Effects of Thermal Contact Resistance and Thomson Heating on the Outputs of Solar Thermoelectric Power Generation System Bimrew Tamrat Admasu and Xiaobing Luo* School of Energy and Power Engineering Huazhong University of Science and Technology Wuhan, Hubei, 430074, China
[email protected] Abstract— We establish a mathematical model with numerical example of solar thermoelectric power generation system. The components’ thermal resistances, thermal contact resistances at the interface between components and Thomson heating are all considered in the model. The inner effects include Seebeck effect, Peltier heating effect, Thomson effect and Joule effect. The thermoelectric material properties are all temperature dependent. From the mathematical model, the junction temperatures, the rate of heat flow at the junctions, the power output from the system and thermal efficiency of the system are formulated. Applying the model to a practical example in engineering, the system is evaluated to identify the effect of thermal contact resistance between components and the Thomson effect. The results indicate that neglecting thermal contact resistance between components and Thomson effect highly influences on the outputs of the system. Keywords— Thomson heating; thermal contact resistance; thermoelectric; temperature dependent material property
I. INTRODUCTION When a thermoelectric generator is inserted between a solar heat flux receiver and a heat sink, part of solar heat flux will be converted into electrical power via the Seebeck effect. Gordon [1], Wu [2-4], Agrawal and Menon [5], Chen and Wu [6], and Nuwayhid, Shihadeh and Ghaddar [7] analyzed the effect of the finite rate heat transfer between the thermoelectric device and its external heat reservoirs on the performance of single-element single stage thermoelectric generators. Crane and Jackson [8] investigated the characteristics of single-stage multi-element thermoelectric generators with the irreversibility of finite rate heat transfer, Joulean heat inside the thermoelectric device, and the heat leak through the thermoelectric couple leg. Elena, Enescu, Marcel, and Stan [9] optimize the thermoelectric power generation system. In most of the published papers, the Thomson heat term was neglected on the assumption that it is relatively small. However, in thermoelectric power generation, the materials are temperature dependent and there will be significant variation of the Seebeck coefficient with temperature, so, the effect of Thomson heat is substantial and should be taken into consideration. The existence of a finite contact resistance is
due principally to surface roughness effects. This will cause temperature drop in the interface between the components and has an impact on the system outputs. II. THEORY AND ANALYSIS A. Basic configeration of the system Solar driven thermoelectric power generation system consists of solar concentrator, heat receiver, thermoelectric module and cooling block. In this paper, Fresnel solar concentrator is used to increase solar intensity and acquiring an increase in thermal flux on the hot junction of the module. This concentrator is chosen for the reasons of its different merits like, simpler structure, light weight, small size, high efficiency and lower cost. The basic configuration of solar thermoelectric power generation system is shown in Fig. 1. B. Thermal resistance network for the system Fig. 2 shows the thermal circuit diagram of the whole system. The assembly of the solar thermoelectric generator can be considered as a one dimensional series composite wall having layers of different materials with different component thickness, surface heat transfer area of the components, thermal contact resistance at the interface between the components and thermal resistance of the components.
Fig. 1. Basic configuration of solar thermoelectric power generation system
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In terms of the equivalent thermal resistance, Re, the heat transfer rate into the hot junction and rejected from the cold junction of the system, respectively can be expressed as:
Fig. 2. Thermal circuit diagram of the system
In the previous papers, researchers didn’t consider the temperature drop at the interfaces between two materials. Even if we have two flat and smooth surfaces, they are far from truly flat or smooth. When these two surfaces are brought into contact with one another, only the peaks make contact. So, there might be temperature drop between the interfaces of the materials that will attribute to the thermal contact resistance. So, this temperature drop should be considered. C. Energy equations Based on the thermal circuit diagram of the system shown in Fig. 2, one can write an equivalent (total) thermal resistance from the hot junction to the heat source of the system (Rt1) as seen in Eq. (1) and from the cold junction to the ambient of the system (Rt2) as it is indicated in Eq. (2). Rt1 Rap Rc1 Rca Rc3 Rcuh Rc5
(1)
Rt2 Rs Rc2 Rcb Rc4 Rcuc Rc6
(2)
The conventional non-equilibrium thermodynamics i.e. without considering some of the external losses is applied to the hot and cold junctions. Five quantities are associated with the various energy transport mechanisms at the hot and cold junctions. The conductive heat losses are Kg (7hj7cj), where Kg is the total thermal conductance of the thermoelectric couples, Thj and Tcj are the hot and cold junction temperatures. The Joulean electrical resistive loss generates internal heat RgI2, where Rg is the total internal electrical resistance of the modules and I is the electric current generating through the thermoelectric couples. Effectively, half of the Joulean heat is generated in each junction. There is heat transfer due to the Peltier effect, DhIThj and DcITcj at the hot and cold junctions, respectively where, Dh and Dc are the Seebeck coefficients at the hot and cold junctions. There is a reversible absorption or liberation of heat in a homogenous thermoelectric module legs exposed to a simultaneous temperature gradient and electrical gradient for temperature dependent thermoelectric materials due to Thomson effect, IW 7hjTcj) where Wis Thomson coefficient. Assuming the Newton’s heat transfer law, the following equations apply at the hot and cold junctions, respectively as
where n is the number of thermoelectric couple legs. Substitute Eq. (3) into Eq. (5) and Eq. (4) into Eq. (6) and combining these equations yields the hot junction temperature, Thj and the cold junction temperature Tcj, respectively as:
where Rap is the heat receiver thermal resistance. Rc1 and Rc2 are the contact thermal resistance for a unit area of the interface between the heat receiver and ceramic cover, and heat sink and ceramic cover, respectively. Rc3 and Rc4 are the contact thermal resistance between the ceramic cover and the copper strip at the heat source and heat sink sides, respectively. Rc5 and Rc6 are the thermal contact resistance between the copper strip and thermoelectric legs at the heat receiver and heat sink sides, respectively. Rca and Rcb are the ceramic cover thermal resistance at the heat source and heat sink sides, respectively. Rcuh and Rcuc are the copper strips thermal resistance at the heat source and heat sink sides, respectively. Rs is the heat sink thermal resistance.
