Advertisement

Heat and Mass Transfer

, Volume 47, Issue 2, pp 119–130 | Cite as

Effects of Knudsen number and geometry on gaseous flow and heat transfer in a constricted microchannel

  • Hossein Shokouhmand
  • Sajjad BighamEmail author
  • Rasool Nasr Isfahani
Original

Abstract

A flow and heat transfer numerical simulation is performed for a 2D laminar incompressible gas flow through a constricted microchannel in the slip regime with constant wall temperature. The effects of rarefaction, creeping flow, first order slip boundary conditions and hydrodynamically/thermally developing flow are assumed. The effects of Knudsen number and geometry on thermal and hydrodynamic characteristics of flow in a constricted microchannel are explored. SIMPLE algorithm in curvilinear coordinate is used to solve the governing equations including continuity, energy and momentum with the temperature jump and velocity slip conditions at the solid walls in discretized form. The resulting velocity and temperature profiles are then utilized to obtain the microchannel C f Re and Nusselt number as a function of Knudsen number and geometry. The results show that Knudsen number has declining effect on the C f Re and Nusselt number in the constricted microchannel. In addition, the temperature jump on wall and slip velocity increase with increasing Knudsen number. Moreover, by decreasing the throttle area, the fluid flow characteristics experience more intense variations in the constricted region. To verify the code a comparison is carried out with available results and good agreement is achieved.

Keywords

Nusselt Number Slip Velocity Knudsen Number Local Nusselt Number Temperature Jump 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a

Amplitude of the wave (m)

k

Thermal conductivity of air (W/m K)

h

Local heat transfer coefficient (W/m2 K)

J

Jacobian of the coordinate transformation

p

Dimensionless pressure

Re

Reynolds number (Re = ρu i L */μ)

Pr

Prandtl number (Pr = ν/α)

Nu

Local Nusselt number

Nu

Fully developed Nusselt number

Kn

Knudsen number

Ma

Mach number

Pe

Peclet number

Ec

Eckert number

Cf

Skin-friction coefficient

cp

Specific heat (J/kg K)

n

Dimensionless normal direction to the wall

s

Dimensionless tangential direction to the wall

q11, q22, q12

Grid parameters

R

Gas constant (J/kg K)

T

Temperature (K)

q

Heat flux

u

Dimensionless velocity component in x-direction

v

Dimensionless velocity component in y-direction

L*

Channel inlet width

x

Dimensionless horizontal coordinate

y

Dimensionless vertical coordinate

Greek symbols

α

Thermal diffusivity (m2/s)

λ

Surface wavelength (m)

ρ

Density of fluid (kg/m3)

μ

Dynamic viscosity (kg/m s)

γ

Ratio of specific heats (cp/cv)

λ

Molecular mean free path (m)

ν

Kinematic viscosity (m2/s)

