External-Mass-Transfer-Limited Immobilized-Enzyme Plate Reactor: 20-Bio-A5 Systems Analysis & Control, Undated paper, Q3

Question 3: External-Mass-Transfer-Limited Immobilized-Enzyme Plate Reactor (a. 15 marks; b. 5 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. Well-mixed batch liquid of volume $V_L$; enzyme immobilized as a monolayer on plates of total area $A_T$; external film mass-transfer coefficient $k_L$; pseudo-first-order surface reaction $(-r_s')=V_m'S_WA_T$.

Find. (a) the two governing differential/flux-balance equations linking $S_L(t)$ (bulk), $S_W$ (surface) and the two rate processes in series; (b) the closed-form $S_L(t)$.

Fig. 3 — two resistances in series: external-film mass transfer ($k_L$, bulk $S_L$ to surface $S_W$) feeding a pseudo-first-order surface reaction ($V_m'$, per unit area $A_T$).

Approach. Write the unsteady bulk-phase substrate balance driven by the film flux, and the pseudo-steady surface balance (film flux = surface reaction rate, since the film itself holds negligible substrate inventory); combine the two resistances in series into one overall first-order rate constant to get $S_L(t)$.

(a) Bulk-phase unsteady balance. The well-mixed bulk liquid loses substrate only by transfer across the film to the plate surface: $$V_L\frac{dS_L}{dt}=-k_LA_T\big(S_L-S_W\big).$$
(a) Surface (film) balance. The film itself stores negligible substrate, so at every instant the flux arriving at the surface equals the rate consumed by the immobilized enzyme (pseudo-steady-state at the surface): $$k_LA_T\big(S_L-S_W\big)=V_m'S_WA_T\ \ \Rightarrow\ \ k_L\big(S_L-S_W\big)=V_m'S_W.$$
Eliminate $S_W$. Solving Step 2 for the surface concentration: $$S_W=\frac{k_L}{k_L+V_m'}S_L.$$
(b) Combine resistances into an overall rate constant. Substituting $S_W$ back into Step 1: $$V_L\frac{dS_L}{dt}=-k_LA_T\left(S_L-\frac{k_L}{k_L+V_m'}S_L\right)=-A_T\underbrace{\left(\frac{k_LV_m'}{k_L+V_m'}\right)}_{k_{\text{eff}}}S_L,$$ i.e. film resistance and surface-reaction resistance add like resistors in series, $1/k_{\text{eff}}=1/k_L+1/V_m'$.
(b) Integrate. $$\frac{dS_L}{S_L}=-\frac{k_{\text{eff}}A_T}{V_L}dt\ \ \Rightarrow\ \ \boxed{S_L(t)=S_0\exp\!\left(-\frac{k_{\text{eff}}A_T}{V_L}t\right),\qquad k_{\text{eff}}=\frac{k_LV_m'}{k_L+V_m'}.}$$

Result	Expression
Bulk balance	$V_L\,dS_L/dt=-k_LA_T(S_L-S_W)$
Surface pseudo-steady balance	$k_L(S_L-S_W)=V_m'S_W$
Effective rate constant	$k_{\text{eff}}=k_LV_m'/(k_L+V_m')$
$S_L(t)$	$S_0\exp(-k_{\text{eff}}A_T t/V_L)$

Concept & Theory

Topic. Series film-diffusion / surface-reaction resistance for an immobilized enzyme, first-order limit.

Key relations. Film flux $=k_LA_T(S_L-S_W)$; surface reaction $=V_m'S_WA_T$; pseudo-steady-state at the surface (no accumulation in the infinitesimally thin film) sets flux = reaction rate; the two resistances combine as $1/k_{\text{eff}}=1/k_L+1/V_m'$, exactly like two resistors in series.

Why this works. Because the reaction is taken as first order (linear in $S_W$), eliminating the unmeasurable surface concentration $S_W$ collapses the two coupled rate processes into a single effective first-order decay of the bulk concentration — the batch reactor behaves exactly like a simple first-order depletion with rate constant $k_{\text{eff}}$.

Common pitfall. Assuming $S_W=S_L$ (ignoring film resistance) overstates the consumption rate; assuming the surface reaction alone controls (using $V_m'$ in place of $k_{\text{eff}}$) understates the resistance whenever $k_L$ is not much larger than $V_m'$ — both resistances must be combined, not the smaller one dropped, unless one is shown to dominate.

Source. Bailey & Ollis, Biochemical Engineering Fundamentals, Ch. 8 (external mass-transfer limitation, immobilized-enzyme systems).

Question 3 of 6: External-Mass-Transfer-Limited Immobilized-Enzyme Plate Reactor

Question 3: External-Mass-Transfer-Limited Immobilized-Enzyme Plate Reactor (a. 15 marks; b. 5 marks)