Thermophotovoltaic System Simulation with Realistic Experimental Considerations

Evan Schlenker*, Zhiguang Zhou†, Peter Bermel†

*Department of Electrical and Computer Engineering, Bucknell University, Lewisburg, PA 17837, USA †School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA

Abstract Thermophotovoltaic (TPV) systems are a promising type of energy generation method that convert heat into electricity via thermal radiation. TPV has potential to benefit the economy, the energy sector, and the environment by converting waste heat from other power generation methods into electricity. Simulations of these systems can play a key role in designing TPV systems and validating their experimental performance. Current simulation tools can model important aspects of TPV systems fairly accurately, but generally make certain simplifying assumptions that are challenging to reproduce in experiments. Developing a simulation tool that accurately captures thermal emission and reflection in complex, realistic geometries will facilitate understanding and further development of TPV systems. An existing tool developed at Purdue, known as TPVtest, has now been modified and streamlined to create a new tool, TPVexpt, to help achieve this goal. New features in TPVexpt include: (1) the input of shunt and series resistance of a PV cell and consideration of the associated losses when calculating power output; and (2) the input of arbitrary size and position rectangular heater, emitter, and PV diode for view factor calculations to accurately model radiative heat transfer. TPVexpt combines an accessible GUI with TPV analysis that accounts for thermal non-idealities and realistic geometries to produce more accurate power and efficiency predictions for TPV systems. Finally, TPVexpt has been partially validated against real-world TPV experiments up to 800 °C; additional work is needed here to verify the generality of this approach, and to aid current and future researchers in advancing TPV technology.

Introduction Thermophotovoltaic (TPV) systems convert heat into electricity via thermal radiation [1-3]. They have potential to operate with high efficiency in compact areas, even below 1 square centimeter. Their solid state operation may also be valuable for long-term or remote commercial or industrial applications, due to low maintenance requirements. Analytical work has suggested that extremely high conversion efficiencies are possible with this technology. A great deal of modeling incorporating new technologies has been carried out to predict the limits of system performance under various constraints. However, in a real TPV system, structures are not as simple as two squares in proximity - thermal emission and reflection can occur from multiple regions in these devices. The pathways grow exponentially with the number of reflections, leading to highly complex calculations [4,5]. Nonetheless, considering these factors correctly can help further the progress to understand the detailed loss mechanism scientifically and to realize more efficient TPV systems in real-world applications. Thus, it is critical to develop a numerical model that considers the various interactions that can take place and calculate only the most important contributions, while estimating the residual error of the calculation.

TPV cells are solid-state p-n junction semiconductor devices that absorb thermal radiation emitted by a heated element and convert said heat into electricity. Thermal energy supplied by various sources such as combustion, the sun, or waste heat is channeled to an emitter. An ideal emitter exhibits blackbody behavior, where the characteristics of the emitted radiation depend only on the temperature of the emitter. The emitter is typically operated in the temperature range of 1000-1573 K, producing primarily mid-infrared radiation. Because of Wien’s and Stefan- Boltzmann’s laws of thermal radiation, increasing the emitter temperature will result in higher energy photon production and potential for greater output power density. The emitted photons propagate towards the PV cell, which should be at a significantly lower temperature than the emitter (approximately 300 K), since the PV cell is most efficient at lower temperatures. Photons with energy higher than the PV cell bandgap (shorter wavelengths) are absorbed by the cell and converted to electricity via the photoelectric effect, while photons with lower energy than the bandgap (longer wavelength) are either not absorbed or converted to heat, which reduces PV operation efficiency and wastes incident energy. In order to achieve high efficiency operation it is necessary to design the TPV system so the majority of emitted photons have energies greater than the PV cell bandgap. Much PV research has been devoted to Si and GaAs semiconductors, which have too large of a bandgap for TPV applications dealing with relatively low energy incident photons, associated with operating temperatures of 1500 K or below. Lower bandgap semiconductors such as GaSb, InGaAs, and InGaAsSb are therefore more prevalent in TPV systems, due to their lower and thus more suitable bandgap energies [2].

