UNSTEADY STATE HEAT CONDUCTION

Transcription

1 MODUL 5 UNADY A HA CONDUCION 5. Inroduion o his poin, hv onsidrd onduiv h rnsfr problms in hih h mprurs r indpndn of im. In mny ppliions, hovr, h mprurs r vrying ih im, nd rquir h undrsnding of h ompl im hisory of h mprur vriion. For mpl, in mllurgy, h h ring pross n b onrolld o dirly ff h hrrisis of h prossd mrils. Annling (slo ool n sofn mls nd improv duiliy. On h ohr hnd, qunhing (rpid ool n hrdn h srin boundry nd inrs srngh. In ordr o hrriz his rnsin bhvior, h full unsdy quion is ndd: τ y z L q (5. hr α is h hrml diffusiviy. ihou ny h gnrion nd onsidring spil vriion of mprur only in -dirion, h bov quion rdus o: τ (5. For h soluion of quion (5., nd o boundry ondiions in -dirion nd on iniil ondiion. Boundry ondiions, s h nm implis, r frqunly spifid long h physil boundry of n obj; hy n, hovr, lso b inrnl.g. non mprur grdin n inrnl lin of symmry. 5. Bio nd Fourir numbrs In som rnsin problms, h inrnl mprur grdins in h body my b qui smll nd insignifin. Y h mprur givn loion, or h vrg mprur of h obj, my b hnging qui rpidly ih im. From q. (5. n no h suh ould b h s for lrg hrml diffusiviy α. A mor mningful pproh is o onsidr h gnrl problm of rnsin ooling of n obj, suh s h hollo ylindr shon in figur 5.. s > Fig. 5.

2 For vry lrg r i, h h rnsfr r by onduion hrough h ylindr ll is pproimly s i i s q ( π ro l (πro l (5.3 ro ri L hr l is h lngh of h ylindr nd L is h mril hinss. h r of h rnsfr y from h our surf by onvion is q h π r l( (5.4 ( o s hr h is h vrg h rnsfr offiin for onvion from h nir surf. quing (5.3 nd (5.4 givs i s s hl Bio numbr (5.5 h Bio numbr is dimnsionlss, nd i n b hough of s h rio rsisn o inrnl h flo Bi rsisn o rnl h flo hnvr h Bio numbr is smll, h inrnl mprur grdins r lso smll nd rnsin problm n b rd by h lumpd hrml piy pproh. h lumpd piy ssumpion implis h h obj for nlysis is onsidrd o hv singl mssvrgd mprur. In h drivion shon bov, h signifin obj dimnsion s h onduion ph lngh, L r o r i. In gnrl, hrrisi lngh sl my b obind by dividing h volum of h solid by is surf r: V L (5.6 A s Using his mhod o drmin h hrrisi lngh sl, h orrsponding Bio numbr my b vlud for objs of ny shp, for mpl pl, ylindr, or sphr. As humb rul, if h Bio numbr urns ou o b lss hn., lumpd piy ssumpion is pplid. In his on, dimnsionlss im, non s h Fourir numbr, n b obind by muliplying h dimnsionl im by h hrml diffusiviy nd dividing by h squr of h hrrisi lngh: α dimnsionl ss im Fo L (5.7

3 5.3 Lumpd hrml piy nlysis h simpls siuion in n unsdy h rnsfr pross is o us h lumpd piy ssumpion, hrin ngl h mprur disribuion insid h solid nd only dl ih h h rnsfr bn h solid nd h mbin fluids. In ohr ords, r ssuming h h mprur insid h solid is onsn nd is qul o h surf mprur. olid (,, V q s ha ( h Fig. 5. h solid obj shon in figur 5. is ml pi hih is bing oold in ir fr ho forming. hrml nrgy is lving h obj from ll lmns of h surf, nd his is shon for simpliiy by singl rro. h firs l of hrmodynmis pplid o his problm is h ou of obj drs of inrnl hrml during im d nrgy of obj during im d No, if Bio numbr is smll nd mprur of h obj n b onsidrd o b uniform, his quion n b rin s ha s [ ] d Vd ( (5.8 or, d has d (5.9 V ( Ingring nd pplying h iniil ondiion (, ( has ln (5. i V ing h ponns of boh sids nd rrrnging, ( b (5. i hr has b /s (5. V i

4 No: In q. 5., b is posiiv quniy hving dimnsion (im -. h riprol of b is usully lld im onsn, hih hs h dimnsion of im. Qusion: h is h signifin of b? Ansr: Aording o q. 5., h mprur of body pprohs h mbin mprur ponnilly. In ohr ords, h mprur hngs rpidly in h bginning, nd hn sloly. A lrgr vlu of b indis h h body ill pproh h surrounding mprur in shorr im. You n visuliz his if you no h vribls in h numror nd dnominor of h prssion for b. As n ris, plo vs. for vrious vlus of b nd no h bhviour. R of onvion h rnsfr ny givn im : Q& ( has ( [ ] ol moun of h rnsfr bn h body nd h surrounding from o : Q m ( [ ] i Mimum h rnsfr (limi rhd hn body mprur quls h of h surrounding: Q m [ ] i 5.4 pil ffs nd h Rol of Anlyil oluions If h lumpd pin pproimion n no b md, onsidrion mus b givn o spil, s ll s mporl, vriions in mprur during h rnsin pross. h ln ll: oluion o h H quion for ln ll ih ymmril Convion Condiions For pln ll ih symmril onvion ondiions nd onsn propris, h h quion nd iniil/boundry ondiions r: τ (, i h ( L, l [ ]

