4.1. Construction of Response Surface Model
The second-order polynomial response surface model was obtained based on
CCF fitting. The specific test design table is shown in
Table 2. The design parameters were
Re,
α,
Z1/
D and
Z2/
D, and the responses were
Nu/
Nu0 and
f/
f0. The coefficients of the response surface model are given in
Table 3.
Figure 6 shows the comparison between the numerically calculated values of the sample points of the experimental design and the corresponding
RSM predicted values. In the figure, the dotted line represents a deviation of ± 10% from the numerical calculation values, the straight line represents the numerical calculation values, and the scattered points represent the
RSM prediction values.
Figure 6a,b presents a comparison of the values when the responses are
Nu/
Nu0 and
f/
f0, respectively. It can be seen from
Figure 6 that the
RSM predicted values are distributed near the numerically calculated values, and the errors are basically less than 10%. After calculation, the root mean square error
RMSE and determination coefficient
R2 of the response surface model can be obtained. The calculation results are shown in
Table 4. When the responses are
Nu/
Nu0 and
f/
f0, the
RMSE of the models is less than 0.25 and
R2 is greater than 0.93, showing that the fitted response surface model has small error and high accuracy.
4.2. Effect of Channel Parameters on Flow and Heat Transfer
The three-dimensional surface and contour map of the
Nu/Nu0 and
f/f0 of the channels are shown in
Figure 7 and
Figure 8, respectively, to reveal the influence of channel parameters on the heat transfer performance and flow performance of the channels with frustums of a cone.
Figure 7a–f show the influence of the combined action of
Re–
α,
Re–
Z1/
D,
Re–
Z2/D,
α–
Z1/D,
α–
Z2/D and
Z1/D–
Z2/D on the response in turn.
As can be seen from
Figure 7a, when
Re is constant, increasing
α increases the
Nu/Nu0 of the channel, while when
α is constant, the increase of the
Nu/Nu0 of the channel along with the increase of
Re is not very significant. When
Re is 5000 and
α is 0°, the
Nu/Nu0 of the channel reaches its minimum value, while when
Re is 15,000 and
α is 30°, the
Nu/Nu0 of the channel reaches its maximum value. As can be seen from
Figure 7b, when
Re is constant, increasing
Z1/
D makes the
Nu/Nu0 of the channel first increase and then decrease. It can be seen from
Figure 7c that the increase in
Z2/
D under different values of
Re and the increase in
Re under different values of
Z2/
D can improve the
Nu/Nu0 of the channel. As can be seen from
Figure 7d,e, when
α is constant, the
Nu/Nu0 of the channel remains basically unchanged with increasing
Z1/
D and
Z2/
D. When
Z1/
D and
Z2/
D remain unchanged, increasing
α can significantly increase the
Nu/Nu0 of the channel. As can be seen from
Figure 7f, when
Z1/
D is constant and
Z2/
D is increased,
Nu/Nu0 of the channel first decreases and then increases. When
Z2/
D is constant, increasing
Z1/
D causes the
Nu/Nu0 of the channel to first increase and then decrease.
As can be seen from
Figure 8a, at low
Re, the
f/f0 of the channel first decreases and then increases with increasing
α, while at high
Re, increasing
α leads to an increase in the
f/f0 of the channel. When
α is constant, the
f/f0 of the channel increases with increasing
Re. It can be seen from
Figure 8b,c that increasing
Re and decreasing
Z1/
D and
Z1/
D result in an increase in the
f/f0 of the channel. As can be seen from
Figure 8d, when
α is constant and
Z1/
D is increased, and when
Z1/
D is constant and
α is increased, the
f/f0 of the channel first decreases and then increases. According to
Figure 8e, when
α is constant, the
f/f0 of the channel decreases with increasing
Z2/
D. When
Z2/
D remains unchanged, the
f/f0 of the channel first decreases and then increases with increasing
α. As can be seen from
Figure 8f, increasing
Z2/
D and
Z1/
D reduces the
f/f0 of the channel.
The above research shows that when analyzing the flow and heat transfer performance of channels with frustums of a cone, the information obtained limited to a fixed channel parameter is not sufficient to describe the performance of the channels. Building the function of channel performance related to channel parameters based on response surface method is of great significance to studying the influence of channel parameters on channel performance and guiding the parameter optimization and structural design of channels with frustums of a cone.
4.3. Sensitivity Analysis of the Channel Parameters
Figure 9 shows the first-order sensitivity index and total sensitivity index of channel parameters when the response is
Nu/Nu0. The first-order parameter sensitivity index represents the influence of a single parameter on the
Nu/Nu0 of the channels. The total sensitivity index represents the combined influence of a single parameter and its interaction with other parameters on the
Nu/Nu0 of the channels. As can be seen from
Figure 9a, when the response is
Nu/Nu0, the first-order sensitivity indexes of the channel parameters from high to low are
α,
Re,
Z2/
D and
Z1/
D. Among them, the changes of
α and
Re have an important influence on the
Nu/Nu0 of the channel. According to
Figure 9b, the total parameter sensitivity indexes are
α,
Re,
Z2/
D and
Z1/
D from high to low, which is the same as the ranking of the first-order sensitivity indexes of the channel parameters. Through calculation, the difference between them is less than 0.02, indicating that the interaction between a single parameter and other parameters of the channels has no significant impact on the
Nu/Nu0 of the channels. In addition, the proportions of
α and
Re in the total sensitivity index are 50.6% and 47.9%, indicating that
α and
Re have a greater impact on the
Nu/Nu0 of the channels compared with
Z1/
D and
Z2/
D.
