4.1. ANOVA Parameter Modeling
The numerical simulation was performed using the parameter design geometry matrix listed in
Table 2. The resulting data were collected and entered into a factorial design table to facilitate analysis of the response surface design fitted model.
In an ANOVA, Adj SS and Adj MS represent the adjusted sum of squares and adjusted mean square, respectively, both of which are used to measure the aberrance in the different components of the model. The F-value is the test statistic. It is used in models or terms to determine whether a term is associated with a response and in lack-of-fit terms to determine whether higher-order terms containing predictor variables are missing from the current model. A sufficiently large F-value indicates that the term or model is highly significant. The Adj SS, Adj MS, and F-values are used by Minitab to calculate the
p-values of the terms. Decisions about the statistical significance of the terms and models are made from the
p-values. The
p-value represents the probability of rejecting the original hypothesis in a hypothesis test. If the
p-value is less than or equal to the predefined significance level α, it indicates a statistically significant relationship between the response characteristics and the variable. Conversely, a
p-value greater than α indicates non-significance. Usually, α is set at 0.05 [
26]. The lack of fit concept in ANOVA is used to evaluate whether the chosen model effectively fits the data and can adequately account for differences between groups. In ANOVA, if the
p-value from the lack of fit test is greater than the predetermined significance level (usually 0.05), then the hypothesis that the model adequately fits the data is not supported by sufficient evidence to be rejected. Conversely, if the
p-value from the lack of fit test is less than the significance level, it indicates that the model is having difficulty explaining the data and may need to consider of a more complex model or other adjustments to better capture the variability present in the data.
Appendix A presents the ANOVA results of the quadratic model for
Nu before elimination. The results indicate that some items are insignificant and can be removed. The lack of fit test also shows insignificance. Specifically, the main item X-direction position (C) is insignificant but its quadratic item is significant. Therefore, the main item X-direction position (C) is retained.
The insignificant terms were removed from the quadratic model for the
Nu using a backward elimination process.
Table 3 shows the resulting ANOVA data of the reduced quadratic model for the
Nu. In
Table 3, the main items’ apex angle (A), baffle angle (B), and
Re (E), the squared items’ baffle angle (B * B), the squared items X-direction position (C * C), and the interaction item baffle angle *
Re (B * E) all have
p-values less than 0.05, indicating their significance. As the squared item for the X-direction position (C) is also significant, the main item X-direction position is retained. As the
p-value for the lack of fit item is greater than 0.05, it indicates that the model does not have a significant lack of fit. Therefore, the second-order model is considered appropriate and there is no need for a more complicated model.
A Pareto standardization effect plot can provide more detailed insight into the relative influence of factors on the response variable, as well as their statistical significance, helping to identify influences that may have been missed in separate ANOVA tests [
37]. The plot represents the effects of each factor with bar graphs, where the length of each bar indicates the magnitude of the influence on the response variable. Impact values are usually normalized for comparison purposes and a reference line is plotted on the graph, typically at a specific level of significance (e.g., 0.05). If the value of a factor in
Figure 9 exceeds this threshold, it indicates that the factor is statistically significant. The greater the excess, the more significant the factor.
Figure 9a depicts the Pareto chart of the standardization effects of the factors. It is observed that
Re has the greatest influence on
Nu, with a value more than twice that of the second most important factor. The other importance factors, in decreasing order of importance, are the baffle angle (B), quadratic items baffle angle (B * B), quadratic items X-direction position (C * C), apex angle (A), interaction item between baffle angle and the
Re (B * E), and finally the X-direction position (C).
Re indicates the state of fluid flow. The higher the
Re, the greater the turbulence intensity, which has the greatest effect on the
Nu. The angle and position of the baffle cause different degrees of fluid mixing between the center of the channel and the corrugated plate. It is important to note that
Re reflects the state of the fluid flow, with a higher value indicating greater turbulence. As a result, the convective heat transfer coefficient will be greater, resulting in a stronger heat transfer capacity.
Appendix B presents the results of the quadratic model for
f in the form of ANOVA before elimination. The main significant items for
f, as shown in
Appendix B, are the apex angle (A), baffle angle (B), quadratic items baffle angle (B * B), and quadratic items X-direction position (C * C).
Table 4 shows the resulting ANOVA data of the reduced quadratic model for the
f. It retains the main effect of the X-direction position (C) since its squared item is significant. In
Appendix B,
Re (E) is non-significant but removing it results in a
p-value from the lack of fit test being less than the significance level.
