Steady Radial Diffusion of Helium Through a Pyrex Tube Wall: 21-Mat-A2 Materials Transport Phenomena, May 2017, Q5

Question 5: Steady Radial Diffusion of Helium Through a Pyrex Tube Wall (25 marks)

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Given. A Pyrex tube wall of inner radius $R_1$ and outer radius $R_2$ (length $L$), separating a natural-gas/helium mixture inside from the ambient outside; helium diffusivity through Pyrex $D_{He}$; the helium concentration dissolved in the Pyrex at the inner wall is $C_1$ (in equilibrium with the gas mixture) and at the outer wall is $C_2$ (in equilibrium with the ambient).

Find. An expression for the steady-state molar rate $W_{He}$ at which helium diffuses out through the tube wall, in terms of $D_{He}$, $C_1$, $C_2$, $R_1$, $R_2$ (and $L$).

Fig. 4 — end-on view of the Pyrex tube wall; helium diffuses radially outward from $R_1$ to $R_2$.

Approach. Apply a steady-state, no-reaction species differential balance on helium in cylindrical coordinates through the tube wall (the same reduction as Appendix Table A.5 supplied with the exam, with $\partial/\partial t=0$, no angular/axial dependence, and $\dot R_{A,G}=0$), then integrate Fick's first law twice across the wall thickness.

Differential balance. Steady, radially-symmetric diffusion with no generation reduces the cylindrical species-continuity equation to $$\frac{1}{r}\frac{d}{dr}\left(rD_{He}\frac{dC}{dr}\right)=0$$
First integration. At steady state the total molar flow through any cylindrical surface of radius $r$ and length $L$ is the same constant $W_{He}$ (no accumulation or reaction between $R_1$ and $r$): $$W_{He}=N_r\,(2\pi r L)=-D_{He}\,(2\pi r L)\frac{dC}{dr}=\text{const.}\;\Rightarrow\;\frac{dC}{dr}=-\frac{W_{He}}{2\pi L D_{He}}\cdot\frac{1}{r}$$
Second integration. Separate variables and integrate $C$ from $C_1$ at $r=R_1$ to $C_2$ at $r=R_2$: $$\int_{C_1}^{C_2}dC=-\frac{W_{He}}{2\pi L D_{He}}\int_{R_1}^{R_2}\frac{dr}{r}\;\Rightarrow\;C_2-C_1=-\frac{W_{He}}{2\pi L D_{He}}\ln\frac{R_2}{R_1}$$
Solve for the leak rate. $$\boxed{W_{He}=\frac{2\pi L D_{He}\left(C_1-C_2\right)}{\ln\!\left(R_2/R_1\right)}}$$ or, per unit tube length, $W_{He}'=W_{He}/L=2\pi D_{He}(C_1-C_2)/\ln(R_2/R_1)$.

Quantity	Expression
He leak rate, $W_{He}$	$\dfrac{2\pi L D_{He}(C_1-C_2)}{\ln(R_2/R_1)}$
He leak rate per unit length, $W_{He}'$	$\dfrac{2\pi D_{He}(C_1-C_2)}{\ln(R_2/R_1)}$

Check

Assumes local equilibrium (no interfacial resistance) between the gas-phase helium and the helium dissolved in the Pyrex at both $r=R_1$ and $r=R_2$, that the wall has reached steady state, and that $C_1>C_2$ (helium partial pressure inside the tube exceeds that outside, so the net flux is genuinely outward as stated).

Concept & Theory

Topic. Steady-state Fickian diffusion through a cylindrical tube wall (radial-only, no reaction) — the mass-transfer analogue of steady radial heat conduction through a pipe insulation layer.

Key relations. Fick's first law $N_r=-D\,dC/dr$; cylindrical differential balance $\tfrac{1}{r}\tfrac{d}{dr}(rD\,dC/dr)=0$; logarithmic-mean form of the resulting flux, $W=2\pi LD(C_1-C_2)/\ln(R_2/R_1)$.

Why this works. In cylindrical geometry the cross-sectional area for radial flow ($2\pi rL$) itself increases with $r$, so a constant molar flow $W$ requires the concentration gradient to fall as $1/r$ — integrating that gradient produces the characteristic logarithmic profile and the $\ln(R_2/R_1)$ resistance term, exactly analogous to the log-mean-radius formula for conduction through pipe insulation.

Common pitfall. Using the planar (flat-slab) form $W=DA(C_1-C_2)/(R_2-R_1)$ instead of the cylindrical logarithmic form — acceptable only as a thin-wall approximation when $R_2/R_1\to1$, not in general.

Source. Geankoplis, Transport Processes and Separation Process Principles, Ch. 6 (steady-state molecular diffusion, cylindrical-wall balances); Welty et al., Fundamentals of Momentum, Heat and Mass Transfer, Ch. 24 (analogous log-mean-radius conduction derivation).

Question 5 of 5: Steady Radial Diffusion of Helium Through a Pyrex Tube Wall

Question 5: Steady Radial Diffusion of Helium Through a Pyrex Tube Wall (25 marks)