Thermoelectric conversion efficiency of a two-dimensional thermoelectric plate of finite-size with a center crack

Abstract

As a potential functional material in the energy field, the evaluation and improvement of thermoelectric conversion efficiency of thermoelectric materials have attracted widespread attention. Based on two kinds of thermoelectric coupling theories with and without considering the Thomson effect, the effect of a center crack on the thermoelectric conversion efficiency is studied for a finite two-dimensional thermoelectric plate by using the singular integral equation. The numerical solution of this thermoelectric coupling problem is obtained by using the Lobatto–Chebyshev direct numerical quadrature method. Then the electric field, temperature field, and thermoelectric conversion efficiency are also obtained numerically. In the numerical discussion, the effects of the structure size, the partially insulated coefficient, and temperature conditions on the thermoelectric conversion efficiency are also analyzed in detail. The comparison confirms that the thermoelectric conversion efficiency predicted by the thermoelectric coupling theory with the Thomson effect is different from that without the Thomson effect.

Appendix A: Solving the crack problem without the Thomson effect

Since the electric field analysis is completely consistent, only the temperature field analysis is given here. The solutions [30] to the temperature governing equation (4b) can be obtained by the Fourier transform

$$\begin{aligned} T^{ \pm } \left( {x,y} \right) & = \frac{{T_{10} - T_{20} }}{h}y + \frac{{T_{10} h_{2} + T_{20} h_{1} }}{h} + \sum\limits_{n = 1}^{\infty } {\sum\limits_{m = 1}^{2} {\sin \left( {\delta^{ \pm } s_{n}^{ \pm } y} \right)} } \exp \left( {s_{n}^{ \pm } x\lambda_{m} } \right)E_{mn}^{ \pm } \\ & \quad + \frac{2}{\pi }\int_{0}^{\infty } {\sum\limits_{m = 1}^{2} {\cos \left( {sx} \right)} } \exp \left( {\delta^{ \pm } sy\gamma_{m} } \right)F_{m}^{ \pm } {\text{d}}s, \\ \end{aligned}$$

(56)

where $s_{{_{n} }}^{ + } = \frac{n\pi }{{h_{1} }},s_{{_{n} }}^{ - } = \frac{n\pi }{{h_{2} }},\gamma_{1} = \lambda_{1} = - 1,\gamma_{2} = \lambda_{2} = 1,\delta^{ \pm } = \left\{ {\begin{array}{*{20}c} 1 & {y > 0} \\ { - 1} & {y < 0} \\ \end{array} } \right.$, $E_{mn}^{ \pm }$ and $F_{m}^{ \pm }$ are the unknown constants.

The energy flux can be expressed as

$$\begin{gathered} j_{uy}^{ \pm } \left( {x,y} \right) = j_{u0}^{\prime } - \sum\limits_{n = 1}^{\infty } {\sum\limits_{m = 1}^{2} {s_{n}^{ \pm } \delta^{ \pm } \cos \left( {\delta^{ \pm } s_{n}^{ \pm } y} \right)} } \exp \left( {s_{n}^{ \pm } x\lambda_{m} } \right)\left[ {\beta \sigma A_{mn}^{ \pm } + \kappa E_{mn}^{ \pm } } \right] \\\quad- \frac{2}{\pi }\sum\limits_{m = 1}^{2} {\gamma_{m} } \delta^{ \pm } \int_{0}^{\infty } {s\cos \left( {sx} \right)} exp\left( {\delta^{ \pm } sy\gamma_{m} } \right)\left[ {\beta \sigma B_{m}^{ \pm } + \kappa F_{m}^{ \pm } } \right]{\text{d}}s, \\ \end{gathered}$$

(57)

$$\begin{gathered} j_{ux}^{ \pm } \left( {x,y} \right) = - \sum\limits_{n = 1}^{\infty } {\sum\limits_{m = 1}^{2} {s_{{_{n} }}^{ \pm } \lambda_{m} \sin \left( {\delta^{ \pm } s_{n}^{ \pm } y} \right)} } \exp \left( {s_{n}^{ \pm } x\lambda_{m} } \right)\left[ {\beta \sigma A_{mn}^{ \pm } + \kappa E_{mn}^{ \pm } } \right] \\ \quad +\frac{2}{\pi }\sum\limits_{m = 1}^{2} {\int_{0}^{\infty } {ssin\left( {sx} \right)} exp\left( {\delta^{ \pm } sy\gamma_{m} } \right)\left[ {\beta \sigma B_{m}^{ \pm } + \kappa F_{m}^{ \pm } } \right]{\text{d}}s}, \\ \end{gathered}$$

(58)

where $j_{u0}^{\prime } = - \beta \sigma \frac{{F_{10} - F_{20} }}{h} - \kappa \frac{{T_{10} - T_{20} }}{h}$.

Similar to the case of thermoelectric coupling theory with considering the Thomson effect, $k_{0} (x)$ is introduced:

$$k_{0} \left( x \right) = \left\{ {\begin{array}{*{20}c} {\frac{\partial }{\partial x}\left[ {T\left( {x, + 0} \right) - T\left( {x, - 0} \right)} \right]} & \quad {\left| x \right| \le a}, \\ 0 & \,\,\,\,{\left| x \right| > a}. \\ \end{array} } \right.$$

(59)

By using the boundary conditions (8d), (9b) and (10b), the singular integral equation expressed by the unknown function $k_{0} (x)$ can be obtained as

$$\frac{{j_{u0}^{\prime } - \beta j_{e0} }}{\kappa } = \frac{1}{\pi }\int_{ - a}^{a} {k_{0} } \left( r \right)\left[ {\frac{1}{{2\left( {r - x} \right)}} + M_{0} \left( {r,x} \right)} \right]{\text{d}}r.$$

(60)

The expression of $M_{0} (r,x)$ in Eq. (60) is consistent with the expression in Eq. (20). The solvability condition for Eq. (60) is

$$\frac{1}{\pi }\int_{ - 1}^{1} {k_{0} \left( r \right){\text{d}}r} = 0.$$

(61)

Once $k_{0} (x)$ is determined by Eqs. (60) and (61), the entire temperature field in the case of thermoelectric coupling theory without considering the Thomson effect can be solved.

Equations (60) and (61) can be standardized by $r = as$ and $x = at$ [31]. The equations can be rewritten as

$$\left\{ \begin{gathered} \frac{{2(j_{u0}^{\prime } - \beta j_{e0} )}}{\kappa } = \frac{1}{\pi }\int_{ - 1}^{1} {K\left( s \right)\left[ {\frac{1}{{a\left( {s - t} \right)}} + \chi \left( {as,at} \right)} \right]{\text{d}}s} \hfill \\ \frac{1}{\pi }\int_{ - 1}^{1} {K\left( s \right){\text{d}}s} = 0 \hfill \\ \end{gathered} \right.$$

(62)

where $K(s) = k_{0} (as)$, $\chi (s,t) = 2M_{0} (as,at)$, $K\left( s \right) = \frac{{2(j_{u0}^{\prime } - \beta j_{e0} )}}{a\kappa }\frac{\vartheta \left( s \right)}{{\sqrt {1 - s^{2} } }}$.