Derive an expression for effectiveness factor for rectangular catalyst particle.
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The effectiveness factor for a spherical catalyst particle
Fixed bed reactors are very common in chemical industries. These reactors are packed with catalyst particles of irregular shape and the reactants enter the reactor from one end while the products are withdrawn from other end. Here, we consider a spherical porous catalytic particle of radius R located at any point in this reactor. This particle is submerged in a gas stream containing the reactant A and the product B. A first order reaction A→B takes place inside the porous catalyst. There are two major mass transfer steps occurring around this particle as shown in Fig. 39.1. In the first step, the mass transfer of A occurs from bulk phase to the surface of the catalyst particle and in the second step, mass transfer occurs inside the catalyst particle where the chemical reaction takes place. In reality, this problem may be difficult to solve as both steps are coupled with each other. Here, we show how to handle both steps separately. Diffusion and chemical reaction inside a porous spherical catalyst particle Here, we solve the second part of the problem, where both components (A and B) are diffusing inside the pours of the particle and the first order chemical reaction (RA= - k1cA) occurs at inside surface of the pours particle. We assume that the concentration of the reactant A is known at the catalyst surface. This assumption actually decouples the problem from the first step, where the components are transferred from bulk phase to the catalyst surface.

Fig 39.1 Diffusion and chemical reaction inside a porous catalyst
Assumption
Pores are uniformly distributed and both components diffuse only in the radial direction. Thus fluxes in θ and Φ directions(Nθ and NΦ)are zero.
System is in steady state.
Total concentration of both components, c and diffusivity of A in B, DAB, are constants.
The first order chemical reaction takes place inside the catalyst with reaction rate, RA, given by

where k1 is the reaction rate constant which depends on surface area, as shown below  or

Here, "a" is the inside surface area per unit volume of the catalyst. It may be noted that as reactant A diffuses inside the catalyst, the concentration of reactant A decreases in the radial direction. Thus,

The equation of continuity for component A may be written as follows

By using the assumptions (1) to (3), we obtain the following simplified form of the equation of continuity in the spherical coordinate system.

where, NAr may be obtained as

At steady state, we may further assume the equimolar counter diffusion. Thus,  and with this the Equation (39.6) may be simplified as,

Substituting  from Fick’s law, we obtain

Equation (39.5) may now be written as

or

or

where  Expending the Equation (39.11), we obtain

The Equation (39.12) is a second order differential equation. To solve it, we use the following transformation

Thus,

and

Substituting these in the Equation (39.12), we finally obtain

or

which has the following solution

The first boundary condition is at  is finite. This leads to the solution that

or

or

where,  The second boundary condition is given here that at  Thus,

From Equations (39.21) to (39.23), we finally obtain the concentration profile of "A" inside the catalyst particle as

The rate of conversion of A into B may now be calculated as,
wAs =Mass flux x Surface area of catalyst or

or

When Equation (39.26) is used in above expression, we finally obtain
To understand the effect of internal mass transfer resistance, we consider the ideal conditions where this resistance is absent and all available area inside the catalyst particle is exposed to the surface concentration cAS. Thus, the molar rate of conversion may now be calculated as

We may define the effectiveness factor η for the catalyst particle as given below

where,  For irregular shaped catalyst particles, the above results may be applied by reinterpreting the value of average radius R'. For a non-spherical particle, the radius R may be redefined as

where, VP and SP are the average volume and average external surface area of the catalyst particles respectively. Thus, the effectiveness factor may now be calculated as:

where, 
It is clear from Equation (39.31) that if effectiveness factor η is plotted as the function of , as shown in Fig. 39.2.

Fig. 39.2 Effectiveness factor η vs  plot.
However, if  is too small which is possible for smaller size catalyst particles, the pressure drop across the reactor usually increases exponentially, which may not be desirable. Thus, the optimum size catalyst particles should be used which have the sufficiently high effectiveness factors but also have smaller pressure drops across the reactor.
Fixed bed reactors are very common in chemical industries. These reactors are packed with catalyst particles of irregular shape and the reactants enter the reactor from one end while the products are withdrawn from other end. Here, we consider a spherical porous catalytic particle of radius R located at any point in this reactor. This particle is submerged in a gas stream containing the reactant A and the product B. A first order reaction A→B takes place inside the porous catalyst. There are two major mass transfer steps occurring around this particle as shown in Fig. 39.1. In the first step, the mass transfer of A occurs from bulk phase to the surface of the catalyst particle and in the second step, mass transfer occurs inside the catalyst particle where the chemical reaction takes place. In reality, this problem may be difficult to solve as both steps are coupled with each other. Here, we show how to handle both steps separately. Diffusion and chemical reaction inside a porous spherical catalyst particle Here, we solve the second part of the problem, where both components (A and B) are diffusing inside the pours of the particle and the first order chemical reaction (RA= - k1cA) occurs at inside surface of the pours particle. We assume that the concentration of the reactant A is known at the catalyst surface. This assumption actually decouples the problem from the first step, where the components are transferred from bulk phase to the catalyst surface.

Fig 39.1 Diffusion and chemical reaction inside a porous catalyst
Assumption
Pores are uniformly distributed and both components diffuse only in the radial direction. Thus fluxes in θ and Φ directions(Nθ and NΦ)are zero.
System is in steady state.
Total concentration of both components, c and diffusivity of A in B, DAB, are constants.
The first order chemical reaction takes place inside the catalyst with reaction rate, RA, given by

where k1 is the reaction rate constant which depends on surface area, as shown below  or

Here, "a" is the inside surface area per unit volume of the catalyst. It may be noted that as reactant A diffuses inside the catalyst, the concentration of reactant A decreases in the radial direction. Thus,

The equation of continuity for component A may be written as follows

By using the assumptions (1) to (3), we obtain the following simplified form of the equation of continuity in the spherical coordinate system.

where, NAr may be obtained as

At steady state, we may further assume the equimolar counter diffusion. Thus,  and with this the Equation (39.6) may be simplified as,

Substituting  from Fick’s law, we obtain

Equation (39.5) may now be written as

or

or

where  Expending the Equation (39.11), we obtain

The Equation (39.12) is a second order differential equation. To solve it, we use the following transformation

Thus,

and

Substituting these in the Equation (39.12), we finally obtain

or

which has the following solution

The first boundary condition is at  is finite. This leads to the solution that

or

or

where,  The second boundary condition is given here that at  Thus,

From Equations (39.21) to (39.23), we finally obtain the concentration profile of "A" inside the catalyst particle as

The rate of conversion of A into B may now be calculated as,
wAs =Mass flux x Surface area of catalyst or

or

When Equation (39.26) is used in above expression, we finally obtain
To understand the effect of internal mass transfer resistance, we consider the ideal conditions where this resistance is absent and all available area inside the catalyst particle is exposed to the surface concentration cAS. Thus, the molar rate of conversion may now be calculated as

We may define the effectiveness factor η for the catalyst particle as given below

where,  For irregular shaped catalyst particles, the above results may be applied by reinterpreting the value of average radius R'. For a non-spherical particle, the radius R may be redefined as

where, VP and SP are the average volume and average external surface area of the catalyst particles respectively. Thus, the effectiveness factor may now be calculated as:

where, 
It is clear from Equation (39.31) that if effectiveness factor η is plotted as the function of , as shown in Fig. 39.2.

Fig. 39.2 Effectiveness factor η vs  plot.
However, if  is too small which is possible for smaller size catalyst particles, the pressure drop across the reactor usually increases exponentially, which may not be desirable. Thus, the optimum size catalyst particles should be used which have the sufficiently high effectiveness factors but also have smaller pressure drops across the reactor.
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