18. Models for Nonideal Reactors

Success is a journey, not a destination.

—Ben Sweetland

Conflicting goals

The choice of the particular model to be used depends largely on the engineering judgment of the person carrying out the analysis. It is this person’s job to choose the model that best combines the conflicting goals of mathematical simplicity and physical realism. There is a certain amount of art in the development of a model for a particular reactor, and the examples presented here can only point toward a direction that an engineer’s thinking might follow.

A Model must

• Fit the data

• Be able to extrapolate theory and experiment

• Have realistic parameters

For a given real reactor, it is not uncommon to use all the models discussed previously to predict conversion and then make comparisons. Usually, the real conversion will be bounded by the model calculations.

The following guidelines are suggested when developing models for nonideal reactors:

1. The model must be mathematically tractable. The equations used to describe a chemical reactor should be able to be solved without an inordinate expenditure of human or computer time.

2. The model must realistically describe the characteristics of the nonideal reactor. The phenomena occurring in the nonideal reactor must be reasonably described physically, chemically, and mathematically.

3. The model should not have more than two adjustable parameters. This constraint is often used because an expression with more than two adjustable parameters can be fitted to a great variety of experimental data, and the modeling process in this circumstance is nothing more than an exercise in curve fitting. The statement “Give me four adjustable parameters and I can fit an elephant; give me five and I can include his tail!” is one that I have heard from many colleagues. Unless one is into modern art, a substantially larger number of adjustable parameters is necessary to draw a reasonable-looking elephant.¹ A one-parameter model is, of course, superior to a two-parameter model if the one-parameter model is sufficiently realistic. To be fair, however, in complex systems (e.g., internal diffusion and conduction, mass transfer limitations) where other parameters may be measured independently, then more than two parameters are quite acceptable.

¹ J. Wei, CHEMTECH, 5, 128 (1975).

Table 18-1 gives some guidelines that will help your analysis and model building of nonideal reaction systems.

TABLE 18-1 A PROCEDURE FOR CHOOSING A MODEL TO PREDICT THE OUTLET CONCENTRATION AND CONVERSION

The Guidelines

When using the algorithm in Table 18-1, we classify a model as being either a one-parameter model (e.g., tanks-in-series model or dispersion model) or a two-parameter model (e.g., reactor with bypassing and dead volume). In Sections 18.1.1 and 18.1.2, we give an overview of these models, which will be discussed in greater detail later in the chapter.

Nonideal tubular reactors

We first consider nonideal tubular reactors. Tubular reactors may be empty, or they may be packed with some material that acts as a catalyst, heat-transfer medium, or means of promoting interphase contact. Until Chapters 16–18, it usually has been assumed that the fluid moves through the reactor in a piston-like flow (i.e., plug flow reactor), and every atom spends an identical length of time in the reaction environment. Here, the velocity profile is flat, and there is no axial mixing. Both of these assumptions are false to some extent in every tubular reactor; frequently, they are sufficiently false to warrant some modification. Most popular tubular reactor models need to have the means to allow for failure of the plug-flow model and insignificant axial mixing assumptions; examples include the unpacked laminar-flow tubular reactor, the unpacked turbulent flow reactor, and packed-bed reactors. One of two approaches is usually taken to compensate for failure of either or both of the ideal assumptions. One approach involves modeling the nonideal tubular reactor as a series of identically sized CSTRs. The other approach (the dispersion model) involves a modification of the ideal reactor by imposing axial dispersion on plug flow.

Here, we could model the real reactor as two ideal PBRs in parallel, with the two parameters being the volumetric flow rate that channels or by passes, υ_b, and the reactor dead volume, V_D. The real reactor volume is V = V_D + V_S with entering volumetric flow rate υ₀ = υ_b + υ_S.

Using the definition of the RTD presented in Section 16.2, the fraction of material leaving the system of three reactors (i.e., leaving the third reactor) that has been in the system between time t and t + Δt is

In Figure 2-9, we saw how tanks in series could approximate a PFR.

Then

In this expression, C₃ (t) is the concentration of tracer in the effluent from the third reactor and the other terms are as defined previously.

By carrying out mass balances on the tracer sequentially for reactors 1, 2, and 3, it is shown on the CRE Web site in the Expanded Material for Chapter 18 that the exit tracer concentration for reactor 3 is

Substituting Equation (18-2) into Equation (18-1), we find that

Generalizing this method to a series of n CSTRs gives the RTD for n CSTRs in series, E(t):

RTD for equal-size tanks in series

Equation (18-4) will be a bit more useful if we put in the dimensionless form in terms of E(Θ). Because the total reactor volume is nV_i, then τ_i = τ/n, where τ represents the total reactor volume divided by the flow rate, υ, we have

where Θ = t/τ = Number of reactor volumes of fluid that have passed through the reactor after time t.

Here, (E(Θ) dΘ) is the fraction of material existing between dimensionless time Θ and time (Θ + dΘ).

Figure 18-3 illustrates the RTDs of various numbers of CSTRs in series in a two-dimensional plot (a) and in a three-dimensional plot (b). As the number becomes very large, the behavior of the system approaches that of a plug-flow reactor.

We can determine the number of tanks in series by calculating the dimensionless variance from a tracer experiment.

As the number of tanks increases, the variance decreases.

