Piece-wise and Composition of Functions to Determine Altitude Using Barometric Pressure Data from an Airplane Flight.

A Barometer, LabPro data collection device and a TI-83plus calculator were used on a USSTRATCOM "Glory Trip" from Offutt Air Base, Bellevue, NE to Minot Air Base, ND and Vandenberg Air Base, CA. The trip was designed to introduce business, press, and educational leaders to military technology, particularly involving the B-52 Bomber and the ICBM (Intercontinental Ballistic Missile). The Barometric Pressure in millibars was collected along with time data on the takeoff of a KC135 Air Refueling Tanker flown by a Nebraska Air Guard crew. The image at the left is taken from the boom operators window showing the refueling boom extended for refueling. Unfortunately, a hydraulic leak on the aircraft forced the cancellation of a schedule air refueling demonstration during the flight. A special thank you is in order to the KC135 crew for allowing the data collection and providing the altitude and position information. The crew is listed in the credits at the bottom of this page.

The data collected at takeoff from Vandenberg Air Base, CA (elevation 367 feet above sea level) is shown at the right. The data was collected using a Barometer, LabPro and TI-83plus calculator. Data was collected every two seconds beginning during taxi and running until the plane reached a cruising altitude of 33,000 feet. Pending the opportunity to repeat the data collection, some of what is written here is speculative pending the opportunity to collect additional data. The opportunity to collect such data is rare since data collection is not permitted on commerical airliners.

Data collection began while the aircraft was rolling toward the active runway. To interpret the data, one must recognize that barometric pressure is an inverse relationship to altitude. As the altitude increases, the barometric pressure will decrease. It should be noted that the graph above does not show the barometric pressure going to zero on the y-axis. The initial barometric pressure is 747 millibars through the first 80 seconds of the data. From 80 seconds to about 123 seconds, the pressure is increasing. We will look at three possible causes for this increase.

The first possible cause for this increase is a decrease in altitude. The complete data set shows an increase of about 32,700 feet in altitude. The increase is about 1/15th of the altitude of the flight or more than 2,000 feet which is not possible on the Vandenberg Air Base.

The second is a change in an active weather system. While the flight was delayed by fog, there were not active weather systems in the area and the pressure change occurred over about 40 seconds which would tend to discount the effect of weather on the barometer.

The third possibility is related to the pressurization of the aircraft. The effect of aircraft pressurization on the equipment used to collect this data has not studied at this time, but is the most likely cause for the increase in pressure. This is supported by looking at the landing data which shows a similar occurrence indicating the possibility of the plane being depressurized.

Building a Piece-wise Function for Pressure vs Time

Assuming that the maximal pressure occurs on the runway of Vandenberg air base and the plane is at cruising altitude prior to the end of the data, we will attempt to write a piece-wise function p(t) for pressure in terms of time. Note: The model will be simplified by assuming a linear relationship from point-to-point.

The graph at the left shows the values at significant points along the graph. From these points, the piece-wise function will be built. The slope is pressure in millibars divided by the time in seconds or mBar/sec. The y-axis shows the pressure in millibars. Therefore, the mathematical model for the section of data from 128 seconds to 272 seconds is: p(t) = -0.757 mBar/sec * t sec + 858 mBar.

The average value for the pressure from 272 seconds to 304 seconds is 652 millibars. During this time, the aircraft was in a level flight path. Following a brief period of level flight, the aircraft ascended to its cruising altitude. The mathematical model for this portion of the flight from 304 seconds to 384 seconds is: p(t) = -0.463 mBar/sec * t sec + 793 mBar.

Note the slope of the line (-0.757 mBar/sec) for the first segment of the flight indicates it is steeper than the slope of the line (-0.463 mBar/sec) for the third portion of the flight where the plane was ascending. This indicates the plane ascended faster during the first portion of the flight.

The final segment of the graph is from 384 seconds to 500 seconds when data collection terminated. The pressure from this section is an average of 615 millibars.

Merging the pressure vs time data, the piece-wise function is:

Altitude vs Pressure

Using the data from the barometer, instrumentation from the plane and know elevations, one can find a mathematical model for altitude in terms of pressure a(p). Then pressure on the runway at Vandenberg Air Base is measured at 761 mBar. The stated elevation is 367 feet above sea level. The cruising altitude according to the instrumentation on the plane was 33,000 feet above sea level where the pressure was measured at 615 mBar. Assuming a linear relationship between altitude and barometric pressure, one can find the equation of the line through the points (761 mBar, 367 feet) and (615 mBar, 33000 feet). The mathematical model is:

a(p) = -223.5 ft/mBar * p mBar + 170,461 ft

Using a Composition of Functions

The two mathematical models may be used as a composition of functions to convert a given time into an altitude. Given two functions f(x) and g(x), the composition is f(g(x)). In this case, we have a(p(t)) which will convert a given time into an altitude.

Example: Find the altitude at 250 seconds.

Finding p(250) will determine the pressure at 250 seconds. Use the equation for time equals 250 seconds.
p(250)= -0.757 mBar/sec * 250 sec + 858 mBar = 668.75 mBar

Using the pressure, find the altitude.
a(668.75) = -223.5 ft/mBar * 668.75 mBar + 170,461 ft = 20,995 ft above sea level.

Therefore, a(p(250 sec)) = 20,995 feet

Rate of Climb

Taking a more detailed look at the data, one can find the pressure change over a period of time. The linear model for the shaded region in the image at the right shows a pressure change of -0.976 mBar/sec. The model for altitude in terms of pressure found previously is: a(p) = -223.5 ft/mBar * p mBar + 170,461 ft This model indicates that each mBar increase in pressure is -223.5 ft change in altitude. Therefore, -223.5 ft/mBar * -0.976 mBar/sec is 218 ft/sec.

The models indicate the plane was climbing at a rate of 218 feet/second during this phase of the flight.

Special Thanks to the Nebraska Air Guard Flight Crew:

More information on the KC-135 Stratotanker is available from the USAF Fact Sheet

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