多孔介质中垂直圆柱内Casson流体热对流的瞬态分析:熵产和热能转移可视化（英文）
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摘要
随着多孔介质在过滤系统、多孔轴承、多孔隔热层、生物力学、地质力学等方面的应用日益广泛,多孔介质中的热传输引起了工程科学领域的广泛兴趣。基于上述应用,本文就熵产对Casson黏塑性流体在多孔介质中均匀加热垂直圆柱上的分层不可压缩边界层流动中传热和动量传递的影响进行了数值研究。采用Darcy定律模拟了各向同性均质多孔介质在低Reynolds数时的体阻力效应,并进行热线可视化。利用适当的变换变量,推导了数学模型并进行归一化。利用隐式有限差分法求解了具有相关边界条件的非线性时变耦合控制方程,该方法有效且无条件稳定。结果表明,随着Darcy数、Casson流体参数、群参数和Grashof数的增大,熵产和Bejan数均增大。除了流线和等温线外,绘制热线为分析二维区域内的传热过程提供了一种很好的方法。结果表明,随着Darcy数的增加,热壁之间的传热线偏差减小。
Thermal transport in porous media has stimulated substantial interest in engineering sciences due to increasing applications in filtration systems, porous bearings, porous layer insulation, biomechanics, geomechanics etc.Motivated by such applications, in this article, a numerical study of entropy generation impacts on the heat and momentum transfer in time-dependent laminar incompressible boundary layer flow of a Casson viscoplastic fluid over a uniformly heated vertical cylinder embedded in a porous medium is presented. Darcy's law is used to simulate bulk drag effects at low Reynolds number for an isotropic, homogenous porous medium. Heat line visualization is also included. The mathematical model is derived and normalized using appropriate transformation variables. The resulting non-linear time-dependent coupled governing equations with associated boundary conditions are solved via an implicit finite difference method which is efficient and unconditionally stable. The outcomes show that entropy generation and Bejan number are both elevated with increasing values of Darcy number, Casson fluid parameter, group parameter and Grashof number. To analyze the heat transfer process in a two-dimensional domain, plotting heat lines provides an excellent approach in addition to streamlines and isotherms. It is remarked that as the Darcy number increases, the deviations of heat lines from the hot wall are reduced.
Thermal transport in porous media has stimulated substantial interest in engineering sciences due to increasing applications in filtration systems, porous bearings, porous layer insulation, biomechanics, geomechanics etc.Motivated by such applications, in this article, a numerical study of entropy generation impacts on the heat and momentum transfer in time-dependent laminar incompressible boundary layer flow of a Casson viscoplastic fluid over a uniformly heated vertical cylinder embedded in a porous medium is presented. Darcy's law is used to simulate bulk drag effects at low Reynolds number for an isotropic, homogenous porous medium. Heat line visualization is also included. The mathematical model is derived and normalized using appropriate transformation variables. The resulting non-linear time-dependent coupled governing equations with associated boundary conditions are solved via an implicit finite difference method which is efficient and unconditionally stable. The outcomes show that entropy generation and Bejan number are both elevated with increasing values of Darcy number, Casson fluid parameter, group parameter and Grashof number. To analyze the heat transfer process in a two-dimensional domain, plotting heat lines provides an excellent approach in addition to streamlines and isotherms. It is remarked that as the Darcy number increases, the deviations of heat lines from the hot wall are reduced.
引文
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[2]LEE H R,CHEN T S,ARMALY B F.Natural convection along slender vertical cylinders with variable surface temperature[J].ASME Journal of Heat Transfer,1988,110:103-108.
[3]MAKANDA G,SHAW S,SIBANDA P.Diffusion of chemically reactive species in Casson fluid flow over an unsteady stretching surface in porous medium in the presence of a magnetic field[J].Mathematical Problems in Engineering,2015:Article ID 724596.
[4]DASH R K,MEHTA K N,JAYARAMAN G.Casson fluid flow in a pipe filled with a homogeneous porous medium[J].International Journal of Engineering Science,1996,34:1145-1156.
