Chemical Reactor Design · 2018. 1. 30. · SpeciesA is our basis of calculation. a a a N A0 is the...

Chemical Reactor DesignChemical Reactor Design

Y W LYoun-Woo LeeSchool of Chemical and Biological Engineering

Seoul National Universityy155-741, 599 Gwanangro, Gwanak-gu, Seoul, Korea [email protected] http://sfpl.snu.ac.kr

第第33章章第第章章Rate LawsRate Laws

and and StoichiometryStoichiometry


yy


化學反應裝置設計化學反應裝置設計化學反應裝置設計化學反應裝置設計

Seoul National University

Objectives

After completing Chapter 3, reader will be able to:

j

Write a rate law and define reaction order and activation energy.Write a rate law and define reaction order and activation energy. Set up a stoichiometric table for both batch and flow systems and

express concentration as a function of conversionexpress concentration as a function of conversion. Calculate the equilibrium conversion for both gas and liquid phase

reactionsreactions. Write the combined mole balance and rate law in measures other

than conversionthan conversion.


Reactor SizeDesign Equations

X dXNtBatch

Levenspiel plot

A XFV 0

AA Vr

Nt00Batch

CSTRA

A

rV

0

X dX

CSTR

Graphical method

X

AA r

dXFV00PFR

For vs. X, the volume of aA

A0

rF

X

AA r

dXFW00PBR

CSTR and the volume of a PFRcan be represented as the shadedareas in the Levenspiel plots.

If we know the molar flow rate to thereactor and the reaction rate as afunction of conversion, then we can

X

AA r

dXFV00

PFR

calculate the reactor volume necessaryto achieve a specific conversion.)( XfrA


Design Isothermal Reactor

Design isothermal reactorCHAPTER 3

Design isothermal reactor

b

Rate LawsbB

aAA CkCr

-rA=f(X)C = f(X) ry

CHAPTER 2

Cj = f(X)

iom

etr

Nj = f(X)

stoi

chi

)( XfFB B

BFC

V= f(X) s

)( XfB


P t 1Part 1

R t LRate Laws


3.1.1 Relative Rates of Reactions

If the rate law depends on more than one species, we MUST relate the concentrations of different species to each other. A stoichiometric table presentsconcentrations of different species to each other. A stoichiometric table presents the stoichiometric relationships between reacting molecules for a single reaction.

aA + bB cC + dD (2 1)

In formulating our stoichiometric table, we shall take species A as our basis of calculation (i e limiting reactant) and then divide through by the stoichiometric

aA + bB cC + dD (2-1)

calculation (i.e., limiting reactant) and then divide through by the stoichiometric coefficient of A. In order to put everything on a basis of “per mole of A.”

dcb DadC

acB

abA (2-2)

The relationship can be expressed directly from the stoichiometry of the reaction.

rrrr DCBA (3-1)

The relationship can be expressed directly from the stoichiometry of the reaction.

dcba (3 1)


3.2 Reaction Order and Rate Law

Let take A as the basis of calculation a species A is one of the reactants that is disappearing as a result of the reaction Th li iti t t i ll hof the reaction. The limiting reactant is usually chosen as our basis for calculation.

The rate of disappearance of A, -rA, depends on temperature and concentration and it can be written as the product of the reaction

t t k dconstant k and

)()()( CCfTkCT ...),()(),( BAAA CCfTkCTr

Rate raw (Kinetic expression) : the algebraic equation that relates –rA to the species concentration


3.2.1 Power Law Models and Elementary Rate Laws

The dependence of the reaction rate r on the concentration ofThe dependence of the reaction rate –rA on the concentration ofthe species is almost without exception determined byexperimental observation.

The order of a reaction refers to the powers to which theconcentrations are raised in the kinetic rate law.

BAAA CCkr (3-3)

order with respect to reactant A order with respect to reactant Bn (=) : the overall order of the reaction


Unit of Specific Reaction Rate

The unit of the specific reaction rate, kA, vary with the order of the reaction.

