Effect of light scattering on the performance of a direct absorption solar collector

Abstract

Recently, a solar thermal collector often employs nanoparticle suspension to absorb the solar radiation directly by a working fluid as well as to enhance its thermal performance. The collector efficiency of a direct absorption solar collector (DASC) is very sensitive to optical properties of the working fluid, such as absorption and scattering coefficients. Most of the existing studies have neglected particle scattering by assuming that the size of nanoparticle suspension is much smaller than the wavelength of solar radiation (i.e., Rayleigh scattering is applicable). If the nanoparticle suspension is made of metal, however, the scattering cross-section of metallic nanoparticles could be comparable to their absorption cross-section depending on the particle size, especially when the localized surface plasmon (LSP) is excited. Therefore, for the DASC utilizing a plasmonic nanofluid supporting the LSP, light scattering from metallic particle suspension must be taken into account in the thermal analysis. The present study investigates the scattering effect on the thermal performance of the DASC employing plasmonic nanofluid as a working fluid. In the analysis, the Monte Carlo method is employed to numerically solve the radiative transfer equation considering the volume scattering inside the nanofluid. It is found that the light scattering can improve the collector performance if the scattering coefficient of nanofluid is carefully engineered depending on its value of the absorption coefficient.

Keywords

direct absorption solar collector / plasmonic nanofluid / light scattering

Introduction

As fossil fuels are running out and environmental pollution is increasing, research on sustainable and renewable energy has been conducted extensively. Among the alternative energy sources, solar energy has drawn much attention as a next generation energy source because it is abundant and environmental-friendly. There are many methods for converting solar energy into useful energy [¹–⁴]. Among those methods, solar thermal collectors have been widely studied for the heating application of buildings [⁵–⁷]. A solar thermal collector is a kind of heat exchangers and is often composed of a flat-plate black-surface absorber and tubes under the absorbing plate. The flat-plate black-surface absorbs the solar irradiation and transfers the absorbed energy to the working fluid flowing in the tubes as a form of thermal energy. The solar collector has considerable heat loss to the outside through radiation and convection heat transfer because of the concentration of heat to the absorbing plate. Furthermore, there exists unavoidable thermal resistance from the absorbing plate to the working fluid. These facts make it difficult to enhance the collector efficiency [⁸, ⁹]. In the 1970s, the concept of a direct absorption solar collector (DASC) was proposed by Minardi and Chuang [¹⁰]. The DASC utilizes black working fluid that can directly interact with the solar radiation instead of the black-surface absorber. Consequently, the thermal resistance and heat loss of the DASC become lower. However, the particles composing this black working fluid are micron-sized so that the DASC has some critical problems such as clogging and dispersion instability.

Since the synthesis of nanoparticles succeeded through fast development in nanotechnology, several types of research have focused on applications of nanoparticles [¹¹–¹⁴]. In particular, a metallic nanoparticle can have outstanding optical properties in a certain wavelength because of localized surface plasmon resonance that is associated with a collective oscillation of free electrons at the surface of the metallic nanoparticle [¹⁵]. Therefore, when metallic nanoparticles are suspended in the working fluid of the DASC, the working fluid can absorb more solar energy than the base fluid itself [¹⁶–²¹].

Optical properties including absorption and scattering coefficients of the working fluid dominantly affect the thermal efficiency of the solar collector. In many studies on DASC, particle scattering was often neglected because scattering efficiency is much lower than absorption efficiency if the particle size is much smaller than the wavelengths of solar radiation [²²–²⁵]. Because the scattering efficiency is proportional to the fourth power of the particle size parameter in the Rayleigh scattering regime [¹⁵], the assumption of ignoring scattering in the visible and the near-infrared spectral regions is reasonable. However, when localized surface plasmon is excited on metallic nanoparticles, absorption and scattering efficiencies of the nanoparticle can be in the similar order [¹⁵]. In fact, when the diameter of the metallic nanoparticle is larger than 60 nm, the scattering efficiency of the particle cannot be ignored compared with the absorption efficiency [²⁶]. According to Jain et al. [²⁷], the scattering efficiency of gold nanoparticle becomes higher than the unity in a specific wavelength range when its diameter is 80 nm. Therefore, scattering must be taken into account to accurately estimate the thermal performance of the DASC employing the plasmonic nanofluid.

The present study aims to analyze the effect of scattering on the performance of the DASC. The scope of this work also includes a parametric study on the absorption coefficient and channel height because scattering does not independently affect the performance of solar collector. In the analysis, the Monte Carlo method is employed to numerically solve the radiative transfer equation considering the volume scattering inside the nanofluid. The results obtained with the Monte Carlo method are volumetric heat generation in the solar collector and solar-weighted absorption coefficient. The calculated heat generation is incorporated into a two-dimensional heat transfer model for the solar collector using Open FOAM.

