15. Diffusion and Reaction

Research is to see what everybody else sees, and to think what nobody else has thought.

—Albert Szent-Gyorgyi

For diffusion through a stagnant film at dilute concentrations, Equation (14-9) becomes

Substituting in Equation (14-2) one obtains

Understanding and modeling diffusion with chemical reaction is not only important for industrial catalysts but also has many other applications. These applications include medicine, cancer treatment using drug laced particulates, and, as shown in the Expanded Material on the CRE Web site (www.umich.edu/~elements/5e/index.html), tissue engineering. In P15-15_B, we discuss the diffusion and reaction of oxygen in cartilage.

We will now discuss solid-gas catalytic reactions and diffusion limitation in catalyst pellets.

In Chapter 14 we discussed external diffusion. In this section we will discuss internal diffusion and develop the internal effectiveness factor for spherical catalyst pellets. The development of models that treat individual pores and pellets of different shapes is undertaken in the problems at the end of this chapter. We will first look at the internal mass transfer resistance to either the reaction products or reactants that occurs between the external pellet surface and the interior of the pellet. To illustrate the salient principles of this model, we consider the irreversible isomerization

that occurs on the surface of the pore walls within the spherical pellet of radius R.

The pores in the pellet are not straight and cylindrical; rather, they are a series of tortuous, interconnecting paths of pore bodies and pore throats with varying cross-sectional areas. It would not be fruitful to describe diffusion within each and every one of the tortuous pathways individually; consequently, we shall define an effective diffusion coefficient so as to describe the average diffusion taking place at any interior position r in the pellet. We shall consider only radial variations in the concentration; the radial flux W_Ar will be based on the total area (voids and solid) normal to diffusion transport (i.e., 4πr²) rather than void area alone. This basis for W_Ar is made possible by proper definition of the effective diffusivity D_e.

The effective diffusivity accounts for the fact that:

1. Not all of the area normal to the direction of the flux is available (i.e., the area occupied by solids) for the molecules to diffuse.

2. The paths are tortuous.

3. The pores are of varying cross-sectional areas.

An equation that relates the effective diffusivity D_e to either the bulk diffusivity D_AB or the Knudsen diffusivity D_K is

The effective diffusivity

where

¹ Some investigators lump the constriction factor and tortuosity into one factor, called the tortuosity factor, and set it equal to . See C. N. Satterfield, Mass Transfer in Heterogeneous Catalysis (Cambridge, MA: MIT Press, 1970), pp. 33–47.

The constriction factor, σ_c, accounts for the variation in the cross-sectional area that is normal to diffusion.² It is a function of the ratio of maximum to minimum pore areas (Figure 15-2(a)). When the two areas, A₁ and A₂, are equal, the constriction factor is unity, and when β = 10, the constriction factor is approximately 0.5.

² See E. E. Petersen, Chemical Reaction Analysis (Upper Saddle River, NJ: Prentice Hall, 1965), Chap. 3; C. N. Satterfield and T. K. Sherwood, The Role of Diffusion in Catalysis (Reading, MA: Addison-Wesley, 1963), Chap. 1.

First, we will derive the concentration profile of reactant A in the pellet.

We now proceed to perform our shell balance on A. The area that appears in the balance equation is the total area (voids and solids) normal to the direction of the molar flux shown by the arrows in Figure 15-3.

Mole balance for diffusion and reaction inside the catalyst pellet

where r_m is some mean radius between r and r + Δr that is used to approximate the volume ΔV of the shell, and ρ_c is the density of the pellet.

The mole balance over the shell thickness Δr is

Mole balance

After dividing by (–4π Δr) and taking the limit as Δr → 0, we obtain the following differential equation

Because 1 mol of A reacts under conditions of constant temperature and pressure to form 1 mol of B, we have Equal Molar Counter Diffusion (EMCD) at constant total molar concentration (Section 14.2.1) and, therefore

The flux equation

where C_A is the number of moles of A per dm³ of open-pore volume (i.e., volume of gas) as opposed to (mol/vol of gas and solids). In systems where we do not have EMCD in catalyst pores, it may still be possible to use Equation (15-7) if the reactant gases are present in dilute concentrations.

After substituting Equation (15-7) into Equation (15-6), we arrive at the following differential equation describing diffusion with reaction in a catalyst pellet

We now need to incorporate the rate law. In the past, we have based the rate of reaction in terms of either per unit volume

or per unit mass of catalyst

When we study reactions on the internal surface area of catalysts, the rate of reaction and rate law are often based on per unit surface area

As a result, the surface area of the catalyst per unit mass of catalyst

S_a [=] (m²/g-cat)

is an important property of the catalyst. The rate of reaction per unit mass of catalyst, , and the rate of reaction per unit surface area of catalyst are related through the equation

S_a: 10 grams of catalyst may cover as much surface area as a football field.

A typical value of S_a might be 150 m²/g of catalyst.

