The increased design space offered by additive manufacturing (AM) can inspire unique ideas and different modeling approaches. One tool for generating complex yet effective designs is found in numerical optimization schemes, but until relatively recently, the capability to physically produce such a design had been limited by manufacturing constraints. In this study, a commercial adjoint optimization solver was used in conjunction with a conventional flow solver to optimize the design of wavy microchannels, the end use of which can be found in gas turbine airfoil skin cooling schemes. Three objective functions were chosen for two baseline wavy channel designs: minimize the pressure drop between channel inlet and outlet, maximize the heat transfer on the channel walls, and maximize the ratio between heat transfer and pressure drop. The optimizer was successful in achieving each objective and generated significant geometric variations from the baseline study. The optimized channels were additively manufactured using direct metal laser sintering (DMLS) and printed reasonably true to the design intent. Experimental results showed that the high surface roughness in the channels prevented the objective to minimize pressure loss from being fulfilled. However, where heat transfer was to be maximized, the optimized channels showed a corresponding increase in Nusselt number.

## Introduction

Growth in the manufacturing industry has encouraged a new design methodology across a variety of disciplines. Where product design was previously dictated by manufacturing constraints, design for high performance can now dominate. In the case of internal cooling schemes for gas turbine components, effective designs minimize the pressure loss while maximizing the heat transfer.

Additive manufacturing (AM), specifically direct metal laser sintering (DMLS), is an attractive manufacturing process for certain components in the hot section of gas turbines: the method can utilize aerospace-grade materials to create geometries unattainable by conventional manufacturing techniques. Such complex geometries can be conceived in myriad ways, but one quantitative way is through numerical optimization algorithms.

Many different optimization techniques exist and can vary greatly in complexity, but all require an objective function to be minimized or maximized. For this study, a commercially available adjoint optimization solver was used, and three different objective functions were posed. The initial geometries were derived from Kirsch and Thole [1], who designed and additively manufactured wavy microchannels of varying wavelengths; two of the wavelengths were chosen for this optimization study, whereby the inlet and exit areas of the microchannels remained constant. The three objectives were to: (1) minimize the pressure loss through the channels, (2) maximize the heat transfer on the channel walls, and (3) maximize the ratio of heat transfer to pressure loss. To that end, a total of six test coupons were manufactured via DMLS for the two wavelengths.

This study aims to provide some insight into the ability to reproduce numerically optimized geometries and to assess the performance of those optimized geometries in the physical domain. First, a detailed analysis of the optimization results will be provided. Next, the as-manufactured channels will be evaluated and compared to the design intent, and to the baseline designs. Finally, motivated by the insights gathered from the numerical results and the knowledge of the as-manufactured geometries, a discussion on the experimental pressure loss and heat transfer results will follow.

## Literature Review

The design of microchannel heat exchangers varies greatly depending on the end use. Wavy channel designs are primarily used for electronics cooling or other low flow rate applications due to the fluid mixing generated by the waves. Wavy channels can be constructed as sinusoidal waves [2,3], converging-diverging periodic sections [4–6], variable amplitude and/or wavelength sections [7,8], or as a series of circular arcs [9]. Each of these designs promote large vortical structures, which increase the heat transfer, yet the penalty in pressure loss is relatively low.

Most wavy channel studies have been performed at Reynolds numbers well into the laminar regime. For that reason, the study by Kirsch and Thole [1] was conceived to test the potential of wavy channels at flow rates more relevant to gas turbine engines. At Reynolds numbers below 5000, the heat transfer was more of a function of the wavelength than of the channels' high surface roughness, indicating that the flow structures promoted by the wavy channels were the dominant heat transfer mechanism in that flow regime.

High surface roughness is a hallmark of most metal additive manufacturing processes [10,11], where external surfaces can be post-processed and smoothed, internal surfaces remain rough. The roughness features that form are dependent on the machine process parameters, such as laser power, hatch distance, layer thickness, and laser scan speed [12–16]. Bacchewar et al. [16] isolated laser power as a strong contributor to surface roughness on downward facing surfaces, or down skins, decreasing the power on those surfaces yielded smoother features. In a similar vein, Wegner and Witt [15] reported that increasing the laser energy density on up skins yielded a smoother face due to evening out the characteristic stair-stepping effect on inclined surfaces. [17]

Characterizing the as-built DMLS part is essential, especially where tight tolerances are required. In the case of microchannels or other small (<3 mm^{3}) features, the natural shrinkage that occurs from the DMLS process can be up to 10% of the part's initial dimensions [18]. A common method for investigating AM parts is to use a computed X-ray tomography (CT) scan, because it is nondestructive in nature. Multiple studies have used this technique [1,10,11,19,20] with success; Stimpson et al. [21] confirmed via scanning electron microscopy that the resolution of the scans was high enough to resolve large roughness features at the scale of this study.

The AM process represents a powerful tool for building parts whose architecture is unrealizable by conventional manufacturing techniques. Designing for AM requires a completely different methodology, one in which optimization may play a pivotal role. Martinelli and Jameson [22] provided a detailed overview of the natural link between optimization and computational aerodynamics; shape optimization for airfoil design, for example, began in the 1970 s.

