10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 5: Continuous Stirred Tank Reactors (CSTRs)
This lecture covers: Reactions in a perfectly stirred tank. Steady State CSTR.
Continuous Stirred Tank Reactors (CSTRs)
Batch
Continuous
Plug Flow Reactor
CSTR
Open system
Steady State
Continuous
Close system
Well mixed
Transient
Open system
Steady State
Well mixed
Figure 2. A batch reactor. Figure 1. A plug flow reactor, and continuous stirred
tank reactor.
Mole Balance on Component A
In-Out+Production=Accumulation
FAo − FA + rAV = 0 (steady state)
V=
FAo − FA
−rA
FA = FAo − X A FAo
In terms of
conversion, XA?
What volume do you need for a certain amount of conversion?
V=
FAo X A
−rA
where rA is evaluated at the reactor concentration. This is the same as the exit
concentration because the system is well mixed.
For a liquid phase with constant P:
FAo = C Ao v0 (v0 =volumetric flow rate)
FA = C Av0
V C Ao X A
=
v0
−rA
V
τ = ← average time a volume element of fluid stays in the reactor
v0
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
τ=
C Ao X A
−rA
Consider: 1st Order Reaction Kinetics
− rA = kC A concentration or conversion?
Æ convert rate law from CA to XA
C A = C Ao (1 − X A )
− rA = kC Ao (1 − X A )
τ=
C Ao X A
XA
⎪⎫
=
⎬ reactor size in terms of conversion and rate constant
kC Ao (1 − X A ) k (1 − X A ) ⎪⎭
Æ rearrange to find how much conversion for a given reactor size
XA =
τ≡
1
≡
k
τk
1+τ k
average reactor residence time
average time until reaction for a given molecule
We can now define a “Damköhler number”
reaction rate − rAoV
, rAo is the reaction rate law at the feed conditions
Da =
=
flow
FAo
For a liquid at constant pressure with 1st order kinetics:
Da = kτ
⇒ XA =
Da
1 + Da
therefore:
As DaÆ, XAÆ1
As Da∞, XAÆ0 (molecule probably leaves before it can react)
For a liquid at constant pressure with 2nd order kinetics:
− rA = kC A2
= kC Ao 2 (1 − X A )
τ=
2
C Ao X A
C Ao X A
XA
=
=
2
2
−rA
kC Ao (1 − X A )
kC Ao 2 (1 − X A )
solving for conversion:
XA =
(1 + 2τ kC Ao ) −
1 + 4τ kC Ao
2τ kC Ao
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture #5
Page 2 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
Da =
kC Ao 2 V
= τ kC Ao
C Ao vo
Thus, conversion can be put in terms of Da.
XA =
(1 + 2 Da ) −
1 + 4 Da
2 Da
How long does it take for a CSTR to reach steady state?
In-Out+Production=Accumulation
FAo − FA + rAV =
dN A
dt
For a liquid at constant density this is:
C Ao − C A + rAτ = τ
dC A
dt
Ænon-dimensionalize
C
Cˆ A = A
C Ao
t
tˆ =
τ
τ C Ao dCˆ A
C Ao − C Ao Cˆ A − kC Ao Cˆ Aτ =
dtˆ
τ
Da
dCˆ A ⎛ P ⎞ ˆ
+ ⎜ 1 + kτ ⎟ C A = 1
⎟
dtˆ ⎜⎝
⎠
dCˆ A
+ (1 + Da ) Cˆ A = 1
ˆ
dt
with initial conditions:
Cˆ A = 0, tˆ = 0
we have the solution:
Cˆ A =
(
1
ˆ
1 − e− (1+ Da )t
1 + Da
)
In nondimensional terms, it exponentially approaches a new steady state with a
characteristic time
τ
1 + Da
.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture #5
Page 3 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
Cˆ A
1
1 + Da
1
tˆ
1 + Da
Figure 3. Approach to steady state in a continuous stirred tank reactor (CSTR).
The time at which ½ of the steady state concentration of CA is achieved is the half
time:
ln(2)
τ
1 + Da
CSTRs in Series
(Liquid and at constant pressure)
CA0
Da2
Da1
CA2
Figure 4. Two tanks in series. The output of the first tank is the input of the second
tank.
1st order reaction kinetics
C A1 =
C A0
1 + Da1
For the second reactorÆ iterate
C A2 =
C A0
(1 + Da1 )(1 + Da2 )
If the CSTRs are identical,
C An =
C A0
(1+ Da )
n
Æ many CSTRs in series looks like a plug flow reactor.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture #5
Page 4 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].