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Diffusion in Spherical CatalystChemical Reaction Engineering

K. NarendiranPhD Scholar

Ganesh A ViswanathanAssistant Professor

2

Catalyst

A Catalyst is a substance that affects the rate of reaction but emerges from the process unchanged.

Catalysis

Catalysis is the process that affects the rate of a chemical reaction by means of a chemical substance

known as a catalyst.

Fick’s Law

Fick's first law provides a constitutive relation for diffusive flux. The relation is based on the

postulation that the transport of species is from regions of high concentration to regions of low

concentration, and as a result the diffusive flux is directly proportional to the concentration gradient.

The molar flux of species A is in the radial direction given by

---------- (1)

where DeA – effective molecular diffusivity; CA – concentration of the species A;

r – radius of the catalyst pellet; NAr – molar flux in the radial direction

AAr eA

dCN Ddr

Diffusion in a Catalyst PelletConsider the first order catalytic reaction , where A, B, k are the reactant, product,

and rate constant, respectively occurring in the pores of a spherical catalyst pellet placed in a

vessel. Species A is transported from bulk into the pellet. Reaction occurs in the pores of the

pellet and product B diffuses out to the vessel. The rate of the reaction with respect to species A

is given by ----------- (2)

where k - rate constant (1/s)CA - concentration of reactant A (moles/m3)-rA - rate of consumption of species A (moles/m2s)

Reactant A

Product BPorous catalyst

pellet

R

External Diffusion Internal

Diffusion

kA B

A Ar kC

3

Vessel with

Species A

Assumptions

• Spherical catalyst particles

• Pseudo – first order, irreversible reaction

• Isothermal conditions

• Steady state, that is, rate of accumulation is zero

• Concentration gradients present only in the radial direction

Mole Balance

The mole balance for species A is

{Molar flux in} – {Molar flux out} + {Rate of generation due to reaction} = Rate of accumulation

---------- (3)

Dividing by 4πr2Δr and setting Δr → 0 gives

---------- (4)

where R – radius of the spherical catalyst pellet4

2 2 2.4 .4 .4 0Ar Ar Ar r rN r N r r r r

2 2

20

( ) ( )1lim Ar Arr r rAr

r N r Nr

r r

2.4Ar rN r

2.4Ar r rN r

r

R

SC

External SurfaceConcentration

0r r R

Using the definition of the first derivatives

---------- (5)

Using the definition of the Fick’s law of diffusion (Eq. 1), the mathematical model becomes

---------- (6)

and the corresponding boundary conditions are

@ ---------- (7)

@ ---------- (8)

Dimensionless Variables

Assuming the bulk concentration of species A, CAS and pellet’s volume to surface ratio, a as scaling

variables, the dimensionless variables are

---------- (9)

where ---------- (10)

A ASC C r R

0AdCdr

0r

3

2

4Volume of sphere 3

Surface area of sphere 4 3

R RaR

ra

A

AS

CCC

5

22

1 ( )Ar Ad r N r

r dr

22

1 0AeA A

dCdD r kCr dr dr

Substitution of the dimensionless variables (Eq. 9) into the model equations(Eqs 6 – 8) leads to

---------- (11)

where is the Thiele modulus

Boundary Conditions

at ---------- (12)

at ---------- (13)

Introduction of the transformation to solve Eq. 11 leads to

---------- (14)

Boundary Conditions ---------- (15)

---------- (16)

22

2

2 0d C dC Cd d

1C 3

0dCd

0

2

eA

kaD

( )( ) uC

6

22

2 0d u ud

(0) 0u

(3) 3u

The general solution of the model (Eq. 14) is

---------- (17)

where a and are integration constants.

Appling the boundary conditions (Eqs 15 and 16), the concentration profile of species A inside

the pellet is given by

---------- (18)

where - Thiele Modulus

- Dimensionless Radius , range of which is [0,3]

C - Dimensionless Concentration

( ) ( ) ( )u Cosh Sinh

( ) 3 ( )( )(3 )

u SinhCSinh

7

Effect of Thiele modulus on the concentration

profile in the catalyst pelletFor large values of , the time for the reaction to occur is

larger than the time required for molecular diffusion of

the species. Therefore, the reactant species is converted

to product before the species can diffuse well into the

catalyst pellet. As a result, the reaction goes to

completion near the surface of the catalyst.

