Deriving Accurate Exit air velocity of rotor

My goal is to derive the equation for exit air velocity as a function of:

1. Pitch angle
2. RPM
4. Constants like ρ

Assumptions:

• Vc is relative velocity of air upstream, and is = 0
• Inflow is unevenly distributed in the form of tip loss
• Out-of-plane velocity Up is much smaller than the in-plane velocity Ut so that U=Ut
• vi=vh (hover)
• Shrouds of any kind are not used to direct airflow

Definition of variables:

• Pitch angle: theta
• Density of the fluid (air in this case): ρ
• induced inflow velocity in the plane of the rotor: vi
• Rotor Thrust: T
• Area of the rotor disk: A=π(R^2)
• Coefficient of thrust: CT

Current work that I have done: Used a combination of equations 2.58, 3.31, and 3.36 to derive an equation for vh as follows:

Next steps: derived torque equation (have some from chapters 2 and 3 but they are not a function of Nb) Or just a more simple equation that outputs the output velocity as a function of rotor pitch, RPM, Diameter, and blade count

References: Principles of Helicopter Aerodynamics by J. Gordon Leishman

Ultimate goal: derive rotor wash / prop blast as a function of:

• vertical distance from rotor
• horizontal distance from center of rotor

Is there anything I don't know enough to ask? feel free to educate me from the ground up, I have only read 3 chapters from a textbook, so I have a lot to learn.

• I think that we cannot give a better explanation than the Leishman himself. Anyway if you need to have the blade's geometry in your equations then you need to use the blade element momentum theory and not the simpler momentum theory that you've been using so far. Commented Nov 12, 2023 at 12:28
• Are there any more books about rotor / prop design that I should read? I've tried to use blade element momentum theory, but have trouble with calculating L (lift force) due to lack of data about the rotors since I only have pitch angle, diameter, blade count, and rpm, thanks! Commented Nov 12, 2023 at 13:09
• Is this an university assignment? What's your background? What are the precise known inputs and requested outputs? Commented Nov 12, 2023 at 13:35
• Its for a project involving a 1x1ft square of metal that is being hit with the prop blast of a variable drone (variable diameter, motor number, rotor diameter, rotor blade number, and rotor blade pitch, motor rpm, and height from 1x1ft square) assuming the center of the drone is also on the center of the square. The goal is to simulate the amount of force or "weight" put on the 1x1ft square from the air produced by the motion of the props. Assuming hover conditions. Commented Nov 13, 2023 at 17:40
• The GUI I did for it so you can get a visual feel for it: imgur.com/a/a6DZ4DZ Commented Nov 13, 2023 at 17:42