L4-1 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Ideal CSTR Design Eq with X A :

L4-1

Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.

Ideal CSTR Design Eq

with XA:

Review: Design Eq & ConversionD

ad

C ac

B ab A

fed A moles reacted A moles

XA

BATCHSYSTEM: A0Aj0jj XNNN

jA0A

jj0TjT XNNNN

FLOW SYSTEM: A0Aj0jj XFFF

jA0A

jj0TjT XFFFF

r

XFV

A

A0A

Vr dt

dXN A

A0A Ideal Batch Reactor

Design Eq with XA:

AX

0 A

A0A Vr

dXNt

AA

0A rdV

dXF Ideal SS PFR

Design Eq with XA:

AX

0 A

A0A r

dXFV

'rdW

dXF A

A0A Ideal SS PBR

Design Eq with XA:

AX

0 A

A0A 'r

dXFW

j≡ stoichiometric coefficient; positive for products, negative

for reactants

L4-2


Review: Sizing CSTRsWe can determine the volume of the CSTR required to achieve a specific conversion if we know how the reaction rate rj depends on the conversion Xj

AA

0ACSTR

A

A0ACSTR X

rF

V rXF

V

Ideal SS CSTR

design eq.

Volume is product of FA0/-rA and XA

• Plot FA0/-rA vs XA (Levenspiel plot)

• VCSTR is the rectangle with a base of XA,exit and a height of FA0/-rA at XA,exit

FA 0 rA

X

Area = Volume of CSTR

X1

V FA 0 rA

X1

X1

L4-3


FA 0 rA

Area = Volume of PFR

V 0

X1FA 0 rA

dX

X1

Area = VPFR or Wcatalyst, PBR

dX'r

FW

1X

0 A

0A

Review: Sizing PFRs & PBRsWe can determine the volume (catalyst weight) of a PFR (PBR) required to achieve a specific Xj if we know how the reaction rate rj depends on Xj

A

exit,AX

0 A

0APFR

exit,AX

0 A

A0APFR dX

r

FV

r

dXFV

Ideal PFR design eq.

• Plot FA0/-rA vs XA (Experimentally determined numerical values) • VPFR (WPBR) is the area under the curve FA0/-rA vs XA,exit

A

exit,AX

0 A

0APBR

exit,AX

0 A

A0APBR dX

r

FW

r

dXFW

Ideal PBR

design eq.

dXr

FV

1X

0 A

0A

L4-4


Numerical Evaluation of Integrals (A.4)Simpson’s one-third rule (3-point):

2102X

0XfXf4Xf

3h

dxxf

hXX 2

XXh 01

02

Trapezoidal rule (2-point):

101X

0XfXf

2h

dxxf

01 XXh

Simpson’s three-eights rule (4-point):

32103X

0XfXf3Xf3Xfh

83

dxxf 3

XXh 03

h2XX hXX 0201

Simpson’s five-point quadrature :

432104X

0XfXf4Xf2Xf4Xf

3h

dxxf 4

XXh 04

L4-5


Review: Reactors in Series

2 CSTRs 2 PFRs

CSTR→PFR

VCSTR1 VPFR2

VPFR2VCSTR1

VCSTR2

VPFR1

VPFR1

VCSTR2

VCSTR1 + VPFR2

≠

VPFR1 + CCSTR2

PFR→CSTR

A

A0

r-

F

i j

CSTRPFRPFR VVV

If is monotonically

increasing then:

CSTRi j

CSTRPFR VVV

L4-6


L4: Rate Laws & Stoichiometry

• Reaction Rates (–rA )

1. Concentration2. Temperature 3. Reversible reactions

• How to derive an equation for –rA [–rA = f(XA)]

1. Relate all rj to Cj

2. Relate all Cj to V or u3. Relate V or u to XA

4. Put together

A

A

XA

A00

dX

rt N

V

A

A

A0F XV

r

AXA

A00 Ar

dXV F

A

A

XA

A00

dX

rW

'F

L4-7


Concentration and Temperature• Molecular collision frequency concentration• Rate of reaction concentration

A A A B-r k T f C ,C ,...

