COMPARATIVE ANALYSIS OF MEANS TO CONTROL THE THERMAL REGIME OF A COOLING THERMOELEMENT WHILE MINIMIZING THE SET OF BASIC PARAMETERS

V l a d i m i r Z a y k o v PhD, Head of Sector Research Institute «STORM» Tereshkovoi str., 27, Odessa, Ukraine, 65076 V l a d i m i r M e s c h e r y a k o v Doctor of Technical Sciences, Professor, Head of Department Department of Informatics Odessa State Environmental University Lvivska str., 15, Odessa, Ukraine, 65016 Y u r i i Z h u r a v l o v Corresponding author PhD, Associate Professor Department of Technology of Materials and Ship Repair National University «Odessa Maritime Academy» Didrikhsona str., 8, Odessa, Ukraine, 65029 E-mail: ivanovich1zh@gmail.com This paper reports a comparative analysis of the thermal regime control means while minimizing a set of basic parameters in various combinations with the indicators of reliability and dynamics of the functioning of a single-stage thermoelectric cooler. The connection has been established between the optimal relative operating current corresponding to the minimum of the set on the relative temperature difference and heat sink capacity of the radiator. The results of calculating the main parameters, reliability indicators, time of entering the stationary mode of operation for various current modes of operation at a fixed temperature diffe rence, thermal load at different geometry of the branches of thermoelements are given. A comparative analysis of the main parameters, indicators of the reliability and operational dynamics of a single-stage cooler under various characteristic current modes of operation has been carried out. Minimizing the set of basic parameters in conjunction with the reliability indicators and operational dynamics of the cooling thermoelement provides a decrease in the refrigeration coefficient up to 40 % compared to the maximum cooling capacity mode, as well as the optimal heat sink capacity of the radiator, the amount of energy expended, the time of entering the stationary mode, the relative intensity of failures. The analysis of the influence of the temperature difference at a predefined thermal load on the relative operating current, the time it takes for the cooler to enter the stationary thermal regime, the heat sink capacity of the radiator, the relative intensity of failures has been performed. The devised method of optimal control over the thermal regime of a single-stage thermoelectric cooler based on minimizing the set of basic parameters makes it possible to search for and select compromise solutions, taking into consideration the weight of each of the limiting factors


Introduction
Strict requirements for the weight and size characteristics and failure rates of heat-loaded onboard equipment for ensuring the thermal regimes of heat-loaded electronic equipment make the use of thermoelectric coolers alternative-free. The inclusion of a thermoelectric cooler in the negative feedback circuit of the control system requires an increase in the dynamic characteristics of the cooler, which fundamentally contradicts the reliability indicators. The relevance of scientific problems is due to the need to find such a current mode of the thermoelectric cooler, which contributes to finding a compromise between interrelated dynamic characteristics and reliability indicators. The practical significance of the results of such studies is to minimize the weight and size indicators and energy consumption of onboard equipment.

Literature review and problem statement
Paper [1] reports the conditions for the functioning of heat-loaded elements of electronic equipment and the means of ensuring their thermal regimes. However, the issues of improving the reliability of thermoelectric systems for ensuring thermal regimes have not been highlighted, although this parameter is decisive under the pulse-periodic mode for heat-loaded elements [2]. The influence of the load on the reliability indicators of the operating range of temperature differences and operating currents is described in [3] but the issues of the impact of the design parameters of the cooler remained unresolved. The connection of the energy interaction of the heat-releasing object with the current modes of the thermoelectric cooler is considered in work [4]. The influence of structural parameters on the reliability indicators of the thermoelectric cooler in harsh operating conditions required studies aimed at improving operational reliability [5]. At the same time, the cited studies are limited to the static modes of operation of thermoelectric coolers although it is known that the rate of change in the temperature field adversely affects the reliability of the contact connection of the thermoelement with the electrode [6]. The dynamic characteristics of thermoelectric coolers were not given sufficient attention because according to these features, they are significantly superior to air and compression systems for ensuring thermal regimes [7]. Active systems for ensuring thermal regimes involve the inclusion of a thermoelectric cooler in the feedback circuit, which significantly increases the importance of the dynamic characteristics of the cooler [8]. The relationship of dynamic characteristics with reliability indicators is a fundamental problem [9], so further research is aimed at studying the influence of energy indicators and structural parameters. Changes in design parameters for the purpose of more effective removal of heat flow [10] and search for ways to increase operational reliability are considered. However, control issues related to the complex influence of the current modes of operation of the thermoelectric cooler, structural parameters in the operating temperature range remained unsolved. At the same time, the expansion of the scope of use of thermoelectric coolers expands the requirements for their control [11][12][13]. The choice of a set of interrelated parameters for controlling the thermoelectric cooler is an important task, resolving which could solve the problem of controlling the thermoelectric system for ensuring the thermal regimes of heat-loaded elements. The set of parameters should provide a compromise between the required dynamic characteristics, permissible reliability indicators of the thermoelectric cooler.

