Non-Newtonian power-law fluid’s thermal characteristics across periodic array of circular cylinders

  • Ram PraveshEmail author
  • Amit Dhiman
  • R. P. Bharti
Technical Paper


The thermal characteristics of incompressible non-Newtonian power-law fluids across periodic array of circular cylinders have been examined using the finite volume-based numerical solver ANSYS-FLUENT for the following ranges of physical parameters: Reynolds number; 1 ≤ Re ≤ 40: Prandtl number; 1 ≤ Pr ≤ 100: power-law index; 0.40 ≤ n ≤ 1.8; and fluid volume fractions ranging from 0.70 to 0.99. The thermal features have been described via isotherm patterns, local and average Nusselt numbers and the Colburn heat transfer factor and found to be strongly dependent over the above physical parameters. It was observed that the dense isotherms with the increasing inertial and viscous diffusion suggest an improvement in the rate of heat transfer across the periodic cylinders. An increase in local Nusselt number was seen with the increasing values of Re and/or Pr across all the fluid volume fractions. Further, the different behavior of the average Nusselt number was noticed because of the shear-thinning and shear-thickening natures. An enhancement of about 97% was noticed in the shear-thinning region between the extreme fluid volume fractions for the highest value of Pr and the lowest values of Re and n. However, in many cases, the enhancement was noticed to be even more than 100%. An empirical correlation for the average Nusselt number and the Colburn heat transfer factor (jH) has been developed to give the additional physical insight of the results. Finally, the comparison was made with the available literature which displayed a good agreement with the present results.


Cylinders Periodic flow Average Nusselt number Fluid volume fractions Prandtl number Power-law fluids 

List of symbols


Specific heat (J/kg K)


Cylinder diameter (m)


Grid sizes (i = 1, 2, 3 and 4)


Convective heat transfer coefficient (W/m2 K)


Colburn heat transfer factor (-)


Thermal conductivity (W/m K)


Cylinder spacing (m)


Flow behavior consistency index (Pa sn)

\(\dot{m}_{i} ,\dot{m}_{o}\)

Mass flow inlet and outlet, respectively (kg/s)


Total number of grid cells (–)


Power-law index (–)


Unit vector (–)


Mean or average Nusselt number (–)


Number of grid points on cylinders (–)


Local Nusselt number (–)


Pressure (Pa)


Prandtl number (–)


Radius of cylinders (m)


Reynolds number (–)


Temperature (K)

Ti& To

Inlet and outlet fluid temperature, respectively (K)


Cylinder wall temperature (K)

\({T_{\infty } }\)

Bulk fluid temperature (K)


Volume averaged velocity (m/s)

Vx, Vy

Directional components of velocities (m/s)

x, y

Coordinates axes (m)

XN and XN

Normalized parameters (–)

Greek symbols


Functional parameter (–)


Maximum grid spacing (–)


Minimum grid spacing (–)


Viscosity (Pa s)


Fluid volume fractions (–)


Solid volume fraction (–)


Density of fluid (kg/m3)


Surface angle (radian)


Extra stress tensor (Pa)


Percent relative change (–)



