Control Volume (Eulerian Approach)

Transcription

1 Thermynamic Systems Systems a fite prtin f matter r restricte prtin f space that we wish t exame Cntrl Mass (Lagrangian Apprach) fixe prtin f matter srrne by real r imagary bnaries matter can nt crss bnary Cntrl Vlme (Elerian Apprach) vlme f space srrne by real r imagary bnaries matter can crss bnary gas a pistn flw thrgh an enge Cnservatin Eqatins -2 Cnservatin Eqatins Gvern mechanics an thermynamics f systems Cntrl Mass Laws mass: can nt create r estry mass (e.g., neglect nclear reactins) mmentm: Newtn s Law, =ma energy: 1st Law f thermynamics, E=Q- W entrpy: 2n Law, S=Q/T+P s Prplsin systems generally emply fli flw nee t write cnservatin laws terms fr Cntrl Vlmes Cnservatin Eqatins -3 1

2 Reynls Transprt Therem - RTT Prvies general frm fr cnvertg cnservatin laws frm cntrl mass t cntrl vlmes Take arbitrary cntrl vlme (); can be mvg mass an energy ( heat Q an wrk W) can crss W cntrl srface Q nˆ m A nˆ () bnaries frces als act n an m ref. frame Clse mass Open = velcity f material crssg ative t mtin f Cnservatin Eqatins -4 RTT Eqatin Take any extensive prperty B, that fllws a cnservatin law an its tensive versin (per mass); can shw B V Replace with apprpriate Strage term Cntrl Mass (rate f crease Cnservatin Law sie ) Always leas t PICO atinship nˆ Net flx f prperty leavg, carrie by flw (tflw - flw) Prctin + Inpt = Change ( time) + Otpt Cnservatin Eqatins -5 2

3 Mass Cnservatin If prperty f terest B ( m) is mass B m, β 1 m m B rm RTT V nˆ : : ( m) ( m) 1V V nˆ nˆ Integral Cntrl Vlme rm f Mass Cnservatin Cnservatin Eqatins -6 Simplifie Mass Cnservatin Unifrm flw (at ) - n variatins acrss flw nˆ A A t tlets lets m m tlets lets A Steay-State 0 V m 1 t m m m 2 tlets lets m m 2 PI=CO Inpt = Otpt Cnservatin Eqatins -7 0 V nˆ 1 m3 m 3 3

4 Cnservatin f Mmentm Lear mmentm B m, β RTT then gives (m) V Use Newtn s Law ( m) ttal frce actg n fli V nˆ Inpt = Change + Ot - In Cnservatin Eqatins -8 mmentm transferre t fli nˆ Cnservatin Eqatins -9 rce Terms Exame ifferent frces that can act n matter r cntrl vlme By frces by n fv e.g., gravity by n with f e.g., pressre, shear,... srface n by frce/mass, i.e, acceleratin Srface frces free srfaces: nt cnnecte t sli by crssg cntrl srface cnnecte srfaces: sli bnaries where there are reactin frces 4

5 Srface rces ree srfaces nrmal stress pn ˆ srface shear shear stress 0 if visci Cnnecte sli srfaces frce n fli is reactin frce (verse) f frce n sli by n fli n sli by Cmbe sli by n fli pa pressre acts ppsite irectin t n n strt n fli Integral Cntrl Vlme rm f Mmentm Cnservatin pn ˆ shear fv V nˆ nˆ nˆ pa Cnservatin Eqatins -10 Cnservatin f Energy Energy: micrscpic + macrscpic frms f energy ternal energy 1 2 B E E Eketic E m 2 2 energy per mass β e e 2 RTT then gives E tt e tt V e tt nˆ Use 1 st Law Thermynamics fr energy cnservatin f cntrl mass Cnservatin Eqatins -11 5

6 1 st Law f Thermynamics Differential frm Int RTT Q E W Q t E W e Q V W Bt wrk is ate t frces ( x) actg n alreay exame sme ks f frces let s ate them t wrk terms t Q e t W t m Q W nˆ Cnservatin Eqatins -12 Relatinship By frces Wrk an rces x W W f V by li frces (stresses) W shear Reactin rces lmp t sefl wrk term, e.g., shaft wrk lw Wrk Sce we let wrk be psitive when ne BY fli W press p nˆ W shaft Cnservatin Eqatins -13 6

7 Cnservatin Eqatins -14 Energy Cnservatin Cmbe t RTT reslt (neglect shear frces) Q W f V p nˆ e V e nˆ Q W shaft shaft Cmbe flw wrk an energy flx f V Similar rm p p e e h p r reference frame mvg with cntrl vlme at cnstant velcity an n by frces 0 PI=CO Q W shaft In - Ot n ˆ ev h nˆ e V Change h nˆ Stagnatin Enthalpy 2 h h 2 Ot - In, f energy mass 7