Figure 10 shows the sensitivity index of the channel parameters when the response is
f/f0, where
Figure 10a and
Figure 10b are the first-order sensitivity index and total sensitivity index, respectively. It can be seen from
Figure 10a that when the response is
f/f0, the first-order sensitivity indexes of the channel parameters are
Re,
Z1/
D,
α and
Z2/
D from high to low. Among them, the first-order sensitivity index of
Re is significantly higher than other channel parameters, and the first-order sensitivity indexes of
α,
Z1/
D and
Z2/
D are basically the same, all distributed around 0.15. According to
Figure 10b, the total sensitivity indexes of the parameters are
Re,
Z1/
D,
α and
Z2/
D from high to low, which is the same as the first-order sensitivity indexes. Through calculation, it can be seen that the difference between the two is less than 0.021, indicating that the interaction between a single parameter and other parameters of the channels has no significant impact on the
f/f0 of the channels. In addition, the proportion of
Re in the total sensitivity index is 57.4%, while the proportions of the total sensitivity coefficient of the other three channel parameters are all about 15%, which indicates that
Re has the greatest impact on the
f/f0 of the channels, while
α,
Z1/
D and
Z2/
D have a fairly small impact.
4.4. Multi-Objective Optimization Results of the Parameters
With the aim of obtaining the maximum values of
Nu/Nu0 and the minimum values of
f/f0 for the channel, the
NSGA-II was used to find the optimal combination of channel parameters in the global range. The population number of the genetic algorithm was 12, the genetic algebra was 40, the crossover probability was 0.9, the mutation probability was 0.1, the crossover distribution index was 10, and the mutation distribution index was 20. The specific settings can be found in Ref. [
24]. After the operation, a total of 481 solutions were generated, of which the Pareto solution set, the set of optimal solutions, had a total of 130 solutions.
Figure 11 shows the solution set of multi-objective optimization. In
Figure 11, the blue dots represent all of the solution sets, and the red curve represents the Pareto front connected by the Pareto solution sets. According to
Figure 11, when the
f/
f0 of the channel is constant, the
Nu/
Nu0 of the channel of the point on Pareto front must be at its maximum. Similarly, when the
Nu/
Nu0 of the channel is constant, the
f/
f0 of the channel of the point on the Pareto front must be at its minimum.
On the basis of the sensitivity analysis of the parameters, the parameter
Re has the greatest impact in the performance of the channel in terms of the flow and heat transfer performance of the channel. Consequently, the K-means clustering algorithm was used to cluster the Pareto solution sets under different values of
Re.
Figure 12 illustrates the results of K-means clustering of the Pareto solution set. As can be seen from
Figure 12, the Pareto solution sets can be divided into four categories—A, B, C and D—under different values of
Re. Without considering the influence of
α,
Z1/
D and
Z2/
D, when
Re increases, the
Nu/
Nu0 and
f/
f0 of the channel under the Pareto solution set increase slightly.
An optimal solution was selected from each of the four categories A, B, C and D in
Figure 12 for comparative analysis. The specific optimal parameter combinations are provided in
Table 5. According to
Table 5, the
Nu/
Nu0 of optimization points A, B, C and D increased by 9.70%, 21.82%, 26.06% and 27.88%, respectively, compared with the reference channel. In addition, the
f/
f0 of the optimization points decreased by 19.89%, 1.05%, –7.85% and –9.42%, respectively, compared with the reference channel. Among them, the
Nu/
Nu0 and
f/
f0 of optimization points A and B were optimized, while the
Nu/
Nu0 of optimization points C and D was considerably improved, but the
f/
f0 had increased moderately. This is because the values of
Re for optimization points C and D are large. When the
Nu/
Nu0 increases,
f/
f0 will also increase. Overall, compared with the reference channel, the
Nu/
Nu0 of the optimized channels increased by 21.36% on average, and the
f/
f0 decreased by 9.16% on average. This shows that the optimization results of the channel parameters in the present study are good, and can serve as a reference for the multi-objective optimization of channels with turbulent structures.
To further explore the influence of channel parameters on the flow and heat transfer performance of the channels,
Figure 13 shows the comparison of surface streamline, temperature distribution, and
Nu distribution of the heat transfer walls of the reference channel and optimization point C.
Figure 13a,c shows the reference channel, and
Figure 13b,d gives the optimization points C. It can be seen from
Figure 13a,b that the high-temperature area of the heat transfer wall is mainly distributed upstream and downstream of the convex and upstream of the bottom of the concave. In comparison, the temperature of the high-temperature region of the optimized channel is lower and the area with high temperature is smaller. There are large vortexes upstream of the bottom of the concave and upstream and downstream of the convex in the reference channel, while the vortexes in the optimized channel are improved. The improvement of the vortexes will reduce the frictional resistance of the channel and the accumulation of airflow, thus reducing the local temperature of the heat transfer wall and reducing the area with high temperature. According to
Figure 13c,d, contrary to the temperature distribution of the heat transfer wall, the high-temperature area had a lower
Nu and the low-temperature area had a higher
Nu. In comparison, the
Nu value of the high-
Nu region of the optimized channel was higher and the area with high
Nu was larger. The results show that the optimized channel improves the vortexes at the bottom of the concave and upstream and downstream of the convex, so that the heat transfer wall of the channel has a lower temperature distribution and a higher
Nu distribution.