rarefaction effect on heat transfer and fluid flow in microchannel

Topic A 2nd International Conference on Energy Conversion and Conservation

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Rarefaction Effect on Heat Transfer and Fluid Flow in Microchannel

1* Hossein Afshar, 2 Seyed Mojtaba Mousavi Nainian, 3 Mehrzad Shams, 4 Goodarz Ahmadi

1 Ph.D Candidate, Mechanical Engineering Department, K.N.Toosi University of Technology Mollasadra Ave. Vanak Sq., Tehran, 19991-43344, Iran

2 Associate Professor, Mechanical Engineering Department, K.N.Toosi University of Technology Mollasadra Ave. Vanak Sq., Tehran, 19991-43344, Iran

3 Associate Professor, Mechanical Engineering Department, K.N.Toosi University of Technology Mollasadra Ave. Vanak Sq., Tehran, 19991-43344, Iran

4 Proffesor, Mechanical and Aeronautical Engineering Department, Clarkson University Potsdam NY, 13699, USA

* E-mail: [email protected]

Keywords: Slip Flow, Heat Transfer, Microchannel, Rarefaction Abstract If the hydrodynamic diameter of a channel is comparable with the mean free path of the gas molecules moving inside the channel, the fluid can no longer be considered to be in thermodynamic equilibrium and a variety of non-continuum or rarefaction effects can occur. To avoid enormous complexity and extensive numerical cost encountered in modeling of nonlinear Boltzmann equations, the Navier-Stokes equations can be solved considering the concepts of slip flow regime and applying slip velocity boundary conditions at the solid walls. In this study, a new slip boundary condition according to the kinetic theory of gases is introduced. Navier-Stokes and energy equations for fluid flow in a microchannel in no-slip and slip flow regimes are solved. Temperature and velocity profiles are evaluated and the effect of rarefaction parameters on heat transfer in the microchannel is discussed.

I. Introduction Even though the balance and the conservation equations of fluid-dynamics are valid in all rarefaction regimes, the solution of the Navier-Stokes equations becomes lacking with increasing rarefaction. In fact, numerical integration of these equations relies also on the computation of shear stress and heat flux. In low-density regimes phenomenological equations of Newton, Fourier and Fick are no longer valid. Furthermore, as the density decreases, the intermolecular collisions in the gas get too few for maintaining the isotropy of the pressure tensor, the conventional no-slip boundary condition can be no longer applied and finally effects such as thermal and pressure diffusion, usually not included in the Navier-Stokes solvers, become more prominent. In recent years, many researchers are interested in small scale flows and many attempts are made in minimization of scales and improvement of the performance of instruments (Latif (2008), by increasing the usage of small scale instruments, understanding the behavior of such flows has become more important. In micro-scale, rarefaction and interferences between fluid and solid surface that causes the violation of no slip boundary condition need to be accounted for in the analysis. In most macro-scale applications, the fluid flow in channels is in turbulent flow regime but in micro-scale and nano-scale applications, most fluid flows are in laminar regime. The Knudsen number which is the ratio of mean free path over flow characteristic length, defines flow characteristics when the flow dimensions approach the molecular mean free path. This non

dimensional quantity, is defined as (1)

cLKn = (1)

Where Lc is the flow characteristic length, (hydraulic diameter in a microchannel), and is the molecular mean free path. Flow regime is defined according to the value of Knudsen number.

Continuum Flow 310Kn Slip Flow 110Kn310 Transitional Flow 10Kn110 Free Molecular FlowO(10)Kn >

In microchannels even though the fluid density whould not rely on low-density regime, but because of the value of length scale which would be in order of mean free path, it could be in slip flow regime. Morini et al. (2005) theoretically investigated the conditions for experimentally evidencing rarefaction effects on the pressure drop. It was demonstrated that for a fixed geometry of the microchannel cross-section, it is possible to calculate the minimum value of the Knudsen number for which the rarefaction effects can be observed experimentally. Hung and Ru (2006) studied the heat transfer characteristics of fluid flow in microchannel by the lattices-Boltzmann method. A nine-velocity model and an internal energy distribution model were used to obtain the mass, momentum and temperature distributions in micro-channel flow. Khadem et al. (2008) performed a two dimensional numerical simulation for incompressible and compressible fluid flow through microchannels in slip flow regime.


