Fitting CSTR Data

The goal is to determine a function, b.T) that can be used to design reactors. 

Suppose the reaction is A B and that the CSTR measurements are all done at the same temperature. Then = should be a function of a and possibly of b. Referring to Equations 12-1.6, here are some choices that contain no more than three adjustable constants  

Models with more constants will give better fits, but this does not justify their use. Statistical tests can help avoid overfitting data, but there may be simpler physical tests Is there any reason to believe that site competition is important Is the reaction essentially irreversible It may be that n = - va provides a good fit to the data so that n ceases to be an adjustable constant. Most importantly, is the residual sum of squares, SS, low enough to represent experimental error  

The choice of a model that contains one, two, or more adjustable constants is a matter of physical judgment combined with mathematics. The mathematical portion is the minimization of the sum of squares  

The predicted rate is calculated by substimting the measured concentrations for (tout and bout into one of Equations 7.7 with assumed values for the model parameters, for example, k, n, and to- The optimization methods of Chapter 6 are then used to adjust the parameter values in order to minimize SS. There will be some residual error  

---

Stevens and Funderburk (5 ) presented theoretical models for particle size distributions based on Smith-Ewart Case II and several other particle growth theories. They concluded that the Smith-Ewart Case II theory containing the Stockmayer modification fit CSTR data for styrene better than other models. 

The reaction of Example 7.4 is not elementary and could involve shortlived intermediates, but it was treated as a single reaction. We turn now to the problem of fitting kinetic data to multiple reactions. The multiple reactions hsted in Section 2.1 are consecutive, competitive, independent, and reversible. Of these, the consecutive and competitive t5T>es, and combinations of them, pose special problems with respect to kinetic studies. These will be discussed in the context of integral reactors, although the concepts are directly applicable to the CSTRs of Section 7.1.2 and to the complex reactors of Section 7.1.4. 

Fig. t2. Concentration profile of blue dextran in effluent of heparinase reactor during washout experiment. Inlet flow rate 120 mL/min. Concentration is normalized by initial concentration of dye in the reactor, and time is normalized by mean residence time in the reactor as determined by best fit to data. Line is theoretical prediction for washout of dye from CSTR [from Bernstein et al. (50)]. 

Arrhenius plot of rate constants extracted from TS-CSTR data as temperature was ramped during a single run. The number of points available is such that the experimental data produces the almost-continuous line shown. The fitted Arrhenius line is shown as a solid line. 

Now we can really see why the CSTR operated at steady state is so different from the transient batch reactor. If the inlet feed flow rates and concentrations are fixed and set to be equal in sum to the outlet flow rate, then, because the volume of the reactor is constant, the concentrations at the exit are completely defined for fixed kinetic parameters. Or, in other words, if we need to evaluate kab and kd, we simply need to vary the flow rates and to collect the corresponding concentrations in order to fit the data to these equations to obtain their magnitudes. We do not need to do any integration in order to obtain the result. Significantly, we do not need to have fast analysis of the exit concentrations, even if the kinetics are very fast. We set up the reactor flows, let the system come to steady state, and then take as many measurements as we need of the steady-state concentration. Then we set up a new set of flows and repeat the process. We do this for as many points as necessary in order to obtain a statistically valid set of rate parameters. This is why the steady-state flow reactor is considered to be the best experimental reactor type to be used for gathering chemical kinetics. 

In some applications, a simple CSTR is not sufficient to describe the residence time distribution of particles. Habermann et al. (1998) described a metallurgical process where the fluidized bed showed a stagnant zone and a two-zone model with a CSTR and a stagnant dead-zone fitted the data much better. For the resi-... 

Experimental data that are most easily obtained are of (C, t), (p, t), (/ t), or (C, T, t). Values of the rate are obtainable directly from measurements on a continuous stirred tank reactor (CSTR), or they may be obtained from (C, t) data by numerical means, usually by first curve fitting and then differentiating. When other properties are measured to follow the course of reaction—say, conductivity—those measurements are best converted to concentrations before kinetic analysis is started. 