The aim and objectives of the study
The aim of this work is to identify the analytical connection of the set of basic parameters with the optimal thermal regime of a single-stage thermoelectric cooler. This will make it possible to minimize the weight and size characteristics of onboard systems for ensuring thermal regimes of heat-loaded electronic equipment.
To accomplish the aim, the following tasks have been set: -to devise a model of a thermoelectric cooler in relation to a set of basic parameters; -to analyze the dynamic characteristics and reliability indicators for the main current modes of thermoelectric coolers.

The study materials and methods
To build and analyze a mathematical model of the thermoelectric cooling device, we applied methods of thermophysical modeling of dynamic systems for ensuring thermal regimes of electronic equipment. The construction of the model is based on the law of conservation of energy; simpli fications assume the homogeneity of the material of thermoelements and the identity of their geometric and thermophysical characteristics; the limitations are to neglect the distortions of the thermal field at the boundaries of removable electrodes [14]. Procedures of model calculations whose correctness is confirmed by the results of experimental studies when performing research and experimental design work are given in [14]. The connection of the dy-namic characteristics of the thermoelectric cooler with the geometry and material of thermoelements, the structural technological features of the cooler, the energy performance indicators in the range of operating temperatures and loads is presented in [15].

5.
The results of studying means to control the thermal mode of cooling thermoelement

1. Thermoelectric cooler model
Among the main parameters of the thermoelectric cooler (TEC), providing a predefined thermal mode of operation, are the structural, energy, operational, and dynamic ones. In particular, the number of thermoelements n, the value of the operating current I, the relative failure rate λ/λ 0 , the probability of trouble-free operation P, the time to enter the stationary mode of operation τ, the heat sink capacity of the radiator αF. This determines the set of interrelated basic parameters that affect the thermal mode of TEC operation. In the rational design of the TEC regime, one should strive to reduce n, I, λ, τ, αF, and increase P, which are interrelated. Therefore, by varying the main parameters (n, I, λ, τ, αF) and minimizing their various combinations, it is necessary to conduct a comprehensive comparative assessment of the main characteristics of TEC. Based on the analysis, choose a mode of operation that would reveal which requirements are prevalent, taking into consideration the weight of each of the limiting factors and ease of control.
To calculate the main parameters, reliability indicators, and time of entering the stationary mode of operation for various current modes of operation, the following ratios from [14] are used. The number of thermoelements n of a singlestage TEC can be determined from the following ratio: where Q 0 is the thermal load value, W; I eT R is the maximal working current, A; e K is the average value of the thermoEMF coefficient of the thermoelement branch, at the end of the cooling process, B/K; R l S K K = ( ) σ is the electrical resistance of thermoelement branch, Ohm; l and S are, respectively, the height and cross-sectional area of the branch of a thermoelement; σ K is the average value of electrical conductivity of a thermoelement branch, S/cm, at the end of the cooling process; T K is the heat-absorbing junction temperature, K; B is the relative operating current at the end of the cooling process; I is the working current value, A; is the relative temperature difference; T is the heat-generating junction temperature K; ∆T T z max . = 0 5 0 2 is the maximal temperature difference, K; z is the average value of efficiency of initial thermoelectric materials in the module, 1/K. The power consumption W K by TEC can be determined from the following expression: Voltage drop U K : (3) The refrigeration coefficient E can be calculated: The relative value of the failure rate λ/λ 0 can be determined from the following expression given in [14]: is the relative heat load; K T is the significant coefficient of low temperatures.
The probability of TEC trouble-free operation P can be determined from the following expression: where t is the assigned resource, hour. The ratio for determining the time of entering the stationary mode of operation τ takes the following form given in [15]: where γ = ( )( ) is the maximum operating current at the beginning of the cooling process, A. Provided that the currents are equal at the beginning and end of the cooling process: Based on the results of the research on minimizing the sets of the main parameters in conjunction with the reliability indicators and dynamics of functioning, a series of current operating modes have been developed [12]. Fig. 1, positions 4-7 represent the dependences of the relative operating current B = f(Θ) on the relative temperature difference Θ for the developed current modes of operation (4) to (7).
Consider several additional current modes of operation of the cooling thermoelement to ensure the completeness of the comparative analysis of functioning for different geometry of We obtain from the condition dK dB = 0: The dependence of the optimal relative operating current B opt , corresponding to the minimum of the set on the relative temperature difference Θ, is presented in Fig. 1, position 2. b) Mode (nαF) min . Using ratio (1) and the expression to determine the heat sink capacity of the radiator: where then expression (12) can be written as: then the nαF set can be represented as: We obtain from the condition dK dB = 0: The dependence of the optimal relative operating current B opt , corresponding to the minimum of the set (nαF) min , on the relative temperature difference Θ, is presented in Fig. 1, position 3. c) Mode Q 0 = 0. Using ratio (1), one can write: The dependence of the optimal relative operating current B opt , corresponding to the mode Q 0 = 0, on the relative temperature difference Θ, is presented in Fig. 1, position 9.

