NUMERICAL INVESTIGATIONS OF HEAT AND MASS TRANSFER IN A SATURATED POROUS CAVITY WITH SORET AND DUFOUR EFFECTS

Transcription

1 NUMERICAL INVESTIGATIONS OF HEAT AND MASS TRANSFER IN A SATURATED POROUS CAVITY WITH SORET AND DUFOUR EFFECTS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy In the Faculty of Engineering and Physical Sciences 2012 KHALED ABDULHUSSEIN JEBEAR AL-FARHANY SCHOOL OF MECHANICAL, AEROSPACE AND CIVIL ENGINEERING

2 Table of Contents Table of Contents TABLE OF CONTENTS... 2 LIST OF FIGURES... 8 LIST OF TABLES ABSTRACT DECLARATION COPYRIGHT STATEMENT ACKNOWLEDGEMENTS PUBLICATIONS AND SUBMITTED PAPERS NOMENCLATURE ACRONYMS Chapter Introduction Background Objective of the research Outline of the thesis

3 Table of Contents Chapter Literature Review Darcy and extended Darcy studies in porous medium Inclined porous medium Variable porosity Conjugate problems Double-diffusive studies Soret and Dufour effects Conclusions from literature review Darcy and non-darcy flow in porous cavities Inclined porous medium Variable porosity Conjugate problems Double-diffusive Soret and Dufour effects Conclusions Chapter Basic Models of Flow in a Porous Medium and the Governing Equations Introduction Continuity equation for porous media Momentum equations for porous media Darcy s law Darcy-Forchheimer s law

4 Table of Contents Darcy-Brinkman s law The generalized model Heat transfer Mass transfer Non-dimensional parameters The model Chapter Numerical Implementation Introduction ALFARHANY code Solution Procedure The alternating-direction-implicit technique (ADI) Grid Convergence criteria Chapter Validation Studies Introduction Validations of convective heat transfer in porous cavities cases Validations of conjugate heat transfer cases Validations of natural convective heat transfer in inclined cavity cases Validations of variable porosity effects on heat transfer cases Validations of double-diffusive natural convection heat and mass transfer cases

5 Table of Contents 5.7. Validations of double-diffusive natural convective heat and mass transfer in inclined cavity cases Validations of natural convection of heat and mass transfer with Soret effects in porous cavity cases Conclusions Chapter Conjugate Natural Convective Heat Transfer in Porous Enclosures Introduction Mathematical formulation: Solution procedure Results Correlation for the average Nusselt number on the left porous wall interface Conclusions Chapter Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium Introduction Mathematical formulation of the model Initial boundary conditions Solution procedure Results Variable porosity effects Wall thickness effects Thermal conductivity ratio effects

6 Table of Contents Lewis number and the buoyancy ratio effects Conclusions Chapter Convective Heat and Mass Transfer in Inclined Porous Cavities Introduction Mathematical modeling Solution procedure Results and conclusions of the square porous cavity cases Results Conclusions Results and conclusions of the rectangular porous cavity cases Results Conclusions Chapter Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Introduction Mathematical formulation of the model: Initial condition Solution procedure Results Soret parameter effects Positive Soret parameter effects Negative Soret parameter effects

7 Table of Contents Dufour parameter effects Conclusions.198 Chapter Conclusions and Future Studies Conclusions Future studies Appendix A A. Journal/Conference Papers Appendix B B. Numerical Methods B.1 Solution algorithm B.2. Discretization and solution of a transport equation B.2.1. Spatial discretization B.2.2. Temporal discretization B.2.3. Calculation of gradients B.2.4. Linearization B.3. Pressure-velocity coupling B.4. Under-relaxation Appendix C C. Post processing BIBLIOGRAPHY

8 List of Figures List of Figures Figure 3-1 The geometry of the model Figure 4-1 Show the first and second step in ADI process Figure 4-2 The variations of the average Nusselt number on the left wall with mesh sizes Figure 4-3 The convergence of the solutions Figure 5-1 The geometry of the model Figure 5-2 The geometry of the model of Saeid [82] Figure 5-3 Comparison of present results with Saeid [82] results for isotherms (left), and Streamlines (right) by using Darcy flow at Ra*=1000, a-(d=0.2, Kr=1), and b-(d=0.1, Kr=10) Figure 5-4 The geometry of the model of Moya et al. [49] Figure 5-5 Comparison of (a) Streamlines and (b) isotherms lines for D=3, Ra*=100 and =10 with the work of Moya et al. [49] work Figure 5-6 The geometry of the experimental work of Seki et al. [22] Figure 5-7 The geometry of the model of Nithiarasu [97]

9 List of Figures Figure 5-8 A comparison of the streamlines, isotherms and iso-concentration lines for Da=10-6, Ra=2*10 8, Le=2, and =0.6 at a-(n=- 0.9), and b-(n=- 1.5) with Nithiarasu s [97] work Figure 5-9 The geometry of the model of Chamkha and Al-Mudhaf [53] Figure 5-10 A comparison of streamlines (left), isotherms (middle) and iso-concentration lines (right) for A=2, Da=10-4, Ra=10 5, =45, Pr=10, Le=10 and N=10 with the work of Chamkha and Al-Mudhaf [53] Figure 5-11 A comparison of Streamlines (left), isotherms (middle) and iso-concentration lines (right) for A=2, Da=10-5, Ra=10 5, =90, Pr=7.6, =0.6, Le=10 and N=10 with the work of Chamkha and Al-Mudhaf [53]. 99 Figure 5-12 The geometry of the model of Khadiri et al. [143] Figure 5-13 A comparison of the present results with Khadiri et al. s [143] results for iso-solutes for Darcy flow at R T =200, Le=10, N=0.1 and different M Figure 5-14 A Comparison of the present results with Khadiri et al. s [143] results for Streamlines, isotherms and iso-concentrations for tricellular flow at R T =200, Le=10, N=0 and M= Figure 6-1 the geometry of the model Figure 6-2 Variation of (a) streamlines, and (b) isotherms lines with nondimensional times for Da=10-6, Ra*=1*10 2, D=0.2 and k r = Figure 6-3 Variation of (a) streamlines, and (b) isotherms with non-dimensional times for Da=10-6, Ra*=1*10 2, D=0.2 and k r = Figure 6-4 Variation of (a) streamlines and (b) isotherms with non-dimensional times for Da=10-6, Ra*=1*10 2, D=0.3 and k r = Figure 6-5 Variation of (a) streamlines and (b) isotherms with non-dimensional times for Da=10-6, Ra*=1*10 2, D=0.3 and k r = Figure 6-6 Variation of (a) streamlines and (b) isotherms with non-dimensional times for Da=10-2, Ra*=1*10 2, D=0.2 and k r =

10 List of Figures Figure 6-7 Variation of (a) streamlines and (b) isotherms with non-dimensional times for Da=10-2, Ra*=1*10 2, D=0.2 and k r = Figure 6-8 Variation of (a) streamlines and (b) isotherms lines with nondimensional times for Da=10-2, Ra*=1*10 2, D=0.3 and k r = Figure 6-9 Variation of (a) streamlines and (b) isotherms lines with nondimensional times for Da=10-2, Ra*=1*10 2, D=0.3 and k r = Figure 6-10 In the steady state, streamlines and isotherms lines for Da=10-2, Ra*=1*10 3, Pr=1.0, k r =1 at (a) D=0.0, (b) D=0.2, (c) D=0.4, and at (d) k r =10, D= Figure 6-11 Variation of average Nusselt numbers in the steady state on the interface left wall of a porous medium with wall thickness at Da=10-7, Pr= Figure 6-12 Variation of Nusselt numbers in the steady state on the interface left wall of a porous medium with Ra* for Da=10-6, Pr=1.0 and D= Figure 6-13 The normal probability of the expected normal value with the residuals Figure 7-1 The geometry of the model Figure 7-2 Streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=100, Le=0.1, N= and =0.36 at (a) variable porosity, (b) uniform porosity Figure 7-3 Streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=100, Le=10, N=- 2.0 and =0.36 at (a) variable porosity, (b) uniform porosity Figure 7-4 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Le=1, N=-2.0 and Ra*=10, 100 and 1000 at (a) variable porosity, (b) uniform porosity. 126 Figure 7-5 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra=100, N=-2.0 and Le=0.1, 1, 10 at (a) variable porosity, (b) constant porosity Figure 7-6 Variation of Nusselt and Sherwood numbers with Ra* for Da=10-6, N=

11 List of Figures Figure 7-7 Streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=100, Pr=1.0, Le=2, N=- 2.0, k r =1 at (a) D=0.2, (b) D=0.3, (c) D=0.4 and at (d) k r =0.1, D= Figure 7-8 Streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=1000, Pr=1.0, Le=1, N=+2.0, k r =1 at (a) D=0.2, (b) D=0.3, (c) D= Figure 7-9 Variation of average Nusselt and Sherwood numbers on the left wall porous interface with wall thickness at Da=10-6, Pr=1.0, Le=2, N= Figure 7-10 Variation of average Nusselt and Sherwood numbers on the left wall porous interface with wall buoyancy ratio at Da=10-6, Pr=1.0, Le=1 and D= Figure 8-1 The geometry of the model Figure 8-2 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-6, Ra*=100, Pr=1.0, =0.36, Le=10.0 and N= -1.0 with different inclination angles Figure 8-3 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-6, Ra*=100, Pr=1.0, =0.36, Le=10.0 and N= -2.0 with different inclination angles Figure 8-4 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-6, Ra*=100, Pr=1.0, =0.36, Le=0.1 and N= -1.0 with different inclination angles Figure 8-5 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-6, Ra*=100, Pr=1.0, =0.36, Le=0.1 and N= 5.0 with different inclination angles Figure 8-6 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=0.1, N= 2.0 for different inclination angles Figure 8-7 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=1.0, N= -5.0 for different inclination angles

12 List of Figures Figure 8-8 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=10.0, N= -5.0 for different inclination angles Figure 8-9 Variation in (a) average Nusselt numbers (Nu) and (b) average Sherwood (Sh) numbers with Ra* for Da=10-4, =30 o and with different Le and N Figure 8-10 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-2, Ra*=1000, Pr=1.0, =0.36, Le=10.0 and N= -1.0 with different inclination angles Figure 8-11 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-2, Ra*=1000, Pr=1.0, =0.36, Le=10.0 and N= -5.0 with different inclination angles Figure 8-12 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-2, Ra*=1000, Pr=1.0, =0.36, Le=0.1 and N= -1.0 with different inclination angles Figure 8-13 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-2, Ra*=1000, Pr=1.0, =0.36, Le=0. 1 and N= 2.0 with different inclination angles Figure 8-14 validation of average Nusselt (Nu) and Sherwood (Sh) numbers on the left porous wall with wall buoyancy ratio (N) with different Ra* at Da=10-2, Pr=1.0, Le=0.1 and =60 o Figure 8-15 Variation of average Nusselt (Nu) and Sherwood (Sh) numbers on the left porous wall with wall buoyancy ratio (N) with different Le at Da=10-2, Ra*=10 2, Pr=1.0 and =60 o Figure 8-16 (a) Stream functions, (b) isotherms and (c) iso-concentration lines for Da=10-4, Ra=5*10 6, Pr=4.5, A= 5, Le=0.1 and N= +2.0 with different angle of inclination Figure 8-17 (a) Stream functions, (b) isotherms and (c) iso-concentration lines for Da=10-4, Ra=5*10 6, Pr=4.5, A= 4, Le=0.1 and N= +2.0 with different angle of inclination

13 List of Figures Figure 8-18 (a) Stream functions, (b) isotherms and (c) iso-concentration lines for Da=10-4, Ra=5*10 6, Pr=4.5, A=2, Le=10 and N= -5.0 with different angles of inclination Figure 8-19 (a) the average Nusselt number and (b) the average Sherwood number for Da=10-4, Ra=5*10 6 with different aspect ratios, Lewis numbers and angle of inclinations Figure 8-20 (a) the average Nusselt Number and (b) the average Sherwood Number for Da=10-4, Ra=5*10 6, Pr=4.5, A=2, N = - 5 with different Lewis numbers and angle of inclinations Figure 8-21 (a) the average Nusselt Number and (b) the average Sherwood Number for Da=10-4, Ra=5*10 6, Pr=4.5, A=2, N =5 with different Lewis numbers and angle of inclinations Figure 9-1 The geometry of the model Figure 9-2 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=1.0, N= +2.0 and D f =0 with different positive Soret effects Figure 9-3 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=5.0, N= +2.0 and D f =0 with different positive Soret effects Figure 9-4 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=5.0, N= -2.0 and D f =0 with different positive Soret effects Figure 9-5 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=10, N= 3.0 and D f =0 for different Soret effects Figure 9-6 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=20, N= 3.0 and D f =0 for different Soret effects Figure 9-7 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=1, N= -1.0 and D f =0 for different Soret effects Figure 9-8 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=0.1, N= -1.0 and D f =0 for different Soret effects

14 List of Figures Figure 9-9 Variation in (a) average Nusselt numbers (Nu) and (b) average Sherwood numbers (Sh) with positive S r for Da=10-6,Ra*=100 with different Le and N Figure 9-10 Variation in (a) average Nusselt numbers (Nu) and (a) average Sherwood numbers (Sh) on the left porous wall with a buoyancy ratio (N) with different Le at Da=10-6, Ra*=100, Pr=0.71, =0.6 and S r = Figure 9-11(a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=1.0, N= +2.0 and D f = 0 with different negative Soret effects Figure 9-12 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=5.0, N= +2.0 and D f = 0 with different negative Soret effects Figure 9-13 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=5.0, N= -2.0 and D f = 0 with different negative Soret effects Figure 9-14 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=10, N= 3.0 and D f =0 for different Soret effects Figure 9-15 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=20, N= 3.0 and D f =0 for different Soret effects Figure 9-16 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=1, N= -1.0 and D f =0 for different Soret effects Figure 9-17 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=0.1, N= -1.0 and D f =0 for different Soret effects Figure 9-18 Variation in (a) average Nusselt number (Nu) and (b) average Sherwood number (Sh) with negative S r for Da=10-6,Ra*=100 with different Le and N Figure 9-19 Variation in (a) average Nusselt number (Nu) and (a) Sherwood number (Sh) on the left porous wall with buoyancy ratio (N) with different Le at Da=10-6, Ra*=200, Pr=0.71, =0.6 and S r =

15 List of Figures Figure 9-20 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-4, Ra*=500, Pr=0.71, =0.6, Le=10, N= 2.0 and D f =0 with different positive Soret effects Figure 9-21 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-4, Ra*=500, Pr=0.71, =0.6, Le=10, N= 2.0 and D f =0 with different negative Soret effects Figure 9-22 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=10, N= 3.0 and S r =-2 with different positive Dufour effects Figure 9-23 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=10, N= -3.0 and S r =2 with different negative Dufour effects Figure 9-24 Variation in (a) average Nusselt numbers (Nu) and (b) average Sherwood (Sh) numbers with positive D f for Da=10-6,Ra * =100, S r =1 with different Le and N Figure 9-25 Variation in (a) average Nusselt numbers (Nu) and (b) average Sherwood (Sh) numbers with positive D f for Da=10-6,Ra * =100, S r =-1 with different Le and N

16 List of Tables List of Tables Table 5-1 Comparison of average Nusselt number at steady state with some previous results for Darcy model (pure heat transfer, N=0) Table 5-2 Comparison of average Nusselt number at steady state with some previous results for Darcy-Brinkman model (N=0, Pr=1) Table 5-3 Comparison of average Nusselt number at steady state with some previous results for Darcy- Forchheimer model (N=0, Pr=1) Table 5-4 Comparison of average Nusselt number at steady state with some previous results for generalized model (N=0, Pr=1, Ra*=10 4 ) Table 5-5 Comparison of variations of average Nusselt Number with α for Ra*= Table 5-6 Geometry and water properties of the experimental work of Seki et al. [22] Table 5-7 Comparison of average Nusselt numbers with some previous studies on variable porosity Table 5-8 A comparison of average Nusselt and Sherwood numbers at steady stat with some previous results for the Darcy model (Le=10, N=0) Table 5-9 A comparison of average Nusselt and Sherwood numbers and maximum stream functions with Khadiri et al. s [143] results for monocellular flow at R T =200, Le=10, N=

17 List of Tables Table 7-1 Average Nusselt and Sherwood numbers at Da=10-6, Pr=1.0, D= Table 8-1 Average Nusselt and Sherwood numbers at Da=10-6, Ra*= Table 9-1 Average Nusselt and Sherwood numbers at Da=10-4, Ra*=100 without Dufour effects Table 9-2 Average Nusselt and Sherwood numbers at Da=10-2, Ra*=500 without Dufour effects

18 Abstract Numerical Investigations of Heat and Mass Transfer in Saturated Porous Cavity with Soret and Dufour Effects Khaled Abdulhussein Jebear Al-Farhany, 2012 Doctor of Philosophy, the University of Manchester The mass and thermal transport in porous media play an important role in many engineering and geological processes. The hydrodynamic and thermal effects are two interesting aspects arising in the research of porous media. This thesis is concerned with numerical investigations of double-diffusive natural convective heat and mass transfer in saturated porous cavities with Soret and Dufour effects. An in-house FORTRAN code, named ALFARHANY, was developed for this study. The Darcy- Brinkman-Forchheimer (generalized) model with the Boussinesq approximation is used to solve the governing equations. In general, for high porosity (more than 0.6), Darcy law is not valid and the effects of inertia and viscosity force should be taken into account. Therefore, the generalized model is extremely suitable in describing all kinds of fluid flow in a porous medium. The numerical model adopted is based on the finite volume approach and the pressure velocity coupling is treated using the SIMPLE/SIMPLER algorithm as well as the alternating direction implicit (ADI) method was employed to solve the energy and species equations. Firstly, the model validation is accomplished through a comparison of the numerical solution with the reliable experimental, analytical/computational studies available in the literature. Additionally, transient conjugate natural convective heat transfer in two-dimensional porous square domain with finite wall thickness is investigated numerically. After that the effect of variable thermal conductivity and porosity investigated numerically for steady conjugate double-diffusive natural convective heat and mass transfer in two-dimensional variable porosity layer sandwiched between two walls. Then the work is extended to include the geometric effects. The results presented for two different studies (square and rectangular cavities) with the effect of inclination angle. Finally, the work is extended to include the Soret and Dufour effects on double-diffusive natural convection heat and mass transfer in a square porous cavity. In general, the results are presented over wide range of non-dimensional parameters including: the modified Rayleigh number (100 Ra* 1000), the Darcy number (10-6 Da 10-2 ), the Lewis number (0.1 Le 20), the buoyancy ratio (-5 N 5), the thermal conductivity ratio (0.1 Kr 10), the ratio of wall thickness to its height (0.1 D 0.4), the Soret parameter (-5 Sr 5), and the Dufour parameter (-2 D f 2). 18

19 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 19

20 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the Copyright ) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trademarks and other intellectual property (the Intellectual Property ) and any reproductions of copyright works in the thesis, for example graphs and tables ( Reproductions ), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP policy (see ), in any relevant Thesis restriction declarations deposited in the University Library, The University library s regulations (see ) and in The University s policy on presentation on Theses. 20

21 Acknowledgements I would like to express my deep gratitude to my supervisor, Professor Ali Turan for his guidance, encouragement and supervision throughout all my studies. I would also like to point out that his help was essential for carrying out my PhD studies and the studies for scientific papers. I am very grateful to the Ministry of Higher Education & Scientific Research of Iraq, for my PhD Scholarship in Manchester University. I also would like to thank the Iraqi cultural attaché in London for their help and support me. I extend my sincere appreciation to my friends Drs Jalal AL-Obaedi and Amr Elbanhawy for their help, advices and useful discussions throughout this study and their comments on the manuscript of this thesis. I am deeply grateful to my dear colleagues and friends in Manchester University, Drs. Sharaf Al-Sharif, Shenghui LEI, Khaled Esteifi, Joseph Dawes and Mr. Yousuf Alhendal, Jieyan Ma, Machimontorn Promtong, Khurram Kafeel and Ayush Saurabh for their support and for many enlightening discussions. I appreciate the kind technical support, given by Dr Simon Hood. Special thanks also go to my colleagues and my friends Drs Tumadhir, Alaa, Ahmed, Mohammed and Hussein and also Mrs Haitham, Adnan and Basim. I wish to thank those who were the source of encouragement in all the years I spent it in Manchester especially all the Iraqi students in Manchester and Salford Universities.

22 Acknowledgements Finally and certainly very importantly, I wish to gratefully acknowledge my parents, my brothers, my sisters and all my family for their endless love and their constant patience and support. In particular, I am deeply indebted to my wife. Thank you all, it could not have happened without you. 22

23 To my parents To my wife and to my children Muntadhar, Ali and my daughter Retaj To my family and my friends

24 Publications and submitted papers The following is a list of the author s publications and Submitted Articles relating to the present thesis, see appendix (A) Journals Publications 1- Al-Farhany, K. and A. Turan, Non-Darcy effects on conjugate doublediffusive natural convection in a variable porous layer sandwiched by finite thickness walls. International Journal of Heat and Mass Transfer, (13-14): p Al-Farhany, K. and A. Turan, Unsteady Conjugate Natural Convective Heat Transfer in a Saturated Porous Square Domain Generalized Model. Numerical Heat Transfer, Part A: Applications, (9): p Al-Farhany, K. and A. Turan, Numerical study of double diffusive natural convective heat and mass transfer in an inclined rectangular cavity filled with porous medium. International Communications in Heat and Mass Transfer, (2): p Conference Publications 1- Al-Farhany, K. and A. Turan, Non-Darcy Effects on Conjugate Natural Convection in Saturated Porous Layer, in Tenth International Congress of Fluid Dynamics (ICFD10)2010. p. ICFD10-EG Al-Farhany, K., A. Turan, and J. Ma. Non-Darcy Effects on Double Diffusive Natural Convection Heat and Mass Transfer in Inclined tall Porous Cavities. in 4th International Symposium on Heat Transfer and Energy Conservation (ISHTEC2012). PP TC th January 2012, Guangzhou, China.

25 Publications and submitted papers 3- Al-Farhany, K., Numerical Study of Double Diffusive Natural Convective Heat and Mass transfer in porous cavities. in 1st conference of engineering since. 1-2 nd October Iraqi cultural attaché, London, UK. Submitted Articles 1- Al-Farhany K, Turan A. " Two-Dimensional Double-Diffusive Natural Convection Heat and Mass Transfer in an Inclined Porous Square Domain Generalized Model ". Submitted to International Journal of Heat and Mass Transfer. 2- Al-Farhany K, Turan A. "Soret and Dufour effects in Natural Convection Heat and Mass Transfer in a Porous Cavity ". Submitted to International Journal of Heat and Mass Transfer. 25

26 Nomenclature Nomenclature A B aspect ratio parameter C concentration, Kg m -3 C non-dimensional concentration, C C Cc C Cc h C 1 cp d parameter specific heat at constant pressure thickness of the solid walls, m D non-dimensional wall thickness, D d L D 1 non-dimensional diameter, D1 ds L Da Darcy number, Da K L 2 D C D TC mass diffusivity the thermodiffusion coefficient D f Dufour parameter, D f CD. TC T. g magnitude of the gravitational acceleration, ms -2 K permeability of the porous medium, m 2 26

27 Nomenclature k k eff k f k p k w thermal conductivity, Wm 1 K 1 effective thermal conductivity of porous medium,wm 1 K 1 thermal conductivity of the fluid, Wm 1 K 1 ratio of fluid to solid thermal conductivity thermal conductivity of the wall, Wm 1 K 1 k r thermal conductivity ratio, k r kw k f L Le length of the cavity, m Lewis number N buoyancy ratio N CC T T N 1 Nu l Nu p P parameter local Nusselt number average Nusselt number pressure, kg.m 1.s 2 non-dimensional pressure, 2 2 P pl Pr Prandtl number, Pr Ra Rayleigh number for porous medium, Ra g TL T 3 Ra * modified Rayleigh number for porous medium, * Ra Ra Da g TKL. T Sh average Sherwood numbers S r Soret parameter, S r TD. CD. CT C t time, s T non-dimensional temperature, T T Tc T h Tc T dimensional temperature, K 27

28 Nomenclature u velocity vector, ms -1 u velocity components in x-direction, ms -1 U non-dimensional velocity components in X-direction, U ul v velocity components in y-direction, ms -1 V x coordinates, m X non-dimensional X-coordinates, X x L Y y coordinates, m Y non-dimensional Y-coordinates, Y y L Greek symbols α effective thermal diffusivity, k c, m 2 s 1 eff p f β T βc coefficient of thermal expansion, K 1 coefficient of concentration expansion, K 1 wall to porous media heat capacity, stream function k k c w p w c eff p f porosity of the porous media, =0.36 kinematic viscosity, m 2 s 1 ρ density, kgm -3 ratio of specific heats non-dimensional time, t L 2 Subscripts C c center of cell cold 28

29 Nomenclature eff f h l p s w effective fluid hot local porous solid practical wall 29

30 Acronyms ADI ALFARHANY CFD MHD QUICK REV SIMPLE SIMPLER Alternating direction implicit method in-house FORTRAN code Computational Fluid Dynamics Magnetohydrodynamic Quadratic Upstream Interpolation for Convective Kinematics representative elementary volume Semi-Implicit Method for Pressure-Linked Equations SIMPLE Revised 30

31 Chapter 1 1. Introduction 1.1. Background A porous medium is usually considered to be composed of a solid matrix and voids. Saturated porous media are, in general, fully filled with one or more fluids, whereas unsaturated porous media are partially filled with the fluid/fluids and the fluid/fluids cannot flow everywhere in the pores because not all the voids are connected. The dynamics of fluids flow through a porous medium is a relatively longstanding topic. The book written by Darcy was published in 1856 and was based on his work on water flow through beds of sands. By that time, Hagen and Poiseuille had already analyzed single-phase flow through pipes. They used the fluid mechanics developed earlier by Navier Stokes, (Poisson and de Saint Venant [1]). Darcy s law has traditionally been used to obtain quantitative information on flow in a porous medium. This law is reliable when the representative Reynolds number is low and where the viscous and pressure forces are dominant. As the Reynolds number increases, deviation from Darcy s law grows due to the contribution of inertial terms to the momentum balance (Kaviany [1] and Bear [2]). It can be shown that, for all

