Practice Problems 4

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AGL vs. MSL

For regulatory purposes, the altitude relative to Mean Sea Level (MSL) is important. The FAA mandates maximum altitudes that cannot be exceeded for high-power rocket flights in terms of MSL. For example, the usual waiver at ROC launches in Lucerne Valley is for 10,000 feet MSL. The waiver for our class launch is 17,500 feet MSL. For modeling and flight performance purposes, the important measurement is the altitude relative to the altitude of the launch site, called Above Ground Level (AGL). This assignment is looking principally at errors in AGL measurements. AGL is always calculated by taking the calculated (not measured) altitude MSL at the launch pad, \(ℎ_{0-MSLcalc}\), and subtracting it from the calculated altitude MSL during flight, e.g.,

\[ ℎ_{AGL}=ℎ_{MSLcalc}−ℎ_{0-MSLcalc} \]

The error in AGL is calculated as:

\[ 𝜖_{AGL}=ℎ_{AGL}−ℎ_{true} \]

Barometric Pressure Variations

The barometric pressure is calculated by measuring the local absolute pressure, and then using the standard atmospheric model to calculate the pressure one would expect at mean sea level. When you read the barometric pressure in a weather report, you always have to correct to your current altitude. Normal atmospheric pressure at mean sea level is 101,325 Pa, 1013.25 millibar (mb), or 1.01325 bar. High pressure regions rarely go above 1050 mb or below 982 mb.

Create a parametric plot that shows the error in MSL altitude from 0 m to 5000 m for \(𝑃_0\)=1050 mb and \(𝑃_0\)=982 mb as parameters Create an additional parametric plot that shows the error in AGL from 0 m to 5000 m for \(𝑃_0\)=1050 mb and \(𝑃_0\)=982 mb as parameters.

Temperature Variations

The Standard Atmospheric model assumes a temperature of 288.15 K (15°C) at mean sea level and a decrease of 6.5 K for every 1000 m gain in altitude. Obtaining data on variations in the lapse rate is challenging, but variations in ground temperature are fairly easy to measure. Lucerne Valley has a seasonal low of about –3°C and a seasonal high of about 38°C. Corrected for MSL, that is a \(𝑇_0\) variation between 2.7°C and 43.7°C.

Create a parametric plot that shows the error in AGL from 0 m to 5000 m for \(𝑇_0=2.7°C\) and \(𝑇_0=43.7°C\) mb as parameters.

NAR requires all official altitude record attempts to have their pressure altimeter data corrected by multiplying the measured AGL by the ratio of the launch site temperature in K to 288.15 K. How well does the correction work for the data in this problem? Where did they get the 288.15 K in the ratio?

Relative Humidity Variations

Water has a molecular weight of 18.01528 g/mol or 0.01801528 kg/mol. Dry air has an average molecular weight of 28.9644 g/mol or 0.0289644 kg/mol. The density of air changes with the relative humidity. The relative humidity is defined as the partial pressure of water in the air divided by the vapor pressure of water at the air temperature.

The molecular weight of an air-water mixture is the mole-weighted average of the molecular weight of air and the molecular weight of water. In practice, it is easiest to calculate \[ 𝑀_{𝑚𝑖𝑥}=\frac{𝑀_{𝑎𝑖𝑟}𝑃_{𝑎𝑖𝑟}+𝑀_{𝑤𝑎𝑡𝑒𝑟}𝑃_{𝑤𝑎𝑡𝑒𝑟}}{𝑃} \]

where \(𝑀_{𝑚𝑖𝑥}\) is the molecular weight of the air-water vapor mixture, \(𝑀_{𝑎𝑖𝑟}\) is the molecular weight of dry air, \(𝑀_{𝑤𝑎𝑡𝑒𝑟}\) is the molecular weight of water vapor, \(𝑃_{𝑎𝑖𝑟}\) is the partial pressure of dry air, \(𝑃_{𝑤𝑎𝑡𝑒𝑟}\) is the partial pressure of water vapor, and 𝑃 is the overall pressure. Without measurements, it is difficult to know how the molecular weight of the air-water mixture varies with altitude. The easiest assumption is to measure the relative humidity (and of necessity, the temperature) at the launch site, calculate the molecular weight and assume it doesn’t change with altitude. Other assumptions require numerical integration of the pressure-altitude equations.

Create a parametric plot that shows the error in AGL from 0 m to 5000 m, for \(𝐻_𝑟's\) of 50%, and 100% at 316.85 K. Don’t correct \(𝑇_0\), just 𝑀.

Put It All Together

The linked data set contains the pressure-versus-time data for the first flight of the DX3 rocket on an F67-9W. Calculate the AGL altitude versus time both using the standard atmospheric model and applying the corrections for \(𝑇_0, 𝑃_0, ℎ_0, and 𝐻_𝑟\).

  • For this flight, how much of a difference did the corrections make?
  • If you were trying for a record altitude flight, which would you prefer: to fly on a cold day or a hot day, under a high pressure or under a low pressure, under a high humidity or under a low humidity?