Most optimization techniques can be grouped into either direct methods (zero order methods) or into gradient-based methods (first order methods) [23]. Direct methods include approaches such as simulated annealing, differential evolution, and genetic algorithms [24–27]. Verstaete et al. [25] combined a conjugate heat transfer analysis and a finite element analysis to perform a parameterized study on the shape of a high pressure turbine blade, including its internal cooling channels. This combination of analysis capabilities provided a robust means of finding the optimum result. However, direct methods can be computationally expensive, especially when the number of design parameters is large.

In gradient-based methods, the determination of an objective function's derivative is required. Efficient calculation of the gradient can reduce the computational effort required to find an optimum, when compared to the effort required from zero-order methods [28]; the adjoint method was specifically derived for this efficient calculation and is widely used for a variety of shape optimization goals [28–30]. Wang and coworkers [31] researched the adjoint method as it applied to finned heat exchangers; fin parameters to be optimized included the width, pitch, height, and length.

Topology optimization, as opposed to shape optimization, changes the distribution of material and not simply its shape [23]. Dede et al. [32] used topology optimization to additively manufacture a heat sink for electronics cooling; the authors' optimized heat sink showed higher heat transfer performance than their baseline. Other topology optimization studies [33,34] have been numerical in nature, but show great promise for future production.

With the exception of the study performed by Dede et al. [32], the combination of numerical optimization schemes with additive manufacturing has not been widely reported in the literature. Our study aims to showcase the capabilities of AM as they relate to a powerful numerical optimization method.

## Numerical Setup

The wavy channel design from which this study derives was developed such that a constant radius of curvature in the channel prevailed; the sign of the radius of curvature switched every period [1]. A top–down image of the channel construction is shown in Fig. 1. A rectangle was swept along the path created by the four circular arcs to form a channel and was kept normal to the channel inlet at all times. The channels were characterized by their wavelength, *λ*, relative to the length of the test coupon, *L*. Two wavelengths from the initial study in Ref. [1] were chosen to be optimized for this study: *λ* = 0.1*L* (Fig. 2(a)) and *λ* = 0.4*L* (Fig. 2(b)). To note, Fig. 2 shows only 40% of the coupon length. Ten periods of the *λ* = 0.1*L* case and 2.5 periods of the *λ* = 0.4*L* case fit in the length of the test coupon.

A commercial computational fluid dynamics solver [35] was used to simulate the pressure loss and heat transfer through the two chosen wavy channel cases. The structured grids were composed in a multiblock pattern using a commercial grid generation program [36]; cell *y*+ values remained near or below one throughout the entire domain. Each model contained one channel and was made up of 1.1 × 10^{6} cells. The steady Reynolds-averaged Navier–Stokes and energy equations were solved using the realizable *k*–*ϵ* turbulence model, which was chosen based on its robustness and economic handling of the governing equations. Especially in this study, where many simulations were to be completed, each with different perturbations to the prior geometry, stable and efficient convergence was the key. The semi-implicit method for pressure linked equations algorithm was chosen as the pressure–velocity coupling scheme, and the spatial discretization of the momentum, turbulent kinetic energy, turbulent dissipation rate and energy was second order.

The numerical setup of the baseline cases mimicked that described by Kirsch and Thole [1], for which a grid sensitivity study was performed on a third wavelength channel, *λ* = 0.2*L*. When the initial grid cell count was doubled, the difference in Δ*P* was −0.1%, and the difference in *Q* was 0.1% between the initial and refined grids.

A velocity boundary condition was imposed at the inlet to the channel, and a pressure boundary condition was imposed at the outlet. To mimic the experimental setup, a constant pressure was held at the inlet to the channel, and the channel top and bottom walls were heated via constant temperature boundary condition.

*J*, for each of the two wavelength channels. Equations (1

*a*)–(1

*c*) show each of the observables. Equation (1

*c*) was chosen due to its proportionality to a commonly used performance factor that was derived from Ref. [37], shown in Eq. (2)

The Adjoint Method section will describe the adjoint formulation and its connection to the flow solver, which was a key component to this study. The sensitivity analysis generated by the adjoint solver encouraged a geometric change that would have been difficult to achieve by a user-controlled parametric study. The inlet and exit cross-sectional areas of the channels were held constant, but each one of the grid nodes outside the inlet and exit represented a degree-of-freedom; this study, where the mesh contained 1 × 10^{6} nodes, therefore contained 1 × 10^{6} degrees-of-freedom. The resulting channel geometries were highly complex and aperiodic. To note, the sensitivities calculated by the adjoint solver were used to inform shape optimization; as opposed to topology optimization; the general trend of the wavy channel composition did not change.

### Adjoint Method.

In a typical engineering optimization problem, the goal is to minimize (or maximize) some objective function by changing a set of design variables; constraints on the problem come in the form of both geometric bounds and fluid dynamic boundary conditions [23,38]. A common means of finding an optimum solution is to employ a gradient-based method, which involves taking the derivative of the objective function with respect to the design variables. One such way to determine this gradient is to perform a sensitivity analysis.