Reaction rate is a function of temperature and concentration

CA : Concentration of A CB : Concentration of B

• As temperature increases, collision frequency increases

• Rate of reaction = f [( CA, CB, ……), (T)]

• At constant temperature : r = f(CA, CB, …….)

Specific rate of reaction, or rate constant, for species A is a function of temperature

L4-8


Elementary Reactions & Rate Laws• Dependence of reaction rate –rA on concentration of chemical species in

the reaction is experimentally determined• Elementary reaction: involves 1 step (only)• Stoichiometric coefficients in an elementary reaction are identical to the

powers in the rate law:

CBA BAAA CCkr

Reaction order:• order with respect to A• order with respect to B• Overall reaction order n =

Zero order: -rA = kA k is in units mol/(volume∙time)

1st order: -rA = kACA k is in units time-1

2nd order: -rA = kACA2 k is in units volume/(mol∙time)

3rd order: -rA = kACA3 k is in units volume2/(mol2∙time)

L4-9


Examples:

• This reaction is not elementary, but under some conditions it follows an elementary rate law

• Forward reaction is 2nd order with respect to NO and 1st order with respect to O2 (3nd order overall)

Overall Stoichiometric Equations• Overall equations describe the overall reaction stoichiometry• Reaction order cannot be deduced from overall equations

Compare the above reaction with the nonelementary reaction between CO and Cl2

2 22NO O 2NO 2

NO NO NO O2r k C C

2 2CO Cl COCl 3 2

CO CO Cl2r kC C

Forward reaction is 1st order with respect to CO and 3/2 order with respect to Cl2 (5/2 order overall)

L4-10


Specific Rate Constant, kAkA is strongly dependent on temperature

Where : A = Pre-exponential factor or frequency factor

(1/time)E = Activation energy, J/mol or cal/molR = Gas constant, 8.314 J/mol K (or 1.987 cal/mol K)T = Absolute temperature, K

Arrhenius Equation E RTAk T Ae

To determine activation energy E, run the reaction at several temperatures, and plot ln k vs 1/T. Slope is –E/R

Taking ln of both sides:

E 1lnk lnA

R T

1/T

ln k -E/R

L4-11


Reversible ReactionskA

k AaA b B c C d D

KC: concentration equilibrium constant (capital K)

a b a bfA A A B fA A A Br k C C r k C C

At equilibrium, the reaction rate is zero, rA=0

Rate of disappearance of A (forward rxn):c d

bA A C Dr k C CRate of generation of A (reverse reaction):

A,net A fA bAr r r r

a b c dA A A B A C Dr 0 k C C k C C

c dC DA

Ca bA A B

C CkK

k C C

Thermodynamic equilibrium relationship

RXC C 1

1

H 1 1K (T) K (T )exp

R T T

KC is temperature dependent (no change in moles or CP):

HRX: heat of reactionIf KC is known for temperature T1, KC for temperature T can be calculated

a b c dA A A B A C Dr k C C k C C

a b c dA A B A C Dk C C k C C

L4-12


L4: Rate Laws & Stoichiometry

• Reaction Rates (–rA )

1. Concentration 2. Temperature 3. Reversible reactions

• How to derive an equation for –rA [–rA = f(XA)]


2. Relate all Cj to V or u3. Relate V or u to XA (Wednesday)

4. Put together (Wednesday)

A

A

XA

A00

dX

rt N

V

A

A

A0F XV

r

AXA

A00 Ar

dXV F

A

A

XA

A00

dX

rW

'F

L4-13



• rA as a function of Cj is given by the rate law• The rate relative to other species (rj) is determined by stoichiometry

D ad

C ac

B ab A “A” is the limiting reagent

ad

r

ac

r

ab

rr DCBA

rj is negative for reactants,

positive for products

In general:j

Aj

rr

j≡ stoichiometric coefficientpositive for products, negative for reactants

ad

ac

1 ab

dcAB

L4-14


For the reaction the rate of O2

disappearance is 2 mol/dm3•s (-rO2= 2 mol/dm3•s).

