SME
Chemical Reaction Engineering · Problem type
CSTRs in series
Use this for “Two CSTRs, each with 5 min space time, run a first-order reaction with k = 0.2 per min. What conversion leaves the second tank?”
Chart
This is an example. Type your own values to start a new problem.
X
=
1
−
1
(1
+
k (min⁻¹)
×
τ (min)
)
X₁
After one tank
worked out (not typed)
Xbig
One big tank
worked out (not typed)
Formulas
X
=
1
−
1
÷
(1 +
k
×
τ
)
n
0.75
=
1
−
1
÷
(1 + 0.2 × 5)
2
X₁
=
k
×
τ
1 +
k
×
τ
0.5
=
0.2
×
5
1 + 0.2 × 5
Xbig
=
n
×
k
×
τ
1 +
n
×
k
×
τ
0.6667
=
2
×
0.2
×
5
1 + 2 × 0.2 × 5
Assumptions
•
First order, isothermal, liquid phase; n equal, well-mixed CSTRs in series.
•
Each tank passes on 1 ÷ (1 + kτ) of the A it receives, so the shares multiply.
•
Xbig is one tank holding all n volumes (space time nτ), for comparison.
Step-by-step
WE KNOW
k = 0.2 min⁻¹, τ = 5 min, n = 2
FIND
conversion after n tanks (X), after one tank (X₁), one big tank (Xbig)
Step 1 · Find conversion after n tanks (X)
Use X = 1 − 1 ÷ (1 + k × τ)ⁿ
Each
tank
leaves
1
1 +
kτ
of
its
feed
unreacted;
n
tanks
leave
that
share
to
the
nth
power.
X
=
1
−
1
÷
(1 + 0.2 × 5)
2
X
=
1
−
1
÷
(1 + 1)
2
X
=
1
−
1
2
2
X
=
1
−
1
4
X
=
1
−
0.25
X
=
0.75
Step 2 · Find after one tank (X₁)
Use X₁ = k × τ ÷ (1 + k × τ)
One
CSTR
balance,
X
=
kτ(1
−
X
),
solved
for
X
.
X₁
=
0.2
×
5
1 + 0.2 × 5
X₁
=
0.2
×
5
1 + 1
X₁
=
0.2
×
5
2
X₁
=
1
2
X₁
=
0.5
Step 3 · Find one big tank (Xbig)
Use Xbig = n × k × τ ÷ (1 + n × k × τ)
The same CSTR result with space time nτ: one tank as big as all of them.
Xbig
=
2
×
0.2
×
5
1 + 2 × 0.2 × 5
Xbig
=
2
×
0.2
×
5
1 + 0.4 × 5
Xbig
=
2
×
0.2
×
5
1 + 2
Xbig
=
2
×
0.2
×
5
3
Xbig
=
0.4
×
5
3
Xbig
=
2
3
Xbig = 0.6667
Check
Put the numbers back in. Both sides should match.
0.75
=
1
−
1
÷
(1 + 0.2 × 5)
2
✓
0.5
=
0.2
×
5
1 + 0.2 × 5
✓
0.6667
=
2
×
0.2
×
5
1 + 2 × 0.2 × 5
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Batch, CSTR and PFR design
Main lesson
Second-order reaction in each reactor
Use this for “A second-order liquid reaction, k = 0.5 L/(mol·min), is fed at 2 mol/L. What space times reach 80% conversion in a CSTR and in a PFR?”
Batch time for a conversion
Use this for “A first-order reaction with k = 0.2 per min runs in a batch reactor. How long until 90% of A has reacted?”
Levenspiel plot: which reactor is smaller
Use this for “20 mol/min of A at 2 mol/L reacts first order, k = 0.2 per min. Which needs less volume for 80% conversion? Read the CSTR as the rectangle and the PFR as the area under FA0 ÷ (−rA).”