Rate laws batch reactors

A common laboratory device is a batch reactor, a nonflow type of reactor. As such, it is a closed vessel, and may be rigid (i.e., of constant volume) as well. Sample-taking or continuous monitoring may be used an alternative to the former is to divide the reacting system into several portions (aliquots), and then to analyze the aliquots at different times. Regardless of which of these sampling methods is used, the rate is determined indirectly from the property measured as a function of time. In Chapter 3, various ways of converting these direct measurements of a property into measures of rate are discussed in connection with the development of the rate law. 

The primary use of chemical kinetics in CRE is the development of a rate law (for a simple system), or a set of rate laws (for a kinetics scheme in a complex system). This requires experimental measurement of rate of reaction and its dependence on concentration, temperature, etc. In this chapter, we focus on experimental methods themselves, including various strategies for obtaining appropriate data by means of both batch and flow reactors, and on methods to determine values of rate parameters. (For the most part, we defer to Chapter 4 the use of experimental data to obtain values of parameters in particular forms of rate laws.) We restrict attention to single-phase, simple systems, and the dependence of rate on concentration and temperature. It is useful at this stage, however, to consider some features of a rate law and introduce some terminology to illustrate the experimental methods. 

The experimental investigation of the form of the rate law, including determination of the rate constants kf and kr, can be done using various types of reactors and methods, as discussed in Chapters 3 and 4 for a simple system. Use of a batch reactor is illustrated here and in Example 5-4, and use of a CSTR in problem 5-2. 

Suppose reaction 8.3-1 with rate law given by equation 8.3-2 is carried out in a constant-volume batch reactor (or a constant-density PFR) at constant 7... 

The single particle acts as a batch reactor in which conditions change with respect to time, This unsteady-state behavior for a reacting particle differs from the steady-state behavior of a catalyst particle in heterogeneous catalysis (Chapter 8). The treatment of it leads to the development of an integrated rate law in which, say, the fraction of B converted, /B, is a function oft, or the inverse. 

For a constant-volume batch reactor operated at constant T and pH, an exact solution can be obtained numerically (but not analytically) from the two-step mechanism in Section 10.2.1 for the concentrations of the four species S, E, ES, and P as functions of time t, without the assumptions of fast and slow steps. An approximate analytical solution, in the form of a rate law, can be obtained, applicable to this and other reactor types, by use of the stationary-state hypothesis (SSH). We consider these in turn. 

Each group of molecules reacts independently of any other group, that is, as a batch reactor. Batch conversion equations for power law rate equations are... 

Based on experimental results and a model describing the kinetics of the system, it has been found that the temperature has the strongest influence on the performance of the system as it affects both the kinetics of esterification and of pervaporation. The rate of reaction increases with temperature according to Arrhenius law, whereas an increased temperature accelerates the pervaporation process also. Consequently, the water content decreases much faster at a higher temperature. The second important parameter is the initial molar ratio of the reactants involved. It has to be noted, however, that a deviation in the initial molar ratio from the stoichiometric value requires a rather expensive separation step to recover the unreacted component afterwards. The third factor is the ratio of membrane area to reaction volume, at least in the case of a batch reactor. For continuous opera-... 

A catalytic hydrogenation is performed at constant pressure in a semi-batch reactor. The reaction temperature is 80 °C. Under these conditions, the reaction rate is lOmmolT s-1 and the reaction may be considered to follow a zero-order rate law. The enthalpy of the reaction is 540 kj moT1. The charge volume is 5 m3 and the heat exchange area of the reactor 10 m2. The specific heat capacity of water is 4.2kJkg 1K 1. 

The gas-phase decomposition of sulfuryl chloride, S02C12 — S02 + Cl2, is thought to follow a first order rate law. The reaction is performed in a constant volume, isothermal batch reactor, and the concentration of S02C12 is measured at several reaction times, with the following results. 

Pseudo- and overall reaction orders. Kinetic textbooks describe other, more complicated methods applicable to other forms of proposed rate equations, mostly for evaluation of results from batch reactors. However, if the development chemist or engineer can commission experiments—as opposed to having to evaluate existing data—he can often save himself much effort by determination of pseudo- and overall reaction orders. For example, for a reaction A + B — product(s) and power-law rate equation —rk = kmCfCB, three series of experiments suggest themselves ... 

A series of batch and mixed flow reactor experiments was performed at pH <3 to determine the effect of SO , Cl , ionic strength, and dissolved O2 on the rate of pyrite oxidation by Fe ". Of these, only dissolved O2 had any appreciable affect on the rate of pyrite oxidation in the presence of Fe. Williamson and Rimstidt (1994) combined their experimental results with kinetic data reported from the literature to formulate rate laws that are applicable over a range panning six order of magnitude in Fe " " and Fe " " concentrations, and for a pH range of 0.5-3.0, when fixed concentrations of dissolved O2 are present ... 

