ANALYSIS OF THE MODEL OF INTERRELATION BETWEEN THE GEOMETRY OF THERMOELEMENT BRANCHES AND RELIABILITY INDICATORS OF THE CASCADE COOLER

Thermoelectric coolers under conditions of elevated thermal loads or under switching modes significantly decrease their reliability inficators. This is due, among others, to thermal stresses in the places where thermoelements and electrodes are coupled. It is obvious that the higher the range of the generated temperature drops, the lower the reliability indicators of thermoelectric systems for maintaining thermal regimes of thermally-loaded radio electronics. Cascade coolers provide for a larger temperature difference compared with single-cascade devices, which is why the requirements to reliability indicators turn out to be more stringent. Stricter requirements to the operation of thermally-loaded equipment and, consequently, to the thermoelectric systems that maintain thermal regimes makes it a relevant task to search for approaches to improve reliability indicators of cascade coolers. In the present studies, we consider structural approaches for enhancing the reliability indicators of two-cascade thermoelectric coolers.


Introduction
Thermoelectric coolers under conditions of elevated thermal loads or under switching modes significantly decrease their reliability inficators.This is due, among others, to thermal stresses in the places where thermoelements and electrodes are coupled.It is obvious that the higher the range of the generated temperature drops, the lower the reliability indicators of thermoelectric systems for maintaining thermal regimes of thermally-loaded radio electronics.Cascade coolers provide for a larger temperature difference compared with single-cascade devices, which is why the requirements to reliability indicators turn out to be more stringent.Stricter requirements to the operation of thermally-loaded equipment and, consequently, to the thermoelectric systems that maintain thermal regimes makes it a relevant task to search for approaches to improve reliability indicators of cascade coolers.In the present studies, we consider structural approaches for enhancing the reliability indicators of two-cascade thermoelectric coolers.

Literature review and problem statement
The issue of operational reliability (failure rate and the probability of non-failure operation) of thermoelectric coolers was addressed in numerous studies, for example [1][2][3].Research is carried out from different perspectives: the impact of technology of the fabrication of devices on reliability indicators [4], protection from the effect of moisture [5], mechanical [6] and climatic impacts [7], thermal load [8], temperature differences [9], operating modes [10].Operational reliability of the thermoelectric devices of the specified design is ensured by the choice of working currents, alignment of energy indicators with temperature drops and thermal load.At the design phase, the potential of reliability is laid out, which may only get worse during operation through the inefficient use of the device.In paper [11], authors rather insufficiently, mainly at the qualitative level, outlined results of the impact of the design of single-stage thermoelectric coolers on reliability indicators.The reliability-oriented quantitative research into a single-cascade thermoelectric cooler, presented by the authors of article [12], cannot be automatically applied to the two-cascade device, because it is necessary to take into account the patterns of temperature distribution in the cascades [13].Given the fact that the single-stage thermoelectric coolers have a limited range of the generated temperature differential, employing the cascading is a required condition in order to increase temperature differential.And this special feature implies undertaking research into defining a connection between re-

The aim and objectives of the study
The goal of present work is to improve reliability indicators of the two-cascade thermoelectric cooler by optimizing the design of thermoelements and their distribution in the cascades.
To accomplish the set goal, the following tasks have been set: -to devise a model that connects reliability indicators of the two-cascade cooler with the geometry of thermoelement branches and their distribution in the cascades for different temperature differential and fixed thermal load; -to analyze the connection between indicators of reliability and the geometry of thermoelements, distribution of thermoelements in the cascades, energy parameters in the operating temperature range of the cooler's functioning.

1. Model of interrelation between reliability indicators and the geometry of thermoelement branches
Interest in the application of the cascade thermoelectric coolers (CTEC) is caused not only by the necessity to achieve the highest possible level of cooling, but also by improving cooling efficiency at a given temperature differential.In some cases, when designing CTEC, a developer may refer to a number of different designs of CTEC, which differ in the quantity of thermoelements n 1 , n 2 in the cascades (n 1 /n 2 ratio) and the geometry of their branches.The geometry of branches is understood as the ratio of height l of the cascade branch cascade to the area of its cross-section S. A designer's task is to choose rationally the geometry of thermoelement branches, taking into account various constraints on dimensions, weight, power consumption, reliability indicators at sequential electrical connection of the cascades.
We shall estimate basic parameters and reliability indicators of the two-cascade TEC of different designs (n 1 /n 2 =var) when used in the cascades of branches of thermoelements with different geometry under condition (l/S) 1 = (l/S) 2 for different temperature differential ∆T under mode (Q 0 /I) max .
In order to solve the set problem, we shall apply ratios [13].
A condition of thermal coupling of the cascades can be written in the form where Q 0 is the thermal load, W; W 1 is the power consumption of the first cascade, W: Q 02 is the refrigeration capacity of the second cascade, W: ( ) n 1 , n 2 is the number of elements in the cascades, pcs.;I max1 , I max2 is the maximum operating current in the cascades, A, R 1 , R 2 is the electric resistance of the thermoelement branch in the cascades, Ohm, ( ) σ are, respectively, averaged values of the coefficient of thermoEMF, V/C, and electrical conductivity, Cm/cm, of the thermoelement branch in the cascades; B 1 , B 2 is the relative operating current in the cascades, B 1 =I/I max1 , B 2 =I/I max2 ; I is the magnitude of operating current, А; T 0 is the temperature of heat-absorbing junction, K; T 1 is the intermediate temperature, K; T is the temperature of the heat-absorbing junction, K; ∆T max1 , ∆T max2 is the maximum temperature differential in the cascades, K, , z z are the averaged efficiency values of the material of thermoelement branches in the cascades, 1/K; Θ 1 , Θ 2 is the relative difference of temperature in the cascades, Refrigeration capacity of the two-cascade TEC is determined by the first cascade: while the sequential electric connection of cascades defines equality of operating currents in the cascades: The total temperature differential on the two-cascade TEC ∆T consists of temperature differentials in cascades ∆T 1 and ∆T 2 : We shall transform expression (1) considering ( 2)-( 6) and obtain a formula to calculate relative refrigeration capacity of the two-cascade TEC (7) where From condition dC 1 /(dB 1 )=0, we shall obtain a ratio to determine the optimal relative operating current of the first cascade B 1opt , corresponding to a maximum of ratio (Q 0 /I) max of the two-cascade TEC with different designs (n 1 /n 2 =var): The value of intermediate temperature T 1 can be obtained taking into account the temperature dependence of parameters of the thermoelement material employing the method of successive approximations [13].Next, it is possible calculate basic significant parameters of the two-cascade cooler, such as B 1 , B 2 , Θ 1 and Θ 2 .
Relative magnitude of the failure rate is calculated in the following way where λ 0 is the nominal failure rate, λ 0 =3•10 -8 , 1/h; C 2 is the relative magnitude of thermal load of the second cascade, C 2 =(Q 0 +W 1 )/(n 2 I 2 max2 R 2 ); K T1 , K T2 are the coefficients of significance, taking into account the impact of reduced temperature [13].
A probability of the failure-free operation P of the two-cascade TEC can be determined from expression where t is the preset resource, t=10 4 h.