2. Analysis of the thermoelectric cooler model relative to the set of basic parameters
The results of calculations of the main parameters, reliability indicators, time of entering the stationary mode of operation for various current modes of operation are given in Table 1. The data were obtained at a temperature difference ∆T = 40 K, the value of the thermal load Q 0 = 0.5 W, T = 300 K, T-T env = 5 K for different geometry of the branches of thermoelements (l/S ratio) l/S = 4.5; 10; 20; 40.  4 -mode (nI ) min ; 5 -mode (nIαF) min ; 6 -mode (nI λ/λ 0 τ) min ; 7 -mode (nI λ/λ 0 ) min ; 8 -mode λ min ; 9 -mode Q 0 = 0 With an increase in the relative operating current B at T = 300 K and a thermal load of Q 0 = 0.5 W and a temperature difference ∆T = 40 K for different geometry of the branches of thermoelements (l/S ratio): -the number of thermoelements n decreases (Fig. 2). The growth of the l/S ratio increases the number of thermoelements n at a predefined relative operating current B. The minimum number of thermoelements n min is provided under the Q 0max mode; n, pcs. 1 -mode Q 0max : 2 -mode (n 2 I ) min ; 3 -mode (nαF) min ; 4 -mode (nI ) min ; 5 -mode (nIαF) min ; 6 -mode (nI λ/λ 0 τ) min ; 7 -mode (nI λ/λ 0 ) min ; 8 -mode λ min -the functional dependence of the refrigeration coefficient E = f(B) has a maximum at B = 0.53 under the (nIλ/λ 0 τ) min mode (Fig. 3). The refrigeration coefficient E does not depend on the geometry of the branches of the thermoelements (l/S ratio). The minimum refrigeration coefficient E min is provided under the (nIλ/λ 0 τ) min mode; -the value of working current I increases (Fig. 4). At a predefined value of the relative operating current B, with an increase in the l/S ratio, the value of the operating current I decreases. The maximum operating current is provided under the Q 0max mode, and the minimum -under the λ min mode;   Continuation of Table 1 -the functional dependence of the voltage drop U = f(B) has a minimum at B = 0.71 under the (nI) min mode (Fig. 5). With the increase in the l/S ratio, the value of the voltage drop U increases at a predefined relative operating current B. Maximum voltage drops U max are provided under the λ min mode; -the functional dependence αF = f(B) has a minimum at B = 0.53 under the (nIλ/λ 0 τ) min mode (Fig. 6). The heat sink capacity of the radiator αF does not depend on the geometry of the branches of the thermoelements (l/S ratio);   6. Dependence of the heat sink capacity of the radiator αF of a single-stage thermoelectric cooler on the relative operating current B at T = 300 K, Q 0 = 0.5 W, ∆T = 40 K, T-T env = 5 K: 1 -mode Q 0max ; 2 -mode (n 2 I ) min ; 3 -mode (nαF) min ; 4 -mode (nI ) min ; 5 -mode (nIαF ) min ; 6 -mode (nI λ/λ 0 τ) min ; 7 -mode (nI λ/λ 0 ) min ; 8 -mode λ min -the relative value of the failure rate λ/λ 0 increases (Fig. 7). With the increase in the l/S ratio, the relative value of the failure intensity λ/λ 0 increases at a predefined relative operating current B.  at T = 300 K, Q 0 = 0.5 W, ∆T = 40 K: 1 -mode Q 0max ; 2 -mode (n 2 I ) min ; 3 -mode (nαF ) min ; 4 -mode (nI ) min ; 5 -mode (nIαF ) min ; 6 -mode (nI λ/λ 0 τ) min ; 7 -mode (nI λ/λ 0 ) min ; 8 -mode λ min The minimum relative failure rate λ/λ 0min is provided under the λ min mode. The maximum relative intensity λ/λ 0 is provided under the mode Q 0max : -the probability of trouble-free operation P is reduced (Fig. 8). As the l/S ratio increases, the