Effect of non-uniform illumination on performance of solar thermoelectric generators

Abstract

Solar thermoelectric generators (STEGs) are heat engines which can generate electricity from concentrated sunlight. The non-uniform illumination caused by the optical concentrator may affect the performance of solar thermoelectric generators. In this paper, a three-dimensional finite element model of solar thermoelectric generators is established. The two-dimensional Gaussian distribution is employed to modify the illumination profiles incident on the thermoelectric generator. Six non-uniformities of solar illumination are investigated while keeping the total energy constant. The influences of non-uniform illumination on the temperature distribution, the voltage distribution, and the maximum output power are respectively discussed. Three thermoelectric generators with 32, 18 and 8 pairs of thermocouples are compared to investigate their capability under non-uniform solar radiation. The result shows that the non-uniformity of the solar illumination has a great effect on the temperature distribution and the voltage distribution. Central thermoelectric legs can achieve a larger temperature difference and generate a larger voltage than peripheral ones. The non-uniform solar illumination will weaken the capability of the TE generator, and the maximum output power decrease by 1.4% among the range of non-uniformity studied in this paper. Reducing the number of the thermoelectric legs for non-uniform solar illumination can greatly increase the performance of the thermoelectric generator.

Keywords

solar thermoelectric generators / non-uniform solar illumination / performance evaluation / solar energy

Introduction

Thermoelectric generator is a device which can directly translate the thermal energy into electricity based on Seebeck effects [¹–⁵]. It has attracted great interest because of its well-known advantages (having no moving component, stability, noiseless modularity and no pollution). The thermoelectric generator can be employed in a wide range of fields because it can generate direct current as long as there exists a temperature difference between the hot side and the cold side. Solar energy, as the heat source for thermoelectric power generation, is a good choice because it is clean and abundant [⁶–¹¹].

However, the direct use of solar energy has a disadvantage because of the low heat flux. To increase the input heat flux, an optical concentrator is usually chosen [¹²–¹⁷]. Baranowski et al. [¹²] have proposed a novel detailed energy balance model for STEGs and predicted the theoretical STEGs efficiencies with today’s materials. The STEGs can achieve an electric efficiency of 15.9% when the incident flux is 100 kW/m². Amatya and Ram [¹³] have theoretically and experimentally investigated a STEGs system using a cheap parabolic concentrator. They have pointed out that the efficiency can reach 3% for a commercial Bi₂Te₃ module with a solar concentration of 66 × suns and a conversion efficiency of 5.6% can be obtained for the novel thermoelectric materials such as n-type ErAs:(InGaAs)_1-x(InAlAs)_x and p-type (AgSbTe)_x(PbSnTe)_1-x when the optical concentration ratio is 120. Chen [¹⁴] has studied the STEGs including optical concentration in addition to thermal concentration. He has found that the STEGs can obtain an efficiency higher than 5% with little or no optical concentration while the thermal concentration is employed to increase the input heat flux.

The utilization of the optical concentrator can increase the heat flux incident to the hot side of TE. However, the optical concentrator inevitably causes the non-uniform of solar radiation no matter what type of optical concentrator is employed. The temperature of the thermoelectric hot side is certainly uneven which may affect the performance of the STEGs. Ming et al. [¹⁸] have investigated the influence of non-uniformity of heat flux on the thermal stress at the STEGs. They have pointed out that the uneven heat flux would significantly increase the thermal stress at the TEG and decrease the life expectancy of the device. Suzuki et al. [¹⁹] have discussed the performance of STEGs under three models of light concentration. They have concluded that the difference between light concentrations has little influence on the total power output. Admasu et al. [²⁰] have divided the hot side of the TE module into four parts to study the performance of STEGs under non-uniform temperature distribution. They have proved, by theory and experiment, that the uneven temperature distribution would weaken the capability of the TE module. However, the investigation of the influence of non-uniform solar irradiance on the performance of STEGs is incomplete, the contour of concentrated sunlight is not taken into consideration, and the square concentrating heat flux or different temperatures of segmented regionsis usually given for the non-uniformity, which is inconsistent with reality.