Theoretical model

For analysis of the scattering effect on the performance of a direct absorption solar collector, optical properties of the nanofluid should be investigated first. In general, dielectric function of the metallic nanoparticle (ε_p) and the base fluid (ε_b) varies depending on the wavelength of incident radiation. If ε_p+2ε_b = 0 at a certain wavelength, localized surface plasmon (LSP) is excited at the surface of metallic nanoparticle [¹⁵]. In such a case, absorption and scattering efficiencies of the nanoparticle can be higher than the unity. Therefore, if carefully engineered metallic nanoparticles supporting the LSP are suspended, the working fluid (i.e., plasmonic nanofluid) can absorb the solar energy more than the case of not involving the LSP [²⁸, ²⁹].

In the present study, the plasmonic nanofluid is employed because its optical properties can be actively controlled by changing the shape and size of metallic nanoparticles. Figure 1 illustrates the absorption and scattering efficiencies of the silica-gold core-shell nanoparticle calculated using the Mie scattering theory. When the core-shell nanoparticle has a 10-nm core radius and a 10-nm shell thickness, the scattering efficiency is negligible compared to the absorption efficiency. On the other hand, the scattering efficiency of larger core-shell nanoparticle (core radius of 80 nm and shell thickness of 10 nm) becomes higher than its absorption efficiency (refer to Fig. 1(b)). It can be also seen from Fig. 1 that the resonance wavelength of the LSP varies with the size and geometry of the nanoparticle [²⁷]. Utilizing controllability of the resonance condition of the LSP, it has been theoretically demonstrated that if the nanofluid is made of the core-shell nanoparticles of various dimensions, the blended plasmonic nanofluid can interact with the light in the solar spectrum [³⁰]. Recently, Jeon et al. [²¹] experimentally demonstrated that a nearly uniform absorption (or scattering) coefficient can be achieved by mixing nanoparticles with different dimensions and adjusting the volume fractions of each type of nanoparticles. Accordingly, the following conditions are assumed for simplicity. First, the magnitude of the scattering and absorption coefficients of plasmonic nanofluid can be tuned independently by changing the size and geometry of nanoparticle suspension as well as particle concentration. Second, the absorption and scattering coefficients can be treated as constant in the solar spectrum (300–2500 nm) by mixing nanoparticles with different sizes and/or geometries (i.e., using a blended plasmonic nanofluid).

Once the optical properties of the plasmonic nanofluid are obtained, the local volumetric heat generation rate, {{\displaystyle \dot{q}}}_{\mathrm{gen}}(y), due to the absorption of the solar radiation in the DASC, can be calculated. Figure 2 shows the schematic of the DASC considered in this work. The DASC consists of a thin (5 mm) glass cover to prevent the evaporation of the nanofluid at the top and the insulation with a mirror at the bottom to increase the optical path length of solar radiation. It is assumed that the solar radiation is normally incident to the DASC. This is to quantitatively analyze the scattering effect under the ideal situation. In the calculation, transmittance and thermal conductivity of the top glass are set to be 90% and \mathrm{1}\hspace{0.17em}\mathrm{W}/(\mathrm{m}\cdot \mathrm{K}), respectively [³¹]. If there is no scattering effect, {{\displaystyle \dot{q}}}_{\mathrm{gen}}(y)can be simply calculated by using Beer’s law [²¹]. Because the volume scattering should be fully taken into account inside the nanofluid, the Monte Carlo method is employed [³⁰, ³²] to solve the radiative transfer equation. In the Monte Carlo simulation, the intrinsic absorption of the base fluid (which could vary spectrally) is neglected based on the assumption that the scattering and absorption associated with the LSP dominantly contribute to optical properties of the plasmonic nanofluid. With constant absorption and scattering coefficients, the plasmonic nanofluid can then be treated as a gray medium. A detailed description of the Monte Carlo simulation can be found in Ref. [³⁰], and thus, they are not repeated here. With {{\displaystyle \dot{q}}}_{\mathrm{gen}}(y), the solar-weighted absorption coefficient, A_m, can be estimated as

(1){{\displaystyle A}}_{\mathrm{m}}={\displaystyle \frac{\mathrm{1}}{{{\displaystyle G}}_t}}{{\displaystyle \int }}_{\mathrm{0}}^H{{\displaystyle \dot{q}}}_{\mathrm{gen}}(y)\mathrm{d}y,

where G_T = 893.54 W/m² is the direct normal irradiance of sunlight obtained by AM1.5. Because the nanofluid is treated as a gray medium, spectral integration is not needed in estimating A_m.

