ANALYSIS OF DYNAMICS AND PREDICTION OF RELIABILITY INDICATORS OF A COOLING THERMOELEMENT WITH THE PREDEFINED GEOMETRY OF BRANCHES

We have investigated the influence of structural and technological elements on the basic parameters, reliability indicators, and the dynamics of operation of thermoelectric cooling devices under various current modes within the operating range of temperature differences. We analyzed the ratios of correlation between the time required to enter a stationary mode and relative intensity of failures in a cooler, and energy indicators, thermoelectric parameters of thermoelements, structural and technological indicators. An analysis of the time required to enter a stationary mode was performed for different modes of operation from the maximum cooling capacity to the minimum failure rate. It is shown that in order to reduce the time required for a cooler to enter a stationary mode, at the predefined geometry of thermoelements and temperature difference, it is necessary to employ the mode of maximum cooling capacity. The quantitative analysis showed that at the predefined geometry of thermoelements branches the time required to enter a stationary working mode does not depend on the number of thermoelements in a thermoelectric cooler. At a difference of temperatures close to the maximum value, the time required to enter a stationary working mode differs slightly for all modes of operation. Comparative analysis of the basic parameters of reliability indicators and dynamical characteristics makes it possible to find compromise solutions when constructing thermoelectric devices taking into consideration the weight of each of the constraints. From a practical point of view, the results obtained suggest that increasing the cooling rate does not require changes to the existing technology for making thermoelectric coolers. Control over performance speed during transition from one stationary state to another state is executed through the selection of current modes in the operation of a thermoelectric device. In this case, there is a possibility to choose the conditions under which reliability indicators match the permissible limit.


Introduction
When designing thermoelectric cooling devices (TED), one of the main requirements in some cases is to ensure the minimum time to enter a stationary working mode.Thus, the application of laser radiation sources with thermoelectric cooling in surgery is characterized by the pulse operating mode with a variation over a wide range of energy at coagulation and tissue cutting.Reliability indicators of the semiconductor laser directly depend on its temperature, that is, on performance speed and reliability of the system that ensures thermal regimes.The dynamic characteristics of TED are defined by the thermophysical properties of the material for thermoelements, for hot and cold electrodes, the mass of thermocouples, constraints for the permissible local temperature gradients that do not promote cracking of the material.However, the cooling rate is influenced by the structural and technological elements at the heat-absorbing weld joint, which are the necessary component of the cooler.To analyze their impact under various current operational modes of TED over the working range of temperatures for the assigned geometry of thermo-elements, it is required to have a dynamic model of the cooler with respect to the structural and technological elements at a heat-absorbing weld joint.Analysis of the model in terms of minimizing the dynamic parameters of a thermoelectric cooler and in relation to the indicators of reliability is of great interest for critical systems, for which these indicators are crucial.

Literature review and problem statement
Modern trends in the development of radio-electronic equipment are associated with a decrease in the elements' weight and dimension parameters, expanding of the temperature range, higher operating frequencies, resulting in the elevated local temperature gradients and decreased indicators of operational reliability [1].Thermally-loaded receiving and transmitting semiconductor elements cannot work without thermal systems that ensure thermal modes since the permissible temperature of their operation does not exceed 80 °С.The most promising, in terms of indicators for performance speed, reliability, weight, and dimensions are the thermoelectric cooling devices [2].Stricter requirements to operating conditions for the thermo-loaded equipment in critical systems call for the same requirements to the system that ensure their thermal regimes.
The pulse character of a thermal load requires maximally fast reaction from thermoelectric coolers, because an uncompensated temperature difference can cause a failure in the thermo-loaded element.Improving the performance speed of a thermoelectric cooling system contradicts the improvement of reliability indicators, because in this case there is a rise in the temperature gradients that reduce the reliability of the product.This is a fundamental problem that cannot be resolved by employing a single approach [3].
The task on improving the reliability indicators of thermoelectric coolers is addressed from various standpoints: through rational design [4], technology and structure of modules fabrication [5], selection of a material for thermoelements [6,7], protection of thermoelements against the effects of external medium [8], designs of thermoelement branches [9], influence of mechanical [10], heat loads [11].The result of such a multifactor approach is the improved resultant probability for a failure-free TED operation.However, stricter requirements to the performance speed of modern equipment necessitate the search for new ways to improve reliability indicators of thermoelectric coolers.
Studying the dynamic characteristics of TED has been addressed by fewer papers [12], which is related to the fact that in terms of performance speed the air and liquid cooling systems are considerably inferior to the thermoelectric ones.Operation under a switch mode is the most complicated because it creates the maximum temperature gradients at places where thermoelectric branches connect to electrodes.Due to the difference in the materials' thermal expansion coefficients, there form the chips and detachments, which reduces reliability indicators of the product.For this reason, the accelerated bench testing of thermoelectric modules is performed precisely under switch modes at which the products' reliability indicators deteriorate by orders of magnitude.
Construction of thermoelectric coolers, for whom the pulse load regime is the working one, is a relevant task, is aimed at ensuring the design of thermo-loaded radio electronic systems operating under extreme conditions of operation.One of the tasks of this problem is to analyze the relation between dynamic characteristics and indicators of reliability and the geometry of thermoelements' branches under various current modes.

