1. Real Reactors Fixed Bed Reactor – 1
(1) The catalyst are held in place and do not move,
(2) Material and energy balance must be conducted for fluid in (a) the interstices of
particles (inter-particle space) and (b) within the particle (intra-particle space),
(3) Reaction occurs only within the catalyst particles,
(4) Reaction in bulk fluid is approximately zero.

2. Real Reactors Fixed Bed Reactor – 2 (5) Catalytic Reaction Steps
(a) transport of reactants and energy from bulk liquid to the catalyst pellet
surface,
(b) transport of reactants and energy from pellet surface to pellet interior,
(c) adsorption of reactants, chemical reaction and desorption of products at
catalytic sites,
(d) transport of products from the pellet interior to the surface,
(e) transport of products into the bulk fluid.
- usually one or at most two of the five steps are rate limiting and dictate,
- most often it is the intra-particle transport step

3. Fixed Bed Reactors Catalyst Bed
Single pellet model is established by averaging the microscopic processes that occur within the intra-particle environment,
An effective diffusion coefficient is used to
represent the information about the
physical diffusion process
and pore structure,
A viable commercial catalyst must have sufficient
active sites to maintain a product formation rate
in the order of 1 mol/L h,
Catalyst pellets usually takes the shape of spheres
( cm), cylinders ( cm O.D. and
L/O.D. = 3-4) and rings (ca. 2.5 cm)

4. Fixed Bed Reactors General Balances Catalyst Particle Material Balance
where

5. Fixed Bed Reactors General Balances Catalyst Particle Energy Balance
where

6. Fixed Bed Reactors Catalyst
Catalyst (usually metal sometimes also metal oxides) is often dispersed onto large surface area support material,
The support is often a refractor, metal oxide such as alumina. Silica, clay, zeolite, carbonaceous (e.g., activated carbon and graphite) are also popular support material.
The support often have surface areas between m2/g.

7. Fixed Bed Reactors Catalyst Pellets – 1
Catalyst pellets are made by tableting and extrusion methods. The latter is the more popular method,
Different pellet shape and size could be obtained by simply changing the extruder head,
The pellet shape and size could be optimized to increase mass transfer rates, while minimizing the pressure drop in the reactor.

8. Fixed Bed Reactors Catalyst Pellets – 2
The pellet void fraction or porosity, where rp is the effective pellet density and Vg is the pore volume,
The pore volume range fro, cm3/g pellet,
The pellet can possess either a uniform pore size or a bimodal pores of two different sizes, a large size to facilitate transport and a small size to contain the active catalyst sites.

9. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 1
Material balance
Steady-state
Spherical coordinate system

10. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 2
Boundary conditions
absence of driving force

11. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 3
Dimensionless equation - 1
characteristic length:
dimensionless length: dimensionless concentration:
concentration scale
length scale

12. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 4
Dimensionless equation – 2
where

13. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 5
Simplification
where

14. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 6
General solution
Specific solution

15. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 7
Concentration profile in pellet

16. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 8
Total productivity in pellet
letting

17. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 9
Effectiveness factor – 1
where
= 1 : the entire pellet volume is reacting at the same high rate because reactant is able to diffuse quickly through the pellet,
= 0 : the pellet reacts at a slow rate, since the reactant is unable to penetrate into the pellet interior.

18. Single Pellet Reaction
First-Order Reaction
(1) Spherical Pellet – 9
Effectiveness factor – 2

19. Single Pellet Reaction
Example – 1
The first order, irreversible reaction took place in a 0.3 cm radius spherical catalyst pellet at T = 450 K.
At 0.7 atm partial pressure of A, the pellet’s production rate is –2.5 x 10-5 mol/g-s, what is the production rate at the same temperature for a 0.15 cm radius catalyst pellet.
Given:

20. Single Pellet Reaction
Example – 2
List the equations for (a) overall productivity, (b) effectiveness factor and (c) Thiele modulus for a first order reaction in a spherical pellet.