A products

sdmmolkkrorder-Zero AA 3)(

}{:

(3-4)

kCkrorderFirst

sdm

1}{:

)(

(3 5)

d

skCkrorder-First AAA

3)(

}{: (3-5)

smoldmkCkrorder-Second AAA

32 )(}{:

(3-6)

smoldmkCkrorder-Third AAA

233 )/(}{: (3-7)


Elementary and Non-elementary Reaction

Kinetic rate rawKinetic rate raw

“Elementary reaction” Non-elementary reaction

2/3

CO + Cl2 COCl2O + CH3OH CH3O + OH

1st order w r t atomic oxygen

2/32ClCOCO CkCr

1st order w r t carbon monoxide

-rO = k CO CCH3OH

1 order w.r.t. atomic oxygen1st order w.r.t. methanoloverall is 2nd order reaction

1 order w.r.t. carbon monoxide3/2 order w.r.t. chorineoverall is 5/2 order reaction

In general first and second order reactions are more commonly observedIn general, first- and second-order reactions are more commonly observed.


Determination of Reaction Rate Law

It is important to remember that the rate laws are determined byIt is important to remember that the rate laws are determined byexperimental observation! They cannot be deduced from reactionstoichiometry.

They are function of the reaction chemistry and not the type ofreactor in which the reactions occur.

Even though a number of reactions follow elementary rate laws atEven though a number of reactions follow elementary rate laws, atleast as many reactions do not. One must determine the reactionorder from the experiments or from literatureorder from the experiments or from literature.


LiteratureTh ti ti f f t d ti dThe activation energy, frequency factor, and reaction order

Floppy disks and CDROMs by National Institute of Standards and Technology (NIST) Floppy disks and CDROMs by National Institute of Standards and Technology (NIST) Standard Reference Data 221/A320 Gaithersburg, MD 20899 Tables of Chemical Kinetics: Homogeneous Reaction,

National Bureau of Standards Circular 510 (Sept. 1951)Suppl. 1 (Nov. 14, 1956), Suppl. 2 (Aug. 5, 1960), Suppl. 3 (Sept. 15, 1961)W hi t D C U S G t P i ti OffiWashington, D.C., U.S. Government Printing Office

Chemical Kinetics and Photochemical Data for Use in Stratospheric Modeling,Evaluate No. 10, JPL Publication 92-20, Aug. 15, 1992, Jet Propulsion, , g , , pLaboratories, Pasadena, CA, USA

International Journal of Chemical Kinetics, Journal of Physical Chemistry Journal of Catalysis, Journal of Applied CatalysisAIChE Journal, Chemical Engineering Science, Korean Journal of Chemical Engineering Chemical Engineering Communications Chemical Engineering Communications Industrial and Engineering Chemistry Research


Example of Rate Law


Example of Rate LawA. First Order Reaction

B. Second Order Reaction


3.3 kA: The specific reaction rate (the rate constant)

The reaction rate constant k is not truly a constant, but is merelyindependent of the concentrations of the species involved in theindependent of the concentrations of the species involved in thereaction.

The quantity k is also referred to as the specific reaction rate(constant). It is almost always strongly dependent on temperature.In gas-phase reactions, it depends on the catalyst and total pressure.In liquid systems, it depends on the total pressure, ionic strengthand choice of solventand choice of solvent.

These other variables normally exhibit much less effect on theese o e v b es o y e b uc ess e ec o especific reaction rate than temperature does with the exception ofsupercritical solvents, such as supercritical water.

In this text, it will be assumed that kA depends only on temperature.Seoul National University

Arrhenius equationq

Specific reaction rate (constant)

Activation energy, J/mol or cal/mol

E

RTA AeTk

)(

1903(44) :Nobel Prize in chemistryfor ionic theory

A )(frequency factor

Absolute Temperature, K

frequency factor

or

pre-exponentialGas constant8.314 J/mol · Kpre exponential

factor mathematical numbere=2.71828…

1.987 cal/mol · K8.314 kPa · dm3/mol · K


Activation energygy

Activation energy E :a minimum energy that must be possessed by reactingmolecules before the reaction will occur.

E The fraction of the collisions betweenl l th t t th h thi i iRTe

molecules that together have this minimumenergy E (the kinetic theory of gases)

Activation energy E is determined experimentally bycarrying out the reaction at several differenty gtemperature.

