lec 2 path2 single reaction

Summary Summary -- Design Equations of Ideal ReactorsDesign Equations of Ideal Reactors

Differential

Equation

Algebraic

Equation

Integral

EquationRemarks

Vrdt

dnj

j )(= ∫=j

jO

n

n j

j

Vr

dnt

)(

Conc. changes with time

but is uniform within the

reactor. Reaction rate

varies with time.

Batch

Conc. inside reactor is

CSTR)( j

jjo

r

FFV

−

−=

Conc. inside reactor is

uniform. (rj) is constant.

Exit conc = conc inside

reactor.

PFRj

j rdV

dF= ∫=

j

jO

F

F j

j

r

dFV

)(

Concentration and

hence reaction rates

vary spatially.

IdealIdeal--Reactor Design EquationReactor Design Equation

Based on Mole Balance for Based on Mole Balance for

SingleSingle--Reaction SystemsReaction SystemsSingleSingle--Reaction SystemsReaction Systems

Consider the general reaction

dDcCbBaA +→+

We will choose A as our basis of calculation

Da

dC

a

cB

a

bA +→+ The basis of calculation is most always the

limiting reactant.

Basis of calculationBasis of calculation

What’s the limiting reactant?!!!

Definition of conversionDefinition of conversion

Da

dC

a

cB

a

bA +→+

How can we quantify how far the reaction proceeds to the right ???

How many moles of C are formed for every moles of A consumed ???How many moles of C are formed for every moles of A consumed ???

batch reactor batch reactor

0A

A0AA N

NNX

−=

flow reactor flow reactor

0A

A0AA F

FFX

−=

XA=

Moles of A reacted

Moles of A fed

Convenient way is given!!! “Conversion”

Further Discussions on ConversionFurther Discussions on Conversion

For irreversible reactions, the maximum value of

conversion, X, is that for complete conversion, i.e. X=1.0.

• Maximum conversion for irreversible reactions

dDcCbBaA +→+

conversion, X, is that for complete conversion, i.e. X=1.0.

• Maximum conversion for reversible reactions

For reversible reactions, the maximum value of

conversion, X, is the equilibrium conversion, i.e. X=Xe.

bBaA+ dDcC +

Consider to example rate equations

A A Br kC C− = 2 2A A Br kC C− = 1

2

AA

A

k Cr

k KC− =

+

Note that

( )Ar f Concentration=

Further Discussions on ConversionFurther Discussions on Conversion

We will discuss this issue in the later courses.

A

For only reaction occurring, Conc. can be expressed in term of X

What is the relationship between X and What is the relationship between X and rrAA ??

We need only -rA = f (X) and FA0 to design a variety of reactors !

The heart of the design of an ideal reactor:

(-rA) as a function of conversion (concentration, partial pressure etc.)

We’ll develop the reactor design equationWe’ll develop the reactor design equation in in

term of conversion term of conversion for single reaction systemfor single reaction system

(-rA) as a function of conversion (concentration, partial pressure etc.)

(We will discuss this issue in the later courses.)

( )A

A

dndt

r V= −

−

Consider to design equation of batch reactor

0 (1 )

( )A A

A

dn Xdt

r V

−= −

−

( )0 1A A An n X= −

Design eq. for batch reactor design

Where;

(-rA) = moles A reacting / (unit

volume) (time)

V = Reactor volume, but really

refers to the volume of fluid in

reactor.

0

( )A A

A

n dXdt

r V=

− 0 ( )AA A

dXn r V

dt= −Or

Design equation of batch reactor in conversion term

Integration form with t = 0 (XA=0) to t=t1 (XA=XA1)

1

0

0 ( )

AX

AA

A

dXt n

r V=

−∫The longer the reactants are

left in the reactor, the greater

will be the conversion.

•• Batch:Batch: Vrdt

dXn AA )(0 −=

? ?

Design eq. for batch reactor design

– The conversion is a function of the time

the reactants spend in the reactor.

– We are interested in determining how – We are interested in determining how

long to leave the reactants in the reactor

to achieve a certain conversion X.

-1/rA

X

Area =00 ( )

t

A At

dX Vt

r n=

= −

∫

Design eq. for batch reactor design (For example)

We are planning to operate a batch reactor to convert A into R.

