Reactor multiple solutions

In Fig. 28, the abscissa kt is the product of the reaction rate constant and the reactor residence time, which is proportional to the reciprocal of the space velocity. The parameter k co is the product of the CO inhibition parameter and inlet concentration. Since k is approximately 5 at 600°F these three curves represent c = 1, 2, and 4%. The conversion for a first-order kinetics is independent of the inlet concentration, but the conversion for the kinetics of Eq. (48) is highly dependent on inlet concentration. As the space velocity increases, kt decreases in a reciprocal manner and the conversion for a first-order reaction gradually declines. For the kinetics of Eq. (48), the conversion is 100% at low space velocities, and does not vary as the space velocity is increased until a threshold is reached with precipitous conversion decline. The conversion for the same kinetics in a stirred tank reactor is shown in Fig. 29. For the kinetics of Eq. (48), multiple solutions may be encountered when the inlet concentration is sufficiently high. Given two reactors of the same volume, and given the same kinetics and inlet concentrations, the conversions are compared in Fig. 30. The piston flow reactor has an advantage over the stirred tank... 

Multiple solutions to equations occur whenever they have sufficient nonlinearity. A familiar example is equHihrium composition calculations for other than A B. The reaction composition in the reaction A i B yields a cubic polynomial that has three roots, although all but one give nonphysical concentrations because thermodynamic equilihrium (the solution for a reactor with f —> co or T — co) is unique. 

McGreavy and THORNTON(23) have developed an alternative approach to the problem of identifying such regions of unique and multiple solutions in packed bed reactors. Recognising that the resistance to heat transfer is probably due to a thin gas film surrounding the particle, but that the resistance to mass transfer is within the porous solid, they solved the mass and heat balance equations for a pellet with modified boundary conditions. Thus the heat balance for the pellet represented by equation 3.24 was replaced by ... 

The classical problem of multiple solutions and undamped oscillations occurring in a continuous stirred-tank reactor, dealt with in the papers by Aris and Amundson (39), involved a single homogeneous exothermic reaction. Their theoretical analysis was extended in a number of subsequent theoretical papers (40, 41, 42). The present paragraph does not intend to report the theoretical work on multiplicity and oscillatory activity developed from analysis of governing equations, for a detailed review the reader is referred to the excellent text by Schmitz (3). To understand the problem of oscillations and multiplicity in a continuous stirred-tank reactor the necessary and sufficient conditions for existence of these phenomena will be presented. For a detailed development of these conditions the papers by Aris and Amundson (39) and others (40) should be consulted. 

Thus, this equation is a stability condition for the adiabatic CSTR. If this condition is not fulfilled, such as in strongly exothermal reactions, there may also be a situation where there are multiple solutions (dashed line in Figure 8.4). In such a case, a small perturbation of one of the process parameters makes the reactor jump from low conversion to high conversion, or reversely, leading to an instable operation. The stability conditions of the CSTR were studied in detail by... 

Lerou and Froment [10] found by calculations that a reactor may ignite under non constant flow conditions while it is still stable if constant flow is assumed. Kalthoff and Vortmeyer [11],(Figure 4) found an improved agreement between measured and calculated ranges of multiple solutions for non -uniform flow. From the previous work therefore can be concluded that non-uniform porosity and flow distributions effect the chemical reactor performance. The question however, whether real improvements are obtained has to be subject to a comparison of experimental results with calculations. 

It is well known that a tubular reactor model with no macromixing (i.e. Per 1) and perfect micromixing 0/ = 0) exhibits no multiple solutions, even in... 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

Equations (260) (263) form a set of differential-algebraic equations which has a unique solution when the two algebraic equations [(261) and (263)] themselves have unique solution of (c) (and Or)) for any fixed cm (and 0lm). Equivalently, the above system has multiple solutions only when Eqs. (261) and (263) evaluated at the reactor exit conditions begin to have multiple solutions. For Le - 1 (typical fluid Lewis numbers vary between 1 and 100), and for y oo, the hysteresis variety for the above set of equations is given by... 

The most important observation that follows from the above analysis of the multi-mode model is that in almost all practical cases, tubular reactor instabilities arise due to mixing/diffusional limitations at the small scales and spread over the reactor. In contrast, pseudohomogeneous models predict (erroneously) that reactor instabilities (ignition, multiple solutions, etc.) arise due to macromixing limitations at the reactor scale. 

We have a first-order homogeneous reaction, taking place in an ideal stirred tank reactor. The volume of the reactor is 20 X 10 3 m3. The reaction takes place in the liquid phase. The concentration of the reactant in the feed flow is 3.1 kmol/m3 and the volumetric flow rate of the feed is 58 X 10 m3/s. The density and specific heat of the reaction mixture are constant at 1000 kg/m3 and 4.184kJ/(kg K). The reactor operates at adiabatic conditions. If the feed flow is at 298 K, investigate the possibility of multiple solutions for conversion at various temperatures in the product stream. The heat of reaction and the rate of reaction are... 

