Exact Solutions of Three Nonlinear Heat Transfer Problems

Transcription

1 Engineering Letters, 9:3, EL_9_3_ Ext Solutions of Three Nonliner Het Trnsfer Prolems Mohmmd Dnish, Shshi Kumr nd Surendr Kumr Astrt In this work, three nonliner het trnsfer prolems nmely, stedy stte het ondution in rod, unstedy ooling of lumped system nd stedy stte het trnsfer from retngulr fin into the free spe y the rdition mehnism, hve een solved nlytilly. Erlier these three prolems were solved y vrious reserhers y using homotopy perturtion, homotopy nlysis nd optiml homotopy nlysis methods nd the pproximte series s were otined. Here, we hve otined ext nlytil s of these three prolems in terms of simple lgeri funtion, Lmert W funtion nd the Guss s hypergeometri funtion, respetively. These ext s gree very well with those otined y the numeril shemes nd re etter thn the reent pproximte s. Moreover, these n lso serve s the yrdstiks for future testing of the pproximte s. Index Terms Het trnsfer, Condution, Convetion, Rdition T I. INTRODUCTION HIS reserh work minly stresses on finding the ext nlytil of three nonliner het trnsfer prolems whih hve nonliner temperture dependent terms. The first prolem represents the stedy stte het ondution proess in metlli rod nd is desried y nonliner BVP (oundry vlue prolem) in seond order ODE (ordinry differentil eqution). Reently, Rji et l. [] hve solved this prolem y using well known pproximte method i.e. HPM (homotopy perturtion method), wheres Sjid nd Hyt [] nd Domirry nd Ndim [3] hve solved the sme prolem y using HPM nd nother very populr pproximte sheme i.e. HAM (homotopy nlysis method). These workers hve otined the results in the form of finite. The seond prolem, onsidered y Gnji [4] using HPM, y Asndy [] using HAM nd y Mrin nd Herișnu [6] using OHAM (optiml HAM), depits the unstedy het onvetion from lumped system. Mnusript reeived August,. Mohmmd Dnish is Assistnt Professor in the Deprtment of Chemil Engineering, Aligrh Muslim University, Aligrh-, Uttrprdesh, Indi nd t present is on study leve t the Deprtment of Chemil Engineering, Indin Institute of Tehnology Roorkee, Roorkee , Uttrkhnd, Indi ( e-mil: mdnish77@rediffmil.om). Shshi Kumr, Assoite Professor, Deprtment of Chemil Engineering, Indin Institute of Tehnology Roorkee, Roorkee , Uttrkhnd, Indi (e-mil: sshifh@iitr.ernet.in). Surendr Kumr, Professor, Deprtment of Chemil Engineering, Indin Institute of Tehnology Roorkee, Roorkee , Uttrkhnd, Indi, Tel: , Fx: (e-mil: skumr@iitr.ernet.in) The relted governing eqution of this prolem is expressed y nonliner initil vlue prolem (IVP) in first order ODE. The s were found in series form. The third prolem desries the stedy stte rditive het trnsfer from retngulr fin into the free spe nd the model eqution is give y nonliner BVP in seond order ODE. This prolem hs lso een reently onsidered y Gnji [4], Asndy [] nd Mrin nd Herișnu [6] y using HPM, HAM nd OHAM, respetively, nd the were found in terms of the trunted series. One should note tht the series s hve hngele degree of ury nd rdius of onvergene, nd re strongly dependent on the numer of terms in the series s well s on the prmeters vlues. Beuse of this, there remins region outside whih the series s strt deviting nd their regulr use eomes limited. Nonetheless, in suh ses efforts re mde either to otin the ext nlytil s or to solve the prolem with the help of some suitle numeril tehnique. Fortuntely, we hve shown tht ll the ove three mentioned prolems re extly solvle in terms of lgeri funtion, Lmert W funtion [7] nd hypergeometri