf Ω1 3kf k

Nanofluids

5.3 Nanofluids under thermal radiation

269

For a more simplified equation of Eq. (5.132) and to eliminate the pressure parameter (ε), it can be derived respect to η:

ð12ϕÞ2:5



  f iv 1 Re 2ffv0 1 2gg0 5 0 1 2 ϕ 1 ðρCNT Þ=ρf ϕ 1



(5.138)

The pressure parameter (ε) can be obtained by Eqs. (5.132) and (5.136), and also the pressure term (P) can be determined by integrating Eq. (5.134) with respect to η as Imtiaz et al. [25] reported. Two main parameters are introduced to discuss the results: skin friction coefficients and Nusselt number. For the lower rotating disk, the shear stress in the radial (τ zr) and tangential directions (τ zθ) is: μf rΩ1 fvð0Þ @u τ zr 5 μnf 5 (5.139) @z z50 ð12ϕÞ2:5 h μf rΩ1 g0 ð0Þ @υ (5.140) τ zθ 5 μnf 5 @z z50 ð12ϕÞ2:5 h Consequently the total shear stress will be: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τ w 5 τ 2zr 1 τ 2zθ

(5.141)

Also, the skin friction factors for lower and upper disks (C1 and C2) are: C1 5 τ w ρf ðrΩ1 Þ2 5

1 Rer ð12ϕÞ2:5

C2 5 τ w ρf ðrΩ2 Þ2 5

1 Rer ð12ϕÞ2:5

z50

(5.142)

z5h

(5.143)

h

h

2  2 i1=2 fvð0Þ 1 g0 ð0Þ

2  2 i1=2 fvð1Þ 1 g0 ð1Þ

where the local Reynolds number (Rer 5 rΩh/υf) and the Nusselt numbers for lower and upper disks can be calculated from: Nux1 5

hqw jz50 kf ðT0 2T1 Þ

(5.144)

Nux2 5

hqw jz5h kf ðT0 2T1 Þ

(5.145)

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5. Nanofluid analysis in different media

In Eqs. (5.144) and (5.145), wall heat flux (qw) and radiative heat flux (qr) will be determined as: @T @T qw 5 2 knf z50 1 qr ; qw 5 2 knf 1 qr (5.146) @z @z z5h z50 z50 z5h z5h 16σ T13 @T 16σ T13 @T qr 5 2 ; qr 5 2 (5.147) 3k @z z50 3k @z z5h z50 z5h Finally the dimensionless forms of the Nusselt numbers will be:   knf 1 Rd θ0 ð0Þ (5.148) Nu1 5 2 kf   knf Nu2 5 2 1 Rd θ0 ð1Þ (5.149) kf To solve the governing equations by least square method (LSM) (as introduced in Chapter 2: Mathematical analysis of nanofluids), the trial functions are considered as: f ðxÞ 5 A1 x 2 ð2A1 1 A2 Þx2 1 ðA1 1 A2 Þx3 1 c1 x2 ðx21Þ2 1 c2 x3 ðx21Þ2 (5.150) gðxÞ 5 1 2 x 1 Ωx 1 c3 xðx 2 1Þ 1 c4 x2 ðx 2 1Þ (5.151)   θðxÞ 5 c5 1 c6 ðx 2 0:5Þ 1 2βγ 1 c5 1 0:5βγ 1 c6 1 βγ 1 1 c6 xðx 2 1Þ   1 2βγ 1 c5 2 2βγ 2 c5 2 βγ 1 c6 2 βγ 2 c6 2 2βγ 1 2 4c6 x2 ðx 2 0:5Þðx 2 1Þ (5.152) By minimization of residuals, the values of ci and consequently the following distributions of f ðxÞ, gðxÞ, and θðxÞ can be calculated for φ 5 0:2; Rd 5 0:3; A1 5 0:7; A2 5 0:8; Ω 5 0:8; γ 1 5 0:4; γ 2 5 0:5: f ðxÞ 5 0:7x 2 2:2x2 1 1:5x3 1 0:0070x2 ðx21Þ2 1 0:0149x3 ðx21Þ2

(5.153)

gðxÞ 5 1 2 0:2x 1 0:0041xðx 2 1Þ 2 0:0025x2 ðx 2 1Þ

(5.154)

θðxÞ 5 0:4696 2 0:0399x 1 0:0001xðx 2 1Þ 2 0:0015x2 ðx 2 0:5Þðx 2 1Þ (5.155) Table 5.6 demonstrates the good accuracy of LSM compared to numerical methods in all profiles and the percentage of errors method is presented via Table 5.7, which confirm the validity of the trial functions. The results of SWCNT and MWCNT are compared in Fig. 5.39, showing that although these two kinds of CNTs have approximately the same tangential velocity profile, SWCNTs have greater temperature

Nanofluids

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5.3 Nanofluids under thermal radiation

TABLE 5.6 The comparison of the results of LSM and numerical solutions for SWCNTswater when φ 5 0:2; Rd 5 0:3; A1 5 0:7; A2 5 0:8; Ω 5 0:8; γ 1 5 0:4; γ 2 5 0:5 [26]. θðxÞ

g(x)

f(x)

Numerical

LSM (Eq. 5.43)

Numerical

LSM (Eq. 5.42)

Numerical

LSM (Eq. 5.41)

0.469547

0.469676

1.0

1.0

0.0

0.0

0

0.465545

0.465665

0.984446

0.979652

0.049514

0.049568

0.1

0.461575

0.461652

0.973178

0.959421

0.064167

0.064256

0.2

0.457647

0.457647

0.962134

0.93929

0.052924

0.053006

0.3

0.453754

0.453654

0.948498

0.919252

0.024723

0.024748

0.4

0.449876

0.449673

0.930656

0.899284

20.011524

20.011594

0.5

0.445991

0.445703

0.908146

0.879373

20.046932

20.047079

0.6

0.442075

0.441736

0.881607

0.859504

20.072667

20.072729

0.7

0.438111

0.437762

0.852748

0.839662

20.079596

20.079514

0.8

0.434095

0.433768

0.824307

0.819832

20.058326

20.058334

0.9

0.430041

0.429735

0.80

0.80

0.0

0.0

1

x

LSM, Least square method; SWCNTs, single walled carbon nanotubes.

TABLE 5.7 data.

Error of applied method compared with numerical outcomes of Table 5.6

Error of θðxÞ (%)

Error of g(x) (%)

Error of f(x) (%)

x

0.027473

0.000

0.000

0

0.025776

0.486974

0.10906

0.1

0.016682

1.413616

0.1387

0.2

0.000

2.374305

0.15494

0.3

0.02204

3.083401

0.10112

0.4

0.04512

3.370956

0.60743

0.5

0.06458

3.168323

0.31322

0.6

0.07668

2.507126

0.08532

0.7

0.07966

1.534568

0.10302

0.8

0.07533

0.54288

0.01372

0.9

0.07116

0.000

0.000

1.00

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5. Nanofluid analysis in different media

FIGURE 5.39 The velocity and temperature profiles of SWCNTswater and the MWCNTswater noanofluids when φ 5 0:2; Rd 5 0:5; A1 5 0:9; A2 5 0:1; Ω 5 0:5; γ1 5 0:7; γ2 5 0:3 [26].

FIGURE 5.40 Effect of the nanoparticle volume fraction on gðηÞ and θðηÞ for the SWCNTswater when Re 5 10; Rd 5 0:5; A1 5 0:6; A2 5 0:3; Ω 5 0:2; γ1 5 0:1; γ2 5 0:4 [26].

values. Fig. 5.40 shows that fluid temperature increases rapidly when ϕ is enlarged due to this fact that by increasing the nanoparticles volume fraction, the thermal conductivity and thermal boundary layer will improve.

Nanofluids

5.3 Nanofluids under thermal radiation

273

5.3.3 Case 3: Ethylene glycol (C2H6O2) carbon nanotubes in rotating stretching channel with nonlinear thermal radiation As the last case in this chapter, consider nonlinear thermal radiation affected on the three-dimensional squeezing flow of CNT NF consisting of SWNT and MWNT nanoparticles and ethylene glycol as base fluid as seen in Fig. 5.41 and detailed in Table 5.8. A rotating channel with fixed and permeable bottom wall is considered for the geometry where the bottom wall is under stretching in the coordinate of y 5 0 along the x-axis with the velocity of Uw 5 ax=ð1 2 ctÞ. Also, the wall velocity along the y-axis is v 5 2 V0 =ð1 2 ctÞ. The channel height changes to the equivalent to high wall velocity along the y-axis, from vh 5 dh=dt. According to the above assumptions, the conditions and governing equations of modeling this problem are [27,28]: @u @v 1 50 @x @y

FIGURE 5.41

Geometry of the problem [27].

Nanofluids

(5.156)

274

5. Nanofluid analysis in different media

TABLE 5.8 Thermophysical properties of base fluids and carbon nanotubes (CNTs) nanoparticles. Physical properties   ρ kg m23   cp J kg21 K21   k W m21 K21

Ethylene glecol

SWCNT

MWCNT

1115

2600

1600

2430

425

769

0.253

6600

3000

MWCNT, Multiwalled carbon nanotubes; SWCNT, single carbon nanotubes.

