To find the volume flow rate in a venturi meter where water is flowing, we use the Bernoulli equation and the principle of continuity. The venturi meter measures the flow rate by noting the difference in fluid pressure before and after a constriction.
Step-by-step Solution:
Use the Bernoulli equation between points A and B in the venturi meter: \(P_A + \frac{1}{2}\rho v_A^2 = P_B + \frac{1}{2}\rho v_B^2\)
The change in height (potential energy change) is negligible in the horizontal setup of the venturi meter, so we focus on velocity and pressure differences.
Using the conservation of mass (continuity equation): \(A_A v_A = A_B v_B\), where \( A_A \) and \( A_B \) are the cross-sectional areas at points A and B.
The pressure difference \(\Delta P = P_A - P_B\) relates to changes in velocity by: \(\Delta P = \frac{1}{2}\rho (v_B^2 - v_A^2)\). Rearranging gives: \(v_B = \sqrt{v_A^2 + \frac{2\Delta P}{\rho}}\).
The volume flow rate \( Q \) is calculated as: \(Q = A_B v_B\).
Assuming the pressure difference measurement from the manometer is in meters of water (converting 5 cm to 0.05 m): \(\Delta P = \rho \cdot g \cdot 0.05\), where \(\rho = 1000 \, \text{kg/m}^3\) and \(g = 9.81 \, \text{m/s}^2\).
Compute the flow rate \( Q \) using the above equations by substituting known values. Let's assume: \( A_A = 2 \, \text{m}^2 \), \( A_B = 1 \, \text{m}^2 \) (hypothetical values for calculation).
Compute \( v_A \) using continuity: \(v_A = \frac{A_B}{A_A} v_B\).
Finally, solve for \( v_B \) and hence \( Q \) using specific calculations.
After solving and substituting all known values, the correct volume flow rate answer is exact or approximated based on given options. From options, the closest answer to an exact calculation or approximation based on setup is:
