Figure 1: Schematic of a fluid element used for deriving the energy equation.

Transcription

1 Driation of th Enrg Eation ME 7710 Enironmntal Flid Dnamics Spring 01 This driation follos closl from Bird, Start and Lightfoot (1960) bt has bn tndd to incld radiation and phas chang. W can rit th 1 st la of thrmodnamics for an opn nstad sstm shon in th figr blo in ords as follos: ratof ratof ratof accmlation intrnal & kintic intrnal & kintic of intrnal & kintic nrg in nrg ot nrg b Adction b Adction 1 nt ratof nt ratof nt ratof nt ratof Work hat addd b hat addd b hat addd b don b th flid Condction Radiation Phas Chang lmnt on srrondings ( ) ( ) ( ) RN RN ( ) Figr 1: Schmatic of a flid lmnt sd for driing th nrg ation. 1

2 1. Rat of Accmlation of Intrnal () and Kintic Enrg ( ) ithin th lmnt: ( ) (1) t. Nt rat of Adction of Intrnal and Kintic Enrg into th olm lmnt: { ( ) ( ) } { ( ) ( ) } { ( ) ( ) } (). Nt rat of Enrg inpt b Condction into th olm lmnt (molclar): ( ) ( ) ( ) () 4. Nt rat of Enrg inpt b Radiation into th olm lmnt: ( Rn Rn ) ( Rn Rn ) ( Rn Rn ) (4) 5. Nt rat of Enrg inpt b Phas Chang into th olm lmnt (Not that this is a Bod Sorc trm). ( L E) (5) Whr L ( J kg -1 ) is th latnt hat of aporiation or condnsation and E (kg m - s -1 ) is th aporation rat or condnsation rat pr nit olm. 6. Nt rat of Work don b th flid lmnt against th srrondings. Rcall, that ork rat don b a forc is th magnitd of th forc mltiplid b th locit in th dirction of th forc, W F, ill rit th ork rats as forcs mltiplid b locitis acting on or flid lmnt. a. Work Against Bod Forcs Rat of doing ork against th graitational forc ( g g g ) (6)

3 b. Work Against Srfacs Forcs i. Rat of Doing ork against th Prssr at th si facs of a olm lmnt { } { } p p p p { ( p) ( p) } (7) Not that hr ar considring or ork rat to b ( p nda) hr n is th otardl pointing normal ctor from th flid lmnt. ii. Rat of doing Work Against th iscos Forcs Figr : Flid lmnt olm indicating th strss sign conntion for th driation.

4 4 { } { } { } (8) Combining Es (1-8), diiding b and taking th limit as 0, 0, and 0 ilds th complt nrg ation E L g g g Rn Rn Rn p p p t p As ith th driation of th momntm ation no tili Ntonian prssions to rlat th locit gradints and strsss, naml or mor compactl this ma b rittn, ij S ij δ ij

5 W no sbtract th Kintic Enrg ation (drid arl in th cors) from th complt nrg ation to ild th Thrmal Enrg Eation: D Dt p R n L E Φ II I III I I (9) Trm I II III I I Phsical intrprtation of ach trm Rat of Gain of intrnal nrg pr nit olm Rat of intrnal nrg inpt b Condction pr nit olm Rrsibl rat of intrnal nrg incras pr nit olm b Comprssion Rat of intrnal nrg inpt b Nt Radiation pr nit olm Rat of intrnal nrg inpt b Phas Chang Irrrsibl rat of intrnal nrg incras pr nit olm b iscos Dissipation Th iscos trm rittn ot in fll is Φ or, Φ S ij S ij. W old no lik to prss th Eation of Thrmal Enrg in trms of tmpratr and hat capacit rathr than intrnal nrg: Rcall from thrmodnamics, that ( α, T) absolt tmpratr. Ths,, hr α is th spcific olm and T th d dα dt α T T α mltipling b th dnsit and considring th sbstantial driatis D Dα C Dt α Dt T Dt 5

6 hrc is th spcific hat of th flid at constant olm. Writing th spcific olm as th inrs of th dnsit and sing th prodct rl of calcls, Dα D 1 1 D 1 Dt Dt Dt Thn for incomprssibl flo hr 0, ha C Dt R n L E Φ' (10) hr th iscos dissipation trm simplifis to Φ or Φ S ij S ij for incomprssibl flo. Using Forir s La of Hat Condction allos s to rit trm II of E. (9) in trms of tmpratr, k dt d k T hr k is th thrmal condctiit of th flid. Ths, k T k T Sbstitting in E. (10) gis s an ation for th chang in tmpratr, C Dt k T R n L E Φ' (11) This ation ilds tmpratr changs from: 1. Hat condction. Radiation dirgnc. Phas chang 4. iscos hating For a constant prssr flid can mak th folloing sbstittion d pdα C p dt hich for an incomprssibl flid lads to 6

7 C p Dt k T R n L E Φ' hich is ssntiall stating that can sitch C and C p. This jstifid hn th prssr trms ar nglctd in a gas flo nrg ation. What rmains is approimatl an nthalp chang. Dt k T 1 R n L E Φ' C p C p C p C p If iscos hating is small and dfin thrmal diffsiit as Dt K T 1 C p R n L E C p K k C p 7