SIMULATION OF NONSTATIONARY HEATING/COOLING PROCESS OF TWO-STAGE THERMOELECTRIC MODULE

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1 SIMULATION OF NONSTATIONARY HEATING/COOLING PROCESS OF TWO-STAGE THERMOELECTRIC MODULE Vainer A.L., Perepeka V.I., Skipidarov S.Ya. ( Scientific research institute «Storm», 7, Tereshkova Str., Odessa, 65078,Ukraine; SCTB NORD, 3, Pishchany kar er Str., Moscow, 09383, Russia) Simulation results of nonstationary heating/cooling processes of two-stage thermoelectric pile are presented in the work. Introduction Successful task solution assigned before the state-of-the-art low-temperature thermoelectricity is substantially determined by the realizability of one of the most interesting properties of Peltier effect. It is based on inertialess properties and fast response of new generation thermoelectric devices is required for its achievement. A mathematical simulation for intensification conditions of reversible pulse operating modes when cooling-heating processes last for seconds was performed to optimize thermoelectric stage cooler in terms of response speed []. Thermopile model is presented in a system of bodies with lumped parameters, i.e. bodies with thermal capacity, in which thermoelectric processes of heat energy absorption or liberation are progressed. These bodies (parts of thermopile) are connected between themselves and environment by thermal conductivities. Such model of stage thermopile is described by the system of regular inhomogeneous nonlinear differential equations that represent the basic physical processes (Peltier, Joule effects and types of heat exchange) under nonstationary operating modes. The modification of finite element method can be adequately and effectively used for correct analysis of such model. It was well-designed in computer physics and adapted by authors for simulation of nonstationary reversing operating modes of thermopiles [-4]. Usable modification of finite element method (finite-balance method) does not require essential expenditures of computer time so the simulation can be realized for many variants of actual initial data. This is the basis of the further optimization of thermoelectric module design. The comparison of calculation variants in terms of prescribed objective function (minimum weight, minimum cost, minimum energy consumption, minimum integrated mass-energy criterion and undoubtedly the basic criterion is a maximum response speed in our case) allows choosing actual initial data and operating modes, which provide the optimization of produced thermoelectric module. The used method makes it possible to obtain the mean integral temperature curves of considered parts of thermoelectric module. Thermal conductivities between module parts are nonlinear coefficients in a differential equation system of model and are computed with a glance of their actual geometry and thermophysical material properties. The values of thermal conductivities between the elements of stage thermopile and also between the elements and environment are computed with a glance of all heat transfer mechanisms generating a specific thermal conductivity (conductive, convection and radiant heat transfer as needed), actual geometry, typology peculiarities and nonlinearity of heat release and heat absorption on cold and hot surfaces. Thus, a computational accuracy of mean integral temperatures, which enables an application of suggested modeling methodology at all development stages, is reached. It is used for ISSN Journal of Thermoelectricity 3, 00 73

2 design statement formation at early stages, as checking calculation at all further stages and for recommendation development for device operating conditions optimization at the closing stage. Simulated results of two-stage thermoelectric module are presented in the paper. Notation conventions 8 Stefan-Boltzmann constant ( = ) B n number of thermoelements, pcs. τ time, s Т temperature, K С thermal capacity of thermopile part (side), J/K F effective emitting area, m thermal conductivity, W/K ε reduced degree of blackness ψ mean angular coefficient, W/K K thermal conductivity of leg of thermoelement, W/K q useful load on the cold junction, W z thermoelectric effectiveness, /K U electric voltage on the leg, V R resistance of leg of thermoelement, Ω Е thermal e.m.f. coefficient, V/K I current, А W power consumption, W U 0 electric voltage on the stage, V. Physical and mathematical model B Physical model of two-stage thermoelectric pile is given in the Figure and is presented as a system of four-bodies with lumped parameters. Body insulating plate with a neighboring to it part of thermoelectric module stage, body half of interstage insulating plate with neighboring part of thermoelectric module stage, body 3 half of interstage insulating plate with neighboring part of thermoelectric module stage, body 4 insulating plate with neighboring to it part of module stage. Fig..Model of two-stage thermopile as a system of bodies with lumped parameters. 74 Journal of Thermoelectricity 3, 00 ISSN

