Coolant flow rate

Optimisation may be used, for example, to minimise the cost of reactor operation or to maximise conversion. Having set up a mathematical model of a reactor system, it is only necessary to define a cost or profit functionOptimisation and then to minimise or maximise this by variation of the operational parameters, such as temperature, feed flow rate or coolant flow rate. The extremum can then be found either manually by trial and error or by the use of a numerical optimisation algorithms. The first method is easily applied with ISIM, or with any other simulation software, if only one operational parameter is allowed to vary at any one time. If two or more parameters are to be optimised this method however becomes extremely cumbersome. 

Both the effects of coolant flow rate and coolant inlet temperature are thus incorporated in the model. 

Case 2. Coolant flow rate is fixed. Here At2 is known, so the tube side and shell side coefficients and area are optimized. Use Equation (/) and (J) to find h0 and hv A0 is then found from Equation (b). 

Consider the case of a proportional controller, which is required to maintain a desired reactor temperature, by regulating the flow of coolant. Neglecting dynamic jacket effects, the reactor heat balance can then be modified to include the effect of the varying coolant flow rate, hj, in the model equation as ... 

We have assumed the overall heat transfer coefficient U is constant It may be a function of the coolant flow rate Fj or the composition of the reaction mass, giving one more variable but also one more equation. 

Heat removed Qr from the coolant flow rate and the inlet and out flow streams. 

Figure 6 shows the coolant flow rate and the feed rate curves vs. the reactor volume. Note that the flow rate of cooling increases rapidly as the reactor size increases, whereas the inlet flow rate increases according to Eq.(39). 

From Figure 16 it follows that the response of the reactor has no offset due to the integrators block. Another interesting aspect is that the incremental jacket temperature is lower than the incremental reactor temperature, so the heat transmission goes from the reactant to the coolant flow rate. The values of the control signals, i.e. the outlet and inlet flow rate deduced from Eq.(50), are shown in Figure 17. 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

In this situation, a periodic variation of coolant flow rate into the reactor jacket, depending on the values of the amplitude and frequency, may drive to reactor to chaotic dynamics. With PI control, and taking into account that the reaction is carried out without excess of inert (see [1]), it will be shown that it the existence of a homoclinic Shilnikov orbit is possible. This orbit appears as a result of saturation of the control valve, and is responsible for the chaotic dynamics. The chaotic d3mamics is investigated by means of the eigenvalues of the linearized system, bifurcation diagram, divergence of nearby trajectories, Fourier power spectra, and Lyapunov s exponents. 

Equations (4) and (8) can be used to simulate the reactor at point P3 of Figure 5 in [1]. Remember that point P2 is unstable, so if the initial conditions are those corresponding to this point, it is easy to show [16], [28], the reactor evolves to points P or P3. Then, two forcing actions on the reactor are considered 1) when the coolant flow rate and the inlet stream temperature are varied as sine waves, and 2) reactor being in self-oscillating mode, an external disturbance in the coolant flow rate can drive it to chaotic behavior. 

It is well known that a nonlinear system with an external periodic disturbance can reach chaotic dynamics. In a CSTR, it has been shown that the variation of the coolant temperature, from a basic self-oscillation state makes the reactor to change from periodic behavior to chaotic one [17]. On the other hand, in [22], it has been shown that it is possible to reach chaotic behavior from an external sine wave disturbance of the coolant flow rate. Note that a periodic disturbance can appear, for instance, when the parameters of the PID controller which manipulates the coolant flow rate are being tuned by using the Ziegler-Nichols rules. The chaotic behavior is difficult to obtain from normal... 

In order to investigate this behavior, we consider the mathematical model of the reactor given by Eq.(l) or (4), and assuming that the reactor is at the steady state corresponding at point P3 of Figure 2 in [1]. The disturbances of the dimensionless inlet stream temperature and the coolant flow rate are the following ... 

Another interesting aspect of the self-oscillating behavior is the following one. If the values of xo,yo) are inside the lobe, an external periodic disturbance of the coolant flow rate can drive the reactor to chaotic behavior. 

In this section we consider a CSTR with a very simple control system formed by two PI controllers. The first controller manipulates the outlet flow rate as a function of the volume in the tank reactor. A second PI controller manipulates the flow rate of cooling water to the jacket as a function of error in reactor s temperature. The control scheme is shown in Figure 12 where the manipulated variables are the inlet coolant flow rate Fj and the outlet flow rate F respectively. 

Exercise 6. Show that the equilibrium point of the model defined by Eq.(34) and the simplified model R given by Eq.(35), i.e. when the dynamics of the jacket is considered negligible, are the same. Deduce the Jacobian of the system (35) at the corresponding equilibrium point. Write a computer program to determine the eigenvalues of the linearized model R at the equilibrium point as a function of the dimensionless inlet flow 4 50. Values of the dimensionless parameters of the PI controller can be fixed at Ktd = 1-52 T2d = 5. The set point dimensionless temperature and the inlet coolant flow rate temperature are Xg = 0.0398, X40 = 0.0351 respectively. An appropriate value of dimensionless reference concentration is C g = 0.245. Does it exist some value of 2 50 for which the eigenvalues of the linearized system R at the equilibrium point are complex with zero real part Note that it is necessary to vary 2 50 from small to great values. Check the possibility to obtain similar results for the R model. 

