Question 6 of 6: C2 — Diffusion with a surface reaction 3A → B on a catalyst
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. EGBC 16-CHEM-B1 Transport Phenomena, May 2018, 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, tank draining, the internal-flow and wetted-wall correlations; F. P. Incropera & D. P. DeWitt, Fundamentals of Heat and Mass Transfer (Wiley) — the Sieder–Tate laminar-tube correlation and cylindrical conduction; C. J. Geankoplis, Transport Processes and Separation Process Principles (Prentice Hall) — the Gilliland–Sherwood wetted-wall correlation and diffusion with heterogeneous reaction; R. H. Perry & D. W. Green, Perry’s Chemical Engineers’ Handbook — transport properties.
Question 6: C2 — Diffusion with a surface reaction 3A → B on a catalyst (25 marks)
Given. A spherical catalyst particle of radius $R_1$ surrounded by a stagnant gas film of thickness $\delta$ (outer edge at $r=R_1+\delta$, where $x_A=X_{A0}$). The surface reaction $3A\to B$ is instantaneous, isothermal, at steady state; total molar concentration $c$ and diffusivity $D_{AB}$ constant.
Find. The local molar rate of A consumption (the reaction rate) in terms of $\delta$ and $X_{A0}$.
Fig. C2: A diffuses inward through the spherical film to the catalyst, where $3A\to B$; B diffuses back out. The $3:1$ stoichiometry makes the film non-equimolar, so a convective drift term rides on top of ordinary diffusion.
Approach. This is a derivation. Fix the flux ratio from stoichiometry, write the combined diffusion–convection flux for A, impose steady-state conservation ($4\pi r^2N_A=$ const) across the spherical film, and integrate from the surface (where the instantaneous reaction drives $x_A\to0$) to the film edge.
Flux ratio from stoichiometry. At the surface $3$ mol A are consumed per mol B produced, and B travels outward while A travels inward, so $N_B=-\tfrac13 N_A$. The total molar flux is $$N=N_A+N_B=N_A\big(1-\tfrac13\big)=\tfrac23 N_A.$$
Combined flux of A. With ordinary diffusion plus the bulk (drift) term, $$N_A=-cD_{AB}\frac{dx_A}{dr}+x_A(N_A+N_B)=-cD_{AB}\frac{dx_A}{dr}+\tfrac23 x_A N_A,$$ so $$N_A\Big(1-\tfrac23 x_A\Big)=-cD_{AB}\frac{dx_A}{dr}.$$
Steady-state conservation in the film. No reaction occurs within the film (only at the surface), so the molar flow of A is constant: $W_A=4\pi r^2N_A=\text{const}$, giving $N_A=W_A/4\pi r^2$. Substituting, $$\frac{W_A}{4\pi r^2}\Big(1-\tfrac23 x_A\Big)=-cD_{AB}\frac{dx_A}{dr}.$$
Separate and integrate across the film. $$\frac{W_A}{4\pi}\int_{R_1}^{R_1+\delta}\frac{dr}{r^2}=-cD_{AB}\int_{0}^{X_{A0}}\frac{dx_A}{1-\tfrac23 x_A},$$ using the instantaneous-reaction surface condition $x_A(R_1)=0$ and film-edge $x_A(R_1+\delta)=X_{A0}$. The left integral is $\dfrac{1}{R_1}-\dfrac{1}{R_1+\delta}=\dfrac{\delta}{R_1(R_1+\delta)}$; the right is $\tfrac32\ln\!\big(1-\tfrac23 X_{A0}\big)$.
Solve for the molar flow (reaction rate). $$\frac{W_A}{4\pi}\,\frac{\delta}{R_1(R_1+\delta)}=\tfrac32 cD_{AB}\,\ln\!\Big(1-\tfrac23 X_{A0}\Big).$$ Because $\ln(1-\tfrac23 X_{A0})<0$, $W_A<0$ (A flows inward). The rate at which A is consumed by the catalyst is $$\boxed{\,\dot n_A=-W_A=6\pi cD_{AB}\,\frac{R_1(R_1+\delta)}{\delta}\,\ln\!\frac{1}{1-\tfrac23 X_{A0}}.}$$ The reaction rate (of $3A\to B$) is $\dot n_A/3$, and in the thin-film limit $\delta\ll R_1$ this reduces to the flat-film form $6\pi cD_{AB}(R_1^2/\delta)\ln[1/(1-\tfrac23X_{A0})]$.
Local rate per unit catalyst surface. Dividing by the surface area $4\pi R_1^2$ gives the local (per-area) rate of A consumption at the surface, $$\boxed{-N_A\big|_{R_1}=\frac{3cD_{AB}}{2\delta}\,\frac{R_1+\delta}{R_1}\,\ln\frac{1}{1-\tfrac23 X_{A0}},}$$ which for $\delta\ll R_1$ becomes the flat-film result $\dfrac{3cD_{AB}}{2\delta}\ln\dfrac{1}{1-\frac23X_{A0}}$. Because the net flux $N=\tfrac23N_A$ points toward the surface together with A, the drift assists transport: the log term exceeds its dilute limit $\tfrac23X_{A0}$.
