Skip to main content
SpringerLink
Log in

Laminar mixing and heat transfer for constant heat flux boundary condition

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

In a recent paper we have investigated mixing and heat transfer enhancement in a mixer composed of two circular rods maintained vertically in a cylindrical tank. The rods and tank can rotate around their revolution axes while their surfaces were maintained at a constant temperature. In the present study we investigate the differences in the thermal mixing process arising from the utilization of a constant heat flux as a boundary condition. The study concerns a highly viscous fluid with a high Prandtl number for which this chaotic mixer is suitable. By solving numerically the flow and energy equations, and using different statistical tools we characterize the evolution of the fluid temperature and its homogenization. Fundamental differences are reported between these two modes of heating or cooling: while the mixing with an imposed temperature results in a homogeneous temperature field, with a fixed heat flux we observe a constant difference between the maximal and minimal temperatures that establish in the fluid; the extent of this difference is governed by the efficiency of the mixing protocol.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Abbreviations

A :

Area (m2)

C p :

Heat capacity (J/kg K)

k :

Thermal conductivity (W/m K)

L :

Wall characteristic length (m)

p :

Pressure (Pa)

\(\dot q\) :

Surface heat flux (W/m2)

R 3 :

Tank radius (m)

R 1, R 2 :

Rod radii (m)

t :

Time (s)

T :

Temperature (K)

U :

Maximum tangential velocity of the walls (m/s)

\({\bf U}\) :

Velocity field (m/s)

Nu :

Nusselt number

Pe :

Péclet number

Pr :

Prandtl number

Re :

Reynolds number

X :

Rescaled dimensionless temperature

\(\varepsilon\) :

Eccentricity (m)

ρ:

Fluid density (kg/m3)

σ:

Standard deviation

τ:

Period of modulation (s)

\(\overset{=}{\tau}\) :

Viscous stress tensor (Pa)

\(\Upomega\) :

Angular velocity (rad/s)

c :

Cell

m :

Mean

f :

Face of a cell

References

  1. Ottino JM (1989) The kinematics of mixing: stretching, chaos, and transport. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  2. Aref H (2002) The development of chaotic advection. Phys Fluids 14:1315–1325

    Article  MathSciNet  Google Scholar 

  3. Jana SC, Sau M (2004) Effects of viscosity ratio and composition on development of morphology in chaotic mixing of polymers. Polymer 45(5):1665–1678

    Article  Google Scholar 

  4. Metcalfe G, Lester DR (2009) Mixing and heat transfer of highly viscous food products with a continuous chaotic duct flow. J Food Eng 95(1):21–29

    Article  Google Scholar 

  5. Caubet S, Le Guer Y, Grassl B, El Omari K, Normandin E (2011) A low-energy emulsification batch mixer for concentrated oil-in-water emulsions. AIChE J 57(1):27–39

    Article  Google Scholar 

  6. Chang HC, Sen M (1994) Application of chaotic advection to heat transfer. Chaos Solitons Fractals 4(6):955–976

    Article  MATH  Google Scholar 

  7. Acharya N, Sen M, Chang HC (2001) Analysis of heat transfer enhancement in coiled-tube heat exchangers. Int J Heat Mass Transf 44(17):3189–3199

    Article  MATH  Google Scholar 

  8. Peerhossaini H, Castelain C, Le Guer Y (1993) Heat exchanger design based on chaotic advection. Exp Therm Fluid Sci 7(4):333–344

    Article  Google Scholar 

  9. Mokrani A, Castelain C, Peerhossaini H (1997) The effects of chaotic advection on heat transfer. Int J Heat Mass Transf 40(13):3089–3104

    Article  Google Scholar 

  10. Gouillart E, Dauchot O, Dubrulle B, Roux S, Thiffeault JL (2008) Slow decay of concentration variance due to no-slip walls in chaotic mixing. Phys Rev E 78(2):026211

    Article  MathSciNet  Google Scholar 

  11. Gouillart E, Thiffeault JL, Dauchot O (2010) Rotation shields chaotic mixing regions from no-slip walls. Phys Rev Lett 104:204502