In Eqs.(33), (34) and (35) the effect of the constrained coolant flow rate due to control valve saturation is not considered, however this limitation can be introduced in the mathematical model of the reactor as follows ... 

This means that the value of the flow rate cooling water is enough to cool down the reactor and consequently, the reactor temperature can reach the desired set point. Therefore, the coolant flow rate reaches a certain steady state value xse, then from the third of Eq.(43) it can be deduced that ... 

In this case it is not possible to reach any value of equilibrium dimensionless coolant flow rate X6e, because when xge is greater than xg ax, it is constrained to the maximum value xe ax due to the flow rate limitation through the control valve. From this moment, the derivative dx /dr) is zero and the flow rate cooling xq remains constant. Consequently, the coolant flow rate cannot decrease the reactor temperature, which reaches a value greater than the set point, and the corresponding reactant concentration will be smaller. From Eq.(43) the set point temperature must be equal to xse, and as a result it is impossible that the reactor temperature would be able to reach the set point temperature Xg, an consequently the control system cannot drive the reactor to the desired equilibrium point. The equilibrium values of dimensionless variables are given by the same Eqs.(45), (46) and (47), but making the substitutions ... 

Eq.(50) shows the variation of the equilibrium dimensionless temperature as a function of the maximum value of the dimensionless coolant flow rate X6max- Plotting XQmax versus X3e a bifurcation curve can be obtained, from which it is possible to determine the value of xsmax which gives a different behavior of the reactor in steady state. It is interesting to note that Eq.(50) is equal to Eq.(47) when we make the substitutions of Eq.(49) into Eq.(47). 

Fig. 15. Dimensionless bifurcation plot when the coolant flow rate is constrained...

However, more precisely, the stability at the different steady sates can be determined by calculating the eigenvalues of the matrix of the linearized model of the CSTR. If there is an eigenvalue with positive real part, the steady state is unstable, and all eigenvalues with negative real part indicates a stable steady state. Thus, by simulation it can be verified that the steady states Pi and P3 are stable and P2 is unstable. This means that it is impossible to reach the point P2 when the coolant flow rate is constrained. 

If p, u > 0 —A > 0 and —A/p > 1 the equilibrium point is unstable, and a Shilnikov orbit may appear. For the reactor, with a value of X50 > 1 and X6max x6max)M (see Figure 15), by simulation it is possible to verify the presence of a homoclinic orbit to the equilibrium point. Figure 17 shows the homoclinic orbit for the model and R, when the steady state has been reached. Note that the Shilnikov orbit appear when the coolant flow rate is constrained. If there is no limitation of the coolant flow rate, a limit cycle is obtained both in models R and R, by simulation. 

It is interesting to note that in chaotic regime, the flow rate outlet stream, which is manipulated by the control valve CVl (see Figure 12), and the reactor volume, are driven by the PI controller to the equilibrium point without chaotic oscillations. However, the other variables have a chaotic behavior as shown in Figure 18. So it is possible to obtain a reactor behavior, in which some variables are in steady state and the others are in regime of chaotic oscillations, due to the decoupling or serial connection phenomena. In this case the control system and the volumetric flow limitation of coolant flow rate through the control valve VC2, are the responsible of this behavior. Similar results can be obtained from model. 

From the study presented in this chapter, it has been demonstrated that a CSTR in which an exothermic first order irreversible reaction takes place, can work with steady-state, self-oscillating or chaotic dynamic. By using dimensionless variables, and taking into account an external periodic disturbance in the inlet stream temperature and coolant flow rate, it has been shown that chaotic dynamic may appear. This behavior has been analyzed from the Lyapunov exponents and the power spectrum. 

The non-linear dynamics of the reactor with two PI controllers that manipulates the outlet stream flow rate and the coolant flow rate are also presented. The more interesting result, from the non-linear d mamic point of view, is the possibility to obtain chaotic behavior without any external periodic forcing. The results for the CSTR show that the non-linearities and the control valve saturation, which manipulates the coolant flow rate, are the cause of this abnormal behavior. By simulation, a homoclinic of Shilnikov t3rpe has been found at the equilibrium point. In this case, chaotic behavior appears at and around the parameter values from which the previously cited orbit is generated. 

The models that examine only stacks focus mainly on the temperature distribution within the stack. As mentioned, there is a much higher temperature gradient in the stack than in a single cell, and it provides design information in terms of coolant flow rate, among other things. - - Also, as mentioned above, transient effects have also been examined. 

If the coolant flow rate is sufficiently high in a jacketed reactor or if boiling liquid is used as the heat transfer fluid, then is constant, as we have assumed implicitly in the previous discussion. 

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