Water is pumped up from the Colorado River to supply Grand...

Water is pumped up from the Colorado River to supply Grand Canyon Village, located on the rim of the canyon. The river is at an elevation of 564 $\mathrm{m}$ , and the village is at an elevation of 2096 $\mathrm{m}$ . Imagine that the water is pumped through a single long pipe 15.0 $\mathrm{cm}$ in diameter, driven by a single pump at the bottom end. (a) What is the minimum pressure at which the water must be pumped if it is to arrive at the village? (b) If 4500 $\mathrm{m}^{3}$ of water are pumped per day, what is the speed of the water in the pipe? Note: Assume the free-fall acceleration and the density of air are constant over this range of elevations. The pressures you calculate are too high for an ordinary pipe. The water is actually lifted in stages by several pumps through shorter pipes.

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Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It is determined by the product of the fluid's density, the gravitational acceleration, and the vertical height difference between two points. This concept is critical when calculating the minimum pressure required to pump a fluid to a higher elevation, as the pump must overcome the weight of the fluid column.

Continuity Equation

The continuity equation in fluid dynamics is a statement of the conservation of mass for a flowing fluid. For an incompressible fluid, it implies that the product of the cross-sectional area of the pipe and the fluid's velocity remains constant along the flow. This principle allows us to relate the volumetric flow rate to the speed of the water in the pipe, which is key when determining required pump specifications.

Flow Rate

Flow rate, defined as the volume of fluid passing through a cross-sectional area per unit time, is a fundamental concept in both fluid dynamics and engineering applications. It is essential for understanding how a specific pumping capacity translates to a given velocity in a pipe with a known cross-sectional area, thereby influencing the design and operation of fluid transport systems.

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