Reference ID: MET-5B03 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The calculation of pressure within a rotating liquid mass is a fundamental requirement in process engineering, particularly for the design and operation of centrifugal separation equipment. In systems such as basket or tubular centrifuges, a liquid body rotating at a constant angular velocity generates a radial pressure gradient due to centrifugal force. This pressure distribution is critical for determining the structural integrity of the centrifuge bowl, calculating the power requirements for rotation, and understanding the phase separation dynamics of multi-component mixtures.
Methodology & Formulas
The pressure distribution is derived from the Navier-Stokes equations under the assumption of steady-state, solid-body rotation. The liquid is treated as an incompressible, Newtonian fluid where the radial pressure gradient is balanced by the centrifugal acceleration.
First, the rotational speed provided in revolutions per minute (RPM) must be converted to angular velocity (\(\omega\)) in radians per second:
\[ \omega = \frac{2 \cdot \pi \cdot N}{60} \]
The pressure difference (\(\Delta P\)) between an inner radius (\(r_{1}\)) and an outer radius (\(r_{2}\)) is calculated based on the fluid density (\(\rho\)) and the angular velocity:
To determine the absolute pressure at the outer wall (\(P_{wall}\)), the reference pressure at the inner radius (\(P_{ref}\)) must be added to the calculated pressure difference:
\[ P_{wall} = P_{ref} + \Delta P \]
The validity of the solid-body rotation assumption is verified by calculating the rotational Reynolds number (\(Re\)), which ensures that viscous effects are appropriately accounted for relative to inertial forces:
\[ Re = \frac{\rho \cdot \omega \cdot r_{2}^{2}}{\mu} \]
Parameter
Condition / Threshold
Engineering Significance
Geometry
\(r_{2} > r_{1} \geq 0\)
Ensures physical consistency of the rotating volume.
Operational Range
\(10 \leq N \leq 10000 \, \text{RPM}\)
Standard industrial range for centrifugal equipment.
Geometric Range
\(0.1 \text{ m} \leq r_{2} \leq 1.0 \text{ m}\)
Typical scale for industrial centrifuge baskets.
Flow Regime
\(Re \leq 10^{7}\)
Ensures stability of solid-body rotation; values exceeding this may indicate turbulence.
To calculate the pressure at any point within a rotating liquid mass in a centrifuge, where the liquid rotates as a solid body, follow these steps:
Define the angular velocity \(\omega\) of the rotating system from the rotational speed \(N\) in RPM.
Establish the radial coordinate \(r\) measured from the axis of rotation.
Apply the governing equation for solid-body rotation of an incompressible fluid: \(P(r) = P_{ref} + \frac{1}{2} \cdot \rho \cdot \omega^{2} \cdot (r^{2} - r_{1}^{2})\), where \(P_{ref}\) is the known pressure at the inner radius \(r_{1}\).
Ensure all units are consistent, using SI units for density (\(\rho\)), radius (\(r\)), and pressure (\(P\)).
In a centrifuge basket, the liquid rotates as a solid body. The pressure increases quadratically with radius, following the equation:
\(P(r) = P_{ref} + \frac{1}{2} \cdot \rho \cdot \omega^{2} \cdot (r^{2} - r_{1}^{2})\).
where \(P_{ref}\) is the pressure at the inner radius \(r_{1}\). The pressure at the outer wall (\(r = r_{2}\)) is the maximum and is critical for mechanical design.
Yes, process engineers must account for mechanical stress and fluid dynamics when designing these systems:
Calculate the maximum pressure at the outer wall to ensure vessel integrity.
Monitor for cavitation if the pressure at the center drops below the vapor pressure of the liquid.
Verify that the structural material can withstand the hoop stress generated by the rotating mass.
Account for potential imbalances that may introduce vibration and dynamic load fluctuations.
Worked Example: Pressure at the Wall of a Centrifuge Basket
A process engineer must assess the pressure at the outer wall of a basket centrifuge used for solid-liquid separation with water at 20°C. The centrifuge rotates at 100 RPM, and the liquid fills the basket from the central axis to an outer radius of 0.5 m. The basket is vented to the atmosphere at the axis. This calculation ensures structural integrity under operating conditions.
Known Input Parameters:
Fluid density, \(\rho = 1000.0 \, \text{kg/m}^3\) (water at 20°C)
Convert rotational speed to angular velocity:
The angular velocity \(\omega\) is calculated from RPM: \(\omega = \frac{2 \pi N}{60}\).
Using \(N = 100.0 \, \text{RPM}\) and \(\pi \approx 3.1415926536\), \(\omega = 10.4720 \, \text{rad/s}\).
Compute the centrifugal pressure difference:
For solid-body rotation, the pressure difference between radii \(r_{2}\) and \(r_{1}\) is:
\(\Delta P = \frac{\rho \omega^2}{2} (r_{2}^{2} - r_{1}^{2})\).
With \(\rho = 1000.0 \, \text{kg/m}^3\), \(\omega = 10.4720 \, \text{rad/s}\), \(r_{2} = 0.5 \, \text{m}\), and \(r_{1} = 0.0 \, \text{m}\):
\(\Delta P = 13707.8 \, \text{Pa}\).
Determine absolute pressure at the wall:
The absolute pressure at the outer wall \(P_{wall}\) is the sum of reference pressure and \(\Delta P\):
\(P_{wall} = P_{ref} + \Delta P = 101325.0 \, \text{Pa} + 13707.8 \, \text{Pa} = 115032.8 \, \text{Pa}\).
Convert pressure to engineering units:
In bar (1 bar = \(10^{5} \, \text{Pa}\)):
\(P_{wall} = 1.150328 \, \text{bar} \approx 1.15 \, \text{bar}\).
Validate solid-body rotation assumption:
The Reynolds number for rotation is \(Re = \frac{\rho \omega r_{2}^{2}}{\mu}\).
With given values, \(Re = 2.6180 \times 10^{6}\). Since \(Re < 1 \times 10^{7}\), viscous effects are negligible, confirming the solid-body rotation assumption.
Final Answer:
The absolute pressure at the outer wall of the centrifuge basket is \(115032.8 \, \text{Pa}\) or \(1.15 \, \text{bar}\). This pressure is used for mechanical design checks.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
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