For small values of , the time required for molecular

diffusion is larger than the reaction rate. Therefore, the

reactant species diffuses into the pellet before the

reaction goes to completion.2

eA

kaD

where

8Dark Blue color of the catalyst represent that reaction occur at surface (large value). Light Blue color of the catalyst representation for lower value i.e. diffusion occur in a catalyst.(In Graph, radius(ξ range of [0,3]) and concentration(C range of [0,1]) is dimensionless)

Color variation from dark to light to represent the concentration of species in the pellet

Higher Concentration

Lower Concentration

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

radius

conc

entra

tion

concentration profile

0.4

0.8

2.0

Color variation from dark to light to represent the concentration of species in the pellet

Conc

entra

tio

n

Radius

Reactant A

(Bulk)

A

AA

A

AA

Assume : Spherical Porous Catalyst – Immersed in Bulk Solution (Reactant A).

During chemical reaction, reactant A is converted into product B in the presence of solid catalyst.

(Graph below the animation shows the concentration profile through out radius of the catalyst)

9

B

B B

B

R

Catalyst

Catalytic diffusion Model

Vessel

Where A – Reactant AB – Product BR – Radius of the Catalyst

This is the overall template of the model; the step wise animation is classified into three cases, that is, effect of radius (slide 10), reaction rate constant (slide 11) and molecular diffusivity (slide 12).

Vessel contains Species - A

Decreasing Radius of the Catalyst

10


Lower Concentration

2

eA

kaD

Effect of varying length scale of the spherical pellet, ‘a’

Radius ‘a’

Reaction Rate constant ‘k’

Effective molecular Diffusivity ‘DeA’

Catalyst

Catalyst

Catalyst

Catalyst

Step - 1 Step - 2 Step - 3 Step - 4


For different radii, find the thiele modulus and generate the dimensionless concentration profile using Eq. 18. Then present the profile using the dimensional radius, that is, r = a.

Dark Blue indicates low concentration of species A

Light Blue indicates high concentration of species A

Tuner

Schematic of the profile as a function of the change in the catalyst dimensions

1

Vessel contains Species - ABulk Solution CAS Bulk Solution CASBulk Solution CASBulk Solution CAS

Decreasing Rate of the Reaction


Lower Concentration

2

eA

kaD

Effect of varying Reaction Rate Constant ‘k’Tuner

Radius ‘a’

11

Catalyst

Catalyst

Catalyst

Catalyst



For different rate constants, k, find the thiele modulus for same r and generate the dimensionless concentration profile using Eq. 18. Then present the profile using the dimensional radius, that is, r = a.





Schematic of the profile as a function of the change in the reaction rate constant ‘k’

2

Vessel contains Species - ABulk Solution CAS Bulk Solution CASBulk Solution CASBulk Solution CAS

Increasing Molecular Diffusivity


Lower Concentration

2

eA

kaD

Effect of varying Molecular Diffusivity ‘DeA’Tuner

Radius ‘a’

12

Bulk Solution CAS

Catalyst

Catalyst

Catalyst

Catalyst







Schematic of the profile as a function of the change in the molecular diffusivity ‘DA’

For different Molecular Diffusivity, DA , find the thiele modulus for same r and generate the dimensionless concentration profile using Eq. 18. Then present the profile using the dimensional radius, that is, r = a.

3

Vessel contains Species - A

Bulk Solution CASBulk Solution CASBulk Solution CAS

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1. Audio support required : N

2. Colour changes to be shown: Y (Specify them in the slides)

3. Is there any process that needs to be shown for a certain time (Please specify):Y (specified in slides)

4. Theory will come in the left panel of the animation.

5. Keywords should come in 'Glossary' section.

6. 'Help' button should give stepwise instruction of how to operate the animation. (User Friendly Desktop)

7. Any other specifications: Graph should also work with respect to catalyst animation.

14

Objective Questions

1. The rate is affected by materials which are neither reactants nor products, such materials are called

as_________

a. Reactant b. Product c. By-product d. Catalyst

2. For a very slow first order reaction, the Thiele modulus is ____

a. Large b. Small c. Zero d. None

3. Thiele Modulus is directly proportional to _______

a. effective diffusivity b. 1/rate constant c. rate constant and effective diameter

d. 1/effective diameter

4. For a very fast first order reaction, the Thiele modulus is ____

a. Large b. Small c. Zero d. None

15

1. H. Scott Fogler, Elements of Chemical Reaction Engineering; Fourth Edition, Pearson

Edition, 2008, 813 – 833.

2. Octave Levenspiel, Chemical Reaction Engineering; third Edition, John Wiley and Sons:

New York, 2004, 376 – 390.

3. James J. Carberry, Arvind Varma, Chemical Reaction and Reactor Engineering, Marcel

Dekker, INC. New York, 1987, 239-260.

4. R. Byron Bird, Warren E. Stewart and Edwin N. Lightfoot, Transport Phenomina, Second

Edition, John Wiley and Sons: New York2005, 563-567.

References

Documents

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