Table 4 shows that keeping
Re results in a lack of fit greater than the significance level, indicating that the model does not lack fit. The
Re is a turning point parameter and is therefore kept.
Furthermore, it has been discovered that the interaction between the baffle angle and Re is of significant importance. The baffle angle has the greatest impact on f, with the other factors decreasing progressively. This is due to the baffle causing the fluid to collide with the wall, resulting in energy loss and a variation in the pressure drop in the pipe. The second factor is the apex angle of the flow channel. In a flow channel with a small apex angle, the fluid flow is deflected more severely, resulting in greater energy dissipation and a greater pressure drop. The position of the channel also affects the disturbance of the upper and lower walls, which ultimately affects the f.
Based on the statistical analysis of the response results, the regression equation for the simulated data is determined using the following model:
The sign in front of each item in the equation determines the effect of that item on a particular response. A positive sign indicates a synergistic effect, while a negative sign indicates an antagonistic effect. Within the range of values for each factor, the equation incorporates the actual value to calculate the corresponding response value. The 32 groups of simulation conditions were taken into the regression equation for comparison and validation; the results are shown in
Figure 10. For
Nu, the root mean square error (RMSE) of the model is 2.081. The smaller the value and closer it is to two, the more acceptable the value. A group of outlier observations exists at NO.32. For
f, the RMSE of the model is 0.098. Three groups of outlier observations exist at NO.27, NO.29, and NO.32. In general, the regression equations for
Nu are better than
f and this result corresponds to
Table 5. Due to the finite degree of fit, they are the optimal model that can be reached. For the cross-corrugated triangular plate heat exchangers with interpolated trapezoidal in this study range, the regression equation is able to predict the simulated values relatively accurately in the majority of scenarios. It could provide guidelines for the design and optimization of cross-corrugated triangular flow channels under non-standard conditions. Moreover, further observations were made by choosing the same
Re (1000) cases in 32 groups of models for the nine NO.4, NO.5, NO.9, NO.10, NO.12, NO.14, NO.16, NO.21, and NO.22 values, respectively.
The model summary in
Table 5 confirms the fitting results. ‘S’ represents the standard deviation of the distance between the data values and the fitted values; the lower the ‘S’ value, the better the model describes the response value. ‘R-sq’ is used to determine how well the model fits the data. The higher the ‘R-sq’ value, the better the model fits the data. The ‘R-sq’ (Adjustment) is the ratio of the number of predictor variables. In cases where you want to compare variables with different numbers of predictors, using the ‘R-sq’ (Adjustment) when comparing variables with different numbers of predictors. The ‘R-sq’ (Projections) measures the extent to which the model predicts the response to new observations. Models with higher predictive ‘R-sq’ values have better predictive ability. In this model, the goodness of fit statistic R-sq, R-sq (Adjustment) and R-sq (Projections) for
Nu are all close to 1, indicating a goodness of fit to the simulated data. The results for
f are also good, although not as strong as for
Nu. 4.2. Analysis of Factors
Figure 11 and
Figure 12 show the main effects of
Nu and
f, respectively, indicating the impact of individual factors on the observed trends. A horizontal line (parallel to the X-axis), indicates no main effect, while a steeper line indicates a greater magnitude of the main effect.
Figure 11 depicts an effect plot of the
Nu. It is clear that the baffle angle and X-direction position factors exhibit a bend in the examined range, reaching a maximum at the top of their parabola, especially at 65° and 0.612H, respectively. The relationship between the factor of apex angle,
Re, and
Nu is linear, with a maximum at 120° and 3000. Cross-corrugated triangular ducts exhibit superior heat transfer performance under these conditions, with a corresponding
Nu fitting value of 36.78. The results were verified by simulation to be 36.54. This value is 1.54 times that of a non-baffled channel with the same apex angle and
Re.
Figure 12 shows an effect plot of the
f. The baffle angle and X-direction position factors reach their minimum values at 15° and 0.875H, respectively. The apex angle and
Re factors are linearly related to
f, reaching their minimum values at their rightmost points, 120° and 3000, respectively, with corresponding
f fitting values of 0.181. The results were verified by simulation to be 0.225. This value is close to the non-baffled channel and even lower than it by 0.007 with the same apex angle and
Re.
When optimizing the performance of cross-corrugated triangular ducts, it is desirable to maximize Nu and minimize f. The response prediction value suggests that the optimal model has an apex angle of 90°, a baffle angle of 52.5°, and a distance of 0.625H in the X-direction with a Re number of 1000. It is important to maintain a balance between Nu and f to achieve optimum performance. At this point, the value of Nu is 16.45 and f is 0.98, resulting in a calculated PEC of 1.22. This represents a significant improvement compared to the case over no baffle.
