C2 — O&sub2; diffusion with zeroth-order consumption in a sphere: 23-Chem-B1 Transport Phenomena, May 2017, Q6

Question 6: C2 — O&sub2; diffusion with zeroth-order consumption in a sphere (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. Steady diffusion of O&sub2; (diffusivity $D$) into a sphere of radius $r_0$ with a constant (zeroth-order) volumetric consumption rate $R_0$ (mol m$^{-3}$s$^{-1}$); the surface concentration is held at $C_0$; spherical symmetry, no convection.

Find. (a) the radial concentration profile $C(r)$; (b) the criterion for an oxygen-free core and an expression locating its radius $r_c$.

Fig. C2: Oxygen diffuses inward and is consumed at a constant rate. If the tissue is large or hungry enough, the concentration hits zero at $r_c$, leaving an anoxic core in which the zeroth-order model no longer holds.

Approach. Reduce the spherical species-continuity equation (Table A.5) to a steady balance of diffusion against the constant sink, integrate for the no-core profile, then split off a reacting outer region when the centre would go anoxic and apply zero-concentration / zero-flux conditions at $r_c$.

Governing equation. With no convection, steady state, constant $D$ and a constant sink $R_0$, the spherical species balance becomes $$D\,\frac{1}{r^2}\frac{d}{dr}\!\Big(r^2\frac{dC}{dr}\Big)=R_0.$$
(a) Integrate for the whole sphere. Two integrations give $C=\dfrac{R_0 r^2}{6D}-\dfrac{A}{r}+B$. Finiteness at the centre forces $A=0$; the surface condition $C(r_0)=C_0$ fixes $B$, giving $$\boxed{C(r)=C_0-\frac{R_0}{6D}\big(r_0^{2}-r^{2}\big).}$$ The profile is convex with its minimum at the centre.
(a→b) Centre concentration and the onset criterion. Setting $r=0$, $C(0)=C_0-\dfrac{R_0 r_0^{2}}{6D}$. A physically impossible negative centre concentration signals an anoxic core; it first appears when $C(0)=0$, i.e. when $$\boxed{\Phi\equiv\frac{R_0\,r_0^{2}}{6D\,C_0}\ge1.}$$ ($\Phi$ is a Thiele-type modulus comparing reaction demand with diffusive supply.)
(b) Reacting region with a dead core. When $\Phi>1$, solve the balance only over $r_cand zero flux at the core edge ($C(r_c)=0$, $dC/dr|_{r_c}=0$, since no O&sub2; crosses into the dead core). These give $$C(r)=\frac{R_0}{6D}\Big(r^{2}+\frac{2r_c^{3}}{r}-3r_c^{2}\Big).$$
(b) Locate $r_c$ from the surface condition. Imposing $C(r_0)=C_0$ yields $\dfrac{R_0}{6D}\big(r_0^{2}+2r_c^{3}/r_0-3r_c^{2}\big)=C_0$, which factors neatly. Writing $\xi=r_c/r_0$, $$\boxed{(1-\xi)^{2}(1+2\xi)=\frac{1}{\Phi}=\frac{6D\,C_0}{R_0\,r_0^{2}}.}$$ This implicit relation gives $r_c=\xi r_0$. It behaves correctly at the limits: $\Phi=1\Rightarrow\xi=0$ (the core just appears), and $\Phi\to\infty\Rightarrow\xi\to1$ (almost the whole tissue is anoxic). For example, $\Phi=2$ gives $\xi=\tfrac12$, i.e. $r_c=r_0/2$.

Result	Expression / Value
(a) Concentration profile (no core)	$C(r)=C_0-\dfrac{R_0}{6D}(r_0^{2}-r^{2})$
(b) Anoxic-core criterion	$\Phi=R_0 r_0^{2}/(6DC_0)\ge1$
(b) Core radius (implicit)	$(1-\xi)^2(1+2\xi)=1/\Phi$, $\ \xi=r_c/r_0$
Illustration ($\Phi=2$)	$r_c=r_0/2$

Concept & Theory

Topic. Reaction–diffusion in a sphere with a zeroth-order sink; the anoxic-core (dead-zone) criterion.

Key relations. Spherical balance $D\,r^{-2}(r^2C')'=R_0$; no-core profile $C=C_0-\tfrac{R_0}{6D}(r_0^2-r^2)$; Thiele-type modulus $\Phi=R_0 r_0^2/(6DC_0)$; core relation $(1-\xi)^2(1+2\xi)=1/\Phi$.

Why this works. A zeroth-order sink is constant while O&sub2; is present, so the equation is linear and integrates in closed form. The core appears exactly when the diffusive supply can no longer meet the fixed demand at the centre; there, both concentration and flux vanish (no O&sub2; enters the dead zone), and those two conditions plus the surface value determine the reacting-region profile and $r_c$.

Common pitfall. Keeping the $1/r$ integration constant for the full sphere (it must vanish for finiteness at the centre) but dropping it in the cored region (there it is essential and set by the zero-flux condition); and reporting a negative centre concentration instead of recognising it as the signal that a core has formed.

Source. Deen, Analysis of Transport Phenomena, Ch. 3–4 (reaction and diffusion; the “core” problem); Bird–Stewart–Lightfoot §18 (diffusion with homogeneous reaction).

Question 6 of 6: C2 — O&sub2; diffusion with zeroth-order consumption in a sphere

Question 6: C2 — O&sub2; diffusion with zeroth-order consumption in a sphere (25 marks)