1. IDEAL GAS: p =  RT (11.1) du = c v dT (11.2) dh= c p dT (11.3) Q + W =  U 1 st and 2 nd LAWS: Tds = du + pdv (11.10a) Tds = h – vdp (11.10b) IDEAL GAS + 1 st + 2 nd LAWS T 2 /  2 (k-1) = const. (11.12a) ds = du/T + pdv/T = c v dT/T + Rdv/v s 2 – s 1 = c v ln (T 2 /T 1 ) + R ln (v 2 /v 1 ) (11.11a) ds = dh/T - vdp/T = c p dT/T + Rdp/p s 2 – s 1 = c p ln (T 2 /T 1 ) + R ln (p 2 /p 1 ) (11.11b) s 2 – s 1 = c v ln (T 2 /T 1 ) + R ln (v 2 /v 1 ) = c v ln(p 2  1 /p 1  2 ) + (c p -c v ) ln (v 2 /v 1 ) s 2 – s 1 = c v ln(p 2 /p 1 ) + c v ln(v 2 /v 1 ) + c p ln (v 2 /v 1 ) - c v ln (v 2 /v 1 ) s 2 – s 1 = c v ln(p 2 /p 1 ) + c p ln (v 2 /v 1 ) (11.11c) h(T) = u(T) + RT; dh = du + RdT; c p dT = c v dT + RdT; c p = c v + R (11.4) Isentropic / ideal T 2 /  2 (k-1) = const. (11.12a) Tp (1-k)/k = const. (11.12b) p/  k = const. (11.12c)

2. 11-3 REFERENCE STATE: LOCAL ISENTROPIC STAGNATION PROPERTIES Since p, T, , u, h, s, V are all changing along the flow, the concept of stagnation conditions is extremely useful in that it defines a convenient reference state for a flowing fluid. To obtain a useful final state, restrictions must be put on the deceleration process. For an isentropic (adiabatic, no friction, no “violent” events) deceleration there are unique stagnation T, p, , u and h properties.

3. Usually changes in stagnation properties can be related to the driving potential of the flow = heat, work, friction, area change. Alternatively can be defined as the static state from which a fluid must be accelerated to attain the actual state of a given flow. isentropic … …

4. REFERENCE STATE: LOCAL ISENTROPIC STAGNATION PROPERTIES 1 2 p 1, T 1,  1, u 1, h 1, s 1, V 1 p 2, T 2,  2, u 2, h 2, s 2, V 2 p 01, T 01,  01, u 01, h 01, s 01 =s 1, V 01 = 0 isentropic p 02, T 02,  02, u 02, h 02, s 02 =s 2, V 02 = 0 isentropic T 0 /T = 1 + M 2 (k-1)/2 p 0 /p = [1 + M 2 (k-1)/2] k/(k-1)  0 /  = [1 + M 2 (k-1)/2] 1/(k-1) If process from 1 to 2 is isentropic then p 01 = p 02, T 01 = T 02 ; etc

5. REFERENCE STATE: LOCAL ISENTROPIC STAGNATION PROPERTIES 1 2 isentropic T 0 /T = 1 + M 2 (k-1)/2 p 0 /p = [1 + M 2 (k-1)/2] k/(k-1)  0 /  = [1 + M 2 (k-1)/2] 1/(k-1) If isentropic process between 1 and 2 and M 2 > M 1 then T 2 <T 1 ; p 2 <p 1 ;  2 <  1

6. REFERENCE STATE: LOCAL ISENTROPIC STAGNATION PROPERTIES 1 2 isentropic If T 01 > T 02 then heating has occurred between 1 and 2 If T 01 < T 02 then cooling has occurred between 1 and 2

7. BE: 1-D, energy equation for adiabatic and no shaft or viscous work (reversible). (p 2 /  2 ) + u 2 + ½ V 2 2 + gz 2 = (p 1 /  1 ) + u 1 + ½ V 1 2 + gz 1 Definition: h = u + pv = u + p/  ; assume z 2 = z 1 h 1 + ½ V 1 2 = h 0 + ½ V 0 2 (let state 2 = stagnation state, 0) Isentropic Process: No heat transfer, No friction, No violent events (reversible)