32 Chapter 1. Introduction investigated media, the axial pressure drop is represented by the sum of two terms; one term being linear in the velocity (viscous contribution) and the other one being quadratic in the velocity (inertial contributions). The inertial contribution is known as Forchheimer s modification of Darcy s law. Basically, the pressure drop occurring in a porous medium is composed of two terms. A similar model for fibrous porous media was proposed by Beavers and Sparrow [3]. A general expression can be obtained by the following formula (Bear [2]). dp dx v (1.1) K It can be seen that the pressure drop is directly proportional to the fluid viscosity and inversely proportional to the permeability of the porous medium. Brinkman [4] studied another significant work for predicting momentum transport in porous media. Brinkman first introduced a term which superimposed the bulk and boundary effects together for flows with bounding walls. By taking into account the porosity effect, the effective viscosity was postulated from experiments performed on beds of spheres to replace the viscosity of fluid in Brinkman's model (1.2) eff where is the porosity. Although studies of flow through a porous medium are an old issue in fluid mechanics, the heat transfer convection in porous media has emerged as a new interest due to new technological developments. Thermal attributes enable applications such as heat dissipation media and recuperation elements. Hence, it has become important to understand the interaction between mass and thermal transport and the resulting effects on the thermo-mechanical characteristics of porous media. The phenomenon of convective heat and mass transfer in saturated porous media has received considerable attention due to its relevance in various applications. Natural and manufactured porous materials have broad applications in engineering processes, heat sinks, catalytic reactors, high breaking capacity fuses, heat exchangers and mechanical energy absorbers. In general, it can be noted that there are many 32

33 Chapter 1. Introduction examples of natural convection through porous media in industrial systems, such as thermal insulation, drying processes, biomedical engineering applications and nuclear reactors (Nield and Bejan [5]). In the past, there have been considerable analytical, numerical and experimental studies undertaken to measure or estimate the overall heat transfer rate of convective heat transfer in porous media. In general, the experimental results are usually described statistically as empirical relationships in terms of dimensionless parameters of Nusselt, Rayleigh, and Prandtl number and permeability Objective of the research The studies of the natural convective heat transfer and the double-diffusive natural convective heat and mass transfer inside a porous medium enclosure have been the subject of an extensive number of experimental, analytical and numerical investigations during the last years using different models. Also, the conjugate heat transfer phenomena in porous media have received increasing attention in recent years and are of practical importance in, for example, high performance insulation for buildings and cold storage installations. Therefore, the effects of the conjugate heat transfer on the double-diffusive natural convective heat and mass transfer in porous cavities are most interested to study. Combined buoyancy driven flows due to both temperature and concentration variations in inclined porous cavities, which tend to be more complex, have not so far been investigated as extensively as the vertical porous medium studies, therefore more researches are needed to study for the inclined porous cavities. The investigations of the Soret and Dufour effects in natural convective heat and mass transfer in porous medium have been presented in limited cases in the last few years and it is a major point for this study. The main objective of this study is the development of an in-house FORTRAN code named ALFARHANY code, for the studies of single fluid flow, heat and mass transfer in an inclined porous enclosure. Darcy model and Darcy extension models including the Darcy-Brinkman model, the Darcy-Forchheimer model and the Darcy- 33

34 Chapter 1. Introduction Brinkman-Forchheimer (generalized) model have been used. Also, the effects of variable thermal conductivity, permeability and the porosity have been undertaken to make ALFARHANY code general. Soret and Dufour effects in double-diffusive natural convective heat and mass transfer as well as the conjugate phenomena have been undertaken. The numerical model adopted is based on the finite volume approach and the pressure velocity coupling is treated using the SIMPLE/SIMPLER algorithm. The whole procedure is explained in detail in the next chapters Outline of the thesis Following this introductory chapter, Chapter 2 presents a literature review of many relevant previous studies. Although this study is concerned with natural convective heat and mass transfer in saturated porous cavities, the literature survey also covers natural convective heat and mass transfer for a plate embedded in a porous medium. Chapter 3 describes the basic models of flow in porous media and the governing equations of two-dimensional unsteady heat and mass transfer in porous cavities including the effects of variable porosity, including the Soret and Dufour effects which are numerically solved in this study. The non-dimensional parameters and the non-dimensional generalized model with general initial and boundary conditions that are used in this work are presented also in this chapter. In Chapter 4, the discretization of the governing equations using finite volume approximation and a SIMPLE/SIMPLER algorithm as well as the alternating direction implicit (ADI) method is presented. An ALFARHANY code was developed for these studies and the whole procedure is explained in detail. Chapter 5 is devoted to the verification of the in-house code with the proposed numerical model. The model validation is firstly accomplished through a comparison of the numerical solution with the reliable experimental, analytical/computational studies available in the literature. This chapter presents a validation of the different cases of natural convection heat transfer, conjugate, variable porosity, angle of inclinations of cavity, double-diffusive natural convection heat and mass transfer and of the Soret and Dufour effects in two-dimensional cavities. 34

35 Chapter 1. Introduction The results that were obtained are presented and discussed in the four following chapters. In the first chapter of these four, Chapter 6 presents a mathematical formulation and the results of unsteady conjugate natural convective heat transfer in a two-dimensional porous square domain with finite wall thickness. A correlation is provided to calculate the average Nusselt number on the internal left wall of the porous cavity as a function of different values of the non-dimensional governing parameters, including the modified Rayleigh number (100 Ra * 1000), the Darcy number (10-7 Da 10-2 ), the thermal conductivity ratio (0.1 Kr 10) and the ratio of wall thickness to its height (0.1 D 0.4). This part of this thesis has been published in the Journal of Numerical Heat Transfer, Part A: Applications (Al-Farhany and Turan [6]). Additionally, the steady state results from this part of the thesis have been published and presented at the Tenth International Congress of Fluid Dynamics (ICFD10) (Al-Farhany and Turan [7]). In Chapter 7, steady conjugate double-diffusive natural convection heat and mass transfer in a two-dimensional variable porosity layer sandwiched between two walls is investigated. The results are presented for a variable porosity porous medium for different values of the non-dimensional governing parameters, including the * modified Rayleigh number ( 10 Ra 1000 ), the buoyancy ratio (-2 N 2), the Lewis number ( 0.1 Le 10 ), the thermal conductivity ratio ( ) and the ratio of wall thickness to its height ( 0 D 0.4 ) where the modified Rayleigh number is given as * Ra Ra Da. This part of the thesis has been published in the International Journal of Heat and Mass Transfer (Al-Farhany and Turan [8]). k r The effect of the inclination angles of porous cavities on double-diffusive natural convection heat and mass transfer are presented into two groups in Chapter 8 for a square and a rectangular cavity respectively. The results are presented for a wide range of non-dimensional parameters, especially for square cavities. The results obtained in the case of a square cavity have been submitted to the International Journal of Heat and Mass Transfer. Additionally, the second part of this chapter has been published in the Journal of International Communications in Heat and Mass Transfer (Al-Farhany and Turan[9]). Moreover, the results obtained in the case of a tall porous cavity from this part of the thesis have been presented at 4th International Symposium on Heat Transfer and Energy Conservation 35

36 Chapter 1. Introduction (ISHTEC2012) (Al-Farhany et al.[10]) and at 1 st conference of engineering since (Al-Farhany [11]). The last part of which presents the results (Chapter 9) is dedicated to the study of Soret and Dufour effects in porous cavities. The results are presented in terms of streamline, isothermal, iso-concentration and average Nusselt and Sherwood number profiles for high porosity (=0.6). This part of the thesis has been submitted to the International Journal of Heat and Mass Transfer. Finally, in Chapter 10, the conclusions of the present studies are presented and are discussed together with some suggestions for future studies. 36

37 Chapter 2 2. Literature Review The effects of fluid and solid porous matrix properties such as porosity, permeability, and thermal conductivity on convective heat and mass transfer in porous media have been received increasing attention in recent years. This is in addition to attention given to Darcy flow and non-darcy flow behaviour and the effects of the Soret, Dufour and conjugate problems. These characteristics are significant in industrial transport processes in porous media, in geophysical, high performance insulation for buildings and cold storage installations, also for nuclear waste disposal, underground pollutant transport, ground-water pollution, and in the mantle flow in the earth's crust as well as several chemical processes. In this chapter, a brief review of previous research into heat and mass transfer in porous medium as reported in the literature is presented. These studies are presented within six groups and these are: Darcy and extended Darcy flow in porous cavities, inclined porous medium, variable porosity, conjugate problems, double-diffusive and finally Soret and Dufour effects.

38 Chapter 2. Literature review 2.1. Darcy and extended Darcy studies in porous medium Over a century ago, a great number of studies on this subject already existed and can be found in the literature. These studies looked at the convective heat transfer in porous media which had used the classical Darcy formulation. This was formulated later by Bear [2] as the macroscopic equation of motion for Newtonian fluids in porous media at a low Reynolds numbers. The flow was showed to be linearly dependent upon the gravitational force and the pressure gradient. The constant in Darcy's equation was proved later by Muskat and Wyckoff [12] and it should be related to the permeability of the porous material. Isotropic porous regimes were derived from the modified Darcian law where the pressure gradient and the velocity vectors are parallel. An equally large body of research has emerged with respect to investigations of convective heat and mass transfer in rectangular porous cavities and vertical surfaces with uniform/non-uniform heat flux and surface temperatures. The classical Darcy formulation has been used as well as the Darcy-Brinkman model, the Darcy-Forchheimer model and the Darcy-Brinkman-Forchheimer model. These areas have been thoroughly reviewed recently by Nield and Bejan [5]. An analytical investigation for a convective flow of a fluid in a rectangular box of porous material heated from below for various box geometries was conducted by Beck [13]. For this particular problem, the critical Rayleigh number coincides between the linear method and energy method. Therefore, the inequalities Ra Ra and Ra Ra c energy c true c true become equalities and subcritical 2 1 (sublinear) instabilities cannot exist, where Ra min b b, and c linear c mn, 2 b m n 2 2 h h For a cubic box ( b 1) the minimum critical Rayleigh 2 number is 4. The analysis which illustrates the invalidity of the inclusion of the nonlinear inertia term in Darcy's law of the form was under taken by Lapwood [14]. This means that the energy method in its present form is thus inappropriate to a range of problems related with fluid flows in porous media. 38

39 Chapter 2. Literature review Latter, Straus and Schubert [15] studied the finite amplitude thermal convection in the nature of two- or three-dimensional convection in cubic boxes of fluid-saturated porous material. It was heated from below for Rayleigh numbers R as large as According to the linear theory, 'strictly' only for critical Rayleigh number R 4.5, is a three-dimensional convection in a cubic box possible [13]. A trivial form of three-dimensional convection can exist for R as small as 2 4 and consists of the super position of orthogonal two-dimensional rolls. On the other hand, Latter, Straus 2 and Schubert s calculations never produced this form of convection for R 4.5. For the used value of the Rayleigh numbers which lies between the critical value of 2 R 4.5 and 150, they showed that there is always a stable form of steady twodimensional convection [16]. Many of the previous numerical studies, such as [17] and [18] indicated that "there would have to be a non-uniqueness in the form of the convection pattern associated with initial conditions" to exist the three-dimensional convection at all of these Rayleigh numbers. The results showed that the Nusselt numbers in two-dimensional flows are smaller than three-dimensional flows for R 97 and the opposite is true for R 97. The differences in Nusselt number between the two dimensions and the three dimensions solution are almost about 12%. Moreover, the results indicated that it was the proper choice for initial conditions makes always possibility to force steady two-dimensional convection or steady three-dimensional convection. Horne [19] investigated the effect of using uniform and non-uniform heating from below in porous media on the stability of natural convective flow in a porous medium. The purpose of his study was to find out the possibility of oscillatory and other unsteady flows, and to look to the conditions under which they may occur. The numerical results were validated with experiments using a uniformly heated case and also, in the Hele Shaw cell, with the results of Caltagirone et al. [20] and from Combarnous and Le Fur [21]. In certain cases, in the uniform heated cases the result shows two different possible modes of flow; one is steady and the other is fluctuating. On the other hand, in the non-uniformly heated case, the solution into a unique mode of flow was forced by the boundary conditions which are regularly oscillatory when there is considerable non-uniformity in the heat input at the lower boundary, as in the case of sufficiently a high Rayliegh number. The type of 39

40 Chapter 2. Literature review boundary conditions and the vertical boundaries employed greatly influence on the natural convective regime of flow through a porous medium. Seki et al. [22] conducted an experimental investigation to examine the effects of the physical properties of porous media and the dimension of a rectangular cavity on convective heat transfer in a confined rectangular cavity packed with porous media. The vertical walls were subjected to different constant temperatures while the horizontal walls were insulated. The experiments were performed by using various diameters of two kinds of solid particles (glass beads and iron balls). Water, transformer oil and ethyl alcohol were used as a working fluid. In addition, he used three rectangular test-cells with different width (22, 57 and 116 mm) in the section area and a constant height (571 mm) and depth (250mm). The porosity obtained in this study was in the range of The experimental work covered a wide range of Prandtl number1 Pr* 200, Darcy-Rayleigh number 1 Ra* 10 and geometrical aspect-ratio 5 HW 26. The results showed that the convective heat transfer through the vertical porous layer was affected significantly by the aspect-ratio in addition to the Prandtl number. The Nusselt number Nu * is correlated by the following relationship Nu* Pr * H W Ra* Good agreement of the experimental data with the numerical results was obtained by 3 Vlasuk [23] for Ra* 10. Therefore, this agreement might mean that the analytical calculation based on Darcy s law is effective in this range. For 5 3 Ra* 10 the analytical results showed an increase greater than that for the experimental result. This difference might indicate that the Darcy s law could not be applied in the 3 physical model of porous layer adopted in the range of Ra* 10. An experimental investigation into transient heat transfer characteristics in a rectangular cavity backed with porous media was undertaken by Inaba and Seki [24]. This study examined the effects of the spherical solid particle diameter (porosity ), the aspect ratio HWand the physical properties of the porous medium on the transient natural convective heat transfer characteristics. The experiments were performed by using four kinds of rectangular test section; 571 mm in height H and 480 mm in depth and with 22, 40, 57 and 116 mm variables in width W in the section area between the heating and cooling parts. The water and the transformer oil 40

41 Chapter 2. Literature review were used as a working fluid. Also, four kinds of spherical solid particles were used; d=11.0 mm from iron balls and d = 1.01, 5.03 and 16.4 mm from glass beads. The vertical walls were kept at different constant temperatures while the other walls were insulated. At the steady state, the results of Nusselt number satisfied the experimental correlation proposed earlier by Seki [22]. The result showed that the thermal steady state in the porous layer was obtained in a shorter time by the involvement of natural convection. It was possible to control the amount of heat stored and the time taken to reach a steady state in the porous layer. This occurred by varying the dimensions of the porous medium, the diameter of solid particles and the combinations of solid particles and fluid. An integral method was used for analysing the influence of temperature-dependent viscosity on free convection in a two-dimensional cavity filled with fluid-saturated porous media by Blythe and Simpkins [25]. Their analysis extended earlier work [26] undertaken on the constant viscosity problem and it dealt with the high Rayleigh number limit. The heat transfer across the cavity is defined by the Nusselt number * Nu L R Nu, where L is cavity aspect ratio, R is Darcy-Rayleigh number, * Nu is reduced Nusselt number. A cogent transformation was found which reduced the core shear distribution to a universal form. A finite-difference and weakly nonlinear theory were conducted by Gary et al. [27]. These were used to describe steady state convective heat transport in water saturated porous media. Two-dimensional convection in a rectangular region was investigated with imposed temperature difference up to 200 o K corresponding to a viscosity ratio of about 6.5. Differences between the computed Nusselt number for the case of constant viscosity (nearly eliminating the temperature difference) and that for a 200 o K difference is, at most, about 7 % for the range of Rayleigh numbers R / R c considered. Recently the effects of temperature that are dependent on kinematics viscosity and density and on the convective stability in a vertical porous regime were experimentally examined by Kwok and Chen [28]. A numerical solution for variable property governing differential equations were looked at by Zhong et al. [29]. This solution was used to determine the variable property effects on the heat transfer and flow behaviour for natural convection flow in a square enclosure for a range of Rayleigh numbers and different wall temperatures. 41

42 Chapter 2. Literature review Boundary layer studies were undertaken by Tong and Subramanian [30] on free convection in vertical porous enclosures. Invariably, it was observed that as the Brinkman effect increases the heat transfer rate reduces because of the velocity decrease at the wall. The effect of the no-slip boundary condition was also looked at analytically and numerically by Vasseur and Robillard [31]. They worked on natural convection in a rectangular porous layer subjected to uniform heat fluxes along the vertical side walls. In the formulation of the problem, the Brinkman-extended Darcy model was used to satisfy the no-slip boundary condition. The analytical solution was based on the modified Oseen linearization method to solve the boundary layer equations. It was found that the boundary effects had a non-negligible weight on the heat transfer and flow field. These effects were more pronounced in high porosity media where the flow rate and the heat transfer are significantly reduced. The flow rate and heat transfer were significantly reduced at the high porosity media. For low porosity, the analytical solution was reduced to the regular Darcy s law solution. Numerical results were reported in the range 20 R 1000, 7 10 Da 10 and 2A 4, where R is the Darcy-Rayleigh number, Da is the Darcy number and A is the aspect ratio. A good agreement was found between the numerical simulation and the analytical predictions of the same phenomenon. This problem has been considered in the past by Bejan [32] on the basis of the boundary layer and on Darcy s approximations. Sen [33] considered a Darcy-Brinkman convective flow in a shallow porous rectangular cavity with adiabatic upper and lower plate boundaries and differentially heated sidewalls. The analysis considered two types of boundaries. Case I imposed no-slip boundary conditions that be used for all rigid boundaries in the cavity. This mathematical treatment paralleled that of Cormack et al. [34]. It is based on the asymptotic limit that the aspect ratio of the cavity goes to zero. In case II, an analysis of natural convection in a shallow porous cavity with a free upper surface was used when the Darcy number was based on the depth of the cavity, and is less than The analyses revealed that both the Darcy and the Brinkman model gave virtually the same result for the Nusselt number. Also, the increase in the heat transfer rate through the cavity was found to occur significantly in the free upper surface, especially when the permeability of the medium is high. 42

43 Chapter 2. Literature review An important non-darcian study has also been presented by Lauriat and Prasad [35] for a heated vertical porous cavity. The inertia and viscous forces on natural convection were examined via the Darcy-Brinkman-Forchheimer model. These results indicated that an asymptotic convection regime exists where the solution is independent of the permeability of the porous matrix, or the Forchheimer and the Darcy numbers. Otherwise, the average Nusselt number decreases when the Forchheimer number increases, while the Nusselt number always increases with the fluid Rayleigh number, the conductivity ratio and the Darcy number. Nithiarasu [36] studied the effect of the differences in existing porous medium flow models on flow and heat transfer in the context of heat flux boundary condition. Buoyancy drove the flow in a non-darcy porous medium subjected to the uniform wall heat flux conditions. A set of generalized porous media equations were solved by using the Galerkin finite element method coupled with the velocity correction procedure. The effects of porosity and nonlinear drag term were isolated by modification to these equations. The results were presented for a wide range of Rayleigh and Darcy numbers. The result showed that the effects of nonlinear drag term and porosity are significant at higher Rayleigh and Darcy numbers. Therefore, when modelling the porous medium flow in a non-darcy regime with low Rayleigh numbers, it is not necessary to include the effects of nonlinear drag term and porosity. On the other hand, it is necessary to include these effects for high Rayleigh numbers models. A numerical investigation for transient free convection in a two-dimensional square cavity filled with a porous medium was conducted by Saeid and Pop [37]. The flow was looked at in the case when one of the cavity vertical walls was suddenly cooled to a constant temperature T c and the other vertical wall was suddenly heated to a constant temperature T h, The horizontal walls were insulated. The finite volume numerical method was used to solve the non-dimensional continuity, Darcy and energy equations. These results were in good agreement with the results obtained by Walker and Homsy [38], Bejan [39], Weber [40], Gross et al. [41] and Manole and Lage [42] and Bankvall [43]. Also, the results showed that the time required to reach a steady state is shorter for high Rayleigh numbers and longer for low Rayleigh numbers. 43

44 Chapter 2. Literature review 2.2. Inclined porous medium The effect of an inclined porous medium on heat transfer has been presented in previous books by Nield and Bejan [5], Ingham and Pop [44] and Vafai [45]. These applications are significant in many of engineering and geophysical applications; industrial transport processes in porous media, geophysical applications, high performance insulation for buildings, solar power collectors and cold storage installations The effect of natural convection by using a constant heat flux on two opposing walls in a thin inclined porous layer were investigated both analytically and numerically by Vasseur et al. [46]. The buoyancy was driven by the flow by using the Boussinesq approximation in a Darcy porous medium subjected to constant heat flux conditions. A finite difference method was used to solve the governing equations. The alternating direction implicit (ADI) method was used to solve the energy equation and the momentum equation was solved by using the successive over-relaxation method (SOR). The results showed that for a tall cavity (1 A ), the maximum heat transfer occurs when the inclined angle is between 90 o and 180 o. Also, the maximum heat transfer occurs at the small values of angle when the Rayleigh number increases. The analytical solution was tested numerically in the range 20 R 50, 0 and 2 A 10, where R is the Rayleigh number, is the inclination angle and A is the aspect ratio. Later, Vasseur et al. s theoretical and numerical methods were used by Sen et al. [47]. The analyses were extended to cover the flows which presented the unicellular convective motions only. Sen et al. studied a multiplicity of solutions for horizontal and vertical inclined cavities. The results presented for Rayleigh numbers Ra 500 with small angles. The same cases were also studied by Báez and Nicolás [48] Moya et al. [49] and Caltagirone and Bories [50]. Natural convection heat mass transfer in an inclined porous layer has been studied experimentally by Inaba et al. [51]. In their studies two opposing walls of a tall rectangular cavity were thermally insulated, while the other walls were kept at different temperatures. The experiments were performed by using four kinds of rectangular test section; 340 mm in height H and 10.4, 22.5, 34 and 68 mm variable 44

45 Chapter 2. Literature review width W in the section area between the heating and cooling parts and 400 mm in depth. Water, transformer oil and ethyl alcohol were used as the working fluid. Furthermore, six kinds of spherical solid particles of various diameters were used; d=9.5 mm from iron balls, d = 5.03 and 10.4 mm from glass beads and the other three kinds were alumina balls of diameter d = 5.1, 10.2 and 22.2 mm. The experimental work covered a wide range of modified Prandtl number 3.1 Pr* 499, the modified Rayleigh number *10 Ra* 3.8*10, the geometrical aspect-ratio 5 HW 32.7, inclination angles and ratios of spherical particle diameter to width of the cavity dw 1.0. Four correlation equations for the heat transfer were presented for small and large inclination angles. Their results showed that for large Rayleigh numbers, the maximum Nusselt number occurs at around = 0, or it occurs at around = 60 for low Rayleigh numbers. Also, the results showed that the contribution of HWwas significantly near = 90 while the contribution of dw on Nu was not significant for small. In a different study, Hsiao [52] numerically investigated natural heat transfer for variable porosity and thermal dispersion effects in an inclined porous cavity. Their results showed that these effects increase the temperature gradient near to the wall. Chamkha and Al-Mudhaf [53] conducted an important Darcy-Brinkman study for double-diffusive natural convection flow in an inclined porous cavity, with the presence of temperature-difference dependent heat generation. The finite difference method was used to solve the governing equations. The results were presented for an aspect ratio where the Prantd1 and the Lewis numbers equal 2, 7.6 and 10, respectively. In general, the results showed that as the cavity inclination angle increased, reductions in the average Nusselt and Sherwood numbers were achieved. However, there was an exception at a critical angle where the Nusselt and Sherwood numbers achieved the maximum magnitude. Another Darcy-Brinkman study was presented by Wang et al. [54] for unsteady natural convective heat transfer in an inclined porous cavity with time-periodic boundary conditions. The constant parameters, Ra* = 10 3, Da = 10 3, Pr = 1, ε = 0.6, and σ = 1 were used in their study. For these parameters, the corresponding maximal time averaged Nusselt number of the right sidewall can be acquired at a 45

46 Chapter 2. Literature review dimensionless oscillating frequency (f) equal to 46.7 and an inclination angle equal () to Later, Wang et al. [55] numerically studied three-dimensional unsteady natural convections in an inclined porous cavity with time oscillating boundary conditions by using The Darcy Forchheimer Brinkman model. Their results showed that, for a small inclination angle (0 75 ), the natural convections inside are stable and near to being two-dimensional. Moreover, if the inclination angle is large (75 90 ), the flow patterns inside are much more complicated and threedimensional multiple roll-cells with different intercrossing angles are established Variable porosity Studies into the effects of porosity on boundary layer heat transfer have tended to fall into one of two categories: variable-porosity studies and simpler porosity variation studies. The former area has been considered by Prasad and Kladias [56] who highlighted that, in addition to the non-darcy inertial and viscous effects, the flow through porous domains become very complex and distorted owing to the large-scale fluctuations in porosity and also permeability. This is particularly prevalent in geophysical porous media which are essentially an isotropic and heterogenous in character. At a stratum or wall region the effects are most pronounced and, consequently, a substantial maldistribution in the hydrodynamic field is produced; this manifests as the phenomenon of "channeling" in the boundary layer. This non- Darcian effect was first noted experimentally by Schwartz and Smith [57] in their study on boundary layer flows in packed beds. Further verification was achieved by Schert and Bishoff [58]. Vafai [59] has shown that variations of porosity are caused by the mechanism of contact between the solid matrixes. The nature of the observed effects is dictated by the porous medium structure/geometry and a specific knowledge of the microstructure of the porous matrix is crucial in making any deductions. The existence of a "high porosity zone" adjacent to the boundary has been documented experimentally by Roblee et al. [60] and Benenati and Brosilow [61] for packed-bed porous media. These investigations have also indicated that porosity varies as a 46

