ANALYSIS OF THE MODEL OF INTERDEPENDENCE OF THERMOELEMENT BRANCH GEOMETRY AND RELIABILITY INDICATORS OF THE SINGLE-STAGE COOLER

The problem of improving reliability of thermoelectric coolers used in electronics thermal condition control systems remains the pressing problem because of permanently toughening requirements to the present-day land-based and on-board equipment. Improvement of reliability indicators of thermoelectric coolers is realized according to various principles at various steps: – in design engineering: according to parametric and design approaches; 23. Nevljudov, I. Sh. Tehnologicheskoe obespechenie tochnosti razmerov pri formoobrazovanii plastmassovyh izdelij [Text] / I. Sh. Nevljudov, S. V. Sotnik // Jelektronnaja komponentnaja baza. Sostojanie i perspektivy razvitija. – 2009. – P. 183–186. 24. Nevljudov, I. Sh. Metod rascheta oformljajushhih detalej formoobrazujushhej osnastki dlja tehnologicheskogo obespechenija zhiznennogo cikla plastmassovyh izdelij RJeA [Text] / I. Sh. Nevljudov, A. A. Andrusevich, S. V. Sotnik // Radiotehnika. – 2009. – Issue 156. – P. 240–243. 25. Shah, V. Spravochnoe rukovodstvo po ispytanijam plastmass i analizu prichin ih razrushenija [Text] / V. Shah. – Sankt-Peterburg: Nauchnye osnovy i tehnologii, 2009. – 746 p. 26. Yang, Y. Injection Molding Process Control, Monitoring, and Optimization [Text] / Y. Yang, X. Chen, N. Lu, F. Gao. – Carl Hanser Verlag, 2016. – 397 p. doi: 10.3139/9781569905937 27. Stanek, M. Optimization of Injection Molding Process by MPX [Text] / M. Stanek, D. Manas, M. Manas, O. Suba // 13-th WSEAS International Conference on Automatic Control, Modelling & Simulation. – 2011. – P. 212–216. 28. Gol’dberg, I. E. Puti optimizacii lit’evoj osnastki. Ee velichestvo lit’evaja forma [Text] / I. E. Gol’dberg. – Sankt-Peterburg: Nauchnye osnovy i tehnologii, 2011. – 360 p.


Introduction
The problem of improving reliability of thermoelectric coolers used in electronics thermal condition control systems remains the pressing problem because of permanently toughening requirements to the present-day land-based and on-board equipment. Improvement of reliability indicators of thermoelectric coolers is realized according to various principles at various steps: -in design engineering: according to parametric and design approaches; with parameters connected with the thermoelectric device (TED) reliability indicators [5,6]. Current mode choice determines energy-related operation conditions [7] which are directly related with the cooler reliability indicators [8]. When the design approach is analyzed, the qualitative aspect of ensuring specified levels of reliability of thermoelectric coolers is only considered [9,10]. At the same time, the model of interrelation between the basic parameters and the reliability indicators was established and developed [11]. This enables a comprehensive estimation of thermoelement branch geometry effect on the single-stage TED reliability for various operation conditions. Potentially, this approach enables choice of optimal thermoelement branch geometry. At the same time, failure rate must decrease and probability of failure-free operation of the cooler must increase taking into account restrictive requirements to size, weight and power consumption. Obtaining of a quantitative relation between thermoelement geometry and reliability indicators of thermoelectric coolers involves studies for various temperature gradients and operation conditions.

The aim and tasks of the study
The aim of this work is to find design approaches to the improvement of reliability indicators through selection of an optimal geometry of thermocouple branches for different operation conditions.
To achieve this objective, solution of the following tasks is necessary: -develop a reliability-oriented model linking reliability indicators with the geometry of thermoelement branches for various temperature gradients and the fixed thermal load for operation modes (Q 0 /I) max and (Q 0 /I 2 ) max ; -determine the possibility of improvement of reliability indicators of the single-stage TEU in modes (Q 0 /I) max and (Q 0 /I 2 ) max by selecting thermocouple branch geometry.