21. Single Pellet Reaction
Example – 2
Solve for Thiele modulus
where
2.125 mol/cm3–s (0.3 cm)2 =
0.007 cm2/s (1.9 x 10-5 mol/cm3)
k (0.3 cm)2
= ( )0.5
0.007 cm2/s

22. Single Pellet Reaction
Example – 2
Solve for overall productivity of a smaller pellet
2.61/s (0.3 cm)2
= ( )0.5
0.007 cm2/s
The smaller pellet has about 60 % better overall productivity!
Note: this is only true when the system is within diffusion-limited regime!

23. Single Pellet Reaction
First-Order Reaction
Other Pellet Geometries – 1
Governing equation

24. Single Pellet Reaction
First-Order Reaction
Other Pellet Geometries – 2
Characteristic Lengths
Dimensionless equations

25. Single Pellet Reaction
First-Order Reaction
Other Pellet Geometries – 3
Effectiveness factor – 1 or

26. Single Pellet Reaction
First-Order Reaction
Other Pellet Geometries – 4
Effectiveness factor – 2

27. Single Pellet Reaction
Other Reaction Orders
Spherical Pellet – 5
Positive reaction orders
Redefining Thiele Modulus

28. Single Pellet Reaction
Other Reaction Orders
Spherical Pellet – 6
Redefining the equations

29. Single Pellet Reaction
Other Reaction Orders
Spherical Pellet – 7
Effectiveness factor as a function of Thiele modulus
n  1

30. Single Pellet Reaction
Other Reaction Orders
Spherical Pellet – 8
Effectiveness factor as a function of Thiele modulus
n < 1

31. Single Pellet Reaction
Other Reaction Orders
Spherical Pellet – 9
Concentration profile within pellet with reaction order less than 1
n = 0

32. Single Pellet Reaction
Other Reaction Orders
Spherical Pellet – 10
Effectiveness factor can be approximated by the analytical solution for first order reaction
n > 0
concentration profile
effectiveness factor
overall productivity

33. Single Pellet Reaction
Other Reaction Orders
Spherical Pellet – 10
Effectiveness factor can be approximated by the analytical solution for first order reaction
n > 0
concentration profile
effectiveness factor
overall productivity

34. Single Pellet Reaction
Hougen-Watson - 1
Find the effectiveness factor for a slab catalyst geometry
(1) Governing equation

35. Single Pellet Reaction
Hougen-Watson - 2
(2) Transformation into dimensionless equation
where (dimensionless adsorption constant)

36. Single Pellet Reaction
Hougen-Watson - 3
(3) Effectiveness factor
(4) Rescaling the Theile modulus

37. Single Pellet Reaction
Hougen-Watson - 4
(5) Effectiveness factor versus Thiele modulus

38. Single Pellet Reaction
External Mass Transfer - 1
Rapid EMT
Slow EMT
<

39. Single Pellet Reaction
External Mass Transfer - 2
(1) The presence of external mass transfer resistance will only affect the boundary condition
(2) Dimensionless boundary conditions x x

40. Single Pellet Reaction
External Mass Transfer - 3
(3) Biot number
(4) Dimensionless equation

41. Single Pellet Reaction
External Mass Transfer - 4
(5) Solving the equation
(6) Concentration profile in spherical pellet
small B means large external
mass transfer resistance
large B means no external mass
transfer resistance

42. Single Pellet Reaction
External Mass Transfer - 5
(7) New definition of effectiveness factor
(8) Effectiveness factor versus Thiele modulus for different Biot numbers
small B means large external
mass transfer resistance
large B means no external mass
transfer resistance

43. Single Pellet Reaction
External Mass Transfer - 6
(9) Effects of external mass transfer resistance
slope -1
slope -2

44. Single Pellet Reaction
External Mass Transfer - 7
(10) Summary

45. Single Pellet Reaction
External Mass Transfer - 8
(11) Observed versus intrinsic kinetic parameters - 1
Reaction-limited
Diffusion-limited

