4.1 LAWS OF MECHANICS - Review

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1 4.1 LAWS OF MECHANICS - Review Ch4 9

2 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: 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. F = m a = d dt de dt = dq dt dw dt ( m v) dm dt sys CV = 0 Ch4 10

0 = dm dt CV CS system Rate of Change of Mass WITHIN the CV Net Mass Flow through CS ρ t dv + ρ ( v n) da = 0 CV CS Special

3 CONTROL VOLUME Conservation of Mass: Rate of change of mass within control volume is equal to net rate of inflow of mass. d dt m = m m Control Volume Streamlines CV in out dv v da Continuity Equation: n d ρ dv Control Surface dt + ρ ( v n) da = 0 = dm dt CV CS system Rate of Change of Mass WITHIN the CV Net Mass Flow through CS ρ t dv + ρ ( v n) da = 0 CV CS Special Case Flow within control volume is steady: d dt m CV = 0 ( ρ A v) in = ρ A v ρ t 0 ( ) out m in = m out ρ( v n)da = 0 CS Ch4 11

4 Control Volume Linear Momentum: Newton s 2 nd Law as applied to system: F = d ( m v) = m a dt Newton s 2 nd Law as applied to Control Volume: Momentum Theorem F = d dt ρ v dv + ρ v( v r n)da = d dt CV CS Special Case m=const. ( m v) system Rate of Change of Linear Momentum WITHIN the CV Net Momentum Flux through CS Note: Above equation applies for a control volume moving at a relative velocity v. For a control volume at rest v r r v : F = d dt ρ v dv + ρ v( v n)da CV F is the sum of all forces exerted by the surrounding on the material occupying the control volume: Body forces Surface forces. CS Ch4 12

5 4.2 THERMODYNAMICS - Review Ch4 13

Isentropic relations describe the flow in the nozzle.

6 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 e = u + e potential + e kinetic dw = dw shaft + dw flow h u + p v u + p ρ m = ρ A v Specific Heat c v du dt v=const. c p dh dt p=const. γ c p c v Ch4 14

7 LAWS OF THERMODYNAMICS 1 st Law of TD (Energy Equation) de = dq dw Expresses the universal law of conservation of energy. dh + v dv = dq dw shaft e pot = const. de pot = 0 No-work reversible interaction dq rev = du + p dv = dh v dp = dh + v dv 2 nd Law of TD (Entropy) ds dq T ds = dq T rev Expresses the universal law of increasing entropy. No work reversible interaction Tds = dh v dp 1 st & 2 nd Laws of TD Spec. Volume v dp + v dv = dq rev Tds Velocity! Ch4 15

8 THERMODYNAMIC PROCESSES Adiabatic Process No heat or other energy crosses system boundary. Adiabatic processes include isentropic and throttling processes. Isobaric Process Constant pressure process. Isothermal Process Constant temperature process. Isochoric/Isometric Process Constant volume process. Q = 0 Δp = 0 ΔT = 0 ΔV = 0 ΔU = W Q = ΔH Q = W Q = ΔU W = 0 Isentropic Process An adiabatic process in which there is no change in system entropy. This is an reversible process. ΔS = 0 Q = 0 Ch4 16

9 PERFECT GAS Definition: Particles posses three translational degrees of freedom. Intermolecular forces are negligible. Equation of State p = ρ R T p v = p ρ = R T = R M T R is universal gas constant: R is gas constant for a particular gas: R = R = R M J mol K 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

10 Equation of State p = ρ R T PERFECT GAS p v = p ρ = R T = R M T Internal Energy Enthalpy u = u( T) du = c v dt h = h( T) dh = c p dt h = u + p v = u + R T dh = du + R dt Specific Heat c p = c v + R c v = R γ 1 γ c p c v c p = γ γ 1 R Speed of Sound Mach Number a 2 = γ R T M = u a = u γ R T Ch4 18

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