Proceedings of the Fifth International Conference on Nanochannels, Microchannels and Minichannels ICNMM2007 June 18-20, 2007, Puebla, Mexico
ICNMM2007-30048 Proceedings of ICNMM07 ASME 2007 5th International Conference on Nanochannels, Microchannels and Minichannels June 18-20, 2007, Puebla, Mexico
ICNMM2007-30048
TRANSPORT PHENOMENA THROUGH GASEOUS MIXTURES IN MICROCHANNELS
Felix Sharipov Deparatmento de F´ısica, Universidade Federal do Parana´ Caixa Postal 19044, Curitiba, Parana´ 81531-990 Brazil Email:
[email protected]
ABSTRACT In practice, one deals with gaseous mixtures more frequently than with a single gas. However, very few papers about the transport phenomena through a mixture of rarefied gases were published. The aim of this work is to present a general approach to calculations of mass, heat and momentum transfer through gaseous mixtures over the whole range of the gas rarefaction. Results on some classical problems such as slip coefficient, Poiseuille flow, Couette flow and heat transfer are given for a gaseous mixture. A comparison with results corresponding to a single gas is carried out. Such a comparison shows the peculiarities of the transport phenomena in mixtures.
q dimensionless heat flux q∗x peculiar heat flux R tube radius r position vector T temperature U velocity of plate u bulk velocity v molecular velocity v0 most probable speed w mean velocity of mixture δ rarefaction parameter Λi j kinetic coefficient µ viscosity ξ small linearization parameter Π dimensionless pressure tensor σP viscous slip coefficient
NOMENCLATURE C molar concentration fα velocity distribution function H distance between plates hα perturbation function JP volume flow rate JT heat flux JC diffusion flux k the Boltzmann constant Lˆ αβ linearized collision operator m molecular mass n number density P pressure Pxy pressure tensor Q collision integral
INTRODUCTION Gaseous mixtures are more complicated than a single gas and have many features. First, gaseous mixture flows are determined by more parameters than flows of a single gas. Besides the gas rarefaction, pressure and temperature gradients, the mixture flows depend on its concentration, species of the mixture components etc. Second, in a mixture an additional driving force, i.e. the concentration gradient, arises. Third, several new phenomena appear in a non-equilibrium mixture, e.g. thermal diffusion, barodiffusion, diffusion thermoeffect, diffusion baroeffect, etc. 1
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where ξ is a small parameter,
In the present paper, a general approach to numerical calculations of gaseous mixtures over the whole range of the Knudsen number is presented. Such a situation is frequently realized in microchannels. To describe a gaseous mixture having N components we have to consider N distribution functions fα (t, r, v), 1 ≤ α ≤ N, which obey the following system of the Boltzmann equations ∂ fα ∂ fα + vα · = ∂t ∂r
M
∑
j=1
Z
w( fα′ fβ′ − fα fβ ) dv′α dv′β dvβ ,
fαM (r, v) = n
α
and m =
1 n α mα , n∑ α
(1) vα ·
Lˆ αβ h =
Z
w fβM h′α + h′β − hα − hβ dv′α dv′β dvβ .
(5)
(6)
To calculate a gas flow for intermediate Knudsen numbers the exact Boltzmann equation should be solved. This equation provides reliable numerical data, but it requires great computational efforts. Although, nowadays it is possible to calculate rarefied gas flows applying the exact Boltzmann equation, the model equations continue to be good tools for practical calculations, because they allow us to reduce essentially the computational efforts. The idea of the model equations is to substitute the exact expression of the collision integral (6) by a simple expression satisfying the main properties of the former. The only question is that: What model equation must be applied to obtain reliable results? An appropriate model must satisfy the following conditions: (i) Conservation laws, i.e.
