Question

A rotameter

• A. Incurs constant and small permanent pressure drop
• B. Incurs constant but very large permanent pressure drop
• C. Is inaccurate for low flow rates
• D. Need not be mounted always vertically

Hint:

Flow Meter Comparison: - Orifice / Venturi Meter \(\implies\) Variable pressure drop, constant flow cross-sectional area. - Rotameter \(\implies\) Constant pressure drop, variable flow cross-sectional area.

Answer 1

The Correct Option is A

Concept: A rotameter is a variable-area flow meter used to measure the volumetric flow rate of liquids or gases in a closed pipe. It consists of a vertically oriented, tapered glass or plastic tube containing a specialized internal float. As fluid flows upward through the tapered tube, it lifts the float. The float stabilizes at a height where the upward buoyant and hydrodynamic drag forces balance the downward force of gravity. Unlike fixed-orifice meters (like orifice plates or venturi meters) where the flow area is constant and pressure drop varies with flow rate, a rotameter varies its flow area while maintaining a nearly constant pressure drop across the float.

Step 1: Analyzing the forces acting on the rotameter float.
When the float stabilizes at a constant height during a steady flow condition, it is in a state of dynamic force equilibrium. The three forces acting on the float are:

• Downward gravity force: \( F_g = V_f \cdot \rho_f \cdot g \)

• Upward buoyant force: \( F_b = V_f \cdot \rho \cdot g \)

• Upward dynamic drag force caused by pressure drop: \( F_d = \Delta P \cdot A_f \)
Where \(V_f\) is float volume, \(A_f\) is cross-sectional area of the float, \(\rho_f\) is float density, and \(\rho\) is fluid density.

Step 2: Deriving the pressure drop relationship.
Setting up the vertical force equilibrium balance equation: \[ F_d + F_b = F_g \implies \Delta P \cdot A_f + V_f \cdot \rho \cdot g = V_f \cdot \rho_f \cdot g \] Solving explicitly for the differential pressure drop (\(\Delta P\)) across the float: \[ \Delta P \cdot A_f = V_f \cdot g \cdot (\rho_f - \rho) \implies \Delta P = \frac{V_f \cdot g \cdot (\rho_f - \rho)}{A_f} \] Because the physical properties of the float (\(V_f, A_f, \rho_f\)) and the fluid density (\(\rho\)) are constants, the pressure drop \(\Delta P\) remains completely constant, regardless of the float's height or the fluid flow rate.

Step 3: Evaluating the scale of the pressure drop.
Because the float is light and the annular clearance area adjusts dynamically to minimize flow resistance, this permanent pressure drop is very small compared to a standard fixed orifice plate. Additionally, because it relies on a vertical gravitational balance, a standard rotameter must always be mounted vertically, making Option (D) incorrect. This confirms Option (A) as the correct choice.