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Chemical Reaction Engineering (CRE) is the field that studies the rates and mechanisms of chemical reactions and the design of the reactors in which they take place. Lecture 17
Chemical Reaction Engineering (CRE) is the field that studies the rates and mechanisms of
chemical reactions and the design of the reactors in which they take place.
Lecture 17
Today’s lectureEnergy Balance Fundamentals
Adibatic reactors
2
Energy Balances, Rationale and Overview
Let’s calculate the volume necessary to achieve a conversion, X, in a PFR for a first-order, exothermic and adiabatic reaction.
The temperature profile might look something like this:T
V
k
V
X
V
Todays Lecture
3
The combined mole balance, rate law and stoichiometry yield:
0A
A
F
r
dV
dX
A1
iA CT
1
T
1
R
Eexpkr
X1CF
T1
T1
RE
expk
dV
dX0A
0A
1i
Energy Balances, Rationale and Overview
4
We cannot solve this Equation because we don’t have X either as a function of V or T. We need another Equation. That Equation is:
X1CF
T1
T1
RE
expk
dV
dX0A
0A
1i
Energy Balances, Rationale and Overview
5
The Energy Balance
User Friendly Equations Relate T and X or Fi
1. Adiabatic CSTR, PFR, Batch or PBR
€
˙ W S = 0 Δ ˆ C P = 0
XEB i CPi
T T0 H o
Rx
X ˜ C PA
T T0 HRx
T T0 H o
Rx XEB
iCPi6
7
Adiabatic
T
XEB
Exothermic
T0
0
T
XEB
Endothermic
T0
0
2. CSTR with heat exchanger, UA(Ta-T) and a large coolant flow rate
XEB
UA
FA 0
T T a
iCPi T T 0
H oRx
T
Ta
Cm
User Friendly Equations Relate T and X or Fi
8
3. PFR/PBR with heat exchange
FA0
T0
CoolantTa
User Friendly Equations Relate T and X or Fi
3A. PFR in terms of conversion
€
dT
dV=
rA ΔHRx T( )
Qg6 7 4 8 4 −Ua T − Ta( )
Qr6 7 4 8 4
FA 0 ΘiCPi + ΔCp X∑( )=
Qg − Qr
FA 0 ΘiCPi + ΔCp X∑( )9
User Friendly Equations Relate T and X or Fi
3B. PBR in terms of conversion
dT
dW
rAHRx T Ua
b
T Ta
FA 0 iCPi Cp X 3C. PBR in terms of molar flow rates
dT
dW
rAHRx T Ua
b
T Ta FiCPi
10
User Friendly Equations Relate T and X or Fi
3D. PFR in terms of molar flow rates
dT
dV
rAHRx T Ua T Ta FiCPi
Qg Qr
FiCPi
4. Batch
dT
dt
rAV HRx UA T Ta N iCPi
11
User Friendly Equations Relate T and X or Fi
5. For Semibatch or unsteady CSTR
€
dT
dt=
˙ Q − ˙ W S − Fi0 CPiT − Ti0( ) + −ΔHRx T( )[ ] −rAV( )( )
i=1
n
∑
N iCPi
i=1
n
∑
6. For multiple reactions in a PFR (q reactions and m species)
dT
dV
riji1
q
HRx ij Ua T Ta
FiCPjj1
m
12 Let’s look where these User Friendly
Equations came from.
Da
dC
a
cB
a
bA
Q W
s
molFi 0
mol
JEi 0
iF
iE
Rate of energy in by flow
Rate of energy out by flow
Heat added to the system
Work done by the system
Rate of energy accumulation
- -+ =
state)steady (at 0 sJ sJ sJ sJdt
dE W Q EFEF sys
ii0i0i
Energy Balance
13
0A
ini
0A
ini
H .,g.e
H
F .,g.e
F
A
outi
A
outi
H .,g.e
H
F .,g.e
F
SW
Q
Energy Balance on an open system: schematic.
1 dt
dEEFEFWQ system
outiiin0i0iS
Energy Balance
14
OK folks, here is what we are going to do to put the above equation into a usable form.
