Of isothermal CSTRs

Fig. 33. Characteristics of isothermal CSTR 1, kinetic reaction rate dependence 2, straight line for substance transfer into environments.

Below, we analyze the operation of isothermal CSTRs with single reactions for different types of chemical reactions. For convenience, we divide the analysis into two sections reactor design and determination of the rate expression. In the former, we determine the size of the reactor or the production rate of a given reactor for a known reaction rate, and, in the latter, we determine the rate expression from experimental reactor operating data. 

We start the analysis of isothermal CSTRs with single chemical reactions and consider a first-order gas-phase chemical reaction ... 

Below, we describe the design formulation of isothermal CSTRs with multiple reactions for various types of chemical reactions (reversible, series, parallel, etc.). In most cases, we solve the equations numerically using mathematical software. In some simple cases, we obtain anal5dical solutions. 

Design and operation of isothermal CSTRs with single leaetions. 

N is the total number of isothermal CSTR reactors considered (a sensitivity variable). Temperatures Tj, concentrations Cj,j and fractions xj j are the state variables considered. MINLP Model Results... 

Kiss et al. [18] studied the multiplicity behaviour of isothermal CSTR / Separation / Recycle systems involving six reaction systems of increasing complexity, including chain-growth polymerisation. Below a critical value of the plant Damkohler number Da < Dd the... 

Tha heat removed includes the heal carried by the sensible heat of the reacting fluid also. If the feed was at a lover temperature than the reactor then q,d = (F/ V ) p c (T - To). For the example it was assumed that T - To 0 for simplicity sake as the isothermal CSTR. 

The general rule is that combinations of isothermal reactors provide intermediate levels of performance compared with single reactors that have the same total volume and flow rate. The second general rule is that a single, piston flow reactor will give higher conversion and better selectivity than a CSTR. Autocatalytic reactions provide the exception to both these statements. 

Example 14.1 shows how an isothermal CSTR with first-order reaction responds to an abrupt change in inlet concentration. The outlet concentration moves from an initial steady state to a final steady state in a gradual fashion. If the inlet concentration is returned to its original value, the outlet concentration returns to its original value. If the time period for an input disturbance is small, the outlet response is small. The magnitude of the outlet disturbance will never be larger than the magnitude of the inlet disturbance. The system is stable. Indeed, it is open-loop stable, which means that steady-state operation can be achieved without resort to a feedback control system. This is the usual but not inevitable case for isothermal reactors. 

ILLUSTRATION 8.6 DETERMINATION OF REQUIRED CSTR VOLUME UNDER ISOTHERMAL OPERATING CONDITIONS—LIQUID PHASE REACTION... 

Calculate the ratio of the volumes of a CSTR and a PFR ( Vst pf) required to achieve, for a given feed rate in each reactor, a fractional conversion (/A) of (i) 0.5 and (ii) 0.99 for the reactant A, if the liquid-phase reaction A - products is (a) first-order, and (b) second-order with respect to A. What conclusions can be drawn Assume the PFR operates isothermally at the same T as that in the CSTR. 

For the kinetics scheme in problem 18-11, and from the information given there, compare L or the following features of a CSTR and a PFR operated isothermally. 

Data obtained in continuous stirred tank reactors have the merits of isothermicity and of an algebraic relation between the variables rather than a differential one. At steady state In a CSTR the material balance on a reactant A is... 

A reaction with rate equation, r = k C2/ (1 + k2C), is to be conducted in an isothermal CSTR, Examine the possibility of the occurrence of more than one steady state conversion. 

The system is sketched in Fig. 3.1 and is a simple extension of the CSTR considered in Example 2.3. Product B is produced and reactant A is consumed in each of the three perfectly mixed reactors by a first-order reaction occurring in the liquid. For the moment let us assume that the temperatures and holdups (volumes) of the three tanks can be different, but both temperatures and the liquid volumes are assumed to be constant (isothermal and constant holdup). Density is assumed constant throughout the system, which is a binary mixture of A and B. 

M perfectly mixed, isothermal CSTR has an outlet weir. The flow rate over the weir proportional to the height of hquid over the weir, h, to the 1.5 power. The weir height is. The cross-sectional area of the tank is A. Assume constant density. 

The equations describing the series of three isothermal CSTRs were developed in Sec. 3.2. 

Example 9Ji, Consider the isothermal CSTR of Example 6.6. The equation describing the system in terms of perturbation variables is... 

Two isothermal CSTRs are conneaed by a long pipe that acts like a pure deadtime of D minutes at the steadystate flow rates. Assume constant throughputs and holdups and a first-order irreversible reaction... 