  @u @u @u 2ω0 1 @p μnf @2 u @2 u 1u 1v 1 2 w1 1 50 (5.157) @t @x @y ρnf @x @y2 1 2 ct ρnf @x2   @v @v @v 1 @p μnf @2 v @2 u 1u 1v 1 2 1 50 (5.158) @t @x @y ρnf @x @y2 ρnf @x2   μnf @2 w @2 w @w @w @w 2ω0 1u 1v 2 w2 1 2 50 (5.159) @t @x @y @y 1 2 ct ρnf @x2 0 1 2 2 μnf μnf @T @T @w @ T @ T 1 @qr @ 1u 1v 2 2 1 2A 2 @t @x @y @y ðρCp Þnf @y ðρCp Þnf @x2 ðρCp Þnf   2    2  2  @u @u @v @w @w 1 3 4 1 1 1 50 @x @x @y @x @y (5.160) where u, v, and w are the components of the velocity in the directions of the x-, y-, and z-axis, respectively. Also, ρnf ,μnf , and ðρCp Þnf are the density, dynamic viscosity, and heat capacitance of the NF, respectively. T indicates the fluid temperature. In Eq. (5.160), qr is the radiative heat flux, and by Ressoland approximation for radiation, the thermal flux is defined as: qr 5 2

4σ @T4 3k @y

(5.161)

where σ and k are the StefanBoltzman constant and mean absorption coefficient, respectively. Assuming a small temperature difference in flow, the Taylor series approximation for T4 in terms of TN is as follows: T 4  4TTN 3 2 3TN 3

Nanofluids

(5.162)

5.3 Nanofluids under thermal radiation

275

After replacing Eqs. (5.161) and (5.162) with Eq. (5.160), the final equation is obtained as follows: 0 1   2 2 μnf @T @T @w @ T @ T 1 16σ @ 3 @T @ A 1u 1v 2 1 2 2 T @t @x @y @y ðρCp Þnf 3k @y @y ðρCp Þnf @x2   2    2  2  μnf @u @u @v @w @w 1 2 1 1 4 50 1 @x @x @y @x @y ðρCp Þnf (5.163) Furthermore, the appropriate boundary conditions for the above equations are given as follows: 8 ax 2 V0 > > ; v5 ; w 5 0; T 5 Tw at y 5 0 > u 5 Uw 5 > < 1 2 ct 1 2 ct sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c v > > ; w 5 0; T 5 Th as y 5 hðtÞ u 5 0; v 5 Vh 5 2 > > : 2 að1 2 ctÞ (5.164) In this case, the thermal conductivity of CNTs is considered based on Maxwell theory. In this model, particle size, volume fraction, thermal conductivity, and temperature play a significant role in increasing the heat transfer of the NF. Thus:       8 N1 5 ρnf 5 ð1 2 φÞρf 1 φρCNT ; N2 5 ρCp nf 5 ð1 2 φÞ ρCp f 1 φ ρCp CNT > > <     μf knf ð1 2 φÞ 1 2φ kCNT =ðkCNT 2 kf Þ ln ðkCNT 1 kf Þ=2kf >     N 5 μ 5 ; N 5 5 4 > nf : 3 kf ð1 2 φÞ 1 2φ kf =ðkCNT 2 kf Þ ln ðkCNT 1 kf Þ=2kf ð12φÞ2:5

(5.165) where μnf indicates the dynamic viscosity of the NF; φ is the volume fraction of the nanoparticle; and ρf and ρCNT denote the fluid and carbon nanotubes densities, respectively. Also, kf and kCNT are the thermal conductivity of base fluid and carbon nanotubes, respectively. The following definitions are given in order to simplify Eqs. (5.156)(5.159), (5.163), and (5.164): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aν T 2 Tw y 0 fðηÞ; w 5 Uw gðηÞ; θðηÞ 5 u 5 Uw f ðηÞ; v 5 ; η5 að1 2 ctÞ hðtÞ Tw 2 Th (5.166) where η is the local similarity variable; f 0 ðηÞ shows the dimensionless velocity function along the x-axis; fðηÞ is the dimensionless velocity function along the y-axis; and gðηÞ and θðηÞ are the rotational velocity and dimensionless temperature functions of the fluid, respectively.

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By replacing Eq. (5.166) with partial differential equations (5.156) (5.159), (5.163), and (5.164), the ODEs are finally obtained with boundary conditions as follows:    S 3fv 1 ηfv0 5 0 f iv 2 ð12φÞ2:5 N1 f 0 fv 2 ffv0 1 2Ωg0 1 (5.167) 2   η gv 1 ð12φÞ2:5 N1 fg0 2 f 0 g 1 2Ωf 0 2 S g 1 g0 5 0 (5.168) 2    2      R N2 Pr S 0 N2 Pr Ec 4f 1 g2 0  2   50 11 θf 2 ηθ 1 θv 1 2 N4 N4 2 N4 ð12φÞ2:5 1 Ecm fv 1 ðg0 Þ (5.169)

8 0 < fð0Þ 5 A; f ð0Þ 5 1; gð0Þ 5 0; θð0Þ 5 1 S 0 : fð1Þ 5 2 ; f ð1Þ 5 0; gð1Þ 5 0; θð1Þ 5 0

(5.170)

The dimensionless numbers and parameters in the above equations are defined as follows: Ω 5 ω0 =a is the rotation parameter; S 5 c=a indicates the squeeze parameter when the upper plate of the channel moves toward the lower stretching one (S . 0), while when the upper plate is moving away from the lower one (S , 0); R 5 16σ T 3 =3k  kf is the radiation parameter; Pr 5 μf ðCp Þf =k is the Prandtl number; Ec 5 v2 =ðh2 ðCp Þf ðTw 2 Th ÞÞ is the 2 Eckert number; Ecm 5 UW =ððCp Þf ðTw 2 Th ÞÞ shows the modified Eckert number; and A 5 V0 =ah refers to the suction parameter. Also, the skin fraction coefficient (Cfx ) and local Nusselt number (Nux ) are defined as [29]: 0  0  1 1 μnf @u=@y μnf @u=@y A ; Cfupper 5 @ A Cflower 5 @ ρnf U02 ρnf U02 y50 0  0  y5hðtÞ  1  1 x @T=@y x @T=@y A 1 qr ; Nulower 5 @ A Nulower 5 @ 1 qr knf ðTw 2Th Þ knf ðTw 2Th Þ y50

y5hðtÞ

(5.171) After simplifying Eq. (5.171), surface drag forces and heat transfer are determined as:   1 1 1=2 1=2 Cf ReX 5 fv ð 0 Þ; Cf Re 5 fvð1Þ; X lower upper N1 ð12φÞ2:5 N1 ð12φÞ2:5   21=2 21=2 Nu ReX 5 2 ðN4 1 RÞθ0 ð0Þ; Nu ReX 5 2 ðN4 1 RÞθ0 ð1Þ lower

upper

(5.172)

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Here, Rex 5 Uw x=ν f is the local Reynolds number. In this section the governing equations are solved using the RungeKutta Fehlberg fourth-fifth order numerical method. The basic principle of this method is to convert the boundary value of the problem to the base values, selecting finite values for ηN , and finally replacing them in the equations. Approximation of the fourth and fifth orders to the solution are:   25 1408 2197 1 k0 1 k2 1 k3 2 k4 ym11 5 ym 1 h (5.173) 216 2565 4109 5   16 6656 28561 9 2 k0 1 k2 1 k3 2 k4 1 k5 ym11 5 ym 1 h (5.174) 135 12825 56430 50 55 Also, in this method for the exact solution of the problem, step sizes should be properly selected. Each step consists of six parts as follows:   8 k0 5 f xm 1 ym > > > 0 1 > > > > h hk0 A > > > k1 5 f @xm 1 ; ym 1 > > 4 4 > > > > > 0 0 1 1 > > > > > > k2 5 f @xm 1 3 h; ym 1 @ 3 k0 1 9 k1 AhA > > > 8 32 32 > > > > < 0 0 1 1 12 1932 7200 7296 A A > k0 2 k1 1 k2 h k3 5 f @xm 1 h; ym 1 @ > > > 13 2197 2197 2197 > > > > 0 0 1 1 > > > > > 439 3860 845 A A > > k0 2 8k1 1 k2 2 k3 h k4 5 f @xm 1 h; ym 1 @ > > 216 513 4104 > > > > > 0 0 1 1 > > > > h 3544 1859 11 A A > > @ @ 8 > > : k5 5 f xm 1 2 ; ym 1 2 27 k0 1 2k1 2 2565 k2 1 4104 k3 2 40 k4 h (5.175) Furthermore, the boundary condition of these equations must be changed from nonlinear to first-order linear differential equations. To consider this, each boundary condition is equivalent to a variable that is defined as follows: f 5 q1 ;

f 0 5 q2 ;

fv 5 q3 ;

fv0 5 q4 ;

g 5 q5 ;

g0 5 q 6 ;

θ 5 q7 ;

θ 0 5 q8 (5.176)

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After that, by replacing new definitions (Eq. 5.176) in Eqs. (5.167) (5.169) and boundary condition of Eq. (5.170), the reduced equations and their boundary condition are equal to:    S 2 ð12φÞ N1 q2 q3 2 q1 q4 1 2Ωq6 1 3q3 1 ηq4 5 0 2

(5.177)

  η q06 1 ð12φÞ2:5 N1 q1 q6 2 q2 q5 1 2Ωq2 2 S q5 1 q6 5 0 2

(5.178)

q04

2:5

      R 0 N2 Pr S N2 Pr Ecð4q21 1 q25 Þ q8 1 50 11 q7 q2 2 ηq8 1 2 2 N4 N4 2 N4 ð12φÞ2:5 1 Ecm ððq3 Þ 1 ðq6 Þ Þ (5.179)

(

q1 ð0Þ 5 A;

q2 ð0Þ 5 s1 ;

q3 ð0Þ 5 1;

q4 ð0Þ 5 s2

q5 ð0Þ 5 0;

q6 ð0Þ 5 s3 ;

q7 ð0Þ 5 1;

q8 ð0Þ 5 s4

(5.180)

In Eq. (5.180), s1 ; s2 ; s3 and s4 are actually equivalent to fð0Þ-0; f 0 ð0Þ-0; gð0Þ-0, and θð0Þ-0 when η-1. The above solution repeats until it converges. The degree of convergence and the solving step are considered to be 1026 and Δη 5 0:001, respectively. As a result, the effect of nanoparticle volume fraction (φ) on the velocity and temperature profiles of the NFs is depicted in Fig. 5.42AF. Firstly, in Fig. 5.42A, it can be observed that increasing the nanoparticle volume fraction leads to an increase in velocity in the y-direction (fðηÞ) in both cases of SWCNT and MWCNT. It must be noted that the MWCNT has increased the velocity profile more than SWCNT, and fðηÞ always has a greater value at S . 0 than at S , 0 due to the MWCNT density, which is lower than for the SWCNT. Secondly, in Fig. 5.42B, in the interval of 0 # η # 0:3, the rate of variation in the velocity in the x-direction is decreasing, but in the interval of η . 0:3, a different treatment is observed. It must be pointed out that when the channel is narrowed (S . 0), the velocity variations are greater than when the channel height is increased (S , 0). The increasing φ also leads to an increase in the rotational velocity (gðηÞ) as seen in Fig. 5.42C and D. Furthermore, the function gðηÞ has a greater value for the MWCNT case than the SWCNT case. Also, gðηÞ has a greater value at S . 0 than S , 0. An inverse relationship between φ and the temperature profile is also observed in Fig. 5.42E and F for all the cases.

