How To Calculate Pressure Altitude

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Have you ever wondered why you can’t open the windows on a plane? You sure do so on a car, a boat or other means of transportation. Aircrafts are different, though. They fly at very high altitudes and therefore have special requirements. 

If you were to visit the remains of the famous Titanic, you would have to dive down almost 4 km to the bottom of the North Atlantic. To achieve this, you would have to use a special type of submarine, since your body alone would not be able to withstand the enormous weight of 4 km of ocean water above you. 

These two situations are alike in that they both occur inside a fluid: in the second case, we have the ocean water, and in the first case we have air —yes, air is a gaseous fluid. In both scenarios, the fluid surrounding you exerts pressure on your body. The difference lies in the amount of pressure: while in the bottom of the ocean you are subjected to huge amounts of pressure due to the water around you, at a usual cruce altitude of 31000 ft (9500 m) there is almost no air to compress your body. These two extreme situations require you to be protected from the environment in pressurized chambers, a.k.a. a submarine or an airplane. Otherwise your body would implode or explode, respectively, since it is built to survive near the sea level. 

These amazing phenomena exist due to very interesting properties of fluids, like the pressure they exert on bodies immersed in them. In our two examples, pressure depends on the immersion depth or flying altitude. This property is used “backwards”, for example, to determine the altitude of a flying airplane, mountains, cities and buildings, through the air pressure measured at their location. The result is known as the pressure altitude, since it is determined using the air’s pressure. 

How to calculate pressure altitude

In order to calculate the pressure altitude h in meters of any location with respect to the sea level, follow these steps: 

  1. Measure the air pressure at the desired location with a reliable device
  2. Convert the result to units of millibars
  3. Plug the result, p, into the following equation: 

What is pressure altitude

Fluids are composed of atoms and molecules which move in all directions and collide with each other at very high speeds. Even when a fluid is at rest, its nanoscopic components possess high kinetic energies. The macroscopic effect of these collisions is what we call its pressure. For example, if you introduce your hand into a pool you will feel a soft force pressing your skin, which is the result of billions of water molecules hitting its surface. Since this force acts upon a specific area, we can define pressure as: 

Where F is the total force acting perpendicularly upon area A due to the fluid’s interactions. The units of pressure are therefore N/m2, also called pascal. Nevertheless, there are many other pressure units, like bar, millibar, atmosfere, psi (pounds per square inch), etc., which are used according to the field of application. For example, in meteorology, units of millimeters of mercury (mm Hg) or millibars, which are equivalent to 133,32 and 100 Pa, respectively, are most likely to be used. If you are having trouble with these units, you can always turn to a converter like this one by Unit Converters. 

A fluid’s pressure acts in all directions. If you introduce an object into a glass of water the force per unit area exerted on it by the liquid has to act in all possible directions with the same magnitude, otherwise there would be a net force acting on the body and it would accelerate.

Pressure also depends on different variables. Imagine you put water inside a tightly closed metal container and put it on the stove. As water inside starts to heat up, its molecules start to move faster and faster, until the liquid eventually evaporates. Water steam is composed of water molecules that have converted heat into a high amount of kinetic energy, and thus travel at very high speeds. These collide against each other and against the container walls, exerting a force on its entire inner area. This force per unit area is the steam pressure inside the container. 

If you increase the power of your stove, the steam will heat up further. This means, the water molecules will move even faster and collide with higher energy against each other and the container walls. These collisions will also happen with a higher frequency. The result is an increase in the fluid’s pressure. In general, the higher the temperature of a fluid, the higher its pressure

This also applies to the Earth’s troposphere, the lower part of the atmosphere, where we all live. It contains air, a fluid composed of a mixture of different gases which molecules possess high kinetic energies, partly due to the Sun’s radiation. The air pressure depends, consequently, on its temperature. In principle, it is higher in hotter regions of the Earth than in cooler regions. Nevertheless, this is not necessarily true in all cases. Let’s see why:

You might think since the Earth is not a closed system as in our previous example, air could easily flow away from it, especially when its temperature increases —for example, on the side the Sun is shining on. Let’s see why this does not happen.

