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Dr.-Eng. Zayed Al-Hamamre
Advance Chemical Reaction Engineering
Heterogeneous Catalysis: Kinetic in Porous
Catalyst Particles
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Content
Introduction
Diffusion Mass Transfer
External Resistance to Mass Transfer
Mass Transfer-Limited Reactions in Packed Beds
Diffusion through a Spherical Catalyst Pellets
Thiele Modulus
Effectiveness Factor
Combining External Mass Transfer with Diffusion
Giving up is the ultimate tragedy
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Heterogeneous reactions are distinguished from homogeneous ones by the different phases
present during reaction.
Introduction
For the design of heterogeneous chemical reactors a special consideration should be taken for
The transfer of matter between phases,
Transport processes play a critical role, capable to have strong influence on the degree of
conversion and the selectivity. The heat and mass transfer coefficient as well as the
exchange area are the parameters that describe the transport rate,
The fluid dynamics and chemistry of the system.
Additional complexity enters into the problem
Complication of the rate expression, and
Complication of the contacting patterns for two-phase systems
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Introduction
The Complications of the Rate Equation.
Since more than one phase is present, the movement of material from phase to phase must be
considered in the rate equation.
The rate expression in general will incorporate mass transfer terms in addition to the usual
chemical kinetics term.
These mass transfer terms are different in type and numbers in the different kinds of
heterogeneous systems; hence, no single rate expression has general application
Thus, in addition to an equation describing the rate at which the chemical reaction proceeds,
one must also provide a relationship or algorithm to account for the various physical processes
which occur.
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Burning of carbon particle in air
Examples
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Aerobic fermentation : Air bubble pass thorough liquid tank to the microbial cell to form
product material.
There are up to seven possible resistance steps, only one involving the reaction
If the steps are in series
If the steps are in parallel
Examples
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Diffusion Mass Transfer Diffusion is the spontaneous intermingling or mixing of atoms or molecules by random
thermal motion.
It gives rise to motion of the species relative to motion of the mixture.
In the absence of other gradients (such as temperature, electric potential, or gravitational
potential), molecules of a given species within a single phase will always diffuse from regions
of higher concentrations to regions of lower concentrations.
The mass transfer flux law is given according to Ficks law by
The molar flux of A. WA is the result of two contributions:
i. JA the molecular diffusion flux relative to the bulk motion of the fluid produced by a
concentration gradient, and
ii. BA the flux resulting from the bulk motion of the fluid,
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Or
where the molar average velocity is
hence
and
Diffusion Mass Transfer
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Diffusion Mass Transfer
Since the flux of A must be constant through
the stagnant film (conservation of mass), the
derivative of the flux with respect to
distance in the film must vanish
With boundary conditions:
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Equimolar Counter Diffusion
Dilute Concentrations
Diffusion Mass Transfer
When the mole fraction of the diffusing solute and the bulk motion In the direction of the
diffusion are small<<
For constant total concentration
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Diffusion Through a Stagnant Gas
Diffusion Mass Transfer
Forced Convection,
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Diffusion Mass Transfer
In a tubular flow reactor
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External Resistance to Mass Transfer
where
Mass Transfer to a Single Particle
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If the external mass-transfer rate is low, the concentrations in the bulk fluid and external
catalyst surface are Significantly different.
At steady state, the molar flux to the boundary equal to convective transport across the
boundary layer
External Resistance to Mass Transfer
Internal diffusion External mass transfer
Multiply by
in which is the Biot number or dimensionless mass-
transfer coefficient
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Example
For dilute concentrations of the solute the radial flux is
Because reaction is assumed to occur instantaneously on the external surface of the pellet,
also
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For the isomerization reaction
Example
If the temperature is sufficiently high, then we have very weak adsorption (i.e., low surface
coverage) of A and B: thus
At steady state
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Example Cont.
Or where
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Rapid Reaction:
The rate of mass transfer to the surface limits the overall rate of reaction.
Example Cont.
kc can be found using several correlations such as
And the surface centration of reactant approaches zero,
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Slow Reaction
Example Cont.
The external resistance decreases as
The velocity across the pellet is
Increased,
The particle size is decreased.
The boundary layer becomes smaller and
the mass transfer coefficient (mass transfer
rate) increases,
And the surface concentration approaches the bulk fluid concentration.
