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Asymptotic Modeling of Flows in Micro-Channel by Using ...

Asymptotic Modeling of Flows in Micro-Channel by Using ...

conditions

conditions defined by Bird [9] are introduced. The model of diffuse reflection with perfect accommodation is applied to describe the interaction of molecules with the channel walls and the Variable Hard Sphere (VHS) model is chosen for the intermolecular collisions. On the micro-channel walls, the temperature T depends on the longitudinal variable with a constant gradient and it varies from T in to T out . The computation is performed with the following numerical values: P out = 100 kPa , T out = 300 K , 110 kPa ≤ P in ≤ 200 kPa , 300 K ≤T in ≤ 400 K (simulations n°1 to n°9 on Table 1). The number of test molecules is about 4 5 7 . 2× 10 , and the number of samples about 7 . 5× 10 . The noise levels for pressure and temperature are deduced from Hadjiconstantinou et al. formulas [10] and are of order 0 .25% and 0 .15% respectively. The following notations are introduced: P i and T i for pressure and temperature values at the centre of the entrance section of the micro-channel ( x = 0 µm ), P o and T o at the centre of the exit section ( x = 10 µm ). The DSMC simulations give, among other output quantities, the values P i , T i , , P o , T o , and the Mach number and the Knudsen number at the centre of the micro-channel ( x = 5 µm , y = 0 µm ). The Knudsen number Kn varies from 0.035 to 0.066 and Mach number M from 0.0014 to 0.22 (Table 1). In addition, in Table 1, we have given three results from literature. The first one, reference [2], corresponds to a numerical result for a micro-channel with l = 4.52 µm and h = 0.226 µm . The second [6] is a simulation result in a channel with l = 10 µm and h =1 µm . Finally, the third one is an experimental result [11] where the dimensions of the channel are l = 5000 µm and h = 4.48 µm . In these last results, the Mach number is small but the Knudsen number is not so small. In short, Mach number may be considered as small and Knudsen number as small or moderate. w Entrance area Solid wall T w Exit area T out T in y h’ 0 h x P in Symmetry axis l ' = 7 µm l = 10 µm l ' = 7 µm P out FIGURE 1. Micro-channel geometry for the DSMC simulations. TABLE 1. Numerical values on the left and right ends and at the centre of the micro-channel P in T in P out T out P i T i P o T o kPa K kPa K kPa K kPa K Kn M Simulation n°1 130 300 100 300 131 299 99 301 0.047 0.075 Simulation n°2 140 300 100 300 141 299 98.8 301 0.045 0.097 Simulation n°3 160 300 100 300 162 298 97.5 302 0.041 0.14 Simulation n°4 180 300 1000 300 182 298 96.5 302 0.038 0.18 Simulation n°5 200 300 100 300 202.5 298 96 302 0.035 0.22 Simulation n°6 130 400 100 300 118 392 101 303 0.061 0.04 Simulation n°7 120 400 100 300 109 391 101 302 0.064 0.02 Simulation n°8 111 400 100 300 101 392 101 302 0.066 0.0014 Simulation n°9 110 400 100 300 100 392 101 302 0.066 0.0033 Karniadakis et al. [2] 360 300 100 300 0.3 0.007 Aktas et al. [6] 133 300 100 300 1.1 0.04 Colin et al. [11] 252 102.6 0.1

ASYMPTOTIC MODELING OF GAS FLOWS IN MICRO-CHANNELS A frame of reference ( O ; x, y ) is attached to the micro-channel with length l and width h (Fig. 1). The walls located at y = ± h / 2 are at rest with the same temperature distribution T w ( x ) . The upstream ( x = 0 ) and downstream ( x = l ) boundary conditions for the pressure and the temperature will be specified further. The Navier- Stokes equations (1) to (4) are written for an ideal gas with shear viscosity µ and thermal conductivity k . The pressure, volumetric mass, temperature, longitudinal and transversal velocities are denoted p , ρ , T , u and v . The ideal gas law is p = rρ T with r = c p − cv where c p and c v are the heat capacities at constant pressure and constant volume. All these physical coefficients µ , k , partial derivatives in the x and y directions. ( u) ( ρ v) = 0 x + y c p and c v are constant. The indexes x and y denote the ρ (1) ⎛ 4 1 ⎞ ( u u v u ) + p − µ ⎜ u + u + v ⎟ = 0 ρ x + y x xx yy xy (2) ⎝ 3 3 ⎠ ⎛ 4 1 ⎞ ( u v v v ) + p − µ ⎜ v + v + u ⎟ = 0 ρ x + y y yy xx xy (3) ⎝ 3 3 ⎠ ⎛ 4 2 4 2 4 2 2 ⎞ ( u T v T ) + p ( u + v ) − ⎜ u + v − u v + u + v + 2 u v ⎟ − k ( T + T ) = 0 ρ c v x + y x y µ x y x y y x y x xx yy (4) ⎝ 3 3 3 ⎠ The ratio γ of heat capacities, the Prandtl number Pr and the local mean free path λ are classically defined by: c p µ c p µ π γ = , Pr = , λ = (5) c k ρ 2 r T v The order one boundary conditions on the two walls y = ± h / 2 [1,2,11] are now written: { u} = λ + , { v} = 0, { T} y=± h / 2 ⎧ ∂ u 3 µ ∂ T ⎫ ⎧ 2γ λ ∂ T ⎫ ⎨m ⎬ y=± h / 2 y=± h / 2 = Tw m ⎨ ⎬ (6) ⎩ ∂ y 4 ρ T ∂ x ⎭ ⎩ γ + 1 Pr ∂ y ⎭ y=± h / 2 y=± h / 2 The characteristic scales for the longitudinal and transversal lengths, the longitudinal and transversal velocities, the pressure, the temperature and the volumetric mass are respectively denoted l and h , U c and V c , P c , T c and ρ c (with Pc = rρcTc ). Then, we write down the balance equations (1) to (4) with the dimensionless quantities x , y , u , v , p , ρ and T defined as x = l x , y = h y , u = U u , v = V v , p = P p , ρ = ρ ρ and T = T T . As usual for gas flows in micro-channels, the small parameter ε = h / l is introduced, as well as the Reynolds number Re , the Knudsen number Kn and the Mach number M : c c c c c h ε = , l V c ρ , c M U c U c γ pc = , 1 µ π Kn = , h ρ 2 r c T c ρ c U c h M π γ Re = = (7) µ Kn 2 Due to the Principe of Least Degeneracy [12], we must keep the two terms in the dimensionless mass law; we α β then obtain Vc = ε U c . In order to look for the degenerate systems, we put: Kn = ε , M = ε . The comparison of the magnitude orders of the different terms in the dimensionless balance equations for momentum and energy, gives one degeneracy with α = 0 and β = 1, among others [7]. This one corresponds to small Mach numbers and to Knudsen numbers of order of unity. From now on, we put M = ε M 0 and Kn = Kn0 where both M 0 and Kn 0 are of order of unity. For the sake of simplicity, we omit the bars to denote the dimensionless variables and we obtain: ( u) ( ρ v) = 0 ρ (8) x + y

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