*J*represent the objective function.

*J*is a function of both the flow variables,

**q**, and the geometry,

*F*, which is a function of the design variables,

**b**. The gradient of

*J*with respect to the design variables is written in Eq. (3) in its expanded form using the chain rule

The quantity $\u2202q/\u2202b$ represents the sensitivity of the flow field to the design variables, which is not easily determined without running the flow solver for every perturbation in every design variable; the number of required simulations, therefore, becomes prohibitive in even moderately complex problems. The advantage of the adjoint method comes in its ability to eliminate this high computational cost. The mathematical approach to the adjoint method will be laid out here briefly.

**R**denote the conservation laws governing the fluid behavior.

**R**is also a function of the flow variables,

**q**, and the geometry,

*F*, and is identically equal to zero; its first-order gradient takes a form similar to that in Eq. (3) and is shown in simplified form in Eq. (4)

**Λ**, is introduced in the form of an arbitrary vector and is multiplied across Eq. (4). Because the goal is to eliminate the quantity $\u2202q/\u2202b$ from Eq. (3), the value for

**Λ**is chosen such that Eq. (5) is satisfied

In Eq. (6), the sensitivity of the flow field to the design variables is removed, and the change in *J* becomes a function of the geometric sensitivity, which is relatively straightforward to calculate, and the adjoint variable. The adjoint variable contains the sensitivity of the flow variables to changes in the geometry, which can be used to inform the design change necessary to achieve the objective function.

In this study, the flow solver and adjoint solver were contained in the same program [35]. Constraints were imposed on the inlet and exit cross-sectional areas, as well as the distance between inlet and exit, which governed the channel length; all channels were required to fit into equally sized test coupons. While the discretization of the flow equations and the formation of the discretized adjoint equations were handled by the program, the steps taken by the user are outlined here:

- (1)
Run the flow solver to convergence: obtain the flow variables,

**q**, by solving the conservation laws,**R**. - (2)
Run the adjoint solver to convergence: obtain the sensitivity of the flow field to geometric variations by solving Eq. (5), the adjoint equations.

- (3)
Modify the geometry based on the sensitivity results to achieve the objective function.

- (4)
Rerun the flow solver and compare the objective function to that from the previous flow solution.

- (5)
Repeat until the objective function has reached a sufficient value or until flow variables show no more sensitivity to geometric changes.

For observable *J*_{1}, these five steps were repeated 14 times; for *J*_{2}, five times; for *J*_{3}, seven times. The adjoint and flow solvers were run for a Reynolds number of 5000 for both wavelength geometries. Typically, the adjoint solver converged near 15,000 iterations, while the flow solver converged in 6000 iterations, where the threshold for convergence was set at 1 × 10^{−9}.

For the *λ* = 0.1*L* case, the observable *J*_{1} was also optimized at a Reynolds number of 15,000. The difference between the optimized geometries at the two Reynolds numbers was small. The following discussion on the optimization results, and the subsequent discussion on the experimental results, assumes that the optimized shape changes roughly apply across a range of Reynolds numbers.

### Optimized Geometries.

The final observables from the six optimization studies are shown in Tables 1 and 2 for the *λ* = 0.1*L* and *λ* = 0.4*L* cases. The percentage difference values are relative to the respective baseline cases. Outlined boxes show the results of the quantity for which the adjoint solver was used to optimize; for comparison, the other two quantities are listed as well. In general, larger differences from the baseline cases were seen for the *λ* = 0.1*L* case than for the *λ* = 0.4*L* case.

Figure 3 shows samples of the geometric changes to the channels as a result of the sensitivity study for the *λ* = 0.1*L* case. The outlines of the channels in Fig. 3(a) are at 50% the channel height. A line plot showing the change in cross-sectional area through the optimized channels, normalized by the baseline computer-aided design cross-sectional area, is in Fig. 3(b); the contours in Figs. 3(c)–3(f) are colored by nondimensional temperature, with velocity vectors overlaid. Nondimensional temperature, *θ*, is defined such that *θ* is equaled to one when the fluid and wall temperatures are equal, thereby making it a measure of heat transfer performance.

The most dramatic changes in wall shape came in the streamwise middle of the channel, with the inlet and exit of the channels showing only slight deviations from the baseline, which can be seen in Fig. 3(b). In general, the changes in cross-sectional area between the *J*_{1} (min(Δ*P*)) and *J*_{2} (max(*Q*)) observables mirrored each other: where the *J*_{1} case showed an increase in the cross-sectional area, *J*_{2} showed a decrease, and vice versa. The *J*_{3} shapes struck a balance between *J*_{1} and *J*_{2}.

A different shape change occurred for each of the ten periods in the channel and occurred predominantly between peaks and troughs in the channels, where the radius of curvature defining the wave switched signs (Figs. 3(c) and 3(d)). At the peaks and troughs, all the optimized channel shapes were the same as the baseline shape (Figs. 3(e) and 3(f)).

