Abstract
We conduct topology optimization of convective heat transfer problems based on the power law type non-Newtonian fluid. A heat transfer maximization problem is studied by using a material distribution based optimization method to optimize configurations of non-Newtonian cooling devices. The key idea of the method is to discern the fluid and the solid domains by a design variable, namely the “material density.” It is updated according to the gradient information obtained from an adjoint-based sensitivity analysis process. The non-Newtonian effects on optimal configurations of thermal devices are numerically investigated. Our results show that more branched flow channels appear in the optimal designs as the pressure difference or heat generation grows. Meanwhile, the dependence of the optimal layout on the power law index is demonstrated and higher power law index can result in more complex configurations and lower flow rate. Compared with the low power law index one, the optimal design of the high power law index problem has much better heat transfer performance on the same condition.
Similar content being viewed by others
References
Alexandersen J, Aage N, Andreasen CS, Sigmund O (2014) Topology optimisation for natural convection problems. Int J Numer Methods Fluids 76:699–721
Alexandersen J, Sigmund O, Aage N (2016) Large scale three-dimensional topology optimisation of heat sinks cooled by natural convection. Int J Heat Mass Transf 100:876–891
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. (Second Edition). Springer - Verlag Berlin Heidelberg GmbH
Borrvall T, Petersson J (2003) Topology optimization of fluids in stokes flow. Int J Numer Methods Fluids 41:77–107. https://doi.org/10.1002/fld.426
Chhabra RP (2006) Bubbles, drops, and particles in non-Newtonian fluids (second edition), 2nd edn. CRC Press, Boca Raton
Chhabra RP, Richardson JF (2008) Non-Newtonian flow and applied rheology (second edition). Elsevier Ltd. https://doi.org/10.1016/B978-0-7506-8532-0.X0001-7
Coffin P, Maute K (2016) A level-set method for steady-state and transient natural convection problems. Struct Multidiscip Optim 53:1047–1067. https://doi.org/10.1007/s00158-015-1377-y
Dede E (2009) Multiphysics topology optimization of heat transfer and fluid flow systems. Proceedings of the COMSOL Conference, Boston
Dede EM (2012) Optimization and design of a multipass branching microchannel heat sink for electronics cooling. J Electron Packag 134:041001. https://doi.org/10.1115/1.4007159
Dhiman AK, Chhabra RP, Eswaran V (2007) Heat transfer to power-law fluids from a heated square cylinder. Numer Heat Transfer, Part A 52:185–201
Duan XB, Ma YC, Zhang R (2008a) Optimal shape control of fluid flow using variational level set method. Phys Lett A 372:1374–1379. https://doi.org/10.1016/j.physleta.2007.09.070
Duan XB, Ma YC, Zhang R (2008b) Shape-topology optimization for Navier-Stokes problem using variational level set method. J Comput Appl Math 222:487–499. https://doi.org/10.1016/j.cam.2007.11.016
Dugas F, Favennec Y, Josset C, Fan Y, Luo L (2018) Topology optimization of thermal fluid flows with an adjoint lattice Boltzmann method. J Comput Phys 365:376–404
Ebrahimi A, Naranjani B, Milani S, Javan FD (2017) Laminar convective heat transfer of shear-thinning liquids in rectangular channels with longitudinal vortex generators. Chem Eng Sci 173:264–274
Esmaeilnejad A, Aminfar H, Neistanak MS (2014) Numerical investigation of forced convection heat transfer through microchannels with non-Newtonian nanofluids. Int J Therm Sci 75:76–86. https://doi.org/10.1016/j.ijthermalsci.2013.07.020
Haertel JHK, Nellis GF (2017) A fully developed flow thermofluid model for topology optimization of 3D-printed air-cooled heat exchangers. Appl Therm Eng 119:10–24. https://doi.org/10.1016/j.applthermaleng.2017.03.030/
Hyun J, Wang S, Yang S (2014) Topology optimization of the shear thinning non-Newtonian fluidic systems for minimizing wall shear stress. Computers & Mathematics with Applications 67:1154–1170. https://doi.org/10.1016/j.camwa.2013.12.013
