Steady-state reactor behavior

This work may be viewed as a continuation of the previous work 02). We want to describe the steady-state discontinuous behavior of the reactor, thus generalizing some of the results obtained by Viswanathan and Aris Q.). We want to derive the necessary and sufficient conditions for the development of the internal discontinuities. Then, we shall see how the geometric methods can be used in analysis and prediction of the discontinuous behavior. 

The steady state kinetic behavior of such an enzyme membrane reactor is only related to the kinetic behavior of enzymes in the gel or in the soluble state. Unless knowledge of intrinsic enzyme kinetics both in the gel and in the soluble form is available, a steady state analysis is not useful for determining whether gel formation has occurred. 

An important consideration in reactor operation is the time dependence of MSS, that is, the transients that develop during start-up or shutdown. Qualitative curves of versus time for an adiabatic CSTR are shown in Figure 13.11 for a simple first-order irreversible reaction. Note that the system converges to the upper or lower steady state (A a,ss,u or A a ssj)-Any apparent approach to the middle unsteady steady state is deceptive because it quickly turns to either of the extreme steady states. The behavior is identical with respect to temperature. 

Recycle reactors are peculiar because their geometric behavior is harder to predict than reactors with bypass—we must find a steady-state reactor effluent concentration C that when mixed with Cj the mixture concentration C is also the feed to the reactor that produces C again. (For PFR trajectories, this result is equivalent to drawing a straight line passing through Cf and C that cuts the PFR trajectory at two points.) This trait of recycle reactors is true for any dimension. 

Parulekar, S.J. and Y.T. Shah. "Steady State Thermal Behavior of an Adiabatic Three Phase Fluidized Bed Reactor-Coal Liquefaction Under Slow Hydrogen Consumption Reaction Regime." Chem. Eng. J. (1981). in press. 

Lee, C. K., Morbidelli, M., and Varma, A. Steady state multiplicity behavior of an isothermal axial dispersion fixed-bed reactor with nonuniformly active catalyst. Chem. Eng. Sci. 42, 1595-1608, 1987. 

Onozaki M, Namiki Y, Ishibashi H, Takagi T, Kobayashi M, Morooka S. Steady-state thermal behavior of coal liquefaction reactors based on NEDOL process. Energy Fuels 14 355-363, 2000b. 

The number of (ission neutrons which are delayed usually is considered as an important factor in reactor safety. However, if reactivity is added as a linear rate function at low initial reactor power, the delayed neutrons have little influence upon the rcactiiity addition ahove prompt critical. Tliey will influence the stability and steady-state operational behavior of th( reactor, though, and are neces.sary to damp the power oscillation following a reactivity addition. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Operation of a reactor in steady state or under transient conditions is governed by the mode of heat transfer, which varies with the coolant type and behavior within fuel assembHes (30). QuaHtative understanding of the different regimes using water cooling can be gained by examining heat flux, q, as a function of the difference in temperature between a heated surface and the saturation temperature of water (Eig. 1). 

Steady state models of the automobile catalytic converter have been reported in the literature 138), but only a dynamic model can do justice to the demands of an urban car. The central importance of the transient thermal behavior of the reactor was pointed out by Vardi and Biller, who made a model of the pellet bed without chemical reactions as a onedimensional continuum 139). The gas and the solid are assumed to have different temperatures, with heat transfer between the phases. The equations of heat balance are ... 

Equations 11 and 12 are only valid if the volumetric growth rate of particles is the same in both reactors a condition which would not hold true if the conversion were high or if the temperatures differ. Graphs of these size distributions are shown in Figure 3. They are all broader than the distributions one would expect in latex produced by batch reaction. The particle size distributions shown in Figure 3 are based on the assumption that steady-state particle generation can be achieved in the CSTR systems. Consequences of transients or limit-cycle behavior will be discussed later in this paper. 

