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CHE-27

Thiele modulus and effectiveness

φ = R √(k/De), η = tanh(φ)/φ for a first-order slab/sphere snapshot.

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DimensionlessThiele 1939

Governing equation

φ=Rk/De,η=tanh⁡φ/φ\varphi=R\sqrt{k/D_e},\quad\eta=\tanh\varphi/\varphi

k∈[0.01,50]k \in \left[0.01, 50\right]

where

R
Pellet radius (mm)
k
Rate constant (1/s)
D_e
Effective diffusivity (mm²/s)
\varphi
Thiele modulus (—)
\eta
Effectiveness (—)

The figure, part by part

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CHE-27 · reactor
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①

Governing law — red

φ=Rk/De,η=tanh⁡φ/φ\varphi=R\sqrt{k/D_e},\quad\eta=\tanh\varphi/\varphi

k∈[0.01,50]k \in \left[0.01, 50\right]

k=0.01k = 0.01φ=0.798\varphi = 0.798

②

Domain of the variable — blue

kmin⁡=0.01,kmax⁡=50k_{\min} = 0.01,\quad k_{\max} = 50

k∈[0.01,50]k \in \left[0.01, 50\right]

k=0.01k = 0.01k=0.01k = 0.01

③

Response curve — green

φ=Rk/De,η=tanh⁡φ/φ\varphi=R\sqrt{k/D_e},\quad\eta=\tanh\varphi/\varphi

k∈[0.01,50]k \in \left[0.01, 50\right]

k=0.01k = 0.01φ=0.798\varphi = 0.798

④

Rate of change — amber

dφdk\frac{\mathrm{d}\varphi}{\mathrm{d}k}

k∈[0.01,50]k \in \left[0.01, 50\right]

k=0.324k = 0.324φ′=−1.046\varphi^{\prime} = -1.046

Lecture brief

Historical brief

From CSTR/PFR mole balances and Arrhenius rates to McCabe–Thiele stages and NTU exchangers, chemical engineering is conservation plus equilibrium. The lab is that design arithmetic. This sheet (CHE-27 — Thiele modulus and effectiveness) is the form associated with Thiele 1939. Working symbols: RR, kk, DeD_e →\rightarrow φ\varphi, η\eta. Thiele compared reaction time to diffusion time in a catalyst pellet. Large φ means a steep intraparticle gradient and low η.

Purpose

Purpose: compute φ\varphi, η\eta from RR, kk, DeD_e in Chemical engineering via φ=Rk/De,η=tanh⁡φ/φ\varphi=R\sqrt{k/D_e},\quad\eta=\tanh\varphi/\varphi φ = R √(k/De), η = tanh(φ)/φ for a first-order slab/sphere snapshot. Use it when a real chemical engineering question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given R=2.000 mmR = 2.000\,\mathrm{mm}, k=2.000 1/sk = 2.000\,\mathrm{1/s}, De=0.050 mm2/sD_e = 0.050\,\mathrm{mm^{2}/s}, the governing relation φ=Rk/De,η=tanh⁡φ/φ\varphi=R\sqrt{k/D_e},\quad\eta=\tanh\varphi/\varphi yields φ=12.649 —\varphi = 12.649\,\mathrm{—}, η=0.079 —\eta = 0.079\,\mathrm{—}. A pellet, a concentration sag, an η bar. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Thiele modulus \varphi12.649 —
  • Effectiveness \eta0.079 —
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CHE-27 · reactor
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Narration of this film

A pellet, a concentration sag, an η bar.

Thiele compared reaction time to diffusion time in a catalyst pellet. Large φ means a steep intraparticle gradient and low η.

Reading speed

Watch on YouTube