Effective Diffusivity in an Immobilized-Enzyme Bead from the Observed Thiele Modulus: 23-Chem-B4 Biochemical Engineering, Undated paper, Q3

Question 3: Effective Diffusivity in an Immobilized-Enzyme Bead from the Observed Thiele Modulus (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.

Convention note (exam Note 1): the observable Thiele modulus is defined here as in Shuler & Kargi, with the characteristic length V_p/S_p = R/3 for a sphere: Φ = (R/3)²r_obs/(D_eS₀). This matches the Thiele modulus φ = (R/3)√(k/D_e) used in that text. The Weisz–Prater form in Fogler and in Bailey & Ollis uses R² instead; with the same data it gives D_e = 1.94×10⁻⁹ m²/s, exactly 9 times larger. That value is about three times the diffusivity of glucose in free water, which is not physically reasonable for diffusion inside a particle, so the R/3 convention is preferred. S₀ is taken as the substrate concentration at the particle surface (no external film resistance).

Given.

Quantity	Symbol	Value
Particle diameter	d_p	1 mm
Particle radius	R	0.5 mm = 5×10⁻⁴ m
Observed reaction rate	r_obs	200 μmol/(cm³ catalyst·min)
Initial (surface) substrate concentration	S₀	100 mol/m³
Observable Thiele modulus	Φ	4.3

Find. The effective diffusivity D_e of the substrate inside the bead (m²/s).

Fig. 2 — immobilized-enzyme bead: strong internal diffusion limitation (Φ=4.3) leaves the particle core substrate-starved (schematic, qualitative profile only — polyline truncated by the plotting window for the steepest part of the curve).

Approach. Convert the observed rate to SI units, then use the observable Thiele modulus. It is built only from quantities measured on the working particle (size, observed rate, surface concentration) plus D_e, so no intrinsic rate constant is needed and it can be solved directly for D_e.

Observed rate in SI units. 1 μmol = 10⁻⁶ mol, 1 cm³ = 10⁻⁶ m³ and 1 min = 60 s, so $$r_{obs}=\frac{200\times10^{-6}\ \text{mol}}{(10^{-6}\ \text{m}^3)(60\ \text{s})}=\boxed{3.333\ \text{mol}/(\text{m}^3\cdot\text{s})}$$
Observable modulus definition. For a sphere the characteristic length is $V_p/S_p=R/3$, and $$\Phi=\eta\,\phi^2=\left(\frac{R}{3}\right)^2\frac{r_{obs}}{D_e\,S_0},\qquad\frac{R}{3}=\frac{5\times10^{-4}}{3}=1.667\times10^{-4}\ \text{m}$$ Unlike the intrinsic modulus $\phi=(R/3)\sqrt{k/D_e}$, this group needs no intrinsic rate constant.
Rearrange and substitute. $$D_e=\frac{(R/3)^2\,r_{obs}}{\Phi\,S_0}=\frac{(1.667\times10^{-4})^2\times3.333}{4.3\times100}=\frac{9.259\times10^{-8}}{430}$$ $$\boxed{D_e=2.15\times10^{-10}\ \text{m}^2/\text{s}}$$
Interpretation. Φ = 4.3 is well above 1, so pore diffusion strongly limits the rate: the substrate is consumed before it can reach the particle centre, leaving a starved core (Fig. 2). The value of D_e is about one-third of the diffusivity of a small sugar such as glucose in free water (about 6.7×10⁻¹⁰ m²/s), which is typical for diffusion through a gel or porous support.

Quantity	Value
Observed rate, r_obs	3.333 mol/(m³·s)
Characteristic length, R/3	1.667×10⁻⁴ m
Effective diffusivity, D_e	2.15×10⁻¹⁰ m²/s
Weisz–Prater (R²) alternative	1.94×10⁻⁹ m²/s

Topic: Internal (pore) diffusion limitation in immobilized-enzyme particles: Thiele modulus, effectiveness factor and the observable modulus.

Key relations: $\phi=(R/3)\sqrt{k/D_e}$ (intrinsic, sphere); $\eta=\dfrac{1}{\phi}\left[\dfrac{1}{\tanh 3\phi}-\dfrac{1}{3\phi}\right]$; $\Phi=\eta\phi^2=(R/3)^2r_{obs}/(D_eS_0)$.

Why this works: The intrinsic rate constant of an immobilized enzyme is rarely known with confidence, because immobilization itself can change it. The observable modulus combines the unknown effectiveness factor and intrinsic modulus into one group made only of measured quantities, so D_e can be found without ever knowing k.

Common pitfall: Forgetting to convert the observed rate from μmol/(cm³·min) to mol/(m³·s), or mixing the (R/3)² and R² definitions of the modulus. The first error puts D_e out by a factor of 60, the second by a factor of 9.

Source: Shuler & Kargi, Bioprocess Engineering, 2nd ed., Ch. 3 (immobilized enzyme kinetics, diffusion limitations); Fogler, Elements of Chemical Reaction Engineering, 4th ed., Ch. 12 (Weisz–Prater criterion).

Question 3 of 5: Effective Diffusivity in an Immobilized-Enzyme Bead from the Observed Thiele Modulus

Question 3: Effective Diffusivity in an Immobilized-Enzyme Bead from the Observed Thiele Modulus (20 marks)