THERMAL MANAGEMENT OF A SET OF THERMOELECTRIC COOLERS CONNECTED IN PARALLEL IN AN UNEVEN TEMPERATURE FIELD

A method of increasing the efficiency of thermoelectric system for providing thermal modes of distributed heat-loaded elements of an on-board information system is considered. It is shown that thermoelectric coolers are most susceptible to thermal effects. In accordance with the reliability model, they are connected in series with heat loaded elements and to a great extent determine their operability. The inclusion of thermoelectric coolers in the feedback loop of a thermal management system places increased demands on the dynamic performance of the coolers. However, dynamics and reliability performance are


Introduction
Modern information systems are based on components with a high concentration of thermal emission per unit area. This is particularly significant for on-board systems where weight, size and energy consumption are critical. Heat-intensive components cannot function without thermal management systems, which are an essential component of information systems. One of the most promising ways of thermal management of electronic components is thermoelectric cooling, which is the most effective over a wide range of operating temperatures. Thermoelectric cooling devices (TEC) allow controlling the value of heat flow by simply changing the operating current value. The main advantages of thermoelectric cooling are high reliability and small overall dimensions, easy operation and rapid response. These advantages are inherently a consequence of the solid-state nature of such coolers, with no moving parts, pumped liquids or gases. The design features of on-board equipment include the dispersed arrangement of heat loaded elements with varying dissipation capacity. Therefore, to ensure a given thermal regime of a number of dispersed thermally loaded and temperature-dependent elements, a group arrangement system of thermoelectric coolers, located on one heat sink and connected in electrical parallel, can be used. In this case it is important to use a number of standard voltages to power the complex and determine the optimum supply voltage, taking into account the limiting factors for mass-size, power, dynamic characteristics and reliability indicators in an uneven temperature field.
Literature Overview Thermal control systems for thermally loaded components are an essential component of modern on-board avionics [1]. Thermoelectric coolers are the most suitable for on-board systems in terms of mass-size and performance characteristics [2]. The main advantage of TEC over air and liquid cooling systems is the ease of control and high dynamic characteristics [3]. At the same time, toughening requirements to dynamics and reliability of thermally loaded equipment [4; 5], assumes their increase also for systems of providing thermal modes [6; 7]. In [8] influence of loading on reliability indices of thermoelectric coolers is investigated, however, influence of design parameters is not considered. In [9] research of influence of design parameters on reliability indicators of thermoelectric coolers is presented. In [10] a relationship between reliability indices and current operating modes of the cooler is analysed, which allowed to choose optimal operating conditions by this criterion. However, for controlling thermoelectric systems, apart from reliability indicators, dynamic characteristics are also important, the relationship between which was not considered in the cited sources [11]. It is known that dynamics unambiguously negatively affects reliability performance, which is a fundamental problem [12], in particular, linear thermal expansion of thermocouple and substrate materials leads to cracking of junction sites [13]. In [14] the relationship of dynamic performance with TEC design, in [15] with the number of thermocouples, in [16] with the current operation modes of the product, however, only for a single cascade cooler. Subsequent works [17; 18] analyzed the possibilities of optimizing thermoelectric control according to complex criteria, including both reliability and dynamics indicators. A relevant development in this direction is the control of thermoelectric coolers when they are connected in parallel in an uneven field. This is aimed at solving the problem of coordinating the reliability and dynamics indicators of operation, as applied to the management of critical systems for providing thermal modes of radio electronic equipment.
Purpose and objectives of the study The aim of the work is to develop a thermal management model for thermoelectric coolers connected in parallel and operating in an uneven temperature field.
In order to achieve this aim it is necessary to solve tasks: 1 To develop a mathematical model of thermoelectric cooler which connects energy, dynamic, reliability and structural parameters.
2 To analyses the developed model to identify optimal modes of thermoelectric cooler operation.

Development of a thermoelectric cooler model
We will use the relations from [19] to calculate the basic parameters, reliability indicators and dynamic characteristics of the TEC.
The voltage drop on the TEC can be determined from the relation: where n -number of thermocouples, pcs; The value of the operating current can be determined from the expression: The number of thermocouples n of a single stage TEC can be determined from the ratio: where Q 0 -is the heat load value, W. The refrigerating factor E can be calculated using the formula: The relative magnitude of the failure rate λ/λ 0 can be determined from the expression [20]: where C Q nI R = 0 2 max -the relative heat load; K T -the coefficient of reduced temperatures. The probability of no-failure operation P of the TEC can be determined from the expression: where t=10 4 hour -is the assigned resource.
The expression for the steady-state operation time τ can be represented as [20]: where J ; m C 0 0 -is the product of the mass and heat capacity of the cooling object. In our case m C 0 0 → 0 (no object); m C i i i ∑ -is the total value of the product of the heat capacity and mass of the constituent structural and technological elements at the heat absorbing junction of the module at a given l/S; index H denotes the starting point in time; index K is the finite point in time; R н -is the electrical resistance of the thermocouple branch at the start of the cooling process, Ohm.
Relative operating current at the start of the cooling process at τ=0: The expression for the relative operating current B can be obtained by substituting (1) into (3), followed by conversion: where A UI Q = max 2 0 -is a relative value depending on the voltage drop U, the heat load value Q 0 , the cooling level temperature T 0 and the geometry of the thermocouple branches (l/S ratio). For a cooling system consisting of M independent elements (TEC), the probability of failure of the i-th element is P i (t), then the total probability of system failure is [19]: Calculations of basic parameters, reliability indices and dynamic characteristics of the complex TEU with parallel electric connection in an uneven temperature field have been carried out. Results have been received for temperatures from T 0 = 295K to T 0 = 250K at various thermal loading from Q 0 = 0.5 W to Q 0 = 15 W, standard values of a supply voltage from U = 6.0 V to U = 24 V, the set geometry of branches of thermoelements l/S, T=300K and are presented in the form of graphic dependences.

