Reactors equations for

Multiphase Reactor Equations For reactant A in phase a this equation becomes... 

It can be readily discerned that the reactor equation for the batch reactor (5.12) and the plug-flow reactor (5.13) are identical. In the former, the concentration changes with time, in the latter, with location. In contrast to the situation in the other two ideal reactors, the residence time T in a CSTR is only an average, as every volume element has a different residence time throughout the reactor. 

The reactor equation for a fixed-bed reactor with plug flow is Eq. (5.13). 

Consequently, if we have the bateh reactor equation for X(t) and measure the RTD experimentally, we ean find the mean conversion in the exit stream. Thus, if we have the RTD and the reaction rate expression in a segregated flow situation, we have sujflcient information to calculate the conversion. 

Conversion if fluid is completely segregated. Tlhe batch reactor equation for a second-order reaction of this type is... 

The H-Oil reactor (Fig. 21) is rather unique and is called an ebullated bed catalytic reactor. A recycle pump, located either internally or externally, circulates the reactor fluids down through a central downcomer and then upward through a distributor plate and into the ebullated catalyst bed. The reactor is usually well insulated and operated adiabatically. Frequently, the reactor-mixing pattern is defined as backmixed, but this is not strictly true. A better description of the flow pattern is dispersed plug flow with recycle. Thus, the reactor equations for the axial dispersion model are modified appropriately to account for recycle conditions. 

Now, our quest for knowledge concerning gas-liquid reactors, if we look at it, began with equation (8-164) so we should feel nearly saturated at this point. In fact, though, there are many other cases considered in the work of Russell et ah, that may be of use in certain applications. We have taken what might be considered the most important, or most frequently encountered for presentation. As in the case for fluid-bed or slurry reactors, we must now determine where the many parameters appearing in the reactor equations for gas-liquid systems originated. But first, an example. 

The most important reactor equations for simple kinetics are summarized in Table... 

The use of this equation in developing batch reactor equations for a typical complex reaction is illustrated in Example 11.1. 

The next step is to combine these equations with the reactor equations for fixed- and fluidized-bed reactors developed in Chapter 12. 

Equation E21.1.5 for the current density for the reaction is the electrochemical equivalent of the rate equation for a conventional reaction. We now solve this equation simultaneously with the reactor equation for the selected reactor, for example. Equation 21.46 for a conventional BR with circulation operated as a PFR, or Equation 21.51 for a BR with circulation operated as a CSTR. Thus the electrode area A. can be estimated as a function of conversion Af, [/4]f, current density a, and reaction time t, t- All electrochemical parameters of the equations can be experimentally determined from polarization studies, The reactor efficiency can then be obtained from ( a and i. ... 

Equation (1) is the general kerjiel form of the reactor equation for the angular flux under the restriction of no extraneous sources, constant ratio of fissionable species, and isotropic emission of fission neutrons. 

Equation (11) has the form of a characteristic value problem whereas (10) does not— i.e., the kernel H characterizes the medium, so to speak, in all of its non-multiplying aspects it is the multiplying or chain character of the reaction in the medium which introduces the homogeneous linear term on the right-hand side of (11), and thus makes the basic reactor equation for all reactors a characteristic value problem. Much of the discussion to be given over the next two days will be based on Equation (11) or a variant thereof. 

This formulation of the reactor equation for thermal, heterogeneous reactors was originally due to Edward Teller [2] however, its full exploitation is due to S. M. Feinberg [3]. 

For all other components than Hi and HiS the Henry laws constant Hi is assumed to be zero. The constants for these two components have been calculated as a function of temperature from HYSYS (a commercial process simulation package) simulations. The simulated oil had the same boiling point distribution as the actual gas oil. From these simulations we also got the oil vapor pressure. The reactor equation for all compounds is then ... 

Suppose that the reactor equations for the critical assembly can be written in the general form... 

Batch Technique. As with river reaeration measurements, tracers can also be put into lakes, estuaries, and oceans to measure the influence of wind on liquid film coefficient. If a volatile tracer has been placed in a lake with a well established mixed layer, for example, the batch reactor equation for a well-mixed tank can be applied ... 