47 Chapter 2. Literature review damped oscillatory function with separation from the boundary wall and this has been thoroughly validated in the literature by Poulikakos and Renken [62], Vafai [63]and Vafai et al. [64].There is a greater porosity at the walls since the solid particles are incapable of packing very efficiently here due to the obstacle. Cheng [65] has experimentally shown that porosity is equal to unity at the wall and varies to core value 5 diameters from the wall. Cheng also discussed how the flow velocity parallel to the solid boundary escalates as the wall is approached, due to the high porosity there, and rises to a maximum before decreasing to zero in order to satisfy physically the condition of "no-slip". This results in a net rise in volumetric flux or the "channeling effect". Lauriat and Vafai [66] have underlined the need to consider the above expression as, at best, a "line-averaged porosity" perpendicular to the solid boundary. Vafai et al. [63] have shown that the overshoot velocity is affected considerably by the magnitude of the pore Reynolds number. The latter approach has received growing attention in recent years. Recently the specific case of high-porosity media was examined by Chen and Lin [67] in their study of Darcy-Brinkman-Forchheimer natural convection from an isothermal surface embedded in a thermally stratified porous medium. Foam metals and fibrous media that have high porosities were studied, the non-darcian effects i.e. Brinkman friction and Forchheimer inertia, being more pronounced here than for low porosity media where Darcy's law only is applicable. A scalar parameter was used for the porosity (similar to that employed in the present study) with a typical value of 0.9. Also a scalar value for porosity was employed and not a porosity function which varies with distance from the wall (i.e. porosity is not a function of a spanwise similarity coordinate as discussed by Vafai and Tien [68].) Nithiarasu et al. [69] studied the effect of porosity on natural convective heat transfer in fluid saturated porous media. In this investigation, a non-darcy model used in tow-dimension square cavity filled with fluid saturated porous medium. The vertical walls were maintained at different temperatures while the horizontal walls were insulated. The governing equations (Nithiarasu et al. [70]) were solved by using Galerkin's finite element method with a Eulerian velocity correction procedure. The results showed that, at a low Darcy number with a low Rayleigh number, there is no difference observed in an average Nusselt number with the variation of porosity. 47

48 Chapter 2. Literature review While, at high Rayleigh numbers, the variation in the average Nusselt number with porosity is significant. On the other hand, porosity substantially affects the flow and heat transfer in porous medium at higher Darcy numbers. A maximum difference of 40% in an average Nusselt number was observed between different porosities. Following on the work of Nithiarasu et al., the effect of variable porosity in natural convective heat transfer in a porous cavity was analyzed by Marcondes et al. [71]. The vertical walls of the cavity are isotherms and the horizontal walls are insulated. This numerical work used the generalized model by Nithiarasu [70] in Darcian and non-darcian regimes. The finite-volume method is used to solve these equations in term real value, and the PRIME method is used to solve the pressure-velocity coupling problems. A wide range of non-dimensional parameters are used in this work, namely the Darcy number number Da, the Rayleigh Ra 6 10, the Prandtl number 0.7 Pr 480, the Aspect ratio 5 A 10 and the thermal conductivity ratio between the modium and the fluid 1R k 6. The correlation equation of average Nusselt numbers is given as: Nu 0.074Ra Pr R Da A h k 2.4. Conjugate problems In the last three decades, a wide range of studies have been published emphasizing the conjugate heat transfer phenomena in a porous media. It is of practical importance in a number of applications, for example, high performance insulation for buildings and cold storage installations. Nield and Bejan [5] presented many reviews of the existing studies on such topics. Reviews can also be found in Ingham and Pop [44, 72, 73], Pop and Ingham[74], Vafai [45] and Al Amiri [75]. Numerical investigations into non-darcian effects on transient conjugate natural convection-conduction heat transfer from a two-dimensional vertical plate fin embedded in a high-porosity medium were carried out by Hung et al.[76]. Air ( Pr 0.7 ) and water ( Pr 5.5 ) with selected values of conjugate convectionconduction parameter (N cc ) were selected in this study. It was used to compute the transient heat transfer characteristics for the conjugate heat transfer problems. Other 48

49 Chapter 2. Literature review parameters used for computations are porosity ( ) =0.89, specific heat ratio ( ) =1.265, aspect ratio ( A r ) =20, permeability ( K ) = Also, 10 was used for an inertial effect and 0was used for no an inertia effect. A cubic spline collocation method was used to solve the coupled nonlinear partial differential equations in the fin and in the saturated porous medium. The results have shown that inertial effects on heat transfer characteristics are negligible at earlier times. On the other hand, the effect becomes increasingly important over longer periods of time. A fluid with a low Prandtl number transfers less heat than a fluid with a high Prandtl number. In addition, the time taken to reach a steady state is less at a high value of Pr and for a fluid with a high value of N cc. Miyamoto et al. [77] analyzed the two-dimensional conjugate heat-transfer problems of free convection from a vertical flat plate. This analysis was used to predict theoretically the temperature and the heat-transfer rate at the solid-fluid interface. Two thermal boundary conditions (constant temperature and constant heat flux) were considered at the outside surface of the flat plate, while the horizontal walls were insulated. Experimental works were conducted, in order to satisfy his analytical solutions at constant heat flux. Three materials were selected; stainless steel (K=16 w/m.k), aluminum (K=240 w/m.k), and glass (K=0.76 w/m.k). These plates were 600 mm in width, 400 mm in height and 5 mm in thickness (for both stainless steel and aluminum) and 6mm for glass. The results showed that the axial heat conduction and the controlled dimensionless parameter K/D in the flat plate insignificantly affects the temperature distribution when the outside surface of the flat plate is maintained at a uniform higher temperature. Whereas the axial heat conduction and the controlled dimensionless parameter KD in the flat plate has significant affects on the temperature distributions for the larger KD. Large values for the controlled dimensionless parameter KD= l0 5 gave a nearly uniform interfacial temperature. A numerical investigation into a two-dimensional steady state conjugate free convection due to a vertical plate of finite extent adjacent to a semi-infinite porous medium was undertaken by Vynnycky and Shigeo [78]. A wide range of nondimensional parameters were used: Rayleigh number Ra, the plate aspect ratio and the thermal conductivity ratio k between the plate and the porous medium. The investigation was used to describe of the effects of non-dimensional parameters on 49

50 Chapter 2. Literature review flow and heat transfer characteristics. The heat momentum equations were solved numerically by using finite difference techniques. The results were in good agreement with the results obtained by Chen and Minkowycz [79]. The result showed that when the thermal conductivity ratio k decreased, the mean Nusselt number became less dependent on the Rayleigh number. Additionally, the mean Nusselt number is essentially independent of the Rayleigh number when k < 1 and Ra > 200 for = 0.25 and when k < 2.5 and Ra > 200 for = 1. Furthermore, as mentioned above, a two-dimensional transient conjugate free convection due to a vertical plate in a porous medium was investigated both analytically and numerically by Vynnycky and Kimura [80] and Kimura et al. [81]. Important contributions were also made by Saeid [82] regarding a steady conjugate natural convection conduction heat transfer in a two-dimensional vertical porous enclosure. A horizontal wall with a finite thickness was heated, while the outer surfaces of the vertical walls were isothermal at different temperatures. The horizontal boundaries were kept insulated. The Darcy model was considered in this study and the finite volume method was used to solve the non-dimensional governing equations. The numerical work covered a wide range of governing 3 parameters. These ranges were the Rayleigh number10 Ra 10, the thermal conductivity ratio 0.1 K r 10 and the ratio of wall thickness to its height 0.02 D 0.4. It was found that the average Nusselt numbers increased when the Rayleigh number and the thermal conductivity increased, while the Nusselt numbers decreased when the wall thickness increased. On the other hand, it was found that, as the wall thickness increased, the average Nusselt numbers increased for a spatial case at a low Ra ( Ra 10 ) and high conductive walls ( K 10 ). For small values of Rayleigh number, the results indicated that, in both the wall and the porous layer, heat is transferred mainly by conduction and the average Nusselt number is approximately constant. r Previously, Al-Amiri et al. [83] had conducted a numerical study into twodimensional steady state conjugate natural convection in a fluid saturated porous cavity bounded by a conducting vertical wall. The finite element method was used to solve the governing equations. The Forchheimer Brinkman-extended Darcy model (generalized model) for momentum equations was used to solve the governing 50

51 Chapter 2. Literature review equations of the fluid inside the porous cavity. The vertical walls had different constant temperatures while the horizontal walls were kept insulated. The results showing the wall interface temperatures, the average Nusselt numbers, the isotherms, and the streamlines were presented for a wide range of dimensionless parameters. 4 6 These parameters included the Rayleigh number10 Ra 10, the Darcy 5 1 number10 Da 10, conjugate wall thickness W 0.2, and porosity , in addition to the wall-to-fluid thermal conductivity ratio1k r 10, the solid-to-fluid thermal conductivity of the porous medium 0.1 K K 100, and aspect ratio 0.25 A 2. The results showed that, s f as the wall thickness increases, the overall Nusselt number is reduced, while the average Nusselt number increases when the Rayleigh number increases. The Nusselt number correlation was mathematically expressed for Da 3 10, 0.9 as follows: Nu W A A k s k f k Ra A r (1.3) These results were in good agreement with the results obtained by Kaminski and Prakash [84], Hribersek and Kuhn [85], Wansophark et al. [86] and Al-Amiri [87] Double-diffusive studies In many industrial and natural applications, flow through the porous medium is driven by the density difference caused by temperature and concentration variations; examples of these applications are insulation problems in which air saturated the porous matrix and water vapour disperses into the medium; other applications include food processing, ground-water pollution, sea-water flow, mantle flow in the earth s crust as well as several chemical processes. The phenomenon of a combined heat and mass transfer in a porous medium is usually referred to as double-diffusive. The reason for the scarcity of work on the subject may be related to its complexity and also to the lack of a general theory. In the last decade, research into this area has mostly been mainly theoretical investigations (Mojtabi and Charrier-Mojtabi [88] ). An overview of the current state of the art of double-diffusive in saturated porous media has been recently presented by Nield and Bejan [5] and Ingham and Pop 51

52 Chapter 2. Literature review [72],[73]. Most of the work available on double-diffusive is concerned with confined porous media and is studied using the Darcy model. In particular, two main configurations have studied cavities with imposed uniform heat and mass fluxes, and imposed uniform temperature and concentration. The use of non-darcian models has only been introduced recently for the explanation of double-diffusive and the use of the Generalised model has been restricted to a few papers by Nithiarasu et al. [89], [90]. For opposing but comparable buoyancy forces due to temperature and concentration, and for certain configurations, the flow can be unsteady as expounded by Nishimura et al. [91] and Weaver and Viskanta [92]. Although these oscillations have been noticed during some experiments (Weaver and Viskanta [92]), available numerical solutions were not able to predict this particular behaviour. Such findings highlight the need for an algorithm based on the time dependent solution of the flow, and the generalized model is probably suitable for the scope envisaged in these investigations. Trevisan and Bejan [93] studied two-dimensional natural heat and mass convection in a vertical porous layer. The temperature and the concentration differences were supplied in a horizontal direction. The control volume numerical method was used to solve the non-dimensional continuity, mass, Darcy and energy equations. Both heat and mass fluxes across the boundaries were calculated by using the power low scheme. The results were presented on streamlines, isotherms, concentration lines, and Sherwood numbers due to a wide range of non-dimensional parameters. Numerical results were reported in the ranges 0.01 Le 100, 5 N 3 and H L 1, where Le is the Lewis number, R H R H, the Darcy-Rayleigh number, N is the buoyancy ratio and H L is the aspect ratio. A good agreement was found between the numerical simulation and the analytical predictions of the same phenomenon. An important Darcy-Brinkman study has also been presented by Goyeau et al. [94] for double-diffusive natural convection in a porous cavity. The horizontal boundaries of the cavity were adiabatic and the vertical walls were maintained at fixed different temperatures and concentrations (T 1, C 1 at the left wall and T 2, C 2 at the right wall). 52

53 Chapter 2. Literature review The finite volume method was used to solve the governing equations. These results were in good agreement with the results obtained by Trevisan and Bejan [93] in a Darcy model. Their numerical results were in excellent agreement with the scaling analysis for mass transfer; while for heat transfer, the boundary layer analysis did not prove to be a suitable method to predict the correct scales of heat transfer. Karimi-Fard et al. [95] numerically studied double-diffusive natural convective heat and mass transport in a square cavity filled with porous medium. The Darcy, Darcy- Forchheimer, Darcy-Brinckman, generalized models were considered. The Soret and Dufour effects were neglected and all the porous and fluid properties assumed to be constant except that of the density. The Boussinesq approximation was used to express the variation of density due to the temperature and concentration variations. The horizontal walls were insulated, while the vertical walls were kept at uniform but different temperatures and concentrations. The control volume method obtained to solve the governing equations. The inertial term was considered as a source term and non-uniform staggered grid discretizetion were used. An iterative conjugate gradient method was used to solve the algebraic equations at each time step. The results from the heat and mass transfer that were obtained were in good agreement with the results obtained by by Goyeau et al. [94] and Trevisan and Bejan [93] in a Darcy model. Karimi-Fard et al. observed that when the Darcy number increased, the Nusselt and Schmidt numbers decreased, whereas they decreased when the inertial parameter increased. Also, the result showed that in double-diffusive natural convection, the boundary effects are important vis-a-vis non-darcy effects, especially for high Prandtl and Schmidt numbers while the inertial effects are almost negligible. Two-dimensional double-diffusive natural convection in a square porous cavity was also studied analytically and numerically by Bourich et al. [96]. The porous cavity was heated isothermally from below and cooled from the upper surface, while a horizontal solutal gradient was subjected to the vertical walls. The Darcy model with the Boussinesq approximation was considered. The Soret and Dufour effects were neglected. A scale analysis was obtained to produce the order of heat and mass transfer. Numerically, the alternate direction implicit (ADI) method employing a finite difference scheme was used to solve the governing equations. The numerical 53

54 Chapter 2. Literature review solutions were used for validate the scale analysis solution. For the wide range of the governing parameters, correlations of Nusselt and Sherwood numbers were proposed. Important studies were also undertaken by Nithiarasu et al. [97] concerning steady double-diffusive natural convection in an enclosure filled with saturated porous media. All the walls were assumed to be impermeable and the vertical walls were subjected to constant different uniform temperatures and concentrations, while the horizontal walls were insulated. A generalized equation was obtained and Galerkin s finite element method, coupled with the Eulerian velocity correction, was used. For pure heat transfer, the results were in good agreement with the Darcy, Darcy- Forchheimer, Darcy-Brinckman, and generalized models in many previous studies. Also, they were in good agreement with the experimental results of double-diffusive convection in a cavity without porous media. The results showed that for high Rayleigh and Darcy numbers, the effects of flow, heat, and mass transfer become significant Soret and Dufour effects Over recent years, coupled heat and mass transfer in a fluid porous medium has attracted considerable attention. The thermal-diffusion (Soret) effect refers to mass flux produced by a temperature gradient and the diffusion-thermo (Dufour) effect refers to heat flux produced by a concentration gradient (Nield and Bejan [5] ). Benano-Molly et al. [98] investigated the effect of using Soret coefficient measurement experiments in unsteady heat and mass transfer in a binary fluid mixture in a rectangular porous medium. The vertical walls were set to a constant (different) temperature while the horizontal walls were adiabatic. The semi-implicit and augmented Lagrangian method based on the Uzawa algorithm were obtained used to solve the governing equation. The results showed that, depending on the Soret number value, multiple convection-roll flow patterns can develop, when thermal and solutal buoyancy forces oppose each other. When evaluating the optimum permeability for solute separation, a discrepancy occurs between numerical and experimental results. 54

55 Chapter 2. Literature review According to Platten and Legros [99], under the effect of thermal diffusion, the mass fraction gradient established is very small. However, it has a disproportionately large influence on hydrodynamic stability relative to its contribution to the buoyancy of the fluid. They also stated that, in most liquid mixtures, the Dufour effect is inoperative, but that this may not be the case in gases. Sovran et al. [100] confirmed this by noting that in liquids the Dufour coefficient is of an order of magnitude smaller than the Soret effect. They concluded that for saturated porous media, the phenomenon of cross diffusion is further complicated because of the interaction between the fluid and the porous matrix and because accurate values for the crossdiffusion coefficients are not available. An important study has also been presented by Ghorayer [101] to investigate the future of computational variation in a two-dimensional porous medium with a binary mixture of C 1 /C 4. The Darcy model with Boussinesq approximation was obtained. The governing equations were solved by using the finite volume method. For the temporal integration, the semi-implicit first order scheme was used. The results showed that the magnitude of the thermal diffusion and the pressure diffusion coefficients might enhance or weaken the convection of the horizontal composition variation. Furthermore, they showed that the compositional variation in a binary mixture may drastically differ from multi-component mixtures due to the crossdiffusion effects. Joly et al. [102] analytically and numerically studied the influence of the Soret effect on the onset of convection in a porous cavity saturated with a binary mixture. The vertical walls were subjected to uniform heat fluxes. The Brinkman-extended Darcy model for momentum equations was used to solve the governing equations. A Newtonian Boussinesq-incompressible fluid, whose density varied according to temperature gradient and concentration gradient, was used to model the binary fluid. The same magnitude of opposed solutal and thermal buoyancy forces was considered in this study. Different non-dimensional parameters were used in this study to present the influences of these parameters on the strength of convection and on the Nusselt and Sherwood numbers. These parameters were the thermal Darcy-Rayleigh number R T, Lewis number Le, Darcy number Da, and aspect ratio A. The resulting linear stability theory (numerical solution) indicated that the supercritical Rayleigh 55

56 Chapter 2. Literature review number R Tc2 depended strongly upon the Darcy number Da, the Lewis number Le, o R and the aspect ratio A of the cavity. Where RTc 2, and R o was a constant Le depending on the Darcy number Da of the porous medium and the aspect ratio A of the cavity. Furthermore, the resulting nonlinear theory (analytical solution) indicated the existence of a subcritical Rayleigh number R Tc1 for the onset of convection. The value of R Tc1, which depended upon Da and Le, was lower than the critical Rayleigh number R Tc2. Two dimensional heat and mass transfer in a porous medium cavity filled with a binary mixture of methane and n-butane were investigated numerically by Faruque et al. [103]. By including the energy equation in the model, they repeated the work of Riley and Firoozabadi [104] and Ghorayeb and Firoozabadi [105]. The governing equations were solved by using the control volume method. A spatial discretization was performed using a second order central scheme. The convection and diffusion terms were approximated by using the power law scheme. In their study Faruque et al. found that, for lateral heat, the Soret effect was weak, whereas it was more pronounced in the bottom heated condition cases; therefore it should not be neglected. The Soret effect in a binary fluid saturating on natural convection within a horizontal porous domain subject to cross fluxes of heat and mass was studied analytically and numerically by Bennacer et al. [106]. The Darcy model was used to solve the governing equations and the density variation was taken into account by the Boussinesq approximation. The governing parameters of the problem were the thermal Rayleigh number R T, the aspect ratio A, the Lewis number Le, the buoyancy ratio N, and the Soret coefficient N S. The analytical solution was based on the parallel flow approximation, and the numerical solution used the control volume approach to solve the governing equations. The results showed a good agreement of the analytical solution with the numerical solution of the full governing equations. When the vertical solutal gradient was destabilizing ( N 0 ), the existence of both natural and antinatural flows was demonstrated. On the other hand, when the vertical solutal gradient was stabilizing ( N 0), multiple study state solutions were possible 56

57 Chapter 2. Literature review (depending on which initial conditions are used). Furthermore, the flow pattern depended strongly on the magnitude of R T, N and N S. The Soret effect on double diffusive natural convection in a square cavity was studied by Mansour et al. [107]. The vertical walls were adiabatic and maintained to constant but different concentrations, while the horizontal walls were maintained at constant temperatures. A finite difference method was used to solve the equations. Alternating direction implicit (ADI) methods were used to solve the energy and species equations. The results showed a strong effect on heat and mass transfer and on the multiplicity solutions in the porous enclosure due to the Soret parameter. Numerical investigations of two-dimensional heat and mass transfer in a heated vertical cavity filled with mixture of methane and n-butane were undertaken by Jiang et al. [108]. This study showed the interaction between buoyancy convection and the soret effect. Thermal, molecular and pressure diffusion coefficients were the functions of temperature, pressure and other properties of the mixture by using the irreversible thermodynamics theory of Shukla and Firoozabadi [109] (Firoozabadi et al. [110]). The control volume method was used to solve the governing equations. The second order centered scheme was used in the space discretization and the power low scheme was used for the non liner convection terms. The results showed that, as the permeability increases, the component separation process increases, it reaches its peak, and then decreases due to the Soret effect. Furthermore, the lighter fluid components migrate to the hot side of the cavity. Important studies were also undertaken by Er-Raki et al. [111] to show the influence of the Soret effect in a porous cavity. Analytical and numerical solutions, based on the Darcy model with the Boussinesq approximation, were used to solve the governing equation in a vertical tall porous cavity subject to horizontal heat and mass fluxes. The governing parameters were the thermal Rayleigh number R T, the buoyancy ratio N, the Lewis number Le, the Soret parameter M, and the aspect ratio A r. The results showed the strong effect of the Soret parameter on the vertical boundary layer thickness. When ( N 0), the boundary layer thickness increased when the Soret parameter increased. Otherwise, the boundary layer thickness decreased when the Soret parameter increased for ( N 0 ). 57

58 Chapter 2. Literature review Later, the effects of Soret and Dufour in double-diffusive free convective heat and mass transfer from a vertical plate embedded in an electrically conditioned saturated fluid was studied analytically by Partha et al. [112]. Along the plate, the uniform magnetic field was forced. The Darcy-Forchheimer model was selected and the governing equations were solved by using the similarity solution technique. The results were presented over a wide range of Lewis numbers. Narayana and Murthy [113] investigated the effect of Soret and Dufour on natural convection from a horizontal flat plate embedded in saturated porous medium. Because of their initial assumptions, the Darcy model was used for the low permeability. The similarity solution was used to solve the governing equations and the wall was assumed to be at a constant temperature and concentration. They observed, from the results, that the Nusselt number increased linearly with an increasing Dufour parameter for aiding buoyancy. However, it decreased nonlinearly with the Lewis numbers. Also, it increased when the Soret parameter increased. On the other hand, the Sherwood number increased nonlinearly with increasing Lewis numbers and decreased linearly with increasing Soret and Dufour parameters. Heat and mass transfer of natural convection about a vertical surface embedded in porous media has been studied analytically by El-Arabawy [114]. A surface temperature distribution was obtained from the effects of the diffusion-thermo and thermal-diffusion. Fourth order Runge-Kutta method alongside the Nachtsheim- Swigert shooting technique was used to solve the set of coupled non-linear equations. The results showed that when the ( in the surface temperature equation x ) decreases the Nusselt number decreases while the Sherwood number increases. Furthermore, for the thermally opposing flow, both the Nusselt and Sherwood number increases when the Lewis number increases. However in the case of thermally assisting flows, when the Lewis number increases, the Nusselt number decreases and the Sherwood number increases. Furthermore, as mentioned above, the thickness of the thermal boundary layer decreases with thermally assisting flows than with the thermally opposing flow. Previously, Tsai and Huang [115] analytically studied the Soret and Dufour effects in natural convection flow over a vertical plate with power low heat flux embedded 58

59 Chapter 2. Literature review in porous media. They transformed the continuity, momentum, energy and concentration into a set of coupled equations. Similarity analysis was used to solve the set of coupled equations. The results showed that the Soret and Dufour effects play a significant role when the fluid through the porous media has a light or medium molecular weight. Also, increasing the Dufort number or increasing the Soret number led to increasing the temperature profile, as well as to decreasing the concentration distributions. Many of mathematical studies presented last year in terms of showing the Soret and Dufour effects on free convection heat and mass transfer from vertical/inclined plate in a porous medium or for the outside flow over a vertical truncated cone conclusions from literature review by Cheng [ ], Malashetty et al.[120] Conclusions from literature review Darcy and non-darcy flow in porous cavities There is wide range of experimental, analytical and numerical studies in natural convective heat transfer in porous cavities. These studies are based on different boundary conditions (e. g. uniform heat flux subjected to vertical walls or from the bottom and constant temperature differences placed upon vertical walls). Also, different models are used to solve the governing equations. In general, these studies show that the Nusselt number is increased when the Rayleigh number Ra and the Prandtl number Pr are increased. Whereas, the Nusselt number Nu are decreased when the aspect ratio A and Darcy number Da are decreased Inclined porous medium The effects of the inclination angles on the convective heat transfer and the doublediffusive natural convective heat and mass transfer have not been studied in wide range of parameters. Different models have been used to solve the governing equations. Also, these studies are based on different boundaries. In general, these studies have shown that for the heat transfer cases and when the large Rayleigh numbers obtained, the maximum Nusselt number occurs at around = 0, or it occurs at around = 60 for low Rayleigh numbers. For the heat and mass transfer 59

60 Chapter 2. Literature review cases in tall porous cavities, the results showed that as the cavity inclination angle increased; reductions in the average Nusselt and Sherwood numbers are achieved Variable porosity In the Darcy model, no difference is observed in the average Nusselt number with variation of porosity, especially at low Rayleigh number. On the other hand, the porosity significantly affects the flow and heat transfer in a porous medium at a higher Darcy number. The non-darcian effects, i.e. Brinkman friction and Forchheimer inertia are more pronounced for high porosity media than for low porosity media Conjugate problems The conjugate heat transfer phenomena in porous media have received increasing attention in recent years and are of practical importance in, for example, high performance insulation for buildings and cold storage installations. Studies have shown that the Nusselt number decreases when the wall thickness increases. Furthermore, the mean Nusselt number becomes less dependent on the Rayleigh number when the thermal conductivity ratio k decreases Double-diffusive There are a wide range of analytical and numerical studies in the double-diffusive of natural convective heat and mass transfer in a porous medium. These studies are based on different types of boundaries. Also, different models have been used to solve the governing equations. In general, these studies have shown that the Nusselt (Nu) and Sherwood (Sh) numbers increase when the Rayleigh number (Ra) and Prandtl number (Pr) increase, whereas the Nu and Sh decrease when Darcy number (Da) increased. Moreover, when the Lewis number (Le) increases, the Sh increases, while the Nu decreases. 60

61 Chapter 2. Literature review Soret and Dufour effects The thermal-diffusion (Soret) effect refers to mass flux produced by a temperature gradient and the diffusion-thermo (Dufour) effect refers to heat flux produced by a concentration gradient. The results showed the strong effect of the Soret parameter on vertical boundary layer thickness. Therefore, boundary layer thickness is increased when the Soret parameter is increased and when the buoyancy ratio is positive. Otherwise, boundary layer thickness is decreased when the Soret parameter is increased for a buoyancy ratio less than zero Conclusions The conclusion reached in this section, together with the previous literature review undertaken, is that steady/unsteady double-diffusive natural convective heat and mass transfer in a two-dimensional constant/variable porosity in a vertical/inclined porous rectangular cavity, with Soret and Dufour effects is a good starting point for this study; in addition to adding in the factor of the thickness of the vertical walls to add the convection-conduction effect (conjugate phenomena effects). This model will be used to design the porous media that will enable optimum heat transfer from one side to another side. The Forchheimer Brinkman-extended Darcy (generalized) model has been selected to solve the governing equations in the saturated porous region. The flow is driven by a combined buoyancy effect due to both temperature and concentration variations. The density variations are described by the Boussinesq approximation. Correlations to evaluate the average Nusselt and Sherwood numbers have been proposed as a function of the Rayleigh number, the Darcy number, the Lewis number, the buoyancy ratio, the Soret and Dufour coefficients, and the thermal conductivity ratio, including a number of physical, geometrical and material property ratios. 61

62 Chapter 3 3. Basic Models of Flow in a Porous Medium and the Governing Equations 3.1. Introduction Fluid flow through a porous medium is a longstanding subject and it has been studied widely using different models of flow in a porous medium. In this chapter, there is a brief introduction to these basic models and the mathematical formulations of the two-dimensional governing equations which have been used in this study are also presented. Moreover, the energy equations with the Dufour effects are also presented in nondimensional form as well as the mass transfer equations including the Soret effects.

