Song K, Song H P, Gao C F. Improving compatibility between thermoelectric components through current refraction. Chinese Physics B, 2018, 27(7): 077304
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Improving compatibility between thermoelectric components through current refraction
Song K, Song H P †, Gao C F
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China
Project supported by the Fundamental Research Funds for the Central Universities, China (Grant No. NS2016008).
Abstract
It is well known that components with dissimilar compatibility factors cannot be combined by segmentation into an efficient thermoelectric generator, since each component needs a unique optimal current density. Based on the complex variable method, the thermal-electric field within a bi-layered thermoelectric composite has been analyzed, and the field distributions have been obtained in closed-form. Our analysis shows that current refraction occurs at the interface, both the refraction angle and current density vary with the incidence angle. Further analysis proves that the current densities in two components can be adjusted independently by adjusting the incidence current density and incidence angle, thus the optimal current density can be matched in both components, and the conversion efficiency can be significantly increased. These results point to a new route for high efficiency thermoelectric composites.
Thermoelectric materials have great advantages due to their specific ability of converting heat directly into electricity, and vice versa, and thus they are widely used in many fields, such as modern industry,[1–4] military,[5,6] and protection of the environment.[7–9] The performance of a thermoelectric material is represented by figure of merit ZT, which is deduced from electric conductivity σ, thermal conductivity κ, and Seebeck coefficient S as[10]
A great deal of effort has been devoted to adjusting the material properties, since a high figure of merit needs high electric conductivity and Seebeck coefficient while low thermal conductivity. For example, the thermoelectric performance of AgPbmSbTem+2 was maximized by optimizing the spark plasma sintering temperature.[11] Zhang et al.[12] enhanced the thermoelectric performance of Se-doped PbTe bulk materials via nanostructuring and multi-scale hierarchical architecture. The thermoelectric property of Cu1.8S was improved via introducing Bi2S3 and Bi2S3Bi core-shell nanorods.[13] The high ZT Si nanowires were processed by a wafer-scale manufacturing technique.[14] Chen et al.[15,16] significantly reduced the thermal conductivity of silicon nanowires (SiNWs) and germanium nanowires. Besides, many researchers focused on developing hybrid thermoelectric composites, since it is rather difficult to satisfy competing requirements on thermoelectric properties for high conversion efficiency in a single-phase material. Mahan and Lyod[17] designed a layered thermoelectric device using semiconductor quantum wells. The thermoelectric properties of alternatively layered films of polyaniline were studied by Yan and Toshima.[18] A bulk thermoelectric cooler was designed by Antonik et al. by employing thin-film thermoelectric materials sandwiched between plastic substrates.[19] The routes for enhanced performance of thermoelectric materials by incorporating nanostructures within the bulk materials was discussed by Rao et al.[20]
Quite a few continuum models have been developed to understand the overall efficiency of thermoelectric composites. Snyder and Ursell[21] deeply discussed the thermoelectric efficiency and compatibility, and proved that components with dissimilar compatibility factors cannot be combined by segmentation into an efficient thermoelectric generator. Song textitet al.[22] analyzed the unavoidable electric current caused by inhomogeneities and its influence on measured material parameters of thermoelectric materials. Rigorous continuum analysis has also been carried out for one-dimensional parallel[23] and layered[24–26] thermoelectrics, wherein the conception of optimal electric current is proposed, and thus the conversion efficiency of a composite can be slightly higher than its constituents as the optimal current density can be matched everywhere by tailoring the material properties. These analyses, however, were based on the model that the electric current always flows through the interface vertically and retains its direction. As such, components with dissimilar compatibility factors (or optimal current density) cannot be combined into an efficient thermoelectric generator, which greatly restricts the component matching of composite thermoelectric materials. It is thus highly desirable if arbitrary components can fit together and work well.
