10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam 1 Review
In-Out+Production=Accumulation
Accumulation=0 at steady state
FA0 − FA + rAV = 0
FA0 = [ A]0 ν 0
FA = [ A]ν
For a liquid phase with constant density:
ν0 =ν
For A → B the reaction moles are the same, so
For A → 2 B ,
ξ [ =]
ν0 =ν
ν0 ≠ν
moles (extent of reaction)
(-) for a reactant and (+) for a product
N rxns
N i = N i 0 + ∑ υi ,nξ n
n =1
Suppose A → B + C
XA =
N A0 − N A
N A0
N A = N A0 (1 − X A )
Thermodynamics
Suppose
Ke = e
k1
ZZZ
X
A + B YZZ
Z 2C
k
−1
− ΔG RT
ΔG = ΔG °f , products − ΔG °f ,reactants
Kc =
[ B ][C ]
[ A]
has units. You need to use standard states, such as 1M, to make it
dimensionless.
Enzyme Catalysis
Energy
without enzyme
with enzyme
S
P
Reaction Progress
Figure 1. Energy diagram for a reaction with and without enzyme.
Cite as: William Green, Jr., and K. Dane Wittrup, 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].
E + S U ES
ES → E + P
Pseudo steady state approximation:
d [ ES ]
=0
dt
[ ES ] = f (other species)
Cell Growth
# cells
N=
volume
N = N0e μ t
Monod kinetics:
μ=
μmax [ S ]
KS + [S ]
YA =
B
ΔA
ΔB
Rate Constants
k (T ) = Ae − Ea
RT
Given k1 and k2, you can calculate k and a different temperature.
CSTRs
V=
FA0 X A
−rA
Incorporates changing
volumetric flow rate
If the reaction is 1st order and it consists of liquids with constant density:
XA
k (1 − X A )
V
volume
τ= =
τ=
ν0
volumetric flow rate
τk
1+ τ k
Da = τ k =Damköhler number: ratio of kinetic effect to volumetric effect or
XA =
2nd order reaction:
ratio of reaction rate to dilution rate
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam1 Review
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τ=
XA
kC A0 (1 − X A )
2
Da = τ kC A0
XA =
1 + 2 Da − 1 + 4 Da
2 Da
For constant density and 1st order reaction:
dN A
dt
dC A
C A0 − C A + rAτ = τ
dt
FA0 − FA + rAV =
Let:
C
Cˆ A = A
C A0
t
tˆ =
τ
Nondimesionalize:
dĈ A
+ (1 + Da)Ĉ A = 1
dtˆ
Solve given Cˆ A = 0 at tˆ = 0
1
ˆ
1 − e− (1+ Da )t
Cˆ A =
1 + Da
(
)
Tanks in series: 1st order reaction
C A, n =
C A,0
(1+ Da )
n
Reactor Design Equations
FA0 X A
−rA
dN A
Batch: rAV =
dt
N A = V [ A] If V changes, then V must remain in the differential
CSTR:
PFR:
V=
dX A −rA
=
Adz FA0
PBR: Pressure drop consideration
If
A( g ) → 2 B( g )
Use
ν 0 ρ0 = νρ
(conservation of mass)
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam1 Review
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Cite as: William Green, Jr., and K. Dane Wittrup, 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].
Introduce ideal gas law
Pm
FT RT
ρ=
FT = FT 0 (1 + ε X )
⎛
⎞
m
= Pν = P = FT RT ⎟
⎜ nRT
ρ
⎝
⎠
Reactor volume:
positive order
reactions
VCSTR
FAo
− rA
VPFR (area under curve)
XA
Figure 2. Levenspiel plot for a CSTR and a PFR for positive order reactions.
Selectivity
CSTR
φinst
Φ overall ΔC A
C Af
C A0
CA
Figure 3. Fractional yield versus concentration. Overall yield times concentration
difference shown for a CSTR.
CSTR
k1
k2
A ⎯⎯
→ P ⎯⎯
→C
k3
A ⎯⎯
→U
( k1 [ A] + k3 [ A])V = FA0 − [ A]ν
Non-ideal reactors
Residence time distribution E(t)
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam1 Review
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Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
reactor
δ (t ) ⎯⎯⎯
→ C (t )
E(t) must have a pulse trace
E (t ) =
C (t )
∞
∫ C (t )dt
0
∫ E (t ) = 1
∞
0
∞
tm = ∫ tE (t )dt
0
Mean residence time, tm, for an:
Ideal CSTR: τ
Ideal PFR: τ
∞
σ 2 = ∫ ( t − tm ) E (t )dt
2
0
Variance, σ2, for an:
Ideal CSTR: τ2
Ideal PFR: 0
⎛
⎛ V
⎜ E (t ) = δ ⎜ t −
⎝ ν
⎝
⎞⎞
⎟⎟
⎠⎠
∞
∫ f (t )δ (t − t )dt = f (t )
0
0
Åproperty of a dirac delta function
0
For a CSTR, E (t ) =
e−t τ
τ
Example 1
z
L
FA0
FA, FB, FC
Figure 4. Schematic of a PFR with inflow of A and outflow of A, B, and C.
r1
r2
A ⎯⎯
→ B ⎯⎯
→C
r1 = k1C A r2 = k2CB
YB =
moles of B produced
moles of A in
=
FB ( L)
FA0
Mole balance on B
1 dFB
= rB = r1 − r2 = k1C A − k2CB
Axs dz
FB = CBν 0
FA0 = C A0ν 0
10.37 Chemical and Biological Reaction Engineering, Spring 2007
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Technology. Downloaded on [DD Month YYYY].