probability of trouble-free operation P decreases at predefined B. The maximum probability of trouble-free operation P max is provided under the mode λ min . The minimum probability of trouble-free operation P is provided under the mode Q 0max ; -the time of entering the stationary mode of operation τ is reduced (Fig. 9). With the growth of the l/S ratio, the time of entering the stationary mode of operation τ decreases. The minimum time for entering the stationary mode of operation τ min is provided under the mode Q 0max ; -the functional dependence of the amount of energy expended N = f (B) has a minimum at B = 0.71 under the mode nI ( ) min . With an increase in the ratio (l/S), the amount of energy spent N decreases (Fig. 10) at a fixed relative operating current B. The maximum amount of energy spent N max corresponds to the λ min mode.
The results of calculations of the main parameters, reliability indicators, time of entering the stationary mode of operation for various current modes of operation of a single-stage TEC at temperature drops from ∆T = 10 K to ∆T = 60 K, thermal load Q 0 = 0.5 W, T-T c = 5 K, l/S = 4.5 K are given in Table 2.  5 -mode (nIαF ) min ; 6 -mode (nI λ/λ 0 τ) min ; 7 -mode (nI λ/λ 0 ) min ; 8 -mode λ min Table 2 Results of calculations of the main parameters, reliability indicators, time of entering the stationary mode of operation With the increase in temperature difference ∆T for different current operating modes at Q 0 = 0.5 W, l/S = 4.5: -the relative operating current B (Fig. 11) increases except for the Q 0max mode (B = 1). With a fixed temperature difference ∆T, the relative operating current B increases from the λ min mode to the Q 0max mode; -the functional dependence of the number of thermoelements n = f(∆T) has a minimum at ∆T = 40 K except for the mode Q 0max , (nI) min , (nIαF) min (Fig. 12). With a fixed temperature difference ∆T, the number of thermoelements n decreases from the λ min mode to the Q 0max mode;  Fig. 11. Dependence of the relative operating current B of the single-stage thermoelectric cooler on the temperature difference ∆T for different current operating modes at T = 300 K, l/S = 4.5, Q 0 = 0.5 V: 1 -mode Q 0max ; 2 -mode (n 2 I ) min ; 3 -mode (nαF ) min ; 4 -mode (nI ) min ; 5 -mode (nIαF ) min ; 6 -mode (nI λ/λ 0 τ) min ; 7 -mode (nI λ/λ 0 ) min ; 8 -mode λ min n, pcs. Fig. 12. Dependence of the number of thermoelements n in a single-stage thermoelectric cooler on the temperature difference ∆T for different current modes of operation at T = 300 K, Q 0 = 0.5 W, l/S = 4.5: 1 -mode Q 0max ; 2 -mode (n 2 I ) min ; 3 -mode (nαF ) min ; 4 -mode (nI ) min ; 5 -mode (nI αF ) min ; 6 -mode (nI λ/λ 0 τ) min ; 7 -mode (nI λ/λ 0 ) min ; 8 -mode λ min -the value of the operating current I (Fig. 13) increases except for the Q 0max mode. Under the mode Q 0max , the value of the operating current I decreases; -the value of the operating current I decreases from the Q 0max mode to the λ min mode; -the refrigeration coefficient E decreases (Fig. 14). difference ∆T for different current operating modes at T = 300 K, Q 0 = 0.5 W: 1 -mode Q 0max ; 2 -mode (n 2 I ) min ; 3 -mode (nαF ) min ; 4 -mode (nI ) min ; 5 -mode (nI αF ) min ; 6 -mode (nI λ/λ 0 τ) min ; 7 -mode (nI λ/λ 0 ) min ; 8 -mode λ min The maximum refrigeration coefficient E max is provided under the mode (nIλ/λ 0 τ) min : -the time of entering the stationary mode of operation τ increases (Fig. 15).

