Question

For a Venturi meter, the flow rate is proportional to:

• A. \( \sqrt{(\Delta P)} \)
• B. \( \Delta P \)
• C. \( \Delta P^2 \)
• D. \( \Delta P^{1/3} \)

Hint:

All obstruction-type flow meters (Venturi meters, Orifice meters, and Flow nozzles) follow the relationship \( Q \propto \sqrt{\Delta P} \).
This is a key characteristic of head flow meters.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the relationship between the volumetric flow rate (\( Q \)) and the measured pressure drop (\( \Delta P \)) across a Venturi meter.
This is a standard fluid mechanics question related to flow measurement devices.

Step 2: Key Formula or Approach:
A Venturi meter is a variable head flow meter that restricts flow to convert pressure energy into kinetic energy.
We apply Bernoulli's equation and the continuity equation between the inlet (point 1) and the throat (point 2):
\[ \frac{P_1}{\rho \cdot g} + \frac{v_1^2}{2 \cdot g} = \frac{P_2}{\rho \cdot g} + \frac{v_2^2}{2 \cdot g} \]
\[ A_1 \cdot v_1 = A_2 \cdot v_2 \]

Step 3: Detailed Explanation:

• Express the inlet velocity in terms of the throat velocity using the continuity equation:
\[ v_1 = v_2 \cdot \left(\frac{A_2}{A_1}\right) = v_2 \cdot \beta^2 \]
where \( \beta = D_2 / D_1 \).

• Substitute \( v_1 \) into Bernoulli's equation:
\[ \frac{P_1 - P_2}{\rho} = \frac{v_2^2 - v_1^2}{2} = \frac{v_2^2 \cdot (1 - \beta^4)}{2} \]

• Solve for the throat velocity \( v_2 \):
\[ v_2 = \sqrt{\frac{2 \cdot \Delta P}{\rho \cdot (1 - \beta^4)}} \]

• The volumetric flow rate \( Q \) is:
\[ Q = C_d \cdot A_2 \cdot v_2 = C_d \cdot A_2 \cdot \sqrt{\frac{2 \cdot \Delta P}{\rho \cdot (1 - \beta^4)}} \]
where \( C_d \) is the coefficient of discharge, which accounts for frictional losses.

• From this equation, we can see that:
\[ Q \propto \sqrt{\Delta P} \]


Step 4: Final Answer:
The volumetric flow rate in a Venturi meter is proportional to the square root of the pressure drop, \( \sqrt{\Delta P} \).