Question

A Pitot tube is mounted on an airplane wing to determine the speed of the plane relative to the air, which has a density of \(1.03 \mathrm{~kg} / \mathrm{m}^{3} .\) The tube contains alcohol and indicates a level difference of \(26.2 \mathrm{~cm} .\) What is the plane's speed relative to the air? The density of alcohol is \(810 \mathrm{~kg} / \mathrm{m}^{3}\).

Short answer

The speed of the plane relative to the air is approximately \(68.4\) m/s

Step-by-step solution

Understand the Pitot Tube Principle

The principle of the Pitot tube is based on Bernoulli's equation, which has a central role in fluid dynamics. When the airplane is at rest, the air pressure is the same inside and outside the tube. But when the plane is moving, the air pressure inside the tube increases. This pressure difference causes the alcohol in the tube to rise.

Use the Bernoulli equation

Adapting the Bernoulli equation, the speed of the airplane can be represented as: \(v = \sqrt{{2 \delta h g \rho_f} / {\rho_a}}\), where \(v\) is the plane's speed, \(\delta h\) is the level difference of alcohol in the tube (measured in meters), \(g\) is the acceleration due to gravity, \(\rho_f\) is the fluid (alcohol) density, and \(\rho_a\) is the air density.

Insert Known Values

Now, we plug in the known values into the equation: \(v = \sqrt{{2 * 0.262 * 9.81 * 810} / {1.03}}\), where the level difference of alcohol in the tube is \(0.262\) m, the acceleration due to gravity is \(9.81 m/s^2\), the density of alcohol is \(810 kg/m^3\), and the density of air is \(1.03 kg/m^3\).

Solve For v

Upon solving the equation, we find that the speed of the plane relative to the air is approximated to \(68.4 m/s\)

Key concepts

Bernoulli's Equation

Understanding Bernoulli's equation is crucial when calculating the speed of an airplane using a Pitot tube. This fundamental principle in fluid dynamics describes the behavior of a moving fluid and how pressure varies within it. Bernoulli's equation asserts that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

Mathematically, Bernoulli's equation can be expressed as follows: \

, where:\

• \(p\) is the fluid pressure,
• \(\rho\) is the fluid density,
• \(v\) is the fluid velocity,
• \(g\) is the acceleration due to gravity, and
• \(h\) is the height above a reference point.

For a Pitot tube, Bernoulli's equation helps us understand the relationship between the speed of the fluid (in this case, air) and the pressure measured within the tube. When applied correctly, it allows us to determine the aircraft's airspeed by inspecting the fluid's behavior inside the tube.

Fluid Dynamics

Fluid dynamics is a sub-discipline of fluid mechanics concerned with the flow of liquids and gases. It covers various situations, from water flowing through pipes to air moving around aircraft wings. A central aspect of fluid dynamics is understanding how the pressure, velocity, and density of a fluid relate to each other, and how they are affected by external forces like gravity or obstacles in their path.

Pitot tubes are practical applications of fluid dynamics. They operate based on the principle that the fluid flow's kinetic energy converts to potential energy when it encounters an object, causing a measurable pressure difference. By calculating this pressure difference with Bernoulli's equation, the speed of the flow can be determined, allowing pilots to know the speed of the airplane relative to the air.

Air Density

Air density, often denoted as \(\rho_a\), is a measure of the mass per unit volume of atmospheric air. It plays a critical role in aerodynamics and is influenced by factors such as temperature, altitude, and humidity. Aircraft performance, including lift, drag, and engine thrust, is heavily affected by air density.

In the context of a Pitot tube airplane speed calculation, air density is a vital component of Bernoulli's equation. The accuracy of the speed determined by the Pitot tube is contingent upon correct air density values. Since air density decreases with altitude, measurements from Pitot tubes become less reliable as the airplane climbs, unless appropriate adjustments are made based on standard atmospheric conditions.

Alcohol Density in Pitot Tube

In a Pitot tube, alcohol or another suitable manometric fluid is used due to its predictable density (\(\rho_f\)) and response to pressure changes. The density of the alcohol used in the tube is lower than that of mercury, another common manometric fluid, making it sensitive enough to measure the small pressure differences encountered in air.

When the airplane moves through the air, the change in air pressure inside the tube causes the alcohol level to displace proportionally. By measuring the difference in height (\(\delta h\)) between the levels of alcohol within the tube and applying the known density of alcohol, the speed of the airplane can be computed using Bernoulli's equation. It's essential for these calculations to use the precise alcohol density; otherwise, the final speed estimation won't be accurate.