3 Structural layout and its electrothermal analogue for simulation of nonstationary cooling and heating processes in two-stage thermoelectric pile are given in the Fig.. Including heater a) b) Fig.. Structural layout (a) and electrothermal model (b) of two-stage pile. Differential equation system for description of nonstationary process of heating or cooling in two-stage thermoelectric pile is as follows 4 T4 T eu T U a KM ( T T) a +Б R R dt T Ta =± ( T T) εψ F( T Ta ) ± n () dτ C C C eu T U + KM ( T T) R 3 lk R = ( T T) + ( T T3) n dt () dτ C C C eu T U 4 + KM ( T4 T3) 3 3 R 34 R = ( T3 T) ( T4 T3) ± n dt dτ C C C eu T U + + KM ( T4 T3) dt4 = ε ψ ± dτ C C C 4 4 T4 Ta 4a Б 34 T4 T R R a ( T T) 4 4F4( T4 Ta ) n (3) (4) ISSN Journal of Thermoelectricity 3, 00 75

4 In designations (±) and ( ) the upper sign refers to heating mode, while the lower to cooling mode. The system makes it possible to calculate the temperature distribution in two-stage pile against the time. Initial data of simulation. In this study there was a simulation of two-stage thermoelectric pile operation in heating and cooling modes of thermocyclic object performed. The required pile parameters for calculation are given in the Table. Table Stage I t a, ºС Ambient temperature 7 z Figure of merit of pile material е, V/K Thermal e.m.f coefficient К, W/K Heat conductivity of thermoelement leg R, Ω Resistance of thermoelement leg L, m Height of thermoelement leg n, pcs Elements number of st thermopile stage 7 U, B Voltage of st thermopile stage С, J/K Thermal capacity of body С, J/K Thermal capacity of body 6.009, W/K Thermal conductivity between body and body a, W/K Thermal conductivity between body and environment , W/K Thermal conductivity between the stages 48 Stage II t, ºС Ambient temperature (body with 40 ºС temperature) 40 z Material figure of merit of nd stage е, V/K Thermo e.m.f. coefficient К, W/K Thermal conductivity of leg of thermoelement R, Ω Resistance of leg of thermoelement L, m Height of leg of thermoelement n, pcs Number of elements of nd thermopile stage 7 U, V Voltage of nd thermopile stage С 3, J/K Thermal capacity of body 6.09 С 4, J/K Thermal capacity of body , W/K Thermal conductivity between body 3 and body a, W/K Thermal conductivity between body 4 and environment 6 76 Journal of Thermoelectricity 3, 00 ISSN

5 Simulation results Simulation results of nonstationary modes of two-stage module are shown in the Fig. 3-. Surface temperature dependencies of the st stage (temperature of body ) on time at different supply currents of stages in heating and cooling modes respectively are shown in the Fig. 3,4. Fig.3. Surface temperature dependence of the st stage ( temperature of body ) on time in heating mode in a range of 40 C 00 C for different supply currents of stof I stages and nd of I stages Fig.4. Surface temperature dependence of the st stage ( temperature of body ) on time in cooling mode in a range of 00 C 40 C for different supply currents of st I and nd I stages There are heating and cooling process thermograms of all thermopile parts (bodies -4) with and without the temperature mode preparation of interstage heat spreader at the expense of nd stage operation within the retention interval after cooling and heating mode in the Fig. 5-8 presented. In the preparation process before the heating mode the body s temperature of thermoelectric pile was brought up to 00 C. Heat sink temperature (body 4) is maintained on a level 70 C. Fig. 5. Heating process thermograms of all thermopile parts (of body -4) in the mode without preparation. τ time, s; T body temperature ; T body temperature ; T 3 body temperature 3, T 4 body temperature 4. Fig. 6. Heating process thermograms of all thermopile parts (of body -4) in the mode with preparation during the retention interval after cooling mode. ISSN Journal of Thermoelectricity 3, 00 77

6 Fig. 7. Cooling process thermograms from00 C of all thermopile parts (of body -4) in the mode without preparation within the retention interval after heating mode up to 00 C. The temperature of interstage heat spreader of thermoelectric pile was brought up to 45 C in preparation process before the cooling mode. Heat sink temperature (body 4) is maintained on a level 70 C (Fig. 8) or 50 C (Fig. 9). Fig. 8. Cooling process thermograms from00 C of all thermopile parts (of body -4) in the mode with preparation within the retention interval after heating mode up to 00 C.Heat sink temperature is 70 C. Fig. 9. Cooling process thermogram from 00 С of all thermopile parts (of body -4) in the mode with preparation within the retention interval after heating mode up to 00 C. Heat sink temperature comes to 50 С. An assurance of maximum allowed interstage temperature of two-stage module in preparation mode before the beginning of heating mode and its minimum allowed value before the beginning of cooling mode is substantiated. This is due to the fact that conductive component of the nd stage influences on heating (cooling) rate under the heating (cooling) of module s work side of the st stage with thermocyclic object. This influence is significant at a certain temperature ranges. The expediency to rise heat sink temperature of the nd stage (within limits) is unobvious, meanwhile the refrigerating capacity (heat effect) increases. Nevertheless, there is a conductive heat flux between hot and cold junctions at the same time enlarging; meanwhile the variation speed of work side temperature of the st stage is not substantially changed. It should be noted that for optimization of constructive solutions of stage thermopiles one shall perform the multiple-choice calculations and analyze the temperature fields with application of the most accurate grid finite element methods (e.g. with bundled software ANSYS). 78 Journal of Thermoelectricity 3, 00 ISSN

