Temperature Distribution in a Cylindrical Core and Coil Assembly with Heat Generation

Transcription

1 Temperature Distributio i a Cylidrical Core ad Coil Assembly with Heat Geeratio P. S. Yeh, Ph.D. Abstract A residetial distributio trasformer cosists of a cylidrical steel tak ad a core-ad-coil assembly placed cocetrically withi the tak. The tak is early filled with coolig fluid, so that the heat geerated iside the core-ad-coil ca be dissipated to the surroudig air. The trasfer of heat is mostly i the radial directio. The power ratig of a residetial trasformer ca be from to kva, with the heat loss at full load from 78 to,77 watts. The diameter of the core-ad-coil ca be from 9 to 47 cm, ad its height from 3 to 5 cm. The preset study is o the mathematical ad umerical aalyses of the temperature variatio withi the core-ad-coil assembly. Due to the cotiuous flow of electrical curret i the coil widig, ad due primarily to the electrical resistace of the widig material, a small fractio of the electrical eergy is coverted ito heat eergy, ad the heat eergy is physically exhibited i the form of a elevated temperature i the widig. To study the operatioal characteristics of the device, especially to prevet the malfuctio or eve the failure of the device, it is importat to kow the temperature variatio withi the device. Uder steady state operatio, two aalyses have bee ivestigated i the preset study, amely: (a) the case where the axial legth of the core-ad-coil is much larger tha the radial dimesio, so that the goverig mathematical equatio is oe-dimesioal; (b) the case where both the axial legth ad the radial dimesio are take ito accout, so that the goverig differetial equatio is twodimesioal. I both cases, mathematical solutios are preseted, ad umerical computatios are performed. It is show that the factors affectig the temperature distributio are: the thermal coductivity of the core-ad-coil materials, the amout of heat geerated withi the coductor, ad the physical dimesios of the core-ad-coil assembly. The oe-dimesioal study provides us the followig coclusios: (a) For the traditioal trasformer desig, a high degree of safety factor has bee provided. May trasformers have bee i cotiuous operatio for more tha thirty years or eve loger. (b) Uless a high operatioal efficiecy is desirable to reduce the loss i electrical eergy, covetioal materials such as copper or alumium provide sufficiet thermal coductivity to prevet excessive temperature rise. (c) Overloadig a trasformer several times over its rated load does ot cause sigificat icrease i temperature. Itroductio A group of electromagetic devices, which employ the priciple of mutual iductio to covert variatios of curret i a primary circuit ito variatios of voltage ad curret i a secodary circuit, such as a trasformer, a electromagetic relay, or a geerator, cosists of a electrical coil which is woud aroud a core made of either a iro, a steel, or a cobalt. The core is usually of cylidrical i cofiguratio, hece the complete assembly is also of cylidrical shape. A graphical represetatio of a residetial distributio trasformer is show i Figure. The core ad coil assembly is also show i graphical form i the same figure. Whe electrical curret flows through the coil, a small percetage of the electrical eergy is lost i the form of heat. These losses cosist of resistace loss i the widigs, hysteresis loss i the core, ad eddy currets i the core. Due to these eergy losses, the temperature i Professor Emeritus of Egieerig, Jacksoville State Uiversity, 5 Maco Drive SE, Jacksoville, AL psyeh@jsucc.jsu.edu ASEE 4 Southeast Sectio Coferece

2 the assembly icreases i compariso with its ambiet, ad if sufficiet provisio is provided for the cotiuous dissipatio of heat, a steady state temperature distributio will evetually be reached. Figure. Graphical Represetatios of a Residetial Distributio Trasformer ad the Core ad Coil Assembly Oe Dimesioal Aalysis Durig the steady state operatio of a electromagetic device, heat is geerated withi the coil assembly. Sice the coil carries electrical curret, ad the trasmissio of a electrical curret is a irreversible process, a small fractio of the electrical eergy is coverted ito heat eergy, which is physically represeted by the icrease i temperature. The rate of heat geeratio i joules per cubic meter per secod, represeted by q e, is give by the followig expressio [], [] : q = ρ I ( J / s m 3 or W / m 3 ) e e where ρ e is the electrical resistivity of the widig, i ohm m, ad I is the curret desity, i A / m. As for the core of the device, there are magetic flux lies passig through the magetic material. However, sice the flux lies represet the alteratio of the atomic structure of the core material due to exteral force (i.e., the mageto-motive force), ad is ot a measure of the flow of charged particles through the core material, therefore, there is o eergy geeratio or loss withi the core. The temperature of the core remais essetially uchaged. A cylidrical core-ad-coil is show graphically i Figure. If the axial dimesio of the relay is much larger tha the radial dimesio, the the temperature variatio ad the eergy flow are primarily i the radial directio. Therefore, the followig ordiary differetial equatio ca be set up for the coil [], [4]: dt dr dt qe + + = r dr k ASEE 4 Southeast Sectio Coferece

