Molecular Flow

Consider the flow of an ideal gas down a uniform pipe of circular cross-section in the limit that the gas pressure is sufficiently low that the mean free path between collisions greatly exceeds the diameter of the pipe. This type of flow is known as molecular flow.

Suppose that the pipe runs along the $z$-axis, and that a pressure difference, ${\mit\Delta} p$, is established between the two ends of the pipe, in order to drive the flow. The temperature of the gas is assumed to be uniform along the pipe. Finally, the length of the pipe, $L$, is assumed to be much greater than its diameter, $d$. Making use of Equation (5.170), the net flux of molecules down the pipe at position $z$ is

$${\mit\Phi}_z(z)= \frac{1}{4}\,\langle v\rangle\left[n(z-d)-n(z+d)\right],$$ (5.274)

where $\langle v\rangle$ is the mean molecular speed [which is constant because it only depends on the temperature; see Equation (5.433)], and $n(z)$ the molecular number density. The right-hand side of the previous equation represents the difference between the particle flux in the $+z$-direction and that in the $-z$-direction. The former flux is characterized by the value the number density calculated at the position at which the molecules moving in the $+z$ direction last collided with the wall of the pipe, which is estimated to be $z-d$. Likewise, the latter flux is characterized by the number density calculated at the position at which the molecules moving in the $-z$ direction last collided with the wall of the pipe, which is estimated to be $z+d$. According to Equation (5.175), the pressure of the gas in the pipe is

$$p(z) = \frac{1}{3}\,n(z)\,m\left\langle v^2\right\rangle,$$ (5.275)

where $m$ is the molecular mass, and $\left\langle v^2\right\rangle$ is the mean square molecular speed (which is also constant because it only depends on the temperature). [See Equation (5.434).] The previous two equations yield

$${\mit\Phi}_z(z) = \frac{3}{4}\,\frac{\langle v\rangle}{m\,\left\langle v^2\right\rangle}\left[p(z-d)-p(z+d)\right]$$

$$\simeq- \frac{3}{2}\,\frac{d\,\langle v\rangle}{m\,\left\langle v^2\right\rangle}\,\frac{dp}{dz}.$$ (5.276)

In a steady state, ${\mit\Phi}_z$ must be uniform along the pipe, which implies that $dp/dz$ is also uniform. Hence, we can write

$$-\frac{dp}{dz}=\frac{{\mit\Delta p}}{L},$$ (5.277)

which yields

$${\mit\Phi}_z = \frac{3}{2}\,\frac{\langle v\rangle}{\left\langle v^2\right\rangle}\,\frac{d}{m\,L}\,{\mit\Delta p}.$$ (5.278)

Now, the cross-sectional area of the pipe is $A= \pi\,d^{\,2}/4$. Hence, the rate of mass flow down the pipe is

$$\dot{M} = {\mit\Phi}_z\,A\,m= \frac{3\pi}{8}\,\frac{\langle v\rangle}{\left\langle v^2\right\rangle}\,\frac{d^{\,3}}{L}\,{\mit\Delta}p.$$ (5.279)

However, in a ideal gas,

$$\langle v\rangle^2 = \frac{8}{3\pi}\left\langle v^2\right\rangle.$$ (5.280)

(See Section 5.5.9.) Thus, we obtain

$$\dot{M} = \frac{d^{\,3}}{\langle v\rangle\,L}\,{\mit\Delta} p.$$ (5.281)

Given that $\langle v\rangle \propto T^{1/2}$ [see Equation (5.244)], where $T$ is the temperature of the gas, we deduce that the mass flow rate due to molecular flow of an ideal gas down a pipe, when a given pressure difference is established between the two ends of the pipe, is proportional to the cube of the pipe diameter, inversely proportional to the length of the pipe, and inversely proportional to the square-root of the temperature.

Now, the standard formula for the mass flow rate of a viscous fluid down a pipe of circular cross-section, which applies to the case under discussion when the mean free path between collisions is much less than the diameter of the pipe, is

$$\dot{M} = \frac{\pi}{128}\,\frac{d^{\,4}\,n\,m}{\eta\,L}\,{\mit\Delta} p.$$ (5.282)

Making use of Equation (5.266), we deduce that the mass flow rate of an ideal gas down a pipe of circular cross-section, in the short mean free path limit, is

$$\dot{M} =\frac{3\pi}{128}\,\frac{d^{\,4}}{\langle v\rangle\,l\,L}\,{\mit\Delta}p,$$ (5.283)

where $l$ is the mean free path. Thus, Equation (5.281) holds when $l\gg d$, and Equation (5.283) holds when $l\ll d$.