Rate Laws Rate Laws and

Chemical Reactor Design Chemical Reactor Design

Y W L

Youn-Woo Lee

School of Chemical and Biological Engineering Seoul National Universityy

155-741, 599 Gwanangro, Gwanak-gu, Seoul, Korea  ywlee@snu.ac.kr  http://sfpl.snu.ac.kr

第

第33章章 第

第 章章

Rate Laws Rate Laws and

and Stoichiometry Stoichiometry

Chemical Reactor Design Chemical Reactor Design

y y

Chemical Reactor Design Chemical Reactor Design

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

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 conversion

express concentration as a function of conversion.

 Calculate the equilibrium conversion for both gas and liquid phase reactions

reactions.

 Write the combined mole balance and rate law in measures other than conversion

than conversion.

Reactor Size

Design Equations



 ^X dX

N Batch t

Levenspiel plot

A X V F ⁰





A

A r V

N t 0 0

Batch

CSTR

A A

V r

 ⁰



CSTR

Graphical method



 ^X

A

A r

F dX V 0 0

PFR

For vs. X, the volume of a

A A0

r F





 ^X

A

A r

F dX W 0 0

PBR CSTR and the volume of a PFR

can be represented as the shaded areas in the Levenspiel plots.

If we know the molar flow rate to the reactor and the reaction rate as a function of conversion, then we can



 ^X

A

A r

F dX V 0 0

PFR

calculate the reactor volume necessary to achieve a specific conversion.

) f ( X r_A 



Design Isothermal Reactor

Design isothermal reactor

CHAPTER 3

Design isothermal reactor

b

Rate Laws

b B a

A

A

kC C

r 



-r

=f(X)

C = f(X)

ry

CHAPTER 2

C_j= f(X)

iomet r

N_j = f(X)

stoich i

) f ( X F_B 

 ^B

B

C F

V= f(X)

s

) f ( X

 

B 

P t 1

Part 1

R t L

Rate 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 presents concentrations of different species to each other. A stoichiometric table presents the stoichiometric relationships between reacting molecules for a single reaction.

aA + bB cC + dD

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

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.”

d c

b D

a C d

a B c

a

A  b   ^(2-2)

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

r r

r

r_A  _B  _C  _D (3-1)

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

d c

b a 

 ^{(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 h

of the reaction. The limiting reactant is usually chosen as our basis for calculation.

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

t t k d constant k and

) (

) ( )

( T , C ) k ( T ) f ( C , C ...)

(

A

T C k T f C C

r  



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

3.2.1 Power Law Models and Elementary Rate Laws

The dependence of the reaction rate r on the concentration of The dependence of the reaction rate –r_A on the concentration of the species is almost without exception determined by experimental observation.

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





 r

k

C

C

 order with respect to reactant A

 order with respect to reactant B

n (=) : the overall order of the reaction

Unit of Specific Reaction Rate

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

A  products

s dm

k mol k

r order

-

Zero _{A A 3}

) } (

{

:     (3-4)

k C

k r

order First

s dm

} 1 { :

) (





 (3 5)

d k s

C k r

order -

First _{A A A}

)3

( } {

:    (3-5)

s mol k dm

C k r

order -

Second _{A A A}

3

2 ( )

} {

:     (3-6)

s mol k dm

C k r

order -

Third _{A A A}

2 3

3 ( / )

} {

:    (3-7)

Elementary and Non-elementary Reaction

Kinetic rate raw Kinetic rate raw

“Elementary reaction” Non-elementary reaction

2 / 3

CO + Cl₂  COCl₂ O^ + CH₃OH  CH₃O^ + OH^

1^st order w r t atomic oxygen

2 / 3 Cl2

CO

CO

kC C

r 



1^st order w r t carbon monoxide

-r_O = k C_O C_CH3OH

1 order w.r.t. atomic oxygen 1^st order w.r.t. methanol

overall is 2^nd order reaction

1 order w.r.t. carbon monoxide 3/2 order w.r.t. chorine

overall is 5/2 order reaction

In general first and second order reactions are more commonly observed In 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 by It is important to remember that the rate laws are determined by experimental observation! They cannot be deduced from reaction stoichiometry.

