ANALYSIS OF RELATIONSHIP BETWEEN THE DYNAMICS OF A THERMOELECTRIC COOLER AND ITS DESIGN AND MODES OF OPERATION

Determining the time that it takes for a thermoelectric cooling device (TED) to enter a stationary working mode over the preset temperature range is an interesting task. This is related to the fact that dynamic indicators for the means that enable heat regimes of thermally loaded elements largely define both functional and reliable capabilities of critical 32. Correction of the operating modes of an induction motor with asymmetrical stator windings at vector control / Zagirnyak M., Kalinov A., Melnykov V., Kochurov I. // 2015 International Conference on Electrical Drives and Power Electronics (EDPE). 2015. doi: 10.1109/edpe.2015.7325303 33. Zagirnyak M., Maliakova M., Kalinov A. Analysis of operation of power components compensation systems at harmonic distortions of mains supply voltage // 2015 Intl Aegean Conference on Electrical Machines & Power Electronics (ACEMP), 2015 Intl Conference on Optimization of Electrical & Electronic Equipment (OPTIM) & 2015 Intl Symposium on Advanced Electromechanical Motion Systems (ELECTROMOTION). 2015. doi: 10.1109/optim.2015.7426958 34. Improvement of compensation method for non-active current components at mains supply voltage unbalance / Al-Mashakbeh A. S., Zagirnyak M., Maliakova M., Kalinov A. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 1, Issue 8 (85). P. 41–49. doi: 10.15587/1729-4061.2017.87316 35. Zagirnyak M., Maliakova M., Kalinov A. Compensation of higher current harmonics at harmonic distortions of mains supply voltage // 2015 16th International Conference on Computational Problems of Electrical Engineering (CPEE). 2015. doi: 10.1109/cpee.2015.7333388 36. Zagirnyak M., Kalinov A., Maliakova M. An algorithm for electric circuits calculation based on instantaneous power component balance // Przegląd elektrotechniczny (Electrical Review). 2011. Issue 12b. P. 212–215. URL: http://pe.org.pl/articles/2011/12b/59.pdf 37. Al-Mashakbeh A. S. O. Modern control design of power system // Australian journal of basic and applied sciences. 2009. Vol. 3, Issue 4. P. 4267–4271. URL: https://www.researchgate.net/publication/294390623_Modern_control_design_of_power_system


Introduction
Determining the time that it takes for a thermoelectric cooling device (TED) to enter a stationary working mode over the preset temperature range is an interesting task. This is related to the fact that dynamic indicators for the means that enable heat regimes of thermally loaded elements largely define both functional and reliable capabilities of critical systems. In this case, only the mass and specific heat of an object are typically accounted for in the process of entering the mode. At the same time, experience has shown that there is a need to additionally take into consideration the heat capacity and mass of structural and technological elements, as well as current operating mode. In terms of operational control, of special interest is the current mode, and in terms of strategic control -the effect of heat capacity on the dynamic characteristics of a thermoelectric cooling device.
Thus, it is a relevant task to create a controllable dynamic system to monitor temperature at a thermally loaded element.

Literature review and problem statement
The issues of enabling thermal modes are integral part of the development of radio electronic equipment whose elements operate under thermally loaded modes [1]. Comparative analysis of compression and solid-state coolers [2] reveals that in terms of weight and dimensions, performance and reliability, thermoelectric coolers have a clear advantage [3]. Improved reliability indicators when designing thermoelectric coolers are achieved by taking into consideration the influence of thermal-physical, electrical properties, chemical activity of the thermoelements' materials when interacting with external environment [4]. Creation of new materials with enhanced thermoelectric efficiency [5] gives rise to new challenges associated with the growing influence of contact resistances, heat conductivity of thermal elements, linear expansion of thermoelement contact with electrode. Specification of requirements to thermoelectric coolers for cooling capacity, energy indicators, weight and dimensions, resulted in the variety of thermoelectric modules [6]. Since such an integrated indicator as reliability depends on the design and manufacturing technology, there are developed methods to investigate indicators of reliability over the entire life cycle, starting at the design stage all the way to operation of thermoelectric coolers [7]. For the on-board systems, the most important is the influence of mechanical and thermal loads. The effect of impact and harmonic mechanical load on the cooler is strengthened by the fact that lower temperatures lead to the worsening of plasticity of the thermoelement soldering with the electrode, and to the increased fragility of a thermoelectric material [8]. Heat load increases temperature gradients, which can lead to the cracking of places where dissimilar materials are connected [9].
Under the non-stationary heat flows, control over coolers for deviation is ineffective. Working out a temperature deviation at the receiving element starts only after the temperature wave reaches the sensor of a thermal control system [10]. Working out a thermal perturbation by the cooler, which is typically described bye integrating link, includes the lag time in the process of transition into a stationary mode, during which temperature of the thermally loaded element may exceed maximum permissible temperature. Proactive control implies launching a cooler prior to the moment when the heat wave reaches the cooler, therefore, it employs more complex algorithms to process data in order to make appropriate decisions [11]. The dynamics of control is directly dependent on the performance efficiency of the controlling element, which, in this case, is the cooler [12]. Studies into the inertia of single-stage thermoelectric devices have shown that it is mainly determined by the ratio of heat capacities of the load and a thermoelectric cooler [13]. At the same time, the model considered does not take into consideration structural and technological elements of the cooler, which are a necessary component of the single-stage thermoelectric cooler. The need to improve performance efficiency of the thermoelectric cooler is in contradiction with the reliability indicators, which requires additional research.