The aim and objectives of the study
The aim of this study is to analyze the impact of structural and technological elements in the cooling device at the specified geometry of thermoelements' branches on duration of the transition process in conjunction with indicators of reliability for various current operation modes of TED, temperature differences, and a thermal load.
To accomplish the aim, the following tasks have been set: -to analyze the dynamical model of TED with respect to the components of the structural and technological elements of the cooler at the specified geometry of thermoelements; -to estimate the influence of dynamical characteristics of TED on reliability indicators for basic modes of operation.

Analysis of the dynamical model of TED with respect to structural and technological elements
The dynamical model under consideration is an analytic representation of actual thermoelectric cooler, where we take into consideration, at the heat-absorbing weld joint, mass m and heat capacity C of the structural and technological elements (STE) while entering a stationary mode of operation in a range of temperature differences DT from DT = 5 K to DT = 60 K for different current operation modes of TED.Paper [12] derived the ratios for determining the time required to enter a stationary working mode, taking into consideration the cooled object (m 0 С 0 ), and sufficiently enough addressed those STE that are taken into account when determining the dynamical characteristics of TED at l/S = 10 cm -1 .
Consider the case when m 0 С 0 →0 and the dynamics of TED is determined only by STE: where where I R H H max 2 is, respectively, the maximum operating current and the electrical resistance of a thermoelement branch at the beginning of the cooling process at τ = 0; I maxK , R K are, respectively, the maximum operating current and the electrical resistance of a thermoelement branch at the end of the cooling process; B K = I/I maxK is the relative operating current at τ; B Н = I/I maxН is the relative operating current at τ = 0; I maxK = e K T 0 /R K is the maximum operating current at τ; I maxН = e Н T/R Н is the maximum operating current at τ = 0; e Н , e K is the thermo emf coefficient of a thermoelement branch, respectively, at the beginning and at the end of the cooling process, V/K; R Н , R K is the electrical resistance a thermoelement branch at the beginning and at the end of the cooling process, Ohm.
Under condition of equality of currents at the beginning and end of a cooling process: T 0 is the temperature of the heat-absorbing weld joint at the end of the cooling process, K; T is the temperature of the heat-absorbing weld joint at the beginning of the cooling process, K, τ = 0; Θ = DT/T max is the relative temperature difference; DT = T-T 0 is the temperature difference in TED, K; DT z T K max .= 0 5 0 2 is the maximum temperature difference, K; z K is the average value for the efficiency of a thermoelectric material in the module at the end of the cooling process, 1/K; I is the working current magnitude, A; K lS is the total magnitude of the product of heat capacity and the mass of components of CTE at the specified geometry of thermoelements branches l/S = 10 cm -1 .
The number n of elements can be determined from the ratio: where Q 0 is the thermal load magnitude, W.
Power consumption W K of TED can be determined from expression: Voltage drop U: Refrigeration coefficient E can be calculated from formula: Relative failure intensity λ/λ 0 and the probability of nonfailure operation P can be determined from expressions [13]: where λ 0 is the rated failure rate, 1/h, obtained experimentally by calculation for the generally accepted manufacturing technology; is the relative heat load; K T1 is the meaningful temperature coefficient.The expressions given above provide an opportunity to analyze the dynamics of TED and to predict reliability indicators at the predefined geometry of thermoelements' branches.