[5]CASSON N.A flow equation for pigment oil suspensions of the printing ink type[M]//Rheology of Disperse Systems.Oxford:Pergamon Press,1956:84-102.
[6]ABBAS Z,SHEIKH M,MOTSA S S.Numerical solution of binary chemical reaction on stagnation point flow of Casson fluid over a stretching/shrinking sheet with thermal radiation[J].Energy,2016,95:12-20.
[7]MUSTAFA M,HAYAT T,POP I,HENDI A.Stagnationpoint flow and heat transfer of a Casson fluid towards a stretching sheet[J].Zeitschrift für Naturforschung A,2012,67:70-76.
[8]MYTHILI D,SIVARAJ R.Influence of higher order chemical reaction and non-uniform heat source/sink on Casson fluid flow over a vertical cone and flat plate[J].Journal of Molecular Liquids,2016,216:466-475.
[9]DAS M,MAHATO R,NANDKEOLYAR R.Newtonian heating effect on unsteady hydromagnetic Casson fluid flow past a flat plate with heat and mass transfer[J].Alexandria Engineering Journal,2015,54:871-879.
[10]NADEEM S,HAQ R U,LEE C.Magnetohydrodynamic flow of a Casson fluid over an exponentially shrinking sheet[J].Scientia Iranica,2012,19:1550-1553.
[11]RAJU C S K,SANDEEP N,SUGUNAMMA V,BABU M J,REDDY J R.Heat and mass transfer in magnetohydrodynamic Casson fluid over an exponentially permeable stretching surface[J].Engineering Science and Technology,2016,19:45-52.
[12]GIANGASPERO G,SCIUBBA E.Application of the entropy generation minimization method to a solar heat exchanger:Apseudo-optimization design process based on the analysis of the local entropy generation maps[J].Energy,2013,58:52-65.
[13]BADESCU V.Optimal paths for minimizing lost available work during usual heat transfer processes[J].Journal of Non-Equilibrium Thermodynamics,2004,29:53-73.
[14]KOCKUM H,JERNQVIST A.Entropy generation in multifield flows:Equations and Examples of Applications[J].Chemical Engineering Research and Design,1998,76:210-222.
[15]BEJAN A.A study of entropy generation in fundamental convective heat transfer[J].ASME Journal of Heat Transfer,1979,101:718-725.
[16]BEJAN A.Second law analysis in heat transfer[J].Energy,1980,5:720-732.
[17]BEJAN A.Second law analysis in heat transfer and thermal design[J].Advances in Heat Transfer,1982,15:1-58.
[18]MAHIAN O,OZTOP H,POP I,MAHMUD S,WONGWISES S.Entropy generation between two vertical cylinders in the presence of MHD flow subjected to constant wall temperature[J].International Communications Heat and Mass Transfer,2013,44:87-92.
[19]MIRZAZADEH M,SHAFAEI A,RASHIDI F.Entropy analysis for non-linear viscoelastic fluid in concentric rotating cylinders[J].International Journal of Thermal Science,2008,47:1701-1711.
[20]MAHIAN O,MAHMUD S,HERIS SZ.Analysis of entropy generation between co-rotating cylinders using nanofluid[J].Energy,2012,44:438-446.
[21]BASSAM A K ABU-H,WALEED N H.Entropy generation due to laminar natural convection over a heated rotating cylinder[J].International Journal of Heat and Mass Transfer,1999,42:4225-4233.
[22]JIA Q,MUHAMMAD M B,MUNAWWAR A A,MOHAMMAD M R,MOHAMED EL-S A.Entropy generation on MHD Casson nano fluid flow over a porous stretching/shrinking surface[J].Entropy,2016,18(4):1-14.
[23]ASMA K,ILYAS K,ARSHAD K,SHARIDAN S.Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium[J].Engineering Science and Technology,2015,18(3):309-317.
