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Chapter 2The Earth’s Atmosphere

Airplanes fly in the earth’s atmosphere and therefore , it is necessary to have a knowledge of the properties of the earth’s atmosphere.

In this chapter, we will study the average characteristics of the earth’s atmosphere in various regions and the International Standard Atmosphere (ISA) which is used for calculation of airplane performance.

2.1 Introduction

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The earth’s atmosphere is a gaseous blanket around the earth which is divided into the five regions based on certain intrinsic features (see Fig.2.1)

– Troposphere

– Stratosphere

– Mesosphere

– Ionosphere or Thermosphere

– Exosphere

There is no sharp distinction between these regions and each region gradually merges with the neighboring regions.

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Fig.2.1 Variation of temperature and pressure in the earth’s atmosphere

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The Troposphere

– This is the region closest to the earth’s surface.

– It is characterized by turbulent conditions of air.

– The temperature decreases linearly at an approximate rate of 6.5 K / km.

– The highest point of the troposphere is called tropopause. The height of the tropopause varies from about 9 km at the poles to about 16 km at the equator.

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The Stratosphere

– This extends from the tropopause to about 50 km.

– High velocity winds may be encountered, but they are not gusty.

– Temperature remains constant up to about 25 km and then increases.

– The highest point of the stratosphere is called the stratopause.

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The Mesosphere

– The mesosphere extends from the stratopause to about 80 km.

– The temperature decreases in this region.

– In the mesosphere, the pressure and density of air are very low, but the air still retains it’s composition as that at sea level.

– The highest point of the mesosphere is called the mesopause.

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The Ionosphere

– This region extends from the mesopause to about 500 km.

– It is characterized by the presence of ions and free electrons.

– Temperature increases from 0ºC to about 2200ºC at the upper limit.

– Some electrical phenomena like the aurora borealis occur in this region.

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The exosphere

– This is the outer fringe of the earth’s atmosphere.

– Very few molecules are found in this region.

– The region gradually merges into the interplanetary space.

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The properties of earth’s atmosphere like pressure, temperature and density vary not only with height above the earth’s surface but also with the location on earth, from day to day and even during the day.

The performance of an airplane is dependent on the physical properties of the earth’s atmosphere.

Hence, for the purposes of comparing

(a) the performance of different airplanes and

(b) the performance of the same airplane measured in flight tests on different days,

a set of values for atmospheric properties have been agreed upon, which represent average conditions prevailing for most of the year, in Europe and

2.2 International Standard Atmosphere (ISA)

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North America. Though the agreed values do not represent the actual conditions anywhere at any given time, they are useful as a reference.

This set of values called the International Standard Atmosphere (ISA) is prescribed by ICAO (International Civil Aviation Organization). It is defined by the pressure and temperature at mean sea level, and the variation of temperature with altitude up to 20 km. With this model of the atmosphere, it is possible to find the required physical characteristics at any altitude.

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Remark:

The actual performance of an airplane is measured in flight tests under prevailing conditions of temperature, pressure and density. Methods are available to deduce, from the flight test data , the performance of the airplane under ISA conditions. When this procedure is applied to various airplanes and performance presented under ISA conditions , then comparison among different airplanes is possible.

2.2 Features of ISA

The main features of the ISA are the standard sea level values and the variation of temperature with altitude.

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The air is assumed as dry perfect gas

The standard sea level conditions are as follows:

– Temperature (T0) = 288.15 K = 15°C

– Pressure (p0) = 101326 N/m2

– Temperature Lapse Rate = 6.5 K/km upto 11 km

= 0 K/km from 11-20km

The region of ISA upto 11 km is referred to as troposphere and the above that, as stratosphere.

Note:

Using the equation of state p = ρ R T, we get the sea level density (ρ0) as 1.225 kg/m3

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For calculation of the variations of pressure, temperature and density with altitude, we use the following equations

– The equation of state p = ρ R T (2.1)

– The hydrostatic equation dp/dh = - ρ g (2.2)

Remark

The buoyancy equation can be easily derived by considering the balance of forces on a small fluid element.

Consider a cylindrical fluid element of area A and height ∆h as shown in the Fig.2.2.

The forces acting in the vertical direction on the element are the pressure forces and the weight of the element.

2.2.2 Variations of properties with altitude in ISA

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Fig 2.2 Equilibrium of a fluid element.

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For vertical equilibrium of the element,

pA – {p + (dp/dh) ∆h}A – ρgA∆h = 0

Simplifying, dp/dh = - ρg

Variations of p and ρ with h:

Substituting for ρ from the eq. (2.1) in eq. (2.2) gives :

dp/dh = -(p/RT)g

or (dp/p) = -g dh/RT (2.3)

Equation (2.3) is solved separately in troposphere and stratosphere, taking into account the temperature variations in each region.