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Cooling of electronic microchips is one important application of microchannels. Webb (2005) introduced microchannels as next generation devices for electronic cooling.The heat flux in these cases can reach to 100 w/cm2. Afshar et al. (2008 and 2009) studied heat transfer and dispersion of nanoparticles in a microchannel. They studied channel adsobtion efficieny and slip effect on dispersion of nanoparticles. Martin and Boyd (2006) modeled the fluid flow in a laminar boundary layer using a slip boundary condition. It was shown that the slip condition changes the boundary layer structure from a self-similar profile to a two-dimensional structure. One of the most important concerns in modelling rarefied flow in a microchannel in slip flow regime is how to apply slip boundary conditions for velocity and temperature. Hettiarachchi et al. (2008) numerically studied Three-dimensional laminar slip-flow and heat transfer in rectangular microchannels having constant temperature walls using the finite-volume method for thermally and simultaneously developing flows. They defined a modified convectiondiffusion coefficient at the wallfluid interface to incorporate the temperature-jump boundary condition. Zhang et al. (2009) established a numerical model for three-dimensional compressible gaseous slip flow in microchannel. They modified gas viscosity based on Knudsen number using Veijolas model due to the increased rarefaction effects in microscale. In this study an analytical method for velocity and themperature boundary conditions in slip flow regime according to rarefied gas dynamics will be presented. II.Governing Equations In two dimensional incompressible fluid flow, continuity and momentum equations are as follows:

0=+

yv

xu

(2)

+

+=

+

2

2

2

2

yu

xu

xp

yuv

xuu

(3)

+

+=

+

2

2

2

2

yv

xv

yp

yvv

xvu

(4)

Energy equation:

)()( 22

2

2

yT

xTk

yTv

xTuc p

+=

+

(5)

In order to obtain velocity and temperature fields, above equations should be solved according to proper boundary conditions. For rarefied flows in slip flow regime, slip boundary conditions should be applied.

wwv

vwrs s

TTn

uuuu

+

==

432

(6)

v is the tangential momentum accommodation coefficient. n and s are directions normal and tangebtial to the boundary. us is the velocity of fluid molecules adjascent to the wall.

wi

riv uu

uu= (7)

ui is the tangential velocity of incident molecules which come from out of the Knudsen layer, ur is the velocity of reflected molecules and uw is the wall velocity.

wT

Tws n

TTT

+

=

Pr122

(8)

T is the energy accommodation coefficient. Ts is the temperature of fluid molecules adjascent to the wall. is the specific heat.

wi

riT ee

ee= (9)

ei is the energy of incident molecules, er is the energy of reflected molecules and ew is the energy of wall molecules. III.Methodology According to the definition of Knudsen number (eq.1), if the characteristic length is comparable to the mean free path of the molecules, assumption of equilibrium will not be valid any more. The non-equilibrium exchange in momentum and energy between molecules is done in Knudsen layer which height is about a mean free path of the molecules (Struchtrup et al. (2007)) so the molecules move toward the wall form out of the Knudsen layer where equilibrium conditions are valid and then reflect. Distance between the centers of molecules from the wall is equal to radius of molecules, so slip occurs in a molecular radius from the wall (figure 1).

By assuming that the temperature of reflected molecules will be equal to the wall temperature, the momentum flux for molecules adjascent to the wall is

ssss uCmn41= (10)

ns is molecular number density, m is weight of molecules and C is the mean thermal velocity which is defined as:

MRTC

8= (11)R is the gas constant and M is the molar weight of the gas. For molecules adjascent to the wall, half of them are coming from out of the kndusen layer and half of them are reflecting from the wall. So the momentum flux can be written as:

rrriiisss uCmnuCmnuCmn 41

41

41 +=

(12)

Substituting ur from equation (7) and C from equation

Knudsen Layer

ui , ei

us , es

Figure 1: Knudsen layer and slip surface near the wall


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(11) to equation (12), noting that sri nnn 21== and EAC

equal to unity, a non-linear equation for velocity and temperature slip is obtained.

