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

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

CONSERVATION OF MASS Relative to CS Inertial 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

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

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

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

Application to Rocket Engines Chemical Energy F 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

Application to Airbreathing Engines Chemical Energy Thermal Energy Kinetic 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

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

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