Chapter 2 Earth s atmosphere (Lectures 4 and 5)

Transcription

1 Chapter 2 Earth s atmosphere (Lectures 4 and 5) Keywords: Earth s atmosphere; International standard atmosphere; geopotential altitude; stability of atmosphere. Topics 2.1 Introduction 2.2 Earth s atmosphere The troposphere The stratosphere The mesosphere The ionosphere or thermosphere The exosphere 2.3 International standard atmosphere (ISA) Need for ISA and agency prescribing it Features of ISA 2.4 Variations of properties with altitude in ISA Variations of pressure and density with altitude Variations with altitude of pressure ratio, density ratio speed of sound, coefficient of viscosity and kinematic viscosity. 2.5 Geopotential altitude 2.6 General remarks Atmospheric properties in cases other than ISA Stability of atmosphere References Exercises Dept. of Aerospace Engg., Indian Institute of Technology, Madras 1

2 2.1 Introduction Airplanes fly in the earth s atmosphere and therefore, it is necessary to know the properties of this atmosphere. This chapter, deals with 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. 2.2 Earth s atmosphere 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). These five regions are: (i) Troposphere, (ii) Stratosphere, (iii) Mesosphere, (iv) Ionosphere or Thermosphere and (v) Exosphere. There is no sharp distinction between these regions and each region gradually merges with the neighbouring regions. Fig.2.1 Typical variations of temperature and pressure in the earth s atmosphere Dept. of Aerospace Engg., Indian Institute of Technology, Madras 2

3 2.2.1 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 The stratosphere This extends from the tropopause to about 50 km. High velocity winds may be encountered in this region, 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 The mesosphere The mesosphere extends from the stratopause to about 80 km. The temperature decreases to about C in this region. In the mesosphere, the pressure and density of air are very low, but the air still retains its composition as at sea level. The highest point of the mesosphere is called the mesopause The ionosphere or thermosphere This region extends from the mesopause to about 1000 km. It is characterized by the presence of ions and free electrons. The temperature increases to about 0 0 C at 110 km, to about C at 150 km and peak of about C at 700 km (Ref.2.1). Some electrical phenomena like the aurora borealis occur in this region 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. 2.3 International Standard Atmosphere (ISA) Need for ISA and agency prescribing it 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. As mentioned in Dept. of Aerospace Engg., Indian Institute of Technology, Madras 3

4 section 1.9, the performance of an airplane is dependent on the physical properties of the earth s atmosphere. Hence, for the purpose 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 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 32 km (Ref.1.11, chapter 2). With these values being prescribed, it is possible to find the required physical characteristics (pressure, temperature, density etc) at any chosen altitude. 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 Features of ISA The main features of the ISA are the standard sea level values and the variation of temperature with altitude. The air is assumed as dry perfect gas. The standard sea level conditions are as follows: Temperature (T 0 ) = K = 15 0 C Pressure (p 0 ) = N/m 2 = 760 mm of Hg Rate of change of temperature: = K/km upto 11 km = 0 K/km from 11 to 20 km = 1 K/km from 20 to 32 km Dept. of Aerospace Engg., Indian Institute of Technology, Madras 4

5 The region of ISA from 0 to 11 km is referred to as troposphere. That between 11 to 20 km is the lower stratosphere and between 20 to 32 km is the middle stratosphere (Ref.1.11, chapter 2). Note: Using the values of T 0 and p 0, and the equation of state, p = ρrt, gives the sea level density (ρ 0 ) as kg/m Variations of properties with altitude in ISA For calculation of the variations of pressure, temperature and density with altitude, the following equations are used. The equation of state p = ρ R T (2.1) The hydrostatic equation dp/dh = - ρ g (2.2) Remark: The hydrostatic 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 Δh as shown in Fig.2.2. Fig 2.2 Equilibrium of a fluid element. The forces acting in the vertical direction on the element are the pressure forces and the weight of the element. For vertical equilibrium of the element, pa {p + (dp /dh) Δh} A ρ g A Δh = 0 Dept. of Aerospace Engg., Indian Institute of Technology, Madras 5

