3.1. Optimization in 2D and Transfer to 3D
First, the 2D structure is optimized on the basis of the reference structure. A FFSF of 15 is used for the optimization, as this represents the best compromise between shape change and mesh quality. This is kept constant for further design iterations.
The exact cost function FLUENT is using is not known; for the 2D multi-objective optimization the goal is to minimize the pressure loss and maximize the fluid outlet temperature, so the optimization objective is:
is the mass-weighted average fluid temperature at the outlet and
the overall pressure loss of the entire fluid domain (refer to
Figure 4).
The optimization can now be carried out using the determined FFSF. The weight factors are
and
to increase the heat transfer at a constant pressure loss; this structure is referred to as A27. The weights were obtained by a “try-and-error” method. The optimization is initially carried out for a Reynolds number of Re = 200.
Figure 11 shows the contour plots of the velocity and temperature for different design iterations with the same weights and the same FFSF. This clearly shows that with increasing design iterations, the fin structures are increasingly drawn out in length and interlock with each other. Furthermore, a more pointed design of the stagnation points can be seen, which, in combination with the entanglement, leads to thin thermal boundary layers. The analysis of the change in geometry also shows that the change in geometry slows down after 140 design iterations.
Furthermore, after 140 design iterations, a manufacturing limit is reached, particularly at the leading and trailing edge of the fin, as also stated in [
33].
Figure 12 illustrates this case. Due to the limited diameter of the laser spot of ~50 µm, the leading and trailing edges of the fins cannot be resolved by the laser and, therefore, are not manufactured accurately. As a consequence, further optimization in this area would not make any sense since it would have no effect on the later experiments.
As a result, for 140 iterations, an increase in the heat transfer rate of 16.1% is achieved for A27, while the pressure loss decreases by 0.5%.
Based on the optimization with the designation A27, further optimizations are now carried out to reduce pressure loss. For the single objective optimization the objective function changes to
with
and a FFSF of 15. In order to prevent this from happening simply by reducing the fin cross-section, any change in shape in the
y-direction is suppressed. The reason for this is that the fin efficiency cannot be taken into account in 2D optimization. If an optimization is carried out with a focus on a lower pressure loss, there is a risk that the fins will be made thinner, which would drastically reduce the fin efficiency. Furthermore, the manufacturability limit would be undercut, especially in the inflow and outflow area of the fin, which would mean that the optimization could no longer be verified experimentally. A total of 17 design iterations are carried out.
Figure 13 shows the fin shapes after design iterations 5, 11 and 17. These structures are referred to as B27, C27 and D27.
The figure shows that the fins become shorter as the number of iterations increases and the overlap decreases, which reduces the pressure losses, but at the same time the product of heat transfer coefficient and heat transfer area also decreases. As a result, a reduction in pressure loss of 14.3% and an increase in the “heat transfer*area” product of 3.7% is achieved for structure D27; see the 2D values in Figure 15.
In the next step, these shapes are converted into three dimensions in order to achieve better comparability, also taking into account the fin efficiency. For this purpose, the optimized 2D geometries are exported as .stl files, converted into a solid body and transferred into 3D, as indicated in
Figure 14. The structures have an inclination angle of 37° to the vertical to enable additive manufacturing; the normal height is 2.8 mm. These 3D structures are then meshed and provided with boundary conditions in accordance with
Figure 5. The calculations are performed with both aluminum and stainless steel to evaluate the influence of different materials.
For the 3D case, no improvement compared to the 3D-reference structure in heat transfer can be determined for the stainless-steel variant, which is between 91 and 95% of the reference structure, see
Figure 15. As a result of the thin thermal boundary layer at the fin tip, there are high heat transfer coefficients, which only lead to low heat flux densities compared to the reference structure due to the smaller solid cross-section at the fin tip. This leads to a greater drop in performance, particularly with structure A27, as the greater overlap of the fin rows results in higher heat transfer coefficients at a position with a small fin cross-section than is the case with the other structures.
An analysis of the pressure loss for stainless steel shows that this increases by 0.2% for variant A27 compared to the 2D variant. For the other variants, a greater decrease can be observed compared to the 2D case, attributed to the smaller temperature differences and, thus, smaller viscosity differences along the fin height and the fluid.
If aluminum is used for finned structures, there is a lower drop in the heat transfer rate compared to the 2D initial structure due to the higher fin efficiency. With the exception of structure D27, a higher heat transfer is achieved for all variants as compared to the reference structure. In the case of aluminum, the areas of the stagnation point can now take advantage of a higher heat transfer and dissipate the heat flux density better.