The rate law

As mentioned previously, at high temperatures, the denominator of the catalytic rate law often approaches 1 as discussed in Section 10.3.7. Consequently, for the moment, it is reasonable to assume that the surface reaction is of nth order in the gas-phase concentration of A within the pellet

where the units of the rate constants for –r_A, , and are

Similarly,

Substituting the rate-law equation (15-9) into Equation (15-8) gives

Differential equation and boundary conditions describing diffusion and reaction in a catalyst pellet

Letting k_n represent the terms under the bracket, differentiating the first term and dividing through by –r²D_e, Equation (15-10) becomes

The boundary conditions are:

1. The concentration remains finite at the center of the pellet

2. At the external surface of the catalyst pellet, the concentration is C_As

With the transformation of variables, the boundary condition

becomes

and the boundary condition

becomes

We now rewrite the differential equation for the molar flux in terms of our dimensionless variables. Starting with

we use the chain rule to write

Then differentiate Equation (15-12) with respect to ψ and Equation (15-13) with respect to r, and substitute the resulting expressions

into Equation (15-14) to obtain

The flux of A in terms of the dimensionless variables, ψ and λ, is

The total rate of consumption of A inside the pellet, M_A (mol/s)

At steady state, the net flow of species A that enters into the pellet at the external pellet surface reacts completely within the pellet. The overall rate of reaction is therefore equal to the total molar flow of A into the catalyst pellet. The overall rate of reaction, M_A (mol/s), can be obtained by multiplying the molar flux into the pellet at the outer surface by the external surface area of the pellet, 4πR²

All the reactant that diffuses into the pellet is consumed (a black hole).

Consequently, to determine the overall rate of reaction, which is given by Equation (15-17), we first solve Equation (15-11) for C_A, differentiate C_A with respect to r, and then substitute the resulting expression into Equation (15-17).

Differentiating the concentration gradient, Equation (15-15), yields

After dividing by C_As/R², the dimensionless form of Equation (15-11) is written as

Then

Dimensionless form of equations describing diffusion and reaction

where

The square root of the coefficient of ψⁿ in Equation 15-19 (i.e., ϕ_n) is called the Thiele modulus (pronounced th –l ). The Thiele modulus, ϕ_n, will always contain a subscript (e.g., n), which refers to the reaction order and distinguishes this symbol from the symbol for porosity, ϕ, used in the Ergun pressure drop equation and defined in Chapter 5, which has no subscript. The quantity is a measure of the ratio of “a” surface reaction rate to “a” rate of diffusion through the catalyst pellet

Thiele modulus

When the Thiele modulus is large, internal diffusion usually limits the overall rate of reaction; when ϕ_n is small, the surface reaction is usually rate-limiting. If, for the first order reaction

A → B

Limiting conditions:

the reaction were surface reaction rate limited with respect to the adsorption of A and desorption of B, and if A and B were weakly adsorbed (i.e., low surface coverage), we can write the apparent first-order reaction rate law per unit volume of pellet as

–r_A = k₁C_A

where k₁ is the rate constant for a single catalyst pellet.

Recalling k₁ = S_aρ_ck″ we could also write the rate in terms of pellet catalytic surface area (mol/m²•s)

The units of are m³/m² s (= m/s).

For a first-order reaction, Equation (15-19) becomes

the Thiele modulus for this first-order reaction is

where

The boundary conditions are

With these transformations, Equation (15-22) reduces to

This differential equation has the following solution (see Appendix A.3):

y = A₁ cosh ϕ₁λ + B1 sinh ϕ₁λ

In terms of ψ,

The arbitrary constants A₁ and B₁ can easily be evaluated with the aid of the boundary conditions. At λ = 0, cosh ϕ₁λ → 1, (1/λ) → ∞, and sinh ϕ₁λ → 0. Because the second boundary condition requires ψ to be finite at the center (i.e., λ = 0), therefore A₁ must be zero.

The constant B₁ is evaluated from B.C. 1 (i.e., ψ = 1, λ = 1) and the dimensionless concentration profile is

Concentration profile

Figure 15-4 shows the concentration profile for three different values of the Thiele modulus, ϕ₁ . Small values of the Thiele modulus indicate surface reaction controls and a significant amount of the reactant diffuses well into the pellet interior without reacting. As a result, the concentration profile is very shallow, with the concentration at the center of the pellet being close to that at the external surface. That is, virtually the entire internal surface is accessible to the reactant concentration C_As. Large values of the Thiele modulus indicate that the surface reaction is rapid and that the reactant is consumed very close to the external pellet surface and very little penetrates into the interior of the pellet. Consequently, if the porous pellet is to be plated with a precious metal catalyst (e.g., Pt), it should only be plated in the immediate vicinity of the external surface when large values of ϕ_n characterize the diffusion and reaction. That is, it would be a waste of the precious metal to plate the entire pellet when internal diffusion is limiting because the reacting gases are consumed near the outer surface. Consequently, the reacting gases would never contact the center portion of the pellet.

For large values of the Thiele modulus, internal diffusion limits the rate of reaction.

η is a measure of how far the reactant diffuses into the pellet before reacting.

The overall rate, , is also referred to as the observed rate of reaction [–r_A (obs)]. In terms of symbols, the effectiveness factor is

To derive the effectiveness factor for a first-order reaction, it is easiest to work in reaction rates of moles per unit time, M_A, rather than in moles per unit time per volume of catalyst (i.e., –r_A)

First, we shall consider the denominator, M_As . If the entire surface were exposed to the concentration at the external surface of the pellet, C_As, the rate for a first-order reaction would be

The subscript s indicates that the rate –r_As is evaluated at the conditions (e.g., concentration, temperature) present at the external surface of the pellet (i.e., λ = 1).

The actual rate of reaction 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.”) We recall Equation (15-17) on page 727 for the actual rate of reaction

The actual rate of reaction

Differentiating Equation (15-27) and then evaluating the result at λ = 1 yields

Substituting Equation (15-30) into (15-17) gives us

We now substitute Equations (15-29) and (15-31) into Equation (15-28) to obtain an expression for the effectiveness factor

Internal effectiveness factor for a first-order reaction in a spherical catalyst pellet

A plot of the effectiveness factor as a function of the Thiele modulus is shown in Figure 15-5. Figure 15-5(a) shows η as a function of the Thiele modulus, ϕ_s, for a spherical catalyst pellet for reactions of zero, first, and second order. Figure 15-5(b) corresponds to a first-order reaction occurring in three differently shaped pellets of volume V_p and external surface area A_p, and where the Thiele modulus for a first-order reaction, ϕ₁, is defined differently for each shape. When volume change accompanies a reaction (i.e., ε ≠ 0), the corrections shown in Figure 15-6 apply to the effectiveness factor for a first-order reaction.