    Article  Google Scholar 

  12. El Omari K, Le Guer Y (2010) Alternate rotating walls for thermal chaotic mixing. Int J Heat Mass Transf 53:123–134

    Article  MATH  Google Scholar 

  13. Ghosh S, Chang HC, Sen M (1992) Heat-transfer enhancement due to slender recirculation and chaotic transport between counter-rotating eccentric cylinders. J Fluid Mech 238:119–154

    Article  MathSciNet  MATH  Google Scholar 

  14. Ganesan V, Bryden MD, H Brenner (1997) Chaotic heat transfer enhancement in rotating eccentric annular-flow systems. Phys Fluids 9:1296–1306

    Article  Google Scholar 

  15. Lefevre A, Mota JPB, Rodrigo AJS, Saatdjian E (2003) Chaotic advection and heat transfer enhancement in Stokes flows. Int J Heat Fluid Flow 24(3):310–321

    Article  Google Scholar 

  16. Mota JPB, Rodrigo AJS, Saatdjian E (2007) Optimization of heat-transfer rate into time-periodic two-dimensional Stokes flows. Int J Numer Methods Fluids 53(6):915–931

    Article  MATH  Google Scholar 

  17. Price TJ, Mullin T, Kobine JJ (2003) Numerical and experimental characterization of a family of two-roll-mill flows. R Soc Lond Proc Ser A 459:117–135

    Article  MATH  Google Scholar 

  18. Young DL, Chiu CL, Fan CM (2007) A hybrid Cartesian/immersed-boundary finite-element method for simulating heat and flow patterns in a two-roll mill. Numer Heat Transf Part B Fundam 51(3–4):251–274

    Article  Google Scholar 

  19. Chiu CL, Fan CM, Young DL (2009) 3D hybrid Cartesian/immersed-boundary finite-element analysis of heat and flow patterns in a two-roll mill. Int J Heat Mass Transf 52(7–8):1677–1689

    Google Scholar 

  20. Jana SC, Metcalfe G, Ottino JM (1994) Experimental and computational studies of mixing in complex Stokes flows: the vortex mixing flow and multicellular cavity flows. J Fluid Mech 269:199–246

    Article  MathSciNet  Google Scholar 

  21. El Omari K, Le Guer Y (2009) A numerical study of thermal chaotic mixing in a two rod rotating mixer. Comput Therm Sci 1:55–73

    Google Scholar 

  22. El Omari K, Le Guer Y (2010) Thermal chaotic mixing of power law fluids in a mixer with alternately-rotating walls. J Non-Newton Fluid Mech 165:641–651

    Article  Google Scholar 

  23. Le Guer Y, El Omari K (2012) Chaotic advection for thermal mixing. In: Aref H, van der Giessen E (eds) Advances in applied mechanics, vol 45. Elsevier, Amsterdam, pp 189–237

  24. Geuzaine C, Remacle JF (2009) Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities. Int J Numer Methods Eng 79(11):1309–1331

    Article  MathSciNet  MATH  Google Scholar 

  25. El Omari K, Kousksou T, Le Guer Y (2011) Impact of shape of container on natural convection and melting inside enclosures used for passive cooling of electronic devices. Appl Therm Eng 31:3022–3035

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire des Sciences de l’Ingénieur Appliquées à la Mécanique et au Génie Electrique (SIAME), Université de Pau et des Pays de l’Adour (UPPA), Bat. D’Alembert, Rue Jules Ferry, BP 7511, 64075, Pau Cedex, France

    Kamal El Omari & Yves Le Guer

Authors
  1. Kamal El Omari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yves Le Guer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kamal El Omari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

El Omari, K., Le Guer, Y. Laminar mixing and heat transfer for constant heat flux boundary condition. Heat Mass Transfer 48, 1285–1296 (2012). https://doi.org/10.1007/s00231-012-0976-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-012-0976-z

Keywords

Search

Navigation