Influence of convection at outer ceramic surfaces on the characterization of thermoelectric modules by impedance spectroscopy

Impedance spectroscopy is a useful method for the characterization of thermoelectric (TE) modules. It can determine with high accuracy the module dimensionless figure of merit (zT) as well as the average TE properties of the module thermoelements. Interpretation of impedance results require the use of a theoretical model (equivalent circuit) which provides the desired device parameters after a fitting is performed to the experimental results. Here we extend the currently available equivalent circuit, only valid for adiabatic conditions, to account for the effect of convection at the outer surface of the module ceramic plates, which is the part of the device where convection is more prominent. This is performed by solving the heat equation in the frequency domain including convection heat losses. As a result, a new element (convection resistance) appears in the developed equivalent circuit, which starts to influence at mid-low frequencies, causing a decrease of the typically observed semicircle in the impedance spectrum. If this effect is not taken into account, an underestimation of the zT occurs when measurements under room conditions are performed. The theoretical model is validated by experimental measurements performed in a commercial module with and without vacuum. Interestingly, the use of the new equivalent circuit allows the determination of the convection heat transfer coefficient (h) if the module Seebeck coefficient is known and an impedance measurement in vacuum is performed, opening up the possibility to develop TE modules as h sensors. On the other hand, if h is known, all the properties of the module (zT, ohmic (internal) resistance, average Seebeck coefficient and average thermal conductivity of the thermoelements, and thermal conductivity of the ceramics) can be obtained from one impedance measurement in vacuum and another measurement at room conditions.


I. INTRODUCTION
2][3][4][5][6][7] In addition, it also allows a complete characterization of these devices if the thermal conductivity of the ceramics is given, providing the ohmic (internal) resistance, the module zT, and the average thermoelements Seebeck coefficient and thermal conductivity. 6,8,9Interpretation of impedance results typically require the use of a theoretical model (equivalent circuit) which provides the desired device parameters after a fitting is performed to the experimental results.We have recently developed the equivalent circuit for suspended TE modules under adiabatic conditions, which consists of the ohmic module resistance connected in series with a parallel combination of two Warburg The impedance function Z=V/I of a TE module under the above considerations is given by, where V(0) and V(L) are the voltages at x=0 and x=L, respectively, I 0 is the electrical current flowing through the device at x=0, R Ω is the total ohmic resistance, which includes the contribution of all the TE legs, the copper interconnects, the wires, and the contact resistances, S is the average Seebeck coefficient of each thermoelement, and T(0) and T(L) are the temperatures at x=0 and x=L, respectively.It should be noted in Equation ( 1) that due to the symmetry of the system with respect to the constant temperature plane (see FIG 1), the temperature difference across the thermoelement can be determined from the temperature value at x=0.
To determine the variation with frequency of T(0), the heat equation of the system in the frequency domain must be solved, 8 where θ is the Laplace transform of the temperature with respect to the initial temperature (θ=L[T-T initial ]), j is the imaginary number, ω is the angular frequency (defined as ω=2πf, where f is the frequency) and α i is the thermal diffusivity of the TE leg (i=TE) or the ceramic (i=C).
The solution of Equation ( 2) and its derivative is given by, where L i is the half the length of the thermoelement (i=TE) or the thickness of the ceramic contact (i=C), ω i is the characteristic angular frequency of each material (being where i 0 is the Laplace transform of the current at x=0 (i 0 =L[I 0 ]), h the convection heat transfer coefficient, η the TE module filling factor, which is given by the ratio of the total area of the TE legs (2NA) to the total area of the ceramic plate, λ C the thermal conductivity of the ceramic, and λ TE the average thermal conductivity for each thermoelement.Equation (5) defines the constant temperature at the half-length plane due to the symmetry of the system.Equations ( 6) and ( 7 As shown in Equation ( 1), the impedance function in the frequency domain can be obtained once θ(0) is known, where the different elements in Equation ( 10) are defined as, The equivalent circuit corresponding to Equation ( 10) is show in FIG 2a.This equivalent circuit includes three new elements in addition to the R Ω , the constant-temperature Warburg (Z WCT ) and the adiabatic Warburg (Z Wa ), which form the previously reported equivalent circuit which discarded convection effects. 8,10The first new element, given in Equation ( 13 For the case of TE modules with ceramic plates (λ C ≈35 WK -1 m -1 ) and conditions where the convection heat transfer coefficient is approximately h<500 WK -1 m -2 , Equation ( 10) can be simplified as, since the impedance of the parallel combination of Z WCT,C and C Conv is much lower than the impedance of the parallel combination of Z Wa and R Conv .The equivalent circuit obtained from Equation ( 16) can be seen in FIG 2b .In any case, it should be taken into account that this approximation is not valid if the convection heat transfer coefficient takes significantly higher values, or the TE module is formed by electrically insulating plates of thermal conductivity significantly lower than typical values of ceramics (e.g.polymers where λ≈0.2 WK -1 m -1 ), although this is not usually the case.
Some simulations of the equivalent circuit elements from FIG 2b were performed to understand the effect of convection in the impedance spectra, which are shown in FIG.3a.As described in our previous article, 10 Z Wa would be obtained in the impedance spectrum in the hypothetical case where there conduction of heat towards the TE elements is not produced (λ TE =0) and convection effects are neglected.In this case, all the Peltier heat diffuses from the junction towards the outer surface of the ceramic (slope 1 line from Z Wa in the higher magnification inset of FIG.3a).Then, once the heat reaches the outer surface, it is accumulated in the ceramic, since it cannot escape due to the adiabatic conditions, producing the capacitive (vertical line) feature in the impedance response.However, when convection effects are considered, part of the heat reaching the outer surface can escape by convection, and the impedance response induced by the ceramic layers is now given by the parallel combination of Z Wa and R conv , which produces the closing of the vertical line at mid-low frequencies, as shown by the green line from FIG. 3a.The effect of convection is not sensed at high frequencies (see FIG. 3a), since for the convection to occur it is required that the heat reaches the outer surface of the ceramics and also certain temperature increase in this material, which does not take place at high frequencies.For this reason the Z Wa element and the parallel combination of Z Wa and R conv provide the same impedance response at mid-high frequencies (see the overlap in the insets of FIG.3a).FIG.3b shows the total impedance response (equivalent circuit in FIG 2b) for the cases of high vacuum (h=0 WK -1 m -2 ) and room conditions (h=50 WK -1 m -2 ). 12 As previously discussed, the convection effect does not make any influence at high frequencies, thus, the impedance response only differ at mid-low frequencies, as observed in FIG.3b.It can be also observed from FIG. 3b that the main difference in the impedance response produced by the convection is a reduction of the semicircle.
This reduction can be quantified by the dc (steady state) limit (ω→0), which is given by , and becomes reduced with respect to the high vacuum (adiabatic) case (R Ω +R TE ) by the presence of R conv .It should be noticed that if the module zT=R TE /R Ω is calculated by impedance spectroscopy 6,13 at room conditions by considering R TE as the difference between the low and high frequency intercepts of the spectrum with the real impedance axis, or using the adiabatic equivalent circuit (R conv =0), a lower value of R TE will be obtained as shown in FIG.3b, consequently producing an underestimation of the zT.

