Lecture 5 Flusso Quasi-Mono-Dimensionale (forma

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1 Lecture 5 Dimensionale forma Text: Motori Aeronautici Mar. 6, 2015 Dimensionale forma Mauro Valorani Univeristà La Sapienza 5.50

2 Agenda Dimensionale forma 1 quasi-monodimensionale

3 quasi-monodimensionale Dimensionale forma 5.52

4 Dimensionale forma Mach number definition M := v a a := γrt 1 dm 2 M 2 dx = 2 dv v dx 1 dt T dx Thermal EOS for a single species ideal gas p = ρrt ; R = R W 1 dp dt p dx 1 T dx 1 ρ Caloric EOS for a single species ideal gas dρ dx = 0 h = c pt ; dh dx = dt cp cp ; cp cv = R ; dx R = γ γ 1 ; γ = cp c v Stagnation Total enthalpy h 0 = c p = h + v 2 2 ; = T 1 + γ 1 M 2 2 dh 0 dx = d cp dx = dh dx + v dv dx = dt cp dx + v dv dx 5.53

5 Mass Steady form 0 = ρ V nda ṁ := ρ V nax = const A Dimensionale forma dṁ dx = 0 0 = 1 da A dx + 1 dv v dx + 1 dρ ρ dx ; v := V n Momentum Steady form F = S 2 V dṁ S 1 V dṁ S Sw =S 1 +S 2 p n + τ t ds G Adp + τ c dx + XAdx = ρvadv ; τ = ρ v 2 f ; X = 0 ; c := 2πR 2 dp dx + τ c = ρv dv dp A dx ; dx + τ c = γpm2 v dv A v 2 dx 1 dp p dx + τ c = γm 2 1 dv p A v dx ; τ p = ρ p f v 2 2 = γm2 v f v = γm2 2 f 1 dp p dx + γm2 2 f c = γm 2 1 dv A v dx ; F := f c A = f 2πR πr = 4 f ; D := 2R 2 D 1 p dp dx + 1 dv γm2 v dx + γm2 2 F =

6 Dimensionale forma d dx [h + 12 ] Q v 2 = ṁ Ẇ ṁ dh dx + v dv dx = dq dw ; dv c p dt dx + v v 2M 2 dm 2 dx + v 2T Q ; dq := ṁ ; dw := ṁ Ẇ dx = v dm 2 2M 2 dx + v dt 2T dx dt = dq dw dx No work exchange in duct: dw = 0 Heat exchange measured as a change in total temperature: dq = c pd c p + v 2 2T dt c p dx + v 2 2T dt dx = cp dt dx + v M2 γrt 2c pt 1 + γ 1 dt M 2 2 dx + γ 1M2 T 2 dm 2 2M 2 dx = cpd dt dx = cp 1 + γ dm 2 M 2 dx = d ; = T M 2 dt dx 1 + γ 1 M T dt dx + δm2 1 dm δm 2 M 2 dx = d ; δ := γ

7 Differential Form Dimensionale forma is a set of 5 ODEs in 5 unknowns: ρ, v, M, p, T A A + v v + ρ ρ = γf M2 + p p + γm2 v = 0 v δm 2 M 1 + δm 2 M + T T = p p T T ρ ρ = 0 2M M = T T + 2v v The processes forcing the changes of the state variables along the duct are: Variable duct shape pos/neg; the cross-section area, A = Ax, is a prescribed function of space: A pos/neg Heat by Friction sempre pos; F := f c A = f 2πR πr 2 = 4 D f x, Re,... > 0 Heat transfer pos/neg; the total temperature, = x, is a prescribed function of space: pos/neg 5.56

8 Solution Dimensionale forma Solution γm 2 p 2 A + A F p M, γ, x = 2A 1 + γ 2δM 2 ρ 2γ 2δM 2 A + A γfm ρ M, γ, x = v v M, γ, x = M M M, γ, x = T M, γ, x = 1 + 2δM δm 2 2A 1 + γ 2δM 2 2 A + A γfm δm 2 2A 1 + γ 2δM δm 2 2 A + A γfm 2 + 2A 1 + γ 2δM 2 T 2δM 2 A + A γδfm 4 + A 1 + γ 2δM δm 2 T γm γm δm 2 T 0 Remark: in general the state of the system defined by the 5 unknowns changes because of the simultaneous action of the 3 driving processes 5.57