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References

279

FIGURE 5.42 Influence of φ on velocity profile (A: fðηÞ, B: f 0 ðηÞ, C, D: gðηÞ), and temperature profile (θðηÞ) (E, F) [27].

References [1] M. Hatami, J. Zhou, J. Geng, D. Song, D. Jing, Optimization of a lid-driven T-shaped porous cavity to improve the nanofluids mixed convection heat transfer, J. Molec. Liquids 231 (2017) 620631. [2] M. Satyajit, S. Sourav, S. Sumon, M.A. Mamun, Combined effect of Reynolds and Grashof numbers on mixed convection in a lid-driven T-shaped cavity filled with water-Al2O3 nanofluid, J. Hydrodyn. 27 (2015) 782794.

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5. Nanofluid analysis in different media

[3] M. Hatami, J. Geng, D. Jing, Enhanced efficiency in Concentrated Parabolic Solar Collector (CPSC) with a porous absorber tube filled with metal nanoparticle suspension, Green Energy Environ. 3 (2) (2018) 129137. [4] M. Sheikholeslami, H.B. Rokni, Nanofluid convective heat transfer intensification in a porous circular cylinder, Chem. Eng. Process. Process Intensif. 120 (2017) 93104. [5] M. Hatami, J. Jin, H.R. Ashorynejad, D. Jing, Uniform magnetic field (UMF) effect on the heat transfer of a porous half-annulus enclosure filled by Cu-water nanofluid considering heat generation, Curr. Nanosci. 14 (3) (2018) 187198. [6] M. Sheikholeslami, S. Soleimani, M. Gorji-Bandpy, D.D. Ganji, S.M. Seyyedi, Natural convection of nanofluids in an enclosure between a circular and a sinusoidal cylinder in the presence of magnetic field, Int. Commun. Heat Mass Transfer 39 (2012) 14351443. [7] A.M. Rashad, M.M. Rashidi, G. Lorenzini, S.E. Ahmed, A.M. Aly, Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium saturated with Cuwater nanofluid, Int. J. Heat Mass Transfer 104 (2017) 878889. [8] M. Sheikholeslam, T. Hayat, A. Alsaedi, Numerical study for external magnetic source influence on water based nanofluid convective heat transfer, Int. J. Heat Mass Transfer 106 (2017) 745755. [9] M. Hatami, J. Zhou, J. Geng, D. Jing, Variable magnetic field (VMF) effect on the heat transfer of a half-annulus cavity filled by Fe3O4-water nanofluid under constant heat flux, J. Magnet. Magnet. Mater. 451 (2018) 173182. [10] M. Sheikholeslami, K. Vajravelu, Nanofluid flow and heat transfer in a cavity with variable magnetic field, Appl. Math. Comput. 298 (2017) 272282. [11] M. Hatami, M. Khazayinejad, D. Jing, Forced convection of Al2O3water nanofluid flow over a porous plate under the variable magnetic field effect, Int. J. Heat Mass Transfer 102 (2016) 622630. [12] S.S. Nourazar, M.H. Matin, M. Simiari, The HPM applied to MHD nanofluid flow over a horizontal stretching plate, J. Appl. Math. 2011 (2011) 117. [13] G. Domairry, M. Hatami, Squeezing Cuwater nanofluid flow analysis between parallel plates by DTM-Pade´ method, J. Molec. Liquids 193 (2014) 3744. [14] A.R. Ahmadi, A. Zahmatkesh, M. Hatami, D.D. Ganji, A comprehensive analysis of the flow and heat transfer for a nanofluid over an unsteady stretching flat plate, Powder Technol. 258 (2014) 125133. [15] J. Jin, D. Song, J. Geng, D. Jing, Time-dependent scattering of incident light of various wavelengths in ferrofluids under external magnetic field, J. Magnet. Magnet. Mater. 447 (2018) 124133. [16] D. Song, D. Jing, B. Luo, J. Geng, Y. Ren, Modeling of anisotropic flow and thermodynamic properties of magnetic nanofluids induced by external magnetic field with varied imposing directions, J. Appl. Phys. 118 (2015) 045101. Available from: https:// doi.org/10.1063/1.4927043. [17] S. Vinod, J. Philip, Role of field-induced nanostructures, zippering and size polydispersity on effective thermal transport in magnetic fluids without significant viscosity enhancement, J. Magn. Mag. Mater. 444 (2017) 2942. Available from: https://doi. org/10.1016/j.jmmm.2017.07.100. [18] C. Rablau, P. Vaishnava, C. Sudakar, R. Tackett, G. Lawes, R. Naik, Magnetic-fieldinduced optical anisotropy in ferrofluids: a time-dependent light-scattering investigation, Phys. Rev. E 78 (2008) 051502. Available from: https://doi.org/10.1103/ PhysRevE.78.051502. [19] P.C. Fannin, S.W. Charles, The study of a ferrofluid exhibiting both Brownian and Neel relaxation, J. Phys. D Appl. Phys. 22 (1989) 187191. Available from: https:// doi.org/10.1088/0022-3727/22/1/027.

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Further reading

281

[20] S.A. Rovers, R. Hoogenboom, M.F. Kemmere, J.T.F. Keurentjes, Relaxation processes of superparamagnetic iron oxide nanoparticles in liquid and incorporated in poly (methyl methacrylate), J. Phys. Chem. C 112 (2008) 1564315646. Available from: https://doi.org/10.1021/jp805631r. [21] H.C. Hamaker, The London-van der Waals attraction between spherical particles, Physica 4 (1937) 10581072. Available from: https://doi.org/10.1016/S0031-8914(37) 80203-7. [22] D. Song, H. Jin, D. Jing, X. Wang, Dynamic properties of polydisperse colloidal particles in the presence of thermal gradient studied by a modified Brownian dynamic model, J. Phys. D Appl. Phys. 51 (10) (2018) 105301. [23] S. Fayolle, T. Bickel, A. Wu¨rger, Thermophoresis of charged colloidal particles, Phys. Rev. E 77 (2008) 041404. [24] M.Y. Lin, H.M. Lindsay, D.A. Weitz, et al., Universal diffusion limited aggregation, J. Phys. Condens. Matter. 2 (1990) 3093. [25] M. Imtiaz, T. Hayat, A. Alsaedi, B. Ahmad, Convective flow of carbon nanotubes between rotating stretchable disks with thermal radiation effects, Int. J. Heat Mass Trans. 101 (2016) 948. [26] S. Mosayebidorcheh, M. Hatami, Heat transfer analysis in carbon nanotube-water between rotating disks under thermal radiation conditions, J. Molec. Liquids 240 (2017) 258267. [27] S.S. Ghadikolaei, Kh Hosseinzadeh, M. Hatami, D.D. Ganji, M. Armin, Investigation for squeezing flow of ethylene glycol (C2H6O2) carbon nanotubes (CNTs) in rotating stretching channel with nonlinear thermal radiation, J. Molec. Liquids 263 (2018) 1021. [28] U. Khan, N.A. Syed, T. Mohyud-Din, Numerical investigation for three dimensional squeezing flow of nanofluid in a rotating channel with lower stretching wall suspended by carbon nanotubes, Appl. Thermal Eng. 113 (2017) 11071117. [29] N. Ahmed, A. As, U. Khan, S.T. Mohyud-Din, Influence of thermal radiation and viscous dissipation on squeezed flow of water between Riga plates saturated with carbon nanotubes, Colloids Surf A Physicochem. Eng. Aspects 522 (2017) 389398.

Further reading R. Di Leonardo, F. Ianni, G. Ruocco, Colloidal attraction induced by a thermal gradient, Langmuir 25 (2009) 42474250.

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C H A P T E R

6 Nanofluid analysis in different applications

6.1 Nanofluids for cooling and heating As mentioned in early chapters, the main purpose of introducing nanofluid (NF) is to present new fluid with different thermal properties to can use them more efficiently in industrial process such as solar panels, microchannels, cooling, heating, etc. In this chapter some examples of applications are introduced and analyzed numerically and experimentally. In the first application of NF in cooling and heating, two cases are discussed: microchannel and HVAC system models.

