Process variable space, optimization

Once a direction is estabflshed for the next poiat ia the space of the variables of optimization (whether by random search, by systematic evaluation of gradients, or by any other methods of making perturbations), it is possible to take a jump ia the directioa of the improvement much greater than the size of the perturbations. This could speed up the process of finding the optimum and reduce computer time. If such a leap is successful, the next iteration may take a bigger leap and so on, until the improvement stops. Then one could reverse the direction and decrease the size of the step until the optimum is found. 

Catalyst library design is considered as an optimization procedure in a multidimensional experimental space. The variables in the multi-dimensional space can be differentiated as follows (i) compositional variables, and (ii) process variables. The term compositional variables have already been discussed. 

Pareto-optimal solutions can be represented in two spaces - objective space (e.g., /i(x) versus /2(x)) and decision variable space. Definitions, techniques and discussions in MOO mainly focus on the objective space. However, implementation of the selected Pareto-optimal solution will require some consideration of the decision variable values. Multiple solution sets in the decision variable space may give the same or comparable objectives in the objective space in such cases, the engineer can choose the most desirable solution in the decision variable space. See Tarafder et al. (2007) for a study on finding multiple solution sets in MOO of chemical processes. 

To optimize process variables, such as space velocity, temperature, and pressure. If activation (for example, reduction or sulfiding) is required, proper procedures are developed at this time. 

Using flow reactors imder steady-state conditions, we can easily collect data for process optimization, record activity, and selectivity and study catalyst life and deactivation processes. If we know the contacting pattern in the reactor, then we can explore the kinetics from the reactor performance equation. All of the flow reactors described previously present data for the average reactor concentration versus time. Generally the activity, selectivity, and stability are presented as a function of different process variables such as temperature, pressure, and space velocity. From conversion we can calculate the rate from the performance equation of the reactor, for example, for a CSTR ... 

It can be argued that an approach based on the output space could overcome this deficiency. A practical problem in working with the output space is its potentially large dimensionality, especially for plantwide problems. If one were to include internal process variables, such as temperature, pressure, level, etc, the dimensionality of output space could become prohibitive even for problems of moderate complexity. Though it is essential to maintain many of these outputs within their acceptable intervals for reasons of safety, corrosion, optimality or other considerations, the focus of an initial operability study should be on a few critical process variables. 

Pareto optimality is a cornerstone concept in the field of optimisation. In single objective optimisation problems, the Pareto optimal solution is unique as the focus is on the decision variable space. The multi-objective optimisation process extends the optimisation theory by allowing single objectives to be optimised simultaneously. The multi-objective optimisation is considered as a mathematical process looking for a set of alternatives that represents the Pareto optimal solution. In brief, Pareto optimal solution is defined as a set of non-inferior solutions in the objective space defining a boimdary beyond which none of the objectives can be improved without sacrificing at least one of the other objectives [17]. 

In real-life problems ia the process iadustry, aeady always there is a nonlinear objective fuactioa. The gradieats deteroiiaed at any particular poiat ia the space of the variables to be optimized can be used to approximate the objective function at that poiat as a linear fuactioa similar techniques can be used to represent nonlinear constraints as linear approximations. The linear programming code can then be used to find an optimum for the linearized problem. At this optimum poiat, the objective can be reevaluated, the gradients can be recomputed, and a new linearized problem can be generated. The new problem can be solved and the optimum found. If the new optimum is the same as the previous one then the computations are terminated. 

To understand the mathematics, consider a large empty space into which a number of production units are to be placed, and assume that the major variable to be optimized is the cost of transporting materials between them. If the manufacturing process is essentially a flow-line operation, then the order in which units should be placed is clear (from the point of view of transport costs), and the problem is simply to fit them into the space available. In a job-shop, where materials are flowing between many or all the production units, the decision is more difficult. All the potential combinations of units and locations... 

In TLC the detection process is static (sepaurations achieved in space rather than time) and free from time constraints, or from interference by the mobile phase, which is removed between the development and detection process. Freedom from time constraints permits the utilization of any variety of techniques to enhance detection sensitivity, which if the methods are nondestructive, nay be applied sequentially. Thus, the detection process in TLC is nore flexible and variable than for HPLC. For optical detection the minimum detectable quantities are similar for both technlqpies with, perhaps, a slight advantage for HPLC. Direct comparisons are difficult because of the differences in detection variables and how these are optimized. Detection in TLC, however, is generally limited to optical detection without the equivalent of refractive... 

This method for optimizing a process parallels the method given in the mapping example. First, some limit must be placed on all variables. Otherwise it would be impossible to cover the entire surface. In the mapping example it was the continental boundaries. Second, for each independent variable a number of specific points that are uniformly spaced and cover its whole range are chosen. The objective... 

Reduced gradient method. This technique is based on the resolution of a sequence of optimization subproblems for a reduced space of variables. The process constraints are used to solve a set of variables (zd), called basic or dependent, in terms of the others, which are known as nonbasic or independent (zi). Using this categorization of variables, problem (5.3) is transformed into another one of fewer dimensions ... 

One of the simplest optimization tasks is aimed to select the proper catalyst combination and the corresponding process parameters. In this case the main task is to create a proper experimental space with appropriate variable levels as shown in Table 1. This experimental space has 6250 potential experimental points (N) (N = 2 x 5 = 6250). This approach has been used for the selection of catalysts for ring hydrogenation of bi-substituted benzene derivatives. The decrease of the number of variable levels from 5 to 4 would result in significant decrease in the value of N (N= 2 X 4 = 2048). 

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