funtion, respetively. These s hve een otined y using simple mthemtil mnipultions e.g. ssuming n impliit form of the or y reduing the eqution into simpler form y dding nd sutrting ertin terms, s elorted in the following setions. Thus found nlytil s re firly helpful sine: (i) Better insight of the tul physil proess is esily gined. (ii) These n strightforwrdly e utilized in finding the preise temperture profiles nd temperture grdients for whole rnge of prmeters' vlues disprte to their pproximte series ounterprts whih hve onvergene relted issues for the entire rnge of prmeters' vlues espeilly for the extreme vlues of prmeters. (iii) One n lso e deploy them to vlidte the ury of other pproximte s. Physil desription of the mentioned proesses, derivtion of respetive model equtions nd the methods to find the ext s re disussed elow. II. PROBLEM : HEAT CONDUCTION IN A METALLIC ROD This prolem minly portrys the stedy ondutive het trnsfer in metlli rod nd prtilly rises in estimting the therml ondutivity of metls e.g. het flow meters [8, 9]. In this prolem, the two ends of the rod re kept t different ut fixed tempertures nd het trnsfer (Advne online pulition: 4 August )

2 Engineering Letters, 9:3, EL_9_3_ tkes ple from higher temperture to the lower y the mehnism of ondution. In this ondution prolem, we ssume tht the therml ondutivity vries linerly with temperture nd there is no het loss to the surrounding from the round surfe of the rod. We onsider rod of length, L nd uniform ross setionl re, A with its end mintined t two different tempertures i.e. T ( x ) T nd T ( x L) T. For these stted ssumptions, the stedy stte energy lne over the rod gives in the following dimensionl eqution nd the ssoited BCs (oundry onditions): d dt A k( T ) () dx dx BCI: T T t x () BCII: T T t x L () T T Where k( T ) k is the temperture T T dependent therml ondutivity of the rod. With the introdution of the following dimensionless vriles, the governing eqution nd the ssoited BCs i.e. ()-(), trnsform into the following equtions i.e. ()-(): x T T, L T T '' ' () BCI: () () BCII: () () Where ' & '' represents the first nd seond order derivtives of with respet to, respetively. Following two different pprohes n e dopted to otin the ext of the ove eqution, s demonstrted elow: A. Approh I A reful inspetion of () shows tht it n onveniently e expressed in the following form: ' ' (3) Integrting the ove eqution two times with respet to, one otins the following qudrti eqution in : C C (4) Where C nd C re the onstnts of integrtion nd hve een found from the ssoited BCs i.e. () & (). Sustituting these vlues in (4) nd solving for, one finds the following two expliit s; two s pper euse of the nonliner nture of the eqution. () () Sine, seond does not stisfy the BCs nd is unrelisti it is, therefore, rejeted. If one expnds () round using Tylor series the following pproximte series is otined If one ompres it with the pproximte HPM [(47)] of Rji et l. [] nd pproximte HAM of Domirry nd Ndim [3] for the onvergene ontrol prmeter h (used therein), n urte ompline is oserved. However, we ould not ompre the results otined y ext with the results of Sjid nd Hyt [] sine no suh term ws provided. However, in this se the results were judged ginst those of Sjid nd Hyt [] y tulting the vlues of temperture grdients t nd (see Tle ). A lose onformity is oserved etween these vlues. The results otined y the urrent i.e. () hve lso een suessfully verified ginst those otined y (47) of Rji et l. [] nd those otined y numeril methods, s shown in Fig.. In this figure it is lerly visile