9 Suppose of zeroing out one driving process at the time: the state of the system will change because of the individual action of one of the 3 driving processes; a table of "influence coefficients" can be constructed that reads as: M M T T Area A A γM2 1+M 2 Friction F Heat γm γM M 2 1+γM2 1+γγM 4 1+M M γM 2 p p γm2 1+M 2 v v γm 2 1+M 2 1 γm 2 1+M 2 2 2M γM 2 1+γM M γM2 1+γM 2 1+M 2 γm γM2 1+M γM2 1+M 2 All depend on Mach number M and ratio of specific heat γ, only All denominators approach zero for M 1; this condition is said to be critical because the product of the influence coefficient times the area change, friction term, or heat addition term becomes infinitely large unless A, F, or vanish as Mach tends to unity Dimensionale forma 5.58

10 Dimensionale forma General definition ds γ R = ds R = Area is isentropic Friction is irreversible Heat addition is irreversible γ dt γ 1 T dp p = M γf + γ d dp 0 γ 1 p M γ γ ds R = 0 ds R = γ 2 M2 F > 0 ds R = γ 1 + δm 2 < / > 0 γ 1 T 0 cooling/heating 5.59

11 At critical conditions M=1, and with all 3 processes active, the relative change of Mach number is zero for: M sonicflowcondition = Numerator[ ] = 0 /.{M = 1} M 2 A = A γf γ Thus, at M = 1, the 3 driving processes are not independent one to another. At critical conditions M=1, and with no friction, the relative change of Mach number is zero for: Solve [sonicflowcondition/.{f = 0}, A ] A = 1 + γa T 0 2 At critical conditions, and no friction, the area change depends on heat addition Solve [ sonicflowcondition/. { T 0 = 0}, A ] Dimensionale forma A = 1 2 γa F At critical conditions, and no heat addition, the area change depends on friction 5.60

12 If lim M 1 f M = lim M 1 gm = 0 or ± and lim M 1 f M/g M exists, then: f M lim M 1 gm = lim f M M 1 g M The differentiation of the numerator and denominator often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be evaluated more easily. D[Numerator[MxM], x] 2δM M 2 A + A γf M γm δm 2 2A + A γf M γm 2 2 A +A 2γM M F + + T 0 + γm 2 F + F + D[Denominator[MxM], x] Dimensionale forma γ 2δM 2 A + 2γ 2δA M M + 2A 1 + γ 2δM

13 Dimensionale forma Area [ Only Solve M D[Numerator[MxM],x] == M /. D[Denominator[MxM],x] { T 0 = 0, T 0 = 0, F = 0, F = 0, A = 0, M = 1 }, M ] {{ } { }} 1 + γa M 1 + γa = 2, M = + A 2 A Reduce [{1 + γa > 0, γ > 1}, {A }] A > 0 Real solutions for M exist only for A >0 when A = 0 = A has a minimum throat For A >0, both positive and negative values of M are possible sub or super sonic flow depending on the downstream boundary conditions 5.62

14 Dimensionale forma [ Solve M D[Numerator[MxM],x] == M /. D[Denominator[MxM],x] { T 0 = 0, F = 0, F = 0, A = 0, A = 0, M = 1 }, M ] M = 1 + γ 2 T 0 2 2, M = + Reduce [{ 1 + γ 2 > 0, > 0, γ > 1 }, { }] > 0 && T 0 < γ 2 T Real solutions for M exist only for T 0 < 0 when T 0 = 0 = has a maximum For T 0 < 0, both positive and negative values of M are possible sub or super sonic flow depending on the downstream boundary conditions 5.63

15 Dimensionale forma [ M D[Numerator[MxM], x] Solve == /. M D[Denominator[MxM], x] { T A 0 = 0, A = 0, F = 0, F = 0, M = 1 }, M ] M = ± 1 + γ 2 A γa T A Reduce [{ 1 + γ 2 A γa } { > 0, A > 0, γ > 1, A, }] T0 A Reals && T 0 < 2 A A + γa when T 0 = 0 && A =0, real solutions for M < 2 A 1 + γ A exist only if: 5.64

16 [ M D[Numerator[MxM], x] Solve == /. M D[Denominator[MxM], x] { T A 0 = 0, A = 0, F = 0, M = 1 }, M ] Dimensionale forma M = ± 1 + γ 2 A + A γ F γt A [{ 0 Reduce 1 + γ 2T0 A + A γ F γt } 0 > 0, A > 0, γ > 1, { F, A, T }] }] 0 T0 F A Reals && T 0 < γa F + 2 A A + γa when T 0 = 0 && A =0 && F = 0, real solutions for M < γ 1 + γ F + 2 A 1 + γ A exist only if: 5.65

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

Section 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow Anderson: Chapter 2 pp. 41-54 1 Equation of State: Section 1 Review p = R g T " > R g = R u M w - R u = 8314.4126