Dominant Variables

Preliminary process optimization. Dominant process variables such as reactor conversion can have a major influence on the design. Preliminary optimization of these dominant variables is often required. 

Laboratory experiments using rodents, or the use of gas analysis, tend to be confused by the dominant variable of fuel—air ratio as well as important effects of burning configuration, heat input, equipment design, and toxicity criteria used, ie, death vs incapacitation, time to death, lethal concentration, etc (154,155). Some comparisons of polyurethane foam combustion toxicity with and without phosphoms flame retardants show no consistent positive or negative effect. Moreover, data from small-scale tests have doubtful relevance to real fine ha2ards. 

Many experimental studies of entrainment have been made, but few of them have been made under actual distillation conditions. The studies are often questionable because they are hmited to the air-water system, and they do not use a realistic method for collecting and measuring the amount of entrainment. It is clear that the dominant variable affecting entrainment is gas velocity through the two-phase zone on the plate. Mechanisms of entrainment generation are discussed in the subsection Liquid-in-Gas Dispersions. ... 

The varianee equation provides a valuable tool with whieh to draw sensitivity inferenees to give the eontribution of eaeh variable to the overall variability of the problem. Through its use, probabilistie methods provide a more effeetive way to determine key design parameters for an optimal solution (Comer and Kjerengtroen, 1996). From this and other information in Pareto Chart form, the designer ean quiekly foeus on the dominant variables. See Appendix XI for a worked example of sensitivity analysis in determining the varianee eontribution of eaeh of the design variables in a stress analysis problem. 

Plotting this data as a Pareto chart gives Figure 3. It shows that the load is the dominant variable in the problem and so the stress is very sensitive to changes in the load, but the dimensional variables have little impact on the problem. Under conditions where the standard deviation of the dimensional variables increased for whatever reason, their impact on the stress distribution would increase to the detriment of the contribution made by the load if its standard deviation remained the same. 

In the absence of TCE and chlorine, the possible active species are holes (h+), anion vacancies, or anions (02 ), and hydroxyl radicals (OH ). At constant illumination and oxygen concentration, we may expect h+, and O2 concentrations to be approximately constant, and the dark adsorption to be a dominant variable. If kh+, or ko2- does not vary appreciably with the contaminant structure, the rate would depend clearly on the contaminant coverage as shown in Figme 2a, and the reaction would therefore occur via Langmuir-Hinshelwood mechanism. (Note only rates with conversions below 95% are correlated here (filled circles), as the 100% conversion data contains no kinetic information). This rate vs. d>r LH plot is smoother than those for koH or koH suggesting that non-OH species (holes, anion vacancies, or O2 ) are the active species reacting with an adsorbed contaminant. 

Below 7 MPa, the dominant variable for the compressibility factor in the PVT equation is the molecular weight of the gas. At this pressure level, the addition of ethane or propane increases the molecular weight of the gas more rapidly than the z factor decreases. Thus there is an advantage to removing ethane, propane, etc. from the gas. 

Probably the most important aspect of reactor design and control for a substantial number of industrial processes involves heat transfer, that is, maintaining stable and safe temperature control. Temperature is the dominant variable in many chemical reactors. By dominant variable, we mean it plays a significant role in determining the economics, quality, safety, and operability of the reactor. The various heat transfer methods for chemical reactors are discussed in a qualitative way in this chapter, while subsequent chapters deal with these issues in detail with several illustrative quantitative examples. 

The basic message is that the essential problem in reactor control is temperature control. Temperature is a dominant variable and must be effectively controlled to achieve the desired compositions, conversions, and yields in the safe, economic, and... 

In a different vein, Kothare et al. (2000) formally defined the concept of partial control on the basis of the practical premise that, in some cases, complex chemical processes can be reasonably well controlled by controlling only a small subset of the process variables, using an equally small number of dominant manipulated variables. An analysis method for identifying the dominant variables of a process was proposed in Tyreus (1999). 

Tyreus, B. D. (1999). Dominant variables for partial control. 1. A thermodynamic method for their identification. Ind. Eng. Chem. Res., 38, 1432-1443. 

