Deriving the Simple Radial Equilibrium Equation
Inviscid Flow:
Steady Axial Inviscid Flow:
Axisymmetric Inviscid Flow:
(Effect of discrete blades not transmitted to flow; the assumption of axial symmetry enables the designer to fix attention on the radial distribution of the gas properties at a station between blade rows by averaging out the gas-property variations in the circumferential direction.)
Steady Axisymmetric Inviscid Flow:
Steady Axial Axisymmetric Inviscid Flow with Negligible Radial Blade Force:
These assumptions reduce the momentum equation to:
Free Vortex
Choosing free vortex where:
Forced Vortex
In a forced vortex, the fluid rotates as a solid body (no shear). The motion can be realized by placing a dish of fluid on a turntable rotating at x radians/second.
Constant Reaction/First Power
n=1, so that
Using the equation above, and assuming 
,
, it can be integrated at station 1 and 2 to find the resulting values of 
at any radius depending on the n values.
Using n=2 as an example in the general whirl velocity,
Integrating from mean radius to any radius r,
, 
n=1
Hence,
Replace
where 
with 
=1
This gives:
Where 
And so 
Degree of Reaction
From the 
definition
Therefore,
.
Substituting 
results in
.
Exponential
n=0, so that
Following the same integration procedure as in n=1 gives:
Where 
And so 
Degree of Reaction
From the definition
Therefore,
Substituting 
gives
.