(2)
2
∑
β=1
Z
ψ(vα ) Lˆ αβ h dvα = 0,
(7)
where ψ(vα ) is the collision invariant, i.e. ψ = 1, mvα , 12 mv2α . (ii) H-theorem, i.e. 2
∑
β=1
Linearized kinetic equation Here, we consider only stationary and weakly nonequilibrium flows so that the distribution function does not depend on time and the Boltzmann equation can be linearized by the standard manner, i.e. the distribution function is represented as |ξ| ≪ 1,
M ∂hα = ∑ Lˆ αβ h, ∂r β=1
where Lˆ αβ h is the linearized collisions operator between species α and β, which in quite general form reads
respectively. The aim of the present work is to show that such an approach works only in the hydrodynamic regime, while in the transitional regime the kinetic equation (1) must be solved. However, for a mixture with a small difference of the molecular masses some quantities can be calculated on the basis of results corresponding to a single gas.
fα (r, v) = fαM (r, v)[1 + ξ hα (r, v)],
(4)
is the Maxwellian and hα = hα (r′ , v′ ) is a perturbation function. Substituting Eq.(3) into the Boltzmann equation (1) we obtain the system of the two linearized Boltzmann equations
where 1 ≤ α ≤ N, vα is a molecular velocity of species α, r is a position vector, t is the time and w(vα , vβ ; v′α , v′β ) is the probability density that two molecules having the velocities v′α and v′β will have the velocities vα and vβ , respectively, after a binary collision between them. One can see that the computational efforts drastically increase if we replace a single gas by a gaseous mixture. So, it is very attractive to use numerical results on a single gas for a gaseous mixture. It is obvious that the cross phenomena mentioned above cannot be described in the frame of the single gas theory, while the direct effects such as the mass flow caused by a pressure gradient and heat flux caused by a temperature gradient can be calculated from numerical data on the corresponding single gas flows. It is possible to offer the following approaches to describe a gaseous mixture based on the data obtained for a single gas. We substitute a gaseous mixture by a single gas having the mean molecular mass. This means, if nα is the number density and mα is the molecular mass of species α, we consider that the ”single” gas has the number density and the molecular mass equal to n = ∑ nα ,
m 3/2 mα v2α α exp − 2πkT 2kT
Z
h(vα ) Lˆ αβ h dvα ≤ 0,
(8)
(iii) and it must provide correct expressions of all transport coefficients, i.e. viscosity µ, thermal conductivity κ, diffusion D 12 , and thermal diffusion ratio aT . Many models were proposed for a gaseous mixture [1–8]. Among them the model satisfying all above mentioned conditions is that proposed by McCormack [4], which is used in the present work.
(3) 2
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Longitudinal se tion of tube I
1.2
σ
y
P C T
II
x
0
I
P
P C T
0
z ` 0
I
II
II
1.1
Cross se tion of tube
y
0
x
0
1 0
Figure 1.
0.2
0.4
Viscous slip coefficient
σP
C
0.6
0.8
1
Figure 2.
Velocity slip coefficient To solve the Navier-Stokes equation, usually, the non-slip boundary condition for the gas velocity on a solid surface is assumed. It is valid when the Knudsen number is so small that the gas rarefaction can be neglected. However, for moderately small Knudsen numbers the gas rarefaction can be taken into account via the velocity slip boundary condition [9–12], which reads µ = σP P
2kT m
1/2
∂ut′ , ∂x′n
n1 , n1 + n2
Scheme of the flow and coordinates
composition significantly and reaches the value σP = 1.2 at C = 0.8. Note, the mean molecular mass m in Eq.(9) also depends on the chemical composition.