6.1.1 Case 1: Design of microchannel heat sink with wavy and straight wall In the first case, two channels, a straight microchannel and a wavy microchannel, are modeled and analyzed using finite volume method (FVM). The straight microchannel dimensions are considered from optimally designed in the literature [1] as presented in Table 6.1. Its base size is 28 3 100 mm and the heat flux of 100 w cm22 is applied from the bottom surface (Fig. 6.1A). To save computational time, only one of the straight channels is taken as the computational domain as depicted in Fig. 6.1B. The geometry dimensions of each channel are as follows: wall thickness Wr =2 5 28 μm; channel width Wc 5 85 μm; channel height Hc 5 700 μm; and cover plate thickness Hr 5 100 μm [2]. In addition to the straight channel, a wavy microchannel heat sink is also considered to be an improved design compared to a rectangular straight microchannel, as presented in Fig. 6.1C. All of the dimensions are considered

Nanofluids DOI: https://doi.org/10.1016/B978-0-08-102933-6.00006-8

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© 2020 Elsevier Ltd. All rights reserved.

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6. Nanofluid analysis in different applications

TABLE 6.1 Optimal geometric dimension for straight microchannel heat sink. Lx (mm)

Ly (μm)

Lz (mm)

Wc (μm)

Wr (μm)

Hc (μm)

δ

10

900

10

85

56

700

100

(A)

(μm)

q W cm22 100

Lz Lx

Ly

y

x

z (B)

q"

(C)

δ

Outlet

Hc

Inlet

δ

Wr/2 Wc Wr/2

FIGURE 6.1 Schematic of microchannel heat sinks for (A) heat sink, (B) straight microchannel, and (C) wavy channel. The meanings of all the symbols can be found in Table 6.1 [2].

as the same of rectangular straight microchannel except the wavy-wall function along the x-axis, which can be introduced by [2]:   x (6.1) y 5 A sin 2π λ



where A and λ are the amplitude and wavelength, respectively. Also, x is the flow direction. To simplify the analysis in this case, the following assumptions have been considered for the Mini-channel based Heat Sink (MCHS) conditions: the working fluid (water) is incompressible, the flow is laminar and fully developed thermally and hydraulically, and the surface of the MCHS is completely insulated.

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285

Based on the above assumptions, the governing equations are as follows [2]: The continuity equation: @u @v @w 1 1 50 @x @y @z

(6.2)

where u, v, and w are the velocity components in the x, y, and z directions, respectively. The momentum equations for the coolant will be: 0 1 0 1 2 2 2 @u @u @u @p @ u @ u @ u ρ f @u 1v 1w A52 1 μf @ 2 1 2 1 2 A @x @y @z @x @x @y @z 0 1 0 1 2 2 2 @v @v @v @p @ v @ v @ v 1 μf @ 2 1 2 1 2 A ρ f @u 1 v 1 w A 5 2 (6.3) @x @y @z @y @x @y @z 0 1 0 1 2 2 2 @w @w @w @p @ w @ w @ w 1v 1w A52 1 μf @ 2 1 2 1 2 A ρ f @u @x @y @z @z @x @y @z where ρf and μf are the density and dynamic viscosity of the coolant, respectively, and p is the coolant pressure. Also, the energy equation for the coolant is: !   @Tf @Tf @Tf @2 T f @ 2 Tf @2 T f 1v 1w 1 1 ρf cpf u (6.4) 5 κf @x @y @z @x2 @y2 @z2 And the energy equation for the solid region:  2  @ Ts @ 2 Ts @2 T s 0 5 κs 1 1 @x2 @y2 @z2

(6.5)

where the thermal properties of water such as density, viscosity, heat capacity, and thermal conductivity are determined to be temperature dependent [3]:   T 1 15:9414 ðT2276:9863Þ2 ρðTÞ 5 1000 1 2 (6.6) 508; 929:2ðT 2 204:87037Þ     T 8:9  23 exp 4700 T21 2 29321 (6.7) μðTÞ 5 1:005 3 10 293 kf ðT Þ 5 2 1:579 1 0:01544T 2 3:515 3 1025 T2 1 2:678 3 1028 T3

(6.8)

The inlet temperature is defined uniformly as 298K while the velocity inlet is specified as 0.6, 0.8, 1, 1.2, and 1.4m s21 , which is related to the

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Reynolds number and ranges from 99 to 232. Constant heat flux 100 W cm22 is applied at the bottom of the MCHS and the pressure outlet matches atmosphere pressure. Symmetrical boundary conditions are assumed on the left and right sides. The boundary conditions are: For inlet: u 5 uin ; v 5 0; w 5 0; T 5 Tin

(6.9)

p 5 p out

(6.10)

u 5 0; v 5 0; w 5 0; T 5 Tin

(6.11)

Outlet:

Coolantsolid interface:

Bottom wall of the heat sink: qw 5 2 ks

@Ts @n

(6.12)

Other solid walls and symmetric boundaries: 2ks

@Ts 50 @n

(6.13)

Based on the Reynolds number definition: Re 5

ρuav Dh μ

(6.14)

where Dh indicates the hydraulic diameter defined as Dh 5 2Hc Wch =ðHch 1 Wch Þ. The j factor as a dimensionless surface heat transfer coefficient can be introduced as: j5

Nu Re Pr1=3



(6.15)

Also, the resistance coefficient is: f5

ΔP A ð1=2Þρu2 L

(6.16)

The comprehensive heat transfer coefficient, β, is introduced to investigate the comprehensive effects of enhanced heat transfer by [4]: β 5 j=f 1=2

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(6.17)

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21,000

water (havg)

h (W m–2 K–1)

20,000 19,000 18,000 17,000 16,000 15,000

1

2

3

4

5

6

Number of cells (million)

FIGURE 6.2 Grid independency examination for the rectangular microchannel [2].

where β is used to judge whether the increase of heat-exchange capability is larger than the increase of flow resistance at the same pressure drop. In this case, the FVM with the semiimplicit method for pressurelinked equation consisted (SIMPLEC) algorithm is used to model pressurevelocity coupling using ANSYS-FLUENT software. The prescribed convergence criterions of velocities and continuity are 1024 , and for energy, it is set to be 1026 . As reported in Ref. [2], the hexagonal structured grid is used, a grid of 1000 3 104 3 20 was applied for the computational domain. To identify the grid independence, four mesh numbers of structured grids are considered for the Reynolds number Re 5 150. Fig. 6.2 shows the results of heat transfer coefficient h obtained from different grid sizes. Because the relative differences of calculated h are less than 0.1% over cell number 4 and 5.5 million cells, so 5 million cells is considered appropriate for all the computational simulations. Also, to validate the accuracy of this modeling, the experimental data of ΔP from Ref. [5] was compared with the simulation of the current model. The results are in excellent agreement and are shown in Fig. 6.3. Sakanova et al. [6] considered constant numbers of Eq. (6.1) as 25, 50, and 75 for A, and 250 and 550 for λ, respectively. In this case, the effect of these numbers on the comprehensive heat transfer coefficient (β) is investigated. For this aim, central composite design (CCD) is applied, which resulted in nine different geometries for three levels for each of these two factors as shown in Table 6.2. Fig. 6.4 demonstrates the streamlines for the nine proposed cases at a flow rate u 5 1 m s21. It can be observed that the maximum velocity in the x direction increases with the increment of wave amplitude at the same wavelength. Furthermore, the vortex is more probably to happen at the larger A and the smaller λ. The obtained β values for the nine proposed cases are also reported in

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Pressure gradient (Pa m–1)

1.00E+08 1.00E+07

Experimental data Simulation data

1.00E+06 1.00E+05 1.00E+04 1.00E+03 1.00E+02 1.00E+01 10

100

1000

10,000

Hydraulic diameter of circular tube (μm)

FIGURE 6.3 Validation of the model accuracy with the data available in Ref. [5]. TABLE 6.2 Variation of β for different central composite design (CCD) cases when Re 5 150. λ

β

Case number

A

1

10

100

0.046034

2

10

550

0.0422877

3

10

1000

0.0428088

4

25

100

0.0414412

5

25

550

0.0477744

6

25

1000

0.0476693

7

40

100

0.0556904

8

40

550

0.0554782

9

40

1000

0.0417381

Table 6.2. It can be seen that the highest heat transfer can be achieved over cases 7 and 8, and the vortex can be observed in case 7. Fig. 6.5 compares the (1) convective heat transfer coefficient and (2) pressure drop of the wavy-enhanced microchannel and straight microchannel. It is clear that the convective heat transfer coefficient of the wavy channels is significantly improved compared to the straight channel. Moreover, it can be observed that the higher heat transfer coefficients occurs for the cases of A 5 40, λ 5 100, A 5 40, λ 5 550, and A 5 25, λ 5 100,. Furthermore, the wavy channel has higher pressure drop than the rectangular channel.

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X velocity

X velocity

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Case 1

A = 10 μm λ = 100 μm

X velocity

Case 2

A = 10 μm λ = 550 μm

X velocity

X velocity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Case 3

A = 10 μm λ = 1000 μm

X velocity

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 2.2 2.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Case 4

A = 25 μm λ = 100 μm

X velocity –0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 2.2 2.4

Case 7

A = 40 μm λ = 100 μm

Case 5

A = 25 μm λ = 550 μm

X velocity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

Case 8

A = 40 μm λ = 550 μm

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Case 6

A = 25 μm λ = 1000 μm

X velocity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Case 9

A = 40 μm λ = 1000 μm

FIGURE 6.4 Velocity vector streamlines along the x direction for the typical section of the nine different geometries of microchannels obtained by response surface methodology (RSM) [2].

