Shape design for power generation system using the finite element and the adjoint variable methods

Asian Congress of Structural and Multidisciplinary Optimization 2020 (ACSMO2020) May 24-28, 2020 / Seoul, Korea

Shape design for power generation system using the finite element and the adjoint variable methods

Takashi Aoki¹, Takahiko Kurahashi^1* and Eiji Katamine²

1Department of Mechanical Engineering, Nagaoka University of Technology, 940-2188, Nagaoka, Niigata, Japan

* Corresponding author: [email protected]

2Department of Mechanical Engineering, National Institute of Technology, Gifu College, 501-0495, Motosu, Gifu, Japan

Abstract In this study, we present some results of the shape optimization analysis in natural convection field for the power generation system. As an example of numerical analysis, we discuss about the power generation system about called the “Solar Updraft Tower” shown in Fig. 1. The “Solar Updraft Tower” is one of the power generation system, and generates the electricity using the convection generated by the heat. The turbine is installed in the “Solar Updraft Tower”

for power generation, and the power generation efficiency is determined by the rotational speed of the turbine. The purpose of this study is to obtain an optimal outer shape of the “Solar Updraft Tower” so as to approach closely the target flow velocity distribution, and some numerical analyses are carried out including the “Solar Updraft Tower”. The performance function is defined by the square error between the flow velocity distribution in the domain and the target flow velocity distribution in target domain. The adjoint variable method is applied to consider the governing equations for minimization problem of the performance function, and the Lagrange function is obtained. As the governing equations, the Navier-Stokes equation, the continuity equation and the heat transfer equation are introduced to represent the natural convection. The stationary conditions are derived by the calculation of the first variation of the Lagrange function. Also, the shape gradient for this minimization problem is derived by the first variation of the Lagrange function. Reshaping is accomplished using a traction method that is proposed as a solution to the domain optimization problems [2]. The traction method is an algorithm for non-parametric shape optimization problems, and the smooth shape can be obtained by using this method. FreeFem++ [3] is used to calculate the optimal shape as the numerical analysis software. The computational result of comparison of initial and optimal shapes is shown in Fig. 2.

Fig. 1 Solar Updraft Tower. [1] Fig. 2 Comparison of initial and optimal shapes.

References

1. J.Schlaich, R.Bergermann, The Solar Updraft Tower: Das Aufwindkraftwerk Motivation and Concept, SCPT 2010 Proc 2010.

2. Azegami, H., Solution to domain optimization problems, Transactions of the Japan Society of Mechanical Engineers, Series A, Vol.60, No.574 (1994), pp.1479-1486 (in Japanese).

3. Hecht, F., New development in FreeFem++. Journal of Numerical Mathematics, Vol.20, No.3-4, (2012), pp.251-265.

65Y15.