Integral and Differential Reactors

Under the assumption of pseudohomogeneity (cf. Table 4.3) and under constant temperature conditions, the kinetic analysis of a process is reduced to the search for functions / (c) and/or /2(c), as specified under types 1 and 2 in Fig. 4.12. The distinctions between reactors with or without a concentration profile is also a decisive factor that is, the distinction between so-called integral and differential reactors is a necessary one. 

As an aid in the precise definition of integral and differential reactors, Fig. 4.13 shows a conversion versus time diagram, and the transfer of these data to a continuous tubular reactor (Moser and Lafferty, 1976). 

A differential reactor works with small concentration differences, for exam- 

Precise kinetic measurements are, however, better (made) using a differentially operated reactor. A small measured change in conversion may be directly related to a reaction rate 

The analytical problems with differential reactors can be overcome by using so-called gradient-free reactors for example, loop reactors (Fig. 4.15). In this 

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Figure 7.14 Integral and differential reactors for experimental rate determination.

Integral and Differential Reactor Data Evaluation Methods... 

The connection between integral and differential reactors and that between integral and differential methods of data evaluations are shown in Fig. 4.16 (after Froment, 1975). Data from integral reactors can be evaluated in the same way as data from differential reactors if the data are first numerically differentiated, or differentiated analytically or, more often, graphically. In cases where integral data are to be evaluated differentially, the following steps should be followed ... 

Steady-state reactors with ideal flow pattern. In an ideal isothermal tubular pZi/g-yZovv reactor (PFR) there is no axial mixing and there are no radial concentration or velocity gradients (see also Section 5.4.3). The tubular PFR can be operated as an integral reactor or as a differential reactor. The terms integral and differential concern the observed conversions and yields. The differential mode of reactor operation can be achieved by using a shallow bed of catalyst particles. The mass-balance equation (see Table 5.4-3) can then be replaced with finite differences ... 

The results Illustrated by Figures 3 and 4 resemble those obtained in the Berty recycle reactor under similar conditions. The space-mean, time average rates for the fixed-bed reactor were only about 50% of those measured in the Berty reactor, because, of course the former reactor achieved conversions high enough for the back reaction to become important. The significance of these observations is that 1) CSTR and differential reactors, widely used for laboratory studies, seem to reflect performance improvements obtainable with fixed-bed, integral reactor which resemble commercial units, and 2) improvement from periodic operation are still observed even tfien reverse reactions become important. 

In this chapter we are concerned only with the rate equation for the i hemical step (no physical resistances). Also, it will be supposed that /"the temperature is constant, both during the course of the reaction and in all parts of the reactor volume. These ideal conditions are often met in the stirred-tank reactor (see-Se c." l-6). Data are invariably obtained with this objective, because it is extremely hazardous to try to establish a rate equation from nonisothermal data or data obtained in inadequately mixed systems. Under these restrictions the integration and differential methods can be used with Eqs. l-X and (2-5) or, if the density is constant, with Eq. (2-6). Even with these restrictions, evaluating a rate equation from data may be an involved problem. Reactions may be simple or complex, or reversible or irreversible, or the density may change even at constant temperatur (for example, if there is a change in number of moles in a gaseous reaction). These several types of reactions are analyzed in Secs. 2-7 to 2-11 under the categories of simple and complex systems. 

Derivation of the integral and differential energy balance equations for flow reactors and reducing these equations to dimensionless forms... 

There are two fundamental types of experimental reactors for measuring solid-catalyzed reaction rates, integral and differential. The integral reactor consists essentially of a tube of diameter less than 3 cm filled with, say, IF g of catalyst. Each run comprises steady-state operation at a given feed rate, and based on several such runs, a plot of the conversion X/ versus IF/F o is prepared. Differentiation of this curve gives the rate at any given (i.e., concentration) as... 

Since kinetic data from a series of DCSTR experiments not only serve as the basis for process development in the DCSTR but also serve to a first approximation for the design of continuous reactors, integral and differential data evaluation methods for the case of a DCSTR will be considered in more detail. 