Development and analysis of the model of interrelation
between reliability indicators and design and energy parameters in modes (Q 0 /I) max and (Q 0 /I 2 ) max

1. The model of interrelation between reliability indicators of the single-stage TED and thermoelement branch geometry
The ratio of thermoelement branch height l to cross-sectional area S called as thermoelement geometry is directly connected with the cooler reliability indicators. By varying operation current value, it is possible to ensure the cooler operation in all operation modes from the maximum cooling power Q 0max to the minimum failure rate λ min . Consider the effect of geometry in the most frequently used modes (Q 0 /I) max and (Q 0 /I 2 ) max for various temperature gradients ΔT=0÷60 K for a given heat load Q 0 =2,0 W.
To solve this problem, use the earlier developed relationships [11].
The cooling power of the thermoelectric cooling unit (TEU) can be represented as: where n is the number of thermocouples; I max =ēT 0 /R is the maximum operating current, А; l R S = σ is electrical resis-tance of the thermoelement branch, Оhm; ē, σ are averaged values of coefficient of thermal electromotive force, B/K and electrical conductivity, Cm/cm of thermoelement branch respectively; T 0 is temperature of the heat-absorbing junction, K; B=I/I max is the relative operating current, relative units; I is the value of operation current, А; is relative temperature gradient, relative units; T is the temperature of the heat-generating junction, K; is the maximum temperature gradient K; z is the averaged thermoelectric efficiency of thermoelement branch, 1/K; is the maximum thermoelectric cooling power, W. The TEU power consumption can be expressed as: The TEU refrigeration coefficient can be written down as: Relative failure rate can be expressed as: is the relative heat load, relative units; K T is the significance coefficient depending on the temperature [11]. The probability of failure-free operation of TEU can be determined by the formula: where t is the assigned resource, hr.

2. Analysis of the simulation results
Calculation of the basic parameters and reliability indicators of the single-stage TEU for different ratios l/S = var in modes (Q 0 /I) max and (Q 0 /I 2 ) max was conducted with the following starting data: -heat load Q 0 =2 W; -temperature of the heat-generating junction T=300 K; -temperature gradient ∆T=10÷60 K; -l 0 =3·10 -8 1/hr, t=10 4 hr.
Calculation results are given in Tables 1, 2.
At an equal heat load Q 0 and temperature gradient ∆T: ng= c o n s t ( 6 ) for various thermoelement branch geometries l/S. 1) Mode (Q 0 /I) max . Analysis of the calculated basic parameters and reliability indicators has shown that with reduction in the ratio l/S at a given temperature gradient ΔT and heat load Q 0 in the mode (Q 0 /I) max : -maximum cooling power γ increases (Fig. 1); -operating current I increases (Fig. 1); -the number n of thermoelements decreases (Fig 2); -the voltage U drop decreases (Fig 2); -the failure rate λ decreases (Fig. 3); -the failure-free operation probability P increases (Fig. 3) at constant values of power consumption W and refrigeration coefficient E.

Тable 1
Calculation data for basic parameters and reliability indicators of the single-stage TEU for various temperature gradients ∆T at Т=300 K, heat load Q 0 =2.0 W in (Q 0 /I) max mode  With the growth of the temperature gradient ΔT at a given heat load Q0 for different values of l/S: -the maximum thermoelectric cooling power γ decreases; -the value of the operation current I increases (Fig. 4); -functional dependence of thermoelement number n in the TEU on ΔT is minimal (Fig. 5); -failure rate λ increases (Fig. 6); -probability P of failure-free operation decreases (Fig. 7). Operation current I increases with reduction of ratio l/S (Fig. 8). Note that the reduction in ratio l/S from 20 to 10 for the mode (Q 0 /I) max at ΔT=40 K and Q 0 =2.0 W makes it possible to reduce failure ratio λ by 50 % and therefore increase the probability P of failure-free operation. Besides, the value of the oper-ation current I increases from 1.75 to 3.5 A, power consumption W and refrigeration coefficient E remain constant (W=5.9 W and E=0.34) and the number of thermoelements halves. 2) Mode (Q 0 /I 2 ) max . Analysis of the calculated values of the basic parameters and reliability indicators has shown that with reduction in the ratio l/S at a given temperature variation ΔT and heat load Q 0 in the mode (Q 0 /I 2 ) max : -the maximum cooling power γ increases (Fig. 9); -the value of the operation current I increases (Fig. 9); -the number of thermoelements n decreases (Fig. 10); -the voltage drop U decreases (Fig. 10); -the failure rate λ decreases (Fig. 11); -the probability P of failure-free operation increases (Fig. 11)at constant values of power consumption W and refrigeration factor E.
With the growth of temperature gradient ΔT at a given heat load Q 0 for various values of l/S: -the maximum thermoelectric cooling power γ decreases; -the value of the operation current I increases (Fig. 12).
-the functional dependence of the number of thermoelements n in the TEU on ΔT has a pronounced minimum (Fig. 13) which can be explained by the growth of the cooling power per one thermoelement (Q 0 /n) for ∆T opt at the point of minimum; -failure rate λ increases (Fig. 14); -probability P of failure-free operation decreases (Fig. 15).