The aim and objectives of the study
The aim of present study is to reduce the time it takes for a thermoelectric cooler to enter a stationary regime by taking into consideration the impact of structural and technological elements of the cooler, as well as operational modes.
To accomplish the aim, the following tasks have been set: -to develop a dynamic model of TED that would account for the structural and technological elements of the cooler; -to perform a reliability-oriented analysis of the model in order to estimate a possibility to control the time it takes for TED to enter a stationary regime.

Development of dynamic model of TED taking into consideration its structural and technological elements
The structural and technological elements on the heat absorbing junction of TED include: -copper switching plates; -a layer of soldering and a nickel coating; -ceramic plate and a metallization layer in line with the switching circuit of thermoelements branches; -a diffusion layer of a semiconductor material.
The following has to be taken into consideration: -condition of the thermoelectric material surface, which is related to the technology of processing and storage conditions [8,9]; -the depth of copper atoms migration in a thermoelectric material.
The thickness of a diffusion layer of the thermoelectric material, a contact area "metal-semiconductor" can be adopted equal to 100-150 μm. We used an aluminum plate with a mass of 1 gram as the object to be cooled. Indicative data on mass and heat capacity of structural and technological elements of TED are given in Table 1. When calculating the volume, for the geometry of thermoelements l/S= =10 cm -1 , we used dimensions of the branch cross-section equal to 2×2 mm at height l=4 mm.
The total magnitude of heat capacity and the mass of TED components can be represented in the form: We shall consider the process of cooling an object in time τ, which is determined by the current mode selected, the magnitude of thermal load Q 0 , branch geometry of the cooling thermoelement (l/S), taking into consideration the temperature dependence of parameters of a thermoelectric material in the module (Fig. 1), as well as specific heat capacity C and thermal diffusivity a (Fig. 2). The dependence of total heat capacity and the mass of structural elements on the geometry of TED branches (l/S) is shown in Fig. 3. The temperature of heat emitting junctions is accepted to be constant and equal to T=300 K due to intensive heat exchange.
where m 0 , C 0 are, respectively, the mass and specific heat capacity of the cooled object; = 0 max eT I R is the maximum operating current, A; , e R are, respectively, the averaged value of coefficient of thermoEMF, V/K, and electrical resistance of the thermoelement branch, Ohm; B=I/I max is the relative operating current; I is the working current magnitude, A; T 0 is the temperature of a heat absorbing junction, K; Θ= =∆T/T max is the relative difference in temperature; ∆T max = = 2 0 0,5 z T is the maximum temperature difference, K; z is the averaged value of the efficiency of a thermoelectric material in module 1/K; ∆T=T-T 0 is the difference in temperature at TED, K; n is the number of thermoelements, pcs.
By solving differential equation (2) under initial conditions τ=0; T=T 0 , we shall obtain where K 1 is the heat transfer coefficient, K 1 =ӕ S/l, W/K; γ = 2 max0 0 2 max1 1 ; I R I R I max0 , R 0 are, respectively, the maximum operating current and electrical resistance of the thermoelement branch at the beginning of the cooling process at τ=0; I max1 , R 1 are, respectively, the maximum operating current and electrical resistance of the thermoelement branch at the end of the process of cooling; ӕ is the coefficient of thermal conductivity.
This formula represents an analytical dependence of the time required to enter a stationary mode on the current operating mode (the magnitude of relative current B), heat load Q 0 (number of thermoelements n), taking into consideration both the mass and the heat capacity of the cooled object m 0 C 0 , and the structural and technological elements of TED at a preset temperature difference ∆T(Θ).
Given that B 1 =I/I max0 ; B 2 =I/I max1 , we shall write: -for mode Q 0max I=I max1 , B 1 =I max1 /I max0 , B 2 =1.0; -for mode λ min where η is the correction factor [9]. We can derive from equation (3) the temperature of heat-absorbing junction T 0 depending on the cooling time τ: Expression (8) describes the relationship for various current operating modes and heat load for the assigned temperature difference, taking into consideration the mass and specific heat capacity of structural and technological elements of TED.