The number of tanks in series is

This expression represents the number of tanks necessary to model the real reactor as n ideal tanks in series. If the number of reactors, n, turns out to be small, the reactor characteristics turn out to be those of a single CSTR or perhaps two CSTRs in series. At the other extreme, when n turns out to be large, we recall from Chapter 2 that the reactor characteristics approach those of a PFR.

where

It is acceptable (and usual) for the value of n calculated from Equation (18-11) to be a noninteger in Equation (5-15) to calculate the conversion. For reactions other than first order, an integer number of reactors must be used and sequential mole balances on each reactor must be carried out. If, for example, n = 2.53, then one could calculate the conversion for two tanks and also for three tanks to bound the conversion. The conversion and effluent concentrations would be solved sequentially using the algorithm developed in Chapter 5; that is, after solving for the effluent from the first tank, it would be used as the input to the second tank and so on as shown on the CRE Web site for Chapter 18 Expanded Materials.

The proof of Equation (18-12) is given in the Expanded Materials on the CRE Web site for Chapter 18.

Tracer pulse with dispersion

To illustrate how dispersion affects the concentration profile in a tubular reactor, we consider the injection of a perfect tracer pulse. Figure 18-5 shows how dispersion causes the pulse to broaden as it moves down the reactor and becomes less concentrated.

Recall Equation (14-14). The molar flow rate of tracer (F_T) by both convection and dispersion is

In this expression, D_a is the effective dispersion coefficient (m²/s) and U (m/s) is the superficial velocity. To better understand how the pulse broadens, we refer to the concentration peaks t₂ and t₃ in Figure 18-6. We see that there is a concentration gradient on both sides of the peak causing molecules to diffuse away from the peak and thus broaden the pulse. The pulse broadens as it moves through the reactor.

Correlations for the dispersion coefficients in both liquid and gas systems may be found in Levenspiel.² Some of these correlations are given in Section 18.4.5.

² O. Levenspiel, Chemical Reaction Engineering (New York: Wiley, 1962), pp. 290–293.

An unsteady state mole balance on the inert tracer T gives

Substituting for F_T and dividing by the cross-sectional area A_c, we have

Pulse tracer balance with dispersion

Once we know the boundary conditions, the solution to Equation (18-14) will give the outlet tracer concentration–time curves. Consequently, we will have to wait to obtain this solution until we discuss the boundary conditions in Section 18.4.2.

The plan

We are now going to proceed in the following manner: First, we will write the balance equations for dispersion with reaction. We will discuss the two types of boundary conditions, closed-closed and open-open. We will then obtain an analytical solution for the closed-closed system for the conversion for a first-order reaction in terms of the Peclet number, Pe (dispersion coefficient) and the Damköhler number. We then will discuss how the dispersion coefficient can be obtained either from correlations in the literature or from the analysis of the RTD curve.

Rearranging Equation (14-16) we obtain

This equation is a second-order ordinary differential equation. It is nonlinear when r_A is other than zero or first order.

When the reaction rate r_A is first order, r_A = –kC_A, then Equation (18-16)

Flow, reaction, and dispersion

is amenable to an analytical solution. However, before obtaining a solution, we put our Equation (18-16) describing dispersion and reaction in dimensionless form by letting ψ = C_A/C_A0 and λ = z/L

D_a = Dispersion coefficient

Da₁ = Damköhler number

The quantity Da₁ appearing in Equation (18-17) is called the Damköhler number for a first-order conversion and physically represents the ratio

Damköhler number for a first-order reaction

The other dimensionless term is the Peclet number, Pe,

in which l is the characteristic length term. There are two different types of Peclet numbers in common use. We can call Pe_r the reactor Peclet number; it uses the reactor length, L, for the characteristic length, so Pe_r ≡ UL/D_a. It is Pe_r that appears in Equation (18-17). The reactor Peclet number, Pe_r, for mass dispersion is often referred to as the Bodenstein number, Bo, in reacting systems rather than the Peclet number. The other type of Peclet number can be called the fluid Peclet number, Pe_f; it uses the characteristic length that determines the fluid’s mechanical behavior. In a packed bed this length is the particle diameter d_p, and Pe_f ≡ Ud_p/ϕD_a. (The term U is the empty tube or superficial velocity. For packed beds we often wish to use the average interstitial velocity, and thus U/Φ; is commonly used for the packed-bed velocity term.) In an empty tube, the fluid behavior is determined by the tube diameter d_t, and Pe_f = Ud_t/D_a . The fluid Peclet number, Pe_f, is given in virtually all literature correlations relating the Peclet number to the Reynolds number because both are directly related to the fluid mechanical behavior. It is, of course, very simple to convert Pe_f to Pe_r: Multiply by the ratio L/d_p or L/d_t . The reciprocal of Pe_r, D_a/UL, is sometimes called the vessel dispersion number.

For open tubes Pe_r ~ 10⁶, Pe_f ~ 10⁴

For packed beds Pe_r ~ 10³, Pe_f ~ 10¹

Substituting for F_A yields

Solving for the entering concentration C_A(0^–) = C_A0

Concentration boundary conditions at the entrance

At the exit to the reaction section, the concentration is continuous, and there is no gradient in tracer concentration.

Concentration boundary conditions at the exit

These two boundary conditions, Equations (18-20) and (18-21), first stated by Danckwerts, have become known as the famous Danckwerts boundary conditions.³ Bischoff has given a rigorous derivation by solving the differential equations governing the dispersion of component A in the entrance and exit sections, and taking the limit as the dispersion coefficient, D_a in the entrance and exit sections approaches zero.⁴ From the solutions, he obtained boundary conditions on the reaction section identical with those Danckwerts proposed.