Solve this problem for Da = 1.19, (iy= 3.56 x 10 , s = 2.53. (Hint Look for multiple solutions.) See Schmidt (2004) and Fogler (2005) for additional information about biological reactors. 

Techniques based on the implicit function theorem have been used to predict the existence of multiple solutions in a CSTR (Chang and Calo, 1979). An extension of catastrophe theory known as singularity theory has also been effectively used to determine the conditions for the existence of multiple solutions in a CSTR and a tubular reactor (Balakotaiah and Luss, 1981, 1982 Witmer et al., 1986). In this subsection, the technique of singularity to find the maximum number of solutions of a single mathematical equation and its application to analysis of the multiplicity of a CSTR are presented (Luss, 1986 Balakotaiah at al., 1985). The details of singularity theory can be found in Golubitsky and Schaeffer (1985). 

The existence of multiple solutions does not ensure that these solutions are physically attainable. In order for these solutions to be physically attainable, they must be stable. The linear stability analysis presented here provides the necessary conditions for the stability. The method of Lyapunov s fimction can also be used to assess the stability and the magnitude of the permissible pertiffbations so that the reactor returns to the steady state. In the case of Unear stability analysis, the eigenvalues of a differential operator determine the stability. An excellent account of the stabihty analysis of chemical reactors can be found in Perlmutter (1972). 

The flnding that a fluidized-bed reactor can operate at more than one steady state (Elnashaie and Cresswell, 1973 Bukur and Amundson, 1975a, b Furusaki et al., 1978 de Lasa et al., 1980), in particular the Kulkami-Ramachandran-Doraiswamy criterion in 1980 for multiple solutions for a first-order reaction... 

The plots of Weisz and Hicks (1962) are reproduced in Figure 7.6. The nature of the curves at high values of /3m suggests multiple solutions. In other words, the reaction can occur at three steady states, two stable and one unstable. We shall not be concerned with this aspect of effectiveness factors, but it is instructive to note that e given by one of the solutions in the multiple steady-state region can be orders of magnitude higher than unity. Instabilities of this kind are essentially local in nature, and are briefly considered in Chapter 12. The reactor as a whole can also exhibit multiple steady states, a feature that is briefly treated in Chapter 13. 

The simplest kinetic reactor model is the CSTR (continuous-stirred-tank reactor), in which the contents are assumed to be perfectly mixed. Thus, the composition and the temperature are assumed to be uniform throughout the reactor volume and equal to the composition and temperature of the reactor effluent However, the fluid elements do not all have the same residence time in the reactor. Rather, there is a residence-time distribution. It is not difficult to provide perfect mixing of the fluid contents of a vessel to approximate a CSTR model in a commercial reactor. A perfectly mixed reactor is used often for homogeneous liquid-phase reactions. The CSTR model is adequate for this case, provided that the reaction takes place under adiabatic or isothermal conditions. Although calculations only involve algebraic equations, they may be nonlinear. Accordingly, a possible complication that must be considered is the existence of multiple solutions, two or more of which may be stable, as shown in the next example. 

In both of these questions, we must begin with knowledge of the final answer before the reactor structure is devised—we must know, beforehand, of all possible solutions (for all possible structures) to understand if multiple solutions exist, and the same information must be known to understand if our structure is globally optimal. 

Multiple Solutions Suppose that various sets of reactors could all achieve the same toluene concentration of 0.09 mol/L, but that some reactor arrangements are able to achieve this concentration in a smaller total reactor volume. Is your design still appropriate Understanding the range of different options available to us is useful in selecting the most appropriate reactor configuration for our needs. 

A large number of case studies is reported in scientific literature dealing with physical equilibria, design purposes for unit operations, reactor stability, and so on. We have included some below to highlight their individual peculiarities. These include heat exchange in a thermal furnace, vapor-liquid equilibrium calculation, multiple solutions in a continuously stirred tank reactor (CSTR) reactor, and critical nuclear reactor size. Certain special cases are also discussed in Section 7.22. 

Nevertheless, both immobilized enzyme reactors and biosensors are used as detection units in flow injection analysis and Hquid chromatography. Some of these flow systems are illustrated in Figure 3. The choice of flow system will be determined both by the number of analytes to be measured and by the complexity of the sample. Flow injection in combination with immobilized enzymes is used for single solute determinations, as in the systems shown in Figure 3A-C. Multiple solute determination requires a separation step whereby the chromatography column is introduced (Figure 3D). There is also an additional need for separation power when interfering matrix components in complex samples need to be eliminated in order to permit accurate quantitation of the analyte. The use of multiple flow lines with immobilized enzyme reactions in flow injection systems has been demonstrated, whereby each flow line measures a... 

Equation (8.123) is the criticality equation for the equivalent stationary reactor. The solution of this equation for a system of given size and composition yields the multiplication constant of the actual reactor. Before dealing with time-dependent systems, we first examine the problem of the stationary, or critical, reactor. 

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