We observe that as the particle diameter becomes very small, ϕ_n decreases, so that the effectiveness factor approaches 1 and the reaction is surface reaction-limited. On the other hand, when the Thiele modulus ϕ_n is large (e.g., 30), the internal effectiveness factor η is small (e.g., 0.1), and the reaction is diffusion-limited within the pellet. Consequently, factors influencing the rate of external mass transport such as fluid velocity will have a negligible effect on the overall reaction rate when the reaction is either internal surface reaction rate limited or internal diffusion limited. For large values of the Thiele modulus, the effectiveness factor can be written as

To express the overall rate of reaction in terms of the Thiele modulus, we rearrange Equation (15-28) and use the rate law for a first-order reaction in Equation (15-29)

For a first-order reaction

Combining Equations (15-33) and (15-34), the overall rate of reaction for a first-order, internal-diffusion-limited reaction is

Note that for catalytic packed beds the rate varies inversely with the particle diameter of the catalyst.

as a function of ε for various values of the Thiele modulus. This plot is given in Figure 15-6.

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 η′ = 0.8η .

How can the rate of reaction be increased?

Correction for volume change with reaction (i.e., ε ≠ 0)

For large values of the Thiele modulus, the effectiveness factor is

Consequently, for reaction orders greater than 1, the effectiveness factor decreases with increasing concentration at the external pellet surface.

The left-hand side is the Weisz–Prater parameter:

Showing where the Weisz–Prater comes from

Substituting for

in Equation (15-59) we obtain

Are there any internal diffusion limitations indicated from the Weisz–Prater criterion?

All the terms in Equation (15-39) are either measured or known. Consequently, we can calculate C_WP to learn if there are any diffusion limitations.

If

No diffusion limitations

there are no diffusion limitations and consequently no concentration gradient exists within the pellet.

However, if

Severe diffusion limitations

internal diffusion limits the reaction severely. Ouch!

You may not be measuring what you think you are.

Measured rate:

Measured rate with apparent reaction order n’

We will now proceed to relate this measured reaction order n’ to the true reaction order n. Using the definition of the effectiveness factor, noting that the actual rate, , is the product of η and the rate of reaction evaluated at the external surface, , i.e.,

Actual rate:

For large values of the Thiele modulus, ϕ_n, where internal mass is limiting, we can use Equation (15-38) to substitute into Equation (15-41) to obtain

Simplifying

We equate the true reaction rate, Equation (15-42), to the measured reaction rate, Equation (15-40), to get

The functional dependence of the reaction rate on concentration must be the same for both the measured rate and the theoretically predicted rate

therefore the measured apparent reaction order n’ (n_Apparent) is related to the true reaction order n (n_True) by

The true and the apparent reaction order

In addition to an apparent reaction order, there is also an apparent activation energy, E_App . This value is the activation energy we would calculate using the experimental data from the slope of a plot of ln () as a function of (1/T) at a fixed concentration of A. Substituting for the measured and true specific reaction rates in terms of the activation energy gives

into Equation (15-43), we find that

Taking the natural log of both sides gives us

where E_T is the true activation energy.

As with the dependence rate on concentration, the temperature dependence must be the same for the analytical rate. Comparing the temperature-dependent terms on the right- and left-hand sides of Equation (15-45), we see that the true activation energy is equal to twice the apparent activation energy.

The true activation energy

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, E_T, would cause the reaction to be much more temperature sensitive, and there is the possibility for runaway reaction conditions causing an explosion to occur.

Important industrial consequence of falsified kinetic runaway reactions. Safety considerations!

Here, both internal and external diffusion are important.

The molar rate of mass transfer from the bulk fluid to the external surface is

where a_c is the external surface area per unit reactor volume (cf. Chapter 14) and ΔV is the volume.

This molar rate of mass transfer to the surface, M_A, is equal to the net (total) rate of reaction on and within the pellet

---

Recall that ρ_c is the density of catalyst pellet, kg per volume of pellet and ρ_B is the bulk density of catalyst, kg-cat per reactor volume.

We now combine the above equations, to obtain total molar flow into all the catalyst in volume ΔV

Combining Equations (15-47) and (15-48), and canceling the volume ΔV, we see the flux to the pellet surface, W_Aza_c, is equal to the rate of consumption of A in and on the catalyst.

For most catalysts, the internal surface area is much greater than the external surface area (i.e., ), in which case we have

Comparing units on the r.h.s. and l.h.s. of Equation (15-49), we find

where is the overall rate of reaction within and on the pellet per unit surface area, is the rate of reaction per mass of catalyst

and –r_A is the overall rate per volume of reactor, i.e.,

then

with the corresponding units for each term in Equation (15-49) shown below.