46. Single Pellet Reaction
External Mass Transfer - 9
(11) Observed versus intrinsic kinetic parameters - 2
Diffusion-limited
Internal mass transfer-limited
External mass transfer-limited

47. Catalyst Pellet
General Balances
(1) Material Balance
where

48. Catalyst Pellet
General Balances
(2) Energy Balance
where

49. Single Pellet Reaction
Nonisothermal Condition - 1
(1) Material Balance
(2) Energy Balance
Practical catalyst pellet usually have high thermal conductivity and therefore heat transfer could
often be neglected.

50. Single Pellet Reaction
Nonisothermal Condition - 2
(3) Solving the two balance equations
for constant properties
therefore

51. Single Pellet Reaction
Nonisothermal Condition - 3
(4) Simplification
defining the dimensionless variables
gives

52. Single Pellet Reaction
Nonisothermal Condition - 4
(5) Dimensionless material balance for nonisothermal pellet
Weisz-Hicks Problem
with boundary conditions

53. Single Pellet Reaction
Nonisothermal Condition - 5
(6) Effectiveness factor
Weisz-Hicks Problem
(7) Rescaling the Theile modulus

54. Single Pellet Reaction
Nonisothermal Condition - 6
(8) Effectiveness factor versus Thiele modulus
Weisz-Hicks Problem
Note: at large Thiele modulus that asymptotes
are the same for all values of g and b.
The effectiveness factor could be larger than 1
for some of the parameter values, which becomes
more pronounced for more exothermic reaction.
The interior temperature of the pellet could be
higher than the surface for exothermic reaction.
Multiple steady-state is possible in the pellet.

55. Single Pellet Reaction
Nonisothermal Condition - 7
(9) Concentration and temperature profiles in pellet
Weisz-Hicks Problem

56. Fixed Bed Reactor FBR Design – 1
Analysis of a fixed bed reactor with a packed bed of catalyst pellets involves:
(1) fluid phase that transports the reactants and products through the reactor,
(2) solid phase where reaction-diffusion processes occurs.

57. Fixed Bed Reactor FBR Design – 2
(1) Coupling between catalyst and fluid
The two phases communicate by exchanging materials and energy
(2) The following assumptions will be made for the analysis of a FBR

58. Fixed Bed Reactor FBR Design – 3 (3) Fluid Phase (a) mole balance
(b) energy balance
(c) pressure drop (Ergun Equation)

59. Fixed Bed Reactor FBR Design – 4 (4) Catalyst pellet (a) mole balance
(b) energy balance

60. Fixed Bed Reactor FBR Design – 5
(5) Coupling between fluid and catalyst phases
(a) mole balance
(b) energy balance

61. Fixed Bed Reactor
FBR Design – 6
(6) Quick summary

62. Fixed Bed Reactor FBR Design – 7 (7) Simple examples
The first order, irreversible reaction took place in a 0.3 cm radius spherical catalyst pellet at T = 450 K.
The feed to the reactor is pure A (12 mol/s, 1.5 atm), the pellet’s production rate is –2.5 x 10-5 mol/g-s. The bed density is given to be 0.6 g/cm3. Assume that the reactor operates isothermally at 450 K. External mass-transfer limitations are negligible.
Given:
Find the FBR volume needed for 97 % conversion of A.

63. Fixed Bed Reactor FBR Design – 8 (7a) FBR design equation
(7b) First order, irreversible reaction
Thiele modulus is independent of concentration
(7c) Effectiveness factor is constant along the axial length

64. Fixed Bed Reactor FBR Design – 9
(7d) Concentration in term of molar flow
(7e) Substituting into the FBR design equation

65. Fixed Bed Reactor FBR Design – 9
(7f) What happen when there is external diffusion resistance
let