Convecton Heat Transfer Tetbook: Convecton Heat Transfer Adran Bean, John Wley & Sons Reference: Convectve Heat and Mass Transfer Kays, Crawford, and Wegand, McGraw-Hll Convecton Heat Transfer Vedat S. Arac and Poul S. Larsen, Prentce-Hall Inc Convecton Heat Transfer Content: 1. Flud Proertes and Conservaton Laws. Eternal/Internal Lamnar Flows. Eternal/Internal Natural Convecton 4. Eternal/Internal Turbulent Flows 5. Hgh Seed Flows Gradng: HW (0%)+Mdterm (0%) + Fnal (0%) + Reort (15%) 1
Wanted frcton force: F f cos 0 da 0 Skn frcton coeffcent C f 1 U S ressure drag: F sn P0 da F form drag coeffcent C 1 U A S Wanted heat transfer rate at the surface no-sl hyothess at wall : ure conducton adacent to the wall Fourer s law: T q0 k y y 0 convecton heat transfer coeffcent: ht T q0 0. e. T k y h T T 0 y 0 local Nusselt number: h Nu k ~ heat transfer rate when n flow heat transfer rate when statonary
Wanted averaged convecton heat transfer coeffcent: h 0 q T 0 avg 1 0 T q d 0 avg total heat transfer rate over (0,) = qd 0 h0 T 0 avg overall Nusselt number: Nu 0 h0 q0 k T avg k Analyss Methods scalng analyss : qualtatve analyss magntude of order related to what arameters and how? ntegral analyss : quanttatve analyss magntudes wth a lttle errors related to what arameters and how? smlarty analyss : eact analyss under model assumtons erturbaton analyss : crtcal analyss near some crtcal ont
Flud Proertes (1) vscosty coeffcent: kg m sec ; m sec Newtonan Fluds: u u u u k k temerature deendence: T, P usually assumed ( Flud Mechancs, Landau & Lfschfz, 1959) gases : as T T r T T c lqud : as T 4
Flud Proertes () thermal conductvty: k W m K ; k m sec c Fourer s Law: q (heat flu, W m ) kt sotroc k usually assumed temerature deendence: k k T, P kr k kc gases : lqud : k k as T as T T r T T c 5
Dmensonless Parameters (1) Reynolds number: nertal force Re UL ~ vscous force L Re ~ L U char. dffuson char. convecton tme tme Re > Re cr turbulent flows () Prandtl number: () Eckert number: momentum dffuson Pr ~ thermal dffuson U Ec c T knetc energy er unt mass ~ enthaly dfference er unt mass Ec << 1 neglgble vscous dssaton Hgh seed flows sgnfcant vscous dssaton Tme Dervatves total dervatve d Vobserver dt t Lagrangan/ materal dervatve D u t Euleran dervatve 6
Conservaton Laws dm dv Let some hyscal quantty er unt mass total amount of quantty wthn the control volume () dv outflow rate of quantty through the control surface (CS) CS u nda Conservat on requres : t q dv CS u nda sources source er unt tme er unt volume qdv Dfferental Form Dvergence Theorem: Conservaton requres: S a nda V a dv V a dv dv u nda sources qdv t CS Conservaton law: sources t dv u dv qdv Consder an nfntesmal control volume dv dv u dv qdv t t u q 7
Mass Conservaton total amount of quantty wthn the control volume () dv t u q mass: 1, no source q 0 t 0 u vector dentt y: a a a D u u u 0 t 1 D 1 D u volume change rate er unt volume momentum u Momentum Conservaton total amount of quantty wthn the control volume () dv Dvergence Theorem: S a nda V a dv V a dv source qdv force actng on the by ts surroundn = body forces + contact forces X dv n da CS X dv g fluds dv q X + 8
Momentum Conservaton t u q momentum u q X + t u u u u t u0 X Newtonan flud: Newtonan flud: u uu X t Energy Conservaton total energy: e (nternal energy er unt mass) body force X V 1 u u V (otental energy) sources eteranl heat generaton Dvergence Theorem: heat dffuson nto/out of CS CS a nda a dv work done on the by ts surroundn gs a u qdv q nda n da u S V V a dv t u q q u 9
total energy equaton: Energy Conservaton u u u u D e uu V q q u u 1 knetc energy equaton: u Du X X V DV V V u 0 u X u X t thermal energy equaton: De q q u u thermal (nternal) energy De q q u the tme change rate of the nternal energy of an nfntesmal control volume = the heat generaton rate + the net heat dffuson rate + ressure work rate done by surroundng flud + vscous dssaton rate 10
Thermal Energy Conservaton De q q u u vscous dssaton rate = rate at whch knetc energy s rreversbly converted to thermal energy by vscosty Newtonan flud: u u u u u u u u 1u u 1u u u u u u u vscous dssaton rate u u u u u u u u 4 u u 8u 1 8u 8u u u 8u1 u u1 u u u 1, 1 1 1 4u 1 u1 u u 4u 1 u1 u u 1 1 1 1 4 u u u u u u, 1 4 u1 u u1 u u u u u 1 1, 1 > 0 11
Temerature-based De q q u he dh de dtds ddtds d (frst law) s( T, ) s s ds dt d T T s dt d T st,, 1 s (, s T) (, ) (, ) ( T, ) ( T, ) ( T, ) T T (Mawell relaton) s dh T dt T d T Temerature-based De q q u s dh T dt T d T hh T, h c T s T T dh c dt 1T d Dh c DT 1T D 1
Temerature-based enthahy he Dh De 1 D 1 D De 1 D u De q q u Dh c DT 1T D DT D D c 1 T q q u u c DT q q T D Temerature c DT 1 1 T T D q q T Assumtons: (1) neglgble comressblty effect 0 () no eternal heat generaton q 0 () neglgble vscous dssaton 0 Ec 1 (4) Fourer s Law: q kt = thermal eanson coeffcent c DT T c u T kt t 1