Matching the properties of the thermal radiation with the PV cell is of great importance in maximizing TPV system efficiency. Selective emitters, filters, and photon recycling can all function to increase TPV efficiency, potentially by orders of magnitude. Selective emitters attached to the radiator allow for spectral shaping, essentially suppressing emission of below bandgap photons and enhancing emission of above gap photons [3]. This can be achieved by implementation of photonic crystals (PhCs). PhCs come in a wide variety of types (1D, 2D, 3D) and materials (semiconductors, rare earth metals). Simulation has shown that using PhCs made from refractory metals (which are ideal for TPV use since they can withstand high temperatures) can significantly improve TPV efficiency when used as a selective emitter [6]. PhCs can also be utilized as a front-side filter deposited on the front of the PV diode that allow energies above the bandgap to pass through to the PV diode and reflect photons with energies below the bandgap [7]. This reflection of low energy photons back to the emitter for reabsorption is known as photon recycling. Spectral utilization in photon recycling may be quantified with filter efficiency, defined as the ratio of above bandgap optical power received by the PV cell to the total amount of optical power absorbed by both the PV cell and the filter [8].

Fabricating materials and testing all combinations of PhC materials, PV materials, filters, and photon recycling methods is tedious and time consuming. Accurate simulations of TPV systems can dramatically accelerate the discovery of higher-performance designs. GaSb has been extensively tested to yield the temperature dependent material parameters that can be used for accurate TPV simulation, including carrier concentration and mobility and absorption coefficient [9]. A program called TPVtest that uses a finite difference time-domain (FDTD) simulation developed by MIT known as MEEP can be used to consider the performance of different emitter materials and properties of TPV systems [6]. TPVtest does not consider thermal emission and reflection in its analysis. TPVtest incorporates a compact model based on

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experiment, but cannot predict the effects of new PV cell architectures. Instead, this can be understood using other simulation tools, such as PC1D, ADEPT, or AMPS. They have previously been used to analyze the effects of Auger recombination (results when excess energy from absorbed photons is given to another carrier and causes thermal vibration loss) and to predict the effects of PV series resistances over a range of temperatures [10].

While the above simulations and models are useful and fairly accurate, there are a number of factors that can cause the simulated/predicted TPV system behavior to differ from experimental results. For example, surface diffusion in a nanostructured selective emitter causes reshaping of features with high curvature, which shifts resonance peaks towards lower wavelengths and eventually decreases their spectral emissivity [11]. Thermal emission and reflection is another source of loss and non-ideal behavior. Geometric and thermodynamic concepts and theory can be used to analyze radiative heat exchange between surfaces with specular reflection [4]. Implementing thermal analysis into a new simulation tool will allow for more accurate TPV simulation.

Developing a tool to accurately model thermophotovoltaic performance in complex and realistic geometries could play an important role in progressing the scientific understanding of the loss mechanisms present in real experiments, and to help optimize existing system designs for improved performance.

Methodology TPVtest is a TPV system simulation tool created in 2013 [12]. The tool employs MEEP, which is a Finite Difference Time Domain (FDTD) method that simulates electromagnetic wave propagation [13]. It also implements the Stanford Stratified Structure Solver (S4), a frequency domain code to solve the linear Maxwell’s equations in layered periodic structures [14]. TPVtest simulations require the view factor from the PV cell to the emitter as a user defined input. This can be very difficult to calculate. It also assumes that the PV cell and emitter are both equivalently sized squares, which is often not the case in a TPV system. The heater is not considered in TPVtest, which often has a major impact on system efficiency. Furthermore, electrical non idealities such as series and shunt resistance as well as some other assumptions such as 100% PV cell active area and constant dark current density are included.

TPVtest works well for ideal simulations, but neglects many of the real-world factors that a researcher would encounter in a TPV system. TPVexpt is a new program developed using the Rappture interface on NanoHUB.org, which is a science and engineering network funded by the NSF and maintained by the Network for Computational Nanotechnology (NCN). TPVexpt is based on TPVtest. It includes modifications such as: view factor calculations, heater implementation including new outputs, series/shunt resistances, PV cell active area, changes in dark current density, as well as cosmetic changes to the GUI.

The view factor from the PV diode to the emitter or heater plays a key role in determining the overall TPV system performance. In radiative heat transfer, the view factor 푭푨→푩 is the proportion of the radiation which leaves surface A that strikes surface B. Thus, its value ranges from 0 to 1. In TPVtest, the view factor is a user-specified input parameter. This implementation

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can be cumbersome because it can be difficult for a user to estimate or calculate the view factor accurately, given the complexity of the view factor calculations, particularly for any geometries differing from two aligned, parallel squares, which encompasses the vast majority of cases. TPVexpt still somewhat simplifies this aspect of a TPV system, but allows the user to input the dimensions of the heater, emitter, and PV diode as well as control the alignment between the components. Figure 1 shows the modeled TPV system geometry.

