Optimization design equations

Optimal design of ammonia synthesis by differential equation solution and a numerical gradient search... 

Omoleye, J. A., Adesina, A. A., and Udegbunam, E. O., Optimal design of nonisothermal reactors Derivation of equations for the rate-temperature conversion profile and the optimum temperature progression for a general class of reversible reactions, Chem. Eng. Comm., Vol. 79, pp. 95-107, 1989. 

The reader will appreciate that the rules for what maximizes what can be quite complicated to deduce and even to express. The safe way is to write the reactor design equations for the given set of reactions and then to numerically determine the best values for reaction time and temperature. An interior optimum may not exist. When one does exist, it provides a good starting point for the more comprehensive optimization studies discussed in Chapter 6. 

In early work in the optimal control theory design of laser helds to achieve desired transformations, the optimal control equations were solved directly, without constraints other than those imposed implicitly by the inclusion of a penalty term on the laser huence [see Eq. (1)]. This inevitably led to laser helds that suddenly increased from very small to large values near the start of the laser pulse. However, physically realistic laser helds should tum-on and -off smoothly. Therefore, during the optimization the held is not allowed to vary freely but is rather expressed in the form [60] ... 

The new pulse design equations for the optimal control of photodissociation may be summarized as... 

Thus, the design equations for a batch reactor for the optimization of a temporal superstructure can be based on differential or algebraic equations. 

Finding the optimal design according to the above definition amounts to finding the minimum of the total holding time, which is a function of all dr 3 s. Mathematically formulated, this involves finding the intermediate dr j-values subject to the following equation ... 

Furthermore, it can be shown that, in the limiting cases of first-order kinetics [Equation (11.35) also holds for this case] and zero-order kinetics, the equal and optimal sizes are exactly the same. As shown, the optimal holding times can be calculated very simply by means of Equation (11.40) and the sum of these can thus be used as a good approximation for the total holding time of equal-sized CSTRs. This makes Equation (11.31) an even more valuable tool for design equations. The restrictions are imposed by the assumption that the biocatalytic activity is constant in the reactors. Especially in the case of soluble enzymes, for which ordinary Michaelis-Menten kinetics in particular apply, special measures have to be taken. Continuous supply of relatively stable enzyme to the first tank in the series is a possibility, though in general expensive. A more attractive alternative is the application of a series of membrane reactors. 

Hill and Robinson (1989) derived an equation to predict the minimum possible total residence time to achieve any desired substrate conversion. They found that three optimally designed CSTFs connected in series provided close to the minimum possible residence time for any desired substrate conversion. 

The third-order D-optimal design is prepared relative to pseudocomponents Zj, Z2 and Z3 and the content of initial components at the design points is determined by Eq. (3.84). Table 3.40 presents the experimental conditions both in terms of pseudocomponents and on the natural scale (per cent). The sample variance here is S 0.53 and the number of degrees of freedom is f=13. From Eq. (3.105) for viscosity at 0 °C the coefficients have been calculated for the third-order regression equation ... 

A fourth-order D-optimal design is produced with reference to pseudocomponents Zi Z2 and Z3 - Table 3.42. The pseudocomponents satisfy the principal condition for Scheffe s designs. The conversion to initial components at any point within the local simplex studied is carried out from Eq. (3.84). According to this design, an experiment is run with mixtures, each observation being repeated twice. Using Eqs. (3.109)-(3.113) the coefficients of fourth-order regression equation are calculated in pseudocomponents... 

This example is taken from Mujtaba and Macchietto (1996). Here, the profit function (Equation 7.27) used by Logsdon et al. (1990) is considered for the multiple separation duties presented in section 7.3.4.2. Using the input data presented in Table 7.3 with (0, = 02 = 0 and tsul = tm2 - 0 hr) the optimal design and operation policies are obtained and the results are presented in Table 7.7. The profit per batch for each separation is calculated by multiplying the profit per hr and the batch time. 

Thermal systems can be completely described using balance equations for mass, energy, and entropy in conjunction with thermophysical property relations and/or equations of state, equipment performance characteristics, thermokinetic or rate equations, and boundary/initial conditions. With the thermal system adequately described, it can be optimized by any current technique. Although the approach presented in this paper is not explicit in Second Law terms, it never-the-less will yield the optimal design and with the appropriate transformations, will yield any desired Second Law quantity. 

The output of the function is the design matrix XN. A typical session for the construction of an exact D-optimal design for n = 3 variables, N = 10 points, and a quadratic model (Equation 8.75) is shown below ... 

The role of Eq. 9.57 and similar equations in the description and manipulation of modern chromatography is very broad, but the details are beyond the scope of this chapter (see Chapter 12). The equation can be used to optimize, design, and understand the behavior of both GC and LC systems. Because the equation is written to be valid for all chromatographic forms, its most important function, from the point of view of this chapter, is to show that a common theoretical framework stretches across all chromatographic methods. This conclusion is a microcosm of the theme of this book, that all separation methods are dominated by common basic features. 

With these particular examples before us we are in a position to give a general algorithm for optimal design in sequences of stirred tanks. The basic equations have been given in Section 3.2 and need not be derived again. They are ... 

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