Independent Study
Atmospheric Effects on Aircraft Gas Turbine Life
Humidity Section
Revised May 23, 2011
Kevin Roberg
1. Introduction to Atmospheric Model
It is said in New England “Don’t like the weather? Wait a minute, it will change.” This is in fact true in
the majority of locations on earth. It is also true that change can be found in any direction, including
up.
This study examines the effect of atmospheric conditions on the life of aircraft gas turbines. It is
impractical, if not impossible, to model the instantaneous structure of the atmosphere during every
flight in the life of an engine. Fortunately, it is possible to describe an average atmosphere which,
over the course of many flights and a long period of time, closely resembles the environment
experienced in operation.
The relationships developed in this study describe average conditions. They will rarely, if ever,
correctly describe instantaneous conditions exactly. Where there structure of the atmosphere can
consistently differ from the average, such as a ground level inversion present each morning or
evening, these effects are described.
2. Basic Atmosphere
2.1. Atmospheric Composition
The atmosphere is composed of a mixture of gasses. For the purposes of analyzing the
atmosphere using the ideal gas law, those gases must be known in order to assign an accurate
gas constant. Dry air, that is the atmosphere excluding water vapor is:
Table 1: Composition of the Atmosphere (Stull, p. 8)
Symbol Name
N2
Nitrogen
Fractional
Molecular
Weight
Molecular Volume
Weight
Fraction
28.01
78.08 21.870208
32
20.95
6.704
O2
Oxygen
Ar
Ne
He
Argon
Neon
Helium
H2
Hydrogen
Xe
Xenon
Carbon
dioxide
131.3 0.000009
44.01
0.035
Methane
16.04
0.00017
CO2
CH4
39.95
20.18
4
2.02
0.93 0.371535
0.0018 0.0003632
0.0005
0.00002
0.00005
1.01E-06
1.182E-05
0.0154035
2.727E-05
N2O
Nitrous
Oxide
44.01
0.00003
CO
Carbon
Monoxide
28.01
0.0035
SO2
Sulfur
Dioxide
64.06 0.000014
O3
Ozone
NO2
Nitrogen
Dioxide
48 0.000012
Totals
46.01 0.000005
100.00
1.32E-05
0.0009804
8.968E-06
5.76E-06
2.301E-06
28.96
Dividing the Universal Gas Constant by the Molecular weight of dry air:
8.314472 𝑘𝑃𝑎 ∙ 𝐾 −1 ∙ 𝑚3 ∙ 𝑘𝑚𝑜𝑙
𝑘𝑔
28.96𝑘𝑚𝑜𝑙
yields the gas constant for dry air:
ℛ𝑑 = 0.287053 𝑘𝑃𝑎 ∙ 𝐾 −1 ∙ 𝑚3 ∙ 𝑘𝑔−1 (Stull, p. 8)
2.2. Atmospheric Structure
The earth’s atmosphere is divided into distinct layers delineated by temperature extremes.
Beginning at the ground where temperatures are warmest, with energy being derived from
absorption of visible light. Temperature decreases with distance from the ground until the
stratosphere is reached. Temperature increases through the statopause until the level where
most ultraviolet light is absorbed, the stratopause. After passing the stratopause temperature
again declines through the mesosphere, until entering the troposphere where most other
radiation is absorbed. (Stull, p. 13)
The troposphere is the portion of the atmosphere nearest the ground in which nearly all clouds
and weather occur (Stull, p. 13). The temperature of the troposphere decreases linearly with
altitude. The top portion of the troposphere is the tropopause. Within the tropopause
temperature is constant. The transition from troposphere nominally occurs at a height of 11
km. The tropopause continues to a height of 20 km. Since aircraft activity using gas turbines is
generally confined to altitudes well below 20 km only the tropopause and troposphere will be
considered in this study.
2.3. Standard Model of the Atmosphere
A standard model of the atmosphere exists for the purpose of modeling atmospheric behavior.
The 1976 U.S. Standard Atmosphere is widely used for modeling the relationships between
temperature, pressure, and height.