6.1.2 Case 2: Nanofluids in the heating process of an heating, ventilation, and air conditioning system model To examine NF application in a heating, ventilation, and air conditioning (HVAC) system, the following process is used. Three kinds of aqueous NFs were prepared using TiO2 (85 nm) and SiO2 (12 nm) and carbon nanotube (CNT) (6.2 nm diameter and 15 μm length) nanoparticles. Solutions with the desired volume concentration of nanoparticles were prepared by mixing the appropriate amounts of distilled water and nanoparticles. The mixtures were stirred for 15 min and sonicated for 3 h. Fig. 6.6 shows a sample of prepared

Nanofluids

100,000 90,000 80,000 70,000 60,000

case 1 case 2 case 5 case 4 case 7 case 8 rectangular channel

(B)

case 3 case 6 case 9

50,000

Δ Pa

havg (W m–2 K–1)

(A)

40,000

106

case 1 case 4 case 7 rectangular

case 2 case 5 case 8 channel

case 3 case 6 case 9

105

30,000 104

20,000 100

125

150

Re

175

200

225

100

125

150

175

200

225

Re

FIGURE 6.5 Comparison of wavy-enhanced microchannel and straight microchannel for convective heat transfer coefficient (A) and pressure drop (B), respectively [2].

6.1 Nanofluids for cooling and heating

291

FIGURE 6.6 Three types of prepared nanofluids [7].

NFs. To obtain the characterization of NFs, the morphology and microstructure of samples were explored by scanning electron microscopy (SEM). As presented in Fig. 6.7, the average particle sizes are 85 and 12 for SiO2 and TiO2, respectively, and 6.2 nm diameter and 15 μm length for CNT. Chapter 2, Mathematical analysis of nanofluids, showed the physical properties of these The  nanoparticles.  effective density ρn f , the effective heat capacity ρCp n f , and the thermal expansion ðρβ Þn f of the NF are as described in Chapter 2, Mathematical analysis of nanofluids [7]. A simple setup is designed to find the effect of NF in a heating model system considering the amount of total dissolved solids (TDS) as shown in the Fig. 6.8. As seen, a double glazing barrel is used for the heated area model where the NF passes between its walls and heats the inner space containing air. Moreover, a high-temperature bath circulator comprised of a pump, heater, and NF tank with a monitor is used in the experiment setup. More information is summarized in Table 6.3. Since TDS indicates the ions in water such as magnesium (Mg21), calcium (Ca21), and carbonate (CO322) in the presence of heating, they will sediment as magnesium bicarbonate (Mg(HCO3)2) or calcium bicarbonate (Ca(HCO3)2), which are undesirable for the pipes. Thus the aim is to retain the TDS at low value to prevent the deposits. As seen in Fig. 6.9A, by heating the NF during the time, a maximum TDS value of SiO2 will be lower than TiO2 and CNT. Also, Fig. 6.9B shows that the CNT has lower TDS values, but during the experiments, they settle more, so it is not suitable for HVAC applications. Furthermore, CNTs have smaller specific heat and are not suitable for heat transfer applications. TiO2 was reported as the best nanoparticle for HVAC applications due to its low energy consumption [7].

Nanofluids

FIGURE 6.7.

Scanning electron microscopy (SEM) pictures of (A) SiO2, (B) TiO2, and (C) carbon nanotube (CNT) nanoparticles [7].

6.1 Nanofluids for cooling and heating

293

FIGURE 6.8 Experimental setup and its schematic diagram [7].

TABLE 6.3

Bath circulator characterization [7].

Company

WiseCircu

Dimensions (mm)

330 3 185 3 385

Heater (W)

800

Capacity (Lit.)

6 

Temperature range ( C)

0100



Temperature accuracy ( C)

6 0.1

Sensors

PT100 21

Pump flow rate (Lit min )

5

Display

Digital LCD Display with Backlight Function

Weight (kg)

11

Power supply

AC 110/220 V, 50/60 Hz

Nanofluids

FIGURE 6.9 Effect of nanoparticles type on total dissolved solid (TDS) (A) during the time and (B) versus nanofluid temperature [7].

295

6.2 Nanofluids in nuclear engineering

6.2 Nanofluids in nuclear engineering The second application of NF presented here is in the cooling process of nuclear rods. The following two geometries are introduced in which heat transfer from tubes by NF is investigated.

6.2.1 Case 1: Turbulent nanofluids flow in pressured water reactor In this case, as shown in Fig. 6.10, an arrangement of fuel rods for a pressurized water reactor (PWR) is considered, which is cooled by Al2O3water NFs. Finding the best nanoparticles and rod diameters/ distance are desired outcomes in this case to reach the maximum heat transfer through Nusselt number maximization [8]. The threedimensional turbulent flow is simulated numerically and the dimensional governing equations are [8,9]: Continuity equation: @ ðρui Þ 5 0 @xi

ði 5 1; 2; 3Þ

(6.18)

Momentum equations:

      @uj @  @p @ @ui @

ρui uj 5 2 1 μ 1 2ρu0i u0 j i 5 1; 2; 3 and i 6¼ j 1 @xj @xi @xj @xi @xj @xj

(6.19) Energy equation:

  @ @ @T ðρui TÞ 5 ðΓ 1 Γ t Þ ði 5 1; 2; 3Þ @xi @xi @xi

where Γ 5 μ=Pr and Γ t 5 μt =Prt and   @uj @ui 0 0 1 ρui u j 5 μt @xj @xi

(6.20)

(6.21)

where turbulent viscosity can be defined by: μt 5 ρcμ k2 =ε

(6.22)

In this case, the RNG kε model (using ANSYS-FLUENT) is selected as the turbulence model and the transport equations for this model are:   @ @ @ @k ðρkÞ 1 ðρkui Þ 5 αk μeff (6.23) 1 Gk 1 Gb 2 ρε 2 YM 1 Sk @t @xi @xj @xj

Nanofluids

(A)

(B) Reactor core

Subchannel Gap Fuel rod

Fuel rod

Fluid

7.1342 mm

7.1342 mm

Fluid

Fluid

Control volume Fuel assembly

FIGURE 6.10

Control volume between fuel rods filled by nanofluids [8].

4.75 mm

297

6.2 Nanofluids in nuclear engineering

TABLE 6.4 volume [8].

Thermal properties of Al2O3water nanofluid at different fraction

Volume of fraction (%)

Thermal conductivity (W m21 K21)

Specific heat (kJ kg21 K21)

1.34

0.675

4.133

8.51

2.78

0.745

4.084

12.33

4.33

0.810

4.012

16.21

Pr

and

  @ @ @ @ε ε ε2 ðρεÞ 1 ðρεui Þ 5 αε μeff 1 C1ε ðGk 1 C3ε Gb Þ 2 C2ε ρ 2 Rε 1 Sε @t @xi @xj @xj k k (6.24)

where Gk and Gb are the generation of turbulence kinetic energy due to mean velocity gradients and the generation of kinetic turbulent energy due to buoyancy forces, respectively. For this equation constants are considered as C1ε 5 1.42, C2ε 5 1.68 [9]. The thermal properties of Al2O3water NF can be found in Table 6.4 or by using the following equations (see also Chapter 1, Introduction to nanofluids, and Chapter 2, Mathematical analysis of nanofluids): 

ρnf 5 ð1 2 ϕÞρbf 1 ϕρp     ρcp nf 5 ð1 2 ϕÞ ρcp bf 1 ϕ ρcp p   μnf 5 123ϕ2 1 7:3ϕ 1 1 μbf   knf 5 4:97ϕ2 1 2:72ϕ 1 1 kbf 

(6.25) (6.26) (6.27) (6.28)

These types of dynamic viscosity (Eq. 6.27) and thermal conductivity (Eq. 6.28) were proposed by Wang et al. (1999) and HamiltonCrosser (1962), respectively, and are introduced in Ref. [9]. For heat transfer analysis, the heat transfer coefficient can be obtained by: q TðzÞW 2 TðzÞb ð 1 L hav 5 hðzÞdz L 0

hðzÞ 5

(6.29) (6.30)

Finally, the nondimensional average Nusselt number (Nu) over the wall can be calculated by: Nuav 5



hav D k0

Nanofluids

(6.31)

298

6. Nanofluid analysis in different applications

Also, Nazififard et al. [9] reported that Nu can be determined by Nu 5 0:085Re0:71 Pr0:35 ; 104 # Re # 5 3 105 ; 6:6 # Pr # 13:9; 0 , ϕ , 10%

(6.32)

The SIMPLEC algorithm is used to deal with the pressurevelocity coupling and the second-order upwind scheme is performed to discretize the convection terms due to its better accuracy. Details of numerical modeling settings are presented in Ref. [8]. Fig. 6.11 displays the mesh independency study for the six different mesh numbers detailed in Table 6.5. Mesh types are QUADS and HEXAS cell kinds as described in details in Table 6.5. As observed in this figure, mesh accuracy is suitable for about 600,000 grid numbers, and also Y 1 is in acceptable range (below 10.0). Fig. 6.12 validates the outcomes of Nusselt number with the available experimental data by Pak and Cho [10]. After this validation, CCD is applied to define different geometries based on a and r variable parameters (half of the rod’s distance and diameter). CCD resulted in nine critical designs as presented in Table 6.6. Figs. 6.13 and 6.14 indicate the temperature and velocity of the NF between the rods in the suggested geometries, respectively. This is totally evident that when the rod’s diameter is larger (or distance between the rods is smaller), thermal/velocity profiles have significant effects on each other. Fig. 6.15 reveals that case 6 has the maximum Nusselt number among the tested cases and Fig. 6.16 confirms that increasing ϕ causes an increase in heat transfer and Nusselt number.

6.2.2 Case 2: Nanoparticles around the heated cylinder in a wavy-wall enclosure In the second case of cooling tubes or cylinders, as presented in Fig. 6.17 [11,12], the enclosure is made of two horizontal flat walls and two vertical wavy walls with inside circular cylinders as the rods. It was reported that the best position of the heated cylinder is the center of the cavity as fixed in the current case [12]. The space between the rod and the enclosure walls is filled by Cuwater NF. The horizontal walls are kept adiabatic while the wavy walls as well as the inner cylinder are kept at constant temperatures (the temperature of inner cylinder is higher). The wavy-wall profile for the left wall is [11]:      1 π 2πY 2 λ 1 λ 1 2 sin 1 ; 0#Y#A (6.33) X5 2 2 A

Nanofluids

(A)

(B) 16 15

155

14 13

Nu

Y+

150 12 11 145 10 9 140 1

2

3

4

5

8

6

1

2

Mesh number

FIGURE 6.11

TABLE 6.5

3

4

5

6

Mesh number

Results of different generated meshes (Table 6.5). (A) Nusselt number and (B) Y 1 [8].

