A water tank (open to the atmosphere) contains water to a...

A water tank (open to the atmosphere) contains water to a depth of $5 \mathrm{m} .$ A 25 -mm-diameter hole is punched in the bottom. Modeling the hole as square-edged, estimate the flow rate (L/s) exiting the tank. If you were to stick a short section of pipe into the hole, by how much would the flow rate change? If instead you were to machine the inside of the hole to give it a rounded edge $(r=5 \mathrm{mm})$, by how much would the flow rate change?

Key Concepts

Torricelli’s Law

This principle states that the speed at which fluid exits an orifice under the force of gravity is given by the square root of twice the product of gravitational acceleration and the height of the fluid above the orifice. Derived from energy conservation and Bernoulli’s equation, Torricelli’s Law provides the ideal exit velocity for fluids discharging from tanks, serving as a fundamental concept in analyzing open-channel and free-surface flows.

Orifice Flow Equation and Discharge Coefficient

The orifice flow equation relates the volumetric flow rate through an opening to the effective area of the hole and the fluid’s exit velocity. However, due to real-world effects such as viscosity and flow separation, a correction factor called the discharge coefficient is incorporated. This coefficient adjusts the ideal flow rate to reflect the actual observed discharge by accounting for deviations from ideal behavior.

Flow Contraction (Vena Contracta)

Upon exiting an orifice, the jet of fluid typically narrows to a region of minimal cross-sectional area known as the vena contracta. This reduction, due to converging streamlines immediately after the orifice, results in an effective flow area that is smaller than the physical opening. Recognizing and quantifying this contraction is critical to accurately predicting the actual flow rate through the orifice.

Effect of Edge Geometry on Flow

The shape and contour of the orifice’s edge, such as whether it is square-edged or rounded, significantly influence the formation of the vena contracta and the overall flow characteristics. Rounded edges tend to promote a smoother flow transition which may modify the contraction coefficient and reduce turbulence, thereby altering the effective flow rate compared to sharp edges.

Influence of Pipe Insertions in Orifices

Introducing an extended, short section of pipe into an orifice modifies the entry and exit conditions of the flow, potentially influencing factors such as friction losses and the development of the vena contracta. While a short pipe may not drastically change the overall flow rate, its presence can alter flow patterns and edge effects, subtly affecting the resulting discharge.

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