PHYSICAL PROCESSES IN ANISOTROPIC THERMOELEMENT AND THEIR FEATURES

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1 J. Nano- Electron. Pys. (009) No3, P SumDU (Sumy State University) PACS numbers: 7.5.Jf, 7.0.Pa, b PHYSICAL PROCESSES IN ANISOROPIC HERMOELEMEN AND HEIR FEAURES V.M. Ìàtyega, O.G. Danalakiy National ecnical University Karkov Polytecnic Institute, Cernivtsi Department, 03-à, Holovna Str., 5808, Cernivtsi, Ukraine OGDanalaki@gmail.com We investigate te galvano-termomagnetic cooling elements of te transverse-type and anisotropic termoelectric cooling elements wit and witout a cooling substrate, te ollow circular-cylindrical galvano-termomagnetic cooling elements and anisotropic cooling elements, te operating principle of wic is based on te idea of direct contact between te ot face of a cooling element and te coolant of a termostat. Keywords: HERMOELEMENS, PHYSICAL PROCESSES, EMPERAURE DISRIBUION, HERMAL EMF. (Received 4 July 009, in final form 7 November 009). INRODUCION e present work is focused on te study of te nature of pysical processes in termoelectric mediums, wic are te basis of discovery of original and important from te practical point of view elements and investigation of te penomena in tese objects. Questions connected wit cascading of te transverse galvano-termomagnetic cooling elements (GM CE) and anisotropic termoelectric cooling elements (A CE) are poorly covered. is idea is based on te assumptions about te one-dimensionality of te temperature and te constancy of te electric field, wic cannot be eld simultaneously. Presented results of te pysical investigations, directed to elucidate tese questions, are te components of termoelectricity and ave te original nature from te point of view bot of te penomena and te practical application. e aim of te present paper is to develop te models of new longitudinal and transverse anisotropic and GM elements and investigation of te pysical processes in tese objects; clarification of te improvement possibilities of te operating caracteristics of usual longitudinal termal elements by teir modernization; elucidation of te nature of te pysical processes in termal elements wit side eat-excange and in loaded anisotropic termal elements (AE). o acieve te desired aim it is necessary to solve te following problems:. Analyze operation of te two-stage A CE based on te one-dimensional temperature model wit te assumption of te constancy of te electric current density from te point of view of te maximum temperature drop, and generalize te results obtained for te case of te model of multistage transverse galvano-termomagnetic cooler (GMC) and anisotropic termoelectric cooler (AC). 43

2 44 V.M. ÌÀYEGA, O.G. DANALAKIY. Analyze te influence of anisotropy of te termal conductivity on te transverse termal emf of AE under condition tat te anisotropy parameter is muc more tan. 3. Investigate te vortex anisotropic termal elements and pysical processes in tese objects. o acieve te aim and to solve te aforesaid problems we ave used te fundamental concepts of termodynamics of te termoelectric penomena and metods of te matematical pysics. In te present work we ave studied te pysical processes in AE from te point of view of te one- and two-dimensional temperature models, te use validity of wic was proved in [3]. One-dimensional temperature model is used wile investigating te operation of cascade AC, and two-dimensional one wile investigating te influence of te anisotropy of termal conductivity on te temperature field of AE. Operation of te loaded AE from te point of view of te conversion efficiency is studied.. ANISOROPIC HERMAL ELEMEN IN COOLING MODE Scematic diagram of te A CE is represented in Fig.. In tis paper we present te investigation results first publised in []. y 0 j 0 0 l x B 3 0 Fig. Scematic representation of A CE or GM CE if sample is in magnetic field perpendicular to te xy-plane. Current leads are made of metal (for example, copper) and contact wit te termostat 4 troug te tin dielectric layer 3 wit ig termal conductivity If cooling element (CE) is long enoug one can consider te temperature as (y) in its midsection. For te electric field component along te x- axis we can write 4 E j + /y, were and are te longitudinal resistivity and te coefficient of te transverse termal emf, respectively, wic are considered to be constant. In accordance wit te continuity equation j j(y). Along te y-axis te electric field conditioned by a temperature gradient also appears: E /y, were is te coefficient of te termal emf along te y-axis, wic is also constant. Electric field sould be te potential one, i.e.,