Another important issue is that assessment of long-life cores requires accurate nuclear data and reactor system analysis codes. Here, some integral and differential experiments may be needed to provide a basis for the validation. 

More common for kinetic studies of heterogeneously catalyzed gas reactions are tubular reactors loaded with catalyst (fixed bed reactor). The tubular reactor displays a simple design and is easy to operate. A simultaneous integral and differential mode of operation can be achieved in a reactor with taps for measuring concentration and temperatures at defined axial positions (Figure 4.11.9). By using a tab reactor, the density of information obtainable during experiments with fixed bed reactors is improved. 

Tank flow reactors are occasionally used to obtain information on reaction velocity constant(s) and order(s) of reaction. These reactors are operated under conditions approaching perfect mixing and steady-state. The same two basic methods of analysis are available the integration and differentiation methods, although there is no need to integrate or differentiate the describing equations. 

The objective of cross-section adjustment is to reduce the fi ctional discrepancy between the measured value, Ek, and calculated value, Ck, of integral property k taking into account the uncertainties in the cross-section data and the integral measurements. For a review of recent work in the subject see [4.41]. The assumption is made that the difference between C k, the value calculated using the adjusted cross-sections, and Ck can be approximated as linearly dependent on the fractional adjustments to the nuclear data parameters, Xi. The values of Xi are found by a least squares fit to the integral and differential cross-section measurements, relative to the uncertainties. The procedure produces the covariance matrix of the adjusted cross-sections from which the fractional accuracy of reactor... 

The plant control system is designed based on the proportional, integral, and differential (PID) control principle (see Sect. 4.4). The reactor behavior has been analyzed against various perturbations with the designed and optimized plant control system. 

Each experimental run gives the reaction rate at the composition of the exit fluid. Tubular reactors can be operated as differential reactors (i.e., at high throughputs and low conversions) or as integral reactors (i.e., at low throughputs and high conversions). Differential reactors give the rate as ... 

Kinetic analysis of the data obtained in differential reactors is straightforward. One may assume that rates arc directly measured for average concentrations between the inlet and the outlet composition. Kinetic analysis of the data produced in integral reactors is more difficult, as balance equations can rarely be solved analytically. The kinetic analysis requires numerical integration of balance equations in combination with non-linear regression techniques and thus it requires the use of computers. 

Differential equations Batch reactor with first-order kinetics. Analytical or numerical solution with analytical or numerical parameter optimisation (least squares or likelihood). Batch reactor with complex kinetics. Numerical integration and parameter optimisation (least squares or likelihood). 

In principle, if the temperatures, velocities, flow patterns, and local rates of mixing of every element of fluid in a reactor were known, and if the differential material and energy balances could be integrated over the reactor volume, one could obtain an exact solution for the composition of the effluent stream and thus the degree of conversion that takes place in the reactor. However, most of this information is lacking for the reactors used in laboratory or commercial practice. Consequently, it has been necessary to develop approximate methods for treating... 

In the case of a PFR, it is usually easier to vary Tin a controllable and measurable way if it is operated as a differential reactor rather than as an integral reactor. In the latter case, it may be difficult to eliminate an axial gradient in Tover the entire length of the reactor. 

Ideal reactors can be classified in various ways, but for our purposes the most convenient method uses the mathematical description of the reactor, as listed in Table 14.1. Each of the reactor types in Table 14.1 can be expressed in terms of integral equations, differential equations, or difference equations. Not all real reactors can fit neatly into the classification in Table 14.1, however. The accuracy and precision of the mathematical description rest not only on the character of the mixing and the heat and mass transfer coefficients in the reactor, but also on the validity and analysis of the experimental data used to model the chemical reactions involved. 

Equations (1) and (2) represent reaction rates and, as such, can represent directly only data from a differential reactor. In many cases, however, data are obtained from an integral reactor. Are the data to be differentiated and compared directly to Eqs. (1) or (2), or are the equations to be integrated with the conservation equations and compared to the integral data ... 

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