( )[ ]wvwivwiiss uTuTuTuT ++= 121 (13)Equation (13) can be sloved by an itertive procedure to obtain velocity and temperature slip instead of using equations (6) and (8) as boundary conditions. Reordering equation (8) leads to have temperature of molecules adjacent to the wall.

siT

Tws

TTTT

+

=Pr1

22 (14)

Ts is the temperature of gas molecules adjacent to the wall, Ti is the temperature of gas molecules a distance of mean free path (Knudesn layer thickness) from the wall. So Ts can be written explicitly in terms of temperature of incoming molecules and wall temperature. ( ) ( )

( ) ( )TTWTiT

sTTT

++++=

22Pr11Pr22 (15)

Combining equations (13) and (15) lead to temperature and velocity slip in term of wall and flow in equilibrium conditions. These relations can be used instead of conventional boundary conditions that use gradients, and can accelerate the numerical convergence. Note that in equation (15), the second grid point in numerical simulation should be placed in a distance of mean free path from the wall. If temperature effect on tangential velocity slip whould be neglected, then velocity profile in the microchannel due to analytical solution can be written as equation (16).

+

= KnHy

Hy

dxdpHyu

v

v

2

22

)(22

(16)

According to equation (16) it can be concluded that as the Knudsen number increases, the maximum velocity at the centre of the microchannel decreases whilst the tangential slip-velocity at the wall increases. The net effect of these changes is to produce a velocity profile which becomes more uniform with increasing Knudsen number. Another interesting feature of the flow redistribution is the fact that the velocity remains invariant with respect to Knudsen number at two locations across the microchannel. It can readily be shown that for flow in the microchannel position of this feature occurs at

321

21 =

Hy (17)

It should be noted that equation (16) and (17) are valid for traditional first order slip boundary condition without consideration of temperature effects. IV.Results Slip flow in a short microchannel with 4 micrometers height and 100 micrometers long (Figure 2) is investigated. Air flow enters the microchannel uniformly with velocity of 0.3 m/s, Temperature of 300K and in the exit; it is

hydro dynamically fully developed. Constant temperature of 500K is imposed to upper and lower walls. Tangential momentum accommodation and energy accommodation coefficients are set to unity. Velocity and temperature distribution in the

microchannel is shown in figures (3) and (4) respectively. As the temperature is constant in upper and lower

walls, so temperature variation occurs just in a small region in the channel entrance and in the rest of microchannel, the temperature is constant. Also temperature and velocity slip can be seen in figures (3) and (4).

Figure 2: Configuration of the microchannel

Figure 3: Slip-Velocity distribution in the microchannel (Kn=.01)

Figure (4): Temperature distribution in the microchannel in slip flow regime (Kn=.01)

Kn=0

Kn=0.01

Figure (5): Velocity profile in x/L=1/70


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Velocity profile in a channel section where temperature

is not fully developed (x/L=1/70) for slip and no-slip flow regimes is shown in figure (5). Velocity profile in a channel section (x/L=3/70) for slip and no-slip flow regimes is shown in figure (6). Temperature effect on slip velocity is obvious in figures (5) and (6). As it is supposed, slip velocity profile is flater than no-slip profile. Figure (6) shows that two positions in the microchannel that velocity remains invariant with respect to Knudsen number are not the positions that are referd to in equation (17). So it can be concluded that as temperature effect is often neglected in equation (6) and so in derivation of equation (17), they can not be introduced as proper statements in slip flow regime. Comparison of figures (5) and (6) shows that constant velocity positions versus Knudsen number move toward the center of the microchannel as the flow passes the entrance region and becomes fully developed. Temperature distribution in different sections of the

microchannel for no-slip boundary condition is shown in figure (7). Air flow temperature in the enterance is 300K and upper and lower walls are at constant temperature of 500K. As shown in figure (7), air temperature inceases as it flows in the channel and after x/L greater than 4/70, temperature becomes uniform equal to 500K. Temperature distribution in different sections of the microchannel for Knudsen number equal to 0.01 is shown in figure (8).

Figure (8) shows that slip temperature decreases along the microchannel. Comparing figures (7) and (8) show that the temperature in the center of the microchannel decreases due to slip. It means that thermal slip increases the thermal entrance region (figure 4) and fluid passes a distance more than no-slip flow to have zero temperature gradient. Temperature distribution in different sections of the microchannel for Knudsen number equal to 0.1 is shown in figure (9).

