Optimization design of heat pipe heat exchanger using response surface methodology

: In this paper, Optimization design of heat pipe heat exchanger (HPHX) is processed utilizing the Response Surface Methodology (RSM). The response surface model was built by regressive analysis using Latin hypercube experimental design method and numerical simulation. Through response surface analysis, it is found that the two input variables affecting the performance of HPHX are the heat pipe pitch and the Inlet and outlet distance. Moreover, the maximum value of the overall performance factor on the response surface is searched using genetic algorithm, and the optimal values of four input variables are obtained.


Introduction
HPHXs are widely used in industrial processes due to their excellent performance, and the design of HPHX is of great importance. The main objective of HPHX design is to achieve higher overall performance. In this paper, combined with numerical simulation, RSM is utilized to perform the optimization design of the HPHX.

Response surface methodology
The basic idea of RSM is to use simple polynomial model to describe the real complex model approximately in a certain value range [1]. Following steps are considered in RSM: 1. Generating a certain number of sample points in the value range of input variables by a reasonable experimental design method. 2. Calculating the values of corresponding output variables by numerical calculation. 3. Establishing an approximate model (response surface) between the input variables and the output variables using polynomial model. 4. Searching the global optimization point using optimization algorithms.

Latin hypercube experimental design method
In order to ensure that the fitted response surface is a reasonable approximation of the actual problem, it is necessary to properly use the experimental design method to generate a set of sample points, on the basis of which the response surface can be efficiently established. This paper will adopt Latin hypercube experimental design method.
Assuming that the output variable y is a  Latin hypercube sampling can be regarded as a compromise process, which combines many desirable characteristics of random sampling and stratified sampling.

Response surface model
The RSM replaces the actual complex model with a simple polynomial function. When the range of input variables is small, the lower-order polynomial model can get a better approximation effect. The lower-order polynomial model can be written in the following form: is obtained by the least square method [3]: 3 Optimization procedure Figure 2 shows a HPHX using 19 heat pipes as heat transfer elements. The performance of the HPHX is affected by many parameters. This paper utilizes RSM to perform the optimization design of the HPHX.

Numerical model
The heat pipes are assumed as solid thermal conductors with a constant thermal conductivity using thermal resistance model [4]. Velocity inlet and pressure outlet are applied to the inlets and outlets respectively. No-slip wall boundary conditions are applied to all wall surfaces.

Input variables
Four input variables are shown in figure 2. And their value range is shown in table 1.

Objective function
The overall performance factor (  ) of a heat exchanger is defined as follows [5]: where  is the dimensionless drag coefficient expressed as follows: The larger the  , the better the overall performance of  Table 2.

Discussion
Apply the least squares method to the data in Table 2 to get the regression equation of the response surface. The regression equation is as follows. A, B, C, D refers to L, Ue, Uc, H respectively. . -.
-.   Figure 4 shows the response surface of  with respect to L and H. And figure 5 shows the response surface of  with respect to Ue and Uc.

Response surface
The overall performance factor increases with the increase of H and L, and there is interaction between them. when H is larger, L has greater influence on the overall performance factor; the larger L is, the greater the influence of H on the overall performance factor. The overall performance factor decreases with the increase of Ue and Uc, and the influence of them is small.

Optimization design of HPHX
In this paper, the optimization design of HPHX takes two steps: the first step is to use genetic algorithm to search the largest value of overall performance factor on the response surface model, and the second step is to use numerical calculation method to verify it. The results are listed in Table 3. The error between the largest value of overall performance factor in RSM ( l η ) and the numerical result ( n η ) is 2.25%, which is in good agreement. The optimal values of L, H, Ue, Uc are 28.78mm, 66.67mm, 1.4134m/s, 12.919m/s.

Conclusions
In this paper, optimization design of HPHX using RSM is performed, the overall performance factor is taken as the optimization goal. The following conclusions are summarized.
(1) The increase of H and L is beneficial to improve the overall performance of the HPHX; The increase of Ue and Uc is not conducive to improving the overall performance of the HPHX. Among them, H and L has a significant impact, and Ue and Uc has a small impact.
(2) The maximum value of  on the response surface is obtained by genetic algorithm, and the optimal values of L, H, Ue, Uc are 28.78mm, 66.67mm, 1.4134m/s, 12.919m/s.