63 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations The conjugate problems are one from the important engineering applications, and for that reason it has been taken into account in addition to the effect of variable porosity and thermal conductivity ratio Continuity equation for porous media As compared with pure fluids, the continuity equation in a porous medium remains unchanged, except that it is formulated in terms of the physical velocity of the fluid as it travels through the pores. The continuity equation is derived based on the conservation of mass law which can be shown by: f t 0 f u (3.1) where f is the fluid density. For an incompressible flow the above equation can be reduced to the following form: u 0 (3.2) 3.3. Momentum equations for porous media Darcy s law Over a century ago, a great number of studies looked at convective heat transfer in porous media which used the classical Darcy formulation (Nield and Bejan[5] and Whitaker [121]). The constant in Darcy's equation was proved later by Muskat and Wyckoff [12] and should be related to the permeability of the porous material. For an isotropic medium, the original form of the equation has been re-written as: p u (3.3) K where K is the permeability, and p is the pressure gradient. 63

64 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations Although flow through many natural occurring porous media can be described by using Darcy s law, it is not valid for all types of situations. Accordingly, many researchers in the past had already noticed the inappropriateness of Darcy s law and had started to propose new models Darcy-Forchheimer s law Many of scientific community agree on the need to modify Darcy s equation. This modification takes effect on the non-linear terms in the momentum equation. Basically, the pressure drop occurring in a porous medium is composed of two terms; one term is being linear in the velocity and the other one is being non-linear in the velocity. The physical explanation for the non-linear term is still not completely understood. General expressions of the non-linear terms are usually introduced through Forchheimer s equation: c p K K F u f u u (3.4) where the u is the magnitude of velocity and c F is a non-dimensional form-drag constant. In the last three decades, different expressions have been proposed for Forchheimer s equation. Ergun s equation (Ergun [122]) is one of the famous forms of the Forchheimer s equation was derived, and appears in this particular form: 2 p 1 u 1 f uu (3.5) L d d s s where d s is the average size of the solid particles. For laminar flow through a packed bed, the second term in the above equation may be dropped, resulting in the Blake-Kozeny equation (Bear [2] and Ergun [122]) : 64

65 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations 1 2 p u 150 L d 3 2 s (3.6) The left term in RHS of equation (3.5), the liner term is equal to left term in the RHS of equation (3.4) when the Carman-Kozeny equation has been used (Kaviany [1]). K 2 3 d s (3.7) Darcy-Brinkman s law A significant work for predicting momentum transport in porous media was undertaken by Brinkman [4] in Brinkman first introduced a term which superimposed the bulk and boundary effects together for flows with bounding walls. A relationship between permeability and porosity was obtained by Brinkman [123], [4]. Where there is large porosity and permeability in the medium, the effect of viscosity becomes great. Brinkman initially assumed the effective viscosity to be equal to the fluid viscosity. He presented a new model which is called Brinkman's equation or Brinkman's extension of Darcy's law. e p= K 2 u u (3.8) where e is the effective viscosity. There are usually two weaknesses pointed out concerning the Brinkman equation. The first one relates to the procedure used by Brinkman to derive it. Actually, the Brinkman model is valid only for high porosity (porosity >0.6) (Nield and Bejan[5]). The second problem relates to the value of the effective viscosity, which is known to be dependent on the geometry of the medium, but which is not experimentally well documented (Kaviany [1]). Porous medium geometry effects on effective viscosity is still subject to investigation (Martys et al. [124] and Alazmi and Vafai [125]). 65

66 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations The generalized model Natural and manufactured porous materials have broad applications in engineering processes. Although natural porous media has low porosity which means that the fluid flow can be solved by using Darcy s law, many manufactured porous materials have high porous media. For high porosity, Darcy law is not valid and the effects of inertia and viscosity force should be taken into account. In general, porous media can have porosity from 0.02 up to 0.99 (Nield [5] and Kaviany [1]), therefore, the generalized model is extremely suitable in describing all kinds of fluid flow in a porous medium. The general form of the momentum equation of incompressible fluid in saturated variable porosity can be derived by averaging the Navier-Stokes equations over the representative elementary volume (REV). The procedure introduced by Whitaker [126] and Tien and Vafai [127] can be used to drive the generalized momentum equation. For fluid saturated variable porosity, the generalized momentum equation can be written as: f u uu t 1 e 2 c f F uu p u u f ( ) 12 ref f g K K (3.9) where f is the fluid density, u is the Darcy velocity vector, g is gravitational acceleration, P is fluid pressure, c F is the Forchheimer s coefficient, and e is the effective viscosity. Ergun s correlation was initially introduced for packed beds. His correlation is used to represent the total drag force of the solid matrix on the fluid. In this case the coefficient c F can be written as: c F 1.75 (3.10) Also, the Carman-Kozeny equation can be used for the permeability, K (equation(3.7)). 66

67 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations Hsu and Cheng [128] derived theoretically the generalized momentum equation and it has been extensively used in the literature (Gartling et al. [129], Hsu and Cheng [128] and Nithiarasu et al. [70], [89], [69, 130]) Heat transfer Heat transfer through a porous medium is a large topic and there are a large number of engineering applications that uses such a factor such as: solidification of binary mixtures, insulation, dehumidification and heat pipes etc. (Kaviany [131]). The energy equation for a saturated porous medium is derived by using the volume averaging procedure including the Dufour effect, and can be written as (Nield and Bejan[5]): c c p eff TC p f T t u (3.11). T. T D C where eff (3.12) k c c p c p (1 ) c p p eff f s f (3.13) k k (1 ) k (3.14) eff f s where T is the temperature, C is the concentration, (c p ) eff, (c p ) s, (c p ) f is the effective, solid and fluid volumetric heat capacity respectively, k eff is the effective conductivity of the saturated porous, is the thermal diffusivity, DTC is the thermodiffusion coefficient, and DTC is the Dufour coefficient. In variable porosity cases, thermal conductivity is used as a function of porosity. Cheng and Hsu [132] s expression of thermal conductivity has been used and it can be obtained by: 67

68 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations k eff 1 p 1 k 2 pb 2 1 k B 1 B 1 B ln k 1k pb k pb 2 1k pb f (3.15) where k p is the ratio of fluid thermal conductivity to solid thermal conductivity, and the parameter B can be defined as a function of the medium porosity by: B (3.16) 3.5. Mass transfer For many applications, especially when using binary fluid/mixture and/or multiphase flow as a working fluid in a porous medium, the species equation needs to be solved as well as the mass, momentum and energy equations. The transport species equation is similar to the heat transfer equation. The study of the components (component i) in the mixture can be defined in terms of its concentration in the medium: C i = m i /V. Specifically, the macroscopic equation for the calculation of concentration can be derived from the volume averaging method, including the Soret effect, under the assumption of local chemical equilibrium. The species equation can be written as (Nield and Bejan [5]): C u. C. DC C DCT T (3.17) t where D c is the effective mass diffusivity coefficient of the component and D CT D is the Soret coefficient. C 3.6. Non-dimensional parameters The non-dimensional procedure is one of the formulas that have commonly been used. It is used for non-dimensionalize governing equations before their solution, especially for complex problems which have several variables. The non-dimensional process involves the choice of different parameters. The effects of the choice of these 68

69 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations parameters on non-dimensional equations are very important in order to show the effect of physical properties and which parameters should be focussed on in order to present these physical properties. The present cases study phenomena that have completely different scales of heat, mass and fluid flow as well as different scales of conjugate heat transfer. The following parameters have been employed to obtain the non-dimensional equations of double-diffusive natural convective heat and mass transfer in inclined enclosures filled with saturated variable porous medium (as shown in Figure 3-1) and they are given as: X x, L Y y, Y L 2* D, L A H, L D d s d, D1 L L U ul, U, Y V vl, V, X P 2 pl, 2 k k 1 k, eff f s C k eff c p f, Pr Le D, t kw C C, k L 2 r, N k T T T T h T T c c C C C C C, c h c f, T, Ra g TL 3 T,, kw cp w k c cp 1 cp f c p s, f eff p f S r TD. CT, CD. C D f CD. TC T. where L, H is the characteristic width and high of the cavity respectively, d is the wall thickness, d s is the solid particles diameter, and c, T are the coefficients of the concentration and thermal expansion respectively. In addition, is the dynamic viscosity of the fluid, x, y are the position vectors and u and v are the velocity, g represents the magnitude of the gravitational vector, T h, T c, C h, and C c are the hot and cold wall temperatures and concentrations respectively. Also, k eff, k w, k f, k s are the thermal conductivity of effective, wall, fluid and the solid, respectively. 69

70 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations The above quantities were used to define the non-dimensional variables: Le is the Lewis number, Ra is the Rayleigh number, Pr is the Prandtl number, and N is the buoyancy ratio. Also, A is the aspect ratio, D is the dimensionless wall thickness, D 1 is the dimensionless diameter, is the wall to porous media heat capacity. Finally, S r and D f are the Soret and Dufour parameters, respectively. Figure 3-1 The geometry of the model 3.7. The model The study of heat and mass transfer through porous media is based on the conservation of various quantities, such as mass, momentum, energy and species. Isotropic, homogeneous, local thermal balance, and saturated with an incompressible fluid has been assumed. The Soret (mass flux due to temperature gradients) and Dufour (heat flux produced by concentration gradients) effects are not negligible. All the properties have been assumed to be constant except for the density. Therefore, 70

71 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations the governing equations were derived and developed in the non-dimensional form, as shown below. The flow is driven by a combined buoyancy effect due to both temperature and concentration variations. The density variations are described by the Boussinesq approximation: o 1 T ( T To ) 1 C ( C Co ) (3.18) where T 1 T PC, is the thermal expansion coefficient. and C 1 C PT, is the concentration expansion coefficient. By using the above non-dimensional parameters, the continuity equation can be written as: U X V Y 0 (3.19) The generalized model has been selected to solve the momentum equations. The purpose for choosing this model is to solve all the kinds of single phase fluid flow in a porous medium by using only one equation including the temporal terms in the momentum equation as well as the inertia and viscose terms. Some authors such as Vafai and Kim [133] and Nield [134] have argued about the inappropriateness of including temporal term in the momentum equation, which was added in order to make the model more generally. In their studies, it is clear that they do have some effect during the development of the boundary layer and the transient solutions. Also, regarding problems of fluid flow in porous media, these terms produce appropriate effects naturally (Gartling et al. [129]). From equation (3.9) and by using the same derivations as presented in Chapter 8 from book of Lewis et al. [135] and also by using the above non-dimensional parameters, the momentum equations in the X and Y-directions can be written in nondimensional variable porosity form as: 71

72 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations X-momentum equation: 1 U 1 U U 1 U V P X Y X U Da 150 Da X Y Ra PrT sin( ) N Ra Pr C sin( ) Pr 1.75 U V U Pr U U 3/2 2 2 (3.20) Y-momentum equation: 1 V 1 V V 1 U V P X Y Y V Da 150 Da X Y Ra PrT cos( ) N Ra Pr C cos( ) Pr 1.75 U V V Pr V V 3/2 2 2 (3.21) Energy equation: U V X Y X Y 2 2 C C Df 2 2 X Y 2 2 T T T T T 2 2 (3.22) Species conservation: U V X Y Le X Y 2 2 C C C 1 C C Le X Y 2 2 S r T T (3.23) The energy equation for the walls: T T T X Y W W W (3.24)

73 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations The heat flux and the temperatures at the solid-porous media interface must be continuous. k w T X w k eff T X (3.25) where the subscript w refers to the wall. For the non-linear matrix resistance, Ergun s correlation (3.5) has been used in the momentum equations. For variable porosity the Chandrasekhara and Vortmeyer [136] exponential relation has been used. N 1 1 X C1 exp D 1 (3.26) where X is the normal dimensionless distance to the hot wall and is the average porosity. C 1 =1.4 and N 1 =5 are the constants used by Hus and Cheng [137] where the solid particles diameter (d s ) equals 5 mm and D1 d s L is the dimensionless diameter. Because of the variation in porosity, the Darcy number can be obtained by: Da d D s L (3.27) In general, the Nusselt number is defined as the ratio between the heat transferred in the convection mode, with respect to that in the conduction if the fluid does not Qcon move,( Nu ), therefore the local Nusselt number ( Nu l ) at the walls can be Q cond defined as follows: Nu l T X D f C X x (3.28) and the average Nusselt number on the wall can be easily obtained from: A hl 1 T 1 C Nu Y D Y k A X A X eff f 0 0 A (3.29) 73

74 Chapter 3. Basic Models of Flow in a Porous Medium and the Governing Equations The Sherwood number represents the concentration gradient on the surface. As with the local Nusselt number, the local Sherwood number ( Sh ) can be defined as the following: l Sh l C X S r T X x (3.30) and the average Sherwood number on the wall can be easily obtained from: A 1 C 1 T Sh Y S Y A X A X r 0 0 A (3.31) 74

75 Chapter 4 4. Numerical Implementation 4.1. Introduction Numerical calculations of the mass, momentum, energy and species equations were solved by using an in-house FORTRAN code, named ALFARHANY code, which was developed for these studies and the whole procedure is explained in detail below. This code is used to solve the simple flow, heat and mass transfer cases in the porous medium as well as the Soret and Dufour effects. The code is based on a finite volume method. Therefore, the solver algorithm, discretization and solution of a transport equation, the pressure-based Solver, the pressure-velocity coupling, under relaxation and the percentage errors are provided briefly in Appendix B. Originally, the alternating direction implicit (ADI) method is a finite difference method. Because of the code is based on a finite volume method, and for that reason the ADI method is presented in details in this chapter.

76 Chapter 4. Numerical Implementation Finally the grid sensitivity convergence criteria are presented in the last section of this chapter ALFARHANY code ALFARHANY code was developed for this research to carry out the twodimensional simulations of heat and mass transfer in inclined porous enclosure with Soret and Dufour effects. The code was based on the finite volume and it can analyze the flow through porous medium under transient and steady state conditions. Also, the code was developed to solve the momentum equations using Darcy model and Darcy extension models including the Darcy-Brinkman model, the Darcy- Forchheimer model and the generalized. For the pressure velocity coupling, this code is able to employ the SIMPLE and SIMPLER algorithm. A second-order central differencing discretization scheme is used for the momentum, the energy and the species equations while, a semi-implicit first-order scheme is used for time step advancement in the momentum equations, and the alternating direction implicit (ADI) technique is adopted for the energy and species equations. The ADI technique was used in order to incorporate the effects of boundary conditions and also to provide further stability as well as refining accuracy due to the fact that the solver is second-order accurate in time steps as well as in X and Y components. The staggered-grid schemes was used in this code similar to the staggered-grid schemes of Patankar [138] Solution Procedure Initially ALFARHANY code needs to be classified which kind of flow through the porous medium needs to be solved by choosing the Darcy model or one of the non- Darcy (Darcy extension) models. Before the code starting solve the equations, it needs to inter the input file of all the non-dimensional variables including: Rayleigh number (Ra), Darcy number (Da), Prandtl number (Pr) and thermal codactivity ratio (k r ). Also, for the mass transfer case studies it needs to inter the Lewis number (Le) and buoyancy ratio (N) as well as the Soret parameter (S r ) and Dufour parameter (D f ). Furthermore, regarding to the 76

77 Chapter 4. Numerical Implementation geometric of the porous cavity it also need to add the aspect ratio (A) and the angle of the inclination () and the wall thickness (D). In general, the code uses these variables to create new temporary and output files and these files will be named depending on the variable itself. The effects of variable thermal conductivity, permeability and variable porosity have been undertaken in this code. Therefore equations(3.15), (3.26) and (3.27) are solved and the magnitude of thermal conductivity, variable porosity and Darcy number can be saved in the center of each control volume. Before the solution of the discretized transport equations starts, it is necessary to impose the boundary conditions as well as initializing the variables including P*,V*,U*, T, and C. Then the discretized of the velocity components can be solved. In terms of using the SIMPLER algorithm, the corrected velocities u ˆ and v ˆ can be solved using the guessed velocities and then using them to obtain the pressure field. Then treating this pressure as guessed pressure P*. The above steps are the main different between the SIMPLER and SIMPLE algorithms. After that for both algorithms, it can be used this P* to calculate the U* and V*. Additionally, the equation for P must be solved to impose continuity. Then the velocities field can be modified based on these pressure corrections. These modified velocities can be used to correct the pressure field when the SIMPLE algorithm obtained and the pressure field must not be corrected when the SIMPLER algorithm are used. These algorithms are presented in detail in Appendix B. Then at the same time step and by using the new values of velocities, the code carries on solving the energy and species equations by using ADI method and temperature and concentration field are computed. This process must be repeated until the steady state is reached. Finally when the convergence criteria are reached, the stream function () and the average Nusselt and Sherwood numbers are calculated and the post processing are presented in details in Appendix C The alternating-direction-implicit technique (ADI) The specific technique has been used to solve the energy and species equations in order to add the effects of boundary conditions. The alternating-direction-implicit 77

78 Chapter 4. Numerical Implementation technique (ADI) is one of the implicit solutions and it is second order accurate in t, X, Y ; O(t) 2, O(x) 2, O(y) 2 John and Anderson [139]. Equations (3.22)-(3.24) have been solved by using this technique to increase the stability of the solution in additional to get more accuracy. The implementation of the solution algorithm to substantially increase the time step is achieved via the two steps process shown in Figure 4-1. For example equation (3.22) (without Dufour effect) can be written in discretized form as: a T a T a T a T a T a T (4.1) o o p p W W E E S S N N P P where 1 Uw a A A X 2 W w w (4.2) 1 U e a A A X 2 E e e (4.3) 1 U s a A A Y 2 S s s (4.4) 1 U n a A A Y 2 N n n a 0 P V (4.5) (4.6) a a a a a a F (4.7) o P W E S N P where F U eae Uw Aw U nan U sas (4.8) A A Y ; A A X ; V X Y (4.9) e w n s In the first step, equation (4.1) is discretized implicitly in the direction of Y-axis over a time interval l+/2, while the terms in the direction of X-axis are kept as 78

79 Chapter 4. Numerical Implementation those are of the old time step as shown in Figure 4-1 (a). The energy equation for the first step: p p W W E E S S N N l 12 l l l 1 2 l 1 2 l a T a T a T a T a T b (4.10) This equation is used to solve for the temperature l 12 T p in the internal domain. The direction of the numerical solution is switched to the Y-direction as shown in Figure 4-1(c). Figure 4-1 (b) displays the second step for the solution of the 1 equations finally yielding temperaturest. The equation (4.1) is derived implicitly l p in the X-direction over a time interval from l+/2 to l+1, while the terms representing the Y-direction refers to the old time l+ /2. The energy equation for the second step: p p W W E E S S N N l 1 l 1 l 1 l 1 2 l 1 2 l 12 a T a T a T a T a T b (4.11) For the second step, the direction of numerical solution in the X-axis is shown in Figure 4-1(d), where the influences of the horizontal boundary conditions influence the solution as compared to the vertical boundary conditions. All these systems of equations are set of equations that can be solved with the efficient Gauss-Seidel iteration Grid Following grid sensitivity tests for the cases of heat transfer in square porous cavity (without the effects of wall thickness). In the porous cavity, different uniform mesh sizes were employed ranging from 41*41 to 121*121. An 81*81 grid for the low Rayleigh and Darcy numbers and a 121*121 grid for the high Rayleigh number were deemed adequate as shown in Figure Convergence criteria The percentage error is used as convergence measure in this study and it can be defined at each node as: 79

80 Chapter 4. Numerical Implementation i i j n1 n i, j i, j j n 1 i, j (4.12) where could be any variable. In general, the steady state was considered to have been arrived at when the changes in U, V, P, T and C (for the mass transfer cases) satisfied this equation. The above equation was also used to obtain the convergence of the solution at each time step when the time effects were taking into account (unsteady state cases). Figure 4-3 shows the convergence of the solutions of the test case of Doublediffusive natural convective heat and mass transfer in a square cavity filled with porous medium for Da=10-6, Ra*=500, and Le=2.0 where N= 2.0. The solution runs for this testing case until a maximum fifteen iterations in each time step or until satisfied the percentage error for all variables. Also, to obtain stable convergence behaviour for the solution, various under-relaxation parameters were used. For the U, V-velocities, pressure, the energy and the species equations, the under-relaxation parameters adopted for the most cases were 0.2, 0.2, 0.3, 0.5 and 0.5, respectively. 80

81 Chapter 4. Numerical Implementation Explicit Implicit l 1 2 t y i+1 y i l t y i- x i1 x i x i+1 (a) first step (b) second step 3 3 y j i x (c) first step direction j i (d) second step direction Figure 4-1 Show the first and second step in ADI process. 81

82 Chapter 4. Numerical Implementation Nu mesh sizes Figure 4-2 The variations of the average Nusselt number on the left wall with mesh sizes. Figure 4-3 The convergence of the solutions 82

83 Chapter 5 5. Validation Studies 5.1. Introduction Numerical solutions can be valuable for all those situations where it is difficult to perform experiments. In addition they can save money. The generalized model of convective heat transfer, natural convection double-diffusive porous medium flows with/without Soret and Dufour effects and the model s solution procedure using the finite volume method and the SIMPLE and SIMPLER algorithm have been presented in the previous chapters. ALFARHANY code developed for these studies and it was validated using previous studies. This chapter presents a validation to the different cases of unsteady natural convective heat transfer, conjugate, variable porosity, inclination angles of the cavity, double-diffusive natural convective heat and mass transfer and Soret effects in two-dimensional cavities. The comparison is carried out for the model using reliable experimental, analytical/computational available studies in the literature. These studies are presented within six groupings and these are: validations of natural convective heat transfer using Darcy and non-darcy flow in porous cavities, variable

84 Chapter5. Validation Studies porosity, conjugate problems, double-diffusive, inclined cavity and, finally, the Soret effects Validations of convective heat transfer in porous cavities cases ALFARHANY code developed for convective heat transfer studies was validated using the previous studies. The model was constructed by using a two-dimensional square cavity filled with an isotropic porous medium as shown in Figure 5-1. The horizontal boundaries of the cavity were adiabatic and the vertical walls were maintained at fixed different temperatures T h and T c. The results for natural convective heat transfer in porous cavity are presented in Table 5-1 to 5-4. The results from this study are in good agreement for most of the cases using Darcy, Darcy-Brinkman, Darcy-Forchheimer and generalized models. Y adiabatic wall Porou medium T h L g T c L adiabatic wall X Figure 5-1 The geometry of the model 84

85 Chapter5. Validation Studies Table 5-1 Comparison of average Nusselt number at steady state with some previous results for Darcy model (pure heat transfer, N=0) Author Nu Ra*=10 Ra*=50 Ra*=100 Ra*=500 Ra*=1000 Walker and Homsy [38] Lauriat and Prasad [140] Trevisan and Bejan [141] Nithiarasu [69] Present work Table 5-2 Comparison of average Nusselt number at steady state with some previous results for Darcy-Brinkman model (N=0, Pr=1) Nu Author Da=10-6 Da=10-2 Ra*=10 Ra*=100 Ra*=1000 Ra*=10 Ra*=100 Ra*=1000 Lauriat and Prasad [35] Nithiarasu [69] Present work

86 Chapter5. Validation Studies Table 5-3 Comparison of average Nusselt number at steady state with some previous results for Darcy- Forchheimer model (N=0, Pr=1) Nu Author Da=10-6 Da=10-2 Ra*=10 Ra*=100 Ra*=1000 Ra*=10 Ra*=100 Ra*=1000 Lauriat and Prasad [35] Nithiarasu [69] Present work Table 5-4 Comparison of average Nusselt number at steady state with some previous results for generalized model (N=0, Pr=1, Ra*=10 4 ) Author Nu Da=10-1 Da=10-4 Lauriat and Prasad [35] Beckerman [142] Nithiarasu [69] Present work

87 wall wall Chapter5. Validation Studies 5.3. Validations of conjugate heat transfer cases As another check on the accuracy of the code for conjugate natural convective heat transfer studies, the effect of the thickness of the finite walls was compared with those in the study undertaken by Saeid [82]. The model comprises a two-dimensional porous cavity sandwiched between two finite thickness walls filled with an isotropic porous medium as shown in Figure 5-2. All the walls were assumed to be impermeable. The horizontal boundaries of the cavity were adiabatic and the vertical walls were maintained at fixed different temperatures with T h at the left wall and T c at the right wall. Constant porosity and pure heat transfer cases were undertaken by using the Darcy model for different wall thermal conductivity ratios and wall thickness to cavity length. Figure 5-3 shows that the isotherms and streamlines of the present study matched fairly well with Saeid's predictions. Also, the average Nusselt numbers of the wall-porous media interface were in good agreement with Saeid with a maximum deviation of about %. y Adiabatic wall Porous medium T h T c H g d Adiabatic wall L d x Figure 5-2 The geometry of the model of Saeid [82] 87

88 Chapter5. Validation Studies Saeid results ( Nu w =1.888, Nu P =1.888) (a) Present results ( Nu w =1894, Nu P =1.893) Saed results ( Nu w =1.023, Nu P =10.238) (b) Present results ( Nu w =1.0251, Nu P =10.250) Figure 5-3 Comparison of present results with Saeid [82] results for isotherms (left), and Streamlines (right) by using Darcy flow at Ra*=1000, a-(d=0.2, Kr=1), and b-(d=0.1, Kr=10). 88

