The thermoelectric cooler (TEC) is a kind of cooling equipment which used to dissipate heat from the devices by Peltier effect. The cooling capacity (Qc) and coefficient of performance (COP) are both significant performance parameters of a thermoelectric cooler. In this article, three-dimensional numerical simulations are carried out by finite element analysis based on the temperature-dependent materials properties. The experimental and geometrical parameters have important effects on the TEC performance which have been analysed, such as electrical current, geometric configuration of thermoelectric leg, Thomson effect, thermal contact resistances and electrical contact resistances. The results show when the Thomson effect is ignored, the maximum difference in the cooling capacity is 7.638 W while the maximum difference in the COP is 0.09. When contact effect is not considered, the maximum difference in the cooling capacity is 22.06 W while the maximum difference in the COP is 0.75. Furthermore, the cooling capacity and COP have also been simultaneously optimized according to the multi-objective genetic algorithm. The best optimal value is obtained making use of TOPSIS (technique for order preference by similarity to an ideal solution) method from Pareto frontier. Investigated on these optimal design parameters which were anticipated to provide real guidance in industry.
I. INTRODUCTION
Recently, it is of more importance to achieve thermal management of electronic devices. As the size of the devices decreases, heat generated by the devices has increased. Excessive heat can affect the life of the device seriously. Many passive cooling methods including natural convection cooling, water cooling, micro-channel cooling and heat pipe cooling to decrease the operating temperature of electronic devices. With the increase of heat fluxes of electronic devices, passive cooling system can’t completely remove the high heat fluxes in time.1 Hence, an effective cooling method need be developed to solve this heat dissipation problem.2
It is a good way to dissipate heat by using thermoelectric coolers, which uses Peltier effect to transfer thermal power from the cold side to the hot side. Thermoelectric devices have many advantages over the passive cooling devices such as no moving parts, high reliability, small volume, low noise and light weight.3 Thermoelectric cooler are connected electrically in series and thermally in parallel, which is composed of three main components: thermoelectric legs, metal electrodes and substrates.4
Although thermoelectric cooler is a good cooling device, high cost and low efficiency limit its wide applications. Researchers have tried to optimize thermoelectric cooler in the past decades. The performance of thermoelectric cooler is enhanced by improving the thermoelectric figure of merit. In addition to this, structure optimization of thermoelectric cooler is also a way to improve performance of thermoelectric cooler.5 Some researchers have studied the optimization of structure. Gong et al.6 analyzed the influence of leg geometry and contact layers on performance of thermoelectric coolers. It is beneficial to the TEC performance with designing smaller leg height and larger leg cross-sectional area. Xuan et al.7 designed two different structures of two-stage thermoelectric coolers. Meanwhile, the two structures were optimized to achieve maximum cooling capacity.
Therefore, optimizing the geometry structure of TEC is an effective method to improve the cooling capacity and COP of TEC. What’s more, many optimization algorithms are widely applied in the geometric design. Cheng et al.8,9 developed a method for optimization of geometric of the single stage and two-stage TEC to obtain the maximum cooling capacity by genetic algorithm. Huang et al.10 established a three dimensions TEC model and a simplified conjugated-gradient algorithm was used to optimize the cooling power of TEC. The geometric parameters have been optimized to maximize the cooling capacity when temperature difference and input current were changing. Rao et al. used the modified teaching learning based optimization (TLBO) to optimize two-stage TEC.11 Amin et al.12 considered cooling capacity and COP of the TEC as double objective functions. They showed that the chemical reaction (CRO) algorithm can be effectively applied in optimal design of two-stage thermoelectric cooler. Lamba et al.13 optimized performance of trapezoidal thermoelectric cooler using genetic algorithm including Thomson effect and thermal contact resistances.
The most of above studies used zero-dimensional model to obtain the analytical solution of TEC performance, however its accuracy is low attribute to many simplifying assumptions. These studies ignore the effect of Thomson effect or assume Thomson effect to be symmetrically distributed in thermoelectric elements in combination with optimization algorithm. Chen et al.14 showed that the cooling capacity of single stage TEC considering the Thomson effect can be improved by 5–7%. Moreover, with the improvement of manufacturing technology, thermoelectric cooler can be scaled down to micro level and achieve higher cooling power. As a result, the effect of contact resistance can’t be ignored. However, in the above optimization studies, contact resistance is mostly ignored. We need to build a complete model considering all thermoelectric coupled effects to accurately predict the TEC performance. The cooling capacity and the COP are two important parameters to describe TEC performance. However, the optimum cooling capacity solution by single objective optimization does not obtain optimum COP and this is true reciprocally,15 so we need to obtain maximum cooling capacity and COP simultaneously by multi-objective optimization algorithm.