EAk 1ll

TRAkA lnln


P t 2Part 2

St i hi tStoichiometry


Reactor SizeDesign Equations

X dXNtBatch

Levenspiel plot

A XFV 0

AA Vr

Nt00Batch

CSTRA

A

rV

0

X dX

CSTR

Graphical method

X

AA r

dXFV00PFR

For vs. X, the volume of aA

A0

rF

X

AA r

dXFW00PBR

CSTR and the volume of a PFRcan be represented as the shadedareas in the Levenspiel plots.

If we know the molar flow rate to thereactor and the reaction rate as afunction of conversion, then we can

X

AA r

dXFV00

PFR

calculate the reactor volume necessaryto achieve a specific conversion.)( XfrA





b

Rate LawsbB

aAA CkCr

-rA=f(X)C = f(X) ry

CHAPTER 2

Cj = f(X)

iom

etr

Nj = f(X)

stoi

chi

V= f(X) s


Example 3-5: Determination of Cj=hj(X)for a Gas Phase Reactionfor a Gas-Phase Reaction

Isothermal

TT

PP

XXabCC B

AB0

0 1)/(

TPX 01


Stoichiometric Table

If the rate law depends on more than one species, we MUST relate theconcentrations of different species to each other. A stoichiometric table presentsconcentrations of different species to each other. A stoichiometric table presentsthe stoichiometric relationships between reacting molecules for a single reaction.

aA + bB cC + dD (2 1)aA + bB cC + dD

rrrr DCBA

(2-1)

(3 1)

I f l ti t i hi t i t bl h ll t k i A b i f

dr

cr

br

ar DCBA

(3-1)

In formulating our stoichiometric table, we shall take species A as our basis of calculation (i.e., limiting reactant) and then divide through by the stoichiometric coefficient of A

db DadC

acB

abA (2-2)

In order to put everything on a basis of “per mole of A.”Seoul National University

3.5 Batch System

t = 0N

t = tN

NA0NB0N NA

NBN

NC0ND0N NC

NDNDdCcBbA

Ni0

NiD

aC

aB

aA

Batch reactors are primarily used for the production ofc e c o s e p y used o e p oduc o ospecialty chemicals and to obtain reaction rate data in order todetermine reaction rate laws and rate law parameters such as k,the specific reaction rate.


3.5 Batch System

t = t

t = 0NA0NB0

At time t=0, we will open the reactor andl b f l f i A B t t

NANBNCN

NB0NC0ND0Ni0

place a number of moles of species A, B,C, D, and I (NA0, NB0, NC0, ND0, and NI0,respectively)

NDNi

DdCcBbA

p y)

Species A is our basis of calculation.D

aC

aB

aA

NA0 is the number of moles of A initially present in the reactor.

NA0X moles of A are consumed as a result of the chemical reaction.

NA0-NA0X moles of A leave in the system.

The number of moles of A remaining in the reactor after conversion X

N N N X N (1 X)NA = NA0 - NA0X = NA0(1-X)Seoul National University

Determination of the number of moles of B

To determine the number of moles ofi B i i t ti t ( ft N X

DdCcBbA

species B remaining at time t (after NA0Xmoles of A have reacted). For every moleof A that reacts, b/a moles of B must D

aC

aB

aA ,

reacted;

The number of moles of B remaining in the system, NBB

moles of A reacted

XNbNN ABB 00 a ABB 00

moles of B disappearedmoles of B initially


Stoichiometric table

To determine the number of moles ofh i i i f

db

each species remaining after NA0Xmoles of A have reacted, we form thestoichiometric table (Table 3 3)

DadC

acB

abA

stoichiometric table (Table 3-3).

This stoichiometric table presents the following information

Column 1: the particular species

This stoichiometric table presents the following information.

Column 1: the particular species

Column 2: the number of moles of each species initially present

Column 3: the change the number of moles brought about by reaction

Column 4: the number of moles remaining in the system at time t.g y


Stoichiometric Table for a Batch System

Remaining Change Initially Species

0000 )( AAAAA XNNNXNNA(mol) (mol) (mol)