A two-dimensional (2-D) model for the solar collector is constructed with Open FOAM [³³]. The flow channel consists of 800000 rectangular meshes with 2000 quarter in the x-direction and 400 quarter in the y-direction. Conditions of the solar collector are set as follows: the mass flow rate per unit width \dot{m}=\mathrm{0.004}\mathrm{kg}/(\mathrm{m}\cdot \mathrm{s}), the channel length L=1 m, and the channel height H=1.5 cm. Here, \dot{m} is set as the reference value of 0.004 \mathrm{kg}/(\mathrm{m}\cdot \mathrm{s}) to make the conditions for obtaining a meaningful collector efficiency and gain temperature with reference to other studies [²¹, ²⁵, ³⁰]. In this situation, Re_H=22.9; thus, the flow in the collector is laminar. Moreover, because the hydrodynamic entrance length L_e=8.2 cm is much smaller than the collector length, the velocity profile in the solar collector can be assumed to be fully developed in the entire portion of the channel. As a result, the energy equation can be independently solved by adopting the parabolic velocity profile of the Poiseuille flow [³⁴]. Thermophysical properties of the nanofluid are assumed to be the same as that of pure water because the particle volume fraction is lower than 0.01% [³⁴, ³⁵].

Results and discussion

To analyze the effect of scattering from multiple perspectives, several simulations are conducted under various conditions of parameters, as summarized in Table 1. At first, the thermal performance of the solar collector is calculated for absorption coefficients in the range of 0.05–1 cm^–1 when scattering is neglected. In the Monte Carlo simulation, the incident solar radiation is treated as photon bundles, and each photon bundle is traced until it is either completely absorbed by the working fluid or escapes from the solar collector (i.e., through the top glass cover). For numerical convergence, 30 million photon bundles are used in each simulation.

Figure 3 demonstrates the calculated solar-weighted absorption coefficient (A_m) and the corresponding efficiency (η) with the absorption coefficient (α) of nanofluid. When α <0.4 cm^–1, more than 20% of the solar radiation cannot be absorbed by the nanofluid; that is, A_m<0.8, as can be seen from Fig. 3(a). Specifically, if α = 0.3 cm^–1, the corresponding A_m = 0.734 and η = 45%. The relatively low collector performance is partially because nearly 26% of the solar radiation is not absorbed by the working fluid. This could be remedied by increasing the optical path length (i.e., increasing either the channel height or the actual traveling distance of photon bundles inside the channel).

As modeled in the Monte Carlo simulation, it can be imagined that the propagation direction of a photon bundle is altered after each scattering event. Therefore, the scattering associated with nanoparticle can increase the optical path length of solar radiation at a given channel height and eventually help improve the thermal performance of the DASC. To illustrate the effect of scattering on the local heat generation rate in the channel, the simulation for the solar collector is conducted using the nanofluid with α = 0.3 cm^–1 while varying its scattering coefficient (σ). It can be seen from Fig. 4 that the local heat generation rate largely depends on the scattering coefficient value. Compared to the case with no scattering, scattering can substantially increase the local heat generation rate, especially near the location of y=0.2H. On the other hand, scattering causes {{\displaystyle \dot{q}}}_{\mathrm{gen}}\left(y\right)in the vicinity of the top glass cover to decrease as compared to the case with no scattering. Because the photon bundles near the top glass cover have more chances to be scattered out from the channel, the local absorption near the top glass cover will be decreased due to back scattering. Therefore, the maximum of absorption will appear inside the channel if back scattering occurs. It can also be seen from Fig. 4 that {{\displaystyle \dot{q}}}_{\mathrm{gen}}\left(y\right)sharply decreases towards the bottom surface of the channel if σ >1.6 cm^–1. This is related to the average propagation length of photon bundle given by L_avg = 1/(α+σ) [²¹]. Statistically, it can be imagined that due to scattering, photon bundles travel in a zigzag way across the channel depth. Therefore, there will be more chances for photon bundles to be absorbed in the upper side of the channel, especially for the case with a larger scattering coefficient. Besides, it is noteworthy that there exists an inflection point of {{\displaystyle \dot{q}}}_{\mathrm{gen}}\left(y\right) near the bottom of the channel for σ >1.6 cm^–1. This is due to the reflection by the mirror at the bottom surface [²¹]. Notice that an inflection point in the local heat generation rate indicates the combined absorption effect of forwardly propagating and backwardly propagating photon bundles. Once the photon bundle reaches the bottom surface, it will be perfectly reflected by the mirror and continuously travels upward inside the channel. For lower values of α, the corresponding L_avg would be larger, leading to the fact that photon bundles will quickly move to the upper region after reflected from the mirror. On the other hand, photon bundles with smaller L_avg values will have more chances to be absorbed near the bottom surface due to scattering. Figure 4 clearly exhibits that light scattering by suspended nanoparticles redistributes the local absorption of photon bundles by the nanofluid.