This is a liquid reaction, the stoichiometry is A+R, and the rate of

reaction is given in below Table. How long must we react each

batch to achieve 90% conversion? The initial concentration of A is 2.0 M


Known and unknown

Specify problem: Determine time to achieve 90% conversion of the bath reactor

0 2.0AC =

0.9X =?V =?t =

Relation between –rA and CA

Mol/l

Basis: (1) Using A as the basis species

(2) Basis calculation based in 1 liter of batch reactor volume

Assumption: (1) Unsteady state operation

(2) Well-mixed condition

(3) Constant volume (because of liquid reaction)

Relation between –rA and CA


( )

0.9

0

0

AA

dXt C

r=

−∫

( )

0.9

0

?A

dXI

r= =

−∫

t C I= × 0.00

5.00

10.00

15.00

20.00

25.00

1/(-rA)

0At C I= ×

Using numerical technique for integration to solve IUsing numerical technique for integration to solve I

14.87I =

2.0 14.87 29.74t = × = min

0.00

0.00 0.20 0.40 0.60 0.80 1.00 xi

Numerical Evaluation of IntegralsNumerical Evaluation of Integrals

Integration with unequal segmentsIntegration with unequal segments

Suitable for experimental data, in-equally space data points

( )b

a

I f x dx= ∫

0 1 11 2( ) ( ) ( ) ( )( ) ( )... n nf x f x f x f xf x f x

I h h h −+ ++≅ + + +0 1 11 2

1 2

( ) ( ) ( ) ( )( ) ( )...

2 2 2n n

n

f x f x f x f xf x f xI h h h −+ ++≅ + + +

where,

( )1i i ih x x −= − or the width of segmentsf(x)

xi

The entering molar flow rate of species A, FA0 (mol/s), is just the product of the

entering concentration, CA0 (mol/dm3), and the entering volumetric flow rate, v0

(dm3/s):

00A0A vCF =

For liquid system, CFor liquid system, CAA00 is commonly given in terms of is commonly given in terms of molaritymolarity, for , for

example, Cexample, CAA00 = = 2 2 mol/dmmol/dm33

RelationshipRelationship ofof molarmolar//volumetric volumetric flowrateflowrate and and

concentrationconcentration

example, Cexample, CAA00 = = 2 2 mol/dmmol/dm

ForFor gasgas system,system, CCAA00 cancan bebe calculatedcalculated fromfrom thethe enteringentering temperaturetemperature andand

pressurepressure using the ideal gas law or some other gas law. For an ideal gas using the ideal gas law or some other gas law. For an ideal gas

lawlaw

0

00A

0

0A0A RT

Py

RT

PC ==

0

00A00A00A RT

PyvCvF ==

yA0 = entering mole fraction of A (-)

P0 = entering total pressure (kPa)

PA0 = yA0P0 = entering partial pressure of A (kPa)

T0 = entering temperature (K)

R = ideal gas constant (=8.314 kPa⋅dm3/mol⋅K)

A gas of pure A at 830 kPa enters a reactor with a volumetric flow

rate, v0, of 2 dm3/s at 500 K. Calculate the entering concentration of

A, CA0, and the entering molar flow rate, FA0.

Solution

300 /20.0)830)(1(

dmmolkPaPy

C A ===

RelationshipRelationship ofof molarmolar//volumetric volumetric flowrateflowrate and and

concentrationconcentration

33

0

000 /20.0

)500)(/314.8(

)830)(1(dmmol

KKmolkPadm

kPa

RT

PyC A

A =⋅⋅

==

smolsdmdmmolvCF AA /4.0)/2)(/2.0( 33000 ===

This feed rate (FA0 = 0.4 mol/s) is in the range

of that which is necessary to form several

mill ion pounds of product per year.

Design Equations for Design Equations for CSTRCSTR

Consider to design equation of CSTR reactor

A

A0A

r

FFV

−−

=

FA0

FAC

0A0A )X1(FFV

−−=

)X1(FF 0AA −=

FA

CACA

CA

A

0A0A

r

)X1(FFV

−−−

=

exitA

A

r

XFV

)(0

−=

Because the reactor is perfectly

mixed, the exit composition from the

reactor is identical to the composition

inside the reactor, and the rate of

reaction is evaluated at the exit

conditions.