³ P. V. Danckwerts, Chem. Eng. Sci., 2, 1 (1953).

⁴ K. B. Bischoff, Chem. Eng. Sci., 16, 131 (1961).

Danckwerts Boundary Conditions

The closed-closed concentration boundary condition at the entrance is shown schematically in Figure 18-8 on page 857. One should not be uncomfortable with the discontinuity in concentration at z = 0 because if you recall for an ideal CSTR, the concentration drops immediately on entering from C_A0 to C_Aexit. For the other boundary condition at the exit z = L, we see the concentration gradient, (dC_A/dz), has gone to zero. At steady state, it can be shown that this Danckwerts boundary condition at z = L also applies to the open-open system at steady state.

Open-open boundary condition

Prof. P. V. Danckwerts, Cambridge University, U.K.

At z = L, we have continuity of concentration and

For the closed-closed system, the Danckwerts boundary conditions in dimensionless form are

Da₁ = τk
Pe_r = UL/D_a

At the end of the reactor, where λ = 1, the solution to Equation (18-17) is

Nomenclature note Da₁ is the Damköhler number for a first-order reaction, τk D_a is the dispersion coefficient in cm²/s Pe_r = UL/D_a

This solution was first obtained by Danckwerts and has been published in many places (e.g., Levenspiel).^5,6 With a slight rearrangement of Equation (18-26), we obtain the conversion as a function of Da₁ and Pe_r.

⁵ P. V. Danckwerts, Chem. Eng. Sci., 2, 1 (1953).

⁶ Levenspiel, Chemical Reaction Engineering, 3rd ed. (New York: Wiley, 1999).

Outside the limited case of a first-order reaction, a numerical solution of the equation is required, and because this is a split-boundary-value problem, an iterative technique is needed.

To evaluate the exit concentration given by Equation (18-26) or the conversion given by (18-27), we need to know the Damköhler and Peclet numbers. The first-order reaction rate constant, k, and hence Da₁ = τk, can be found using the techniques in Chapter 7. In the next section, we discuss methods to determine D_a by finding the Peclet number.

1. Laminar flow with radial and axial molecular diffusion theory

2. Correlations from the literature for pipes and packed beds

3. Experimental tracer data

Three ways to find D_a

At first sight, simple models described by Equation (18-14) appear to have the capability of accounting only for axial mixing effects. It will be shown, however, that this approach can compensate not only for problems caused by axial mixing, but also for those caused by radial mixing and other nonflat velocity profiles.⁷ These fluctuations in concentration can result from different flow velocities and pathways and from molecular and turbulent diffusion.

⁷ R. Aris, Proc. R. Soc. (London), A235, 67 (1956).

where U is the average velocity. For laminar flow, we saw that the RTD function E(t) was given by

In arriving at this distribution E(t), it was assumed that there was no transfer of molecules in the radial direction between streamlines. Consequently, with the aid of Equation (16-47), we know that the molecules on the center streamline (r = 0) exited the reactor at a time t = τ/2, and molecules traveling on the streamline at r = 3R/4 exited the reactor at time

The question now arises: What would happen if some of the molecules traveling on the streamline at r = 3R/4 jumped (i.e., diffused) onto the streamline at r = 0? The answer is that they would exit sooner than if they had stayed on the streamline at r = 3R/4. Analogously, if some of the molecules from the faster streamline at r = 0 jumped (i.e., diffused) onto the streamline at r = 3R/4, they would take a longer time to exit (Figure 18-9). In addition to the molecules diffusing between streamlines, they can also move forward or backward relative to the average fluid velocity by molecular diffusion (Fick’s law). With both axial and radial diffusion occurring, the question arises as to what will be the distribution of residence times when molecules are transported between and along streamlines by diffusion. To answer this question, we will derive an equation for the axial dispersion coefficient, D_a, that accounts for the axial and radial diffusion mechanisms. In deriving D_a, which is often referred to as the Aris–Taylor dispersion coefficient, we closely follow the development given by Brenner and Edwards.⁸

⁸ H. Brenner and D. A. Edwards, Macrotransport Processes (Boston: Butterworth-Heinemann, 1993).

Molecules diffusing between streamlines and back and forth along a streamline

The convective–diffusion equation for solute (e.g., tracer) transport in both the axial and radial direction can be obtained by combining Equation (14-3) with the diffusion equation (cf. Equation (14-11)) applied to the tracer concentration, c, and transformed to radial coordinates

where c is the solute concentration at a particular r, z, and t, and D_AB is the molecular diffusion coefficient of species A in B.

We are going to change the variable in the axial direction z to z*, which corresponds to an observer moving with the fluid

A value of z* = 0 corresponds to an observer moving with the average velocity of the fluid, U. Using the chain rule, we obtain

Because we want to know the concentrations and conversions at the exit to the reactor, we are really only interested in the average axial concentration, , which is given by

Consequently, we are going to solve Equation (18-30) for the solution concentration as a function of r and then substitute the solution c (r, z, t) into Equation (18-31) to find (z, t). All the intermediate steps are given on the CRE Web site in the Professional Reference Shelf, and the partial differential equation describing the variation of the average axial concentration with time and distance is

where D* is the Aris-Taylor dispersion coefficient

Aris-Taylor dispersion coefficient

That is, for laminar flow in a pipe

D_a ≡ D*

Figure 18-10 shows the dispersion coefficient D* in terms of the ratio D*/U (2R) = D*/Ud_t as a function of the product of the Reynolds (Re) and Schmidt (Sc) numbers.