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TABLE 1.
P Qhj Qcj
(9)
where Qhj and Qcj are the rate of heat transfer at the hot and cold junctions, respectively. The efficiency of solar driven thermoelectric power generation system is the ratio of the power output from the system into the power input to the system and one can derive it as follows Kte (QhjQcj)Qhj
(10)
where Kte is the efficiency of the system. Equations (7) to (10) are the major results of this paper. They reflect the effects of heat receiver temperature (T1), ambient temperature (Ta), internal total electrical resistance of the module legs (Rg), total thermal conductance of the module legs (Kg), Seebeck coefficient (D), Thomson coefficient (W) electrical current (I),and equivalent thermal resistances (Rt1) and (Rt2), thermal contact resistances at the interface between the components (Rc), thermal resistance of each components, number of thermoelectric couple legs (n) on the outputs of solar thermoelectric power generation system. [
III.
Units
Source
Kcu
398
Wm-1K-1
[10]
Kc
35.8
Wm-1K-1
[10]
Kg
2.486
Wm-1K-1
[14]
Rg
2.754x10-5
ȍ
[15]
Rc5 = Rc6
8.5x10-5
K m2W-1
[11] /assumed
Rc1=Rc2 =Rc3=Rc4
1.5x10-7
K m2W-1
[12, 14]/ assumed
tabulated in Table 1. In the analysis, comparisons between the solar thermoelectric power generation system that includes the thermal contact resistance at the interface between components and Thomson effect, and that excludes those parameters mentioned above are done. Fig. 3 shows, the effect of Thomson heating and thermal contact resistance as the number of thermoelectric couple legs increase. As seen on the power versus the thermoelectric couple legs and the change in temperature versus the thermoelectric couple legs curves, neglecting the Thomson heating effect that have been created on a thermoelectric module legs having a temperature gradient along with the direction of the electric current and the thermal contact resistance at the interface between the components of the solar thermoelectric power generation brings over estimation of the system outputs. Fig. 4 shows the power output and efficiency versus Thomson coefficient of the thermoelectric modules. One can see that when Thomson heating coefficient increases, the power output and the total system efficiency decrease. Thermoelectric power generation will have higher outputs if the temperature difference between the hot and cold junction is higher. If there is Thomson heating it will be difficult to maintain substantial heat differential. Due to that the change in 55
1.05
50
NUMERICAL EXAMPLES AND ANALYSIS
0.90
45
W & Rt ,c
40
0
0.75
35 0.60
30 25
0.45
20
W & Rt ,c ! 0
15
Power [W]
Numerical calculations were performed in order to analyze the solar thermoelectric power generation system. In the calculation, T1 450K and Ta 300K are set. The temperature dependent Seebeck coefficient and Thomson coefficient might be obtained from the temperature polynomial function for the thermoelectric material properties [13]. So, Dh 4.023u10V K-1 and Dc 3.588u10V K-1 are estimated from the average hot junction and cold junction temperatures, respectively. The Thomson coefficient W 8.2646u10V K-1 is evaluated at the average temperature, Tave 77a 2.The thermal contact resistance at the interface between components, thermal conductivity of copper (Kcu), thermal conductivity of ceramic cover (Kc) and other physical properties of the components are
Values
Change in junction temperature [k]
The rate of heat transfer at the hot junction and the rate of heat transfer at the cold junction can be obtained by substituting the value of Thj and Tcj into eqs. (3) and (4) or (5) and (6). From that, one can derive the power output of the solar thermoelectric power generation system as follows:
Parameters
0.30
10
0.15
5 0.00 0
5
10
15
20
25
30
35
Thermoelectric couple legs [n]
Fig. 3. Change in junction temperature & power versus thermoelectric couple legs
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junction heat flow rates, the system power output and the system efficiency are derived.
