C2 — Deriving the diffusion equation in spherical coordinates: 23-Chem-B1 Transport Phenomena, December 2019, Q6

Question 6: C2 — Deriving the diffusion equation in spherical coordinates (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 stationary medium ($\mathbf u=0$), no chemical reaction ($\dot R_{A}=0$), and constant molecular diffusivity $D$. Find. The transient diffusion (Fick’s second law) equation written out fully in spherical coordinates $(r,\theta,\phi)$.

Fig. C2: Spherical polar coordinates. A differential control volume has faces of area $r^2\sin\theta\,d\theta\,d\phi$ (radial), $r\sin\theta\,dr\,d\phi$ ($\theta$), and $r\,dr\,d\theta$ ($\phi$). Balancing the diffusive flux across these curved faces produces the geometric $2/r$ and $\cot\theta$ terms.

This is a derivation rather than a numerical problem, so the argument is developed as connected reasoning from the species conservation law down to the fully-expanded operator.

Start from the species continuity equation. For component A the general conservation of moles is $$\frac{\partial C_A}{\partial t}+(\nabla\cdot\mathbf N_A)=\dot R_A,$$ where $\mathbf N_A=-D\,\nabla C_A+C_A\mathbf u$ is the combined diffusive-plus-convective molar flux (Appendix Table A.5).
Apply the stated simplifications. A stationary medium sets $\mathbf u=0$ (no convection), and no reaction sets $\dot R_A=0$. With constant $D$ the flux is purely diffusive, $\mathbf N_A=-D\nabla C_A$, and the balance collapses to Fick’s second law: $$\frac{\partial C_A}{\partial t}=D\,\nabla^2C_A\quad\Longleftrightarrow\quad\frac1D\frac{\partial C_A}{\partial t}=\nabla^2C_A.$$
Write the Laplacian in spherical coordinates. In $(r,\theta,\phi)$ the Laplacian operator is $$\nabla^2C_A=\frac1{r^2}\frac{\partial}{\partial r}\!\left(r^2\frac{\partial C_A}{\partial r}\right)+\frac1{r^2\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial C_A}{\partial\theta}\right)+\frac1{r^2\sin^2\theta}\frac{\partial^2C_A}{\partial\phi^2}.$$ This is exactly $\nabla\cdot(\nabla C_A)$ evaluated with the spherical control-volume face areas of Fig. C2.
Expand the radial term. Differentiating the product, $$\frac1{r^2}\frac{\partial}{\partial r}\!\left(r^2\frac{\partial C_A}{\partial r}\right)=\frac1{r^2}\!\left(2r\frac{\partial C_A}{\partial r}+r^2\frac{\partial^2C_A}{\partial r^2}\right)=\frac{\partial^2C_A}{\partial r^2}+\frac2r\frac{\partial C_A}{\partial r}.$$ The $2/r$ term is the geometric consequence of the radial area growing as $r^2$.
Expand the polar ($\theta$) term. $$\frac1{r^2\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial C_A}{\partial\theta}\right)=\frac1{r^2\sin\theta}\!\left(\cos\theta\frac{\partial C_A}{\partial\theta}+\sin\theta\frac{\partial^2C_A}{\partial\theta^2}\right)=\frac1{r^2}\frac{\partial^2C_A}{\partial\theta^2}+\frac{\cot\theta}{r^2}\frac{\partial C_A}{\partial\theta}.$$
Assemble the result. The azimuthal term is already expanded, so collecting all three contributions gives $$\boxed{\frac1D\frac{\partial C_A}{\partial t}=\frac{\partial^2C_A}{\partial r^2}+\frac2r\frac{\partial C_A}{\partial r}+\frac1{r^2}\frac{\partial^2C_A}{\partial\theta^2}+\frac{\cot\theta}{r^2}\frac{\partial C_A}{\partial\theta}+\frac1{r^2\sin^2\theta}\frac{\partial^2C_A}{\partial\phi^2},}$$ which is precisely the equation to be shown.

Contribution	Expanded form
Radial	$\partial^2C_A/\partial r^2+(2/r)\,\partial C_A/\partial r$
Polar ($\theta$)	$(1/r^2)\partial^2C_A/\partial\theta^2+(\cot\theta/r^2)\partial C_A/\partial\theta$
Azimuthal ($\phi$)	$(1/r^2\sin^2\theta)\,\partial^2C_A/\partial\phi^2$
Full equation	$\tfrac1D\,\partial C_A/\partial t=\nabla^2C_A$ (spherical, as boxed above)

Concept & Theory

Topic. Reduction of the species-conservation equation to Fick’s second law and expansion of the spherical Laplacian.

Key relations. Species continuity $\partial C_A/\partial t+\nabla\cdot\mathbf N_A=\dot R_A$; diffusive flux $\mathbf N_A=-D\nabla C_A$ (stationary medium); spherical Laplacian $\nabla^2=\frac1{r^2}\partial_r(r^2\partial_r)+\frac1{r^2\sin\theta}\partial_\theta(\sin\theta\,\partial_\theta)+\frac1{r^2\sin^2\theta}\partial_\phi^2$.

Why this works. Dropping convection and reaction leaves only accumulation balanced by net diffusion, i.e. the transient heat-equation analogue for concentration. The conservative (divergence) form automatically carries the metric factors $r^2$ and $\sin\theta$ from the curved control-volume faces; differentiating them out produces the first-derivative “geometric” terms $2/r$ and $\cot\theta$ that distinguish spherical from Cartesian diffusion.

Common pitfall. Writing $\nabla^2C_A$ as a bare sum of second derivatives (the Cartesian form) and losing the $2/r$ and $\cot\theta$ terms; or differentiating the product terms incorrectly. Keep $D$ inside or outside the divergence consistently — here $D$ is constant so it factors out cleanly.

Source. Bird, Stewart & Lightfoot, Transport Phenomena, 2nd ed., §19.1 and Appendix B (equations of change for multicomponent systems in curvilinear coordinates).

Question 6 of 6: C2 — Deriving the diffusion equation in spherical coordinates

Question 6: C2 — Deriving the diffusion equation in spherical coordinates (25 marks)