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Diffusion in Spherical Catalyst Chemical Reaction Engineering K. Narendiran PhD Scholar Ganesh A Viswanathan Assistant Professor 1

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 2

Porous catalyst pellet Diffusion in a Catalyst Pellet Consider 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 B Porous catalyst pellet R External Diffusion Internal Diffusion Vessel with Species A 3

4 Assumptions Mole Balance 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 pellet External Surface Concentration 4

5 Dimensionless Variables 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) 5

Substitution of the dimensionless variables (Eq 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) 6

7 ---------- (17) The general solution of the model (Eq. 14) is ---------- (17) where a and b 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 f - Thiele Modulus x - Dimensionless Radius , range of which is [0,3] C - Dimensionless Concentration 7

Effect of Thiele modulus on the concentration profile in the catalyst pellet For large values of f, 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 f, 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. Higher Concentration Lower Concentration Color variation from dark to light to represent the concentration of species in the pellet Color variation from dark to light to represent the concentration of species in the pellet where 8 Dark Blue color of the catalyst represent that reaction occur at surface (large f value). Light Blue color of the catalyst representation for lower f 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)

9 Catalytic diffusion Model Catalyst A A B B R A A B B Reactant A Vessel contains Species - A Reactant A (Bulk) Vessel A A 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). Where A – Reactant A B – Product B R – Radius of the Catalyst Concentration 9 Radius 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)

10 1 Catalytic diffusion Model Tuner Effect of varying length scale of the spherical pellet, ‘a’ Radius ‘a’ Reaction Rate constant ‘k’ Effective molecular Diffusivity ‘DeA’ Tuner Schematic of the profile as a function of the change in the catalyst dimensions Vessel contains Species - A Higher Concentration Lower Concentration Step - 1 Step - 2 Step - 3 Step - 4 Catalyst Bulk Solution CAS Catalyst Bulk Solution CAS Catalyst Bulk Solution CAS Catalyst Bulk Solution CAS Decreasing Radius of the Catalyst Dark Blue indicates low concentration of species A Light Blue indicates high concentration of species A 10 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 = xa.

Decreasing Rate of the Reaction Catalytic diffusion Model 2 Effect of varying Reaction Rate Constant ‘k’ Tuner Radius ‘a’ Reaction Rate constant ‘k’ Effective molecular Diffusivity ‘DeA’ Schematic of the profile as a function of the change in the reaction rate constant ‘k’ Step - 1 Step - 2 Step - 3 Step - 4 Vessel contains Species - A Higher Concentration Lower Concentration Catalyst Bulk Solution CAS Catalyst Bulk Solution CAS Catalyst Bulk Solution CAS Catalyst Bulk Solution CAS Decreasing Rate of the Reaction Dark Blue indicates low concentration of species A Light Blue indicates high concentration of species A 11 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 = xa.

Increasing Molecular Diffusivity Catalytic diffusion Model 3 Effect of varying Molecular Diffusivity ‘DeA’ Tuner Radius ‘a’ Reaction Rate constant ‘k’ Effective molecular Diffusivity ‘DeA’ Schematic of the profile as a function of the change in the molecular diffusivity ‘DA’ Vessel contains Species - A Step - 1 Step - 2 Step - 3 Step - 4 Higher Concentration Lower Concentration Bulk Solution CAS Bulk Solution CAS Bulk Solution CAS Bulk Solution CAS Catalyst Catalyst Catalyst Catalyst Increasing Molecular Diffusivity Dark Blue indicates low concentration of species A Light Blue indicates high concentration of species A 12 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 = xa.

13 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. 13 15 15

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 14

References 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. 15