σT

Energy accommodation coefficient

σν

Momentum accommodation coefficient

θ

Dimensionless temperature

ξ

Curvilinear horizontal coordinate

η

Curvilinear vertical coordinate

τ

Shear stress

Subscripts

ave

Mean value

w

Surface conditions

i

Inlet conditions

s

Fluid property near the wall

Superscripts

C

Contravariant velocities

tang

Tangential direction

*

Returns to dimensional parameters

References

  1. 1.
    Galvis E, Jubran BA, Xi F, Behdinan K, Fawaz Z (2008) Numerical modeling of pin–fin micro heat exchangers. Heat Mass Transf 44:659–666CrossRefGoogle Scholar
  2. 2.
    Graur IA, Méolans JG, Zeitoun DE (2006) Analytical and numerical description for isothermal gas flows in microchannels. Microfluid Nanofluid 2:64–67CrossRefGoogle Scholar
  3. 3.
    Jiji LM (2008) Effect of rarefaction, dissipation, and accommodation coefficients on heat transfer in microcylindrical couette flow. ASME J Heat Transf 130:385–393CrossRefGoogle Scholar
  4. 4.
    Jie D, Diao X, Cheong KB, Yong LK (2000) Navier–Stokes simulations of gas flow in micro devices. J Micromech Microeng 10:372–379CrossRefGoogle Scholar
  5. 5.
    Chen CS, Kuo WJ (2004) Heat transfer characteristics of gaseous flow in long mini- and microtubes. Numer Heat Transf Part A Appl 46:497–514CrossRefGoogle Scholar
  6. 6.
    Larrode FE, Housiadas C, Dreossinos Y (2000) Slip-flow heat transfer in circular tubes. Int J Heat and Mass Transf 43:2669–2680zbMATHCrossRefGoogle Scholar
  7. 7.
    Kavehpour HP, Faghri M, Asako Y (1997) Effects of compressibility and rarefaction on gaseous flows in microchannels. Numer Heat Transf Part A Appl 32:677–696CrossRefGoogle Scholar
  8. 8.
    Dennis SCR, Smith FT (1980) Steady flow through a channel with a symmetrical constriction in the form of a step. Proc R Soc Loud A 372:393–414zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Vradis G, Zalak V, Bentson J (1992) Simultaneous, variable solutions of the incompressible steady Navier-Stokes equations in general curvilinear coordinate systems. Trans ASME I J Fluids Engng 114:299–305CrossRefGoogle Scholar
  10. 10.
    Wang CC, Chen CK (2004) Forced convection in a wavy-wall channel. Int J Heat Mass Transf 47:3877–3887CrossRefGoogle Scholar
  11. 11.
    Cheng RT-S (1972) Numerical Solution of the Navier-Stokes Equations by the Finite Element Method. Phys Fluids 15:2098–2105zbMATHCrossRefGoogle Scholar
  12. 12.
    Deshpande MD, Giddens DP, Mabon RF (1976) Steady laminar flow through modelled vascular stenoses. J Biomech 9:165–174CrossRefGoogle Scholar
  13. 13.
    Ahmed SA, Giddens DP (1983) Flow disturbance measurements through a constricted tube at moderate Reynolds number. J Biomech 16:955–963CrossRefGoogle Scholar
  14. 14.
    Arkilic EB, Breuer KS, Schmidt MA (1994) Gaseous flow in microchannels. ASME Appl Microfabrication Fluid Mech 197:57–66Google Scholar
  15. 15.
    Harley JC, Huang Y, Bau HH, Zemel JN (1995) Gas flow in micro-channels. J Fluid Mech 284:257–274CrossRefGoogle Scholar
  16. 16.
    Beskok A, Karniadakis GE, Trimmer W (1996) Rarefaction and compressibility effects in gas microflows. ASME J Fluids Eng 118:448–456CrossRefGoogle Scholar
  17. 17.
    Chen CK, Cho CC (2007) Electro-kinetically-driven flow mixing in microchannels with wavy surface. J Colloid Interf Sci 312:470–480CrossRefGoogle Scholar
  18. 18.
    Kennard EH (1938) Kinetic theory of gasses. McGraw-Hill, New YorkGoogle Scholar
  19. 19.
    Gombosi TI (1994) Gas kinetic theory. Cambridge University Press, New YorkGoogle Scholar
  20. 20.
    Gad-el-Hak M (2002) The MEMS Handbook, CRC Press LLC, Boca RatonGoogle Scholar
  21. 21.
    Karniadakis GE, Beskok A, Aluru N (2004) Micro flows and nanoflows fundamental and simulation. Springer, USAGoogle Scholar
  22. 22.
    Kandlikar S, Garimella S, Li D, Colin S, King MR (2006) Heat transfer and fluid flow in minichannels and microchannels, Elsevier, BritainGoogle Scholar
  23. 23.
    Liou WW, Fang Y (2006) Microfluid mechanics principal and modeling. McGraw-Hill, New YorkGoogle Scholar
  24. 24.
    Morini GL, Spiga M, Tartarini P (2004) The rarefaction effect on the friction factor of gas flow in microchannels. Superlattices and microstructures 35:587–599CrossRefGoogle Scholar
  25. 25.
    van Rij J, Ameel T, Harman T (2009) The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer. Int J Therm Sci 48:271–281CrossRefGoogle Scholar
  26. 26.
    Patankar SV (1972) A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J Heat Mass Transf 15:1787–1806zbMATHCrossRefGoogle Scholar
  27. 27.
    Spalding DB (1972) A novel finite difference formulation for differential expressions involving both first and second derivatives. Int J Numer Methods Eng 4:551–559CrossRefGoogle Scholar
  28. 28.
    Hoffman KA (1989) Computational fluid dynamics for engineers. Eng Educ Sys, AustinGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Hossein Shokouhmand
    • 1
  • Sajjad Bigham
    • 1
    Email author
  • Rasool Nasr Isfahani
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of TehranTehranIran

Personalised recommendations