Cooling of a hot metal forging. , dt dt

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1 Tranen Conducon Uneady Analy - Lumped Thermal Capacy Model Performed when; Hea ranfer whn a yem produced a unform emperaure drbuon n he yem (mall emperaure graden). The emperaure change whn he yem condered unform. The old ha a large hermal conducvy wh a low hea lo/gan. The hea lo/gan beween he old and urroundng flud large compared o he hea ranfer whn he old elf. Example: Ho meal forgng of a old meal whch nally a T, quenched by placng n a flud bah a a emperaure of T, T <T. Coolng of a ho meal forgng The emperaure of he old expeced o drop o he flud emperaure a a reul of convecon a he oldflud nerface. If he old emperaure aumed o be unform a any nan, (emperaure graden whn he old negleced), an energy balance wll gve, E or ha ( T T) Vc [c p pecfc hea capacy] ou E Wh ranformaon of varable, = T - T, and dθ =, d d Vc p d ha d The equaon may be negraed from me = 0 wht(0) = T, Vc p d d ha 0 Inegrang gve Vc p ln or T T ha exp exp ha T T Vc p = τln ( T T T a T ) p d m = V nmg.fkm/feb2011/march2014/march2016 Page 1

2 T T ha T T exp Vc b Knowng he emperaure of he old a me, he rae of hea ranfer a ha me can be deermned from Newon law of coolng, Q () = ha [T() T ] W The oal amoun of hea ranfer beween he body and he urroundng medum over he me nerval = 0 o he change n energy conen of he body, Q = mc p [T() T ] kj The amoun of hea ranfer reache upper lm when he body reache he urroundng emperaure T, hu he maxmum hea ranfer beween he body and urroundng, Q max = mc p [T T ] kj The dfference beween he old and flud emperaure decay exponenally o zero a approache nfny Tranen emperaure repone lumped capacance old for dfferen hermal me conan. The hermal me conan appearng n (4.4) defned a 1 Vc RC ha ( ) here R he reance o convecon hea ranfer, C he lumped hermal capacance of he old. Q. Wha wll happen when R or C ncreae? Now, he oal hea ranfer may alo be obaned from / Vc / / Q qd ha d ha e d ha e d Vc e ha 0 we ge [can you perform he negraon?] Q ( Vc) 1 exp nmg.fkm/feb2011/march2014/march2016 Page 2

3 where Q cloely relaed o he change n energy n he old, Q E Q pove for quenchng (old experence a decreae n energy). Valdy of he Lumped Capacance Mehod The mehod mple and convenen o ue for ranen heang/coolng problem So, le u conder a urface wh emperaure T,1, whle he oher urface expoed o a flud a T, where T <T,1. The urface expoed o he flud urface a T,2 wh T <T,2 <T,1. An energy balance on he urface gve (Hea lo due o conducon) = (hea ganed by convecon) ka L (T,1 T,2 ) = ha(t,2 T )whch upon rearrangemen gve T,1 T,2 = (L/kA) = R cond = hl = B T,2 T (1/hA) R conv k The Bo number - dmenonle - preen n conducon problem aocaed wh urface convecon effec - reflecve of he emperaure drop whn a old relave o he emperaure dfference beween ha old urface and he urroundng flud. When B<<1, reaonable o aume a unform emperaure drbuon whn he old a any me durng he ranen proce. When B<<1, he reance o conducon whn he old >> he reance o convecon acro he flud boundary layer. Conder a plane wall whch nally a a emperaure T uddenly mmered n a flud of T <T, conderng 1-dmenonal n x where we are nereed o know abou T(x,), 3 condon may occur Tranen emperaure drbuon for dfferen B number n a plane wall ymmercally cooled by convecon. Q. Can you decrbe he phycal phenomena n each of he hree cae? I mperave ha you deermne B number fr when you are faced wh a ranen hea ranfer problem. nmg.fkm/feb2011/march2014/march2016 Page 3