8. h 1 + ½ V 1 2 = h 0 + ½ V 0 2 h 1 – h 0 = ½ V 0 2 - ½ V 1 2 C p = dh/dT (ideal gas) C p = (h 1 -h 0 )/(T 1 -T 0 ) C p (T 1 -T 0 ) = ½ V 0 2 – ½ V 1 2 If pick stagnation conditions V 0 = 0 -C p T o = -C p dT 1 - ½ V 1 2

9. C p T o = C p T + ½ V 2 C p – C v = R; R/C p = 1- 1/k C p = R/(1-1/k) = kR/(k-1) [kR/(k-1)]T o /T = [kR/(k-1)] + ½ V 2 /T T o /T = 1 + ( 1 / 2 V 2 ) / (T[kR/(k-1)]) T o /T = 1 + {(k-1)/2} V 2 /(kRT) M 2 = V 2 /c 2 = V 2 /[kRT] T o /T = 1 + {(k-1)/2} M 2 (11.20b) STEADY, 1-D, ENERGY EQUATION FOR ADIABATIC FLOW OF A PERFECT GAS

10. Ideal Gas & Isentropic: T 0 /  0 (k-1) = T/  (k-1) (11.12a) (T o /T) = (  0 /  ) (k-1) (  0 /  ) = (T 0 /T) 1/(k-1) T o /T = 1 + {(k-1)/2} M 2 (11.20b)  o /  = (1 + {(k-1)/2} M 2 ) 1/(k-1) (11.20c) Ideal gas and isentropic (isentropic = adiabatic + reversible)

11. T 0 p 0 (1-k)/k = Tp (1-k)/k (11.12b) T 0 /T = (p/p 0 ) (1-k)/k p/p 0 = (T 0 /T) k/(1-k) p 0 /p = (T 0 /T) k/(k-1) T o /T = 1 + {(k-1)/2} M 2 p o /p = (1 + {(k-1)/2} M 2 ) k/(k-1) (11.20a) Ideal gas and isentropic (isentropic = adiabatic + reversible)

12. Local isentropic stagnation properties for an ideal gas: Book derives from differential control volume using continuity and momentum eqs. and isentropic relations for ideal gas.

13. 11-4 Critical Conditions Whereas stagnation conditions are useful as reference conditions for p, T, s and , not very useful for velocity (by definition = 0). A useful reference for speed is the critical speed, defined as the speed attained when the flow is either accelerated or decelerated to M = 1 (instead of zero) isentropically. Properties at critical speed are designated by an asterisk. V * = c *

14. isentropic … … V* = c* Critical Conditions Can be asked to find V * for a particular stagnation condition, Even though V at stagnation = 0.

15. p o /p * = [(k+1)/2] k/(k-1)  o /  * = [(k+1)/2] 1/(k-1) T o /T * =(k+1)/2

16. p o /p * = [(k+1)/2] k/(k-1)  o /  * = [(k+1)/2] 1/(k-1) T o /T * =(k+1)/2 for ideal gas V * = c * = (kRT * ) 1/2 = (kR2T o /(k + 1)) 1/2 V * = (kRT * ) 1/2 (in terms of critical temperature) V * = (kR2T o /(k + 1)) (in terms of stagnation temperature) Can be asked to find V * for a particular stagnation condition, Even though V at stagnation = 0.

17. Review of Equations in Chapter 11

18. IDEAL GAS: p =  RT (11.1) du = c v dT (11.2) dh= c p dT (11.3) Q + W =  U 1 st and 2 nd LAWS: Tds = du + pdv (11.10a) Tds = h – vdp (11.10b) IDEAL GAS + 1 st + 2 nd LAWS T 2 /  2 (k-1) = const. (11.12a) ds = du/T + pdv/T = c v dT/T + Rdv/v s 2 – s 1 = c v ln (T 2 /T 1 ) + R ln (v 2 /v 1 ) (11.11a) ds = dh/T - vdp/T = c p dT/T + Rdp/p s 2 – s 1 = c p ln (T 2 /T 1 ) + R ln (p 2 /p 1 ) (11.11b) s 2 – s 1 = c v ln (T 2 /T 1 ) + R ln (v 2 /v 1 ) = c v ln(p 2  1 /p 1  2 ) + (c p -c v ) ln (v 2 /v 1 ) s 2 – s 1 = c v ln(p 2 /p 1 ) + c v ln(v 2 /v 1 ) + c p ln (v 2 /v 1 ) - c v ln (v 2 /v 1 ) s 2 – s 1 = c v ln(p 2 /p 1 ) + c p ln (v 2 /v 1 ) (11.11c) h(T) = u(T) + RT; dh = du + RdT; c p dT = c v dT + RdT; c p = c v + R (11.4) Isentropic / ideal T 2 /  2 (k-1) = const. (11.12a) Tp (1-k)/k = const. (11.12b) p/  k = const. (11.12c)

