Question 6 of 7: Bulk-solution concentration for diffusion into a biofilm
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2014 — 04-Chem-A3 Mass Transfer Operations. Three-hour, open-book exam; any non-communicating calculator permitted. Format: seven questions in three parts — answer one of Q1–Q2 (Part A), one of Q3–Q4 (Part B) and two of Q5–Q7 (Part C); four questions of equal value constitute a complete paper. All seven are solved below for completeness. Every property datum (vapour pressures, diffusivities, Henry constants) is stated in the question, so each answer is self-contained.
Reference texts: Geankoplis, Transport Processes and Separation Process Principles (4th ed., Prentice Hall) — molecular diffusion, convective mass transfer and gas absorption; Welty, Wicks, Wilson & Rorrer, Fundamentals of Momentum, Heat and Mass Transfer (6th ed., Wiley) — boundary-layer mass transfer and diffusion with reaction; Incropera & DeWitt, Fundamentals of Heat and Mass Transfer (species diffusion, sphere in a stagnant medium); Treybal, Mass-Transfer Operations (3rd ed.) — packed-tower and wetted-wall design; supporting data from Perry's Chemical Engineers' Handbook (9th ed.).
Question 6: Bulk-solution concentration for diffusion into a biofilm (Part C — equal value)
Given. Species A diffuses from a well-mixed solution into a reactive biofilm; the concentration profile inside the film is given. With no external resistance, the film’s outer face ($z=0$) sits at equilibrium with the bulk solution through a partition relation.
Quantity
Value
Rate constant $k_1$
$1.5\times10^{-4}$ s⁻¹
Diffusivity $D_{AB}$
$1.26\times10^{-5}$ cm²/s
Film thickness $L$
0.2 cm
Outer-surface concentration $C_{A0}$
0.4 mol/L
Partition relation
$C_A=0.4\,C_{A\infty}$
Find. The bulk-solution concentration $C_{A\infty}$ (and, for context, the surface flux into the film).
Approach. Evaluate the given profile at the outer face to identify the surface concentration, then invert the equilibrium/partition relation to recover the bulk concentration; the cosh profile is only needed for the flux.
Concentration at the outer surface. Setting $z=0$ in the profile, $\cosh[m(L-0)]/\cosh[mL]=1$, so $C_A(0)=C_{A0}=0.4$ mol/L. This is the biofilm-side concentration at the face touching the solution.
Apply the partition equilibrium. With negligible solution-side resistance, the film face is in equilibrium with the adjacent bulk, $C_A=0.4\,C_{A\infty}$. Substituting the surface value: $$0.4=0.4\,C_{A\infty}\ \Rightarrow\ \boxed{C_{A\infty}=1.0\ \text{mol/L}}.$$
Context — flux into the film. The film modulus is $m=\sqrt{k_1/D_{AB}}=\sqrt{1.5\times10^{-4}/1.26\times10^{-5}}=3.45\ \text{cm}^{-1}$, so $mL=0.690$. The surface flux is $$N_A\big|_{z=0}=D_{AB}\,C_{A0}\,m\tanh(mL)=(1.26\times10^{-5})(4.0\times10^{-4})(3.45)(0.598)=1.04\times10^{-8}\ \text{mol/cm}^2\!\cdot\!\text{s},$$ using $C_{A0}=0.4$ mol/L $=4.0\times10^{-4}$ mol/cm³.
Quantity
Result
Film-face concentration $C_A(0)$
0.4 mol/L $=C_{A0}$
Bulk-solution concentration $C_{A\infty}$
1.0 mol/L
Film modulus $mL$
0.690
Surface flux $N_A|_{z=0}$
$1.04\times10^{-8}$ mol/cm²·s
Check
Because the mass-transfer resistance in the solution is negligible, there is no concentration drop between the bulk and the film face — the entire jump is the equilibrium partition. The partition coefficient (0.4) less than unity means A prefers the solution, so the bulk concentration (1.0 mol/L) exceeds the film-face value (0.4 mol/L), as found. The cosh profile and modulus $m$ are not needed for $C_{A\infty}$ itself; they only quantify the flux.