The effect of scattering on the mean optical path length of photon bundles is shown in Fig. 5(a). Here, the mean optical path length is estimated in the Monte Carlo simulation as the average of the travel distance of all photon bundles until they are either fully absorbed by the working fluid or escape from the collector channel. In the calculation, if more than 99% of the energy of a photon bundle is absorbed, the photon bundle is regarded as completely absorbed by the nanofluid. Notice that even though there is no scattering, the mean optical path length is longer than 2H (i.e., round trip distance of the channel). This is because the top cover glass reflects some of the photon bundles back to the inside of the solar collector. As the scattering coefficient increases, the mean optical path length is also increased and can be twice of that with no scattering. The increased mean optical path length of photon bundles will result in enhanced absorption of the solar radiation. However, the mean optical path length starts decreasing if σ further increases from 1.7 cm^–1. This is primarily due to the back scattering of photon bundles near the top glass cover. Due to the back scattering, photon bundles cannot propagate deeper into the channel but leave the collector before being considerably absorbed by the nanofluid. Therefore, there exists an optimum value of the scattering coefficient for enhancing the absorption of solar radiation by the nanofluid.

Figure 5(b) depicts the solar-weighted absorption coefficient of the solar collector with respect to the scattering coefficient. To elucidate the positive and negative aspects of light scattering, the portion of increased (or decreased) absorption due to increased (or decreased) mean optical path length of photon bundles as compared to the case of no scattering are calculated. Consistent with Fig. 5(a), the portion of increased absorption increases as the scattering coefficient increases; however, its increase rate slowly decreases towards the higher scattering coefficient. The reason for this is that the absorption mean free path is set to be a constant of 1/α ≈ 3.33 cm. That is, the mean optical path length will be bounded due to the fixed absorption mean free path. On the other hand, the portion of decreased absorption keeps increasing with the scattering coefficient due to back scattering. Similar to the mean optical path length in Fig. 5(a), for a given α value, there exists an optimum value of the scattering coefficient for enhancing the solar-weighted absorption coefficient. For the nanofluid with α = 0.3 cm^–1, the solar-weighted absorption coefficient can be increased from 73.4% (no scattering) to 85.6% by introducing σ = 1.4 cm^–1.

Figure 6 plots the collector efficiency and gain temperature with the scattering coefficient at α = 0.3 cm⁻¹. The collector efficiency exhibits a similar trend to the solar-weighted absorption coefficient in Fig. 5(b). When σ > 0.8 cm^–1, however, {{\displaystyle \dot{q}}}_{\mathrm{gen}}\left(y\right)becomes larger in the upper region of the channel (refer to Fig. 4), causing a greater heat loss via convection and radiation from the top glass cover, suggesting that the location of the maximum heat generation affects the collector efficiency. In fact, the collector efficiency depends on various factors, such as heat generation rate, heat loss, and mass flow rate per unit length. As a result, when σ = 1.5 cm^–1, the collector efficiency increases by 7.2% point and the gain temperature increases by 3.8 K compared with the case of no scattering.

It should be noted that if the scattering by nanoparticle suspensions is negligible, the energy of photon bundles will exponentially decay along the channel depth. As shown in Fig. 7(a), in the case of no scattering, the maximum heat generation always appears at the top glass cover, causing the maximum temperature to appear also at the top surface. Therefore, the heat loss (via convection as well as radiation) from the top glass cover will increase continuously with the absorption coefficient if there is no scattering. However, if scattering occurs inside the nanofluid, the maximum of heat generation shifts to the inside of the channel (see Fig. 4), which, in turn, lowers the temperature of the top surface. Therefore, the heat loss to the outside can be smaller than the case with no scattering. Furthermore, even if the scattering coefficient increases, the heat generation rate in the vicinity of the top glass cover does not change considerably, as noted from Fig. 4. Consequently, the heat loss from the top glass cover will be saturated even though the scattering coefficient continuously increases. For instance, the total heat loss at α = 0.8 cm^–1 and σ = 0.3 cm^–1 is 248 W/m² in Fig. 7(b). On the other hand, in Fig. 7(c), the total heat loss at α = 0.3 cm^–1 and σ = 2 cm^–1 is 224 W/m², which is about 10% lower than the previous case. Consequently, in regards to reducing the heat loss from the top glass cover, increasing the scattering coefficient can be advantageous as compared to increasing the absorption coefficient.

