1. Flow inside turbomachines
Contents:
Equation of balance of angular momentum for a control volume
One-dimensional equation of torque on turbomachines shaft
Euler equation for turbomachines
Bernoulli equation for steady relative flow

2. Balance of angular momentum for a control volume
In a turbomachine, the inlet cross section S1 and the exit cross section S2 of the control volume are fixed and surfaces of revolution S1 S2
Control Volume
In a turbomachinery, the inlet cross area and the exit cross area of the fluid through the control volume are fixed and surfaces of revolution

3. Balance of angular momentum for a control volume
Sum of applied forces over a Control Volume (CV):
Outlet momentum flow rate
Inlet momentum flow rate
Accumulation rate
of momentum inside CV
Resul
Steady State Condition
2nd Law of Newton:
(force over a fluid particle)

4. Balance of angular momentum for a control volume
In turbomachines, we are interest in moments of forces and torques: o P F r
Angular momentum:
In turbomachinery studies,
Taking the derivative:
=0 (colinear vectors)

5. Balance of angular momentum for a control volume
By resemblance :
It comes (replacing for ):
Momentum with respect to the origin resulting from the forces applied to the CV

6. Rotor of a turbomachine
Control Surfaces
S1 and S2 – surfaces of revolution
Inner wall of the rotor and inner walls of the casing

7. Cylindrical Coordinates
Unit vectors
Velocity
We are looking for axial moments
Replacing
for
and
for

8. Assuming steady flow (constant N and flow rate Q) it turns out:

9. Torque on the axle by all the forces applied on the CV
Torque by the CV on the rotor
Moment of forces over the stator wall (revolution surface)
Pressure
Shear Stresses
Moment of forces over S1 and S2 – revolution surfaces
Moment of weight W
For one dimensional flow:

10. Pumps, fans, compressors
Torque on the rotor
Same sign (turbine)
power
Opposite signs
(Pumps, fans, compressors)
Energy per unit mass
Assuming positive values of
Turbines
Pumps, fans, compressors
Euler Equations for turbomachinery (1754)
Applicable to compressive or incompressible flow with or without viscous effects

11. r - distance to the rotation axle
Fixed reference
Absolute velocity
Relative velocity
Rotor’s reference
Transport Velocity
r - distance to the rotation axle

12. (from the velocity triangle)
Turbines
Euler Equation
Energy Conservation
But
(from the velocity triangle)
Bernoulli Equation for steady state relative flow conditions
Valid for: Turbines, pumps, fans and compressors
One dimensional flow
With or without viscous effects
Compressible or incompressible flow

13. Extra therm
Fictitious potencial energy associated to centrifugal forces
Incompressible Flow:
Losses

14. Radial pumps, fans and compressors
Flow with increasing
Centrifugal flow
Radial Turbines