Correlation coefficients and exponents















  1. 1.
    Ram RP, Bharti RP, Dhiman AK (2016) Forced convection flow and heat transfer across an in-line bank of circular cylinders. Can J Chem Eng 94:1381–1395CrossRefGoogle Scholar
  2. 2.
    Vijaysri M, Chhabra RP, Eswaran V (1999) Power-law fluid flow across an array of infinite circular cylinders: a numerical study. J NonNewt Fluid Mech 87:263–282CrossRefGoogle Scholar
  3. 3.
    Gowda YTK, Narayana PA, Seetharamu KN (1998) Finite element analysis of mixed convection over in-line tube bundles. Int J Heat Mass Trans 41:1613–1619CrossRefGoogle Scholar
  4. 4.
    Mauret E, Renaud M (1997) Transport phenomena in multi-particle systems-I. Limits of applicability of capillary model in high voidage beds—application to fixed beds of fibers and fluidized beds of spheres. Chem Eng Sci 51:1807–1817CrossRefGoogle Scholar
  5. 5.
    Kiljanski T, Dziubinski M (1996) Resistance to flow of molten polymers through filtration screens. Chem Eng Sci 51:4533–4536CrossRefGoogle Scholar
  6. 6.
    Zukauskas A (1987) Convective heat transfer in cross-flow: handbook of single-phase convective heat transfer. Wiley, New York (chapter 6) Google Scholar
  7. 7.
    Nishimura T (1986) Flow across tube banks. Ency Fluid Mech 1:763–785Google Scholar
  8. 8.
    Nishimura T, Itoh H, Ohya K, Miyashita H (1991) Experimental validation of numerical analysis of flow across tube banks for laminar flow. J Chem Eng Jpn 24:666–669CrossRefGoogle Scholar
  9. 9.
    Nishimura T, Itoh H, Ohya K, Miyashita H (1993) The influence of tube layout on flow and mass transfer characteristics in tube banks in the transitional flow regime. Int J Heat Mass Transf 36:553–563CrossRefGoogle Scholar
  10. 10.
    Roychowdhury DG, Das SK, Sundararajan T (2002) Numerical simulation of laminar flow and heat transfer over banks of staggered cylinders. Int J Num Methods Fluids 39:23–40CrossRefGoogle Scholar
  11. 11.
    Gamrat G, Marinet MF, Stephane LP (2008) Numerical study of heat transfer over banks of rods in small Reynolds number cross-flow. Int J Heat Mass Transf 51:853–864CrossRefGoogle Scholar
  12. 12.
    Chen CJ, Wung TS (1989) Finite analytic solution of convective heat transfer for tube arrays in cross-flow: part-II—heat transfer analysis. J Heat Transf (ASME) 111:641–648CrossRefGoogle Scholar
  13. 13.
    Martin AR, Saltiel C, Shyy W (1998) Frictional losses and convective heat transfer in sparse, periodic cylinder arrays in cross flow. Int J Heat Mass Transf 41:2383–2397CrossRefGoogle Scholar
  14. 14.
    Mandhani VK, Chhabra RP, Eswaran V (2002) Forced convection heat transfer in tube banks in cross flow. Chem Eng Sci 57:379–391CrossRefGoogle Scholar
  15. 15.
    Abrate S (2002) Resin flow in fiber preforms. Appl Mech Rev 55:579–599CrossRefGoogle Scholar
  16. 16.
    Ghosh UK, Upadhyay SN, Chhabra RP (1994) Heat and mass transfer from immersed bodies to non-Newtonian fluids. Adv Heat Transf 25:251–319CrossRefGoogle Scholar
  17. 17.
    Chhabra RP, Richardson JF (1999) Non-Newtonian flow in the process industries. Butterworth-Heinemann, OxfordGoogle Scholar
  18. 18.
    Chhabra RP (1999) In advances in the flow and rheology of non-Newtonian fluids. Elsevier, Amsterdam (chapter 39) Google Scholar
  19. 19.
    Soares AA, Ferreira JM, Chhabra RP (2005) Flow and forced convection heat transfer in crossflow of non-Newtonian fluids over a circular cylinder. Ind Eng Chem Res 44:5815–5827CrossRefGoogle Scholar
