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Development of New Thermodynamic Cycles
[report]

D. Yogi Goswami

2002
unpublished

In general, optimization is a way to find the best solution to a problem of any kind. We do a lot of optimization in our daily life. For example, we always search for the best product, which meets our demands, with the lowest possible price. In engineering, optimization is a very powerhl tool. There is usually more than one acceptable solution to an engineering problem. The goal of an engineer is to find the best one. When there are more than a few acceptable solutions, it may be simply
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... y be simply impossible to compare them by hand. More sophisticated mathematical methods have been developed to help find the best solution. Introduction to Optimization Mathematical Formulation Mathematically speaking, optimization is the minimization or maximization of a function subject to a set of constraints on its variables. For a thermodynamic cycle, the optimization objective usually is to find the maximum efficiency or work output. Constraints would be like approach temperatures in a heat exchanger. Engineering optimization problems, despite their diversity, have amazingly the same mathematical formulation. It can be written as: min Ax) s.t. h(x) = 0 g(x) 5 0 1 4 (4.1) Where x is the vector of a set of continuous real variables; f i s the objective function, a function of x that we want to minimize; h(x) = 0 and g(x) 5 0 are sets of equality constraints and inequality constraints, respectively. If an optimization problem is to maximize rather than minimizef, we can easily accommodate this change by minimizing -f in the formulation (4.1). A well-defined optimization problem features a number of variables greater than the number of equality constraints, which implies that there exist degrees of freedom upon which we optimize. Formulation (4.1) contains a number of classes of optimization problems, by appropriate consideration or elimination of its elements. If a problem doesn't have constraints on the variables, then formulation (4.1) becomes an unconstrained optimization problem. Otherwise it's a constrained optimization problem. Among constrained optimization problems, if both the objective function and all the constraints are linear functions of x, then formulation (4.1) becomes a linear programming (LP) problem. If at least one of the constraints or the objective function is nonlinear fbnction, formulation (4.1) becomes a nonlinear programming (NLP) problem. A special case of constrained optimization is bound-constrained optimization. In this special case, all constraints are boundary constraints on x ( L

doi:10.2172/803214
fatcat:ir524tx3uvbyvmhnxrpihiuacu