[24]RAMACHANDRA P V,SUBBA R A,ANWAR B O.Flow and heat transfer of Casson fluid from a horizontal circular cylinder with partial slip in non-Darcy porous medium[J].Journal of Applied&Computational Mathematics,2013,2(2):1-12.
[25]HARI R,KATARIA,HARSHAD R P.Radiation and chemical reaction effects on MHD Casson fluid flow past an oscillating vertical plate embedded in porous medium[J].Alexandria Engineering Journal,2016,55:583-595.
[26]KIMURA S,BEJAN A.The heatline visualization of convective heat transfer[J].ASME Journal of Heat Transfer,1983,105:916-919.
[27]BEJAN A.Convection heat transfer[M].New York:John Wiley and Sons,1984.
[28]RANI H P,REDDY G J.Heatline visualization for conjugate heat transfer of a couple stress fluid from a vertical slender hollow cylinder[J].International Communications in Heat and Mass Transfer,2013,48:46-52.
[29]RANI H P,REDDY G J,KIM C N,RAMESHWAR Y.Transient couple stress fluid past a vertical cylinder with Bejan’s heat and mass flow visualization for steady state[J].ASME Journal of Heat Transfer,2015,137:032501.
[30]MOREGA A.The heat function approach to the thermomagnetic convection of electroconductive melts[J].Revue Roumaine des Sci Tech.Electrot et Energ,1998,33:359-368.
[31]MABOOD F,ABDEL-RAHMAN R G,LORENZINI G.Effect of melting heat transfer and thermal radiation on Casson fluid flow in porous medium over moving surface with magnetohydrodynamics[J].Journal of Engineering Thermophysics,2016,25:536-547.
[32]RANI H P,JANARDHANA R G,KIM C N.Transient analysis of diffusive chemical reactive species for couple stress fluid flow over vertical cylinder[J].Applied Mathematics and Mechanics(English Edition),2013,34:985-1000.
[33]QING J,BHATTI M M,ABBAS M A,RASHIDI M M,ALIM EL-S.Entropy generation on MHD Casson nanofluid flow over a porous stretching/shrinking surface[J].Entropy,2016,18:1-24.
[34]BUTT A S,ALI A,MEHMOOD A.Numerical investigation of magnetic field effects on entropy generation in viscous flow over a stretching cylinder embedded in a porous medium[J].Energy,2016,99:237-249.
[35]MAHDAVI M,SAFFAR-AVVAL M,TIARI S,MANSOORIS.Entropy generation and heat transfer numerical analysis in pipes partially filled with porous medium[J].International Journal of Heat and Mass Transfer,2014,79:496-506.
[36]BEJAN A.Entropy Generation Minimization[M].New York,USA:CRC Press,1996.
[37]REDDY G J,KUMAR M,KETHIREDDY B,CHAMKHAA J.Colloidal study of unsteady magnetohydrodynamic couple stress fluid flow over an isothermal vertical flat plate with entropy heat generation[J].Journal of Molecular Liquids,2018,252:169-179.
[38]REDDY G J,HIREMATH A,KUMAR M.Computational modeling of unsteady third-grade fluid flow over a vertical cylinder:A study of heat transfer visualization[J].Results in Physics,2018,8:671-682.
[39]REDDY G J,KETHIREDDY B,KUMAR M,RANI H P,GORLA R S.Effect of Prandtl number for Casson fluid flow over a vertical cylinder:Heatline approach[J].International Journal of Applied and Computational Mathe matics 2018,4:1-19.
[40]SHEHZAD S A,HAYAT T,ALSAEDI A.Three-dimensional MHD flow of Casson fluid in porous medium with heat generation[J].Journal of Applied Fluid Mechanics,2016,9:215-223.
[2]LEE H R,CHEN T S,ARMALY B F.Natural convection along slender vertical cylinders with variable surface temperature[J].ASME Journal of Heat Transfer,1988,110:103-108.
