Part of the water supply for the South Rim of Grand Canyon...

Part of the water supply for the South Rim of Grand Canyon National Park is taken from the Colorado River $[54] .$ A flow rate of 600 gpm, taken from the river at elevation $3734 \mathrm{ft}$, is pumped to a storage tank atop the South Rim at $7022 \mathrm{ft}$ elevation. Part of the pipeline is above ground and part is in a hole directionally drilled at angles up to $70^{\circ}$ from the vertical; the total pipe length is approximately $13,200 \mathrm{ft}$. Under steady flow operating conditions, the frictional head loss is 290 ft of water in addition to the static lift. Estimate the diameter of the commercial steel pipe in the system. Compute the pumping power requirement if the pump efficiency is 61 percent.

Key Concepts

Darcy-Weisbach Equation

The Darcy-Weisbach equation is a fundamental relation in fluid mechanics that quantifies the frictional head loss along a pipe due to viscous effects. It expresses the head loss as a function of the flow velocity, pipe length, diameter, and a friction factor that accounts for the pipe's roughness and flow regime, providing a means to estimate the necessary pipe diameter and design pump requirements for a given system.

Continuity and Flow Rate

The continuity principle in fluid dynamics states that the mass flow rate is constant along a streamline, so the product of cross?sectional area and velocity remains constant for incompressible fluids. This concept is essential for estimating the internal flow velocity based on a measured volumetric flow rate, and hence can be used to relate flow characteristics to the pipe's diameter.

Frictional Head Loss

Frictional head loss refers to the energy loss due to friction between the fluid and the walls of the pipe as well as internal turbulence. It is a crucial factor in designing piping systems because it affects the pressure requirements of pumps. This loss can be estimated using empirical relationships like the Darcy-Weisbach equation, which relates head loss to the pipe diameter, length, flow velocity, and friction factor.

Pump Efficiency and Hydraulic Power

Pump efficiency is a measure of how effectively a pump converts input power into useful hydraulic power to move fluids against gravitational and frictional forces. Determining the hydraulic power requirement involves calculating the work needed to overcome both the static lift (elevation gain) and the frictional head loss. After estimating the necessary hydraulic power, incorporating the pump efficiency allows one to determine the actual power input required, which is critical for system design and energy consumption calculations.

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