1. Mode Q 0max
Results of calculation of the basic parameters, taking into consideration the temperature dependence [2] of the time required to enter a stationary working mode and the reliability indicators for mode Q 0max (B K = 1.0;B Н = B K (I maxK /I maxН ) at T = 300 K, l/S = 10 cm -1 ; DT = 5; 10; 20; 30; 40; 50; 60 K; 175 10 4 J/K and at various thermal load Q 0 , are given in Table 1.

Table 1
Calculation of basic parameters under mode  An increase in the temperature difference DT at m 0 С 0 → 0, where С 0 is the heat capacity of the object, leads to: -an increase in the time required to enter a stationary working mode τ (Fig. 1, point 1) that does not depend on heat load Q 0 ; -a decrease in the magnitude of working current I (Fig. 2, point 1) that does not depend on heat load Q 0 ; -a decrease in refrigeration coefficient E (Fig. 3, point 1) that does not depend on heat load Q 0 ; -an increase in voltage drop U for different heat load Q 0 (Fig. 4); the voltage drop increases with an increase in the heat load; -a decrease in the number of thermoelements n for different heat load Q 0 (Fig. 5); -an increase in failure rate λ/λ 0 for different heat load Q 0 (Fig. 6); the failure rate increases with an increase in the heat load; -a decrease in failure-free operation probability P for different heat load Q 0 (Fig. 7); -the failure-free operation probability decreases with an increase in the heat load.175 10 4 J/K and at different heat load Q 0 are in Table 2.
An increase in temperature difference DT at m 0 С 0 → 0 leads to: -an increase in the time required to enter a stationary mode τ (Fig. 1, point 2) that does not depend on heat load Q 0 ; -a decrease in the magnitude of working current I (Fig. 2, point 2) that does not depend on heat load Q 0 ; -a decrease in refrigeration coefficient E (Fig. 3, point 2) that does not depend on heat load Q 0 ; -an increase in voltage drop U (Fig. 8) for different heat load Q 0 ; the voltage drop increases with an increase in the heat load; -the functional dependence of thermoelements number n = f(DT) has a minimum at DT = 20 K for different heat load Q 0 (Fig. 9); the number of thermoelements increases with an increase in the heat load; -an increase in failure rate λ/λ 0 for different heat load Q 0 (Fig. 10); the failure rate increases with an increase in the heat load; -a decrease in failure-free operation probability P for different heat load Q 0 (Fig. 11); -the failure-free operation probability decreases with an increase in the heat load.
An increase in temperature difference DT at m 0 С 0 → 0 leads to: -an increase in the time required to enter a stationary mode τ (Fig. 1, point 3) that does not depend on heat load Q 0 ; -a decrease in the magnitude of working current I (Fig. 2, point 3) that does not depend on heat load Q 0 ; -a decrease in the refrigeration coefficient E (Fig. 3, point 3) that does not depend on heat load Q 0 ; -an increase in voltage drop U (Fig. 12) for different heat load Q 0 ; the voltage drop increases with an increase in the heat load Q 0 ; -the functional dependence of the thermoelements number n = f(DT) has a minimum at DT = 35 K for different heat load Q 0 (Fig. 13); the number of thermoelements grow with an increase in the heat load;

Continuation of Table 2
-an increase in the failure rate λ/λ 0 for different heat load Q 0 (Fig. 14); the failure rate grows with an increase in the heat load; -a decrease in the failure-free operation probability P for different heat load Q 0 (Fig. 15); -the failure-free operation probability decreases with an increase in the heat load.
Table 3 Calculation of basic parameters under mode (Q 0 /I 2 ) max at T = 300 K, l/S = 10, regime (Q 0 /I 2 ) max , 175 10 4 J/K and different heat load Q 0 are given in Table 4.
An increase in temperature difference DT at m 0 С 0 → 0 leads to: -the functional dependence τ = f(DT) has a minimum at DT = 30 K (Fig. 1, point 3) that does not depend on heat load Q 0 ; -an increase in working current I (Fig. 1, point 4) that does not depend on heat load Q 0 ; -a decrease in refrigeration coefficient E (Fig. 3, point 4) that does not depend on heat load Q 0 ; -an increase in voltage drop U (Fig. 16) for different heat load Q 0 ; the voltage drop increases with an increase in the heat load; -the functional dependence of thermoelements number n = f(DT) has a minimum at DT = 40 K (Fig. 17) for different heat load Q 0 (Fig. 13); the number of thermoelements grows with an increase in the heat load; -an increase in failure rate λ/λ 0 for different heat load Q 0 (Fig. 18); the failure rate increases with an increase in heat load; -a decrease in failure-free operation probability P for different heat load Q 0 (Fig. 19); -the failure-free operation probability decreases with an increase in heat load.To run a comparative analysis of the basic parameters, indicators of reliability, and the time required to enter a stationary working mode for various current regimes, we shall employ calculations data given in Table 5 at temperature difference DT = 40 K, heat load Q 0 = 1.0 W, and the geometry of thermoelements' branches l/S = 10 cm -1 .