tht the pproximte temperture profile otined y Rji et. l. [] devites ppreily even for moderte vlues of nd eomes redundnt for lrger vlues of. Although not shown, the sme hrteristis n lso e sried to the HAM of Domirry nd Ndim [3] for the onvergene ontrol prmeter h. On the ontrry, no devition is oserved in the present, even for higher vlues of. One notes from Fig. tht s vries from to, the temperture of the rod tends to reh the higher temperture ( ) nd thus sertin the ft tht with the inrese in therml ondutivity the temperture of the rod lso rises. Dimensionless Temperture, Ɵ(ξ) B. Approh II β =,,, Dimensionless Length, ξ In this pproh we ssume tht the derivtive ' is funtion of only i.e. ' p( ), in other words, the of () exhiits n impliit form i.e. f ( ) β=: Eq. (47), [] β=: Eq. (47), [] β=: Eq. (47), [] β=: Eq. (47), [] Fig. Dimensionless temperture profiles long the length of the rod (prolem ), solid lines: ext ; filled irle: numeril. d p Consequently, '', where p (still unknown) is d funtion of, only. It is useful to mention tht this pproh is quite helpful whenever the independent vrile is sent in the onerned eqution. Repling (Advne online pulition: 4 August )

3 Engineering Letters, 9:3, EL_9_3_ ' & '' in () y the ove respetive definitions, one otins: p d p (6) d Now, sustituting p y nd fter little ltertions the ove eqution redues to the following first order liner ODE: y ' y (7) Solving the ove first order liner ODE y integrting ftor method one finds: C y (8) Or p d d C Where C is onstnt of integrtion. Integrting the ove (9) one more, one finds the expression for [note tht the eqution elow is similr, in form, to the (4)]: C C (9) () C is nother onstnt of integrtion nd C nd C re evluted from the ssoited BCs, like in the first pproh. Sustituting the vlues of these onstnts in () nd solving for, one rrives t the following two s whih re extly sme s those given in () & (). () () Seond does not stisfy the BCs so disrded. Rest of the disussion remins sme s presented in Approh I. TABLE I COMPARISON OF DIMENSIONLESS TEMPERATURE GRADIENT AT BOTH THE ENDS OF THE ROD (PROBLEM ) S. No. β Numeril θ'() Sjid & Hyt [] Ext () Numeril θ'() III. PROBLEM : COOLING OF A LUMPED SYSTEM This prolem represents the temporry ooling of lumped system the speifi het of whih vries linerly with temperture. In rel world, this prolem rises in the ooling of heted stirred vessels nd ooling of eletroni omponents with high therml ondutivity et [8]. HPM, HAM nd OHAM s of this prolem hve een found y Gnji [4], Asndy [] nd Mrin nd Herișnu [6], respetively nd the s were otined Ext () / - 7/ 7/ 7/ in the form of series. The prolem is stted s: t the outset of the experiment, system with density, volume V nd het trnsfer re A, is exposed to surrounding t different temperture ( T ) nd het is trnsferred from the system to the surrounding y onvetion. The leding model eqution is derived y pplying the unstedy energy lne over the system nd is desried y the following nonliner IVP (initil vlue prolem) in first order ODE: dt V( T) ha( T T ) () dt IC: T () T () T T Where ( T ) is the het pity of T T the system showing liner dependeny on temperture nd h is the onstnt het trnsfer oeffiient. With the ssistne of the following dimensionless quntities, () & () ttin the dimensionless form given y (3) & (3), respetively. hat T T, V T T ' (3) IC: () (3) A simple rerrngement of the ove (3) yields: ' ' (4) Integrting (4) with respet to results in: Log[ ] C () Where C is the onstnt of integrtion nd using IC, it is found to e C. Sustituting k the so found vlue