Gas flow through a long tube Consider two reservoirs containing the same binary gaseous mixture and connected by a tube of length ℓ and radius R as is shown in Fig.2. Let PI , TI and CI be the pressure, temperature and concentration, respectively, of the mixture in the left container and PII , TII and CII be the pressure, temperature and concentration, respectively, in the right one. Let us denote the longitudinal gradients of pressure, concentration and temperature as
(9)
where ut′ is the tangential velocity of the gas on a solid surface, x′n is the normal coordinate directed to the gas, µ is the gas viscosity, P is the local pressure, T is the local temperature, m is the molecular mass of the gas and k is the Boltzmann constant. The dimensionless quantity σP is the viscous slip coefficient. So, the boundary condition (9) means that the gas velocity is not equal to zero on a solid boundary but its tangential component depends on the velocity profile near the surface. Numerical calculations of the slip coefficient for the mixture He-Ar were carried out in Ref. [13] and for a single gas in Ref. [14]. Note, for a single gas σP = 1.018. A dependence of σP on the molar concentration is presented in Fig.1. Here, the molar concentration of mixture C is defined as: C=
0
R
for mixture He-Ar vs concentration
C
ut′
z
ξP =
R ∂P , P ∂x
ξT =
R ∂T , T ∂x
ξC =
R ∂C , C ∂x
(11)
respectively. Here, P, T and C are pressure, temperature and concentration of mixture in a given cross section. Since we assumed ℓ >> R, these gradients are very small |ξP | ≪ 1;
|ξT | ≪ 1;
|ξC | ≪ 1
(12)
and can be used as the small parameter of the linearization. Moreover, the assumption ℓ ≫ R allows us to neglect the end effects and to consider only the longitudinal components of the heat flow vector q′ and of the hydrodynamic (bulk) velocity u′ of the mixture.
(10)
where nα (α = 1, 2) is the number density of gaseous species α. It can be seen that the slip coefficient σP depends on the chemical 3
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The following rates through a cross section of the tube are defined volume flow rate JP = −2πn
Z R 0
3
Λ
PP
2.5
wx rdr,
(13) 2
the heat flux JT = −
2π kT
Z R 0
q∗x rdr,
1.5
(14)
1 0.001
the diffusion flux JC = −2πn1
Z R 0
(u1x − u2x)rdr,
and q∗x is so called the peculiar heat flow vector defined as (17)
(18)
PR , µv0
v0 =
2kT m
1/2
.
ΛPP
vs δ: solid line - single gas, circles - mixture
" # 8 ∆m 2 3 ΛPP = √ 1 + C(1 − C) 8 m1 3 π
at δ = 0.
at ∆m ≪ m1 , (22)
i.e. the deviation from the single gas value is always positive and has the order of the square of the relative difference of the molecular masses.
(19)
Numerical calculations for the mixture He-Ar were carried out in Ref. [15] and for a single gas in Ref. [16]. A comparison between the data for the coefficient ΛPP is presented in Fig.3. Note the rarefaction parameter is defined as δ=
10
(21) This equation can be used to√estimate the maximum deviation from the value of ΛPP = 8/(3 π) corresponding to a single gas. If the molecular masses are close to each other, i.e. m2 = m1 + ∆m and ∆m ≪ m1 , then Eq.(21) is reduced to
All kinetic coefficients Λ′i j have the same dimension and for the result presentation it is convenient to use the reduced kinetic coefficient defined as 2 m 1/2 ′ Λi j . Λi j = nπR2 2kT
1
= 0.5
" 1/2 # 8 m m 1/2 ΛPP = √ C + (1 − C) m1 m2 3 π
Since the gradients (11) are small, the fluxes (13)-(15) can be written as ′ ′ ′ JP ΛPP ΛPT ΛPC ξP JT = Λ′TP Λ′TT Λ′TC ξT . JC Λ′CP Λ′CT Λ′CC ξC
δ
It can be seen that in the transition (δ ∼ 1) and free molecular regimes the chemical composition affects significantly the flow rate. In the free molecular regime (δ = 0) the kinetic coefficient ΛPP can be obtained analytically
(16)
Coefficient
He-Ar with C
where wx is the x component of the mean velocity of the mixture
5 q∗x = q′x − P(wx − ux ), 2
0.1
(15) Figure 3.
