Question 6 of 6: C2 — Steady diffusion through a spherical membrane wall
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. EGBC 16-CHEM-B1 Transport Phenomena, December 2017, 3 hours, open-book (one textbook permitted, no loose notes). Six problems in three sections (A Fluid Mechanics, B Heat Transfer, C Mass Transfer), each worth 25 marks; candidates attempt one problem from each section plus a fourth from any section, and only the first four in the answer book are marked. All six problems are solved below as a complete study resource. Appendix A of the paper supplies the equations of change (continuity, Navier–Stokes, energy and species-continuity tables) in rectangular, cylindrical and spherical coordinates; these are quoted rather than re-derived.
Reference texts: R. B. Bird, W. E. Stewart & E. N. Lightfoot, Transport Phenomena (2nd ed., Wiley) — the equations of change and the differential momentum / energy / species balances; J. R. Welty, C. E. Wicks, R. E. Wilson & G. L. Rorrer, Fundamentals of Momentum, Heat and Mass Transfer (Wiley) — pipe friction, the transfer analogies, boundary-layer and film coefficients; F. P. Incropera & D. P. DeWitt, Fundamentals of Heat and Mass Transfer (Wiley) — internal-flow convection and the sphere/flat-plate correlations; F. M. White, Fluid Mechanics (McGraw-Hill) — the three-reservoir branching-pipe problem; R. H. Perry & D. W. Green, Perry’s Chemical Engineers’ Handbook — transport properties.
Question 6: C2 — Steady diffusion through a spherical membrane wall (25 marks)
Given. A spherical membrane wall $R_i\le r\le R_O$; steady state; no chemical reaction and no bulk convection; diffusivity $D$ constant. The core holds the solute at a fixed concentration $C_i$ at the inner surface, and the surrounding bath is at zero concentration; partition coefficient unity means membrane-face concentrations equal the adjacent bulk values.
Find. (a) the governing equation; (b) the boundary conditions; (c) $C(r)$ in the membrane; (d) the diffusive flux at the outer surface $r=R_O$.
Fig. C2: Radial diffusion through a spherical membrane wall. With no reaction and steady state, $r^2\,dC/dr$ is constant, so the profile varies as $1/r$; the outward flux is largest at the inner face and falls as $1/r^2$.
Approach. Reduce the spherical species-continuity equation (Table A.5) to its steady, no-reaction, radial form; that integrates twice to a two-constant profile fixed by the inner and outer concentrations; differentiating and applying Fick’s law at $R_O$ gives the surface flux.
(a) Governing equation. Steady state ($\partial C/\partial t=0$), no convection, no reaction, spherical symmetry ($C=C(r)$) reduce the species balance to $$\boxed{\frac{1}{r^2}\frac{d}{dr}\!\left(r^2\frac{dC}{dr}\right)=0\quad\Longleftrightarrow\quad \frac{d}{dr}\!\left(r^2\frac{dC}{dr}\right)=0.}$$
(b) Boundary conditions. With unit partition coefficient the membrane faces equal the adjacent bulk concentrations: $$C(R_i)=C_i\quad(\text{inner, core}),\qquad C(R_O)=0\quad(\text{outer, bath}).$$
(c) Integrate the ODE. First integral: $r^2\dfrac{dC}{dr}=A$ (a constant), so $\dfrac{dC}{dr}=\dfrac{A}{r^2}$ and $$C(r)=-\frac{A}{r}+B.$$ The $1/r$ dependence is the hallmark of source-free spherical diffusion.
(c) Fix the constants. Imposing the two conditions, $-\dfrac{A}{R_i}+B=C_i$ and $-\dfrac{A}{R_O}+B=0$. Subtracting gives $A\!\left(\dfrac{1}{R_O}-\dfrac{1}{R_i}\right)=C_i$ and $B=\dfrac{A}{R_O}$, whence $$\boxed{\,C(r)=C_i\,\frac{R_i\,(R_O-r)}{r\,(R_O-R_i)}=C_i\,\frac{1/r-1/R_O}{1/R_i-1/R_O}.}$$ It satisfies $C(R_i)=C_i$ and $C(R_O)=0$ by construction.
(d) Flux at the outer surface. Fick’s law (outward positive) gives $N_r=-D\,\dfrac{dC}{dr}=-D\dfrac{A}{r^2}$. With $A=\dfrac{C_i}{1/R_O-1/R_i}=-\dfrac{C_iR_iR_O}{R_O-R_i}$, $$\boxed{\,N_r\big|_{r=R_O}=\frac{D\,C_i\,R_i}{R_O\,(R_O-R_i)}\ \ (>0,\ \text{outward}).}$$ The corresponding total molar release rate is $W=4\pi R_O^2\,N_r|_{R_O}=\dfrac{4\pi D\,C_i\,R_iR_O}{R_O-R_i}$, which (correctly) is independent of $r$.