We have shown that in order to calculate the time necessary to achieve a given conversion X in a batch system, or to calculate the reactor volume needed to achieve a conversion X in a flow system, we need to know the reaction rate as a function of conversion. In tins chapter we show how this functional dependence is obtained. First there is a brief discussion of chemical kinetics, emphasizing definitions, which illustrates how the reaction rate depends on the concentrations of the reacting species. This discussion is followed by instructions on how to convert the reaction rate law from the concentration dependence to a dependence on conversion. Once this dependence is achieved, we can design a number of isothermal reaction systems. 

Batch reactors are used primarily to determine rate law parameters for homo, geneous reactions. This determination Ls usually achieved by measuring coa centration as a function of time and then using either the differential, integral, or least squares method of data analysis to determine the reaction order, a, and specific reaction rate, k. If some reaction parameter other than concentration i s monitored, such as pressure, the mole bMance must be rewritten in terms of the measured variable (e.g., pressure). 

Recalling Example 5-1, the combined mole balance and rate law for a constant-volume batch reactor can be expressed in the form... 

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... 

The corresponding combined batch reactor mole balances and rate laws are For the initiator ... 

Liquid Phase. For liquid-phase reactions in which there is no volume change, concentration is the preferred variable. The mole balances are shown in Table 4-5 in terms of concentration for the four reactor types we have been discussing. We see from Table 4-5 that we have only to specify the parameter values for the system (CAo,Uo,etc.) and for the rate law (i.e., ifcyv. .3) to solve the coupled ordiaaiy differential equations for either PFR, PBR, or batch reactors or to solve the coupled algebraic equations for a CSTR. 

A sinister gentlemen is interested in producing methyl perchlorate in a batch reactor. The reactor has a strange and unsettling rate law. [2nd Ed. P4-2S]... 

To outline the procedure used in the differential method of analysis, we consider a reaction carded out isothermally in a constant-volume batch reactor and the concentration recorded as a function of time. By combining the mole balance with the rate law given by Equation (5-1), we obtain... 

A combination of the mole balance on a constant-volume batch reactor tad the rate law gives... 

For a constant-volume batch reactor the combined mole balances and rate laws for disappearance of ethane (PI) and the formation of ethylene (PS) are... 

E Related material This problem uses the mole balances developed in Chapter 1 for a batch reactor and the stoichiometry and rate laws... 

Numeroxis reactions are performed by feeding the reactants continuously to cylindrical tubes, either empty or packed with catalyst, with a length which is 10 to 1000 times larger than the diameter. The mixture of unconverted reactants and reaction products is continuously withdrawn at the reactor exit. Hence, constant concentration profiles of reactants and products as well as a temperature profile are established between the inlet and the outlet of the tubular reactor (see Fig. 8.10). This requires, in contrast to the batch reactor, the application of the law of conservation of mass over an infinitesimal volume element, d V, or mass element, dW, of the reactor. For a tubular reactor with a fixed catalytic bed, it is more convenient to relate the production rates to the catalyst mass, rather than to the reactor volume. 

The describing equation for chemical reaction mass transfer is obtained by applying the conservation law for either mass or moles on a time rate basis to the contents of a batch reactor. It is best to work with moles rather than mass since the rate of reaction is most conveniently described in terms of molar concentrations. The describing equation for species A in a batch reactor takes the form... 

The equation describing temperature variations in batch reactors is obtained by applying the conservation law for energy on a time-rate basis to the reactor contents. Since batch reactors are stationary (fixed in space), kinetic and potential effects can be neglected. The equation describing the temperature variation in reactors due to energy transfer, subject to the assumptions in its development, is... 

The rate equation (i.e.. rate law) fur is an algebraic equation that is solely a function of the properties of the reacting materials and reaction conditions (e.g., species cuncentralfon. temperature, pre.ssure, or type of catalyst, if any) at a point in the system. The rale equation is independent of the type of reactor (e.g., batch or continuous flow) in which the reaction is carried out. Howeier, because the properties and reaction conditions of the reacting materials may vary with position in a chemical reactor, can in turn he a function of position and can vary from point to point in the system. 

Overview. In Chapter 2, we showed that if we had the rate of reaction as a function of conversion, = /(X), we could calculate reactor volumes necessary to achieve a specified conversion for flow systems and the time to achieve a given conversion in a batch system. Unfortunately, one is seldom, if ever, given = yiX) directly from raw data. Not to fear, in this chapter we will show how to obtain the rate of reaction as a function of conversion. This relationship between reaction rate and conversion will be obtained in two steps. In Step 1, Part 1 of this chapter, we define the rate law, which relates the rate of reaction to the concentrations of the reacting species and to temperature. In Step 2, Part 2 of this chapter, we define concentrations for fiow and batch systems and develop a stoichiometric table so that one can write concentrations as a function of conversion. Combining Steps 1 and 2, we see that one can then write the rate as a function conversion and use the techniques in Chapter 2 to design reaction systems. 

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