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Ch. 11: Introduction to Compressible Flow

Ch. 11: Introduction to Compressible Flow. When a fixed mass of air is heated from 20 o C to 100 o C, what is change in…. p 2 , h 2 , s 2 ,  2 , u 2 , Vol 2 100 o C. STATE 2. p 1 , h 1 , s 1 ,  1 , u 1 , Vol 1 20 o C. STATE 1. …. Constant s? constant p? constant volume?….

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Ch. 11: Introduction to Compressible Flow

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  1. Ch. 11: Introduction to Compressible Flow When a fixed mass of air is heated from 20oC to 100oC, what is change in…. p2, h2, s2, 2, u2, Vol2 100oC STATE2 p1, h1, s1, 1, u1, Vol1 20oC STATE1 …. Constant s? constant p? constant volume?…

  2. Ch. 11: Introduction to Compressible Flow When a fixed mass of air is heated from 20oC to 100oC – What is the change in enthalpy? Change in entropy (constant volume)? Change in entropy (constant pressure)? If isentropic change in pressure? If isentropic change in density?

  3. IDEAL, CALORICALLY PERFECT GAS p = RT [R=Runiv/mmole] (11.1) du = cvdT (11.2) u2- u1 = cv(T2 – T1) (11.7a) dh = cpdT(11.3) h2- h1 = cp(T2 – T1) (11.7b)

  4. h = u + pv IDEAL GAS h = u + RT dh = du + RdT IDEAL GAS du = cvdT & dh = cpdT cpdT = cvdT + R dT cp – cv = R Eq. (11.4)

  5. IDEAL GAS cp - cv = R(11.4) k  cp/cv([k=]) (11.5) cp = kR/(k-1)(11.6a) cv = R/(k-1) (11.6b)

  6. always true dq + dw = du ds = q/T|rev Tds = du - pdv = dh – vdp Ideal calorically perfect gas – constant cp, cv p =  RT; cp = dh/dT; cv = du/dT s2 – s1 = cvln(T2/T1) - Rln(2/1) s2 – s1 = cpln(T2/T1) - Rln(p2/p1)

  7. Ideal / Calorically Perfect Gas s2 – s1 = cvln(T2/T1) - Rln(2/1) s2 – s1 = cpln(T2/T1) - Rln(p2/p1) Handy if need to find change in entropy

  8. Ideal / Calorically Perfect Gas Cv = du/dT; Cp = dh/dT; p = RT = (1/v)RT Tds = du + pdv = dh –vdp ds = du/T + RTdv/T ds = cvdT/T + (R/v)dv s2 – s1 = cvln(T2/T1) + Rln(v2/v1) s2 – s1 = cvln(T2/T1) - Rln(2/1)

  9. Ideal / Calorically Perfect Gas Cv = du/dT; Cp = dh/dT; p = RT = (1/v)RT Tds = du + pdv = dh –vdp ds = du/T + RTdv/T ds = cvdT/T + (R/v)dv Note: don’t be alarmed that cv and dv in same equation! cv = du/dT is ALWAYS TRUE for ideal gas

  10. Ideal / Calorically Perfect Gas Cv = du/dT; Cp = dh/dT; p = RT = (1/v)RT Tds = du + pdv = dh –vdp ds = dh/T – vdp/T ds = CpdT/T - (RT/[pT])dp s2 – s1 = Cpln(T2/T1) - Rln(p2/p1)

  11. Ideal / Calorically Perfect Gas Cv = du/dT; Cp = dh/dT; p = RT = (1/v)RT Tds = du + pdv = dh –vdp ds = dh/T – vdp/T ds = CpdT/T - (RT/[pT])dp Note: don’t be alarmed that cp and dp are in same equation! cp = dh/dT is ALWAYS TRUE for ideal gas

  12. Isentropic Ideal / Calorically Perfect Gas 2/1 = (T2/T1)1/(k-1) p2/p1 = (T2/T1)k/(k-1) (2/1)k = p2/p1; p2/2k = const c = kRT Handy if isentropic

  13. s2 – s1 = Cvln(T2/T1) - Rln(2/1) If isentropic s2 – s1 = 0 ln(T2/T1)Cv = ln(2/1)R cp – cv = R; R/cv = k – 1 2/1 = (T2/T1)cv/R = (T2/T1)1/(k-1) assumptions ISENROPIC & IDEAL GAS & constant cp, cv

  14. s2 – s1 = cpln(T2/T1) - Rln(p2/p1) If isentropic s2 – s1 = 0 ln(T2/T1)cp = ln(p2/p1)R cp – cv = R; R/cp = 1- 1/k p2/p1 = (T2/T1)cp/R = (T2/T1)k/(k-1) assumptions ISENROPIC & IDEAL GAS & constant cp, cv