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Diffusion from the Bulk to the External Transport
: The diffusion coefficient
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Reactant Concentration Profiles
kr
kr
kr
Reactant concentration profiles around a catalyst
pellet for reaction control and for external mass
transfer control
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An interphase effectiveness factor, is defined as the reaction rate based on surface conditions
divided by the rate that would be observed in the absence of diffusional limitations:
Interphase effectiveness factor
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ExampleDilute A diffuses through a stagnant liquid film onto a plane surface consisting of B, reacts there
to produce R which diffuses back into the mainstream. Develop the overall rate expression for the
L/S reaction
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The flux of A to the surface is
Example Cont.
Reaction is first order with respect to A
At steady state the flow rate to the surface is equal to the reaction rate at the surface (steps in
series).
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The individual rate steps on the same basis (unit surface of burning particle, unit volume of
fermenter, unit volume of cells, etc.).
Definitions
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Definitions
And
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Mass Transfer-Limited Reactions in Packed Beds
For the mass transfer-limited reaction
Curried out in PBR
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Mass Transfer-Limited Reactions in Packed Beds
The molar flow rate of A in the axial direction is
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If the flow rate through the bed is very large, the axial diffusion can be neglected,
Or
Mass Transfer-Limited Reactions in Packed Beds
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Mass Transfer-Limited Reactions in Packed Beds
At steady state
Hence
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Mass Transfer-Limited Reactions in Packed Beds
In most mass transfer-limited reactions, the surface concentration is negligible with respect to
the bulk concentration
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Mass Transfer-Limited Reactions in Packed Beds
To determine L the reactor length necessary to achieve a conversion X
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Diffusion through a Spherical Catalyst Pellets The pores in the pellet are not straight and
cylindrical;
They are a series of tortuous, interconnecting
paths of pore bodies and pore throats with
varying cross-sectional areas.
Effective diffusion coefficient is used to describe the average diffusion taking place at any
position r in the pellet
where
accounts for the variation in the cross-sectional area that is
normal to diffusion
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Diffusion through a Spherical Catalyst Pellets
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Diffusion Effect of pore size on the diffusivity of gas molecules
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Diffusion through a Spherical Catalyst Pellets For the reaction
The material balance is
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Diffusion Through a Spherical Catalyst Pellets
But
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Diffusion Through a Spherical Catalyst Pellets
The boundary conditions are
whereThiele modulus
φ21 is a measure of the ratio of "a" surface reaction rate to "a" rate of diffusion through the
catalyst pellet.
When the Thiele modulus is large, internal diffusion usually limits the overall rate of reaction;
when φn is small, the surface reaction is usually rate-limiting.
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Diffusion Through a Spherical Catalyst Pellets
Reactant concentration
gradients in a sphere for a
first-order reaction.
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Diffusion Through a Spherical Catalyst Pellets
Reaction rate limitations.
Moderate effect of diffusion on the average rate
the concentration at the center is almost zero, and the reaction rate will be very
low in the central part of the catalyst (regime of diffusion controlled reaction)
Pore diffusion limitation
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Reactant concentration profiles around and
within a porous catalyst pellet for the cases
of reaction control, external mass transfer
control, and pore diffusion control. Each of
these situations leads to different reaction
rate expressions.
Diffusion Through a Spherical Catalyst Pellets
Quiz: If the initial concentration of species A is doubled, how will the Thiele modulus
change?
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For large pellets, it takes a long time for the reactant A
to diffuse into interior compared to the time it takes for
the reaction to occur on the interior surface
The reactant is only consumed n the exterior surface of
the pellet and the catalyst near the center of the pellet
wasted catalyst
For very small pellets it takes very little time lo
diffuse into and out of the pellet interior and, as a
result, internal (fusion no Longer limits the rate of
reaction.
The rare of reaction
Diffusion Through a Spherical Catalyst Pellets
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Remarks For a first order reaction take place within a spherical catalyst pellet
In the Thiele modulus,
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The Rate of Reaction At steady state, the amount of reactant entering the particle must equal that consumed by the
reaction
The overall rate is the diffusion flux into the pellet:
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Effectiveness Factor To measure how much the reaction rate is lowered because of the resistance to pore diffusion,
define the quantity the effectiveness factor as
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Effectiveness Factor The internal effectiveness factor for a first-order reaction in a spherical catalyst pellet
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Effectiveness Factor
apparent
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Effectiveness Factor
Generally
The characteristic length aFor 1st order reaction
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We observe that as the particle diameter becomes very small, φn decreases, so that the
effectiveness factor approaches 1 and the reaction is surface-reaction- limited.
