Consider the pipe flow from the water tower of Prob. lem...

Consider the pipe flow from the water tower of Prob. lem 8.83 . To increase delivery, the pipe length is reduced from 600 ft to 450 ft (the flow is still fully turbulent and $f=0.035$ ). What is the flow rate?

Key Concepts

Darcy-Weisbach Equation

This fundamental equation in fluid mechanics relates the energy (or head) loss due to friction within a pipe to the flow velocity, pipe length, diameter, and friction factor. It forms the basis for calculating pressure drops and flow rates in piping systems by linking these variables through a power law relationship, which is especially useful in engineering applications involving turbulent flow.

Turbulent Flow

Turbulent flow refers to a flow regime characterized by chaotic, eddying motion, which significantly enhances mixing and increases friction losses compared to laminar flow. When a flow is fully turbulent, as assumed in this problem, the friction factor tends to be relatively constant, thereby simplifying the analysis and prediction of pressure drops and flow rates in the pipe.

Friction Factor

The friction factor is a dimensionless parameter that quantifies the resistance to flow in a reaction pipe due to friction. It depends on factors like the pipe’s roughness and the nature of the flow (laminar or turbulent). In fully turbulent flows, as assumed in this problem, the friction factor remains approximately constant, allowing engineers to use it directly in the Darcy-Weisbach equation without additional corrections.

Flow Rate Scaling in Pipe Systems

Flow rate in a pipe system can be influenced by changes in the pipe length, among other parameters. Since the frictional pressure drop is directly proportional to the pipe length (when all other factors remain constant), reducing the length results in lower head loss and higher flow velocity. This concept is crucial for understanding how modifications to a system, such as shortening a pipe, can enhance the delivery of fluids.

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