Design Equations for Design Equations for PFRPFR

PFR Design EquationsPFR Design Equations

FA FA +dFdV

bA B

a+

l mL M

a a+

AA r

dV

dF=

)X1(FF 0AA −= dXFdF 0AA −=

dXFdF 0AA −=A0A r

dV

dXF −= differential form

∫ −=

X

0A

0A r

dXFV integral form

REACTOR DIFFERENETIAL ALGEBRAIC INTEGRAL

FORM FORM FORM

Vrdt

dXN AAO )(−= ∫ −=

X

AAO Vr

dXNt

0BATCHBATCH

Design Equations Design Equations in Terms of Conversionin Terms of Conversion

)( AAO rdV

dXF −= ∫ −=

X

AAO r

dXFV

0

CSTRCSTR

PFRPFR

ExitA

AO

r

XFV

)(

)(

−=

Levenspiel Plots:

Illustration of Reactor Sizing for

Single-Reaction SystemsSingle-Reaction Systems

Octave Levenspiel

(PhD 1952): Octave Levenspiel also obtained an MS from

Oregon State and served as a faculty member for 25 years

until he retired in 1991. He published over 100 papers and

proceedings, two of which have been listed as "Citation

Classics." He was awarded 1977 American Institute of

Chemical Engineers W.K. Lewis Award, the 1979 R.H.

Wilhelm Award, and the 2003 Founders Award and Gold

Medal, the highest honor given by the society. In 2000 he

Wilhelm Award, and the 2003 Founders Award and Gold

Medal, the highest honor given by the society. In 2000 he

was inducted into the National Academy of Engineering. He

also received two honorary doctorates , one from France.

He is considered to be one of the founders of He is considered to be one of the founders of He is considered to be one of the founders of He is considered to be one of the founders of

Chemical Reaction Engineering.Chemical Reaction Engineering.Chemical Reaction Engineering.Chemical Reaction Engineering.

SOURCE: http://engr.oregonstate.edu/oregonstater/fame/1998/che/octavelevenspiel.html

Levenspiel Plots: Sizing of CSTR

•• CSTR:CSTR:

– We are interested in determining the size of

the reactor to achieve a certain conversion X.

XFV

][×=

exitAACSTR r

XFV

)(

][0 −×=

-1/rA

X

Area =1

( )A

Xr

×−

0CSTR AV F Area= ×

Evaluate VCSTR by Levenspiel Plot

Levenspiel Plots: Sizing of PFR

•• PFR:PFR:

– We are interested in determining the size of

the reactor to achieve a certain conversion X.

1dX× ∫

=

=PFRxx

A dXF

V 0

X

)(

1

Ar−

XPFRdX

( )A

dXr

×− ∫

= −=

x A

APFR dX

rV

0

0

Area =1

( )A

dXr

×−∑

0

1

( )

PFRX

AX

dXr=

×−∫

0PFR AV F Area= ×

Evaluate VPFR by Levenspiel Plot

or

The isomerization reaction

A→B

was carried out adiabatically in the liquid phase and the data in below table

were obtained.

X (-rA) kmol/l.hr

0.95 0.10

0.90 0.30

Calculate the volume of each of the reactors

for an entering molar flow rate of species A of

50 kmol/hr.

For example:

0.90 0.30

0.85 0.50

0.80 0.60

0.75 0.50

0.70 0.25

0.65 0.10

0.60 0.06

0.50 0.05

0.35 0.05

0.00 0.04

50 kmol/hr.

