SME
Chemical Reaction Engineering · Problem type
Effectiveness factor of a pellet
Use this for “A first-order reaction (k = 10 s⁻¹) runs in spherical pellets 3 mm in radius with Dₑ = 10⁻⁶ m²/s. Find the Thiele modulus and the effectiveness factor.”
Chart
η(9.487) ≈ 0.2829
This is an example. Type your own values to start a new problem.
φ
=
R (mm)
×
k (s⁻¹)
÷
Dₑ (m²/s)
η
=
3
÷
φ
2
×
(
φ
×
coth(
φ
)
−
1)
Formulas
φ
=
R
×
√(
k
Dₑ
)
9.487
=
0.003
×
√(
10
1 ×
10
-6
)
η
=
3
φ
2
×
(
φ
×
coth(
φ
)
−
1)
0.2829
=
3
9.487
2
×
(9.487
×
coth(9.487)
−
1)
Formulas are worked in m. The step-by-step shows the conversions.
Assumptions
•
A first-order, isothermal reaction in a spherical pellet, with the surface at the bulk concentration.
•
The Thiele modulus φ = R√(k ÷ Dₑ) compares the reaction rate with diffusion into the pores.
•
η is the actual rate over the rate if the whole pellet saw the surface concentration; for large φ, η ≈ 3 ÷ φ.
Step-by-step
WE KNOW
R = 3 mm, k = 10 s⁻¹, Dₑ = 1 × 10⁻⁶ m²/s
FIND
Thiele modulus (φ), effectiveness factor (η)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m).
R = 3 mm = 0.003 m (1 m = 1000 mm)
Step 2 · Find Thiele modulus (φ)
Use φ = R × √(k ÷ Dₑ)
The
root
of
k
Dₑ
is
one
over
the
length
diffusion
reaches
before
the
reaction
uses
A
up;
R
over
that
length
is
φ
.
φ
=
0.003
×
√(
10
10
-6
)
φ
=
0.003
×
3,162.3
φ
=
9.487
Step 3 · Find effectiveness factor (η)
Use η = 3 ÷ φ² × (φ × coth(φ) − 1)
Solve diffusion with reaction in a sphere and compare its rate with the rate at the surface concentration.
η
=
3
9.487
2
×
(9.487
×
coth(9.487)
−
1)
η
=
3
9.487
2
×
(9.487
−
1)
η
=
3
9.487
2
×
8.487
η
=
3
90
×
8.487
η
=
0.03333
×
8.487
η
=
0.2829
Check
Put the numbers back in. Both sides should match.
9.487
=
0.003
×
√(
10
10
-6
)
✓
0.2829
=
3
9.487
2
×
(9.487
×
coth(9.487)
−
1)
✓
Topic check
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