  20. 20.
    Spelt PDM, Selerland T, Lawrence CJ, Lee PD (2005) Flow of inelastic non-Newtonian fluids through arrays of aligned cylinders, Part I (creeping flows). J Eng Maths 51:57–80CrossRefGoogle Scholar
  21. 21.
    Spelt PDM, Selerland T, Lawrence CJ, Lee PD (2005) Flow of inelastic non-Newtonian fluids through arrays of aligned cylinders, Part II (Inertial effects for square arrays). J Eng Maths 51:81–97CrossRefGoogle Scholar
  22. 22.
    Bruschke MV, Advani SG (1993) Flow of generalized Newtonian fluids across a periodic array of cylinders. J Rheology 37:479–493CrossRefGoogle Scholar
  23. 23.
    Mangadoddy N, Bharti RP, Chhabra RP, Eswaran V (2004) Forced convection in cross-flow of power-law fluids over a tube bank. Chem Eng Sci 59:2213–2222CrossRefGoogle Scholar
  24. 24.
    Bharti RP, Chhabra RP, Eswaran V (2007) Steady forced convection heat transfer from a heated circular cylinder to power-law fluids. Int J Heat Mass Transf 50:977–990CrossRefGoogle Scholar
  25. 25.
    Patil RC, Bharti RP, Chhabra RP (2008) Forced convection heat transfer in Power law liquids from a pair of cylinders in tandem arrangement. Ind Eng Chem Res 47:9141–9164CrossRefGoogle Scholar
  26. 26.
    Slattery JC (1972) Mass momentum and energy transfer in continua. McGraw-Hill, New YorkGoogle Scholar
  27. 27.
    Happel J (1964) An analytical study of heat and mass transfer in multi-particle systems at low Reynolds numbers. AIChE J 10:605–611CrossRefGoogle Scholar
  28. 28.
    Dhotkar BN, Chhabra RP, Eswaran V (2000) Flow of non-Newtonian polymeric solutions in fibrous media. J Appl Polym Sci 76:1171–1185CrossRefGoogle Scholar
  29. 29.
    Shibu S, Chhabra RP, Eswaran V (2001) Power-law fluid flow over a bundle of cylinders at intermediate Reynolds numbers. Chem Eng Sci 56:5545–5554CrossRefGoogle Scholar
  30. 30.
    Prasad DVN, Chhabra RP (2001) An experimental investigation of the cross-flow of power law liquids past a bundle of cylinders and in a bed of stacked screens. Can J Chem Eng 79:28–35CrossRefGoogle Scholar
  31. 31.
    Chhabra RP, Comiti J, Machac I (2001) Flow of non-Newtonian fluids in fixed and fluidized bed’s. Chem Eng Sci 56:1–27CrossRefGoogle Scholar
  32. 32.
    Malleswara Rao TV, Chhabra RP (2003) A note on pressure drop for the cross flow of power-law liquids and air/power-law liquid mixtures past a bundle of circular rods. Chem Eng Sci 58:1365–1372CrossRefGoogle Scholar
  33. 33.
    Chhabra RP, Dhotkar BN, Eswaran V, Satheesh VK, Vijaysri M (2000) Steady flow of Newtonian and dilatant fluids over an array of long circular cylinders. J Chem Eng Japan 33:832–841CrossRefGoogle Scholar
  34. 34.
    Ferreira JM, Chhabra RP (2004) An analytical study of drag and mass transfer in creeping power-law flow across tube banks. Ind Eng Chem Res 43:3439–3450CrossRefGoogle Scholar
  35. 35.
    Adams D, Bell KJ (1968) Fluid friction and heat transfer for flow of sodium carboxy-methyl cellulose solutions across banks of tubes. Chem Eng Prog Symp Ser 64:133–145Google Scholar
  36. 36.
    Goel MK, Gupta SN (2015) The effect of neighbouring tubes on heat transfer coefficient for Newtonian and non-Newtonian fluids flowing across tube bank. Int J Mech Eng Robot Res 4:12–23Google Scholar
  37. 37.
    Panda SK (2017) Two-dimensional flow of power-law fluids over a pair of cylinders in a side by side arrangement in the laminar regime. Braz J Chem Eng 34:507–530CrossRefGoogle Scholar
  38. 38.
