DEVELOPMENT OF A MODEL FOR PREDICTING THE RELIABILITY INDICATORS IN THE DESIGN OF CASCADE THERMOELECTRIC COOLERS

The application of cascade thermoelectric devices (CTED) is predetermined not only by the need to achieve a maximally possible level of cooling, but also to improve cost efficiency. In some cases, a designer has a number of chosen structures of CTED at his disposal, constructed based on the standardized modules. It is necessary to determine maximally possible energy effectiveness at the assigned temperature differential, to select such current mode in the operation of CTED that it would match the maximum of energy effectiveness of CTED of the assigned design (regime Еmax). The relevance of providing the maximum of energy effectiveness is caused by the need to reduce the mass-and-size indicators in the systems that provide thermal modes of the thermally loaded elements.


Introduction
The application of cascade thermoelectric devices (CTED) is predetermined not only by the need to achieve a maximally possible level of cooling, but also to improve cost efficiency.In some cases, a designer has a number of chosen structures of CTED at his disposal, constructed based on the standardized modules.It is necessary to determine maximally possible energy effectiveness at the assigned temperature differential, to select such current mode in the operation of CTED that it would match the maximum of energy effectiveness of CTED of the assigned design (regime Е max ).The relevance of providing the maximum of energy effectiveness is caused by the need to reduce the mass-and-size indicators in the systems that provide thermal modes of the thermally loaded elements.

Literature review and problem statement
Expansion in the scope of application of thermoelectric coolers [1,2] leads to more stringet requirements to energy and reliability indicators.Since the refrigeration capacity of coolers depends first of all on the thermoelectric effectiveness of material of thermoelements, considerable efforts of designers are concentrated in this direction [3,4].The most impressive results are achieved in the field of nano-technologies [5,6], however, there is still a long way to go before the implementation of industrial production of such materials.Furthermore, improving the effectiveness of thermoelectric materials does not fully solve the given problem, since the reliability of functioning of coolers is not a less important indicator [7,8].The capability to resist mechanical impacts is one of the components of coolers reliability [9].A transition to the planar technologies in the production of thermoelectric coolers [10,11] and the corresponding reduction in the mass-and-size indicators does not resolve the task either.New problems occur that are related to the influence of resistances of thermoelements and the increased thermal conductivity [12].Moreover, when we take into account the existing market for thermoelectric coolers [13], then it becomes obvious that it is necessary to search for the ways of improving the energy and reliability indicators of the existing thermoelectric modules.The models of interrelation between the indicators of thermoelectric effectiveness and reliability, presented in [14], as well as the influence of re-

The aim and tasks of the study
The aim of present work is to develop a model that would make it possible to evaluate the efficiency of functioning and predicting the reliability indicators of a two-stage TED of the chosen design.
To achieve this objective, it is necessary to solve the following tasks: -to develop a model of interrelation between reliability indicators of CTED and the design and energy indicators under the mode of the highest energy effectiveness; -to analyze the model to identify conditions for improving the efficiency of CTED of different designs.