For small value of φn
o The concentration of reactant does not drop appreciably within the pore;
o Pore diffusion offers negligible resistance.
o It means either a short pore, slow reaction, or rapid diffusion, all three factors tending to
lower the resistance to diffusion
For large value of φn
o when φn is large (about 30), the internal effectiveness factor η is small ( η << 1) and the
reaction is diffusion- limited within the pellet
o The reactant concentration drops rapidly to zero on moving into the pore,
o Diffusion strongly influences the rate of reaction (regime of strong pore resistance).
Thiele Modulus
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Effectiveness Factor For large values of the Thiele modulus,
the effectiveness factor for a first order reaction can be written as
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Other Catalyst Shapes
In which B.Cs
The characteristic length a
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Other Catalyst Shapes
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Effectiveness Factor
η
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Effectiveness Factor
Spherical pellet
Spherical pellet
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The overall rate of reaction in terms of φn and η
Effectiveness Factor
i. Decrease the radius of reaction be R (make pellets smaller);
ii. Increase the temperature;
iii. Increase the concentration; and
iv. Increase the internal surface area
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Effectiveness Factor
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For a reaction to take place within a spherical catalyst pellet, with
then
Nth Order Reaction
shell balance equation on A
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For large values of the Thiele modulus, the effectiveness factor is
Nth Order Reaction
An approximate value for the Thiele modulus can be given as
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Nth-Order Reaction with Other Catalyst Shape
The reaction-diffusion equation
For the reaction
In which The characteristic
length a
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Nth-Order Reaction with Other Catalyst Shape
The equation for the effectiveness factor in
a slab is the simplest and will be used for
all pellet shapes with the appropriate
Thiele modulus.
η
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Combining External Mass Transfer with Diffusion For finite external mass transfer, the dimensionless model and boundary conditions for a
spherical catalyst pellet are
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With decreasing B, corresponding to slower external mass transfer the concentration profile in the pellet
becomes more uniform and the dimensionless surface concentration decreases. The lower concentration
leads to lower reaction rates.
1st order reaction in a spherical pellet with φ=1
Biot number
Combining External Mass Transfer with Diffusion
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First-order reaction in a spherical pellet.
As B goes to infinity, then the reaction proceeds without external mass-transfer limitations
Combining External Mass Transfer with Diffusion
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The controlling mechanisms for pellet reaction rate given finite rates of internal
diffusion and external mass transfer.
Combining External Mass Transfer with Diffusion
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Example
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Example Cont.
Assuming the porosity and tortuosity to be 0.5 and 4, respectively
0.745 cm
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Example
Use the production rate and pellet parameters for the 0.3 cm pellet to find the unknown values
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Example Cont.
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The smaller pellet is half the radius of the larger pellet,
Example Cont.
Decreasing the pellet size increases the production rate by almost 60%.
This is possible only when the pellet is in the diffusion-limited regime.
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Reactor Containing Porous Catalyst
Expanded views of a fixed-bed reactor
The pellet volume consists of both
void and solid. The pellet void
fraction (or porosity) is
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Reactor Containing Porous Catalyst
Plug Flow
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Mixed Flow
Reactor Containing Porous Catalyst
For a Reactor Containing a Batch of Catalyst
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Use the data presented in the previous example to
Example
the Thiele modulus
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Example Cont.
For an ideal gas mixture
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Example Cont.
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At steady state, the transport of the reactant.(s) from the bulk fluid to the external surface of
the catalyst is equal to the net rate of reaction of the reactant within and an the pellet.
For the case when the external and internal resistance to mass transfer to and within the pellet
are of the same order of magnitude
Overall Effectiveness Factor
And
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Overall Effectiveness Factor
Also
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Example
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Example Cont.
But
If the flow rate through the bed is very large, the
axial diffusion (dispersion) can be neglected, i.e., if
Then
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Example Cont.
For
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Example Cont.