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.

15

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

17

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

19

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

22

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

23

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

24

€

˙ 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

25

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

11

C

BAA2) Rate Laws:

Example Adiabatic PFR

26

A ↔ B

XCC

X1CC

0AB

0AA

3) Stoichiometry:

Pii

0X

0 C

XHTT

4) Energy Balance:

27

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

30