[3]MAKANDA G,SHAW S,SIBANDA P.Diffusion of chemically reactive species in Casson fluid flow over an unsteady stretching surface in porous medium in the presence of a magnetic field[J].Mathematical Problems in Engineering,2015:Article ID 724596.
[4]DASH R K,MEHTA K N,JAYARAMAN G.Casson fluid flow in a pipe filled with a homogeneous porous medium[J].International Journal of Engineering Science,1996,34:1145-1156.
[5]CASSON N.A flow equation for pigment oil suspensions of the printing ink type[M]//Rheology of Disperse Systems.Oxford:Pergamon Press,1956:84-102.
[6]ABBAS Z,SHEIKH M,MOTSA S S.Numerical solution of binary chemical reaction on stagnation point flow of Casson fluid over a stretching/shrinking sheet with thermal radiation[J].Energy,2016,95:12-20.
[7]MUSTAFA M,HAYAT T,POP I,HENDI A.Stagnationpoint flow and heat transfer of a Casson fluid towards a stretching sheet[J].Zeitschrift für Naturforschung A,2012,67:70-76.
[8]MYTHILI D,SIVARAJ R.Influence of higher order chemical reaction and non-uniform heat source/sink on Casson fluid flow over a vertical cone and flat plate[J].Journal of Molecular Liquids,2016,216:466-475.
[9]DAS M,MAHATO R,NANDKEOLYAR R.Newtonian heating effect on unsteady hydromagnetic Casson fluid flow past a flat plate with heat and mass transfer[J].Alexandria Engineering Journal,2015,54:871-879.
[10]NADEEM S,HAQ R U,LEE C.Magnetohydrodynamic flow of a Casson fluid over an exponentially shrinking sheet[J].Scientia Iranica,2012,19:1550-1553.
[11]RAJU C S K,SANDEEP N,SUGUNAMMA V,BABU M J,REDDY J R.Heat and mass transfer in magnetohydrodynamic Casson fluid over an exponentially permeable stretching surface[J].Engineering Science and Technology,2016,19:45-52.
[12]GIANGASPERO G,SCIUBBA E.Application of the entropy generation minimization method to a solar heat exchanger:Apseudo-optimization design process based on the analysis of the local entropy generation maps[J].Energy,2013,58:52-65.
[13]BADESCU V.Optimal paths for minimizing lost available work during usual heat transfer processes[J].Journal of Non-Equilibrium Thermodynamics,2004,29:53-73.
[14]KOCKUM H,JERNQVIST A.Entropy generation in multifield flows:Equations and Examples of Applications[J].Chemical Engineering Research and Design,1998,76:210-222.
[15]BEJAN A.A study of entropy generation in fundamental convective heat transfer[J].ASME Journal of Heat Transfer,1979,101:718-725.
[16]BEJAN A.Second law analysis in heat transfer[J].Energy,1980,5:720-732.
[17]BEJAN A.Second law analysis in heat transfer and thermal design[J].Advances in Heat Transfer,1982,15:1-58.
[18]MAHIAN O,OZTOP H,POP I,MAHMUD S,WONGWISES S.Entropy generation between two vertical cylinders in the presence of MHD flow subjected to constant wall temperature[J].International Communications Heat and Mass Transfer,2013,44:87-92.
[19]MIRZAZADEH M,SHAFAEI A,RASHIDI F.Entropy analysis for non-linear viscoelastic fluid in concentric rotating cylinders[J].International Journal of Thermal Science,2008,47:1701-1711.
[20]MAHIAN O,MAHMUD S,HERIS SZ.Analysis of entropy generation between co-rotating cylinders using nanofluid[J].Energy,2012,44:438-446.
[21]BASSAM A K ABU-H,WALEED N H.Entropy generation due to laminar natural convection over a heated rotating cylinder[J].International Journal of Heat and Mass Transfer,1999,42:4225-4233.
[22]JIA Q,MUHAMMAD M B,MUNAWWAR A A,MOHAMMAD M R,MOHAMED EL-S A.Entropy generation on MHD Casson nano fluid flow over a porous stretching/shrinking surface[J].Entropy,2016,18(4):1-14.
