Reactors integral, differential, mixed

Differential (flow) reactor Integral (plug flow) reactor Mixed flow reactor Batch reactor for both gas and solid... 

The catalyst may be held in a packed bed and the reactants passed over the catalyst. A packed bed flow reactor is commonly called a fixed bed reactor and the term plug-flow is also used to indicate that no attempt is made to back-mix the reaction mixture as it passes through the catalyst bed. The main modes of operation of a flow reactor are differential involving a small amount of reaction so that the composition of the mixture is approximately constant throughout the catalyst bed, or integral involving a more substantial amount of reaction such that the composition of material in contact with the final section of the catalyst bed is different from that entering the bed. 

If the compositions vary with position in the reactor, which is the case with a tubular reactor, a differential element of volume SV, must be used, and the equation integrated at a later stage. Otherwise, if the compositions are uniform, e.g. a well-mixed batch reactor or a continuous stirred-tank reactor, then the size of the volume element is immaterial it may conveniently be unit volume (1 m3) or it may be the whole reactor. Similarly, if the compositions are changing with time as in a batch reactor, the material balance must be made over a differential element of time. Otherwise for a tubular or a continuous stirred-tank reactor operating in a steady state, where compositions do not vary with time, the time interval used is immaterial and may conveniently be unit time (1 s). Bearing in mind these considerations the general material balance may be written ... 

Gas-solid (catalytic) Differential reactor Integral reactor Mixed (Carberry) reactor Mic roreactor Fluid-bed reactor Single-pellet reactor (Chapter 7)... 

This is a very complicated expression that contains both a derivative and an integral. The integral term is included to account for the possibility that the reactor is not well mixed, and the rate of reaction will then be an average based on the concentration everywhere within the reactor. The differential term accounts for the reahty that certain components may be accumulated within the reactor, depending on the design of the system. 

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 ... 

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... 

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. 

Since the differential and mixed flow reactors give the rate directly they are more useful in analyzing complex reacting systems. The test for anything but a simple kinetic form can become awkward and impractical with the integral reactor. 

Various laboratory reactors have been described in the literature [3, 11-13]. The most simple one is the packed bed tubular reactor where an amount of catalyst is held between plugs of quartz wool or wire mesh screens which the reactants pass through, preferably in plug flow . For low conversions this reactor is operated in the differential mode, for high conversions over the catalyst bed in the integral mode. By recirculation of the reactor exit flow one can approach a well mixed reactor system, the continuous flow stirred tank reactor (CSTR). This can be done either externally or internally [11, 12]. Without inlet and outlet feed, this reactor becomes a batch reactor, where the composition changes as a function of time (transient operation), in contrast with the steady state operation of the continuous flow reactors. 

It should be noted that if a differential mass, energy and electron balance approach were followed to describe the reactor s behavior one would obtain a set of non-linear, partial, integro-differential equations. However, since the reactor is simulated by a two-dimensional array of mixing cells, a large number of algebraic equations result in which there are no differentials and in which double integrals are replaced by double summations. 

To derive the overall kinetics of a gas/liquid-phase reaction it is required to consider a volume element at the gas/liquid interface and to set up mass balances including the mass transport processes and the catalytic reaction. These balances are either differential in time (batch reactor) or in location (continuous operation). By making suitable assumptions on the hydrodynamics and, hence, the interfacial mass transfer rates, in both phases the concentration of the reactants and products can be calculated by integration of the respective differential equations either as a function of reaction time (batch reactor) or of location (continuously operated reactor). In continuous operation, certain simplifications in setting up the balances are possible if one or all of the phases are well mixed, as in continuously stirred tank reactor, hereby the mathematical treatment is significantly simplified. 

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. 

A mixed-flow reactor (MFR), also known as the continuous stirred tank reactor (CSTR), is fuUy mixed at the molecular level, and the composition of the exiting stream is identical to that within the reactor. In this case, the material balance of Equation (11.12) is applied for the entire reactor and not just for a differential element as in a PFR. No integration is needed the eqnation becomes... 

However the region at the left of the point P is not convex. But we may continue from the point P with a PFR, as indicated in Fig. 8.27B. The computation is done by simply Integrating the differential equations of a PFR, this time input concentrations supplied by the exit of the CSTR. The new augmented region is convex. This time all three conditions are fulfilled. No other mixed reactors can be found above the boundary that could give a higher amount of B. This is the final Attainable Region. 

Coupled mass and thermal energy balances are required to analyze the nonisother-mal response of a well-mixed continuous-stirred tank reactor. These balances can be obtained by employing a macroscopic control volume that includes the entire contents of the CSTR, or by integrating plug-flow balances for a differential reactor under the assumption that temperature and concentrations are not a function of spatial coordinates in the macroscopic CSTR. The macroscopic approach is used for the mass balance, and the differential approach is employed for the thermal energy balance. At high-mass-transfer Peclet numbers, the steady-state macroscopic mass balance on reactant A with axial convection and one chemical reaction, and units of moles per time, is... 

For such chemical reactors which are characterized by an ideally well mixed reaction volume, therefore allowing the assumption of a uniform temperature distribution, a transfer of this differential heat balance to an integral balance is possible by multiplying it with the volume of the system balanced. This way the general integral heat balance for well mixed reactors is obtained ... 

Under such simplified description of the streamer development, we could model the subsequent evolution of the gas phase in a standard way, using the continuity equations for each chemical species and solving a system of mono-dimensional first-order differential equations easily and quickly tackled by numerical integration (Riccardi, 2000). From a chemical engineering point of view, indeed it means that the model can be formulated as a well-mixed reactor (Benson, 1982). The gas-phase composition in the reactor is determined by the chemical reactions among the reactive species and the transport processes. The time evolution of the concentration of the different N sp>ecies in the gas phase is determined by integrating each balance equation for the density nk of the Id spiecies ... 

A convenient way of operating a differential reactor at integral conversions is to use a fully mixed reactor in which a constant concentration... 

---