Rate constants from batch reactor data

Example 7.5 Suppose the consecutive reactions 2A B C are elementary. Determine the rate constants from the following experimental data obtained with an isothermal, constant-volume batch reactor ... 

The rates of liquid-phase reactions can generally be obtained by measuring the time-dependent concentrations of reactants and/or products in a constant-volume batch reactor. From experimental data, the reaction kinetics can be analyzed either by the integration method or by the differential method ... 

We can also use nonlinear regression to determine the rate law parameters from concentr on-time data obtained in batch experiments. We recall that the combined rate law-stoichiometty-mole balance for a constant-volume batch reactor is... 

To determine reaction rate parameters from the experimental data, the following differential equation was used to describe the reaction system in a constant-volume batch reactor assuming a pseudo-first-order equation for propylene epoxidation ... 

The gas-phase production of methanol from carbon monoxide and hydrogen is carried out in a small constant-volume batch reactor under isothermal conditions and the pilot-plant operator measures the total pressure within the reactor vs. time for subsequent reaction-rate data analysis. A stoichiometric feed of carbon monoxide and hydrogen is introduced to the reactor at time t = 0, and the total pressure is 3 atm. Sketch (he raw data as total pressme vs. time. Be sure to indicate the appropriate equation that describes the shape of the curve. 

EP.2 The reaction occurs in a batch reactor and determines the specific rate constant from the following experimental data. 

Example. Fitting the rate constant from multiresponse dynamic batch reactor data... 

Use of the routines that perform these calculations is demonstrated below for the example of fitting the rate constant k for the reaction A - - B C from the combined batch reactor data of Table 8.1 and Table 8.3. 

The data given below are typical of the polymerization of vinyl phenylbutyrate in dioxane solution in a batch reactor using benzoyl peroxide as an initiator. The reaction was carried out isothermally at 60 °C using an initial monomer concentration of 73 kg/m3. From the following data determine the order of the reaction and the reaction rate constant. Note that there is an induction period at the start of the reaction so that you may find it useful to use a lower limit other than zero in your integration over time. The reaction order may be assumed to be an integer. 

If the rate equation is to be employed in its integrated form, the problem of determining kinetic constants from experimental data from batch or tubular reactors is in many ways equivalent to taking the design equations and working backwards. Thus, for a batch reactor with constant volume of reaction mixture at constant temperature, the equations listed in Table 1.1 apply. For example, if a reaction is suspected of being second order overall, the experimental results are plotted in the form ... 

The classical procedures used by the chemist or engineer to obtain polymerization rate data have usually involved dilatometry, sealed ampoules, or samples withdrawn from model reactors—batch, tubular, and CSTR s alone or in various combinations. These rate data, together with data on molecular weight can be used to obtain the chain initiation constant and certain ratios such as kp2/kt and ktr/kp. Some basic relationships are shown in Figure 5. To determine individual rate constants such as kp and kt, other techniques are needed. For example, by periodic photochemical initiation it is possible to obtain kp/kt. If the ratio kp2/kt (discussed above) is also known, kp and kt can each be calculated. Typical techniques are described by Flory (20). 

Propose a generalized rate expression for testing the data. Analysis of rate data by the differential method involves utilizing the entire reaction-rate expression to find reaction order and the rate constant. Since the data have been obtained from a batch reactor, a general rate expression of the following form may be used ... 

A batch reactor by its nature is a transient closed system. While a laboratory batch reactor can be a simple well-stirred flask in a constant temperature bath or a commercial laboratory-scale batch reactor, the direct measurement of reaction rates is not possible from these reactors. The observables are the concentrations of species from which the rate can be inferred. For example, in a typical batch experiment, the concentrations of reactants and products are measured as a function of time. From these data, initial reaction rates (rates at the zero conversion limit) can be obtained by calculating the initial slope (Figure 3.5.1b). Also, the complete data set can be numerically fit to a curve and the tangent to the curve calculated for any time (Figure 3.5. la). The set of tangents can then be plotted versus the concentration at which the tangent was obtained (Figure 3.5.1c). 

This chapter focuses attention on reactors that are operated isotherraally. We begin by studying a liquid-phase batch reactor to determine the specific reaction rate constant needed for the design of a CSTR. After iilustrating the design of a CSTR from batch reaction rate data, we carry out the design of a tubular reactor for a gas-phase pyrolysis reaction. This is followed by a discussion of pressure drop in packed-bed reactors, equilibrium conversion, and finally, the principles of unsteady operation and semibatch reactors. 

Figure 11.10. Rate constants for reactions of ozone and OH radicals with solutes, (a, b) Examples of rate constants for direct reactions of ozone with organic and inorganic solutes versus pH (data selected), is related to k by the yield factor, (c) Rate constants for reactions of OH radicals with different solutes. (A03)37 is the required amount of decomposed ozone, which results in the elimination of the quoted substrate to 37% of the initial value (batch-type or plug-flow reactor). This scale is calibrated for eutrophic lakewater (Lac de Bret, DOC = 4 mg liter"[HCOf] = 1.6 mM, pH = 8.3. The latter changes proportionally to the DOC of water). (From Hoignd, 1988.)...

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. 

The information required here is not concentration versus time, but rate of reaction versus concentration. As will be seen later, some types of chemical reactors give this information directly, but the constant-volume, batch systems discussed here do not [ What does it profit you, anyway —F. Villon], In this case it is necessary to determine rates from conversion-time data by graphical or numerical methods, as indicated for the case of initial rates in Figure 1.25. In Figure 1.27 a curve is shown representing the concentration of a reactant A as a function of time, and we identify the two points Cai and Ca2 for the concentration at times q and t2- The mean value for the rate of reaction we can approximate algebraically by... 

Table 2 summarizes the mass transfer coefficients determined for this model for batch tubes, stirred batch, and flowthrough reactor coi gurations and shows that the diffusive mass transfer coefficient kd increases in this order of reactor type. However, one would expect the mass transfer coefficient to follow such a pattern as flow is increased. On the other hand, although the rate constants for conventional models based on only chemical reaction can also be fit to data from these three reactor types, rate constants for these models should only depend on temperature, and these variations would not be expect. Thus, coupling mass transfer to reaction appears to provide a more meaningful explanation for the effects of flow on performance, but further work is needed to Mly develop and evaluate this approach. 

The operational conditions, that is, the concentration of substrate and enzyme, the temperature range, and the reactor configuration are summarized in Table 13.2. The activation energy of the reaction, E, was typically obtained for a batch reactor and compared with that calculated for a CSMR. The data obtained in the CSMR at steadystate enabled us, by using a semi-log plot of reaction rate versus time, to identify a first-order mechanism of enzyme deactivation and to determine both its first-order deactivation constant, kj, and the reaction rate at time zero, r, for each substrate and temperature. It was thus possible to compare the effect of the operational parameters on the activity and stability of these two enzymes. From the Arrhenius plot of these Tq, the E,-values were determined for each substrate, and were found to match the values obtained in the batch reactors. 

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