orthogonal functions

In this paper, optimization of CU element profile of Ljungstrom in power plant has been investigated in order to increase its performance, reduce pressure drop and reach minimum price. First, a three-dimensional model was developed and then the desired model is simulated. To optimize the proplem, geometry parameters rewrite in form of unknown coefficient vectors by using orthogonal functions and then by using adjoint method, optimal value of geometric coefficients, plates angle and flow velocity are obtained. Objective function is defined based on the maximum Nusselt number, minimum pressure drop coefficient and minimum price and according to them, the design parameters are determined. So, with angles between 20 and 70 degrees for different Reynolds numbers, the optimal coefficients of the profile are determined and variations of the coefficients with ratio of the heat transfer to the pressure drop are investigated. Results show that when pressure drop is more important, small Reynolds number and angle are more efficient, and when heat transfer is important, large Reynolds number and angle are effective. For angles about 70 degrees and Reynolds number about 8000, maximum of function is obtained. At angles greater than 70 and at Reynolds numbers greater than 8000, although the heat transfer coefficient increases, the flow experiences high pressure drop, which reduces objective function.

The problem of solving optimal control of Singular problems in the classic method has a complexity that is solved by approximation of the equations in the problem with orthogonal bases instead of solving the dynamic equation system of a set of static problems. In return for a more relaxed solution, it will face some errors in the computation .however, it has an appropriate precision. Legendre and Fourier series are presented using the specifications of the Fourier transform of Legendre and Fourier series . In this algorithm, the state variables, and the state - derivative variables and the control vector are extended by the orthogonal basis of Legendre and Fourier series with unknown coefficients. in order to compute optimal control vector and optimal path of linear Singular systems with quadratic cost function , we are introduced by using the properties of orthogonal functions introduced by the coefficients and .using the proposed method , the system dynamics are converted into algebraic equations and the problem of dynamic optimization of dynamic space has been mapped to static space optimization problem with quadratic cost function and linear constraints . First, it is used to solve the problem using an orthogonal basis of raw material and then the problem solving with orthogonal basis of Fourier series is repeated. Finally, the application and effectiveness of the proposed method are presented.