3 PHYSICAL PROCESSES IN ANISOROPIC HERMOELEMEN 45 E y E, x terefore E const since E does not depend on x. So, we conclude: if temperature is one-dimensional tan E const. In tis approximation te energy conservation law can be written in te form ( ) d E d Z Z 0, () dy dy were Z /( ) is te anisotropic termoelectric efficiency, and are te resistivity and te termal conductivity along te x- and y-axes, respectively. Equation () sould be considered jointly wit te boundary conditions: (0) 0, ( ). General solution of equation () as a form ln( Z ( E / ) y A) B, (3) Z Z( E / ) y A were A and B are te constants of integration, wic are found from te boundary conditions (). We will assume tat te top face of A CE (Fig. ) is adiabatically isolated from te environment. is can be acieved if CE, for example, is placed in te vacuum. Condition of te adiabatic isolation is te following: j y y 0 (). (4) Using conditions () and (4) and expression (3) we will obtain te equation Z0 ln( Z0 Z Z( E / ) ) Z, (5) Z Z z( E / ) 0 wic connects and E. In general case it does not ave an analytical solution wit respect to. In te case wen Z( 0 ) << we find from (5) Z0 0 Z0 ln ZE / Z Z( E / ). In tis case te optimal value of te parameter C Z(E / )/ is equal to C Z 0, terefore te minimum temperature will be te following: ln( Z0 ) min. Z If Z 0 << tan Z. min 0 0 /

4 46 V.M. ÌÀYEGA, O.G. DANALAKIY Approximation j const for te one-dimensional temperature distribution leads to te following expression for min [3]: min Z0, Z wic olds for any Z. For small Z wen Z 0 << we obtain min 0 Z 0 /. In [3] approximation j const is preferred, since it can be easily realized experimentally. 3. SEADY-SAE EMPERAURE REGIME OF HE WO-SAGE ANISOROPIC HERMAL ELEMEN IN COOLING MODE It was mentioned above, tat investigations of cascading of te transverse coolers [4, 5] are not convincing enoug: pysical models of separate and cascading coolers, wic are te basis of tese investigations, are too far from te reality. Separate galvano-termomagnetic (GM) or anisotropic termoelectric (A) elements of te squared sape are placed one above te oter. Here we suppose tat te eat extracted by eac element is te termal load of termal elements. Wile calculating te maximum temperature decrease a number of assumptions sould be made, and te main ones are te following: te same electrostatic field is applied to eac CE; tere is an electric contact between te separate termal elements, and as it seems, it does not influence te current and temperature distributions. Fig. Cascade transverse GM or A element operating in cooling mode Calculations performed on te basis of te mentioned simplifications lead to tat te maximum temperature decrease is acieved by te exponential cange in te side generatrixes subject to x (Fig. ). Experiments wit GMC sowed tat te exponential cross-section gives deeper cooling tan te same GMC of te squared sape. In [5] te investigation results of cascade transverse GMC are generalized for te case of AC. In tis paper te two-stage AC from te point of view of te maximum temperature [6] is studied in detail. Scematic diagram of te two-stage AC is represented in Fig. 3. It consists of te separate A CE and A CE, wic are connected in te termal ratio in suc a way tat te eat extracted on te bottom face of A CE is te termal load of A CE. ere is no electric contact between te bottom face of A CE and te top face of A CE. At te same time termal contact between tem is considered to be te perfect one. In electric ratio te A CE and A CE are connected as it as sown in Fig. 3.