2.3.1.Geopotential Height
In modeling the atmosphere meteorologists apply adjustments to account for the variation
in acceleration due to gravity (Stull, p. 12) and centrifugal force (Dutton, p. 65) due to
differences in distance from the center of the earth. The adjustment for geopotential
height is given by:
𝑍=
980.6160[1−2.64×10−3 𝑐𝑜𝑠2𝜙+5.9×10−6 𝑐𝑜𝑠2 2𝜙]𝑐𝑚/𝑠2 ∙𝑎
980𝑐𝑚/𝑠2
𝑧
) (Dutton,
𝑎+𝑧
(
p. 65)
Where 𝑎 is equal to the average radius of the earth, 6356.766 km, 𝜙 represents latitude,
and 𝑧 is altitude above sea level. Using the above equation shows the difference between
0 and 90 degrees latitude is 0.53%. Similarly, the difference between true height and
geopotential height at 15 km is 0.24%.
2.3.2.Temperature
The 1976 Standard Atmosphere provides equations defining temperature curves for
standard atmospheric temperature at altitudes up to 51 km. The vast majority of aircraft
activity takes place below 15 km, so two of these equations are of interest. First, for
altitudes less than 11 km, i.e. within the troposphere:
𝐾
𝑇 = 288.15𝐾 − (6.5 𝑘𝑚
)𝐻 (Stull, p. 13)
Note that 288.15K is equal to 15°C, which is standard sea level temperature. (Stull, p. 7).
𝐾
The value 6.5 𝑘𝑚
is the temperature lapse rate, which is fairly constant regardless of
location and season (Dutton, p. 83). For altitudes between 11 km and 20 km in the
standard atmosphere, and area of atmosphere known as the tropopause, temperature is
constant:
𝑇 = 216.65𝐾 (Stull, p. 13)
The tropopause begins when the temperature reaches 216.65K, so it is evident considering
the constant lapse rate that the tropopause will begin lower than 11 km when sea level
temperatures are below 15°C, and higher than 11 km when sea level temperatures are
above 15°C (Dutton, p. 83). Thus the equations for standard atmosphere can be easily
adapted to any atmosphere by adjusting the ground level temperature:
𝐾
𝑇 = 𝑇𝑠𝑙 − (6.5𝑘𝑚) ∙ 𝐻
The maximum altitude for which the above is valid is:
216.65𝐾
𝐻=
𝐾
𝑇𝑠𝑙 − (6.5𝑘𝑚
)
Beyond this altitude the remainder of the atmosphere of interest to this paper will be at
216.65K.
These equations are applicable for most day and night conditions but at night, in cases
where the surface is cooler than that atmosphere, such as winter or marine conditions
temperature may increase with altitude briefly before resuming the normal stratospheric
pattern. This condition is called an inversion and should be accounted for if examining
winter conditions or low level marine areas. (Dutton, p. 67)
2.3.3.Pressure
The pressure of the Standard Atmosphere is described by equations dependent on the
temperatures calculated above. The equation for pressure in the troposphere is:
288.15𝐾 −5.255877
)
𝑇
P=(101.325𝑘𝑃𝑎) ∙ (
(Stull, p. 13)
And the pressure in the tropopause is given by:
P=(22.632𝑘𝑃𝑎) ∙ 𝑒 −0.1577∙(𝐻−11𝑘𝑚)
Again, it is desirable to generalize these equations for an atmosphere with arbitrary sea
level temperature. This has already been accomplished for the stratospheric pressure
equation by generalizing its temperature equation. The pressure equation for the
tropopause must be modified by replacing the constant 11 km with the maximum
troposphereic altitude calculated above.