Different mesh and grid numbers for mesh independency study [8].

No. mesh type

1

2

3

4

5

6

NODES

244,621

469,044

623,574

893,850

961,116

1,371,378

QUADS

51,624

72,818

84,200

102,004

105,850

127,740

HEXAS

218,160

431,970

580,800

842,160

907,500

1,306,800

Y1

14.9778

11.1223

9.13261

9.13261

9.13261

9.13261

300

6. Nanofluid analysis in different applications

600 CFD_1.34% CFD_2.78% Exp_1.34% Exp_2.78%

500

Nu

400

300

200

100

0

20,000

40,000

60,000

80,000

Re

FIGURE 6.12 Validation of the results by experimental work [8] by Pak and Cho [10]. TABLE 6.6 Different geometries proposed by central composite design (CCD) for rod diameter and distance [8]. Geometry no.

a (mm)

r (mm)

1

6

4

2

8

3

3

8

4

4

7

3

5

7

5

6

6

5

7

8

5

8

6

3

9

7

4

And the right-side wall is symmetric of the left side by y-axis as seen in the Fig. 6.17. Here, λ indicates the ratio of a/W, and A is the aspect ratio 5 H/W 5 1.5 5 4D, where W is the average width of the enclosure. The governing equations are: @u @v 1 50 @x @y

Nanofluids

(6.34)

6.2 Nanofluids in nuclear engineering

FIGURE 6.13

Different temperature profiles for nanofluids cooling the fuel rods [8].



 2  @u @u @p @ u @2 u 1v 1μ ρf u 1 52 @x @y @x @x2 @y2

 2  @v @v @p @ v @2 v 1μ 1 52 ρf u 1 v @x @y @y @x2 @y2

    2 φ 2 φc ρp 2 ρf0 g 1 1 2 φc ρf0 ðT 2 Tc Þg u

301

(6.35)

(6.36)

 2  ðρcÞp @T @T @ T @2 T 1v 5α 1 1 2 2 @x @y @x @y ðρcÞf "

 ( 2  2 )# @φ @T @φ @T DT @T @T 1 3 DB 1 1 @x @x @y @y @x @x Tc





(6.37)

Nanofluids

FIGURE 6.14 Different velocity profiles for nanofluids cooling the fuel rods [8]. 1100 1000 900

Nu

800 700 Geo. 1 Geo. 2 Geo. 3 Geo. 4 Geo. 5 Geo. 6 Geo. 7 Geo. 8 Geo. 9

600 500 400 300 200 20,000

40,000

60,000

Re

FIGURE 6.15 nanofluid [8].

Effect of different geometries on Nusselt number for 4.33% Al2O3

580 1.34% Nanofluid 2.78% Nanofluid 4.33% Nanofluid

800

1.34% Nanofluid 2.78% Nanofluid 4.33% Nanofluid

560

Nu

Nu

600

540

400

200

520 20,000

40,000

Re

FIGURE 6.16

60,000

20,000

40,000

Re

Effect of different nanoparticle volume fraction on Nusselt number for geometry number 8 [8].

60,000

304

6. Nanofluid analysis in different applications

FIGURE 6.17 Enclosure geometry and boundary conditions [11].

2

  2

@φ @φ @ φ @2 φ DT @ T @2 T 1v 5 DB u 1 2 1 1 2 @x @y @x2 @y @x2 @y Tc

(6.38)

Continuity, momentum under Boussinesq approximation, and energy equations for the laminar and steady state natural convection in a twodimensional form were presented in Eqs. (6.34)(6.38). The boundary conditions are defined as: T 5 Th ; φ 5 φh on the inner cylinder boundary T 5 Tc ; φ 5 φc on the outer wavy boundaries @T=@n 5 @φ=@n 5 0 on two flat insulation boundaries ψ 5 0 on all the solid boundaries; no slip conditions

(6.39)

Here, the stream function and vorticity are defined as: u5

@ψ ; @y

v52

@ψ ; @x

w5

@v @u 2 @x @y

(6.40)

The nondimensional variables used in this case are considered as: X5

x y wL2 ψ T 2 Tc φ 2 φc ; Y5 ; Ω5 ; Ψ 5 ; Θ5 ; Φ5 L L α α Th 2 Tc φh 2 φc

(6.41)

By applying these dimensionless parameters the equations will be:      2  @Ψ @Ω @Ψ @Ω @ Ω @2 Ω @Θ @Θ 2 2 Nr 1 5 Pr (6.42) 1 Pr Ra @Y @X @X @Y @X2 @Y2 @X @X

Nanofluids

6.2 Nanofluids in nuclear engineering

305

     2  2 ! @Ψ @Θ @Ψ @Θ @2 Θ @2 Θ @Φ @Θ @Φ @Θ @Θ @Θ 2 5 2 1 Nt 1 1 Nb 1 @Y @X @X @Y @X2 @Y2 @X @X @Y @Y @X @Y

     2  @Ψ @Φ @Ψ @Φ 1 @2 Φ @2 Φ Nt @ Θ @2 Θ 2 1 2 1 1 5 @Y @X @X @Y Le @X2 @Y @Y2 Nb Le @X2



@2 Ψ @2 Ψ 1 52Ω @X2 @Y2

(6.43) (6.44) (6.45)

where the thermal Rayleigh number, the thermophoretic parameter, the buoyancy ratio number, the Brownian motion parameter, Prandtl number, and Lewis number of the NF are defined as in as [13]. The boundary conditions in dimensionless forms are (Fig. 6.17): Θ 5 1; Φ 5 1 on the inner cylinder boundary Θ 5 0; Φ 5 0 on the outer wavy boundaries @Θ=@n 5 @Φ=@n 5 0 on two flat insulation boundaries Ψ 5 0 on all solid boundaries

(6.46)

The local Nusselt number on the cold circular wall can be stated as: Nuloc 5 2

@Θ @n

(6.47)

where n is the direction normal to the outer cylinder surface. The average number on the cold circular wall is calculated by: ð 1 L Nuave 5 Nuloc ðξ Þdξ (6.48) L 0 By using the FEM commercial code (FlexPDE), the solution of the governing equations is achieved and the local/average Nusselt numbers for wavy-wall and cylinder wall are calculated when Pr 5 7, Ra 5 10,000, Nt 5 0.5, Nb 5 0.5, Le 5 2, and Nr 5 3. Table 6.7 shows the nine different cases proposed by CCD for different values of A and λ. In this table, Nuw and NuC denote the average Nusselt number on the wavy-wall and cylinder wall, respectively. Figs. 6.186.20 show the difference for isothermal lines, streamlines, and nanoparticle volume fraction lines. It must be pointed out that in all nine cases, cylinder diameter is constant (D 5 0.5). These figures show that greater nanoparticle concentration values (like heat lines) are found under the cylinder due to natural convection and gravity. The main reason nanoparticles move down is due to the gravity and natural convection of fluid as demonstrated in Eq. (6.36). Fig. 6.21 depicts the average Nusselt numbers for these cases, which shows that the Nu for the cylinder wall is smaller than for the wavy wall for all the cases and cases 7 and 9 have the maximum Nu values [11].

Nanofluids

306

6. Nanofluid analysis in different applications

TABLE 6.7 Proposed cases by central composite design (CCD) and calculated Nusselt numbers for walls when Pr 5 7, Ra 5 10,000, Nt 5 0.5, Nb 5 0.5, Le 5 2, and Nr 5 3 [11]. Case number

λ

A

Nuw

NuC

1

0.4

2.5

13.41

10.82

2

0.25

2.5

13.35

10.57

3

0.1

2.5

12.57

9.81

4

0.1

1

8.53

6.75

5

0.4

1

9.79

7.48

6

0.25

1

8.91

6.85

7

0.25

4

14.71

11.76

8

0.1

4

13.72

10.74

9

0.4

4

14.69

12.12

6.3 Nanofluids in renewable energies One of the most common applications of NFs is in renewable energies such as solar energy. NF application and its treatment depend on the type of solar collector used. The following two main types of collectors using NFs are investigated.

6.3.1 Case 1: Nanofluid-based Concentrating Parabolic Solar Collector As depicted in Fig. 6.22, consider a Nanofluid-based Concentrating Parabolic Solar Collector (NCPSC) under constant heat flux and variable magnetic field effect. For the magnetic source, a magnetic wire is located vertically on the xy-plane at the point ða; bÞ. The collector is filled with Fe3O4water NF and the components of the magnetic field intensity ðH x ; Hy Þ and the magnetic field strength H are measured in the following form [14]: Hx 5

  γ 1 y2b 2 2 2π ðx2aÞ 1 ðy2bÞ

Hy 5 2 H5

γ 1 ðx 2 a Þ 2π ðx2a Þ2 1 y2b 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ 1 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hx 1 Hy 5  2ffi 2π 2 ðx2a Þ 1 y2b

Nanofluids

(6.49) (6.50)

(6.51)

6.3 Nanofluids in renewable energies

307

FIGURE 6.18 Isotherm lines of the nine central composite design (CCD) proposed cases when Pr 5 7, Ra 5 10,000, Nt 5 0.5, Nb 5 0.5, Le 5 2, and Nr 5 3 [11].

where γ indicates the magnetic field strength at the source and (a,b) is the position where the source is located [in this case (0,0)]. Constant heat flux, constant low temperature Tc, and insulated due to symmetric are the boundary condition of the outer wall, inner wall and two above side walls, respectively. The governing equations for a steady and incompressible two-dimensional laminar NF flow are [15]: @v @u 1 50 @y @x

Nanofluids

(6.52)

308

6. Nanofluid analysis in different applications

FIGURE 6.19 Streamlines of the nine central composite design (CCD) proposed cases when Pr 5 7, Ra 5 10,000, Nt 5 0.5, Nb 5 0.5, Le 5 2, and Nr 5 3.

