lecture3a

# 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?
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.
• 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
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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