Jensen KE, Szabo P, Okkels F (2012) Topology optimization of viscoelastic rectifiers. Appl Phys Lett 100. https://doi.org/10.1063/1.4728108
Kim SJ (2004) Methods for thermal optimization of microchannel heat sinks. Heat Transfer Eng 25:37–49
Koga AA, Lopes ECC, Nova HFV, de Lima CR, Silva ECN (2013) Development of heat sink device by using topology optimization. Int J Heat Mass Transf 64:759–772. https://doi.org/10.1016/ijheatmasstransfer.2013.05.007
Kondoh T, Matsumori T, Kawamoto A (2012) Drag minimization and lift maximization in laminar flows via topology optimization employing simple objective function expressions based on body force integration. Struct Multidiscip Optim 45:693–701. https://doi.org/10.1007/s00158-011-0730-z
Kreissl S, Pingen G, Maute K (2011) Topology optimization for unsteady flow. Int J Numer Methods Eng 87:1229–1253. https://doi.org/10.1002/nme.3151
Kulkarni K, Afzal A, Kim KY (2016) Multi-objective optimization of a double-layered microchannel heat sink with temperature-dependent fluid properties. Appl Therm Eng 99:262–272
Kurnia JC, Sasmito AP, Mujumdar AS (2014) Laminar heat transfer performance of power law fluids in coiled square tube with various configurations. Int Commun Heat Mass Transfer 57:100–108. https://doi.org/10.1016/j.icheatmasstransfer.2014.07.016
Li P, Xie Y, Zhang D (2016a) Laminar flow and forced convective heat transfer of shear-thinning power-law fluids in dimpled and protruded microchannels. Int J Heat Mass Transf 99:372–382
Li P, Zhang D, Xie Y, Xie G (2016b) Flow structure and heat transfer of non-Newtonian fluids in microchannel heat sinks with dimples and protrusions. Appl Therm Eng 94:50–58. https://doi.org/10.1016/j.applthermaleng.2015.10.119
Li S-N, Zhang H-N, Li X-B, Li Q, Li F-C, Qian S, Joo SW (2017) Numerical study on the heat transfer performance of non- Newtonian fluid flow in a manifold microchannel heat sink. Appl Therm Eng 115:1213–1225. https://doi.org/10.1016/j.applthermaleng.2016.10.047
Liu XM, Zhang B, Sun JJ (2015) An improved implicit re-initialization method for the level set function applied to shape and topology optimization of fluid. J Comput Appl Math 281:207–229. https://doi.org/10.1016/j.cam.2014.12.017
Martinez DS, Garcia A, Solano JP, Viedma A (2014) Heat transfer enhancement of laminar and transitional Newtonian and non-Newtonian flows in tubes with wire coil inserts. Int J Heat Mass Transf 76:540–548. https://doi.org/10.1016/j.ijheatmasstransfer.2014.04.060
Matsumori T, Kondoh T, Kawamoto A, Nomura T (2013) Topology optimization for fluid-thermal interaction problems under constant input power. Struct Multidiscip Optim 47:571–581. https://doi.org/10.1007/s00158-013-0887-8
Mukherjee S, Biswal P, Chakraborty S, Dasgupta S (2017) Effects of viscous dissipation during forced convection of power-law fluids in microchannels. Int Commun Heat Mass Transfer 89:83–90
Oignet J, Hoang HM, Osswald V, Delahaye A, Fournaison L, Haberschill P (2017) Experimental study of convective heat transfer coefficients of CO2 hydrate slurries in a secondary refrigeration loop. Appl Therm Eng 118:630–637. https://doi.org/10.1016/j.applthermaleng.2017.02.117
Okkels F, Bruus H (2007) Scaling behavior of optimally structured catalytic microfluidic reactors. Phys Rev E 75. https://doi.org/10.1103/PhysRevE.75.016301
Olesen LH, Okkels F, Bruus H (2006) A high-level programming-language implementation of topology optimization applied to steady-state Navier-stokes flow. Int J Numer Methods Eng 65:975–1001. https://doi.org/10.1002/nme.1468
Panda S, Chhabra RP (2011) Laminar forced convection heat transfer from a rotating cylinder to power-law fluids. Numerical Heat Transfer, Part A 59:297–319
Pingen G, Maute K (2010) Optimal design for non-Newtonian flows using a topology optimization approach. Comput Math Appl 59:2340–2350. https://doi.org/10.1016/j.camwa.2009.08.044