Fig. 1. Coordinate system for rectangular model of TPV system components [15]. Surface A1 represents the PV diode. Surface A2 represents the heater or emitter, depending on the calculation.

Equation 1 shows the view factor F1-2 calculation from surface A1 to surface A2 [15].

(1)

The results of the view factor calculations from Equation 1 were verified by analyzing special case scenarios (e.g. different sized center-aligned squares or identical center-aligned rectangles) [16,17]. Results from Equation 1 were exactly the same as from other implementations of the special case scenarios. As such, the view factor calculation in TPVexpt is accurate and verified to the greatest extent possible given the lack of prior coded implementations of Equation 1.

Equation 1 can be used to calculate the view factor for the uncovered heater area. If the heater is larger than the emitter (as is the case in most TPV systems), there will be uncovered heater material around the emitter. This area can be broken up into different rectangle zones, an example of which is shown in Figure 2.

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Fig. 2. Example of rectangular divided uncovered heater area. The back rectangle represents the heater (dark blue). The light blue rectangle represents the emitter.

This uncovered area is usually a different temperature and material than the emitter. This results in significant differences when performing power calculations between the PV cell and either the emitter or uncovered heater. By summing the view factors from the PV diode to each rectangle (shown in Figure 2) and accounting for the material and temperature differences in the heater, it is possible to calculate the power contribution from the heater.

Substantial modification to the TPVtest code was necessary in order to accommodate the heater power calculations. The heater is treated as a graybody emitter (blackbody radiation pattern) with a constant emittance of 0.9. Power is calculated the same way as for the emitter, but using the heater-specific emittance spectrum, temperature, and view factor. The ISC generated from the heater is added to the ISC generated from the emitter to find the total ISC. Similarly, the power emitted from the heater is added to the emitter power to calculate the total emitted power in the system. Total emitted power (W) is an output for TPVexpt. Waste heat (mW) is also a new output. In this context, waste heat accounts for power absorbed by the cell but not converted into electricity, as well as a fraction (20%) of the non-absorbed emitted power. This factor is chosen as a compromise, since it is pessimistic to include all the non-absorbed emitted heat as waste heat, but unrealistic to assume that all heat is used. Equation 2 shows the formula for the waste heat Pwaste, where Pabs is the total power absorbed by the PV diode, Pelectric is the power generated by the PV cell, and Pemit is the total power emitted from the heater and emitter.

푃푤푎푠푡푒 = (푃푎푏푠 − 푃푒푙푒푐푡푟𝑖푐) + 0.2[푃푒푚𝑖푡 − (푃푎푏푠 − 푃푒푙푒푐푡푟𝑖푐)] (2)

TPVexpt also considers electrical non-idealities by accommodating series and shunt resistance inputs for the PV diode. Series resistance in a solar cell has three causes: the flow of current through the emitter and base of the solar cell, the contact resistance between the metal contact and the silicon, and the resistance of the top and rear metal contacts [18]. Non-ideal shunt resistance results from manufacturing defects. Figure 3 shows a model of a PV diode with the aforementioned resistances incorporated.

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Fig. 3. PV diode model incorporating both series and shunt resistances [18].

Basic circuit analysis of the above figure suggests that in order to maximize output power, RS must be kept low, while RSH must be kept high. Deviations from the ideal cases (RS = 0 Ω, RSH = ∞ Ω) reduces the power output of the PV cell by affecting the fill factor. The fill factor is defined as the ratio of the maximum power from the solar cell to the product of VOC and ISC. Equation 3 shows the calculation for the ideal fill factor FF0, where 푣푂퐶 is the normalized open circuit voltage of the PV cell.

푣푂퐶 − ln (푣푂퐶 + 0.72) 퐹퐹0 = (3) 푣푂퐶 + 1

Equations 4, 5, 6, and 7 are used to calculate the fill factor FF that accounts for series/shunt resistances [18].

푉푂퐶 푅퐶퐻 = (4) 퐼푆퐶

푅푆 푟푆 = (5) 푅퐶퐻

푅푆퐻 푟푆퐻 = (6) 푅퐶퐻

2 2 푟푆 푉푂퐶 + 0.7 퐹퐹0 푟푆 퐹퐹 = 퐹퐹0 {(1 − 1.1푟푆) + } {1 − [(1 − 1.1푟푆) + ]} (7) 5.4 푉푂퐶 푟푆퐻 5.4

Equation 8 shows how FF (and thus the series/shunt resistances) affects the efficiency, η.