6 Simplifying, dp /dh = - ρ g Variations of pressure and density with altitude 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. For example, in the troposphere, the variation of temperature with altitude is given by the equation T = T 0 λ h (2.4) where T 0 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 (T 0 λ h) (2.5) Taking g as constant, Eq.(2.5) can be integrated between two altitudes h 1 and h 2. Taking h 1 as sea level and h 2 as the desired altitude (h), the integration gives the following equation, the intermediate steps are left as an exercise. (p/p 0 ) = (T/T 0 ) (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/ρ 0 ) = (T/T 0 ) (g/λr)-1 (2.7) Thus, both the pressure and density variations are obtained once the temperature variation is known. As per the ISA, R = m 2 sec -2 K and g = m/s 2. Using these and λ = K/m in the troposphere yields (g/rλ) as Thus, in the troposphere, the pressure and density variations are : (p/p 0 ) = (T/T 0 ) (2.8) (ρ/ρ 0 ) = (T/T 0 ) (2.9) Note: T= h; h in m and T in K. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 6

7 In order to obtain the variations of properties in the lower stratosphere (11 to 20 km altitude), the previous analysis needs to be carried-out afresh with λ = 0 i.e., T having a constant value equal to the temperature at 11 km (T = K). From this analysis the pressure and density variations in the lower stratosphere are obtained as : (p / p 11 ) = (ρ / ρ 11 ) = exp { -g (h ) / RT 11 } (2.10) where p 11, ρ 11 and T 11 are the pressure, density and temperature respectively at 11 km altitude. In the middle stratosphere (20 to 32 km altitude), it can be shown that (note in this case λ = K / m): (p / p 20 ) = (T / T 20 ) (2.11) (ρ / ρ 20 ) = (T/ T 20 ) (2.12) where p 20, ρ 20 and T 20 are pressure, density and temperature respectively at 20 km altitude. Thus, the pressure and density variations have been worked out in the troposphere and the stratosphere of ISA. Table 2.1 presents these values. Remark: 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) Variations with altitude of pressure ratio, density ratio, speed of sound, coefficient of viscosity and kinematic viscosity The ratio (p/p 0 ) is called pressure ratio and is denoted by δ. Its value in ISA can be obtained by using Eqs.(2.8),(2.10) and (2.11). Table 2.1 includes these values. The ratio (ρ / ρ 0 ) is called density ratio and is denoted by σ. Its values in ISA can be obtained using Eqs.(2.9),(2.10) and (2.12). Table 2.1 includes these values. The speed of sound in air, denoted by a, depends only on the temperature and is given by: a = (γ RT) 0.5 (2.13) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 7

8 where γ is the ratio of specific heats; for air γ = 1.4. The values of a in ISA can be obtained by using appropriate values of temperature. Table 2.1 includes these values. The kinematic viscosity ( ) is given by: = μ / ρ where μ is the coefficient of viscosity. The coefficient of viscosity of air (μ) depends only on temperature. Its variation with temperature is given by the following Sutherland formula. 3/2-6 T μ = 1.458X10 [ ] T+110.4, where T is in Kelvin and μ is in kg m-1 s -1 (2.14) Table 2.1 includes the variation of kinematic viscosity with altitude. Example 2.1 Calculate the temperature (T), pressure (p), density (ρ ), pressure ratio (δ ), density ratio (σ ), speed of sound (a), coefficient of viscosity (μ ) and kinematic viscosity ( ) in ISA at altitudes of 8 km, 16 km and 24 km. Solution: It may be noted that the three altitudes specified in this example, viz. 8 km, 16 km and 24 km, lie in troposphere, lower stratosphere and middle stratosphere regions of ISA respectively. (a) h = 8 km Let the quantities at 8 km altitude be denoted by the suffix 8. In troposphere: T=T0 -λh Where, T 0 = K, λ =0.0065K/m Hence, T 8 = = K From Eq.(2.8) p p = δ 8 = T/T 0 = / = Or p = = N/m ρ 8 = p 8/ RT 8 = = kg/m σ 8 = ρ 8/ρ 0 = /1.225 = Dept. of Aerospace Engg., Indian Institute of Technology, Madras 8