5 No: On spil vribiliy of mprur is inludd, hr is isn of svn diffrn indpndn vribls. (,, i,, h,, α Ho my h funionl dpndn b simplifid? h nsr is Non-dimnsionlision. firs nd o undrsnd h physis bhind h phnomnon, idnify prmrs govrning h pross, nd group hm ino mningful non-dimnsionl numbrs. Non-dimnsionlision of H quion nd Iniil/Boundry Condiions: h folloing dimnsionlss quniis r dfind. Dimnsionlss mprur diffrn: * Dimnsionlss oordin: Dimnsionlss im: h Bio Numbr: * * α Fo L hl Bi solid L i h soluion for mprur ill no b funion of h ohr non-dimnsionl quniis * f ( *, Fo, Bi oluion: C p ζ C * n n * ( Fo os( ζ n 4sinζ ζ nζ n n n n ζ n sin( ζ n n h roos (ignvlus of h quion n b obind from bls givn in sndrd boos. i Bi

6 h On-rm Approimion Fo >. Vriion of mid-pln ( * mprur ih im ( Fo * C p( ζ Fo i From bls givn in sndrd boos, on n obin Vriion of mprur ih loion ( * nd im ( Fo : * ( ζ * * os Chng in hrml nrgy sorg ih im: Q s sinζ * Q Q ζ Q V ( i nd ζ s funion of Bi. C Cn h forgoing rsuls b usd for pln ll h is ll insuld on on sid nd onvivly hd or oold on h ohr? is Cn h forgoing rsuls b usd if n isohrml ondiion ( s i insnnously imposd on boh surfs of pln ll or on on surf of ll hos ohr surf is ll insuld? Grphil Rprsnion of h On-rm Approimion: h Hislr Chrs Midpln mprur:

7 mprur Disribuion Chng in hrml nrgy org Assumpions in using Hislr hrs: l Consn i nd hrml propris ovr h body l Consn boundry fluid by sp hng l impl gomry: slb, ylindr or sphr Limiions: l Fr from dgs l No h gnrion (Q l Rlivly long fr iniil ims (Fo >.

8 Rdil ysms Long Rods or phrs Hd or Coold by Convion Bi hr / Fo α / r imilr Hislr hrs r vilbl for rdil sysms in sndrd boos. Imporn ips: y nion o h lngh sl usd in hos hrs, nd lul your Bio numbr ordingly. 5.5 Numril mhods in rnsin h rnsfr: h Fini Volum Mhod Considring h sdy onvion-diffusion quion: ( φ div( φ u div ( Γ grd φ φ h im nd onrol volum ingrions giv: CV ( φ d ( ( dv n φ u da d n Γ grd φ da d Unsdy on-dimnsionl h onduion: A A CV φdv d ( (

9 Considr h on-dimnsionl onrol volum. Ingrion ovr h onrol volum nd ovr im inrvl givs: CV v CV d dv d dv d dv R-rin ( d V d A A dv d If h mprur nod is ssumd o prvil ovr h hol onrol volum, pplying h nrl diffrning shm, hv: ( ( d V d A A V An ssumpion bou h vriion of, nd ih im. By gnrlizing h pproh by mns of ighing prmr bn nd : [ ] d I ( hrfor, R-rrnging: [ ] [ ] Comprd ih sndrd form: [ ] [ ] [ ] b ( hr ( b hn, h rsuling shm is plii. hn <, h rsuling shm is implii. hn, h rsuling shm is fully implii.

10 hn /, h rsuling shm is h Crn-Niolson.

11 plii shm [ ] [ ] [ ] b ( h sour rm is linrisd s nd s p p u b h plii disrision: [ ] u ( hr h shm is bsd on brd diffrning nd is ylor sris runion rror ury is firs-ordr ih rsp o ims. All offiin mus b posiiv in h disrisd quion: ( > or ( > or > or ( < I boms vry pnsiv o improv spil ury. his mhod is no rommndd for gnrl rnsin problms. Nvrhlss, providd h h im sp siz is hosn ih r, h plii shm dsribd bov is ffiin for simpl onduion lulions.

12 Crn-Niolson shm [ ] [ ] [ ] b ( / b hr ( p p u b h mhod is implii nd simulnous quions for ll nod poins nd o b solvd h im sp. All offiin mus b posiiv in h disrisd quion: > or ( < his is only slighly lss rsriiv hn h plii mhod. h Crn-Niolson mhod is bsd on nrl diffrning nd hn i is sond-ordr ur in im. o, i is normlly usd in onjunion ih spil nrl diffrning.

13 h fully implii shm [ ] [ ] [ ] b ( hr A sysm of lgbri quions mus b solvd h im lvl. h ury of h shm is firs-ordr in im. h im mrhing produr srs ih givn iniil fild of mprur. h sysm is solvd fr sling im sp. All offiins r posiiv, hih ms h implii shm unondiionlly sbl for ny siz of im sp. h implii mhod is rommndd for gnrl purpos rnsin lulions bus of is robusnss nd unondiionl sbiliy.