HEAT TRANSFER BETWEEN PARALLEL PLATES VIA ... - CiteSeerX

HEAT TRANSFER BETWEEN PARALLEL PLATES VIA KINETIC THEORY IN THE WHOLE RANGE OF THE KNUDSEN NUMBER S. Pantazis, D. Valougeorgis Department of Mechanical & Industrial Engineering, University of Thessaly, Volos, Greece

Abstract The conductive heat transfer problem through a rarefied gas confined between parallel plates maintained at different temperatures is investigated. The formulation is based on the non-linear Sakhov kinetic model subject to Maxwell diffuse-specular reflection and it is solved by the discrete velocity method. Results are presented for the heat flux, the density and temperature profiles as well as for the particle distribution function in the whole range of the Knudsen number and for various temperature differences. Some interesting physical findings are observed at large temperature ratios. Also the extension of validity of the linearized kinetic analysis is discussed. The accuracy of the results is validated in several ways including the recovery of the analytical solutions at the free molecular and continuum limits and successful comparison with corresponding linearized results at small temperature ratios.

1 Introduction In many occasions the design and optimization of micro-devices include the study of heat transfer effects through gases in non-equilibrium conditions. It is well known that Navier-Stokes-Fourier solutions, subject to velocity slip and temperature jump boundary conditions are valid for Knudsen numbers less than 0.1 ( Kn ≤ 0.1 ). When the flow and transport phenomena are far from local equilibrium ( Kn > 0.1 ), which is the case in several miniaturized systems, alternative approaches such as molecular dynamics, kinetic theory (Cercignani, 1988) and DSMC (Bird, 1994) must be implemented. Over the years, kinetic type solutions, based on the Boltzmann equation or suitable kinetic models, have been shown to be very reliable and computationally efficient in handling nonequilibrium (or otherwise rarefied) gas systems (Sharipov & Seleznev, 1998). Here, we apply a kinetic formulation and solution to a very simple physical problem, that of conductive heat transfer in a rarefied gas placed between two parallel plates, which are maintained at different temperatures (Loyalka & Thomas, 1982, Sharipov et al., 2007). This problem has been studied extensively, obtaining very accurate solutions in the whole range of the Knudsen number. Even more, it has been used as a benchmark problem to judge the efficiency of several semianalytical and numerical methods solving kinetic equations (Bassanini, et al., 1967; Thomas, et al., 1973; Thomas & Valougeorgis, 1985; Clause & Mareschal, 1988). However, most of this work is based on linearized kinetic solutions under the restrictive assumption of small temperature differences between the two walls. Only few results exist in the literature for the more general case of arbitrary large temperature ratios between the plates and almost exclusively all of them are based on the BGK model with purely diffuse reflection at the walls (Willis, 1963). In the present work the non-linear Sakhov model (Sakhov, 1968) subject to Maxwell diffuse-specular boundary conditions is applied and it is solved by the discrete velocity method (Naris et al., 2005, Naris & Valougeorgis, 2005). Numerical results are presented for the heat flux, the density and temperature profiles as well as for the distribution function in the whole range of the Knudsen number from the free molecular through the transition and slip regime all the way up to the continuum limit. In

addition, the dependency of the results on the plate temperature ratio is examined in detail and some interesting physical findings are observed at large temperature differences. Also, the extension of validity of the linearized kinetic analysis based on small differences is discussed.

2 Formulation The task is to describe the state of a stationary monoatomic gas confined between two infinite parallel plates fixed at yˆ = ∓ H / 2 , maintained at constant temperatures Ti = T0 ± ΔT / 2 , i = 1, 2 respectively, while T0 is a reference temperature. Then, the temperature ratio is defined as

T1 / T2 = (1 + β ) / (1 − β ) , where β = ΔT / 2T0 . Since a kinetic approach is applied, the basic

unknown is the distribution function f = f ( yˆ , ξ ) , which for this specific problem depends on the spatial variable yˆ denoting the vertical distance between the plates and the molecular velocity ξ = (ξ x , ξ y , ξ z ) . The distribution function obeys the Sakhov model equation

P ⎪⎧ ∂f =− ⎨f − fM μ⎪ ∂yˆ ⎩ where

ξy

⎡ ⎛ mξ 2 5 ⎞ ⎤ ⎫⎪ 2m ˆ ξ q − ⎟⎥ ⎬ ⎢1 + y⎜ 2 ⎢⎣ 15n ( k BT ) ⎝ 2k BT 2 ⎠ ⎥⎦ ⎪⎭

(1)