What is the rate of formation of NO2?

2 22NO O 2NO

jA

j

rHint: r

2

2

NOO

rr

2 1 2 2O NO2 r r

2 2NO NO3 3

mol mol2 2 r 4 r

dm s dm s

rNO2 = 4 mol/dm3•s

L4-15


2a. Relate all Cj to V (Batch System)

BAAA CCkr Reaction rate is a function of Cj:

How is Cj related to V and XA? Batch: jj

N molC

V L

D ad

C ac

B ab A

B0 A0 AB

B

bN N X

N aCV V

C0 A0 AC

C

cN N X

N aCV V

AA

NC

V A0 A0 A

AN N X

CV

Put NA in terms of XA:

D0 A0 AD

D

dN N X

N aCV V

Do the same for species B, C, and D:

Cj is in terms of XA and V. But what if V varies with XA? That’s step 3a!

A0Aj0jj XNNN

L4-16


2a. Additional Variables Used in Textbook

B0 A0 AB

B

bN X

N aC

N

V V

Book uses term Θi:

A0

0

0

i

Ai

0 iC

C

N

N

So species Ni0 can be removed from the equation for Ci

A0

A0 A

AB0A

0

0

B

NbX

N

N N1 aC

N

VMultiply numerator by NA0/NA0:

A

B

B

B

A

AA0

0

BV

bX

XCba

Ca

NC

D ad

C ac

B ab A

L4-17


T

0 0 0 T0 0

ZN RTPVP V Z N RT

3a. Relate V to XA (Batch System)Volume is constant (V = V0) for:

• Most liquid phase reactions• Gas phase reactions if moles reactants = moles products

2 2 2CO g H O g CO g H g

If the volume varies with time, assume the equation of state for the gas phase:At time t: PV = ZNTRT and at t=0: P0V0 = Z0NT0RT0

P: total pressure, atm Z: compressibility factorNT: total moles T: temperature, K

R: ideal gas constant, 0.08206 dm3∙atm/mol∙K

d c b change in total # moleswhere = 1

a a a Moles A reacted

Want V in terms of XA. First find and expression for V at time t:

NT at time t is:

0 T0

0 0 T0

P NT ZV V

P T Z N

T j T0 A0 Aj

d c bN N N 1 N X

a a a T T0 A0 AN N N X

What is NT at t?

L4-18


TA

T0

N1 X

N

3a. Relate V to XA (continued)

T T0 A0 AN N N X d c b change in total # moleswhere = 1

a a a Moles A reacted

0 T0

0 0 T0

P NT ZV V

P T Z N

Can we use the eq. for NT above to find an expression for NT/NT0?

A0A0

T0 = =mole fraction of A iniS tubstitut ially pre:

Ny

Nesent

A0S eubsti xpanstute ion : ry facto

T T00

0A

T 00 T0

P T ZPlug : into

N NV V

P T1 X

N NZ

00

0A

0

P T ZV V

P T1

ZX

T0TA

T0 T0

A0

T0

N

N

NNX

N N

TA

T0A0

N1 y X

N

L4-19


T T0

AT0

N NX

N

What is the meaning of ε?

