Mean Free Path — Question 10

Question 10

A channel size sets the pressure needed for a chosen Knudsen number

At 300 K, a dilute gas has $\lambda p=6.00\times10^{-3}\,\mathrm{Pa\,m}$. A channel has characteristic width $L=10.0\,\mu$m. Define $\mathrm{Kn}=\lambda/L$. For this exercise adopt the design criterion $\mathrm{Kn}\le.0100$ for a continuum approximation.

1. Find the minimum pressure satisfying the criterion and the Knudsen number at 100 Pa.

2. For a channel one-tenth as wide, find the new minimum pressure. Explain whether reducing pressure improves the adopted continuum approximation.

Question 10 - Solution

A channel size sets the pressure needed for a chosen Knudsen number

(a) Method and calculation

$$\mathrm{Kn}=\frac{6.00\times10^{-3}}{p(10^{-5})}=\frac{600\,\mathrm{Pa}}p.$$$$p_{\min}=600/.0100=60000\,\mathrm{Pa},\quad\mathrm{Kn}(100\,\mathrm{Pa})=6.00.$$

(b) Interpretation and further calculation

$$L\to L/10\ \Rightarrow\ p_{\min}\to10p_{\min}=600000\,\mathrm{Pa}.$$$$p\downarrow\ \Rightarrow\lambda\uparrow\ \Rightarrow\mathrm{Kn}\uparrow.$$Lower pressure makes molecule-wall travel more important relative to intermolecular collisions. The numerical cutoff is the stipulated design convention, not a sharp universal phase boundary.

Physical check

Any high-pressure extrapolation also requires the given dilute-gas relation to remain applicable.