−0.1577∙(𝐻−
P=(22.632𝑘𝑃𝑎) ∙ 𝑒
216.65𝐾
𝐾 )
𝑇𝑠𝑙 −(6.5
)
𝑘𝑚
2.3.4.Pressure Altitude
Ambient pressure is used to determine altitude in aircraft. Different methods are used
depending on intended altitude. In all jurisdictions there exists a pressure altitude above
which standard atmospheric pressure is used as a baseline for determining altitude. In the
United States that altitude is 18,000 ft above mean sea level (MSL) Below 18,000 ft MSL
altitude is adjusted to actual pressure at a station within 100 nautical miles of aircraft
position. (Federal Aviation Administration, pp. 7-2-1)
Since this paper is primarily concerned with the local atmosphere an aircraft experiences,
altitude will be normalized to local conditions.
The formula used for calibrating altimeters is:
ℎ𝑝 =
518.4°𝑅
𝑝 0.1903
−( )
[1
]
0.00357°𝑅/𝑓𝑡
𝑝0
(Leishman, p. 213)
This is simply the standard atmosphere equation from above re-arranged and using English
units with the exception that true height, not geopotential height, is used. Given this, and
the minimal differences between geopotential and true height within altitudes of interest,
true height will be used in this paper.
It should be noted that the above equation does not compensate for temperature. If the
temperature is above standard for the indicated altitude an aircraft is higher than the
altimeter indicates. Alternately an aircraft is lower than indicated if the outside
temperature is lower than standard. (Federal Aviation Administration, pp. 7-2-3) This
effect is illustrated in Figure 1 below.
Atmospheric Pressure by Altitude
140
Pressure (kPa)
120
100
80
0°C
60
15°C
40
30°C
20
0
0
5
10
15
20
Altitude (km)
Figure 1: Atmospheric Pressure by Altitude
The discussion thus far has not considered the effect of density. Though in any column of
air pressure will decrease with temperature as altitude increases, a low temperature
column of air will tend to have a higher than standard sea level pressure as a result of
higher air density. The converse is true of warm air. As a result Figure 1 shows impossibly
small atmospheric pressures at sea level. To achieve a more accurate result actual, rather
than calculated, sea level pressure must be used as a starting point. This is done by
replacing 101.325 kPa in the pressure equation with an appropriate value to yield the
actual sea level atmospheric pressure. For example, to achieve a more realistic sea level
pressure of 101.327 kPa at 0°C the equation becomes:
288.15𝐾 −5.255877
)
𝑇
P=(134.2𝑘𝑃𝑎) ∙ (
The resulting curve is shown against the standard curve in Figure 2.
2.5
100
2
Pressure (kPa)
120
80
1.5
60
1
40
0.5
20
0
0
-1
Divergence (kPa)
Atmospheric Pressure by Altitude Adjusted to
Sea Level Actual
0°C
15°C
Divergence
-0.5
1
3
5
7
9
11
13
15
Altitude (km)
Figure 2 Atmospheric Pressure Adjusted to Sea Level Actual
This adjustment, equivalent to setting an altimeter to an airfield barometer measurement
produces a slow deviation from the correct value as altitude increases which is in line with
cold temperature error data supplied to pilots as shown in Table 2.
Table 2 Cold Temperature Error Table (Federal Aviation Administration, pp. 7-2-4)
2.3.5.Density Altitude
Though density altitude will not be used extensively in this paper it is an important value in
aviation and is helpful to clarify the discussion above. Warm air is generally less dense than
cold air, consequently warm air usually has a higher pressure than cold air. Confusion can
arise since this is the opposite of the altitude relationship. It must be recalled that the
temperature altitude relationships described above only hold within a column of air with
constant sea level pressure. Altitude calculations must be continually corrected to sea level
pressure.
Density altitude refers to an alternate method of computing altitude using air density
rather than pressure. This method is generally used on the ground for the purpose of
calculating required take off distance. As temperature rises, density altitude increases.
Density is important to flight dynamics, but is less important to the concerns of this paper.
3. Moisture in the Atmosphere
Water vapor can make up from 0 to 4% of air by volume. The amount of water in the air affects the
atmosphere in diverse ways. As a result there are diverse measures of the quantity used to examine
various effects. For the purposes of this paper certain measures are of more utility than others. In
this section the measures will be examined and their respective advantages discussed.