Class I methods for single-stage processes have been discussed by several authors (5,6,7). While a detailed discussion will not be given here, it is worth noting that the issues of composition lag and pairing of equations with their dominant variables arise in essentially the same ways, and are as important, as for multi-stage processes. 

Figure 2.13 illustrates the variation of the economic potential during flowsheet synthesis at different stages as a function of the dominant variable, reactor conversion. EPmin is necessary to ensure the economic viability of the process. At the input/output level EP2 sets the upper limit of the reactor conversion. On the other hand, the lower bound is set at the reactor/separation/recycle level by EP3, which accounts for the cost of reactor and recycles, and eventually of the separations. In this way, the range of optimal conversion can be determined. This problem may be handled conveniently by means of standard optimization capabilities of simulation packages, as demonstrated by the case study of a HDA plant [3]. 

In the SR method, temperatures are the dominant variables and are found by a Newton-Raphson solution of the stage energy balances. Compositions do not have as great an influence in calculating the temperatures as do heat effects or latent heats of vaporization. The component flow rates are found by the tridiagonal matrix method. These are summed to get the total rates, hence the name sum rates. 

We can draw a very useful general conclusion from this simple binary system that is applicable to more complex processes changes in production rate can be achieved only by changing conditions in the reactor. This means something that affects reaction rate in the reactor must vary holdup in liquid-phase reactors, pressure in gas-phase reactors, temperature, concentrations of reactants (and products in reversible reactions), and catalyst activity or initiator addition rate. Some of these variables affect the conditions in the reactor more than others. Variables with a large effect are called dominant. By controlling the dominant variables in a process, we achieve what is called partial control. The term partial control arises because we typically have fewer available manipulators than variables we would like to control. The setpoints of the partial control loops are then manipulated to hold the important economic objectives in the desired ranges. 

Hence a goal of the plantwide control strategy7 is to handle variability in production rate and in fresh reactant feed compositions while minimizing changes in the feed stream to the separation section. This may not be physically possible or economically feasible. But if it is, the separation section will perform better to accommodate these changes and to maintain product quality, which is one of the vital objectives for plant operation. Reactor temperature, pressure, catalyst/initiator activity, and holdup are preferred dominant variables to control compared to direct or indirect manipulation of the recycle flows, which of course affect the separation section. 

In Chaps. 4 and 6 we discuss specific control issues for chemical reactors and distillation columns. We shall then have much more to say about the important concepts of dominant variables and partial control. Much of the material in those chapters centers on the control of the units individually. However, we also try to show how plantwide control considerations may sometimes alter the control strategy for the unit from what we would normally have in an isolated system. 

There are situations where reactor temperature is not a dominant variable or cannot be changed for safety or yield reasons. In these cases, we must find another dominant variable, such as the concentration of the limiting reactant, flowrate of initiator or catalyst to the reactor, reactor residence time, reactor pressure, or agitation rate. 

Once we identify the dominant variables, we must also identify the manipulators (control valves) that are most suitable to control them. The manipulators are used in feedback control loops to hold the dominant variables at setpoint. The setpoints are then adjusted to achieve the desired production rate, in addition to satisfying other economic control objectives. 

When the setpoint of a dominant variable is used to establish plant production rate, the control strategy must ensure that the right amounts of fresh reactants are brought into the process. This is often accomplished through fresh reactant makeup control based upon liquid levels or gas pressures that reflect component inventories. Wre must keep these ideas in mind when we reach Steps 6 and 7. 

However, design constraints may limit our ability to exercise this strategy concerning fresh reactant makeup, An upstream process may establish the reactant feed flow sent to the plant. A downstream process may require on-demand production, which fixes the product flowrate from the plant. In these cases, the development of the control strategy becomes more complex because we must somehow adjust the setpoint of the dominant variable on the basis of the production rate that has been specified externally. We must balance production rate with what has been specified externally. This cannot be done in an open-loop sense, Feedback of information about actual internal plant conditions is required to determine the accumulation or depletion of the reactant components. This concept was nicely illustrated by the control strategy in Fig. 2.16, In that scheme we fixed externally the flow of fresh reactant A feed. Also, we used reactor residence time (via the effluent flowrate)... 