Once the Reynolds number is calculated, D_a can be found.

in dimensionless form, discuss the different types of boundary conditions at the reactor entrance and exit, solve for the exit concentration as a function of dimensionless time (Θ = t/τ), and then relate D_a, σ², and τ.

For a pulse input, C_T0 is defined as the mass of tracer injected, M, divided by the vessel volume, V. Then

Initial condition

The initial condition is

The mass of tracer injected, M, is

Equation (18-34) has been solved numerically for a pulse injection, and the resulting dimensionless effluent tracer concentration, ψ_exit, is shown as a function of the dimensionless time Θ in Figure 18-13 for various Peclet numbers. Although analytical solutions for ψ can be found, the result is an infinite series. The corresponding equations for the mean residence time, t_m, and the variance, σ², are⁹

⁹ See K. Bischoff and O. Levenspiel, Adv. Chem. Eng., 4, 95 (1963).

and

which can be used with the solution to Equation (18-34) to obtain

¹⁰ O. Levenspiel, Chemical Reaction Engineering, 2nd ed. (New York: Wiley, 1972), p. 277.

Effects of dispersion on the effluent tracer concentration

Calculating Pe_r using t_m and σ² determined from RTD data for a closed-closed system

Consequently, we see that the Peclet number, Pe_r (and hence D_a), can be found experimentally by determining t_m and σ² from the RTD data and then solving Equation (18-39) for Pe_r.

For an open-open system, the boundary conditions at the entrance are

F_T (0^–, t) = F_T (0⁺, t)

Then, for the case when the dispersion coefficient is the same in the entrance and reaction sections

Open at the entrance

Because there are no discontinuities across the boundary at z = 0

At the exit

Open at the exit

There are a number of perturbations of these boundary conditions that can be applied. The dispersion coefficient can take on different values in each of the three regions (z < 0, 0 < z < L, and z > L), and the tracer can also be injected at some point z₁ rather than at the boundary, z = 0. These cases and others can be found in the supplementary readings cited at the end of the chapter. We shall consider the case when there is no variation in the dispersion coefficient for all z and an impulse of tracer is injected at z = 0 at t = 0.

For long tubes (Pe_r > 100) in which the concentration gradient at ± ∞ will be zero, the solution to Equation (18-34) at the exit is¹¹

¹¹ W. Jost, Diffusion in Solids, Liquids and Gases (New York: Academic Press, 1960), pp. 17, 47.

Valid for Pe_r > 100

The mean residence time for an open-open system is

Calculate τ for an open-open system.

where τ is based on the volume between z = 0 and z = L (i.e., reactor volume measured with a yardstick). We note that the mean residence time for an open system is greater than that for a closed system. The reason is that the molecules can diffuse back into the reactor after they diffuse out at the entrance. The variance for an open-open system is

Calculate Pe_r for an open–open system.

We now consider two cases for which we can use Equations (18-39) and (18-46) to determine the system parameters:

Case 1. The space time τ is known. That is, V and υ₀ are measured independently. Here, we can determine the Peclet number by determining t_m and σ² from the concentration–time data and then use Equation (18-46) to calculate Pe_r. We can also calculate t_m and then use Equation (18-45) as a check, but this is usually less accurate.

Case 2. The space time τ is unknown. This situation arises when there are dead or stagnant pockets that exist in the reactor along with the dispersion effects. To analyze this situation, we first calculate mean residence time, t_m, and the variance, σ², from the data as in case 1. Then, we use Equation (18-45) to eliminate τ² from Equation (18-46) to arrive at

We now can solve for the Peclet number in terms of our experimentally determined variables σ² and . Knowing Pe_r, we can solve Equation (18-45) for τ, and hence V. The dead volume is the difference between the measured volume (i.e., with a yardstick) and the effective volume calculated from the RTD.

Finding the effective reactor voume

¹² K. Elgeti, Chem. Eng. Sci., 51, 5077 (1996).

or

Equivalency between models of tanks-in-series and dispersion

where

where U is the superficial velocity, L the reactor length, and D_a the dispersion coefficient.

For the conditions in Example 18-1, we see that the number of tanks calculated from the Bodenstein number, Bo (i.e., Pe_r), Equation (18-49), is 4.75, which is very close to the value of 4.35 calculated from Equation (18-11). Consequently, for reactions other than first order, one would solve successively for the exit concentration and conversion from each tank in series for both a battery of four tanks in series and for five tanks in series in order to bound the expected values.

In addition to the one-parameter models of tanks-in-series and dispersion, many other one-parameter models exist when a combination of ideal reactors is used to model the real reactor shown in Section 18.7 for reactors with bypassing and dead volume. Another example of a one-parameter model would be to model the real reactor as a PFR and a CSTR in series with the one parameter being the fraction of the total volume that behaves as a CSTR. We can dream up many other situations that would alter the behavior of ideal reactors in a way that adequately describes a real reactor. However, it may be that one parameter is not sufficient to yield an adequate comparison between theory and practice. We explore these situations with combinations of ideal reactors in the section on two-parameter models.

The reaction-rate parameters are usually known (e.g., Da), but the Peclet number is usually not known because it depends on the flow and the vessel. Consequently, we need to find Pe_r using one of the three techniques discussed earlier in the chapter.