For the problems mentioned above, the present study is based on the following three aspects. Firstly, to determinate TEC performance accurately, the 3D multi-physics TEC model is established and carried out by COMSOL Multiphysics 5.4 including Seebeck effect, Peltier effect, Joule heating, Fourier’s heat conduction, Thomson effect and contact effect. Secondly, the optimization of geometric parameters and the applied currents, which has been illustrated to be a feasible and effective way to improve the cooling performance of the TEC, the influence of the applied current and the geometric parameters on the optimal TEC performance are also discussed. Thirdly, two objective functions, the cooling capacity and the COP, are selected for objective function of the system. The optimal values of the design parameters for the TEC are solved using NASA-II (the non-dominated sorting genetic algorithm) optimization method. Afterwards, a TOPSIS is used to determine the best optimal solution from the Pareto frontier. It is conducive to design a TEC that meets performance requirements in common way. The optimization approach used and the solution results are anticipated to provide a favor to the real TEC design.
II. MODELS AND METHODS
A. Physical model
The overall model of TEC is presented in Fig. 1, including thermoelectric legs, copper electrodes and ceramic plates. The TEC model consists of seventy-one pairs of P-N legs connected with copper conductors. Ceramic substrates are attached on both sides of the TEC model. We used Bi0.5Sb1.5-xCdxTe3(x=0.01) for p-type legs and commercially hot extruded Bi2Te3-based materials for n-type legs. Temperature dependent thermoelectric properties are shown in Fig. 2.16 In this study, the structure parameters are presented in Table I.
Temperature dependent material properties; (a) Thermal conductivity (b) conductivity coefficient (c) Seebeck coefficient.
Temperature dependent material properties; (a) Thermal conductivity (b) conductivity coefficient (c) Seebeck coefficient.
Structure parameters of TEC model.
Parameter . | Value . |
---|---|
Substrate thickness (mm) | 0.8 |
Copper thickness (mm) | 0.1 |
Area of the legs (mm2) | A=w×l |
Leg length (mm) | h |
Leg gap distance (mm) | 1.0 |
Parameter . | Value . |
---|---|
Substrate thickness (mm) | 0.8 |
Copper thickness (mm) | 0.1 |
Area of the legs (mm2) | A=w×l |
Leg length (mm) | h |
Leg gap distance (mm) | 1.0 |
B. Governing equations
The governing equations explain the TEC theory. The energy equation described in Eq. (1), and the continuity equation of electric current in Eq. (2).
where is the density, is specific heat capacity, is temperature, is heat flux, is Joule heating, is electric field intensity, is electric current intensity, is electric permittivity matrix.
Under steady-state condition:
The relationship between electric field and electric scalar potential is expressed in Eq. (5):
The constitutive equation is expressed in matrix form as:
where is Peltier coefficient matrix, is thermal conductivity matrix, is electric conductivity matrix, is Seebeck coefficient matrix. Inserting Eqs. (1)–(5) in Eqs. (6) and (7) we get:
Generally, the COP is defined as,
where P is the input power and Qc is cooling capacity.
C. Boundary conditions
The following assumptions were used to simplify some negligible factors, which have little influence on the results:
Except for the hot end and cold end of TEC, other surfaces were assumed to be thermal isolated.
Convection heat loss between thermoelectric legs and radiation losses were not considered.
The N-type and the P-type legs have same cross-sectional area and leg length.
Temperatures of the hot end and the cold end of TEC were fixed. The cold end temperature was set at 300K while the hot end temperature was set as 320K.
It is noted that the effects of ceramic plates and copper electrodes on performance of thermoelectric cooler, which were negligibly small.
D. Validation
To verify the simulation model, we established the same model as indicated in the article.16 The electrical contact resistance of 1.0×10-9 Ω·m2 and the thermal contact resistance of 1.8×10-4 m2·K·W-1 between thermoelectric legs and metal have been considered. These are within the contact resistances range reported in literatures.17–19 Under the same parameters, the results of simulation and experiment showed good consistency as shown in Fig. 3. The discrepancy could be attributed to the heat loss by convective and radiative heat transfer to the ambient and the lateral surface heat convection of the thermoelectric legs. The heat loss increased with temperature difference between hot end and cold end. The heat loss was ignored in simulation, while the heat loss existed in fact. Therefore, the difference between the simulation and experimental results for output power and efficiency increase with the temperature difference. The maximum difference between the simulation and experimental results for power was found to be less than 6.0% and for efficiency was found to be less than 2.0%. Hence, the simulation model established in this article was feasible.