0000 )( ABBAB XNabNNXN

abNB

0000 )( ACCAC

dd

XNacNNXN

acNC

0000

)(

)( ADDAD

NNNinertI

XNadNNXN

adND

00)( III NNNinertI

XNab

ac

adNNNTotal ATTT 000 1

aaa

XNNN ATT 00 Seoul National University

The total number of moles per mole of A reacted

DdCcBbA aaa

The total number of moles in the system, NTy , T

XNbcdNN 1

XNN

XNaaa

NN

AT

ATT

00

00 1

bcd Change in the total number of moles1

ab

ac

ad

Change in the total number of moles

mole of A reacted





b

Rate LawsbB

aAA CkCr

-rA=f(X)C = f(X) ry

CHAPTER 2

Cj = f(X)

iom

etr

Nj = f(X)

stoi

chi

V= f(X) s


3.5.1. Concentration of each species

VXN

VNC AA

A0 )1(

bB

aAA CkCr

VXNabN

VNC ABB

B00 )/(

CA =f(X)

XNacNNC

VVACC

C00 )/(

Nj = f(X)V= f(X)

XNadNNC

VVC

ADD

C

00 )/(

V= f(X)

VVC ADD

D00 )(


3.5.1. Concentration of each species

yCNyy

CC

NN

A

i

A

i

A

ii

0

0

0

0

0

0

XNNC AAA

)1(0

XabNXNabNNC

VVC

BAABBB

A

)/()/( 000

XacNXNacNNC

VVVC

CAACCC

B

)/()/( 000

XadNXNadNNC

VVVC

DAADDD

C

)/()/( 000

VVVC D


3.5.2 Volume as a function of conversion

)/()/()( XbNXNbNXN )(

)/()()/(

)()( 000

XVXabN

XVXNabN

XVXNC BAABB

B

We need V(X) to obtain C =f(X)We need V(X) to obtain CB=f(X)- For liquids, volume change with reaction is negligiblewhen no phase changes are taking place (V=V0)when no phase changes are taking place. (V V0)

- For gas-phase reactions, the volumetric flow rate mostoften changes during the course of the reaction due to aoften changes during the course of the reaction due to achange in the total number of moles or in temp. or pressure.


3.5.2 Constant-Volume Batch Reaction Systems

Constant volume system (=constant density system):

(1) The lab bomb calorimeter reactor: the volume within the vesselis fixed and will not change. V=V0is fixed and will not change. V V0

(2) A constant-volume gas-phase isothermal reaction occurs whenth b f l f d t l th b f l fthe number of moles of products equals the number of moles ofreactants. (Ex: water-gas shift reaction, CO+H2O CO2+H2)

(3) For liquid-phase reactions taking place in solution, the solventusually dominates the situation. As a result, changes in thedensity of the solute do not affect the overall density of thedensity of the solute do not affect the overall density of thesolution significantly and therefore it is essentially a constant-volume reaction process: Most liquid-phase organic reactions,except polymerization.


Constant Volume Batch Reactor


Constant Volume System

For Liquid phase reactions (or isothermal and isobaric gas-phasereactions with no change in the total number of moles)reactions with no change in the total number of moles)

DadC

acB

abA

aaa

BAAA CCkr -rA = f(C)

b

Eq. (3-26)

X

abXCkr BAAA )1(2

0-rA = f(X)

Levenspiel plot


Example 3-2: Liquid-Phase Reaction

Soap consists of the sodium and potassium salts of various fattyacids as oleic(C18=), stearic(C18), palmitic(C16), lauric(C12),and myristic(C14) acids. The saponification for the formation ofsoap from aqueous caustic soda and glyceryl stearate is as followsoap from aqueous caustic soda and glyceryl stearate is as follow.

3N OH (C H COO) C H 3C H COON C H (OH)3NaOH + (C17H35COO)3C3H5 3C17H35COONa + C3H5(OH)3

Letting X represent the conversion of sodium hydroxide (themole of sodium hydroxide reacted per mole of sodium hydroxidei i i ll ) i hi i bl i hinitially present), set up a stoichiometric table expressing theconcentration of each species in terms of its initial concentrationand the conversion of Xand the conversion of X.