3 where T is the temperature i the coil, i Kor o C, r is the radial coordiate i, ad k is the thermal coductivity of the coil, i W / m K. A closed form solutio ca be obtaied for this liear differetial equatio, ad the solutio is give below: m T e qk r C = l k r + C ( 3) 4 I Equatio (3), the itegratio costats C ad C ca be evaluated usig the boudary coditios specified as follows: At r = r, the temperature is T = T C At r = r, the temperature is T = T C After satisfyig these boudary coditios, the costats ad C are give by the followig expressio: C k qe C T T r r k r r = [( ) ( )] ( 4) l ( / ) 4 C qe T r T r r r k r r r = r l l + l l 5 l ( / ) { 4 [ ]} Equatio (3) represets a temperature T which is a trascedetal fuctio of the radial coordiate r. The equatio idicates that there are three physical quatities which affect the temperature. These are the thermal coductivity k, the amout of heat geeratio q e, ad the radial locatio r. These factors are aalyzed i the followig: (a) I Figure, based o a typical 5-kVA trasformer operatig at full load with a heat geeratio of, (W / m 3 ), the temperature T as a fuctio of the radial distace r is plotted. The ier ad outer diameters of the core ad coil are 5. cm ad 3. cm, respectively. The cylidrical tak of the trasformer is 45. cm i diameter ad 64. cm i height. The total heat loss at the rated load is 35 W [8]. It ca be see that the thermal coductivity k practically has o effect o the temperature distributio. The thermal coductivities used i the computatio correspod to those of silver ( k = W / m K ), copper ( k = 38. W / m K ), alumium ( k = 6. W / m K ) ad iro ( k = 486. W / m K ). Eve though there is early te times differece i thermal coductivities betwee the highest ad the lowest values, i.e., 48.7/48.6 = 8.69, the umerical results idicate that the largest temperature differece at ay give radial locatio is.5 o C. ASEE 4 Southeast Sectio Coferece

4 Figure. Temperature vs. Radial Coordiate with Thermal Coductivity as Parameter (b) I Figure 3, at a much higher heat loss, hece at a much higher operatig load, which is times higher tha the rated value, the effect of thermal coductivity o the temperature variatio is clearly show. Whe the coductivity is reduced to 3.8 ( W / m K), which is times less tha that of the copper, the temperature begis to icrease as the radial distace icreases. It reaches a maximum value of o C at r =.5 m. It the gradually decreases to the outer boudary value of 5. o C. However, if the coductivity is times less tha or times larger tha that of the copper, the effect o the temperature variatio is ot sigificat. (c) I Figure 4, at a fixed thermal coductivity which is times smaller tha that of the copper, the effect of the amout of heat geeratio o the temperature is show. Obviously, the maximum temperature icreases as the heat geeratio icreases. But o maximum temperature exists eve whe the heat loss is times larger tha that of full load. Two Dimesioal Aalysis I the more realistic situatio, the variatio of temperature i both the radial ad the axial directios should be cosidered. The goverig differetial equatio for such a case is give by the followig relatio [], [4]: T r + + T T r r z qe + = ( 6) k To solve this partial differetial equatio, let a ew depedet variable θ be itroduced such that the followig relatioship is true: ASEE 4 Southeast Sectio Coferece

5 Figure 3. Temperature vs. Radial Coordiate at times Overload Figure 4. Temperature vs. Radial Coordiate at Various Overload ASEE 4 Southeast Sectio Coferece

