1 Turbomachinery Lecture 2a - Conservation of Mass, Momentum - Bernoulli Equation - Performance Parameters

Fluid Flow Microscopic: atomic- molecular motion Macroscopic: continuum motion approach 2

3 Eulerian Control Volume Approach Lagrangian Approach: –Follow fluid particle or droplet on path [called path line]. –Fluid particle is an aggregate of discrete particles of fixed identity. Focus on particles as they move through flow. –Each particle is labeled by its original position. –Examples are particle tracking for sprays and coatings, continuum mechanics, oceanographics [flow meters drift along prevailing currents]

4 Eulerian Control Volume Approach Eulerian or Control Volume Approach: –Watch a fixed point in space, not one particle, as time proceeds. –CV is designated in space and the bondary known. The amount and identity of the matter in CV may change with time, but shape is fixed. –Property field, e.g. V=velocity=V(x,y,z,t) [streamline] is defined. –Path lines and streamlines are identical in a steady flow.

5 Eulerian Control Volume Approach Eulerian – Lagrangian –Extensive property: dependent on mass in volume under consideration: N –Intensive property: independent of mass in volume under consideration: n

6 Eulerian Control Volume Approach Elemental Fixed Volume Cube

Brief Math Review Cartesian, cylindrical coordinate systems Unit vectors [i, j, k, n] Dot products Grad, curl operators Green’s integral theorem Ordinary, partial, and material derivatives Be careful, many variables have same symbol –V=volume, V=velocity, etc. 7

8 Eulerian Control Volume Approach Scalar convection –Substantial derivative [for a coordinate system] of an extensive property N –Consider some arbitrary extensive property N of fluid associated with CV –n = N per unit mass and  dV is the elemental mass Differentiate along particle path  a Lagrangian process where total or Stokes derivative reflects the change in N over time and space

9 Scalar Conservation I II III Time rate of change convection Time=t Time=t+  t

10 Conservation Laws

11 Conservation Laws 2D Steady Flow No. equations = 5 No. unknowns = , u, v, w, p, h 0 Therefore need additional relations to close system p

12 Conservation Laws 2D Steady Flow of Energy

13 Conservation Laws 2D Steady Flow u1u1

14 Conservation Laws 2D Steady Flow p

Mass to Continuity 15

16 Bernoulli’s Equation

17 Bernoulli’s Equation Bernoulli - Steady - Inviscid - Irrotational or on streamline -Incompressible What is meant by incompressible?

V p0p0 psps Pitot – Kiel Head Pressure Probe

19 Inviscid Momentum Equation Adding additional force terms (gravity, magnetism, etc.) on flow, gives more general form of inviscid, integral momentum equation: Equation is basis of C.V. approach to many problems Note - for steady flow calculate force on immersed object from flow variables on surface of C.V.!!! To solve unsteady and/or viscid flows must integrate throughout the volume - orders of magnitude more difficult!

20 Steady Inviscid Momentum Equation Integral form of Inviscid Momentum Equation is outward normal from surface area. In 2D:

21 Steady Inviscid Momentum Equation For cylindrical surface, use cylindrical coordinates  r y x

22 Steady Inviscid Momentum Equation Substituting into vector equation: Writing this as two scalar equations:

23 Steady Inviscid Momentum Equation The math works - but don't lose common sense. Still: Pressure force is positive to right, acceleration is positive to right, so:

24 Steady Inviscid Momentum Equation Examples: –Circular cylinder in flight: Compressible flow homework –Circular cylinder in duct: Compressible flow homework –Jet Engine In Flight

25 Inviscid Momentum Equation Example: Application to Jet Engine in Flight AB0cancels IJ BC0 CD0cancels EF DE0 EF0cancels CD FG0 GH0 HI0 IJ0cancels AB JA0

26 Inviscid Momentum Equation Summing terms: Jet engine control volume chosen to eliminate Same result for any control volume fully enclosing the engine Generally cannot eliminate for internal flows

Effect of Ambient Pressure [p atm ]

28 Uninstalled (Ideal) Thrust So From Force Considerations (Control Volume Analysis), the Uninstalled (Ideal) Thrust for an Engine is: Why now pressure term? What is effect of P ambient term?

29 Specific Fuel Consumption The Rate of Fuel Used by the Propulsion System per Unit of Thrust: –uninstalled: –installed: So

30 3D Steady Flow Energy Equation If flow velocity brought to zero adiabatically, apply Gibbs equation at stagnation properties Energy based on 1 st law Was derived earlier

31 Other Important Equations