1. Development of a reliability-oriented model of CTED
In order to solve the set problem, we shall use known relationships [16].The refrigeration capacity Q 0 of a two-cascade CTED can be written in the form where I max1 is the maximum operating current, A; I max1 =e 1 T 0 /R 1 ; n 1 is the quantity of thermoelements in the first cascade, pieces; T 0 is the temperature of the heat-absorbing joint of the first cascade, K; e 1 is the coefficient of thermal EMF of the branch of thermoelements of the first cascade, V/K; R 1 is the electrical resistance of the branch of thermoelement of the first cascade, Ohm; B 1 is the relative operating current of the first cascade, rel.un., В 1 =I/I max1 ; θ 1 is the relative temperature drop in the first cascade, rel.un.
where T 1 is the intermediate temperature, K; ΔT max1 is the maximum temperature differential in the first cascade, K. Sequential electrical connection of cascades defines the equality of operating currents in the cascades, which can be written in the form where B 2 is the relative operating current of the second cascade, rel.un., B 2 =I/I max2 ; where e 2 is the coefficient of thermal EMF of the branch of thermoelements of the second cascade, V/K; R 2 is the electri-cal resistance of the branch of thermoelement of the second cascade, Ohm.
A general temperature differential on a two-stage CTED can be written in the form where ΔT 1 is the temperature differential in the first cascade, K, ΔT 1 =T 1 -T 0 ; ΔT 2 is the temperature differential in the second cascade, K, ΔT 2 =T-T 1 ; θ 2 is the relative temperature differential in the second cascade, rel.un., where ΔT max2 is the maximum temperature differential in the second cascade, K.
A condition of the thermal joining of cascades can be written in the form where n is the quantity of thermoelements in the second cascade, pieces Refrigeration coefficient of a two-stage CTED can be written in the form where W 1 is the power of consumption of the first cascade, W, where W 1 is the power of consumption of the second cascade, W, By using relations ( 1)-( 7), refrigeration coefficient can be written in the form where has a maximum for different designs of TED (n 1 /n 2 ) and temperature differentials ΔT=60 K; 70 K; 80 K; 90 K at T=300 K, n 1 =9, l 2 /s 2 =l 1 /s 1 =10.With an increase in temperature ΔT, the optimum magnitude of relative operating current B 1 shifts toward larger values.= we shall obtain a relation for determining the optimum magnitude of relative operating current B 1 , corresponding to the maximum of refrigeration coefficient E N of the TED of assigned design (n 1 /n 2 ) and to the temperature differential ΔT: Presented relation ( 9) makes it possible to determine the magnitude of optimum relative operating current B 1 that provides for the maximum refrigeration coefficient E N at the given values of ratio n 1 /n 2 and temperature differential ΔT.
Next, we determine relative temperature differentials in cascades θ 1 и θ 2 , using a successive approximation method, taking into account the temperature dependence of parameters (one-two approximations siffice): and, according to expression (1), refrigeration capacity Q 01 for the assigned design (n 1 /n 2 ) of TED in regime E max at the assigned Δ T.
In accordance with [16], for a two-stage TED, the magnitude of relative failure rate can be written in the form where λ 0 is the nominal failure rate, 1/h; С 1 , С 2 are the relative thermal load of the first and second cascades, rel.un., K is the coefficient of significance taking into account the effect of reduced temperature [16].The analytical model obtained provides the possibility to analyze a relation between the relative failure rate and the energy and design indicators of a thermoelectric cooler in the working range of functioning temperature.

Table 1
Results of calculation of the basic parameters and indicators of reliability of two-cascade TED of different designs under the mode E max at ΔT=60 K Results of calculation of the basic parameters and indicators of reliability of two-cascade TED of different designs under the mode E max at ΔT=70 K Table 3 Results of calculation of the basic parameters and indicators of reliability of two-cascade TED of different designs under the mode E max at ΔT=80 K Results of calculation of the basic parameters and indicators of reliability of two-cascade TED of different designs under the mode E max at ΔT=90 K  under the mode E max on the ratio n 1 /n 2 at T=300 K; Fig. 2. Dependence of general refrigeration coefficient E and on the cascades ε 1 and ε 2 , refrigeration capacity Q 01 and C 1 of two-stage thermoelectric coolers under the mode E max on the ratio n 1 /n 2 at T=300 K; ΔT=60 K; n 1 =9; (l/s) і =10: At the decrease in ratio n 1 /n 2 at the assigned value of temperature differential ΔT=70 K: -the magnitude of intermediate temperature T 1 decreases (Fig. 5,  -refrigeration coefficient E has a maximum at n 1 /n 2 =0.37 (Fig. 6, pos.1); -refrigeration coefficient of the first cascade ε 1 increases (Fig. 6, pos.2), and of the second cascade ε 2 -decreases (Fig. 6, pos.3); -refrigeration capacity Q 01 (Fig. 6, pos.4) and its relative magnitude C 1 (Fig. 6, pos.5) increase; -the total magnitude of failure rate λ Σ increases (Fig. 7, pos.3); -failure rates of the first λ 1 and the second λ 2 cascades also increase (Fig. 7, pos. 1, 2); -total power of energy consumption W increases, (Fig. 7, pos.4); -the total probability of failure-free operation P decreases (Fig. 8, pos.3); -the probability of failure-free operation of the first P 1 and the second cascade P 2 decreases (Fig. 8, pos. 1, 2).under the mode E max on the ratio n 1 /n 2 at T=300 K; Fig. 6.Dependence of general refrigeration coefficient E and by the cascades ε 1 and ε 2 , refrigeration capacity Q 01 and C 1 of two-cascade thermoelectric coolers under the mode E max on the ratio n 1 /n 2 at T=300 K; ΔT=70 K; n 1 =9; (l/s) і =10: At the decrease in ratio n 1 /n 2 at the assigned value of temperature differential ΔT=80 K: -the magnitude of intermediate temperature T 1 decreases (Fig. 9, pos.1); -relative operating current of the first cascade B 1 and of the second cascade B 2 increases (Fig. 9, pos.2, 3); -relative drop in temperature of the first cascade θ 1 decreases (Fig. 5, pos.4), and of the second cascade θ 2 increases (Fig. 9, pos.5); -the magnitude of operating current I increases (Fig. 9, pos.6); -refrigeration coefficient E has a maximum at n 1 /n 2 =0.23 (Fig. 10, pos.1), in this case the refrigeration coefficients of the first cascade ε 1 and of the second cascade ε 2 are equal to each other: ε 1 =ε 2 =0.22 (Fig. 10, pos.2, 3); -refrigeration capacity Q 01 (Fig. 10, pos.4) and its relative magnitude C 1 (Fig. 10, pos.5) increase; -the total magnitude of failure rate λ Σ increases (Fig. 11, pos.3), in this case the failure rates of the first λ 1 and of the second λ 2 cascades also increase (Fig. 11, pos. 1, 2), but not equally; -total power of energy consumption W Σ increases (Fig. 11, pos.4); -the total probability of failure-free operation P Σ decreases (Fig. 12, pos.3), in this case the probability of failure-free operation of the first (P 1 ) and of the second (P 2 ) cascades also decreases (Fig. 12, pos. 1, 2).At the decrease of ratio n 1 /n 2 at the assigned value of temperature differential ΔT=90 K: -the magnitude of intermediate temperature T 1 decreases (Fig. 13  The given qualitative description of the energy indicators of a cooler depending on the ratio of number of thermoelements in the cascades allows us to estimate the ways of designing the two-cascade thermoelectric devices with improved reliability.
Fig. 15.Dependence of total failure rate λ Σ and the probability of failure-free operation P S and of each cascade separately λ 1 and λ 2 and P 1 and P 2 of two-cascade thermoelectric coolers under the mode E max on the ratio n 1 /n 2 at T=300 K; ΔT=80

