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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps

En Fang
• Corresponding author
• School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221008, Jiangsu Province, China
• Other articles by this author:
/ Xiaojie Wu
• School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221008, Jiangsu Province, China
• Other articles by this author:
/ Yuesen Yu
• School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221008, Jiangsu Province, China
• Other articles by this author:
/ Junrui Xiu
• School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221008, Jiangsu Province, China
• Other articles by this author:
Published Online: 2017-03-12 | DOI: https://doi.org/10.1515/phys-2017-0004

## Abstract

In this paper, a numerical model is developed by combining thermodynamics with heat transfer theory. Taking inner and external multi-irreversibility into account, it is with a complementary equation for heat circulation in air gaps of a steady cooling system with commercial thermoelectric modules operating in refrigeration mode. With two modes concerned, the equation presents the heat flowing through air gaps which forms heat circulations between both sides of thermoelectric coolers (TECs). In numerical modelling, a TEC is separated as two temperature controlled constant heat flux reservoirs in a thermal resistance network. In order to obtain the parameter values, an experimental apparatus with a commercial thermoelectric cooler was built to characterize the performance of a TEC with heat source and sink assembly. At constant power dissipation, steady temperatures of heat source and both sides of the thermoelectric cooler were compared with those in a standard numerical model. The method displayed that the relationship between Φf and the ratio ${\Phi }_{\text{c}}^{\prime }/{\Phi }_{\text{c}}$ was linear as expected. Then, for verifying the accuracy of proposed numerical model, the data in another system were recorded. It is evident that the experimental results are in good agreement with simulation(proposed model) data at different heat transfer rates. The error is small and mainly results from the instabilities of thermal resistances with temperature change and heat flux, heat loss of the device vertical surfaces and measurements.

PACS: 44.05.+e

## 1 Introduction

The direct energy conversion between heat and electricity based on thermoelectric effects is attractive for many applications as an alternative to traditional methods in power generation and heat transmission [1, 2]. Thermoelectric modules (TEMs) have outstanding advantages such as solid-state, compact, light-weight, containing neither moving parts nor refrigerants, maintenance free, highly reliable and quiet operation, environmental friendly performance, and so on [3, 4]. The use of TEMs for cooling, heating and power generation in a simple and reliable way has been comprehensively discussed in many industrial fields [5]. TEMs can be divided into two types, namely thermoelectric coolers (TECs) and thermoelectric generators (TEGs) [68]. Since conventional air cooling techniques for high power electronic packages are reaching the limits in term of cooling capacity, various efforts have been devoted to enhance the cooling design and performance [9]. Among them, the use of TECs, in combination with air cooling or liquid cooling approaches, is gaining more and more attention. The major thermal benefit of this development trend lies in the fact that a negative temperature gradient and thus reduced thermal resistance could be generated from the use of TECs [10, 11].

Bell [12] points out two important pathways that will lead to additional applications of thermoelectric devices. One is to promote the intrinsic efficiencies of TE materials. The other is to improve the way that existing TEMs are currently used by modeling and analyzing of thermoelectric systems [13, 14]. For TECs, the primary mission should be to keep the heat resource working in a certain temperature range. Thermoelectric devices could not be used independently. Heat sources and exchangers should be connected with to dissipate and absorb heat [15]. The commonly used approach is the iterative method as given in [11].

A numerical optimization of a TEC was presented by Xuan [16]. The results indicated that the construction cost of a TEC was closely related to the cooling power density; whereas the running cost was inversely proportional to COP. High efficiency was obtained when the length of the thermo elements was large [17]. In addition, the power generated declined with the cross-sectional area of the thermo elements, whereas efficiency showed the opposite trend [6].