5 PHYSICAL PROCESSES IN ANISOROPIC HERMOELEMEN 47 y H j j 0 Fig. 3 Scematic diagram of te two-stage AC e model described ere differs from te one presented in [5] by te fact tat, firstly, instead of te condition of te constancy of te electric field te condition j const is taken. At first sigt te condition E const is stricter. However, it is difficult to realize tis condition. Condition j const is satisfied easier: for tis it is necessary to produce te corresponding current leads [3]. Secondly, cascading, wic is known from [5], foresees te identity of te cooling coefficients of te separate termal elements tat is beneat criticism as well. Coosing A CE and A CE long enoug, i.e., assuming H/L << and l/l <<, were and l H are te sizes of CE and CE along te y-axis, L is te size along te x-axis (Fig. 4), one can consider tat in te midsection of AE te temperature will depend on te y only. We also suppose tat te kinetic coefficients of materials of AE and AE do not depend on te temperature. is assumption is confirmed if te operating temperature range is not very wide. Generalized equation of te termal conductivity in te steady-state case under condition tat te current densities in CE and CE are constant will be written in te form di bi dy were i, is a number of CE, bi iji i. Boundary conditions are te following: L x 0, (5) (0) 0, (), (), (H) H. (6) General solution of equation (5) as te form i y by i Ay i Bi, (7) were A i and B i are te constants of integration. Using te boundary conditions (6) and solution (7) we can find te expressions for te temperature distributions in CE

6 48 V.M. ÌÀYEGA, O.G. DANALAKIY 0 y 0 by b y, H H y by H y bh l. l We will obtain temperatures and H from te conditions b 0 H a k bl H al, b H alh 0, ii j were ai, k, i is te transverse termal emf. i l ese conditions indicate, respectively, te continuity of te eat flow at te interface between A CE and A CE and te adiabatic isolation of te top face of A CE. Using tem we find H bl al, (8) kbl al 0 kbl b. (9) a al al kal Let us consider two cases. ) a > 0, a > 0. In order tat cooling takes place, i.e., H < 0, it is necessary to suggest in (8) and (9) tat j < 0 and j < 0, tat is to direct currents in te negative direction of te x-axis. In tis case we can state about te parallel connection of A CE and A CE. Let us suppose tat k <<,, and. en we will obtain for H H 0 I c I c, I c I c I c were I and I are te currents in A CE and A CE, respectively, c is te CE tickness. In te last expression we neglect te quantities of te second order of smallness I /(c), I /(c) and [I /(c) ]k. Furter investigations are to fill in te optimal currents I and I, wic, generally speaking, sould be different. For numerical estimation we take I I I. Let 0 4 V/K, 0 W/(cmK), 0 3 Omcm, c cm, I 0 A, K, ten find H 39 K. For one-stage A CE in accordance wit formula

7 PHYSICAL PROCESSES IN ANISOROPIC HERMOELEMEN 49 min Z0, Z were Z /() is te anisotropic termoelectric Q-factor. At Z 0 3 K we ave min 65 K. So, cooling is deeper for te two-stage AC. ) a < 0, a > 0. In tis case it is necessary to cange te current direction in A CE by te opposite one. It is appropriate mention ere about te series connection of A CE and A CE. aking te same values of te kinetic coefficients, sizes and currents as in te first case, we will obtain H 39 K. We sould note, tat te given calculations ave only to sow tat in te case of AC cascading leads to deeper cooling. 4. MODEL OF HE CASCADE RANSVERSE HERMOELECRIC AND GALVANO-HERMOMAGNEIC COOLERS Cascade transverse cooler consists of te separate rectangular CE (Fig. 4), te sizes of wic are cosen in suc a way as to acieve te usual cascade cooling wile CE are arranged one above te oter [4]. In tis case te eat extracted on te ot face of eac element is a termal load of termal elements. y L N N N- N- k k- n 0 0 Fig. 4 Scematic diagram of te cascade AC Electrical part of te cascade cooler is te following: separate termal elements of te cooler are not only in te perfect termal contact, but also in te electric one. Electric contact, as considered in te cited publications, does not influence te potential and temperature distributions and gives te possibility to use te only one power source for all termal elements. Suc idea of cascading leads to tat te transverse cooler sould ave te side faces of a special sape. Experimental investigations sow tat suc a cooler as te rigt of existence: for example, it gives deeper cooling tan te same cooler of te squared sape [7, 8]. But a number of questions arise wile analyzing te model of cascade transverse cooler, wic give rise to doubts about its validity.

