chap4_9_18 - AAE 439 4.1 LAWS OF MECHANICS - Review Ch4 9...

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Unformatted text preview: AAE 439 4.1 LAWS OF MECHANICS - Review Ch4 9 AAE 439 SYSTEM System: Moving Fluid Definitions: "System" is defined as an arbitrary quantity of mass of fixed identity. "Surrounding is everything external to this system. "Boundary" separates surroundings from system. Case: CV Control Volume is at rest. Control Volume is moving with a velocity. All Laws of Mechanics are written for a system stating what happens when there is an interaction between the system and its surroundings. Conservation of Mass Linear (Angular) Momentum Energy Equation m sys = const. d F = ma = mv dt dE dQ dW = - dt dt dt ( ) dm dt = 0 sys Ch4 10 AAE 439 CONTROL VOLUME Conservation of Mass: Rate of change of mass within control volume is equal to net rate of inflow of mass. d mCV = m in - m out dt Control Volume Streamlines dV dA Continuity Equation: v n dm d dV + CS ( v n) dA = 0 = dt dt CV system Net Mass Flow through CS Control Surface Rate of Change of Mass WITHIN the CV dV + ( v n) dA = 0 t CV CS 0 t Special Case Flow within control volume is steady: d mCV = 0 dt Av CS ( v n)dA = 0 m in = m out ( ) = ( A v) in out Ch4 11 AAE 439 Control Volume 2nd Linear Momentum: d Newton's Law as applied to system: F= m v = ma dt Newton's 2nd Law as applied to Control Volume: Momentum Theorem ( ) Special Case m=const. d d F = v dV + v( v r n) dA = mv dt dt CV CS Rate of Change of Linear Momentum WITHIN the CV Net Momentum Flux through CS ( ) system Note: Above equation applies for a control volume moving at a relative velocity v r . For a control volume at rest v r v : d F = v dV + v( v n) dA dt CV CS F is the sum of all forces exerted by the surrounding on the material occupying the control volume: Body forces Surface forces. Ch4 12 AAE 439 4.2 THERMODYNAMICS - Review Ch4 13 AAE 439 ENERGY & WORK Rocket Motor Functions Chemical rocket performance comprises two primary parts: Form of Energy Kinetic Energy Thermodynamics define relationships between forms of energy. Isentropic relations describe the flow in the nozzle. Total Energy Total Work Enthalpy Continuity Specific Heat e = u + e potential + ekinetic dw = dw shaft + dw flow h u + pv u + m = Av du cv dT v=const. dh cp dT p=const. p cp cv Ch4 14 AAE 439 LAWS OF THERMODYNAMICS de = dq - dw dh + v dv = dq - dw shaft epot = const. depot = 0 1st Law of TD (Energy Equation) Expresses the universal law of conservation of energy. No-work reversible interaction dq rev = du + p dv = dh - v dp = dh + v dv dq T dq ds = T rev 2nd Law of TD (Entropy) ds Expresses the universal law of increasing entropy. No work reversible interaction Tds = dh - v dp 1st & 2nd Laws of TD v dp + v dv = dq rev - Tds Spec. Volume Velocity ! Ch4 15 AAE 439 THERMODYNAMIC PROCESSES Q=0 U = -W Adiabatic Process No heat or other energy crosses system boundary. Adiabatic processes include isentropic and throttling processes. Isobaric Process p = 0 T = 0 V = 0 S = 0 Q = H Q=W Q = U W =0 Q=0 Constant pressure process. Isothermal Process Constant temperature process. Isochoric/Isometric Process Constant volume process. Isentropic Process An adiabatic process in which there is no change in system entropy. This is an reversible process. Ch4 16 AAE 439 PERFECT GAS Definition: Particles posses three translational degrees of freedom. Intermolecular forces are negligible. Equation of State p = RT pv = p = RT = J mol K T M is universal gas constant: = 8.314 M R is gas constant for a particular gas: R = Characteristics: Pressure is a result of kinetic energy exchange with molecules in movement. As T rises, kinetic energy goes up, pressure goes up. Perfect gas approximation is good at low p and high T. More accurate "equations of state" (e.g., van der Waals equation) can be used. Ch4 17 AAE 439 PERFECT GAS p = RT u=u T h=h T Equation of State Internal Energy Enthalpy pv = p = RT = ( ) T M du = c v dT dh = c p dT ( ) h = u + pv = u +RT dh = du + R dT Specific Heat cp = cv + R cv = R -1 cp cv cp = -1 R Speed of Sound Mach Number a2 = R T M= u u = a RT Ch4 18 ...
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This document was uploaded on 01/15/2012.

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