1 Introduction

Solar heat is one promising source of renewable energy thanks to its multiple advantages, such as abundancy, cleanliness, high power density, access easiness, and zero greenhouse gas generation [16]. The heat is generated from our sun and emits out in the form of thermal radiation, i.e., randomly polarized light. The intensity spectrum is close to that from a blackbody surface at 5762 K following the Planck’s law [2]. Its spectral intensity is unevenly distributed, and the bandwidth is wide. Majority of the solar power is within the visible and infrared regions. Total solar power reaching the earth is about 1.7 × 1011 MW [15]. Although the amount is huge, usage of solar heat is not straightforward. The installed global capacity is less than 5 GWp based on recent statistics [18]. One critical issue is the ineffective heat absorption because of concurrent loss from the surface of absorbers. Temperature there is much higher than that of ambient, and the re-emission loss via thermal radiation could be significant. Only wavelength-selective absorbers are able to remedy this drawback [32].

The most popular way of realizing wavelength-selectivity is carefully-tailored surface topography, for example, adding structures over a substrate. Rephaeli and Fan numerically developed a structured tungsten slab with subwavelength periodicity, and the structure consists of a square lattice array of pyramids [26]. Diem et al. [8] added submicron tungsten wires on a silicon nitride substrate to realize a wide-angle perfect absorber. Wu et al. [35] presented a solar heat collection system using a nanoimprint-patterned film of plasmonic structures, which were composed of tungsten rectangular pillars on an aluminum nitride substrate. Wang et al. [34] developed black absorbers using tungsten rectangular nano-structures on a silicon dioxide film and tungsten substrate. In the same year, Hu et al. [14] developed a metamaterial absorber consisting of a chromium circular-shaped ring resonator embedded in a dielectric layer and a chromium ground plane. Cheng et al. [5] presented a photoexcited switchable absorber composed of X-shaped gold structures and a silicon pad. Nunes et al. [24] demonstrated a selective surface composed of black chromium (Cr/Cr2O3) deposited on substrates of AISI 304 stainless steel for solar heat absorption. Tian et al. proposed a one-dimensional HfO2/Al2O3–W nanocomposites/W/Al2O3/W multilayered photonic structure as potential wavelength-selective thermal devices. They theoretically investigated the emission properties of the proposed Mie-resonance metamaterials [31]. Liang et al. [19] recently designed and numerically investigated a dielectric filled two-dimensional nickel grating for wide-angle and polarization-independent absorption in the visible range.

Development of wavelength-selective absorbers has been facilitated with multiple design methodologies. For example, utilizing unique physical mechanisms could enhance absorption at target wavelengths. Successful attempts included generation of interference within a thin film [2, 27, 29], resonance of an electrical circuit [7, 11, 20], and band structure of photonic crystals [3, 10]. Additional efforts are necessary for solar heat absorbers using physical mechanisms. These mechanisms need to occur within the broad spectral range of interest.

Other works performed numerical experiments with optimization strategies. One-factor-a-time was intuitive and simple, while the optimization process could end up with a local maximum/minimum easily [27]. Game theory approach was fun and able to identify a satisfactory absorber within a well-defined searching region [13]. Both quality of merit [23] and Taguchi method [4, 17] were utilized to develop absorbers when design objectives were quantitatively defined. On the other hand, the genetic algorithm was capable of tuning all design factors provided computational resources were sufficient [6].

Aforementioned experiments are a series of runs in which purposeful changes are made to the input variables so that reasons for changes are observed in the performance. Each absorber can be visualized as a combination of materials, volume ratio, surface morphology, and micro-/nano-structures that convert sunlight into heat. These variables and material properties are called factors, and some of them are controllable [22]. Two types of factors are often considered in developing absorbers. The design factors are selected for study in the numerical experiments. The noise factors vary naturally and uncontrollably in the process which can be controlled for purposes of an experiment. The conversion efficiency is a common example of design objective.

Previous works mainly studied influences from each of multiple design factors. However, those from an interaction between two design factors were never looked into. Impacts from dimension tolerance and sunlight orientation variation as noise factors were never simultaneously inspected. Performance of developed absorbers then becomes vulnerable to continuously varying sunlight as well as mismatch between designed and real dimensions. The aim of this study is to eliminate above concerns and approach a high-quality solar heat absorber further. Topography of the absorber is defined with five design factors. They are tailored for effective heat absorption, i.e., maximizing the ratio of gained heat over total solar heat. The robustness of performance under noise interferences is taken into account. Studied noise includes the polarization state of incidence, relative orientation between the absorber and direct sunlight, and dimension variation of the absorber. Numerical models are built to evaluate performance of every considered absorber. Subsequently, the design of experiments is employed to quantitatively investigate main effects induced by each design factor and interactions between any two of them. Finally, superiority of developed absorbers on the basis of the absorption effectiveness and robustness is exhibited.

