多孔介质平板通道传热模型的两种求解方法
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摘要
基于Brinkman-Darcy扩展模型和非局部热平衡模型,考虑液相和固相含有内热源的情况,建立了多孔介质平板通道传热的一般模型.分别采用直接法和间接法将液相与固相能量方程解耦,进而求得充分发展传热条件下的多孔介质温度场.与直接解耦法相比,间接解耦法可在原始边界条件下求解二阶微分方程,更加简单易行.通过对无量纲温度表达式系数以及温度分布的比较,验证了两种求解方法的等价性.在两种极限情形下,间接法所得温度分布解析解与现有文献结果相当吻合,这也在一定程度上证明了所建模型更具一般性.参数分析表明,液固两相温差随着Biot数或有效导热系数比的增大而减小,Nusselt数随着内热源比的增大而减小.
A general heat transfer model of a parallel plate channel filled with a porous medium was constructed based on the Brinkman-Darcy extended model and the local thermal non-equilibrium model in viewof the internal heat sources in fluid and solid phases.The temperature field of the porous medium under the fully developed heat transfer condition was respectively formulated with the direct and indirect decoupling methods of solving the fluid-phase and solid-phase energy equations.Compared to the direct decoupling method,the indirect one is more convenient to be employed to solve the2nd-order differential equations under the original boundary conditions.The equivalence of the 2 decoupling methods was verified through comparison of the coefficients in the dimensionless temperature expressions and the temperature distributions between them.A good agreement was found between the temperature distributions obtained with the indirect decoupling method and those reported in the previous literatures in2 limit cases,meanwhile the better generality of the proposed model was also proved to some extent.The parametric study shows that the temperature difference between the fluid and solid phases decreases with the Biot number or the effective thermal conductivity ratio,and the Nusselt number decreases with the internal heat source ratio.
A general heat transfer model of a parallel plate channel filled with a porous medium was constructed based on the Brinkman-Darcy extended model and the local thermal non-equilibrium model in viewof the internal heat sources in fluid and solid phases.The temperature field of the porous medium under the fully developed heat transfer condition was respectively formulated with the direct and indirect decoupling methods of solving the fluid-phase and solid-phase energy equations.Compared to the direct decoupling method,the indirect one is more convenient to be employed to solve the2nd-order differential equations under the original boundary conditions.The equivalence of the 2 decoupling methods was verified through comparison of the coefficients in the dimensionless temperature expressions and the temperature distributions between them.A good agreement was found between the temperature distributions obtained with the indirect decoupling method and those reported in the previous literatures in2 limit cases,meanwhile the better generality of the proposed model was also proved to some extent.The parametric study shows that the temperature difference between the fluid and solid phases decreases with the Biot number or the effective thermal conductivity ratio,and the Nusselt number decreases with the internal heat source ratio.
引文
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[19]Nouri-Borujerdi A,Noghrehabadi A R,Rees D A S.Onset of convection in a horizontal porous channel with uniform heat generation using a thermal nonequilibrium model[J].Transport in Porous Media,2007,69(3):343-357.
[20]Nield D A,Kuznetsov A V,Xiong M.Effect of local thermal non-equilibrium on thermally developing forced convection in a porous medium[J].International Journal of Heat and Mass Transfer,2002,45(25):4949-4955.
[21]许飞,郭钢,胡康,韩松燕,马晓楠,李再顺.稠油油藏中含内热源多孔介质的传热研究[J].石油化工应用,2012,31(2):28-32.(XU Fei,GUO Gang,HU Kang,HAN Song-yan,MA Xiaonan,LI Zai-shun.The research on heat transfer of combustible porous meida with inner heat source[J].Petrochemical Industry Application,2012,31(2):28-32.(in Chinese))
[22]Keangin P,Vafai K,Rattanadecho P.Electromagnetic field effects on biological materials[J].International Journal of Heat and Mass Transfer,2013,65:389-399.
[23]Mahjoob S,Vafai K.Analytical characterization of heat transfer through biological media incorporating hyperthermia treatment[J].International Journal of Heat and Mass Transfer,2009,52:1608-1618.
[2]Cheng P.Heat transfer in geothermal systems[J].Advances in Heat Transfer,1979,14:1-105.
[3]Kim S J,Jang S P.Effects of the Darcy number,the Prandtl number,and the Reynolds number on local thermal non-equilibrium[J].International Journal of Heat and Mass Transfer,2002,45(19):3885-3896.
[4]Kim S J,Vafai K.Analysis of natural convection about avertical plate embedded in a porous medium[J].International Journal of Heat and Mass Transfer,1989,32(4):665-677.
[5]Celli M,Barletta A,Storesletten L.Local thermal non-equilibrium effects in the Darcy-Bénard instability of a porous layer heated from below by a uniform flux[J].International Journal of Heat and Mass Transfer,2013,67:902-912.