푉 퐼 퐹퐹 휂 = 푂퐶 푆퐶 (8) 푃𝑖푛

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Clearly, as FF decreases, (as RSH and RS deviates from the ideal parameters) the efficiency of the TPV system decreases as well. The default values in TPVexpt of series and shunt resistance are 0 Ω and 10,000 Ω, respectively. Users can input a “0” into the shunt resistance to ignore the effects of shunt resistance entirely.

For a PV cell, only certain parts of the surface contribute to generation of ISC. This is due to the dark lines necessary for fabrication of a PV cell, which can cause shading losses. These losses directly affect the ISC produced by the cell, but do not significantly affect the total absorbed power of the cell. TPVexpt users can input the active area of a PV cell in cm2. The “area factor”, or AF, calculated in Equation 9 is then multiplied into the existing ISC calculation to calculate the correct ISC.

퐴푐푡𝑖푣푒 푎푟푒푎 × 100 (푐푚2) 퐴퐹 = (9) 퐶푒푙푙 푙푒푛𝑔푡ℎ (푚푚) × 퐶푒푙푙 푤𝑖푑푡ℎ (푚푚)

This calculation takes place before the series and shunt resistance calculations so that the proper ISC is used for calculating the fill factor.

Temperature can have a significant effect on the dark current density. The default room temperature (300 K) dark current density in TPVexpt is 1 μA/cm2. Equation 10 has been implemented in TPVexpt to account for the effect of temperature on dark current I0, where Tdiode is the PV cell temperature in K, Eg is the PV cell bandgap, and k is Boltzmann’s constant [18].

3 1 1 푇푑𝑖표푑푒 (퐸푔⁄푘)(− + ) 퐼 = 퐼 ( ) 푒 푇푑푖표푑푒 300 (10) 0 0(푟표표푚 푡푒푚푝) 3003

In addition to more realistic experimental considerations over TPVtest, TPVexpt contains updates to the GUI. The tabs for the emitter, filter, and system have been converted to a phase setup. In this case, the user can progress through the pages by clicking the buttons at the bottom or the numbered buttons at the top. The user can click the fourth tab to initiate the simulation without progressing through the rest of the input parameters. The output will now display in an entire window, allowing more space for analysis of the results.

The new inputs include reuse emittance.txt, heater temperature, heater dimensions and positioning, emitter dimensions and positioning, PV diode active area, PV diode temperature, PV diode dimensions and positioning, and series and shunt resistance. Total emitted power, heat loss, and light generated current have been added as outputs, while cutoff wavelength has been removed due to feedback from current TPVtest users. Many new images detailing the TPV system layout and dimensions are included. The images have been added as .html files to reduce the size and complexity of the example files that need to contain the tool information.

Results and Discussion TPVexpt combines a clear graphical user interface with realistic conditions to provide accurate analysis of a TPV experimental setup. Figure 4 shows the first two tabs of the GUI.

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(a) (b)

Fig. 4. Images of the first two phases of TPVexpt. Users can access the menu in (b) by clicking the “Filter” button seen at the bottom of (a), or by clicking the button “2 filter” at the top of (a).

The inputs in the menus in Figure 4 are unchanged from TPVtest, except for the “Resuse emittance.txt” slider, which allows the user to use the last calculated emittance spectrum to save time when performing system analysis with the same emitter. The photonic crystal diagram that used to exist in Figure 4(a) is replaced with a TPV system diagram, followed by some brief introductory information for the tool.

Figure 5 shows the different tabs in the “System” phase, which contain the majority of the changes from TPVtest.

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(a) (b) (c)

Fig. 5. “System” phase for TPVexpt. In (a), the user can enter parameters specific to the heater and emitter. In (b), the user can enter both physical and electrical PV cell properties. In (c), the user can alter the alignment and positioning in the system.

Figure 6 shows an example of the emittance curve for an ErAlGa selective emitter, along with a list of the other available outputs after running a simulation with TPVexpt.

Fig. 6. Available outputs of TPVexpt. Currently displaying emittance curve.

It is important to verify that TPVexpt produces the correct simulation results. This can be done by comparing output data with experimental data. Measuring efficiency for a physical TPV system can be

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difficult. Thus, recent experimental ISC results obtained by graduate mentor Zhiguang Zhou at Purdue’s Birck Nanotechnology Center for an ErAlGa TPV setup were compared with the ISC outputs of TPVexpt. The experimental setup being analyzed included a heatshield that was flush with the surface of the emitter, covering the entire heater. It was simulated by using a custom heater emittance file with a constant emittance of 0.1. There was no filter in the system. Modifications to the default TPVexpt inputs are as follows: material choice = ErAlGa, type of filter = rugate, high index = 1, low index = 1, number of materials = 1, heater length = 35.2 mm, heater width = 25 mm, and emitter thickness = 0 mm.