Discussion of results of analyzing the use of basic parameters
Our results are explained, first of all, by the revealed analytical connection between the design parameters of the thermoelectric cooler and the operational indicator of the failure intensity, current operating modes, and the probability of trouble-free operation indicator, entering the steady mode, and the operating temperature difference. Studying analytical functions is much simpler than experimental dependences and makes it possible, already at the design stage, to identify their extreme values (Fig. 6, 10, 13, 19). The nature of changes in the weight and size indicators of the system for ensuring thermal regimes (Fig. 6), the influence of energy indicators on the intensity of failures (Fig. 7), temperature differences on the time of entering the stationary mode (Fig. 15) makes it possible to identify compromise design solutions.
To resolve the issue of ensuring the specified reliability when controlling the cooling capacity of the thermoelectric cooler, a comparative analysis of the main parameters, the indicators of reliability and dynamics of the functioning of a single-stage TEC under various characteristic current modes of operation was carried out. The obtained results showed the possibility of optimal control over the thermal mode of operation by selecting the thermal mode of operation, taking into consideration the weight of each of the limiting factors. A distinctive feature of using a set of basic parameters instead of controlling each of them is to increase the response speed of the system with TEC in the feedback chain and optimize energy indicators. Thus, the minimization of the set in conjunction with the reliability indicators and dynamics of the functioning of the thermoelement provides: -the maximum refrigeration coefficient E max and the minimum heat sink capacity of the radiator αF min under the mode (nIλ/λ 0 τ) min and does not depend on the geometry of the branches of thermoelements; -the minimum amount of energy N min expended and the minimum voltage drop U min under the mode (nI) min ; -the minimum time of entering the stationary mode of operation τ min and the minimum number of thermoelements n min under the mode Q 0max ; -the minimum operating current value I min , the minimum relative failure rate (λ/λ 0 ) min , and, therefore, the maximum probability of trouble-free operation P max and the maximum drop voltage U max under the mode λ min .
Our study of operational limitations showed that with an increase in temperature difference ∆T at thermal load Q 0 = 0.5 W and l/S = 4.5: -the relative operating current B, the value of the operating current I, the amount of energy spent N, the heat sink capacity of the radiator αF are increased; -there is an increase in the time of entering the stationary mode of operation τ, the relative failure rate λ/λ 0 , and the number of thermoelements n for the modes Q 0max , (n 2 I) min , (nαF) min ; -there is a decrease in the refrigeration coefficient E, the probability of trouble-free operation P, the cooling capacity Q 0 at a predefined number of thermoelements n under the mode Q 0max , (n 2 I) min , (nαF) min .
The practical significance of the present research is to optimize control over thermoelectric coolers in the system for providing heat-loaded elements. This is achieved due to the fact that for the operating modes (nI) min , (nIαF) min , (nIλ/λ 0 τ) min , (nIλ/λ 0 ) min , λ min the functional dependence Q 0 = f(Θ) has a maximum for different Θ, and the dependence n = f(∆T) has a minimum for different ∆T. The devised method of optimal control over the thermal regime of a single-stage TEC based on minimizing the set of basic parameters makes it possible to search for and select compromise solutions, taking into consideration the weight of each limiting factor.
The structural limitations of the use of the proposed model have been analyzed, which showed that with an increase in the l/S ratio at a predefined temperature difference of ∆T = 40 K and a thermal load of Q 0 = 0.5 W: -there is an increase in the number of thermoelements n, voltage drop U, which leads to an increase in the relative failure rate λ/λ 0 ; -the value of the operating current I decreases, and, as a result, the probability of trouble-free operation P.
Further research could address minimizing the failure rates of the thermoelectric cooler in the operating range of temperature changes and changing operating temperature conditions.

Conclusions
1. A mathematical model of the connection of the set of basic parameters with the structural, energy, time, and reliability indicators of a single-stage thermoelectric cooler has been built, which provides a choice of optimized solutions depending on the design goals. A distinctive feature of the model is that the set of parameters includes physically heterogeneous values: structural, energy, operational, temporal, which, in combination, produce a new result -improving the quality of control.
2. The mathematical model was analyzed in the operating range of temperature differences ∆T from 0 to 60 K and the structural parameters of the geometry of thermoelements l/S = 4.5; 10; 20; 40. Extreme values for the refrigeration coefficient (Fig. 3), energy expended (Fig. 10), heat dissipation capacity of the radiator (Fig. 6), cooling capacity (Fig. 19) have been determined, providing for the possibility of optimal control over the thermal mode of operation. It is shown that the gain in the refrigeration coefficient under the mode (nIλ/λ 0 τ) min , in comparison with the maximum cooling capacity regime, amounts to 1.4 times.