Estimation of Internal Effectiveness Factor (η):

CHC3170: Chemical Reaction Engineering II, Unit III: Diffusion and Reactions within a

Catalyst

Estimation of Internal Effectiveness Factor (η):

From figure 3.5 it is conclusive that the concentration of reactant varies with in the

catalyst pellet. Therefore, to counter this effect of concentration variation, we

introduce an Internal Effectiveness Factor which will take care of the variation.

1. Isothermal First Order Catalytic Reactions:

The internal effectiveness factor is defined as the ratio of –

It can also be understood as “the measure of how far the reactant diffuses into

the pellet before reacting”.

In terms of symbols,

To derive the effectiveness factor for a first-order reaction, we will first convert

the above equation in terms moles per unit time (MA) so as to make the calculation

easy and convenient. Therefore,

Considering the denominator, MAs, first. If the entire surface were exposed to the

concentration at the external surface of the pellet, CAs, the rate for a first-order

reaction would be –

where, subscript s denotes that the rate -rAs is evaluated at the conditions (e.g.,

concentration, temperature) present at the external surface of the pellet (i.e., λ =

1).

------ (3.28)

------ (3.29)


Catalyst

Now, taking the actual rate of reaction into consideration. This is the rate at which

the reactant diffuses into the pellet at the outer surface; that is, all of A that

diffuses into the pellet at the outer surface reacts and no A diffuses back out. (It

behaves as a “black hole.”)

From equation 3.17, the actual rate of reaction in dimensionless form can be

given as (in terms of MA) –

Also, from equation 3.27,

Now, differentiating equation 3.27 written above w.r.to λ and evaluating the

result at λ = 1, we get –

Substituting equation 3.30 in above written equation 3.17 we get

Now, substituting equations 3.29 and 3.31 in equation 3.28, we get the expression

for internal effectiveness factor as –

------ (3.30)

------ (3.31)

------ (3.32) Internal Effectiveness Factor

for a first-order reaction in a

spherical catalyst pellet.


Catalyst

A plot of the effectiveness factor as a function of the Thiele modulus is shown

in Figure 3.5.

Figure 3.5(a) shows η as a function of the Thiele modulus, ϕs, for a spherical

catalyst pellet for reactions of zero, first, and second order.

Figure 3.5(a): Effectiveness factor plot for nth-order kinetics on spherical catalyst particles.

Figure 3.5(b) corresponds to a first-order reaction occurring in three differently

shaped pellets of volume Vp and external surface area Ap, and where the Thiele

modulus for a first-order reaction, ϕ1, is defined differently for each shape.

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. On the other hand, when the Thiele modulus ϕn is large (e.g., ~30), the

internal effectiveness factor is small (e.g., ~0.1), and the reaction is diffusion-

limited within the pellet.


Catalyst

Figure 3.5(b): First-order reaction in different pellet geometrics.

For large values of the Thiele modulus, the effectiveness factor can be written as:

Now, expressing overall rate of reaction in terms of Thiele Modulus, first re-

arrange equation 3.28 as –

Also, for a first-order reaction,

------ (3.33)

------ (3.34)


Catalyst

Combining the equations 3.33 and 3.34, the overall rate of reaction for a first-

order, internal-diffusion-limited reaction is –

NOTE: From the above expression, it is conclusive that for catalytic packed beds,

the rate varies inversely with the particle diameter of the catalyst.

2. Effectiveness Factors with Volume change with Reaction:

When there is volume change (ε ≠ 0), a correction factor is used to rectify this

change in volume. The correction is obtained from a plot of the ratio of effective

factors –

as a function of ε for various values of the Thiele Modulus. The plot is given by

figure 3.6.

For example, if the Thiele modulus were 10 for the gas-phase reaction A → 2B

with (ε = 1) then the effectiveness factor with volume change would be ηʹ = 0.8η.

Figure 3.6: Effectiveness factor ratios for the first-order kinetics on spherical catalyst pellets

for various values of the Thiele Modulus of a Sphere, ϕs, as a function of volume change


Catalyst

For larger values of Thiele Modulus, the effectiveness factor η is given as:

Again, from the above expression, it is conclusive that for reaction order greater

than 1, effectiveness factor decreases with increase in concentration at the

external pellet surface.

3. Effectiveness Factor for Non-Isothermal Process:

The preceding discussion of effectiveness factors is valid only far isothermal

conditions. When a reaction is exothermic and non-isothermal, the effectiveness

factor can be significantly greater than 1 as shown in Figure 3.7.

Figure 3.7: Non-isothermal Effectiveness Factor

------ (3.35)


Catalyst

Values of η greater than 1 occur because the external surface temperature of the

pellet is less than the temperature inside the pellet where the exothermic reaction

is taking place. Therefore, the rate of reaction inside the pellet is greater than the

rate at the surface. Thus, because the effectiveness factor is the ratio of the actual

reaction rate to the rate at surface conditions, the effectiveness factor can be

greater than 1, depending on the magnitude of the parameters β and γ. The

parameter γ is sometimes referred to as the Arrhenius number, and the parameter

β represents the maximum temperature difference that could exist in the pellet

relative to the surface temperature TS.

