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Thermoelectric coolers (TECs) use the Peltier effect for thermal management of electronic devices. They offer high reliability and low noise operation but limited in use due to low performance. In the present work, through the use of a genetic algorithm (GA), two singleobjective optimizations associated with two separate objectives are carried out, aiming maximization of cooling capacity and maximization of the coefficient of performance (COP) of TEC with space restrictions. Interfacial thermal resistance and electrical contact resistance are taken into consideration to obtain a more realistic model. This paper presents a new approach to finding appropriate solutions by optimally arranging the length of ntype and ptype thermoelectric (TE) elements, the crosssectional area of TE elements, and input electric current. To validate the GA predictions, threedimensional steadystate TEC models are prepared, and finiteelement simulations are carried out using ANSYS^{®}. Close agreement between the GA and ANSYS^{®} has been observed. This study provides a new mathematical optimization model that is more realistic and is quite close to the physical construction of TEC modules manufactured by industry.
thermoelectric cooler, optimization, genetic algorithm, finiteelement method, ANSYS workbench, cooling capacity, COP
The solidstate thermoelectric (TE) technology attract great attention of the researchers because of its potential use as green energy conversion devices. The Peltier effect of thermoelectric technology offers direct conversion of electrical energy into temperature difference. Conversely, the Seebeck effect of TE technology provides the conversion of thermal energy of temperature differential into electric power [1]. A thermoelectric cooler (TEC) dissipates the heat and removes the hotspots of the electronic devices in an environmentfriendly manner using the Peltier effect. A TEC could be installed easily within a restricted space due to its practical manufacturing possibility in small sizes. Thermoelectric coolers must be appropriately designed and manufactured to meet the necessary performance requirements. Two essential performance parameters of a TEC are the cooling capacity and the coefficient of performance. The cooling capacity of thermoelectric coolers ranges from milliwatts to watts depending on the requirements. The maximum cooling effect or higher COP for a thermoelectric cooler can be achieved through upgraded TE materials and improved device design.
The efficiency of TE materials increases with a material property known as figure of merit (Z). The term Z is defined as α^{2}/RK, where α is the Seebeck coefficient, R is the electrical resistance, K is the thermal conductance. With absolute temperature (T), the dimensionless figure of merit (ZT) is used to characterize TE materials. A higher value of ZT corresponds to better cooling performance. Hicks et al. described that the value of ZT could be enhanced by reducing the dimensions of thermoelectric materials [2, 3]. At room temperature, Venkatasubramanian et al. [4] reported a ZT∼2.4 for ptype Bi_{2}Te_{3}/Sb_{2}Te_{3} superlattice devices. Peak ZT values of different TE materials are attainable at different temperatures. Over the past two decades, significant progress in maximizing ZT has been made in developing thermoelectric materials [510].
With the significant ongoing efforts to improve TE materials, the researchers also focus on designing and assembling the TECs. The investigations established that the geometric structure of thermoelectric elements affects the performance of thermoelectric coolers [1115]. Huang et al. [16] combined a three dimensional TEC model with a simplified conjugategradient technique. They reported that at a fixed temperature difference and fixed current, a substantial value of the total area of TE elements with small element length can maximize cooling capacity. Yang et al. [17] reported that microthermoelectric coolers operating in a transient regime could provide a better cooling effect. Nain et al. [18] reported that a suitable value of dimensionless current can enhance the performance of TEC. Paretooptimal solutions were obtained for different settings of temperature ratio. Shen et al. [19] reported that a twosegmented TE element structure can reduce the joule heating effect from 50% to 35% on the cold side. The results showed a remarkable 118.1% improvement in maximum cooling capacity. Nain et al. [20] optimized cooling capacity and COP performance of TEC using dimensional structural parameters as design variables. The geometrical parameters were optimized to find Paretooptimal solutions. Jeong [21] reported that the COP of TEC can be increased by optimal values of current and length of thermoelements. Lee [22] proposed a dimensional analysis approach to find out the optimal design of TE devices with feasible mechanical constraints. Mijangos et al. [23] reported a novel design of asymmetrical legs to enhance the performance of TE devices.
