1.Blowdown of a Pressurized Tank

Transcription

1 .Blowdown of a ressurized Tank This experimen consiss of saring wih a ank of known iniial condiions (i.e. pressure, emperaure ec.) and exiing he gas hrough a choked nozzle. The objecive is o es he heory developed which predics he ime required for he ank o be evacuaed (blowdown). One of he criical issues o be resolved from his es is o characerize he expansion process in some manner. Tha is, we will make a couple of assumpions in he following derivaion regarding he emperaure in he ank. We can use experimenal daa o deermine he validiy of our assumpions. Consider firs a ank sysem consising of a ank of pressurized air, a nozzle of known shape, and piping connecing he nozzle o he ank. We will sar wih he unseady mass conservaion equaion wrien in inegral form: d + nds ˆ 0 (.) The surfaces making up he boundary of he conrol volume in his case are he walls of he ank and piping plus he area of he nozzle hroa, i.e.: S S San k + Spipe + Shroa (.) Noe ha no air is flowing across eiher he ank or piping walls, herefore nˆ 0 an k nˆ 0 pipe (.3)

2 A he nozzle hroa we will assume ha he air flow is one-dimensional and perpendicular o he nozzle hroa cross-secion. The velociy vecor is hen parallel o he hroa area uni vecor n n (.4) ˆhroa ˆan k The second erm in (.) is now non-zero a he hroa only and evaluaed here i becomes nds ˆ A (.5) S Turning our aenion now o he firs erm in (.), we know ha he conrol volume is no changing wih ime, so ha he parial derivaive can hen be brough inside he inegral d d (.6) We will also assume ha he densiy in he ank is uniform over he enire ank volume. Tha is, a any given insan in ime, he densiy does no vary from one poin in he ank o anoher. Using his, (.6) can be evaluaed as d d (.7) Subsiuing (.7) and (.5) ino (.) we ge he equaion governing he blowdown of he ank: + A 0 (.8) The velociy hrough he hroa can be relaed o he hermodynamic properies a he hroa: M a γ RT (.9)

3 Now we noe ha if he flow hrough he nozzle is choked, hen he properies a he nozzle hroa will remain consan. Moreover, hese properies can be relaed o oal condiions (i.e. condiions in he ank) by assuming an isenropic expansion in he nozzle and applying he isenropic relaions a he hroa. Noe ha assuming he ank condiions are equal o he oal condiions in he ank neglecs any losses in he pipes. Since he flow is choked, Mach number a he hroa is uniy, and he isenropic relaions reduce o T T γ + (.0) γ + Applying (.9) and (.0) o (.8) gives, afer some simplificaion, he governing differenial equaion in erms of ank variables only γ + RA T + ( γ ) / + γ 0 (.) Equaion (.) is a single differenial equaion wih wo unknowns, and T, assuming ha we know he properies of he gas, he volume of he ank, and he area of he nozzle hroa. We mus close he sysem by adding an exra equaion or making an addiional assumpion. The approach aken here is o make wo differen assumpions regarding he expansion in he ank and compare resuls wih he experimens. I is no clear wha we can assume abou he densiy in he ank, so we will make an assumpion abou he ank emperaure and solve for he densiy. Isohermal Expansion Our firs assumpion will be o assume ha he emperaure in he ank is consan (isohermal). This is mos accurae for siuaions in which he ank wall is hin and expansion is slow. In his case, any drop in emperaure inside he ank would creae a hea flux from a consan emperaure reservoir, say he room, and he ank emperaure would warm back up o room emperaure. The isohermal approximaion assumes ha his process is insananeous and he emperaure of he ank says effecively consan. Since he emperaure is now known, we have a firs order differenial equaion for he ank densiy of he form dx A Bx 0 d + (.) where A and B are consans. 3

4 Equaion (.) has a soluion of he form x ( B/ A ) ce (.3) The consan c is deermined from iniial condiions, i.e. c x(0) (0). Applying he soluion (.3) o (.) gives he ank densiy as a funcion of ime: e λ (.4) where γ + ( γ ) λ γ RA T γ + / (.5) In he lab i is mos convenien o measure he ank gauge pressure. The soluion can be rewrien in erms of gauge pressure by using he equaion of sae RT RT g g + RT + RT am am (.6) where g is he ank gauge pressure and am is he amospheric pressure. Subsiuing (.6) ino (.4) gives + e (.7) λ g g am am Equaion (.7) is valid for he isohermal blowdown of a ank which exis hrough a choked nozzle. For all oher siuaions his equaion is an approximaion. Noe ha our original equaion, (.) made no assumpion as o wheher he flow is viscous or inviscid, however recall ha (.0) assumed an isenropic expansion hrough he diffuser of he nozzle and negleced losses in he pipe, so (.7) is also subjec o hese condiions. For a given nozzle and exi pressure, he oal pressure corresponding o firs criical ( s ) in he nozzle can be found. This is he poin a which he choked flow assumpion firs becomes invalid. Subsiuing he value for s ino (.7) and solving for can hen give he ime o firs criical and an esimaion of he blowdown ime of he ank. 4

5 Isenropic Expansion An isenropic expansion is effecively he opposie of an isohermal expansion. In his case, we assume ha here is no hea ransfer hrough he walls of he ank. Tha is he expansion is adiabaic. The heory of hermodynamics has hen shown ha he densiy and emperaure can be relaed by T T 0 T 0 or γ γ T We can now eliminae he ank emperaure in (.). Afer some simplificaion, he equaion becomes γ + RA T + (.8) ( γ γ γ + ) / + γ (.9) Clearly we can deermine he iniial ank condiions and he only unknown lef is once again he ank densiy. Equaion (.9) is a firs order non-linear differenial equaion of he form for n any number. This equaion has a soluion of he form dx n Dx 0 d + (.0) n x D n + c (.) where c is again deermined from iniial condiions. If we apply (.9) o (.), we ge a soluion for he densiy in he ank as a funcion of ime. In our case γ+ n (.) 5

6 The resuling soluion is γ + γ ( ) RA ( γ γ γ γ ) / T + c + (.3) If we se 0, we can deermine he unknown consan c c γ γ (.4) Noe ha we can facor (.4) ou of (.3). In his case, (.3) becomes γ + γ γ γ RA ( γ ) / 0 T + + (.5) We again wish o wrie he final soluion in erms of he ank gauge pressure. The isenropic relaion beween pressure and densiy is γ g + am g am γ (.6) The final soluion for he ank gauge pressure is γ γ + γ γ γ RA / g g + am + am ( γ ) T + (.7) Once again, assuming he ank pressure a firs criical is known, his equaion can be solved for he ime o firs criical and he ank blowdown ime. 6