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where C cu term shows the transient behavior of the thermal capacitance of the interface as heat flows through, R c is the ohmic
contact resistance, [tanh(β cu d cu )/λ cu S cu β cu ] is the contribution of the copper electrodes. We can therefore calculate the first
harmonic contribution to the sample voltage by the following equation:
1
tanh( βd)
+ Zw
[1 − sech(
βd)]
2 λSβ
ρ d
| Z | = α ws T
+
(6)
⎡ 1
⎤
S
0.5⎢
+ λSβZw
⎥ tanh( βd)
+ 1
⎣λSβZw
⎦
A fitting routine was developed using Mathematica software in order to simulate the experimental data and define the
Seebeck coefficient, thermal conductivity, electrical resistivity of the block and the contact resistances of the interfaces.
Figure 2: Complete thermal model
Results and Discussion
Figure (3) shows the electrical response of the thermoelectric element and the simulation curve. The low frequency
regime is representative of both the Seebeck and ohmic contribution. As frequency increases, the heat cannot diffuse into the
material and the thermal wave variation becomes less and less important, and thus, at high frequencies, the ohmic part of the
voltage only remains. After having plotted each term of equation (7) versus frequency, in figure 4, we observe that the
contribution of the copper electrodes is almost negligible. Thus, after having simplified equation (7) to Z w -1 =C cu +R c -1 we
have calculated |Z|. The quadrupole model fitting to the data, gives values for the thermal and electrical properties of the
element, which as can be seen in Table 1, are in accordance with those reported in literature [7].
|Z| (Ohms)
0,0105
0,0100
0,0095
0,0090
0,0085
0,0080
0,0075
0,0070
0,0065
0,0060
0,0055
0,0050
0,0045
1E-3 0,01 0,1 1 10 100
Zw (K/W)
5000
4000
3000
2000
1000
Z w
C cu
tanh( βcu
⋅dcu)
RC
+
λcu
⋅S
⋅ βcu
1E-3 0,01 0,1 1 10 100 1000 10000
Table 1: Results
λ
(W/mK)
α ws
(mV/K)
ρ
(Ωm)
R C
(K/W)
1,72
222
0,99⋅10 -5
frequency(Hz)
frequency f (Hz)
Figure 3:Experimental vs simulation
Figure 4: frequency contribution on Z w
Conclusions
The quadrupole method is an explicit method of the representation of heat transfer through multi-materials. It is based on
2x2 matrices that allow finding a linear relationship between the Laplace temperature and heat flux transformations at
boundaries (θ in , φ in ) and (θ out , φ out ) of the considered medium. Using an AC electrical measurement, the frequency-domain
response of a common thermoelectric element has been obtained and has been successfully modeled.
415,6
Acknowledgement
It is acknowledged the financial support of the project entitled "Application of Advanced Materials Thermoelectric
Technology in the Recovery of Wasted Heat from automobile exhaust systems" by the Greek Secretariat of Research and
Development under the bilateral framework with Non-European countries (Greece-USA)
References
[1] Dilhaire S, Patino-Lopez L.D, Grauby St, Rampoux J.M, Jorez S, “Determination of ZT on PN thermoelectric couples by
AC electrical measurements”, ICT Conference Proceedings, 321, (2002).
[2.] Downey A.D, Timm E, Poudeu P.F.P, Kanatzidis M.G, Shock H, Hogan T.P, “Application of Transmission Line theory
for Modeling of a Thermoelectric Module in Multiple Configurations for Electrical Measurements”, MRS Symp. Proc. 886,
F10-07.1, (2006).
[3] Maillet D, “Thermal Quadrupoles, Solving the Heat Equation through Integral Transforms’’, Wiley & Sons, LTD, (2000)
[4] Becavin C, “Mesure des proprietes thermelectriques en regime harmonique”, Stage de Master, CPMOH, Groupe Cox,
Bordeaux
[5] Downey A.D, Hogam T., “Circuit model of a thermoelectric module for AC electrical measurements”, ICT Conference
Proceedings, 79, (2005)
[6] Patino-Lopez, “Caractersation des proprietes thermoelectriques en regime harmonique”, Phd Thesis, (2004)
[7] Row D.M , CRC Handbook of thermoelectrics, CRC Press (1995)
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