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For example, in the troposphere, the variation of temperature with altitude is given by the equation

T = T0 – λ h (2.4)

where T0 is the sea level temperature, T is the temperature at the altitude h and λ is the temperature lapse rate in the troposphere.

Substituting from eq.(2.4) in eq.(2.3) gives

(dp/p) = - gdh/R(T0 – λ h) (2.5)

Equation (2.5) can be integrated between two altitudes h1 and h2.

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Taking h1 as sea level and h2 as the desired altitude (h), the integration gives

(p/p0) = (T/T0)(g/λR) (2.6)

Where T is the temperature at the desired altitude (h) given by eq.(2.4).

Equation(2.6) gives the variation of pressure with altitude.

The variation of density with altitude can be obtained using Eq.(2.6) and the equation of state. The resulting variation of density with temperature in the troposphere is given by

(ρ/ρ0) = (T/T0)(g/λR)-1 (2.7)

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Thus both the pressure and density variations are obtained once the temperature variation is known.

As per the ISA, R = 287.05 m2sec-2K and

g = 9.806536 m/s2.

Using these and λ = 0.0065 K/m in the troposphere yields (g/Rλ) as 5.256.

Thus, in the troposphere, we get the pressure and density variations as

(p/p0) = (T/T0)5.256 (2.8)

(ρ/ρ0) = (T/T0)4.256 (2.9)

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In order to obtain the variations in the stratosphere, we need to carry out the previous analysis with

λ = 0 i.e., T having a constant value of temperature at 11 km (T = 216.65 K).

From this analysis we obtain the following pressure and density variations in the stratosphere,

(p2/p1) = (ρ2/ρ1) = exp{g(h1-h2)/RTs} (2.10)

Where Ts is the temperature at 11 km.

Thus, the pressure and density variations have been worked out in the troposphere and the stratosphere of ISA.

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Remarks:

i) Using Eqs.(2.1) and (2.2) the variations of pressure and density can be worked out for other variations of temperature with height (see exercise 2.1).

ii) The ratio (ρ/ρ0) is denoted by σ and the ratio (p/p0) is denoted by δ.

iii) The speed of sound in air, denoted by ‘a’depends only on the temperature and is given by:

a = ( γRT)0.5 (2.11)

where γ is the ratio of specific heats; for air γ = 1.4

iv) The kinematic viscosity ( ν ) is given by:

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ν = µ / ρ where µ is the coefficient of viscosity.

v)The coefficient of viscosity of air (µ) depends only on temperature. Its variation with temperature is given by the following Sutherland formula

(2.12)

Where T is in Kelvin and µ is in kg m-1 s-1.

vi) Variations of atmospheric properties in standard atmosphere are presented in Table (2.1).

3 / 261.458 10 [ ]

110.4TX

Tµ −=

+

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General remarks

i ) Atmospheric properties in cases other than ISA.

It will be evident from chapters 4 and 5 that the engine characteristics and the airplane performance depend on atmospheric characteristics. Noting that ISA only represents average atmospheric conditions, other atmospheric models have been proposed as guidelines for extreme conditions in artic and tropical regions. Figure 2.3 shows temperature variations with altitude in artic and tropical atmospheres along with ISA. It is seen that the artic minimum atmosphere has following features . (a) The sea level temperature in -500C (b) The temperature increases at the rate of 10 K per km up to 1500 m altitude.

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Fig 2.3 Temperature variations in arctic minimum, ISA and tropical maximum atmospheres(Adapted from Ref.1.5, Chapter 3)

Temperature 0C

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(c ) The temperature remains constant at -350C up to 3000 m altitude (d) Then the temperature decreases at the rate of 4.72 K per km up to 15.5 km altitude (e) The tropopause in this case is at 15.5 km and the temperature there is -940c.

The features of the tropical maximum atmosphere are as follow.

(a) Sea level temperature in 450C .

(b) The temperature decreases at the rate of 6.5 K per km up to 11.54 km and then remains constant at -300C.

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Note:

(a) The local temperature varies with latitude but the sea level pressure (po) depends on the weight of air above and is taken same at all the places i.e. 101326 N/m2. Knowing po and To , and the lapse rate (s), the pressure temperature and density in tropospheres of arctic minimum and tropical maximum can be obtained using Eqs. (2.4), (2.6) and (2.7) . Equation (2.10) gives pressure and density in stratosphere (see also exercise 2.1) .

(b) Some airlines/ air forces may prescribe intermediate values of sea level temperature e.g. ISA +150C or ISA +200C. The variations of pressure , temperature and density with altitude can be worked out from the aforesaid equations.

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ii) Stability of atmosphere:

We generally assume that the air mass is stationary . However some packets of air mass may acquire motion due to local changes. For example due to absorption of heat, from sun , by the earth’s surface, an air mass adjacent to the surface may become lighter and buoyancy may cause it to rise. If the atmosphere is stable , a rising packet of air must come back to its original position. On the other hand if the air packet remains in the disturbed position, then the atmosphere is neutrally stable. If the rising packet continue to move up then the atmosphere is unstable .