Continuation of Table 4
Table 5 Comparative analysis of basic parameters, indicators of reliability, and the time required to enter a stationary working mode for various current regimes of TED An analysis of calculation results reveals that an increase in relative working current B K leads to: -a decrease in time required to enter a stationary working mode τ from τ = 14.0 s under mode λ min to τ = 6.4 s under mode Q 0max by 2.2 times (Fig. 20 Within the framework of the adopted model, the estimation of the time required by a thermoelectric device itself to enter a stationary working mode, as well as the basic parameters and indicators of reliability for different modes of operation in a range of working temperature differences at the predefined geometry of thermoelements branches l/S = 10, is as follows: -from τ = 0.The results obtained suggest that the dynamical characteristics of TED could be controlled through the variation of current regime.An increase in the working current simultaneously with an increase in the rate of a transition process lead to an increase in mechanical stresses due to temperature gradients of linear expansion of dissimilar materials of the thermoelement and the substrate.The result is a deterioration of reliability indicators of coolers.The difference in relative values for the growth in performance and a decrease in reliability indicators shows the need for an external control over current regimes of TED depending on the instantaneous spectrum of heat load frequencies.That would allow for TED operation under the modes of maximum currents at worse indicators of reliability only in the regions of the highest frequencies of the heat load, passing subsequently over to modes with lower currents and higher reliability indicators.The absence of such results prevented setting a task to control indicators of reliability in the control analogs of thermoelectric devices that ensure thermal modes of radio electronic equipment, thereby making it the task for further research.

Conclusions
At the predefined geometry of thermoelements and the absence of objects to cool by a thermoelectric cooler: -the time required to enter a stationary mode does not depend on the number of thermoelements, it is slightly different at the maximum temperature difference, and in a transition from the mode of operation at minimum failure rate λ min to the mode of maximum cooling capacity Q 0max , it reduces by 2.2 times; -under a dynamical load of TED, the mode Q 0max , in order to ensure the minimum duration of the transition process, must be as short as possible, because, when compared to the mode λ min , relative failure rate increases almost by an order of magnitude.

Fig. 12 .Fig. 13 .
Fig. 12. Dependence of voltage drop U in a single-stage TED on temperature difference DT at T = 300 K, l/S = 10 cm -1 for different heat load Q 0 under mode (Q 0 /I 2 ) max

Fig. 16 .Fig. 18 .
Fig. 16.Dependence of voltage drop U in a single-stage TED on temperature difference DT at T = 300 K, l/S = 10 cm -1 for different heat load Q 0 under mode λ min , point 1); -a decrease in the number of thermoelements n from n = 24 pcs.under mode λ min to n = 7.8 pcs.under regime Q 0max , that is by 3 times (Fig. 20, point 2); -an increase in the magnitude of working current I from I = 2,11 А under mode λ min to I = 5.0 А under mode Q 0max , that is by 2.4 times (Fig. 20, point 3); -the functional dependence of refrigeration coefficient E = f(B) has a maximum at B = 0.5 (Fig. 20, point 4); -a decrease in voltage drop U from U = 1.4 V under mode λ min to U = 0.91 V under mode Q 0max , that is by 35 %; -an increase in failure rate λ/λ 0 from λ/λ 0 = 0.71 under mode λ min to λ/λ 0 = 8.0 under mode Q 0max , that is by 11 times (Fig. 21, point 2); -a decrease in failure-free operation probability P from P = 0.99920 under mode λ min to P = 0.9927 under mode Q 0max (Fig. 21, point 2).