Lecture (10)

Reactor Sizing and Design

1. General Mole Balance Equation

Mole balance on species j at any instance in time t ;

…………..4.1

Fj0 = Entering molar flow rate of species j (mol/time) Fj = Exiting molar flow rate of species j (mol/time)

Gj = Rate(total rate) of generation(formation) of species j (mol/time)=rj .V V = Volume (e.g. m3)

rj = rate of generation(formation) of species j (mole/time.vol) Nj = number of moles of species j inside the system Volume V (mole)

If rj varies with position in the system,

Then general mole balance:-

system within j of

onaccumulati of rate

system ofout j of

flow of rate

rxnby systemin j of

generation of rate

system into j of

flow of rate

dt

dN j

jjjo FGF

6V5V

4V

3V

2V

1V

3jr

1jr

4jr 5jr6jr

2jr

m

i

iij

m

i

ijj

jj

VrG

Vr

1

,

1

,

11,1,

G

G

0 ,m VLet

V

jj dVrG

VsystemVolumn

…………4.2

From this general mole balance equation we can develop the design equations for the various types of industrial reactors: batch, semi-batch. and continuous-flow reactors.

o Operate under unsteady state o Neither inflow nor outflow of reactants or products

If the reaction mixture is perfectly mixed so: o Constant rate of reaction throughout the reactor volume o Composition ≠ f (Position)

o Composition =f (time) ideal restrictions

o Temperature ≠ f (Position)

o Temperature ≠ f (time)

Mole Balance

REACTOR SIZING AND DESIGN

PART ONE

Batch Reactor

dt

dNdVr

j

j

V

jjo FF

0, joj FF

Isothermal Operation

...............................4.3

………………………..4.4

Let's consider the isomerization of species A in a batch reactor

As the reaction proceeds. the number of moles of A decreases and the number of moles of B increases, as shown in Figure below

The time t necessary to reduce the initial number of moles NAo to a final number of mole NA can be estimated as : from equation 4.4 ………………4.4

integrating with limits that at :

t = 0 NA = NA0 ← stat of reaction and at t = t NA = NA reaction time (end of reaction ) we obtain

…………..4.5

dt

dNdVr

j

j

V

jjo FF

dt

dNdVr

jV

j

fedmoles

reactedmolesmoles

A of

A of

0at t

fedinitially

A of

consumedor reacted

A of moles

BA

dt

dNVr

j

j

dt

dNVr A

A

Vr

dNdt

A

A

0A

A

N

N A

A

Vr

dNt

XN A

0consumedor reacted

A of moles

number of mole NA remain un-reacted after time t ,

Sub in equation 4.5 and 4.4

……….4.6 …4.7

Differential form Integral form Batch Reactor Design Equation Used in the Interpretation of m Lab Rate Data

XNNN AAA 00

consumedor reacted

A of moles

0at treactor the

tofedinitially

A of

ttime

at (remain)reacter in

A of moles moles

XNN AoA 1

Ao

AAo

N

NNX

Vrdt

dXN AAo

Vrdt

dNA

A 0A

A

N

N A

A

Vr

dNt

XNN AoA 1

tX

A

AoVr

dXNt

0

Space time or Mean Residence Time= is the time necessary to process one reactor mmmmmmmmmmmmmmmmmmm volume of fluid based on entrance conditions.

tB=t+tD

At constant volume batch reactor

i.e constant density reaction mixture.

NAo = CAo * V → then; equations 4.4 and 4.5 become ( ) :

….……….4.8….(Reaction Time)

Evaluation of Reaction Time Graphically:

From equation 4.7 plot vs. X and evaluate the area under the curve

to estimate reaction time

X1 X X

Or

From equation 4.7 plot vs. CA and evaluate the area under the curve

to estimate reaction time

dt

dCr A

A

A

Ao

C

CA

A

r

dCt

V

NC i

i

Ar

1

Ar

1

Ar

1

tX

A

AoVr

dXNt

0

AreaV

Nt Ao *

Area

Ar

1

A

Ao

C

CA

A

r

dCt

CA CA CAo

Example

Evaluation of Reaction Time Numerically:

Need to size reactors or calculate reaction time

o For the reactions in which the rate depends only on the concentration of

one species then

First order and Irreversible :-

,

Second order and Irreversible :-

,

,

BAAA kCr

A

Ao

A

Ao

C

CA

AC

CA

A

C

dC

kkC

dCt

1

Ao

A

C

C

kt ln.