In this work, the problem of current refraction at the interface is studied. We attempt to derive a method that can greatly improve the compatibility of the components. Our results show that the current density in each component can be adjusted independently by adjusting the incidence current density and incidence angle, therefore components with dissimilar compatibility factors can also be combined into efficient thermoelectric generators. These results point to a new route for high efficiency thermoelectric composites.
2. Two-dimensional thermoelectric problems
2.1. Governing equations of thermoelectric material
We consider the coupled transports of heat and electricity in a thermoelectric medium, within which the electric current density J is coupled with the heat flux JQ through the Seebeck coefficient S[27,28]
where T and ϕ are the temperature and the electric potential, respectively. The total energy is transported by both electrons and heat in thermoelectric materials, and can be derived as
We limit ourselves to systems wherein both charges and energy are conserved, such that both current density and energy flux are divergence-free,
It is worth pointing out that the heat flux is not divergence-free due to Joule heating.
2.2. Boundary conditions
In our system, the temperature, electric potential, normal electric current, and normal energy flux are all continuous across the interface, thus
in which the symbol ⊥ donates normal to the interface, and the subscripts 1 and 2 represent the quantities within lower and upper materials, respectively.
2.3. General solutions of two-dimensional thermoelectric problem
Substituting Eq. (2) into Eq. (5) leads to
Then, ϕ + ST can be expressed as
where Re[f[z]] is the real part of f[z], and z = x + iy. Substituting Eq. (12) into Eq. (2), we have
in which and denote the conjugate and derivative of f[z], respectively. Combining Eqs. (13) and (14), we have
Substituting Eq. (3) into Eq. (6), and noticing Eq. (5), we can simplify the relationship between the temperature and electric current density as
Noticing Eq. (15), ▽2T can be expressed by f[z] as
Solving Eq. (17), we obtain
where g[z] is an arbitrary complex function, and N is an arbitrary real constant which donates the uniform temperature field. The electric potential can be derived by substituting Eq. (18) into Eq. (12) as
Finally, the thermal flux and energy flux can be deduced from Eqs. (3), (4), (15), (18), and (19) as
Up to this point, the temperature/electric potential fields as well as the electric/thermal/energy flux have been expressed by two unknown functions f[z] and g[z]. As such, if the two analytic complex functions f[z] and g[z] can be derived, the fields of temperature, electric potential, heat flow, energy flux, and electric current can then be determined, and the problem is completely solved.
3. Solutions for a bi-layered thermoelectric medium
To be specific, we consider a bi-layered thermoelectric medium, as shown in Fig. 1. The thermoelectric properties of layers 1 and 2 are σ1, S1, κ1 and σ2, S2, κ2, respectively. Our goal is to determine the refraction law of the electric current. To this end, at the far field in phase 1, the medium is subjected to imposed electric current , , and energy flux , . Furthermore, the two analytic functions introduced earlier, f[z] and g[z], are denoted as f1[z], g1 [z] in phase 1 and f2 [z], g2 [z] in phase 2 to be distinguished from each other.
Fig. 1. (color online) Schematic diagram of a bi-material thermoelectric medium.
By noticing the far field loadings, analytic functions fi [z] and gi [z] can be expressed as
where A0 and B0 are arbitrary real constants. According to the far field loadings and Eqs. (15) and (21), the unknown coefficients A1 and C1 can be determined as and . Due to this, the temperature fields and electric potential fields in media 1 and 2 can be rewritten as
where N* is a real constant which represents a uniform temperature field in the upper material.
In addition, the electric current density, energy flux, and thermal flux can be rewritten as
Substituting Eqs. (26)–(35) into interfacial conditions Eqs. (7)–(10), and comparing the coefficients of the same power of zk, we obtain the following coefficients:
This set of equations solves the problems completely, leading to the full determination of the field distributions inside materials 1 and 2.