3.1. Measures of Water Content
3.1.1.Relative Humidity
Perhaps the most commonly heard measure of water content in the atmosphere is
Relative Humidity. Relative humidity is a comparison of the amount of water in the air to
the amount that can remain in vapor form under present conditions. That is:
𝑅𝐻
𝑒
𝜌
𝑟
= = ≈
100% 𝑒𝑠 𝜌𝑠 𝑟𝑠
Where e represents partial pressure, 𝜌 represents absolute humidity, and r represents
mixing ratio. Each of these measures will be discussed in more detail. The subscript s
represents saturation. This is the maximum amount of water that can remain in vapor
form the present temperature. So as temperature varies, the same absolute amount of
water vapor will result in different relative humidities. Relative humidity provides a
measure of how easily water, or perspiration will evaporate so it is a useful measure of
comfort to include in weather reports. However, if the actual amount of water in the
atmosphere is of primary concern, relative humidity is a cumbersome measurement
because temperature must also be specified to determine the absolute quantity of water.
3.1.2.Dew Point
If relative humidity measures the amount of water vapor relative to the maximum amount
that can exist at the current temperature, then there must be a temperature at which that
same amount of water can no longer be held as vapor. This temperature is the dew point.
The dew point is 100% relative humidity for a given quantity of air. Dew point is easily
measured by chilling a mirror until dew forms, and provides a value that can be converted
into the amount of water in the atmosphere without having to know temperature. (Stull,
p. 102) If current temperature is known, dew point provides a measure of relative
humidity either own its own or by calculating saturation for that temperature.
While dew point is a useful measurement technique, it can be converted into values that
are more useful for computation.
3.1.3.Partial Pressure
Partial pressure is the portion of atmospheric pressure contributed by water vapor. It is
independent of the other gasses in the air. The maximum partial pressure, saturation
pressure, is determined by:
𝑒 = 0.611𝑘𝑃𝑎 ∙ 𝑒𝑥𝑝 [
0.611𝑘𝑃𝑎∙(𝑇−273.16𝐾)
] (Stull,
𝑇−35.86𝐾
p. 98)
The dew point temperature can be entered to determine the partial pressure of water
vapor in the atmosphere. If current temperature is entered the formula yields the
saturation partial pressure. These values can be used to determine relative humidity.
Partial pressure is a convenient measure because it includes both water vapor quantity
information and temperature information. It can also be used to calculate several other
values.
3.1.4.Absolute Humidity
Absolute humidity is a measure of the quantity of water vapor in a given volume. It can be
calculated from partial pressure as follows:
𝜌𝑣 =
𝑒
ℜ𝑣 ∙𝑇
𝑒
𝑃
= 𝜀 ∙ 𝜌𝑑 (Stull, p. 99)
Where ℜ𝑣 is the water vapor gas constant, 0.4615 𝑘𝑃𝑎 ∙ 𝐾 −1 ∙ 𝑚3 ∙ 𝑘𝑔−1 (Stull, p. 446)
3.1.5.Water Vapor Mixing Ratio
Water vapor mixing ratio is the ratio of the mass of water vapor to the mass of dry air. This
makes it useful for performing calculations within the ideal gas equation. The water vapor
mixing ratio is given by:
𝜀∙𝑒
𝑟 = 𝑃−𝑒 (Stull, p. 99)
Where:
𝜀=
ℜ𝑑
ℜ𝑣
= 0.622 (Stull, p. 99)
3.1.6.Virtual Temperature
Virtual temperature is a modification to the ideal gas law used by meteorologists to
account for reduced density due to water vapor in air. (Dutton, p. 258) Virtual
temperature is defined by:
𝑇𝑣 = (1 + 0.61 ∙ 𝑟)𝑇 (Stull, p. 8)
The total mass of water in a column of air can be represented in the same was as the mass
of a column of air was treated in developing the Standard Model of the Atmosphere
(Dutton, p. 258). As such, water vapor pressure i.e. partial pressure, will vary according to
the same rules as air pressure, and it is appropriate to substitute 𝑇𝑣 into the generalized
tropospheric temperature equations developed above.