9 a 8 = (γ RT 8 ) 0.5 = = m/s From Eq.(2.14): T μ 8 = = = kg m s T = μ 8/ρ 8 = / = m /s Remarks: (i) The values calculated above and those in Table 2.1 may differ from each other in the last significant digit. This is due to the round-off errors in the calculations. (ii) Consider an airplane flying at 8 km altitude at a flight speed of 220 m/s. The Mach number of this flight would be: 220/ = (iii) Further if the reference chord of the wing (c ref ) of this airplane be 3.9 m, the Reynolds number in this flight, based on c ref, would be: Vcref R e = = = (iv) For calculation of values at 16 km altitude, the values of temperature, pressure and density are needed at the tropopause viz. at h=11 km. Now T = = K p = / = N/m ρ 11 3 = 22632/ = kg/m 6 (b) h = 16 km In lower stratosphere Eq.(2.10) gives : p p ρ = = exp-gh /RT ρ Consequently, p p ρ = = exp / = ρ Dept. of Aerospace Engg., Indian Institute of Technology, Madras 9

10 Or p 16 ρ 16 = = N/m = = kg/m 2 3 δ 16 = / = σ 16 = /1.225 = a = = m/s μ 16 = = kg m s = / = m /s Remark : To calculate the required values at 24 km altitude, the values of T and p are needed at h = 20 km. These values are : T 20 = p p = exp / = Or p 20 = = N/m 2 (c) h = 24 km 24 T = = K From Eq.(2.11): p p = T /T Or p 24 = / = N/m ρ = / = Hence, δ 24 = / = and σ 24 = /1.225 = a = = m/s 24 Dept. of Aerospace Engg., Indian Institute of Technology, Madras 10

11 μ 24 = = kg m s = / = m /s Answers: h (km) T (K) p (N/m 2 ) δ =p/p ρ kg/m σ = ρ/ρ a (m/s) μ kg m s x x x m/s x x x Geopotential altitude The variations of pressure, temperature and density in the atmosphere were obtained by using the hydrostatic equation (Eq.2.2). In this equation g is assumed to be constant. However, it is known that g decreases with altitude. Equation (1.1) gives the variation as: R g=g ( ) 0 R+h G Where R is the radius of earth and h G is the geometric altitude above earth s surface. Thus the values of p and ρ obtained by assuming g = g are at an altitude 0 slightly different from the geometrical altitude (h G ). This altitude is called geopotential altitude, which for convenience is denoted by h. Following Ref.1, the geopotential altitude can be defined as the height above earth s surface in Dept. of Aerospace Engg., Indian Institute of Technology, Madras 11

12 units, proportional to the potential energy of unit mass (geopotential), relative to sea level. It can be shown that the geopotential altitude (h) is given, in terms of geometric altitude (h G ), by the following relation. Reference 1.13, chapter 3 may be referred to for derivation. R h G = h R-h It may be remarked that the actual difference between h and h G is small for altitudes involved in flight dynamics; for h of 20 km, h G would be km. Hence the difference is ignored in performance analysis. 2.6 General remarks: Atmospheric properties in cases other than ISA It will be evident from chapters 4 to 10 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 arctic and tropical regions. Figure 2.3 shows the temperature variations with altitude in arctic and tropical atmospheres along with ISA. It is seen that the arctic minimum atmosphere has the following features. (a) The sea level temperature is C (b) The temperature increases at the rate of 10 K per km up to 1500 m altitude. (c) The temperature remains constant at C 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 c. The features of the tropical maximum atmosphere are as follows. (a) Sea level temperature is 45 0 C. (b) The temperature decreases at the rate of 6.5 K per km up to km and then remains constant at C. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 12

13 Fig 2.3 Temperature variations in arctic minimum, ISA and tropical maximum atmospheres (Reproduced from Ref.1.7, Chapter 3 with permission of author) Note: (a) The local temperature varies with latitude but the sea level pressure (p 0 ) depends on the weight of air above and is taken same at all the places i.e N/m 2. Knowing p 0 and T 0, and the temperature lapse rates, 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). (see also exercise 2.1). (b) Some airlines/ air forces may prescribe intermediate values of sea level temperature e.g. ISA C or ISA C. The variations of pressure, temperature and density with altitude in these cases can also be worked out from the aforesaid equations Stability of atmosphere It is generally assumed that the air mass is stationary. However, some packets of air mass may acquire motion due to local changes. For example, due Dept. of Aerospace Engg., Indian Institute of Technology, Madras 13