Pizzolato A, Sharma A, Maute K, Sciacovelli A, Verda V (2017a) Design of effective fins for fast PCM melting and solidification in shell-and-tube latent heat thermal energy storage through topology optimization. Appl Energy 208:210–227. https://doi.org/10.1016/j.apenergy.2017.10.050
Pizzolato A, Sharma A, Maute K, Sciacovelli A, Verda V (2017b) Topology optimization for heat transfer enhancement in latent heat thermal energy storage. Int J Heat Mass Transf 113:875–888
Poh HJ, Kumar K, Chiang HS, Mujumdar AS (2004) Heat transfer from a laminar impinging jet of a power law fluid. Int Commun Heat Mass Transfer 31:241–249. https://doi.org/10.1016/s0735-1933(03)00229-x
Romero JS, Silva ECN (2017) Non-newtonian laminar flow machine rotor design by using topology optimization. Struct Multidiscip Optim 55:1711–1732. https://doi.org/10.1007/s00158-016-1599-7
Shojaeian M, Karimzadehkhouei M, Kosar A (2017) Experimental investigation on convective heat transfer of non-Newtonian flows of Xanthan gum solutions in microtubes. Exp Thermal Fluid Sci 85:305–312. https://doi.org/10.1016/j.expthermflusci.2017.02.025
Siddiqa S, Begum N, Hossain MA, Gorla RSR (2017) Natural convection flow of a two-phase dusty non-Newtonian fluid along a vertical surface. Int J Heat Mass Transf 113:482–489
Singh A, Kishore N (2018) Laminar mixed convection of non-Newtonian nanofluids flowing vertically upward across confined circular cylinders. J Therm Sci Eng Appl 10:14
Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373
Svanberg K (2010) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12:555–573
Tahiri A, Mansouri K (2017) Theoretical investigation of laminar flow convective heat transfer in a circular duct for a non-Newtonian nanofluid. Appl Therm Eng 112:1027–1039
Wang XD, An B, Xu JH (2013) Optimal geometric structure for nanofluid-cooled microchannel heat sink under various constraint conditions. Energy Convers Manag 65:528–538. https://doi.org/10.1016/j.enconman.2012.08.018
Yaji K, Yamada T, Kubo S, Izui K, Nishiwaki S (2015) A topology optimization method for a coupled thermal-fluid problem using level set boundary expressions. Int J Heat Mass Transf 81:878–888. https://doi.org/10.1016/j.ijheatmasstransfer.2014.11.005
Yaji K, Yamada T, Yoshino M, Matsumoto T, Izui K, Nishiwaki S (2016) Topology optimization in thermal-fluid flow using the lattice Boltzmann method. J Comput Phys 307:355–377
Yoon GH (2010) Topological design of heat dissipating structure with forced convective heat transfer. J Mech Sci Technol 24:1225–1233. https://doi.org/10.1007/s12206-010-0328-1
Zhang B, Liu XM (2015) Topology optimization study of arterial bypass configurations using the level set method. Struct Multidiscip Optim 51:773–798. https://doi.org/10.1007/s00158-014-1175-y
Zhang B, Liu XM, Sun JJ (2016) Topology optimization design of non-Newtonian roller-type viscous micropumps. Struct Multidiscip Optim 53:409–424. https://doi.org/10.1007/s00158-015-1346-5
Zhao X, Zhou M, Sigmund O, Andreasen CS (2018) A “poor man’s approach” to topology optimization of cooling channels based on a Darcy flow model. Int J Heat Mass Transf 116:1108–1123. https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.090
Zhou SW, Li Q (2008) A variational level set method for the topology optimization of steady-state Navier-Stokes flow. J Comput Phys 227:10178–10195. https://doi.org/10.1016/j.jcp.2008.08.022
Acknowledgments
The authors are grateful to Professor X.-P Chen for helpful discussions and language editing.
Funding
B.Z. is supported by the Fundamental Research Funds for the Central Universities (No. G2018KY0306). L.G. is supported by the National Natural Science Foundation of China (No. 51790512).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Responsible Editor: Anton Evgrafov
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, B., Gao, L. Topology optimization of convective heat transfer problems for non-Newtonian fluids. Struct Multidisc Optim 60, 1821–1840 (2019). https://doi.org/10.1007/s00158-019-02296-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-019-02296-6