For the observables *J*_{2} and *J*_{3}, where heat transfer was to be maximized, higher angular velocity was seen throughout the channel as compared to the baseline. The walls tended to bow outward (Fig. 3(c)), then bow inward immediately after rounding a peak or a trough in the channel (Fig. 3(d)). This alternating pattern worked to draw flow from the center of the channel toward the upper and lower endwalls. The consequences of these shape changes can be seen at locations (*e*) and (*f*), where the vectors showed coherent flow patterns in the *J*_{2} and *J*_{3} cases that differed considerably from the flow patterns in the baseline case.

In looking at the nondimensional temperature contours in Figs. 3(c)–3(f), the highest nondimensional temperatures were found where the vortices met the side walls. The fluid motion caused by the channel waviness created an impingement-like effect for every turn in the channel; this effect was exacerbated in the *J*_{2} and *J*_{3} cases, where the walls' protrusions into the channel provided a larger impingement surface. Figure 3(d) shows this event most clearly. Near the end of the channel (Figs. 3(e) and 3(f)), the heat transfer performance at the channel midspan was notably higher for the *J*_{2} and *J*_{3} cases than for the baseline and *J*_{1} cases.

The solution to minimizing pressure loss, observable *J*_{1}, came in minimizing the amount of backflow in the channel. As the flow navigated the channel waves, the direction of centripetal force exerted on the fluid particles switched signs every period. As the force direction changed, the direction of the vortices in the channel switched as well, which caused a small amount of flow to move backward. This behavior is evident in Fig. 3(d) in looking at the velocity vectors for the baseline study. The cluster of vectors seen in the slice of the baseline study was not present for the *J*_{1} case; the movement of the walls for *J*_{1} eased the transition for the vortical structures through each subsequent period. However, the difference in vector patterns for the *J*_{1} and baseline channels at locations (*e*) and (*f*) was small, indicating that the optimizer was more closely focused on the locations between channel peaks and troughs. The wall shape changes were not as significant for the *J*_{1} case and indeed, the degree to which *J*_{1} was satisfied was less than *J*_{2} and *J*_{3} (Table 1).

Figure 4 shows the optimization results for the *λ* = 0.4*L* case in a manner similar to Fig. 3; Fig. 4(a) shows a top–down view of the channel outlines, taken at 50% the channel height, Fig. 4(b) shows the change in cross-sectional area through the optimized channels, normalized by the baseline CAD cross-sectional area, and the contours in Figs. 4(c)–4(f) show nondimensional temperature with velocity vectors overlaid. Much like the shorter wavelength channels, the most significant shape changes came between the peaks and troughs in the channels (Figs. 4(c) and 4(d)); the shape of all three optimized channels matched the baseline channel at each peak and trough (Figs. 4(e) and 4(f)).

For the same observable at any streamwise location, the general shape transformations between the *λ* = 0.1*L* case and *λ* = 0.4*L* case were markedly different, as were the changes in cross-sectional area through the optimized channels relative to the baseline channels. Unlike for the *λ* = 0.1*L* baseline case, the *λ* = 0.4*L* baseline channel structure allowed for the formation of Dean vortices, which are characteristic in flows through curved channels [9]. The shape changes induced by the optimizer, therefore, revolved around either enhancing those vortical structures (in the *J*_{2} and *J*_{3} observables) or diminishing them (in the *J*_{1} observable).

Comparing the contours in Figs. 4(c) and 4(d) shows that the shape changes for each of the observables were similar, but occurred in different streamwise locations depending on the main objective. For example, the *J*_{1} slice at location (*b*) exhibited a similar shape as the *J*_{2} and *J*_{3} slices at location (*c*). Where heat transfer was to be maximized, in observables *J*_{2} and *J*_{3}, the leeward wall bowed outward, in the direction of the fluid motion (Fig. 4(c)); following the peak in the channel, that same wall curved inward, again in the direction of the fluid motion (Fig. 4(d)). This wall movement facilitated the formation of vortices, the result of which can be seen at locations (*e*) and (*f*) for the *J*_{2} and *J*_{3} cases. When compared to the baseline flow structure at those locations, the vortex patterns for *J*_{2} and *J*_{3} showed fuller vortical structures that were more centered in the spanwise (*z*) dimension. The temperature contours in Fig. 4 confirmed that the shape changes related to maximizing heat transfer achieved the objective: from location (*d*) until the end of the channel, the *J*_{2} and *J*_{3} cases showed higher heat transfer performance than the *J*_{1} and baseline cases. However, instead of providing an impingement surface to maximize the heat transfer, like in the *λ* = 0.1*L* geometry, the wall movements simply relocated and strengthened the vortical structures that were already present in the baseline case.

An analysis of the results for the *J*_{1} observable shows that the leeward wall bowed inward at location (*c*) in Fig. 4, then outward at location (*d*), which was the exact opposite behavior seen for the *J*_{2} and *J*_{3} cases. Increasing the cross-sectional area of the channel after the fluid rounded the peak lowered the flow momentum and discouraged the formation of vortices. In Figs. 4(e) and 4(f), the size of the vortices in the *J*_{1} channel was much smaller than in the baseline, suggesting that diminishing the vortical structures was the key to achieving the pressure loss objective. The temperature contours show the lowest heat transfer effectiveness for the *J*_{1} case, consistent with the results seen in Table 2.