Discussion of results of the analysis of relation between the number of elements and the energy and reliability indicators
An analysis of calculated data revealed that there is an optimum ratio n 1 /n 2 , corresponding to the maximum of refrigeration coefficient E at the assigned temperature differential ΔT.
In the point of the maximum of refrigeration coefficient E we observe the equality of values of relative temperature differential θ 1 and θ 2 and refrigeration coefficients ε 1 and ε 2 in the cascades.Results of the calculations are given in Table 5.With an increase in the temperature differential ΔT for different designs of TED (n 1 /n 2 =1.0; 0.67; 0.5; 0.33; 0.2; 0.1): -relative operating currents in the first (B 1 ) and the second (B 2 ) cascades increase; -the magnitude of operating current I increases as well; -the intermediate temperature T 1 decreases; -relative temperature differentials in the first (θ 1 ) and the second (θ 2 ) cascades increase; -refrigeration coefficient E decreases, in this case refrigeration coefficient of the first cascade ε 1 and of the second ε 2 decrease; -refrigeration capacity Q 01 and its relative magnitude C 1 decrease; -the total power of energy consumption W Σ grows; -total voltage drop U Σ grows; -the total magnitude of failure rate λ Σ grows; -the total probability of failure-free operation P Σ decreases.

Conclusions
1. We developed a model of the relation between indicators of reliability of a cascade thermoelectric cooler and the distribution of the number of thermoelements in the cascades of a thermoelectric cooler, temperature differential, the refrigeration capacity and thermal load.Its special feature is in providing for the possibility to design the structural and energy indicators of a cooler in accordance with a criterion of the minimum failure rate.
2. We carried out an analysis of the model under regime of the highest energy efficiency, which demonstrated a possibility to evaluate the operational efficiency of a cascade cooler, to predict the optimum values of refrigeration coefficient at the assigned temperature differential and the relation of the number of elements in cascades under varied conditions of operation.

Table 5
Results of the calculation of basic parameters and indicators of reliability of two-cascade TED of different designs under themode E max at different values of temperature differential Ʃ , W U Ʃ , V λ Ʃ /n 1 λ 0 λ Ʃ 10 -8 , 1/h P