Diffusion-limited combustion of a carbon sphere: 23-Chem-A3 Heat and Mass Transfer, December 2014, Q2

Question 2: Diffusion-limited combustion of a carbon sphere (Part A — equal value)

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 carbon sphere burns in air; oxygen (A) diffuses radially inward through the surrounding gas to the surface, where an infinitely fast reaction consumes it. Because one mole of CO₂ leaves for every mole of O₂ that arrives, the transport is equimolar counterdiffusion — there is no net molar (Stefan) flow.

Quantity	Value
Sphere radius $r_0$	1 mm $=1.0\times10^{-3}$ m
Gas temperature $T$	1500 K
Total pressure $P$	1.0 atm $=101{,}325$ Pa
O₂ diffusivity $D_{AB}$	$1.8\times10^{-4}$ m²/s
Bulk O₂ mole fraction	0.21 (air)

Find. The steady-state molar consumption rate of oxygen, $W_{O_2}$ (kmol/s).

Figure 2 — Equimolar counterdiffusion around a burning carbon sphere: one O₂ diffuses in for each CO₂ that diffuses out, so the net molar flux is zero and the classic $Sh=2$ stagnant-sphere result applies. An infinite surface reaction rate drives the surface O₂ concentration to zero.

Approach. For equimolar counterdiffusion from an infinite stagnant medium to a sphere, integrate Fick’s law radially to get the closed-form molar flow $W_A=4\pi r_0 D_{AB}(c_{A\infty}-c_{A,s})$, with $c_{A,s}=0$.

Bulk oxygen concentration. Air is 21 mol% O₂, so $$c_{A\infty}=0.21\,\frac{P}{RT}=0.21\times\frac{101{,}325}{(8.314)(1500)}=1.71\ \text{mol/m}^3.$$
Surface concentration. An infinitely fast reaction consumes O₂ the instant it arrives, so $c_{A,s}=0$; the full bulk concentration is the driving force.
Radial diffusion to a sphere. Solving $\dfrac{d}{dr}\!\left(r^2\dfrac{dc_A}{dr}\right)=0$ with $c_A(r_0)=c_{A,s}$, $c_A(\infty)=c_{A\infty}$ gives the standard stagnant-sphere result $$W_A=4\pi r_0 D_{AB}\,(c_{A\infty}-c_{A,s}).$$ This is the $Sh=2$ limit: $k_c(2r_0)/D_{AB}=2$.
Evaluate the oxygen consumption rate. $$W_{O_2}=4\pi(1.0\times10^{-3})(1.8\times10^{-4})(1.71-0)=3.86\times10^{-6}\ \text{mol/s}=\boxed{3.86\times10^{-9}\ \text{kmol/s}}.$$ Carbon is consumed at the same molar rate (1:1 stoichiometry).

Quantity	Result
Bulk O₂ concentration $c_{A\infty}$	1.71 mol/m³
Effective Sherwood number	$Sh=2$
O₂ consumption rate $W_{O_2}$	$3.86\times10^{-9}$ kmol/s

Concept & Theory

Topic. Steady diffusion with heterogeneous reaction — the diffusion-controlled burning sphere.

Key relations. Sphere in a stagnant medium: $W_A=4\pi r_0 D_{AB}(c_{A\infty}-c_{A,s})$; low-flux limit $Sh=k_c d/D_{AB}=2$; equimolar counterdiffusion (no Stefan drift because $N_{CO_2}=-N_{O_2}$).

Why this works. Since one CO₂ molecule replaces each O₂ molecule at the surface, the mixture’s total molar flux is zero and the drift-flux (log-mean) correction of Stefan diffusion vanishes — ordinary Fick diffusion applies. An infinitely fast surface reaction makes the process wholly diffusion-limited, so the surface concentration collapses to zero and the entire bulk concentration is available as the driving force.

Common pitfall. Applying a Stefan log-mean factor (valid for evaporation into a stagnant inert). Here the counterdiffusing CO₂ cancels the bulk flow, so no such factor appears. A second trap is forgetting that $Sh=2$ is a diameter-based number, so $k_c=D_{AB}/r_0$, not $D_{AB}/(2r_0)$.

Source. Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, §14 (species diffusion, sphere); Welty et al., §26 (diffusion with heterogeneous reaction).

Question 2 of 7: Diffusion-limited combustion of a carbon sphere

Question 2: Diffusion-limited combustion of a carbon sphere (Part A — equal value)