Analysis of temporal and reliability indicators of the model for different operation modes of TED
The results of calculation of basic parameters and the time required for TED to enter a stationary mode, parameters of reliability for current modes of operation Q 0max , (Q 0 /I) max , E max and λ min at T=300 K; ∆T=40 K; l/S=10 cm -1 ; Table 2. Here τ 0 is the time required to enter a stationary mode, calculated with respect to the mass and heat capacity of the object, τ¢-with respect to the mass and heat capacity, of both the object and the elements of TED design. Table 2 Results of calculation of basic parameters and indicators of reliability for a single-stage TED, obtained at the following original data:  Fig. 4-7 show that an increase in heat load Q 0 (the number of thermoelements n in TED) at the assigned temperature difference ∆T for various current modes of operation leads to the following: -the time required to enter a stationary mode τ reduces; -the number of thermoelements n increases; -voltage drop U grows. An analysis of data given reveals that the time required to enter a mode τ¢ increases compared to τ 0 . For example, at thermal load Q 0 =5.0 W: -under mode Q 0max τ 0 =9.0 s; τ¢=15.9 s, that is, it is increased by 77 %; -under mode (Q 0 /I) max τ 0 =8.6 s; τ¢=16.7 s, that is, it is increased by 94 %; -under mode E max τ 0 =7.7 s; τ¢=17.7 s, that is, it is increased by 130 %; -under mode λ min τ 0 =6.1 s; t¢=20 s, that is, it is increased by 228 %.   With a decrease in the time required to enter stationary mode τ, the intensity of failures λ/λ 0 increases for different modes of operation ( Fig. 9, afor modes Q 0max and (Q 0 /I) max , Fig. 9, bfor modes E max and l min ) -due to the increase in the number of thermoelements n in TED.
An analysis of the results of estimating the temporal process of TED entering a stationary mode of operation makes it possible to consider the relation between the mass and heat capacity of the object (m 0 C 0 ) and the mass and heat capacity of structural elements at the heat absorbing junction of TED ( ∑ i i i n m C ). The relation can be written in the form: In this case, the relative magnitude of the time required to enter a stationary mode ≥ t t ¢ 0 1,5. The time required for TED to enter a stationary mode with respect to the mass and heat capacity of structural and technological elements exceeds the time required to enter a stationary mode with respect to the mass and heat capacity of the object by not larger than 50 %; b) 1.5≥f≥0.75, that is, there is an approximate match between the mass and heat capacity of the object, and the mass and heat capacity of structural and technological elements of TED. In this case, the magnitude t t ¢ 0 is in the range of 2.2≥ t t ¢ 0 ≥1.7, that is, the time required to enter a stationary mode with respect to the mass and heat capacity of structural and technological elements may exceed the time required to enter a stationary mode without taking them into consideration by the magnitude of 70 to 120 %; c) f<<1, that is, the mass and heat capacity of the object are much smaller than the mass and heat capacity of structural and technological elements. In this case, the magnitude ≥ t t ¢ 0 2,2, that is, the time required to enter a stationary mode with respect to the mass and heat capacity of structural and technological elements is much longer, by 2-10 times, than the time required to enter a stationary mode without taking them into consideration. Dependence of relative magnitude of the time required for a single-stage TED to enter a stationary mode t t ¢ 0 on the magnitude of f at T=300 K; ∆T=40 K; l/S=10 cm -1 is shown in Fig. 10. It should be noted that a given dependence applies to all the considered modes of operation. In accordance with expression (8), we shall estimate the temperature of a heat-absorbing junction T 0 and other basic parameters for a single-stage TED for the operation modes Q 0max , (Q 0 /I) max , E max and λ min . Initial conditions: T=300 K; ∆T=40 K; l/S=10 cm -1 with and without taking into consideration the mass and heat capacity of TED structural and technological elements for various thermal load Q 0 . Calculated data are given in Tables 3-6, where T 0 , ∆T max , Θ, λ/λ 0 and P are those without taking into consideration the structural and technological elements; ¢ 0 , T D ¢ max , T Θ¢, ( ) ¢ l l and ¢ P are those taking into consideration the structural and technological elements.   Fig. 11-14 show the time-temperature dependences of a heat-absorbing junction T 0 and failure rate λ/λ 0 of a single-stage TED for different modes of operation and varying heat load Q 0 at T=300 K; ∆T=40 K; l/S= =10 cm -1 .
Thus, for example, at equal thermal load Q 0 =0.5 W the time to reach the set temperature of T 0 =260 K at T= =300 K and the mass and heat capacity of the object      An analysis of the estimation data reveals that the λ min mode ensures minimum time required to enter a stationary mode at minimal failure rate λ/λ 0 for a varying heat load Q 0 .
It follows from Fig. 15 that with a growth of thermal load Q 0 for different modes of operation and the preset cooling temperature level T 0 =260 K and the geometry of thermoelement branches l/S at T=300 K: -the magnitude of relative time required to enter a stationary mode β increases, with the greatest magnitude β observed under the mode of λ min ; -the magnitude of relative failure rate λ/λ 0 increases, with the largest magnitude of failure rate λ/λ 0 observed under the mode of Q 0max , and the lowest is under the mode of λ min . Fig. 15. Dependence of relative magnitudes of the failure rate l/l 0 and the time required to enter a stationary mode b=(τ¢-τ 0 )/τ 0 of a single-stage TED on thermal load Q 0 at T=300 K; ∆T=40 K; l/S=10 cm -1 ; Q 0 =0.5 W; l 0 =3×10 - 8