III. EXPERIMENTAL VALIDATION
In  3b).As expected from the above theoretical analysis, both measurements overlap at the highest frequencies (inset of FIG. 4) and the differences due to the effect of convection appear at the mid-low frequencies.
Since h≈0 WK -1 m -2 for the measurement under high vacuum, the fitting to this result was performed considering R conv =0, which allows obtaining R Ω , R TE , ω TE , R C (R C =4NS 2 T initial L C /(λ C A)), and ω C , whose values are shown in Table I.The resistances R TE and R C depend on the properties of the materials, which do not change with the pressure of the environment.
These two resistances were fixed when the fitting was performed to the high vacuum measurement, otherwise the fitting is not possible and very high errors will be obtained.All the fitting results are shown in Table I.As it can be seen from Equation ( 13), h can be obtained from R conv if the average Seebeck coefficient of the module is known.In order to determine the value of h in this way, the Seebeck coefficient of the module was measured from an open-circuit voltage vs. temperature difference curve, obtaining a value of 192.14 μVK -1 (see Table I).Using this value of S, a convection heat transfer coefficient h=40.12WK -1 m -2 was obtained, which although somewhat higher than the typical values for natural convection at ambient pressure (2-25 WK -1 m -2 12 ) it is not very far and in the same order of magnitude.A possible reason for the deviation observed could be attributed to convection effects that are also produced in other parts of the TE module, such as the surface at the side of the ceramics where the thermoelements are attached that is not covered by them.It should be noted that the mentioned approach to determine h when the properties of the TE module are known can potentially offer the possibility to use TE modules as h sensors.
On the other hand, if the value of h of the measurement system is known (40.12 WK -1 m -2 in our case), the Seebeck coefficient can be obtained from the convection resistance if measurements under high vacuum (which provides R TE and R C ) and at room conditions are performed.Once S is known from R conv all the average properties of the TE module can be obtained (λ TE from R TE , R Ω , zT, and λ C from R C ), without the need of knowing the thermal conductivity of the ceramic plates, which was a requirement in a previous approach. 6,8s it was mentioned above, if the measurement at room conditions had been used to determine zT, a value of R TE =1.25 would have been obtained, instead of the correct value from the vacuum measurement (1.33).This provides a zT=0.718which is 6% lower than the accurate vacuum result (0.764).Hence, in order to accurately determine zT from measurements under room conditions, the new model developed here (FIG 2b) should be employed.