When conversion is complete (XA=1):

Tf T0 A

T0

N N Change in total # moles at X 1

N total moles fed

The expansion factor, , is the fraction of change in V per mol A reacted that is caused by a change in the total number of moles in the system

A0A0

T0

Nd c bexpansion factor: y 1

a a a N

TA

T0

N1 X

N If we put the following

equation in terms of ε:

TA

T0

N1 X

N

T T0

T0 A

N N

N X

00 A

0 0

P T ZV V

Z1

P TX where

L4-20


4a. Put it all together (batch reactor)

Batch:

0

j j0 jj

A0 A

0 A0 0

V P T

N N N X

ZV 1

P T

C

XZ

j A0 0 0

A 0

0 Aj

j T ZP1 X P T Z

CC

XC

For a given XA, we can calculate Cj and plug the Cj into –rA=kCjn

jjC

N

V

j0 A Aj

j 0CN

V

N Xj j j A AN N N X 0 0

00 A

0 0

P T ZV V 1 X

P T Z

0

0i0

i

V

NC

What about flow systems?

L4-21


2b. Relate all Cj to u (Flow System)

How is Cj related to uand Xj?

Flow:j

jF mol s mol

CL s Lu

BAAA CCkr

Reaction rate is a function of Cj:

D ad

C ac

B ab A

B0 A0 AB

B

bF F X

F aCu u

C0 A0 AC

C

cF F X

F aCu u

AA

FC

u A0 A0 A

AF F X

Cu

Put FA in terms of XA:

D0 A0 AD

D

dF F X

F aCu u

Do the same for species B, C, and D:

We have Cj in terms of XA and u, but what if u varies with XA? That’s step 3b!

A0Aj0jj XFFF

L4-22


3b. Relate u to XA (Flow System)Start with the equation of

state for the gas phase:

TT

NPC

ZRT V

What is CT0 at the entrance of the reactor?

T0 0T0

0 0 0

F PC

Z RTu

T

T0 0 0 0 0

F ZRT 1 P

F Z RT 1 Puu

TPV ZN RT

Rearrange to put in terms of CT, where CT = NT/V:

TT

FC

u

Can we relate CT to u? T

1F ZRT

Pu

0T0

T0 0 0

PF Z TF Z T P

u u

Rearrange to put in terms of u:

Put in terms of u0: T0 0 0 00

1F Z RT

Pu

Use these 2 equations to put uin terms of known or measurable quantities

TF PZRTu

L4-23


3b. Relate u to XA (continued) T T0 A0 A subst F F F X: and simplifyitute in

0

0T0 0

T0 A0

0

A PZ TF Z T P

F F Xu u

When conversion is complete (XA=1):

Tf T0 A

T0

N N Change in total # moles at X =1

N total moles fed

00

T0

T

0 0

PZ TF Z T P

Fu u

A0 substitute y:

A00

0 A0 0

PZ Ty1 X

Z T Pu u

A 00

0

A

T 0

0

0

X PZ T1

Z T

F

F P

u u

A0 0A0 A0 A0

A0 A0T0 T0 T0 0 T0

Simplify witN VF N F

y yF N N V F

Because : huu

00 A

0 0

PZ T1 X

Z T Pu u

L4-24


4b. Put it all together (flow reactor)

Flow:

0

j j0 j

0 A0 0

0 Aj

A

P T Z1 X

P

CF

T

F F X

Z

uu

j A0 0 0

A 0

0 Aj

j T ZP1 X P T Z

CC

XC


jjC

F

u

j0 A Aj

j 0CF

V

F Xj j j A AF F F X 0 0

00 A

0 0

P T Z1 X

P T Zu u

0

0i0

iFC

u

This is the same equation as that for the batch reactor!

L4-25


4. Summary: Cj in terms of Xj

Batch:

j j0 j A0 Aj

00 A

0 0

N N N XC

V P T ZV 1 X

P T Z

j0 j A0 A 0 0j

A 0

C C X T ZPC

1 X P T Z

j0j0

0

NC

V

Flow:

j j0 j A0 Aj

00 A

0 0

F F F XC

P T Z1 X

P T Z

u

u j0

j00

FC

u

j0 j A0 A 0 0j

A 0

C C X T ZPC

1 X P T Z

This is the same equation as that for the batch reactor!


Documents

L4-1 Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign. Ideal CSTR Design Eq with X A :