The relationship for the rate of mass transport to the external catalyst surface is

Again, comparing units on the l.h.s. and r.h.s.

where k_c is the external mass transfer coefficient (m/s). Because internal diffusion resistance is also significant, not all of the interior surface of the pellet is accessible to the concentration at the external surface of the pellet, C_As . We have already learned that the effectiveness factor is a measure of this surface accessibility [see Equation (15-41)]:

Assuming that the surface reaction is first order with respect to A, we can utilize the internal effectiveness factor to write

Recall that

We need to eliminate the surface concentration from any equation involving the rate of reaction or rate of mass transfer, because C_As cannot be measured by standard techniques. To accomplish this elimination, we use Equations (15-49), (15-50), and (15-51) in order to equate the mass transfer rate of A to the pellet surface, –W_Ara_c, to the rate of consumption of A within the pellet, ηk₁C_As

W_Ar a_c = ηk₁C_As

Then substitute for W_Ar a_c using Equation (15-50)

Solving for C_As, we obtain

Concentration at the pellet surface as a function of bulk gas concentration

Substituting for C_As in Equation (15-51) gives

In discussing the surface accessibility, we defined the internal effectiveness factor η with respect to the concentration at the external surface of the pellet, C_As, as

We now define an overall effectiveness factor that is based on the bulk concentration

Two different effectiveness factors

Dividing the numerator and denominator of Equation (15-54) by k_c a_c, we obtain the net rate of reaction, –r_A, in terms of the bulk fluid concentration, which is a measurable quantity:

The actual rate of reaction is related to the reaction rate evaluated at the bulk concentration of A. Consequently, the overall rate of reaction in terms of the bulk concentration C_Ab is

where

Overall effectiveness factor for a first-order reaction

Note that the rates of reaction based on surface and bulk concentrations are related by

where

The actual rate can be expressed in terms of the rate per unit volume, –r_A, the rate per unit mass, , and the rate per unit surface area, , which are related by the equation

Recall that is given in terms of the catalyst surface area (m³/m² · s), is given in terms of catalyst mass (m3/g-cat · s), and k₁ is given in terms of reactor volume (1/s)

We saw in Chapter 14 that as the velocity of the fluid increases, the external mass transfer coefficient k_c increases (cf. 14-49). Consequently, for large flow rates resulting in large values of the external mass transfer coefficient k_c, we can neglect the ratio in the denominator

High flow rates of fluid

and the overall effectiveness factor approaches the internal effectiveness factor

Dial soap

⁶ D. E. Mears, Ind. Eng. Chem. Process Des. Dev., 10, 541 (1971). Other interphase transport-limiting criteria can be found in AIChE Symp. Ser. 143 (S. W. Weller, ed.), 70 (1974).

Is external diffusion limiting?

Here, we measure (obs), C_Ab, ρ_b, R and n, and then calculate k_c to determine MR, where

Mears proposed that when

MR < 0.15

external mass transfer effects can be neglected and no concentration gradient exists between the bulk gas and external surface of the catalyst pellet. This proposal by Mears was endorsed unanimously by the Jofostan legislature. The mass transfer coefficient can be calculated from the appropriate correlation, such as that of Thoenes–Kramers, for the flow conditions through the bed.

Mears also proposed that the bulk fluid temperature, T, will be virtually the same as the temperature at the external surface of the pellet when

T_b ≅ T_s

and the other symbols are as in Equation (15-62).

We shall perform a balance on species A over the volume element, ΔV, neglecting any radial variations in concentration and assuming that the bed is operated at steady state. The following symbols will be used in developing our model:

A mole balance on ΔV, the volume element (A_c Δz), yields

Mole Balance

Dividing by A_c Δz and taking the limit as Δz 0 yields

Combining Equation (14-5) and (14-7), we get

Also, writing the bulk flow term in the form

B_Az = y_Ab (W_Az + W_Bz) = y_Ab cU = UC_Ab

Equation (15-64) can be written in the form

Now we will see how to use η and Ω to calculate conversion in a packed bed.

The term D_AB (d²C_Ab/dz²) is used to represent either diffusion and/or dispersion in the axial direction. Consequently, we shall use the symbol D_a for the dispersion coefficient to represent either or both of these cases. We will come back to this form of the diffusion equation when we discuss dispersion in Chapter 18. The overall reaction rate, –r_A, is a function of the reactant concentration within the catalyst. This overall rate can be related to the rate of reaction of A that would exist if the entire surface were exposed to the bulk concentration C_Ab through the overall effectiveness factor Ω

For the first-order reaction considered here

Substituting this equation for –r_A into Equation (15-65), we form the differential equation describing diffusion with a first-order reaction in a catalyst bed

Flow and first-order reaction in a packed bed

As an example, we shall solve this equation for the case in which the flow rate through the bed is very large and the axial diffusion can be neglected. Young and Finlayson have shown that axial dispersion can be neglected when⁷

⁷ L. C. Young and B. A. Finlayson, Ind. Eng. Chem. Fund., 12, 412.

Criterion for neglecting axial dispersion/diffusion

where U₀ is the superficial velocity, d_p the particle diameter, and D_a is the effective axial dispersion coefficient. In Chapter 18 we will consider solutions to the complete form of Equation (15-67).

Neglecting axial dispersion with respect to forced axial convection

Equation (15-67) can be arranged in the form

With the aid of the boundary condition at the entrance of the reactor

C_Ab = C_Ab0     at z = 0

Equation (15-69) can be integrated to give

Conversion in a packed-bed reactor

The conversion at the reactor’s exit, z = L, is

Variation of reaction rate with system variables

The correlation for the mass transfer coefficient, Equation (14-66), shows that k_c is directly proportional to the square root of the velocity and inversely proportional to the square root of the particle diameter

We recall from Equation (E15-3.5), a_c = 6(1 – ϕ)/d_p, that the variation of external surface area with catalyst particle size is

We now combine Equations (15-72), (15-73), and (15-74) to obtain

Consequently, for external mass transfer–limited reactions, the rate proportional to the velocity to the one-half power is inversely proportional to the particle diameter to the three-halves power.

From Equation (14-72), we see that for gas-phase external mass transfer– limited reactions, the rate increases approximately linearly with temperature.

