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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