3/ 2

⎛ m ⎞ ⎡ mξ 2 ⎤ f = n⎜ exp ⎟ ⎢− ⎥ ⎝ 2π k BT ⎠ ⎣ 2 k BT ⎦ is the local Maxwellian distribution, n ( yˆ ) = ∫ ∫ ∫ fd ξ x d ξ y d ξ z M

T ( yˆ ) =

m 3nk B

∫ ∫ ∫ξ

2

f dξ x dξ y dξ z

(2)

(3) (4)

and m (5) ξ 2ξ y d ξ x d ξ y d ξ z ∫ ∫ ∫ 2 are the number density, temperature and heat flux respectively. By applying the conservation of energy it is easily deduced that the heat flux is constant at any position between the plates ( qˆ ( yˆ ) = const. ). Finally, m and k B denote the molecular mass and the Boltzmann constant qˆ ( yˆ ) =

respectively. At this stage it is convenient to introduce the dimensionless quantities fυ3 yˆ qˆ ξ n T and q = y= , c= , g= 0 , ρ= ,τ = n0 P0υ0 H n0 T0 υ0

(6)

where H is the distance between the plates, υ0 = 2k BT0 / m is the most probable molecular 1/ 2

velocity, n0 =

∫ n ( y ) dy

is an average density which specifies the density level, g = g ( y, c ) is the

−1/ 2

reduced distribution function, ρ , τ and q are the dimensionless number density, temperature and heat flux respectively, while P0 = n0 k BT0 is a reference pressure. Even more,

δ0 =

P0 H

=

π 1

(7) μ0υ0 2 Kn0 is the reference rarefaction parameter and it is proportional to the inverse reference Knudsen number, while μ0 is the gas viscocity at temperature T0 . Then, equation (1) becomes

⎧⎪ ⎡ 4 qc y ⎛ c 2 5 ⎞ ⎤ ⎫⎪ ∂g M cy = −δ 0 ρ τ ⎨ g − g ⎢1 + − ⎟⎥ ⎬ 2 ⎜ ∂y ⎣ 15 ρτ ⎝ τ 2 ⎠ ⎦ ⎭⎪ ⎩⎪ where g M = ρ / (πτ )

3/ 2

(8)

exp ⎡⎣ −c 2 / τ ⎤⎦ is the local Maxwellian in terms of dimensionless quantities.

Since the problem is one-dimensional in the physical space the two components of the molecular velocity in the x and z directions may be eliminated by a projection procedure. To eliminate the independent variables cx and cz we define

φ ( y, c y ) = ∫ ∫ g ( y, c ) dcx dcz

and ψ ( y, c y ) = ∫ ∫ ( cx2 + cz2 ) g ( y, c ) dcx dcz

(9) (10)

and operate accordingly on equation (8) to reduce after some routine manipulation the coupled set of non-linear integro-differential equations ∂φ cy = −δ 0 ρ τ (φ − φ M ) (11) ∂y and ∂ψ cy = −δ 0 ρ τ (ψ −ψ M ) . (12) ∂y Equations (11) and (12) must be solved for the unknown reduced distributions φ and ψ , which they depend only on two independent variables reducing significantly the required computational effort. Also, the functions φ M and ψ M are

φ

M

2 ⎡ ρ 4 qc y ⎛ c y 3 ⎞ ⎤ 2 =⎢ + ⎜ − ⎟⎟ ⎥ exp ⎡⎣ −c y / τ ⎤⎦ 5/ 2 ⎜ ⎣⎢ πτ 15 πτ ⎝ τ 2 ⎠ ⎥⎦

(13)

and ⎡ ρ τ 4 qc y ⎛ c y2 3 ⎞ ⎤ 2 ψ =⎢ + ⎜ − ⎟⎟ ⎥ exp ⎡⎣ −c y / τ ⎤⎦ . 3/ 2 ⎜ ⎣⎢ π 15 πτ ⎝ τ 2 ⎠ ⎦⎥ Finally, the dimensionless macroscopic quantities in terms of φ and ψ are M

ρ = ∫ φ dc y

τ=

2 3ρ

(14)

(15)

∫ (ψ + c φ ) dc 2 y

y

and q = q y = ∫ c y (ψ + c y2φ ) dc y .

