2. Consider the slow, steady flow of a uniform gas in a horizontal cylindrical capillary of radius R. A force balance yields the following expression for the velocity in the direction of the flow (x-direction): r² apg 4μ & +C₁ where C₁ is a constant of integration, is the applied pressure gradient, and r is the radial apg ax coordinate. dv dr where ß is known as the slip friction coefficient. It is well known that gas molecules may "slip" at the wall of the capillary, such that the velocity of the gas is non-zero. That is: Bv₂ = μ- (1) at r= R (2)/n(a) Use equations (1) and (2) to develop a modified Hagen-Poiseuille equation for gas flow in a capillary (b) By employing the theory of statistical mechanics, the following momentum equation may be developed for the average velocity of a uniform gas in a horizontal capillary of radius R: [DK R²] aps + ps 8μ êx Here D* is known as the Knudsen diffusion coefficient. It is proportional to the mean molecular velocity (V) of the gas and the capillary radius (D* ==RV). Is equation (3) consistent with your answer to part (a)? Show all work. V (c) Consider the porous medium depicted in Figure 1. It is composed of a collection of capillaries of three sizes, R₁ and R₂ in the x direction and R3 in the direction. N, is the number of capillaries of type i (i = 1, 2, or 3) per unit cross-sectional area of the medium (perpendicular to capillary i). R₂ R₁ y Figure 1 R₂ (3)/nBased upon equation (3), develop a vector equation that describes the volumetric flux of gas per unit area (g) in this porous medium under an imposed pressure gradient (VP). Note: you may assume that the weight of the gas is negligible.

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