of C in (), provides the following ext nlytil. Log[ ] (6) Due to the ove impliit form of, it hs to e found for eh nd every y solving (6) with the help of some suitle itertive numeril sheme. This feture limits the repeted use of the ove formul. Keeping this in view, we now develop, from (6), the expliit form. A onstnt term Log[ ] is dded nd sutrted in (6) nd fter performing little modifition, (7) is otined. Log e Log (7) Eqution (7) n e further expressed s: e e (8) The L.H.S. of (8) n e repled y the Lmert W funtion (implemented s ProdutLog funtion in some mthemtil softwres e.g. Mthemti). A Lmert W y funtion is silly the inverse funtion of x ye i.e. y= Lmert(x) nd is symolized y y W ( x). In generl, the domin nd rnge of the funtion is the set of omplex vlues however, for x [, ) Lmert W funtion yields single rel vlues. For x (, ), Lmert W funtion e does not evlute to ny rel vlue wheres, for x [,) it omputes two rel vlues. Now, with this e (Advne online pulition: 4 August )

4 Engineering Letters, 9:3, EL_9_3_ funtion ville, the trnsient dimensionless temperture profile is given y: ProdutLog e (9) Expnding round y using Tylor series, yields the following expnsion whih hrmonizes with the (8) of Gnji [4] nd (9) of Asndy [] for h. e e e e 4e 3 e 3... Fig. ompres tht the trnsient temperture profiles otined y the present (9), HPM otined y Gnji [4] nd those otined y numeril sheme. It is ler tht the present mth very well with the numeril wheres, the s otined y Gnji [4] show onsiderle disrepnies exept for where the (3) eomes liner. Fig. lso supports the ft tht with the inrese in, the speifi het inreses whih in turn uses the derese in temperture grdient. Extending the omprison, the initil rtes of temperture hnge, given y the following (), hve lso een found using (9) nd plotted in Fig. 3 long with those otined y Asndy []. '() () Aury is evident y the overlpping profiles. Similr omprisons with the OHAM of Mrin nd Herișnu [6] hve een voided due to their more involved expression. However, it n e shown tht our present, eing ext in form, is superior to the pproximte of Mrin nd Herișnu [6]. Dimensionless Temperture, Ɵ(t) Ɵ'() β = -.,,.,.3. β = -.: Eq. (8), [4] β = : Eq. (8), [4]. β =.: Eq. (8), [4]. β = : Eq. (8), [4] Dimensionless Time, t Fig.. Trnsient profile of the dimensionless temperture (prolem ), solid lines: ext ; filled irle: numeril β Fig. 4, [] Fig. 3. Initil rte of hnge of dimensionless temperture vs. (prolem ) solid lines: ext ; filled irle: numeril IV. PROBLEM 3: STEADY STATE RADIATIVE HEAT TRANSFER FROM A RECTANGULAR FIN This prolem represents the stedy stte het trnsfer from retngulr fin to the free spe y the rdition mehnism. Suh situtions pper in the ooling of the heted prts of the spe vehiles. This prolem, too, hs een tkled y Gnji [4], Asndy [] nd Mrin nd Herișnu [6] with the help of HPM, HAM nd OHAM, respetively nd the s were otined in the form of series. We onsider retngulr fin hving ross setionl re A, perimeter P, length L nd the onstnt therml ondutivity nd emissivity s k nd, respetively. The fin se is mintined t higher temperture T nd the fin is trnsmitting the het energy into the spe y the mode of rdition. It is ssumed tht the stedy stte is previling nd the negligile het trnsfer tkes ple from fin end []. Keeping these ssumptions in view, the governing model eqution is derived y pplying the stedy energy lne over the fin element nd is desried y the following nonliner BVP in seond order ODE: d dt 4 4 A k P ( T Ts ) () dx dx BCI: T T t x L (t fin se) () dt BCII: dx t x (t fin end) () It is worthwhile to note tht the spe temperture n very well e repled y the solute zero temperture i.e. T [4-6]. Tking this ft into ount nd defining the s following dimensionless vriles, the ove equtions re onveniently expressed into the dimensionless form given y () - (). 