1. Replace Ui by Ui=Hi-PVi
2. Express Hi in terms of heat capacities
3. Express Fi in terms of either conversion or rates of reaction4. Define ΔHRx
5. Define ΔCP
6. Manipulate so that the overall energy balance is either in terms of the User Friendly Equations.
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Assumptions:
mol
mV~
V~
PFV~
PF work flow
shaft work work flowW
KEPEUE
3
ii0i00i
iiii
=0 =0Other energies small compared to internal
Intro to Heat Effects
16
H i U i P ˜ V i
Recall:
€
Fi0∑ U i0 − Fi∑ U i + ˙ Q − − Fi0∑ P0˜ V i0 + FiP∑ ˜ V i + ˙ W S[ ] =
dE sys
dt
€
Fi0∑ U i0 + P0˜ V i0[ ]
H i 06 7 4 8 4 − Fi∑ U i + P ˜ V i[ ]
H i6 7 4 8 4 + ˙ Q − ˙ W S =
dE sys
dt
Intro to Heat Effects
€
Fi0∑ H i0 − Fi∑ H i + ˙ Q − ˙ W S =dE sys
dt
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Substituting for
€
˙ W
General Energy Balance:
dt
dEHFHFWQ system
ii0i0iS
For Steady State Operation:
0HFHFWQ ii0i0iS
18
Intro to Heat Effects
Flow Rates, Fi
For the generalized reaction:
Da
dC
a
cB
a
bA
€
FA = FA 0 1− X( )
€
FB = FA 0 ΘB −b
aX
⎛
⎝ ⎜
⎞
⎠ ⎟
In general,
€
Fi = FA 0 Θi +υ iX( )
a
d ,
a
c ,
a
b ,1 DCBA
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Intro to Heat Effects
€
Fi0H i0 = FA 0 Θi∑∑ H i0
20
€
FiH i = FA 0 Θi +υ iX( )∑∑ H i = FA 0 Θi∑ H i + FA 0X υ iH i∑ΔH Rx6 7 8
€
˙ Q − ˙ W S + FA 0 Θi H i0 − H i( ) + FA 0XΔHRx∑( ) = 0
Intro to Heat Effects
€
H i T( ) = H i0 TR( ) + CPiTR
T
∫ dT
Enthalpy of formation at temperature TR
€
H i0 − H i = H i0 + CP T0 − TR( ) − H i + CP T − TR( )[ ]
€
H i0 − H i = CPi T − T0( )
υ iH i∑ = υ iH i0∑ + υ iCPi∑ T − TR( )
Heat of reaction at temperature T
Intro to Heat Effects
21
For No Phase Changes
€
→ H i T( ) = H i0 TR( ) + CPi T − TR( )
Constant Heat Capacities
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Intro to Heat Effects
€
iH i∑ = υ iH i0∑ + υ iCPi∑ T − TR( )
€
HR T( ) = ΔHRο TR( ) + Δ ˆ C P T − TR( )
€
iˆ C Pi∑ = Δ ˆ C P =
d
aˆ C PD +
c
aˆ C PC −
b
aˆ C PB − ˆ C PA
ABCDRx HHa
bH
a
cH
a
dH
PAPBPCPDP CCa
bC
a
cC
a
dC
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Intro to Heat Effects
Substituting back into the Energy Balance
€
˙ Q − ˙ W S − FA 0X ΔHRο TR( ) + Δ ˆ C P T − TR( )[ ] − FA 0 Θi
˜ C Pi T − Ti0( )∑ = 0
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€
˙ Q − ˙ W S + FA 0 Θi H i0 − H i( ) + FA 0XΔHRx∑( ) = 0
Intro to Heat Effects
X
T0
T
Adiabatic Energy Balance:
T T0 X HR
TR ˆ C P T TR i
˜ C Pi X ˆ C PT0
X HR T0 i
˜ C Pi X ˆ C P
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0A
A
F
r
dV
dX1) Mole balance:
T
1
T
1
k
Hexpkk 0C
T
1
T
1
R
Eexpkk
k
CCkr
2
0X
2CCP
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C
BAA2) Rate Laws:
Example Adiabatic PFR
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A ↔ B
XCC
X1CC
0AB
0AA
3) Stoichiometry:
Pii
0X
0 C
XHTT
4) Energy Balance:
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Example Adiabatic PFR
First need to calculate the maximum conversion which is at the Adiabatic Equilibrium.
A ↔ B
Pii
X
C
XHTT
0
0
T
XC Adiabatic equilibrium conversion
€
Xeq =KC
1+ KC28
Example Adiabatic PFRA ↔ B
We can now form a table. Set X, then calculate T, -VA, and FA0/-rA, increment X, then plot FA0/-rA vs. X:
FA0/-rA
X29
End of Lecture 17
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