Many heterogeneous reactions are diffusion limited.

When internal diffusion limits the rate of reaction, we observe from Equation (15-42) that the rate of reaction varies inversely with particle diameter, is independent of velocity, and exhibits an exponential temperature dependence that is not as strong as that for surface-reaction-controlling reactions. For surface-reaction-limited reactions, the rate is independent of particle size and is a strong function of temperature (exponential). Table 15-1 summarizes the dependence of the rate of reaction on the velocity through the bed, particle diameter, and temperature for the three types of limitations that we have been discussing.

Very Important Table

The exponential temperature dependence for internal diffusion limitations is usually not as strong a function of temperature as is the dependence for surface reaction limitations (cf. Section 15.4). If we would calculate an activation energy between 8 and 24 kJ/mol, chances are that the reaction would be strongly diffusion-limited. An activation energy of 200 kJ/mol, however, suggests that the reaction is reaction rate–limited.

Source: Satterfield, C. N. AIChE Journal., 21, 209 (1975); P. A. Ramachandran and R. V. Chaudhari, Chem. Eng., 87(24), 74 (1980); R. V. Chaudhari and P. A. Ramachandran, AIChE Journal., 26, 177 (1980). With permission of the American Institute of Chemical Engineers. Copyright © 1980 AIChE. All rights reserved.

TABLE 15-2 APPLICATIONS OF THREE-PHASE REACTORS

The multiphase reactors discussed in this edition of the book are the slurry reactor, fluidized bed, and the trickle bed reactor. The trickle bed reactor, which has reaction and transport steps similar to the slurry reactor, is discussed in the first edition of this book and on the CRE Web site along with the bubbling fluidized bed. In slurry reactors, the catalyst is suspended in the liquid, and gas is bubbled through the liquid. A slurry reactor may be operated in either a semibatch or continuous mode.

where ϕ₁ is the Thiele modulus. For a first-order reaction

2. The effectiveness factors are

3. For large values of the Thiele modulus for an n^th order reaction

4. For internal diffusion control, the true reaction order is related to the measured reaction order by

The true and apparent activation energies are related by

5. A. The Weisz–Prater Parameter

The Weisz–Prater criterion dictates that

B. The Mears Criterion for Neglecting External Diffusion and Heat Transfer

There will be no external diffusion limitations if

And there will be no temperature gradients if

1. Web Example 15-1 Application of Diffusion and Reaction to Tissue Engineering (also see P15-15_B)

2. 15.3.3 Effectiveness Factor for Nonisothermal First-Order Catalytic Reactions

3. Problem 15-12_C Diffusion and Reaction in Spherical Catalyst Pellets

4. Additional Homework Problems

• Learning Resources

1. Summary Notes

2. Solved Problems

• Professional Reference Shelf

R15.1. Slurry Reactors

A. Description of the Use of Slurry Reactors

Example R15-1 Industrial Slurry Reactor

B. Reaction and Transport Steps in a Slurry Reactor

C. Determining the Rate-Limiting Step

1. Effect of Loading, Particle Size, and Gas Adsorption

2. Effect of Shear

Example R15-2 Determining the Controlling Resistance

D. Slurry Reactor Design

Example R15-3 Slurry Reactor Design

R15.2. Trickle Bed Reactors

A. Fundamentals

B. Limiting Situations

C. Evaluating the Transport Coefficients

R15.3. Fluidized Bed Reactors

A. Descriptive Behavior of the Kunii-Levenspiel Bubbling Bed Model

B. Mechanics of Fluidized Beds

Example R15-4 Maximum Solids Hold-Up

C. Mass Transfer in Fluidized Beds

D. Reaction in a Fluidized Bed

E. Solution to the Balance Equations for a First-Order Reaction

Example R15-5 Catalytic Oxidation of Ammonia

F. Limiting Situations

Example R15-6 Calculation of the Resistances

Example R15-7 Effect of Particle Size on Catalyst Weight for a Slow Reaction

Example R15-8 Effect of Catalyst Weight for a Rapid Reaction

R15.4. Chemical Vapor Deposition Reactors

A. Chemical Reaction Engineering in Microelectronic Processing

B. Fundamentals of CVD

C. Effectiveness Factors for Boat Reactors

Example R15-9 Diffusion Between Wafers

Example R15-10 CVD Boat Reactor

Q15-2 Suppose someone had used the false kinetics (i.e., wrong E, wrong n)? Would the catalyst weight be overdesigned or underdesigned? What are other positive or negative effects that occur?

(a) Example 15-1. Effective Diffusivity. Make a sketch of a diffusion path for which the tortuosity is 5. How would your effective gas-phase diffusivity change if the absolute pressure were tripled and the temperature were increased by 50%?

(b) Example 15-2. If possible, determine the percent of the total resistance for (1) internal diffusion and (2) for reaction rate for each of the three particles studied. Apply the Weisz-Prater criteria to a particle 0.005 m in diameter.

(c) Example 15-3. Overall Effectiveness Factor. (1) Calculate the percent of the total resistance for the resistance of external diffusion, internal diffusion, and surface reaction. Qualitatively, how would each of your percentages change (2) If the temperature were increased significantly? (3) If the gas velocity were tripled? (4) If the particle size were decreased by a factor of 2? How would the reactor length change in each case? (5) What length would be required to achieve 99.99% conversion of the pollutant NO?

What if...

(d) you applied the Mears and Weisz–Prater criteria to Examples 15-4 and 15-3? What would you find? What would you learn if kcal/mol, h = 100 Btu/h · ft² · °F, and E = 20 kcal/mol?