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

Even though a number of reactions follow elementary rate laws at Even though a number of reactions follow elementary rate laws, at least as many reactions do not. One must determine the reaction order from the experiments or from literature

order from the experiments or from literature.

Literature

Th ti ti f f t d ti d

The 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 Offi

Washington, 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 , , p

Laboratories, Pasadena, CA, USA

 International Journal of Chemical Kinetics, Journal of Physical Chemistry

 Journal of Catalysis, Journal of Applied Catalysis

 AIChE 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 Law

Example of Rate Law

A. First Order Reaction

B. Second Order Reaction

3.3 k

: The specific reaction rate (the rate constant)

The reaction rate constant k is not truly a constant, but is merely independent of the concentrations of the species involved in the independent of the concentrations of the species involved in the reaction.

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 strength and choice of solvent

and 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 e specific reaction rate than temperature does with the exception of supercritical solvents, such as supercritical water.

Arrhenius equation q

Specific reaction rate (constant)

Activation energy, J/mol or cal/mol

 E A T Ae RT

k ( )  ^

for ionic theory

A ( )

frequency factor

Absolute Temperature, K frequency factor

or

pre-exponential

Gas constant 8.314 J/mol · K pre exponential

factor mathematical number e=2.71828…

1.987 cal/mol · K

8.314 kPa · dm³/mol · K

Activation energy gy

Activation energy E

a minimum energy that must be possessed by reacting molecules before the reaction will occur.

E The fraction of the collisions between

l l th t t th h thi i i

e

Activation energy E is determined experimentally by carryingy g out the reaction at several different temperature.

 

  A E

k 1

l

l 



 



 

 A R T

k

ln

ln

P t 2

Part 2

St i hi t

Stoichiometry

Reactor Size

Design Equations



 ^X dX

N Batch t

Levenspiel plot

A X V F ⁰





A

A r V

N t 0 0

Batch

CSTR

A A

V r

 ⁰



CSTR

Graphical method



 ^X

A

A r

F dX V 0 0

PFR

For vs. X, the volume of a

A A0

r F





 ^X

A

A r

F dX W 0 0

PBR CSTR and the volume of a PFR

can be represented as the shaded areas in the Levenspiel plots.

If we know the molar flow rate to the reactor and the reaction rate as a function of conversion, then we can



 ^X

A

A r

F dX V 0 0

PFR

calculate the reactor volume necessary to achieve a specific conversion.

) f ( X r_A 



Design Isothermal Reactor

Design isothermal reactor

CHAPTER 3

Design isothermal reactor

b

Rate Laws

b B a

A

A

kC C

r 



-r

=f(X)

C = f(X)

ry

CHAPTER 2

C_j= f(X)

iomet r

N_j = f(X)

stoich i

V= f(X)

s

Example 3-5: Determination of C

=h

(X) for a Gas Phase Reaction

for a Gas-Phase Reaction

Isothermal



 



 

 









 

T T P

P X

X a C b

C_{B A0} ^{B 0}

1

) / (



 

 1 X P₀ T

Stoichiometric Table

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

aA + bB cC + dD

aA + bB cC + dD

r r

r

r_{A B C D}

(2-1)

(3 1)

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

d r c

r b

r a

r_{A B}  _C  _D

 

 ^(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

d

b D

a C d

a B c

a

A  b   _(2-2)

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

3.5 Batch System

t = 0 N

t = t N N

N

N N

N

N N

N

N N

N

D

N

C d B c

A  b  

N

N

a D a C

a B

A   

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

3.5 Batch System

t = t t = 0

N_A0 N_B0

At time t=0, we will open the reactor and

l b f l f i A B ^{t t}_N

A

N_B N_C N N_B0

N_C0 N_D0 N_i0

place a number of moles of species A, B, C, D, and I (N_A0, N_B0, N_C0, N_D0, and N_I0, respectively)

N_D N_i

d D c C

b B

A   

p y)

Species A is our basis of calculation.

a D a C

a B

A   

N_A0 is the number of moles of A initially present in the reactor.

