Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY. Paper CIT06-097 ABRÃO MACANDI DONGALA Unversdade do Estado do Ro de Janero, UERJ, Faculdade de Engenhara. Rua São Francsco Xaver, 54 0550-03 Ro de Janero, RJ, Brasl. abramado@hotmal.com LEON MATOS RIBEIRO DE LIMA Unversdade do Estado do Ro de Janero, UERJ, Faculdade de Engenhara. Rua São Francsco Xaver, 54 0550-03 Ro de Janero, RJ, Brasl. leonmatos@hotmal.com ROGÉRIO MARTINS SALDANHA DA GAMA Departamento de Engenhara Mecânca, Unversdade do Estado do Ro de Janero, UERJ. Rua São Francsco Xaver, 54 0550-03 Ro de Janero, RJ, Brasl. rsgama@terra.com.br Abstract. In ths work a systematc procedure s proposed for smulatng the conducton heat transfer process n a sold wth a strong dependence of the thermal conductvty on the temperature. Such knd of nherently nonlnear partal dfferental equaton, subjected to classcal lnear boundary condtons, wll be solved wth the ad of a Krchoff Transform by means of a sequence of very smple lnear problems. The proposed procedure provdes the exact soluton of the problem and nduces fnte dmensonal approxmatons that can be used for computatonal smulatons. Some typcal cases are smulated by means of a fnte dfference scheme. Keywords: Temperature dependent thermal conductvty, Krchoff transform, heat transfer, nonlnear equatons system.. Introducton Most of the conducton heat transfer phenomena are descrbed under the assumpton of temperature ndependent thermal conductvty. Such hypothess s mathematcally convenent because, n general, gves rse to lnear partal dfferental equatons. Nevertheless, the thermal conductvty s always a temperature dependent functon. Many tmes, neglectng such dependence (assumng constant thermal conductvty, we have an nadequate mathematcal descrpton of the conducton heat transfer process. The man objectve of ths work s to provde a relable and systematc procedure for descrbng the conducton heat transfer n a rgd sold, wth temperature dependent thermal conductvty, subjected to lnear boundary condtons (Newton s law of coolng. Ths procedure s exact and employs only the tools utlzed n problems n whch the thermal conductvty s assumed to be a constant. The frst step conssts of employng the Krchoff transform n order to change the orgnal problem nto another one consstng of a lnear partal dfferental equaton subjected to a nonlnear boundary condton. The second step conssts of regardng the new problem as the lmt of a sequence of lnear problems. So, the problems wth temperature dependent thermal conductvty wll be regarded as the lmt of a sequence whose elements are soluton of lnear problems.. Governng Equatons Let us consder a rgd, opaque and sotropc body at rest wth doman represented by bounded open set Ω wth boundary Ω. The steady-state heat transfer process nsde ths body s mathematcally descrbed by the followng ellptc partal dfferental equaton
Proceedngs of the ENCIT 006, ABCM, Curtba PR, Brazl Paper CIT06-097 ( kgradt + q = 0 dv n Ω ( where T, q and k denote, respectvely, the temperature feld, the nternal heat generaton rate (per unt volume and the thermal conductvty. In ths work the thermal conductvty s assumed to be a functon of the local temperature. In other words, ( k = k T ( Assumng that body boundary and the envronment exchanges energy accordng to Newton s law of coolng, the boundary condtons assocated wth equaton ( are gven by ( kgradt n = h( T T on Ω (3 n whch n s the unt outward normal (defned on Ω, h s the convecton heat transfer coeffcent and T s a temperature of reference The resultng problem (nonlnear s gven by dv ( kgradt 0 ( kgradt n= h( T T n Ω on Ω (4 3. The Krchoff Transform Snce the thermal conductvty s always postve-valued, the new varable ω (Krchoff transform defned by ω= T k ( ξ dξ = f ( T (5 T 0 beng an nvertble functon of T. Ths defnton allows us to wrte gradω= k grad T (6 So, the orgnal problem can be rewrtten as follows dv( gradω 0 n ( gradω n= h( fˆ ( ω T Ω on Ω (7 where T f ( ω. Although above problem remans nonlnear, ths nonlnearty takes place only on the boundary, not n the partal dfferental equaton. It s to be notced that, snce the thermal conductvty s everywhere postve, dω dt k > 0 > 0 and > 0 everywhere (8 dt dω so, the temperature s a strctly ncreasng functon of ω. For nstance, f we have k = constant f T < T0 k = (9 k = constant f T T0 t s easy to show that
Proceedngs of the ENCIT 006, ABCM, Curtba PR, Brazl Paper CIT06-097 ˆ ω ω ω T0 (0 k k k k T f ( = + + + 4. Constructng the Soluton from a Sequence of Lnear Problems The soluton of dv ( gradω ( gradω n= h( fˆ 0 ( ω n T Ω on Ω ( can be represented by the lmt of the nondecreasng sequence [,,,...] obtaned from the soluton of the lnear problems below dv ( grad ( grad + β = h( fˆ + + ( T n whose elements are 0 n = α + β on Ω ( 0 α n whch α s a, suffcently large, constant and 0 0. So, ( Ω It s remarkable that, for each, the functon + s the unknown and the functon f s always known n (, beng evaluated from the followng equaton = T 0 f ( k ( ζ dζ s known. For each spatal poston, the root of the above equaton s unque. Ths unqueness s supported by equaton (8.. In reference The constant α must be large enough for ensurng that, at any pont of Ω, + [] such result s proven as well as t s provded an upper bound for the constant α [3]. For the problem consdered n ths work t s suffcent to choose α such that h α (4 k MIN where k s the mnmum value of the thermal conductvty. MIN 5. Convergence The lmt of the sequence [,,,...] 0, denoted here by the problem. To prove ths asserton, let us begn showng that (3 exsts and s, n fact, a soluton of s a soluton of (7. In other words dv ( grad ( grad n= h( fˆ 0 n Ω ( T on Ω (5
