lecture3a - Fundamental Concepts and Basic Definitions How...

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Fundamental Concepts and Basic Definitions How does a rocket work? Gemini 3 - Titan II • Derivation of thrust for a rocket engine • Basic definition of rocket parameters • Rocket equation Reading: Sutton and Biblarz Chapter 2, 4.1-4.4
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Rocket propulsion relies on expulsion of mass to generate thrust. The source for this thrust is momentum exchange. Rocket Fundamental Force balance? Equation of motion? Performance parameter?
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Reynold’s transport theorem or Newton’s second law applied to control volume Derivation of Thrust for a Rocket Engine F −> = d dt mV = Momentum flow across C.S. + Rate of Change of Momentum within C.V. • Momentum flow across control surface m C . S . ∫∫ V v n V n V The mass flow out of the C.V. across an elemental piece of the control surface, ds m = ρ V ds m C . S . ∫∫ V = ρ V ds C . S . ∫∫ V Conservation of momentum
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∫∫ ∫∫∫ + = cs cv s d V x dv x t dt d r r ρ χ ( ) ∫∫∫ ∫∫∫ + = cv cv dv V x dv x t dt d r r () dv V x x t dt d cv ∫∫∫ + = r r Using the Gauss Divergence Theorem, the surface flux integral can be written as a volume integral: Since the control volume itself is arbitrary: The Lagrangian derivative is just also the time derivative following the fixed mass Reynold’s Transport Theorem This equation, referred to as Reynold’s Transport Theorem , relates the Lagrangian time derivative of χ (i.e. d/d t ) for the control mass to the Eulerian time derivative of χ for the fixed control volume.
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• Rate of Momentum Change within Control Volume t ( mV −> ) Within an elemental piece dv of the control volume m = ρ dv dV n V n V t ( ) C . V . = t ρ dvV C . V . ∫∫∫ = t V dv C . V . ∫∫∫ • And total net rate of momentum change within the control volume is Derivation of Thrust for a Rocket Engine
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Newton’s second law applied to control volume F −> = d dt mV = ρ V ds C . S . ∫∫ V + t V dv C . V . ∫∫∫ Resolution of Forces Acting on Control Volume - Body Forces (electo-magnetic, gravitational, buoyancy etc.) - Pressure Forces - Viscous or frictional Forces F body = f b dv C . V . ∫∫∫ F pressure =− p () C . S . ∫∫ dS (Minus sign Because pressure Acts inward) F fric = | f × dS C . S . ∫∫ | dS S C . S . Derivation of Thrust for a Rocket Engine
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Steady, inviscid flow, body forces negligible Ideal rocket model assumption: ρ f b −> dv C . V .
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This note was uploaded on 12/12/2010 for the course MECH 351 taught by Professor Chekhov during the Fall '10 term at Concordia Canada.

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lecture3a - Fundamental Concepts and Basic Definitions How...

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