1/h for different modes of operation
We shall consider a possibility of reducing the time required for a single-stage TED to enter a stationary mode by increasing the number of thermoelements n for a preset heat load Q 0 and temperature difference ∆T: at T=300 K; ∆T=40 K; Q 0 =0.5 W; l/S=10 cm -1 .
Results of the calculations are given in Table 7.
With the increasing number of thermoelements n for a preset heat load Q 0 and temperature difference ∆T (Fig. 16): -relative operating current B and the magnitude of operating current I decrease; -voltage drop U grows; -functional dependence of cooling coefficient E=f(n) has a maximum; -the time required to enter a stationary mode τ 0 and τ' is reduced; the time required to enter a stationary mode τ', taking into consideration the structural and technological elements, increases compared to τ 0 : for example, at n=25 pcs., τ'=61 s; τ 0 =41 s, that is, τ' increases by 53 % (Fig. 17); -functional dependence of the failure rate λ/λ 0 =f(n) has a minimum at n λmin ; at n>n λmin , failure rate λ/λ 0 increases (Fig. 18); -relative magnitude of the time required to enter a stationary mode β=(τ'-τ 0 )/τ 0 increases (Fig. 18).  It should be noted that with an increase in the relative working current B for a preset heat load Q 0 =0.5 W and a temperature difference ∆T=40 K: -the magnitude of operating current I increases; -the magnitudes of voltage drop U and the number of thermoelements n decrease; -functional dependence of the cooling coefficient E=f(n) has a maximum at current under the mode of E max (Fig. 19); -failure rate λ/λ 0 grows; -relative magnitude β decreases (Fig. 20); -functional dependence of the time required to enter a stationary mode without taking into consideration the structural and technological elements τ 0 and taking into consideration the structural and technological elements τ' has a flat maximum at B=0.8 (Fig. 21). Thus, it is possible, given the preset value of the time required to enter a stationary mode τ, to determine graphically the magnitude of relative working current B for the assigned temperature difference ∆T and the magnitude of thermal load Q 0 (Fig. 21).

