Turbomachinery Aero-Thermodynamics

47 Pages·2015·2.3 MB·English

Turbomachinery Aero-Thermodynamics
Aero-Thermodynamics 2D – Losses
Alexis. Giauque1
1LaboratoiredeM´ecaniquedesFluidesetAcoustique
EcoleCentraledeLyon
Ecole Centrale Paris, January-February 2015
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Evaluation
Evaluation for sessions 1 & 2
QROC I (20 mins)
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And now what are the stakes and technologies?
Miniaturizing technology
Electricity production – Feed the robots needs
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Table of Contents
1. Euler theorem for turbomachines
Naive derivation
Rothalpy
Formal derivation
2. Velocity triangles
3. Losses in axial compressors
Introduction
Profile losses
Effect of the incidence angle
Other types of losses
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Euler theorem for turbomachines – Naive derivation
Let’s consider the following axial machine
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Euler theorem for turbomachines – Naive derivation
The torque experienced by a physical system is equal to the temporal
dM
change of its angular momentum. i.e. C = θ(1).
dt
If we apply this relation to the sys-
tem on the left, assuming velocities
are uniform in 1 and 2, we obtain that
dM −dM
C = θ2 θ1
dt
dm V r −dm V r
C = 2 θ2 2 1 θ1 1
dt
1The angular momentum of a rotating mass is defined as M =mV r
θ θ
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Euler theorem for turbomachines – Naive derivation
C = m˙(V r −V r )
θ2 2 θ1 1
Cω = m˙(V r ω−V r ω)
θ2 2 θ1 1
P = m˙(V U −V U )
u θ2 2 θ1 1
∆w = ∆(V U)
u θ
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Euler theorem for turbomachines – Naive derivation
Euler theorem for turbomachines
This relation is fondamental. It relates the changes in velocity directions
and intensity (aerodynamics) to the effective work (thermodynamics).
It applies to all kind of turbomachines (axial,radial,mixed).
∆w = ∆h = ∆(V U)
u 0 θ
Note! Thanks to the representation of the velocity vectors we can learn
about the work exchange.
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Euler theorem for turbomachines – A few comments
This fundamental equation brings a few important comments
Note! ∆w > 0 for a compressor and ∆w < 0 for a turbine. In some
u u
textbooks dedicated to turbines, the relation is multiplied by -1 to
have positive quantities...Be careful!
Note! M = f(ρ,V ,r). Modifying the radius of a stage between 1
theta θ
and 2 will therefore lead to potentially more power delivered (turbine)
or a higher compression rate (compressors). This effect is the main
reason for the development of centrifugal compressors and turbines.
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Rothalpy
The rothalpy is defined as
I = h −UV
0 θ
This definition comes in handy because of the Euler equation for
turbomachines which states that thought the rotor
∆h = ∆(UV )
0 θ
We therefore have
∆(h −UV ) = ∆I = 0
0 θ
Rothalpy
The rothalpy I = h −UV is conserved through a turbomachinery stage.a
0 θ
aNote however that the rothalpy is not a thermodynamic variable per se, it
depends of the frame of reference.
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