The simplest model of time-dependent behavior of a neutron population in a reactor consists of the point kinetics differential equations, where the space-dependence of neutrons is disregarded. The safety of reactors is greatly enhanced inherently by the existence of delayed neutrons, which come from radioactive decay rather than fission. The differential equations for the neutron population, n, and delayed neutron emitters, are... 

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Example Consider the equation for convection, diffusion, and reaction in a tiihiilar reactor. 

This is the equation for a plug flow reactor. It can be derived directly from the rate equations with the aid of Laplace transforms. The sequences of second-order reactions of Figs. 7-5n and 7-5c required numerical integrations. 

Tubular flow reaclors operate at nearly constant pressure. How the differential material balance is integrated for a number of second-order reactions will be explained. When n is the molal flow rate of reactant A the flow reactor equation is... 

This is to be solved simultaneously with the flow reactor equation, Eq. (7-84). Alternatively, dV can be eliminated from Eq. (7-88) for a direct relation between P and n . 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

Find the conditions for minimum V fV when conversion is 80%. The flow reactor equation is... 

The above equations for heat transfer apply when there is no heat generation or absorption during the reaction, and the temperature difference between the solid and the gas phase can be simply defined tliroughout the reaction by a single value. Normally this is not the case, and due to the heat of the reaction(s) which occur tlrere will be a change in the average temperature with time. Furthermore, in tire case where a chemical reaction, such as the reduction of an oxide, occurs during the ascent of tire gas in the reactor, the heat transfer coefficient of the gas will vary with tire composition of tire gas phase. 

This model v/as used by Atwood et al (1989) to compare the performance of 12 m and 1.2 m long tubular reactors using the UCKRON test problem. Although it was obvious that axial conduction of matter and heat can be expected in the short tube and not in the long tube, the second derivative conduction terms were included in the model so that no difference can be blamed on differences in the models. The continuity equations for the compounds was presented as ... 

From diese various estimates, die total batch cycle time t(, is used in batch reactor design to determine die productivity of die reactor. Batch reactors are used in operations dial are small and when multiproducts are required. Pilot plant trials for sales samples in a new market development are carried out in batch reactors. Use of batch reactors can be seen in pharmaceutical, fine chemicals, biochemical, and dye industries. This is because multi-product, changeable demand often requues a single unit to be used in various production campaigns. However, batch reactors are seldom employed on an industrial scale for gas phase reactions. This is due to die limited quantity produced, aldiough batch reactors can be readily employed for kinetic studies of gas phase reactions. Figure 5-4 illustrates die performance equations for batch reactors. 

Assuming that the reactions are first order in a constant volume batch reactor, the rate equations for components A, B, C, and D, respectively, are ... 

Equation 5-247 is a polynomial, and the roots (C ) are determined using a numerical method such as the Newton-Raphson as illustrated in Appendix D. For second order kinetics, the positive sign (-r) of the quadratic Equation 5-245 is chosen. Otherwise, the other root would give a negative concentration, which is physically impossible. This would also be the case for the nth order kinetics in an isothermal reactor. Therefore, for the nth order reaction in an isothermal CFSTR, there is only one physically significant root (0 < C < C g) for a given residence time f. 

From the mass balance equation for a batch reactor... 

Omoleye, J. A., Adesina, A. A., and Udegbunam, E. O., Optimal design of nonisothermal reactors Derivation of equations for the rate-temperature conversion profile and the optimum temperature progression for a general class of reversible reactions, Chem. Eng. Comm., Vol. 79, pp. 95-107, 1989. 

It is illustrated in Section 3.4.4 by tracing the paths for leaking engine compression and applied to fault tree construction for the FFTF reactor Fullwood and Erdmann, 1974). The method involves writing Boolean equations for all paths whereby hazardous material may be released. It is primarily useful for enumerating release paths, but not for what started the release It was used to enumerate the possible paths for stealing nuclear bomb material from a facility. 

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