For conduction and convection, the heat only flows from the end at high temperature to low. In electronic devices, the reliability and lifetime of electronic components decrease greatly with temperature rise due to power dissipation. Although a variety of cooling means have been used, such as natural convection, forced air convection and liquid exchangers, heat pipe exchangers, and so on, the electronic components are still often the parts with the highest temperature in electronic devices. The TECs work as heat pumps converting heat flux from the cold end to hot continuously at the expense of electric energy. They can be conveniently controlled by DC power suppliers.

In this paper, a numerical model with the complementary equation is established according to various equations concerning with cooling system and thermoelectric coolers.

## 2 Numerical model on a cooling system with TECs

A TEC is a solid-state heat pump placed between heat sources and heat sinks. It normally consists of an array of 2N pellets from dissimilar semiconductor materials (p and n type) that make up N thermoelectric couples which are electrically by conducting strips in series and thermally in parallel [18, 19]. The p and n legs are joined by metal interconnects with ohmic contact. The conducting strips of the thermoelectric elements are fixed at a thermal conducting and electrical insulating ceramic plate [20]. The module is packaging by thermally insulating epoxy resin. The thermoelectric elements in most commercial thermoelectric modules are not closely arranged, that is, air gaps exist in the module which lead to heat microcirculations. The heat convection (or conduction) and radiative heat transfer occur between two ceramic plates in the area that the thermoelectric elements do not occupy.

## 2.1 Assumptions

To simplify the cooling system with TECs, the following assumptions are made:

1. The numerical model is for a steady-state system;

2. The TEC is single-stage constructed from N thermoelectric couples which are identical and joined electrically in series and thermally in parallel [6];

3. All the cooling means such as convection, conduction and radiation are included in the computational domain as a one-dimension thermal resistance varying with heat flux and temperatures in simplification;

4. Heat transfer at a finite rate, and electrical resistive losses, are necessarily irreversible processes and unavoidable in a thermoelectric device [5];

5. The ceramic plates work as favorable heat conductors and electrical insulators.

## 2.2 Standard Model

One-dimensional thermal resistance network of the whole cooling system with a single-stage TEC is shown in Figure 1. The TEC is separated as two temperature controlled constant heat flux reservoirs in thermal resistance network. The positive directions of heat flow expected in the scenario are shown in Figure 1. Presenting the problems in electronic circuit terms helps to understand its functionality, and facilitates the solving of cooling problems without the need for expertise in thermal engineering [21].

Figure 1

One-dimension thermal resistance network of a cooling system with a single-stage TEC

According to the non-equilibrium thermodynamics, the inner effects of the thermoelectric elements include Seebeck effect, Fourier effect, Joule effect and Thomson effect. The increment rate of inner energy of the infinitesimal is zero at steady-state, so one can obtain the energy conservation equation as follows: $Φc=NαITc−12I2R−K(Th−Tc)Φh=Φc+NαI(Th−Tc)+I2RΦh=Th−TaRhΦc=Ploss−Tj−TaR′Tj−Tc=ΦcRc$(1)

## 2.3 Proposed Model with complementary equation

In standard models, the heat flux ${\Phi }_{\text{c}}^{\prime }$ absorbed by the semiconductor nodes is assumed to be of an equivalent amount to that flows through the cold side Φc. But in fact, due to the existences of constriction thermal resistance, they are not equal. Comparing the heat flow in TECs against the air flow in an unsealed duct with two exhaust fans, two modes are maintained in Figure 2.

Figure 2

Two modes of heat flow in TECs

Thus, $Φc=Φc′+Φf+Φl$(2) Where, Φf is the heat flowing through air gaps which forms heat circulations between both sides of TECs. It should be a piecewise function depending on the relationship of Φc and ${\Phi }_{\text{c}}^{\prime }.$ Φl is the leakage heat through the thermal insulation material. The conduction properties and insulation characteristics of the materials should have significant effect on Φf. It is also related to the thickness of ceramic plates and conduction trips, the area ratio of N(Sn +Sp)/S, and so on. As these parameters vary little in a certain range, in view of the air flow in an unsealed duct with two exhaust fans, the piecewise function is proposed to be linear. And the positive directions of heat flow are defined as in Figure 2(a). $Φf=δ1Φc′Φc+Φp1,Φc≥(Φc′+Φl)δ2Φc′Φc+Φp2,Φc<(Φc′+Φl)$(3)