(16)

(17)

The interaction between the particles and the walls is modelled according to Maxwell’s diffusespecular boundary conditions: 3/ 2 ⎡ ⎤ ⎛ m ⎞ ξ2 + − exp − fi ( ξ ) = α ni ⎜ (18) ⎢ ⎥ + (1 − α ) fi ( ξ ) , for ξ y 0 ⎟ 2 / k m T ( ) B i ⎝ 2π k BTi ⎠ ⎣ ⎦ In equation (18), the subscript i = 1, 2 denotes the quantities at the left and the right wall respectively, f i + and f i − denote departing and arriving particle distributions at the corresponding wall, 0 < α ≤ 1 is the accommodation coefficient ( a = 1 corresponds to purely diffuse reflection) and ni are two parameters to be specified by the condition of no penetration at each wall. By introducing the dimensionless quantities the corresponding boundary conditions become

ρi

⎡ c2 ⎤ − > exp (19) ⎢ − 1 ± β ⎥ + (1 − α ) gi ( c ) , for c y < 0 , 3/ 2 ⎡⎣π (1 ± β ) ⎤⎦ ⎣ ⎦ while after applying the projection procedure the following boundary conditions are deduced: ⎡ c y2 ⎤ ρi − φi+ ( c y ) = α exp (20) ⎢− ⎥ + (1 − α ) φi ( c y ) , for c y 0 , 1/ 2 ⎢⎣ 1 ± β ⎥⎦ ⎡⎣π (1 ± β ) ⎤⎦ gi+ ( c ) = α

1/ 2 ⎡ 1 ± β ) ρi ( =α exp −

c y2 ⎤ − ψ ( cy ) ⎢ ⎥ + (1 − α )ψ i ( c y ) , for c y 0 , ± 1 π 1/ 2 β ⎣⎢ ⎦⎥ By applying the no penetration condition at the walls, the quantities ρi are specified by + i

ρi =

2 π 1± β

∫

∞

0

φi− c y dc y .

(21)

(22)

The problem is described by the kinetic equations (11) and (12), coupled by the moments (15)-(17) and subject to boundary conditions (20) and (21), where the parameters ρi at the two walls are given by (22). It is seen that the problem is specified in terms of three dimensionless parameters, namely the rarefaction parameter δ 0 , the temperature ratio β and the accommodation coefficient α . In the next section the implemented computational scheme is described and then in the fourth section results are provided in terms of the three parameters.

3 Computational scheme The implemented computational scheme has been extensively applied to solve linear kinetic equations describing several non-equilibrium systems in a very efficient and accurate manner (Naris et al, 2004). Here, it is accordingly extended to the case of non-linear kinetic equations. The kinetic equations (11) and (12) are discretized in the molecular velocity and physical spaces. In particular, in the molecular velocity space the discretization is performed by the discrete velocity method, where the continuum spectrum c y ∈ ( −∞, ∞ ) is replaced by a suitable set of discrete velocities c ym , m = 1, 2,..., M . We choose c ym to be the roots of the Legendre polynomials of order M . By performing this discretization in the velocity space, equations (11) and (12) are reduced to a set of 2 x M first order ordinary differential equations. In the physical space the distance y ∈ [ −1/ 2,1/ 2] is divided in equal intervals and the discretization at each interval i = 1, 2,..., I is performed by the diamond-difference scheme (Lewis & Miller, 1984). This is a second order central difference scheme, which has been used in solving elliptic integro-differential equations. In addition, the macroscopic moments of the reduced distributions defined by equations (15)-(17) are estimated by a Gauss-Legendre quadrature. Then, the discretized problem is solved in an iterative manner consisting of two steps. In the first step, the kinetic equations are solved for the unknown distributions φim and ψ im assuming that the macroscopic quantities ρi , τ i and qi at the right hand side of the equations are known. In the second step, updated estimates of the macroscopic quantities are computed based on the moments of the distribution functions. The iterative procedure is ended when the convergence criterion applied on the macroscopic quantities is satisfied. It is important to note that at each iteration the system of algebraic equations is solved by a marching scheme and no matrix inversion is required. For each discrete velocity the distribution functions are computed at each node explicitly marching

through the physical domain. Following this procedure, supplemented by a reasonably dense grid and an adequately large set of discrete velocities we are able to obtain grid-independent results with modest computational effort.