Analytical solutions to dispersion with reaction can only be obtained for isothermal zero- and first-order reactions. We are now going to use COMSOL to solve the flow with reaction and dispersion with reaction.

We are going to compare two solutions: one which uses the Aris–Taylor approach and one in which we numerically solve for both the axial and radial concentration using COMSOL. These solutions are on the CRE Web site.

Case A. Aris-Taylor Analysis for Laminar Flow

For the case of an nth-order reaction, Equation (18-15) is

where is the average concentration from r = 0 to r = R, i.e.,

If we use the Aris-Taylor analysis, we can use Equation (18-15) with a caveat that and λ = z/L we obtain

where

For the closed-closed boundary conditions we have

Danckwerts boundary conditions

For the open-open boundary conditions we have

Equation (18-53) is a nonlinear second-order ODE that is solved on the COMSOL on the CRE Web site.

Case B. Full Numerical Solution

To obtain profiles, C_A(r,z), we now solve Equation (18-51)

First, we will put the equations in dimensionless form by letting ψ = C_A/C_A0, λ = z/L, and ϕ = r/R. Following our earlier transformation of variables, Equation (18-52) becomes

Equation (18-56) gives the dimensionless concentration profiles for dispersion and reaction in a laminar-flow reactor. The Expanded Material on the CRE Web site gives an example, Web Example 18-2, where COMSOL is used to find the concentration profile.

Creativity and engineering judgment are necessary for model formulation.

A tracer experiment is used to evaluate the model parameters.

In determining the suitability of a particular reactor model and the parameter values from tracer tests, it may not be necessary to calculate the RTD function E (t). The model parameters (e.g., V_D) may be acquired directly from measurements of effluent concentration in a tracer test. The theoretical prediction of the particular tracer test in the chosen model system is compared with the tracer measurements from the real reactor. The parameters in the model are chosen so as to obtain the closest possible agreement between the model and experiment. If the agreement is then sufficiently close, the model is deemed reasonable. If not, another model must be chosen.

The quality of the agreement necessary to fulfill the criterion “sufficiently close” again depends on creativity in developing the model and on engineering judgment. The most extreme demands are that the maximum error in the prediction not exceed the estimated error in the tracer test, and that there be no observable trends with time in the difference between prediction (the model) and observation (the real reactor). To illustrate how the modeling is carried out, we will now consider two different models for a CSTR.

The model system

The Duct Tape Council of Jofostan would like to point out the new wrinkle: The Junction Balance.

The bypass stream and effluent stream from the reaction volume are mixed at the junction point 2. From a balance on species A around this point

We can solve for the concentration of A leaving the reactor

Let α = V_s/V and β = υ_b/υ₀. Then

For a first-order reaction, a mole balance on V_s gives

Mole balance on CSTR

or, in terms of α and β

Substituting Equation (18-60) into (18-58) gives the effluent concentration of species A:

Conversion as a function of model parameters

We have used the ideal reactor system shown in Figure 18-14 to predict the conversion in the real reactor. The model has two parameters, α and β. The parameter α is the dead zone volume fraction and parameter β is the fraction of the volumetric flow rate that bypasses the reaction zone. If these parameters are known, we can readily predict the conversion. In the following section, we shall see how we can use tracer experiments and RTD data to evaluate the model parameters.

Model system

Tracer balance for step input

The conditions for the positive-step input are

A balance around junction point 2 gives

The junction balance

The model system

As before

Integrating Equation (18-62) and substituting in terms of α and β gives

Combining Equations (18-63) and (18-64), the effluent tracer concentration is

We now need to rearrange this equation to extract the model parameters, α and β, either by regression (Polymath/MATLAB/Excel) or from the proper plot of the effluent tracer concentration as a function of time. Rearranging yields

Consequently, we plot ln [C_T0/(C_T0 – C_T)] as a function of t. If our model is correct, a straight line should result with a slope of (1 – β)/τα and an intercept of ln [1/(1 – β)].

Evaluating the model parameters

Other Models. In Section 18.7.1 it was shown how we formulated a model consisting of ideal reactors to represent a real reactor. First, we solved for the exit concentration and conversion for our model system in terms of two parameters, a and . We next evaluated these parameters from data on tracer concentration as a function of time. Finally, we substituted these parameter values into the mole balance, rate law, and stoichiometric equations to predict the conversion in our real reactor.

To reinforce this concept, we will use one more example.

The model system

υ₁ = βυ₀

Two parameters: α and β

and let α represent that fraction of the total volume, V, occupied by the highly agitated region:

V₁ = αV

Then

V₂ = (1 – α)V

Two parameters: α and β

The space time is

As shown on the CRE Web site Professional Reference Shelf R18.2, for a first-order reaction, the exit concentration and conversion are

and

Conversion for two-CSTR model

where C_A1 is the reactor concentration exiting the first reactor in Figure 18-17(b).

Unsteady-state balance of inert tracer

and C_T1 is the measured tracer concentration existing the real reactor. The tracer is initially dumped only into reactor 1, so that the initial conditions C_T10 = N_T0/V₁ and C_T20 = 0.

Substituting in terms of α, β, and τ, we arrive at two coupled differential equations describing the unsteady behavior of the tracer that must be solved simultaneously.

See Appendix A.3 for method of solution

Analytical solutions to Equations (18-71) and (18-72) are given on the CRE Web site, in Appendix A.3 and in Equation (18-73), below. However, for more complicated systems, analytical solutions to evaluate the system parameters may not be possible.

where

By regression on Equation (18-73) and the data in Table E18-2.2 or by an appropriate semilog plot of C_T1/C_T10 versus time, one can evaluate the model parameters α and β.