3 T x PT L,, T L ka And the () - () eome d 4 d () BCI: t (t fin se) () d BCII: d t (t fin end) () To solve the ove BVP, the sme pproh hs een followed s dopted previously for the of prolem, nd here lso, it is ssumed tht the derivtive d d is d funtion of only i.e. p( ) where p is yet to e d d p found. This ssumption leds to '' d '' in () y this reltion, one otins: d p. Repling 4 (3) d Now, repling p with y, the (3) ttins the following first order liner ODE: dy 4 d (4) (Advne online pulition: 4 August )

5 Engineering Letters, 9:3, EL_9_3_ Integrting the ove eqution, one finds y C () C is onstnt of integrtion nd n e evluted with the help of BCII i.e. () nd is found to e C ; where is the unknown dimensionless temperture t the fin se. Sustituting this vlue of C in (), one gets d y (6) d A minor rerrngement of the ove eqution yields d d (7) Integrting the ove eqution etween the limits presried y the BCs I & II, following definite integrl is found. d d (8) The integrtion of the ove eqution gives the following result. 6 HG F,,, i 6 3/ 7 (9) The unknown is omputed y solving the following nonliner eqution whih hs een otined y foring (9) to stisfy the unutilized BCI i.e. t. 6 HG F,,, 6 i (3) 3/ 7 Where [ z] nd HG [,,, ] F z re the well known Gmm nd the Guss' Hypergeometri funtions, respetively nd re defined s follows []: z t z t e dt [ ] [ ] HG F z t t tz dt [,,, ] ( ) ( ) [ ] [ ] Gnji [4], Asndy [] nd Mrin nd Herișnu [6] hve solved this prolem y using HPM, HAM nd OHAM, respetively nd s re otined in terms of the series. For omprison purposes, the two terms HPM nd HAM s of Gnji [4] nd Asndy [] re reprodued elow, however, euse of omplexity in the expression of Mrin nd Herișnu [6], it hs not een onsidered here. 4 x x 6x Gnji 6 x x Asndy h h( h) (3) 4 x 6x h (3) 6 Figs. 4 &, plot the dimensionless temperture profiles otined y the ove pproximte series s, the urte numeril sheme s well s those otined y the presently otined ext i.e. (9) & (3). It n e noted tht in Fig. the sme vlue of the prmeter hve een tken s those onsidered in [4] nd [] i.e..7. It n e seen in Fig. 4 tht the profile otined y Gnji [4] devites to some extent with the numeril wheres the profile otined y the ext nlytil depits n exellent mthing with its numeril ounterprt. Dimensionless Temperture, Ɵ(ξ) Dimensionless Temperture, Ɵ(ξ) ε =.7: Eq. (4), [4] ε =.: Eq. (4), [4] Dimensionless Length, ξ Fig. 4. Dimensionless temperture profiles long the length of the fin (prolem 3), solid lines: ext ; filled irle: numeril ε =.7: seond order HAM, [] ε =.7: fifth order HAM, [] Dimensionless Length, ξ Fig.. Dimensionless temperture profiles long the length of the fin (prolem 3), solid lines: ext ; filled irle: numeril Similrly, in Fig., the two terms HPM of Gnji [4] yields divergent results wheres, the two term HAM of Asndy [] show minor devitions with the numerilly otined urte profile. However, the five term HAM otined y Asndy [] mthes well with the numeril. In ontrst to this, the ext nlytil i.e. (9) & (3) re in omplete greement with the numeril. It n e verified tht the devitions in the series s of Gnji [4] nd Asndy [], will inrese with the inrese in the vlue (Advne online pulition: 4 August )