Stearic acid Seoul National University


(C17H35COO)3C3H5 Glyceryl stearate

C17H35COONa

Na

C3H5(OH)3Glycerine=Glycerol


Example 3-2 Stoichiometric Table

3NaOH + (C17H35COO)3C3H5 3C17H35COONa + C3H5(OH)3

11

ionConcentratRemainingChangeInitiallySpecies

DCBA31

31

3

)1()1()(

)(mol/m(mol) (mol) (mol) ionConcentrat Remaining Change Initially Species

XCXNXNNA

0000

0000

33)(

31

)1()1()(

BABAAB

AAAAXCXNXNNB

XCXNXNNA

0000

0000

)(1)()()(

333

BABAAD

CACAACXCXNXNND

XCXNXNNC

000

0000

)(33

)(3

III

BABAAD

CNNinertI

CNXNND

00 0 TTT NNNTotal Seoul National University

3.6 Flow Systems

FA0F

FAF

Entering Leaving

DadC

acB

abA

FB0FC0FD0FI0

FBFCFDFIFI0 FI

Molar flow rate

litermoles

eliters/timmoles/time

AA

FCDefinition ofconcentration

for flow system(3-27)

litereliters/timvolumetric flow rate

00000 iiiii

yCvCF

00000 AAAAi yCvCF


3.6.1 Equations for Concentrations in Flow Systems

Batch System Flow System

DadC

acB

abA

XNabNNV

XNVNC

ABB

AAA

00

0

)/(

)1(

XFabFFX

FFC AAA

0

)/()1(

XNacNNC

VXNabN

VNC

ACCC

ABBB

00

00

)/(

)/(

XFacFFC

XFabFFC

ACCC

ABBB

00

00

)/(

)/(

VXNadN

VNC

VVC

ADDD

C

00 )/(

XFadFFC

C

ADDD

C

00 )/(VV


Stoichiometric Table for a Flow System

rate Effluent Change rate Feed

X)(FFX)(FFA 1(mol/time) (mol/time) (mol/time)

reactor from reactor in reactor to Species

BABAABB

AAAA

XabΘFFX)(F

abFΘFB

X)(FFX)(FFA

0000

000 1

CACAACC XacΘFFX)(F

acFΘFC

aa

0000

DADAADD XadΘFFX)(F

adFΘFD 0000

IAIAII ΘFFFΘFI(inert) 000

XFbcdFFFTotal 1 XF

aaaFFFTotal ATTT 000 1

Table 3-4 XFFF ATT 00 Seoul National University

3.6.2 Liquid-Phase Concentrations

FA0F

FAF

Entering Leaving

DadC

acB

abA

FB0FC0FD0FI0

FBFCFDFIFI0 FI

)1()1(0 XCXFFC AA baCkCr

)(

)1()1(

0

00

XbCC

XCXC

BAB

AA

BAA CkCr

(3-29))(0 X

aCC BAB

CA = f(X)

FA = f(X)v=f(X)


3.6.3 Change in the Total Number of Moles with Reaction in the Gas Phasewith Reaction in the Gas Phase

(1) Gas-phase reactions that do not have an equal number of product( ) p q pand reactant moles. In flow systems where this type of reaction occurs,the molar flow rate will be changing as the reaction progress. Thevolumetric flow rate will also change due to molar flow change

N + 3H 2NH

volumetric flow rate will also change due to molar flow change.

N2 + 3H2 2NH3

(2) The combustion chamber of the internal-combustion engine

(3) The expanding gases within the breech and barrel of a firearm as it p g gis fired.

breech : 포미 (砲尾)

barrel : 총신(銃身)


Batch Reactor with Variable Volume )()(

XVXNC B

B

Individual concentrations can be determined by expressing the volume V forbatch system (or volumetric flow rate v for a flow system) as a function ofy ( y )conversion using the following equation of state.

RTZNPV T (3-30)

T = temperature, KP = total pressure, atm (kPa; 1 atm=101.3 kPa)Z ibilit f tZ = compressibility factorR = gas constant = 0.08206 dm3·atm/gmol · K

00000 RTNZVP T

This equation is valid at any point in the system at any time. At time t=0,

(3-31)

0 TNZTP

Dividing (3-30) by (3-31) and rearranging yields

We need this!

