The adiabatic CSTR

Temperature control in an isothermal continuous stirred tank reactor 8.3.1 The adiabatic CSTR 

Probably the main advantage of using a stirred tank reactor is the relatively easy temperature control that is possible with this reactor type. This applies both to the semi-batch and the continuous operation modes. 

A chemical reaction of the type A P with a heat effect of 10 J/kmol is carried out in an org mic solvent, with a specific heat of 2.10 J/kg K and a density of 10 kg/m. All these physical constants are averages in the temperature range concerned. What is the maximum reactant concentration in order to keep the adiabatic temperature rise under 100 K From eq. (8.2) it follows then that the maximum concentration is then 2 kmol/m.  

This adiabatic way of operation seems very simple just cool the reactor feed sufficiently before it enters the reactor. The mixing takes care of a rapid equalization of temperature. However there are some disadvantages  

The instability follows from eqs. (8.1) and (8.2). A small variation in the conditions, e.g., the feed concentration may cause the temperature to rise slightly, so that the rate constant goes up also. This results in a higher reaction rate, causing the temperature to rise more, and so forth. However, the temperature rise will be check by the decrease of the reactant concentration. This is demonstrated by the the non-steady state heat balance for a first order reaction in a CSTR  

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This is a straight line whose slope is IJ and which intersects the T axis at X = 0 where Tq (Figure 6-3). It is evident that there may be one or three intersections of these curves (the large dots in the figure), and these represent one or three-steady states in the adiabatic CSTR. 

Figure 63 Dimensionless heat removal curve R(T) versus T for the adiabatic reactor plotted along with the heat generation curve X(T). There can be one or three intersections corresponding to one or three possible temperatures in the adiabatic CSTR, depending on Tq.

Figure 6-12 Transients in the adiabatic CSTR for the irreversibe reaction of the previous example. The upper panels show X(t) and T(t), while the lower panel displays X(T) for the same curves shown in the upper panel. The system converges on one of the stable steady states but never on the middle unstable steady state.

Before we proceed, note that these equations look identical in form to the adiabatic CSTR equations of Chapter 6,... 

It is clear that in a certain region of the parameter s = q p Cp, the adiabatic CSTR has three steady states with R(T) = G(T), marked by (1), (2), (3) in Figure 3.2. From a steady-state analysis point of view, the two steady states (1) and (3) are stable, while... 

As in Section 3.1 for the adiabatic CSTR problem, we again start with a generic MATLAB fzero.m based root finder to try to settle the issues of multiplicity in the nonadiabatic CSTR case. The MATLAB m file solveNadiabxy. m below finds the values for y (up to three values if a lies in the bifurcation region) that satisfy equation (3.12) for the given values of a, / , 7, Kc, and yc using MATLAB s root finder fzero. 

Compared to solveadiabxy. m for the adiabatic CSTR case in Section 3.1, the above MATLAB function solveNadiabxy. m depends on the two extra parameters Kc and yc that were defined following equation (3.9). It uses MATLAB s built-in root finder fzero.m. As explained in Section 3.1, such root-finding algorithms are not very reliable for finding multiple steady states near the borders of the multiplicity region. The reason - as pointed out earlier in Section 1.2 - is geometric the points of intersection of the linear and exponential parts of equations such as (3.16) are very shallow, and their values are very hard to pin down via either a Newton or a bisection method, especially near the bifurcation points. 

Our final MATLAB m file in this section rounds out our efforts just as Figure 3.10 did for the adiabatic CSTR problem. It uses the plotting routine for Figure 3.18 in conjunction with a MATLAB interpolator to mark and evaluate the (multiple) steady state(s) graphically for nonadiabatic, nonisothermal CSTR problems. 

It is worthwhile to explore the differences between the results of NadiabNisoalgraphsol and solveNadiabxy from the beginning of this section. For the nonadiabatic case such an exploration will duplicate what we have already learned about the differences between our f zero based and our graphics based steady-state finders for the adiabatic CSTR case in Section 3.1 see the exercises below. 

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... 

Linearized or asymptotic stability analysis examines the stability of a steady state to small perturbations from that state. For example, when heat generation is greater than heat removal (as at points A— and B+ in Fig. 19-4), the temperature will rise until the next stable steady-state temperature is reached (for A— it is A, for B+ it is C). In contrast, when heat generation is less than heat removal (as at points A+ and B— in Fig. 19-4), the temperature will fall to the next-lower stable steady-state temperature (for A+ and B— it is A). A similar analysis can be done around steady-state C, and the result indicates that A and C are stable steady states since small perturbations from the vicinity of these return the system to the corresponding stable points. Point B is an unstable steady state, since a small perturbation moves the system away to either A or C, depending on the direction of the perturbation. Similarly, at conditions where a unique steady state exists, this steady state is always stable for the adiabatic CSTR. Hence, for the adiabatic CSTR considered in Fig. 19-4, the slope condition dQH/dT > dQG/dT is a necessary and sufficient condition for asymptotic stability of a steady state. In general (e.g., for an externally cooled CSTR), however, the slope condition is a necessary but not a sufficient condition for stability i.e., violation of this condition leads to asymptotic instability, but its satisfaction does not ensure asymptotic stability. For example, in select reactor systems even... 

The adiabatic CSTR volume is less than the PFR volume... 

Let s calculate the adiabatic CSTR volume necessary to achieve 40% conversion. Do you think the CSTR will be larger or smaller than the PFR The mole balance is... 

Equations (5.48) and (5.49) are similar to those of the adiabatic CSTR, but with different physical meaning of the parameters. Those two equations can be decoupled into the single equation ... 

The steady state model of the adiabatic CSTR-Separator-Recycle system is obtained by setting /3= 0 and dropping the time derivatives in Eqs. 13.7 and 13.8. After some algebraic manipulations, the model can be reduced to one equation with one state variable X (the recycle of A is pure component so that z, =1) ... 

When the stability limits represented by A and B are marked on a number of constant t lines, as shown in Figure 13.8, the area enclosed by them can be identified as the region of instability of an adiabatic CSTR on the Xp T plane. Fortunately, however, the practical limits of operation of such a reactor usually lie outside this region, and hence the adiabatic CSTR is almost always in the stable region. 

Let us consider as an example the adiabatic CSTR and an irreversible first-order reaction at constant volume. The reaction rate is ... 

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