A case history for terephthalic acid

Figure 18-17(a) describes a real PFR or PBR with channeling that is modeled as two PFRs/PBRs in parallel. The two parameters are the fraction of flow to the reactors [i.e., β and (1 β)] and the fractional volume [i.e., α and (1 – α)] of each reactor. Figure 18-17(b) describes a real PFR/PBR that has a backmix region and is modeled as a PFR/PBR in parallel with a CSTR. Figures 18-18(a) and (b) on page 884 show a real CSTR modeled as two CSTRs with interchange. In one case, the fluid exits from the top CSTR (a) and in the other case the fluid exits from the bottom CSTR (b). The parameter β represents the interchange volumetric flow rate, β^v₀, and a the fractional volume of the top reactor, aV, where the fluid exits the reaction system. We note that the reactor in Figure 18-18(b) was found to describe extremely well a real reactor used in the production of terephthalic acid.¹³ A number of other combinations of ideal reactions can be found in Levenspiel.¹⁴

¹³ Proc. Indian Inst. Chem. Eng. Golden Jubilee, a Congress, Delhi, 1997, p. 323.

¹⁴ Levenspiel, O. Chemical Reaction Engineering, 3rd ed. (New York: Wiley, 1999), pp. 284–292.

Models for Nonideal Reactors

a. Zero adjustable parameters

(1) Segregation model

(2) Maximum mixedness model

b. One adjustable parameter

(1) Tanks-in-series model

(2) Dispersion model

c. Two adjustable parameters: real reactor modeled as combinations of ideal reactors

2. Tanks-in-series model: Use RTD data to estimate the number of tanks in series,

For a first-order reaction

3. Dispersion model: For a first-order reaction, use the Danckwerts boundary conditions

where

For a first-order reaction

4. Determine D_a

a. For laminar flow, the dispersion coefficient is

b. Correlations. Use Figures 18-10 through 18-12.

c. Experiment in RTD analysis to find t_m and σ².

For a closed-closed system, use Equation (S18-6) to calculate Pe_r from the RTD data

For an open-open system, use

5. If a real reactor is modeled as a combination of ideal reactors, the model should have at most two parameters.

6. The RTD is used to extract model parameters.

7. Comparison of conversions for a PFR and CSTR with the zero-parameter and two-parameter models. X_seg symbolizes the conversion obtained from the segregation model and X_mm is that from the maximum mixedness model for reaction orders greater than one.

Cautions: For rate laws with unusual concentration functionalities or for nonisothermal operation, these bounds may not be accurate for certain types of rate laws.

1. W18.2.1 Developing the E-Curve for T-I-S

2. Web Example 18-1 Equivalency of Models for a First Order Reaction

X_T-I-S = X_seg = X_mm

3. Sloppy Tracer Inputs

4. Case A Aris-Taylor Analysis for LFR

5. Web Example 18-2 Dispersion with Reaction

6. Web Example 18-2 (COMSOL)

7. Web Problem 18-12_C

8. Web Problem 18-14_D

9. Web Problem 18-17_D

10. Web Problem 18-18_B

11. Web Problem 18-19_C

12. Web Problem 18-20_B

• Learning Resources

1. Summary Notes

• Living Example Problems

1. Example 18-3 CSTR with Bypass and Dead Volume

• Professional Reference Shelf

R18.1. Derivation of Equation for Taylor-Aris Dispersion

R18.2. Real Reactor Modeled as two Ideal CSTRs with Exchange Volume

Example R18-1 Two CSTRs with interchange

Q18-2 What if you were asked to design a tubular vessel that would minimize dispersion? What would be your guidelines? How would you maximize the dispersion? How would your design change for a packed bed?

Q18-3 What if someone suggested you could use the solution to the flow-dispersion-reactor equation, Equation (18-27), for a second-order equation by linearizing the rate law by lettering . (1) Under what circumstances might this be a good approximation? Would you divide C_A0 by something other than 2? (2) What do you think of linearizing other non-first-order reactions and using Equation (18-27)? (3) How could you test your results to learn if the approximation is justified?

(a) Example 18-1. Vary D_a, k, U, and L. To what parameters or groups of parameters (e.g., kL²/D_a) would the conversion be most sensitive? What if the first-order reaction were carried out in tubular reactors of different diameters, but with the space time, τ, remaining constant? The diameters would range from a diameter of 0.1 dm to a diameter of 1 m for kinematic viscosity v = μ/ρ = 0.01 cm²/s, U = 0.1 cm/s, and D_AB = 10^–5 cm²/s. How would your conversion change? Is there a diameter that would maximize or minimize conversion in this range?

(b) Example 18-2. How would your answers change if the slope was 4 min^–1 and the intercept was 2 in Figure E18-2.2?

(c) Example 18-3. Download the Living Example Polymath Program. Vary α and β, and describe what you find. What would be the conversion if α = 0.75 and β = 0.15?

P18-2_B The gas-phase isomerization

is to be carried out in a flow reactor. Experiments were carried out at a volumetric flow rate of υ₀ = 2 dm³/min in a reactor that had the following RTD

E(t) = 10 e^–10t min^–1

where t is in minutes.

(a) When the volumetric flow rate was 2 dm³/min, the conversion was 9.1%. What is the reactor volume?

