lecture9b - MECH 7221 MECH 6251/498D Non-Ideal Nozzle Flow...

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1 MECH 7221 Non-Ideal Nozzle Flow Analysis • Non-ideal effects in real nozzle flow Objectives Reading assignment: Sutton and Biblarz Chapter 6 Saturn V engine
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2 Nozzle Heat Loss Correction Consider Nozzle with P 0 c , T 0 c leaving combustion chamber • Assume nozzle is shock-free • What happens at to exit plane conditions with heat loss in the nozzle? • From earlier Analysis (1) … exit conditions without heat loss (2) … exit conditions with heat loss T 0 2 = T 0 1 1 c p q nozzle m
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3 Nozzle Loss Correction • Quasi 1-D momentum equation p 1 + ρ 1 V 1 2 = p 2 + ρ 2 V 2 2 p 1 + p 1 R g T 1 V 1 2 = p 2 + p 2 R g T 2 V 2 2 p 1 1 + γ M 1 2 = p 2 1 + γ M 2 2 p 2 p 1 = 1 + γ M 1 2 1 + γ M 2 2 (1) … exit conditions without heat loss (2) … exit conditions with heat loss
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4 Nozzle Loss Correction • Quasi 1-D continuity equation Plug into momentum eqn. ρ 1 V 1 = ρ 2 V 2 1 2 = V 2 V 1 T 2 T 1 = p 2 p 1 1 2 = p 2 p 1 V 2 V 1 = 1 + γ M 1 2 1 + γ M 2 2 V 2 V 1 T 2 T 1 = 1 + γ M 1 2 1 + γ M 2 2 M 2 M 1 2 (1) … exit conditions without heat loss (2) … exit conditions with heat loss
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5 Nozzle Loss Correction • But from definition for stagnation conditions T 2 T 1 = 1 + γ M 1 2 1 + γ M 2 2 M 2 M 1 2 • and from earlier (1) … exit conditions without heat loss (2) … exit conditions with heat loss M 2 M 1 1 + γ M 1 2 1 + M 2 2 2 = T 02 T 0 1 × 1 + 1 2 M 1 2 1 + 1 2 M 2 2 T 2 T 1 = T 0 1 1 + γ− 1 2 M 1 2 T 0 2 1 + 1 2 M 2 2 M 2 2 1 + 1 2 M 2 2 1 M 2 2 2 = T 0 2 T 0 1 M 1 2 1 + 1 2 M 1 2 1 M 1 2 2
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6 Nozzle Loss Correction • Let • Solve for M 2 M 2 2 1 + γ− 1 2 M 2 2 1 M 2 2 2 = T 0 2 T 0 1 M 1 2 1 + 1 2 M 1 2 1 M 1 2 2 F ( M 1 ) = T 02 T 0 1 M 1 2 1 + 1 2 M 1 2 1 + γ M 1 2 2
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7 Nozzle Loss Correction • Regroup in terms of M 2 M 2 2 1 + γ− 1 2 M 2 2 1 + γ M 2 2 2 = T 02 T 0 1 M 1 2 1 + 1 2 M 1 2 1 + γ M 1 2 2 M 2 2 1 + 1 2 M 2 2 F ( M 1 )1 + γ M 2 2 2 = 0 F ( M 1 ) = T 02 T 0 1 M 1 2 1 + 1 2 M 1 2 1 + γ M 1 2 2
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8 Nozzle Loss Correction • Sub in and collect terms in powers of M 2 • Quadratic Equation, … but quadratic in M 2 2 γ− 1 2 M 2 4 + M 2 2
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lecture9b - MECH 7221 MECH 6251/498D Non-Ideal Nozzle Flow...

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