## Geometric Characterization

The optimized channels from the numerical study were duplicated to fill a test coupon; 20 channels for the *λ* = 0.1*L* case and 18 channels for the *λ* = 0.4*L* case fit in the spanwise dimension of the test coupons. All coupons, including both baseline studies, were built layerwise at a 45-deg angle using DMLS; the machine parameters were set to those recommended for the chosen material [39], which was stock Inconel 718 powder. The baseline coupons were included on the same build plate in order that the performance of the optimized geometries could be directly compared to the baseline; any effects from variabilities in the DMLS process were therefore negated.

Figure 5 shows the build orientation of the test coupons, along with relevant dimensions. The test coupons were 25.4 × 25.4 × 1.5 mm in size; the aspect ratio of the rectangular channels was two, with the channels spaced in the spanwise direction at *S*/*D _{h}*=2.0. Channel hydraulic diameter was nominally 0.68 mm. Support structures were fixated on the coupon flanges, as well as on the bottom-most coupon wall and served not only to provide physical support for the build layers, but also to conduct heat away from the part toward the build plate during the build.

To determine how well the optimized features were produced, the internal surfaces of each coupon were evaluated using a CT scanner. The resolution, or voxel size, of the CT scan image was 35 *μ*m, although the software used to analyze the CT scan data allowed for the determination of a part's surface to be resolved within 3.5 *μ*m. The internal and external surfaces of each coupon were determined using algorithms within the software, which compared local grayscale values in each voxel to distinguish between material and background. Once the surfaces were identified, 2D slices from the CT scan were analyzed to determine the cross-sectional area, perimeter, and surface area of each channel.

Figure 6 shows selected results from the DMLS channels for *λ* = 0.1*L*; Fig. 6(a) shows the change in the cross-sectional area of the DMLS-optimized channels, normalized by the DMLS baseline channel cross-sectional area, for half of the coupon length (0.1 < *x*/*L* < 0.6). Figure 6(b) shows a 2D slice of all DMLS channels atop one another at the same streamwise location as in Fig. 3(d).

A comparison between Figs. 6(b) and 3(d) shows that the wall shapes for all cases stayed relatively true to the optimized design: the flow constriction called for by the optimizer in the *J*_{2} and *J*_{3} cases was achieved, as was the slight curve in both channel side walls for the *J*_{1} case. Additionally, in looking at Fig. 6(a), the trend in cross-sectional area at *x*/*L* ≈ 0.23 for the three optimized cases resembled the design intent (Fig. 3(b)). The cross-sectional area measured for the *J*_{2} case was much smaller than the *J*_{1} and *J*_{3} cases, whose measured cross-sectional areas were nearly equal. However, the ordinate extrema of the line plots in Fig. 6(a) are far greater than in Fig. 3(b), which indicates that the DMLS process was unable to reproduce the nuanced differences between the CAD baseline and optimized channels.

Figure 7 shows selected results from the CT scans of the *λ* = 0.4*L* DMLS channels. Figure 7(a) shows the change in cross-sectional area through the DMLS optimized coupons, normalized by the DMLS baseline cross-sectional area (for 0.1 < *x*/*L* < 0.6), and Fig. 7(b) shows a 2D slice of all DMLS channels at the same location as in Fig. 4(d). Unlike for Fig. 6(a), both the magnitude and the general trend in cross-sectional area of the optimized channels relative to the baseline were similar to the plot from Fig. 4(b). At *x*/*L* ≈ 0.3, the cross-sectional area of the *J*_{1} case was the largest of the three cases, followed by the *J*_{3} case, then by the *J*_{2} case. The channel outlines in Fig. 7(b) support this trend and largely match those seen in Fig. 4(d); the large negative feature of the *J*_{1} optimized result built well, as did the large positive features called for by *J*_{2} and *J*_{3} optimized results.

Table 3 shows relevant measured dimensions of the eight channels from this study, compared with the dimensions from the design intent. The measured hydraulic diameters and inlet cross-sectional areas built slightly smaller than the intended CAD, but the surface area of the channels nearly matched the intent.

Roughness levels in each coupon, denoted as *R _{a}*, are also included in Table 3 and were quantified by measuring the distance between each point on the CT scanned model and surfaces fit to the model. In general, the

*λ*= 0.1

*L*cases exhibited larger roughness features than the

*λ*= 0.4

*L*cases. However, the difference among the optimized channels for the same wavelength was minimal; the optimized features did not affect the levels of roughness in the channels relative to the baseline.

As expected, neither the baseline nor the optimized DMLS matched their corresponding CAD models perfectly. Overall, however, optimized features as small as 50 *μ*m (10% of the channel width) built successfully, which is on the order of the build layer thickness. Additionally, both positive and negative features were built with equal success on both upward and downward facing surfaces. The failure of optimized features to build was related to their location on the channel walls. Given that the DMLS process could not produce sharp corners, as evidenced by the 2D slices in Figs. 6(b) and 7(b), any optimized features near the corners were not able to be resolved.

