ASME231 Atmsheri hermdynamis NC A& State U Deartment f Physis Dr. Yuh-Lang Lin htt://meslab.rg ylin@nat.edu Leture 10 Adiabati Presses (Se.3.5 f Hess) [Classial equatin editr: 0 dq ] Definitin: If a thermdynami system hanges its hysial state (i.e.,, and/r ) withut any heat being either added t r extrated frm the system, the hanging ress is said t be adiabati. Mathematially, adiabati resses an be simly defined by dq 0 Ntie that the first law f thermdynamis an be written in varius frms, suh as dq du dw, dq vd d, dq dhd, dq d d. herefre, the first law f thermdynamis fr an adiabati ress redues t the fllwing, r v d d 0 (3.5.1) 1
d d 0. (3.5.2) Eq. (3.5.2) nstrains the relatins amng the three state variables,, and fr adiabati resses. We als knw that these three state variables are nstrained by the equatin f state fr ideal gases, r. (3.5.3) Substituting (3.5.3) int (3.5.2), we get d d 0. (3.5.4) Dividing Eq. (3.5.4) by leads t, r d d 0, ( / ) d d 0, (3.5.5) Integrate (3.5.5) frm the initial state (, ) t any state (, ), we have 2
3 0. d d his leads t. aking antilg n bth sides, we have. / (3.5.6) Eq. (3.5.6) is alled the Pissn's equatin, whih is a very imrtant equatin fr adiabati resses. It exresses the temerature hange as ressure hanges in an adiabati ress. If the temerature at the riginal ressure level is, then the final temerature after this adiabati ress will beme / (3.5.7)
where =/ is a nstant, in units f hpa (mb), and the wuld-be temerature is alled tential temerature. Fr the dry air, =287/1004=0.286. As an be seen frm Eqs. (3.5.6) and (3.5.7), is a nstant (i.e. at = ) fr an insulated gaseus system f fixed msitin, i.e. fr an adiabati ress. hus, if an air arel is brught adiabatially t a higher level (smaller ), then the temerature f this air arel will dr due t exansin. On the ther hand, the temerature will inrease when the air arel is brught dwn ( larger) due t mressin. Eq. (3.5.6), i.e. the Pissn's equatin, an be exressed in different frms, suh as: nstant, nstant, 1 nstant, (3.5.8) where =/ and = / v are nstants. Claim: is a nservative rerty fr adiabati resses, i.e., d=0 if dq=0. Prf: Starting frm the definitin (3.5.7), ( ) 4
aking the derivative n the abve equatin, d d (3.5.9) whih leads t r d d d, d ( ) d d, (3.5.10) Cmaring (3.5.10) with the 1st law dq d d, we btain d dq 0, Sine bth and annt be zer, therefre d must be zer, i.e., tential temerature is nserved fr adiabati resses. Eq. (3.5.9) may als be used t evaluate the hange in tential temerature during a nn-adiabati r diabati ress. 5
In the real atmshere, many resses an ntribute t heating, suh as (a) surfae radiative heating r ling (b) latent heat due t hase hange (ndensatin, evaratin, freezing, sublimatin, et.) () fritin (d) transfer f sensible heat by turbulene (e.g., frm the ean), et. S, reisely seaking, many real atmsheri resses are nt adiabati. Let's examine the abve diabati heating resses. (a) is imrtant in the lwer atmshere sine the mst f the radiative energy the atmshere reeives is frm the earth (g wave radiatin). adiatin uld be imrtant n t f a stratus lud, t; (b) latent heat is imrtant fr mist air and in regins f strng vertial mtin (nvetin); () fritin is imrtant near the surfae in the lanetary bundary layer (PBL); (d) is imrtant als near the surfae. S, mst f the diabati heat resses ur either near the surfae r in/near the nvetive regin. hus, exet in thse regins, the atmshere an be treated as arximately adiabati. he adiabati ress an als be reresented in a - diagram. Befre we d this, let us nsider an isthermal 6
ress, whih an be reresented by a urve f ==nstant. Nw, let draw a few f these istherms and nsider an adiabati ress by mressing a ylinder filled with air frm vlume 2 t 1. Due t the mressin, the temerature f the air in the ylinder during this adiabati ress will inrease, s the adiabati urve will be steeer than the istherms. Fig.10.1: - diagram. 1 < 2 < 3 are istherms. a b: isthermal ress (==nstant) a : adiabati ress / 1.4 [ v nstant (steeer)] 7