Simulation Investigation of High-Efficiency Solar ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 6, JUNE 2015

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Simulation Investigation of High-Efficiency Solar Thermoelectric Generators With Inhomogeneously Doped Nanomaterials Shanhe Su and Jincan Chen Abstract—By introducing an inhomogeneously doped nanostructure thermoelectric generator (TEG) with a deltalike electronic density of states (DOS), a novel model of the solar TEG (STEG) with high conversion efficiencies is established. The STEG is composed of a flat-panel collector without optical concentration, but with thermal concentration, and a TEG in a vacuum enclosure. Reversible electron transport is achieved by designing quantum-confined electrons in thermoelectric materials. The maximum efficiency calculated herein is much larger than that previously reported. Influences of the current density, thermal conductivity, and energy band width of the electronic DOS on the performance are revealed. The optimization problems of the system are discussed. Results show that the STEG made of nanostructure materials, permitting the electron transport to approach energy-specific equilibrium, possesses great potential to increase solar energy conversion efficiencies. Index Terms—Inhomogeneous doping, nanomaterial, narrow band width, optimum design, solar thermoelectric generator (STEG).

I. I NTRODUCTION

S

OLAR thermoelectric generators (STEGs), as a new type of low-cost high-efficiency solar conversion technology, constitute promising devices of power generation [1]–[3]. Compared with conventional solar thermal electric devices, STEGs enjoy evident advantages, such as absence of moving parts, little noise in operation, small-scale applications, and longterm stabilities [4], [5]. Previous studies on solar thermoelectric energy converters [6]–[8] were subject to lower limits on their efficiencies due to the inherent irreversibility from the structure and material design such that they failed to attract much attention. However, latest breakthroughs have been developed and have shown promising results for terrestrial applications [3], [9]–[11]. Maximum efficiency of 4.6%, higher than that of previously reported flat-panel STEGs, can be obtained through Manuscript received May 16, 2014; revised September 18, 2014 and October 17, 2014; accepted October 26, 2014. Date of publication November 20, 2014; date of current version May 8, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 11175148, in part by the 973 Program under Grant 2012CB619301, and in part by the Fundamental Research Fund for the Central Universities of China under Grant 201312G007. (Corresponding author: Jincan Chen.) The authors are with the Fujian Key Laboratory of Semiconductor Materials and Applications and the Department of Physics, Xiamen University, Xiamen 361005, China (e-mail: [email protected]; jcchen@ xmu.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2014.2371433

the use of vacuum enclosures, which reduce convective heat losses, flat-panel collectors, which provide large thermal concentration, and high-performance thermoelectric materials [3]. A fundamental issue in thermoelectric generators (TEGs) is lowness of conversion efficiencies. For broader applications of STEGs, it is crucial to develop high-performance thermoelectric materials, which exhibit properties of low thermal conductivities and electrical resistivities in addition to high thermopowers [12]. Nanostructure thermoelectric materials with dramatically higher efficiencies have attracted considerable interest [13], [14]. Low-dimensional thermoelectric materials with a delta-like electronic density of states (DOS) have been found to increase the Seebeck coefficient without a corresponding reduction in the electrical conductivity [14]–[16] and lower the thermal conductivity contributed by electrical carriers [17]. In light of this idea, a nanostructure TEG with inhomogeneous doping has been proposed to achieve reversible electron transports in finite energy ranges when occupations of states in reservoirs are equalized [18], [19]. When electrons diffuse through a material in a relatively narrow energy band with a continuous spatial variation in the electrochemical potential and the temperature, thermoelectric materials can achieve high figure of merits [18]. Such thermoelectric materials boast great potential for practical applications in novel STEGs. In this paper, by combining a flat-panel collector and highperformance thermoelectric nanomaterials in a vacuum enclosure, the model of a high-efficiency flat-panel STEG is proposed. This paper is organized as follows. In Section II, a model of the STEG is briefly described. The design of a solar collector is discussed based on a detailed energy balance. An inhomogeneously doped TEG with a delta-like DOS is constructed. In Section III, the finite-difference method in one dimension is used to derive the power output and overall efficiency of the STEG. In Section IV, influences of the current density, thermal conductivity, and energy band width of the electronic DOS on the performance of the STEG are disclosed. The optimization problems of the system are discussed in detail. Finally, several important concluding remarks are made.