4 Results The numerical results presented here have been obtained using I = 401 nodes and M = 96 discrete velocities. The convergence criterion has been set equal to 10−6 and the results are considered accurate up to at least three significant figures. In Table 1, numerical results for the dimensionless heat flux q defined by equation (17) have been tabulated for 0 ≤ δ ≤ 1.5 × 10 2 and for small, moderate and large values of the ratio T1 / T2 . It is seen that for δ ≤ 1.5 , as T1 / T2 is increased, the heat flux is also increased up to some value of T1 / T2 and then as T1 / T2 is further increased the heat flux is reduced. The ratio T1 / T2 at which the maximum heat flux is observed depends on δ . This is an unexpected behaviour of q in terms of T1 / T2 and it is not easily justified physically. For δ ≥ 15 , the heat flux in increased monotonically with T1 / T2 . Next, in terms of δ , it is seen that for T1 / T2 ≤ 7 the heat flux is decreased as δ is increased, i.e. as we are moving from the free molecular to the continuum regimes. This is expected and has been also observed in linear analysis with small temperature ratios. However here, for T1 / T2 ≥ 10 , as δ is increased, the heat flux is initially increased and then at some δ is reduced. It is noted that for δ = 0 and δ = 1.5 ⋅10 2 , which correspond to the free molecular and hydrodynamic limits, there is very good agreement with the corresponding analytical results presented by Willis (1963) and Sharipov et al. (2006). Results on the dimensionless heat flux, based on linear kinetic analysis, are presented in Table 2, for the same set of the parameters δ and T1 / T2 . It is seen that for small and moderate T1 / T2 there is a qualitative agreement between linear and non-linear results. Also, for small values of T1 / T2 there is good quantitative agreement. Of course, for moderate and large T1 / T2 there are discrepancies and as expected the linearized kinetic analysis is not capable of producing accurate results. In particular, at small δ and large T1 / T2 the linear results are erroneous. Therefore, linear analysis can not capture the non-monotonic behaviour of q at large temperature ratios, which has been previously observed. However, it may be argued that the range of applicability of the linear analysis is wider than expected. In Figure 1, the heat flux is plotted in terms of δ for various values of the accommodation coefficient α . It is seen that as the portion of specular reflection is increased (i.e. α is reduced) the heat flux is decreased. This becomes more evident at rarefied atmosphere (small δ ). Temperature profiles, based on linear and nonlinear analysis, are presented in Figure 2 for δ = 1.5 ⋅10−4 , 1.5 and δ = 1.5 ⋅10 2 , with T1 / T2 = 3 ( β = 0.5) . The linear results are always symmetric about y = 0 . This is not the case for the non-linear analysis. In the non-linear results the temperature jumps at the hot plate ( y = −1/ 2 ) are much larger than the corresponding ones at the cold plate ( y = 1/ 2 ) . These results are indicative for other values of δ and T1 / T2 .

Table 1: Non-linear dimensionless heat fluxes for various values of δ and T1 / T2 T1 1 + β = δ T2 1 − β 0.0 1.5(-4) 1.5(-1) 1.5 1.5(+1) 1.5(+2)

1.01 5.643(-3) 5.641(-3) 5.231(-3) 3.571(-3) 9.917(-4) 1.220(-4)

1.1 5.637(-2) 5.636(-2) 5.227(-2) 3.569(-2) 9.914(-3) 1.218(-3)

1.5 2.222(-1) 2.222(-1) 2.064(-1) 1.414(-1) 3.952(-2) 4.864(-3)

3 5.058(-1) 5.058(-1) 4.742(-1) 3.324(-1) 9.675(-2) 1.203(-2)

7 6.142(-1) 6.142(-1) 5.892(-1) 4.383(-1) 1.394(-1) 1.773(-2)

10 5.982(-1) 5.983(-1) 5.818(-1) 4.485(-1) 1.494(-1) 1.920(-2)

100 2.830(-1) 2.832(-1) 3.200(-1) 3.675(-1) 1.643(-1) 2.237(-2)

Table 2: Linear dimensionless heat fluxes for various values of δ and T1 / T2 T1 1 + β = δ T2 1 − β 0.0 1.5(-4) 1.5(-1) 1.5 1.5(+1) 1.5(+2)

