Governing equations in TURBOMACHINERY STAGES
Rotating flow passages, Euler work
Alessandro Corsini,
Department of Mechanical & Aerospace Engineering
Sapienza University of Rome, Italy
Introduction
Turbomachinery are devices in which energy transfer occurs between a flowing
fluid and a rotating element due to dynamic action, and results in a change in
pressure and momentum of the fluid.
Mechanical energy transfer occurs inside or outside of the turbomachine, usually in
a steady-flow process.
The energy transfer can go from the rotor to the fluid (power absorbing or
macchine operatrici) like in a compressor or a pump or vice-versa (powerproducing or macchine motrici) like in a turbine. The classical distinction of
machine operatrici/motrici doesn’t have a straightforward translation, they directly
refer to compressor and turbines.
Turbomachinery can be classified in other ways, such as the main direction of the
fluid inside the impeller (axial or radial) and the properties of the fluid
(incompressible or compressible).
Turbomachinery stage
Walter Hossli
Scientific American
Vol. 220, No. 4 (April 1969)
Turbomachinery stage
Axial
configurations
Turbomachinery stage
Radial
configurations
FLOW REGIMENS IN FLUIDMACHINERY
Euler Work Equation or the Conservation of Moment of Momentum
Momentum equation
• Newton’s second law of motion relates the sum of external forces acting on
the fluid to its acceleration.
• If we consider a system of mass m, the sum of all surface forces F acting on
m along the x direction, is given by (36):
𝑑
∑𝐹! =
𝑚𝑐!
𝑑𝑡
(36)
• For a control volume (inlet - 1, outlet - 2 ), where the fluid enters and exits
with a meridian velocity c1 and c2 respectively, the momentum equation
becomes:
∑𝐹! = 𝑚̇ 𝑐!! − 𝑐!"
(37)
Momentum of momentum equation
• It is possible to derive a similar form of the momentum equation for the
moments of forces.
cu2
• The sum of all the moments, along the A axis is given by:
𝜏" = 𝑚
𝑑
rc#
𝑑𝑡
cu1
(37)
where cu is the tangential velocity
• For a control volume, the flow enters at radius r1 with tangential velocity cu1
and exits at radius r2 with cu2 velocity.
The equilibrium of momentum becomes:
𝜏" = 𝑚̇ 𝑟$ 𝑐%$ − 𝑟& 𝑐%&
(38)
Euler work equation
• For a pump or compressor rotor running with angular velocity W, the rate at
which the rotor does work on the fluid is given by:
c
u2
𝜏" Ω = 𝑚̇ Ω𝑟$ 𝑐%$ − Ω𝑟& 𝑐%&
(39)
𝜏" Ω = 𝑚̇ 𝑈$ 𝑐%$ − 𝑈& 𝑐%&
(40)
where U = W r is the blade speed.
cu1
Euler work equation
• For a pump or compressor rotor running with angular velocity W, the rate at
which the rotor does work on the fluid is given by:
c
u2
𝜏" Ω = 𝑚̇ Ω𝑟$ 𝑐%$ − Ω𝑟& 𝑐%&
(39)
𝜏" Ω = 𝑚̇ 𝑈$ 𝑐%$ − 𝑈& 𝑐%&
(40)
cu1
where U = W r is the blade speed.
• Thus, the work done on the fluid per unit of mass, namely the Euler’s pump
equation, is computed by:
𝑊̇
𝜏" Ω
𝑊' =
=
= 𝑈$ 𝑐%! − 𝑈& 𝑐%" > 0
(41)
𝑚̇
𝑚̇
• In a turbines, work is extracted from the fluid, therefore the signs are
inverted in the Euler’s turbine equation:
𝑊̇
𝜏" Ω
𝑊' =
=
= 𝑈& 𝑐%" − 𝑈$ 𝑐%! > 0
𝑚̇
𝑚̇
(42)
Euler work equation in adiabatic machines
• For any adiabatic turbomachine, the Euler work equation reads:
𝑊' = (ℎ(& − ℎ($ ) 𝑎𝑠 𝑠𝑢𝑐ℎ (ℎ(& − ℎ($ ) = 𝑈$ 𝑐%! − 𝑈& 𝑐%" (43)
Δℎ( = Δ 𝑈𝑐%
(44)
• When U = 0 the total enthalpy is constant since a stationary blade cannot
transfer any work to or from a fluid.
• Remark 0. In adiabatic condition the total enthalpy change measures the
mechanical work
Euler work equation in adiabatic machines, remarks
• For any adiabatic turbomachine, the Euler work equation reads:
𝑊' = (ℎ(& − ℎ($ ) = 𝑈$ 𝑐%! − 𝑈& 𝑐%"
(43)
Δℎ( = Δ 𝑈𝑐%
(44)
• Equations (43) and (44) are the general form of Euler equations.
• Remark 1. Those equations are valid for adiabatic flow for any streamline
through the rotating (or moving) blade rows of a turbomachine.
• Remark 2. It is applicable to both viscous and inviscid flow, since the torque
provided by the fluid on the blades can be exerted by pressure forces or
frictional forces.
• Remark 3. It is strictly valid only for steady flows but it can also be applied to
time-averaged unsteady flow provided the averaging is done over a long
enough time period.
Euler work equation in adiabatic machines, remarks
• For any adiabatic turbomachine, the Euler work equation reads:
𝑊' = (ℎ(& − ℎ($ ) = 𝑈$ 𝑐%! − 𝑈& 𝑐%"
(43)
Δℎ( = Δ 𝑈𝑐%
(44)
• Equations (43) and (44) are the general form of Euler equations.
• Remark 4. Euler work equation can also reads in a different form explicitly
correlating the thermo-fluid dynamic evolution within the blade passage to
kinetic energy variation.
• Remark 5. Euler equation bridges the dynamics (performance) to the kinematics
of the fluid along prescribed streamlines, this in turn is providing the explicit
link to the geometry of the turbomachine.