An air-pipe friction experiment consists of a smooth brass...

An air-pipe friction experiment consists of a smooth brass tube with $63.5 \mathrm{mm}$ inside diameter, the distance between pressure taps is $1.52 \mathrm{m}$. The pressure drop is indicated by a manometer filled with Meriam red oil. The centerline velocity $U$ is measured with a pitot cylinder. At one flow condition, $U=23.1 \mathrm{m} / \mathrm{s}$ and the pressure drop is $12.3 \mathrm{mm}$ of oil. For this condition, evaluate the Reynolds number based on average flow velocity. Calculate the friction factor and compare with the value obtained from Eq. 8.37 (use $n=7$ in the power-law velocity profile).

Key Concepts

Pitot Tube Measurement

A Pitot tube is an instrument used to measure the fluid velocity in a flow field by converting the kinetic energy of the flow into a measurable pressure difference. In pipe flow experiments, the centerline velocity is often measured with a Pitot tube, and adjustments are made using a velocity profile to obtain the average velocity, which is required for accurate Reynolds number and friction factor calculations.

Darcy-Weisbach Equation

The Darcy-Weisbach equation is a fundamental relationship in fluid mechanics that relates the pressure drop or head loss due to friction along a pipe to the flow velocity, pipe length, diameter, and friction factor. This equation is widely used in the engineering design and analysis of fluid systems to estimate energy losses and ensure efficient transport of fluids.

Power-Law Velocity Profile

The power-law velocity profile is an empirical relationship used to describe how velocity varies across the cross-section of a pipe, particularly in turbulent flows. It relates the centerline velocity to the average velocity through an exponent that characterizes the distribution's shape. This profile is essential when correcting point-measured velocities for use in global flow calculations.

Reynolds Number

The Reynolds number is a dimensionless quantity that characterizes the relative influence of inertial forces to viscous forces in a flow. In pipe flow, it is calculated using the average velocity, the characteristic diameter of the pipe, and the fluid properties. This number helps determine whether the flow is laminar, transitional, or turbulent, which in turn influences the frictional behavior in the system.

Friction Factor

The friction factor is a dimensionless number used to quantify the resistance or friction losses in a pipe due to the flow of a fluid. It is central to the Darcy-Weisbach equation, which relates the pressure drop in a pipe to the friction factor, pipe length, diameter, and average velocity. The friction factor depends on the flow regime and pipe characteristics, and it is key in designing and analyzing actual piping systems.

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