In this paper, a three-dimensional finite element model of solar thermoelectric generators is established. The two-dimensional Gaussian distribution is employed to modify the illumination profiles incident on thermoelectric generator. Six non-uniformities of solar illumination are investigated while keeping the total energy constant. The influences of non-uniform illumination on temperature distribution, voltage distribution, and maximum output power are respectively discussed. Three thermoelectric generators with 32, 18 and 8 pairs of thermocouples are compared to investigate their capability under non-uniform solar radiation.

Theoretical model

Geometry structure

Figure 1 shows the schematic diagram of the STEGs which contains an optical concentrator, a TE module, a heat sink added to the thermoelectric cold side, a selective absorber coating at the hot side of the TE module, and a glass enclosure. A Fresnel lens is selected to concentrate sunlight, which can theoretically focus the light on a point. A selective absorber is employed to effectively convert the incident light to thermal energy without large radiative heat loss. The TE generator is placed in the glass enclosure. On the one hand, the glass enclosure can protect the selective absorber from dust and rain. On the other hand, the convective heat loss will be weakened and more solar energy can be used. The structure of the TE generator is given in Fig. 2, which consists of ceramic plates, copper electrodes, and 32 pairs of thermocouples. The thermocouples are connected in series, and there exists an external resistance between the anode and the cathode which constitutes a complete circuit. The upper surfaces of the p/n semiconductor legs are defined as the TE hot side and the lower surfaces as the TE cold side.

Governing model of solar thermoelectric generators

The heat conduction equation is expressed as [²¹]

(1)\rho C_P{\displaystyle \frac{\partial T}{\partial t}}+\nabla \cdot \overrightarrow{q}=\dot{q},

where r is density, C_P is specific heat capacity, T is temperature, t is time, \overrightarrow{q} is heat flux vector, and \dot{q} is heat generation rate. The heat flux vector \overrightarrow{q} can be calculated by [²²]

(2)\overrightarrow{q}=sT\overrightarrow{J}-k\nabla T,

where s and k are respectively Seebeck coefficient and the thermal conductivity of thermoelectric material, and \overrightarrow{J} is electric current density vector which is given as [²²]

(3)\overrightarrow{J}=\sigma (\overrightarrow{E}-s\nabla T),

where s is the electrical conductivity of thermoelectric material and \overrightarrow{E} is electric field intensity vector which is defined as

(4)\overrightarrow{E}=-\nabla \phi ,

where j is electric scalar potential.

The continuity equation of electric charge is

(5)\nabla (\overrightarrow{J}+{\displaystyle \frac{\partial \overrightarrow{D}}{\partial t}})=\mathrm{0},

where \overrightarrow{D} is electric flux density vector.

The coupled equations of thermoelectricity can be obtained from Eqs. (4)–(5). The STEGs is simulated under the condition of steady-state. Therefore, the coupled equations can be written as [²³]

(6)\nabla \cdot (sT\overrightarrow{J})-\nabla \cdot (k\nabla T)=\dot{q},

(7)\nabla \cdot (\sigma s\nabla T)+\nabla \cdot (\sigma \nabla \phi )=\mathrm{0.}

Material properties

When trying to simulate the temperature and the electric properties of the STEGs under non-uniform illumination, the parameters of thermoelectric semiconductors are temperature dependent according to the actual TE generator. The properties of the Bi₂Te₃ commercial TE module (GM200-71-14-16) are employed [¹⁹] which are illustrated in Fig. 3. When the temperature of the TE module exceeds 500K, the properties of the temperature dependent Bi₂Te₃ are obtained by the linear extrapolation.

Non-uniform illumination

A Fresnel lens is selected to concentrate sunlight, which can theoretically focus the light on a point. The heat flux in the middle of the absorber surface is the highest and the lowest on the edge. The two-dimensional Gaussian distribution is employed to modify the illumination profiles incident on the thermoelectric generator. The distribution of the heat flux can be expressed as

(8)G(x,y)={\displaystyle \frac{\omega }{\mathrm{2}\pi D^{\mathrm{2}}}}{\mathrm{e}}^{-\frac{\mathrm{1}}{\mathrm{2}{D}^{\mathrm{2}}}(x^{\mathrm{2}}+y^{\mathrm{2}})},

where G is the solar radiation density after optical concentrating; p is circumference ratio; D² is the variance of Gaussian distribution, which controls the shape of the Gaussian distribution; and w is the normalization factor that ensures that the total input power is equivalent to the one of uniform radiation, which can be calculated by