1

kt

AoA eCC

2

AA kCr BA

A

Ao

A

Ao

C

CA

AC

CA

A

C

dC

kkC

dCt

22

1

AoA CCkt

111

ktC

CC

Ao

AoA

1

)( AA Cfr

)(CAfrA

AreaAreat

nth order and Irreversible :-

,

Example

Bimolecular Reactions

o when the rate law depends on more than one species , we must relate the

concentrations of the different species to eac2h other "as a function of

conversion ". This relationship is most easily established with

the aid of a Stoichiometric table.

In formulating our stoichiornetsic table, we shall take species A component as our basis of calculation (i.e.. limiting reactant) and then divide through by the stoichiometric coefficient of A , in order to put everything on a basis of "pet mole of A ".

Stoichiornetsic table presents the following information

o Column I: the particular species o Column 2: the number of moles of each species initially present o Column 3: the change in the number of moles brought about by reaction o Column 4: the number of moles remaining in the system at time t o Column 5: concentrations as a function of conversion of each species

• Consider the general reaction;

Stoichiometry set up of equations with A as basis

The rate law is :

n

AA kCr BA

11

1

1

n

Ao

n

A CCn

kt

nn

AoAoA tkCnCC 1

1111

Da

dC

a

cB

a

bA

C

d

D

c

Cb

B

a

AAAK

CCCCkr

)(XfrA

Constant Volume (Constant Density)

liquid-phase and some of gas phase reaction system fall into this category.

Stoichiometric Table Batch System

Specie Initial Change Remaining Concentration A NAo

-NAo X

NA = NAo(1 – X)

AC XCA 10

B NBo = NAo B

-(b/a)NAo X

NB = NAo[B –(b/a)X] BC

X

a

bC BA0

C NCo = NAo C

+(c/a)NAo X

NC = NAo[C +(c/a)X] CC

X

a

cC CA0

D NDo = NAo D

+(d/a)NAo X

ND = NAo[D +(d/a)X] DC

X

a

dC DA0

I NI = NAo

NI = NAo I

IoC

NTo = NAo i NT = NTo +NAoX

Where

i = Nio/NAo = Cio/CAo= yio/yAo

= (d/a) + (c/a) – (b/a) - 1

• Express table in terms of concentrations

– Concentration (batch):

V

NC i

i

0VV

Xa

bCX

a

b

V

N

V

NC

XCV

XN

V

NC

BABAB

B

AAA

A

0

0

0

0

0

0 11

Mole balance equation and the rate law are coupled and then solved

Example

Variable Volume (Variable Density, but with Constant T and P )

Individual concentration can be determined by expressing the volume for

batch system as a function of conversion using the equation of state:

PV=ZNTRT………..at any time in the reaction

PoVo=ZoNToRTo……at any time =0;when reaction is initiated

Then,

0

0

00

0Z

Z

P

P

T

T

N

NVV

T

T………………….4.9

Change in the total number of moles during reaction in gas phase reaction system,

but with constant temperature and pressure, and the compressibility factor will not

change significantly during the course of the reaction ,

0

0

T

T

N

NVV

Where NT = NTo +NAoX

= (d/a) + (c/a) – (b/a) – 1

= (change in total number of mole) / (mole of A reacted)

XN

N

N

N

T

Ao

T

T 00

1

0T

Ao

AoN

Ny

Xa

dCX

a

d

V

N

V

NC

Xa

cCX

a

c

V

N

V

NC

DADAD

D

CACAD

C

0

0

0

0

0

0

Ao

T

Ao yN

N

0

…………………………4.10a.

Then

XN

N

T

T 10

XN

NN

T

ToT

0

………………….…………….4.10b

At complete conversion i.e X=1 , NT= NTf ; therefore ,

0T

ToTf

N

NN ………………………………….4.11

= (change in total number of mole for complete conversion ) / (total moles fed)

Then the volume as a function of conversion :

XVV 10 …………………………………….4.12

Concentration at variable volume or density

Specie

V

NC A

A V

XN A

10

)1(

10

XV

XN

o

A

)1(

10

X

XCA

V

NC B

B

V

XN B (b/a)- B0

)1(

(b/a)- B0

XV

XN

o

B

)1(

(b/a)- B0

X

XCB

V

NC C

C

V

XNCo (c/a) C

)1(

(c/a) C

XV

XN

o

Co

)1(

(c/a) C

X

XCCo

V

NC D

D

V

XND (d/a)- D0

)1(

(d/a)- D0

XV

XN

o

D

)1(

(d/a)- D0

X

XCD

V

NC I

I V

N IAo )1( XV

N

o

IAo

)1( X

C IAo

Example

Chemical reactors can liberate or absorb very large amounts of energy , and the handling of

this energy is a major concern in reaction engineering. It is important to estimate the

temperature increase or decrease in an adiabatic reactor in which no heat is add or

removed, and exothermic reactor and also the composition of the reaction mixture at any

time.