2 Design objectives and factors

Figure 1 plots G1.5, the solar irradiation spectrum AM1.5 [30] to specify providing the incident solar heat employed here. The corresponding zenith angle θ of direct sunlight is 48.2°. Oscillations within the spectrum are caused by absorption and scattering in the atmosphere. Wavelength λ of most (> 98%) incident radiation is between 0.3 and 2.1 μm. Absorbing solar heat within the spectral region becomes the first objective for absorber development. On the other hand, only large absorption is not sufficient to guarantee ideal solar heat absorbers because simultaneous loss from its surface may be also large. The loss is mainly re-emission because temperature of the absorber is much higher than that of ambient [15]. For example, the surface temperature is 773 K, melting temperature of a phase change material CaCl2/NaCl [1]. The material directly contacts the absorber and saves absorbed energy using the latent heat. Reducing the re-emission loss is also needed for absorption effectiveness. For demonstration convenience, 773 K is assigned as the absorber temperature hereafter. Figure 1 also plots spectral emissive power from a blackbody surface at 773 K (E773). The two spectra, G1.5 and E773, cross each other at the wavelength λ = 1.73 μm. Spectral power of incoming solar heat is larger than emission loss at short wavelengths, but emission loss is relatively large at long wavelengths. Therefore, spectral absorptance of an ideal absorber is unity (αλ = 1) at λ < 1.73 μm for large absorption. The spectral emittance is null (ελ = 0) at λ > 1.73 μm for little re-emission. The spectral absorptance of an ideal solar heat absorber is plotted as αideal. Notice that Kirchhoff’s law [2] is applied here.

Fig. 1
figure 1

Direct solar irradiance at AM1.5 (G1.5), blackbody radiation spectrum at T = 773 K (E773), and wavelength-selective spectral absorptance (αλ) for an ideal solar heat absorber

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Figure 2 shows proposed topography for solar heat absorbers. The absorber surface contains two-dimensionally periodic rectangular structures. Defining their dimensions along the z-axis is thickness d. On the other hand, Λx and Λy are periods along the x- and y-axis, respectively. Corresponding filling ratios along the x- and y-axis are fx and fy, respectively. The filling ratio f is defined by ratio between the side length of a nano-structure w and period Λ, i.e., f = w/Λ. These five dimensions are design factors that are tailored for absorption effectiveness and robustness. A dimensional error is a noise factor possibly caused by fabrication tolerance or thermal expansion. Schematic of directly incident sunlight is also shown in Fig. 2. Orientation of the sunlight is defined using the zenith angle θi and azimuth angle ϕi, where subscript i refers incident wave. Polarization state is determined by the polarization angle ψi. It is the angle between electric field \(\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }\) and normal vector \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {n}\) on the plane of incidence. Because all of three angles are varying constantly and naturally, they are included as noise factors, too.

Fig. 2
figure 2

Schematic of proposed absorber under incidence. The absorber surface contains periodic rectangular structures. The incidence direction of place wave is defined with the zenith angle (θi) and azimuth angle (ϕi), while polarization state is specified with the polarization angle (ψi)

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Copper is selected as the absorber material because of its high temperature sustainability, thermal conductivity [2], and fabrication easiness. In fact, Cu nano-structures have recently exhibited promising values for solar heat absorbers [9, 33]. Wavelength-dependent optical constants of Cu in Ref. [25] after appropriate interpolation were employed in our numerical model. Only the spectral range from 0.3 to 9.52 μm is taken into account for emission loss because of limited data availability. Emitting power within the spectral range is 82.7% for E773. The portion is sufficient for performance evaluation. In the following numerical experiments, spectral absorptance αλ is obtained from programs based on the rigorous coupled-wave analysis [4, 12]. Inputs include both incidence and absorber topography, while convergence of programs have been re-validated.