While ${\Phi }_{\text{c}}\ge \left({\Phi }_{\text{c}}^{\prime }+{\Phi }_{\text{l}}\right),\phantom{\rule{thinmathspace}{0ex}}{\Phi }_{\text{f}}\phantom{\rule{thinmathspace}{0ex}}\ge \phantom{\rule{thinmathspace}{0ex}}0$ and vice versa. As mentioned before, with the complementary equation for the heat flowing through air gaps, the model is proposed as, $Φc−Φc′−Φl=δ1Φc′Φc+Φp1,Φc≥(Φc′+Φl)δ2Φc′Φc+Φp2,Φc<(Φc′+Φl)Φc′=NαITc−12I2R−K(Th−Tc)Φh=Φc+NαI(Th−Tc)+I2RΦhRh=Th−TaΦc=Ploss−Tj−TaR′Tj−Tc=ΦcRc$(4)

The proposed numerical model may be applied to the calculation of characteristic parameters, temperature prediction and further optimization study for thermoelectric coolers. The simulation results can be used as feasibility and effectiveness reference by employing component package and device cooling.

## 3.1 System characterization with TEC in refrigeration

In order to realize the output performance following control of this case study, a cooling process with a TEC was considered where film resistor temperature sensors were located at different places, as shown in Figure 3.

Figure 3

Setup to measure the temperatures of heat source and TEC

A sandwiched structure (from top to bottom) of thermal insulation layer/patch heater/thermal silicone grease/copper plate/thermal silicone grease/TEC/thermal silicone grease/heat sink was introduced. The CH-J404020 is one of the patch heater modules available from Company MCH, which was attached to a copper plate on the cold side of the TEC. The TEC 12706AJ (N = 127) is manufactured by Wanhao Technology. A water cooled exchanger was used to absorb the heat from the hot side of TEC. The coolant (water) was supplied from a water-circulating chiller. In order to reduce thermal contact resistance, a thermal interface material, thermal silicone grease (Aobaili ABL3601) was smeared uniformly on every contact surface. Figure 4 shows the schematic diagram of the experimental cooling system. The assembly was insulated with heat-insulated foam to minimize the parasitic heat load from the ambient except for the heat sink and is held under compression between four aluminum latches for proper contact between surfaces at a constant pressure. Temperatures on the heater (Tj), cold side (Tc) and hot side (Th) of the TEC, as well as the ambient temperature (Ta), are measured using film resistor temperature sensors which are held in place with thermally conductive silicone grease. The sensors have an uncertainty of ± 0.5 K. Two programmable DC power supplies (Chroma 62100H-450) are employed for heater and TEC, respectively. The voltage across and current through the patch heater are measured to determine the heat dissipation in the cooling system. The uncertainty in the measurements of voltage and current are ± 1 mV and ± 1 mA, respectively.

Figure 4

Schematic diagram of the experimental setup of water cooling system The programmable DC supply for TEC worked as a constant current power

The programmable DC supply for TEC worked as a constant current power supply, while the other one as a constant voltage source for patch heater (Uheater = 20V). All the temperatures at different currents of the TEC with this water cooled exchanger were recorded.

Then, the programmable DC supply for TEC worked as a constant current power supply (ITEC = 0.5A), while the other one as a constant voltage source for patch heater. By replacing the water cooled exchanger with aluminum radiator, the experimental temperatures at different heat transfer rate with aluminum heat sink were also recorded.

## 3.2 System characterization with TEM for thermal conduction

Before the system performance testing, a performance characterization should be achieved by means of testing the parameters, such as the equivalent thermal resistance of contact thermal resistance, constriction thermal resistance and convection thermal resistance, and so on.