Model analysis
As the supply voltage U of the TEC complex increases with total heat load Q 0Σ =35 W in an irregular temperature field: -the value A UI Q = max 2 0 (Fig. 1) for different thermocouple branch geometry l/s increases. As the l/s ratio increases, the value A decreases for a given supply voltage U: -the number of thermocouples n increases (Fig. 2  p. 1); -the total operating current I Σ decreases (Fig. 2  item 2); -the refrigerating factor E decreases (Fig. 2 item 3); -the amount of consumed energy N increases ( Fig. 3 item 1); -the required heat dissipation capacity of the heat sink increases αF ( fig. 3 item 2); -the ramp-up time τ increases (Fig. 3 item 3); -the relative failure rate λ/λ 0 increases ( Fig. 4 item 1); -the probability of no-failure operation P decreases ( Fig. 4 item 2).
The analysis of results of investigation of basic parameters of the complex TEC at application of standard voltages U has shown the necessity of application of current operation modes close to the mode Q 0 =0 (b<0). This leads to an increase in the number of thermocouples n, dimensions and mass of the TEC complex even when using the geometry of thermocouple branches (ratio l/s = 4.5). Therefore, in the following we will consider the possibility of applying the characteristic current modes of operation of the TEC complex for different geometry of thermocouple branches (ratio l/s = 4.5, 10,20) for the values of supply voltage obtained by calculation.
Calculations of basic parameters, dynamic characteristics and reliability indicators of TEС complex, consisting of 6 elements of radio-electronic equipment were performed. Used power dissipation power from Q 0 =0.5W to Q0=15W, different level of tem- perature cooling from T 0 =295K to T 0 =250K, different characteristic current operation modes, different geometry of thermoelement branches (ratio l/s =4. 5,10,20). Graphical dependencies were plotted and the analysis was carried out.
As the average operating current B of the TEC complex increases for different thermocouple branch geometries (l/s ratio) and characteristic current operating modes: -the value of the operating current I increases (Fig. 5). As the l/s ratio increases, the operating current I decreases with a fixed relative operating current B (for the different characteristic current operating modes); -the number of thermocouples n decreases (Fig. 6). As the l/s ratio increases, the number of thermocouples n increases with a fixed relative operating current B; -the functional dependence of voltage drop U=f(B) on the relative operating current B has a minimum at B = 0.52 (mode (nI) min for different geometry of thermocouple branches (l/s ratio) (Fig. 7). As the l/s ratio increases, the voltage drop U increases at a fixed relative operating current B; -the functional dependence of the refrigerant E=f(B) on the relative operating current B has a maximum at B = 0.32 for the current mode (nI λ⁄λ 0 τ) min and is independent of the geometry of the thermocouple branches (l/s ratio) (Fig. 8); -the functional dependence of the heat dissipation capacity of the heat sink αF = (B) on the relative operating current B has a minimum at B = 0.32 in the mode (nI λ⁄λ 0 τ) min and is independent of the geometry of the thermocouple branches (l/s ratio) (Fig. 9); -the steady-state time τ decreases (Fig. 10). As the l/s ratio increases, the steady-state time τ decreases at a fixed relative operating current B. The minimum steady-state time τ min is achieved at Q 0max ; -the functional dependence of the amount of spent energy N=f(B) on the relative operating current B has a minimum at B = 0.52 in (nI) min mode (Fig. 11). As the l/s ratio increases, the amount of energy expended N decreases at a fixed relative operating current B; -the relative failure rate λ⁄λ 0 increases (Fig. 12). As the l/s ratio increases, the relative failure rate λ⁄λ 0 at a fixed relative operating current B; -the probability of failure-free operation P decreases (Fig. 13). As the ratio l/s increases, the probability of failure P decreases for a fixed relative operating current B.
Discussion of the results of the analysis For clarity and ease of comparative analysis, all calculated data are shown in Table 1.
When selecting nominal supply voltage for the complex with parallel electric connection of TEC it is necessary to take into account the limiting requirements: for operating current I, number of thermoe- lements n, cooling factor E, power consumption W, and, therefore, dimensions and weight of heat sink αF, failure rate λ⁄λ 0 and dynamics of operation τ. At the same time, it is necessary to estimate the weighting of each of the limiting factors and choose an acceptable variant of the complex design. The proposed approach allows a rational design of the TEC complex with selection of the most acceptable variants.
Conclusions A model of thermal mode support system based on a set of thermoelectric coolers with parallel electric connection has been developed to control the thermal mode of a number of temperature-dependent elements of radio electronic equipment. The model is designed to operate with different power dissipation in an uneven temperature field for different supply voltages and thermocouple vertex geometries.
Comparative analysis of basic parameters, reliability indices and dynamic characteristics of thermoelectric cooler complex for different supply voltages has been carried out.
Analysis of the results has shown the possibility of selecting the supply voltage rating taking into account the limitations on mass-size, energy, dynamic and reliability characteristics for different geometries of thermocouple vertebrae.