The students of Phi Gamma Delta are putting a kiddy pool on...

The students of Phi Gamma Delta are putting a kiddy pool on a porch attached to the second story of their house and plan to fill it with water from a garden hose. The kiddy pool has a diameter of $5 \mathrm{ft}$, and is $2.5 \mathrm{ft}$ deep. The porch is 18 ft above the faucet. The garden hose is very smooth on the inside, has a length of $50 \mathrm{ft}$, and a diameter of $5 / 8$ in. If the water pressure at the faucet is 60 psi, how long will it take to fill the pool? Neglect minor losses.

Key Concepts

Flow Rate and Volume Calculations

Determining the time to fill the pool involves calculating the volumetric flow rate based on the water velocity and the hose’s cross-sectional area, and then relating this flow rate to the pool’s volume. The pool, approximated as a cylindrical volume, requires a known amount of water to be filled. Comparing the flow rate with the pool’s volume yields the filling time, illustrating the direct interplay between fluid dynamics and volumetric analysis.

Unit Conversion and Consistency

Accurate analysis in fluid mechanics problems relies on consistent units across all calculations. Converting pressure (psi), lengths (feet, inches), and areas (cross-sectional area of the hose and volume of the pool) into a consistent unit system is essential. This step is crucial to avoid errors and correctly apply equations such as the Bernoulli equation and Darcy?Weisbach equation, ensuring meaningful and accurate results.

Continuity Equation

The continuity equation in fluid mechanics expresses the principle of conservation of mass by equating the product of the cross?sectional area and the fluid velocity at any two points along a streamline. This key concept is used to determine the volumetric flow rate out of the hose by relating the hose's outlet diameter and the velocity calculated from Bernoulli’s Principle, ensuring that the amount of water delivered per unit time is consistent.

Hydrostatic Pressure and Elevation Head

Hydrostatic pressure considers the change in pressure associated with gravity acting on a fluid column. The elevation head is the potential energy per unit weight of the fluid due to its height above a reference level. When fluids flow from a higher-pressure source to a higher elevation, as in this problem, the available pressure energy is reduced by the gravitational potential energy required to lift the water, which is a key step in determining the effective pressure to drive the flow.

Bernoulli's Principle

This principle explains the conservation of energy for flowing fluids. It states that the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline when no energy is added or lost. In problems like this, Bernoulli’s Principle is used to relate the pressure at the water faucet (including its pressure head) to the velocity of the water flow at the outlet, after accounting for changes in elevation and frictional losses.

Darcy?Weisbach Equation

This equation is used to calculate the frictional losses (head loss) in a pipe due to viscous effects. It relates the head loss in a pipe to the length, diameter, flow velocity, and friction factor of the pipe. In this context, even though minor losses (such as losses from fittings) are neglected, the Darcy?Weisbach equation helps estimate the energy lost due to friction over the long, smooth garden hose, which in turn affects the water velocity and ultimately the flow rate.

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