Cooled CSTR

In cases where the isothermal CSTR is cooled by the jacket, the heat balance comprises three terms  

If we assume the initial conversion is zero, the heat balance is 

This equation, together with the mass balance FA0-XA = ( rA) V, calculates the jacket temperature (Tc) required to maintain the reactor temperature at the desired level T while obtaining a conversion XA [1]. As an example, for a first-order reaction, by combining the mass balance in Equation 8.3 and the heat balances we find  

Since the mass flow rate is related to the volume flow rate, we can write 

Figu re 8.2 Heat balance of a cooled CSTR, with the cooling term (straight line) and the S-shaped heat release rate curve. The working point is at the intercept. 

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There are three resistances/coefhcients that must be considered in a jacket-cooled CSTR. There is a film coefficient hin at the inside wall of the vessel, a thermal conductivity km of the metal walls and a him coefficient hout at the outside surface of the wall ... 

Figure 3.24 gives the response to a 20% increase in feed flowrate. Figure 3.25 gives a direct comparison of the coil-cooled CSTR and the jacket-cooled CSTR with a jacket thickness of 0.025 m. Even with this very small jacket holdup, the coil-cooled system has tighter temperature control. 

The volume of the liquid in the shell (total shell volume minus tube volume) is typically equal to the tube volume. A circulating cooling water system is assumed, and a high circulation rate of the process liquid is assumed. So the temperature in the shell is Tc, and the temperature in the tubes is TR. The linear and nonlinear models are the same as for the jacket-cooled CSTR except the volume and area of the heat exchanger are used instead of the jacket volume and area. 

The reactor is the jacket-cooled CSTR with an irreversible, exothermic, liquid-phase reaction A —> B, which was considered in Section 3.1. In that section the flowrate of the cooling water Fj to the jacket was the manipulated variable for the reactor temperature controller (TR <— Fj control). In this section we explore the use of the flowrate of the fresh feed F() to control reactor temperature (TR <— F0 control). 

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... 

We will take a closer look at one of the simplest systems conceivable, a constant-volume and -density, cooled CSTR with a first-order, irreversible reaction A - B. While this model is quite simple it still contains most of the relevant issues surrounding an open-loop, nonlinear reactor. Referring to Fig. 4.5, this system can be described by one component balance and one energy balance ... 

For the specific jacket-cooled CSTR process considered in this section and the next, a simple heuristic approach can be used to incorporate quantitatively the limitations of controllability into the steady-state design. The idea is to specify a design criterion that ensures good controllability. In the reactor temperature control problem we use the criterion of a specified ratio of the maximum heat removal rate to the heat removal rate at design conditions. This simple approach is easily understood by designers and operators, and it requires no dynamic simulation or control analysis. We illustrate its usefulness in the following section to determine the besf reactor operating temperature. 

With its use the unsteady heat balance of the cooled CSTR using dimensionless numbers is obtained ... 

The cooled SBR has great similarity to the cooled CSTR as far as balancing is concerned. The most important difference is the reaction volmne which changes with time. 

When all start-up effects v diich are discussed in the following section have leveled off, the cooled CSTR operates in a steady state. The safety assessment has to check whether the cooling capacity is adequate to ensure the removal of all heat produced in such a way that the desired process temperature is not exceeded, as well to evaluate... 

Fig. 4-10. Possible steady state operating conditions for the cooled CSTR...

The easiest way to obtain such plots is to use the so-called coupling equation and the stationary mass balance of the CSTR given in Equ. (4-29). The coupling equation is derived by first transforming the imsteady heat balance of the cooled CSTR into the stationary balance ... 

Fig. 4-12. Possible steady state operating conditions for the cooled CSTR Parameters E/R = 10500 K, To = 310 K, AT,j = 40 K,...

To enable the calculation of Figures 4-12 and 4-13, an additional step is necessary. The equation for the steady state mass balance as well as the coupling equation of the cooled CSTR each contain a parameter which depends on the residence time the Damkoehler and the Stanton number, respectively. To eliminate this problem it is best to solve the coupling equation for the Stanton number and to divide the resulting equa-... 

Fig. 4-16. Cooled CSTR with statically unstable operating points...

The safely technical assessment of the unsteady operating behaviour of a cooled CSTR is an important element of the overall evaluation of the normal operating conditions. In the case of the CSTR, a non-steady state behaviour always exists during startup and shut-down. As these phases turn out to be critical their discussion and interpretation shall be supported graphically. 

A dramatic overshoot of the internal temperature is clearly observed. Temperature levels are reached which are close to 80 K higher than the later steady state temperature. Whether or not the reactor will ever reach this steady state operation is very questionable, as the probability that consecutive and decomposition reactions become so dominant at 400 K and higher that a final runaway will occur is very high. The results from this example can be combined to a first rule for the safe start up of cooled CSTR processes ... 

Fig. 4-19. Start-up behaviour of a cooled CSTR -start-up widi X=0 and T=To...

The necessary procedure can be derived from the classical stability theory for oscillating systems. The unsteady mass and heat balances for the cooled CSTR may be expressed in a very general form... 

Fig. 4-23. Cooled CSTR with statically stable but dynamically unstable operating points Parameters E/R = 10500 K, In Da., = 32.5, ATad = 135 K, St = 1.5...

In principle, the safety assessment of a process performed in a cooled CSTR under normal operating conditions is fully completed if start-up conditions as well as static and dynamic stability have been evaluated. The assessment performed this way yields unambiguous yes/no statements. 

Fig. 4-25 and 4-26. Cooled CSTR transition fiom operating point 1... 

Fig. 4-27. Cooled CSTR with only one tangential point of the curves for steady state solutions and the dynamic stability limit...

Finally a further recommendation for the start-up of cooled CSTR processes shall be given. The oscillating transient behaviour during start-up which was demonstrated here may be avoided if the CSTR is started up as an SBR with constant cooling temperature until the desired operating point is reached. At this time the feed of the second component is then switched on. The design principles to be obeyed for this procedure are discussed in Section 4.3.1.4. 

At this point the remark made in Section 4.1.3.1 about an optimized start-up strategy for the cooled CSTR shall be explained. The safety technical assessment procedure for the cooled isoperibolic SBR has demonstrated that in the case of correct design a prediction of the maximum reaction temperature is easily possible. This can be utilized for the optimization of the start up of the CSTR. The later steady state operating temperature of the CSTR is defined as the set value for the maximum SBR process temperature. In a next step one of the two reactants of the CSTR process is charged initially. Then the reactor is started as a semibatch process by feeding the second reactant. When the maximum temperature is reached, the feed of the initially charged reactant is started, and the feed streams are adjusted in such a way that the Stanton number of the CSTR is established. This way the initial oscillations are elegantly avoided. 

Cooled CSTR For a cooled CSTR, equations (4.10.59)-(4.10.61) lead to ... 

Figure4.10.27 Heat production function (sigmoidal curve) and heat removal lines for an exothermic irreversible first-order reaction in a cooled CSTR.

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