7 Response speed comparisons of one and two-stage modules (Fig. 0 and ) show the advantage of stage system. The process acceleration by 5-35% is possible at the same consumed power with its help. New reserves are also observed here: transfer to common interstage plate or its giving up at all. Fig.0. Thermogram of typical operating mode of -stage pile without preparation (variant ). Operating mode thermogram of -stage pile with preparation (variant ). Operating mode thermogram of one stage pile (var.3). τ, с Variant Fig..Diagram of thermopile variants with different constructive and mode parameters, which differ by time of transient cooling process in a range of 95 C to 45 C: Variant -stage pile (operation without preparation), Variant stage pile (operation with preparation), Variant 3 -stage pile (operation with preparation, U >), Variant 4 -stage pile (operation with preparation, U <), Variant 5 -stage pile (operation with preparation, n *), Variant 6 -stage pile (n = 6), Variant 7 stage pile (n = ) Design-technological prerequisites for maximum response speed provision of stage module Critical lowering of all thermal capacities of details and design components is a mandatory priority at high-speed stage module designing. The first consequence of this is an application of one (common) interstage high heat-conducting ceramic plate (at its minimal sufficient thickness) from beryllium oxide (preferentially) or aluminum oxide on which thermoelements of upper and lower stage are assembled from the both sides simultaneously. An aberration from such requirement results in a sharp decrease of response speed of stage modules. It is reasonable to produce the first and the second stage to match the adjoining areas on sizes ISSN Journal of Thermoelectricity 3, 00 79

8 and contours. The necessary cascading coefficient thereto can be reached not only at the expense of thermoelements number increment, but also by means of respective lowering of thermoelements height of second (lower, basic) stage. The number of thermoelements and their sectional view remain equal to analogous characteristics of first stage thermoelements. It is desirable to simulate the temperature field of a whole area including fields along such legs length of thermoelements to refine and optimize the operating modes of stage module. This shall show how to change configurations (forms) and parameters of legs of thermoelements and yield a new digits of response speed and heat flow reduction (e.g. by 0-30% of a Nominal it was carried out in computations). Most probably, we can expect for requirements reformulation, which relates not only to legs configuration but also to fast-responding (high-speed) thermoelements. As for legs configuration, they can be formed as a right parallelepiped or have a changed contour along such length (a species of waist ) to accelerate the response speed. The thermoelement itself can be composite with realizable distributed Peltier effect in it. General approach therein is connected with detection of temperature field real picture of stage thermoelement. Such task solution requires tens of factors accounting and is connected with a large amount of simulations. Surely, an experimental picture taking of temperature field of high-speed stage would be notable, but it is too early to expect this for the present. It is necessary to refine (to approximate to actual model) the mathematical model of module with a glance of all electrical-thermal connecting and functional components for real preparation mode optimization of stage module onto fast-responding. And one shall consider eight - ten bodies instead of four bodies (of four differential equations) or what is more reasonably use a grid detailed model, which is realized in ANSYS software product. Conclusions Analysis of obtained results shows that acceleration of heating (cooling) for object thermocycling can be reached at the expense of:. Current increment in acceptance limits via stage modules (graphs in the Fig.3 and 4).. Provision of maximum allowed interstage temperature of two-stage module in the preparation mode before the beginning of heating mode and its minimum allowed value before the beginning of cooling mode. 3. Increment of heat sink temperature of nd stage from 50 to 70 С (Fig. 8, 8a) 4. Maximum allowed lowering of thermal capacity of thermocycling object, which is obvious. Therefore, there is the solution for nonstationary problem of stage cooler heating/cooling enabling an effective usage of inertialess feature of Peltier effect suggested. References. Anatychuk L.I. Thermoelements and thermoelectric devices. K.: Naukova dumka Publ, p.. Korn G., Korn T. Mathematics handbook (for scientists and engineers). M.: Nauka Publ., p. 3. Kakhaner D., Mouler K., Njesh S. Numerical methods and software. M.: Mir Publ., Mathews D., Fink K. Numerical methods. Application of МАТLAB. «Williams» publishing house 3 rd edition с. 5. Vainer A.L., Perepeka V.I. Nonstationary operating mode of thermopiles // Thermoelectricity Publ p. Submitted Journal of Thermoelectricity 3, 00 ISSN

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