(e) your internal surface area decreased with time because of sintering (see Section 10.7). Describe how your effectiveness factor would change and the rate of reaction change with time if k_d = 0.01 h^–1 and η = 0.01 at t = 0? Explain, being as quantitative as possible when you can.

(f) you were to assume that the resistance to gas absorption in the CRE Web site Professional Reference Shelf R15.1 was the same as in Professional Reference Shelf R15.3 and that the liquid-phase reactor volume in Professional Reference Shelf R15.3 was 50% of the total? Could you determine the limiting resistance? If so, what is it? What other things could you calculate in Professional Reference Shelf R15.1 (e.g., selectivity, conversion, molar flow rates in and out)? Hint: Some of the other reactions that occur include

(g) the temperature in the CRE Web site Example R15.2 were increased? How would the relative resistances in the slurry reactor change?

(h) you were asked for all the things that could go wrong in the operation of a slurry reactor? What would you say?

P15-2_B Concept problem: The catalytic reaction

takes place within a fixed bed containing spherical porous catalyst X22. Figure P15-2_B shows the overall rates of reaction at a point in the reactor as a function of temperature for various entering total molar flow rates, F_T0 .

(a) Is the reaction limited by external diffusion?

(b) If your answer to part (a) was “yes,” under what conditions of those shown (i.e., T, F_T0) is the reaction limited by external diffusion?

(c) Is the reaction “reaction-rate-limited”?

(d) If your answer to part (c) was “yes,” under what conditions of those shown (i.e., T, F_T0) is the reaction limited by the rate of the surface reactions?

(e) Is the reaction limited by internal diffusion?

(f) If your answer to part (e) was “yes,” under what conditions of those shown (i.e., T, F_T0) is the reaction limited by the rate of internal diffusion?

(g) For a flow rate of 10 g mol/h, determine (if possible) the overall effectiveness factor, Ω, at 360 K.

(h) Estimate (if possible) the internal effectiveness factor, η, at 367 K. (Ans.: η = 0.86)

(i) If the concentration at the external catalyst surface is 0.01 mol/dm³, calculate (if possible) the concentration at r = R/2 inside the porous catalyst at 367 K. (Assume a first-order reaction.)

Additional information:

P15-3_B Concept problem: The reaction

is carried out in a differential packed-bed reactor at different temperatures, flow rates, and particle sizes. The results shown in Figure P15-3_B were obtained.

(a) What regions (i.e., conditions d_p, T, F_T0) are external mass transfer–limited?

(b) What regions are reaction rate–limited?

(c) What region is internal-diffusion-controlled?

(d) What is the internal effectiveness factor at T = 400 K and d_p = 0.8 cm? (Ans.: η = 0.625)

P15-4_A Concept problem: Curves A, B, and C in Figure P15-4_A show the variations in reaction rate for three different reactions catalyzed by solid catalyst pellets. What can you say about each reaction?

P15-5_B A first-order heterogeneous irreversible reaction is taking place within a spherical catalyst pellet that is plated with platinum throughout the pellet (see Figure 15-3). The reactant concentration halfway between the external surface and the center of the pellet (i.e., r = R/2) is equal to one-tenth the concentration of the pellet’s external surface. The concentration at the external surface is 0.001 g mol/dm³, the diameter (2R) is 2 × 10^–3 cm, and the diffusion coefficient is 0.1 cm²/s.

(a) What is the concentration of reactant at a distance of 3 × 10^–4 cm in from the external pellet surface? (Ans.: C_A = 2.36 × 10^–4 mol/dm³.)

(b) To what diameter should the pellet be reduced if the effectiveness factor is to be 0.8? (Ans.: d_p = 6.8 × 10^–4 cm. Critique this answer!)

(c) If the catalyst support were not yet plated with platinum, how would you suggest that the catalyst support be plated after it had been reduced by grinding?

P15-6_B The swimming rate of a small organism [J. Theoret. Biol., 26, 11 (1970)] is related to the energy released by the hydrolysis of adenosine triphosphate (ATP) to adenosine diphosphate (ADP). The rate of hydrolysis is equal to the rate of diffusion of ATP from the midpiece to the tail (see Figure P15-6_B). The diffusion coefficient of ATP in the midpiece and tail is 3.6 × 10^–6 cm²/s. ADP is converted to ATP in the midsection, where its concentration is 4.36 × 10^–5 mol/cm³. The cross-sectional area of the tail is 3 × 10^–10 cm².

(a) Derive an equation for diffusion and reaction in the tail.

(b) Derive an equation for the effectiveness factor in the tail.

(c) Taking the reaction in the tail to be of zero order, calculate the length of the tail. The rate of reaction in the tail is 23 × 10^–18 mol/s.

(d) Compare your answer with the average tail length of 41 μm. What are possible sources of error?

P15-7_B A first-order, heterogeneous, irreversible reaction is taking place within a catalyst pore that is plated with platinum entirely along the length of the pore (Figure P15-7_B). The reactant concentration at the plane of symmetry (i.e., equal distance from the pore mouth) of the pore is equal to one-tenth the concentration at the pore mouth. The concentration at the pore mouth is 0.001 mol/dm³, the pore length (2L) is 2 × 10^–3 cm, and the diffusion coefficient is 0.1 cm²/s.

(a) Derive an equation for the effectiveness factor.

(b) What is the concentration of reactant at L/2?

(c) To what length should the pore length be reduced if the effectiveness factor is to be 0.8?

(d) If the catalyst support were not yet plated with platinum, how would you suggest the catalyst support be plated after the pore length, L, had been reduced by grinding?

P15-8_A A first-order reaction is taking place inside a porous catalyst. Assume dilute concentrations and neglect any variations in the axial (x) direction.