Literature reports several studies on performance optimization of TEC [13, 15, 2429]. However, in the current study, the two performance parameters, namely, cooling capacity and coefficient of performance, are optimized as two singleobjective optimization problems. So far, the standard approach has been to choose either a set of geometric design variables or operating design variables. In this paper, a combined set of three design variables, electric current, length of ntype and ptype TE elements and crosssectional area of TE elements is chosen in both optimization problems. The optimization algorithm mathematical model is customized to handle the presence of ceramic substrate, copper contacts, electric contact resistances at the interface, and heat sink, which are essential parts in the fabrication of a TEC module in industrial applications. It is a new aspect of modelling TEC. The geometry of the thermoelectric element plays a vital role in the performance of the thermoelectric cooler. However, tight geometric space constraints are found in many telecommunications and other scientific applications. The TEC is used for cooling electronic devices where space restrictions are quite prevalent. Hence, consideration of performance optimization of TEC with space restrictions is a very valid assumption. The genetic algorithm is used to maximize the cooling capacity and COP of a TEC with space restrictions in two different optimization problems. The optimization results are validated through finiteelement simulations using ANSYS^{®}.
The general schematic diagram of a practical singlestage thermoelectric cooler is shown in Figure 1 (a). A thermoelectric cooler (TEC) consists of many thermoelectric (TE) elements. These thermoelectric elements are assembled electrically in series. Copper tabs are used to interconnect ntype and ptype elements. This array configuration is sandwiched between two thermally conducting ceramic substrates. Figure 1 (b) is an exploded view diagram of a practical TEC system.
The basic unit of the physical model of a TEC is a thermocouple (pair of ntype and ptype semiconductor thermoelectric elements). The number of pairs of thermoelectric elements may vary from several to hundreds. On the one hand, the manufacturing cost of TEC is high, and on the other hand, many TE materials are highpriced. Further, to predict the performance of a TEC with a heat sink, knowing the temperature at important points is quite difficult. Also, the thermal resistances in the heat sink, copper conductors, and ceramic substrates play a significant role in the total resistance to heat flow in the TEC system. These issues make the performance optimization problem challenging to solve. In this work, the effects of electrical contact resistance and thermal resistance are included. The impact of Joule heat and thermal conduction are included as well.
In this work, to simplify the investigation considering thermal resistances, a thermalresistance model has been developed. This model includes thermal resistance of copper tabs, ceramic substrates, and cold side heat sink for developing a more realistic TEC model. A thermocouple and the developed thermal resistance model for this work is shown in Figure 2.
Figure 1. (a) Singlestage TEC (b) Exploded view of TEC
Figure 2. (a) Thermoelectric couple (b) Thermal resistance model
By applying the electrical analogy of the heat flow to the thermal resistance model shown in Figure 2(b), the temperatures at the TEC hot surface and the cold surface can be expressed as
$T_{h}=Q_{h}\left(R_{h s}+R_{c r}+R_{c u}\right)+T_{a}$ (1)
$T_{c}=T_{c o}Q_{c}\left(R_{c r}+R_{c u}\right)$ (2)
where, T_{h }and T_{c }are the hot and cold side temperatures (K) of ntype and ptype elements. Q_{h }is the heat rejection rate (W) from the hot side. Q_{c }is the heat absorption rate at the cold side (W), which is referred to as the cooling capacity in common usage. T_{co }and T_{ho} are the temperatures (K) at the cold surface and hot surface of TEC, respectively. R_{hs }is the thermal resistance (℃/W) of the heat sink attached to the hot side of TEC, R_{cr }is the thermal resistance (℃/W) of the ceramic substrates, and R_{cu }is the thermal resistance (℃/W) of the copper tabs. T_{a }is the ambient temperature (K).
In the current study, some reasonable assumptions are considered.
A constant electric current pass through the circuit of dissimilar semiconductors. The heat is pumped to one of the two sides. It results in making one side cool and another side hot. A heat sink attached externally to the hot side ceramic substrate dissipates heat to the ambient environment. A thermoelectric couple produces cooling or heating effect depending on the direction of the electric current. Eq. (3) and Eq. (4) shows the heat energy balance at the cold and the hot side of the thermoelectric cooler. T_{c} and T_{h} correspond to the temperature at TE elementcopper conductor interface at the cold side and hot side, respectively, and used with the same reference in each referred equation of this paper.