N_A0X moles of A are consumed as a result of the chemical reaction.

N_A0-N_A0X 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)

N

= N

- N

X = N

(1-X)

Determination of the number of moles of B

To determine the number of moles of

i B i i t ti t ( ft N X

d D c C

b B

A   

species B remaining at time t (after N_A0X moles of A have reacted). For every mole

of A that reacts, b/a moles of B must _D

C a B a

A  a  

, reacted;

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

moles of A reacted

X b N

N

N





a

B

B 0 0

moles of B disappeared moles of B initially

Stoichiometric table

To determine the number of moles of

h i i i f

d b

each species remaining after N_A0X moles of A have reacted, we form the

stoichiometric table (Table 3 3) _D

a C d a B c

a

A b  

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

0 0

0

0 ( _A ) _{A A A}

A N X N N N X

N A

(mol)

(mol)

(mol)







0 0

0

0 ( _A ) _{B B A}

B N X

a N b

N X

a N N b

B   

0 0

0

0 ( _A ) _{C C A}

C

d d

X a N

N c N

X a N

N c

C   

0 0

0 0

) (

)

( _{A D D A}

D

N N

N inert

I

X a N

N d N

X a N

N d D









0

) 0

(inert N_I N_I N_I

I 

X a N

b a

c a

N d N

N

Total _{T0 T T0} 1 _A0



 



   



  a a a 

X N

N

N   

The total number of moles per mole of A reacted

d D c C

b B

A   

a a

a

The total number of moles in the system, Ny , _TT

X b N

c N d

N     1 X

N N

X a N

a N a

N

A T

A T

T

0 0

0

0 1









   





b c

d

 1





 a

b a

c a

 d

mole of A reacted

Design Isothermal Reactor

Design isothermal reactor

CHAPTER 3

Design isothermal reactor

b

Rate Laws

b B a

A

A

kC C

r 



-r

=f(X)

C = f(X)

ry

CHAPTER 2

C_j= f(X)

iomet r

N_j = f(X)

stoich i

V= f(X)

s

3.5.1. Concentration of each species

V

X N

V

C

N

( 1  )





a A

A kC C

r 



V

X N

a b

N V

C

N

 ( / )





X N

a c

N C N

V V

A C

C C 0

 ( / )





X N

a d

N C N

V C V

A D

D C

0

0

 ( / )

V= f(X)

V

C

V

( )





3.5.1. Concentration of each species

y C

N

y y C

C N

N

A i A

i A

i i

0 0 0

0 0

0  





X N

C_A N ^{A A}₀ (1 )









N X

N a b N

C N

V C V

B A

A B B

B A

) / ( )

/

( _{0 0}

0  

 









N X

N a c N

C N

V V

C V

C A

A C

C C B

) / ( )

/

( _{0 0}

0  

 









N X

N a d N

C N

V V

C V

D A

A D D

D C

) / ( )

/

( _{0 0}

0  

 



 V V V

C_D   

3.5.2 Volume as a function of conversion





) / ) (

(X N b N X N b X

N





) (

) / ( )

(

) / ( )

(

)

( _{0 0 0}

X V

X a b N

X V

X N

a b N

X V

X

C_B N ^{B B A A} ^B 

 





We need V(X) to obtain C =f(X) We need V(X) to obtain C_B=f(X)

- For liquids, volume change with reaction is negligible when 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 most often changes during the course of the reaction due to a often changes during the course of the reaction due to a change 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 vessel is fixed and will not change. V=V₀

is fixed and will not change. V V₀

(2) A constant-volume gas-phase isothermal reaction occurs when

th b f l f d t l th b f l f

the number of moles of products equals the number of moles of reactants. (Ex: water-gas shift reaction, CO+H₂O  CO₂+H₂) (3) For liquid-phase reactions taking place in solution, the solvent

usually dominates the situation. As a result, changes in the density of the solute do not affect the overall density of the density of the solute do not affect the overall density of the solution 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-phase reactions with no change in the total number of moles)

reactions with no change in the total number of moles) a D C d

a B c

a

A  b   a a

a

B A

A

A

k C C

r 



-r_A = f(C)



 b

Eq. (3-26)

 

 



  





 X

a X b

C k

r

( 1 )

-r_A = f(X)

Levenspiel plot

Example 3-2: Liquid-Phase Reaction

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

i i i ll ) i hi i bl i h

initially present), set up a stoichiometric table expressing the concentration of each species in terms of its initial concentration and the conversion of X

and the conversion of X.

