Understanding Water Flow Rate and Funnel Effectiveness in Glass Funnel

Determining the Flow Rate of Water through the Funnel

Water at 20°C is flowing through a vertical glass funnel that consists of a 20 cm high cylindrical reservoir and a 1 cm diameter, 40 cm high pipe. The funnel is always maintained full by the addition of oil from a tank. Assuming the entrance effects are negligible, the flow rate of water through the funnel is determined by the principle of continuity.

The cross-sectional area of the cylindrical reservoir (A₁) and the pipe (A₂) can be calculated using the formula for the area of a circle: A = πr². Since the diameter of the reservoir is assumed to be equal to the diameter of the pipe (1 cm), the radius of the reservoir is 0.5 cm. Therefore, A₁ = π(0.5 cm)². The radius of the pipe is also 0.5 cm, so A₂ = π(0.5 cm)².

The principle of continuity states that A₁v₁ = A₂v₂, where v₁ and v₂ are the velocities of the water in the reservoir and the pipe, respectively. Given that the height of the water column in the reservoir is 20 cm, the velocity of water exiting the funnel (v₂) is 20 cm/s.

Using the calculated cross-sectional area of the pipe and the velocity of the water, the flow rate of water through the funnel is Q = A₂v₂ = π(0.5 cm)²(20 cm/s) = 4.16 cm³/s.

Calculating the Funnel Effectiveness

The "funnel effectiveness" is the ratio of the actual flow rate through the funnel to the maximum flow rate for the "frictionless" case. In this case, the funnel effectiveness is 1, indicating that the actual flow rate matches the maximum flow rate for a frictionless case.

The maximum flow rate for the "frictionless" case would be solely determined by the cross-sectional area of the pipe and the velocity of the water in the pipe, which is A₂v₂. Therefore, the funnel effectiveness can be calculated as the actual flow rate (Q) divided by the maximum flow rate for the "frictionless" case (A₂v₂).

The funnel effectiveness = Q / (A₂v₂) = 4.16 cm³/s / (π(0.5 cm)²(20 cm/s)) = 1.