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Axial velocity at exit of vane profiles. [V2M ]
From the author’s Engineering Collection, included in the ETSII4 module (ETI4 on the CL Library)

This program calculates the axial velocity at a radius r of an impeller for axial pumps, characterized by the ideal head performance equation: Ht = a + b r ; and within the boundaries of the blades.

The general expression for the axial velocity is given below:

Vm^2 = Vmi^2 + 2gb(r-ri) – 2b (g/w)^2 [a(1/ri - 1/r) + b ln (r/ri) ]

Where Vmi (a constant value) is the exit axial velocity at the root of the blade, and can determined as a boundary condition when the flow is known, by means of the following numerical integration:

Q = 2pi INTG { r Vm(r) } dr ; between Ri and Re

Thus this program will first use a numerical integration routine to obtain the value of Vmi, and with it, it’ll take upon solving for r in the Vm equation. The nested arrangement explains the long execution times, as the integral needs to be calculated at each iteration of the root finder!

Both the numeric integration and root-finding routines are included in the ETSII4 module for your convenience, so there are no additional dependencies.

Example.

Calculate the exit axial velocity at radius r= .25 for an axial pump with the following characteristics: