Considering a homogeneous thermoelectric medium with the length of L. The temperature and electric potential at two ends of the medium are T_l, ϕ_l, and T_h, ϕ_h, respectively as shown in Fig. 1.

Fig. 1.

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	Fig. 1. Homogeneous thermoelectric medium.

Assuming that the material parameters are independent of temperature, and substituting Eq. (10) into Eq. (12), we have 13 The solution of Eq. (13) is 14 Substituting Eq. (14) into Eq. (6), yields 15 So we have 16 Noticing Eq. (10), the undetermined coefficient c₁ can be derived as17 where .

Then the temperature field and electric current field can be expressed as 18 19

Equations (18) and (19) reveal a physical image that the distributions of temperature and electric potential are determined by electric current density J and energy flux J_E, if we ignore the constants c₂ and c₃ that represent uniform temperature and electric potential field, which means that the diversification of temperature/electric potential is only dependent on the electric current/energy flux for a certain thermoelectric material. More importantly, equations (18) and (19) still hold even the component is embedded in thermoelectric composites, as they are obtained directly from governing equations (6)–(12).

On the other hand, electric current and energy flux can be determined by boundary conditions as 20 21

Now consider a multilayered thermoelectric medium with material parameters σ_i, ε_i, and κ_i, and length L_i ( . The temperature and electric potential are continuous at the interface, and represented as T_i and ϕ_i at the right end of each layer, as shown in Fig. 2.

Fig. 2.

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	Fig. 2. (color online) Multilayered thermoelectric medium.

Obviously, the total temperature and electric potential differences at two ends of the multilayered thermoelectric medium consist of temperature and electric potential differences between layers, thus 22

As shown in the above section, the temperature and electric potential differences between layers are only dependent on the electric current and energy flux. Noting that electric current and energy flux are uniform in one-dimensional multilayered thermoelectric medium, the differences in temperature and electric potential at two ends of phase-i can be expressed from Eqs. (18) and (19) as 23 24 where the parameters b_i can be calculated from 25 It is seen that b_i also depends on J and J_E as well as the boundary conditions T_l and ϕ_l.

Finally, substituting Eqs. (23)–(24) into Eq. (22), yields 26 The only two unknown quantities J and J_E can be obtained from Eq. (26), then temperature and electric potential in each layer can be derived from Eqs. (18) and (19). This set of equations solves the problems completely, leading to the full determination of the field distributions in each layer.

For bilayered medium, equation (26) can be reduced to 27 Solving Eq. (27), the electric current density J and energy flux J_E can be deduced as28 29 where The results are consistent with the Yang Y et al.ʼs results.^[29]

The Peltier effect implies the heating or cooling existing at the junction of two different thermoelectric materials, which can be considered as the back-action counterpart to the Seebeck effect. When electric current flows through a junction between two thermoelectric materials, heat may be absorbed or released at the junction as shown in Fig. 3. The heat flux changed by Peltier effect is equal to^[31] 30 where is the Peltier coefficient of phase-i, and the relationship between Peltier and Seebeck coefficients is 31

Fig. 3.

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	Fig. 3. (color online) Two-layered thermoelectric medium.

As both the charges and energy are conserved in a current system, the change of heat flux at the junction should be provided by electric energy . Substituting Eqs. (30) and (31) into Eqs. (8) and (9) yields 32 where is the electric potential difference at the junction. Therefore the electric potential difference caused by Peltier effect at interface can be expressed as 33 Noticing this, the inherent relation of multilayered thermoelectric medium in Eq. (22) can be rewritten as 34 From Eq. (33), the total electric potential jump generated by Peltier effect at junction can be expressed as 35

Like Eq. (25), the parameter b_i while considering Peltier effect can be calculated by36 Substituting Eqs. (23), (24), and (35) into Eq. (34), and noticing the boundary condition, we have 37 Similarly, the only two unknown quantities J and J_E can be obtained from Eq. (37), then temperature and electric potential in each layer can be derived from Eqs. (18) and (19) and interface conditions. Thus temperature is continuous at the interface while electric potential difference at interface is . This set of equations solves the problems completely, leading to the full determination of the field distribution in each layer.

The conversion efficiency η in a thermoelectric medium is^[32] 38 where J_Q0 is the initial heat flux. The η can be optimized with respect to current density for a given temperature,^[32] and the maximum conversion efficiency in homogeneous thermoelectric material can be expressed as39

The maximum energy conversion efficiency for an arbitrary multilayered thermoelectric medium is too complicated to be expressed analytically. However, for the case of attaching a thin thermoelectric film to the right end of the homogeneous material, if the interface Peltier effect is taken into consideration, the maximum energy conversion efficiency can be obtained as 40 where is the Seebeck coefficient of the attached film.

The optimal electric current density of this medium can also be obtained as 41 where L is the length of the medium

By comparing Eq, (39) with Eqs. (40) and 41we can find out that the Peltier effect has a remarkable influence on both the maximum conversion efficiency and the optimal electric current density.

A three-layered thermoelectric medium is used to demonstrate the analysis by using the material parameters shown in Table 1. The temperature loads at two ends of the medium are taken to be and .

Table 1.

Table 1.

Table 1. Material parameters of thermoelectric samples.[33–35] . Sample BiSbTe 27700 213 0.7 Bi2Te3 110000 200 1.6 Bi0.5Sb1.5Te3 38000 250 0.85

Table 1. Material parameters of thermoelectric samples.[33–35] .

The energy conversion efficiency of three-layered thermoelectric medium is dependent on the electric flux density while the temperatures at two ends are fixed. The ϕ_l is taken to be zero without loss of generality, then the energy conversion efficiency in Eq. (38) can be rewrite as 42

The maximum conversion efficiency of a three-layered thermoelectric medium and its constituents are analyzed in Fig. 4. As we can see, when considering the interface Peltier effect, the maximum conversion efficiency of the composite is significantly higher than those of all the constituents under the same temperature difference.

Fig. 4.

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	Fig. 4. (color online) Maximum conversion efficiencies of a thermoelectric composite and its constituents.

The plots of energy conversion efficiency versus electric current density under two different interface conditions are shown in Fig. 5. It can be seen that the energy conversion efficiency is significantly improved when considering Peltier effect at the interface, and the relevant maximum efficiency increases from 0.11 to 0.17. Besides, the optimal electric current density increases from to .

Fig. 5.

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	Fig. 5. (color online) Energy conversion efficiencies versus electric current density.

To evaluate such observations, we also show the field distributions under two different interface conditions, including the electric potential, temperature, electric energy and heat flux in Fig. 6. Numerical simulations is also made to verify the correctness of our results, since interface conditions in software cannot be modified, we only give the simulation results under traditional interface conditions. From Fig. 6(a), it is seen that the consideration of Peltier effect at the interface results in much bigger electric potential difference at the same electric current density, which leads to much higher electric energy and energy conversion efficiency. On the other hand, there is little difference in temperature distribution when considering Peltier effect at the interface as shown in Fig. 6(b). From Figs. 6(c) and 6(d) we can see that both the electric energy and thermal flux jump at the interface when considering the interface Peltier effect. The electric energy jumps up at the interface, leading to higher energy conversion efficiency, while the thermal flux jumps down due to energy conservation. Besides, numerical simulations under traditional interface conditions fit well to our analytical solutions.

Fig. 6.

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	Fig. 6. (color online) Comparisons among fields under two different interface conditions and numerical simulations. (a) optimum electric current density, (b)temperature, (c) electric energy, (d) thermal flux.

Finally we point out that a thermoelectric material with temperature dependent properties can be approximated as a multilayered thermoelectric composite, and each layer has constant property. Therefore the solutions obtained above also offer a path to the analysis of temperature dependent properties, especially when the materials are subjected to large temperature gradient so that the material properties cannot be considered as being unchanged.