4 The error aocaed wh ung he lumped capacance mehod mall f B = hl c k < 0.1 The erm L c he characerc lengh, a rao of he old volume o he urface area, L c V/A. () L c = L for a plane wall of hckne 2L, () L c = R 0 /2 for a long cylnder, () L c = R 0 /3 for a phere. Try o oban he 3 relaonhp above froml c V/A.Wha aumpon were made? Wha happen when hee aumpon were no aken? Now, knowng ha L c V/A, now expreed a ha h hlc k hlc ha or 2 2 B Fo Vc clc k c Lc k Lc Vc where Fo (Fourer number) 2 L c Fourer number a dmenonle me, ubung no gve T T exp( B Fo) T T General Lumped Capacance Analy Tranen conducon n a old may occur when here convecon hea ranfer o/from an adjonng flud. a emperaure dfference beween he old and he urroundng ga or vacuum caung a radaon exchange change n nernal hermal energy change n emperaure. a hea flux appled a a poron or all or he old urface a hermal energy generaed whn he old.e pang elecrcal curren Conder a uaon where hermal condon whn a old may be nfluenced mulaneouly by convecon, radaon, an appled urface hea flux, and an nernal hea generaon. Inally ( = 0): T T T urr Then q and q are naed, c, V, T(0) = T urroundng, T urr q rad q E g, E q conv T, h A,h A (c,r) Conrol urface for general lumped capacance analy nmg.fkm/feb2011/march2014/march2016 Page 4

5 Or from The mpoed hea flux q and he convecon-radaon hea ranfer occur a muually excluve poron of he urface, A,h and A (c,r), repecvely. Convecon-radaon hea ranfer aumed o be from he urface. The urface, may dffera,h A (c,r). Applyng an energy balancea any nan, q A h E g q conv q, ( rad ) A ( c, r) Vc d q A h E 4 4, g [ h( T T ) ( T Tur ) A ( c, r) ] Vc d Equaon a non-lnear, 1 -order, nonhomogeneou, ODE ha canno be negraed o oban an exac oluon! Equaon may be mplfed under hee condon; () Cae I: No mpoed hea flux or hea generaon, no convecon or convecon may be negleced 4 4 ( T Tur ) A ( c, r) Vc d () Cae 2: Radaon may be negleced and h ndependen of me d a b 0 d wherea = (ha,c /Vc) and b = [(q A,h + E g )/Vc] Q. Can you come up wh a oluon for each of he cae above? Spaal Effec In cae where he lumped capacance mehod napproprae, and he emperaure graden whn he old no longer neglgble, we need o olve he dffuon equaon Conder a plane wall (em-nfne old) where only 1 paal coordnae needed for he ranen analy, wh no hea generaon and conan hermal conducvy, we have 2 T T = 1 x 2 α (*) Two BC and one IC are needed o oban a unque oluon. If we conder he plane wall whch nally(=0) a T experencng convecon hea ranfer a he wall, T, h T, h x BC #1: T(x,0) = T, T = 0 x x=0 BC #2: k T = h[t(l, ) T x ] x=l ymmery BC! convecve BC nmg.fkm/feb2011/march2014/march2016 Page 5

6 Noe ha he emperaure n he wall depend on many parameer, T = T(x,, T, T, L, k,, h) The above problem may be olved () Analycally - Non-dmenonalzng fr, - Drecly ung eparaon of varable mehod () Numercally If equaon non-dmenonalzed, θ θ θ = T T T T x x L α L 2 Fo where he ame dmenonle Fourer number. Equaon (*) now become 2 θ θ x 2 = Fo T(x,0) = T, * (x *,0) = T T T T =1 BC #2: wh he BC now (BC #2: a = 0* = 0) T θ = 0 x x=0 x 2 = 0 x =0 k T = h[t(l, ) T x ] θ x=l x hl = x = x L =L =1 k θ (1, ) = Bθ (1, ) L The equaon may be olved ung mehod of eparaon of varable, aumng ha * (x *,Fo) = X(x * )F(Fo) Whou non-dmenonalzng, 2 T x = 1 T α may be olved alo hrough eparaon of varable mehod, aumng T(x,) = X(x)() reulng n 1 2 X X x 2 = 1 = conan Q. In boh cae, non-dmenonalzng or whou non-dmenonalzng, wha would be he approprae eparable conan? Shape Facor for mul-dmenonal conducon and complex geomere For 2 urface each mananed a conan emperaure of T 1 and T 2 repecvely, he eady rae of conducon beween hee 2 urface gven by, Q = Sk(T 1 T 2 ) where S he conducon hape facor S ha he dmenon of lengh and k he hermal conducvy of he medum beween he urface. S depend on he geomery of he yem only ee Table 3.5 for common urface. The conducon hape facor relaed o he hermal reance, R = 1 ks Check ou wha happen when he hermal conducvy, k = 1! nmg.fkm/feb2011/march2014/march2016 Page 6