UNSADY HA RANSFR Mny h rnsfr problms 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: 1, or hr = is h hrml diffusiviy
A hd/o body i is suddnly posd o fluid ih non h rnsfr offiin. ihr vlu h mprur givn im, or find im for givn mprur. Fig. 5.1 Q: Ho good n pproimion ould i b o sy h nnulr ylindr is mor or lss isohrml? A: Dpnds on h rliv imporn of h hrml onduiviy in h hrml irui omprd o h onviv h rnsfr offiin.
Bio No. Bi Dfind o dsrib h rliv rsisn in hrml irui of h onvion omprd Bi hl L / A 1/ ha Inrnl onduion rsisn ihin solid rnl onvion rsisn body surf L is hrrisi lngh of h body Bi : No onduion rsisn ll. h body is isohrml. Smll Bi: Conduion rsisn is lss imporn. h body my sill b pproimd s isohrml Lumpd pin nlysis n b prformd. Lrg Bi: Conduion rsisn is signifin. h body nno b rd s isohrml.
rnsin h rnsfr ih no inrnl rsisn: Lumpd rmr Anlysis Vlid for Bi<.1 ol Rsisn= R rnl + R inrnl Solid G: d d ha m p BC: i Soluion: l, hrfor d d ha m p
Lumpd rmr Anlysis ln i i i i i ha m p ha m p m p ha - o drmin h mprur givn im, or - o drmin h im rquird for h mprur o rh spifid vlu. No: mprur funion only of im nd no of sp!
) p( V ha L Bi L L hl V ha 1 1 hrml diffusiviy: (m² s -1 ) Lumpd rmr Anlysis
Fo Lumpd rmr Anlysis Dfin Fo s h Fourir numbr (dimnsionlss im) L nd Bio numbr Bi = p(-bi*fo) hl h mprur vriion n b prssd s C hr L is hrrisi lngh sl : rl o h siz of h solid invlovd in h problm for mpl, L r o (hlf - rdius) hn h solid is ylindr. L r o 3 (on - hird rdius) hn h solid is sphr L L (hlf hinss) hn h solid is pln ll ih L hinss
Spil ffs nd h Rol of Anlyil Soluions h ln ll: Soluion o h H quion for ln ll ih Symmril Convion Condiions (, ) = i 1 (,) l i h ( L, ), h = -L *=/L, h =+L
h ln ll: No: On spil vribiliy of mprur is inludd, hr is isn of svn diffrn indpndn vribls. 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 nondimnsionl numbrs.
Dimnsionlss mprur diffrn: Dimnsionlss oordin: Dimnsionlss im: h Bio Numbr: * L Fo L hl Bi * solid * i i h soluion for mprur ill no b funion of h ohr non-dimnsionl quniis * f ( *, Fo, Bi ) * Soluion: * C p Fo os C n n 4sin sin n n1 n n n n h roos (ignvlus) of h quion n b obind from bls givn in sndrd boos. n n n Bi
h On-rm Approimion Fo. Fo Vriion of mid-pln mprur ih im ( * ) * C 1 p 1 i From bls givn in sndrd boos, on n obin C1 nd 1 s funion of Bi. Vriion of mprur ih loion ( * ) nd im ( Fo ): * * os 1 * Fo Chng in hrml nrgy sorg ih im: s Q sin 1 * Q Q 1 1 Q V i
Numril Mhods for Unsdy H rnsfr Unsdy h rnsfr quion, no gnrion, onsn, ondimnsionl in Crsin oordin: S h rm on h lf hnd sid of bov q. is h sorg rm, rising ou of umulion/dplion of h in h domin undr onsidrion. No h h q. is pril diffrnil quion s rsul of n r indpndn vribl, im (). h orrsponding grid sysm is shon in fig. on n slid.
(d) ) (d) ) Ingrion ovr h onrol volum nd ovr im inrvl givs CV v CV d SdV d dv d dv d V S d A A dv d
If h mprur nod is ssumd o prvil ovr h hol onrol volum, pplying h nrl diffrning shm, on obins: n d V S d A A V No, n ssumpion is md bou h vriion of, nd ih im. By gnrlizing h pproh by mns of ighing prmr f bn nd 1: f f d n 1 S f f n n n n n ) (1 Rping h sm oprion for poins nd,
; ; ; ; Upon r-rrnging, dropping h suprsrip n, nd sing h quion ino h sndrd form f (1 f ) f (1 f ) f f (1 ) (1 ) b b S h im ingrion shm ould dpnd on h hoi of h prmr f. hn f =, h rsuling shm is plii ; hn < f 1, h rsuling shm is implii ; hn f = 1, h rsuling shm is fully implii, hn f = 1/, h rsuling shm is Crn-Niolson.
Vriion of ihin h im inrvl for diffrn shms f=.5 f= n f=1 +D plii shm Linrizing h sour rm s nd sing f = ( Su ) For sbiliy, ll offiins mus b posiiv in h disrizd quion. Hn, ( S )
) ( ) ( h bov limiion on im sp suggss h h plii shm boms vry pnsiv o improv spil ury. Hn, his mhod is gnrlly no rommndd for gnrl rnsin problms. Crn-Niolson shm Sing f =.5, h Crn-Niolson disrision boms: b S 1 ) ( 1 p p u S S b 1 ; ; ; ;
For sbiliy, ll offiin mus b posiiv in h disrizd quion, rquiring ) ( h Crn-Niolson shm only slighly lss rsriiv hn h plii mhod. I is bsd on nrl diffrning nd hn i is sond-ordr ur in im. h fully implii shm Sing f = 1, h fully implii disrision boms: S ; ; ;
Gnrl rmrs: 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 h slr. h sysm is solvd fr sling im sp Δ. For h implii shm, ll offiins r posiiv, hih ms i unondiionlly sbl for ny siz of im sp. Hn, h implii mhod is rommndd for gnrl purpos rnsin lulions bus of is robusnss nd unondiionl sbiliy.