000

00

T

T

NN

ZZ

TT

PP

VV

(3-32)


Stoichiometric Table for a Batch System

Remaining Change Initially Species

0000 )((mol) (mol) (mol)

AAAAA XNNNXNNA

0000 )( ABBAB XNabNNXN

abNB

0000 )( ACCAC

dd

XNacNNXN

acNC

00

0000

)(

)(

III

ADDAD

NNNinertI

XNadNNXN

adND

00)( III NNNinertI

XNab

ac

adNNNTotal ATTT 000 1

aaa

XNNN ATT 00 Seoul National University

Volume as a function of conversionvariable-volume batch reactor

The total number of moles in the system, NT

XNNN (3 33)XNNN ATT 00

We divide through by NT0

(3-33)

XXyXNN

NN

AT

A

T

T 111 00

0

0(3-34)

where yA0 is mole fraction of A initially present. If all the species in thegeneralized reaction are in the gas phase, then

(3-35)

0

01 AA y

NN

ab

ac

ad

( ) 0TNaaa

Eq (3-32) now becomes

)1(00

00

000

00 X

ZZ

TT

PP

VNN

ZZ

TT

PP

VVT

T

(3-37)


Volume as a function of conversionvariable-volume batch reactor

In gas-phase systems that we shall be studying, the temperature andpressure are such that the compressibility factor will not changepressure are such that the compressibility factor will not changesignificantly during the course of the reaction; hence Z0~Z. For a batchsystem the volume of the gas at any time t is

0 )1( TXP

VV

(3-38)

00 )1(

TX

PVV

Eq (3-38) applies only to a variable-volume batch reactor. If the reactoris a rigid steel container of constant volume, then of course V=V0. For ais a rigid steel container of constant volume, then of course V V0. For aconstant-volume container, V=V0, and Eq. (3-38) can be used tocalculate the pressure inside the reactor as a function of temperature andconversionconversion.


Volume as a function of conversionvariable-volume flow system

To derive the concentration of the species in terms of conversion for avariable-volume flow system, we shall use the relationships for the totalvariable volume flow system, we shall use the relationships for the totalconcentration. The total concentration at any point in the reactor is

PF (3-39)ZRT

PFC TT

At the entrance to the reactor

00

0

0

00 RTZ

PFC T

T

At the entrance to the reactor

(3-40)000

Taking Eq (3-40)/Eq(3-39) and assuming Z~Z0,

0

0

00 T

TPP

FF

T

T (3-41) 00 TPFT


Concentration as a function of conversionvariable-volume flow systemvariable-volume flow system

We can express the concentration equation of species j for a flow system in terms

TPFFFF

C jTjj 00

of conversion:

TPF

TT

PP

FF

CT

jT

T

T

jjj

0

00

0

0

000

(3-42)

TT

PP

FF

CCT

jTj

0

00

TPFT 0

n

FFFFFFF

The total molar flow rate is just the sum of the molar flow rates of each of the species in the system and is

j

jIDCBAT FFFFFFF1


Concentration as a function of conversionvariable-volume flow systemvariable-volume flow system

One of the major objective of this chapter is to learn how to expressany given rate law –rA as a function of conversion. The schematicdiagram in Figure 3-6 helps to summarize our discussion on this

i tpoint.

The concentration of B expressed as a function of conversion inThe concentration of B expressed as a function of conversion inboth flow and batch systems, for various conditions of temperature,pressure, and volume.


Expressing concentration as a function of conversion

DadC

acB

abA

B

BFC

VNC B

B VNC B

B FC B

B

Liquid Phase Gas Phase

Flow Batch FlowBatch

V

TPNNC

TT

PP

NNVV

V

TB

T

T

00

0

0

00

0 0VV

TPFFC

TT

PP

FF

TBT

T

000

0

00

No Phase ChangeConstantVolumeVV

TT

PP

NNCC

TPVNC

T

BTB

TB

0

00

00

TT

PP

FFCC

TPFC

T

BTB

TB

0

00

00

XbCC BAB 0

0VV

X

aCC BAB 0

0y A TPXvv 0)1(

No Phase Change or No Semipermeable Membranes

00

1

PC

ab

ac

ad

T

TT

PP

X

XabC

C

TPXvv

BA

B0

0

0

00

1

)1(

0

000

00

RTPy

C

RTC

AA

T

TPX 01

isothermal Isothermal + No P

0

0

1 PP

X

XabC

CBA

B

X

XabC

CBA

B

1

0No P

pp112 Seoul National University


From Table 3-3, the total molar flow rate can be written in terms of X

XFFF ATT 00 (3-43)

0

0

0

000 T

TPP

FXFF

T

AT

0

000

0

0

0

00

00

11TT

PP

XyTT

PP

XFF

TPF

AT

A

T

(3-44)000T

0

00 1

TT

PP

X

(3-45)