Achieving steady-state operation in a continuous tank reactor system can be difficult. Particle nucleation phenomena and the decrease in termination rate caused by high viscosity within the particles (gel effect) can contribute to significant reactor instabilities. Variation in the level of inhibitors in the feed streams can also cause reactor control problems. Conversion oscillations have been observed with many different monomers. These oscillations often result from a limit cycle behavior of the particle nucleation mechanism. Such oscillations are difficult to tolerate in commercial systems. They can cause uneven heat loads and significant transients in free emulsifier concentration thus potentially causing flocculation and the formation of wall polymer. This problem may be one of the most difficult to handle in the development of commercial continuous processes. 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

There is an interior optimum. For this particular numerical example, it occurs when 40% of the reactor volume is in the initial CSTR and 60% is in the downstream PFR. The model reaction is chemically unrealistic but illustrates behavior that can arise with real reactions. An excellent process for the bulk polymerization of styrene consists of a CSTR followed by a tubular post-reactor. The model reaction also demonstrates a phenomenon known as washout which is important in continuous cell culture. If kt is too small, a steady-state reaction cannot be sustained even with initial spiking of component B. A continuous fermentation process will have a maximum flow rate beyond which the initial inoculum of cells will be washed out of the system. At lower flow rates, the cells reproduce fast enough to achieve and hold a steady state. 

Free Enzymes in Flow Reactors. Substitute t = zju into the DDEs of Example 12.5. They then apply to a steady-state PFR that is fed with freely suspended, pristine enzyme. There is an initial distance down the reactor before the quasisteady equilibrium is achieved between S in solution and S that is adsorbed on the enzyme. Under normal operating conditions, this distance will be short. Except for the loss of catalyst at the end of the reactor, the PFR will behave identically to the confined-enzyme case of Example 12.4. Unusual behavior will occur if kfis small or if the substrate is very dilute so Sj Ej . Then, the full equations in Example 12.5 should be (numerically) integrated. 

In industry, as well as in a test reactor in the laboratory, we are most often interested in the situation where a constant flow of reactants enters the reactor, leading to a constant output of products. In this case all transient behavior due to start up phenomena have died out and coverages and rates have reached a constant value. Hence, we can apply the steady state approximation, and set all differentials in Eqs. (142)-(145) equal to zero ... 

Analysis of CSTR Cascades under Nonsteady-State Conditions. In Section 8.3.1.4 the equations relevant to the analysis of the transient behavior of an individual CSTR were developed and discussed. It is relatively simple to extend the most general of these relations to the case of multiple CSTR s in series. For example, equations 8.3.15 to 8.3.21 may all be applied to any individual reactor in the cascade of stirred tank reactors, and these relations may be used to analyze the cascade in stepwise fashion. The difference in the analysis for the cascade, however, arises from the fact that more of the terms in the basic relations are likely to be time variant when applied to reactors beyond the first. For example, even though the feed to the first reactor may be time invariant during a period of nonsteady-state behavior in the cascade, the feed to the second reactor will vary with time as the first reactor strives to reach its steady-state condition. Similar considerations apply further downstream. However, since there is no effect of variations downstream on the performance of upstream CSTR s, one may start at the reactor where the disturbance is introduced and work downstream from that point. In our generalized notation, equation 8.3.20 becomes... 

The design q>roblem can be approached at various levels of sophistication using different mathematical models of the packed bed. In cases of industrial interest, it is not possible to obtain closed form analytical solutions for any but the simplest of models under isothermal operating conditions. However, numerical procedures can be employed to predict effluent compositions on the basis of the various models. In the subsections that follow, we shall consider first the fundamental equations that must be obeyed by all packed bed reactors under various energy transfer constraints, and then discuss some of the simplest models of reactor behavior. These discussions are limited to pseudo steady-state operating conditions (i.e., the catalyst activity is presumed to be essentially constant for times that are long compared to the fluid residence time in the reactor). 

In this work we attempt to measure kinetics data in a time short compared with the response time of the catalyst stoichiometry. An alternative is to measure kinetics in a true steady state, i.e., to increase the line-out time at each reactor condition until hysteresis is eliminated. The resulting apparent reaction orders and activation energies would be appropriate for an industrial mathematical model of reactor behavior. 

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. 

Some aspects of reactor behavior are developed in Chapter 5, particularly concentration-time profiles in a BR in connection with the determination of values of and k2 from experimental data. It is shown (see Figure 5.4) that the concentration of the intermediate, cB, goes through a maximum, whereas cA and cc continuously decrease and increase, respectively. We extend the treatment here to other considerations and other types of ideal reactors. For simplicity, we assume constant density and isothermal operation. The former means that the results for a BR and a PFR are equivalent. For flow reactors, we further assume steady-state operation. 

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