89 Chapter5. Validation Studies 5.4. Validations of natural convective heat transfer in inclined cavity cases As a further check on the accuracy of ALFARHANY code for natural convective heat transfer in inclined porous cavity, the effects of the inclination angle were compared with the result from Moya et al. [49]. The model comprised a two-dimensional porous rectangular cavity filled with an isotropic porous medium as shown in Figure 5-4. The two opposing walls were assumed to be impermeable and they were maintained at fixed different temperatures while the other two boundaries of the cavity were adiabatic. The Darcy model was used to solve the governing equations. Table 5-5 shows that the results of the average Nusselt numbers on the hot wall for the modified Rayleigh number (Ra*)=100 are in good agreement with Moya et al. [49] and with Caltagirone and Bories [50]. Moreover, Figure 5-5 shows that the isotherms and the streamlines of the present study match fairly well with Moya et al. s predictions for Aspect ratio(d)=3, Ra*=100 and the incitation angle () =10. Figure 5-4 The geometry of the model of Moya et al. [49]. 89

90 Chapter5. Validation Studies Table 5-5 Comparison of variations of average Nusselt Number with α for Ra*=100. D α Moya et al. [49] Nu Caltagirone and Bories [50] Present work

91 Chapter5. Validation Studies Moya et al. predictions Present study (a) Streamline Moyaet al. predictions Present study (b) Isotherms Figure 5-5 Comparison of (a) Streamlines and (b) isotherms lines for D=3, Ra*=100 and =10 with the work of Moya et al. [49] work 91

92 Chapter5. Validation Studies 5.5. Validations of variable porosity effects on heat transfer cases The effects of porosity variations on boundary layer heat transfer are important and this factor was mentioned in the previous studies. These effects were added into this study s ALFARHANY code. In order to check on the accuracy of the code with the variable porosity effects, a model was undertaken by using a rectangular cavity filled with an isotropic porous medium as shown in Figure 5-6. Two different studies from the experimental work of Seki et al. [22] had been used as a comparison with available experimental, analytical and computational studies in the literature. For the case I; the height of the rectangle was 580 mm and the width was 116 mm, while the height and width of the rectangle for case II were 570 mm and 57 mm, respectively. adiabatic wall T h H T c adiabatic wall L Figure 5-6 The geometry of the experimental work of Seki et al. [22] 92

93 Chapter5. Validation Studies Water was selected to be used as a working fluid and spherical solid particles - from glass beads - of 16.4mm and 5.07mm diameter were used with porosity equal to 0.46 and 0.39 for case I and case II respectively as shown in Table 5-6. The horizontal boundaries of the cavity were adiabatic and the vertical walls were maintained at fixed different temperatures. Table 5-6 Geometry and water properties of the experimental work of Seki et al. [22] fluid d p L D 1 A k f K eff Pr Case I water 16.4 mm 116 mm Case II water 5.07 mm 57 mm Table 5-7 shows that the present results are in good agreement with the experimental and numerical studies for variable porosity pure heat transfer cases. Table 5-7 Comparison of average Nusselt numbers with some previous studies on variable porosity. Nu Ra Seki [22] Darcy Seki[22] Vafai [59] Marcondes [71] present work 6.432* Case I 9.262* * * * Case II 3.519* *

94 Chapter5. Validation Studies 5.6. Validations of double-diffusive natural convection heat and mass transfer cases ALFARHANY code developed for double-diffusive natural convective heat and mass transfer in a porous cavity was validated using previous studies. The Nithiarasu [97] model has been used for these comparisons. The model was constructed by using a two-dimensional square cavity filled with an isotropic porous medium as shown in Figure 5-7. The horizontal boundaries of the cavity were adiabatic and the vertical walls were maintained at fixed different temperatures and concentrations, T h and C h at the left wall and T c and C c at the right wall. The Brinkman- Forchheimer-extended Darcy (generalized) model was used to solve the governing equations. The code was undertaken to validate two different cases. The first case was undertaken by using the Darcy model without the effect of buoyancy ratio (N=0) and with the Lewis number (Le) equal to 10 and the modified Rayleigh number (Ra*) equal to 100 and 200. The results are presented for the average Nusselt and Sherwood numbers as shown in Table 5-8. The results given here are in good agreement with previous studies. Y- axes adiabatic wall L Porous Medium T h C h H g T c C c X- axes adiabatic wall Figure 5-7 The geometry of the model of Nithiarasu [97]. 94

95 Chapter5. Validation Studies The second case was undertaken by using the generalized model. The result was presented in streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=2*10 8, Le=2, and =0.6 for two different buoyancy ratios (N=- 0.9and N=- 1.5) as shown in Figure 5-8. The results showed that the isotherms, streamlines and isoconcentration lines of the present study match fairly well with Nithiarasu s [97] predictions. Table 5-8 A comparison of average Nusselt and Sherwood numbers at steady stat with some previous results for the Darcy model (Le=10, N=0) Author Ra*=100 Ra*=200 Nu Sh Nu Sh Karimi-Fard et al. [95] Goyeau et al.[94] Trevisan and Bejan [93] Present work

96 Chapter5. Validation Studies Pesent work Nithiarasu work Present work Nithiarasu work (a) Figure 5-8 A comparison of the streamlines, isotherms and iso-concentration lines for Da=10-6, Ra=2*10 8, Le=2, and =0.6 at a-(n=- 0.9), and b-(n=- 1.5) with Nithiarasu s [97] work. (b) 96

97 Chapter5. Validation Studies 5.7. Validations of double-diffusive natural convective heat and mass transfer in inclined cavity cases The effects of the inclination angle of the cavity are important to study. This parameter has been added into ALFARHANY code. For a check on the accuracy of the code for double-diffusive natural convection heat and mass transfer in inclined porous cavity cases the model was compared with that of Chamkha and Al-Mudhaf [53]. A two-dimensional porous rectangular cavity filled with an isotropic porous medium was used as shown in Figure 5-9. The two opposing walls were assumed to be impermeable and they were maintained at fixed different temperatures and concentrations while the other boundaries of the cavity were adiabatic. The Darcy model was used to solve the governing equations. Isotropic, homogeneous, local thermal balance, and saturated with an incompressible fluid has been assumed. Figure 5-10 and Figure 5-11 show that the streamlines, isotherms and iso-concentration lines of the present study matched fairly well with Chamkha and Al-Mudhaf s [53] predictions. Figure 5-9 The geometry of the model of Chamkha and Al-Mudhaf [53]. 97

98 Chapter5. Validation Studies Chamkha and Al-Mudhaf predictions Present study Figure 5-10 A comparison of streamlines (left), isotherms (middle) and iso-concentration lines (right) for A=2, Da=10-4, Ra=10 5, =45, Pr=10, Le=10 and N=10 with the work of Chamkha and Al-Mudhaf [53] 98

99 Chapter5. Validation Studies Chamkha and Al-Mudhaf predictions Present study Figure 5-11 A comparison of Streamlines (left), isotherms (middle) and iso-concentration lines (right) for A=2, Da=10-5, Ra=10 5, =90, Pr=7.6, =0.6, Le=10 and N=10 with the work of Chamkha and Al-Mudhaf [53]. 99

100 Chapter5. Validation Studies 5.8. Validations of natural convection of heat and mass transfer with Soret effects in porous cavity cases The last validation of the ALFARHANY code looking at the Soret effects on doublediffusive natural convective heat and mass transfer in a porous cavity and was validated with the results from the work of Khadiri et al. [143]. The model was constructed by using a two-dimensional square porous matrix saturated with a binary mixture as shown in Figure The vertical boundaries of the cavity were adiabatic and the horizontal walls were maintained at fixed but different temperatures and concentrations. The Darcy model with Soret effects was used to solve the governing equations. Figure 5-13 presents a comparison of the present study s results with Khadiri et al. s [143] results for iso-solutes at thermal Rayleigh number (R T ) =200, Lewis number (Le)=10, Buoyancy ratio (N)=0.1 and with different parameters characterizing the Soret effects (M). The results showed that the iso-solutes of the present study match fairly well with Khadiri et al. s predictions for all kinds of flow. Table 5-9 shows the average Nusselt and Sherwood numbers and the maximum stream functions on the bottom wall which are in good agreement with the results of Khadiri et al.. Also, for tricellular flow, the results shows that the streamlines, isotherms and iso-solutes of the present study match fairly well with Khadiri et al. s work, as shown in Figure T F, S F L Adiabatic wall L Porous Medium g Adiabatic wall T c, S c Figure 5-12 The geometry of the model of Khadiri et al. [143]. 100

101 Chapter5. Validation Studies Khadiri et al. s work present work (a) (b) Iso-solutes for monocellular flow for (a) M =0 and (b) M =-5 Khadiri et al. s work present work (a) (b) Iso-solutes for bicellular flow for (a) M =0 and (b) M =10 Figure 5-13 A comparison of the present results with Khadiri et al. s [143] results for iso-solutes for Darcy flow at R T =200, Le=10, N=0.1 and different M. 101

102 Chapter5. Validation Studies Table 5-9 A comparison of average Nusselt and Sherwood numbers and maximum stream functions with Khadiri et al. s [143] results for monocellular flow at R T =200, Le=10, N=0.1. Khadiri et al. [143] results Present results M Nu Sh Max Nu Sh Max streamlines isotherms iso-solutes (a) Khdiri et al. s work stremlines isotherms iso-solutes (b) present wok Figure 5-14 A Comparison of the present results with Khadiri et al. s [143] results for Streamlines, isotherms and iso-concentrations for tricellular flow at R T =200, Le=10, N=0 and M=0 102

103 Chapter5. Validation Studies 5.9. Conclusions This chapter presents a validation of ALFARHANY code for the different cases of unsteady natural convective heat transfer, conjugate, variable porosity, inclination angles of the cavity, double-diffusive natural convective heat and mass transfer and Soret effects in two-dimensional cavities. The comparison is carried out for the model using reliable experimental, analytical/computational available studies in the literature. These studies are presented within six groupings and these are: validations of natural convective heat transfer using Darcy and non-darcy flow in porous cavities, variable porosity, conjugate problems, double-diffusive, inclined cavity and, finally, the Soret and Dufour effects. The results from this study were in good agreement for most of the cases using Darcy, Darcy-Brinkman, Darcy-Forchheimer and generalized models. In general, the results shows that the streamlines, isotherms as well as the iso-concentrations lines (for mass transfer case studies) of the present study match fairly well with the previous studies and also, the average Nusselt numbers and the average Sherwood (for mass transfer case studies) were in good agreement with the previous studies 103

104 Chapter 6 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures 6.1. Introduction In the past, a wide range of studies have been published in natural convection in porous enclosures especially for the porous enclosures which were heated from the side. It is of practical importance in a number of engineering applications, such as industrial cold-storage installations and insulations for buildings. In fact, there are limited studies are made with the effect of the finite wall thickness. Most of these studies used the Darcy model. Therefore, unsteady natural convective heat transfer for all kind of single-fluid flow regime in porous medium by using the generalized model with the effects of finite wall thickness is interested to study. This chapter presents a mathematical formulation and the results of unsteady conjugate natural convective heat transfer in two-dimensional porous square domain

105 wall wall Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures with finite wall thickness. This part of this thesis has been published in the Journal of Numerical Heat Transfer, Part A: Applications (Al-Farhany and Turan [6]) Mathematical formulation: The model was constructed by using a two-dimensional square porous cavity sandwiched between two finite thickness walls comprising an isotropic porous medium as shown in Figure 6-1. The horizontal boundaries of the cavity were adiabatic and the outer surfaces of the vertical walls were maintained at fixed different temperatures T h and T c. The generalized model for constant porosity was used to solve the governing equations in the porous region. The medium was assumed to be isotropic, homogeneous, in local thermal balance incorporating an incompressible fluid. All the properties have been assumed to be constant except for the density. The flow was driven by a buoyancy force due to temperature variations only. The density variations were described by the usual Boussinesq approximation. Y Adiabatic wall Porous medium T h T c H g D Adiabatic wall L D X Figure 6-1 the geometry of the model 105

106 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures The two-dimensional continuity and momentum equations in the X and Y directions and the energy equation for non-dimensional unsteady natural convection can be written as: continuity equation: U X V Y 0 (6.1) X-momentum equation: 1 U 1 U 1 U U V 2 2 X Y U X Da 150 Da X Y P Pr 1.75 U V U Pr U U 3/2 2 2 (6.2) Y-momentum equation: 1 V 1 V 1 V U V 2 2 X Y P Pr 1.75 U V V Pr V V V Ra PrT 3/2 2 2 Y Da 150 Da X Y (6.3) Energy equation: U V X Y X Y 2 2 T T T T T 2 2 (6.4) and the energy equation at the walls: T T T X Y W W W 2 2 (6.5) 106

107 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures Equations (6.1)-(6.5) are solved using non-dimensional initial boundary conditions: Tw at X 0 U V 0 X k r T X (6.6) Tw at X 1 U V 0 X k r T X (6.7) at X D T 1 (6.8) h at X 1 D T 0 (6.9) c T at Y 0 U V 0 0 Y (6.10) at Y T A U V 0 0 Y (6.11) The initial conditions used in this study are: p 1, T 0, U 0, V 0 In this investigation, a homogenous porous medium with porosity equal to 0.36 was used to solve the problem and this was taken as constant everywhere in the cavity Solution procedure A finite volume approach has been used to solve the non-dimensional governing equations (6.1) to (6.5). The transport processes and the corresponding equations were strongly coupled due to the conjugate effect (convection-conduction) between the walls and the porous media. The pressure velocity coupling was treated via the SIMPLE algorithm tailored for the porous media. A second-order central differencing discretization scheme was used for the momentum and the energy equations, and a semi-implicit first-order scheme was used for time step advancement in the momentum equations, while the alternating direction implicit (ADI) technique was adopted for the energy equation (John and Anderson [139]). The ADI technique was used in order to incorporate the effects of boundary 107

108 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures conditions and also to provide further stability as well as refining accuracy due to the fact that the solver is second order accurate in, X and Y. The inertial term in the momentum equations was considered a source term. In the porous cavity, different uniform mesh sizes were tested. An 81*81 grid for the low Rayleigh and Darcy numbers and a 121*121 grid for the high Rayleigh number were deemed adequate. For the low Darcy number (Da1*10-4 ), a grid of 81*81 was adopted and it was found that this grid resolution was adequate for good accuracy. To obtain stable convergence behaviour for the solution, various under-relaxation parameters were used. For the U, V-velocities, pressure and the energy equations, the under-relaxation parameters adopted were 0.2, 0.2, 0.3 and 0.5, respectively. The steady state was considered to have been arrived at when the changes in U, V, P and T satisfied the equation(4.12). This equation was also used to obtain the convergence of the solution at each time step Results In this section, the predictions are displayed in terms of streamlines, isotherms and average Nusselt number profiles. A correlation is provided to calculate the average Nusselt number on the internal left wall of the porous cavity as a function of different values of the non-dimensional governing parameters, including the modified Rayleigh number (100 Ra * 1000), Darcy number (10-7 Da 10-2 ), thermal conductivity ratio (0.1 k r 10) and the ratio of wall thickness to its height (0.1 D 0.4). In Figure 6-2 the streamlines and isotherms at different time steps are presented for Da=10-6, Ra*=1*10 2, D=0.2 and k r =1. Early on in the procedure ( = 0.039), it can be seen in Figure 6-2 (b) that the isotherms are parallel through the left hot wall due to conduction heat transfer and nearly parallel near the left wall of the porous medium. A small increase in the isotherms near the left upper corner as a result of the fluid rising due to the influence of buoyancy force can be observed. Figure 6-2 (a) shows the streamlines. For instances early in the transient at = 0.039, it can be seen that the recirculation flow region resides closer to the left wall. Furthermore, the fluid rises up near the hot left wall for no more than 20% of the cavity length and 108

109 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures subsequently returns downward near the cooled right wall. Shortly after that when = 0.195, the convective heat transfer becomes more important as the flow has been established in the porous cavity. It is easy to observe from the isotherms that the temperature variation increases near the top insulated wall and closer to the right wall, while it remains almost constant near the left bottom corner. Also the recirculation region has migrated from the hot left wall towards the direction of the cold right wall. The same behaviour can be seen at = 0.39 and this continues until the steady state is achieved. The effects of heat transfer and fluid flow on the isotherms and streamlines are similar to those in Figure 6-3 for Da=10-6, Ra*=1*10 2, D=0.2 and k r =5. The difference is that, for cases of high thermal conductivity ratio K r, the recirculation region strength in the porous cavity is increased. This is due to the thermal field in the porous cavity, which is influenced by the increasing temperature gradients in the horizontal direction with increases in the thermal conductivity ratio, as well as decreases in the time required to reach the steady state. For wall thickness D=0.3, the streamlines and isotherms are presented in Figure 6-4 and Figure 6-5 for Da=10-6, Ra*=1*10 2, and k r =1 and 5, respectively. It is clear that the recirculation strength in the porous medium is lower with thick walls. Also the isotherms show that most of the temperature variation still lies in the wall thickness due to conduction in the wall. Therefore, the time required to reach the steady state increases when the wall thickness increases. Similar behaviour as shown above for Da=10-6 can be seen in Figure 6-6 to Figure 6-9 for Da=10-2 and different k r and D. It is clear that when the Darcy number increases, the time to reach the steady state will increase especially for low thermal conductivity ratio and thick walls. Additional predictions are displayed for high Rayleigh numbers in streamlines and isotherms for steady state in Figure 6-10 for Da=10-2, Ra*=1*10 3, Pr=1.0, k r =1, 10 and different wall thickness. This figure displays essentially the same behaviour as above. The circulation strength is lower with thick walls as shown in Figure 6-10 (a) - (c) and also, the isotherms reveal that most of the temperature variation still lies in the wall thickness due to conduction in the wall. The average Nusselt numbers on the interface left porous wall decrease 109

110 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures =0.039 =0.195 =0.390 =steady state (a) Streamlines =0.039 =0.195 =0.390 =steady state (b) Isotherms Figure 6-2 Variation of (a) streamlines, and (b) isotherms lines with nondimensional times for Da=10-6, Ra*=1*10 2, D=0.2 and k r =1. =0.039 =0.195 =0.390 =steady state (a) Streamlines =0.039 =0.195 =0.390 =steady state (b) Isotherms Figure 6-3 Variation of (a) streamlines, and (b) isotherms with non-dimensional times for Da=10-6, Ra*=1*10 2, D=0.2 and k r =5. 110

111 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures =0.039 =0.195 =0.390 =steady state (a) Streamlie =0.039 =0.195 =0.390 =steady state (b) Isotherms Figure 6-4 Variation of (a) streamlines and (b) isotherms with non-dimensional times for Da=10-6, Ra*=1*10 2, D=0.3 and k r =1. =0.039 =0.195 =0.390 =steady state (a) Stralines =0.039 =0.195 =0.390 =steady state (b) Isotherms Figure 6-5 Variation of (a) streamlines and (b) isotherms with non-dimensional times for Da=10-6, Ra*=1*10 2, D=0.3 and k r =5. 111

112 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures when the wall thickness is increased as shown in Figure Such conclusions imply that wall conduction heat transfer provides a dominant influence in a porous cavity. Moreover, the average Nusselt numbers increase when the Rayleigh numbers and the thermal conductivity ratio increase, as shown in Figure 6-11 and Figure In Figure 6-10 (c) and (d) the streamlines and isotherms are presented for k r = 1, 10 respectively to show the effect of the thermal conductivity ratio. It can be seen that, for cases displaying increases in the thermal conductivity ratio, the circulation strength is increased. For the low thermal conductivity ratios there are no significant changes in the Nusselt number behaviour even with a high Rayleigh number as shown in Figure Correlation for the average Nusselt number on the left porous wall interface Correlations are extremely important for practical engineering approximation design and development activities in order to describe the dependent variable as a function of the appropriate relevant independent variables covering the domain. For the steady state, the average Nusselt number on the left porous wall interface is correlated in a form given by: Nu a* Da b Ra c * D d * k e r (6.12) Hundreds of simulations were run to calculate the average Nusselt number for steady state on the internal left wall of the porous cavity. A statistical program (STATISTICA version 8.0) has been used to arrive at this correlation equation. For the whole range of the parameters, the correlation equation finally derived is: Nu 0.098* Da Ra * D * k r (6.13) The variance R 2 is also called Proportion of variance accounted for (Hill and Lewicki [32]) for this correlation equals The variance can be explained, effectively is much closer to one and that means that this correlation equation covers all the data in one representative equation. Moreover, Figure 6-13 shows the normal probability of the expected normal value with the residuals. 112

113 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures 6.6. Conclusions Unsteady conjugate natural convective heat transfer in a two-dimensional porous cavity sandwiched between two finite thickness walls comprising an isotropic porous medium has been investigated numerically. The outer surfaces of the vertical walls were maintained at fixed different temperatures, while the horizontal boundaries of the cavity were adiabatic. The generalized model with the Boussinesq approximation was used to solve the governing equations in the saturated porous region. A finite volume approach was used to solve the non-dimensional governing equations and the SIMPLE algorithm was employed in the porous media domain to solve the pressure velocity coupling. The results were presented at different time steps for different values of the non-dimensional governing parameters, including the modified Rayleigh number (100 Ra* 1000), Darcy number (10-7 Da 10-2 ), thermal conductivity ratio (0.1 k r 10) and the ratio of wall thickness to its height (0.1 D 0.4). The results showed that as the Darcy number increases, the average Nusselt number is reduced and the time required to reach the steady state is longer whereas the time required to reach the steady state is shorter for a high Rayleigh number and longer for a low Rayleigh number. Furthermore, when the wall thickness increases, the overall Nusselt number is reduced, whereas the average Nusselt number increases when the thermal conductivity ratio increases. On the other hand, the time required to reach the steady state is longer for thick walls and low thermal conductivity ratios and/or is shorter for thin walls and/or high thermal conductivity ratios. A correlation to evaluate the average Nusselt numbers on the interface left wall was given in equation(6.13). 113

114 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures =0.039 =0.195 =0.390 =steady state (a) Streamlines =0.039 =0.195 =0.390 =steady state (b) Isotherms Figure 6-6 Variation of (a) streamlines and (b) isotherms with non-dimensional times for Da=10-2, Ra*=1*10 2, D=0.2 and k r =1. =0.039 =0.195 =0.390 =steady state (a) Streamlines =0.039 =0.195 =0.390 =steady state (b) Isotherms Figure 6-7 Variation of (a) streamlines and (b) isotherms with non-dimensional times for Da=10-2, Ra*=1*10 2, D=0.2 and k r =5. 114

115 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures =0.039 =0.195 =0.390 =steady state (a) Streamlines =0.039 =0.195 =0.390 =steady state (b) Isotherms Figure 6-8 Variation of (a) streamlines and (b) isotherms lines with nondimensional times for Da=10-2, Ra*=1*10 2, D=0.3 and k r =1. =0.039 =0.195 =0.390 =steady state (a) Streamlines =0.039 =0.195 =0.390 =steady state (b) Isotherms Figure 6-9 Variation of (a) streamlines and (b) isotherms lines with nondimensional times for Da=10-2, Ra*=1*10 2, D=0.3 and k r =5. 115

116 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures (a) Streamlines isotherms (b) Streamlines isotherms (c) Streamlines isotherms (d) Streamlines iotherms Figure 6-10 In the steady state, streamlines and isotherms lines for Da=10-2, Ra*=1*10 3, Pr=1.0, k r =1 at (a) D=0.0, (b) D=0.2, (c) D=0.4, and at (d) k r =10, D=

117 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures Figure 6-11 Variation of average Nusselt numbers in the steady state on the interface left wall of a porous medium with wall thickness at Da=10-7, Pr=1.0 Figure 6-12 Variation of Nusselt numbers in the steady state on the interface left wall of a porous medium with Ra* for Da=10-6, Pr=1.0 and D=

118 Chapter 6. Conjugate Natural Convective Heat Transfer in Porous Enclosures Figure 6-13 The normal probability of the expected normal value with the residuals 118

119 Chapter 7 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium 7.1. Introduction The phenomenon of combined heat and mass transfer in a porous medium is usually referred to as double-diffusive. The reason for the scarcity of work on the subject may be related to its complexity and to the lack of a general theory. Furthermore, the effects of variable porosity on double-diffusive natural convection in porous cavities have not been studied before. Therefore, it is interesting to study the effects of variable porosity on double-diffusive natural convective heat and mass transfer. For many engineering applications, for example thermal insulation and nuclear reactors, the porous cavity will almost be sandwiched between two finite thickness walls. Therefore, the effects of the thermal conductivity ratio (the ratio of the thermal

120 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium conductivity of the wall to the thermal conductivity of the porous media) had been considering. This chapter presents the results of unsteady conjugate double-diffusive natural convective heat and mass transfer in two-dimensional variable porosity layer sandwiched between two walls cases. The comparison is carried out for the model using well experimental, analytical/computational available studies in literature. This part of the thesis has been published in the International Journal of Heat and Mass Transfer (Al-Farhany and Turan [8]) Mathematical formulation of the model The model comprised a two-dimensional porous cavity sandwiched between two finite thickness walls filled with an isotropic porous medium and is shown in Figure 7-1. All the walls were impermeable. The horizontal boundaries of the cavity were adiabatic and the vertical walls were maintained at fixed different temperatures and concentrations T h and C h at the left wall and T c and C c at the right wall. The generalized model was used to solve the governing equations. The medium was assumed to be isotropic, homogeneous, in local thermal balance and saturated with an incompressible fluid. The Soret and Dufour effects were assumed to be negligible. All the governing equations and the non-dimensional parameters are presented in Chapter 3. For the momentum equations, the angle of inculcation is equal to zero (=0). Additionally, as a result of neglecting the Soret and Dufour effects, the energy and the species equations become: energy equation: U V X Y X Y 2 2 T T T T T 2 2 (7.1) species equation: U V X Y Le X Y 2 2 C C C 1 C C 2 2 (7.2) 120

121 wall wall Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium y Adiabatic wall Porous media T h C h C c T c, H g D Adiabatic wall L D x Figure 7-1 The geometry of the model 7.3. Initial boundary conditions Usually, using the non-dimensional form in the numerical analysis reduces the number of variables in the equations as well as assisting the transformation of the equations and the numerical equations. The governing equations are solved by using the non-dimensional initial boundary conditions: at X D T 1 (7.3) h at X 1 D T 0 (7.4) c Tw at X 0 U V 0 C h 1 X k r T X (7.5) Tw at X 1 U V 0 Cc 0 X k r T X (7.6) 121