$Q_{c}=2 N\left[I \alpha T_{c}\frac{k A\left(T_{h}T_{c}\right)}{L}\frac{1}{2} I^{2}\left(\frac{\rho L}{A}+2 \frac{r_{c}}{A}\right)\right]$ (3)
$Q_{h}=2 N\left[I \alpha T_{h}\frac{k A\left(T_{h}T_{c}\right)}{L}+\frac{1}{2} I^{2}\left(\frac{\rho L}{A}+2 \frac{r_{c}}{A}\right)\right]$ (4)
where, thermoelectric material properties α, ρ, k are the Seebeck coefficient (V/K), electrical resistivity (Ωm) and thermal conductivity (W/mK), respectively. r_{c} is the electrical contact resistance (Ωm^{2}). L and A are the length (m) and crosssectional area (m^{2}) of ntype and ptype thermoelectric elements, respectively. I is the supplied electric current (A), and N is the total number of thermoelectric couples. There are three essential terms on the right side of Eq. (3) and Eq. (4). The first terms, IαT_{c }and IαT_{h, }represent the Peltier heat at the cold junction and hot junction, respectively. The second heat transfer term kA (T_{h }−T_{c})/L is due to thermal conduction. The third term ½ I^{2} (ρL/A+2r_{c}/A) represents the Joule heat generation.
The selection of thermoelectric materials directly affects the performance of TEC. The material properties of thermoelectric elements are temperature dependent. Bismuth telluride (Bi_{2}Te_{3}) is the popular thermoelectric material used in thermoelectric coolers. The material properties of Bi_{2}Te_{3} used in this work are given below, as specified by Fraisse et al. [30]. T_{ave} is the average of T_{c} and T_{h}.
$\alpha=\left(22224+930.6 T_{\text {ave }}0.9905 T_{\text {ave }}^{2}\right) \times 10^{9}$ (5)
$\rho=\left(5112+163.4 T_{\text {ave }}+0.6279 T_{\text {ave }}^{2}\right) \times 10^{10}$ (6)
$k=\left(62605277.7 T_{\text {ave }}+0.4131 T_{\text {ave }}^{2}\right) \times 10^{4}$ (7)
Cooling capacity (Q_{c}) is one of the significant performance indexes of TEC, which is used in this study. The Coefficient of Performance (COP) is another crucial performance index of thermoelectric coolers. Both performance indexes are considered in the current study. COP is the ratio of cooling capacity to power consumption and defined by the following equation.
Coefficient of Performance, $\mathrm{COP}=\frac{Q_{c}}{P}$ (8)
The input electric power (P), as shown in Figure 2(b), can be calculated by the following relationship.
Input Electric Power $, P=Q_{h}Q_{c}$ (9)
The costcompetitive and highperformance TEC system will pave the way for a promising future of such green devices.
The various geometrical, material and operational parameters affect the cooling performance of the thermoelectric cooler. Besides, the restricted maximum area of cooling devices, which depends on its application in electronic devices, is a significant constraint for TEC design. Performance optimization is vital to enhance the use of thermoelectric coolers in realworld applications. In this study, the objective is to maximize the cooling capacity of TEC with space restrictions. This paper presents a new approach by selecting electric current, length of ntype and ptype TE elements and crosssectional area of TE elements as design variables.
3.1 Optimization of cooling capacity of TEC
The singleobjective optimization problem for maximization of the cooling capacity of TEC is formulated mathematically as:
$\left\{\begin{array}{ll} & \text {Maximize } Q_{c} \\ \text {Subjectto} & \\ I_{\min } \leq I \leq I_{\max } \\ L_{\min } \leq L \leq L_{\max } \\ A_{\min } \leq A \leq A_{\max }\end{array}\right.$ (10)
Further, the total number of thermoelectric couples (N) is a dependent design variable. Its value depends on the crosssectional area of ntype and ptype thermoelectric elements and computed using Eq. (11).
$N=\frac{\text {Available area}(S) \text { of } T E C \times \text { packaging density}}{2 \times A}$ (11)
The optimization problem, as mentioned in Eq. (10) has been solved using some specific values of parameters. Table 1 lists the values of the parameters and properties used in this work.
Table 1. Values of parameters and properties
Description 
Parameter 
Value 
Cold surface temperature Ambient temperature Heat sink thermal resistance Electrical contact resistance Available C.S. area of TEC Packaging density Ceramic thermal conductivity Copper thermal conductivity 
T_{co} T_{a} R_{hs} r_{c} S PD k_{cr} k_{cu} 
293.15 K or 20℃ 298.15 K or 25℃ 0.10 ℃ /W 1 x 10^{8} Ω m^{2} 15 mm x 15 mm 80% 35.3 W/m°C 386 W/m°C 
Genetic algorithm (GA) is an evolutionary algorithm based on natural genetics. The genetic algorithm begins with the creation of a population of possible solutions (called individuals). Based on the value of the objective function, each member of the population is assigned a fitness value. To evolve better solutions, new generations are created by undergoing selection, recombination, and mutation of solutions. The fitness of the new generation is evaluated. This cycle is repeated over generations until the stopping criterion is met. The objective of GA is to search for an appropriate solution for the design problems. This involves maximization or minimization of the objective function.
