A Study about One-Dimensional Steady State. Heat Transfer in Cylindrical and. Spherical Coordinates

Transcription

1 Appled Mathematcal Scences, Vol. 7, 03, no. 5, HIKARI Ltd, A Study about One-Dmensonal Steady State Heat ansfe n ylndcal and Sphecal oodnates Lesson Patck Santos José Osma Mano Júno Fedeal Unvesty of Itajubá, ampus of Itaba , Itaba, MG, Bazl Maco Donsete de ampos Fedeal Unvesty of of Mato Gosso Insttute of Exact and Eath Scences, , Baa do Gaças, M, Bazl Estane lao Romão Fedeal Unvesty of Itajubá, ampus of Itaba , Itaba, MG, Bazl estane3@yahoo.com.b opyght 03 Lesson Patck Santos et al. hs s an open access atcle dstbuted unde the eatve ommons Attbuton Lcense, whch pemts unestcted use, dstbuton, and epoducton n any medum, povded the ognal wok s popely cted. Abstact In ths atcle wee developed one-dmensonal steady state heat tansfe equatons n cylndcal and sphecal coodnates, neglectng o not the vscous dsspaton. We used second ode appoxmatons fo the development of a computatonal code. Some applcatons wee made and eos wee analyzed. Keywods: Heat tansfe, cylndcal and sphecal coodnate systems, second-ode appoxmaton fo tempeatue

2 68 Lesson Patck Santos et al. Intoducton Heat tansfe poblems ae pesent n seveal applcatons n ndusty and envonment. As an engneeng pactce, gew out of themodynamcs at aound the tun of the 0th centuy. hs aose because of the need to deal wth the desgn of heat tansfe equpment equed by emegng and gowng ndustes. Ealy applcatons ncluded steam geneatos fo locomotves and shps, and condenses fo powe geneaton plants. Late, the apdly developng petoleum and petochemcal ndustes began to eque ugged, lage-scale heat exchanges fo a vaety of pocesses [3]. Statng n the late 950s, at least thee unelated developments apdly changed the heat exchange ndusty. he fst was wth espect to heat-exchange desgn and szng, the geneal avalablty of computes pemtted the use of complex calculaton pocedues that wee not possble befoe. he second was the development of nuclea enegy ntoduced the need fo pecse desgn methods, especally n bolng heat tansfe. Fnally, the enegy css of the 970s seveely nceased the cost of enegy, tggeng a demand fo moe-effcent heat utlzaton [3]. As a esult, heat-tansfe technology suddenly became a pme ecpent of lage eseach funds, especally dung the 960s and 980s []. he man goal of the analyss of many tansfe heat poblems s to obtan the tempeatue pofle wthn the system and tansfe ate n cetan condtons o, altenatvely, the condtons needed, such as dmensons, shape, and flow measuement, among othes, to acheve a heat ate o tempeatue dstbuton, o both. he smulaton of heat tansfe can be done computatonally, the pocesses va numecal soluton of the govenng equatons. hs compute smulaton s wdely used n fundamental eseach and n ndustal applcatons esultng n that, nceasngly; new and bette numecal methods ae developed to mpove the accuacy, effcency, and ange of applcablty. In ths wok one-dmensonal steady state heat tansfe equaton n cylndcal and sphecal coodnates wee developed, neglectng o not the vscous dsspaton, usng second ode appoxmatons fo the development of a computatonal code. Numecal Fomulaton he heat conducton equatons n cylndcal and sphecal coodnate systems ae gven by Eq. (a-b q& q& (a (b

3 One-dmensonal steady state heat tansfe 69 whee, and q&, epesents the tempeatue, the adus and the heat tansfe ate, espectvely. onsdeng the followng appoxmatons and ( and substtutng n the pevous equatons, we have: q &!(a q &!! (b 3 Numecal Applcatons In the followng applcatons wee consdeed the geneal analytc soluton fo the cylndcal and sphecal coodnate, espectvely: B A ln( ( (3a B A ( (3b whee A and B ae constants. Applcaton : In ths case, wee consdeed the followng bounday condtons (0.50 and (, whee s constant. Fom Eq. (3a-b and usng the bounday condtons, we obtan, espectvely, ln( / (ln( ( and / ( (. he Fg. (a-b show the adal tempeatue pofle. A compason of esults obtaned usng the computatonal code wth the analytcal soluton s shown n able. able : Maxmum eo fo vaous values of Eo - cylndcal coodnates 7.59E-05.5E-04.7E E E-04 Eo - sphecal coodnates 8.94E-07.78E-06.6E E E-06