Таble 2
Calculation data of the basic parameters and reliability indicators of the single-stage TEU for various temperature gradients ∆T at Т=300 K, heat load Q 0 =2.0 W in the mode (Q 0 /I 2 ) max  With the reduction of ratio l/S, operation current I increases (Fig. 16), the refrigeration coefficient does not change (E=0.38) and the number of thermoelements decreases by 2 times.
Note that for the mode (Q 0 /I 2 ) max at ΔT=40 K and Q 0 =2.0 W, reduction of the ratio l/S from 20 to 10 enables reduction of the failure rate λ by 50 %, and hence increase in the probability P of the failure-free operation. At the same time, the value of the operation current I increases from 1.24 to 2.48 A. and the power consumption W and refrigeration coefficient E remain constant (W=5.24 W and E=0.38) and the number of thermoelements is halved.

Discussion of the results of analysis of the influence of the branche geometry on performance of the single-stage TEU
Analysis of the calculation data for the mode (Q 0 /I) max at ΔT=40 K and Q 0 =2.0 W has shown that reduction in the thermoelement branch ratio l/S of the single-stage TEU from 20 to 10 results in the following: -2-fold increase in the maximum cooling power γ; -2.1-fold reduction in the required number n of thermoelements; -2-fold increase in the operating current I value; -about a 2-fold reduction in the magnitude of the voltage U drop; -2-fold reduction in the failure rate λ; -the failure-free operation probability P increases. Besides, refrigeration coefficient E=0.34, the relative operation current B=0.71, power consumption W=5.9 W.
For the mode (Q 0 /I) max, the following occurs at various fixed values of l/S and Q 0 =2.0 W with an increase in the temperature variation ΔT from 20 to 40 K: -14 % lower maximum cooling power γ; -functional dependence n=f(ΔT) has a pronounced minimum which can be explained by the maximum cooling power at an optimum ΔT; -47 % increase in operation current I; -2 times increase in the value of the voltage drop U; -9.4 times higher failure rate λ at l/S=20; -reduced probability P of the failure-free operation; -53 % increase in the relative operation current B; -3 times increase in the power consumption W; -2.9 times reduced refrigeration coefficient E; -2.3 times increase in relative temperature gradient Θ.
Analysis of calculation data for the mode (Q 0 /I 2 ) max at ΔT=40 K and Q 0 =2.0 W has shown that with reduction from 20 to 10 in the ratio l/S of the thermoelement branch of the single-stage TEU, the following occurs: -2 times increase in the maximum cooling capacity γ; -2 times decrease in the required number n of thermoelements; -2 times increase in the value of the operation current I; -2 times reduced magnitude of voltage U drop; -2 times reduced failure rate λ; -probability P of the failure-free operation increases. Besides, refrigeration coefficient E=0.38, relative operation current B=0.5, power consumption W=5.24 W.
For the mode (Q 0 /I 2 ) max, the following occurs at various fixed values of l/S and Q 0 =2.0 W with an increase in the temperature gradient ΔT from 20 to 40 K: -14 % reduced maximum cooling power γ; -functional relation n/f(ΔT) has a pronounced minimum which can be explained by maximum cooling power at an optimal gradient ΔT; -2.2 times higher operation current I; -1.6 times higher voltage drop U; -38 times higher failure rate λ; -reduced probability P of failure-free operation; -2.3 times increase in the relative operation current B; -3.6 times increase in power consumption W; -3.6 times reduced refrigeration coefficient E; -2.3 times higher relative temperature gradient Θ.

Conclusions
1. We proposed the model of interconnection of the reliability indicators and the basic parameters of the single-stage TEU during variation of the thermoelement branch geometry for various temperature gradients and a fixed heat load in modes (Q 0 /I) max and (Q 0 /I 2 ) max . The model makes it possible to design TEU at l/S=var with consideration of restrictive re-quirements to size, weight, power consumption and reliability with a possibility of choice of a compromise design.
2. The possibility is defined of a significant increase in the reliability indicators of the single-stage TEU both in (Q 0 /I) max and (Q 0 /I 2 ) max modes by choosing thermoelement branch geometry with a smaller ratio l/S for the specified values of temperature gradient, heat load and power consumption.