Structural and Multidisciplinary Optimization

, Volume 60, Issue 5, pp 1821–1840 | Cite as

Topology optimization of convective heat transfer problems for non-Newtonian fluids

  • Bin ZhangEmail author
  • Limin Gao
Research Paper


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.


Topology optimization Non-Newtonian fluid Thermal-fluid Convective heat transfer Power law model Sensitivity analysis 



The authors are grateful to Professor X.-P Chen for helpful discussions and language editing.

Funding information

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).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Alexandersen J, Aage N, Andreasen CS, Sigmund O (2014) Topology optimisation for natural convection problems. Int J Numer Methods Fluids 76:699–721MathSciNetCrossRefGoogle Scholar
  2. 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 100:876–891CrossRefGoogle Scholar
  3. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
  4. Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. (Second Edition). Springer - Verlag Berlin Heidelberg GmbHGoogle Scholar
  5. Borrvall T, Petersson J (2003) Topology optimization of fluids in stokes flow. Int J Numer Methods Fluids 41:77–107. MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chhabra RP (2006) Bubbles, drops, and particles in non-Newtonian fluids (second edition), 2nd edn. CRC Press, Boca RatonCrossRefGoogle Scholar
  7. Chhabra RP, Richardson JF (2008) Non-Newtonian flow and applied rheology (second edition). Elsevier Ltd.
  8. Coffin P, Maute K (2016) A level-set method for steady-state and transient natural convection problems. Struct Multidiscip Optim 53:1047–1067. MathSciNetCrossRefGoogle Scholar
  9. Dede E (2009) Multiphysics topology optimization of heat transfer and fluid flow systems. Proceedings of the COMSOL Conference, BostonGoogle Scholar
  10. Dede EM (2012) Optimization and design of a multipass branching microchannel heat sink for electronics cooling. J Electron Packag 134:041001. CrossRefGoogle Scholar
  11. Dhiman AK, Chhabra RP, Eswaran V (2007) Heat transfer to power-law fluids from a heated square cylinder. Numer Heat Transfer, Part A 52:185–201CrossRefGoogle Scholar
  12. Duan XB, Ma YC, Zhang R (2008a) Optimal shape control of fluid flow using variational level set method. Phys Lett A 372:1374–1379. MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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. MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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 365:376–404MathSciNetCrossRefGoogle Scholar
  15. 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 173:264–274CrossRefGoogle Scholar
  16. 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. CrossRefGoogle Scholar
  17. 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. CrossRefGoogle Scholar
  18. 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. MathSciNetCrossRefzbMATHGoogle Scholar
  19. Jensen KE, Szabo P, Okkels F (2012) Topology optimization of viscoelastic rectifiers. Appl Phys Lett 100. CrossRefGoogle Scholar
  20. Kim SJ (2004) Methods for thermal optimization of microchannel heat sinks. Heat Transfer Eng 25:37–49CrossRefGoogle Scholar
  21. 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. CrossRefGoogle Scholar
  22. 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. MathSciNetCrossRefzbMATHGoogle Scholar
  23. Kreissl S, Pingen G, Maute K (2011) Topology optimization for unsteady flow. Int J Numer Methods Eng 87:1229–1253. MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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 99:262–272CrossRefGoogle Scholar
  25. 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. CrossRefGoogle Scholar
  26. 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 99:372–382CrossRefGoogle Scholar
  27. 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. CrossRefGoogle Scholar
  28. 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. CrossRefGoogle Scholar
  29. 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. MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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. CrossRefGoogle Scholar
  31. 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. CrossRefzbMATHGoogle Scholar
  32. 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 89:83–90CrossRefGoogle Scholar
  33. 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. CrossRefGoogle Scholar
  34. Okkels F, Bruus H (2007) Scaling behavior of optimally structured catalytic microfluidic reactors. Phys Rev E 75.
  35. 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. MathSciNetCrossRefzbMATHGoogle Scholar
  36. Panda S, Chhabra RP (2011) Laminar forced convection heat transfer from a rotating cylinder to power-law fluids. Numerical Heat Transfer, Part A 59:297–319CrossRefGoogle Scholar
  37. Pingen G, Maute K (2010) Optimal design for non-Newtonian flows using a topology optimization approach. Comput Math Appl 59:2340–2350. MathSciNetCrossRefzbMATHGoogle Scholar
  38. 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. CrossRefGoogle Scholar
  39. 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 113:875–888CrossRefGoogle Scholar
  40. 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. CrossRefGoogle Scholar
  41. Romero JS, Silva ECN (2017) Non-newtonian laminar flow machine rotor design by using topology optimization. Struct Multidiscip Optim 55:1711–1732. MathSciNetCrossRefGoogle Scholar
  42. 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. CrossRefGoogle Scholar
  43. 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 113:482–489CrossRefGoogle Scholar
  44. Singh A, Kishore N (2018) Laminar mixed convection of non-Newtonian nanofluids flowing vertically upward across confined circular cylinders. J Therm Sci Eng Appl 10:14Google Scholar
  45. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373MathSciNetCrossRefGoogle Scholar
  46. Svanberg K (2010) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12:555–573MathSciNetCrossRefGoogle Scholar
  47. 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 112:1027–1039CrossRefGoogle Scholar
  48. 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. CrossRefGoogle Scholar
  49. 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. CrossRefGoogle Scholar
  50. 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 307:355–377MathSciNetCrossRefGoogle Scholar
  51. Yoon GH (2010) Topological design of heat dissipating structure with forced convective heat transfer. J Mech Sci Technol 24:1225–1233. CrossRefGoogle Scholar
  52. Zhang B, Liu XM (2015) Topology optimization study of arterial bypass configurations using the level set method. Struct Multidiscip Optim 51:773–798. MathSciNetCrossRefGoogle Scholar
  53. Zhang B, Liu XM, Sun JJ (2016) Topology optimization design of non-Newtonian roller-type viscous micropumps. Struct Multidiscip Optim 53:409–424. MathSciNetCrossRefGoogle Scholar
  54. 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. CrossRefGoogle Scholar
  55. 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. MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Engineering Mechanics, School of Mechanics, Civil Engineering and ArchitectureNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.School of Power and EnergyNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

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