Figure 13 shows a contour plot of
Nu against each factor, allowing for exploration of the desired response values and operating conditions. The plot illustrates the relationship between the two significant predictor variables and the response variable
Nu. The darker colors indicate higher values and better quality. Using the interaction between the apex angle and baffle angle as an example (see
Figure 13a) and holding the other variables constant,
Nu is expected to be above 28 for an apex angle greater than 98° and a baffle angle between 46° and 80°. Conversely, when the apex angle is between 60° and 110° and the baffle angle is less than 22°,
Nu is just below 18, which is a difference of 1.6 times in the results. The remaining part of the
Nu plot exhibits differences ranging from 1.3 to 2.3 times.
Figure 14 shows a plot of the contours
f against each factor. To ensure the smallest possible expected value of
f, the optimal range for
f is limited to the light green and blue areas. For example, when considering the interaction between the apex angle and baffle angle (see
Figure 14a), while holding the other variables constant,
f should be less than 0.4 when the apex angle is between 85° and 120° and the baffle angle is less than 20°. Conversely, when the apex angle is between 60° and 85° and the baffle angle is 40° and 80°,
f should be over 0.9, which represents a difference of 2.3 times in the results. The remainder of the
f plot also exhibits a difference ranging from 1.5 to 2.0 times.
To visualize the influence of the baffle inclination on the heat transfer and pressure drop characteristics, a comparison is made with the existing vertical trapezoidal and triangular baffles under the conditions of the geometry with the best comprehensive performance (apex angle of 90°, baffle inclination angle of 52.5°, baffle positions of 0.625H and 0.5L). The
Re was chosen for the range of overlap (1000–2700).
Figure 15 and
Figure 16 show the comparison results for
Nu and
f, respectively. It can be seen from the figure that the inclined baffle attenuates both
Nu and
f compared to the vertical baffle, while the comprehensive performance needs to be judged from
Figure 17. A significant increase in
PEC can be observed for the inclined baffle at low
Re. This is because the inclined baffle has a minor difference in
Nu compared to the vertical baffle, while
f is far less than that of the vertical baffle. As the
Re increases, the increase in
PEC is no longer evident. As soon as the intersection point (near 1750) is exceeded, the comprehensive performance is lower than that of the vertical baffle. Thereafter, the gap in
Nu gradually becomes wider, while the change in
f is slight, so the performance is below that of the vertical baffle. Furthermore, it is observed that the trapezoidal baffle is generally ahead of the triangular baffle.
4.3. Velocity Field Distribution
Figure 18 shows the velocity field distribution from the fifth to the eighth fluid cell at a
Re of 1000. For NO.5, it is evident that when the baffle is in the lower half and inclined at 15°, the fluid in the upper half flows through the channel in an advective manner with minimal disturbance. A slight disturbance is only observed near the corrugated groove, where a clockwise vortex appears in the groove. By placing the same baffle on the upper part of the NO.22 channel, it reduces the flow rate of the upper fluid and causes more fluid to flow to the lower part. Although more vortices are generated at the corrugation, the speed is not as high as before. In NO.16, when the baffle is inclined 90°, vortices become visible at the rear of the baffle. The density of vortices in the slot increases and the collision between the fluid and the baffle intensifies, resulting in more adequate mixing. This results in a notable rise in pressure difference, which subsequently increases
f, making it more probable for vortices to form at larger baffle angles. When the baffle is located in the lower half, a significant backflow phenomenon arises in the NO.14 slot, with some of the fluid flowing back to the front through the gap between the baffle and the slot. The flow patterns of models with 90 and 120° apex angles are similar to those of the 60°.
The computation shows that the highest PEC value among the nine cases is NO.12 with 1.22, followed by NO.10 with 1.21. It is obvious that both cases have significant vortices in the channel. In NO.12, the baffle is positioned in the center, with a larger area. This results in a uniform discharge of fluid in both the upper and lower parts of the flow channel, which produces noticeable vortices on the lower wall surface of the groove. The gas rotates along the groove wall, effectively promoting the mixing of the middle and bottom fluid. This intensifies the internal friction and collision of the fluid, thus increasing energy consumption. The most optimal gas separation results are achieved at this junction. In NO.10, the channel height is at its minimum and the vertically oriented baffle accelerates the fluid flow in its vicinity, resulting in a greater dependence of heat transfer on particle movement and a thinner hydrodynamic and thermal boundary. Additionally, the vortices created disturbances in the boundary layer, further thinning it.