    Asif M, Dhiman A (2018) Analysis of laminar flow across triangular periodic array of heated cylinders. J Braz Soc Mech Sci Eng 40(350):1–24Google Scholar
  39. 39.
    Raju CSK, Kumar RVMSSK, Varma SVK, Madaki AG, Prasad PD (2018) Transpiration effects on MHD flow over a stretched cylinder with Cattaneo-Christov heat flux with suction or injection. Arab J Sci Eng 43:2273–2280CrossRefGoogle Scholar
  40. 40.
    Raju CSK, Hoque MM, Priyadharshini Mahanthesh B, Gireesha BJ (2018) Cross diffusion effects on magnetohydrodynamic slip flow of Carreau liquid over a slandering sheet with non-uniform heat source/sink. J Braz Soc Mech Sci Eng 40:222CrossRefGoogle Scholar
  41. 41.
    Upadhya SM, Mahesha Raju C S K, Shehzad SA, Abbasi FM (2018) Flow of Eyring-Powell dusty fluid in a deferment of aluminum and ferrous oxide nanoparticles with Cattaneo-Christov heat flux. Powder Technol 340:68–76CrossRefGoogle Scholar
  42. 42.
    Upadhya SM, Mahesha Raju C S K (2018) Comparative study of Eyring and Carreau fluids in a suspension of dust and nickel nanoparticles with variable conductivity. Eur Phys J Plus 133:156CrossRefGoogle Scholar
  43. 43.
    Prasad PD, Raju CSK, Varma SVK, Shehzad SA, Madaki AG (2018) Cross diffusion and multiple slips on MHD Carreau fluids in a suspension of microorganisms over a variable thickness sheet. J Braz Soc Mech Sci Eng 40:256CrossRefGoogle Scholar
  44. 44.
    Bird RB, Stewart WE, Lightfoot EN (2002) Transport Phenomena, 2nd edn. Wiley, New YorkGoogle Scholar
  45. 45.
    Patankar SV, Liu CH, Sparrow EM (1977) Fully developed flow and heat transfer in ducts having stream-wise periodic variations of cross-sectional area. J Heat Transf 99:180–186CrossRefGoogle Scholar
  46. 46.
    Sivakumar P, Bharti RP, Chhabra RP (2006) Effect of power-law index on critical parameters for power-law flow across an unconfined circular cylinder. Chem Eng Sci 61:6035–6046CrossRefGoogle Scholar
  47. 47.
    Sivakumar P, Bharti RP, Chhabra RP (2007) Steady flow of power-law fluids across an unconfined elliptic cylinder. Chem Eng Sci 62:1682–1702CrossRefGoogle Scholar
  48. 48.
    Daniel A, Dhiman A (2013) Aiding-buoyancy mixed convection from a pair of side-by-side heated circular cylinders in power-law fluids. Ind Eng Chem Res 52:17294–17314CrossRefGoogle Scholar
  49. 49.
    Kawase Y, Ulbrecht J (1981) Drag and mass transfer in non-Newtonian flows through multi-particle systems at low Reynolds number. Chem Eng Sci 36:1193–1205CrossRefGoogle Scholar
  50. 50.
    Kawase Y, Ulbrecht J (1981) Motion of the mass transfer from an assemblage of solid spheres moving in a non-Newtonian fluid at high Reynolds numbers. Chem Eng Comm 8:233–245CrossRefGoogle Scholar
  51. 51.
    Zhu J (1995) Drag and mass transfer for flow of a Carreau fluid past a swarm of Newtonian drops. Int J Multiphase Flow 21:935–940CrossRefGoogle Scholar
  52. 52.
    Satish MG, Zhu J (1992) Flow resistance and mass transfer in slow non-Newtonian flow through multi-particle systems. J Appl Mech (ASME) 59:431–437CrossRefGoogle Scholar
  53. 53.
    Eckert ERG, Sohengen E (1952) Distribution of heat transfer coefficients around circular cylinders in cross flow at Reynolds number from 20 to 500. Trans ASME 74:343–347Google Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Chemical EngineeringInstitute of Engineering and TechnologyLucknowIndia
  2. 2.Department of Chemical EngineeringIndian Institute of Technology RoorkeeRoorkeeIndia

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