Temperature-concentration phase planes

In reactor startup it is often very important /tow temperature and concentrations approach their steady-state values. For example, a significant overshoot in temperature may cause a reactant or product to degrade, or the overshoot may be imacceptable for safe operation. If either case were to occur, we would say that the system exceeded its practical stability limit. Although we can solve the imsteady temperature-time and concentration-time equations numerically to see if such a limit is exceeded, it is often more insightful to study the approach to steady state by using the temperature-concentration phase plane. To illustrate these concepts we shall confine our analysis to a liquid-phase reaction carried out in a CSTR. 

The subscript indicates that the property is in the low-temperature region. The results are summarized by the boundary lines (thick lines) on the temperature-concentration phase plane in Figure 2.20. The liquid-liquid phase separation line (coexistence curve) in the low-temperature region is shown by the solid line in the figure. 

Rg. 2.20 Scaling laws of polymer solutions shown on the temperature-concentration phase plane. 

Jacobson and Stockmayer [15] showed the fraction of chains and rings on the temperature-concentration phase plane, and found very interesting phenomena that are analogous to Bose-Einstein condensation. When the parameter B exceeds a certain critical value, 100% rings are formed below a critical concentration of polymers. In fact, when p = 1, we have the coupled equations... 

At a second-order SmA-SmC phase transition, the symmetries are different but the layer spacing is the same. Fluctuations can drive a line of second-order SmA-SmC phase transitions to an N-SmA-SmC mul-ticritical point (see Fig. 4) [53]. Competing N-SmA and N-SmC fluctuations pull the N phase under the SmA phase in the temperature-concentration phase plane, leading to the Nj-e-SmA-SmC multicritical point [16]. High-resolution studies, as a function of both concentration and pressure, resolve the fluctuation-driven N-SmA/ N-SmC step (see, e.g. Fig. 5) into a universal spiral (Fig. 12) [16] around the N-SmA-SmC and the Nre-SmA-SmC multicritical points. Loosely speaking, the N-SmA transition line is dominated by N-SmA fluctuations, and the N-SmC transition line is dominated by Brazovskii fluctuations [54] that drive the N-SmC transition to lower temperatures compared to the N-SmA transition [18]. 

Figure 3.17. Phase-plane representations of reactor stability. In the above diagrams the point -I- represents a possible steady-state solution, which (a) may be stable, (b) may be unstable or (c) about which the reactor produces sustained oscillations in temperature and concentration.

Program THERM solves the dynamic model equations. The initial values of concentration and temperature in the reactor can be changed after each run using the ISIM interactive commands. The plot statement causes a composite phase-plane graph of concentration versus temperature to be drawn. Note that for comparison both programs should be used with the same parameter values. 

Using THERMPLO, locate the steady states. With the same parameters, verify the steady states using THERM. To do this, change the initial conditions (A and TR) using VAL and GO. Plot as a phase-plane and also as concentration and temperature versus time. 

Program REFRIG2 calculates the dynamic behaviour and generates a phase-plane plot for a range of reactor concentrations and temperatures. 

Figure 5.55. Plot of the concentration-temperature phase-plane, corresponding to the run in Fig. 5.54 using REFRIG2.

Fig. 3 Plots of the concentration-temperature phase-planes, corresponding to the conditions of Fig. 2 using REFRIG2.

Gray fit Yang (Ref 1), a mathematical model was proposed to unity the chain and thermal mechanisms of explosion. It was shown that the trajectories in the phase plane of the coupled energy and radical concentration equations of an explosive system will oive the time-dependent behavior of the system when the initial temperature and radical concentration are given. In the 2nd paper of the same investigators (Ref 2), a general equation for explosion limits (P—T relation) is derived from a unified thermal and chain theory and from chis equation, the criteria of explosion limits for either the pure chain or pure thermal theory can be deduced. For detailed discussion see Refs... 

For such phase space representations with one variable, or indeed with any number of independent concentrations, there are a number of rules which the trajectories must obey. In particular, trajectories cannot cross themselves, except at singular points (the stationary states) or if they form closed orbits (such as limit cycles or some other forms we will introduce later). Also, the trajectories cannot pass over singular points. The first of these rules is perhaps most easily shown for two-dimensional systems, where we have a two-dimensional phase plane. Let us assume that the rate equations for the two independent concentrations (or concentration and temperature), x and y, can be written in the form... 

The system of equations of non-steady combustion was analyzed on the concentration-temperature phase plane with the help of methods developed in the... 

To compare reactions with different time constants it is useful to plot them as trajectories in a multi-dimensional phase space whose coordinates are the species concentrations and the temperature. Fig. 2 shows trajectories projected onto the temperature vs. [r] plane for reactions with identical initial fuel and air concentrations but different initial radical concentrations and temperature. Trajectories beginning at the left had no initial radicals, and the trajectory starting at 1200 K is represented in Fig. 1. The exponential increase of [r] to [r]e is isothermal so it appears horizontal in Fig. 2. The knee of the curve represents the relatively flat portion of Fig. 1 where [r] is approximately [R]e. As the temperature increases [r] remains approximately equal to [R]e, which lies to the left of the dashed line due to consumption of fuel and oxygen. 

Fig. 4.16. Carbon monoxide-oxygen phase plane. The solid lines give the trajectories corresponding to a starting temperature of 1000 K and fuel-to-air ratios 0.5, 1.0 and 1.5, respectively, up to a simulation time of 0.01 s. The upper ends of the lines belong to the initial unburned mixture. For the preparation of the repro-model the initial composition of the mixture was uniformly distributed between = 0.5 and 1.5. Dots represent the values of CO and O2 concentrations used for the generation of the repro-model. Only a part of the 30,000 data points are plotted on the figure for clarity.

Startup of a CSTR (Figure S-1) and the approach to the steady state (CD-ROM). By mapping out regions of the concentration-temperature phase plane, one can view the approach to steady state and learn if the practical stability limit is exceeded. 

Approach to the Steady-state Phase-Plane Plots and Trajectories of Concentration versus Temperature... 

Example 2.2.8 is reviewed here using the phase plane analysis. For this purpose the independent variable is eliminated and temperature T is solved as a function of C, the concentration. 

Phase Plane Plots. 1 e transform the temperature and concentration profiles into a phase plane. 

Figures (E9-S.I) and

In the phase plane the intersections of curves 1 and 2 determine potential steady-state operating points dCp /dt = dT/dt = ()) , it is seen that regardless of the precise location of a (Ca, T) coordinate representing a starting condition in the six regions, only the two outer steady states are approached. Whether the upper or lower state is the one attained in steady operation is strictly a function of the signs of the concentration and temperature derivatives. The middle steady-state is an unstable one the reactor will not spontaneously move to this condition on startup. The consequences of this have already been illustrated for an experimental system in Figure 6.2. 

A limit separating safe fix>m unsafe reactor behaviour can be deduced by discussing the so-called phase plane set between temperature and relative concentration of the added component. This is shown in Figure 4-52. 

It can be seen that the trajectories form a loop in the two safety technically critical processes c and d. They are generated because in the course of time first the maxima of added component and temperature are passed before the concentration of the added component proceeds through a minimum. In the case of the safe process, the two extremes with respect to the added component are passed before the maximum of the temperature is reached. Safe and critical reactor behaviour become separated by the unique case of a singularity formation in the phase plane, characterized by the simultaneous occurrence of mass balance minimum and heat balance maximum. This is represented by case b. 

---