Proceedngs of the ENCIT 006, ABCM, Curtba PR, Brazl Paper CIT06-097 Snce β s gven by β h f = ( T α we have that ( and (5 concde. So, s a soluton. Now, takng nto account that the sequence s nondecreasng and has an upper bound, we ensure the convergence, once that the soluton of (5 belongs to the same space of the solutons of ( for each [4] e [5]. 6. An Example Let us consder the followng problem (sphercal body wth unform heat generaton, surrounded by the same medum, wth h= and T = - n some system of unts d dt r k 0 for 0 r + = < (7 dt k = T at r= dr n whch t s assumed that T represents an absolute temperature and k = 3T +. The soluton s easly reached and gven by ( r / T = + + 3 9 Now, let us employ the proposed procedure. Wth the Krchoff transform, the problem yelds d dω r 0 for 0 r + = < / dω 4 ω = f ( ω = + at r dr 3 + = 9 3 The lnear procedure for reachng the elements of the sequence s represented as follows d d 0 0 r + for r + = < d+ = α + + β at r= dr 4 β = + α 3 + 9 3 or, smply as d d 0 0 r + + = for r< / d+ 4 = α+ α + + at r= / dr 3 9 3 Snce the problem makes sense only for T > 0, we have that k MIN >. So, we can work wth any α /. We shall use α = 3! The general soluton of equaton (6 (8 (9 (0 (
Proceedngs of the ENCIT 006, ABCM, Curtba PR, Brazl Paper CIT06-097 d d 0 0 r + + = for r< ( s r + = + C+ 6 for 0 r< (3 where the constant C +, for > 0, s obtaned from the boundary condton. In other words, 4 = 3 C+ 3 C C 3 + 6 + 6 + + + 3 9 3 6 Then, C+ = C+ + C 3 3 3 The constant C s obtaned from 5 = 3 C C 3 + = 6 8 If we employ α = 0, then C+ = C + + C 0 3 3 and the constant C s obtaned from = 0 C C 3 + = 6 60 Table presents a comparson between the values obtaned for / C wth two dstnct values of α (4 (5 (6 (7 (8 = 3 4 5 0 0 50 00 500 000 α = 3 α = 0 C = C = 0.3708367838 0.45039985346 0.584033696 0.57707808559 0.67955775 0.800503959 0.940947974 0.998754569 0.999995078 0.999999985099 0.999999985099 0.995859053 0.54395070 0.3090837348 0.460059896 0.60744856563 0.394597507 0.445758444 0.6793455535 0.8660859 0.9998455009 0.99999994984 Table A comparson between results obtaned wth α = 3 and α = 0. It can be proven that C =. Hence, the lmt of the sequence s gven by r = + for 0 r< (9 6 The soluton ω s exactly the lmt of the sequence. Takng nto account that
Proceedngs of the ENCIT 006, ABCM, Curtba PR, Brazl Paper CIT06-097 / 4 ω f ( ω = T = + 3 + 9 3 we have the followng result (concdent wth the exact soluton, prevously obtaned 4 r T = + + + 3 9 3 6 / (30 (3 7. A computatonal numercal smulaton example A computatonal support has been developed parallel to the mathematcal modelng n order to provde numercal results of the proposed method. All the programs were made n MATLAB. Now, a computatonal smulaton for a typcal problem wll be shown. The physcal characterstcs of the problem are descrbed by the followng parameters: convecton heat transfer coeffcent h = 5, heat generaton q = 0, reference temperature T = 0. The conducton s processed n a rectangular flat plate. The spatal doman s descrbed n a rectangular cartesan coordnates system. The plate s represented by a rectangular elements mesh for Fnte Dfferences Method applcaton, and the lnear equatons system s solved by the Gauss-Sedel teratve method. The temperature dependence of the thermal conductvty s gven by: k k = k = 45 T < = 0 T In ths smulaton we assume α = 8. The followng fgures show the teratve evoluton of the smulaton.. Fgure. frst teraton result Fgure. seventh teraton result
Proceedngs of the ENCIT 006, ABCM, Curtba PR, Brazl Paper CIT06-097 Fgure 3. thrteenth teraton result Fgure 4. seventeenth teraton result Fgure 5. twenty thrd teraton result Fgure 6. thrteth and fnal result Fgure 7. plane dvdng the temperature feld n two regons of constant thermal conductvty
Proceedngs of the ENCIT 006, ABCM, Curtba PR, Brazl Paper CIT06-097 8. Fnal remarks The presented procedure proved to be a smple, but effcent subsdy for solvng problems of conducton heat transfer wth temperature dependent thermal conductvty, by means of classc tool employed on heat transfer lnear problems, such as fnte dfferences. 9. References Slattery, J. C., 97, Momentum, energy and mass transfer n contnua, McGraw-Hll, Tokyo. Gama, R. M. S., An a pror upper bound estmate for conducton heat transfer problems wth temperaturedependent thermal conductvty (to appear Int. J. Non-Lnear Mech. Gama, R. M. S., 000, An upper bound estmate for a class of conducton heat transfer problems wth nonlnear boundary condtons, Int. Comm. Heat Mass Transfer, vol. 7, pp.955-964. Helmberg, G., 974, Introducton to spectral theory n Hlbert space, North-Holland. John, F., 98, Partal dfferental equatons, Sprnger-Verlag. Wyle, C. R., 960, Advanced Engneerng Mathematcs, McGraw-Hll.