flow inside turbomachines contents: equation of balance of angular momentum for a control volume ...

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

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

Balance of angular momentum for a control volume

Sum of applied forces over a Control Volume (CV):

12 SS

dwVdwVdVdt

dF

dsnVdw

Outlet momentum flow rate Inlet momentum flow rate

Accumulation rate of momentum inside CV

Steady State Condition

Vmdt

dF

2nd Law of Newton:

(force over a fluid particle)

Balance of angular momentum for a control volume In turbomachines, we are interest in moments of

forces and torques:

FrT

o

PF

r

Angular momentum: VmrH

Taking the

derivative: Vm

dt

drVm

dt

rd

dt

Hd

V

=0 (colinear vectors)

F

FrT

dt

HdT

Balance of angular momentum for a control volume By resemblance :

12 SS

dwVdwVdVdt

dF

Vm

dt

dF

Vmrdt

d

dt

HdT

It comes (replacing for ):V

Vr

12 SS

dwVrdwVrdVrdt

dT

Momentum with respect to the origin resulting from the forces applied to the CV

Rotor of a turbomachine

Control Surfaces

S1 and S2 – surfaces of revolution

Inner wall of the rotor and inner walls of the casing

Cylindrical Coordinates

Unit vectors

Velocity

We are looking for axial moments

Replacing for and for

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

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:

Torque on the rotor

Same sign (turbine)

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

power

Fixed reference

Rotor’s reference

Absolute velocity

Relative velocity

r - distance to the rotation axle

Transport Velocity

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

Extra therm Fictitious potencial energy associated to centrifugal forces

Incompressible Flow:

Losses

Flow with increasing

Radial pumps, fans and compressors

Centrifugal flow

Radial Turbines