Stable operating point

A cascade of three continuous stirred-tank reactors arranged in series, is used to carry out an exothermic, first-order chemical reaction. The reactors are jacketed for cooling water, and the flow of water through the cooling jackets is countercurrent to that of the reaction. A variety of control schemes can be employed and are of great importance, since the reactor scheme shows a multiplicity of possible stable operating points. This example is taken from the paper of Mukesh and Rao (1977). 

The parameters used in the program give a steady-state solution, representing, however, a non-stable operating point at which the reactor tends to produce natural, sustained oscillations in both reactor temperature and concentration. Proportional feedback control of the reactor temperature to regulate the coolant flow can, however, be used to stabilise the reactor. With positive feedback control, the controller action reinforces the natural oscillations and can cause complete instability of operation. 

The problem of ignition and extinction of reactions is basic to that of controlling the process. It is interesting to consider this problem in terms of the variables used in the earlier discussion of stability. When multiple steady-state solutions exist, the transitions between the various stable operating points are essentially discontinuous, and hysteresis effects can be observed in these situations. 

In order to assist convergence in this circuit, the UIC statement was included in the. TRAN simulation. This statement helps in circuits where a steady-state operating point may not exist, multiple stable operating points exist, or it is difficult for SPICE to determine the correct operating point. 

An enzyme membrane reactor allows continuous transketolase-catalyzed production of L-erythrulose from hydroxypyruvate and glycolaldehyde with high conversion, stable operational points, and good productivity (space-time yield) of 45 g (L d) 1, thus best overcoming transketolase deactivation by substrates (Bongs, 1997). 

Figure 2.4 Semenov diagram the intersections S and I between the heat release rate of a reaction and the heat removal by a cooling system represent an equilibrated heat balance. Intersection S is a stable operating point, whereas I represent an instable operating point. Point C corresponds to the critical heat balance.

Hydrodynamic considerations lead to the contention that there are two stable operating points in reactive extrusion - one with a large, fully filled length and a high conversion, and one with a small, fully filled length and a low conversion. Severe... 

Whether or not multiple steady states will appear, and how large the deviation of the effectiveness factors between both stable operating points will be, is determined by the values of the Prater and Arrhenius numbers. Effectiveness factors above unity generally occur when p > 0 (exothermal reactions). However, for the usual range of the Arrhenius number (y = 10-30), multiple steady states are possible only at larger Prater numbers (see Fig 13). For further details on multiple steady states, the interested reader may consult the monograph by Aris [6] or the works of Luss [69, 70]. 

Before we can demonstrate the connection between process control and Eq. (A.20), we need to introduce the concept of Lyapunov functions (Schultz and Melsa. 1967). Lyapunov functions wnre originally designed to study the stability of dynamic systems. A Lyapunov function is a positive scalar that depends upon the system s state. In addition, a Lyapunov function has a negative time derivative indicative of the system s drive toward its stable operating point where the Lyapunov function becomes zero. Mathematically we can describe these conditions as... 

In cases of multiplicity (that is more than one solution to Equation 4.10), of the three intersection points only those at the lowest and highest pellet temperature represent stable conditions. These are therefore called the stable operating points. The intermediate intersection point is referred to as an unstable operating point. 

The value of the upper stable operating point is fairly insensitive to variations in gas velocity and the degree of conversion this is valid not only for one piece of catalyst but equally well for a large collection of particles. This can be understood as follows. The mass and heat transfer coefficients are related by the Chilton-Colbum analogy ... 

Figure 4.4 Relative heat production and heat withdrawal rates versus dimensionless pellet temperature 0p A) exothermic reactions, 0p, 0t and An > 0 B) endothermic reactions, 0p, 0t and An < 0 ( ) stable operating point (o) unstable operating point.

If the velocity and thus 0 are increased again, the pellet reenters the region of multiplicity for 0 = 0.0035. In this case the pellet remains at the higher stable operating point... 

Here the external-temperature difference is of little iml)ortance because the rate is relatively low. For both conversion levels the results apply to a stable operating point corresponding to Aj in Fig. 10-4. 

After that the liquid reactant feed and the hydrogen gas are supplied to the reactor in the desired ratio. The reaction starts, the reaction mixture heats up and at reaching a temperature somewhat below the boiling point of the solvent at the set reactor pressure, the evaporation will become so high that a stable operating point is reached. Solvent vapours are condensed and returned to the reactor, the liquid phase leaves the top of the catalyst bed via an overflow. For stopping the reaction the catalyst bed is washed out with pure solvent and the catalyst is kept covered with liquid. 

It is readily argued that the intersection of coalescence and generation rates in Figure 11 leads to a stable steady state. If the system is perturbed away from this point, it naturally returns. Consider a small, positive perturbation in the local liquid saturation. The coalescence rate then declines, and the foam texture becomes finer. This change causes an increased flow resistance that returns the liquid saturation back to the stable operating point. The converse negative saturation perturbation is similarly argued to be stable. 

The lower limit of the conversion achievable at a stable operating point decreases as the flow rates increase (Eq. 13.22, 13.23) and increases (Eq. 13.20, 13.21). Hence,... 

Figure 13.26 First-order reaction in adiabatic CSTR - Separator - Recycle system Conversion achievable at stable operating points...

However, operation of the CSTR at 301 K is not very attractive because the conversion of reactant A predicted is only 1.4%. Similar analysis reveals that Tipper is also a stable operating point because small changes in reactor temperature above or below Topper produce an imbalance between Grx(T) and R(T) that shifts the steady-state operating point back to Topper. For example (see Figure 5-1) ... 

---