𝐾
𝑇𝑣 = 𝑇𝑣_𝑠𝑙 − (6.5𝑘𝑚) ∙ 𝐻
By using this temperature a slightly different pressure altitude relationship is obtained.
Humidity is not taken into account in altimeter calibration, this is a minor source of error.
Care must be taken to remember that virtual temperature is not actual temperature. It is a
construct to be used in conjunction with the ideal gas law to adjust for water in the gas
mixture.
3.2. Humidity/Altitude Relationship
Published data shows that the humidity altitude relationship depends on weather conditions
and does not follow a general equation. Such data is shown in Figure 3 (Chiang & Subrata
Kumar Das, 2009).
Figure 3 Temperature and Humidity over Chung-Li Taiwan December 2003 and May 2004
However, is data is averaged over a longer period of time relative humidity begins to appear
more constant from the surface up to the tropopause. Figure 4 shows the average of data
collected over a 12 year period (Tomasi et al, 2004) .
Figure 4 Averaged Relative Humidity Readings Taken at Terra Nova Bay Antarctica
For the purpose of an average analysis, if we assume relative humidity remains constant as
altitude increases then:
𝑒 = 𝑅𝐻 ∙ 𝑒𝑠
The resulting mixing ratio profile appears as shown in Figure 5.
Mixing Ratio Profile Assuming
Constant RH
Altitude (km)
16
14
12
10
8
6
4
2
0
0.001
0.01
0.1
10% RH
50% RH
100% RH
1
10
100
Mixing Ratio (g/kg)
Figure 5 Calculated Mixing Ratio Profile
Comparison with the averaged measured profile shown in Figure 4 demonstrates reasonable
correspondence in overall shape and magnitude allowing for some smoothing across the tropopause.
Figure 6: Annual Mean Mixing Ratio in Middle Latitudes (Dutton, p. 87)
If the humidity profile for the atmosphere at a specific time is desired then actual
atmospheric soundings are required. If averaged are to be used assuming a constant
relative humidity up to the tropopause appears to be a reasonable assumption.
In a still atmosphere with no precipitation the amount of water, measured either by
absolute humidity or mixing ratio, will remain stable throughout daily temperature
variations. If local temperature drops below the dew point, dew will form. In cases when
inversions occur due to a relatively cool surface, low lying fog may form though
moisture remains in vapor form at higher elevations. As discussed above, when
analyzing winter conditions or low lying marine areas it must be recalled that such a
situation can occur.
3.3. Conclusion
Thus far a model has been developed that can be adjusted for average local temperature,
pressure and humidity. At this point in our study we are able to replicate with reasonable
accuracy ambient temperature, pressure, and water vapor content from sea level to 15 km.
Sources:
Chiang, C.-W., & Subrata Kumar Das, J.-B. N.-x.-l. (2009, August). Simultaneous measurement of
humidity and temperature in the lower troposphere over Chung-Li, Taiwan. Journal of
Atmospheric and Solar-Terrestrial Physics, 71(12), 1389-1396.
Dutton, J. A. (1986). The Ceasless Wind. Mineola, New York, United States of America: Dover
Publications.
Federal Aviation Administration. (2011). Aeronautical Information Manual: Official Guide to Basic Flight
Information and ATC Procedures (Change 2 ed.). Washington D.C., United States of America:
Department of Transportation.
Leishman, J. G. (2006). Priciples of Helicopter Aeodynamics (2nd ed.). New York, NY, USA: Cambridge
University Press.
Stull, R. B. (2000). Meteorology for Scientists and Engineers (2nd ed.). Belmont, CA, USA: Brooks/Cole.
Tomasi et al, C. (2004). Mean vertical profiles of temperature and absolute humidity from a 12-year
radiosounding data set at Terra Nova Bay (Antarctica). Atmosperic Research(71), 139-161.