Heat generation curve

Volumetric heat generation increases with temperature as a single or multiple S-shaped curves, whereas surface heat removal increases linearly. The shapes of these heat-generation curves and the slopes of the heat-removal lines depend on reaction kinetics, activation energies, reactant concentrations, flow rates, and the initial temperatures of reactants and coolants (70). The intersections of the heat-generation curves and heat-removal lines represent possible steady-state operations called stationary states (Fig. 15). Multiple stationary states are possible. Control is introduced to estabHsh the desired steady-state operation, produce products at targeted rates, and provide safe start-up and shutdown. Control methods can affect overall performance by their way of adjusting temperature and concentration variations and upsets, and by the closeness to which critical variables are operated near their limits. 

With a high heat removal rate, corresponding to an almost vertical line, as was the case in the experiments in the CSTR, the full heat generation curve could be measured. An intersection could be achieved between the heat generation curve and the very steep heat removal line at the point where the non-existent middle point was, but this was just one of the many stable solutions possible and not an unstable point. ... 

Heat balances occur at the intersection of the heat generation curve and the heat removal line (points C and D). Stable operation will occur at point C. A reaction temperature lower than point C will result in self-heating up to point C because the heat generation rate exceeds the heat removal rate. At temperature Tb, the heat removal rate exceeds the heat generation rate, so the reaction temperature will fall until point C is reached. Although point D is a heat balance point, no stable operation is possible here a temperature slightly lower than that at point D will result in a decrease in reactor temperature to... 

The conversion achieved in the adiabatic reactor, x, can be calculated by the method outlined in Sect. 3.2.2 and the two sides of eqn. (192) are plotted separately as functions of the reactor inlet temperature, Tj, in Fig. 27. The left-hand side, which is proportional to the heat absorbed in the exchanger, is linearly dependent on T while the right-hand side, which is proportional to the heat generated in the reactor, is a function of X, which itself depends on V/F. For a particular reactor, V is fixed and thus there is a family of heat generation curves each corresponding to a particular value of F. 

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.

This is the example from the previous chapter, and k( T) was determined previously. X(T) is shown in Figure 6-9 for T = 1 mm. As T increases so that the curves move into the situation where multiple roots emerge, the system does not jump to them because it is already in a stable steady state. Only when Tq is so high that the heat removal line becomes tangent to the heat generation curve does the lower intersection disappear. The system then has no alternative but to jump to the upper intersection. A similar argument holds in the decrease in Tq from the upper steady state. 

For a CSTR system, Figure 4 (A-2) shows maximally three steady states and Figure 6 (A-2) maximally five, with the possibility of fewer steady state for other specific y parameter values. This maximal steady-state number depends on the shape of the heat generating curve G(y). For different heat generation functions there may possibly be seven, nine, or any odd maximal number of steady states for certain values of y. 

Figure 8-21 Variation of heat generation curve with space-time.

The most customary assumption is that of negligible reactant consumption (i.e., y = 1) such that Eq. (1) disappears and one is only concerned with the relationship between the heat generation curve... 

This is illustrated in Fig. 1 below (with parameters B = I, y = 3 and 6c = —1.75) for two particular heat removal lines (the upper dashed line with a = 0.968 and the lower dashed line with a = 0.199). Three basic patterns arise one safe, stable steady state (to the left of the upper dashed line), multiple steady states (in between the two dashed lines, where the lower steady state is stable and safe, the intermediate steady state is unstable, and there may be a stable but unsafe steady state at high temperature) and an unsafe region (to the right of the lower dashed line). The upper, unsafe steady state will only exist when 6/y >> 1 (which is usually not the case for industrially relevant reactions). In this limit, the heat generation curve levels off asymptotically to a value of 5exp(7). It is also noted that the classical result obtained by Semenov for safe operation in the limit of high activation energy (i.e., 7 oo) when 0c = 0 is (x/B > e. 

Fig. 1 Multiple steady states and their classifications in a batch reactor with negligible reactant consumption. The solid line is the heat generation curve, with B = y = 3, and 6c = —1.75. The upper dashed line is the heat removal line with a = 0.968 and the lower dashed line is the heat removal line with a = 0.199. [Data from Eq. (3).]...

If this maximum slope is greater than (1 -f- k), the slope of the heat removal line, there will be the possibility of more than one intersection of the heat removal line and the heat generation curve, as shown in Fig. 7.5. As long as... 

Fig. 7.5 Possible intersections of the heat generation curve with the heat removal line.

Consider what happens to a given reactor when the feed temperature is very slowly increased and everything else is held constant. This means that the heat generation curve is fixed and the heat removal line moves parallel to itself. Suppose that the maximum slope of the E. curve is greater than the slope of the Er lines. Then in Fig. 7.6 the E curve is fixed and A, B, C, D, and E represent five typical E - lines for five feed temperatures. 

As before the left-hand side gives the heat removal line and the right the heat generation curve, and there are now a much greater variety of shapes possible for the latter. Suppose, for example, that both reactions are exothermic but that 1 2 We have seen that the functions k /(l d- Ok ... 

What is the slope of the heat removal line Deduce that a = dv/dr, the slope of the heat generation curve, is given by p(l — v /t and has its greatest value when v + r =... 

Exercise 7.3,4. Discuss the possible form of the heat generation curve for the first order reactions A —> B —> C when the second reaction is endothermic. 

Exercise 7.3.6. Show that if the greatest production (i.e., greatest value of is required from a given reactor for a single exothermic reaction, then the heat removal line should pass through the highest point of the heat generation curve (see Fig. 7.10). 

By contrast, at the intermediate steady state the slope of the heat generation curve is greater than that of the heat removal line. Hence a slight increase in temperature produces a net heat generation which would tend to drive the temperature even higher. Conversely, a slight drop of temperature induces a net removal of heat, which would cause the temperature to fall even more. In this sense, we say that the intermediate steady state is unstable. 

The middle points 5 and 8 in Figures 8-19 and 8-20 represent unstable steady-state temperatures. Consider the heat removal line d in Figure 8-19 along with the heat-generated curve which Is replotted in Figure 8-21. 

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