Many analytical and numerical methods for pipe and duct optimization have been developed during the last century. A comprehensive survey of existing numerical duct optimization methods was conducted by Tsal and Adler (1987). The first optimization method was developed by Grashoff in 1875 for a single pipeline. Several of the calculation procedures for duct optimization attempt to minimize total cost by establishing optimum velocities or friction rates. These procedures are based on the classical calculus minimization technique of setting the first derivative to zero in order to find the diameter of the pipe or duct or to determine air velocity.
The classical method of optimization for multi-path district heating systems was first applied by Shifrinson (1937) ref198 and for multi-path duct systems by Lobaev (1959) ref190. These techniques are impractical for manual calculation. According to Tsal and Adler (1987), analytical approaches can be effectively used only to identify trends in system behavior. A comprehensive analysis of a multi-path duct system was published by Bouwman (1982) ref162.
Computer-aided numerical optimization methods are divided into two categories, discrete methods, (coordinate descent, dynamic programming, and T-Method) and continuous methods (penalty function, Lagrange multipliers, reduced gradient, and quadratic search).
The coordinate descent method is the most common technique for duct optimization [Tsal and Chechik, 1968]. A number of different techniques are based on this method for selecting initial conditions, searching for the next duct section to be changed, and "freezing" selected diameters. Dynamic programming is one of the most powerful methods for multi-path tree network optimization [Tsal and Chechik, 1968]. The penalty function method transforms constrained problems into non-constrained problems, by adding penalty coefficients to the objective function. This method has been applied to different networks by Tsal and Chechik (1968). Another way to directly optimize a network is by the Lagrange multipliers method [Zanfirov, 1933 ref223; Bertschi ref161, 1969; Kovaric, 1971 ref187; Stoecker et al., 1971]. The modified Lagrange multipliers method has been applied for network optimization by Murtagh (1972) ref194. The reduced gradient method [Arklin and Shitzer, 1979 ref151] is one of the best computerized-techniques for application to rectangular duct optimization. It performs nonlinear optimization with equality constraints and then applies the Newton-Raphson technique to find an optimum solution. A technique called quadratic search was introduced for a concave problem optimization by Leah et al. (1987) ref189. It was applied for chilled water system optimization.
Many of these methods can find the minimum of an unconstrained concave problem, but most fail to yield a solution that can be successfully used in practice. In general, the objective function is not uniformly concave. An example in Tsal and Adler (1987) explains this phenomenon. There is no analytical or numerical method that can easily find the global minimum and satisfy all duct system constraints.