1 wx = (n1 u1x + n2 u2x ) n
0.01
Couette flow Let us consider a binary gaseous mixture between two parallel plates fixed at x′ = ±H/2. The equilibrium state is perturbed by a motion of the plates in the y′ direction with velocities ±U/2, respectively. Thus, H is the distance between the plates and U is their relative velocity. We assume that the plate velocities are small compared with the characteristic molecular velocity of the
(20) 4
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0.3
1
Π
q 0.8 0.2 0.6 0.4 0.1 0.2 0 0.01
0.1
1
δ
0 0.01
10
Figure 4. Pressure tensor vs δ: solid line - single gas, circles - mixture
Figure 5.
He-Ar with C = 0.5
with C
mixture, i.e U ≪ v0 . Thus, the ratio ξ = U/v0 is used as the small linearization parameter. We are interested in the pressure tensor Pxy between the plates. The results are presented in terms of dimensionless tensor introduced as Π=
Pxy v0 . 2P0 U
(24)
Numerical calculations for the mixture He-Ar were carried out in Ref. [17] and for a single gas in Ref. [18]. A comparison between these data is presented in Fig.4. It can be seen that like the previous case in the transition (δ ∼ 1) and free molecular regimes the chemical composition affects significantly the pressure tensor. The analytical expression of the pressure tensor in the free molecular regime reads m 1/2 1 m1 1/2 2 + (1 − C) Π= √ C m m 2 π
Heat flux vs δ: solid line - single gas, circles - mixture He-Ar
∆T . T0
(27)
Note, the dimensionless heat flux q is introduced by such way that it coincides with that for a single gas in the limit cases C0 = 0 and C0 = 1. Numerical calculations for the mixture He-Ar and for a single gas were carried out in Ref. [19]. A comparison between these data is presented in Fig.5. Like both previous cases, in the transition (δ ∼ 1) and free molecular regimes the chemical composition affects significantly the heat flux. In the free molecular regime (δ = 0) the heat flux q is ob-
at δ = 0. (25)
at ∆m ≪ m1 .
10
= 0.5
q′x = q P0 v0
In case of mixture with a small difference of the molecular masses Eq.(25) is simplified as " # 1 ∆m 2 1 Π = √ 1 − (1 − C)C 8 m1 2 π
δ
Heat transfer Consider a binary gaseous mixture confined between two parallel plates fixed at x′ = ±H/2. Let P0 , T0 and C0 be equilibrium pressure, temperature and molar concentration of the mixture, respectively. The equilibrium state is perturbed by small deviations of the temperatures of the plates, i.e. the left plate (x′ = −H/2) temperature is T0 + ∆T /2, while that of the right plate (x′ = H/2) is T0 − ∆T /2, where ∆T ≪ T0 . The heat flux q′x between the plates over the whole range of the gas rarefaction is calculated. The rarefaction parameter δ is defined by Eq.(24). The results are given in terms of the dimensionless heat flux q related to the dimension one q′x as
(23)
P0 H . µ v0
1
It can√be seen the deviation from the single gas value, i.e. Π = 1/(2 π), is always negative and it is proportional to (∆m/m1 )2 .