6 Engineering Letters, 9:3, EL_9_3_ of, however, this is not true for the urrently derived ext. The true profiles signify the shrp derese in temperture with the inrese in the prmeter. This oservtion is in ompline with the physis of the prolem. V. CONCLUSION In this work, the three nonliner het trnsfer prolems of prtil interests hve een solved in n ext mnner nd the s re found in terms of elementry lgeri nd trnsendentl funtions. These prolems represent stedy stte het ondution in solid rod, the unstedy ooling of lumped prmeter system nd the stedy stte rditive het trnsfer from retngulr fin to the spe, respetively. The orresponding ext s hve een otined in terms of simple lgeri funtion, Lmert W funtion nd Guss s hypergeometri funtion, respetively. These nlytil s mth well with their numeril ounterprts nd re found to e finer thn the erlier otined pproximte s. From these ext s one n get etter piture of the physil proess unlike their pproximte lterntives; moreover, these n e pretty useful in judging the ury of other pproximte s nd re vlid for ll prmeter rnges. ACKNOWLEDGMENT M. Dnish is thnkful to his prent institution A.M.U., Aligrh-, U.P., Indi for kindly grnting study leve to pursue reserh t I.I.T. Roorkee, Roorkee-47667, Uttrkhnd, Indi. [W/m.K 4 ] Stephn-Boltzmnn onstnt (= ) [-] dimensionless time [-] dimensionless distne REFERENCES [] A. A. Rji, D. D. Gnji, H. Therin, Applition of homotopy perturtion method in nonliner het ondution nd onvetion equtions, Phys. Lett. A. vol. 36, pp. 7-73, 7. [] M. Sjid, T. Hyt, Comprison of HAM nd HPM methods in nonliner het ondution nd onvetion equtions, Nonliner Anlysis RWA, vol. 9, pp. 96-3, 8. [3] G. Domirry, N. Ndim, Assessment of homotopy nlysis method nd homotopy perturtion method in nonliner het trnsfer eqution, Int. Commun. Het Mss Trnsfer, vol. 3, pp. 93-, 8. [4] D. D. Gnji, The pplition of He's homotopy perturtion method to nonliner equtions rising in het trnsfer, Phys. Lett. A. vol. 3, pp , 6. [] S. Asndy, The pplition of homotopy nlysis method to nonliner equtions rising in het trnsfer, Phys. Lett. A. vol. 36, pp. 9-3, 6. [6] V. Mrin, N. Herișnu, Applition of optiml homotopy symptoti method for solving nonliner equtions rising in het trnsfer, Int. Commun. Het Mss Trnsfer, vol. 3, pp. 7-7, 8. [7] M. Dnish, Sh. Kumr, S. Kumr, Approximte expliit nlytil expressions of frition ftor for flow of Binghm fluids in smooth pipes using Adomin deomposition method, Commun. Nonliner S. Num. Simul., vol. 6, pp. 39-,. [8] A. Bejn, A. D. Kruss, Het trnsfer hndook, first ed., New Jersey: John Wiley & Sons, In., 3. [9] M. Dnish, Sh. Kumr, S. Kumr, Ext nlytil s of three het trnsfer models, Leture notes in Engineering nd Computer Siene: Proeedings of the World Congress on Engineering, WCE, 6-8 July,, London, U.K., pp-. [] A. D. Krus, A. Aziz, J. Welty, Extended Surfe Het Trnsfer, first ed., New York: John Wiley & Wiley,. [] M. Armowitz, I. Stegun, Hndook of Mthemtil Funtions, first ed., New York: Dover, 964. NOMENCLATURE A [m ] het trnsfer re A [m ] ross-setionl re,, [-] onstnts [J/kg.K] speifi het t temperture T ( T ) [J/kg.K] speifi het t temperture T C, C [-] onstnts of integrtion h [J/s.m.K] het trnsfer oeffiient k [J/s.m.K] therml ondutivity t temperture T k( T ) [J/s.m.K] therml ondutivity t temperture T L [m] length of rod p [-] funtion of t [s] time T [K] temperture T s [K] rdition sink temperture u [-] dummy vrile V [m 3 ] volume x [m] distne vrile y [-] funtion of z [-] dummy vrile Greek letters [-] dimensionless prmeter for k( T ) nd ( T ) [-] emissivity [-] ondution rdition prmeter [-] dimensionless temperture [kg/m 3 ] density (Advne online pulition: 4 August )