## Experimental Setup

A bench-top rig, a cross section of which is shown in Fig. 8, was used to measure the pressure loss and heat transfer performance of the eight DMLS coupons. Flow was governed by a commercial mass flow controller [40], and air was used as the working fluid. A constant pressure was held at the inlet to the test section; to achieve target Reynolds numbers between 300 and 15,000, the pressure downstream of the test section was adjusted.

Static pressure taps were located in the upstream and downstream nylon pieces of the test facility to measure the pressure drop through the channels. A loss coefficient of zero was assumed at the coupon inlet, while a loss coefficient of one was assumed at the outlet to account for the test section expansion. In the friction factor calculation, the fluid density was obtained via the ideal gas law, and the channel velocity, *U*, was calculated from the known mass flow rate through the system. The channel length was measured as the length that the fluid navigated.

For heat transfer tests, a heated copper block provided a constant temperature boundary condition on the test coupon walls. Heat into the system was set by power supplies connected to electrical resistance surface heaters, which were adhered to the copper blocks. The coupon surface temperature was calculated using a 1D conduction analysis, a full description of which can be found in Ref. [11]; for each test, the power supply voltages were set such that the temperature on both top and bottom walls of the test coupon were equal. Using the coupon surface temperature, as well as thermocouple measurements at the inlet and outlet of the test section, a log mean temperature difference could be calculated, and the convective heat transfer coefficient was then obtained using Eq. (7).

The term $\u2211Qloss$ from Eq. (7) represented the combined conduction losses through the test facility. Conduction losses were quantified by placing thermocouples in the copper blocks, the rigid foam, and the Nylon test pieces; at low Reynolds numbers, conduction losses neared 15% of the total heat into the system but decreased to 2% for Reynolds numbers over 8000. For each test, an energy balance was performed and matched to within 15% of the measured net heat input to the test rig; for Reynolds numbers above 4000, the energy balance was within 11% of the calculated heat input.

### Uncertainty Analysis.

To quantify experimental uncertainty, the methods proposed by Kline and McClintock [17] were applied to all measured and calculated quantities. The largest source of overall uncertainty for friction factor tests came in the size of the pressure transducer used to measure the pressure drop across the coupon. At low Reynolds numbers (<500) for the *λ* = 0.4*L* cases, which represented the worst case scenario for the entire test matrix, friction factor uncertainty neared 17%. However, for all flow tests above a Reynolds number of 4000, overall friction factor uncertainty was below 7%. The precision uncertainty, calculated using a 95% confidence level, ranged between 1.5% and 2% across the entire extent of Reynolds numbers.

Uncertainty in Nusselt number was driven by the calculation of the coupons' surface temperatures. Uncertainty in the thickness of the thermal paste was a contributing factor, as was the uncertainty in the thickness of the coupon top and bottom walls. Both the uncertainty in surface temperature and Nusselt number were below 6% for all coupons. Precision uncertainty for Nusselt number was 3% across all Reynolds numbers.

## Results and Discussion

Results will be presented first in the form of a pressure loss analysis, followed by a discussion on the heat transfer performance of the optimized channels relative to their respective baselines. Due to the large number of DMLS process parameters and the high sensitivity of microchannels to small variations in those process parameters, the data to be presented in this paper come only from coupons manufactured on the same build plate. While the design of the baseline wavy channels originated in Kirsch and Thole [1], data from that initial study will not be presented here.

Additionally, the channel dimensions used in the upcoming discussions will be those at the inlet to each coupon, as calculated from the CT scan data. The optimizer worked to achieve each objective relative to the bulk channel properties, because the inlet and exit areas were kept constant during the simulations. Therefore, to most accurately understand how well the optimized DMLS channels achieved their intended goals, the aperiodic changes in cross-sectional area that occurred beyond the channel inlet will not be taken into account in the geometric scaling parameters. A friction factor and Nusselt number were calculated for the numerical results in the same manner. Where applicable, the numerical results will also be included in the upcoming discussions.

To validate the results from the test facility, an aluminum test coupon containing cylindrical channels was machined; the channels were reamed smooth to achieve nearly zero relative roughness. Data from the smooth channels will be presented in the upcoming results, along with smooth channel correlations: laminar theory (64/Re) and the Colebrook formula for the friction factor tests, and the Gnielinski correlation for the heat transfer tests.

### Pressure Loss Performance.

Figure 9 shows friction factor for each of the DMLS optimized channels, along with the DMLS baseline coupons and smooth benchmarking coupon, versus Reynolds number. The friction factor results from the numerical simulations are included as well. Given that the simulations modeled smooth-walled microchannels, the numerical results showing significantly reduced friction factors relative to their DMLS channel counterparts was to be expected. The experimental results exhibited friction factors near five times those from the simulations. This discrepancy can be attributed to both the high surface roughness in the channels and the fact that flow through wavy channels is inherently unsteady [9]; the steady simulations most likely failed to capture all pertinent flow characteristics.