Genetic algorithm is a populationbased optimization approach to find optimal or nearoptimal solutions. In terms of quality and robustness of solutions, GA's capability has been widely recognized for providing excellent results on classic discrete and continuous optimization problems. The genetic algorithm's performance depends on many genetic parameters such as population size, crossover, and mutation rate. GA parameters play an important role, and a different combination of parameters may lead to a significant GA performance change. The smaller population size helps faster convergence than larger population sizes. The decision on various GA parameters and operators are usually selected based on recommendations made by GA researchers.
The realvariable GA employing SBX operator created by Deb and Agarwal is used in this study [31]. Table 2 lists the values of the GA parameters like population size, crossover, mutation & number of generations that are used in the present study. The results are reported after multiple runs of GA converged to the same best solution.
Table 2. Values of GA parameters
Parameter 
Value 
Population size Crossover probability Mutation probability Number of generations 
50 0.80 0.25 1000 
3.2 Optimization of Coefficient of Performance (COP) of TEC
The objective of the second optimization problem is the maximization of the coefficient of performance of TEC. The design variables are the same as those selected in the previous problem. The fixed values of the parameters and properties are identical to the values used in the previous problem and described in Table 1. The thicknesses of ceramic substrates and copper tabs have the same values of 0.2 mm and 0.1 mm, respectively. This new problem is mathematically expressed as:
$\left\{\begin{array}{l}\text { Maximize COP } \\ \text { Subjectto } \\ \qquad \begin{array}{l}I_{\min } \leq I \leq I_{\max } \\ L_{\min } \leq L \leq L_{\max } \\ A_{\min } \leq A \leq A_{\max }\end{array}\end{array}\right.$ (12)
The goal of this optimization problem is to find the design variables within the variable bounds that result in the maximum COP of the device.
3.3 Optimization procedure
To apply the genetic algorithm to the optimization problems described in Eq. (10) and Eq. (12), the fitness evaluation of solution vectors is required. However, the procedure for evaluating fitness function is slightly tricky for this problem. The unknown values of T_{h} and T_{c }are initially guessed for approximately estimate Q_{c} and Q_{h}_{. }The initial guess for T_{h} and T_{c} satisfies TEC's prevailing temperature conditions, i.e., T_{h }>T_{a} and T_{c }<T_{co}. In principle, these conditions must be satisfied. The initial guess will be iteratively modified and reach the exact value. Eq. (1) and Eq. (2) are used to calculate new values of T_{h} and T_{c }that are termed as T_{hn} and T_{cn}. These are updated repeatedly to corresponding new values until the difference in old values and new values are negligible. Then the values of Q_{c} and Q_{h} are accepted.
A flowchart for GA implementation for these two optimization problems are given in Figure 3.
The brief steps of the fitness evaluation procedure for a population individual (solution vector) followed in this work are described below.
Figure 3. Flowchart for GA implementation
In the first segment of present work, cooling capacity Q_{c}, the first performance index of TEC is maximized. The algorithm of this study is coded in C language. The GA source code is developed by Deb and used in this work [32]. Multiple runs of 1000 generations have been repeated, and the best run is reported in Table 3 on which algorithm converged several times during various runs.
Table 3. Result of GA based optimization for maximum Q_{c}
Optimized 
Optimal Values of Design Variables 

Q_{c} 
I 
L 
A 
N 




(Dependent) 
8.476807 W 
2.993 A 
1.0 mm 
1.607 mm^{2} 
56 
At optimal values of design variables, the corresponding values of T_{h} and T_{c} are found at 28.59℃ and 19.78℃, respectively. The hot surface temperature (T_{ho}) of TEC is 27.84℃. The heat rejection rate (Q_{h}) at the hot side is 28.401 W. For the maximized Q_{c}, the value of COP obtained is 0.425. It can be observed that L is hitting lower bound while other parameters have optimal values without hitting any bound of the permitted range.
To optimize the second performance index of TEC, the coefficient of performance (COP) is maximized. The boundary conditions and assumptions are similar to those considered during the optimization of Q_{c}. This optimization problem is solved using the same parameters of GA, as mentioned in Table 2. The steps to implement GA in this problem are similar to those used in the optimization of cooling capacity and shown with the help of a flowchart in Figure 3. Several runs of 1000 generations have been performed to reach solutions with the highest quality, and the best run is reported in Table 4. It is worth mentioning that GA converged to the same results in various runs.