(b) Compare CSTR and PFR volume at 65%

conversion

(a) Compare CSTR and PFR at 90%

conversion

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0.00 0.20 0.40 0.60 0.80 1.00

For example:

CSTR: CSTR: 9090% conversion% conversion

exitA

A

r

XFV

)(0

−=

kmol

hrl

hr

kmolV

.2.229.050 ××=

1000=V liter

-1/rA

X

PFR: PFR: 9090% conversion% conversion

( )∫ −=X

AA r

dXFV

0

0

∫ −×=9.0

0

50Ar

dX

hr

kmolV

78.91634.1850 =×=V liter

X 1/(-rA)

0.95 25.00

0.90 22.22

0.85 20.00

0.80 16.67

0.75 10.00

0.70 4.35

0.65 6.25

0.60 10.00

0.50 16.67

0.35 25.00

0.00 33.33

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0.00 0.20 0.40 0.60 0.80 1.00

For example:

CSTR: CSTR: 6565% conversion% conversion

liter

( )0AA exit

XV F

r=

−

.50 0.65 6.25

kmol l hrV

hr kmol= × ×

203V =

-1/rA

XX 1/(-rA)

0.95 25.00

0.90 22.22

0.85 20.00

0.80 16.67

0.75 10.00

0.70 4.35

0.65 6.25

0.60 10.00

0.50 16.67

0.35 25.00

0.00 33.33

PFR: PFR: 6565% conversion% conversion

( )∫ −=X

AA r

dXFV

0

0

∫ −×=65.0

0

50Ar

dX

hr

kmolV

75307.1550 =×=V liter

X

Ideal Reactors in SeriesIdeal Reactors in Series

- X1 at point i=1 is the conversion

achieved in the CSTR

- X2 at point i=2 is the total

conversion achieved in at this

point in the CSTR and the PFR

- X3 is the total conversion

achieved by all three reactors

For reactors in series, the conversion X is the total number of moles

of A that have reacted up to that point per mole of A fed to the first

reactor.

only be used when the feed stream only

enters the first reactor in the series and

t h e r e n o s i d e s t r e a m s e i t h e r

fed or withdrawn.

Xi=

Total moles of A reacted up to point i

Moles of A fed to the first reactor

achieved by all three reactors


CSTRs in SeriesCSTRs in Series

Reactor 1:

0AF1AF

2AF

1AX2AX

1Ar−

2Ar−

0VrFF 11A1A0A =+−

)X1(FF 10A1A −=1

1A

0A1 X

r

FV

−=

Reactor 1:

Reactor 2:0VrFF 22A2A1A =+−

)X1(FF 20A2A −=)XX(

r

FV 12

2A

0A2 −

−=


-1/rA

11

01 X

r

FV

A

ACSTR

−=−

)( 120

2 XXF

V ACSTR −

=−

21 −−− += CSTRCSTRtCSTR VVV

-1/rA1

-1/rA2

CSTRs in Series: CSTRs in Series: LevenspeilLevenspeil plotplot

X

)( 122

2 XXr

VA

CSTR −

−

=−

( )

−×

−+

×

−= 12

201

10

11XX

rFX

rFV

AA

AA

( ) ( )2010 −− ×+×= CSTRACSTRA AreaFAreaFV

X1 X2

***To achieve the same overall conversion, the total volume for two CSTRs in series is less than that required for one CSTR


CSTRs in SeriesCSTRs in Series ((For Example)For Example)

For the two CSTRs in series, 60% conversion is achieved in the first reactor.

What is the volume of each of the two reactors necessary to achieve 90% overall conversion of entering species A

Calculate the volume of each of the reactors

for an entering molar flow rate of species A of

50 kmol/hr.

X (-rA) kmol/l.hr

0.95 0.10

0.90 0.30 50 kmol/hr.0.90 0.30

0.85 0.50

0.80 0.60

0.75 0.50

0.70 0.25

0.65 0.10

0.60 0.06

0.50 0.05

0.35 0.05

0.00 0.04

0 50 /AF kmol hr=1 0A AF F=

2 0A AF F=

1 0.6AX =2 0.9AX =

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

CSTRs in SeriesCSTRs in Series ((For Example)For Example)

21 −−− += CSTRCSTRtCSTR VVV

CSTRCSTR--11: : 6060% conversion% conversion

exitA

ACSTR r

XFV

)(10

1 −=−

kmol

hrl

hr

kmolVCSTR

.106.0501 ××=−

3.633=

-1/rA

X

1000sin =− gleCSTRV liter

0.00

0.00 0.20 0.40 0.60 0.80 1.00

kmolhr

3001 =−CSTRV

CSTRCSTR--22: : 9090% conversion% conversion

( )exitAACSTR r

XXFV

−−

=−12

02

( )kmol

hrl

hr

kmolVCSTR

.22.226.09.0502 ×−×=−

3.3332 =−CSTRV liter

X 1/(-rA)

0.95 25.00

0.90 22.22

0.85 20.00

0.80 16.67

0.75 10.00

0.70 4.35

0.65 6.25

0.60 10.00

0.50 16.67

0.35 25.00

0.00 33.33

literX


PFRs in SeriesPFRs in Series

21 −− += PFRPFRtotal VVV

It is immaterial whether you place two plug-flow reactors in

series or have one continuous plug-flow reactor; the total reactor

volume required to achieve the same conversion is identical.