II. M ODEL D ESCRIPTION OF A STEG In Fig. 1, the STEG primarily consists of a solar collector on the top and a nanostructure TEG at the bottom. The device is encapsulated in a vacuum enclosure to prevent both convective and conductive losses. A high-performance flat-panel solar absorber can convert the solar radiation into the thermal energy

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the right-hand side, i.e., QTE , is the heat flux transferred to the thermoelectric couple, given by QTE = ITh [Sp (L) − Sn (L)] dTp dTn −κp (L)Ap − κn (L)An dx x=L dx x=L

Fig. 1. Schematic of an STEG consisting of a solar collector and a nanostructure TEG.

Fig. 2.

Curve of temperature-dependent thermal emissivity.

and can also act as the thermal concentrator by conducting heat onto the thermoelectric couple. The TEG consists of a pair of n/p-type thermoelectric materials using silicon-based quantum dot superlattice or superlattice nanowire. In the following, the thermal and electrical performance of individual parts and overall device will be discussed. 1) Modeling of a Solar Collector: The solar collector is composed of an optical concentrator and a solar absorber. The collector is capable of generating large thermal concentration, but not optical concentration. The absorptance of the solar absorber α is selected as 0.95 [3]. The effective thermal emissivity ε is dependent on the temperature, as shown in Fig. 2 [20]. Under the assumption that the temperature of the solar absorber is uniform, the energy balance for the plate solar absorber can be expressed as τ αCopt GA = QTE + Aε(Th )σ Th4 − Ta4

(1)

where τ is the transmittance of the optical concentrator, Copt is the optical concentration, G is the intensity of the solar radiation flux, A is the area of the solar collector, Th is the temperature of the solar absorber, and Ta is the temperature of environment. The term on the left-hand side in (1) is the overall solar radiation absorbed by the solar absorber. The first term on

(2)

where S is the Seebeck coefficient, κ is the thermal conductivity, L is the length of thermoelectric legs, and I is the electric current. The symbol A is the cross-sectional area; and subscripts n and p denote the n- and p-type legs of the TEG, respectively. In the following numerical calculation, the forward direction is defined from the cold side to the hot side by assuming that the energy flows in one dimension. The second term on the right-hand side in (1) is the radiative heat exchange between the absorber and the environment. The collector efficiency is given by ε(Th )σ Th4 − Ta4 QTE = τα − (3) ηS = Copt GA Copt G which is defined as the ratio of QTE to the concentrated solar radiation intensity. 2) Nanostructure TEG With Inhomogeneous Doping: The performance of a TEG is characterized by a dimensionless figure of merit, i.e., ZT = [σS 2 /(κel + κph )]T [21]–[23]. Irreversibilities greatly limit the efficiency and economic applications of thermoelectric devices. There exists a challenge in developing high-efficiency materials because the electrical conductivity σ, electronic thermal conductivity κel , and Seebeck coefficient S are all coupled with one another [24]. Increasing S generally decreases σ, whereas increasing σ proportionally elevates κel in bulk materials [25]. Humphrey and Linke first established a novel model of the TEG that can be operated reversibly [18], [19]. When the nanostructure thermoelectric material, which exhibits quantum confinement of carriers, is used, σ and κel are able to become independent of each other, and an exceptionally high figure of merit can be achieved. In this system, electrons are localized in a narrow band with the width ΔE centered on energy E0 and diffuse through the material with a continuous spatial variation in electrochemical potential μ(x) and temperature T (x), as shown in Fig. 1. Electrons can be operated reversibly when occupations of states f = 1/{exp[(E − μ(x))/kB T (x)] + 1} at every position become equal to one another. When the central energy E0 is invariant, thermoelectric material can be designed with inhomogeneous doping to keep [E0 − μ(x)]/T (x) uniform along the n- and p-type legs [18]. The performance of the nanostructure TEG can be calculated by using the Boltzmann transport equation under relaxationtime approximations [18], [19]. For a particular position, the Seebeck coefficient S(x), electrical conductivity σ(x), and electronic thermal conductivity κel (x) are given by S(x) = −K1 (x)/eT (x)K0 (x) σ(x) = e2 K0 (x) K2 (x) − K12 (x)/K0 (x) κel (x) = T (x)

(4) (5) (6)