Comparisons between numerical results and experimental results for (a) power output (b) efficiency.
Comparisons between numerical results and experimental results for (a) power output (b) efficiency.
III. OPTIMIZATION ALGORITHM
A. Multi-objective optimization
In this article, we apply the NSGA-II20 to simultaneously optimize both the COP and cooling power in MATLAB (Matrix Laboratory). Pareto frontier provides a balance between different optimization objectives. The parameter settings of NSGA-II algorithm used in experiments are presented in Table II. The flowchart of whole optimization process is illustrated in Fig. 4. The range of values of optimization parameters is selected as:
Parameters’ settings used in optimization.
NSGA-II parameters’ setting . | . |
---|---|
Number of Generation | 100 |
Population size | 100 |
Initial population | Uniform random |
Crossover probability | 0.01 |
Mutation probability | 0.9 |
Crossover distribution index | 10 |
Mutation distribution index | 20 |
NSGA-II parameters’ setting . | . |
---|---|
Number of Generation | 100 |
Population size | 100 |
Initial population | Uniform random |
Crossover probability | 0.01 |
Mutation probability | 0.9 |
Crossover distribution index | 10 |
Mutation distribution index | 20 |
B. Decision making methods
The TOPSIS method was selected as the decision-making method. TOPSIS is a method to sort evaluation objects according to their proximity to idealized targets. Using this method, the positive ideal solution is the maximum COP and cooling capacity, the negative ideal solution is reverse. The best solution should be closest to the positive ideal solution while furthest away from the negative ideal solution. The flow of the TOPSIS method is shown below:
Build a decision matrix with m alternatives and n objectives.
Decision matrix was normalized through Euclidian approach as:
Build the weighted normalized matrix :
Obtain the positive ideal solution F+ and the negative ideal solution F-:
where J1 is profitability index, J2 is lossy index.
Calculate the distance from each target to the positive ideal solution and negative ideal solution:
Compute the relative proximity to the ideal solution:
Compute the point on Pareto frontier with maximum
IV. RESULTS AND DISCUSSION
A. Effect of Thomson effect
First, the leg area and leg length were set to 2.25 mm2 and 1.5 mm, respectively. The cold end temperature was fixed at 300K while the hot end temperature was fixed at 320K, respectively. The material properties of TEC given in Table III.21 The contact resistance was ignored. To prove the effect of Thomson effect, a comparison between different analytical results and the simulation result for the cooling capacity and COP. The Thomson effect was ignored in the simplified model22 and that is assumed to be uniform distributed on both sides of the leg and mean temperature is used to calculate Thomson coefficient in the improved simplified model,23 respectively. It can be noted that the simplified and the improved simplified models give identical results. It can be explained by the small temperature difference between the hot side and cold side, leading to low Thomson effect. The analytical solution is higher than the simulation solution as shown in Fig. 5. The maximum difference in the cooling capacity is 7.638 W, while the maximum difference in the COP is 0.09. Compared to simulation model, any analytical model cannot consider Thomson coefficient when the Seebeck coefficient is assumed to be constant. Therefore, the accuracy of analytical model is lower than that of simulation model.
Thermoelectric data used for the simulations.21
Parameter . | Expression evaluation . |
---|---|
Thermal conductivity | |
Electrical conductivity | |
Seebeck coefficient | |
Thomson coefficient |
Parameter . | Expression evaluation . |
---|---|
Thermal conductivity | |
Electrical conductivity | |
Seebeck coefficient | |
Thomson coefficient |
Simplified model:
The Seebeck coefficient , the conductivity coefficient and the thermal conductivity of thermoelectric legs are kept constant and estimated from the mean temperature . Where N is the number of TEC thermocouples, is the heat flux of hot side, is total electrical resistance of the TEC module, is total thermal conductance of the TEC module, I is electrical current and is temperature difference between two sides.
Improved simplified model:
The conductivity coefficient , the thermal conductivity and the Thomson coefficient of thermoelectric legs are evaluated from the mean temperature . The temperature of each end is used to determine and , respectively.