Table 4. Result of GA optimization for maximum COP
Optimized 
Optimal Values of Design Variables 

COP 
I 
L 
A 
N 




(Dependent) 
4.11 
0.283A 
2.0mm 
1.956mm^{2} 
45 
With this maximum COP, the corresponding Q_{c} is obtained as 0.745992 W. The corresponding values of T_{h} and T_{c} are 25.11℃ and 19.97℃, respectively. The hot surface temperature (T_{ho}) of the thermoelectric cooler is 25.09℃. The heat rejection rate (Q_{h}) at the hot side is 0.927 W. The optimal values of I and A design variables are unique, while the optimal value of L is hitting the upper boundary. It can be seen that COP increased significantly, and cooling capacity is just 8.8% of Max. Q_{c} obtained, as mentioned in Table 4. It is found that a design variable L hits its lower bound for high Q_{c} while for high COP, L hits its upper bound.
From these two results, it is well established that maximization of Q_{c} and maximization of COP are obtained at a different set of design parameters. Also, maximum Q_{c }does not ensure providing optimal COP and viceversa. This means that these objectives are conflicting. The resolution of these conflicting design objectives will be Pareto solutions through multiobjective optimization if there is no specific objective interest. It will be useful to determine a set of solutions that will allow the decisionmaker to choose among them according to the application's requirement.
The results of this study show that it is possible to improve the cooling capacity or COP of the thermoelectric coolers with these design variables to be competitive with compressorbased cooling devices. The complex impacts of electrical contact resistance and thermal resistance deteriorate the TEC performance. These factors always need to be included in the model for optimization and analysis.
Finiteelement simulation is a computational method for solving complex engineering problems of the realworld. The finite element simulations are performed to validate the optimization results of GA. ANSYS^{®} is a useful, commonpurpose finiteelement method tool. It is used to solve a broad range of engineering problems numerically. Hence ANSYS^{® }is used in the current study. The Thermalelectric module of ANSYS^{®} is capable of providing simultaneous solutions of thermal and electrical fields. The present work makes use of the thermalelectric module for the steadystate analysis of the TEC model. A threedimensional nonlinear finiteelement model is setup. The model in this work is set up with one pair of ntype and ptype elements as per the GA result. A new approach to incorporate the effect of electric contact resistance on the performance of TEC is used in the present study. The finiteelement simulation includes four additional geometric parts termed as ‘Contact’ and used for modelling of the electric contact resistance effect. These parts have material properties as per the thermoelectric behaviour of electrical contact resistance. The contact geometries are created at each end of the TE elements. The complete schematic of the TEC model for Finiteelement simulation to validate GA results is shown in Figure 4.
Figure 4. Schematic of TEC for finiteelement simulation to validate GA results
5.1 Finiteelement simulation for maximum Q_{c}
To validate GA predictions for maximum Q_{c}, the length of ntype and ptype elements is taken as 1.0 mm, as reported in Table 3. The TE elements are of the square crosssection of 1.27 mm. The distance between ntype and ptype elements is 0.31 mm. The material properties for the simulation are computed at average (T_{ave}) of T_{h} and T_{c} values obtained during the GA based optimization of Q_{c}. The finiteelement simulation input parameters of the modelled TEC are given in Table 5.
To model adiabatic heat transfer from the exposed surfaces of TEC, a small convection loss of 0.000001 W/mK was applied on all surfaces except the ones on which boundary conditions mentioned in Table 5 are specified. The computationally generated mesh, electric voltage, and temperature distribution across the finiteelement model of the thermoelectric cooler are shown in Figure 5.
Table 5. Finiteelement simulation input parameters for maximum Q_{c}
Description 
Parameter 
Value per pair of TE Elements 
Cooling Capacity Current Temperature (hot side of TEC) 
Q_{c} I T_{ho} 
0.1514 W 2.993 A 27.84℃ 
Figure 5. (a) Mesh (b) Voltage distribution (c) Temperature distribution in the finiteelement model for maximum Q_{C}
The parameters obtained from finiteelement simulation are compared with the GA results and reported in Table 6. It is observed that the results for a single pair of TE elements from GA simulation and those obtained from finiteelement simulation are in close agreement. The finiteelement simulation result represents a 3D solution based on a numerical technique, while GA results are based on 1D analytical equations. Hence, the optimization result obtained by GA is verified through the solutions of the thermalelectric module of ANSYS^{®}.
