Heat removal curve

FIGURE 5.6 Heat balance in a CSTR (a) heat generated by reaction (b) heat removed by flow and transfer to the environment (c) superposition of generation and removal curves. The intersection points are steady states, (d) Superposition of alternative heat removal curves that give only one steady state. 

If we want to run the reactor at the steadystate temperature %, the heat-removal curve must be modified by changing the parameters of the system (or by adding a feedback controller, as we will show in the next part of this book) to make the Qn curve intersect the Qq curve at 7 with a slope greater than (dQJdT r ) as sketched in Fig. 6.9c. 

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 6-7 Heat removal curves in which Ta is increased or decreased. One or three intersections with X(T) indicates that one or three steady states occur.

Show that the intermediate steady state in a CSTR will always be unstable by considering the slopes of the heat generation and heat removal curves. 

Figure 4.6 Heat generation and heat removal curves for CSTR.

Bexp(0/(1 + 0/y)) and the heat removal line (x 6 — 9c). It has been shown that bifurcations or critical conditions exist when the heat generation and heat removal curves are tangent to each other, such that the following pair of equations is satisfied ... 

As shown in Figure 2.2-13, the intersection of the heat-production and heat-removal curves gives a stationary reactor temperature of672 K, which corresponds to a conversion of 90 %. 

They considered contact distillation, i.e. concentration-driven diffusion process, to possess a maximum at adiabatic conditions (curve 1) and thermal distillation to increase rapidly with both heat supply and heat removal (curve 2) as the result of fractional vaporization or condensation respectively. The total effect (curve 3) was regarded as a sum of the two separate effects. 

A steady-state analysis can also predict the stability of a chemical reactor. Such an analysis is carried out by plotting the heat production and heat removal as a function of reactor temperature. Usually the heat production curve is sigmoid while that for heat removal is linear. Any intersection of the two curves results in a steady-state condition. Figure 4.11 shows that three intersections are possible (Van Heerden, 1953). If the slope of the heat production curve is greater than that of the heat removal curve, then the steady state is unstable. This is illustrated by point B in Figure 4.11. If the slope of the heat production curve is less than that of the heat removal curve, then the steady state is stable. This is illustrated by points A and C. 

From the nature of the terms, we see that q is a linear function of T, whereas q is a nonlinear function of T. Figure 1.8a shows the generation curve specifically for a first-order irreversible reaction, along with the heat-removal curves with different initial temperatures, Tq, whereas Figure 1.8b shows a first-order irreversible reaction with the heat-removal... 

Stability will be assured if heat is removed by boiling one or more of the ingredients in the reaction, since this makes the system almost isothermal. On the other hand, if heat Is removed by a mechanism like evaporation of liquid into a dry gas stream. Its flow may change very little with temperature. In this case the slope of the heat-removal curve would be slight, and the reactor could be expected to be unstable. 

The first condition is known as the slope condition a steady state solution is unstable whenever the slope of the heat generation curve exceeds that of the heat removal curve, e.g., the steady state A in Figure 8.5. It is seen from the figure that when three steady states exist, at most two are stable. The second condition has no simple physical meaning and is implied by the slope condition whenever ... 

The stability of such a coil, in which resistance increases as any local temperature rise occurs, must be examined. Figure 5 is an illustration of stable versus unstable operation. Local heat production, for a particular coil, is plotted against conductor temperature, and three lines representing heat removal rate (at some fixed coolant temperature) are shown. Intersection of heat production and heat removal curves represent possible operating conditions. 

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