Immobilized-Enzyme Packed-Bed Column — Specific Surface Area and Effective Diffusivity: 23-Chem-B4 Biochemical Engineering, December 2014, Q2

Question 2: Immobilized-Enzyme Packed-Bed Column — Specific Surface Area and Effective Diffusivity (20 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.

Quantity	Symbol	Value
Column diameter	D_col	0.20 m
Column (bed) height	H	2.0 m
Mass of catalyst particles	m	18 kg
Particle diameter, part (a)	d_p	2 mm
Particle density	ρ_p	1500 kg/m³
Observed reaction rate, part (b)	r_obs	200 µmol·cm^-3·min^-1
Bulk substrate concentration	S₀	100 mol/m³
Observable Thiele modulus	φ_obs	4.3
Bead diameter, part (b)	d_p,b	1 mm

Find. (a) The specific external surface area α of the packed bed, in cm^-1; (b) the effective substrate diffusivity D_e inside the 1 mm bead, in m²/s.

Approach. (a) Get the packed-bed volume from column geometry and the total particle volume from mass/density; a sphere's surface-to-volume ratio 6/d_p then converts particle volume directly to total particle surface area, which is divided by bed volume for α. (b) Rearrange the observable Thiele modulus $\phi_{obs}=(V_p/S_p)\sqrt{r_{obs}/(D_eS_0)}$ — built from the OBSERVED (measured) rate rather than an intrinsic rate constant, so it needs no separate kinetic-constant estimate — for D_e.

Fig. 2 — immobilized-enzyme packed-bed column: the specific surface area α converts particle surface area into an area-per-unit-reactor-volume basis.

(a) Packed-bed volume from column geometry. $$V_{bed}=\frac{\pi}{4}D_{col}^2H=\frac{\pi}{4}(0.20)^2(2.0)=\boxed{0.06283\ \text{m}^3}$$
Total particle volume from the mass/density basis. $$V_p=\frac{m}{\rho_p}=\frac{18}{1500}=0.01200\ \text{m}^3$$
Total particle surface area via the sphere identity area/volume = 6/d_p. For a sphere, $S_p/V_p=(\pi d_p^2)/(\tfrac{\pi}{6}d_p^3)=6/d_p$, so the TOTAL surface area of all particles follows directly from their total volume without needing the particle count individually: $$S_{p,tot}=V_p\times\frac{6}{d_p}=0.01200\times\frac{6}{0.002}=\boxed{36.0\ \text{m}^2}$$ (equivalently, N=V_p/(πd_p³/6)≈2.86×10⁶ particles, each of area πd_p²≈1.257×10^-5 m².)
Specific surface area α = total particle area / bed volume. $$\alpha=\frac{S_{p,tot}}{V_{bed}}=\frac{36.0}{0.06283}=572.9\ \text{m}^{-1} =\boxed{5.73\ \text{cm}^{-1}}$$
(b) Convert the observed rate to SI. 1 µmol·cm^-3·min^-1 numerically equals 1 mol·m^-3·min^-1 (the 10^-6 mol/µmol and 10^-6 m³/cm³ factors cancel), so: $$r_{obs}=\frac{200}{60}=3.333\ \text{mol}\cdot\text{m}^{-3}\cdot\text{s}^{-1}$$
Characteristic diffusion length of the sphere. $$\frac{V_p}{S_p}=\frac{R}{3}=\frac{0.5\times10^{-3}}{3}=1.667\times10^{-4}\ \text{m}$$
Rearrange the observable Thiele modulus for D_e. Squaring $\phi_{obs}=(V_p/S_p)\sqrt{r_{obs}/(D_eS_0)}$ and solving for D_e: $$D_e=\frac{(V_p/S_p)^2\,r_{obs}}{S_0\,\phi_{obs}^2} =\frac{(1.667\times10^{-4})^2(3.333)}{(100)(4.3)^2}=\boxed{5.01\times10^{-11}\ \text{m}^2/\text{s}}$$ This is a plausible effective diffusivity for a substrate inside a gel-type immobilization matrix (typically 10^-10–10^-11 m²/s, well below the free-solution value).

Quantity	Value
Bed volume, V_bed	0.0628 m³
Total particle volume, V_p	0.0120 m³
Total particle surface area	36.0 m²
(a) Specific surface area, α	5.73 cm^-1 (573 m^-1)
V_p/S_p for the 1 mm bead	1.667×10^-4 m
(b) Effective diffusivity, D_e	5.01×10^-11 m²/s

Topic: Packed-bed reactor geometry (specific surface area) and internal diffusion characterization via the observable Thiele modulus.

Key relations: sphere area/volume = 6/d_p; α = total particle area/bed volume; observable Thiele modulus $\phi_{obs}=(V_p/S_p)\sqrt{r_{obs}/(D_eS_0)}$, built from the MEASURED rate r_obs rather than an intrinsic rate constant.

Why this works: Because φ_obs uses the already-diffusion-affected observed rate (rather than an intrinsic kinetic constant that would itself need to be back-calculated from an effectiveness factor), it lets an engineer estimate D_e directly from a single measured rate at known bulk concentration — no separate kinetic experiment at negligible particle size is required.

Common pitfall: Forgetting the sphere's characteristic length is R/3 (not R or d_p) when inverting for D_e, and mismatching time units between the rate (per minute) and the modulus convention (per second) — both silently introduce large multiplicative errors in D_e.

Source: Bailey & Ollis, Biochemical Engineering Fundamentals, 2nd ed., Ch. 5 (immobilized enzyme kinetics, generalized/observable Thiele modulus); Shuler & Kargi, Bioprocess Engineering, 2nd ed., Ch. 8 (packed-bed reactor geometry).

Question 2 of 5: Immobilized-Enzyme Packed-Bed Column — Specific Surface Area and Effective Diffusivity

Question 2: Immobilized-Enzyme Packed-Bed Column — Specific Surface Area and Effective Diffusivity (20 marks)