6 e θ = T T = θ + + qk zh z ort T qe k zh z ( 7) I Equatio (7), the term qz e ( h z)/ k represets the particular solutio of the differetial equatio give i Equatio (6). The, Equatio (6) ca be trasformed ito the followig equatio: 8 r r r θ + θ = r z Based o the separatio of variables techique, which is applied very ofte i the solutio of partial differetial equatio, the solutio of θ ca be represeted by the product of two parts, amely, θ ad θ, where θ is a fuctio of r oly, ad θ is a fuctio of z oly. Therefore, the followig relatioship ca be writte: θ = θ () r θ () z () 9 I terms of θ () r ad θ () z, Equatio (8) becomes the followig: θ r d θ θ θ dr r d d + = ( ) dr dz The last equatio ca be rearraged ito the followig form: d r dr r d θ d θ = = m θ dr θ dz where m is a iteger costat. It is called the separatio costat. Notice that Equatio () cosists of two ordiary differetial equatios, oe of which, θ () r, cotais r as the idepedet variable, the other oe, θ () z, cotais z as the idepedet variable. Based o the regular procedure for the solutio of a differetial equatio, the solutio for θ () z is give by the followig equatio: θ ( z) = C 3 cosh( mz) + C 4 sih( mz) Where C 3 ad C 4 are the itegratio costats. The solutio for θ () r is give below: θ ( r) = C 5 J ( mr) + C 6 Y ( mr) ( 3) C 5 6 J ( mr ) Where ad C are both costats of itegratio, is the zero-order Bessel fuctio of the first kid, ad Y( m r) is the zero-order Bessel fuctio of the secod kid [3]. These fuctios are defied as follows, ad the graphical represetatios of these fuctios are show i Figures 5 ad 6: J r r 4 r 6 r 8 ( r) (!) ( ) (!) ( ) ( 3!) ( ) ( 4!) ( ) ( ) = + + = (!) = r ( 4) ASEE 4 Southeast Sectio Coferece

7 Y r J r r r r r = { [ l +. ] + (!) ( ) (!) ( ) ( + ) + ( 3!) ( ) ( + + π 3 ) ( 4!) ( r ) ( 3 4 ) } For the solutio of θ, Equatio (9) therefore ca be writte as follows: θ = [ C cosh( mz) + C sih( mz)] [ C J ( mr) + C Y ( mr)] The zeroth-order Bessel fuctio of the secod kid, Y( mr), approaches a large egative value as the argumet mr approaches zero, this implies that the temperature is udefied ear the ceter of the cylider. Hece the costat m or the variable r ca ot be zero. The boudary coditios to be applied to Equatio (6) are listed below: At r = r, the temperature is T = T, or θ = θ = T T ( q / k) z ( h z) e At r = r, the temperature is T = T, or θ = θ = ( q / k) z ( h z) e At z =, the temperature is T = T, or θ = At z = h, the temperature is T = T, or θ = Whe these coditios are applied to Equatio (6), the costats follows: C3, C4, C5ad C6 are determied to be as C = C G C = C G C C = G where G, G, ad G are give i the followig: G = Yo( mr)/ Jo( mr) ( ) G = sih( mh) / cosh( mh) G 3 T T ( qe / k) z ( h z) = [ G cosh( mz) + sih( mz)] [ G J ( mr) + Y ( mr)] ASEE 4 Southeast Sectio Coferece