Discussion of results of analysis of the time required to enter a stationary mode, energy and reliability indicators of a single-stage TED
The analytical expressions obtained allow us to determine: -the time required to enter a stationary mode taking into consideration structural and technological elements on the heat-absorbing junctions of a single-stage TED for different modes of operation Q 0max ; (Q 0 /I) max ; E max ; λ min , heat load Q 0 and temperature difference ∆T=40 K; -the temperature of a heat-absorbing junction T 0 depending on time with respect to structural and technological elements for different modes of operation of a single-stage TED and thermal load Q 0 .
An analysis of these expressions shows that if heat capacity and mass of an object are much smaller than the heat capacity and mass of structural and technological elements f<<1, the time required to enter a mode is much longer than the time required to enter mode τ 0 without taking into consideration these factors.
With an increase in heat load Q 0 : -the time required to enter stationary mode τ, with respect to the mass and heat capacity of structural and technological elements , t¢ and without taking them into consideration τ 0 , is reduced; in both cases, the mass and heat capacity of the cooled object are taken into account; -voltage drop U and the required number of thermoelements n increase; -relative magnitude β=(t¢-τ 0 )/τ 0 of the time required to enter a stationary mode grows. The largest gain in the time required to enter a stationary mode is observed under the λ min mode, the lowestunder the Q 0max mode.
With a decrease in the time required to enter stationary mode τ: -failure rate λ/λ 0 grows; -the probability of failure-free operation P for different modes of operation decreases, both with and without taking in to consideration the mass and heat capacity of structural and technological elements.
With an increase in the number of thermoelements n at the assigned heat load Q 0 and temperature difference ∆T: -the time required to enter stationary mode τ 0 and τ' is reduced; -the relative magnitude of β grows; -functional dependence of failure rate λ/λ 0 has a minimum at n=10 pcs.
With a growth in the relative working current B at a preset temperature difference ∆T=40 K and thermal load Q 0 : -the magnitude of working current I grows; -voltage drop U and the number of thermoelements n decrease; -functional dependence of the cooling coefficient E=f(B) has a maximum at B=0.55 (the E max mode); -failure rate λ/λ 0 grows, therefore, the probability of failure-free operation P decreases; -the relative magnitude of β decreases; -the time required to enter stationary mode τ 0 and τ' increases, both with and without taking into consideration the structural and technological elements.
There is a flat maximum of dependence τ=f(B) for the E max mode.
The results obtained could form the basis for the development of control algorithms over dynamic characteristics of single-stage thermoelectric coolers during work with a nonstationary thermal load for the criterion of minimum relative failure rate.

1.
We have developed an analytical model for the relation between a cooling time of a single-stage thermoelectric cooler with current modes of operation, heat load in the range of working temperature difference, taking into consideration the impact of structural and technological components of the device.
2. The results of analysis of dynamic characteristics, energy and reliability indicators of a single-stage TED showed the possibility to control the time required to enter a stationary mode. Structural control, enabled by selecting the number and geometry of TED thermoelements, and the mass and heat capacity of the load makes it possible to reduce the time required for TED to enter a stationary mode by up to 2.5 times. Operational control, executed by changing working current of the cooler, makes it possible to reduce the time required to enter a stationary mode by up to 3 times.

Introduction
The construction and reconstruction of high-pressure waterworks sets a number of scientific and engineering tasks that require a new approach to their solution. One of them is to design reliable and economical culverts, able to work both in the construction and operational periods, making it possible to combine the spillway and energy flow channels. Damping of the excess energy of idle flows is one of the most important tasks when creating hydraulic spillway systems. The choice of the method for damping the kinetic energy of the flow significantly affects the overall layout of the hydraulic engineering structure.
This task becomes the most urgent in the transition to the construction of high-pressure hydraulic systems, which requires studying the phenomena associated with highspeed water flows, their interaction and the development of fundamentally new designs of spillway structures. The hydraulic sections that are used to solve the problems of transit water flows through such structures have been developed. When designing spillways in high-pressure waterworks, it is necessary to take into account the features of the interaction of high-speed flows with solid boundaries and the air environment. It is essential to take into account the probability of various wave processes, a possible local pressure drop, phenomena of aeration and cavitation and their consequences, as well as peculiarities of energy damping. It is important to ensure ventilation in the case of gravity and partial pressure in closed conduits, as well as take into account other phenomena of hydraulic nature. The resulting hydrodynamic loads under these phenomena are transferred to the building structures, and they must be taken into account in the design, construction and operation of spillway systems.
One of the promising areas for solving these and a number of other problems is the use of swirling water flows in hydrotechnical facilities. The so-called counter-vortex flows of liquid and gas and consideration of the prospects for their practical application have been studied at Moscow State University of Civil Engineering (MGSU, Russia) for several years.

Literature review and problem statement
The creation of effective designs of spillway structures for hydrotechnical and hydropower facilities ensures sustainable performance of the entire complex. The design and construction of such systems first and foremost solves the 11. Egorov V. I. Exact methods for solving heat conduction problems. Sankt-Peterburg: SPb. GU ITMO, 2006. 48 p.
13. Zaykov V., Mescheryakov V., Zhuravlov Yu. Analysis of the possibility to control of the inertia of the thermoelectric cooler //