Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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Chapter 3The Standard Atmosphere

1.Structure Of The Atmosphere

●Atmosphere is the air envelope surrounding the earth

● Air is a mixture of several gases, where Oxygen

represents (21%) ,Nitrogen (78%), and other gases

(1%) of the total volume.

●Up to about 90 km altitude, composition of air is the

same (the homosphere(

● Above about 90 km, composition varies with altitude

(the heterosphere) as shown in figure.

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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●The atmosphere is divided into several layers

according to temperature variation with altitude Why do we study the atmosphere?

● For the design and performance determination of any flying vehicle (airplane, missile, satellite,….etc), knowledge of the vertical distribution of pressure, temperature, and density of air is required.●The properties of the real atmosphere never remain constant at any particular time or place.● Consequently, a hypothetical model is employed as an approximation to what may be expected. This model is known as the standard atmosphere.

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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• The Standard atmosphere depends on the measurement of the mean value of temperature with altitude combined with the hydrostatic equation which gives the variation of p with the altitude .

2.The Hydrostatic Equation

Force diagram for the hydrostatic equation

Fluid element of air at rest +ve hG

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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• For equilibrium of the air element, the some of

the forces in any direction must be zero.• Resolving in the vertical direction:

• p(1)(1) – (p+dp)(1)(1) – ρ (1)(1) g dhG = 0

• Then - dp – ρ g dhG = 0

• Or dp/dhG = - ρ g Hydrostatic equation

• This equation means that the pressure decreases with the increase of height.

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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What is the altitude? ●There are six (6) different altitudes:

1-Geometric Altitude hG is the height above sea-level.

2-Absolute Altitude ha is the height measured from the center of the earth.

ha = hG + R where R is the radius of earth ( R = 6356.766 km)

The absolute altitude is important for space flight because g varies with ha .

●According to Newton’s law of gravity,

g = go [R/ha ]2 = go [R/(R + hG) ]2

where go is g at sea-level where 3-Geopotential Altitude h is a fictitious (hypothetical)

altitude used to facilitate the calculations ( h ≠ hG )

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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ha

R

hG

Surface of earth

)Sea level(

Definition of altitude

By definition : dp = - ρ g dhG = - ρ go dhThen : dh/dhG = g / go

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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• What is the relation between h & hG?

• dh = R2/(R+hG)2 dhG

• Integrate: 0 ∫h dh =

0 ∫hG R2/(R+hG)2 dhG

• The result is: h = [R/(R+hG)] hG

• At hG = 6.5 km , h is less than hG by about 0.1% only.

• At hG = 65 km , h is less than hG by about 1% only.

The Standard Atmosphere (SA)

• It is defined in order to relate flight tests, wind-tunnel tests, and the general design and performance of flying vehicle ( aircraft, missile, satellite,….etc ) to a common reference.

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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• It gives mean values of p, T, ρ, and other properties as function of altitude.

• These values are obtained from experimental balloons and sounding-rocket measurements with a mathematical model of the atmosphere ( based on the equation of state and the hydrostatic equation).

• Several different standard atmospheres exist, due to using different experimental data in the models, but the differences are insignificant below 30 km (100,000 ft), which is the domain of contemporary airplanes.

■ Definition Of The Standard Atmosphere (SA)

●The standard atmosphere (SA) is defined by:

1. Conditions at sea-level:

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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po = 1.01325 x 105 N/m2 = 2116.2 Ib/ft2

To = 288.16 oK = 518.60 oR

ρo = 1.225 kg/m2 = 0.002377 slug/ft2

go = 9.807 m/s2 = 32.17 ft/s2

2. A defined variation of T with altitude, based on experimental evidence, as shown in Figure.

# Given T = T(h), then p = p(h) and ρ = ρ(h) can now be determined by calculation.

# The atmospheric layers of the SA are either isothermal or gradient layers, as shown in Figure.

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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Temperature distribution in the standard atmosphere

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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Atmospheric properties variation of SA

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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(A) Calculation in an Isothermal Layer:

h1

h T,p,ρ

T1,p1,ρ1

Isothermal Layer

Base of Isothermal Layer

From the hydrostatic equation: dp = - ρ go dh

Substitute ρ from the equation of state: p = ρ RT dp/p = - (go/RT) dh (1)

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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• Integrate: p1

∫p dp/p = -(go/RT)

h

1

∫h dh

• Hence, ln (p/p1) = -(go/RT) (h – h1)

• Or : (p/p1) = e -(go/RT)(h – h1)

= (ρ/ρ1) (2)

• Because , ( p/p1) = (ρT/ρ1T1) = (ρ/ρ1)

from equation of state

• and T = T1 (3)

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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(B) Calculation in a Gradient Layer:

h T,p,ρ

Base of layer h1T1,p1, ρ1

The temperature variation is linear, (T-T1)/(h-h1) = dT/dh = a a is the temperature lapse rate ( +ve or –ve) T = T1 + a (h – h1) (4)Then, dh = dT/a and substitute into Eqn.(1) dp/p = - (go/aR) dT/T

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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• Integrate: p1

∫p dp/p = -(go/aR)

T1∫T dT/T

• Hence, ln p/p1 = - (go/aR) ln T/T1

• Then, p/p1 = (T/T1) -go/aR (5)

• From equation of state:

• ( p/p1) = (ρT/ρ1T1) = (ρ/ρ1)(T/T1)

• Hence, (ρ/ρ1) = (T/T1) -[(go/aR)+1]

(6)

• Where T is calculated from Eq.(4)

Prof. Galal Bahgat Salem Aerospace Dept., Cairo University

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• With the Figure of variation of T with h and Eqs.(2),(3),(4),(5), and (6) a Table of values for the Standard Atmosphere is constructed and is given in Appendix A for SI units and Appendix B for English Engineering Units (see textbook “Introduction to Flight”, Anderson, pages 709-729)

• 4.Pressure Altitude hp:The altitude in the Standard Atmosphere corresponding to a particular static air pressure.

• 5.Temperature Altitude hT :The altitude in the Standard Atmosphere corresponding to a particular air temperature.

• 6.Density Altitude hρ : The altitude in the Standard Atmosphere corresponding to a particular air density.