Control Volume Derivation


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1 School of eospace Engineeing Conol Volume 1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass conseaion, ou conol mass law was m sys CM In inegal fom (inegaing oe conol mass), V CM
2 School of eospace Engineeing Conol Mass/Conol Volume Consie geneal conol mass an conol olumes ha ae moing in ime; coincie a ime Wan o see how o cone CM law o law CM (, ) V el elaie elociy of maeial cossing s iffeenial aea elemen on s el CM( ) Mus inegae 3 inegal oe ime epenen omain Conol Volume  Copyigh 1 by Jey M. Seizman. ll ighs esee. CM() () ( ) Conol Sufaces
3 School of eospace Engineeing Time Deiaie of 3 Inegal Sa wih sana limi alue efiniion of eiaie el CM( ) ( ) bu CM CM () Conol Volume 3 (, ) () CM (, ) V ( ) V (, ) lim lim Copyigh 1 by Jey M. Seizman. ll ighs esee. CM V ( ) (, ) ( ) ( ) (, ) (, ) V V CM V () (, ) Shae Re gion V (, ) ( ), V () () CM 1 (, ) V CM() () Shae Re gion V
4 School of eospace Engineeing Time Deiaie of 3 Inegal (con ) Bu [ ] em is efiniion of eiaie (, ) V (, ) V ( ) () lim V Use V el n wih a eco nomal o an poine ouwa Shae egion em becomes 1 lim Shae Re gion n (, ) V lim ( n) ( n) of a el el () (in) (ou) () el n el Conol Volume 4 Copyigh 1 by Jey M. Seizman. ll ighs esee.
5 School of eospace Engineeing Conol Volume Fom of Mass Conseaion Fom peious wo equaions, we hae CM V () () () ( n) pplying mass conseaion (LHS) V () () V el el ( n) Poucion ae of mass, Conol Volume 5 P & mass P & mass Copyigh 1 by Jey M. Seizman. ll ighs esee. Time ae of change of mass insie m Ne ouwa mass flow ae cossing m& oules m& Don hae o know mass isibuion in
6 School of eospace Engineeing Simplificaions Unifom flow (a ) n () i.e.,m& ( ) el el oules Woking in fame of efeence whee no moing el el SeaySae () oules V m& oules m& Conol Volume 6 Copyigh 1 by Jey M. Seizman. ll ighs esee.
7 School of eospace Engineeing Simplificaions (con ) Tansien, inegae oe fie ime 1 m V () ( ) ( ), m V,1 V m 1 m& m oules oules m m& oules 1 m Change of mass in beween 1 an Ne amoun of mass eneing beween 1 an Conol Volume 7 Copyigh 1 by Jey M. Seizman. ll ighs esee.
8 School of eospace Engineeing InClass Poblems 1. Niogen a kpa an 5 C flows hough a 35 mm iam. pipe a m/s. Fin he mass flow ae of niogen hough he pipe.. Liqui wae enes he squae uc shown wih an aeage elociy 1 mm m/s. Deemine he aeage elociy 45º an mass flow ae a he ei. 3. i flows in a cicula pipe wih a elociy of m/s. oun he pipe, in an annulus, is a n flow of ai, wih a elociy of 4 m/s. Boh flows ehaus ino a 15 cm iam. pipe. If he flow a e is i unifom, eemine he flow e elociy a e. ssume he D cm ai ensiy is consan. D1 cm Conol Volume 8 Copyigh 1 by Jey M. Seizman. ll ighs esee.
9 Conol Volume 9 Copyigh 1 by Jey M. Seizman. ll ighs esee. School of eospace Engineeing Diffeenial Fom of Mass Conseaion Fo Quasi1D, Seay Flow ssume flow elociy is 1D (only aiaion in ) in nonconsan aea, iffeenial olume m& consan ( ) Compae o coninuiy ( ) ( ) ( ) y ( ) ( ) sicly, y z z 1
10 School of eospace Engineeing Reynols Tanspo Theoem Poies geneal fom fo coneing fom CM o conseaion laws Fo gien eensie popey B, wih inensie esion β (somehing pe mass), ha follows a conseaion law can show B CM ( n) Will also lea o a PICO elaionship βv β el Replace wih appopiae Conol Mass Conseaion Law Poucion Inpu Change (in ime) Oupu Conol Volume 1 Copyigh 1 by Jey M. Seizman. ll ighs esee.
11 Conol Volume 11 Copyigh 1 by Jey M. Seizman. ll ighs esee. School of eospace Engineeing Reynols Tanspo Theoem: Eample Eample, linea momenum B m, β RTT hen gies (m) V CM ( n) Use Newon s Law e.g., gaiy (m) Fon F boy F CM on P& momenum Fon V el e.g., pessue (p), shea sess suface on ( P& ) ( n) el momenum
12 School of eospace Engineeing Momenum Conseaion: 1D Flow Fo seay inisci flow, no boy foces F F F sheasess on F boy on F sessesfom soli p n oules p n m & V m & ( n) p11 p pamb( 1) m & m & ( p1 pamb ) 1 ( p pamb ) m & m & p 1 1 n el n p am n F p Conol Volume 1 Copyigh 1 by Jey M. Seizman. ll ighs esee.
13 School of eospace Engineeing Momenum Conseaion: 1D Flow Diffeenial fom τ τ τ p L L L p p p p p ( p p)( ) p p m& ( ) m & p m & p p Fo seay, no boy foces an shea sess efine o be in  iecion L p peimee F p y,sies τ n pp/ n θ / pp  / ˆ Tem comes fom p p ( n ˆ ) y p p sin θ sin θ Conol Volume 13 Copyigh 1 by Jey M. Seizman. ll ighs esee.
14 School of eospace Engineeing Flow Raes an Flues Flow ae of popey ha is caie by he mass cossing he conol suface (in some iecion) is e.g., B β m& m & ( el n) ( kg s), ( N) fo1 D flow wih saionay The flu of ha same popey is gien by B& e.g., β ( n) ( massflu,kg / s / m ), ( momenumflu, N / m ) fo nonmoing el Conol Volume 14 Copyigh 1 by Jey M. Seizman. ll ighs esee.
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