8 Figure 5. Bessel Fuctios of the First Kid Figure 6. Bessel Fuctios of the Secod Kid ASEE 4 Southeast Sectio Coferece

9 Geeral Forms of Bessel Fuctio Bessel Fuctio of the first kid of order i ca be expressed i a geeral form as follows [4], [7]: r i+ ( ) Ji( r) = i = 3,,,, ( 3)!( i+ )! = Bessel Fuctio of the secod kid of order i ca be expressed i a geeral form as follows: r Yi( r) = { Ji( r)[l ] π = r i+ ( ) i+ i r i i m m!( i )! [ ] + ( )! + } +! m= m= = i+ i For =, replace m + m by m ( 4) m= m= I several published books, the mathematical equatios for Bessel fuctios of order oe ad higher cotai some errors, therefore, the correct forms of these fuctios are listed below. The graphical represetatios of these fuctios are show i Figures 5 ad 6. For i=: m= r r 3 r 5 r 7 r 9 J( r) = 5!! ( )!! ( ) +! 3! ( ) 3! 4! ( ) + 4! 5! ( ) Y r J r r r r r 3 = { [ ] [ ()!! ( ) + l!! ( ) ( + + ) + π For i=: J !! 3 3 r 3!! ( r ) ]} ( 6 ) 3 4 r r 4 r 6 r 8 r ( r) = ( ) 7!! ( ) 3!! ( )! 4! ( ) 3! 5! ( ) ! 6! ( ) + Y r J r r r r = { [ ] [ + ] [ l!! ( ) ( + + ) + π !! r 3 3! 4! ( ) ( 4 r ) ASEE 4 Southeast Sectio Coferece

10 !! ( r ) ( ) ]} ( 8 ) Summary O the study of the temperature variatio i a electromagetic core ad coil, the oe dimesioal mathematical aalysis is a idealized simplificatio of the more complex two dimesioal problem, but it provides a closed form mathematical solutio, ad some quatitative umerical results of the physical problem ca be obtaied. It is show that the factors affectig the temperature distributio are: the thermal coductivity of the core-ad-coil materials, the amout of heat geerated withi the coductor, ad the physical dimesios of the core-ad-coil assembly. The oedimesioal study provides us the followig coclusios: (a) For the traditioal trasformer desig, a high degree of safety factor has bee provided. May trasformers have bee i cotiuous operatio for more tha thirty years or eve loger. (b) Uless a high operatioal efficiecy is desirable to reduce the loss i electrical eergy, covetioal materials such as copper or alumium provide sufficiet thermal coductivity to prevet excessive temperature rise. (c) Overloadig a trasformer several times over its rated load does ot cause sigificat icrease i temperature. Whe the two-dimesioal effect is take ito accout, the mathematical solutio ca oly be expressed i terms of ifiite series ad trascedetal fuctios. The umerical results ca be obtaied by the summatio of successive terms, util a term is reached where the umerical value is very small, so that ay followig terms ca be dropped. The author is workig o a FORTRAN program to perform a detailed umerical aalysis of the preset problem. Refereces. Bird, R. Byro, Stewart, Warre E., Lightfoot, Edwi N. Trasport Pheomea, 96, page 67. Joh Wiley & Sos, Ic. New York.. Scheider, P. J. Coductio Heat Trasfer, 955, page 6. Addiso-Wesley Publishig Compay, Readig, Massachusetts. 3. Broshtei, I. N., Semedyayev, K. A., Had Book of Mathematics, pages 4-4. Va Nostrad Reihold Compay, New York. 4. Carslaw, H. S., ad Jaeger, J. C. Coductio of Heat i Solids. Pages 6, 9, 3, 4, 488. Oxford Uiversity Press. 5. Duffy, Dea G. Advaced Egieerig Mathematics, 998. Pages 37, 38. CRC Press, New York. 6. Morse,Philip M., ad Feshbach, Herma. Methods of Theoretical Physics Page 94. McGraw-Hill Book Compay, Ic. 7. Hilderbrad, Fraces B. Advaced Calculus for Applicatios. 96. Pages Pretice Hall, Ic. 8. Yeh, P. S. ad Wishma, A. F. Heat Trasfer Problems o a Direct-buried URD Trasformer. Coferece Paper, IEEE Paper No. 69 CP 663-PWR, Jue, 969. ASEE 4 Southeast Sectio Coferece

11 P. S. Yeh P. S. Yeh received his B.S. degree from the Natioal Taiwa Uiversity, i Taipei, Taiwa, the M.S. degree from the Uiversity of Illiois i Urbaa-Champaig, Illiois, ad the Ph.D. degree from Rutgers Uiversity i New Bruswick, New Jersey, all i Mechaical Egieerig. He is a professor emeritus i egieerig at Jacksoville State Uiversity, i Jacksoville, Alabama. His major areas of teachig ad research are i fluid mechaics, thermodyamics, heat trasfer, computer-aided desig, computer programmig, acoustics, ad evirometal egieerig. ASEE 4 Southeast Sectio Coferece