[6]Vafai K,Sozen M.Analysis of energy and momentum transport for fluid flow through a porous bed[J].Journal of Heat Transfer,1990,112(3):690-699.
[7]Quintard M,Whitaker S.Two-phase flow in heterogeneous porous media(I):the influence of large spatial and temporal gradients[J].Transport in Porous Media,1990,5(4):341-379.
[8]Marafie A,Vafai K.Analysis of non-Darcian effects on temperature differentials in porous media[J].International Journal of Heat and Mass Transfer,2001,44(23):4401-4411.
[9]李明春,田彦文,翟玉春.非热平衡多孔介质内反应与传热传质耦合过程[J].化工学报,2006,57(5):1079-1083.(LI Ming-chun,TIAN Yan-wen,ZHAI Yu-chun.Coupled processes of chemical reaction,heat and mass transfer in non-thermal equilibrium porous medium[J].Journal of Chemical Industry and Engineering,2006,57(5):1079-1083.(in Chinese))
[10]杨骁,刘雪梅.多孔介质平板通道发展传热中非局部热平衡时的温度分布特征[J].应用数学和力学,2006,27(8):978-986.(YANG Xiao,LIU Xue-mei.Temperature profiles of local thermal nonequilibrium for thermal developing forced convection in a porous medium parallel platechannel[J].Applied Mathematics and Mechanics,2006,27(8):978-986.(in Chinese))
[11]Yang K,Vafai K.Analysis of temperature gradient bifurcation in porous media—an exact solution[J].International Journal of Heat and Mass Transfer,2010,53(19/20):4316-4325.
[12]Yang K,Vafai K.Analysis of heat flux bifurcation inside porous media incorporating inertial and dispersion effects—an exact solution[J].International Journal of Heat and Mass Transfer,2011,54(25):5286-5297.
[13]何颖,邵宝东,程赫明.等边三角形微通道内层流的流动特性和换热特性的研究[J].应用数学和力学,2014,35(3):313-321.(HE Ying,SHAO Bao-dong,CHENG He-ming.Laminar flow and heat transfer in equilateral triangle micro-channels[J].Applied Mathematics and Mechanics,2014,35(3):313-321.(in Chinese))
[14]Buonomo B,Manca O,Lauriat G.Forced convection in micro-channels filled with porous media in local thermal non-equilibrium conditions[J].International Journal of Thermal Sciences,2014,77:206-222.
[15]Dukhan N,Al-Rammahi M A,Suleiman A S.Fluid temperature measurements inside metal foam and comparison to Brinkman-Darcy flow convection analysis[J].International Journal of Heat and Mass Transfer,2013,67:877-884.
[16]刘伟,明廷臻.管内核心流分层填充多孔介质的传热强化分析[J].中国电机工程学报,2008,28(32):66-71.(LIU Wei,MING Ting-zhen.Analysis for heat transfer enhancement in the core flow of a tube filled with porous media at different layers[J].Proceedings of the CSEE,2008,28(32):66-71.(in Chinese))
[17]Qu Z G,Xu H J,Tao W Q.Fully developed forced convective heat transfer in an annulus partially filled with metallic foams:an analytical solution[J].International Journal of Heat and Mass Transfer,2012,55(25):7508-7519.
[18]公维平,曹玉荣.多孔介质强化传热的理论与实验研究[J].水动力学研究与进展,2003,18(3):276-282.(GONG Wei-ping,CAO Yu-rong.Thoretical and experimental investigation on heat transfer augmentation in porous medium[J].Journal of Hydrodynamics,2003,18(3):276-282.(in Chinese))
[19]Nouri-Borujerdi A,Noghrehabadi A R,Rees D A S.Onset of convection in a horizontal porous channel with uniform heat generation using a thermal nonequilibrium model[J].Transport in Porous Media,2007,69(3):343-357.
[20]Nield D A,Kuznetsov A V,Xiong M.Effect of local thermal non-equilibrium on thermally developing forced convection in a porous medium[J].International Journal of Heat and Mass Transfer,2002,45(25):4949-4955.
[21]许飞,郭钢,胡康,韩松燕,马晓楠,李再顺.稠油油藏中含内热源多孔介质的传热研究[J].石油化工应用,2012,31(2):28-32.(XU Fei,GUO Gang,HU Kang,HAN Song-yan,MA Xiaonan,LI Zai-shun.The research on heat transfer of combustible porous meida with inner heat source[J].Petrochemical Industry Application,2012,31(2):28-32.(in Chinese))
[22]Keangin P,Vafai K,Rattanadecho P.Electromagnetic field effects on biological materials[J].International Journal of Heat and Mass Transfer,2013,65:389-399.
[23]Mahjoob S,Vafai K.Analytical characterization of heat transfer through biological media incorporating hyperthermia treatment[J].International Journal of Heat and Mass Transfer,2009,52:1608-1618.