Manufacturer-data-only-based modeling and optimized design of thermoelectric harvesters operating at low temperature gradients
Introduction
Thermoelectric harvesters (TEH), converting potential energy of a temperature gradient ΔT between heat source and ambient air into electrical energy, have been widely investigated and used for decades [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]]. Nevertheless, harvesting from low (less than 100 °C) and sub-low (less than 10 °C) temperature gradients are relatively new trends [13], owing to increased amount of devices with ultra-low power consumption. The most popular application is probably powering of wireless sensors [3,7,[14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]], and wearable devices [[32], [33], [34], [35], [36], [37], [38], [39], [40], [41]].
This study focuses on typical TEH topology, shown in Fig. 1, consisting of thermoelectric module (TEM) and heat sink, providing convective heat exchange with ambient air. Under low-temperature gradients, heat sink thermal resistance is a main factor limiting TEH performance. The lower the thermal resistance, the better the heat exchange ability of the heat sink with ambient air and the higher the amount of harvested energy. In case of microwatt-scale generation, the use of fans is impossible (since harvested power is insufficient to run a fan) and the less effective free convection is used. The need for minimization of TEH dimensions (i.e. geometric area decrease) and cost (use of cheap materials with high thermal resistance) also leads to the fact that heat sink thermal resistance will probably be high and thus non-negligible in power calculations [42].
The main contribution of this study includes:
- 1)
Developing a 4-parameter TEH model in the form of V–I characteristics, suitable for load matching and maximum power point tracking (MPPT);
- 2)
Representing the revealed V–I curve as a Maclaurin series within feasible definition boundaries, allowing to simplify the model in accordance with required accuracy;
- 3)
Providing a simplified linear model of a TEH, acceptable for low power harvesting applications;
- 4)
Developing an electrical Thévenin equivalent circuit, based on the proposed analytical model, useful for design of interfacing power converter and MPPT [13,15,37,[51], [52], [53], [54], [55], [56]];
- 5)
Suggesting a normalized dimensionless expression, predicting maximum output TEH power at matched loading for given thermoelectric material figure of merit (FOM) and environmental conditions;
- 6)
Proposing a method for matching thermal and electrical TEH components for output power maximization.
Analytical calculations are verified by means of laboratory measurements conducted on modules from various manufacturers, with different numbers of thermoelectric pairs, under different environmental conditions and with different heat sinks.
The rest of the paper is organized as follows. Physical processes of a TEH are reviewed in Section 2. Full and simplified TEH models are derived in Section 3, where a method for correct TEM selection, allowing to obtain maximum output power at matched load, is also proposed. A method for TEM parameters extraction from manufacturer’s data is presented in Section 4. Analytical and experimental results of applying the proposed methodology to commercial off-the-shelf devices are presented in Section 5. The paper is concluded in Section 6.
Section snippets
Background
Classical TEH model is formed by a system of equations, describing main physical processes taking place in the device. These processes include conductive heat transfer, convective heat transfer, and Seebeck/Thomson/Peltier energy conversion actions. Referring to Fig. 1, the difference ΔT = T1 – T2 is converted into electromotive force due to Seebeck effect. On the contrary, Peltier effect tends to reduce the temperature gradient across the TEM. Joule heating of semiconductor elements due to
TEH modeling and matching with heat sink
According to Fig. 4 and [[57], [58], [59], [60], [61], [62], [63], [64]], energy balance at hot and cold TEM surfaces is described byandrespectively, where qj [W] is the heat dissipated by internal electrical resistance, qk [W] is the heat transferred from surface 1 toward surface 2 of the TEM by conduction, q1 [W] is the heat accepted by the TEM at surface 1 from the heat source, q2 [W] is the heat released from surface 2 to
Obtaining TEH model parameters
TEM parameters αm, Θm, and Rm, may be extracted from a set of standard experimental results, typically provided by the manufacturer. A method of parameters extraction [58] and estimation of corresponding uncertainty imposed by manufacturer data tolerances is described next.
Analytical results
Tables 3 and 4 summarize values and corresponding estimation errors (obtained utilizing (25)–(28)) of open circuit voltage, equivalent resistance and maximum electrical power harvested by six different TEHs feeding a matched load, consisting of optimal TEM and heat sinks with two different thermal resistances. Each TEH includes one TEM with 40 mm × 40 mm working area and one heat sink. Six TEHs are constructed by combining one of TEMs listed in Table 1, with one of two heat sinks with thermal
Discussion and conclusions
In this study, full and simplified models of a thermoelectric harvesters were developed. It was shown that the simplified model assuming temperature-independent parameters could be used with a high degree of accuracy at low currents and low temperature gradients. Moreover, it was demonstrated that the simplified model can be represented by a Thevenin equivalent circuit with linear output characteristics suitable for use by common electrical circuit simulators. A methodology of obtaining model
CRediT author statement
Simon Lineykin, Conceptualization, Methodology, Investigation, Supervision, Writing - original draft. Kareem Maslah, Investigation, Validation. Alon Kuperman, Supervision, Writing-Reviewing and Editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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