The relative importance of the scattering coefficient to different values of the absorption coefficient for the system performance is also investigated. Figure 8 plots the collector efficiency and the gain temperature with the scattering coefficient. The collector efficiency increases and then decreases as the scattering coefficient increases. Figure 8 suggests that the collector efficiency is more sensitive to the scattering coefficient if α<0.3 cm^–1. When α= 0.1 cm^–1, for instance, the collector efficiency increases from 24.4% (no scattering) to 40.1% (with σ = 1.8 cm^–1). However, the effect of scattering on the collector performance becomes insignificant for the nanofluid with higher values of the absorption coefficient. The reason for this is that when α > 0.6 cm^–1, the solar-weighted absorption coefficient is already above 0.9 (see Fig. 3); that is, most of the solar radiation can be effectively absorbed by the nanofluid without any scattering effect. As a result, the absorption and scattering coefficients should be carefully designed to maximize their effectiveness with the lowest possible cost (i.e., minimum volume fraction of nanoparticles). Interestingly, the value of σ that leads to the maximum of collector efficiency decreases as increases.

The effect of the channel height on the performance of solar collector is considered in Fig. 9. For non-scattering case, the solar-weighted absorption coefficient will mainly depend on the factor of e^–αH. Therefore, for the solar collectors having the same value of αH, the resulting A_m is nearly independent of the channel height if σ = 0 (i.e., no scattering). Slight differences at σ = 0 are caused by the fact that the temperature distribution inside the channel depends on α instead of αH, leading to different values of heat loss from the top cover glass. For the nanofluid with non-zero values of σ, the effect of scattering on the collector efficiency depends on the channel height. It is found that the solar collector with larger values of H is more sensitive to scattering. For example, the collector efficiency with H = 2 cm increases faster with the scattering coefficient than the case with H = 1 cm. The reason for this is that, at a given αH value, a larger H corresponds to a lower α. As noted from Fig. 8, the effect of scattering becomes more significant for the lower values of α. Moreover, at a given σ, the probability of photon bundles to be scattered would be higher when H becomes larger.

Figure 10 quantitatively demonstrates how much enhancement can be made with σ = 1 cm^–1 as compared to the case of no scattering for different values of α. As noted earlier, the scattering by nanoparticle suspension helps the nanofluid absorb more solar energy than the case of no scattering. As can be seen from Fig. 10(a), for α <0.4 cm^–1, the increase of A_m due to scattering can be more than 10%. On the other hand, if α = 0.6 cm^–1, it is difficult to increase the solar-weighted absorption coefficient any more through scattering. The trend of collector efficiency shown in Fig. 10(b) mostly looks similar to that of the solar-weighted absorption coefficient. However, differently from A_m, for the case of α = 1 cm^–1, the efficiency with σ = 1 cm^–1 becomes higher by 0.3% than that with σ = 0 cm^–1 due to the heat loss from the top glass cover. Recall that the nanofluid with α = 1 cm^–1, substantial {{\displaystyle \dot{q}}}_{\mathrm{gen}}(y)occurs in the vicinity of the glass cover (refer to Fig. 7(a)). It can be seen from Fig. 10(b) that the solar collector with α = 0.3 cm^–1and σ = 1 cm^–1can be equivalent in terms of the collector efficiency to the system with α = 0.5 cm^–1 and σ = 0 cm^–1.

Conclusions

In the present study, the effects of scattering on the performance of direct absorption solar collector were numerically investigated. If the suspended nanoparticle is made of metal, the scattering cross-section of metallic nanoparticles could be comparable to their absorption cross-section depending on the particle size, especially when the localized surface plasmon is excited. It is shown that the light scattering by nanoparticle suspension could increase the mean optical path length of photon bundles inside the collector channel, which, in turn, enhances the solar-weighted absorption coefficient. Moreover, in regards to reducing the heat loss from the top glass cover, increasing the scattering coefficient can be advantageous as compared to increasing the absorption coefficient. However, due to increased back scattering with the scattering coefficient, there exists an optimum value of the scattering coefficient for enhancing the absorption of solar radiation by the nanofluid. It is demonstrated that the solar collector with α = 0.3 cm^–1 and σ = 1 cm^–1 can be equivalent in terms of the collector efficiency to the system with α = 0.5 cm^–1 and σ = 0 cm^–1. The results of this study will provide guidance for designing direct absorption solar collectors using plasmonic nanofluids.

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