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**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.

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**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

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**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)

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**Fixed Bed Reactors General Balances Catalyst Particle Material Balance**

where

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**Fixed Bed Reactors General Balances Catalyst Particle Energy Balance**

where

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**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.

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**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.

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**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.

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**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.

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**Single Pellet Reaction**

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

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**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:

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**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.

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**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

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**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!

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**Single Pellet Reaction**

First-Order Reaction Other Pellet Geometries – 1 Governing equation

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

Other Reaction Orders Spherical Pellet – 6 Redefining the equations

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

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

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**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

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**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

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**Single Pellet Reaction**

Hougen-Watson - 1 Find the effectiveness factor for a slab catalyst geometry (1) Governing equation

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**Single Pellet Reaction**

Hougen-Watson - 2 (2) Transformation into dimensionless equation where (dimensionless adsorption constant)

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**Single Pellet Reaction**

Hougen-Watson - 3 (3) Effectiveness factor (4) Rescaling the Theile modulus

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**Single Pellet Reaction**

Hougen-Watson - 4 (5) Effectiveness factor versus Thiele modulus

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**Single Pellet Reaction**

External Mass Transfer - 1 Rapid EMT Slow EMT <

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**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

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**Single Pellet Reaction**

External Mass Transfer - 3 (3) Biot number (4) Dimensionless equation

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**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

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**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

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**Single Pellet Reaction**

External Mass Transfer - 6 (9) Effects of external mass transfer resistance slope -1 slope -2

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**Single Pellet Reaction**

External Mass Transfer - 7 (10) Summary

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**Single Pellet Reaction**

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

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**Single Pellet Reaction**

External Mass Transfer - 9 (11) Observed versus intrinsic kinetic parameters - 2 Diffusion-limited Internal mass transfer-limited External mass transfer-limited

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Catalyst Pellet General Balances (1) Material Balance where

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Catalyst Pellet General Balances (2) Energy Balance where

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**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.

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**Single Pellet Reaction**

Nonisothermal Condition - 2 (3) Solving the two balance equations for constant properties therefore

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**Single Pellet Reaction**

Nonisothermal Condition - 3 (4) Simplification defining the dimensionless variables gives

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**Single Pellet Reaction**

Nonisothermal Condition - 4 (5) Dimensionless material balance for nonisothermal pellet Weisz-Hicks Problem with boundary conditions

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**Single Pellet Reaction**

Nonisothermal Condition - 5 (6) Effectiveness factor Weisz-Hicks Problem (7) Rescaling the Theile modulus

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**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.

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**Single Pellet Reaction**

Nonisothermal Condition - 7 (9) Concentration and temperature profiles in pellet Weisz-Hicks Problem

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**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.

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**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

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**Fixed Bed Reactor FBR Design – 3 (3) Fluid Phase (a) mole balance**

(b) energy balance (c) pressure drop (Ergun Equation)

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**Fixed Bed Reactor FBR Design – 4 (4) Catalyst pellet (a) mole balance**

(b) energy balance

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**Fixed Bed Reactor FBR Design – 5**

(5) Coupling between fluid and catalyst phases (a) mole balance (b) energy balance

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

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**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.

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**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

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**Fixed Bed Reactor FBR Design – 9**

(7d) Concentration in term of molar flow (7e) Substituting into the FBR design equation

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**Fixed Bed Reactor FBR Design – 9**

(7f) What happen when there is external diffusion resistance let

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