4. The refraction law of electric current, heat flux, and energy flux
We now consider the field distributions within media 1 and 2, specifically the current density, energy flux, heat flux, temperature, and potential given by Eqs. (26)–(35). The spatial variations of these field variables can be better appreciated by an algebraic expansion of these equations using the nonzero coefficients derived in Section 3. In particular, the electric current density of media 1 and 2 are determined from Eqs. (30) and (31) as
It can be seen that the current is uniform within media 1 and 2, while the density and direction of the current in these two media are different. The incident angle in material 1 is defined as
Then the refracted angle in material 2 can be expressed as
Thus, the refraction law of electric current is
which indicates that the refraction angle of the current depends not only on σ2/σ1, but also on .
The refractive index, on the other hand, can be expressed by θ1 and θ2 as
Equation (41) clearly shows that the current densities within media 1 and 2 are different from each other, and their ratio varies with the incident angle θ1, since θ2 also varies with θ1, as expressed in Eq. (39). Using the refraction of the current, we can control the current density in each medium. For example, we maintain the incident current density equal to the optimal current density of medium 1, and then adjust the current density in medium 2 by adjusting the incidence angle. Finally, the current density in each medium can approach the optimal current density in this medium.
An important calculation result is shown in Fig. 2, with the material parameters detailed in Table 1, which clearly reveals that the refractive index |J2/J1| varies with the incidence angle θ1 in a wide range, thus providing enough space to adjust the current density in medium 2.
Material parameters of thermoelectric samples.[29,30]
.
Sample
σ/S·m−1
S/μV · K1
κ/W·m−1 · K−1
BiSbTe
27700
213
0.7
Bi2Te3
110000
200
1.6
Table 1.
Material parameters of thermoelectric samples.[29,30]
.
From Eqs. (33) and (35), the refracted angle of heat flux and energy flux can be expressed as
Although the expression of the heat flux refraction angle (Eq. (42)) is significantly different from that of the current refraction angle (Eq. (39)), the numerical results in Fig. 3 show that there is only a slight difference between them. The heat flux refraction also affects the energy conversion efficiency. The inconsistency between the heat flux and current direction usually leads to the decrease of efficiency, but this effect is weak. Generally speaking, the current refraction plays a major role in improving the efficiency.
Fig. 3. (color online) Refracted angles of electric current θ2 and heat flux versus incident angle θ1.
5. Conversion efficiency improved by current refraction
The conversion efficiency is also calculated to demonstrate the analysis. In classical bi-layered thermoelectric systems, the electric current flows through the interface vertically and retains its direction, which leads to a uniform electric current density in the whole system. In our new system, as shown in Fig. 4(b), the electric current refraction occurs at the interface, both the density and the direction of the current change, and the shape of the system is also altered to adapt to the refraction of the current. In the classical bi-layered thermoelectric system, the conversion efficiency can be expressed as[17]
We can compare the conversion efficiency of the new bi-layered thermoelectric system with the maximum conversion efficiency of the classical bi-layered thermoelectric system in Fig. 5, with y1 = y2 = 0.01 m, l = 0.0025 m, Tc = 300 K, and Th = 500 K, where θ1 is the incidence angle, and |J∞| equals to the optimal current density of medium 1. It is again evident that the current densities in different components can be adjusted independently, which is impossible for the classic bi-layered thermoelectric system. By adjusting the incidence current density and incidence angle, the upper bound of the conversion efficiency is significantly improved. Besides, as the incident angle increases, the conversion efficiency first rises to the maximum and then falls down, which indicates the existence of the optimal incidence angle.
Fig. 5. (color online) Conversion efficiency versus θ1 at Tc = 300 K and Th = 500 K when J1 is the optimal electric current in the lower material.
6. Conclusion
The problem of current refraction at the interface is studied in this paper. The relationships between the incidence angle, the refraction angle, and the refractive index are discussed in detail. We also propose a new model based on the conception of current refraction, wherein the current densities in two components can be adjusted independently, thus the optimal current density can be matched in both components, and the conversion efficiency can be significantly increased. In the new model, components with dissimilar compatibility factors can also be combined into efficient thermoelectric generators. These results point to a new route for high efficiency thermoelectric composites.
