Question 5 of 6: C1 — Diffusion-limited drug release from a gel capsule
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format: National Exams, December 2014 — 04-CHEM-B1 Transport Phenomena; 3 hours, open book. Six problems in three sections (A Fluid Mechanics, B Heat Transfer, C Mass Transfer); candidates attempt one from each section plus a fourth from any section (four of six marked, 25 marks each). The worked solutions below cover all six problems. A summary of the conservation equations (continuity, Navier–Stokes, energy, species) is provided as Appendix A in the paper and is quoted throughout rather than re-derived.
Reference texts: R. S. Brodkey & H. C. Hershey, Transport Phenomena — A Unified Approach (McGraw-Hill) — the paper’s own reference and source of the appended conservation-equation tables; R. B. Bird, W. E. Stewart & E. N. Lightfoot, Transport Phenomena (2nd ed., Wiley) — 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) — annular flow, variable-conductivity conduction and diffusion; F. P. Incropera & D. P. DeWitt, Fundamentals of Heat and Mass Transfer (Wiley) — internal-flow convection ($Nu=3.66$) and transient conduction/diffusion; C. J. Geankoplis, Transport Processes and Separation Process Principles (Prentice Hall) — Stefan-tube / evaporating-drop mass transfer.
Question 5: C1 — Diffusion-limited drug release from a gel capsule
Given. A spherical gel capsule (radius $R$) loaded uniformly with drug $A$ at $C_{A0}=50\ \text{mg/cm}^3$; the surrounding fluid is a perfect sink. Release is diffusion-limited within the gel.
Find. (a) a sketch and five mass-transfer assumptions; (b) Fick’s flux law at the surface; (c) the governing PDE for $C_A(r,t)$ with boundary/initial conditions.
Fig. C1: The gel bead releases drug $A$ by radial diffusion. The interior starts uniform at $C_{A0}$; the surface is held at zero concentration because the drug is swept away instantly, driving an outward flux that decays as the bead depletes.
Approach. This is a transient, spherically symmetric diffusion problem with no convection and no reaction inside the gel; reduce the spherical species-continuity equation (Appendix A, Table A.5) and close it with a perfect-sink surface and a uniform start.
(a) Physical picture and assumptions. See Fig. C1. Reasonable mass-transfer assumptions are: (i) spherical symmetry — $C_A=C_A(r,t)$ only, no $\theta,\phi$ dependence; (ii) constant, uniform diffusivity $D_A$ of drug in the gel; (iii) dilute solute, so Fick’s law holds and bulk (convective) flow inside the gel is negligible ($u=0$); (iv) no chemical reaction or degradation of the drug within the gel ($R_A=0$); (v) perfect-sink surface, $C_A(R,t)=0$ (drug consumed/swept away instantly); (vi) uniform initial loading $C_A(r,0)=C_{A0}$; (vii) the gel neither swells nor dissolves (fixed $R$) and temperature is constant.
(b) Fick’s flux at the surface. With a dilute solute and no bulk flow, the molar (or mass) flux of $A$ leaving the bead is pure diffusion, evaluated at $r=R$: $$\boxed{\,N_A\big|_{r=R}=-D_A\,\frac{\partial C_A}{\partial r}\bigg|_{r=R}\,}.$$ The steep interior gradient at the surface sets the instantaneous release rate; the total release is $4\pi R^2 N_A|_R$.
(c) Governing PDE. Take the spherical species-continuity equation and drop convection ($u=0$) and generation ($R_{A,G}=0$), with constant $D_A$: $$\boxed{\,\frac{\partial C_A}{\partial t}=D_A\,\frac{1}{r^2}\frac{\partial}{\partial r}\!\left(r^2\frac{\partial C_A}{\partial r}\right)=D_A\!\left(\frac{\partial^2 C_A}{\partial r^2}+\frac{2}{r}\frac{\partial C_A}{\partial r}\right)}.$$
Boundary and initial conditions. Uniform start, symmetric centre, perfect-sink surface: $$C_A(r,0)=C_{A0}\ (0\le r<R);\quad \frac{\partial C_A}{\partial r}\bigg|_{r=0}=0;\quad C_A(R,t)=0 .$$ The solution is governed by the dimensionless mass Fourier number $\mathrm{Fo}=D_A t/R^2$.