Absorption effectiveness and robustness are our two design objectives. The absorption effectiveness of a solar heat absorber was often evaluated using total absorptance αt [21] as given in the following:

$$\alpha_{t} = \frac{{\int_{0}^{\infty } {\alpha_{\lambda } G_{1.5} } d\lambda }}{{\int_{0}^{\infty } {G_{1.5} } d\lambda }}$$
(1)

where the numerator and denominator are the whole absorbed heat and all incident solar heat, respectively. But αt in Eq. (1) does not count the loss from re-emission and is impractical due to unavailability of αλ at extremely short or long wavelengths. As a result, quantitative evaluation on absorption effectiveness of absorbers at 773 K in this work utilizes the response variable Q below:

$$Q = \frac{{\int_{{0.3\;\mu {\text{m}}}}^{{9.52\;\mu {\text{m}}}} {\alpha_{\lambda } \left( {G_{1.5} - E_{773} } \right)} d\lambda }}{{\int_{{0.3\;\mu {\text{m}}}}^{{9.52\;\mu {\text{m}}}} {G_{1.5} } d\lambda }}$$
(2)

where the numerator and denominator are the net absorbed heat and incident solar heat, respectively. Integrations are replaced with summation in our work without losing accuracy. Q is a dimensionless number determined by αλ of each absorber. The minimum and maximum of αλ is − 5.47 and 0.98, respectively. A positive and large Q illustrates that the absorber gains solar heat effectively. And it can be modified for absorbers working at temperature other than 773 K.

Table 1 lists controllable design factors and their levels employed in our design of experiments. Five control factors A, B, C, D, and E are Λx, Λy, fx, fy, and d, respectively. Each factor has three levels for optimization, while defining these levels considers fabrication feasibility and sufficient coverage. Levels of factor A and B both are 300 nm, 1200 nm, and 2100 nm. The minimum (Λ = 300 nm) is identical to the shortest wavelength of interest. The maximum (Λ = 2100 nm) is the same as upper wavelength limit of G1.5. They are correlated to wavelengths because tailoring absorptance is often realized using structure size comparable to radiation wavelengths. Middle value of these two periods (Λ = 1200 nm) is the other level of both factors. Levels for C and D share 0.3, 0.6, and 0.9. On the other hand, levels for E are 200 nm, 500 nm, and 800 nm. Notice that the feasibility of nano-structures is taken into account when levels are selected.

Table 1 Controllable design factors and their levels in design of experiments
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Table 2 shows five noise factors and their three levels. They are studied for absorption robustness of optimized absorbers. These noises originate from dimensional errors and essence of sunlight. The former brings relative period error a (ΔΛ/Λ), filling ratio error b (Δf), and relative thickness error c (Δd/d). Three levels for each noise are − 0.1, 0, and 0.1. All of noise d, e, and f are angles, but their levels are different. The levels of noise d (zenith angle, θi) are 0°, 30°, and 60° because θi decreases from 90° (sunrise) monotonically to 0° and then increases to 90° (sunset). Each level covers about equal day time. On the other hand, the three levels of noise e (azimuthal angle, ϕi) are 0°, 45°, and 90°. Although ϕi ranges from 0° to 360°, symmetry of four quadrants has been considered in defining levels. ϕi = 0° and ϕi = 90° are associated with the solar motion on the y–z and x–z planes, respectively. ϕi = 45° refers to the third plane (x = y), which equally splits the aforementioned two planes. The noise f (polarization angle, ψi) symbolizes the polarization of incidence. The TM (transverse magnetic) and TE (transverse electric) waves are at ψi = 0° and ψi = 90°, respectively. These two are extremes of polarization state for an electromagnetic wave, and ψi = 45° refers to middle state between the two.

Table 2 Noise factors and their levels in design of experiments
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3 Screening experiments

Conducting screening experiments is to identify factor(s) significantly influencing the absorption effectiveness and robustness. The screening experiments can also study impacts from two-factor interactions [28]. Base for optimization experiments is then set up, and Table 3 lists an orthogonal table for our screening experiments. Total 16 experiments are performed according to a 25−1 fractional factorial design with an accuracy of V(5). The correlation E = ABCD is assigned to five design factors here for reducing calculation loading. Thereafter, interferences from noise factors are added to obtain the response variable Q. A fractional factorial design of 36−4 with an accuracy of III(3) is constructed using four relationships among noises. The relationships are c = mod(2a + b, 3), d = mod(b + c, 3), e = mod(c + d, 3), and f = mod(d + 2e, 3) Rushing et al. [28]. As a result, 9 sets of environment are ready for programs to obtain Q from each of the aforementioned 16 experiments. The average \(\bar{Q}\) over 9 Q’s is able to diminish bias on the response variable in reality. Furthermore, variation among these Q’s provides an estimate of performance variation. \(\bar{Q}\) and the standard deviation S are also listed in Table 3 for further analysis. For example, the mean effect of factor A is the difference between the average of Q from even number of experiments and the counterpart from odd number of experiments. Its value is − 0.05 = [(0.18 − 0.38 − 0.39 + 0.13 − 1.14 + 0.10 + 0.10 − 0.55) − (0.22 + 0.17 + 0.21 − 1.21 + 0.14 − 0.17 − 0.94 + 0.08)]/8. The standard deviation for factor A is calculated using − 0.05 and \(\bar{Q}\). Similarly, effect of another factor or any two-factor interaction can also be calculated.