B. Effect of contact resistance
The geometric of this model were the same as the parameters above. Temperature dependent TE properties are shown in Fig. 2. The electrical contact resistance of 1.0×10-9 Ω·m2 and the thermal contact resistance of 1.8×10-4 m2·K·W-1 between thermoelectric legs and metal have been considered. Fig. 6 compare the cooling capacity and COP with and without contact resistance. The results show that the contact resistance has a great impact on performance of thermoelectric cooler. With the increase of current, the performance difference of TEC becomes larger and larger with and without contact resistance. The maximum difference in the cooling capacity is 22.06 W. The maximum difference in the COP is 0.75. Adding contact resistance resulted in extra electrical resistance and thermal resistance. The maximum cooling capacity and COP of TEC were reduced. Therefore, it is important to reduce the contact resistance using different ways, such as searching proper metal connection for different thermoelectric legs, increasing loading pressure, using the thermal treatment to the contact points and using more advanced micromanufacturing techniques.
C. Effect of search variables
Individual parameter including the current, the leg length and leg area is analysed to reveal their impact on TEC performance, as shown in Fig. 7. The cold end temperature was fixed at 300K, the hot end temperature was fixed at 320K, respectively.
The influence of (a)-(b) electrical current; (c)-(d) leg length; (e)-(f)cross-sectional area of the legs on TEC performance.
The influence of (a)-(b) electrical current; (c)-(d) leg length; (e)-(f)cross-sectional area of the legs on TEC performance.
The initial structure of the TEC has h=1.5 mm, A=2.25 mm2. Figs. 7(a) and (b) confirms that cooling capacity and COP increase first and then decrease as the current increases. The current to achieve the maximum cooling capacity is larger than 6 A, while the current to achieve the maximum COP is less than 3 A. It is impossible to find an optimal current to achieve the maximum of both at the same time. Consequently, we needed to comprehensively consider in practice when design structure of TEC.
Figs. 7(c) and (d) shows the influence of leg length on TEC performance when current changes. With the decrease of leg length, cooling capacity doesn’t improve significantly. It can be explained that internal resistance of TEC decreases with the length of the TEC legs decreases. The proportion of contact resistance increases, and its effect on performance increases. Besides, the cooling capacity of the TEC with contact resistances becomes weak when the leg length decreases. Because of the contact resistance, additional Joule heat is generated on the contact surface at the cold end, reducing cooling capacity. Therefore, the maximum cooling capacity values for different leg length are between 11 W and 12 W.
Figs. 7(e) and (f) shows the influence of cross-sectional area of legs on TEC performance when current changes. The contribution of these contact resistances become larger as the cross section of legs becomes larger. They degrade the TEC performance by decreasing the maximum COP and maximum cooling capacity. The maximum cooling capacity reaches 19.15 W when the cross-sectional area of leg is 4 mm2. Compared with leg length, the cross-sectional area of legs has little effect on cooling capacity of TEC.
D. Optimization results
The solution of double objective optimization with conflicting objective functions which COP and cooling capacity is illustrated in Fig. 8. The Pareto front is obtained by NSGA-II, which is set of non-dominated solution. The best optimal point is decided by TOPSIS method.24 In this research, the maximum cooling capacity and maximum COP are desirable. It is found that the cooling capacity decreases monotonically with the increase of COP. What’s more, the cooling capacity ranges from 4.391 W to 18.353 W and COP ranges from 0.379 to 1.095, respectively. An ideal solution is selected for point (1.095, 18.353), a Nadir solution is selected for point (0.379, 4.391). A TOPSIS solution is listed in Table IV.
Pareto frontier with TOPSIS Solution, Nadir Solution and Ideal Solution.
V. CONCLUSIONS
In summary, thermoelectric cooler is optimized using a NSGA-II, and the maximum cooling capacity and maximum COP are considered simultaneously. TOPSIS method is used to choose the best optimal solution from the Pareto frontier. Based on this research, the following conclusions are proved:
The influence of contact resistance and Thomson effect on the TEC performance has been shown. The accuracy of analytical model that Thomson effect is simplified is lower than that of no simplified simulation model. The contact resistance degrades the system performance.
The influence of design parameters such as input current, leg structure on the TEC performance has been shown. It is clear that these variables have considerable impact on COP and cooling capacity.
The relationship between the COP and cooling power is reverse, the NSGA-II algorithm can be effectively used to optimize these opposite objectives. Designer can choose appropriate parameters according to specific requirements from the Pareto solution.
It can be useful for designing the practical TEC system precisely.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant No. 11605145); the Natural Science Foundation of Tianjin City (Grant No. 18JCQNJC03700); and the Science & Technology Development Fund of Tianjin Education Commission for Higher Education (Grant No. 2018KJ210, 2017ZD06, 2018ZD15).