(b) When the volumetric flow rate was 0.2 dm³/min, the conversion was 50%. When the volumetric flow rate was 0.02 dm³/min, the conversion was 91%. Assuming the mixing patterns don’t change as the flow rate changes, what will the conversion be when the volumetric flow rate is 10 dm³/min?

(c) This reaction is now to be carried out in a 1-dm³ plug-flow reactor where volumetric flow rate has been changed to 1 dm³/min. What will be the conversion?

(d) It is proposed to carry out the reaction in a 10-m-diameter pipe where the flow is highly turbulent (Re = 10⁶). There are significant dispersion effects. The superficial gas velocity is 1 m/s. If the pipe is 6 m long, what conversion can be expected? If you were unable to determine the reaction order and the specific reaction rate constant in part (b), assume k = 1 min^–1 and carry out the calculation!

P18-3_B The second-order liquid-phase reaction

is to be carried out isothermally. The entering concentration of A is 1.0 mol/dm³. The specific reaction rate is 1.0 dm³/mol·min. A number of used reactors (shown below) are available, each of which has been characterized by an RTD. There are two crimson and white reactors, and three maize and blue reactors available.

(a) You have $50,000 available to spend. What is the greatest conversion you can achieve with the available money and reactors?

(b) How would your answer to (a) change if you had an extra $75,000 available to spend?

(c) From which cities do you think the various used reactors came from?

P18-4_B The elementary liquid-phase reaction

is carried out in a packed-bed reactor in which dispersion is present. What is the conversion?

Additional information:

(Ans.: X = 0.15)

P18-5_A A gas-phase reaction is being carried out in a 5-cm-diameter tubular reactor that is 2 m in length. The velocity inside the pipe is 2 cm/s. As a very first approximation, the gas properties can be taken as those of air (kinematic viscosity = 0.01 cm²/s), and the diffusivities of the reacting species are approximately 0.005 cm²/s.

(a) How many tanks in series would you suggest to model this reactor?

(b) If the second-order reaction is carried out for the case of equimolar feed, and with C_A0 = 0.01 mol/dm³, what conversion can be expected at a temperature for which k = 25 dm³/mol · s?

(c) How would your answers to parts (a) and (b) change if the fluid velocity was reduced to 0.1 cm/s? Increased to 1 m/s?

(d) How would your answers to parts (a) and (b) change if the superficial velocity was 4 cm/s through a packed bed of 0.2-cm-diameter spheres?

(e) How would your answers to parts (a) to (d) change if the fluid was a liquid with properties similar to water instead of a gas, and the diffusivity was 5 × 10^–6 cm²/s?

P18-6_A Use the data in Example 16-2 to make the following determinations. (The volumetric feed rate to this reactor was 60 dm³/min.)

(a) Calculate the Peclet numbers for both open and closed systems.

(b) For an open system, determine the space time τ and then calculate the % dead volume in a reactor for which the manufacturer’s specifications give a volume of 420 dm³.

(c) Using the dispersion and tanks-in-series models, calculate the conversion for a closed vessel for the first-order isomerization

with k = 0.18 min^–1.

(d) Compare your results in part (c) with the conversion calculated from the tanks-in-series model, a PFR, and a CSTR.

P18-7_A A tubular reactor has been sized to obtain 98% conversion and to process 0.03 m³/s. The reaction is a first-order irreversible isomerization. The reactor is 3 m long, with a cross-sectional area of 25 dm². After being built, a pulse tracer test on the reactor gave the following data: t_m = 10 s and σ² = 65 s². What conversion can be expected in the real reactor?

P18-8_B The following E(t) curve was obtained from a tracer test on a reactor.

The conversion predicted by the tanks-in-series model for the isothermal elementary reaction

was 50% at 300 K.

(a) If the temperature is to be raised 10˚C (E = 25,000 cal/mol) and the reaction carried out isothermally, what will be the conversion predicted by the maximum mixedness model? The T-I-S model?

(b) The elementary reactions

were carried out isothermally at 300 K in the same reactor. What is the concentration of B in the exit stream predicted by the maximum mixedness model?

(c) For the multiple reactions given in part (b), what is the conversion of A predicted by the dispersion model in an isothermal closed-closed system?

P18-9_B Revisit Problem P16-3_C where the RTD function is a hemicircle. What is the conversion predicted by

(a) The tanks-in-series model? (Ans.: X_T-I-S = 0.447)

(b) The dispersion model? (Ans.: X_Dispersion = 0.41)

P18-10_B Revisit Problem P16-5_B.

(a) What combination of ideal reactors would you use to model the RTD?

(b) What are the model parameters?

(c) What is the conversion predicted for your model?

(d) What is the conversion predicted by X_mm, X_seg, X_T-I-S, and X_Dispersion?

P18-11_B Revisit Problem P16-6_B.

(a) What conversion is predicted by the tanks-in-series model?

(b) What is the Peclet number?

(c) What conversion is predicted by the dispersion model?

P18-12_D Let’s continue Problem P16-11_C.

(a) What would be the conversion for a second-order reaction with kC_A0 = 0.1 min^–1 and C_A0 = 1 mol/dm³ using the segregation model?

(b) What would be the conversion for a second-order reaction with kC_A0 = 0.1 min^-1 and C_A0 = 1 mol/dm³ using the maximum mixedness model?

(c) If the reactor is modeled as tanks in series, how many tanks are needed to represent this reactor? What is the conversion for a first-order reaction with k = 0.1 min^–1 ?