   2  @u @u @P @ u @2 u 1u 1 μnf ρnf v 1 52 2 σnf B2y u 1 σnf Bx By v (6.53) @y @x @x @x2 @y2    2  @v @v @P @ v @2 v 52 1 ρnf β nf γ ðT 2 Tc Þ 1 μnf ρnf v 1 u 1 1 σnf Bx By u 2 σnf B2x v @y @x @y @x2 @y2

(6.54)

Nanofluids

6.3 Nanofluids in renewable energies

309

FIGURE 6.20 Nanoparticle volume fraction of the nine central composite design (CCD) proposed cases when Pr 5 7, Ra 5 10,000, Nt 5 0.5, Nb 5 0.5, Le 5 2, and Nr 5 3 [11].



ρCp

0



1

0

1

  @v @T 1 u @T A 5 knf @@ T 1 @ T A 1 σnf uBy 2vBx 2 2 2 @y @x @x @y 8 9  2  2 = < @u2 @v @v @u 1 1 μnf 2 12 1 : @x @x @y @x ; 2

nf

2

By 5 μ0 Hy (6.55)

Nanofluids

310

6. Nanofluid analysis in different applications

nd Nr = 3 [11] 16 Wavy wall Cylinder wall

14

Average Nu

12 10 8 6 4 2 0

0

1

2

3

4

5 6 7 Case number

8

9

10

FIGURE 6.21 Average Nusselt number for described walls and different cases when Pr 5 7, Ra 5 10000, Nt 5 0.5, Nb 5 0.5, Le 5 2, and Nr 5 3 [11].

FIGURE 6.22 Geometry [14], boundary conditions, and generated mesh considered in this study [14].

Nanofluids

6.3 Nanofluids in renewable energies

311

where ρnf, (ρCp)nf αnf, β nf, μnf, knf, and σnf are defined as presented in Chapters 1, Introduction to nanofluids, and Chapter 2, Mathematical analysis of nanofluids. By introducing the following dimensionless variables: T 2 TC uR vR a b y x ; U5 Θ 5  00 ; V5 ; a5 ; β5 ; Y5 ; Χ 5 ; α α R R R R q R=kf f f

P5





(6.56)

  H; Hx ; Hy pR γ ; ; H0 5 H ða; 0Þ 5  2 ; H; Hx; Hy 5 2πjbj H0 ρf αf 2

Eqs. (6.52)(6.55) can be transferred to dimensionless forms: @V @U 1 50 (6.57) @Y @X " #  μnf =μf @2 U @U @U @2 U V 1U 51 Pr 1 @Y @X @X2 @Y2 ρnf =ρf " # (6.58) @P σnf =σf 2 2 ; 2 Ha Pr Hy U 2 Hx Hy V 2 @X ρnf =ρf " #  μnf =μf @2 V @V @V @2 V V 1U 5 Pr 1 @Y @X @Y2 ρnf =ρf @X2 " # " # β nf σnf =σf 2 @P 2 2 Ha Pr 2 Hy V 2 Hx Hy U 1 Ra Pr Θ; @Y ρnf =ρf βf 2

(6.59)

3

knf  2  @Θ @Θ 6 @2 Θ kf 7 @ Θ 7 1V 56 U 1 @X @Y 4ðρCp Þnf 5 @X2 @Y2 ρC ð p Þf 2 3 σnf

6 σ f 7 2 7 1 Ha Ec6 4ðρCp Þnf 5 UHy 1VHx ðρCp Þf 2 3 (   μnf  2  ) 6 μf 7 @U 2 @V @U @V 2 6 7 1 2 1 Ec4 12 1 ðρCp Þnf 5 @X @Y @X @Y ðρCp Þf 2

(6.60)

Nanofluids

312

6. Nanofluid analysis in different applications

With dimensionless parameters:

00 sffiffiffiffiffi q R gβ f R3 kf υf σf   ; ; Pr 5 ; Ha 5 Rμ0 H0 Raf 5 α μ αf υ f f f

αf μf Ec 5 h  q00 R i ρCp f kf R2

(6.61)

The stream function, vorticity, local, and average Nusselt numbers are described as: Ω5

ωR2 ψ @ψ @ψ @u @v ; u5 ; ω52 1 ; ψ5 ; ν 52 αf @x @y @y @x αf

(6.62)

On the cold side walls Θ 5 0:0 αf 5 0:0

On all walls

On all the insulated walls

@Θ 5 0:0 @n

On the heat flux

@Θ 5 2 1:0 @n

Nuloc 5

Nuave 5

  knf 1 j k f Θ Ri

1 πRi

(6.63)

(6.64)

ðπ Nulox rdθ

(6.65)

0

Fig. 6.23 validates the results of the FEM for a half-circular geometry by comparison with the results of Chamkha et al. [16]. As seen, the values of temperature and streamlines are in good agreement and completely symmetric, respectively. Furthermore, the results show good agreement with the FDM outcomes presented by Rashad et al. [17] for the average Nusselt number for different Hartmann numbers. Fig. 6.24 shows the effect of nanoparticle volume fraction (ϕ) on temperature and streamlines. Moreover, to find a reliable treatment for the effect of this parameter on local Nusselt number, Fig. 6.25 is presented. Based on the results, it can be concluded that increasing the ϕ enhances the heat transfer.

Nanofluids

FIGURE 6.23 Validation code for temperature (A) and streamlines (B) compared to Chamkha et al. [16].

FIGURE 6.24 streamlines [14].

Effect of nanoparticle volume fraction (ϕ) on the temperature and

314

6. Nanofluid analysis in different applications

FIGURE 6.25 Effect of ϕ on the local Nusselt number [14].

FIGURE 6.26 (A)Schematic of the nanofluid-based direct absorber solar collector and (B) sample mesh generated and boundary condition [18].

6.3.2 Case 2: Wavy direct absorption solar collector filled with different nanofluids As depicted in Fig. 6.26A, a sinusoidal direct absorption solar collector (DASC) with amplitude of wave Am and number of wave λ is considered and filled with Al2O3water NF. To find the effect of the

Nanofluids

315

6.3 Nanofluids in renewable energies

cross-section geometry of a channel (Am and λ), nine different cases based on the CCD are modeled as shown in Table 6.8. L 5 1 and H 5 0.2 are the length and average height of this physical model, respectively. Other assumptions of the problem are given in Refs. [18,19]. A sample generated mesh for this case is shown in Fig. 6.26B. The governing equations for steady, laminar, and natural convection inside the solar collector in the form of NavierStokes and energy equation are used from Nasrin and Alim [20] as: The continuity equation: @u @v 1 50 @x @y

(6.66)

The momentum equations in the x and y directions:    2  @u @u @p @ u @2 u ρnf u 1v 1 μnf 1 2 52 @x @y @x @x2 @y    2  @v @v @p @ v @2 v 1 μnf 1 2 1 gðρβ Þnf ðT 2 TC Þ 52 ρnf u 1 v @x @y @y @x2 @y

(6.67) (6.68)

and the energy equation: u

TABLE 6.8

 2  @T @T @T @2 T 1v 5 αnf 1 @x @y @x2 @y2

(6.69)

Different geometry parameters for the solar collector [18].

Case number

Am

λ

Average Nu

1

0.06

2.00

2.66

2

0.04

4.00

3.98

3

0.06

4.00

3.97

4

0.04

6.00

2.64

5

0.02

2.00

2.66

6

0.02

6.00

2.65

7

0.04

2.00

2.66

8

0.06

6.00

2.63

9

0.02

4.00

3.98

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316

6. Nanofluid analysis in different applications

to solve the governing equations in nondimensional forms, the following scales are used to find the nondimensional governing equations [1820]: x L y Y5 L uL U5 νf vL V5 νf

X5

(6.70)

pL2 ρf v2f T 2 Tc θ5 Tw 2 Tc P5

The nondimensional governing parameters considered in the present case are: Re 5

ρf u0 L μf

Pr 5

vf αf

Ra 5

gβ f L3 ðTw 2 Tc Þ vf α f

(6.71)

By introducing the abovementioned parameters, the nondimensional forms of the governing equations will be: @U @V 1 50 @X @Y

(6.72)

  ρf @P vnf @2 U @U @U @2 U U 1V 52 1 1 (6.73) @X @Y @Y2 ρnf @X vf @X2   ρf @P vnf @2 V @V @V @2 V Ra ð1 2 φÞðρβ Þf 1 φðρβ Þs 1V 52 1 1 θ 1 U @X @Y @Y2 Pr ρnf @Y vf @X2 ρnf β f 

U

αnf 1 @2 θ @θ @θ @2 θ 1V 5 1 @X @Y @Y2 αf Pr @X2

Nanofluids



(6.74) (6.75)

6.4 Nanofluid in industry

317

As the boundary conditions, the bottom wavy plate is assumed in a constant temperature, the top flat plate in constant heat flux due to solar radiation, and the other walls are insulated. The local Nusselt number on the wavy absorber can be stated as: Nuloc 5 2

knf @θ kf @n

(6.76)

where n indicates the direction normal to the wavy surface. The average number on the cold circular wall is calculated by: ð 1 L Nuave 5 Nuloc ðξ Þdξ (6.77) L 0 To solve the governing equations by the FEM, it is assumed that ϕ 5 0.05, Pr 5 6.2, and Ra 5 10e5; for the upper plate, due to solar radiation, constant heat flux (dθ/dx 5 21) is considered as the boundary condition; and for the bottom wavy plate and side walls constant temperature (θ 5 0) and insulated wall (dθ/dx 5 0) is assumed, respectively. After the numerical solutions for the cases of Table 6.8, Figs. 6.27 and 6.28 are depicted for streamlines and isotherm lines, respectively. It is evident that when the numbers of the waves are 24 or the wave amplitude is small, the generated vortex is extended and the number of produced vortexes is directly related to the wave numbers (Fig. 6.27). As seen in Fig. 6.28, the maximum temperature occurs on the top plate due to the solar heat flux and the main cause of the heat transfer from the top plate to the bottom plate is the temperature difference between the isotherm layers.

6.4 Nanofluid in industry The two main phenomena usually observed in industry are the condensation of NF and NF flow and heat transfer between parallel plates such as bearings, etc. In this section, these two applications are modeled and analyzed numerically.

6.4.1 Case 1: Condensation of nanofluids In this case, consider a vertical wall of temperature Tw, which is lower than the saturation temperature Tv (see Fig. 6.29, which shows that condensation occurs when steam is cooled to lower its saturation temperature). The usual assumptions of Nusselt’s theory are assumed and boundary layer estimate is validated considering small thickness of the film layer and the condense film moving downward under the

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6. Nanofluid analysis in different applications

FIGURE 6.27 Contours of streamlines for the nine different cases [18].

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6.4 Nanofluid in industry

FIGURE 6.28

Contours of isotherm lines for the nine different cases [18].

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319

320

FIGURE 6.29.

6. Nanofluid analysis in different applications

Schematic of the problem [21].

gravity effect. Turkyilmazoglu [22] reported that for a single-phase model of this problem, velocity shape factor (u*), Nusselt (Nu1), and film thickness (δ1) of NF can be obtained from Ref. [21]: 2 k 31=2 nf

kf u  5 4μnf 5

(6.78)

μf

2

knf μnf kf μf

31=4

6 7 δ 1 5 4 2 5 ρ

(6.79)

nf

ρf knf kf



knf Nu1 5 5 δ1 kf

3=4 "

ρnf ρf

#1=2 "

μnf μf

#21=4 (6.80)

where thermal properties of NFs are as defined in Chapter 2, Mathematical analysis of nanofluids. In the two-phase model, it is assumed that the nanoparticle concentration varies through the boundary layer from a preassigned gradient at the wall to a fixed concentration level at the outer edge of the boundary layer, given by φv. By this description, the hydrodynamic and thermal equations of NFs in a condense film (under the usual boundary layer approximation) are:   d du μnf (6.81) 5 2 ρnf g dy dy

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321

6.4 Nanofluid in industry

    d dT dϕ DT dT dT knf 1 50 1 ρs cps DB dy dy dy Tv dy dy

(6.82)

  d dϕ DT dT DB 1 50 dy dy Tv dy

(6.83)

with the boundary conditions as: uð0Þ 5 0; DB

Tð0Þ 5 Tw ;

duðδÞ 5 0; dy

dϕð0Þ DT dTð0Þ 1 5 0; dy Tv dy

ϕðδÞ 5 ϕv ;

TðδÞ 5 Tv

(6.84)

The above model is basically a combination of the Nusselt and Buongiorno models, taking into account thermophoretic and Brownian diffusion mechanisms. Also, the NF properties are estimated to be: μnf μf ρnf ρf



1 5 F1 ðϕÞ 1 2 52 ϕ

(6.85)

5 1 2 ϕ 1 ϕr 5 F3 ðϕÞ

(6.86)

knf 3ðk 2 1Þϕ 5 F2 ðϕÞ 511 k12 kf

(6.87)

where k 5 ks =kf ; r 5 ρs =ρf . By considering the dimensionless variables as: u5

ρf gδ2 μf

U;

θ5

T 2 Tw ; ΔT

η5

y δ

(6.88)

The governing equations can be changed to:   d du F1 5 2 F3 dη dη     ρs cps DB dϕ d dθ dθ dθ F2 1A 50 1 dη dη dη dη dη kf   d dϕ dθ 1A 50 dη dη dη

Nanofluids

(6.89)

(6.90)

(6.91)

322

6. Nanofluid analysis in different applications

and boundary conditions are: θð0Þ 5 0;

duð1Þ 5 0; dη

dϕð0Þ dθð0Þ 1A 5 0; dη dη

ϕð1Þ 5 ϕv ;

uð0Þ 5 0;

θð1Þ 5 1

(6.92)

where A 5 ðDT =DTB ÞðΔT=Tv Þ indicates the ratio of thermophoretic effects to Brownian diffusion effects of NF. Turkyilmazoglu [22] reported that Nusselt number (Nu2) and film thickness (δ2) of NF for the multiphase model can be obtained by:   F2 ðϕð0ÞÞθ0 ð0Þ 1=4 δ2 5 (6.93) F4 Nu2 5 ½F2 ðϕð0ÞÞθ0 ð0Þ

3=4

Ð1

½F4 1=4

(6.94)

where F4 5 3 0 F3 Udη. For analyzing this problem using the least square method (LSM), three coupled equations (Eqs. 6.896.91) should be solved by considering the boundary conditions as Eq. (6.92). Thus three residual functions (that must satisfy the boundary conditions) are assumed as [21]:  

η η3 uðηÞ 5 c4 η 1 2 1 c5 η 1 2 (6.95) 2 3 θðηÞ 5 η 1 c1 ηðη 2 1Þ 1 c2 η2 ðη 2 1Þ

(6.96)

ϕðηÞ 5 ϕv 1 ðc3 1 Ac1 2 AÞðη 2 1Þ 1 c3 ηðη 2 1Þ

(6.97)

By substituting the residual functions, R1(c1c5, η), R2(c1c5, η), and R3(c1c5, η), into the LSM main equation, a set of equations with five equations will appear and coefficients c1c5 will be determined after solving this system of equations. For instance, when the particle is Ag (see Table 6.9 for details), ϕv 5 0:05 and A 5 0.1: uðηÞ 5 1:4285η 2 0:7666η2 1 0:0349η3

(6.98)

θðηÞ 5 η 1 0:0859ηðη 2 1Þ 2 0:0005η ðη 2 1Þ

(6.99)

ϕðηÞ 5 0:1491 2 0:0991η 2 0:0076ηðη 2 1Þ

(6.100)

2

Fig. 6.30 (as the outcomes of single-phase modeling) shows the results of the boundary layer thickness, Nusselt number, and velocity profile, respectively. It can be observed that the condensate film thickness is reduced with the increase of the nanoparticle concentration. However, Ag displays the most significant effect of the film thickness

Nanofluids

TABLE 6.9

Thermal properties of base fluid (water) and nanoparticles [21].

Properties Heat capacitance Density Thermal conductivity

Unit 21

J kg

21

K

23

kg m

21

Wm

21

K

21

Water

Al2O3

Cu

TiO2

CuO

Ag

4179

765

385

686.2

531.8

235

997.1

3970

8933

4250

6320

10,500

0.613

40 24

401 25

8.95 25

76.5 25

Thermal expansion coefficient

K

2.1 3 10

0.85 3 10

1.67 3 10

0.9 3 10

Dynamic viscosity

Ns m22

0.001003







429 25

1.8 3 10

1.89 3 1025 

324

6. Nanofluid analysis in different applications

FIGURE 6.30 Boundary layer thickness, Nusselt number, and velocity profile for single-phase nanofluid modeling [21].

reduction. This reduction leads to increase in Nusselt number, which corresponds to enhanced heat transfer efficiency. Generally, the effect of nanoparticle concentration is greater thermal conductivity (like silver), which results in a thinner momentum boundary layer and accelerates the fluid particles within the layer. Fig. 6.31 demonstrates similar results

Nanofluids

6.4 Nanofluid in industry

325

FIGURE 6.31 Two-phase modeling of nanofluids for (A) A 5 0.1, φv 5 0:05, (B) Ag, φv 5 0:05; and (C) TiO2, φv 5 0:05 [21].

for two-phase modeling, where also the effect of constant parameter, A, on temperature and nanoparticle concentration is depicted. It is clear that when the nanoparticle volume fraction increases, heat transfer can be improved leading to lower temperature for the NFs. Boundary-layer thicknesses and Nusselt numbers for this two-phase modeling for

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326

FIGURE 6.32

6. Nanofluid analysis in different applications

Boundary layer thickness and Nusselt number for two-phase modeling

when A 5 0.1 [21].

different nanoparticles are presented in Fig. 6.32, which reveal that TiO2 and Ag have the maximum boundary layer thicknesses and Nusselt numbers, respectively [22].

6.4.2 Case 2: Heat transfer of nanofluids between parallel plates For the second case of industrial application, consider NFs rotating between two horizontal plates around the y-axis with a constant velocity Ω. The distance between the plates is constant as L. Also, a uniform transverse magnetic field with intensity B is applied through the plates. The governing equations for mass, momentum, energy, and mass transfer in two-phase flow of the NF are [21]: @u @v @w 1 1 50 @x @y @z

Nanofluids

(6.101)

327

6.4 Nanofluid in industry

   2  @u @u @p @ u @2 u 1v 1 2Ωw 5 2 1μ ρf u 1 2 2 σB2 u (6.102) @x @y @x @x2 @y    2  @v @v @p @ v @2 v 1μ 1 2 ρf u 1 v 52 (6.103) @x @y @y @x2 @y    2  @w @w @ w @2 w 1v 2 2Ωw 5 μ 1 (6.104) 2 σB2 w ρf u @x @y @x2 @y2  2  @T @T @T @ T @2 T @2 T 1v 1w 5α 1 2 1 2 u @x @y @z @x2 @y @z 2 8 9 3