0.90
0.88 0.89
0.86
0.87
0.86
Efficiency
Power [W]
0.88
0.84
0.85
0.84 -5 -4 -4 -4 -4 -4 -4 -4 5.0x10 1.0x10 1.5x10 2.0x10 2.5x10 3.0x10 3.5x10 4.0x10
0.82
-1
Thomson coefficient [VK ] Fig. 4. Power output & efficiency versus Thomson effect 0.082 0.6
0.080
W & Rt ,c
0.5
0.078
0
0.076 0.074 0.072 0.3
0.070 0.068
0.2
W & Rt ,c ! 0
0.066
0.1 0.0
Voltage [V]
Power [W]
0.4
0.064 0.062 0
1
2
3
4
5
6
7
8
9
Current [A]
Fig. 5. Power output & voltage output versus current
temperature between the hot and cold junction might be minimized. Fig. 5 shows the solar thermoelectric power output and voltage versus the electric current for n=16 as an example. As electric current increases the power output increases but when we neglect the Thomson effect and thermal contact resistance at the interface, the power output is higher. One can also realize that the voltage output of the system has an inverse relation with the electric current. The two cases were evaluated with the same parameters. One can envisage that if we ignore the Thomson heating effect and the thermal contact resistance at the interface between two components of the system, the voltage output relatively increases. So, we have to consider them when we do the practical analysis. IV.
Applying the model to a practical example in engineering, it is found that neglecting the Thomson heating effect and thermal contact resistance between components highly influences the output of solar driven thermoelectric power generation system. So, these parameters should be considered in the design and application of practical solar thermoelectric power generation system in order to get the optimum benefits from the design and application. REFERENCES [1] JM. Gordon, “Generalized power versus efficiency characteristics of heat engines: the thermoelectric generator as an illustration,” Am J Phys 1991; 59 (5):551–5. [2] C. Wu, “Heat-transfer effect on the specific-power availability of heat engines,” Energy Convers Mgmt 1993; 34 (10):1239–47. [3] C. Wu, “Specific power analysis of thermoelectric OTEC plants,” Ocean Eng 1993; 20(4):433–42. [4] C. Wu, “Analysis of waste-heat thermoelectric-power generators,” Appl Therm Eng 1996; 16(1): 63–9. [5] DC. Agrawal and VJ. Menon, “The thermoelectric generator as an endorevesible Carnot engine,” J Phys D: Appl Phys 1997; 30(2):357-9. [6] J. Chen and C. Wu, “Analysis of the performance of a thermoelectric generator,” Trans ASME J Energ Resour Technol 2000; 122(1):61–3. [7] Rida Y. Nuwayhid, Alan Shihadeh, and Nesreen Ghaddar, “Development and testing of a domestic woodstove thermoelectric generator with natural convection cooling,” Energy Conversion and Management 46 (2005) 1631–1643 [8] DT. Crane and GS. Jackson, “Optimization of cross flow heat exchangers for thermoelectric waste heat recovery,” Energy Convers Manage 2004; 45(6):1565e82. [9] Elena-OtiliaVirjoghe, Diana Enescu, Marcel Ionel, and Mihail-Florin Stan, “Numerical simulation of thermoelectric system,” Latest trends on systems (volume ii) [10] Incropera, DeWitt, Bergmann, and Lavine, Fundamentals of heat and mass transfer. 6th Ed., 2007 [11] Wulf Glatz, Etienne Schwyter, Lukas Durrer, and Christofer Hierold, “Bi2Te3-based flexible micro thermoelectric generator with optimized design. Journal of micro-electromechanical systems,” Vol. 18, NO. 3, JUNE 2009 [12] A. Lahmar, T.P. Nguyen, D. Sakami, S. Orain, Y. Scudeller and F. Danes, “Experimental investigation on the thermal contact resistance between gold coating and ceramic substrates,” Thin Solid Films 389 (2001) 167–172 [13] ANSYS 11.0 Documentation (2007) [14] Min Chen, Lasse Rosendahl, Inger Bach Thomas Condra and John Pedersen, “Notes on computational methodology and tools of thermoelectric energy systems,” EUA4X#31 Conference MASCOT07, IAC-CNR, Roma, Italy [15] S. Orain, Y. Scudeller, S. Garcia, and T. Brousse, “Use of genetic algorithms for the simultaneous estimation of thin films thermal conductivity and contact resistances,” Int. J. Heat Mass Transfer 44 (2001) 3973–3984.
CONCLUSTION
A mathematical model of solar thermoelectric power generation system is presented in this paper. Practically, thermoelectric materials are temperature dependent. So, in the modeling the Thomson heating effect is considered. Besides, the thermal contact resistance at the interface between the two components of the system and the thermal resistance of the components are included. The analytical formulae describing the hot and cold junction temperatures, the hot and cold
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