# Section 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow

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1 Section 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow Anderson: Chapter 2 pp

2 Equation of State: Section 1 Review p = R g T " > R g = R u M w - R u = J/ K-(kg-mole) - R g (air) = J/ K-(kg-mole) Relationship of R g to specific heats, c p = c v + R g = c p c v Internal Energy and Enthalpy h = e + Pv c v = de " dt v c p = " dh dt p 2

3 Chapter 1 Review (cont d) First Law of Thermodynamics, reversible process de = dq pdv dh = dq + vdp First Law of Thermodynamics, isentropic process (adiabatic, reversible) de = pdv dh = vdp 3

4 Chapter 1 Review (cont d) Second Law of Thermodynamics, nonadiabatic, reversible process " s 2 s 1 = c p ln 2 T 2 T 1 ' R g ln " p 2 p 1 ' Second Law of Thermodynamics, isentropic process (adiabatic, reversible) p 2 = T 2 p 1 " T 1 ' ' (1 p 2 p 1 " = ' 2 " ' 1 ( 4

5 Chapter 1 Review (concluded) Speed of Sound for calorically Perfect gas c = R g T Mathematic definition of Mach Number M = V R g T 5

6 Concept of a Control Volume Arbitrary Volume (C.V), fixed in space, with fluid flux across its boundaries Outer surface of C.V is known as Control Surface () System defined as the Mass Within the control volume at Some Instant in time t Mass within control volume Must OBEY Laws of mass And momentum conservation min mout 6

7 Conservation of Mass At any instant in time, the mass contained within the C.V. is conserved t ( ) = m m C.V """ C.V. in m """ C.V. out Within an elemental piece dv of the control volume dm C.V = dv And the total Contained mass is m C.V = """ c.v. dv min dv mout 7

8 Conservation of Mass (continued) The mass flow out of the C.V. across an elemental piece of the control surface, ds, is d m out " = V out cos ( )ds And the incremental mass-flow into the C.V. is: d m in " = V in cos ' ( )ds v min n ds dv mout Integrating over the entire control surface m C.V. = '' "V > > ds "V > > ds '' = ( (t ) ''' c.v. "dv* 8

9 Conservation of Mass (concluded) One Dimensional Steady Flow (mass flow in = mass flow out) t c.v. ' "dv) ( = 0 = * *> "V*> ds ' ( + " 1 V 1 A 1 = " 2 V 2 A 2 continuity equation 9

10 Conservation of Momentum Newton s second law applied to control volume > " F = d dt mv > = Momentum flow across + Rate of Change of Momentum within C.V. Momentum flow across control surface m V > " But from earlier analysis, across an incremental Surface area m = V "> "> ds m (( V "> "> "> = V ds ' ' n V v (( V "> V n 10

11 Conservation of Momentum (cont d) Rate of Momentum Change within Control Volume t (mv"> ) n dv n But from earlier (continuity) analysis For an incremental volume, dv V V m = dv And total net rate of momentum change within the control volume is t (mv"> )C.V. = t dvv "> ' ((( = C.V. ((( C.V. t V "> ' dv 11

12 Conservation of Momentum (cont d) Then Newton s second law becomes > " F = d dt mv > = > 'V> ds (( V > ) + ((( Resolution of Forces Acting on Control Volume C.V. )t 'V > dv - Body Forces (electo-magnetic, gravitational, buoyancy etc.) - Pressure Forces " > F = body - Viscous or frictional Forces " > F = pressure ((( C.V. '' " > ' f b dv ( p) ds > (Minus sign Because pressure Acts inward) > F = > f ' ds > fric (( > ds S 12

13 Conservation of Momentum (continued) Collected terms for conservation of Momentum "> f b dv " ( p) ds "> + "> f ds "> ' ( C.V. V "> "> ds ' ( V "> ) + C.V. )t V "> ' ( dv "> ds S = Steady, In-viscous Flow, Body Forces negligible C.V. "> f b dv " p V "> "> ds ' ( ( ) ds "> + "> f ds "> ' ( V "> ) + C.V. )t V "> ' ( dv "> ds S = "" ( p) ds > = V > > ds ' "" V > 13

14 One-Dimensional Flow Conservation of Momentum (cont d) p 1 V 1 A 2 p 2 V 2 A 2 "" ( p) ds > = V > > ds ' "" V > No flow across solid boundary ( p 2 A 2 p 1 A 1 )i _ x = " 2 V 2 A 2 V 2 i _ x " 1 V 1 A 1 V 1 i _ x p 1 + " 1 V 1 2 ( ) A 1 = p 2 + " 2 V 2 2 ' A 2 14

15 One-Dimensional Flow Conservation of Momentum (concluded) p 1 V 1 p 2 V 2 A 2 "" ( p) ds > = V > > ds ' "" V > A 2 ( 2 p V ) 2 1 A 1 = " p V 2 A 2 j = p j R g T j = ' p j ' R g T j ( ) ( ) p 2 = A ' M p 1 A ' M 2 = ' p j c p ( ' V 1 j ) * No flow across solid boundary 2 c 1 2 +, - A ( 1 = p 2 1+ ' V 2 ) * 2 c 2 2 +, - A 2 15

16 Conservation of Energy From First law of thermodynamics (reversible process) (corresponding to inviscid fluid) de dt de dt = dq dt + dw dt Rate of change of fluid energy as it flows thru C.V. dq dt dw dt Rate of external heat addition to C.V. Rate of work done to fluid inside of C.V. 16

17 de dt Conservation of Energy (cont d) Rate of change of fluid energy as it flows thru C.V. Earlier thermodynamic discussions were for static fluid flow thru C.V. is not static Kinetic Energy terms must also Be captured following process similar to Momentum flow through C.V. de dt = t C.V. " e + V 2 2 ( dv ' ' ( ))) + "V *> *> )) d S e + V 2 2 ( ' Rate of change of energy with C.V Energy Flow Across 17

18 dq dt Conservation of Energy (cont d) ((( Rate of heat addition to C.V. q C.V. " " dv ' dw dt Rate of work done to fluid inside of C.V. dw dt d dt (F dx) = F V dx = direction " of " motion Consider: 1) Pressure Forces, 2) Body Forces 18

19 Conservation of Energy (cont d) 1) Pressure Forces (pd S > ) V > "" 2) Body Forces C.V. ( "> f dv) V "> 19

20 Conservation of Energy (cont d) Collected Energy Equation Scalar equation 20

21 Conservation of Energy (concluded) Steady, Inviscid quasi 1-D Flow, Body Forces negligible 21

22 Conservation of Energy (concluded) Steady, Inviscid quasi-1-d Flow, Body Forces negligible 22

23 Conservation of Energy (concluded) Steady, Inviscid quasi 1-D Flow, Body Forces negligible 23

24 Conservation of Energy (concluded) Steady, Inviscid quasi 1-D Flow, Body Forces negligible, collected Divide through by massflow 24

25 Conservation of Energy (concluded) but from continuity, equation of state 25

26 Conservation of Energy (concluded) but from continuity, equation of state h = e + Pv = e +R g T 26

27 Summary Continuity (conservation of mass) Steady quasi One-dimensional Flow 27

28 Summary (continued) Newton s Second law-- Time rate of change of momentum Equals integral of external forces Steady, Inviscid quasi 1-D Flow, Body Forces negligible 28

29 Conservation of Energy-- Summary (concluded) Steady, Inviscid quasi 1-D Flow, Body Forces negligible 29

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