Figure 7 shows the results of the experimental short circuit current analysis.

Fig. 7. Results of experimental ISC - Theater verification.

The green line is the direct ISC outputs from TPVexpt, where the assumption that the “heater temperature” is 150 °C below the emitter pertains to the heater shield likely being a much lower temperature from the physical heater. This assumption is used for all the curves in Figure 7. In the experimental setup, only the heater temperature itself can be accurately measured due to physical restrictions and spatial confinements. The heater shield temperature cannot be measured; however, altering the “heater temperature” input in TPVexpt used to represent the heater shield had little effect on the shape of the output curve and thus is not an important factor in the final results.

The emitter temperature is also difficult to measure. The assumption that the experimental emitter temperature was equal to the measured experimental heater appears to be valid until approximately 600 °C, where the predicted ISC begins to deviate substantially from the experimental ISC. Further investigation and research suggested that as the heater temperature rises, so does the temperature gradient between the heater and the emitter. This means that the

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temperature measured for the heater can no longer be used as an accurate input for the emitter temperature in TPVexpt.

The temperature gradient is modeled by assuming that the emitter receives heat from the heater by conduction and loses it exclusively through thermal radiation. This gives rise to Equation 11:

4 푘(푇ℎ푒푎푡푒푟 − 푇푒푚𝑖푡푡푒푟) = 휎휀푎푣푒(푇푒푚𝑖푡푡푒푟)(푇푒푚𝑖푡푡푒푟 ) (11) where k is the effective thermal conductivity that takes the thermal contact resistance at each interface into account; 푇ℎ푒푎푡푒푟 is the heater temperature; 푇푒푚𝑖푡푡푒푟 is the emitter temperature; 휎 is the Stefan-Boltzmann constant and 휀푎푣푒 is the spectral averaged emitter emittance. In the red curve, this temperature gradient is modeled according to the prior equation. This calculation matches more closely than before, when the gradient was not considered, but different k values can still cause mismatches at other data points. A value of 0.29 was chosen, since the accuracy of the match is close up to 650 °C and closer for the higher temperatures from when the gradient was not considered.

While the temperature gradient between the heater and the emitter is observed, it does not have as strong a temperature dependence as expected. Therefore, it is necessary to consider other possible causes of the ISC - Theater roll-over visible in the experimental data that occurs beyond 600 °C. It is suspected that the impact of the series resistance of the wired solar cell may be magnified at higher temperatures [18]. Typically, generated current from a PV diode can be expressed as shown in Equation 12:

푞(푉+퐼푅푆) 푛푘푇 퐼 = 퐼푙 − 퐼0(푇푑𝑖표푑푒) (푒 푑푖표푑푒 − 1) (12) where Tdiode is the PV diode temperature, Il is the light generated current, n is the ideality factor, RS is the series resistance and V is the applied voltage to the PV diode. The ISC is measured at V = 0. If RS is sufficiently large, ISC < Il. This discrepancy will expand at higher cell temperature since I0 increases at higher temperatures.

In the case where k = 0.29, the cell temperature Tdiode that can most accurately reproduce the measured ISC - Theater curve was identified. The actual Tdiode values are preliminary in nature, and would need to be confirmed by future experiments. On the other hand, the extremely close fit observed in the blue line in Figure 7 is indicative of the likely suitability of the overall modeling approach.

Future improvements to the experimental setup will allow further verification for the accuracy of TPVexpt. Nevertheless, TPVexpt is certainly much more accurate than TPVtest up to temperatures of at least 800 °C.

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Conclusions TPVexpt successfully models TPV system behavior in a graphical user interface, which streamlines and improves the capabilities of the prior TPVtest tool to accurately simulate experimentally relevant TPV systems. TPVexpt will be a valuable tool in identifying trends and sources of error in TPV experiments, their causes, and possible mitigation strategies.

Acknowledgements I would like to thank Peter Bermel, Zhiguang Zhou, Mike McLennan, Tanya Faltens, and Vicki Leavitt for valuable discussions. This work was supported by NSF Award EEC 1227110 - Network for Computational Nanotechnology Cyberplatform.

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