Objective function, design process

Uncertainties in amounts of products to be manufactured Qi, processing times %, and size factors Sij will influence the production time tp, whose uncertainty reflects the individual uncertainties that can be presented as probability distributions. The distributions for shortterm uncertainties (processing times and size factors) can be evaluated based on knowledge of probability distributions for the uncertain parameters, i.e. kinetic parameters and other variables used for the design of equipment units. The probability of not being able to meet the total demand is the probability that the production time is larger than the available production time H. Hence, the objective function used for deterministic design takes the form ... 

Optimization problems in process design are usually concerned with maximizing or minimizing an objective function. The objective function might typically be to maximize economic potential or minimize cost. For example, consider the recovery of heat from a hot waste stream. A heat exchanger could be installed to recover the waste heat. The heat recovery is illustrated in Figure 3.1a as a plot of temperature versus enthalpy. There is heat available in the hot stream to be recovered to preheat the cold stream. But how much heat should be recovered Expressions can be written for the recovered heat as ... 

Step 6. You should always be aware of the sensitivity of the optimal answer, that is, how much the optimal value of C changes when a variable such as D changes or a coefficient in the objective function changes. Parameter values usually contain errors or uncertainties. Information concerning the sensitivity of the optimum to changes or variations in a parameter is therefore very important in optimal process design. For some problems, a sensitivity analysis can be carried out analytically, but in others the sensitivity coefficients must be determined numerically. 

Constraints in optimization arise because a process must describe the physical bounds on the variables, empirical relations, and physical laws that apply to a specific problem, as mentioned in Section 1.4. How to develop models that take into account these constraints is the main focus of this chapter. Mathematical models are employed in all areas of science, engineering, and business to solve problems, design equipment, interpret data, and communicate information. Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in a usable form. For the purpose of optimization, we shall be concerned with developing quantitative expressions that will enable us to use mathematics and computer calculations to extract useful information. To optimize a process models may need to be developed for the objective function/, equality constraints g, and inequality constraints h. 

Objective functions that allow only discrete values of the independent variable ) occur frequently in process design because the process variables assume only specific values rather than continuous ones. Examples are the cost per unit diameter of pipe, the cost per unit area for heat exchanger surface, or the insulation cost considered in Example 1.1. For a pipe, we might represent the installed cost as a function of the pipe diameter as shown in Figure 4.2 [see also Noltie (1978)]. For... 

Snapshots illustrate specific example situations. Figure 6.18 shows snapshots depicting the state of our spreadsheet before and after an operation. (The thicker lines and bold type represent the state after the operation.) Notice that because we are dealing with a requirements model here, we show no messages (function calls) between the objects they will be decided in the design process. Here we re concerned only with the effects of the operation invoked by the user. This is part of how the layering of decisions works in Catalysis We start with the effects of operations and then work out how they are implemented in terms of collaborations between objects. 

Use case specifications document functional requirements. The next step is to design the partial system that the current iteration is supposed to deliver. The gap between requirements and design is not trivial, and a bridge between the two is desired. This bridge is what object-oriented analysis is about. The domain analysis object model is not the final design. However, it provides a starting point for the design process. 

We start by studying the steady-state design and economics of a process with a single adiabatic reactor. The design considers the entire plantwide process reactor, heat exchangers, gas recycle compressor, preheat furnace, condenser, and separator. The economic objective function is total annual cost, which includes annual capital cost (reactor, catalyst, compressor, and heat exchangers) and energy cost (compressor work and furnace fuel). 

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