  15. 2/1 = (T2/T1)1/(k-1) p2/p1 = (T2/T1)k/(k-1) (2/1)k = p2/p1 p2/2k = p1/1k = constant assumptions ISENROPIC & IDEAL GAS & constant cp, cv

  16. Ch. 11: Introduction to Compressible Flow When a fixed mass of air is heated from 20oC to 100oC – What is the change in enthalpy? • h2 – h1 = Cp(T2- T1)

  17. Ch. 11: Introduction to Compressible Flow When a fixed mass of air is heated from 20oC to 100oC – Change in entropy (constant volume)? • s2 – s1 = Cvln(T2/T1)

  18. Ch. 11: Introduction to Compressible Flow When a fixed mass of air is heated from 20oC to 100oC – Change in entropy (constant pressure)? s2 – s1 = Cpln(T2/T1)

  19. Ch. 11: Introduction to Compressible Flow When a fixed mass of air is heated from 20oC to 100oC – If isentropic change in density? 2/1 = (T2/T1)1/(k-1)

  20. Ch. 11: Introduction to Compressible Flow When a fixed mass of air is heated from 20oC to 100oC – If isentropic change in pressure? p2/p1 = (T2/T1)k/(k-1)

  21. Stagnation Reference (V=0) (refers to “total” pressure (po), temperature (To) or density (o) if flow brought isentropically to rest)

  22. 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 and no friction) deceleration there are unique stagnation To, po, o, uo, so,ho (Vo=0) properties .

  23. 1-D, energy equation for adiabatic and no shaft or viscous work Eq. (8.28); hlT = [u2-u1]- Q/m (p2/2) + u2 + ½ V22 + gz2 = (p1/1) + u1 + ½ V12 + gz1 0 Definition: h = u + pv = u + p/; assume z2 = z1 h2 + ½ V22 = h1 + ½ V12 = ho + 0 ho – h1 = ½ V12 Isentropic process

  24. 1-D, energy equation for adiabatic and no shaft or viscous work (8.28, hlT = [u2-u1]- Q/m) ho - h1 = ½ V12 ho – h1 = cp (To – T1) ½ V12= cp (To – T1) ½ V12+ cpT1 = cp To To = {½ V12+ cpT1}/cp T0 = ½ V12/cp + T1 = ½ V2/cp + T

  25. T0 = ½ V12/cp + T = T (1 + V2/[2cpT]) cp = kR/(k-1) T0 = T (1 + V2/[2kRT/{(k-1)}) T0 = T (1 + (k-1)V2/[2kRT]) c2 = kRT T0 = T (1 + (k-1)V2/[2c2]) M = V2/ c2 T0 = T (1 + [(k-1)/2] M2)

  26. To/T = 1 + {(k-1)/2} M2 Steady, no body forces, one-dimensional, frictionless, ideal, calorically perfect, adiabatic, isentropic

  27. /o = (T/To)1/(k-1) To/T = 1 + {(k-1)/2} M2 /o = (1 + {(k-1)/2} M2 )1/(k-1) Steady, no body forces, one-dimensional, frictionless, ideal, calorically perfect, adiabatic, isentropic

  28. p/p0 = (T/To)k/(k-1) To/T = 1 + {(k-1)/2} M2 p/p0 = (1 + {(k-1)/2} M2)k/(k-1) Steady, no body forces, one-dimensional, frictionless, ideal, calorically perfect, adiabatic, isentropic

  29. Ideal & constant cp & cv p =  RT; cp = dh/dT; cv = du/dT s2 – s1 = cvln(T2/T1) - Rln(2/1) s2 – s1 = cpln(T2/T1) - Rln(p2/p1) Ideal & constant cp & cv & isentropic 2/1 = (T2/T1)1/(k-1); p2/p1 = (T2/T1)k/(k-1); p2/2k = const; c = kRT Ideal & constant cp & cv & isentropic + … p0/p = (1 + {(k-1)/2} M2)k/(k-1); o/ = (1 + {(k-1)/2} M2 )1/(k-1) To/T = 1 + {(k-1)/2} M2

  30. p0/p = (1 + {(k-1)/2} M2)k/(k-1); o/ = (1 + {(k-1)/2} M2 )1/(k-1) To/T = 1 + {(k-1)/2} M2 • Stagnation condition not useful for velocity • Use critical condition – when M = 1, V* = c* • (critical speed is the speed obtained when flow is • isentropically accelerated or decelerated until M = 1) • At critical conditions, the isentropic stagnation quantities become: p0/p* = (1+{(k-1)/2} 12)k/(k-1) = {(k+1)/2}k/(k-1) o/ = (1+{(k-1)/2} 12 )1/(k-1) = {(k+1)/2}1/(k-1) To/T = 1 + {(k-1)/2} 12 = (k+1)/2

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