To control the economic objectives we must have measurements and manipulated variables. However, in the example reactors we have looked at so far it should be clear that tve have only a limited number of manipulated variables, especially after we have taken care of the heat management issues. How is it then possible to achieve any level of economic control of a reactor The answer lies in a concept introduced by Shinnar (1981) called partial control. In short it means that only a few dominant variables in the process (e.g., temperatures, key components) are identified, measured, and controlled by feedback controllers. Then, by varying the setpoints for the dominant variables, it becomes possible to position the process such that all the important economic variables stay within acceptable ranges. We will elaborate more on this important concept in the next section but first we introduce the classification of reactor variables used by Shinnar. 

Control all dominant variables to setpoint with feedback controllers using manipulators with a rapid response. This ensures unit control. 

A few comments about the method are warranted. The controlled (dominant) variables, Ycd, should be measured such that they belong to the set Yd for rapid control. Similarly, the manipulators in the feedback control loops should belong to the set, Ud. The feedback controllers should have integral action (PI controllers). These can be tuned with minimal information (e.g., ultimate gain and frequency from a relay test). The model Ms is usually quite simple and can be developed from operating data using statistical regressions. This works because the model includes all the dominant variables of the system, Y d, as independent variables by way of their setpoints, Y. The definition of domi-... 

On the surface it might appear that partial control does not require a first-principles model for its implementation. After all, M is a regression model and controller tuning is based on relay-feedback information. For simple systems this may be correct. However, for most industrially relevant systems it is not intuitively obvious what constitutes the dominant variables in the system and how to identify appropriate manipulators to control the dominant variables. This requires nonlinear, first-principles models. The models are run off-line and need only contain enough information to predict the correct trends and relations in the system. The purpose is not to predict outputs from inputs precisely and accurately, but to identify dominant variables and their relations to possible manipulators. 

Let s look at some examples. First consider the vinyl acetate reactor discussed in Chap. 11. It is a plug-flow system with external cooling. To satisfy the heat balance we have already proposed to close one loop around the reactor, namely between the reactor exit temperature and the coolant temperature (steam pressure). This provides us with one setpoint, Y cf, that we can use to meet economic objectives, YP. provided exit temperature is a dominant variable. 

Before we investigate the dominance aspects of the vinyl acetate reactor we ask What are the economic objectives As a minimum we would like to control the production rate of vinyl acetate, TIP. and the selectivity SEL to vinyl acetate. Therefore, any dominant variable should have a significant impact on these economic objectives. Figure 4.31 show s how the objectives vary with reactor exit temperature. It is clear that we have identified a dominant variable and that it is possible to set the reactor exit temperature such that both the production rate and the selectivity fall within certain ranges for given values of the feed conditions, W, and a given level of catalyst activity, Us. This is the meaning of partial control. 

To complete the picture w7e should investigate if there are more dominant variables in the system. We have already touched on this issue in the section on reaction rates. There we noted that the acetic acid concentration to the reactor is not dominant. We can also argue that the ethylene partial pressure is not likely to be a dominant variable since ethylene enters the reactor in large excess. However, oxygen is the limiting component and it plays a role in the main reaction as wrell as in the side reaction. Oxygen therefore affects the economic objectives and is considered dominant. Feedback control of the oxygen concentration to the reactor is necessary if we wrant complete control of the unit. 

Following the arguments around the HDA reactor, we conclude that inlet temperature should be a dominant variable. However, in this case it is not strongly dominant due to the low activation energies. In other words, kx and k do not vary much with temperature. Instead, the most dominant variable is C, the concentration of product along the reactor. The component B enters the reactor at a low concentration and dominates the rate of both reactions at least until most of the reactant A is consumed. Furthermore, the main reaction is autocatalytic in B such that the rate of formation of B depends upon it own concentration. Unit control therefore requires that we find a manipulated variable that allows us to vary the inlet concentration of B to the reactor. Without such a loop the output from the reactor would depend entirely on minute variations in the feed concentration, which would be unpredictable. Tb... 

We pointed out earlier in this chapter that simplicity is the key to controllability. What we mean by simplicity is that the list of dominant variables should be kept as short as possible. It is easy to construct reactors where the number of dominant variables far exceed the number of manipulated variables or where the dominant variables interact in a complicated way. 

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