4 630 Lesson Patck Santos et al. (a (b Fgue : Radal tempeatue pofle fo mesh wth 0.05; nnost n (a cylndcal coodnates and (b sphecal coodnates. Applcaton : In ths applcaton, usng the Eq. (3a-b and the followng bounday condtons ( and (, whee s constant, we obtan, espectvely, ( 5ln( and ( (.5 / (.5. he adal tempeatue pofle s shown n next Fg. (a-b. (a (b Fgue : Radal tempeatue pofle fo mesh wth 0.05; nnost n (a cylndcal coodnates and (b sphecal coodnates. A compason of esults obtaned usng the computatonal code wth the analytcal soluton s shown n able. able : Maxmum eo fo vaous values of Eo - cylndcal coodnates.00e-0.00e-0.00e-0.00e-0.0e-0 Eo - sphecal coodnates 3.85E E E E E-0

5 One-dmensonal steady state heat tansfe 63 he eos obtaned n ths second applcaton wee elatvely hgh. In ode to educe t, the mesh was efned and was taken a value fo the constant by nceasng the numbe of nodes n the ntal condton. able 3: Maxmum eo fo Eo - cylndcal coodnates 9.58E E-04 Eo - sphecal coodnates.88e E-04 Applcaton 3: In ths applcaton, usng the Eq. (3a-b and the followng bounday condtons ( 0.5 and ( (, whee s constant, we obtan, espectvely ( 0.5ln( ( 0.5 and ( (0.5 /. he adal tempeatue pofle s shown n next Fg. (3a-b. (a (b Fgue 3: Radal tempeatue pofle fo mesh wth 0.05; nnost n (a cylndcal coodnates and (b sphecal coodnates. A compason of esults obtaned usng the computatonal code wth the analytcal soluton s shown n able 4. able 4: Maxmum eo fo vaous values of Eo - cylndcal coodnates 4.09E-03 4.E E E E-03 Eo - sphecal coodnates 6.E E E E E-03 he eos obtaned n ths second applcaton wee elatvely hgh. In ode to educe t, the mesh was efned and was taken a value fo the constant by nceasng the numbe of nodes n the ntal condton.

6 63 Lesson Patck Santos et al. able 5: Maxmum eo fo Eo - cylndcal coodnates.4e E-04 Eo - sphecal coodnates.5e-04.88e-04 Applcaton 4: In ths case, wee consdeed the geneal analytcal solutons whch epesent a themal convecton system ( ( q & / 4 ( B / A and ( ( q & / 4 B ln A and the followng bounday condtons (0.5 0 and (. he followng fgue shows gaphcally the behavo of the tempeatue vaaton as a functon of adus. (a (b Fgue 4: Radal tempeatue pofle fo mesh wth 0.05; nnost n (a cylndcal coodnates and (b sphecal coodnates. A compason of esults obtaned usng the computatonal code wth the analytcal soluton s shown n able 6. able 6: Maxmum eo fo vaous values of q&. q& Eo - cylndcal coodnates 6.30E-0.6E-0.89E-0.5E-0 3.5E-0 Eo - sphecal coodnates 3.8E-07.3E-06.98E-07.07E E-07 4 onclusons he ental Dffeence Method poved to be an excellent technque fo solvng poblems of heat conducton n cylndcal and sphecal geometes, except wth heat geneaton n cylndcal coodnates. In futue wok, we wll test ths technque n two and thee dmensonal domans.

7 One-dmensonal steady state heat tansfe 633 Acknowledgements he pesent wok was suppoted by the Natonal ouncl of Scentfc Development and echnology, NPq, Bazl and the Natonal ouncl fo the Impovement of Hghe Educaton, APES, Jovens alentos paa ênca pogam. Refeences [] J.aboek, Pocess Heat ansfe, hem. Eng., New Yok, 000. [] L.heodoe, Heat tansfe applcatons fo the pactcng engnee, John Wley & Sons, Hoboken, 0. [3] L. heodoe; F. Rcc and. V. Vlet, hemodynamcs fo the pactcng engnee, John Wley & Sons, Hoboken, 009. Receved: August, 03