Table 3 Construction of the \(2_{\text{V}}^{5 - 1}\) design with the defining relation E = ABCD
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Table 4 lists the mean effects and standard deviation of each. Left columns show those from a design factor, while right columns specify those from two-factor interactions. Overall, \(\left| {\bar{Q}} \right|\)(magnitudes of \(\bar{Q}\)) from the design factor E, the two-factor interaction AC, and the two-factor interaction BD are larger than others. They dominantly influence Q of proposed absorbers. The negative \(\bar{Q}\) from E means that Q gets small as the level of E increases. The positive \(\bar{Q}\) from AC means that Q enlarges when levels of factors A and C simultaneously increase or decrease. Conversely, Q diminishes when a level of the two factors increases while the other decreases. The largest magnitude of S is caused by factor E only. As the level of factor E increases, variation of Q gets significant.

Table 4 Effect estimated from the screening experiment
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Figure 3 shows semi-normal probability plot of effects from screening experiments. Magnitudes of both effects, \(\left| {\bar{Q}} \right|\) and \(\left| S \right|\), from each factor and two-factor interactions are marked. Dashed lines are regression curves with a cumulative probability less than 75%. The corresponding factor or interaction is not dominant to an effect if the mark is nearby the curve. For an outlier away from the regression curve, its corresponding factor or interaction influences the effect significantly. Accordingly, the factor E, interaction AC, and interaction BD all strongly influence \(\left| {\bar{Q}} \right|\). One may confuse that none of A, B, C, or D can lay impact on \(\left| {\bar{Q}} \right|\), while their interaction can. The confusion can be easily clarified once the absorber surface profile is recalled. The two-factor interaction is actually w, the side length of nano-structure. wx is determined only when levels of A and C are both set. The same explanation can be applied to interaction BD. For \(\left| S \right|\), E is the only critical factor. When the level of E increases, variation of absorption effectiveness enlarges. Influences from other factors or interactions are relatively trivial. To sum up, absorption effectiveness and robustness of the proposed absorbers are dominated by E, AC, and BD. The next step is to perform optimization experiments using two sets of factors, ACE and BDE.

Fig. 3
figure 3

Normal probability plot of effects from the screening experiment

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4 Optimization experiments

Figure 4a and b show two main effects \(\bar{Q}\) and \(S\) from optimization experiments using ACE and BDE, respectively. Each experiment employs a full factorial design [22], and each factor has three levels. A data point is the average of effects from experiments with the same level for a specified factor. Three points are obtained, and they are then connected to display a non-linear relationships between an effect and design factor. A dashed horizontal line serves as a benchmark. It is the average effect from all of the experiments. Large deviation between a solid line and the dashed line symbolizes dominance of a design factor on the effect.

Fig. 4
figure 4

Main effects from optimization experiments: a ACE, b BDE

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Figure 4a shows that \(\bar{Q}\) decreases monotonically and significantly with levels of E. But the correlation between \(\bar{Q}\) and A is trivial as the correlation between \(\bar{Q}\) and C. The difference between E and the other two design factors assures the single dominance of E on \(\bar{Q}\). Similarly, E is the only dominant factor on S. But S monotonically increases with levels of E. Figure 4b also shows most of the aforementioned findings. The only exception is the correlation between E and S. As level of E increases from 1 to 2, S decreases a little. Because absorption effectiveness desires a large \(\bar{Q}\) and absorption robustness needs a low S, the optimized level of E is 0.

Figure 5 shows influences on main effects from two-factor interactions. Influences from AC, AE, and CE are plotted in Fig. 5a. Those from BD, BE, and DE are shown in Fig. 5b. A data point is obtained when levels of two involved factors are assigned. Three points are marked identically when the same level is assigned to a factor. Points are then connected with straight line sections to display correlations between an interaction and an effect. For instance, lines connecting red triangles exhibit influences from AC as the level of C is fixed to 0. At the same time, the level of factor A varies from 0 to 2. If lines cross each other, AC is strong enough to influence the main effect. Otherwise, the two-factor interaction is trivial. Examples include interactions AE and CE when the level of E is 0. Furthermore, connected lines in this example are close to horizontal lines. Changes in main effects are negligible regardless of A and C. Such little dependence on level of an insensitive factor will be investigated later.