14 to absorption of solar radiation 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 continues to move up then the atmosphere is unstable. Reference 1.7, 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. Reference 2.1 Gunston, B, The Cambridge aerospace dictionary Cambridge University Press (2004). Dept. of Aerospace Engg., Indian Institute of Technology, Madras 14

15 Exercises 2.1 On a certain day the pressure at sea level is 758 mm of mercury ( N / m 2 ) and the temperature is 25 o C. The temperature is found to fall linearly with height to -55 o C at 12km and after that it remains constant upto 20 km. Calculate the pressure, density and kinematic viscosity at 8km and 16km altitude. (Hint : When the temperature variation is linear, Eqs. (2.6) and (2.7) can be used to obtain the pressure and density at a chosen altitude by using appropriate values of p 0, T 0, ρ 0 and λ. As regards the constant temperature region, an equation similar to Eq (2.10) can be used; note that, in this exercise, the tropopause is at 12 km altitude) [Answers: p 8 = 36,812 N/m 2, 8 = kg/m 3, 8 = x 10-5 m 2 /sec, p 16 = N/m 2, 16 = kg/m 3, 16 = x 10-5 m 2 /sec] Remark : Due to round off errors in calculations, the student may get the numerical values which are slightly different from those given as answers. Values within 0.5% of those given as answers can be regarded as correct. 2.2 If the altimeter in an airplane reads 5000m, on 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 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 the correct altitude when the atmospheric conditions differ from ISA. To solve this exercise, obtain the pressure corresponding to 5000 m 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 this exercise, the pressure at the sea level on that day is N/m 2. 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.) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 15

16 [Answers: 5152 m, 22.3 m]. 2.3 An altimeter calibrated according to ISA reads an altitude of 3,600 m. If the ambient temperature is 6 0 C, calculate the ambient density. [Answer: kg/m 3 ]. 2.4 During a 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 ( 0 C) 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 the indicated altitudes. To get the difference between these two altitudes (Δh), obtain pressures at 1300 and 1600 m heights in ISA. Let the difference in pressures be Δp. Calculate density at the two altitudes using corresponding pressures and temperature. Take average of the two densities ( avg ). Using Eq. (2.2) : Δh -Δp / { avg x g} ) [Answer: 311 m] Remark: The difference between the actual altitudes (311 m) 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 take care of any uncertainty, 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 3.2 km. Assuming ISA conditions and the mean chord of the wing to be 1.5 m, obtain the Reynolds number, based on wing mean chord, and the Mach number in this flight. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 16

17 [Answers: R e = 4.83 x 10 6, M = 0.186] speed Altitudrature Tempe- of Kinematic Pressure δ Density σ sound viscosity (m) (K) (N/m 2 ) (p/p o ) (kg/m 3 ) (ρ/ρ o ) (m/s) (m 2 /s) E E E E E E E E E E E E E E E E E E E E E E E E-005 Table 2.1 Properties in ISA (Cont ) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 17

18 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-005 Table 2.1 Properties in ISA (Cont ) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 18

19 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-005 Table 2.1 Properties in ISA (Cont ) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 19

20 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-004 Table 2.1 Properties in ISA (Cont ) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 20

21 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-004 Table 2.1 Properties in ISA (Cont ) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 21

22 E E E E E E E E E E E E E E E E E E E E E-003 Table 2.1 Properties in ISA Note: Following values / expressions have been used while preparing ISA table. 2-2 R= m sec K g= m/s 2 Sutherland formula for viscosity: 3/2-6 T μ = 1.458X10 [ ] T Dept. of Aerospace Engg., Indian Institute of Technology, Madras 22

23 In troposphere (h = 0 to m): T= h. p = [ h] ρ = [ h] In lower stratosphere (h = to km): T= K. p = exp { (h-11000)} ρ = exp { (h-11000)} In middle stratosphere (h = to km): T = h p = [ (h-20000)] ρ = [ (h-20000)] Dept. of Aerospace Engg., Indian Institute of Technology, Madras 23