# 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

This gives:

**Degree of Reaction**

Therefore,

**Exponential**

n=0, so that

Following the same integration procedure as in n=1 gives:

### Degree of Reaction

Therefore,