10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 8: The Plug Flow Reactor
rA = −k [ A]
2
X A FAo = −rAV
X A FAo
nd
VCSTR =
(2 order reaction)
2
2
k [ A]0 (1− X A )
treact . =
VBatch
XA
k [ A]0 (1− X A )
( [ A] ) = ?
FAo =
0
moles A
VBatch =
time
FAo
[ A]0
=
VBatch [ A]0
treact + td
⎡
⎤
XA
⎢t d +
⎥
k [ A]0 (1 − X A ) ⎦⎥
⎣⎢
Assume XA=90%
If treact>td then
VBatch =
2FAo ⋅ 0.9
[ A]0 k [ A]0 (1− 0.9 )
VCSTR =
0.9FAo
k [ A]0
2
≤
1.8FAo
k [ A]0
2
Figure 1. Three tanks in series.
[ A]CSTR = [ A]in + rA
[ ]
[ A]in
If rA = − k A
[ A]out =
1+ Da
V
v0
[ A]in
=
1+
kV
v0
If n CSTRs are in series:
each volume =
V
n
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].
[ A]out =
[ A]in
⎛ kV ⎞
1+ ⎜
⎟
⎝ nv0 ⎠
n
Æimproves productivity:
concentration of A in 1st one is higher than would be in one large CSTR
= [ A]0 e
[ A]out
Batch
− kV
v0
kV
= 3 ⇒ 95% conversions
v0
[ A]0
out
=
[ A]CSTR
⎛ 3⎞
⎜1+ ⎟
⎝ n⎠
series
n
N
1
10
100
XA
.75
.93
.948
PFR
Figure 2. Diagram of a plug flow reactor.
Plug Flow Reactor (behaves like an infinite number of infinitely small CSTRs)
FAin − FAout + rA ( ΔV ) = 0 CSTR
⎛ FAin − FAout ⎞
⎜
⎟ = −rA
ΔV
⎝
⎠
dFA
= −rA design equation for PFR
dV
dFA
= r ( C A ,CB ,...)
dV
= −kC ACB (for example)
FA = C Av0
dFA
F F
= −k A B
dV
v0 v0
dY
= F ( t ,Y ) where t is replaced by V.
This can be expressed as:
dt
Example:
ZZZ
A + B YZ
ZX
ZC
k
k
rev
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 8
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].
⎛ kFA FB krev FC ⎞
+
⎜−
⎟
v02
v0 ⎟
⎜
⎛ FA ⎞
d ⎜ ⎟ ⎜ kFA FB krev FC ⎟
FB = ⎜ −
+
⎟
dV ⎜⎜ ⎟⎟ ⎜
v0 ⎟
v02
FC ⎠
⎝N
⎜ kF F k F ⎟
Y
⎜⎜ + A2 B − rev C ⎟⎟
v0
v0 ⎠
⎝
F ( t ,Y )
z
Figure 3. Diagram of a plug flow reactor showing flow in the z-direction.
dV = area ⋅ dz
Mass flow rate is constant
( v ρ A ) = const.
ρ = ∑ CiWi
dρ
=0
dz
d ( vρ A)
dv
dA
dρ
+ Av
=0
= ρ A + ρv
dz
dz
dz
dz
For a liquid,
Rearrange:
⎛ 1 dA 1 d ρ ⎞
dv
= −v ⎜
+
⎟
dz
⎝ A dz ρ dz ⎠
dA
dρ
=0
= 0 and for a liquid
For a normal pipe
dz
dz
dv
= 0 ⇒ v = v0
Therefore:
dz
(We can’t assume this for gases!)
For a PFR:
dFA
= rA
dV
FA = v [ A]
d ( vC A )
dV
= rA
For liquids, v is constant so we can take it out of the differential.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 8
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].
v dC A
, for liquids
area dz
dC A area
=
rA
dz
v0
rA =
Instead of treact we have zreact!
area ⋅length
v0
XA
area ⋅ z
=
=
v0
k [ A]0 (1− X A )
t pipe =
t PFR
Flow is driven by the pressure drop across the pipe.
P0
Pfinal
Figure 4. Diagram of a pipe showing pressure upstream and downstream.
PV = NRT
P
∑ Ci = RT
⎫
Fi ⎪
⎬ turns F's into concentrations
∑n Fn ⎪⎭
ρ = ∑ CiWi , Wi is molecular weight of i.
P
Ci =
RT
v=
mass flowrate
← const.
ρ (z)
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 8
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].