Firstly, the patch heater was encapsulated bilateral symmetry by thermal insulation layers. $R′=2(Tj−Ta)Ploss$

Temperatures of the patch heater above ambient at different applied heat load are presented in Figure 5. The thermal resistance of insulation layer is 59.11 K/W as slope. The value of R-square in Figure 5 is 0.9966 which imply the accuracy of the fitting lines that explain our measurement results are more than 99%. Due to high insulation capability of the layer, the thermal resistance value changes little with the heat flux and temperature.

Figure 5

Temperatures of the patch heater above ambient at different applied heat load

Then, the test apparatus mentioned before were tested. The TEM was treated as a conduction module without being powered. As the assembly was insulated with heat-insulated foam, little heat dissipated from vertical sides to ambient, that is to say, ΦcΦh. $Φc=Φh=Ploss−Tj−TaR′Rc=Tj−TcΦcRh=Th−TaΦhK=ΦcTc−Th$

The tested temperature differences of (TjTc), (TcTh) and (ThTa) are plotted against the applied heat load in Figure 6. The equivalent thermal resistance Rc and Rh could be obtained by curve fitting. Hence the slope of the linear fit in Figure 6(a) provides the thermal interface resistance and constriction resistance. In Figure 6(b), the thermal conductivity of the module is characterized. Figure 6(c) and (d) represent the combined heat sink and water cooled exchanger with parasitic resistance in parallel, respectively.

Figure 6

Tested temperatures and corresponding thermal resistance/conductivity against the applied heat load

## 4 Results

A total of 51 samples was collected and then analyzed for modeling. Figure 7 shows the measured and simulated (standard model) curves of temperatures versus current of the TEC at the same power dissipation with water cooling system. Based on the standard model without the microcirculation heat Φf and leakage heat Φl, the theoretical results in Figure 7(a) and (b) are quite different from those in experiments. It shows the necessity of proposing an accurate model to predict the temperature behavior of TECs. In Figure 7(c), the results of the experiment and simulation were highly consistent due to the accuracy of Rh and Φh.

Figure 7

Temperatures for different currents of the TEC with water cooled exchanger at Uheater = 20V

Figure 8 shows the linear relationship between the temperature differences and current of TEC with water cooled exchanger at constant power dissipation. The temperature differences decrease from positive to negative with TEC current increasing. While the current of TEC remains about 4.4A, the experimental junction temperature is the same as the predicted one. The maximum absolute value of temperature difference in test range is 11.55 K.

Figure 8

Temperature differences between experimental and theoretical Tj for different currents of the TEC with the water cooled exchanger

In view of the proposed model in formula (4), based on the measured temperatures, the heat flows in every thermal path are calculated. The factor (Φf + Φl) is shown in Figure 9.

Figure 9

Calculated Φf + Φl for different ${\Phi }_{\text{c}}^{\prime }$

In this case, the method of least squares nonlinear regression and subsection function are introduced to deal with data. As mentioned before, Φf should be a piecewise function of ${\Phi }_{\text{c}}^{\prime }$ at constant Φc, $Φf+Φl=−0.88279Φc′+15.52372,Φc≥(Φc′+Φl)−1.57414Φc′+29.33509,Φc<(Φc′+Φl)$(5)

In the proposed model, there is a turning point from which Φf changes from positive to negative with increasing ${\Phi }_{\text{c}}^{\prime }$. Compared to the segmental fitting equations, the intersection point (19.98, −2.11) is the turning point of subsection.

Thus, at the turning point for this experimental system, Φf = 0W, ${\Phi }_{\text{c}}^{\prime }$ =19.98W, Φl = −2.11W and Φc = ${\Phi }_{\text{c}}^{\prime }$ + Φl = 17.87W. $Φf=−15.78Φc′Φc+17.63,Φc≥(Φc′+Φl)−28.13Φc′Φc+31.45,Φc<(Φc′+Φl)$(6)

Then the water cooled exchanger was replaced with aluminum radiator. The differences between experimental and calculated (proposed model) temperatures at different heat transfer rate are shown in Figure 10.