For the pressure loss, similar behavior can be observed. The calculated relative pressure losses for all structures are close to those of the 2D variants. Due to the higher temperatures and, thus, higher dynamic viscosities in the fluid boundary layer along the fin height, higher fluid friction and pressure losses are present.
3.2. 3D Optimization
The optimizations carried out in advance are limited to 2D shaping, which means that optimizations with regard to fin efficiency and 3D flow control are not taken into account. However, optimizations of the fin efficiency are of great importance for fin materials with lower thermal conductivity, as for stainless steel in this case, as these influence the effectiveness of the heat exchanger accordingly. Furthermore, 3D optimization can reduce unfavorable areas for pressure loss and heat transfer along the fin height. Therefore, a 3D optimization should follow, in which the fin contour also varies along the fin height. The initial structure for the 3D optimization is the reference structure; see
Figure 5 with the corresponding boundary conditions.
For the 3D optimization, the multi-objective function is:
is the area-weighted average wall temperature along the fins and the base surface and the overall pressure loss of the entire fluid domain. Due to the boundary condition “constant heat flux” at the outer wall, the overall heat flow remains constant. To improve the structure in terms of heat transfer, the inner wall temperature has to be reduced to obtain an improvement in convective heat transfer.
A mesh with very high quality is used for 3D optimization; the minimum orthogonal quality is 0.17, which allows a higher number of optimizations. If the optimization is started with a lower orthogonal quality, negative cell volumes occur prematurely and the calculation is aborted. The meshing is carried out with the “Fluent Meshing” program. Following the optimization, re-meshing is performed again to eliminate the influence of the mesh distortion following the optimization.
A total of four optimization directions are carried out with different weightings of pressure loss and the area-averaged wall temperature; the weightings of the two optimization values are listed in
Table 6. A total of 10 iterations with a freeform scale factor of 25 are carried out for each weighting pair. If the orthogonal quality falls below 0.12, the mesh is optimized using Fluent’s “mesh improve” function, which raises the minimum orthogonal quality again.
Figure 16 shows the trend of the mean internal wall temperature, the product of heat transfer and surface area, as well as the pressure loss and its relative change for structure 0dp1ht.
The curves show that the average inner wall temperature decreases with increasing design iteration while the product of the heat transfer coefficient and surface area increases accordingly to a 3% improvement.
The analysis of the pressure loss shows that it increases slightly over the number of iterations to a maximum of 0.15%.
Figure 17 shows the change from the initial structure to the final iteration step. The comparison shows that there are changes to the fin root, fin flank and fin tip compared to the initial structure. There are slight constrictions at the base of the fin in the area of the stagnation point and the detachment area, which increases the flow cross-section in this area and thus reduces the local pressure loss and simultaneously increases the heat transfer, as part of the mass flow is shifted to the lower area of the fins. There are also corresponding constrictions in the area of the fin tip; at these points, areas of the fins that are of secondary importance for heat transfer are removed, while this benefits the pressure loss due to the larger flow cross-section. At the same time, this benefit of lower pressure loss can now be used to optimize heat transfer at a more effective location. This occurs, for example, in the lower area of the fin flank, where slight thickening occurs compared to the reference case. This leads to higher velocities in this area and thus to higher heat transfer rates. Due to the thickening, the heat-conducting cross-section of the fin also increases, which reduces the average fin temperature. The constrictions in the area of the base of the fin are more pronounced, and the thickening in the lower area of the fin flank is smaller. This leads to larger flow cross-sections and, therefore, lower pressure losses, while the heat transfer is reduced only minimally.
The results for the different optimizations depending on the different weights, as well as the different design iterations, are shown in
Figure 18. The analysis of the design iterations for 0dp1ht shows that an improvement in heat transfer can be achieved quickly with the first design iterations 1–5, while the improvements are significantly smaller from the 6th iteration onwards. An improvement of the heat transfer at constant pressure loss is thus obviously increasingly difficult, which from an optimization point of view, indicates a possible local optimum.
A similar result is determined for the weight-pair 04dp06ht. For the first seven design iterations, the improvement of the heat transfer is almost equally distributed. For the last three design iterations, the improvement in heat transfer decreases, and the reduction in pressure loss is greater instead than for the first seven iterations. It appears there is also an increasing difficulty in further optimization of the heat transfer. For the structures 06dp04ht and 1dp0ht, with a focus on pressure loss optimization, this trend is not shown, and the improvement for the different design iterations is equally distributed.
The Colburn j-factor and the Fanning f-factor for different Reynolds numbers are shown in
Figure 19.