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Reference 1.5 , chapter 3 analyses the problem of atmospheric stability and concludes that if the temperature lapse rate is less than 9.75 K per km, then the atmosphere is stable. It is seen that the three atmospheres, representing different conditions , shown in Fig.2.3 are stable.

iii) Geopotential altitude

The variations of pressure , temperature and density in the atmosphere were obtained by using the hydrostatic equation (Eq.2.2)

ρ= −dp gdh

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In this equation ‘g’ is assumed to be constant. However we know that ‘g’ decreases with altitude. Equation (1.1) gives the variation as

where ‘R’ is the radius of earth and ‘h’ is the altitude above earth’s surface.

If this variation of ‘g’ is takes into account , then the values of pressure and density obtained earlier (with g=g0) would occur at a slightly lower height which is called geopotential altitude. It is defined as the height above earth’s surface in units, proportional to the potential energy of

0R( )=+

g gR h

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unit mass (geopotential) relative to sea level (see Ref.2.1) . Reference 1.4 chapter 1 shows that the geopotential altitude (hg) is given, in terms of geometric altitude (h), by the following relation.

The actual difference between h and hg is small for altitudes involved in flight mechanics; for h of 20 km , hg would be 19.942 km. Hence the difference is ignored.

gRhh

R h=

+(2.13)

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Reference

2.1 Gunston , B , “The Cambridge aerospace dictionary” Cambridge university press (2004).

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Exercises

2.1 On a certain day the pressure at sea level is 758 mm of mercury (101059 N/m2) and the temperature is 250C. The temperature is found to fall linearly with height to -55OC at 12km and after that it remains constant. Calculate the pressure, density and kinematic viscosity at 8km and 16km altitude.

[Answer:

p8 = 36,812 N/m2, ρ8 =0.5238 kg/m3, ν8 = 3.002 x 10-5 m2/sec,

p16 = 10897 N/m2, ρ16 =0.1740 kg/m3,

ν16 = 8.218 x 10-5 m2/sec]

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2.2 If the altimeter in an airplane reads 5000m, an the day described in exercise 2.1, what is the altitude of airplane above mean sea level? What would be the indicated altitude after landing on an aerodrome at sea level?

(Hint: An altimeter is an instrument which senses the ambient pressure and indicates height in ISA corresponding to that pressure. It does not read correct altitude when the atmospheric conditions differ from ISA. To solve this exercise, obtain the pressure corresponding to 5000m altitude in ISA. Then find the altitude corresponding to this pressure in the atmospheric conditions prevailing as in exercise 2.1. As regards the second part of

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this exercise, the pressure at the sea level on that day is 101059 N/m2. When the airplane lands at sea level, the altimeter would indicate altitude, in ISA, corresponding to this pressure. In actual practice the air traffic control would inform the pilot about the local ambient pressure and the pilot would adjust zero reading of his altimeter.)

[Answers: 5152 m, 22.3 m].

2.3 An altimeter calibrated according to ISA reads an altitude of 3,600m.If the ambient temperature is –6o C, calculate the ambient density.

[Answer: 0.86 kg/m3]

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2.4 During flight test for climb performance the following readings were observed at two altitudes:

Record Number 1 2

Indicate altitude (m) 1,300 1,600

Ambient Temperature (0C) 16 14

The altimeter is calibrated according to ISA. Obtain the true difference of height between the two indicated altitudes.

(Hint: Note that the ambient temperatures are different from those in ISA at 1300 and 1600 m altitudes. Hence the actual altitudes are different from ISA values. To get the difference between these two altitudes(∆h), obtain pressures at 1300

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and 1600 m heights in ISA. Let the difference inpressures be ∆p. Calculate density at the two

altitudes using corresponding pressures and temperature. Take average of the two densities (ρavg). Using eq.(2.2)

∆h ≈ -∆p / {ρavg x g}. )

[Answer: 312m]Remark :The difference between the actual altitudes(312m) and the indicated altitudes (300 m) is small. Since altimeters of all the airplanes are calibrated using ISA, the difference between indicated altitudes and actual altitudes of two airplanes will be small. To

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take care of any uncertainties the flight paths of two airplanes are separated by several hundred meters. However With the availability of Global Positioning System (GPS) the separation between two airplanes can be reduced.

2.5 A light airplane is flying at a speed of 220 kmph at an altitude of 3.2km. Assuming ISA conditions and the mean chord of the wing to be 1.5m, obtain the flight Mach number and Reynolds number.

[Answer: Re = 4.83 x 106, M = 0.186]

37Table 2.1 Properties of Standard Atmosphere

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