If a proportional controller is used in the three-isothermal CSTR process, a controller gain of 22.6 gives a phase margin of 45°. A gain of 20 gives a maximum closedloop log modulus of +2 dB with a closedloop resonant frequency of 1.1 radian per minute. 

We took the 4- sign on the square root term for second-order kinetics because the other root would give a negative concentration, which is physically unreasonable. This is true for any reaction with nth-order kinetics in an isothermal reactor There is only one real root of the isothermal CSTR mass-balance polynomial in the physically reasonable range of compositions. We will later find solutions of similar equations where multiple roots are found in physically possible compositions. These are true multiple steady states that have important consequences, especially for stirred reactors. However, for the nth-order reaction in an isothermal CSTR there is only one physically significant root (0 < Ca < Cao) to the CSTR equation for a given T. ... 

In this chapter we consider the performance of isothermal batch and continuous reactors with multiple reactions. Recall that for a single reaction the single differential equation describing the mass balance for batch or PETR was always separable and the algebraic equation for the CSTR was a simple polynomial. In contrast to single-reaction systems, the mathematics of solving for performance rapidly becomes so complex that analytical solutions are not possible. We will first consider simple multiple-reaction systems where analytical solutions are possible. Then we will discuss more complex systems where we can only obtain numerical solutions. 

The CSTR is, in many ways, the easier to set up and operate, and to analyse theoretically. Figure 6.1 shows a typical CSTR, appropriate for solution-phase reactions. In the next three chapters we will look at the wide range of behaviour which chemical systems can show when operated in this type of reactor. In this chapter we concentrate on stationary-state aspects of isothermal autocatalytic reactions similar to those introduced in chapter 2. In chapter 7, we turn to non-isothermal systems similar to the model of chapter 4. There we also draw on a mathematical technique known as singularity theory to explain the many similarities (and some differences) between chemical autocatalysis and thermal feedback. Non-stationary aspects such as oscillations appear in chapter 8. 

For instance, if we consider the simple case of the adiabatic non-isothermal CSTR, the stationary-state condition is given by eqn (7.27). Writing x for the extent of reaction, we have... 

Now that the recipes for locating the various changes in the qualitative form of the stationary-state locus have been presented, we can go on to examine the origin of the behaviour in the cubic autocatalytic system with the additional uncatalysed step, and for the non-adiabatic non-isothermal CSTR which has been asserted in previous sections. 

Three model kinetic schemes have been studied relatively intensively with periodic forcing the first-order non-isothermal CSTR of chapter 7 the Brusselator model, which is closely related to the cubic autocatalysis of chapters 2 and 3 and the surface reaction model discussed in 12.6. We will use the last of these to introduce some of the general features. 

The specific models we will analyse in this section are an isothermal autocatalytic scheme due to Hudson and Rossler (1984), a non-isothermal CSTR in which two exothermic reactions are taking place, and, briefly, an extension of the model of chapter 2, in which autocatalysis and temperature effects contribute together. In the first of these, chaotic behaviour has been designed in much the same way that oscillations were obtained from multiplicity with the heterogeneous catalysis model of 12.5.2. In the second, the analysis is firmly based on the critical Floquet multiplier as described above, and complex periodic and aperiodic responses are observed about a unique (and unstable) stationary state. The third scheme has coexisting multiple stationary states and higher-order periodicities. 

In figure 2b, there are clearly folds in the left-hand side of the 3/2 and 2/1 resonance horns. This phenomenon had not (when we observed it) been seen in other forced oscillators such as the Brusselator model (Kai Tomita 1979) and the non-isothermal cstr (Kevrekidis et al. 1986), although it may have been missed in previous numerical studies that did not use arc-length continuation. It is however also to be found in unpublished work of Marek s group. The cusp points at M and L are quite different from the apparent cusp ... 

Continuous reactions in a non-isothermal CSTR-I. Multiplicity of steady states (with P. Cicarelli). Chem. Eng. Sci. 49,621-631 (1994). 

Floquet et al. (1985) proposed a tree searching algorithm in order to synthesize chemical processes involving reactor/separator/recycle systems interlinked with recycle streams. The reactor network of this approach is restricted to a single isothermal CSTR or PFR unit, and the separation units are considered to be simple distillation columns. The conversion of reactants into products, the temperature of the reactor, as well as the reflux ratio of the distillation columns were treated as parameters. Once the values of the parameters have been specified, the composition of the outlet stream of the reactor can be estimated and application of the tree searching algorithm on the alternative separation tasks provides the less costly distillation sequence. The problem is solved for several values of the parameters and conclusions are drawn for different regions of operation. 

Remark 1 The reactor network consists of ideal CSTRs and PFRs interconnected in all possible ways (see superstructure of reactor network). The PFRs are approximated as a cascade of equal volume CSTRs. The reactors operate under isothermal conditions. 

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