[23]ASMA K,ILYAS K,ARSHAD K,SHARIDAN S.Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium[J].Engineering Science and Technology,2015,18(3):309-317.
[24]RAMACHANDRA P V,SUBBA R A,ANWAR B O.Flow and heat transfer of Casson fluid from a horizontal circular cylinder with partial slip in non-Darcy porous medium[J].Journal of Applied&Computational Mathematics,2013,2(2):1-12.
[25]HARI R,KATARIA,HARSHAD R P.Radiation and chemical reaction effects on MHD Casson fluid flow past an oscillating vertical plate embedded in porous medium[J].Alexandria Engineering Journal,2016,55:583-595.
[26]KIMURA S,BEJAN A.The heatline visualization of convective heat transfer[J].ASME Journal of Heat Transfer,1983,105:916-919.
[27]BEJAN A.Convection heat transfer[M].New York:John Wiley and Sons,1984.
[28]RANI H P,REDDY G J.Heatline visualization for conjugate heat transfer of a couple stress fluid from a vertical slender hollow cylinder[J].International Communications in Heat and Mass Transfer,2013,48:46-52.
[29]RANI H P,REDDY G J,KIM C N,RAMESHWAR Y.Transient couple stress fluid past a vertical cylinder with Bejan’s heat and mass flow visualization for steady state[J].ASME Journal of Heat Transfer,2015,137:032501.
[30]MOREGA A.The heat function approach to the thermomagnetic convection of electroconductive melts[J].Revue Roumaine des Sci Tech.Electrot et Energ,1998,33:359-368.
[31]MABOOD F,ABDEL-RAHMAN R G,LORENZINI G.Effect of melting heat transfer and thermal radiation on Casson fluid flow in porous medium over moving surface with magnetohydrodynamics[J].Journal of Engineering Thermophysics,2016,25:536-547.
[32]RANI H P,JANARDHANA R G,KIM C N.Transient analysis of diffusive chemical reactive species for couple stress fluid flow over vertical cylinder[J].Applied Mathematics and Mechanics(English Edition),2013,34:985-1000.
[33]QING J,BHATTI M M,ABBAS M A,RASHIDI M M,ALIM EL-S.Entropy generation on MHD Casson nanofluid flow over a porous stretching/shrinking surface[J].Entropy,2016,18:1-24.
[34]BUTT A S,ALI A,MEHMOOD A.Numerical investigation of magnetic field effects on entropy generation in viscous flow over a stretching cylinder embedded in a porous medium[J].Energy,2016,99:237-249.
[35]MAHDAVI M,SAFFAR-AVVAL M,TIARI S,MANSOORIS.Entropy generation and heat transfer numerical analysis in pipes partially filled with porous medium[J].International Journal of Heat and Mass Transfer,2014,79:496-506.
[36]BEJAN A.Entropy Generation Minimization[M].New York,USA:CRC Press,1996.
[37]REDDY G J,KUMAR M,KETHIREDDY B,CHAMKHAA J.Colloidal study of unsteady magnetohydrodynamic couple stress fluid flow over an isothermal vertical flat plate with entropy heat generation[J].Journal of Molecular Liquids,2018,252:169-179.
[38]REDDY G J,HIREMATH A,KUMAR M.Computational modeling of unsteady third-grade fluid flow over a vertical cylinder:A study of heat transfer visualization[J].Results in Physics,2018,8:671-682.
[39]REDDY G J,KETHIREDDY B,KUMAR M,RANI H P,GORLA R S.Effect of Prandtl number for Casson fluid flow over a vertical cylinder:Heatline approach[J].International Journal of Applied and Computational Mathe matics 2018,4:1-19.
[40]SHEHZAD S A,HAYAT T,ALSAEDI A.Three-dimensional MHD flow of Casson fluid in porous medium with heat generation[J].Journal of Applied Fluid Mechanics,2016,9:215-223.
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