From figure 3.7, it can be observe that multiple steady states (MSSs) can exist for

values of the Thiele Modulus less than 1 and when β is greater than approximately

0.2. There will be no MSSs when the following criterion is fulfilled –

4(1+ β) > βγ

4. Weisz-Prater Criterion for Internal Diffusion Limitations:

It uses measured values of the rate of reaction, -rʹA (obs), to determine if

internal diffusion is limiting the reaction.

For first order reaction, this criterion is given by:

where, CWP is the Weisz-Prater Parameter.

All the terms in equation 3.39 are known terms, hence CWP can be calculated

easily.

If, CWP « 1, then there are no diffusion limitations. Also there is no existence

of any concentration gradient within the pellet.

If, CWP » 1, then internal diffusion limits the reaction significantly.

------ (3.39)

------ (3.36)


Catalyst

Example: Estimating the Thiele Modulus and Effectiveness Factor

The first-order reaction

A B

was carried out over two different-sized pellets. The- pellets were contained in

spinning basket reactor that was operated at sufficiently high rotation speeds, the

external mass transfer resistance was negligible. The results of two experimental

runs made under identical conditions are as given in Table 3.1.

(a) Estimate the Thiele modulus and effectiveness factor for each pellet.

(b) How small should the pellets be made to virtually eliminate all internal diffusion

resistance?

Table 3.1: Data from a Spinning Basket Reaction

Solution:

(a) From equation 3.32, we can write –

Now, combining the above expression (i) with equation 3.39 we get:

Let subscripts 1 and 2 represents the runs 1 and 2.

Applying the above derived equation on runs 1 and 2 and taking their ratio will

results in –

The terms ρc , De, and CAs cancel out because the runs were carried out under

identical conditions. The Thiele modulus is –

------ (i)

------ (ii)

------ (iii)

------ (iv)


Catalyst

Taking the ratio of Thiele Modulus for runs 1 and 2

or,

substituting value of ϕ11 from eqn (vi) in eqn (iii) above and evaluating -rʹA and

R for runs 1 and 2 gives

Solving eqn (ix) for ϕ12 will give –

Therefore, corresponding effectiveness factors from eqn 3.32 or eqn (i) are –

(b) Next we will calculate the particle radius needed to virtually eliminate internal

diffusion control (say, η = 0.95)

From equation 3.32,

solving eqn (x) for ϕ13 will give, ϕ13 = 0.9,

------ (v)

------ (vi)

------ (vii)

------ (ix)

------ (x)


Catalyst

Therefore, from eqn (v),

This implies that, a particle size of 0.55m is required to virtually eliminate

diffusion control.

NOTE: This example shows, how, with only two measurements and some

assumptions, one can determine internal diffusion limitations for the two pellet sizes,

and predict the pellet size necessary to completely eliminate internal diffusion.

Falsified Kinetics:

Sometimes the measured values of reaction order and activation energy are not the

actual or real ones. For example, consider the case in which we obtain reaction-rate

data in a differential reactor at two different temperatures, T1 and T2, where

precautions are taken to virtually eliminate external mass transfer resistance (i.e.,

CAs = CAb). From these data, we construct a log-log plot of the measured rate of

reaction as a function of the gas-phase concentration, CAs (Figure 3.7). The slope of

this plot is the apparent reaction order nʹ and the measured rate law takes the form –

Figure 3.7: Determining the apparent reaction order (-rA = ρb (-rʹA)).

------ (3.40)


Catalyst

Relation of measured reaction order n' to the true reaction order n:

The relationship will be established using the definition of the effectiveness factor,

that is the actual rate -rA, is the product of η and the rate of reaction evaluated at the

external surface, kn, Cn

As, i.e.,

For large values of the Thiele modulus, ϕn, where internal mass is limiting, we can

use equation 3.35 to substitute the value of η into equation 3.41 –

on simplification, the above expression reduces to –

Now, on equating the true reaction rate (equation 3.42) with measured reaction rate

(equation 3.40) we get –

Also, the functional dependence of the reaction rate on concentration must be the

same for both the measured rate and the theoretically predicted rate, therefore:

therefore the measured apparent reaction order nʹ (nApparent) is related to the true

reaction order n (nTrue) by –

------ (3.41)

------ (3.42)

------ (3.43)

------ (3.44)


Catalyst

Similarly, like apparent reaction order, we have apparent activation energy (EApp)

also which is related to true activation energy (ET) in the following manner:

This measurement of the apparent reaction order and activation energy results

primarily when internal diffusion limitations are present and is referred to as

disguised or falsified kinetics.

Serious consequences could occur if the laboratory data were taken in the

disguised regime and the reactor were operated in a different regime.

For example, what if the particle size were reduced so that internal diffusion

limitations became negligible? The higher activation energy, ET, would cause

the reaction to be much more temperature sensitive, and there is the possibility

for runaway reaction conditions causing an explosion to occur.

------ (3.46)

Estimation of Internal Effectiveness Factor (η):

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