Gasoline flows in a long. underground pipeline at a constant...

Gasoline flows in a long. underground pipeline at a constant temperature of $15^{\circ} \mathrm{C}$. Two pumping stations at the same elevation are located $13 \mathrm{km}$ apart. The pressure drop between the stations is $1.4 \mathrm{MPa}$. The pipeline is made from 0.6-m-diameter pipe. Although the pipe is made from commercial steel, age and corrosion have raised the pipe roughness to approximately that for galvanized iron. Compute the volume flow rate.

Key Concepts

Fluid Properties

Key fluid properties such as viscosity and density play an essential role in determining the Reynolds number and, consequently, the flow regime and friction factor. In practical applications, accurate values for these properties are necessary to compute the pressure drop and the corresponding volume flow rate using the Darcy-Weisbach equation.

Pipe Roughness

Pipe roughness is a measure of the irregularities and texture of the interior surface of the pipe. It significantly influences the friction factor in turbulent flow regimes. As the roughness increases, the friction factor generally increases, leading to a greater pressure drop for a given flow rate. This concept is crucial in the design and analysis of pipelines subject to real-world factors such as corrosion or material wear.

Reynolds Number

Reynolds number is a dimensionless parameter that indicates whether the flow in the pipe is laminar or turbulent. It is defined based on the fluid density, velocity, characteristic length (such as the pipe diameter), and viscosity. For high Reynolds numbers, the flow becomes turbulent, and the friction factor depends on both the Reynolds number and the pipe roughness.

Friction Factor

The friction factor is a dimensionless number used in the Darcy-Weisbach equation to quantify the resistance to flow caused by the pipe's surface roughness and flow conditions. In turbulent flow, the friction factor is typically determined using the Moody diagram or empirical correlations, taking into account the relative roughness of the pipe.

Darcy-Weisbach Equation

This equation is a fundamental relationship in fluid mechanics that links the pressure drop (or head loss) due to friction in a pipe to the flow velocity, pipe length, diameter, fluid density, and the friction factor. It is used to analyze and compute the losses that occur as a fluid flows through a pipe, especially in turbulent regimes.

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