ν 0 dCB
= k1C A − k2CB
Axs dz
ν 0 dC A
= −k1C A
Axs dz
dC A −k1 Axs
=
dz
CA
ν0
−k A
ln C A = 1 xs z + φ
ν0
Initial condition at z=0 gives:
ln C A0 = 0 + φ
⎡ −k A ⎤
C A = C A0 exp ⎢ 1 xs z ⎥
⎣ ν0
⎦
⎡ −k A
kA
dCB k2 Axs
+
CB = 1 xs C A0 exp ⎢ 1 xs
dz
ν0
ν0
⎣ ν0
Axs
ν0
⎤
z⎥
⎦
z [ = ] time, call it τ
This is the time it takes for something to flow to the end of the reactor (of length z).
ν0
G
= V velocity
Axs
dCB
+ k2CB = k1C A0 e− k1τ
dτ
kτ
Integrating factor: e 2
d
⎡⎣CB ek2τ ⎤⎦ = k1C A0 e( k2 − k1 )τ
dτ
kC
k −k τ
CB ek1τ = 1 A0 e( 2 1 ) + φ
k2 − k1
Initial condition: z=0, CB=0
B
kC
0 = 1 A0 + φ
k2 − k1
kC
CB (τ ) = 1 A0 ⎡⎣e− k1τ − e− k2τ ⎤⎦
k2 − k1
1 dFC
= + r2
Axs dz
1 dFA
= −r1
Axs dz
1 dFB
= r1 − r2
Axs dz
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam1 Review
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Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
1 ⎡ dFA dFB dFC ⎤
+
+
=0
Axs ⎢⎣ dz
dz
dz ⎥⎦
FA0 = FA + FB + FC
FC = FA0 − FA + FB
F
find
A
C
B
τ∗
τ
Figure 5. Graphs of flow rates of A, B, and C as a function of residence time.
(
)
dCB k1C A0
=
−k1e− k1τ * + k2 e− k2τ * = 0
dτ
k2 − k1
k1e− k1τ * = k2 e− k2τ *
ln k1 − k1τ * = ln k2 − k2τ *
k
ln 1 = ( k1 − k2 )τ *
k2
k
ln 1
k2
τ* =
k1 − k2
dA( x)
A( x) 0
dx
L’Hopital’s rule: lim
= do lim
x → x* B ( x )
x → x*
dy
0
dB ( x)
dx
dx
Find τ* for k1=k2
1
k1 1 1
= =
k1 → k2
k1 → k2 1
k2 k
A L
A z
τ * = xs
τ = xs
lim τ * = lim
ν0
ν0
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam1 Review
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Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
L=
τ *ν 0
Axs
= length of reactor
Example 2
If A → 2 B (Assume negligible pressure drop)
r = kC A
z
L
FA0
Figure 6. Schematic of a PFR.
dFA
= −rA = − kC A
dV
ν0 ↔ν
PV = nRT
( Ftotal = n , ν = V )
RT
−1
ν = Ftotal
= Ftotal Ctotal
P
Ftotal = FA + FB
FA = C Aν = y A Ftotal
dFA
FA
P
= − ky ACtotal = − k
dV
FA + FB RT
dFB
FA
P
= +2ky ACtotal = +2k
dV
FA + FB RT
If P or T changes, you need other equations.
Derivation of E(t) for a CSTR
N 0δ (t )
Figure 7. Schematic of a CSTR.
No reaction:
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam1 Review
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Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
dN
= N 0δ (t ) −ν C
dt
N
= N 0δ (t ) −ν
V
dN ν
+ N = N 0δ (t )
dt V
⎛ν ⎞
t⎟
⎝V ⎠
d ⎛
⎛ν ⎞⎞
⎛ν ⎞
⎜ N exp ⎜ t ⎟ ⎟ = exp ⎜ t ⎟ N 0δ (t )
dt ⎝
⎝V ⎠⎠
⎝V ⎠
⎛ν ⎞
⎛ν ⎞
N exp ⎜ t ⎟ = N 0 ⋅ exp ⎜ 0 ⎟ = N 0 + φ
⎝V ⎠
⎝V ⎠
Integrating factor: exp ⎜
Initial condition: t=0, N=N0 Æ φ=0
⎛ ν ⎞
N = N 0 exp ⎜ − t ⎟
⎝ V ⎠
ν 1
=
V τ
⎛ −t ⎞
C = C0 exp ⎜ ⎟
⎝τ ⎠
C
E (t ) = ∞
∫ Cdt
0
∞
∞
∫ Cdt = ∫ C0e
0
−t
τ
0
−t
dt = C0 ⎡ −τ e
⎣⎢
−t
τ
∞
⎤ = C ⎡ −τ ( 0 − 1) ⎤ = C τ
0 ⎣
0
⎦
⎦⎥ 0
−t
Ce τ e τ
=
E (t ) = 0
τ
C0τ
Long-chain approximation
k2
1
ZZZ
X
E + S YZZ
→P+ E
Z ES ⎯⎯
k
k
−1
tRNA ⎯⎯
→E
k3
k4
E ⎯⎯
→ Deactivate
Enzyme propagates a long time before it is destroyed.
LCA: k3 tRNA = k 4 E
[
]
[ ]
st
(assume 1 order)
If there are other steps, add them into the equation
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam1 Review
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Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
k5
ES ⎯⎯
→ destruction
k3 [tRNA] = k4 [ E ] + k5 [ ES ]
Suppose there is a production term
k6
C ⎯⎯
→E
Add another term
k6 [C ] + k3 [tRNA] = k4 [ E ] + k5 [ ES ]
F
dX
= A0 , − rA = kC A
Adz −rA
This is a single differential equation in terms of X. Use for PFR with gas flow.
ρ0ν 0 = ρν
CA =
C A0 (1 − X ) T0 P
1+ ε X
P0T
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Exam1 Review
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Cite as: William Green, Jr., and K. Dane Wittrup, 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].