IV. CONCLUSIONS
A new theoretical model (equivalent circuit) to interpret impedance spectroscopy measurements of TE modules has been developed.The new model includes convection effects at the outer surface of the module ceramic plates, which introduces three new elements (R Conv , Z WCT,C and C Conv ) in the equivalent circuit.However, for standard commercial TE modules the equivalent circuit can be simplified when h<500 WK -1 m -2 , only requiring to introduce the convection resistance (R conv ).This new element, which depends on the convection heat transfer coefficient h, influences the mid-low frequency part of the spectrum, producing a reduction of the typically observed semicircle.The theoretical model was experimentally validated by performing impedance spectroscopy measurements under high vacuum and at room conditions (ambient pressure) to a commercial TE module.The experimental measurements were found to be in agreement with the predicted results from the theoretical model and allowed the determination of the convection heat transfer coefficient, which was in the same order of magnitude than literature values.
The determination of the convection heat transfer coefficient is possible if the module average Seebeck coefficient is known and an impedance measurement in vacuum is performed, which opens up the possibility to develop TE modules as h sensors.On the other hand, if h is known, all the properties of the TE module (zT, ohmic (internal) resistance, thermal conductivity of the ceramics, average Seebeck coefficient and average thermal conductivity of the thermoelements) can be obtained from one impedance measurement in vacuum and another measurement at room conditions.A final analysis also showed that an underestimation of the module zT of 6% can be produced if the new equivalent circuit is not employed when the characterization of the TE module is performed at room conditions.

FIG 1 .
FIG 1. Thermal model employed in the theoretical analysis.A positive value of both the current and the Seebeck coefficient is considered.The arrows indicate the direction of the heat fluxes appearing at the boundaries.For the Peltier heat, the arrows point out of the junction when the electrons absorb heat from the lattice.The solid line depicts qualitatively a possible thermal profile.The dotted line shows the plane where the temperature remains constant at any time and the dashed line indicates the initial temperature.
) show the energy balance at x=0 and x=-L C , respectively, where the convective heat flow is included in the latter.It should be noticed that the filling factor in this equation accounts for the convection produced in the area of the ceramic which differs from the area of the TE legs (2NA).In commercial modules η can take values around 0.3, which represents a significant part of the ceramic outer surface where convection occurs which should be taken into account.Finally, Equation (8) shows the temperature continuity at the contacts, Using these boundary conditions and Equations (3) and (4), θ(0) can be determined,

8 FIG 2 .
FIG 2. (a) Equivalent circuit obtained when convection effects are considered at the outer surface of the ceramics in a thermoelectric module.(b) Simplified equivalent circuit for standard commercial thermoelectric modules.The equivalent circuit elements framed in the dotted line are related to the ceramic plates.The ones framed by the solid line in grey correspond to the thermoelement.If the opposite case is considered, i.e. a huge convection (h→∞) at the outer surface of the ceramics occurs, R conv →0 and Z Cconv →∞, thus, the equivalent circuit in FIG 2a reduces to the parallel combination of Z WCT,C with Z WCT , connected in series with R Ω .In this case, the Z WCT,C element represents the diffusion of heat within the ceramic from the junction, which is completely removed by the effect of the convection when it reaches the outer surface of the ceramic, producing no temperature change at this surface (constant-temperature boundary).10

FIG. 4 .
FIG. 4. Experimental impedance spectra of a suspended commercial thermoelectric module at room (ambient pressure) and high vacuum conditions (dots) and their corresponding fittings to the equivalent circuit of FIG 2b (lines).The convection resistance was not considered for the vacuum case (h≈0 WK -1 m -2 ).The quantity of 13.8 mΩ was subtracted to the real impedance part of the vacuum measurement in order to match the ohmic resistances (RΩ) and obtain a clearer comparison.FIG. 4 shows the experimental measurements and their corresponding fitting to the equivalent circuit of FIG 2b.Zview software was used to perform the fittings.It can be observed that the results obtained are quite similar to the simulations performed using the theoretical model (FIG.3b).As expected from the above theoretical analysis, both measurements and C 1,i and C 2,i are constants.

TABLE I .
Fitting parameters obtained from the fittings to the experimental measurements in FIG.4of a commercial thermoelectric module under high vacuum and ambient pressure conditions (no vacuum).The errors provided from the fitting are given in brackets.The Seebeck coefficient was experimentally obtained and the convection heat transfer coefficient calculated from the no vacuum fitting result.