Transient Heat Conduction

Transcription

1 ansen Hea Conducon In geneal emeaue of a body vaes wh me as well as oson. umed Sysem nalyss Ineo emeaues of some bodes eman essenally unfom a all mes dung a hea ansfe ocess. he emeaue of such bodes ae only a funcon of me = (). he hea ansfe analyss based on hs dealzaon s called lumed sysem analyss. Consde a body of abay shae of mass m volume V suface aea densy ρ and secfc hea C nally a a unfom emeaue. h = () Sold body m (mass) V (volume) ρ (densy) (nal em) (suface aea) Q = h [ ()] Fg. : umed sysem analyss. me = he body s laced no a medum a emeaue ( > ) wh a hea ansfe coeffcen h. n enegy balance of he sold fo a me neval d can be eessed as: hea ansfe no he body dung d = he ncease n he enegy of he body dung d h ( ) d = m C d Wh m = ρv and change of vaable d = d( ) we fnd: d VC d Inegang fom = o = b VC e b s M. Baham ENSC 388 (F9) ansen Conducon Hea ansfe

2 () b 3 b b b 3 > b > b Fg. : emeaue of a lum sysem. Usng above equaon we can deemne he emeaue () of a body a me o alenavely he me equed fo he emeaue o each a secfed value (). Noe ha he emeaue of a body aoaches he amben emeaue eonenally. lage value of b ndcaes ha he body wll aoach he envonmen emeaue n a sho me. b s ooonal o he suface aea bu nvesely ooonal o he mass and he secfc hea of he body. he oal amoun of hea ansfe beween a body and s suoundngs ove a me neval s: Q = m C [() ] Eleccal nalogy he behavo of lumed sysems shown n Fg. can be neeed as a hemal me consan VC b R C whee R s he essance o convecon hea ansfe and C s he lumed hemal caacance of he sold. ny ncease n R o C wll cause a sold o esond moe slowly M. Baham ENSC 388 (F9) ansen Conducon Hea ansfe

3 o changes n s hemal envonmen and wll ncease he me esond equed o each hemal equlbum. VC RC Ceon fo umed Sysem nalyss Fg. 3: hemal me consan. umed sysem aomaon ovdes a gea convenence n hea ansfe analyss. We wan o esablsh a ceon fo he alcably of he lumed sysem analyss. chaacesc lengh scale s defned as: V c non dmensonal aamee he Bo numbe s defned: hc B h B c c B h convecon a he suface of he body conducon whn he body conducon essance whn he body convecon essance a he suface of he body he Bo numbe s he ao of he nenal essance (conducon) o he eenal essance o hea convecon. umed sysem analyss assumes a unfom emeaue dsbuon houghou he body whch mles ha he conducon hea essance s zeo. hus he lumed sysem analyss s eac when B =. I s geneally acceed ha he lumed sysem analyss s alcable f B. M. Baham ENSC 388 (F9) ansen Conducon Hea ansfe 3

4 heefoe small bodes wh hgh hemal conducvy ae good canddaes fo lumed sysem analyss. Noe ha assumng h o be consan and unfom s an aomaon. Eamle hemocoule juncon whch may be aomaed by a shee s o be used fo emeaue measuemen n a gas seam. he convecon hea ansfe coeffcen beween he juncon suface and he gas s nown o be h = 4 Wm.K and he juncon hemohyscal oees ae = Wm.K C = 4 Jg.K and ρ = 85 gm 3. Deemne he juncon damee needed fo he hemocoule o have a me consan of s. If he juncon s a 5 C and s laced n a gas seam ha s a C how long wll ae fo he juncon o each 99 C? ssumons:. emeaue of he juncon s unfom a any nsan.. Radaon s neglgble. 3. osses hough he leads by conducon ae neglgble. 4. Consan oees. leads Gas seam = 5 C h = 4 Wm.K hemocoule juncon = 5 C = Wm.K C = 4 Jg.K ρ = 85 gm 3 d Soluon: o fnd he damee of he juncon we can use he me consan: D VC hd 6 Reaangng and subsung numecal values one fnds D =.76 mm. Now we can chec he valdy of he lumed sysem analyss. Wh c = 3 3 C M. Baham ENSC 388 (F9) ansen Conducon Hea ansfe 4

5 hc 4 B.35. umed analyss s OK. B <<. heefoe he lumed aomaon s an ecellen aomaon. he me equed fo he juncon o each = 99 C s b e b VC ln b 5. s ansen Conducon n age Plane Walls ong Cylndes and Shees he lumed sysem aomaon can be used fo small bodes of hghly conducve maeals. Bu n geneal emeaue s a funcon of oson as well as me. Consde a lane wall of hcness a long cylnde of adus and a shee of adus nally a a unfom emeaue. Inally a = Inally a = h Inally a = Plane wall ong cylnde Shee Fg. 4: Schemac fo smle geomees n whch hea ansfe s one dmensonal. M. Baham ENSC 388 (F9) ansen Conducon Hea ansfe 5

6 We also assume a consan hea ansfe coeffcen h and neglec adaon. he fomulaon of he one dmensonal ansen emeaue dsbuon () esuls n a aal dffeenal equaon (PDE) whch can be solved usng advanced mahemacal mehods. Fo lane wall he soluon nvolves seveal aamees: = ( α h ) whee α = ρc. By usng dmensonal gous we can educe he numbe of aamees. B o fnd he emeaue soluon fo lane wall.e. Caesan coodnae we should solve he alace s equaon wh bounday and nal condons: Bounday condons: Inal condon: () = So one can we: whee X h B (b) X Bo numbe Foue numbe h dmensonless dsance () (a) dmensonless emeaue he geneal soluon o he PDE n Eq. () wh he bounday condons and nal condons saed n Eqs. () s n he fom of an nfne sees: n e n n cos X able Cengle s boo lss soluons fo lane wall cylnde and shee. hee ae wo aoaches:. Use he fs em of he nfne sees soluon. hs mehod s only vald fo Foue numbe >.. Use he Hesle chas fo each geomey as shown n Fgs. 5 6 and 7. n M. Baham ENSC 388 (F9) ansen Conducon Hea ansfe 6

7 M. Baham ENSC 388 (F9) ansen Conducon Hea ansfe 7 Usng he Fs em Soluon he mamum eo assocaed wh mehod s less han %. Fo dffeen geomees we have:. whee sn e e cos e J shee cylnde wall whee and λ can be found fom able Cengel boo. Usng Hesle Chas hee ae hee chas Fgs. 5 o 7 one assocaed wh each geomey:. he fs cha s o deemne he emeaue a he cene a a gven me.. he second cha s o deemne he emeaue a ohe locaons a he same me n ems of. 3. he hd cha s o deemne he oal amoun of hea ansfe u o he me.