SU AND CHEN: INVESTIGATION OF HIGH-EFFICIENCY STEGs WITH INHOMOGENEOUSLY DOPED NANOMATERIALS

where Kn is defined as Kn (x) =

df β(E) [E − μ(x)]n − dE dE

By employing the first-order approximation, (8a) and (8b) can be arranged into [31], [32] (7)

with β(E) being the electron transmission function [19], [26], [27]. In the numerical calculation, β(E) = β is assumed to remain constant over the narrow band, but vary with ΔE such that σ is equal to 5 × 105 Ω−1 · m−1 at the hot junction of the TEG. This arrangement is to isolate effects on the ZT (due to the width of the DOS, i.e., ΔE) from those on variations of the overall number of available states for electrons [18]. This procedure is similar to that in [18] and [19], except that, in the present analysis, the temperature of the hot junction varies under different operating conditions. The following procedures can be used to manufacture thermoelectric materials capable of reversible electron transport. 1) Nanostructured materials, such as silicon-based quantum dot superlattices or superlattice nanowires, with the DOS in a narrow miniband are fabricated via sputtering and annealing [15], [28]. The DOS can be estimated by means of photoluminescence spectroscopy. 2) Through the inhomogeneous doping of thermoelectric materials, the Seebeck coefficient is observed to be spatially invariant in the presence of a temperature gradient across the material using the scanning probe technique [29]. N- and p-type materials can be obtained by doping with phosphorus and boron via a process of diffusion, respectively [30]. Main challenges stem from that the doping gradient affects the temperature distribution across the material, subsequently affecting the doping gradient. Relatively large numbers of samples should be done until a spatially invariant Seebeck coefficient is obtained.

III. N UMERICAL P ROCEDURE Here, we will present the technique to determine temperature distribution along the legs of the TEG employing finite-difference method. This numerical strategy is capable of attaining rapid convergence and searching the optimal device performance. According to Fig. 1 and [31], Domenicali’s equation for the energy balance over a segment of the discretized thermoelectric element and the heat flux density equation are two basic equations to determine the temperature distribution along the TEG [31]–[33]. For the n-type leg, these equations are expressed by ∂ ∂x

κn (x)

∂Tn (x) ∂x

=−

∂Sn (x) Jn2 + Jn Tn (x) σn (x) ∂x

qn (x) = Jn Tn (x)Sn (x)−κn (x)

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(8a)

∂Tn (x) (8b) ∂x

where Jn is the current density of the n-type leg, and κn (x) comprises the thermal conductivity contributions of the lattice κn,el (x) and electronic κn,ph . Same equations can be applied to the p-type element.

dTn (x) = cn1 (x)Tn (x) + cn2 (x)qn (x) (9a) dx dqn (x) = cn3 (x)Tn (x) + cn4 (x)qn (x) + cn5 (x) (9b) dx where cn1 (x) = Jn Sn (x)/κn (x), cn2 (x) = −1/κn (x), cn3 (x) = Jn2 Sn2 (x)/κn (x), cn4 (x) = −Jn Sn (x)/κn (x), and cn5 (x) = Jn2 (x)/σn (x). Upon using the finite-difference method, the thermoelectric leg L is divided into equally spaced grid points at distance Δx = L/N apart, where N is the number of nodes. Equations (9a) and 9(b) are approximated by the difference form at grid points xi = iL/N, (i = 1, . . . , N ), i.e., Tni − Tni−1 = cn1,i Tλi + cn2,i qni (10a) Δx qni+1 − qni = cn3,i+1 Tni+1 +cn4,i+1 qni+1 +cn5,i+1 (10b) Δx at node i. Based on (10a), heat flux densities at nodes i and i + 1 for one leg can be expressed as qni = qni+1 =

1 − cn1,i Δx 1 Tni − Tni−1 cn2,i Δx cn2,i Δx

(11a)

1 − cn1,i+1 Δx 1 Tni+1 − Tni . cn2,i+1 Δx cn2,i+1 Δx

(11b)

By using the preceding equations, the terms of heat flux densities can be eliminated, and the governing equation for the temperature Tn,i at node i is expressed as an1,i Tn,i−1 + an2,i Tn,i + an3,i Tn,i+1 = bn,i

(12)

where bn,i = cn5,i+1 Δx, an1,i = 1/cn2,i Δx, an2,i = cn4,i+1 / cn2,i+1 − 1/cn2,i+1 Δx−(1−cn1,i Δx)/cn2,i Δx, an3,i = (1 − cn1,i+1Δx)/cn2,i+1Δx−cn3,i+1 Δx−(1−cn1,i+1 Δx)cn4,i+1 / cn2,i+1 , and cnm,i (m = 1−5) are the material properties defined in (9) and (10) at node i. By combining (1) and (2) and the Dirichlet boundary condition at the heat sink [34], i.e., Tn (0) = Tc , (12) can be applied to each segment and iteratively solved to obtain the desired temperature profile. Thus, the heat flux density transferred into node i can be calculated by qn,i =

1 − cn1,i+1 Δx 1 Tn,i+1 − Tn,i . cn2,i+1 Δx cn2,i+1 Δx

(13)

On the basis of the temperature, heat flux characteristics, and (4)–(7), the power density of the n-type leg is given by [32], [35] ⎞ ⎛ L L (x) 1 dT n dx + Jn dx⎠ . (14) Pn = Jn ⎝ Sn (x) dx σn (x) 0

0

The same equations can be applied to the p-type leg. For simplicity, the properties of the p-type leg are assumed to be the same as those of the n-type leg.