The gases that compose the troposphere do not “flee” from the Earth due to the force of gravity, which attracts them towards its surface. This way, gravity acts as a kind of “container” for our troposphere. The difference with our previous example is simply that gravity does not form a solid wall which air molecules collide with, but a force that counteracts their outwards motion. In this case, it also generates a gradual decrease in the air’s density, meaning the number of molecules available to interact per unit volume decreases as their distance to the Earth’s surface increases. One consequence of this is that, at higher altitudes, collisions become less frequent and less energetic. This has two main consequences: heat becomes harder to transfer, which makes the troposphere at higher altitudes generally colder than at lower altitudes; and the air pressure becomes lower, since there are simply less molecules colliding with the area upon which it is measured. 

On the other hand, the amount of air molecules per unit volume increases as we approach the Earth’s surface. This makes collisions between them and with other objects more frequent and more energetic. Consequently, air can transfer heat easier, thus making the troposphere generally warmer at lower altitudes. Additionally, its pressure is greater than at higher altitudes, due to more particles with higher kinetic energies colliding with the area used to measure it. In general, the higher the altitude, the lower the air pressure in the troposphere

The variation of air pressure and altitude is therefore inversely proportional. This does not mean it is linear, though. The following chart shows how atmospheric pressure varies as a function of altitude, measured with respect to the sea level. 

As explained before, the air pressure in our atmosphere can vary a lot depending on where you measure it and on the weather conditions. Nevertheless, scientists use an approximation for this value called the standard atmosphere, which equals 101.325 Pa, and is usually written as 1 atm. This is approximately the average value of air pressure at sea level, and should be used only as a reference.

Since altitude and pressure correlate, you could —in principle— determine a body’s altitude based on the air pressure around it. To achieve this, we first need to have a mathematical relation between these two variables. The resulting height above sea level calculated through this comparison to the variation of air pressure with altitude is therefore called the pressure altitude

Pressure altitude formula

In general, pressure varies inside a fluid depending on the depth at which it is measured. As mentioned before, atmospheric pressure varies with altitude, as does the pressure experienced by a person diving deep into the ocean. For a fluid of uniform density, the pressure, p, varies with height as expressed by the following equation: 

Where is the fluid’s density, g the gravitational acceleration (9,8 m/s2), and h the height measured with respect to the point where a reference pressure, p0, is determined. If we measure altitude with respect to the sea level, then p0=1 atm. This equation is clearly independent of the fluid’s temperature, and it only applies when density is constant throughout the entire altitude h.

As we mentioned before, our atmosphere does not comply with these requirements: its density varies with altitude and its pressure is highly temperature-dependent. So, in order to describe the variation of pressure with altitude reliably, we need a more complex mathematical expression that takes all these variables into account. This is a more precise relation published by the National Weather Service of the United States of America, which is valid up to an altitude of 11 km above sea level: 

Where LT is the standard temperature lapse rate (-0,0065 K/m) —which indicates the variation of temperature with altitude in the atmosphere—, T0 is the temperature at sea level (measured in Kelvin), Mair is the molar mass of the air (0,029 kg/mol), and R the universal gas constant (8,31 J/mol K). Equations 3 and 4 have the same basic structure: the pressure at height h=0 is reduced as h increases. To see this more clearly, keep in mind LT is a negative number. 

If we solve equation 4 for the height, we obtain: 

If we use the pressure and temperature at sea level as references, then we can calculate the altitude of any point by measuring the atmospheric pressure around it. This is what pilots used to do when flying in older aircraft: current values of the atmospheric pressure and temperature of airports near to their location would be collected, and the pressure at the flying height would be precisely measured. These values were then plugged into equation 5 to calculate the altitude above said airports. Currently more advanced techniques are used to determine an airplane’s flying altitude. 

In order to easily apply equation 5 you can use this calculator built by Midé Engineering Solutions. 

Sources: 

Unit Converters. Pressure Converter. Consulted on June 19th at: https://www.unitconverters.net/pressure-converter.html

National Geographic Society. Altitude. Consulted on June 19th at: https://www.nationalgeographic.org/encyclopedia/altitude/

Mide Technology Corporation. Air Pressure at Altitude Calculator. Consulted on June 20th at: https://www.mide.com/air-pressure-at-altitude-calculator

How To Calculate Pressure Altitude

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