∫∫ −+

−=

2

1

1

00 0

X

XA

A

X

AA r

dXF

r

dXF


PFRs in Series: PFRs in Series: LevenspeilLevenspeil plotplot

-1/rA21 −−− += PFRPFRtPFR VVV

∫ −=−

1

001

X

AAPFR r

dXFV

∫X dX

1−PFRV2−PFRV

X

∫ −=−

2

101

X

XA

APFR r

dXFV

∫∫ −+

−=−

2

1

1

00 0

X

XA

A

X

AAtPFR r

dXF

r

dXFV

• The overall conversion of two PFRs in series is the same as

one PFR with the same total volumn.

( ) ( )2010 −−− ×+×= PFRAPFRAtPFR AreaFAreaFV

• To achieve the same overall conversion, the total

volume for two CSTRs in series is less than that

required for one CSTR. -1/rA

X


• The overall conversion of two PFRs in series is the same

as one PFR with the same total volumn.

• CSTRs in series :A PFR can be modelled using a number

of CSTR in series

– useful in modelling catalyst decay in a packed-bed reactor

– modelling transit heat effects in PFRs.

X

-1/rA

X


Combined CSTRs and Combined CSTRs and PFRs in SeriesPFRs in Series

CSTRCSTR--PFR in seriesPFR in series

PFRCSTRtotal VVV +=

∫ −=

2

10

X

XA

APFR r

dXFV

11

0 Xr

FV

A

ACSTR

−=

( ) ∫ −+

−=

2

1)(0

10

X

X AA

AAtotal r

dXF

r

XFV


Combined CSTRs and Combined CSTRs and PFRs in SeriesPFRs in Series

PFRPFR--CSTR in seriesCSTR in series

CSTRPFRtotal VVV +=

∫ −=

1

00

X

AAPFR r

dXFV

( )122

0 XXr

FV

A

ACSTR −

−=

( )

−−

+−

= ∫2

2

120

0

0 )( AA

X

AAtotal r

XXF

r

dXFV


Combined CSTRs and PFRs in Series (For Example)

The isomerization of butane

nC4H10→iC4H10

was carried out adiabatically in the liquid phase and the data in below table

were obtained.It is real data for a real reaction carried out

adiabatically, and the reactor scheme X -rA

[kmol/m3.hr]adiabatically, and the reactor scheme

shown as below:

Calculate the volume of each of the reactors for an entering molar flow

rate of n-butane of 50 kmol/hr.

A

[kmol/m3.hr]

0 39

0.2 53

0.4 59

0.6 38

0.65 25

Space Time & Residence Time

Time is of EssenceTime is of Essence

• Two types of time-parameters are commonly used in chemical reaction engineering

– space time

– residence time

• Space time is often used as a scaling parameter in reactor design

Mole balance for “flow reactor”Space time and space velocity of flow reactor

Space time

0

Vτ

ν≡

** Sometime called “mean resident time”

Reactor volume

Volumetric flow rate at entrance

Actual Residence

Time:

The time actually

spent by fluid

inside the reactor.

Space velocity

** Sometime called “mean resident time”

0 1SV

V

ντ

≡ =

** Measured at STP condition

•LHSV - Liquid Hourly Space Velocity

•GHSV - Gas Hourly Space Velocity

Illustration of difference between space time (Illustration of difference between space time (ττττττττ) and ) and

residence time (residence time (ttresres))

The Pop Corn Example

Under what practical conditions do we expect space

time = residence time ?

Class Assignment 2

(1)

(2)

(3)

Class Assignment 2

(4)

Class Assignment 2

lec 2 path2 single reaction

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