(9)\omega ={\displaystyle \frac{G_{\mathrm{0}}{A}_s}{{\displaystyle \underset{A_s}{\iint }{\displaystyle \frac{\mathrm{1}}{\mathrm{2}\pi D^{\mathrm{2}}}}{\mathrm{e}}^{-\frac{\mathrm{1}}{\mathrm{2}{D}^{\mathrm{2}}}(x^{\mathrm{2}}+y^{\mathrm{2}})}\mathrm{d}x\mathrm{d}y}}},

where G₀ is the density of uniform radiation and A_s is the area of selective absorber.

The absorbed heat flux of the thermoelectric hot side is defined as

(10)Q(x,y)=\tau \alpha G(x,y),

where t is the transmittance of glass and a is the absorptance of the solar selective absorber.

The TE generator studied has 32 pairs of thermocouples, and the size of the selective absorber is 16 × 16 mm². The transmittance of the glass and the absorptance of the solar selective absorber employ the typical values (t = 0.94, a = 0.95). Figure 4 shows the heat flux distribution of the thermoelectric hot side when six non-uniformities (from Cases I to VI) are given. For the uniform radiation, the solar radiation density after concentrating is 10000 W/m², and the uniform heat flux is 8930 W/m², which show that, with the decrease of the Gaussian distribution variance, the non-uniformity increases. When the variance of the Gaussian distribution is small enough, all the heat will be concentrated on the middle of the TE generator, while the heat fluxes of the edge parts are zero.

Output power calculation

A commercial finite element method (FEM) calculation software is employed to calculate the temperature distribution and the electrical properties. To confirm the grid-independence of the grid, three cases with grid numbers of 25380, 40630, and 64517 are tested while the boundary conditions are the same. When the solar irradiation density is uniform (10000 W/m²) and the external resistance is 5 W, the calculating average temperatures of thermocouples at the hot side for three cases are respectively 324.578 K, 324.582 K, and 324.584 K. The external voltages of the TE generator are respectively 339.430 mV, 339.435 mV, and 339.435 mV. The deviation is tiny for three cases and can be neglected. Therefore, for faster calculating speed, the grid with the number of 25380 is employed for the simulation that follows.

In the simulation, an external resistance is connected to the anode and cathode of the TE generator to constitute a complete circuit. The I-V curve of the STEGs is calculated by changing the external resistance, and the power output can be calculated by

(11)P={\displaystyle \frac{V^{\mathrm{2}}}{r_{\mathrm{e}}}},

where V is the voltage difference between TE anode and cathode, and r_e is external resistance. The parameters for the simulation are listed in Table 1.

Results and discussion

Influence of non-uniform illumination on temperature

Figure 5 depicts the temperature distribution of the TE generator under six non-uniform solar illuminations. The average solar radiation density is 50 kW/m². The total absorbed thermal energy by the TE generator is 11.43 W. The radiation heat loss is considered on the upper surface of the TE generator. The emissivity of the selective absorber e is equal to 0.05. The convective heat loss and the radiation heat loss of other surfaces are ignored because of the glass enclosure and insulation layer. The convection heat transfer coefficient h is set to 10000 W/(m²∙K) on the lower surface of the TE module. As it can be found in Fig. 5(a) and (b), the temperature distribution in the case of D² = 2 × 10^-4 has little difference from the uniform radiation which demonstrates that the small non-uniformity has little effect on the TE generator. This may result from the thermal expansion capacity of the ceramic plate. With the increase of the non-uniformity of the solar illumination, the temperature distribution has a distinct change, as demonstrated in Fig. 5(c)–(f). The rising solar illumination non-uniformity dramatically increases the central area temperatures in the TE generator while the marginal region temperatures have different degrees of reduction. The large non-uniformity of ceramic plate temperature also will lead to an uneven hot side temperature of thermocouples because the thermal expansion capacity of the ceramic plate cannot meet the demand.

