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thermoforming of a plastic and for ignition to occur in an engine. Thus, both minimum and maximum values of the design variables may be involved in constraints. Many constraints in thermal systems arise because of conservation laws, particularly those related to mass, momentum, and energy. For instance, under steady-state conditions, mass inflow to the system must equal mass outflow. This condition gives rise to an equation that must be satisfied by the relevant design variables, thus restricting the values that may be employed in the search for an optimum. Similarly, energy balance considerations are important in thermal systems and may limit the range of temperatures, heat fluxes, dimensions, etc., that may be used. Several such constraints are often satisfied during modeling and simulation because the governing equations are based on conservation principles. In this way, the objective function being optimized already considers these constraints. In such cases, only the additional limitations that define the boundaries of the design domain remain to be considered. 5.4.4 Optimization methods There are several methods for optimization described as follows: 5.4.4.1. Classical optimization Classical optimization techniques are useful in finding the optimum solution or unconstrained maximum or minimum of continuous and differentiable functions. Some specifications for classical optimization can be selected based on this understanding, as described below: These are analytical methods that make use of differential calculus in locating the optimum solution. Classical methods have limited scope in practical applications as some involve objective functions which are not continuous and/or differentiable. These methods assume that the function is differentiable twice with respect to the design variables and that the derivatives are continuous. Three main types of problems can be handled by classical optimization techniques: Single variable functions. Multivariable functions with no constraints. Multivariable functions with both equality and inequality constraints. In problems with equality constraints the Lagrange multiplier method can be used. If the 64PDF Image | Energy Systems for Multigeneration Purposes
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