Counterintuitive example on relation between ZT and thermoelectric efficiency

The thermoelectric figure of merit ZT, which is defined using electrical conductivity, Seebeck coefficient, thermal conductivity, and absolute temperature T, has been widely used as a simple estimator of the conversion efficiency of a thermoelectric heat engine. When material properties are constant or slowly varying with T, a higher ZT ensures a higher maximum conversion efficiency of thermoelectric materials. However, as material properties can vary strongly with T, efficiency predictions based on ZT can be inaccurate, especially for wide-temperature applications. Moreover, although ZT values continue to increase, there has been no investigation of the relationship between ZT and the efficiency in the higher ZT regime. In this paper, we report a counterintuitive situation by comparing two materials: although one material has a higher ZT value over the whole operational temperature range, its maximum conversion efficiency is smaller than that of the other. This indicates that, for material comparisons, the evaluation of exact efficiencies as opposed to a simple comparison of the ZTs is necessary in certain cases.


I. INTRODUCTION
Thermoelectric technology has attracted much attention because of the strong demand for eco-friendly energy harvesting [1]. As a thermoelectric heat engine does not contain any moving parts and has a small volume, it can be highly applicable for energy harvesting if the conversion efficiency is sufficient. Over the past decades, the dimensionless thermoelectric figure of merit = ( 2 ⁄ ) has been considered as a good estimator for maximum thermoelectric conversion efficiency, where α, ρ, κ, and T are the Seebeck coefficient, electrical resistivity, thermal conductivity, and absolute temperature, respectively [2, 3,4]. Consequently, the discovery of high-ZT thermoelectric materials has been central to the achievement of high-performance thermoelectric devices.
The ZT-based efficiency theory follows from the constant property model (CPM), in which all thermoelectric properties (TEPs: α, ρ, and κ) are considered to be T-independent [4]. In this case, the temperature distribution inside a one-dimensional ideal thermoelectric engine is uniquely determined as a parabolic polynomial [5]. As a result, the hot-side heat flux and the generated power are analytically determined. Finally, the thermoelectric efficiency (η) under the operating temperature between the hot-side temperature TH and the cold-side temperature TC is bounded above by max = In this paper, we report a counterintuitive example of relations between ZT and thermoelectric efficiency. We find two distinct sets of thermoelectric property (TEP) curves, where one set of TEPs has higher ZT curves over the whole operating temperature range, but its maximum conversion efficiency is smaller than of the other set. Our finding highlights the mathematical inexactness of ZT in efficiency prediction, especially for high ZT (~20).

II. THEORETICAL AND COMPUTATIONAL METHOD
We consider an ideal thermoelectric heat engine containing a one-dimensional single where The thermoelectric efficiency is defined as the ratio of the external power delivered (P) to the hot-side heat flux (QH). Thus, the efficiency ( ), at a given relative resistance = , is computed using the exact temperature distribution ( ) as [3, 4] .

Equation 2
Then, the maximum efficiency max , which satisfies the relation ( ) ≤ max for all ≥ 0, is searched. Note that a positive indicates that the heat engine is in power generation mode. To determine ( ), we solve the 2 nd order differential equation for a one-dimensional leg given as where = / . Here, the temperature satisfies the boundary conditions of ( = 0) = and ( = ) = .

III. RESULTS
The analysis considered a one-dimensional thermoelectric heat engine with a leg length of 1 mm and a leg cross-sectional area of 1 mm 2 , operating at = 900 K and = 300 K. When the electrical circuit of the heat engine is open, only thermal current flows from the hot to the cold side.
If the material has a non-zero Seebeck coefficient, it generates electrical voltage. When the circuit is closed, the induced voltage generates an electrical current and the power is delivered to the outside load resistance.
Two imaginary thermoelectric materials, mat1 and mat2, were considered for the thermoelectric leg. We assumed that the materials have linear TEP curves for the Seebeck coefficient, electrical resistivity, and thermal conductivity (see Table 1 and Figure 1). We computed the maximum thermoelectric efficiency by solving the thermoelectric differential equation for temperature distribution [3, 4, 5]. Table 2 and Figure 2 show the computed ideal thermoelectric efficiency as a function of = . Each TEP curve set has a single maximum value. The maximum efficiencies of mat1 and mat2 are computed as 48.585% and 47.422%, respectively.
Our finding indicates that efficiency evaluation is important when evaluating a material's thermoelectric performance. As higher figure of merit ZT materials continue to be developed, highly accurate efficiency calculation methods, or exact efficiency evaluation, will be required to properly assess their thermoelectric application, especially over wide temperature ranges. The

IV. CONCLUSION
In conclusion, we have found a counterintuitive example in the relation between ZT and thermoelectric efficiency in the higher ZT regime. Whereas ZT is widely accepted as a good estimator for thermoelectric material efficiency in the lower ZT regime, a higher maximum    (blue line), where γ is ⁄ and γ is the optimal load resistance to maximize the efficiency.