122 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium T C at Y 0 U V 0 0 Y Y (7.7) at Y T C A U V 0 0 Y Y (7.8) 7.4. Solution procedure A finite volume approach was used to solve the dimensionless governing equations. The transport processes and the corresponding equations are strongly coupled, since the conjugate effect (convection-conduction) between the walls and the porous media and the variation of density due to temperature and concentration overwhelmingly influence the evolution of the physicochemical environment. The porosity and the Darcy number are calculated in each cell using equations (3.25) and (3.26) in the porous media domain and are then retained as constant for all the cases. The pressure velocity coupling is treated using the SIMPLE algorithm tailored for a porous media domain by Patankar [138]. A second order central discretization scheme is used for the momentum, energy and species equations, a semi-implicit first order scheme is used for time advancement in the momentum equations and the ADI method is adopted for the energy and species equations (John and Anderson [139]). The inertial term in the momentum equations is considered a source term. In the porous media, different mesh sizes were tested. An 81*81 grid for the low Lewis and Rayleigh numbers and a 121*121 grid for the high Lewis and Rayleigh numbers were deemed adequate. To obtain stable convergence behaviour for the solution, various under-relaxation parameters were used. For the U-velocity, the V-velocity, pressure, energy and the concentration equations, the under-relaxation parameters adopted were 0.2, 0.2, 0.3, 0.5 and 0.5 respectively. The steady state was considered to have been reached when the changes in U, V, T and C satisfied the equation(4.12) Results In this section, the results are presented for a variable porosity porous medium for different values of the non-dimensional governing parameters, including the 122

123 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium * modified Rayleigh number ( 10 Ra 1000 ), the buoyancy ratio (-2 N 2), the Lewis number ( 0.1 Le 10 ), the thermal conductivity ratio ( ) and the ratio of wall thickness to its height ( 0 D 0.4 ) where the modified Rayleigh number is given as * Ra Ra Da. k r Variable porosity effects Figure 7-2 and Figure 7-3 show the effect of variable porosity at Da=10-6, Ra*=100 and N=- 2.0 where Le is 0.1 and 10, respectively. For constant porosity (=0.36) and for variable porosity ( =0.36) and D 1 =0.088, it is observed that the vortex strength increases when variable porosity is used. These influences on the vortex strength are proportional to the Lewis number increasing. For a small Lewis number (Le=0.1) the effect of variable porosity on the streamline behaviour is small, while the effect of variable porosity on the streamline behaviour is very pronounced at high Lewis numbers (Le=10). The flow is driven by the buoyancy force due to the effect of both temperature and concentration variations. For low Lewis number situations, most of the buoyancy driver is due to the effect of temperature variations, while similar effects are provided by the concentration field at high Lewis numbers. As a result, the effects of variable porosity are clearly seen on the isotherms at low Lewis numbers, while there is virtually any influence on the iso-concentration contours. On the other hand, the effects of variable porosity are clearly visible on the isoconcentration lines, while very little effect is portrayed on the isotherms at the high Lewis numbers. Figure 7-4 shows the effect of Rayleigh number at Da=10-6, Le=1 and N=- 2.0 on the horizontal and vertical velocity profiles at the mid vertical and horizontal planes of the cavity for uniform and variable porosity respectively. It is clear that the velocities increase when the Rayleigh number increases. There are no effects considered on the U-velocity profiles at the mid vertical plane of the cavity. This is primarily due to the variation of porosity near the hot wall (the porosity will change for no more than 5 percent of the cavity length) so there are no effects of porosity variations on the flow at X=0.5. Also, it can be seen that the same behaviour persists in Figure 7-5 with variable Lewis numbers. On the other hand, at the mid horizontal plane of the cavity (Y=0.5), the variation of the V-velocity along the X-axis for 123

124 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium variable porosity shows relatively more uniform porosity near the left vertical wall, while the opposite behaviour is observed near the right vertical wall especially for high Rayleigh number cases. Figure 7-5 presents the variation in the horizontal and vertical velocity profiles at the mid vertical and horizontal planes of the cavity for uniform and variable porosity respectively at Da=10-6, Ra*=100, N=- 2.0 and different Lewis numbers. In general, (a) Streamlins isotherms iso-concentration (b) Streamlines isotherms iso-concentration Figure 7-2 Streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=100, Le=0.1, N= and =0.36 at (a) variable porosity, (b) uniform porosity. 124

125 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium (b) Streamlines sotherms iso-concentration Figure 7-3 Streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=100, Le=10, N=- 2.0 and =0.36 at (a) variable porosity, (b) uniform porosity. the velocities decrease when the Lewis number increases for the same Rayleigh number. The Lewis number increases implies a similar increase in the effective mass diffusivity coefficient vis-à-vis the effective thermal diffusivity coefficient and is due to the ensuing driving force for the flow field (buoyancy ratio negative) as the velocities consequently decrease. Figure 7-6 shows that the average Sherwood number increases when the Rayleigh and Lewis numbers are increased. Also, the Nusselt number increases when the Rayleigh number increases, whereas it decreases when the Lewis number increases for a low Rayleigh number and increases for the high Rayleigh numbers. These attributes need additional investigations for different cases employing extended parameters. 125

126 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium Wall thickness effects To show the effects of wall thickness (D), streamlines, isotherms and isoconcentration lines are presented in Figure 7-7 for Da=10-6, Ra*=100, Pr=1.0, Kr=1.0 Le=2 and N=- 2.0 and also in Figure 7-8 for Da=10-6, Ra*=1000, Pr=1.0, Kr=1.0, Le=1 and N= The flow circulation strength in a porous medium is lower with thick walls as shown in Figure 7-7 and Figure 7-8 (a) - (c). The average Nusselt and Sherwood numbers on the left wall porous interface decrease when the wall thickness is increased as shown in Figure 7-9. This means that the conduction of heat transfer is dominant in the porous cavity. For a high Rayleigh number the Nusselt number significantly reduces with increasing wall thickness for D ranging up to 0.3. For a wall thickness exceeding 0.3, there are no significant changes in the Nusselt or the Sherwood numbers. (a)variable porosity (b)uniform porosity Figure 7-4 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Le=1, N=-2.0 and Ra*=10, 100 and 1000 at (a) variable porosity, (b) uniform porosity. 126

127 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium (a)variable porosity (b)constant porosity Figure 7-5 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra=100, N=-2.0 and Le=0.1, 1, 10 at (a) variable porosity, (b) constant porosity. (a) (b) Figure 7-6 Variation of Nusselt and Sherwood numbers with Ra* for Da=10-6, N=

128 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium (a) Streamlines isotherms iso-concentration (b) Streamlines isotherms iso-concntration (c) Streamlines iotherms iso-concentration (d) Streamlines isotherms iso-conentration Figure 7-7 Streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=100, Pr=1.0, Le=2, N=- 2.0, k r =1 at (a) D=0.2, (b) D=0.3, (c) D=0.4 and at (d) k r =0.1, D=

129 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium Thermal conductivity ratio effects In Figure 7-7 (c) and (d) the streamlines, isotherms and iso-concentration are presented for Da=10-6, Pr=1.0, Ra*=100, N=- 2.0, Le=2.0, D=0.2 and k r = 1, 0.1 respectively to show the effect of the thermal conductivity ratio. It can be seen that, for increases in the thermal conductivity ratio, the circulation strength of the fluid in the porous cavity is correspondingly magnified. This is due to the thermal field in the porous cavity, as influenced by the increasing temperature gradients in the horizontal direction with increases in the thermal conductivity ratio. Table 7-1 shows that the Nusselt number increase as thermal conductivity increases. Additionally, for the negative buoyancy ratio, the Sherwood number decreases when the thermal conductivity ratio increases, while it increases when the thermal conductivity ratio increases for the positive buoyancy ratio Lewis number and the buoyancy ratio effects As mentioned previously, when the Lewis number increases, the Nusselt number decreases and the Sherwood number increases, as shown in Table 7-1 and Figure 7-6 for Da=10-6, N=-2.0 with different Ra*. Reduction of the Lewis number naturally implies an increase of the effective thermal diffusivity coefficient to the effective mass diffusivity coefficient. Figure 7-10 for Da=10-6, Pr=1.0, Le=1.0, D=0.3 and different Rayleigh numbers, shows that the Nusselt and Sherwood numbers increase when the buoyancy ratio increases (for N -1). This is due to the net buoyancy effects as influenced by the competing thermal and concentration gradients. 129

130 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium (a) Streamlines isotherms iso-concentration (b) Streamlines isotherms iso-concentration (c) Streamlnes isotherms iso-concetration Figure 7-8 Streamlines, isotherms and iso-concentration lines for Da=10-6, Ra*=1000, Pr=1.0, Le=1, N=+2.0, k r =1 at (a) D=0.2, (b) D=0.3, (c) D=

131 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium Figure 7-9 Variation of average Nusselt and Sherwood numbers on the left wall porous interface with wall thickness at Da=10-6, Pr=1.0, Le=2, N=-2.0. Figure 7-10 Variation of average Nusselt and Sherwood numbers on the left wall porous interface with wall buoyancy ratio at Da=10-6, Pr=1.0, Le=1 and D=

132 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium Table 7-1 Average Nusselt and Sherwood numbers at Da=10-6, Pr=1.0, D=0.2. Ra* Le N k r =0.1 k r =1 k r =10 Nu Sh Nu Sh Nu Sh

133 Chapter 7. Variable Porosity Effects on Convective Heat and Mass Transfer in Porous Medium 7.6. Conclusions Results have been presented for a variable porosity medium with different values of the non-dimensional governing parameters, including the modified Rayleigh number (10 Ra * 1000), the buoyancy ratio (-2 N 2), the Lewis number (0.1 Le 10), the thermal conductivity ratio (0.1 k r 10) and the ratio of wall thickness to its height (0.1 D 0.4). In general, the Nusselt number increases when the Rayleigh number increases, whereas it decreases when the thermal conductivity ratio, the Lewis number and the wall thickness increase. With regard to the buoyancy ratio effects, the Nusselt number increases when the buoyancy ratio increases for (N -1) and also when it decreases for (N -1). The same behaviour is displayed by the Sherwood number. Furthermore, the Sherwood number increases when the Rayleigh and Lewis numbers increase, although it decreases when the thermal conductivity ratio decreases. 133

134 Chapter 8 8. Convective Heat and Mass Transfer in Inclined Porous Cavities 8.1. Introduction The effect of an inclined porous medium on heat transfer has been presented widely in previous studies. There is a limited studies had been for double-diffusive natural convective heat and mass transfer in inclined porous cavities. There are many significant applications in many of engineering and geophysical applications; industrial transport processes in porous media, geophysical applications, high performance insulation for buildings, solar power collectors and cold storage installations. This chapter presents the results of steady double-diffusive natural convective heat and mass transfer in a two-dimensional inclined square/rectangular porous cavity. These results are presented within two groups. The first group of results for inclined

135 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities square porous cavity and this part of works has been submitted to International Journal of Heat and Mass Transfer. The second group of results for inclined rectangular porous cavity and this part of works had been published in International Communications in Heat and Mass Transfer Journal (Al-Farhany and Turan [9]) Mathematical modeling The model comprised a two-dimensional inclined square cavity filled with a porous medium, as shown in Figure 8-1. All the walls were impermeable. Two opposing walls of the cavity were adiabatic and the other two walls were maintained at fixed different temperatures and concentrations; T h and C h at the left wall and T c and C c at the right wall. The generalized model was used to solve the governing equations. The medium was assumed to be isotropic, homogeneous, in local thermal balance and saturated with an incompressible fluid. The Soret and Dufour effects were assumed to be negligible, while all the properties have been assumed to be constant except the density. Figure 8-1 The geometry of the model. 135

136 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities The two-dimensional continuity, energy and momentum equations in the X and Y directions, and the species equations for unsteady double diffusive natural convection for constant porosity in inclined cavity are: Continuity equation: U X V Y 0 (8.1) X-momentum equation: 1 U 1 U 1 U U V 2 2 X Y U X Da 150 Da X Y Ra PrT sin( ) N Ra Pr C sin( ) P Pr 1.75 U V U Pr U U 3/2 2 2 (8.2) Y-momentum equation: 1 V 1 V 1 V U V 2 2 X Y V Y Da 150 Da X Y Ra PrT cos( ) N Ra Pr C cos( ) P Pr 1.75 U V V Pr V V 3/2 2 2 (8.3) energy equation: U V X Y X Y 2 2 T T T T T 2 2 (8.4) species equation: 2 2 C C C C C U V X Y Le X Le Y 2 2 (8.5) Also the average Nusselt and Sherwood numbers on the vertical walls for aspect ratio equal 1 can be found as: 136

137 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities Nu 1 T Y (8.6) X 0 1 C Sh Y (8.7) X 0 where Nu is the average Nusselt number, Sh is the average Sherwood number. Equations (8.1)-(8.5) are solved using non-dimensional initial boundary conditions: at X 0 U V 0 1 C T 1 (8.8) h h at X 1 U V 0 0 C T 0 (8.9) h h at Y 0 U V 0 T Y C Y 0 (8.10) for rectangular porous cavity: at Y A U V 0 T C 0 Y Y (8.11) for square porous cavity, therefore: the cavity height (H) equal the cavity length (L), at Y 1 U V 0 T Y C Y 0 (8.12) 8.3. Solution procedure The non-dimensional governing equations (8.2) -(8.5) have been solved by using a finite volume approach and the pressure velocity coupling is treated using the SIMPLER algorithm. A second order central discretization scheme is used for the momentum, energy and species equations, while a semi-implicit first order scheme is used for time advancement in the momentum equations. The inertial term in the momentum equations is considered a source term. Different mesh sizes were tested. 137

138 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities A grid size of 81*81 was used for the low modified Rayleigh number and 121*121 was used for the high modified Rayleigh number and it was deemed adequate for square cases while a grid size of 100*(A*100) was used for rectangular cases. To obtain stable convergence behaviour for the solution, various under-relaxation parameters were used. For the U-velocity, the V-velocity, energy and the species equations, the under-relaxation parameters adopted are 0.3, 0.3, 0.5 and 0.5 respectively Results and conclusions of the square porous cavity cases Results In this section, the results are presented in terms of streamlines, isotherms, and isoconcentration, and U-velocity, V-velocity, average Nusselt and Sherwood number profiles. The predictions include different values for the non-dimensional governing parameters, specifically the modified Rayleigh number (100 Ra* 1000), the Darcy Number (10-6 Da 10-2 (, the angle of inclination of the cavity (0 o 90 o ), the Lewis number (0.1 Le 10), and the buoyancy ratio (-5 N 5) while the Prandtl number is taken as Pr=1.0. Stream functions, isotherms and iso-concentration lines are presented in Figure 8-2 for Da=10-6, Ra*=100, and Le=10.0 where N= -1.0, with different inclination angles ().Where there is no inclination angle (=0 o ), the secondary circulations appear and the counter clockwise flow is homogenised and symmetric on the main diagonal of the porous cavity. This happens because the buoyancy ratio, which is a ratio of fluid density contributions by the concentration to the temperature variations, is equal to -1. The maximum positive stream functions are about 1.20 and the maximum negative stream functions were about For isotherms where the inclination angle is equal to 0, it is observed that the variations of isotherms near the insulated walls are higher than in the middle of the cavity. For the left half of the cavity (0 < X < 0.5), and near the left bottom corner, it is observed that the isotherms are increase in the isotherms up to that are near the middle of the cavity height and then decreased until the left top corner. This 138

139 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities behaviour occurred as a result of the fluid rising due to the influence of the buoyancy force and the effect of the secondary circulations. It also happened with the other isotherms lines until the middle of the cavity length with one difference. It is clear that the maximum isotherms are not in the middle of the cavity length but they are almost near places where there is no movement in the fluid flow (when the stream functions are equal to zero). The same behaviour could be seen in the left half of the cavity but in the opposite direction. Also, for the iso-concentration, it could be seen that the iso-concentration field is more sensitive than the isotherms and that occurred due to the considered Lewis number (Le=10). The maximum (minimum) isoconcentration appeared near the left (right) wall and around the maximum positive stream functions and most of the iso-concentration variations were near the maximum negative stream functions which are around the main diagonal of the cavity. By increasing the inclination angle ( > 0 o ), it is observed that the positive vortex strength increased when the angle of the inclination is increased until =60 o and after that the positive vortex strength decreased when the angle of the inclination was increased. Also, the left positive secondary circulations decreased until they disappeared. For =45 o, the maximum positive stream functions and the maximum negative stream functions were about 3.85 and respectively. Furthermore, the negative vortex strength is decreased when the angle of the inclination is increased until is around 45 o and then it increased with the increasing of the inclination angle. Also, it could be seen that the isotherms are nearly parallel near the left wall and increased in the isotherms field near the left upper corner as a result of the fluid rising due to the influence of the buoyancy force. As noted above, the maximum isoconcentration variations are near the maximum negative stream functions and the iso-concentration lines are nearly parallel near the left high concentration wall and then increased in the iso-concentration fields near the insulated top wall. This behaviour is clearly observed with an increase of the angle of inclination for both isotherm and iso-concentrations fields. For cases of =90 o, different behaviours appeared. It is observed that no vortex appeared and that happened because of low flow intensity. 139

140 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities =0 =30 =45 =60 =90 Figure 8-2 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-6, Ra*=100, Pr=1.0, =0.36, Le=10.0 and N= -1.0 with different inclination angles. 140

141 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities The heat transfer occurred due to natural conduction through the porous media rather than natural convection. These effects are very clear in the isotherms (middle) and iso-concentration (right) lines at =90 o. It showed that the isotherm lines are parallel and they changed regularly from the hot horizontal bottom wall to the cold horizontal top wall and also that the iso-concentration lines are parallel and they changed regularly from the high concentration horizontal bottom wall to the low concentration horizontal top wall. For the opposed flow, the vortex strength is very small due to the effects of mass transfer especially for the high Lewis numbers. Figure 8-3 shows the stream functions, isotherms and iso-concentration lines for Da=10-6, Ra*=100, and Le=10.0 where N= -2.0, with different inclination angles (). When there is no inclination angle (=0 o ), the main flow is counter clockwise with maximum negative stream function of about -1.6 and the secondary clockwise circulations appeared in the left bottom corner and in the right top corner with maximum stream function of about The secondary clockwise circulations appeared as a result of the fluid rising due to the influence of the temperature variation which is more than the effect of mass variations in the buoyancy force at these regions. For an isotherms field when the inclination angle was equal to 0, it is observed that the maximum isotherms field are near the left wall from the bottom and then decreased in the Y- direction until the left top corner. As for the iso-concentration field, as mentioned above, it is more sensitive than the isotherms and that occurred because of the considered Lewis number (Le=10). Maximum iso-concentration appeared near the left bottom corner and minimum concentration appeared near the right top corner. Elsewhere, most of the iso-concentration variations are near the maximum negative stream functions which are around main the diagonal of the cavity. In terms of an increase in the inclination angle ( > 0 o ), it is observed that the positive vortex strength increase when the angle of the inclination is increased until =30 o and then it decrease until disappear. For =45 o, the maximum positive stream functions and the maximum negative stream functions are about and respectively. Furthermore, the negative vortex strength is decrease when the angle of the inclination is increased. 141

142 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities =0 =30 =45 =60 =90 Figure 8-3 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-6, Ra*=100, Pr=1.0, =0.36, Le=10.0 and N= -2.0 with different inclination angles. 142

143 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities When around 45 o and more, two negative vortexes appear. Also for =90 o, no vortex appears. Furthermore, it is observed that the isotherms are nearly parallel near the left wall and decrease in the isotherms field near the left upper corner. This behaviour is increased when the angle of the inclination is increased until all the isotherms lines are become horizontally parallel when =90 o. With regard to the isoconcentration lines, the maximum (minimum) iso-concentration appeared near the left (right) wall and around the maximum positive stream functions and most of the iso-concentration variations are near the maximum negative stream functions which are around the main diagonal of the cavity for an angle of inclination around 30 o. The iso-concentration lines became more parallel an increase in the inclination angle. For =90 o, it could be seen that the iso-concentration lines are parallel and they changed regularly from the high concentration horizontal bottom wall to the low concentration horizontal top wall. For a low Lewis number, Figure 8-4 shows the stream functions, isotherms and isoconcentration lines for Da=10-6, Ra*=100, Pr=1, and Le=0.1 with different inclination angles () where N= For a low Lewis number, the flow has almost been driven due to mass transfer rather than heat transfer. When the buoyancy ratio is equal to -1, Figure 8-4 shows that no significant heat and mass transfer appears for a low inclination angle especially when the inclination angle is equal to zero (=0 o ). From the stream function lines, it is clear that the vortex strength of the fluid is very small. Furthermore, the concentration boundary layer is nearly equal to the isotherms boundary layer and it clearly can be seen that the isotherm and isoconcentration lines are parallel and both the Nusselt and Sherwood numbers are equal to 1. For > 0 o, it can be observed that the positive vortex strength increases when the angle of the inclination is increased until =60 o and subsequently the positive vortex strength decreases when the angle of the inclination is increased. Also, the negative secondary circulations decrease until they disappear. For =45 o, the maximum positive stream functions and the maximum negative stream functions are about 2.0 and respectively. Also, it can be seen that the isotherms are nearly parallel near the left wall and increase in the isotherms field near the left upper corner as a result of the fluid rising due to the influence of the buoyancy force. 143

144 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities =0 =30 =45 =60 =90 Figure 8-4 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-6, Ra*=100, Pr=1.0, =0.36, Le=0.1 and N= -1.0 with different inclination angles. 144

145 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities The maximum isotherms boundary layer can be seen at =60 o ; therefore the Nusselt number is also at its maximum at =60 o, as shown in Table 2. For the cases of heated and concentrated from below (=90 o ), a multiplicity of solutions happened (monocellular and bicellular flows), most of these solution results are in bicellular flow as shown in the Figure 8-4. For Le=0.1, the bicellular flow solution was chosen for all the cases of =90 o. In the case of the isotherms, it is observed that the isotherms boundary layers near the insulated walls are thinner than the isotherms boundary layers near the middle of the bottom wall. Because of this, the fluid flows in the porous cavity rise up from the middle of the hot bottom wall towards the cold top wall. As a result of the effects of the thick cold boundary layer near the top wall, the vertical fluid flow velocities decreased until they are at a minimum near the cold wall, subsequently the fluid flow was driven in two different horizontal directions to achieve the bicellular flows. Table 8-1 Average Nusselt and Sherwood numbers at Da=10-6, Ra*=10 2. Le N =0 =30 =45 =60 =85 Nu Sh Nu Sh Nu Sh Nu Sh Nu Sh

146 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities For a high buoyancy ratio, the stream functions, isotherms and iso-concentration lines are presented in Figure 8-5 for Da=10-6, Ra*=100, Pr=1, and Le=0.1with different inclination angles () where N= A high positive buoyancy ratio means that the fluid rises up due to the influence of the temperature and mass variations which is more than the effect of the temperature variations in the buoyancy force. The results clearly show that when the angle of the inclination increases, the vortex strength of the fluid increases until =45 o that subsequently the vortex strength decreases when the angle of the inclination increases. It can also be observed that there is only a main clockwise flow (no secondary flow) at a maximum positive stream function of about at =45 o. The recirculation region has migrated from the hot left wall towards the direction of the cold right wall. In Figure 8-5, with regard to the isotherm lines, it can be seen that the isotherms boundary layer is too thin near the left wall and it increases near the left upper corner. Also, it is very clear that the isotherms are more sensitive than the isoconcentrations and this happens due to the considered Lewis number (Le=0.1). Furthermore, with regard to the iso-concentrations lines, it can be seen that the isoconcentrations are nearly parallel near the left bottom wall and there is an increase in the iso-concentrations near the left upper corner as a result of the fluid rising due to the influence of the buoyancy force. In the case of a low Lewis number (Le=0.1) the effect of the inclination angles are not significant on either the isotherm or the isoconcentrations field. As mentioned previously, when =90 o bicellular flows occur as shown in Figure 8-5, for Da=10-6, Ra*=100, Pr=1, Le=0.1 and N= For the isotherms, it is observed that the isotherms boundary layers near the insulated walls are thicker than the isotherms boundary layers near the middle of the bottom wall. This is because the fluid flows in the porous cavity rise up near the insulated walls from the hot bottom wall towards the cold top wall. As a result of the effects of the thick cold boundary layer near the top wall, the vertical fluid flow velocities decrease until they are at a minimum near the cold wall; on the other hand, the horizontal velocity increases and then decreases to achieve minimum velocity near the middle of the cavity in order to achieve the bicellular flows. 146

147 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities =0 =30 =45 =60 =90 Figure 8-5 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-6, Ra*=100, Pr=1.0, =0.36, Le=0.1 and N= 5.0 with different inclination angles. 147

148 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities Figure 8-6 presents the variation in the horizontal and vertical velocity profiles at the mid vertical and horizontal planes of the cavity respectively at Da=10-6, Ra*=500, Le=0.1, N= 2.0 and at different inclination angles. It is clear that the U-velocities increase when the inclination angle increases until =60 o and then it decreases when the inclination angle increases. There are no effects considered on the U-velocity profiles at the mid vertical plane of the cavity. On the other hand, at the mid horizontal plane of the cavity (Y=0.5), the variation of the V-velocity along the X-axis for different inclination angles shows that the V-velocities also increase when the inclination angle increases until =60 o and then decrease when the inclination angle increases. For the opposing flow, Figure 8-7 and Figure 8-8 present the variation in the V-velocity and U-velocity profiles at the mid horizontal and vertical planes of the cavity respectively at Da=10-6, Ra*=500, N= -5.0 and at different inclination angles for Le=1 and 10, respectively. For Lewis number unity, the U-velocity profiles shows that for =0 o, the U-velocity profile is very sharp and it smoothly decreases when the inclination angle increases. In the case of the V-velocity profile, the V- velocity decreases when the inclination angles increase. Also, it can be clearly observed that most of the high negative V-velocities are near the hot wall, the high positive V-velocities are near the cold wall (counter clockwise flow) and the minimum V-velocities are in the middle of the porous cavity. Additionally, it can be seen that the same behaviour persists in Figure 8-8 for a Lewis number equal to 10 except in the effects of the secondary circulations. In general, for a positive buoyancy ratio, both the U and V-velocities increase when the angle of inclination increases until =60 o and then they decreases when the inclination angle increases. For the opposing flow (negative buoyancy ratio) the maximum velocities are at =0 o and they decrease when the inclination angles decreases. It can also be observed that the velocity decreases when the Lewis number increases for the same modified Rayleigh number. The Lewis number increases imply a similar increase in the effective mass diffusivity coefficient vis-à-vis the effective thermal diffusivity coefficient and is due to the ensuing driving force for the flow field (buoyancy ratio negative).the velocities consequently decrease. 148

