Transport Phenomena in Gases
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4.1 Transport Coefficients of Gases Transport coefficients characterize the connection between fluxes and weak gradients of some quantities. The diffusion coefficient D for atoms or molecules of a gas is the proportionality coefficient between the flux j of these atoms and the gradient of their concentration c, that is, j = −DN∇c,
(4.1)
where N is the total number density of gas atoms or molecules. If the concentration of atoms of a given type is small (ci 1), i.e. this component is a small admixture to a gas, this formula may be represented in the form j = −Di ∇Ni ,
(4.2)
where Ni is the number density of atoms of a given component. The thermal conductivity of a gas κ is defined as the proportionality coefficient between the thermal flux q and the temperature gradient ∇T as q = −κ∇T .
(4.3)
The viscosity coefficient η is the proportionality coefficient between the friction force F per area unit of a moving gas and the gradient of a gas velocity. In the frame of reference where the direction of the average gas velocity w is x and the average velocity varies in the direction z, the friction force is proportional to the quantity ∂wx /∂z and acts on the surface xy. Correspondingly, the viscosity coefficient is defined as ∂wx , (4.4) F = −η ∂z and this definition holds true both for a gas and for a liquid. Transport coefficients in a gas are determined by collision processes. Because the elastic cross section in an atomic gas or in a molecular gas at not high temperatures is significantly larger than that of inelastic processes, the transport coefficients in a gas are expressed through average cross sections of elastic collisions. The simple connection between these quantities takes place in the Chapman–Enskog ap-
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4 Transport Phenomena in Gases
proximation [185, 186], which corresponds to an expansion over a small numerical parameter. The accuracy of the first Chapman–Enskog approximation is several percent. The diffusion coefficient D of a test atom or molecule in a gas is given in the first Chapman–Enskog approximation [185, 186] √ 1 ∞ −t 2 ∗ μg 2 3 πT e t σ (t) dt, t = , σ ≡ Ω (1,1) (T ) = . (4.5) D= √ 2 0 2T 8N 2μσ Here T is the gas temperature expressed in energetic units, N is the number density of gas atoms, μ is the reduced mass of a test atom and a gas atom, σ ∗ (g) is the diffusion cross section of the collision of two atomic particles at a relative velocity g of collision, and the brackets mean an average over atom velocities with the Maxwell distribution function. The thermal conductivity coefficient κ in the first Chapman–Enskog approximation is given by [185, 186] √ 25 πT (4.6) κ= √ , 32σ2 m where m is the mass of an atom or molecule, an average of the cross section of elastic collision is determined by the following formula: ∞ (2,2) σ2 ≡ Ω (T ) = t 2 exp(−t)σ (2) (t) dt, 0 μg 2 1 − cos2 ϑ dσ. (4.7) t= , σ (2) (t) = 2T The viscosity coefficient η of the first Chapman–Enskog approximation is determined by the formula [185, 186] √ 5 πT m , (4.8) η= 24σ2 where the average cross sect
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