Heat and Mass Transfer - Advances in Science and Technology Applications

Figure 18. Flow chart of optimization procedure.

Figure 19. CCD model. (a) Two factors. (b) Three factors.

by axial points are set in CCD. The numerical results for DOE runs are utilized in reflecting the behavior of responses with geometrical and flow parameters.

#### 5.1.2 NSGA-II algorithm

Nondominated sorting genetic algorithm II (NSGA-II) combined with the multiobjective optimization is adopted in this study. The advantages of NSGA-II are a uniformly distributed Pareto-optimal front, which can suitably detect Paretooptimal front for multi-objective problems, decrease time consuming, and present solutions with a single run.

Figure 20 shows the NSGA-II flowchart. As specified in Figure 18, the RSM has been employed to determine the fitness functions in the optimization algorithm. As well, cross over and mutation contained in genetic operators are used in order to generate a new population. Finally, the optimization process is wrapped up with condition of repetitions number.

This algorithm uses two functions including nondominated sorting function and crowding distance function, respectively. This subprogram takes population members as input, ranks them, and puts them into fronts in proportion to their ranks. Crowding distance function has been designed to avoid the accumulation of

Heat and Mass Transfer in Outward Convex Corrugated Tube Heat Exchangers DOI: http://dx.doi.org/10.5772/intechopen.85494

population members in a limited distance. On the other hand, there are no blank intervals in the domain by using crowding distance function. The function is applied for comparison between members of a front that has equal ranks. Compared to the previous and the next member and also the first and the last member of the population, the normalization Euclidean distance of each solution of the front is for each reference point. Normalization is applied to avoid the problem that the objectives are in the different scale.

#### 5.2 Analysis of variance

ANOVA is one statistical analysis method used to evaluate the fitness of regression models, perform significance testing, and construct simplified regression models between design factors and objective functions. Tables 1 and 2 are ANOVA for Nuc and fc. According to the values of R<sup>2</sup> , the fitting degree of the RSM is estimated. The F value and P value indicate the influencing significances of the model terms, judging the significant degree of each model term for the global sensitivity analysis. If the model term is the most significant, the corresponding P value is minimum, and F value is maximum. Generally, the terms having a P value >0.05 are considered insignificant and are removed from the models.

#### 5.3 Regression model of responses

The regression response surface models are described in quadratic polynomial form. Coefficients in the models are determined based on a series of statistical and mathematical methods. The models evaluate the objective functions G including Nuc/Nus, fc/fs, and η, which are expressed as:

$$\begin{aligned} G &= b\_0 + b\_1 \cdot Re + b\_2 \cdot Y + b\_3 \cdot CR + b\_{1,2} \cdot Re \cdot Y + b\_{1,3} \cdot Re \cdot CR \\ &+ b\_{2,3} \cdot Y \cdot CR + b\_{1,1} \cdot Re^2 + b\_{2,2} \cdot Y^2 + b\_{3,3} \cdot CR^2 \end{aligned} \tag{20}$$


#### Table 1.

Analysis of variable (ANOVA) for Nuc.

In our optimum work, the regression response surface models for evaluating Nu<sup>c</sup> and fc are expressed as:

$$\begin{aligned} Nu\_c &= 64.89 - 51.44p/D + 1026.68H/D - 34.95r/D + 0.0046Re - 377.83p/D \cdot H/D \\ &- 1.82(E - 04)p/D \cdot Re + 0.0042H/D \cdot Re + 20.83(p/D)^2 - 8.12(E - 09)Re^2 \\ &\tag{21} \\ f\_c &= 0.077 - 0.05p/D + 1.21H/D - 0.037r/D - 7.76(E - 07)Re - 0.57p/D \cdot H/D \\ &+ 0.012p/D \cdot r/D - 0.38H/D \cdot r/D - 4.41 \ (E - 06)H/D \cdot Re \end{aligned} \tag{22}$$

#### 5.4 Response surface analysis

We applied 2D response surface contour plots to describe the regression response surface model, in order to display the interaction influence of each pair of design variables on the required responses. From the 2D response surface contour plots, the regulation of objective functions with changing design variables can be clearly observed, distinguished by contour plot color. Figures 21 and 22 show the 2D surface plots of the combined effects for the standard deviation of Nuc and fc. It


Heat and Mass Transfer in Outward Convex Corrugated Tube Heat Exchangers DOI: http://dx.doi.org/10.5772/intechopen.85494

Table 2. Analysis of variable (ANOVA) for fc.

Figure 21.

Response surfaces contour plots of combined effects for Nuc.

Figure 22. Response surfaces contour plots of combined effects for fc.

can be observed that the decrease of p/D, the increase of H/D, the decrease of r/D, and the increase of Re result in the augment of Nu. Moreover, it can be also seen that the decrease of p/D, the decrease of H/D, the increase of r/D, and the decrease of Re result in the weak of fc.

#### 5.5 Pareto front

By inspecting the numerical results of Nuc/Nus and fc/fs, it is found that these two responses are varied with the changes of the design parameters. There must exist design parameters corresponding to the optimal objective functions. The goal of optimization for a corrugated tube subjected the design constrains of structural limitation in this study is to find the optimal values of designing parameters to maximize Nuc/Nus and minimize fc/fs. In this study, the multi-objective optimization is executed by NSGA-II. The results for Pareto-optimal curve are shown in Figure 23, which clearly reveal the conflict between the two responses, Nuc/Nu<sup>s</sup> and fc/fs. Any changed design parameter that increases Nuc/Nus leads to an increase of fc/fs. It is worth noting that the minimum values of fc/fs with Nuc/Nus for various points on Pareto optimal front. Therefore, the reported results are applicable for a problem with one objective function (fc/fs) and specific constraint (the value of

Figure 23. Pareto-optimal curve.

selected or input Nuc/Nus). This means that the presented multi-objective optimization method provides a general optimal solution in simplified form, and one may obtain an optimum design (minimum of fc/fs and maximum of η) with a specified Nuc/Nus.