In this case the gas rarefaction δ is defined as δ=
0.1
(26) 5
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tained analytically " 1/2 # m 1 m 1/2 + (1 − C) q= √ C m1 m2 π
[4] McCormack, F. J., 1973. “Construction of linearized kinetic models for gaseous mixture and molecular gases.”. Phys. Fluids, 16, pp. 2095–2105. [5] Garz´o, V., Santos, A., and Brey, J. J., 1989. “A kinetic model for a multicomponent gas”. Phys. Fluids A, 1(2), pp. 380–383. [6] Garz´o, V., and Lopez de Haro, M., 1992. “Kinetic models for diffusion in shear flow”. Phys. Fluids A, 4(5), pp. 1057– 1069. [7] Marin, C., and Garz´o, V., 1997. “Kinetic model for momentum transport in a binary mixture”. In Rarefied Gas Dynamics, C. Shen, ed., 20st Int. Symp., China, 1996, Pekin University Press, pp. 91–96. [8] Grigoryev, Y. N., and Meleshko, S. V., 1998. “BobylevKrook-Wu models for multicomponent gas mixture”. Physical Review Letters, 81(1), pp. 93–95. [9] Kennard, E. H., 1938. Kinetic Theory of Gases. McGrawHill Book Company, Inc., New York. [10] Kogan, M. N., 1969. Rarefied Gas Dynamics. Plenum, New York. [11] Ferziger, J. H., and Kaper, H. G., 1972. Mathematical Theory of Transport Processes in Gases. North-Holland Publishing Company, Amsterdam. [12] Cercignani, C., 1988. The Boltzmann Equation and its Application. Springer-Verlag, New York. [13] Sharipov, F., and Kalempa, D., 2003. “Velocity slip and temperature jump coefficients for gaseous mixtures. I. Viscous slip coefficient”. Phys. Fluids, 15(6), pp. 1800–1806. [14] Sharipov, F., 2003. “Application of the Cercignani-Lampis scattering kernel to calculations of rarefied gas flows. II. Slip and jump coefficients”. Eur. J. Mech. B / Fluids, 22, pp. 133–143. [15] Sharipov, F., and Kalempa, D., 2002. “Gaseous mixture flow through a long tube at arbitrary Knudsen number”. J. Vac. Sci. Technol. A, 20(3), pp. 814–822. [16] Sharipov, F., and Seleznev, V., 1998. “Data on internal rarefied gas flows”. J. Phys. Chem. Ref. Data, 27(3), pp. 657– 706. [17] Sharipov, F., Cumin, L. M. G., and Kalempa, D., 2004. “Plane Couette flow of binary gaseous mixture in the whole range of the Knudsen number”. Eur. J. Mech. B/Fluids, 23, pp. 899–906. [18] Cercignani, C., and Pagani, C. D., 1966. “Variational approach to boundary value problems in kinetic theory”. Phys. Fluids, 9(6), pp. 1167–1173. [19] Sharipov, F., Cumin, L. M. G., and Kalempa, D., 2007. “Heat flux through a binary gaseous mixture over the whole range of the knudsen number”. accepted in Physica A.
at δ = 0. (28)
If the molecular masses are close to each other, i.e. m2 = m1 + ∆m and ∆m ≪ m1 , then Eq.(28) is reduced to " # 3 1 ∆m 2 q = √ 1 + C(1 − C) 8 m1 π
at ∆m ≪ m1 ,
(29)
i.e. like the flow rate, the deviation of the heat flux from the single gas value is positive and has the order of the square of the relative difference of the molecular masses.
Conclusion Several classical problems of gas dynamics were considered on the basis of the Boltzmann kinetic equation for a gaseous mixture. Numerical results on the viscous slip coefficient, flow rate, pressure tensor and heat flux were analyzed. It was shown that in the transition regime, i.e. at δ ∼ 1, and in the free molecular one (δ = 0) the approach based on results obtained for a single gas does not work if the molecular masses of species are quite different. It was shown that the deviations of the flow rate, pressure tensor and heat flux from their values for a single gas in the free molecular regime have the order of the square of the relative difference of the molecular masses. The deviations of the mass and heat fluxes are positive, while that for the pressure tensor is negative. Thus, if the molecular mass difference is small some quantities can be calculated on the basis of results corresponding to a single gas. However, reliable results for a mixture with a significant difference of the molecular masses must be obtained on the basis of the kinetic equation, which is solved for each specific mixture.
ACKNOWLEDGMENT The authors acknowledge the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq, Brazil) for the support of his research.
REFERENCES [1] Sirovich, L., 1962. “Kinetic modeling of gas mixture”. Phys. Fluids, 5(8), pp. 908–918. [2] Morse, T. F., 1964. “Kinetic model equations for a gas mixture”. Phys. Fluids, 7(12), pp. 2012–2013. [3] Hamel, B. B., 1965. “Kinetic model for binary gas mixture”. Phys. Fluids, 8(3), pp. 418–425. 6
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