Table 6. Comparison of results for maximum Q_{c}
Parameter 
GA 
ANSYS^{®} 
Remarks 
T_{co} Q_{h} P 
20℃ 0.507 W 0.356 W 
20.25℃ 0.508 W 0.357 W 
Value per pair of TE elements 
In this segment, the finiteelement simulation for maximum COP is performed with ANSYS^{® }software. The steadystate TEC model consists of TE elements with 2.0 mm length, as reported in Table 4. The TE elements are of the square crosssection of 1.4 mm. The distance between ntype and ptype elements is 0.38 mm. The temperaturedependent material properties are calculated based on the average of T_{h,} and T_{c} found during GA based optimization of COP. The input parameters of the TEC model for finiteelement simulation are given in Table 7.
Table 7. Finiteelement simulation input parameters for maximum COP
Description 
Parameter 
Value per pair of TE Elements 
Cooling Capacity Current Temperature (hot side of TEC) 
Q_{c} I T_{ho} 
0.0165 W 0.283 A 25.09℃ 
The threedimensional steadystate TEC model is created, and predictions of GA based optimization are tested for maximum COP. For this simulation, the mesh, electric voltage, and temperature distribution are shown in Figure 6. The finiteelement simulation results agree well with the GA results. The parameters for GA and finiteelement simulation results have been compared and reported in Table 8.
Figure 6. (a) Mesh (b) Voltage distribution (c) Temperature distribution in the finiteelement model for maximum COP
The ANSYS^{® }result is consistent with GA based optimization results for the maximization of COP. Hence, the optimization result is verified through the solutions of the thermalelectric module of ANSYS^{®}.
Table 8. Comparison of results for maximum COP
Parameter 
GA 
ANSYS^{®} 
Remarks 
T_{co} Q_{h} P 
20℃ 0.021 W 0.004 W 
19.61℃ 0.021 W 0.004 W 
Value per pair of TE elements 
This paper presents an effective method with a new analytical model to improve cooling capacity and coefficient of performance of thermoelectric cooler for a specific need. In order to analyze more than one factor simultaneously, the thermoelectric cooler's current and geometric parameters were set to be variables. The described study emphasized to find out the optimal values of current, length of ntype and ptype TE elements and crosssectional area of TE elements within size restrictions on space. It was found that length, the crosssectional area of thermoelectric elements, and input electric current had a great influence on the TEC performance. Performance optimizations to maximize cooling capacity and to maximize COP were successfully performed by the genetic algorithm. The use of this stochastic optimization algorithm based on natural genetics theory proved to be the right option. The genetic algorithm successfully converged to the same optimal results over several runs. The finiteelement simulations through ANSYS^{®} validated the GA result.
The work suggests that these design variables should be appropriately selected in practical application. Results revealed that the relationship between the coefficient of performance and cooling capacity is inverse. The maximum cooling capacity does not provide optimum COP and viceversa. The smaller length of thermoelectric elements facilitates maximum cooling capacity whereas greater length of elements obtains maximum coefficient of performance. The best performance requires specific values of electric current and crosssectional area of TE elements as per the objective requirements. The appropriate optimum results can be achieved for any space restriction. This study can guide the TEC designers working for some specific cooling targets. The use of microprocessorbased control of input power parameters to get an optimal cooling with the best possible COP under dynamic conditions needs to be explored.
A COP I k L N P PD Q_{c} Q_{h} R_{hs} R_{cr} R_{cu} S 
crosssectional area of TE elements, m^{2} coefficient of performance electric current, A thermal conductivity, W/m K length of thermoelectric element, m number of thermoelectric couples power input, W packaging density heat absorption rate at the cold side, W heat rejection rate from the hot side, W thermal resistance of heat sink, ℃/W thermal resistance of ceramic, ℃/W thermal resistance of copper, ℃/W available crosssectional area of TEC, m^{2} 
T_{c} T_{h} T_{co} T_{ho} T_{a} T_{ave} Z 
temperature at the cold side of elements, K temperature at the hot side of elements, K temperature at the cold surface of TEC, K temperature at the hot surface of TEC, K ambient temperature, K average of T_{c }and T_{h}, K figure of merit, 1/K 
Greek symbols 

a ρ 
Seebeck coefficient, V/K electrical resistivity, Ω m 
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