(a) Derive an equation for both the internal and overall effectiveness factors for the rectangular porous slab shown in Figure P15-8_A.

(b) Repeat part (a) for a cylindrical catalyst pellet where the reactants diffuse inward in the radial direction. (C-level problem, i.e., P15-8_C(b).)

P15-9_B The irreversible reaction

is taking place in the same porous catalyst slab shown in Figure P15-8_A. The reaction is zero order in A.

(a) Show that the concentration profile using the symmetry B.C. is

where

(b) For a Thiele modulus of 1.0, at what point in the slab is the concentration zero? For ϕ₀ = 4?

(c) What is the concentration you calculate at z = 0.1 L and ϕ₀ = 10 using Equation (P15-9.1)? What do you conclude about using this equation?

(d) Plot the dimensionless concentration profile ψ = C_A/C_As as a function of λ = z/L for ϕ₀ = 0.5, 1, 5, and 10. Hint: there are regions where the concentration is zero. Show that λ_C = (1 – 1/ϕ₀) is the start of this region where the gradient and concentration are both zero. [L. K. Jang, R. L. York, J. Chin, and L. R. Hile, Inst. Chem. Engr., 34, 319 (2003).]

Show that for λ_C ≤ λ > 1.

(e) The effectiveness factor can be written as

where z_C (λ_C) is the point at which both the concentration gradients and flux go to zero, and A_c is the cross-sectional area of the slab. Show for a zero-order reaction that

(f) Make a sketch for η versus ϕ₀ similar to the one shown in Figure 15-5.

(g) Repeat parts (a) to (f) for a spherical catalyst pellet.

(h) What do you believe to be the point of this problem?

P15-10_C The second-order decomposition reaction

is carried out in a tubular reactor packed with catalyst pellets 0.4 cm in diameter. The reaction is internal-diffusion-limited. Pure A enters the reactor at a superficial velocity of 3 m/s, a temperature of 250°C, and a pressure of 500 kPa. Experiments carried out on smaller pellets where surface reaction is limiting yielded a specific reaction rate of 0.05 m⁶/mol · g-cat · s. Calculate the length of bed necessary to achieve 80% conversion. Critique the numerical answer. (Ans.: L = 2.8 x 10^–5 m)

Additional information:

P15-11_C Derive the concentration profile and effectiveness factor for cylindrical pellets 0.2 cm in diameter and 1.5 cm in length. Neglect diffusion through the ends of the pellet.

(a) Assume that the reaction is a first-order isomerization. Hint: Look for a Bessel function.

(b) Rework Problem P15-10_C for these pellets.

P15-12_B Extension of Problem P15-7_B. The elementary isomerization reaction

is taking place on the walls of a cylindrical catalyst pore (see Figure P15-7_B.) In one run, a catalyst poison P entered the reactor together with the reactant A. To estimate the effect of poisoning, we assume that the poison renders the catalyst pore walls near the pore mouth ineffective up to a distance z₁, so that no reaction takes place on the walls in this entry region.

(a) Show that before poisoning of the pore occurred, the effectiveness factor was given by

where

(b) Derive an expression for the concentration profile and also for the molar flux of A in the ineffective region, 0 < z < z₁, in terms of z₁, D_AB, C_A1, and C_As . Without solving any further differential equations, obtain the new effectiveness factor η’ for the poisoned pore.

P15-13_B Falsified Kinetics. The irreversible gas-phase dimerization

is carried out at 8.2 atm in a stirred contained-solids reactor to which only pure A is fed. There are 40 g of catalyst in each of the four spinning baskets. The following runs were carried out at 227°C:

The following experiment was carried out at 237°C:

(a) What are the apparent reaction order and the apparent activation energy?

(b) Determine the true reaction order, specific reaction rate, and activation energy.

(c) Calculate the Thiele modulus and effectiveness factor.

(d) What pellet diameter should be used to make the catalyst more effective?

(e) Calculate the rate of reaction on a rotating disk made of the catalytic material when the gas-phase reactant concentration is 0.01 g mol/L and the temperature is 227°C. The disk is flat, nonporous, and 5 cm in diameter.

Additional information:

P15-14_B Derive Equation (15-39). Hint: Multiply both sides of Equation (15-25) for nth order reaction; that is,

by 2dy/dλ, rearrange to get

and solve using the boundary conditions dy/dλ = 0 at λ = 0.

P15-15_B Applications of Diffusion and Reaction to Tissue Engineering. The equations describing diffusion and reaction in porous catalysts also can be used to derive rates of tissue growth and have been studied by Professor Kristi Anseth and her students at the University of Colorado. One important area of tissue growth is in cartilage tissue in joints such as the knee. Over 200,000 patients per year receive knee joint replacements. Alternative strategies include the growth of cartilage to repair the damaged knee.

One approach is to deliver cartilage-forming cells in a hydrogel to the damaged area such as the one shown in Figure WP15-1.1 on the CRE Web site.

Here, the patient’s own cells are obtained from a biopsy and embedded in a hydrogel, which is a cross-linked polymer network that is swollen in water. In order for the cells to survive and grow new tissue, many properties of the gel must be tuned to allow diffusion of important species in and out (e.g., nutrients in and cell-secreted extracellular molecules such as collagen out). Because there is no blood flow through the cartilage, oxygen transport to the cartilage cells is primarily by diffusion. Consequently, the design must be such that the gel can maintain the necessary rates of diffusion of nutrients (e.g., O₂) into the hydrogel. These rates of exchange in the gel depend on the geometry and the thickness of the gel. To illustrate the application of chemical reaction engineering principles to tissue engineering, we will examine the diffusion and consumption of one of the nutrients, oxygen.