Calculate bout IcLtn for the reversible reaction in Example 5.2 in a CSTR at 280 K and 285 K with F=2h. Suppose these results were actual measurements and that you did not realize the reaction was reversible. Fit a first-order model to the data to find the apparent activation energy. Discuss your results. 

The goal is to determine a functional form for (a, b,. .., T) that can be used to design reactors. The simplest case is to suppose that the reaction rate has been measured at various values a,b,..., T. A CSTR can be used for these measurements as discussed in Section 7.1.2. Suppose J data points have been measured. The jXh point in the data is denoted as S/t-data aj,bj,..., Tj) where Uj, bj,..., 7 are experimentally observed values. Corresponding to this measured reaction rate will be a predicted rate, modeii p bj,7 ). The predicted rate depends on the parameters of the model e.g., on k,m,n,r,s,... in Equation (7.4) and these parameters are chosen to obtain the best fit of the experimental... 

When the rate data come from a CSTR, the analysis was indicated in Section 3.3.4. Other data of (Ca, t) can be fitted by an algebraic equation and differentiated to obtain the derivative. Then the analysis is continued by Equation 3.10. 

A reaction, A = 2R, is conducted in a 5 liter CSTR with an inlet concentration Ca0 = 1.0 mol/liter. Data of temperature, flow rate and product concentration are in the first three columns of the table. Find the rate equation to fit. 

A first-order liquid-phase reaction takes place in a baffled stirred vessel of 2 volume under conditions when the flow rate is constant at 605 dm min and the reaction rate coefficient is 2.723 min the conversion of species A is 98%. Verify that this performance lies between that expected from either a PFR or a CSTR. Tracer impulse response tests are conducted on the reactor and the data in Table 6 recorded. Fit the tanks-in-series model to this data by (A) matching the moments, and (B) evaluating N from the time at which the maximum tracer response is observed. Give conversion predictions from the tanks-in-series model in each case. 

The observations in kinetic studies are usually concentration or conversion measurements. CSTRs and differential PFRs yield rate data directly, so the experimentally observed rate can be directly fitted on the rate expression and its parameters estimated. 

It has become customary to classify evaluation methods as "differential" or "integral." These terms stem from a time when practically all experiments were conducted in batch reactors, so that rates had to be found by differentiation of concentration-versus-time data, and the calculation of concentrations from postulated rate equations required integration. The terms do not fit the work-up of data from gradientless reactors such as CSTRs, in which rates and concentrations are related to one another by algebraic equations requiring no calculus, and are therefore avoided here. 

The model based on S-E Case 2 kinetics has been quite successful in handling steady-state data for styrene emulsion polymerizations in a CSTR. One or more of the mechanisms described above, however, generally cause other monomer systems to deviate from this simple model. The nature of these deviations varies among the different monomers. If published literature data are fitted to equations of the type listed below one can obtain values for the exponents a, h, and c. 

Figure 6.17 Normalized intrinsic viscosity [r ]/[)7]o for a dilute solution of poly(y-benzyl-L-glutamate) (PBLG) = 208,000) in m-cresol. The line is a calculation for the rigid-dumbbell model, with the relaxation time t = lj6Dro adjusted to the value 10- sec to obtain a fit. The stress tensor for a suspension of rigid dumbbells is given by Eq. (6-36) with Cstr replaced by k T/Dro-(From Bird et al. 1987 data from Yang 1958, Dynamics of Polymeric Liquids, VoL 2, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

In this section we focus on the three main types of ideal reactors BR, CSTR, and PFR. Laboratory data are usually in the form of concentrations or partial pressures versus batch time (batch reactors), concentrations or partial pressures versus distance from reactor inlet or residence time (PFR), or rates versus residence time (CSTR). Rates can also be calculated from batch and PFR data by differentiating the concentration versus time or distance data, usually by numerical curve fitting first. It follows that a general classification of experimental methods is based on whether the data measure rates directly (differential or direct method) or indirectly (integral of indirect method). Table 7-13 shows the pros and cons of these methods. 

---