8 50 V.M. ÌÀYEGA, O.G. DANALAKIY Condition of te constancy of te electric field in cascades does not old since wen constructing suc CE te bulk current leads made of materials wit ig electric conductivity, and so, wit ig termal conductivity are required, tat will inevitably lead to te termal interaction wit environment. is interaction canges te temperature distribution in CE, i.e., te distribution ceases to be one-dimensional, tat, in turn, leads to violation of te condition of te constancy of te electric field, and as a result, to denial of te cascading idea. Cascading can be more realistic if separate rectangular termal elements were electrically isolated from eac oter, but at te same time were in te perfect termal contact wit eac oter. In tis case te condition j i const fulfills better tan te condition E i const. Moreover, currents in eac termal element are independent. e cascading sceme can be represented as follows. In Fig. 4 we present te middle part (cross-section) of te long enoug cooler, for wic temperature is one-dimensional, i.e., it depends on te y only. Eac termal element as its own current density j k. en under te condition tat te kinetic coefficients, and are constant, te task for AC is te following: dk dy b 0, (0) k were k is te temperature in te k-t termal element, bk kj k k. Equation (0) as to be solved wit te boundary conditions (0) 0, (N) N, () wic sould be supplemented by te join conditions of te temperatures and eat flows at te interface between CE. us, for example, te eat flow and te temperature sould be continuous between te (k ) and k-t termal elements tat is matematically expressed by te conditions d d dy dy k k k kkjk Sk k kj k k Sk, k k, were S k is te base area of te k-t termal element. Here sould be as many conditions as interfaces. Difference between S k and S k leads to tat te side surfaces will ave te form, wic is different from te plane one. As seen from te aforesaid, tis task is not a simple one, and te more number of separate CE, te more complicated its solution is. o our opinion, te model proposed describes in more detail te real pysical situation. Obviously, te mentioned task sould be solved using a computer, setting te areas S k, material constants and currents. If program is successful, computer calculations can lead to te maximum temperature drop at te fitted optimal currents in separate CE. Experimental realization of te proposed cascade cooler is not a simple task: it is necessary to provide te reliable termal leads between separate termal elements and teir independent supply as well. As known, in practice one can be restricted by a small amount of cascades as it is done for te usual Peltier coolers.

9 PHYSICAL PROCESSES IN ANISOROPIC HERMOELEMEN 5 5. CONCLUSIONS. From te point of view of one-dimensional temperature model under te condition of te constancy of te electric current in separate A CE te maximum temperature drop of te two-stage anisotropic cooler is found. Calculation results are generalized for te case of multistage anisotropic cooler.. Influence of te strong anisotropy of te termal conductivity on te temperature field of anisotropic termal element subject to its sizes and value of te temperature drop is studied. REFERENCES. M.P. Lenyuk, O.G. Danalakiy, Visnyk Donbas koi derzavnoi masynobudivnoi akademii. Progresyvni teknologii No64, 4 (009).. E.Kamke, Spravocnik po obyknovennym differentsial nym uravneniyam (Per. s nem.) (M.: Nauka: 97). 3. V.M. Ìàtyega, O.G. Danalakiy, Visnyk VPI 3 No3, 63 (008). 4. L.I. Anatycuk, ermoelementy i termoelektriceskie ustroistva: Spravocnik (K.: Nauk. dumka: 979). 5. E.V. Osipov, verdotel naya kriogenika (K.:Nauk. dumka: 977). 6. V.V. Gutz, Naukovy visnyk Cernivets kogo universytetu. Fizyka. Elektronika No79, 07 (000). 7..C. Harman, VDI Bericte 3, 667 (983). 8. C.F. Kooi, R.B. Horst, K.F. Cuff, J. Appl. Pys. 39 No9, 457 (968).