MAE 5350: Gas Turbines

1. MAE 5350: Gas Turbines Integral Forms of Mass and Momentum Equations Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. Kinematic Properties: Two ‘Views’ of Motion • Lagrangian Description • Follow individual particle trajectories • Choice in solid mechanics • Control mass analyses • Mass, momentum, and energy usually formulated for particles or systems of fixed identity • ex., F=d/dt(mV) is Lagrangian in nature • Eulerian Description • Study field as a function of position and time; not follow any specific particle paths • Usually choice in fluid mechanics • Control volume analyses • Eulerian velocity vector field: • Knowing scalars u, v, w as f(x,y,z,t) is a solution

3. CONSERVATION OF MASS • This is a single scalar equation • Velocity doted with normal unit vector results in a scalar • 1st Term: Rate of change of mass inside CV • If steady d/dt( ) = 0 • Velocity, density, etc. at any point in space do not change with time, but may vary from point to point • 2nd Term: Rate of convection of mass into and out of CV through bounding surface, S • 3rd Term (=0): Production or source terms Relative to CS Inertial

4. Integral vs. Differential Form • Integral form of mass conservation • Apply Divergence (Gauss’) Theorem • Transform both terms to volume integrals • Results in continuity equation in the form of a partial differential equation • Applies to a fixed point in the flow • Only assumption is that fluid is a continuum • Steady vs. unsteady • Viscous vs. inviscid • Compressible vs. incompressible

5. Summary: Incompressible vs. Constant Density • Two equivalent statements of conservation of mass in differential form • In an incompressible flow • Says particles are constant volume, but not necessarily constant shape • Density of a fluid particle does not change as it moves through the flow field • Incompressible: Density may change within the flow field but may not change along a particle path • Constant Density: Density is the same everywhere in the flow field

6. MOMENTUM EQUATION: NEWTONS 2nd LAW Inertial Relative to CS • This is a vector equation in 3 directions • 1st Term: Rate of change of momentum inside CV or Total (vector sum) of the momentum of all parts of the CV at any one instant of time • If steady d/dt( ) = 0 • Velocity, density, etc. at any point in space do not change with time, but may vary from point to point • 2nd Term: Rate of convection of momentum into and out of CV through bounding surface, S or Net rate of flow of momentum out of the control surface (outflow minus inflow) • 3rd Term: • Notice that sign on pressure, pressure always acts inward • Shear stress tensor, t, drag • Body forces, gravity, are volumetric phenomena • External forces, for example reaction force on an engine test stand • Application of a set of forces to a control volume has two possible consequences • Changing the total momentum instantaneously contained within the control volume, and/or • Changing the net flow rate of momentum leaving the control volume

7. Application to Rocket Engines F Chemical Energy Rocket Propulsion (class of jet propulsion) that produces thrust by ejecting stored matter • Propellants are combined in a combustion chamber where chemically react to form high T&P gases • Gases accelerated and ejected at high velocity through nozzle, imparting momentum to engine • Thrust force of rocket motor is reaction experienced by structure due to ejection of high velocity matter • Same phenomenon which pushes a garden hose backward as water flows from nozzle, gun recoil Thermal Energy Kinetic Energy

8. Application to Airbreathing Engines Chemical Energy Kinetic Energy Thermal Energy • Flow through engine is conventionally called THRUST • Composed of net change in momentum of inlet and exit air • Fluid that passes around engine is conventionally called DRAG

9. Thrust Definitions • Use of conservation of mass and momentum in control volume form to derive governing equation for “net uninstalled thrust” • See Section 3.1 • Textbook notation: Fn)uninstalled • Pay close attention to control volume choices • Control volume option 1: pages 113 – 119 • Control volume option 2: pages 121 - 124 • Understand each term, including the ram drag and pressure mismatch • Can divide into nozzle contribution and inlet contribution • Nozzle contribution is called “gross thrust” • Textbook notation: Fg • Uninstalled Thrust: thrust produced by engine if it had zero external losses • Installed Thrust: actual propulsive force transmitted to aircraft by engine

10. Efficiency Summary • Overall Efficiency • What you get / What you pay for • Propulsive Power / Fuel Power • Propulsive Power = TUo • Fuel Power = (fuel mass flow rate) x (fuel energy per unit mass) • Thermal Efficiency • Rate of production of propulsive kinetic energy / fuel power • This is cycle efficiency • Propulsive Efficiency • Propulsive Power / Rate of production of propulsive kinetic energy, or • Power to airplane / Power in Jet

11. MOMENTUM EQUATION: NEWTONS 2nd LAW Relative to CS Inertial • This is a vector equation in 3 directions • 1st Term: Rate of change of momentum inside CV or Total (vector sum) of the momentum of all parts of the CV at any one instant of time • If steady d/dt( ) = 0 • Velocity, density, etc. at any point in space do not change with time, but may vary from point to point • 2nd Term: Rate of convection of momentum into and out of CV through bounding surface, S or Net rate of flow of momentum out of the control surface (outflow minus inflow) • 3rd Term: • Notice that sign on pressure, pressure always acts inward • Shear stress tensor, t, drag • Body forces, gravity, are volumetric phenomena • External forces, for example reaction force on an engine test stand • Application of a set of forces to a control volume has two possible consequences • Changing the total momentum instantaneously contained within the control volume, and/or • Changing the net flow rate of momentum leaving the control volume

12. Momentum Equation: Mid-Air Refueling • An F-4 Phantom is being refueled in mid-air • Refueling boom enters at an angle of 30° from the F-4 flight path • Fuel flow rate through the boom is 20 kg/s at a velocity of 30 m/s relative to the two aircraft • Density of jet fuel is less than water, and is about 700 kg/m3 • What additional lift force is necessary to overcome force on F-4 fighter due to momentum transfer during refueling?