Consistent with the results from Kirsch and Thole [1], all *λ* = 0.1*L* cases showed a higher friction factor than the *λ* = 0.4*L* cases due to the channel construction; the shorter wavelength and smaller radii of curvature created a stronger propensity for flow separation and yielded a higher pressure loss over the longer wavelength channels, even when pressure loss was to be minimized.

For a given wavelength, the trends among the optimized channels relative to their baseline differed considerably. The stark increase in friction factor from the *λ* = 0.1*L J*_{2} case was prominent in Fig. 9, averaging a 50% increase over the baseline. Both the *J*_{1} and *J*_{3} observables, however, showed nearly equal friction factor to the baseline study. These results indicate that while the *J*_{1} objective was not achieved, the *J*_{3} objective showed promise. Requesting that the numerical optimizer account for both pressure loss and heat transfer translated well to the physical domain.

By contrast, the *J*_{1} objective was wholly unfulfilled for the *λ* = 0.4*L* case, with the *J*_{1} coupon yielding a measurably higher friction factor than its baseline coupon. Both the *J*_{2} and *J*_{3} cases also showed increased friction factors over their baseline coupon, as was predicted by the optimizer.

To highlight the performance of each of the optimized channels relative to their baseline designs, Fig. 10 shows the friction factor augmentation from the optimized channels over their baselines. An augmentation of one is specified in Fig. 10 with a dotted line. The failure to achieve the *J*_{1} objectives is explicitly discernable in the augmentation plots. Additionally, the performance of the *J*_{3} objective relative to the *J*_{2} objective is clear: when the optimizer sought to increase the ratio of heat transfer to pressure drop, the resulting friction factor was measurably lower than when only heat transfer was to be maximized.

As previously discussed, each of the optimized geometries were built relatively true to their optimized design. While not all optimized wall features were reproduced perfectly, distinctly different geometries emerged from the DMLS build process that largely resembled their numerically generated counterparts. The fact that neither wavelength's *J*_{1} objective was achieved implies that the large roughness features inside the channel were the more dominant effect on flow structure, instead of the wall shape. Channel wall movements that worked to reduce the backflow in the channel (*λ* = 0.1*L* case) or to diminish vortical structures (*λ* = 0.4*L* case) failed to work as the optimizer had predicted due to the large, irregular roughness features in the channel.

However, where the vortical structures were to be strengthened without regard to flow losses, in the *J*_{2} cases, the resultant friction factors were measurably higher than the other two objectives. Especially for the shorter wavelength, the dramatic changes in wall shape for the *J*_{2} objective undoubtedly complicated the flowfield relative to the channels from the baseline and the *J*_{1} objective, which negatively impacted the friction factor.

### Heat Transfer Performance.

Heat transfer results are shown in Fig. 11 for all DMLS coupons, along with the results from the smooth benchmarking coupon. Additionally, heat transfer results from the numerical simulations are presented. The discrepancies between the simulations and the experiments were smaller in Fig. 11 than in Fig. 9, with the experimental results averaging 1.5–2 times the Nusselt number values from the simulations. Similarly, the spread in Nusselt number across all DMLS optimized channels was much less than the spread in friction factor seen from Fig. 9. In fact, the heat transfer performance of all *λ* = 0.4*L* cases were within 9% of each other, with the *J*_{2} case marginally outperforming the others.

The *J*_{2} case for the *λ* = 0.1*L* wavelength, however, showed heat transfer performance around 20% higher than the baseline channels. These results implied that the shape changes induced by the optimizer achieved their goals; creating stronger vortical structures would positively influence the heat transfer while negatively affecting the pressure loss, which is what the results support.

Figure 12 more clearly shows the difference between the optimized channels and the baseline channels; Fig. 12(a) shows heat transfer augmentation from the optimized over the baseline channels for the *λ* = 0.1*L* cases, while Fig. 12(b) shows the same augmentation for the *λ* = 0.4*L* cases. Much like the friction factor results, the *J*_{3} objectives performed more similarly to the *J*_{1} cases than to the *J*_{2} cases. This observation further supports the claim that the DMLS build process was able to reproduce the geometric differences between the *J*_{2} and *J*_{3} cases aimed at mitigating the strength of the vortical structures.

### Augmentation Results.

To fully interpret the impact of the numerical optimizer on the experimental results, the following discussion will focus on the combined friction factor and heat transfer augmentation. Figure 13 shows heat transfer augmentation against friction factor augmentation to the one third power, representative of the performance factor mimicked by the objective *J*_{3}, max(*Q*/Δ*P*^{1/3}). The results of the numerical study are included as well. A much larger spread in the augmentation data was seen for the *λ* = 0.1*L* case than the *λ* = 0.4*L* case, which was expected given the changes in observables shown in Tables 1 and 2. The success of the optimizer to reduce the pressure drop (and by extension, friction factor) is evident in Fig. 13 from the numerical results, as is the failure of the DMLS process to follow suit.

