Effectiveness Factor from Particle-Size Experiments: 23-Chem-A4 Chemical Reactor Engineering, Undated paper, Q4

Question 4: Effectiveness Factor from Particle-Size Experiments (25 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. First-order reaction, isothermal porous spheres, constant $D_{A,eff}$; $k$ and $D_{A,eff}$ independent of size; no external resistance.

Particle radius $R$ (mm)	2	0.5	0.15
Measured rate (mol A/L cat·s)	2.5	8.9	20.1

Find. The internal effectiveness factor $\eta$ for each particle radius.

Figure 4 — In a large pellet the reactant is used up before it reaches the core ($\eta\ll1$). Even the smallest pellet tested still has a noticeable internal gradient ($\eta\approx0.76$).

Approach. None of the pellets can be assumed free of diffusion limits; the rate is still rising steeply as the radius falls. So the intrinsic rate is unknown. For a first-order sphere the observed rate is $r_{obs}=\eta(\phi)\,r_{int}$ with $\eta=\dfrac{3}{\phi^2}(\phi\coth\phi-1)$ and $\phi=R\sqrt{k\rho_c/D_{A,eff}}$ (Fogler Ch. 15). Since $k$ and $D_{A,eff}$ do not depend on size, $\phi\propto R$. That leaves two unknowns, the ratio $\phi/R$ and $r_{int}$. A ratio of two rates removes $r_{int}$ and fixes $\phi/R$, and the third rate is a check.

Size ratio removes the intrinsic rate. For radii $R_1$ and $R_2$: $\dfrac{r_1}{r_2}=\dfrac{\eta(\phi_1)}{\eta(\phi_2)}$ with $\phi_1/\phi_2=R_1/R_2$.
Pair 2 mm / 0.5 mm. $\eta(4\phi_{0.5})/\eta(\phi_{0.5})=2.5/8.9=0.281$. Solving numerically gives $\phi/R=14.1\ \text{mm}^{-1}$. In the strong-diffusion limit $\eta\approx3/\phi$ the ratio would be exactly 0.25, so the 0.5 mm pellet is only partly limited.
Pair 0.5 mm / 0.15 mm (check). $\eta(\phi_{0.5})/\eta(0.3\phi_{0.5})=8.9/20.1=0.443$, which gives $\phi/R=15.6\ \text{mm}^{-1}$. The two estimates agree within 10%, so the data are consistent with a single Thiele modulus.
Best fit to all three rates. Minimizing the relative error over the three points gives $\phi/R=15.3\ \text{mm}^{-1}$ and $r_{int}=26.3\ \text{mol/L}\cdot\text{s}$. The Thiele moduli are $\phi=30.6,\ 7.66,\ 2.30$, and the fitted model reproduces all three measured rates within 1%.
Effectiveness factors. $$\boxed{\eta_{2\,\text{mm}}=0.095,\qquad \eta_{0.5\,\text{mm}}=0.34,\qquad \eta_{0.15\,\text{mm}}=0.76}$$ As a hand check on the largest pellet, the strong-diffusion asymptote $\eta\approx3/\phi=3/30.6=0.098$ agrees with the exact value.

The physical reading: the 2 mm pellet does less than a tenth of the work its catalyst could do. Even the 0.15 mm powder loses about a quarter of its activity to pore diffusion. An intrinsic rate of about 26 mol/L·s would only be reached with particles well below 0.1 mm radius ($\phi<1$ requires $R<0.065$ mm).

Check

A tempting shortcut is to call the smallest particle diffusion-free and divide by 20.1. That gives $\eta=0.124,\ 0.443,\ 1.00$, which is inconsistent with the data: if $\eta_{0.15}=1$ then $\phi_{0.15}<1$, so $\phi_{0.5}<3.3$ and $\eta_{0.5}$ would have to exceed 0.6, not 0.44. The simultaneous fit above removes that contradiction.

Particle radius	Thiele modulus $\phi$	Effectiveness factor $\eta$
2 mm	30.6	0.095
0.5 mm	7.66	0.34
0.15 mm	2.30	0.76
Intrinsic rate	26.3 mol A/(L cat·s)

Concept & Theory

Topic. Internal (pore) diffusion in catalyst pellets: Thiele modulus and effectiveness factor.

Key relations. $\eta=r_{obs}/r_{int}=\dfrac{3}{\phi^2}(\phi\coth\phi-1)$ for a first-order sphere; $\phi=R\sqrt{k\rho_c/D_{eff}}\propto R$; limits $\eta\to1$ for $\phi\ll1$ and $\eta\to3/\phi$ for $\phi\gg1$.

Why this works. Reactant diffuses in while it reacts. When the reaction is fast compared with diffusion, only an outer layer of the pellet works, and that layer's thickness does not depend on the pellet size. Its share of the pellet volume therefore scales as $1/R$. Because $k$ and $D_{eff}$ are the same for every size, one value of $\phi/R$ must describe all the pellets, which is what lets two or three rate measurements fix both unknowns.

Common pitfall. Assuming the smallest pellet tested has $\eta=1$ without checking that its rate has stopped changing with size. Here the rate is still rising 2.3-fold from 0.5 to 0.15 mm, so no plateau has been reached.

Source. Fogler, Elements of Chemical Reaction Engineering, Ch. 15 (Thiele modulus, internal effectiveness factor); Levenspiel, Chemical Reaction Engineering, Ch. 18.

Question 4 of 5: Effectiveness Factor from Particle-Size Experiments

Question 4: Effectiveness Factor from Particle-Size Experiments (25 marks)