0TP



The concentration of species j is

XF

XFvFF Ajjj

00 )(

jF XvF jjA 0

TP

j

jC

0

00 1

TT

PP

X

XvF

0

00

0

1TT

PPX

XvFC jjA

j (3-46)

0TP


Concentration as a function of conversionMultiple gas-phase reaction and membrane reactorMultiple gas-phase reaction and membrane reactor

TPF

CC j 0

TPF

CCT

jTj

0

00

Substituting for Fj and FT in terms of conversion in Eq. (3-45) yields

TPXvF jjA 00

TT

PP

XFFXvF

CCAT

jjATj

0

000

00

Dividing numerator and denominator by FT0, we have

TT

PP

XFFXv

FF

CCTA

jj

T

ATj

0

0000

00 )/(1


Concentration as a function of conversionMultiple gas-phase reaction and membrane reactorMultiple gas-phase reaction and membrane reactor

Recalling y 0=FA0/FT0 and CA0=yA0CT0, then

(3 46)

Recalling ya0 FA0/FT0 and CA0 yA0CT0, then

TPXvC jjA0 (3-46)

TT

PP

XXvC

C jjAj

0

0

0

1

where vi is the stoichiometric coefficient, which is negative for reactants and positive for productsreactants and positive for products.

For example, for the reaction

DadC

acB

abA

vA = -1, vB = -b/a, vC = c/a, vD = d/a, and j=Fj0/FA0. pp114 Seoul National University

Concentration as a function of conversioni bl l fl tvariable-volume gas flow system

Table 3-4

AA XFFC 0 )1( 00 )1( PTXFA

00

)1( PTXC A

BABB

A

XabFFC

C

0 )/( 00

00

)1()/(

)1(

PP

TT

XXabF

PTX

BA

00

00

1)/(

)1(

PP

TT

XXabC

PTX

BA

A

CACC

B

XacFFC 0 )/(

0

0

0

0

00

)1()/(

)1(

PP

TT

XXacF

PTX

CA

0

00

0

1)/(

1

PP

TT

XXac

C

PTX

CA

DADD

XadFFC 0 )/( 0

0

0

0

00

)1()/(

)1(

PP

TT

XXadF

PTX

DA

0

00

0

1)/(

1

PP

TT

XXadC

PTX

DA

IAII

FFC 0

0

0

0

0

)1( PP

TT

XF IA

0

00

0

1 PP

TT

XC IA

pp114 Seoul National University

Example 3-4: Manipulation of the Equation Cj=fj(X)

Show under what conditions and manipulation the expression for CB forShow under what conditions and manipulation the expression for CB for a gas flow system reduces to the following equation in Table 3-5.

00 1

)/(PP

TT

XXabCC B

AB

0

0 1 PTXAB

pp114


Example 3-4: Manipulation of the Equation Cj=hj(X)

XbFF BAB 0

aBAB 0

0 TP

B

BFC

0

00 1

TT

PP

X

(3-45)

0

0

1

)/(TPX

XabFC BA

B (E3-4.4)

0

00 1

TPX

TT

PP

XXabCC B

AB0

0 1)/(

TPX 01pp115



Isothermal

TT

PP

XXabCC B

AB0

0 1)/(

TPX 01



BAA CkCr A B C BAA

28.00 Ay


FC0=0

11 XF211

211

20

000XF

FXFFF ATATT


= 227+273


Concentrations as a function of conversion

Li iti t tLimiting reactant


3221 SOOSO

0.25

322 2SOOSO

dm3 )

0.20

N2

(mol

/d

0.15

tratio

n

0.10 SO2SO3

once

nt

0.05O2

C

0.00

Conversion, X0.0 0.2 0.4 0.6 0.8 1.0


Reaction rates as a function of conversion

322 21 SOOSO

CkCr BAA

22

0 )1()5.0)(1(

XXXkC B

A

2)14.01()5.054.0)(1(2

XXX


3221 SOOSO

60

322 2SOOSO

Levenspiel Plot

50

2 )5.0)(1( XXkC

CkCr

B

BAA

p

r A)

402

20

)5.054.0)(1(2

)1())((

XX

XkC B

A

(-1/r

20

30 2)14.01())((

X

10

20

0.0 0.2 0.4 0.6 0.8 1.00

Conversion, XSeoul National University

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