2. Analysis of results of modeling the energy, design, and reliability indicators of the cooler
Calculations of basic relevant parameters and indicators of reliability of the two-cascade TEC were conducted under the mode of maximum refrigeration capacity at assigned current (Q 0 /I) max for different configurations of the thermoelement branches.Conditions: (l/S) 1 =(l/S) 2 =l/S=var at the averaged efficiency value of thermoelectric module m z =(2,4-2,5)•10 -3 1/K, for different values of ratio of the number of thermoelements in the cascades n 1 /n 2 , temperature differential ∆T at T=300 K and thermal load Q 0 =1.0 W. Results of the calculations are summarized in Table 1-4.
In this case, it should be noted that the relative operating currents B 1 and B 2 , relative temperature differentials Θ 1 and Θ 2 , refrigeration coefficient E, relative thermal load C 1 and C 2 are not dependent on the geometry of thermoelement branches in the cascades.It follows from the constructed charts that failure rates and the probability of failure-free operation display clearly pronounced extrema, which can be applied when designing the two-cascade thermoelectric cooling devices with enhanced reliability.
Basic parameters and indicators of reliability of the two-cascade cooler at ∆T=60 K, T=300 K, Q 0 =1.0 W, N=2, (l/S ) 1 =(l/S ) 2 =l/S=var for different values of ratio n 1 /n 2 under mode (Q 0 /I ) max l/S R 1 •10   Basic parameters and indicators of reliability of the two-cascade cooler at ∆T=80 K, T=300 K, Q 0 =1.0 W, N=2, (l/S ) 1 =(l/S ) 2 =l/S=var for different values of ratio n 1 /n 2 under mode (Q 0 /I ) max   Research into influence of the geometry of thermoelement branches in the cascades of thermoelectric cooler on the basic parameters and reliability indicators was performed for the regime of maximum refrigeration capacity at the assigned current, different temperature differential DТ=60 K; 70 K; 80 K; 90 K and thermal load Q 0 =1.0 W. In contrast to [13], where the geometry of branches of the cascades is fixed (l/S) 1 =(l/S) 2 =10, and, therefore, operating currents of the cascades are also fixed, we examined the variant (l/S) 1 =(l/S) 2 =var=40; 20; 10; 4.5; 2.0.
It follows form the conducted research that at the preset values of ∆T and n 1 /n 2 , a decrease in the ratio l/S leads to increased operating current and reduced number of thermoelements.Given the sequential order of turning on the thermoelements in a cooler, the probability of failure-free operation for which equals to the product of probabilities of separate elements, a reduction of the quantity of thermoelements leads to a growth of the probability of failure-free operation and reduction in the failure rate.
As far as the failure rate is concerned, at lowering l/S from 20 to 10, the value of λ/λ 0 decreases by more than 2 times at ∆T=60 K and n 1 /n 2 =0.5 (from 6.0 to 2.5), at ∆T=70 K and n 1 /n 2 =0.5 (from 16.3 to 8.0), at ∆T=80 K and n 1 /n 2 = 0.33 (from 60 to 30), at ∆T=90 K and n 1 /n 2 =0.intensity on the distribution of thermoelements in the cascades demonstrate clearly expressed extrema in the operating temperature range, which makes it possible to receive additional gain on the mass and weight parameters of designed products by reducing the required number of thermoelements and by the optimization of energy distribution in the cascades.

Conclusions
1. We designed a reliability-oriented model of the two-cascade thermoelectric cooling device that links reli-ability indicators of the cooler and the geometry of thermoelements, distribution of thermoelements in the cascades, differences of temperatures in the cascades, operating currents, and thermal load.
2. It is shown that with a decrease in the geometry of thermoelements in the cascades from 10 to 2, the total number of thermoelements decreases and the failure rate reduces by more than twice.Joint application of the variation of geometry and the optimized distribution of thermoelements in the cascades within the operating range of temperature differential makes it possible to lower the failure rate up to 10 times and to reduce mass and weight parameters of the two-cascade coolers.

Fig. 1 .
Fig. 1.Dependence of intermediate temperature T 1 of the two-cascade cooler on the ratio n 1 /n 2 for different values of DТ at Т=300 K; Q 0 =1.0 W under mode (Q 0 /I) max