Schl f Aespace Engeeg Cnsevat Equats Gven mechanics an themynamics f systems Cntl Mass Laws mass: can nt ceate esty mass (e.g., neglect nuclea eacts mmentum: Newtn s Law, ma enegy: 1st Law f themynamics, EδQ- δw entpy: 2n Law, SδQ/TδP s Ppuls systems geneally emply flui flw nee t wite cnsevat laws tems f Cntl Vlumes Cnsevat Equats -1 Cpyight 2001 by Jey M. Seitzman. All ights eseve. Schl f Aespace Engeeg Reynls Tanspt Theem - RTT Pvies geneal fm f cnvetg cnsevat laws fm cntl mass t cntl vlumes Take abitay cntl vlume (; can be mvg mass an enegy ( heat Q an wk W can css W cntl suface Q nˆ m A ρu el nˆ ( bunaies u el u fces als act ef. fame m n an Open Clse mass u el velcity f mateial cssg elative t mt f Cnsevat Equats -2 Cpyight 2001 by Jey M. Seitzman. All ights eseve. 1
Schl f Aespace Engeeg RTT Equat Take any extensive ppety B, that fllws a cnsevat law an its tensive ves β (pe mass; can shw B ρβv ρβ Replace with apppiate Stage tem Net flux f ppety leavg Cntl Mass (ate f cease, caie by flw Cnsevat Law sie (utflw - flw Always leas t PICO elatship Puct Input Change ( time Output el A Cnsevat Equats -3 Cpyight 2001 by Jey M. Seitzman. All ights eseve. Schl f Aespace Engeeg Mass Cnsevat If ppety f teest B ( m is mass B m, 1 m m m RTT ρβv ρβ : : ( m B ( m ρ1v ρ1 0 0 ρv ρ el el A el A A Integal Cntl Vlume m f Mass Cnsevat Cnsevat Equats -4 Cpyight 2001 by Jey M. Seitzman. All ights eseve. 2
Schl f Aespace Engeeg Simplifie Mass Cnsevat Unifm flw (at - n vaiats acss flw ( el nˆ A ρu ela ρ u ( t 0 A Steay-State 0 ρ V t ρv utlets m utlets ρ( uel nˆ m utlets lets lets m m PICO Input Output A lets ρu el m 1 A m 2 m m 1 2 m 3 m 3 Cnsevat Equats -5 Cpyight 2001 by Jey M. Seitzman. All ights eseve. Cnsevat Equats -6 Cpyight 2001 by Jey M. Seitzman. All ights eseve. Schl f Aespace Engeeg Cnsevat f Mmentum Lea mmentum RTT then gives (mu Use Newtn s Law ( m u ttal fce actg n flui P B mu, mmentum u ρuv P ρu A el puct f mmentum mmentum ρuv ρu el Puct Change Out - In A 3
Cnsevat Equats -7 Cpyight 2001 by Jey M. Seitzman. All ights eseve. Schl f Aespace Engeeg ce Tems Exame iffeent fces that can act n matte u cntl vlume By fces e.g., gavity n by by n ρfv with f e.g., pessue, shea,... suface n by fce/mass, i.e, acceleat Suface fces fee sufaces: nt cnnecte t sli by cssg cntl suface cnnecte sufaces: sli bunaies whee thee ae eact fces Schl f Aespace Engeeg Suface ces ee sufaces nmal stess suface fee pna ˆ σ shea stess A shea fee 0 if visci Cnnecte sli sufaces fce n flui is eact fce (vese f fce n sli by n flui Cmbe sli by n flui pen n sli by pa pessue acts ppsite iect t n n stut n flui Integal Cntl Vlume m f Mmentum Cnsevat pna ˆ σsheaa ρfv ρuv ρu( uel nˆ A pen nˆ σ nˆ pa Cnsevat Equats -8 Cpyight 2001 by Jey M. Seitzman. All ights eseve. 4
Schl f Aespace Engeeg Cnsevat f Enegy Enegy: micscpic macscpic fms f enegy tenal enegy 1 2 B E E Eketic E mu 2 2 u enegy pe mass e e 2 RTT then gives ( E tt tt V tt el A Use 1 st Law Themynamics f enegy cnsevat f cntl mass Cnsevat Equats -9 Cpyight 2001 by Jey M. Seitzman. All ights eseve. Schl f Aespace Engeeg 1 st Law f Themynamics Diffeential fm E δq δw E δq δwut Q W Int RTT Q W ut V But wk is elate t fces ( x actg n aleay exame sme ks f fces let s elate them t wk tems ut ut m el Q W A Cnsevat Equats -10 Cpyight 2001 by Jey M. Seitzman. All ights eseve. 5
Relatship By fces Schl f Aespace Engeeg Wk an ces x W u W lui fces (stesses W shea σ ua React ces pen lump t useful wk tem, e.g., shaft wk Sce we let wk be psitive when ne BY flui by ρf uv W pess p( u nˆ A lw Wk pen W shaft Cnsevat Equats -11 Cpyight 2001 by Jey M. Seitzman. All ights eseve. Cnsevat Equats -12 Cpyight 2001 by Jey M. Seitzman. All ights eseve. Schl f Aespace Engeeg Enegy Cnsevat Cmbe t RTT esult (neglect shea fces Q W ρf uv p A V Q W shaft shaft Cmbe flw wk an enegy flux ρf uv pen Simila m p p ρ ρ e ρh ρ ρ p u uel na ˆ V pen efeence fame mvg with cntl vlume at cnstant velcity an n by fces 0 PICO Q W shaft In - Out ( ρh ( uel nˆ V Change ρh A el A Stagnat Enthalpy 2 u h h 2 A Out - In, f enegy mass 6