Our examination of diffusion and reaction in catalyst pellets showed that in many cases the reactant concentration near the center of the particle was virtually zero. If this condition were to occur in a hydrogel, the cells at the center would die. Consequently, the gel thickness needs to be designed to allow rapid transport of oxygen.

Let’s consider the simple gel geometry shown in Figure P15-15_B. We want to find the gel thickness at which the minimum oxygen consumption rate is 10^–13 mol/cell/h .

The cell density in the gel is 10¹⁰ cells/dm³, the bulk concentration of oxygen C_A0 (z = 0) is 2 x 10^–4 mol/dm³, and the diffusivity, D_AB, is 10^–5 cm²/s.

(a) Show that the dimensionless form of concentration and length, ψ = C_A/C_A0, and λ = z/L, differential mole balance on O₂ gives

(b) Show the dimensionless O₂ concentration profile in the gel is

where

(c) Solve the gel thickness when the concentration at z = 0 and C_A = 0.1 mmole/dm³.

(d) How would your answers change if the reaction kinetics were (1) first order in the O₂ concentration with k₁ = 10^–2 h^–1?

(e) Carry out a quasi-steady-state analysis using Equation (E15-1.19) along with the overall balance

to predict the O₂ flux and collagen build-up as a function of time.

(f) Sketch ψ versus λ at different times.

(g) Sketch λ_c as a function of time. Hint: V = A_cL. Assume α = 10 and the stoichiometric coefficient for oxygen to collagen, υ_c, is 0.05 mass fraction of cell/mol O₂. A_c = 2 cm².

P15C-2 Use the references given in Ind. Eng. Chem. Prod. Res. Dev., 14, 226 (1975) to define the iodine value, saponification number, acid number, and experimental setup. Use the slurry reactor analysis to evaluate the effects of mass transfer and determine if there are any mass transfer limitations.

LAPIDUS, L., AND N. R. AMUNDSON, Chemical Reactor Theory: A Review, Upper Saddle River, N.J.: Prentice Hall, 1977.

In addition, see

ARIS, R., Elementary Chemical Reactor Analysis. Upper Saddle River, NJ: Prentice Hall, 1989, Chap. 6. Old, but one should find the references listed at the end of this reading particularly useful.

FOGLER, JOSEPH J., AKA JOFO, A Chemical Reaction Engineers Guide to the Country of Jofostan. To be self published, hopefully by 2020.

LUSS, D., “Diffusion—Reaction Interactions in Catalyst Pellets,” p. 239 in Chemical Reaction and Reactor Engineering. New York: Marcel Dekker, 1987.

The effects of mass transfer on reactor performance are also discussed in

COLLINS, FRANK C., AND GEORGE E. KIMBALL, “Diffusion Controlled Reaction Rates,” Journal of Colloid Science, Vol. 4, Issue 4, August 1949, Pages 425-437.

SATTERFIELD, C. N., Heterogeneous Catalysis in Industrial Practice, 2nd ed. New York: McGraw-Hill, 1991.

2. Diffusion with homogeneous reaction is discussed in

ASTARITA, G., and R. OCONE, Special Topics in Transport Phenomena. New York: Elsevier, 2002.

Gas-liquid reactor design is also discussed in

BUTT, JOHN B., Reaction Kinetics and Reactor Design, 2nd ed. Boca Raton, FL: CRC Press, 2000.

SHAH, Y. T., Gas–Liquid–Solid Reactor Design. New York: McGraw-Hill, 1979.

3. Modeling of CVD reactors is discussed in

DOBKIN, DANIEL, AND M. K. ZUKRAW, Principles of Chemical Vapor Deposition. The Netherlands: Kluwer Academic Publishers, 2003.

HESS, D. W., K. F. JENSEN, and T. J. ANDERSON, “Chemical Vapor Deposition: A Chemical Engineering Perspective,” Rev. Chem. Eng., 3, 97, 1985.

4. Multiphase reactors are discussed in

RAMACHANDRAN, P. A., and R. V. CHAUDHARI, Three-Phase Catalytic Reactors. New York: Gordon and Breach, 1983.

RODRIGUES, A. E., J. M. COLO, and N. H. SWEED, eds., Multiphase Reactors, Vol. 1: Fundamentals. Alphen aan den Rijn, The Netherlands: Sitjhoff and Noordhoff, 1981.

RODRIGUES, A. E., J. M. COLO, and N. H. SWEED, eds., Multiphase Reactors, Vol. 2: Design Methods. Alphen aan den Rijn, The Netherlands: Sitjhoff and Noordhoff, 1981.

SHAH, Y. T., B. G. KELKAR, S. P. GODBOLE, and W. D. DECKWER, “Design Parameters Estimations for Bubble Column Reactors” (journal review), AIChE J., 28, 353 (1982).

The following Advances in Chemistry Series volume discusses a number of multiphase reactors:

FOGLER, H. S., ed., Chemical Reactors, ACS Symp. Ser. 168. Washington, DC: American Chemical Society, 1981, pp. 3–255.

5. Fluidization

KUNII, DAIZO, AND OCTAVE LEVENSPIEL, Fluidization Engineering, 2nd ed. (Butterworths Series in Chemical Engineering Deposition). Stoneham, MA: Butterworth-Heinemann, 1991.

In addition to Kunii and Levenspiel’s book, many correlations can be found in

DAVIDSON, J. F., R. CLIFF, and D. HARRISON, Fluidization, 2nd ed. Orlando: Academic Press, 1985.

YATES, J. G., Fundamentals of Fluidized-Bed Chemical Processes, 3rd ed. London: Butterworth, 1983.