Figure (9) shows that rarefaction has more effect on

Kn=0.01

Kn=0

Figure (6): Velocity profile in x/L=3/70

x/L=3/70

x/L=1/70

x/L=2/7

Figure (7): No-slip temperature distribution in different sections of the microchannel

x/L=3/70x/L=2/70

x/L=1/70

Figure (8): Slip-temperature distribution in different sections of the microchannel

(Kn=0.01)

x/L=1/70

x/L=2/70

x/L=3/70

Figure (9): Slip-temperature distribution in different sections of the microchannel


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the temperature distribution in thermal entrance region. Comparing figures (7), (8) and (9) show that by increasing the Knudsen number, mean temperature in channel sections decreases. It can be concluded that by increasing the Knudsen number, the rate of microchannel heat transfer decreases. V.Conclusions New relations which are introduced as boundary conditions for slip flow regime can be applied easily in numerical and analytical solutions in comparison to conventional relations. Velocity is found explicitly in terms of temperature slip, so it accelerates the convergence in numerical simulations. In derivation of above mentioned slip boundary conditions, equations and relations of kinetic theory of gases are used which are not limited to Knusden numbers between 0.001 and 0.1. So it can be concluded that these boundary conditions are more general.

Nomenclature

Kn : Knudsen number : Mean free path of the molecules cL : Characteristic length

u : Fluid velocity in x direction v : Fluid velocity in y direction : Dynamic viscosity T : Fluid Temperatue : Fluid density pC : Fluid specific heat

v : Tangential momentum accommodation coefficient

us : Velocity of fluid molecules adjascent to the wall

ui : Tangential velocity of incident molecules which come from out of the Knudsen layer

ur : Velocity of reflected molecules from the walluw : Velocity of the wall T : Energy accommodation coefficient

Ts : Temperature of fluid molecules adjascent to the wall : Specific heat

ei : Energy of incident molecules er : Energy of reflected molecules ew : Energy of wall molecules

s : Momentum flux of molecules adjacent to the wall

ns : Molecular number density m : Weight of molecules C : Mean thermal velocity R : Gas constant M : Molar weight of the gas L : Length of the microchannel H : Height of the microchannel

References Afshar H., Shams M., Nainian S.M.M, Ahmadi G., Microchannel Heat Transfer and Dispersion of Nanoparticles in Slip Flow Regime with Constant Heat Flux, International

Communications in Heat and Mass Transfer, 36, 10601066, (2009) Afshar H., Shams M., Nainian S.M.M, Ahmadi G., Two phase Analysis of Heat Transfer and Dispersion of Nano Particles in a Microchannel, Proceedings of 2008 ASME Summer Heat Transfer Conference, August 10-14, Jacksonville, Florida USA, (2008) Hakak Khadem M., Shams M., and Hossainpour S., Direct simulation of roughness effects on rarefied and compressible flow at slip flow regime, International Communications in Heat and Mass Transfer, 36, 88-95, (2009) Hettiarachchi H.D.M., Golubovic M., Worek W. M., Minkowycz W.J. , Three-dimensional laminar slip-flow and heat transfer in a rectangular microchannel with constant wall temperature, International Journal of Heat and Mass Transfer, 51, 50885096,( (2008) Hung W. Ch., Ru Y., A numerical study for slip flow heat transfer, Applied Mathematics and Computation, 173, 12461264, (2006) Latif M. J., Effect of Rarefaction, Dissipation, and Accommodation Coefficients on Heat Transfer in Microcylindrical Couette Flow, Journal of Heat Transfer, Vol. 130, (2008) Martin M.J., Boyd L.D., Momentum and Heat Transfer in a Laminar Boundary Layer with Slip Flow, JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER, 20, 4, OctoberDecember, (2006) Morini G. L., Lorenzini M., Spiga M., "A criterion for experimental validation of slip-flow models for incompressible rarefied gases through microchannels", Microfluid Nanofluid, 1, 190196, (2005) Struchtrup H., Thatcher T. and Torrilhon M., Couette flow solution for regularized 13 moment equations, Rarefied Gas Dynamics: 25-th International Symposium, Novosibirsk, (2007) Webb R., Next generation devices for electronic cooling with heat rejection to the air, Journal of Heat Transfer, 127, (2005) Zhang T.T., Jia L., Wang Z.C., Li C.W., Slip flow characteristics of compressible gaseous in microchannels, Energy Conversion and Management 50, 16761681, (2009)

rarefaction effect on heat transfer and fluid flow in microchannel

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