(d) If the reactor is modeled by a dispersion model, what are the Peclet numbers for an open system and for a closed system? What is the conversion for a first-order reaction with k = 0.1 min^–1 for each case?

(e) Use the dispersion model to estimate the conversion for a second-order reaction with k = 0.1 dm³/mol·s and C_A0 = 1 mol/dm³.

(f) It is suspected that the reactor might be behaving as shown in Figure P18-12_B, with perhaps (?) V₁ = V₂. What is the “backflow” from the second to the first vessel, as a multiple of υ₀ ?

(g) If the model above is correct, what would be the conversion for a second-order reaction with k = 0.1 dm³/mol ·min if C_A0 = 1.0 mol/dm³?

(h) Prepare a table comparing the conversion predicted by each of the models described above.

P18-13_B A second-order reaction is to be carried out in a real reactor that gives the following outlet concentration for a step input:

For 0 ≤ t < 10 min, then C_T = 10 (1–e^–.1t)

For 10 min ≤ t, then C_T = 5+10 (1–^e–.1t)

(a) What model do you propose and what are your model parameters, α and β?

(b) What conversion can be expected in the real reactor?

(c) How would your model change and conversion change if your outlet tracer concentration was as follows?

For t ≤ 10 min, then C_T = 0

For t ≥ 10 min, then C_T = 5+10 (1–e^{–0.2(t–10)})

υ₀ = 1 dm³/min, k = 0.1 dm³/mol ⋅ min, C_A0 = 1.25 mol/dm³

P18-14_B Suggest combinations of ideal reactors to model the real reactors given in problem P16-2_B(b) for either E(θ), E(t), F(θ), F(t), or (1 – F(θ)).

P18-15_B The F-curves for two tubular reactors are shown in Figure P18-15_B for a closed–closed system.

(a) Which curve has the higher Peclet number? Explain.

(b) Which curve has the higher dispersion coefficient? Explain.

(c) If this F-curve is for the tanks-in-series model applied to two different reactors, which curve has the largest number of T-I-S, (1) or (2)?

U of M, ChE528 Mid-Term Exam

P18-16_C Consider the following system in Figure P18-16_C used to model a real reactor:

Describe how you would evaluate the parameters α and β.

(a) Draw the F- and E-curves for this system of ideal reactors used to model a real reactor using β = 0.2 and α = 0.4. Identify the numerical values of the points on the F-curve (e.g., t₁) as they relate to τ.

(b) If the reaction A → B is second order with kC_A0 = 0.5 min^–1, what is the conversion assuming the space time for the real reactor is 2 min?

U of M, ChE528 Final Exam

P18-17_B There is a 2-m³ reactor in storage that is to be used to carry out the liquid-phase second-order reaction

A and B are to be fed in equimolar amounts at a volumetric rate of 1 m³/min. The entering concentration of A is 2 molar, and the specific reaction rate is 1.5 m³/kmol • min. A tracer experiment was carried out and reported in terms of F as a function of time in minutes as shown in Figure P18-17_B.

Suggest a two-parameter model consistent with the data; evaluate the model parameters and the expected conversion.

U of M, ChE528 Final Exam

P18-18_B The following E-curve shown in Figure P18-18_B was obtained from a tracer test:

(a) What is the mean residence time?

(b) What is the Peclet number for a closed-closed system?

(c) How many tanks in series are necessary to model this nonideal reactor?

U of M, Doctoral Qualifying Exam (DQE)

P18-19_B A first-order reaction is to be carried out in the reactor with k = 0.1 min^–1.

Fill in the following table with the conversion predicted by each type of model/reactor.

DOUGLAS, J. M., “The effect of mixing on reactor design,” AIChE Symp. Ser. 48, vol. 60, p. 1 (1964).

ZWIETERING, TH. N., Chem. Eng. Sci., 11, 1 (1959).

2. Modeling real reactors with a combination of ideal reactors is discussed together with axial dispersion in

LEVENSPIEL, O., Chemical Reaction Engineering, 3rd ed. New York: Wiley, 1999.

WEN, C. Y., and L. T. FAN, Models for Flow Systems and Chemical Reactors. New York: Marcel Dekker, 1975.

3. Mixing and its effects on chemical reactor design have been receiving increasingly sophisticated treatment. See, for example:

BISCHOFF, K. B., “Mixing and contacting in chemical reactors,” Ind. Eng. Chem., 58 (11), 18 (1966).

NAUMAN, E. B., “Residence time distributions and micromixing,” Chem. Eng. Commun., 8, 53 (1981).

NAUMAN, E. B., and B. A. BUFFHAM, Mixing in Continuous Flow Systems. New York: Wiley, 1983.

4. See also

DUDUKOVIC, M., and R. FELDER, in CHEMI Modules on Chemical Reaction Engineering, vol. 4, ed. B. Crynes and H. S. Fogler. New York: AIChE, 1985.

5. Dispersion. A discussion of the boundary conditions for closed-closed, open-open, closed-open, and open-closed vessels can be found in

ARIS, R., Chem. Eng. Sci., 9, 266 (1959).

LEVENSPIEL, O., and K. B. BISCHOFF, Adv. in Chem. Eng., 4, 95 (1963).

NAUMAN, E. B., Chem. Eng. Commun., 8, 53 (1981).

6. Now that you have finished this book, suggestions on what to do with the book can be posted on the kiosk in downtown Riça, Jofostan.