A large open water tank has a horizontal cast iron drainpipe...

A large open water tank has a horizontal cast iron drainpipe of diameter $D=1$ in. and length $L=2 \mathrm{ft}$ attached at its base. If the depth of water is $h=3$ ft, find the llow rate $(\mathrm{gpm})$ if the pipe entrance is (a) reentrant, (b) square-edged, and (c) rounded $(r=0.2 \text { in. })$

A large open water tank has a horizontal cast iron drainpipe of diameter $D=1$ in. and length $L=2 \mathrm{ft}$ attached at its base. If the depth of water is $h=3$ ft, find the llow rate $(\mathrm{gpm})$ if the pipe entrance is
(a) reentrant,
(b) square-edged, and
(c) rounded $(r=0.2 \text { in. })$

Key Concepts

Torricelli's Law

This principle states that the velocity of a fluid exiting an orifice under the force of gravity is proportional to the square root of the vertical distance (height) between the free surface and the orifice. It provides the basis for estimating the velocity of the water leaving the tank and is fundamental in determining the flow rate in tank discharge problems.

Bernoulli’s Equation

Bernoulli’s Equation is an expression of the conservation of energy for flowing fluids. In tank discharge problems, it is used to relate pressures, velocities, and elevations along a streamline from the free surface of the tank to the exit of the drainpipe, helping in the derivation of outlet velocity under ideal conditions.

Discharge and Contraction Coefficients

These coefficients account for the losses and flow contractions that occur at the entrance of the pipe, especially when the entrance geometry deviates from ideal. The discharge coefficient (or contraction coefficient) adjusts the ideal flow estimate by incorporating factors such as the reentrant, square-edged, or rounded profile of the pipe entrance, which influence the actual flow rate.

Minor Losses in Pipe Flow

Minor losses refer to the additional energy losses that occur due to local disturbances in a flow system, such as those caused by fittings, contractions, or entrance geometries. In the context of the pipe entrance, differences in design (reentrant, square-edged, rounded) produce different minor losses, thereby affecting the overall discharge rate.

Flow Rate Calculation

The calculation of flow rate combines the ideal fluid velocity derived from principles like Torricelli’s Law with adjustments for real-world effects introduced by factors like contraction and minor losses. This involves using the effective cross-sectional area of the discharge, the actual exit velocity, and appropriate conversion factors to obtain the rate in practical units such as gallons per minute (gpm).

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