Fig. 5
figure 5

Interaction effects from optimization experiments: a ACE, b BDE

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Plots of AC and counterparts of BD both show crossings. These crossings confirm dominant influences from AC and BD on Q. These plots further help selecting the best level for the four factors. Firstly, the largest \(\bar{Q}\) and smallest S are generated when levels of A and C are both 1 (middle level). Secondly, the largest \(\bar{Q}\) occurs if levels of B and D are both 0 (low level). As a result, optimized levels for A, B, C, D, and E of the ideal absorber are 1, 0, 1, 0, and 0, respectively. In our following discussion, these optimized levels define the absorber considered in Case 1. Its detailed αλ spectra under noise interferences will be plotted and compared with those of a flat plate as well as absorbers in another two cases. Case 2 examines if an insensitive factor A can provide dimensional tolerance without much deteriorating absorber performance. The proposed absorber sets level 0 for A, while the other four factors are identical to those of the absorber in Case 1. Case 3 looks into influences from the dominant design factor E. The proposed absorber sets level 1 for E, while the other four factors are the same as those of the absorber in Case 1.

Figure 6 plots αλ within the spectral range of 0.3 μm ≤ λ ≤ 9.52 μm for a flat copper plate as well as absorbers in Cases 1, 2, and 3. Each sub-figure shows results of numerical experiments under interference from a set of noise factors. Because total nine sets are investigated, the number of sub-figure is nine. Insects in each specify Q of the four absorbers. In general, Q of absorbers in Cases 1 and 2 is larger than that of a flat Cu plate. The well-designed absorbers indeed can outperform a plate under various noise interference. But if an absorber is not well designed, nano-structured surfaces may be worse than a plate. For example, the absorber in Case 3 is not suitable for solar heat application. Because the profile difference between absorbers in Case 1 and Case 3 is just a level of a factor, caution really needs to be paid during profile optimization.

Fig. 6
figure 6

Spectral absorptance of a flat copper plate, the optimal absorber in Case 1 (ABCDE = 10100), absorber in Case 2 (ABCDE = 00100), and absorber in Case 3 (ABCDE = 10101)

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Table 5 lists the mean and standard deviation of response variable Q in Fig. 6 as \(\bar{Q}\) and S, respectively. The first finding is that \(\bar{Q}\) for Cases 1 and 2 are both large. Absorbers in the two cases can acquire solar heat more effectively than the other two. Second, because S in the two cases are smaller than that in Case 3, absorption robustness is assured. Third, both \(\bar{Q}\) and S are larger in Case 2 than in Case 1. The absorber in Case 2 can acquire solar heat more effectively than that in Case 1. But the absorber in Case 1 can resist noise interferences better than that in Case 2. Due to this trade-off, the two absorbers are regarded equally satisfactory on the basis of absorption effectiveness and robustness. Since both absorbers perform well, design flexibility from the insensitive factor A is also confirmed. Both level 0 and level 1 of A can give a good solar heat absorber. Fourth, \(\bar{Q}\) becomes negative in Case 3. This negative value implies that re-emission loss is larger than absorbed heat for the absorber. Its temperature cannot maintain at 773 K in this case. Fifth, S for Case 3 is larger than that for Case 1 because shifting the level of E from 0 to 1 deteriorates absorption robustness. Nano-structures need to be carefully designed. Last but not least, the smallest S among four absorbers comes from the flat plate. One reason is null interference from noises a, b, and c. The other is weak interferences from noise d, e, and f thanks to isotropic surface morphology.

Table 5 Effect estimated from nine spectral absorptance spectra in Fig. 6
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5 Conclusions

This work successfully developed solar heat absorbers via design of experiments. Settings of five controllable design factors are optimized, and variability transmitted from six noise factors are minimized. Critical impacts come from the structure thickness as well as the interaction between length and width. Studied noise factors include sunrise and sunset (θi), relative orientation between the sun and absorber (ϕi), incident light polarization state (ψi), and dimensional errors (ΔΛ/Λ, Δf, and Δd/d). Topography of two satisfactory absorbers is identified after numerical experiments and optimization. One absorber has better absorption effectiveness, while the other has better robustness. These absorbers are composed of periodic and rectangular Cu nano-structures to facilitate absorbing, transferring, and even storing thermal energy. Their absorption effectiveness and robustness are quantitatively validated. The same design methodology can facilitate development of absorbers using other materials or working at temperature different from 773 K. These absorbers shall boost utilization of solar heat and diminish consumption of fossil fuels.