1.01 5.642(-3) 5.641(-3) 5.231(-3) 3.571(-3) 9.917(-4) 1.218(-4)

1.1 5.642(-2) 5.641(-2) 5.231(-2) 3.571(-2) 9.917(-3) 1.218(-3)

1.5 2.257(-1) 2.257(-1) 2.093(-1) 1.428(-1) 3.967(-2) 4.873(-3)

3 5.642(-1) 5.641(-1) 5.231(-1) 3.571(-1) 9.917(-2) 1.218(-2)

7 8.463(-1) 8.462(-1) 7.847(-1) 5.356(-1) 1.487(-1) 1.827(-2)

10 9.230(-1) 9.229(-1) 8.558(-1) 5.842(-1) 1.622(-1) 1.993(-2)

100 1.106 1.106 1.025 6.999(-1) 1.944(-1) 2.388(-2)

Figure 1: Heat flux in terms of δ for various values of α and T1/T2=3 (β=0.5)

Figure 2: Temperature profiles for various δ, with T1/T2=3 (β=0.5)

Figure 3: Reduced distribution functions ϕ (left) and ψ (right) in terms of the molecular velocity c y at several locations between the plates for δ=1.5 and T1/T2=3 (β=0.5)

Finally, in Figure 3, typical profiles of the reduced distribution functions φ and ψ in terms of the molecular velocity c y are provided at y = ±1/ 2 and y = 0 for δ = 1.5 and T1 / T2 = 3 . The parts of the distribution for c y > 0 and c y < 0 correspond to particles moving from left to right (hot to cold) and vice versa respectively. The discontinuity at c y = 0 is evident and it is much stronger at the walls than in the center of the slab. Also, in general the discontinuities become stronger as δ is increased and/or T1 / T2 is increased.

References Bassanini, P., Cercignani, C. and Pagani, C.D., 1967, Comparison of kinetic theory analyses of linearized heat transfer between parallel plates, Int. J. Heat Mass Transfer, 10, 447-460 Bird, G. A., 1994, Molecular gas dynamics and the direct simulation of gas flows, Oxford University Press Cercignani, C., 1988, The Boltzmann Equation and its Application, Springer, New York Clause, P.J. and Mareschal, M., 1988, Heat transfer in a gas between parallel plates: Moment method and molecular dynamics, Physical Review A, 38, 8, 4241-4252 Lewis, E. E. and Miller W. F. Jr., 1984, Computational Methods of Neutron Transport Theory, Wiley, New York Loyalka S. K. and Thomas, J. R., 1982, Heat transfer in a rarefied gas enclosed between parallel plates: Role of boundary conditions, Physics of Fluids, 25, 7, 1162-1164 Naris, S., Valougeorgis, D., Sharipov, F. and Kalempa, D., 2004, Discrete velocity modelling of gaseous mixture flows in MEMS, Superlattices and Microstructures, 35, (3-6), 629-643 Naris, S. and Valougeorgis, D., 2005, The driven cavity flow over the whole range of the Knudsen number, Physics of Fluids, 17, 9, 907106.1-907106.12 Naris, S., Valougeorgis, D., Sharipov, F. and Kalempa, D., 2005, Pressure, temperature and density driven micro flows of gas mixtures in rectangular ducts, Physics of Fluids, 17 (10), 100607.1100607.12 Sharipov, F., Cumin, L.M.G. and Kalempa, D., 2007, Heat flux between parallel plates through a binary gaseous mixture over the whole range of the Knudsen number, Physica A, 378, 183-193 Sharipov F. and Seleznev V., 1998, Data on internal rarefied gas flows, J. Phys. Chem. Ref. Data, 27, 3, 657-706 Shakhov, E.M., 1968, Generalization of the Krook kinetic equation, Fluid Dyn., 3, 95 Thomas, J.R.Jr., Chang, T.S. and Siewert, C.E., 1973, Heat transfer between parallel plates with arbitrary surface accommodation, Physics of Fluids, 16, 12, 2116-2120 Thomas, J. R. and Valougeorgis, D., 1985, The FN method in kinetic theory: II. Heat transfer between parallel plates, Transport Theory and Statistical Physics, 14, 497-512 Willis, D.R., 1963, Heat transfer in a rarefied gas between parallel plates at large temperature ratios, Rarefied Gas Dynamics, 1, Acad. Press, New York-London