The temperature difference between the hot and cold side of thermoelectric legs is crucial for the generating capacity of the TE module. Figure 6(a) demonstrates the influence of the non-uniformity of the solar illumination on the temperature distribution of TE legs taking Case VI as an example. The temperature profiles inside a TE leg increase linearly with the height. There are few differences between the cold sides of all TE legs because of the good cooling at the lower side of the TE module. However, temperatures at the hot side of the central legs are nearly 500 K while the ones of the edge legs are less than 400 K. The significant temperature difference will inevitably affect the power output, the thermal stress, and the life expectancy of the TE module. As it can be found in Fig. 6(b), the average temperatures at the hot side of the six cases are almost the same. The maximum temperature at the hot side dramatically increases when the solar illumination non-uniformity rises while the minimum temperature at the hot side decreases. For Case VI, the maximum and minimum temperatures at the hot side are respectively 499.1 K and 393.6 K. The non-uniform solar illumination also has a great effect on the temperature difference at the hot side. The maximum temperature differences at the hot side for the six cases are 0.3 K, 4.5 K, 34.1 K, 62.2 K, 85.1 K, and 105.5 K respectively. Therefore, it may be concluded that concentrating the sunlight on the TE upper surface can increase the temperature differences of intermediate TE legs but decrease the ones of peripheral semiconductor legs.

Influence of non-uniform illumination on power generation

Figure 7 gives the voltage distribution of the TE generator when uniform radiation G₀ = 5 × 10⁴ W/m², r_e = 1.84 W, and h = 1 × 10⁴ W/(m²∙K). The voltage distributions for the six cases of non-uniform solar illumination are similar, taking the case of uniform radiation as an example. The TE generator studied has 64 p/n legs (8 rows and 8 columns) which are connected in series. The voltage of cathode is zero. The total voltage gradually increases with the number of TE legs, and it reaches the maximum voltage at the anode of the TE module. To investigate the influence of the non-uniformity of the solar illumination on the generated voltage, the generated voltage by each leg in the module is calculated and the results of Cases I (uniform radiation) and VI (D² = 2 × 10^-6) are displayed in Fig. 8. The results show that all the thermoelectric legs generate the similar voltages for Case I. One of the n-type semiconductor legs generates a voltage of 16.8 mV while all the p-type semiconductor legs generate a voltage of 16.5 mV because the n-type legs have a larger Seebeck coefficient than the p-type ones for the current temperature. Figure 8(b) exhibits the generated voltage of every TE leg when a non-uniform solar illumination (Case IV) is employed. The voltages of the central legs are significantly higher than those of the peripheral ones. The maximum voltage of the legs can reach 28.3 mV while the minimum generated voltage of a TE leg is 10.7 mV. The great difference between the generated voltage caused by the non-uniform solar illumination makes the peripheral legs not so important. The total voltages of Cases I and VI are 1062.7 mV and 1055.5 mV respectively. This means that although concentrating the sunlight can increase the generated voltages of central TE legs, the ones of peripheral legs also decrease while the total voltage has a small decline.

Figure 9 demonstrates the influence of non-uniform solar illumination on the internal resistance, open circuit voltage, and maximum output power of the TE generator. Six cases (I to VI) of non-uniformities are calculated when the average solar radiation densities of all cases are equal to 50 kW/m² and the convective heat transfer coefficients are 10000 W/(m²∙K). Both the internal resistance and the open circuit voltage of the TE generator gradually decrease with the increasing non-uniformity of solar illumination. Declines are caused by the fact that the non-uniform solar illumination dramatically enlarges the temperatures of the central TE legs compared with the case of uniform radiation discussed in Sub-section 3.1. The Seebeck coefficients and electrical conductivities of p-type and n-type semiconductor legs decrease with the temperature in the temperature range of TE. Therefore, concentrating sunlight will inevitably decrease the internal resistance and the open circuit voltage of the TE generator. As is well-known that the TE generator can achieve a maximum output power when the external resistance is equivalent to the internal resistance, the maximum output power of TE generator can be calculated by