Figure 10

Temperatures for different heat transfer rate with aluminum heat sink at iTEC =0.5A

In Figure 10, the prediction temperatures are very consistent with that observed in test at constant current of TEC (iTEC = 0.5A). There is a maximum difference between theoretical and experimental junction temperatures up to 4K when experimental Tj reaches 364.6 K.

## 5 Discussion and conclusion

Electricity is consumed by thermoelectric effect with ${\Phi }_{\text{c}}^{\prime }$ to maintain system balance. But in fact, there are lots of air gaps between semiconductor legs which lead to the appearance of heat microcirculations between both sides of TEMs. Due to the constriction resistance, conducting strips and air gaps, there is a difference between thermal power absorbed by the semiconductor nodes and the heat fluxing through the ceramic plate on the cold side, which leads to the error between theoretical and experimental Tj as in Figure 7.

High precision of the model can be obtained by the introduction of a correct function $\begin{array}{}{\Phi }_{\text{c}}={\Phi }_{\text{c}}^{\prime }+{\Phi }_{\text{f}}+{\Phi }_{\text{l}}.\end{array}$ And the heat flowing through air gaps Φf can be treated as a linear piecewise function of $\begin{array}{}{\Phi }_{\text{c}}^{\prime }/{\Phi }_{\text{c}}.\end{array}$ The coefficient δ is related to the thickness of ceramic plates and conduction trips, the area ratio of N(Sn +Sp)/S, the conduction properties and insulation characteristics of materials, and so on. The linear relationship is approximate because there are factors (such as heat convections and radiative heat transfer) which caused inaccuracy in thermal resistance measurements.

For the prediction model of junction temperature, it is very difficult to obtain accurate thermal resistance values, which varies with ambient temperature, heat flux, device layout and other factors in the cooling system with TECs. The way to obtain the thermal resistance values by theoretical analysis can be achieved in other papers. In order to verify the validity of the proposed model, thermal resistance values with experimental method are adopted. Figure 6 illustrates temperature distribution in heat flow path including TEC (Tj, Th, Tc and Ta) related to applied heat load. The results show that Rc decreases with the increase of heat flux according to exponential decay. And the thermal conductivity K increases as the reciprocal of according thermal resistance.

The model is derived basing on the structure of the TEMs, so it can be used also in TEGs. In TEGs, there are initiative heat absorptions to generate power. The correction function Φf works only in one mode $\begin{array}{}{\Phi }_{\text{c}}\ge {\Phi }_{\text{c}}^{\prime },\end{array}$ and remains linear to Φc as in TECs.

It is worth mentioning that in most research articles on TEMs, the thermal conductivity is only effected by semiconductor materials, $\begin{array}{}K\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\lambda {A}_{\text{x}}/L\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{\lambda }_{\text{N}}{A}_{\text{x}N}/{L}_{\text{N}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\lambda }_{\text{P}}{A}_{\text{x}P}/{L}_{\text{P}}{\rho }_{\text{N}}{L}_{\text{N}}/{A}_{\text{x}N}.\end{array}$ In this case, the thermal conductivity K, representing the heat transferring from the hot surface to cold through conduction element (not only semiconductor materials but also devices within the package) due to the temperature difference, should be considered. In fact, the thermal conductivity with only semiconductor materials caused a significant error between theoretical and experimental data. The total thermal conductivity tested by experiments is more than one hundred times than the theoretical value of semiconductor materials.

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Accepted: 2016-11-14

Published Online: 2017-03-12

Citation Information: Open Physics, Volume 15, Issue 1, Pages 27–34, ISSN (Online) 2391-5471,

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