149 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities Figure 8-6 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=0.1, N= 2.0 for different inclination angles. Figure 8-7 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=1.0, N= -5.0 for different inclination angles. 149

150 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities Figure 8-8 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=10.0, N= -5.0 for different inclination angles. (a) (b) Figure 8-9 Variation in (a) average Nusselt numbers (Nu) and (b) average Sherwood (Sh) numbers with Ra* for Da=10-4, =30 o and with different Le and N. 150

151 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities The effect of the Rayleigh number on the average Nusselt and the average Sherwood numbers is presented in Figure 8-9 (a) and (b), respectively for Da=10-4 and =30 o with a different Lewis and buoyancy ratio. These figures show that, when the Rayleigh number increases, the average Nusselt number increases. Additionally, for a positive buoyancy ratio, the average Nusselt number increases when the Lewis number increases until Le=1 and then the average Nusselt decreases when the Lewis number increases. Moreover, it decreases when the Lewis number increases for a negative buoyancy ratio as shown in Table 2. Also, the average Sherwood number increases when the Rayleigh and Lewis numbers are increased. At a constant Rayleigh number, the reduction of the Lewis number naturally implies a relatively high mass diffusivity value. When the Lewis number is decreased, the thickness of the thermal boundary layers near the hot and cold walls becomes thinner than the concentration boundary layers. Therefore, when the Lewis number decreases, the heat transfer increases and the mass transfer decreases. Also, the average Nusselt and Sherwood numbers increase when the buoyancy ratio increases (for N -1), and they also increase when the buoyancy ratio decreases (for N -1), as shown in Table 8-1. Table 8-1 presents the effect of inclination angles on the average Nusselt and the average Sherwood numbers for Da=10-6, Ra*=10-2 at different Lewis numbers and buoyancy ratios. It shows that for a positive buoyancy ratio, the maximum average Nu and Sh occur when the inclination angle is between 30 o and 45 o. This means, both the average Nu and Sh increase when the angle of inclination increases until around =45 o, and then they decrease when the angle of inclination is increased. For the opposing flow (negative buoyancy ratio), both the average of Nu and Sh decrease when the angle of inclination decreases. The same behaviour can be seen for low Darcy numbers. Figure 8-10 to Figure 8-13 present the stream functions (left), isotherms (middle) and iso-concentrations (right) lines for Da=10-2, Ra*=1000, Pr=1 with different Lewis and buoyancy ratios. As mentioned above, the average of Nu and Sh increases when the Ra* increases and they also increase when the N 1as shown in Figure 8-14 for Da=10-2, Le=0.1, Pr=1 and =60 o. Figure 8-15 shows the effects of the Lewis numbers on the average Nu and Sh numbers for Da=10-2, Ra*=100, Pr=1 and =60 o. The results show that 151

152 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities the average Nu increases when the Le decreases except when the buoyancy ratios are positive and Le<1. Additionally, the average Sh increases when the Le increases Conclusions Double-diffusive natural convective heat and mass transfer in an inclined porous square have been investigated numerically. Two opposing walls of the cavity were adiabatic while the other two walls were maintained at fixed but different temperatures and concentrations. The generalized model was used to solve the governing equations and the pressure velocity coupling was treated using the SIMPLER algorithm. The results are presented in terms of streamline, isothermal, iso-concentration and U-velocity, V-velocity, average Nusselt and Sherwood number profiles. The predictions included different values in the non-dimensional governing parameters, specifically, the angle of inclination of the cavity (0 o 90 o ), the Lewis number (0.1 Le 10), the buoyancy ratio (-5 N 5), the modified Rayleigh number (100 Ra* 1000) and the Darcy number (10-6 Da 10-2 (, while the Prandtl number is taken as Pr=1.0. The results show that, when opposite buoyancy forces are considered (N -1), the convection in the porous cavity is always a multiplicity of steady solutions with flow fields of two, three and four flow cells. Also, it can be observed that the positive vortex strength increases when the angle of the inclination is increased until =45 o and subsequently the positive vortex strength decreases when the angle of the inclination is increased. Also, the same behaviour can be seen for the positive buoyancy forces. Furthermore, the negative vortex strength decreases when the angle of the inclination is increased. For cases of =90 o, it can be observed that no vortex appears and this occurs because of the low flow intensity. On the other hand, when a positive buoyancy ratio is considered the convection in the porous cavity is always in a single cell and it occurs in a clockwise flow. Moreover, at =90 o the bicellular flows occur. In general, the results show that, for a positive buoyancy ratio (N>0), the Nu and Sh increase when the angle of inclination increases until around =45 o, and then they decrease when is increased. For an opposing flow (negative buoyancy ratio), both 152

153 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities the Nu and Sh decrease when the angle of inclination is decreased. Also, the average Nusselt and Sherwood numbers increase when the Rayleigh number increases, while they decrease when the Darcy number increases. For a positive buoyancy ratio (N>0), the Nu increases when the Le increases until Le is equal to 1 and then the Nu decreases when the Le increases. On the other hand, the Nu decreases when the Le increases for a negative buoyancy ratio. Moreover, the Sherwood number increases when the Lewis number increases. It was also observed that the Nu and Sh numbers increase when the N 1 increases. 153

154 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities =0 =30 =45 =60 =90 Figure 8-10 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-2, Ra*=1000, Pr=1.0, =0.36, Le=10.0 and N= -1.0 with different inclination angles. 154

155 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities =0 =30 =45 =60 =90 Figure 8-11 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-2, Ra*=1000, Pr=1.0, =0.36, Le=10.0 and N= -5.0 with different inclination angles. 155

156 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities =0 =30 =45 =60 =90 Figure 8-12 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-2, Ra*=1000, Pr=1.0, =0.36, Le=0.1 and N= -1.0 with different inclination angles. 156

157 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities =0 =30 =45 =60 =90 Figure 8-13 Stream functions (left), isotherms (middle) and iso-concentration (right) lines for Da=10-2, Ra*=1000, Pr=1.0, =0.36, Le=0. 1 and N= 2.0 with different inclination angles. 157

158 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities Figure 8-14 validation of average Nusselt (Nu) and Sherwood (Sh) numbers on the left porous wall with wall buoyancy ratio (N) with different Ra* at Da=10-2, Pr=1.0, Le=0.1 and =60 o. Figure 8-15 Variation of average Nusselt (Nu) and Sherwood (Sh) numbers on the left porous wall with wall buoyancy ratio (N) with different Le at Da=10-2, Ra*=10 2, Pr=1.0 and =60 o. 158

159 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities 8.5. Results and conclusions of the rectangular porous cavity cases Results In this section, the results are presented in terms of streamline, isothermal, isoconcentration and average Nusselt and Sherwood number profiles. The predictions include different values of the non-dimensional governing parameters, specifically, angle of inclination of the cavity (0 85), Lewis number (0.1 Le 10), and the buoyancy ratio (-5 N 5) while Rayleigh number, Darcy Number and the Prandtl number are taken as 5*10 6, 10-4 and 4.5, respectively. The effects of the angle of inclination (), stream functions, isotherms and isoconcentration lines are presented in Figure 8-16 for Da=10-4, Ra=5*10 6, Pr=4.5, A=5, Le=0.1 and N= Figure 8-16 (a) shows that the vortex strength of the fluid in the porous medium is higher where there is no inclination angle (=0) and it decreases when the angle of inclination increases. The recirculation region has migrated from the hot left wall towards the direction of the cold right wall. Also, secondary circulations appear for a higher angle of inclination and they achieve a maximum at (=85). For isotherm lines, it can be seen in Figure 8-16 (b) that the isotherms are nearly parallel near the left wall and there is an increase in the isotherms near the left upper corner as a result of the fluid rising due to the influence of the buoyancy force. This behaviour increases with an increase in the angle of inclination especially at (=85) when a multi flow appears. The same behaviour can be seen in Figure 8-16 (c) for iso-concentration lines. The same behaviour can be seen in Figure 8-17 when the aspect ratio is equal to 4 and for the Da=10-4, Ra=5*10 6, Pr=4.5, Le=0.1 and N= Figure 8-18, where Da=10-4, Ra=5*10 6, Pr=4.5, A=2, Le=10 and N=-5.0, also presents the stream functions, isotherms and iso-concentration lines. For the opposing flow, (N=-5) the vortex strength is very small due to the effects of mass transfer especially for high Lewis numbers. For high Lewis numbers, most of the buoyancy force is due to the effect of concentration variations whereas it is due to the effect of temperature variation at low Lewis numbers. As a result, in the case of 159

160 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities increases in the angle of inclination, the isotherms and iso-concentration lines are nearly parallel. For (=85), the heat transfer occurs due to natural conduction through the porous media rather than due to natural convection. The effect of the Lewis number and the buoyancy ratio for Da=10-4, Ra=5*10 6, Pr=4.5, =60 and different aspect ratios are shown in Figure = 0 = 30 = 45 = 60 = 85 a b c Figure 8-16 (a) Stream functions, (b) isotherms and (c) iso-concentration lines for Da=10-4, Ra=5*10 6, Pr=4.5, A= 5, Le=0.1 and N= +2.0 with different angle of inclination. 160

161 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities When the Lewis number increases the Nusselt number decreases and the Sherwood number increases. Furthermore, the average Nusselt and Sherwood numbers increase when the buoyancy ratio increases (for N -1) and they also increase when the buoyancy ratio decreases (for N -1), as shown in Figure = 0 = 30 = 45 = 60 = 85 a b c Figure 8-17 (a) Stream functions, (b) isotherms and (c) iso-concentration lines for Da=10-4, Ra=5*10 6, Pr=4.5, A= 4, Le=0.1 and N= +2.0 with different angle of inclination. 161

162 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities In general, the average Nusselt and Sherwood numbers decrease when the aspect ratio increases. To show the effects of the inculcations angle, the average Nusselt and Sherwood numbers are presented in Figure 8-20 for Da=10-4, Ra=5*10 6, Pr=4.5, A=2, N=-5 and also in Figure 8-21 for Da=10-4, Ra=5*10 6, Pr=4.5, A=2, N=+5 and different Lewis numbers. These two figures show that the maximum Nu and Sh occurs when =0 and they decrease when the angle of inclination increases. = 0 = 30 = 45 = 60 = 85 a b c Figure 8-18 (a) Stream functions, (b) isotherms and (c) iso-concentration lines for Da=10-4, Ra=5*10 6, Pr=4.5, A=2, Le=10 and N= -5.0 with different angles of inclination. 162

163 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities (a) average Nusselt Number (b) average Sherwood Number Figure 8-19 (a) the average Nusselt number and (b) the average Sherwood number for Da=10-4, Ra=5*10 6 with different aspect ratios, Lewis numbers and angle of inclinations. 163

164 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities (a) average Nusselt Number (b) average Sherwood Number Figure 8-20 (a) the average Nusselt Number and (b) the average Sherwood Number for Da=10-4, Ra=5*10 6, Pr=4.5, A=2, N = - 5 with different Lewis numbers and angle of inclinations. 164

165 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities (a) average Nusselt Number (b) average Sherwood Number Figure 8-21 (a) the average Nusselt Number and (b) the average Sherwood Number for Da=10-4, Ra=5*10 6, Pr=4.5, A=2, N =5 with different Lewis numbers and angle of inclinations 165

166 Chapter 8. Convective heat and Mass Transfer in Inclined Porous Cavities Conclusions Numerical steady of double-diffusive natural convective heat and mass transfer in a two-dimensional inclined rectangular porous medium has been investigated. Two opposing walls of the cavity are maintained at fixed but different temperatures and concentrations; while the other two walls are adiabatic. The generalized model has been used to solve the governing equations. The results are presented for different values of angle of inclination of the cavity (0 85), Lewis number (0.1 Le 10), and the buoyancy ratio (-5 N 5) while Rayleigh number, Darcy Number and the Prandtl number are taken as 5*10 6, 10-4 and 4.5, respectively. While a number of relevant results have been presented in this paper, work is under way for additional studies incorporating for example, high Rayleigh number and Darcy number cases. The results show that as the aspect ratio increases, the average Nusselt and Sherwood numbers is reduced, although they decrease when the angle of inclination increases. For the buoyancy ratio effects, the Nusselt number increases when the buoyancy ratio increases for (N -1) and also when its decreases for (N -1). This is the same behaviour for Sherwood number. Furthermore the Nusselt number decreases when the Lewis number increases and the Sherwood number increases when the Lewis number increases. 166

167 Chapter 9 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium 9.1. Introduction In the last few years, the investigations Soret effect on double-diffusive porous medium flows and the coupled heat and mass transfer in a fluid porous medium has attracted considerable attention In fact, there have been limited studies made on reviewing the Soret and Dufour effects in heat and mass transfer through a porous medium and most of these studies used the Darcy model. Therefore, a natural investigation of two-dimension doublediffusive natural convection heat and mass transfer with Soret and Dufour effects in a porous medium by using the generalized model is interesting to study.

168 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium This chapter presents the results of the Soret and Dufour effects on double-diffusive natural convection heat and mass transfer in porous cavities. Porosity is assumed to be constant and equal to 0.6. This part of the research has been submitted as a paper to the International Journal of Heat and Mass Transfer Mathematical formulation of the model: The model is performed by using a two-dimensional porous cavity filled with an isotropic porous medium as shown in Figure 9-1. All the walls are impermeable. The left wall is maintained at a constant high temperature (T h ) and a high concentration (C h ) while the right wall is maintained at a constant low temperature (T c ) and at a low concentration (C c ). Furthermore, the horizontal boundaries of the cavity are adiabatic. An isotropic, homogeneous, local thermal balance is assumed, as is saturation with an incompressible fluid. All the governing equations and the nondimensional parameters are presented in Chapter 3. Due to using a constant porous medium and a vertical cavity (=0), the momentum equations will not be the same for the variable porosity and they become: X-momentum conservation: 1 U 1 U 1 U P Pr U V U 2 2 X Y X Da Da X Y U V U Pr U U 3/2 2 2 (9.1) Y-momentum conservation: 1 V 1 V 1 V P Pr U V V 2 2 X Y Y Da Da X Y Ra Pr ( T N C ) U V V Pr V V 3/2 2 2 (9.2) 168

169 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Y adiabatic wall Porous medium T h C h C c T c, L g L adiabatic wall X Figure 9-1 The geometry of the model Initial condition The governing equations are solved by using the non-dimensional initial boundary conditions: at X 0 U V 0 1 C T 1 (9.3) h h at X 1 U V 0 0 C T 0 (9.4) h h at Y 0 U V 0 T Y C Y 0 (9.5) at Y 1 U V 0 T C 0 Y Y (9.6) 169

170 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium 9.4. Solution procedure A finite volume approach has been used to solve the dimensionless governing equations. The main steps of the pressure velocity coupling are treated with the SIMPLER algorithm in the porous media domain. The second order central discretization scheme is used for the momentum, energy and species equations, and the semi-implicit first order scheme is used for the time step in the momentum, energy and species equations. The inertial term in the momentum equations is considered a source term. A grid size of 120*120 was used for all the case studies and it was deemed adequate. The under-relaxation parameters of the U and V-velocities, the energy and the species equations were adopted as Results A wide range of non-dimensional parameters are used to present the Soret and Dufour effects in double-diffusive natural convective heat and mass transfer in porous square cavities which were supplied to heat and mass transfer from the side. These parameters are: the modified Rayleigh number (100 Ra* 500), the Darcy Number (10-6 Da 10-2 ), the Lewis number (0.1 Le 20), the buoyancy ratio (-3 N +3), the Soret parameter (-5 S r +5), and the Dufour parameter (-2 D f +2). The results are presented in terms of streamline, isothermal, isoconcentration and U-velocity, V-velocity, average Nusselt and Sherwood number profiles. The Prandtl number (Pr) and the porosity () are assumed to be constant and equal to 0.71 and 0.6 respectively Soret parameter effects To show the effects of Soret parameter ( S r ) the results are presented into two groups. The first group of results show when the Soret parameter is positive (0 S r 5) and the second group of results are for the negative Soret parameter (-5 S r <0). 170

171 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Positive Soret parameter effects In Figure 9-2 the streamlines, isotherms and iso-concentration are presented for Da=10-6, Ra*=200, Pr=0.71, =0.6, N=+2.0, Le=1.0 where D f =0 with different a Soret parameter ( S r ). As mentioned in the previous chapters, a high positive buoyancy ratio means that the fluid rises up due to the influence of the temperature and mass variations which is more than the effect of the temperature variations in the buoyancy force. Where there is no Soret effects supplied ( S r =0), the recirculation region migrates from the hot left wall towards the direction of the cold right wall and that occurs as a result of the fluid rising due to the influence of the buoyancy force. The results clearly show that when the Soret parameter increases, the vortex strength of the fluid increases and it achieves a maximum (stream function equal to 27.90) when S r =5. For isotherms where the Soret parameter is equal to 0, it can be seen that the isotherms lines are nearly parallel near the left wall and the isotherms field has increased near the left upper corner. Also, it can clearly be seen that the isoconcentrations behaviour is nearly the same as the behaviour of the isotherms and that happens due to the considered Lewis number (Le=1). Moreover, the isoconcentrations' field near the horizontal walls are thicker in comparison with the isotherms' field and this happens because the buoyancy ratio is more than one. When the Soret effects are supplied (Soret parameter more than 0), the heat transfer directly affects the mass transfer. Due to the increase in mass transfer gradient, the buoyancy force increases and then it is implicitly affected to the heat transfer. For the reasons mentioned above, it can be seen that when the Soret parameter increases, the isotherms boundary layers near the vertical walls are thin and also that the isotherms field near the left top corner is small. Also, for the iso-concentration, it can be seen that the iso-concentration field is more sensitive than the isotherms, as mentioned before, especially for high S r. When the Soret parameter is equal to 2, it can be observed that the variations in iso-concentration in the bottom half near the hot left wall are higher than the variations in the iso-concentration in the upper half. On the other hand, the opposite behaviour can be seen near the cold right wall. 171

172 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Moreover, the concentrations gradient in the centre of the cavity increases when the Soret parameter is increased. For S r =5.0, it can be seen that new iso-concentration lines appear with magnitudes more/less than one/zero. These numbers refer to the mass source (for iso-concentration >1) and refer to the mass sink (for isoconcentration < 1). The same behaviour was pointed out and presented in an early study by Khadiri et al. [143]. For a high Lewis number, the vortex strength is small due to the effects of mass transfer especially for the low Soret parameters. Figure 9-3 shows the stream functions, the isotherms and the iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, N=+2.0, Le=5.0 where D f =0 with differents r. When the Soret parameter is equal to 0, the main flow is clockwise due to used positive buoyancy ratio and the maximum stream function of about 7.53 which is very small comparing with the case where the Le=1 (stream function equal to 13.37). As mentioned before, the vortex strength of the fluid increases when the stream function is equal to when the S r increases. The maximum S r =5. For isotherms, it can be observed that there are no significant effects on the isotherms due to increases in the Soret parameter. In general, the Lewis number increases imply a similar increase in the effective mass diffusivity coefficient vis-à-vis the effective thermal diffusivity coefficient. Therefore the flow is driven due to influence of the mass variations in the buoyancy force. As a result, the variations of the iso-concentration near the bottom left corner are higher than in the upper left corner. The thin boundary layer near the hot left wall refers to a high mass transfer which that means the Sh increases when the S r increases. In the case of an opposed flow (negative buoyancy ratio), the vortex strength is smaller than the vortex strength of the positive buoyancy ratio when the other parameters are constant and this happens due to the effects of mass transfer especially for a high Lewis number. Figure 9-4 shows the stream functions, the isotherms and the iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, N=-2.0, Le=5.0 where D f =0 with differents r. 172

173 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium S r = 0 S r = 1 S r = 2 S r = 5 a b c Figure 9-2 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=1.0, N= +2.0 and D f =0 with different positive Soret effects. 173

174 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium S r = 0 S r = 1 S r = 2 S r = 5 a b c Figure 9-3 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=5.0, N= +2.0 and D f =0 with different positive Soret effects. 174

175 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium S r = 0 S r = 1 S r = 2 S r = 5 a b c Figure 9-4 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=5.0, N= -2.0 and D f =0 with different positive Soret effects. 175

176 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Where there are no Soret effects ( S r =0), the flow is counter clockwise with a maximum negative stream function about It can be observed that the negative vortex strength increases when the Soret parameter is increased. For S r =5, the maximum negative stream functions is about In the case of isotherms field where the Soret parameter is equal to 0, it can be observed that the maximum isotherms field is nearest to the left wall from the bottom and then decreases in the Y-direction until the left top corner. Also, for the iso-concentration lines, the maximum (minimum) iso-concentration gradients are near the left top (right bottom) corner. For an increasing Soret parameter S r > 0), as mentioned before, it can be seen that the isotherms gradients as well as the iso-concentration gradients increase near the vertical walls and they decrease in the cavity core. Figure 9-5 presents the variation in the horizontal and vertical velocity profiles at the mid vertical and horizontal planes of the cavity respectively at Da=10-6, Ra*=500, Le=10, N= 3.0 where D f =0 and at different Soret parameters. At the mid horizontal plane of the cavity (Y=0.5), the variation in the V-velocity along the X-axis for different Soret parameters shows that the V-velocities increase when the Soret parameter increases. Also, it can be seen that there are no effects on the V-velocity profiles at the mid vertical plane of the cavity. On the other hand, the variation in the U-velocity along the Y-axis for different Soret parameters at the mid vertical plane of the cavity (X=0.5) shows that the U-velocities increase when the Soret parameter increases. To show the effect of a high Lewis number, the variation in the horizontal and vertical velocity profiles at the mid vertical and horizontal planes of the cavity respectively at Da=10-6, Ra*=500, Le=20, N= 3.0 where parameter are presented in Figure 9-6. For Le=20 and D f =0 and different Soret S r =5, the maximum positive U and V-velocities are about 72 and 190 respectively while for Le=10 and S r =5 the maximum positive U and V-velocities are about 90 and 295 respectively. Therefore, for a high Lewis number both the U and V-velocities decrease when the Lewis number increases for the same modified Rayleigh number. 176

177 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium In the case of an opposed flow, Figure 9-7 and Figure 9-8 present the variation in the V- velocity and U-velocity profiles at the mid horizontal and vertical planes of the cavity, respectively, at Da=10-6, Ra*=500, N= -1.0 and D f =0 with different Soret parameters for Le=1 and 0.1 respectively. When Le=1, the U-velocity profiles show that, for high Soret parameter effects, the U-velocity near the bottom horizontal wall increases very sharply in the Y direction through the boundary layer until it achieves the maximum. After that the U-velocity smoothly decreases until it becomes zero when the height of the cavity is about 20%; then it decreases to achieve the maximum negative U-velocity at around 42% of the cavity height. On the other hand, for the second half of the cavity, the same behaviour can be seen near the upper horizontal wall but it is in the opposite direction. The same behaviour can also be seen for all U-velocities with different Soret parameters. Also, the V-velocity profiles shows that, for high Soret parameter effects, the V-velocity near the left vertical wall decreases very sharply in the X direction through the boundary layer and then it smoothly decreases until it becomes zero when the length of the cavity is about 21%; then it increases to achieve maximum positive V-velocity around 35% of the cavity length. On the other hand, for the right half of the cavity, the same behaviour can be seen near the right vertical wall but it is in the opposite direction. The same behaviour can also be seen for all U-velocities with different Soret parameters. The multiple maximum/minimum U-and V-velocities refer to the emergence of the negative secondary circulations. The counter clockwise secondary circulations flow appears near the left (right) bottom (top) corners and the clockwise main circulations flow is homogenised and symmetrical on the secondary diagonal of the porous cavity. This occurs because the buoyancy ratio is equal to -1. In general, when the Soret parameter increases, the U and V-velocities increase. Furthermore, for a Lewis number less than one, the velocity increases when the Lewis number decreases for the same buoyancy ratio and this happens because the flow has almost been driven due to mass transfer rather than heat transfer. 177

178 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Figure 9-5 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=10, N= 3.0 and D f =0 for different Soret effects. Figure 9-6 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=20, N= 3.0 and D f =0 for different Soret effects. 178

179 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Figure 9-7 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=1, N= -1.0 and D f =0 for different Soret effects. Figure 9-8 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=0.1, N= -1.0 and D f =0 for different Soret effects. 179

180 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium (a) (b) Figure 9-9 Variation in (a) average Nusselt numbers (Nu) and (b) average Sherwood numbers (Sh) with positive S r for Da=10-6,Ra*=100 with different Le and N. (a) (b) Figure 9-10 Variation in (a) average Nusselt numbers (Nu) and (a) average Sherwood numbers (Sh) on the left porous wall with a buoyancy ratio (N) with different Le at Da=10-6, Ra*=100, Pr=0.71, =0.6 and S r =1. 180

181 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium The effect of the Soret parameter on the average Nusselt and the average Sherwood numbers is presented in Figure 9-9 (a) and (b), respectively, for Da=10-6, Ra*=100 with different Le and N. These graphs show that, when the Soret parameter increases, the average Nusselt number increases. Furthermore, when the Lewis number is more than unity, the average Nusselt number decreases when the Lewis numbers are increased, even if the buoyancy ratio is positive or negative. Moreover, the average Sherwood number increases when the Lewis number increases. At a constant heat transfer (a constant modified Rayleigh number), the increase in the Lewis number implies a relatively low mass diffusivity value. When the Lewis number is increased, the thickness of the thermal boundary layers near the hot and cold walls becomes thicker than the concentration boundary layers. Therefore, when the Lewis number increases the heat transfer decreases and the mass transfer increases. As mentioned previously, when the Lewis number increases, the average Nusselt number decreases and the average Sherwood number increases, as shown in Figure 9-10 for Da=10-6, Ra*=100, Pr=0.71, =0.6 and S r =1. Furthermore, for a constant Lewis number, the average Nusselt and Sherwood numbers increase when the buoyancy ratio increases (for N > -1), and they also increase when the buoyancy ratio decreases (for N < -1) Negative Soret parameter effects Streamlines, isotherms and iso-concentration are presented in Figure 9-11 for Da=10-6, Ra*=200, Pr=0.71, =0.6, N=+2.0, Le=1.0 where D f =0 with different negative Soret parameters. For a low negative Soret parameter, the main flow is clockwise and there are no secondary circulations. The results clearly show that when the Soret parameter decreases (or increases in a negative direction), the vortex strength of the fluid decreases and it achieves a minimum (stream function equal to 9.48) when the S r =-2. Also, when the negative Soret parameter decreases, the nondimensional concentration decreases. For a high negative Soret parameter, the nondimensional concentration can be negative as mentioned in previous studies (Postelnicu [144, 145] Bourich et al. [146] and Mansour et al. [107]). Negative non- 181