(12)P_{\max }={\displaystyle \frac{V_{\mathrm{oc}}^{\mathrm{2}}}{\mathrm{4}{r}_{\mathrm{TE}}}},

where V_oc is open circuit voltage and r_TE is the internal resistance of the TE generator. The effect of non-uniformity of solar illumination on the maximum output power of TE is given in Fig. 9(b). All of the maximum output power of the cases of uneven solar radiation are less than that of Case I (uniform solar radiation). In general, raising the non-uniformity of the solar radiation will weaken the capability of the TE generator except Case IV. The reason for this may be that the output power of the TE generator is resulted from the interaction of various factors including the Seebeck coefficient, electrical conductivity, and temperature distribution. Although the open circuit voltage decreases when the non-uniformity of the solar radiation changes from Cases III to IV, the internal resistance decline slightly which leads to the increase of the output power. However, the non-uniformity of solar radiation has little effect on the output power of the TE generator. The maximum output power only decreases by 1.4% when solar radiation changes from Cases I to VI. This means that it is not necessary to pursue uniform solar radiation for designing a solar TE generators system.

Influence of number of thermoelectric legs

As discussed in Sub-sections 3.1 and 3.2, concentrating the sunlight on the TE upper surface can increase the temperature differences of intermediate TE legs but decrease the ones of peripheral semiconductor legs. This makes the peripheral legs contribute little voltage to the total voltage but has a considerable resistance which may dramatically decrease the maximum output power of the TE generator. Therefore, it is necessary to investigate the influence of the number of thermoelectric legs on the total performance of STEG when a non-uniform solar radiation is employed. Figure 10(a) gives the heat flux distribution on the hot side of TE when solar radiation density is 10 kW/m² and the variance of Gaussian distribution is 4 × 10^-7. The non-uniform solar radiation is used to calculate three TE generators with different semiconductor legs (32, 18 and 8 pairs of thermocouples). For three TE generators, the material properties and sizes are all the same and the cooling conditions at the lower side are all 10000 W/(m²∙K). Moreover, all the solar energy can be absorbed by three the TE generators having different receiving area because the non-uniformity of the solar illumination is large enough. Figure 10(b) illustrates the calculating results of the three generators under non-uniform solar illumination of Fig. 10(a). The open circuit voltages of the three TE generators are very close. The reason for this is that, when the number of thermocouples decreases, the temperature difference between the hot side and the cold side of the TE generator increases. However, the change in the number of thermocouples has a great influence on the thermoelectric generator internal resistance. The internal resistances of the TE generators with 32, 18 and 8 pairs of thermocouples are 1.81 W, 1.12 W and 0.56 W respectively. The short circuit current has an opposite tendency compared with the resistance when the number of thermocouples changes, which can be expressed by Ohm’s law. The maximum output power of the TE generator with eight pairs of thermocouples is the largest among the three TE generators studied when the same non-uniform radiation is employed. The maximum output powers of the three generators are respectively 29.8 mW, 51.5 mW, and 102.4 mW. The maximum output power increases by 73.5% when the number of TE legs changes from 32 to 18 and by 244.9% when it changes from 32 to 8. Therefore, when the sunlight is concentrated on a small area, a small size of the TE generator with less thermocouples should be employed.

Conclusions

A three-dimensional finite element model of solar thermoelectric generators was established. The two-dimensional Gaussian distribution was employed to modify the illumination profiles incident on the thermoelectric generator. Six non-uniformities of solar illumination were investigated while keeping the total energy constant. The influences of non-uniform illumination on the temperature distribution, the voltage distribution, and the maximum output power were respectively discussed. The result shows that the non-uniformity of the solar illumination has a great effect on the temperature distribution and the voltage distribution. The central thermoelectric legs can achieve a larger temperature difference and generate a higher voltage than the peripheral ones. The non-uniform solar illumination will weaken the capability of the TE generator, and the maximum output power decreases by 1.4% among the range of non-uniformity studied. Three thermoelectric generators with 32, 18 and 8 pairs of thermocouples were compared to investigate their capability under non-uniform solar radiation. Reducing the number of thermoelectric legs for non-uniform solar illumination can greatly increase the performance of the thermoelectric generator. Under the same non-uniform solar radiation, the maximum output power increases by 73.5% when the number of TE legs changes from 32 to 18 and by 244.9% when it changes from 32 to 8.

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