The curve shows that the optimized variants also show a better heat transfer with simultaneously reduced pressure loss than the reference structure at other Reynolds numbers apart from the optimized variant. The relative improvement or reduction continues to match the values at the optimization point with a tolerance of around 10% for all Reynolds numbers investigated.
This results in the following coefficients in
Table 7 for Equations (14) and (15) for the Colburn j-factor and the Fanning f-factor, respectively.
Two of these 3D-optimized structures are also examined experimentally in order to investigate the accuracy of the calculation in this optimization test.
3.3. Experimental Testing
The optimizations made in advance are to be verified by means of experiments. Therefore, two versions with a focus on increased heat transfer (0dp1ht) and reduced pressure loss (1dp0ht) are manufactured. The structure 1dp0ht will also be analyzed in terms of surface roughness and manufacturing accuracy.
The analysis of the surface roughness in
Figure 20 yields values of approximately 25 µm on the top side of the fin structures and 63 µm on the bottom side on average.
Due to the small duct dimensions of 380 µm, the roughness leads to a recurring constriction of the flow, which could result in a higher pressure loss.
This is further intensified by the general manufacturing deviation. An X-ray microscope analysis shows that the deviation between the manufactured test object and the CAD model is approximately +20–40 µm in all spatial directions (see
Figure 21 for the X-ray image). This leads to smaller channels and distances between the fins and thus to smaller flow cross-sections.
The smallest channel width between two rows of fins is approximately 350 µm for 0dp1ht and 380 µm for 1dp0ht in the 3D model, when considering the manufacturing accuracies from the 1dp0ht X-ray analysis, this leads to 23% (0dp1ht) and 21% (1dp0ht) smaller distances, resulting in 270 µm (0dp1ht) and 300 µm (1p0ht). This has a direct effect on the flow cross-section and thus on the hydraulic diameter, which ultimately results in higher pressure losses. Using an updated hydraulic diameter with the corrected minimum flow cross-section, a corrected f-factor can be determined by applying Equation (20).
The X-ray analysis also reveals that the predicted missing geometry in
Figure 12 is present at the leading and trailing edge of the fins, which influences the heat transfer coefficients, due to the changed stagnation point and the thermal boundary layer in this region.
Figure 22 shows the relative deviation between the experiment and the numerical calculation. For the structure 0dp1ht, the agreement between the experiment and the calculation is between 0.9 and 1.07. This means there is an underestimation of small Reynolds numbers and an overestimation of higher Reynolds numbers. Compared to other investigations of additively manufactured heat exchangers (see introductory chapter), the agreement between the experiment and the calculation can be described as good to very good. However, the fluctuation compared to the improvement in heat transfer is large, meaning that the theoretical improvements in heat transfer can only be mapped to a limited extent due to manufacturing inaccuracies in the manufacturing process. The analysis of the f-factor initially shows a very large deviation of a factor of ~2.5.
The analysis of the j-factor of the structure 1dp0ht shows a significantly better agreement, which ranges between 0.97 and 1.03. As with 0dp1ht, there is a tendency to underestimate small Reynolds numbers and overestimate larger Reynolds numbers. In the experiment, the f-factor is initially overestimated by approximately 1.6.
The uncertainties of the Colburn j-factor and the Fanning f-factor are summarized in
Table 8 for both structures. For the Colburn j-factor the uncertainty ranges between 7% for large and 12% for small Reynolds numbers, and for the Fanning f-factor between 4.4–10% for large and small Reynolds numbers. The uncertainty of the Reynolds number is around 5.1–6.1%.
If the f-factors of the measurements are corrected for the new flow cross-section due to manufacturing deviations (and not considering possible acceleration due to roughness), the differences between the experimental and numerical values for 0dp1ht and 1dp0ht can be reduced significantly to less than 10% on average.
This also indicates that possible deviations due to the difference between the optimization (three rows of fins) and the fin arrangement in the heat exchanger (~14 rows of fins) are not the main cause of the existing deviations and at most cause small differences.
The causes for the underestimation and overestimation of the j-factors cannot be determined directly. However, it is assumed that the underestimation at low Reynolds numbers is due to possible detachment areas caused by the surface roughness. If the Reynolds number is increased, there is a slight overestimation, which could also be caused by the roughness and the associated increase in turbulence.
The experimental investigations show that although the optimizations made can be reproduced by 3D printing, the limits of manufacturing accuracy are also reached, as the fluctuations of the experiment sometimes exceed the potential for improvement, especially in terms of heat transfer improvement.