MIT Hea ranfer Dynamc mdel 4.3./SG nvecn and cndcn and lmped mdel. Hea cnvecn If we have a rface wh he emperare and a rrndng fld wh he emperare a where a hgher han we have a hea flw a Φ h [W] () where he area f he rface and h he ceffcen f hea ranfer. a h Fgre : Hea cnvecn If he rrndng fld crclaed by a fan r pmp we are alkng f frced cnvecn. If he fld mn a rel f byancy frce, we are alkng f free (r naral) cnvecn. The hery fr he calclan f he ceffcen f hea ranfer he man bjec f he dcplne f Hea Tranfer. gd nrdcn can be een n Hea and Ma Tranfer by Incrpera and DeW, Wley,. Fr many prpe ne can e emaed vale - ypcal vale are gven n able : Prce h [W/(m K)] Free cnvecn Gae -5 Lqd 5- Frced cnvecn Gae 5-5 Lqd - Table : Typcal vale f he cnvecn hea ranfer ceffcen. Hea cndcn nder a ld cylnder a hwn n fgre. he ne end he emperare kep n and n he her end a and >. The rd nlaed n de,.e. hea nly ranmed n he hrnal drecn, frm he h end he clder end. k L Fgre : Hea cndcn n a ld cylnder (r fla plae wh hckne L)
MIT Hea ranfer Dynamc mdel 4.3./SG The hea flw can be calclaed a k Φ [W] () L where k he ceffcen f cndcvy fr he maeral ed, L he lengh and he cr ecnal area f he cylnder. Typcal vale fr k gven n able Maeral k [W/(m K)] Mneral fber.3-.5 Wd.-.6 Sanle Seel 3- Seel 4-6 lmnm bber 33 Table : Typcal vale f he cnvecn hea ranfer ceffcen 3. Thermal reance we wll ee laer n h paper cnvenen make an analgy elecrc crc. Eqan () can be rewren Φ a [W] (3) cnv where cnv he cnvecve reance hea ranfer. Y mgh remember Ohm Law ayng ha Vlage = eance me rren r U =. Or = U/ whch he ame a (3) f we be he hea flw wh he elecrc crren, he vlage drp by he emperare drp and he elecrc reance by he hermal reance. h cnv [K/W] (4) Fr cnvecn we have cnv = /(h ).e. he hgher h and/r he lwer reance, whch gve meanng. Fr he hea cndcn prblem we can make he ame analgy gvng he reance hea ranfer by cndcn L k cnd [K/W] (5) We wll e h analgy laer draw eqvalen hermal crc wh lmped capacance and reance elemen. 4. Mlple lmped parameer mdel Nw le ry cmbne he w prncple, ncldng he law f energy balance:
MIT Hea ranfer Dynamc mdel 3 4.3./SG Fgre 3 hw a mdel f a dble wall nlaed n he pace beween he w hermal heavy brck wall. h h Fgre 3: Dble wall (belw: n elecrc mdel f he wall) The elecrc analgy hw ha he prblem can be lked lke an elecrc crc wh 3 reance and w capacr We nw ame, ha he emperare n he w brck wall are alm even drbed, e. hee w wall can be lked lke lmped. Energy balance fr he nner wall, n. gve: Φ Φ (6) where Φ h (7) k Φ (8) ρ (9) c Energy balance fr he er wall, n. gve: Φ Φ () where Φ h () () ρ c Thee eqan can be lved n Smlnk, ee fgre 4
MIT Hea ranfer Dynamc mdel 4 4.3./SG Fgre 4: Mdel f he dble wall [ ] PHI [W] 3 Pl f emperare 5 5 5 5 5 Pl hea flw 5 5 5 me, a [h] PHI PHI PHI Fgre 5: el, baed n nal emperare = Daa fr he mdel: h_ = 8; % [W/(mK)] ef. f hea ranfer, nner h_ = ; % [W/(mK)] ef. f hea ranfer, er = ; % [m] rea k =.5; % [W/(m*K)] ef. f hea cndcn =.; % [m] Thckne, nlan _ = ; % [ ] Inner emperare _ = ; % [ ] Oer emperare c_ = 88; % [J/(kgK)] Spec. hea cap., nner wall c_ = 88; % [J/(kgK)] Spec. hea cap., er wall rh_ = 45; % [kg/m3] Deny, nner wall rh_ = 45; % [kg/m3] Deny, er wall _ =.; % [m] Thckne, nner wall _ =.; % [m] Thckne, er wall m_ m_ =rh_*_*; % [J/K] Hea capacy, nner wall =rh_*_*; % [J/K] Hea capacy, er wall n=; % Inal emperare, nner wall n=; % Inal emperare, er wall The ln f a gven nal vale prblem hwn n fgre 5. 5. Sae-Space frmlan pecal way f preenng he eqan called ae pace. On vecr frm lk lke h x x B y x D The bld ymbl ndcae array r marce
MIT Hea ranfer Dynamc mdel 5 4.3./SG Her x =, a clmn vecr repreenng he emperaren n he fve ne. a w dmennal vecr repreenng he amben emperare g. The y vecr an p vecr here he w emperare. Ung he example abve, wh he fve ne, we ge: If we defne h and k and h Then we have frm (6) (9) and frm () (): Then and B l n hen and D In h frmlan he Smlnk mdel mple: Fgre 6: Sae-Space frmlan
MIT Hea ranfer Dynamc mdel 6 4.3./SG The cnen n he Sae-Space blck a fllw _ = -(/(_*_)+/(_*_)); _ = /(_*_); _ = /(_*_); _ = -(/(_*_)+/(_*_)); B_ = /(_*_); B_ = ; B_ = ; B_ = /(_*_); =[ ; ] B =[B_ B_; B_ B_] =[ ; ] D =[ ; ] % Spdaa: mme =5*36; m('m wall', mme) fgre() h=pl(a/36, (:,), 'r-',a/36, (:,), 'b-'); e(h,'lnewdh',) grd % æer grd ax([-nf nf 3]) xlabel('a [h]') ylabel(' [ ]') legend('_','_') Pl % Sae-Space frmlan % 3.3./SG cle all clear clc 3 5 h_ = 8; % [W/(mK)] ef. f hea ranfer, nner h_ = ; % [W/(mK)] ef. f hea ranfer, er = ; % [m] rea k =.5; % [W/(m*K)] ef. f hea cndcn =.; % [m] Thckne, nlan _ = ; % [ ] Inner emperare _ = ; % [ ] Oer emperare c_ = 88; % [J/(kgK)] Spec. hea cap., nner wall c_ = 88; % [J/(kgK)] Spec. hea cap., er wall rh_ = 45; % [kg/m3] Deny, nner wall rh_ = 45; % [kg/m3] Deny, er wall _ =.; % [m] Thckne, nner wall _ =.; % [m] Thckne, er wall [ ] 5 5 5 5 5 a [h] m_ m _ =rh_*_*; % [J/K] Hea capacy, nner wall =rh_*_*; % [J/K] Hea capacy, er wall = c_*m_; % [J/K] Hea capacy, nner wall = c_*m_; % [J/K] Hea capacy, er wall _ = /(h_*); % [K/W] Thermal reance, nner cnvecn _ = /(k*); % [K/W] Thermal reance, nlan _ = /(h_*); % [K/W] Thermal reance, er cnvecn n=; % Inal emperare, nner wall n=; % Inal emperare, er wall