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# One flow and infinite number of stator and rotor

One Dimensional Flow in Axial
Compressors

The flow through axial compressor is essentially three
dimensional as flow properties and velocities are function of r, ? and z.  in this module, the flow is assumed to be one
dimensional to avoid the complicated flow analysis which require great effort
for analyzing.

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The flow properties and velocities are assumed to vary in
the z direction (i.e. meridional planes) in axial compressors. A meridional
plane is any plane passing by the axis of rotation of the machine. Meridional
for is a two-dimensional flow based on the assumption of asymmetry of flow and
infinite number of stator and rotor blades. The flow properties are assumed to
be constant in r direction. DR.GALAL BOOK

A s we have discussed before the Stage of compressor consist
of a work transfer component with negative work, a decelerating stator and
sometime an accelerating stator may precede the rotor (inlet guide vanes).
photo of 3 component of stages 0 1 2 3

The velocity components
of the working fluid can
be expressed in three velocity vectors, absolute, radial and relative velocity.
Assuming the radial velocity equal zero. The air approaches the rotor with an
absolute velocity C1 at an angle ?1 from the axial direction. Combining
the absolute velocity vectorially with the blade speed U gives the relative velocity W1 at an angle ?1. After passing through the
rotor, which increases the absolute velocity of the air, the fluid leaves the
rotor with a relative velocity W2 at an angle ?2 determined by the blade outlet
angle. The fluid leaving the rotor is consequently the air entering the stator
where a similar change in velocity will occur. Here the relative velocity W2
will be diffused and leaving the stator with a velocity C3 at
an angle ?3.

The velocity vectors and associated velocity diagram for a
typical stage are shown in Fig {number }

Saravanamutto, HIH,
Rogers, GFC och Cohen, H. Gas Turbine Theory Fifth Edition, Pearson Prentice Hall, 2001.

From Euler turbine equation  and Referring to the
velocity triangle in the figure it’s easy to get:

Combining the Euler equation with 1st law
of thermodynamics

We got:

The term I is therefore a constant for the rotor and
is known as the relative total enthalpy.

The flow is assumed
adiabatic and there is no work transfer in the stator, then the total enthalpy
is constant.

In the Figure 7.3 the
thermodynamic states are displayed on the T-S diagram. The distances that
represent the absolute and relative kinetic energies are also shown. The relative
stagnation enthalpy across the rotor remains constant. In the rotor rothalpy
has properties analogous to stagnation enthalpy in the stator. KORPELLA

Stage Efficiency

Is the ratio between isentropic work of compressor to
actual work of compressor

Degree of reaction

Is a measure of the enthalpy rise in the rotor to the
total enthalpy rise in the stage. The degree of reaction R indicates the
portion of energy transferred in the rotor blading. It may be defined based on
the actual enthalpy rise or the isentropic enthalpy rise.

or

Although that there are some differences in the
mentioned expressions of the degree of reaction, but all of them give the same
concept. It’s an important parameter especially for axial machines which defines
the class of machine with particular characteristics, since it’s fixed, the
shape of velocity diagrams and blade arrangement are also fixed. For axial
compressor the degree of reaction is usually 0.5 or higher as the flow is
decelerated through compressor hence high flow turning can’t be carried out. To
increase energy transfer across the stage an increase in rotational speed of
compressor is required.

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