As anticipated from the isolated experimental friction factor and heat transfer results in the Pressure Loss Performance and Heat Transfer Performance sections, the high friction factor seen by the *J*_{2} objectives was not offset by a high heat transfer augmentation; the *J*_{2} cases, therefore, showed poor overall performance in Fig. 13. Additionally, the *J*_{1} and *J*_{3} cases for the *λ* = 0.4*L* wavelength show a higher penalty in friction factor augmentation than benefit to heat transfer when compared with the *λ* = 0.4*L* baseline.

Successful representations of the intended optimized goals were seen for the *λ* = 0.1*L J*_{1} and *J*_{3} objectives. Both observables yielded a higher heat transfer augmentation for the same friction factor augmentation as their baseline channel. Specifically for the *J*_{3} observable, the heat transfer augmentation was consistently 15% higher than the baseline study.

## Conclusions

A numerical optimization study was performed on two different configurations of wavy microchannels characterized by their wavelengths: *λ* = 0.1*L* and *λ* = 0.4*L*. Three objective functions were posed for each channel that reflected common goals across internal cooling schemes: (1) minimize the pressure loss, (2) maximize the heat transfer, and (3) maximize the ratio of heat transfer to pressure loss. Computational results showed that each of the objective functions was successfully realized.

The channel shapes that resulted from the optimizer differed considerably depending on the objective function and on the wavelength of the channel. Where pressure loss was to be minimized for the *λ* = 0.1*L* case, the channel walls sought to decrease the flow separation that occurred in the baseline design. For the *λ* = 0.4*L* case, the optimizer worked to mitigate the strength of the vortices that formed in the baseline. To maximize the heat transfer for both wavelengths, the optimizer generated channel wall shapes that encouraged the formation of and strengthened the vortical structures.

Each of the numerically optimized channels, along with their baseline channels, were built using DMLS. The channels were evaluated nondestructively to understand how well the optimized channel shapes were able to be reproduced. In general, the success or failure of an optimized feature to be built lied with its location in the channel. Near the sharp corners, which are difficult to resolve at the scale of these channels, optimized features did not build. However, any features greater than the build layer thickness near midheight of the channel were reproduced successfully, regardless of whether the surface were upward- or downward-facing.

Experimental results showed that the objective to minimize pressure loss was not achieved for either wavelength. The large roughness features that are characteristic of the DMLS process dominated the flow structure more than the shapes of the walls. On the other hand, where heat transfer was to be maximized without regard for channel losses, the wall shapes successfully enhanced the strength of the vortical structures in the channel and strongly influenced the flowfield in the channels. Both wavelength channels saw an increase in both the friction factor and the heat transfer in the channels whose shape was aimed at maximizing heat transfer. However, the penalty in pressure loss was far higher than the benefit in heat transfer.

The best overall performance was seen by the objective to maximize the ratio of heat transfer to pressure loss. While the objective to minimize pressure loss itself was not accomplished, forcing the optimizer to account for pressure loss while maximizing the heat transfer had strong implications for the experimental results. For the same friction factor augmentation as the *λ* = 0.1*L* baseline channel, the *λ* = 0.1*L* channels optimized for heat transfer and pressure loss exhibited a 15% increase in heat transfer augmentation.

Much work still needs to be done to understand the role that shape optimization has in additively manufactured microchannels. While these experiments represent an extremely small segment of possible internal cooling schemes, lessons learned from these results can be applied more broadly. Surface roughness strongly affects the pressure loss and heat transfer performance of DMLS microchannels, but the wall shape also exerts an unmistakable influence. Both heat transfer and pressure loss should be taken into account for any internal cooling optimization scheme. As progress continues in both the manufacturing industry and in numerical optimization methods, further research can continue to delve into the natural link between numerical optimization and additive manufacturing.

## Acknowledgment

The fabrication and the CT scans of the coupons for this study would not have been possible without Corey Dickman and Griffin Jones at Penn State's CIMP-3D as well as Jacob Snyder from the START Lab. The authors are incredibly grateful for their efforts.

## Funding Data

- •
National Science Foundation Graduate Research Fellowship Program (Grant No. DGE1255832).

## Nomenclature

*A*=_{c}cross-sectional area

*A*=_{s}wetted surface area

**b**=design variables

*D*=_{h}hydraulic diameter, 4·

*A*·_{c}*p*^{−1}*f*=Darcy friction factor

*F*=mathematical description of geometry

*h*=convective heat transfer coefficient

*H*=channel height

*J*=objective function or observable

*k*=thermal conductivity

*L*=length

- Nu =
Nusselt number,

*h*·*D*·_{h}*k*_{air}^{−1} *p*=perimeter

*P*=static pressure

- pf =
performance factor

*q*=flow variables

*Q*=heat transfer rate

**R**=conservation laws

*R*=_{a}arithmetic mean surface roughness, $1n\u2211i=1n|zsurf-zmeas|$

- Re =
Reynolds number,

*U*·*Dh*·*ν*^{−1} *S*=spanwise distance

*T*=static temperature

*U*=fluid velocity

*W*=channel width

*y*+ =inner wall coordinates, y+ = y·u

_{τ}·ν^{−1}