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23-Chem-B1 Transport Phenomena · Undated paper

Question 6 of 6: C2 — Helium leaking through a Pyrex tube wall

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper. National Examination (Engineers Canada) — 16-Chem-B1 Transport Phenomena, May 2019. Open book, 3 hours. Six 25-point problems in three sections — A Fluid Mechanics (A1, A2), B Heat Transfer (B1, B2), C Mass Transfer (C1, C2); one problem from each section must be attempted, plus a fourth from any section. All six problems are solved and fully worked below.

Reference texts: R. B. Bird, W. E. Stewart & E. N. Lightfoot, Transport Phenomena (2nd ed., Wiley) — the equations of change, the power-law slit-flow momentum balance and the film/annular diffusion balances (A2, B2, C2); J. R. Welty, C. E. Wicks, R. E. Wilson & G. L. Rorrer, Fundamentals of Momentum, Heat and Mass Transfer (Wiley) — pipe friction, the Moody chart, and the external-flow convection/mass-transfer correlations (A1, B1, C1); F. P. Incropera & D. P. DeWitt, Fundamentals of Heat and Mass Transfer (8th ed., Wiley) — Churchill–Chu free convection, horizontal-plate correlations, slug-flow internal convection and air properties (B1, B2); C. J. Geankoplis, Transport Processes and Separation Process Principles (Prentice Hall) — boundary-layer mass transfer and the film model (C1); R. H. Perry & D. W. Green, Perry’s Chemical Engineers’ Handbook — transport properties.

Question 6: C2 — Helium leaking through a Pyrex tube wall (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A cylindrical Pyrex wall, inner radius $r_1$ and outer radius $r_2$ (the figure’s $R_1$, $R_2$), tube length $L$ (not drawn on the figure, so $W_{\text{He}}/L$ is the rate per unit length); steady radial diffusion of dissolved helium through the (solid) wall; diffusivity $D$ (He in Pyrex); interfacial helium concentrations $c_1$ at $r_1$ and $c_2$ at $r_2$ ($c_1>c_2$). Find. The total molar leak rate $W_{\text{He}}$.

natural gas (helium source) Pyrex wall r₁ r₂ He diffuses radially outward: c₁ → c₂
Fig. C2: Cross-section of the Pyrex tube. Helium dissolves into the inner wall at concentration $c_1$, diffuses radially through the wall of length $L$, and desorbs at the outer surface ($c_2$). Steady mass conservation makes the total molar flow $W_{\text{He}}$ the same across every cylindrical surface.

Approach. Apply a steady, radial species balance in cylindrical coordinates: with no reaction the total molar flow through any cylinder of radius $r$ is constant. Insert Fick’s law and integrate across the wall.

  1. Steady radial balance. No reaction and steady state give $\dfrac{1}{r}\dfrac{d}{dr}\bigl(r\,N_r\bigr)=0$, so $r\,N_r=\text{const}$. Equivalently, the total leak $W_{\text{He}}=N_r\,(2\pi rL)$ is independent of $r$.
  2. Fick’s law in the solid. Helium is dilute in the glass (no bulk flow), so $N_r=-D\,\dfrac{dc}{dr}$. Then $$W_{\text{He}}=-D(2\pi rL)\frac{dc}{dr}=\text{const}\ \Longrightarrow\ \frac{dc}{dr}=-\frac{W_{\text{He}}}{2\pi DL}\frac{1}{r}.$$
  3. Integrate across the wall. From $r_1$ ($c=c_1$) to $r_2$ ($c=c_2$): $$c_2-c_1=-\frac{W_{\text{He}}}{2\pi DL}\ln\!\frac{r_2}{r_1}.$$
  4. Solve for the leak rate. $$\boxed{\;W_{\text{He}}=\frac{2\pi D L\,(c_1-c_2)}{\ln(r_2/r_1)}.\;}$$ The concentration profile itself is logarithmic, $c(r)=c_1-(c_1-c_2)\dfrac{\ln(r/r_1)}{\ln(r_2/r_1)}$.
QuantityResult
Molar leak rate$W_{\text{He}}=\dfrac{2\pi DL(c_1-c_2)}{\ln(r_2/r_1)}$
Concentration profile$c(r)=c_1-(c_1-c_2)\dfrac{\ln(r/r_1)}{\ln(r_2/r_1)}$
Local flux at radius $r$$N_r=\dfrac{D(c_1-c_2)}{r\ln(r_2/r_1)}$
Check — interfacial concentrations vs. gas pressures

The answer is expressed in the interfacial concentrations of helium dissolved in the Pyrex, $c_1$ and $c_2$, as the question requests. If instead the helium partial pressures $p_1,p_2$ in the gas phases are known, use the solubility (Henry-type) relation $c=S\,p$ to write $W_{\text{He}}=2\pi DSL(p_1-p_2)/\ln(r_2/r_1)$, i.e. in terms of the permeability $\mathcal P=DS$. The thin-wall limit ($r_2\to r_1$) recovers the plane-slab result $W\to 2\pi r_1 L\,D(c_1-c_2)/(r_2-r_1)$ since $\ln(r_2/r_1)\approx(r_2-r_1)/r_1$.

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