Example 3-2: Liquid-Phase Reaction

(C₁₇H₃₅COO)₃C₃H₅ Glyceryl stearate

C₁₇H₃₅COONa

Na

C₃H₅(OH)₃

Example 3-2: Liquid-Phase Reaction

Example 3-2 Stoichiometric Table

3NaOH + (C₁₇H₃₅COO)₃C₃H₅  3C₁₇H₃₅COONa + C₃H₅(OH)₃ 1

1

ion Concentrat Remaining

Change Initially

Species

D C

B

A 3

1 3

1  



3

) 1

( )

1 ( )

(

) (mol/m (mol)

(mol)

(mol)

ion Concentrat Remaining

Change

Initially

Species

X C

X N

X N

N A

0 0

0 0

0 0

0 0

3 ) 3

3 ( 1

) 1

( )

1 ( )

(

B A

B A

A B

A A

A

A X

X C N

X N

N B

X C

X N

X N

N A



 



 



 



 









0 0

0 0

0 0

0 0

) 1 (

) (

) (

) (

3 3

3

B A

B A

A D

C A

C A

A

C X

X C N

X N

N D

X C

X N

X N

N C



 



 







   



0 0

0

0 0

0 0

) (

3 ) 3

3 (

I I

I

B A

B A

A D

C N

N inert

I

C N

X N

N

D 

 

 



 

0

0 0 _{T T}

T N N

N

Total 

3.6 Flow Systems

F_A0 F

F_A F

Entering Leaving

a D C d

a B c

a

A  b   F_B0

F_C0 F_D0 F_I0

F_B F_C F_D F_I

F_I0 F_I

Molar flow rate

liter moles e

liters/tim

moles/time 

 



A

C F

Definition of concentration

for flow system (3-27)

liter e

liters/tim



volumetric flow rate

0 0

0 0

0 i i i

i i

y C

v C

F   





0 0

0 0

0 A A A

A

i

F C v C y



3.6.1 Equations for Concentrations in Flow Systems

Batch System

Flow System

a D C d a B c

a

A  b  

X N

a b N

N

V X N

V C N

A B

B A A A

0 0

0

) / (

) 1

(



 

 

 

 

X F

a b F

F

F X C_A F^{A A0}

) / (

) 1

(

X N

a c N

C N

V

X N

a b N

V C N

A C

C C

A B

B B

0 0

0 0

) / (

) / (

 







 





 

 

X F

a c F

C F

X F

a b F

C F

A C

C C

A B

B B

0 0

0 0

) / (

) / (

V

X N

a d N

V C N

V C V

A D

D D C

0 0  ( / )



 

 

 

 

 

X F

a d F

C F C

A D

D D C

0 0 ( / )

V

V  

Stoichiometric Table for a Flow System

rate Effluent

Change

rate

Feed

X) (

F F

X) (F

F

A 1

(mol/time)

(mol/time)

(mol/time)

reactor from

reactor

in reactor

to Species

B A

B A

A B B

A A

A A

a X Θ b

F F

X) a (F

F b Θ F

B

X) (

F F

X) (F

F A

0 0

0 0

0 0

0 1



 



 













C A

C A

A C

C X

a Θ c

F F

X) a (F

F c Θ F

C

a a

0 0

0

0 



 



 











D A

D A

A D

D X

a Θ d

F F

X) a (F

F d Θ F

D _{0 0 0 0} 



 



 











I A I

A I

I Θ F F F Θ

F

I(inert) ₀  ₀  ₀

X b F

c F d

F F

Total 1

   



 F X

a a

F a F

F

Total _{T0 T T0} 1 _A0

 

   





X F

F

F   