19. IDEAL GAS + 1 st + 2 nd LAWS + isentropic 0 = c v ln (T 2 /T 1 ) + R ln (v 2 /v 1 ) (11.11a) 0 = ln (T 2 /T 1 ) Cv + ln (v 2 /v 1 ) R ; 0 = ln[(T 2 /T 1 ) Cv (v 2 /v 1 ) R ] 1 = (T 2 /T 1 ) Cv (v 2 /v 1 ) R ; 1 = (T 2 /T 1 ) (v 2 /v 1 ) R/Cv c p = c v + R (11.4); R/c v = k-1 (T 2 /T 1 ) = (v 1 /v 2 ) (k-1) ; T 1 v 1 (k-1) = T 2 v 2 (k-1) = T 1 /  1 (k-1) = T 2 /  2 (k-1) = const. (11.12a)  s = 0

20. T 1 v 1 (k-1) = T 2 v 2 (k-1) = T 1 /  1 (k-1) = T 2 /  2 (k-1) = const. (11.12a)  = p/(RT); T/[p/(RT)] (k-1) = constant; TT k-1 /p k-1 = constant T k p 1-k = constant Tp (1-k)/k = const. (11.12b)  s = 0

21. T 1 v 1 (k-1) = T 2 v 2 (k-1) = T 1 /  1 (k-1) = T 2 /  2 (k-1) = const. (11.12a) T = p/(  R); [p/(  R)]/  (k-1) = constant; p/  k = pv -k = const. (11.12c)  s = 0

22. Book’s Derivation of Stagnation Properties

23. Compressible flow in an infinitesimal stream tube (1-dimensional)

24. inertial frame next slide

25. p + dp/2

26. F Sx = -dpA From Continuity Equation ~

27. If incompressible one can integrate to get Bernoulli’s Equation. For ideal gas and isentropic p/  k = const.

28. p/  k = C;  = (p/C) 1/k So dp/  = dp/(p/C) 1/k dp/  = C 1/k p -1/k dp

29. From previous slide From p/  k = C; ideal & isentropic p 1/k +(k-1)/k = p k/k = p

30. (V 2 /2)(  /p)([k-1]/k) + 1 = (p o /p) (k-1)/k p o /p = {(V 2 /2)(  /p)([k-1]/k) + 1} k/(k-1) IDEAL GAS p =  RT M2M2 (11.20a) Can calculate p o at any point in flow field providing know static pressure p and M at that point.

31. What are static, dynamic and stagnation pressures? The thermodynamic pressure, p, used throughout the present and previous chapters refers to the static pressure (a bit of a misnomer). This is the pressure experienced by a fluid particle as it moves. The dynamic pressure is defined as ½  V 2. The stagnation pressure is obtained when the fluid is decelerated to zero speed through an isentropic process (no heat transfer, no friction). For incompressible flow: p o = p + ½  V 2

32. IDEAL GAS: p =  RT T o /T 1 = (p o /  o )(  1 /p 1 ) Isentropic: p/  k = const T o /T 1 = (p o /p 1 ) {  1 /  o } T o /T 1 = (p o /p 1 ) {p 1 /p o } 1/k T o /T 1 = (p o /p 1 ) {p o /p 1 } -1/k T o /T 1 = (p o /p 1 ) (k-1)/k (11.20a)

33. IDEAL GAS: p =  RT Isentropic: p/  k = const p o /  o k =p 1 /  1 k  1 k /  o k = p 1 /p o  1 /  o = {p 1 /p o } 1/k  o /  1 = {p o /p 1 } 1/k (11.20a)