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Fig. 3. Three-dimensional graph of the efficiency η varying with the solar thermal concentration Cth and the current density J.

The total efficiency of the TEG is given by ηTE =

Pn An + Pp Ap . QTE

(15)

The overall efficiency of the STEG is then determined by η=

Pn An + Pp Ap QTE Pn An + Pp Ap = = ηS ηTE Copt GA Copt GA QTE (16)

which is simply the collector efficiency multiplied by the thermoelectric efficiency. IV. P ERFORMANCE E VALUATION AND PARAMETRIC O PTIMUM D ESIGN In the following discussion, the intensity of the solar radiation flux G is 1 kW· m−2 , the transmittance of the optical concentrator τ is 0.94, the solar collector is nonconcentration, i.e., Copt = 1, the length L is 1 cm, the central energy E0 for electron transport is 1 eV, and the environment temperature Ta and the heat sink temperature Tc are assumed to be 300 K. Thermal concentration Cth in the calculation is defined as the ratio of the absorber area A to the cross-sectional area Ap + An of the thermoelectric couple. By using a flat-panel collector whose area is larger than that of the cross-sectional area of the thermocouple, the density of the heat flux passing through each thermoelectric leg is enhanced. On the basis of the aforementioned analyses, the 3-D graph of the efficiency η varying with the thermal concentration Cth and the current density J can be generated, as shown in Fig. 3, where ΔE = 10 meV, and κph = 0.5 W/mK. We note that current densities J = Jp = Jn , because it has been ideally assumed that the thermoelectric device is symmetric with Ap = An . As shown in Fig. 3, through the use of a flat-panel collector with nonoptical concentration and high-performance thermoelectric nanomaterials with inhomogeneous doping in a vacuum enclosure, the STEG can achieve a high energy conversion efficiency. The maximum efficiency is 14.6% when Cth = 45. A physical explanation of the apparent peaking efficiency at a particular Cth value is that the collector efficiency ηS is a monotonically decreasing function of Cth , but the thermoelectric efficiency ηTE is a monotonically increasing

Fig. 4. (a) Thermoelectric efficiency ηTE , collector efficiency ηS , and the overall efficiency η of the STEG and (b) absorber temperature Th as a function of the thermal concentration Cth .

function of Cth , so that the efficiency η of the STEG is not a monotonic function of Cth , as shown in Fig. 4(a). For purposes of further illustrating the effects of the thermal concentration Cth on ηS , ηTE , and η, the curve of the absorber temperature Th varying with Cth is shown in Fig. 4(b). The reason that Th increases with Cth can be simply explained as follows. When Cth is small, implying the fact that the cross-sectional area of the thermoelectric couple is large, the thermal energy can quickly transfer from the absorber to the generator, so that Th decreases. With the increase in Cth , the heat transfer from the absorber to the generator diminishes, so that Th rises. Clearly, the higher the absorber temperature Th is, the smaller the collector efficiency ηS and the larger the thermoelectric efficiency ηTE . This phenomenon results in the appearance of the peaking efficiency of the STEG. The efficiency obtained will enable STEGs to compete with solar thermal technologies traditionally using conventional heat engines to produce electric power. Furthermore, STEGs are solid-state heat engines with no moving parts, greatly increasing reliabilities, lifetimes, and applicabilities for small and large scales. We note that the value of Cth at the maximum efficiency is much smaller than that reported in [3], meaning that the proposed model may largely reduce the thermal radiation losses of a collector and is beneficial to the applications of STEGs. Unless specifically mentioned, Cth = 45 will be adopted in the following discussion. Efficiency curves of the STEG varying with current density for different lattice thermal conductivities κph are plotted in Fig. 5, where ΔE = 10 meV. It is shown in Fig. 5 that the maximum efficiency increases as κph decreases. The figure of merit is found to increase with the decrease in the lattice


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Fig. 5. Efficiency versus current density curves for given lattice thermal conductivities κph .

