Equation For Free Surface Of Liquid In Rotating Cylinder At Constant Pressure Calculator

Equation For Free Surface Of Liquid In Rotating Cylinder At Constant Pressure:

\[ Z_s = h_o - \left( \frac{\omega_{Liquid}^2}{4 \cdot [g]} \right) \cdot \left( R^2 - 2 \cdot r_p^2 \right) \]

Height of Free Surface of Liquid without Rotation (h₀):

Angular Velocity of Rotating Liquid (ω):

rad/s

Radius of Cylindrical Container (R):

Radius at any given Point (rₚ):

Distance of Free Surface from Bottom of Container (Zₛ):

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1. What is the Equation for Free Surface of Liquid in Rotating Cylinder?

The equation calculates the distance of the free surface from the bottom of a cylindrical container when the liquid inside is rotating at constant angular velocity. It describes the parabolic shape formed by the liquid surface due to centrifugal forces.

2. How Does the Calculator Work?

The calculator uses the following equation:

\[ Z_s = h_o - \left( \frac{\omega_{Liquid}^2}{4 \cdot [g]} \right) \cdot \left( R^2 - 2 \cdot r_p^2 \right) \]

Where:

\( Z_s \) — Distance of free surface from bottom of container (m)
\( h_o \) — Height of free surface of liquid without rotation (m)
\( \omega_{Liquid} \) — Angular velocity of rotating liquid (rad/s)
\( [g] \) — Gravitational acceleration (9.80665 m/s²)
\( R \) — Radius of cylindrical container (m)
\( r_p \) — Radius at any given point (m)

Explanation: The equation accounts for the parabolic depression of the liquid surface caused by centrifugal forces during rotation, with the minimum at the center and maximum at the walls.

3. Importance of Free Surface Calculation

Details: Understanding the free surface profile is crucial for designing rotating machinery, centrifugal separators, and analyzing fluid behavior in rotating systems. It helps predict liquid distribution and potential spillage or dry-out conditions.

4. Using the Calculator

Tips: Enter all values in meters and radians per second. Ensure the radius at any point (rₚ) is less than or equal to the container radius (R). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: Why does the liquid surface form a parabola when rotating?
A: The centrifugal force pushes liquid outward, creating a pressure gradient that balances with gravity, resulting in a parabolic free surface.

Q2: What happens at the center of rotation?
A: At the center (rₚ = 0), the surface reaches its minimum height due to the maximum centrifugal force effect.

Q3: How does angular velocity affect the surface profile?
A: Higher angular velocities create deeper parabolic depressions with greater height differences between center and wall.

Q4: Are there limitations to this equation?
A: This equation assumes constant angular velocity, incompressible fluid, and neglects surface tension effects.

Q5: What practical applications use this principle?
A: Centrifugal pumps, rotating machinery lubrication systems, centrifugal separators, and artificial gravity systems in space.