182 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium dimensional concentrations appear when the effect of the negative Soret term in the species equation is more than the convection and diffusion terms. Therefore, for a high negative Soret parameter, the main flow is clockwise with a maximum stream function of about 9.71 and the secondary clockwise circulations appear in the left top corner and in the right bottom corner with a maximum stream function of about The secondary clockwise circulations appear as a result of the fluid rising due to the influence of the temperature variation which is more than the effect of the mass variations in the buoyancy force in these regions. When the Soret parameter decreases, the non-dimensional concentration decreases due to the negative effects of the Soret parameter term in the species equation. For this reason the buoyancy force decreases. Also, it can be seen that the isoconcentration field increases near the right upper corner and it is decreases near the left lower corner and this occurs due to the effect of the heat transfer gradients. For the reasons mentioned above, it can be seen that when the negative Soret parameter decreases, the isotherms field near the left top and the right bottom corners increases and also the isotherms gradients in the centre of the cavity increases. When the Lewis number increases for the same Rayleigh number and a positive buoyancy ratio, the flow will be driven due to the influence of the mass variations in the buoyancy force. For this reason the velocities decrease. The stream functions, isotherms and iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, N=+2.0, Le=5.0 where D f =0 with different S r are presented in Figure In general, the main flow is clockwise due to a used positive buoyancy ratio and the maximum stream function for Le=5 is smaller than the maximum stream function of Le=1 for the same Soret parameter. It can clearly be seen that, when the decreases, the vortex strength of the fluid decreases and the minimum positive stream function is equal to 5.00 and the negative stream function is equal to when the S r =-5. As mentioned previously, there are no significant effects on either the isotherms or the iso-concentrations lines when the negative Soret parameter decreases. In general, both the isotherms and iso-concentrations gradients increase when the Lewis number increases. S r In the case of an opposed flow, a different behaviour can been seen for a negative Soret parameter. Figure 9-13 shows the stream functions, the isotherms and the 182

183 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, N=-2.0, Le=5.0 where D f =0 with different negative Soret parameters. When the Soret parameter is equal to -1, the main flow is counter clockwise with a maximum negative stream function of about and the secondary clockwise circulations appear in the left bottom corner and in the right top corner with a maximum stream function of about The secondary clockwise circulations appear as a result of the fluid rising due to the influence of the temperature variation which is more than the effect of the mass variations in the buoyancy force in these regions. For an isotherms field where S r =-1, it can be observed that the maximum isotherms field is near the left wall from the bottom and then decreases in the Y- direction until the left top corner. As for the iso-concentration field, as mentioned above, it is more sensitive than the isotherms and this occurs because of the considered Lewis number (Le=5). The maximum iso-concentration field appears near the left bottom corner and the minimum concentration field appears near the right top corner. Elsewhere, most of the iso-concentration variations are near the maximum negative stream functions which are around the main diagonal of the cavity. In terms of a decreasing Soret parameter, it can be observed that the positive vortex strength increases when the Soret parameter is decreased until of S r = -2 and then it decreases with the decreasing S r. Furthermore, the negative vortex strength increases when the Soret parameter is increased. ForS r = -5, the maximum positive stream functions and the maximum negative stream functions are about 0.45 and -3.25, respectively. Furthermore, it can be observed that the isotherms boundary layers thin near the upper half of the left wall and they are thick near the lower half of left wall. Also, the opposite behaviour can be seen near the right wall. When the Soret parameter decreases, the isotherms boundary layers increases due to a decreases in the buoyancy force. With regard to the iso-concentration lines, the maximum (minimum) iso-concentration appears near the left (right) wall which is around the maximum positive stream functions and most of the iso-concentration variations are near the maximum negative stream functions which are around the main diagonal of the cavity. 183

184 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium S r = 0 S r =-1 S r =-2 S r =-5 a b c Figure 9-11(a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=1.0, N= +2.0 and D f = 0 with different negative Soret effects. 184

185 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium S r = 0 S r =-1 S r =-2 S r =-5 a b c Figure 9-12 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=5.0, N= +2.0 and D f = 0 with different negative Soret effects. 185

186 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium S r = 0 S r =-1 S r =-2 S r =-5 a b c Figure 9-13 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=5.0, N= -2.0 and D f = 0 with different negative Soret effects. 186

187 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Figure 9-14 to Figure 9-17 present the variation in the horizontal and vertical velocity profiles at the mid vertical and horizontal planes of the cavity, respectively, for Da=10-6, Ra*=500, D f = 0 for different Le and N with different negative Soret parameters. The U and V-velocities have the same behaviour when the Soret parameters are positive, except when the buoyancy ratio is equal to -1. In the cases of a negative buoyancy ratio, the main flow in the cavity is counter clockwise and there is a chance that secondary clockwise circulations will appear. It can clearly be seen that, for the positive buoyancy ratios, the U and the V-velocities decrease when the Soret parameter decreases whereas they increase when the Soret parameter decreases for a negative buoyancy ratio. Moreover, as mentioned before, when the Lewis number increases, the U and V-velocities decrease. The effect of a negative Soret parameter on the average Nusselt and the average Sherwood numbers is presented in Figure 9-18 (a) and (b), respectively, for Da=10-6, Ra*=100 with different Le and N. These figures show that, when the Soret parameter decreases, the average Nusselt number decreases. Furthermore, the average Nusselt number decreases when the Lewis numbers increase even if the buoyancy ratio is positive or negative. Moreover, when the Soret parameter decreases, the average Sherwood number decreases and it increases when Lewis numbers increase. As mentioned previously, when the Lewis number increases, the average Nusselt number decreases and the average Sherwood number increases, as shown in Figure 9-19 for Da=10-6, Ra*=200, Pr=0.71, =0.6 and S r =1. Furthermore, for a constant Lewis number, the average Nusselt and Sherwood numbers increase when the absolute buoyancy ratio ( N 1) increases. In general, the minimum average Nusselt and Sherwood numbers are found when N=-1 for the same Lewis number and the same modified Rayleigh number and they are equal to 1 when Le=1.0. When a negative Soret parameter is applied, the minimum average Sherwood number will be less than 1. Therefore, it can be seen that a negative magnitude arises from the average Sherwood number especially when N=-1 and for low Lewis numbers with high negative Soret parameters. 187

188 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Figure 9-14 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=10, N= 3.0 and D f =0 for different Soret effects. Figure 9-15 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=20, N= 3.0 and D f =0 for different Soret effects. 188

189 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Figure 9-16 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=1, N= -1.0 and D f =0 for different Soret effects. Figure 9-17 U-velocity at X=0.5 and V-velocity at Y=0.5 for Da=10-6, Ra*=500, Le=0.1, N= -1.0 and D f =0 for different Soret effects. 189

190 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium (a) Figure 9-18 Variation in (a) average Nusselt number (Nu) and (b) average Sherwood number (Sh) with negative S r for Da=10-6,Ra*=100 with different Le and N. (b) (a) (b) Figure 9-19 Variation in (a) average Nusselt number (Nu) and (a) Sherwood number (Sh) on the left porous wall with buoyancy ratio (N) with different Le at Da=10-6, Ra*=200, Pr=0.71, =0.6 and S r =

191 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium For high Darcy numbers, Figure 9-20 and Figure 9-21 presents the stream functions, isotherms and iso-concentration lines for Da =10-4, Ra* =500, Pr = 0.71, = 0.6, N=2.0, Le=10.0 where respectively. D f =0 with different positive and negative Soret parameters, When there are no Soret effects supplied ( S r =0), the recirculation region migrates from the hot left wall towards the direction of the cold right wall and the flow is clockwise with a maximum stream function of about For isotherms lines, it can be seen that the isotherms boundary layer is thin near the left wall and it increases near the left upper corner as a result of the fluid rising due to the influence of the buoyancy force. The iso-concentration field, as mentioned before, is more sensitive than the isotherms for high Lewis numbers. It can be seen that the isoconcentrations lines are nearly parallel near the walls and the maximum isoconcentration field appears near the left top corner and minimum concentration field appears near the right bottom corner. Elsewhere, most of the iso-concentration variations are near the wall and there are no significant iso-concentration variations in the cavity core. This happens due to the effect of high Rayleigh and Lewis numbers. In terms of increasing the Soret parameter, Figure 9-20 shows that the vortex strength increases when the Soret parameter increases and the maximum stream function is about 9.38 when S r = 5. Also it can be seen that both the isotherms and iso-concentrations gradients increases when the Soret parameter increases. Also, the same behaviour can be seen with the negative Soret parameter. Figure 9-20 shows that when the negative Soret parameter decreases the vortex strength decreases and the maximum stream function is about 7.90 when S r = -5. Table 9-1 presents the effect of Soret parameters on the average Nusselt and the average Sherwood numbers for Da=10-4, Ra*=100 without Dufour effects at different Lewis numbers and buoyancy ratios. In the case of a positive Soret parameter, the results show that when Lewis number unity, the Nu increases when the Soret parameter increases. Moreover, for a high Lewis number, it can also be seen that when the Soret parameter increases, the Nu 191

192 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium increases for all positive and negative buoyancy ratio cases except when the N=-1. For high Lewis numbers, most of the buoyancy force is due to the effect of concentration variations and when the Lewis number increases, the Soret effects term in the species equation decreases for constant buoyancy ratio. For these reasons the Nu decreases when the Soret parameter increases. Moreover, for all kinds of flow, when the Soret parameter increases, the average Sherwood number increases. Additionally, for negative Soret parameter the results show that for a positive buoyancy ratio, when the Soret parameter decreases, the Nusselt number increases. Furthermore, for an opposing flow (buoyancy ratio> -1), the Nu decreases when the S r decreases until S r =-2 and then the Nu increases and achieves a maximum at S r =-5. It can also be seen that, for a buoyancy ratio equal to -1, the Nusselt number increases, when the Soret parameter decreases. In general, the average Sherwood number decreases when the Soret parameter decreases. For high negative Soret parameters, the effects of the negative Soret term in the species equation are more than the convection and diffusion terms. For this reason, a negative Sherwood number can be seen especially when the buoyancy ratio is closer to -1 (which is almost the minimum Sh), as mentioned in previous studies (Postelnicu [144, 145] Bourich et al. [146] and Mansour et al. [107]). As mentioned previously, the average Nusselt and Sherwood numbers increase when the buoyancy ratio increases (for N > -1), and they also increase when the buoyancy ratio decreases (for N < -1). Moreover, when the Lewis number increases, the average Sherwood number increases while the average Nusselt number decreases. The same behaviour in average Nusselt and Sherwood numbers can be seen in Table 9-2 for Da=10-2, Ra*=500 without Dufour effects at different Lewis numbers and buoyancy ratios. In general, when the Darcy number increases the the average Nusselt and Sherwood numbers increase. 192

193 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium S r = 0 S r = 1 S r = 2 S r = 5 a b c Figure 9-20 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-4, Ra*=500, Pr=0.71, =0.6, Le=10, N= 2.0 and D f =0 with different positive Soret effects. 193

194 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium S r = 0 S r =-1 S r =-2 S r =-5 a b c Figure 9-21 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-4, Ra*=500, Pr=0.71, =0.6, Le=10, N= 2.0 and D f =0 with different negative Soret effects. 194

195 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Table 9-1 Average Nusselt and Sherwood numbers at Da=10-4, Ra*=100 without Dufour effects. Le N S r =-5 S r =-2 S r =0 S r =2 S r =5 Nu Sh Nu Sh Nu Sh Nu Sh Nu Sh

196 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Table 9-2 Average Nusselt and Sherwood numbers at Da=10-2, Ra*=500 without Dufour effects. Le N S r =-5 S r =-2 S r =0 S r =2 S r =5 Nu Sh Nu Sh Nu Sh Nu Sh Nu Sh

197 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium Dufour parameter effects To show the effects of the Dufour parameter ( D ), the results presented in this section are for the Dufour parameter (-2 f Df 2). For positive Dufour effects, the streamlines, isotherms and iso-concentration are presented in Figure 9-22 for Da=10-6, Ra*=200, Pr=0.71, =0.6, N=+3.0, Le=10 where S r =-2 with different Dufour parameters ( D ). Where there are just negative Soret effects (Dufour effect f is equal to 0), the recirculation region migrates from the hot left wall towards the direction of the cold right wall and that occurs as a result of the fluid rising due to the influence of the buoyancy force. The results clearly show that when the Dufour parameter increases, the vortex strength of the fluid increases and it achieves a maximum (stream function equal to 13.50) when the D f =2. For the isotherms, it can clearly be seen that the high isotherms lines are nearly parallel near the left wall and the isotherms field increases near the left upper corner and that the opposite behaviour can be seen near the right wall for low isotherms lines. For high Lewis numbers, the flow is driven by the effect of the concentration variations and when the Dufour parameter increases, the Dufour effects on the heat transfer increases. As a result, high isotherms (more than one) and low isotherms (less than zero) appear. It refers to the heat source when the isotherms >1 and for the heat sinks when the isotherms < 1. In general, when the Dufour parameter increases, the heat source increases and the heat sinks decrease. Due to the Soret effects, the iso-concentration lines have different behaviours. As mentioned above, for high Lewis numbers most of the buoyancy force is due to the effect of the concentration variations. For this reason, the high iso-concentration gradients are near the walls and there is no significant change in the cavity core. When the Dufour parameter increases, there is no significant change in the isoconcentration lines. For the negative Dufour effects, the streamlines, isotherms and iso-concentration are presented in Figure 9-23 for Da=10-6, Ra*=200, Pr=0.71, =0.6, N=-3.0, Le=10 for S =2 with different negative Dufour parameter ( D ). The results clearly show that r f 197

198 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium the main flow is counter clockwise and when the Dufour parameter increases, the negative vortex strength of the fluid increases and it achieve a maximum (negative stream function equal to -8.25) when the D f =-2. For the isotherms, it can clearly be seen that the high isotherms lines are almost parallel near the right wall and the isotherms field increases near the right upper corner and that the opposite behaviour can be seen near the left wall for high isotherms lines. As mention before, when the Dufour parameter decreases, the heat source increases and the heat sinks decrease. Alos, because of the high Dufour effects on the heat transfer (for a high Lewis number), there are significant impact on the iso-concentration. The effect of the positive Dufour parameter on the average Nusselt and the average Sherwood numbers is presented in Figure 9-24 (a) and (b) respectively for Da=10-6, Ra*=100 with different Le and N for S r =1. These figures show that, when the Dufour parameter increases, the average Nusselt number increases. Furthermore, the average Nusselt number increases when the Lewis numbers are increased, even if the buoyancy ratio is positive or negative. Moreover, the average Sherwood number increases even when the positive Dufour parameter increase s as well as when the Lewis numbers increases. In case of a negative Dufour parameter, Figure 9-25 (a) and (b) present the variation of the average Nusselt and the average Sherwood numbers respectively for Da=10-6, Ra*=100 with different Le and N for S r =-1. It is observed that when the negative Dufour parameter decreases, the average Nusselt decreases. Also, it is decreased when the Lewis number increases. Additionally, the average Sherwood number increases when the Dufour parameter increases. On the other hand, the average Sherwood number increases when the Lewis numbers increase Conclusions Two-dimensional double-diffusive natural convection heat and mass transfer with Soret and Dufour effects in porous square have been investigated numerically. The verticals walls were maintained at fixed but different temperatures and 198

199 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium D f = 0 D f =1 D f =1.5 D f =2 a b c Figure 9-22 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=10, N= 3.0 and r S =-2 with different positive Dufour effects. 199

200 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium D f = 0 D f =-1 D f =-1.5 D f =-2 a b c Figure 9-23 (a) Stream functions, (b) isotherms, and (c) iso-concentration lines for Da=10-6, Ra*=200, Pr=0.71, =0.6, Le=10, N= -3.0 and r S =2 with different negative Dufour effects. 200

201 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium (b) (b) Figure 9-24 Variation in (a) average Nusselt numbers (Nu) and (b) average Sherwood (Sh) numbers with positive D f for Da=10-6,Ra * =100, S r =1 with different Le and N. (a) (b) Figure 9-25 Variation in (a) average Nusselt numbers (Nu) and (b) average Sherwood (Sh) numbers with positive D f for Da=10-6,Ra * =100, S r =-1 with different Le and N. 201

202 Chapter 9. Soret and Dufour Effects on Heat and Mass Transfer in Porous Medium concentrations while the horizontal walls were adiabatic. The generalized model was used to solve the governing equations and the pressure velocity coupling was treated using the SIMPLER algorithm. The results were presented in terms of streamline, isothermal, iso-concentration and U-velocity, V-velocity, average Nusselt and Sherwood number profiles. The Prandtl number (Pr) and the porosity () are assumed to be constant and equal to 0.71 and 0.6 respectively. The predictions included different values in the non-dimensional governing parameters, specifically, the modified Rayleigh number (100 Ra* 500),the Darcy Number (10-6 Da 10-2 ),the Lewis number (0.1 Le 20), the buoyancy ratio (-3 N +3), the Soret parameter (-5 S r +5), and the Dufour parameter (-2 D f +2). The results were presented into two groups (positive and negative Soret parameter) in order to show the effects of Soret parameter ( S r ). The results show that, when and positive Soret parameter increases, the vortex strength of the fluid increases and the isotherms boundary layers near the vertical walls are thin. It also can be seen that the isoconcentration field is more sensitive than the isotherms especially for highs r. It can be seen high iso-concentration lines appear with magnitudes more/less than one/zero. These numbers refer to the mass source (for iso-concentration >1) and refer to the mass sink (for iso-concentration < 1). Additionally, when the Soret parameter increases, the average Nusselt and Sherwood numbers increases. Furthermore, the average Nusselt number decreases when the Lewis numbers are increased, while the average Sherwood number increases when the Lewis numbers are increased. The result also shows that, when the positive Dufour parameter increases, the average Nusselt and Sherwood number increases. On the other hand, for the negative Dufour parameter, the average Nusselt decreases when the negative Dufour parameter decreases while the average Sherwood number increases when the negative Dufour parameter decreases. Furthermore, for a constant Lewis number, the average Nusselt and Sherwood numbers increases when the absolute buoyancy ratio ( N 1) increases. 202

203 Chapter Conclusions and Future Studies Conclusions Numerical investigations of double-diffusive natural convective heat and mass transfer in porous cavities with Soret and Dufour effects were considered in this thesis. The Darcy model and Darcy extension models, including the Darcy-Brinkman model, the Darcy-Forchheimer model and the generalized model what were presented briefly in Chapter 3 as well as the non-dimensional parameters and the overall model (using generalized model) including the energy and the species equations with the Soret and Dufour effects. An in-house FORTRAN code was developed for these studies to carry out the twodimension simulations of double-diffusion of natural convective heat and mass transfer in saturated porous enclosure with Soret and Dufour effects. The code was based on the finite volume and it can analyze the flow through porous medium under transient and steady state conditions. Two algorithms were employed in this code for the pressure velocity coupling and these are the SIMPLE and the SIMPLER algorithms. A second order central differencing discretization scheme was used for the momentum, the energy and the species equations while, a semi-implicit first

204 Chapter 10. Conclusions and Future Studies order scheme was used for time step advancement in the momentum equations, and ADI technique was adopted for the energy and species equations. This code named ALFARHANY code, and the whole procedure was explained in detail in Chapter 4. The current code tested and validated with the experimental, analytical, computational studies available in the literature. The validation and the comparisons for unsteady natural convective heat and mass transfer, conjugate, variable porosity, inclination angles of the cavity, and Soret and Dufour effects in two-dimensional saturated porous cavities were presented within six groupings in Chapter 5. The results from this validation were in good agreement with the available previous studies. Based on the study of conjugate natural convective heat transfer in a twodimensional porous square domain with finite wall thickness, it can be concluded that when the Darcy number increases, the average Nusselt number is reduced and the time required to reach the steady state is longer whereas it is shorter for a high Rayleigh number. Furthermore, when the wall thickness increases, the overall Nusselt number is reduced, whereas the average Nusselt number increases when the Rayleigh number and the thermal conductivity ratio increases. On the other hand, the time required to reach the steady state is longer for thick walls and low thermal conductivity ratios. For steady stat, a correlation to evaluate the average Nusselt numbers on the interface left wall for the whole range of the parameters was founded and presented as: Nu 0.098* Da Ra * D * k r Infect to present the effect of variable porosity, steady conjugate double-diffusive natural convection heat and mass transfer in a two-dimensional variable porosity layer sandwiched between two walls were investigated and presented In Chapter 7. The results were presented for the thermal conductivity ratio ( 0.1 k r 10 ), the ratio of wall thickness to its height (0.1 D 0.4), the modified Rayleigh number (10 Ra* 1000 ), the buoyancy ratio (-2 N 2), the Lewis number ( 0.1 Le 10 ). As a result, the effects of variable porosity are clearly seen on the iso-concentration lines while very little effect is portrayed on the isotherms at the high Lewis number. 204

205 Chapter 10. Conclusions and Future Studies On the other hand, the effects of variable porosity are clearly visible on the isotherms at low Lewis numbers, while there is hardly any influence on the iso-concentration lines. In general, the Nusselt and Sherwood numbers increases when the Rayleigh number increases, whereas they decreases when the thermal conductivity ratio and the wall thickness increases. Furthermore, when the Lewis number increases, the Nusselt number decreases while the Sherwood numbers increases. Furthermore, for a constant Lewis number, the average Nusselt and Sherwood numbers increases when the absolute buoyancy ratio ( N 1) increases. The work presented in Chapter 8 is devoted to solution of some cases studies of inclined porous cavities. The results of double-diffusive natural convection heat and mass transfer in inclined porous cavities were presented for square and rectangular cavities. For square porous cavities, A wide range of parameters had been used including; the modified Rayleigh number (100 Ra* 1000) and the Darcy number (10-6 Da 10-2 (, the Lewis number (0.1 Le 10), the buoyancy ratio (-5 N 5), and the angle of inclination of the cavity (0 o 90 o ), while the Prandtl number is taken as Pr=1.0. The results show that, when (N>0), the convection in the porous cavity is always in a single cell. Moreover, at =90 o the bicellular flows occur. Also, the Nu and Sh increases when the angle of inclination increases until around =45 o, and then they decreases when is increased. Different behaviour was observed regarding to Lewis number. It was observed that, the Nu increases when the Le increases until Le is equal to 1 and then the Nu decreases when the Le increases. On the other hand, the Nu decreases when the Le increases for a negative buoyancy ratio. When (N<-1), the convection in the porous cavity is always a multiplicity of steady solutions with flow fields of two, three and four flow cells. For cases of =90 o, it can be observed that no vortex appears. Also it shows that both the Nu and Sh decrease when the angle of inclination is decreased. For rectangular porous cavities, the result was presented for Ra=5*10 6, Da =10-4 and Pr =4.5 with different values of Lewis number (0.1 Le 10), and the buoyancy ratio (-5 N 5) and the inclined angle of the porous cavity (0 85). The results show that as the aspect ratio increases, the average Nusselt and Sherwood numbers is reduced, although they decreases when the angle of inclination increases. Moreover, the multiplicity circulations appear for a higher angle of inclination and they increase when the aspect ratio and the angle of 205

206 Chapter 10. Conclusions and Future Studies inclination increases especially for (=85). It can be seen that for aspect ratio was equal to 4, five circulations appeared and seven circulations appeared when the aspect ratio was equal 5 for the same other variables. The more complicated matter of Soret and Dufour effects on heat and mass transfer was presented in Chapter 9. The results were presented in terms of different variables including: the modified Rayleigh number (100 Ra* 500), the Darcy Number (10-6 Da 10-2 ),the Lewis number (0.1 Le 20), the buoyancy ratio (-3 N +3), as well as the Soret parameter (-5 S r +5), and the Dufour parameter (-2 D f +2) while the Prandtl number (Pr) and the porosity () are assumed to be constant and equal to 0.71 and 0.6, respectively. The results show that, when the positive Soret parameter increases, mass source and mass sink appear and the average Nusselt and Sherwood numbers increases. Furthermore, both of the average Nusselt and Sherwood number increases when the positive Dufour parameter increases. On the other hand, when the negative Dufour parameter decreases, the average Nusselt decreases while the average Sherwood number increases Future studies This study has contributed to the understanding of the Soret and the Dufour effects on double-diffusive natural convection heat and mass transfer in cavity filed with saturated porous medium, by providing detailed data on the heat, mass and flow fields within porous cavity, differentially heated and concentrated porous cavities. As usual, some problems and questions have arisen during the present work, and need further investigations. During this study, and as mentioned before, an in-house FORTRAN code was developed using the finite volume method. Due to using a second-order central differencing discretization scheme for all terms - including the convective term- in the momentum, the energy and the species equations, in some case studies the solution was unstable and could not be converged. For that reason, and to make ALFARHANY code more stable, the upwind and/or QUICK schemes are recommended to be added for the convective term. In general, for the enclosed cavity, most of the convection issues that can occur in the core of the cavity due to the fine mesh, for that reason, a too fine mesh in the core of the cavity could be 206

207 Chapter 10. Conclusions and Future Studies produce a diverging solution. Therefore we are recommended to extend the code to use a non-uniform mesh. There is a useful area for future studies highlighted and these are: Extending the present studies to add the effect of variable porosity and the angle of inclinations with Soret and Dufour effects. Extending the present studies to add the effect of two/multi-phase flow in the porous medium. The effect of the mass flux due to temperature gradients (Soret) and the heat flux produced by concentration gradients (Dufour) are strongly coupled especially, when both of the Soret and Dufour numbers are positive or negative. Therefore, a mathematical study to create a single equation from these equations is recommended. Providing experimental data is very important for validating the computational results of the MHD convection in a porous medium. Therefore, the experimental and CFD studies of MHD convection heat and mass transfer are interested to undertaken with wide numbers of parameters studies. The results of variable porosity show that, when the variable porosity was used the heat and mass transfer increased comparing with the cases of the constant porosity at the same parameters. Therefore, the design of porosity/permeability plays an important role to guide (optimized) the heat and/or mass transfer in porous medium. A CFD study of natural convective heat and/or mass transfer in porous cavities using the percolation theory is interested to study. 207

208 Appendix A A. Journal/Conference Papers In this section, the first page of the journal/conference papers published/submitted during this study is provided.

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213 Appendix A. Journal/Conference Papers 213