Fig. 6. Efficiency η varying with the band width ΔE. The current density J has been selected to obtain a peaking efficiency at each point.

thermal conductivity, indicating an increase in the thermoelectric conversion efficiency. In Fig. 5, the apparent shift of optimum J values for different κph is observed as well. The smaller κph is, the less the heat transfer through the thermal conduction. According to (1) and (2), a large current is required to maintain the energy balance of the solar absorber. The lattice thermal conductivity κph is selected in the following figures to be 0.5 W/mK, which is the same as that adopted in [19]. The effect of the band width ΔE on the efficiency η is illustrated in Fig. 6, where the current density J has been optimally selected. Results show that η increases with the decrease in ΔE. The phenomenon can be explained by Fig. 7(a) and (b). As ΔE decreases, the electronic thermal conductivity κel is dramatically reduced, and electrical conductivity σ is improved. In addition, the Seebeck coefficient S(x) becomes spatially invariable across the thermoelectric material in the presence of a spatially varying chemical potential μ(x) if the DOS is a delta function at E0 (ΔE → 0). Electron transport at E0 is reversible such that the entropy increase is reduced when an electron moves through the material. As a result, the decrease in ΔE will increase ZT , which can reach 3.2 even at low temperatures, as shown in Fig. 7(c). Fig. 8 presents curves of the power density varying with the current density J for different band widths ΔE. For the symmetric dimension, both the elements of the TEG possess the same power density. It can be seen that, as J increases, P first increases, then reaches a maximum value, and decreases gradually afterward. This variation can be explained by the

Fig. 7. (a) Electronic thermal conductivity. (b) Electrical conductivity. (c) Figure of merit across the n-type material for different band widths ΔE.

Fig. 8. Power density and voltage varying with the current density J for different band widths ΔE.

J–V characteristics also shown in Fig. 8, because the voltage of the TEG decreases, whereas the current density increases, implying the existence of a maximum value for P . It is also shown in Fig. 8 that the lower the band width ΔE is, the larger the power density.

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ACKNOWLEDGMENT The authors would like to thank Prof. T. Shih of the University of California for his helpful discussion. R EFERENCES

Fig. 9. (a) Thermoelectric efficiency ηTE , relative efficiency ηTE /ηC , and absorber temperature Th as a function of the current density for different band widths ΔE. (b) Relative efficiency ηTE /ηC as a function of power density P for difference band widths ΔE.

Fig. 9(a) shows the thermoelectric efficiency ηTE , the relative efficiency ηTE /ηC , and the absorber temperature Th versus the current density, where the Carnot efficiency ηC is determined by the absorber temperature Th and environment temperature Ta . The former drops with the increase in the current density due to the Peltier effect, because the heat flux generated by this effect at x = L is proportional to the current density, whereas structure parameters are given. The smaller the band width ΔE is, the lower the thermal irreversible losses for electron transport and the larger the thermoelectric efficiency ηTE and the relative efficiency ηTE /ηC . Fig. 9(b) shows that the curves of the relative efficiency ηTE /ηC varying with the power density P are closed-loop lines passing through the zero point. By using inhomogeneous doping and a minimized DOS, not only the power density but also the thermoelectric efficiency is greatly elevated. When ΔE = 10 meV, the thermoelectric efficiency ηTE can attain 52.06% of the Carnot efficiency ηC at the absorber temperature 512.66 K. The conversion efficiency of the TEG can be further enhanced by the optimal design of the central energy E0 and structure parameters. V. C ONCLUSION Based on a nonoptical-concentration flat-panel collector and an inhomogeneously doped TEG in a vacuum enclosure, the performance of a new type of STEG has been discussed. It can be concluded that inherent irreversible effects will be reduced by decreasing the lattice thermal conductivity and energy band width of the DOS. We can adopt the STEG made of nanostructure materials with smaller thermal concentration to obtain theoretical efficiency larger than those previously reported with nonoptical concentration.

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Shanhe Su received the B.S. degree from Lanzhou University, Lanzhou, China, in 2010. He is currently working toward the Ph.D. degree at Xiamen University, Xiamen, China. He has authored or coauthored more than ten papers published in international journals such as Solar Energy Materials and Solar Cells, Applied Energy, and Energy. His research interests include solar cells and thermoelectric devices.

Jincan Chen received the Ph.D. degree from the University of Amsterdam, Amsterdam, The Netherlands, in 1997. He is currently a Professor with the Department of Physics, Xiamen University, Xiamen, China. He has authored or coauthored over 300 papers published in 65 international journals. His research fields mainly include solar cells, thermoelectric devices, fuel cells, Brownian heat engines, and quantum thermodynamic cycles.