# Thermoelectric generators

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y
x
M
M
Tc
Th
N-type
P-type
H
H
L
RL
I
Fig. 1 Schematic diagram of thermoelectric generator
Thermoelectric generators (TEG) made up of N-type and P-type thermoelectric material can directly convert heat energy into electricity. For the convenience of fabrication, the N-type and P-type thermoelectric materials have the same length, thickness and width (along z-axis), denoted as L, H and W. The bending moment applying at the TEG is M. Temperatures at the hot side (y = 0) and cold side (y = L) of TEGs are Th = 520 K and Tc = 300 K. The external electric resistance is RL, and electric current in the closed circuit is I, where I=jA. The thermal conductivity, electrical resistivity, coefficient of thermal expansion, seebeck coefficient, Young’s modulus and Poisson’s ratio of thermoelectric material is denoted as ki, ri, ai, Si, Ei and ui (i =N, P). Here, thermoelectric materials are modeled as bismuth telluride (Bi2Te3) whose material properties and geometric dimensions are listed in Table I [1-3].
Table I material properties and dimensions of Bi2Te3 [1-3]

ki [W/(mK)]
ri [W m]
ai [10-5/K]
Si [mV/K]
Ei [GPa]
ui
L [mm]
H [mm]
W [mm]
N-type
2.2
1.19´10-5
1.68
-200
47
0.4
11.9
2.6
11.9
P-type
0.91
1.16´10-5
1.21
200
62.35
0.23
11.9
2.6
11.9
The temperature filed of thermoelectric material is , in which, j is the electric current density defined as [4] j = (SP–SN)(Th–Tc) / [AP(RI + RL)], RI represents electric resistance of thermoelectric generator, . Under the hypotheses RL = RI and there is only temperature loadings, what are axial forces in thermoelectric generator and interlaminar shear stress at the interface between N-type and P-type thermoelectric material.
For this problem, the temperature distribution is obtained from the solution of  and is
(6)
Objective 1: calculating the stresses in the PN junction if both ends of them are fixed,
and compare your solutions with Ansys solutions
Hint:
Calculate the temperature and stresses in the beam for the following geometry and environmental parameters:
L is between 11.9 mm and 26.0 mm, j0 is between  and , Th=310 K (this is approximately the human body temperature), Tc is between 273 K and 305 K.
Plot the distribution of temperature along x for different values of L, j0 and Tc. Plot the distribution of stress vs j0 for different values of L and Tc. Study the effects of beam thickness L, the applied electric flux j0 and the environment temperature Tc.

Objective 2: Study the output power of the PN junction and compare the results with Ansys solution.
Hint:
The electric current I through the TEG can be obtained as
(3)
where Th and Tc are the temperatures at the end of thermoelectric legs, RI is the total electric resistance of p- and n-type legs. The power out P can be calculated by . Study the effects of beam thickness L and the environment temperature Tc.
References
[1] Gao JL, Du QG, Zhang XD, et al.. Thermal stress analysis and structure parameter selection for a Bi2Te3-based thermoelectric module. Journal of Electronic Materials, 2011, 40: 884-888.
[2] Al-Merbati AS, Yilbas BS, Sahin AZ. Thermodynamics and thermal stress analysis of thermoelectric power generator: influence of pin geometry on device performance. Applied Thermal Engineering, 2013, 50: 683-692.
[3] Chavez R, Angst S, Hall J, et al.. High temperature thermoelectric device concept using large area PN junctions. Journal of Electronic Materials, 2014, 43: 2376-2383.
[4] Ahmet ZS, Bekir SY. The thermoelement as thermoelectric power generator: Effect of leg geometry on the efficiency and power generation. Energy Conversion and Management, 2013, 65: 26-32.
[5] Suhir E. An approximate analysis of stresses in multilayered elastic thin films. Journal of Applied Mechanics-Transactions of thee ASME, 1988, 55: 143-148.

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