Energy Balance

+ =

)( VrTH Ar )( TTUAQ a

dt

dTCCV iip,

T = reaction temperature K

Ta= wall temperature K

TR= reference temperature K

A = heat transfer area m2

Cpi = specific heat KJ/Kmol

U = overall heat transfer KJ/s.m2.K

Non-Isothermal Operation

Heat Generated by

Reaction

tion

Heat Addition and

Removal by wall

Heat Accumulated by

Reaction

rH =enthalpy change in the reaction per mole of Areacting

The number of moles of species i at any X is = XNN iiAi 0

Then energy balance is :

dt

dTNCTTUAVrTH iipaAr ,)()(

………………….4.13

Energy and mole balance equations with the rate law are coupled and then solved

Mole balance equation

rH is calculated as

T

TpR

o

rrR

dTCTHTH

The rate law is required as a function of temperature and composition

Variable Volume (Variable Density ,T and/or P)

"Variable T in non-isothermal"

The volume for batch system as a function of conversion as :-

0

0

00

0Z

Z

P

P

T

T

N

NVV

T

T

0

0

0

0 1Z

Z

P

P

T

TXVV

If the compressibility factor will not change significantly during the course of the

reaction Zo=Z

P

P

T

TXVV 0

0

0 1

Concentration at variable volume (density , T and/or P )

Specie

dt

dTCpXCNTTUAVrTH ipiAaAr ,0)()(

Vrdt

dXN AAo

V

NC A

A V

XN A 10

o

o

o

A

P

P

T

T

XV

XN

)1(

10

o

oA

P

P

T

T

X

XC

)1(

10

V

NC B

B

V

XN B (b/a)- B0

o

o

o

B

P

P

T

T

XV

XN

)1(

(b/a)- B0

o

oB

P

P

T

T

X

XC

)1(

(b/a)- B0

V

NC C

C

V

XNCo (c/a) C

o

o

o

Co

P

P

T

T

XV

XN

)1(

(c/a) C

o

oCo

P

P

T

T

X

XC

)1(

(c/a) C

V

NC D

D

V

XND (d/a)- D0

o

o

o

D

P

P

T

T

XV

XN

)1(

(d/a)- D0

o

oD

P

P

T

T

X

XC

)1(

(d/a)- D0

V

NC I

I V

N IAo

o

o

o

IAo

P

P

T

T

XV

N

)1(

o

oIAo

P

P

T

T

X

C

)1(

Example

A batch reactor is usually well mixed, so that may neglect the special variation in

temperature and species concentration .

Batch reactors operated adiabatically are often used to determine the reaction orders, activation energies, and specific reaction rates of exothermic reactions by monitoring the temperature-time trajectories for different initial conditions.

In adiabatic operation of a batch reactor

0Q

dt

dTNCVrTH iipAr ,)(

………………………….4.14

Energy and mole balance equations with the rate law are coupled and then

solved:

Adiabatic Operation of a Batch Reactor

dt

dTCpXCNVrTH ipiAAr ,0)(

;Where To = initial temperature

Example

The highest conversion that can be achieved in reversible reactions is the equilibrium conversion XEB. For endothermic reactions, the equilibrium conversion increases with increasing temperature up to a maximum of 1.0. For exothermic reactions, the equilibrium conversion decreases with increasing temperature Figure ( ) show the variation of the concentration equilibrium constant as a function of temperature for an exothermic reaction the corresponding equilibrium conversion XEB as a function of temperature.

Figure ( ) show the variation of the concentration equilibrium constant and equilibrium conversion as a function of temperature for an exothermic reaction.

TH

TTCX

r

oipi

)(,

CpXC

XTHTT

ipi

ro

,

Equilibrium Conversion

To determine the maximum conversion that can be achieved in an exothermic reaction carried out adiabatically, we find the intersection of the equilibrium conversion as a function of temperature ,with temperature –conversion relationships from the energy balance

……………..4.15

Graphical solution of equilibriurn and energy balance equations to obtain the adiabatic temperature

and the adiabatic equilibriurn

conversion XEB.

Example

TH

TTCX

r

oipi

EB

)(,