Earth Atmosphere

SOLO HERMELIN

Updated: 08.11.12

1

Table of Content

SOLO

2

Earth Atmosphere

3

Earth Atmosphere

The Earth Atmosphere might be described as a Thermodynamic Medium in a Gravitational Field and in Hydrostatic Equilibrium set by Solar Radiation. Since Solar Radiation and Atmospheric Reradiation varies diurnally and annually and with latitude and longitude, the Standard Atmosphere is only an approximation.

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Earth Atmosphere

The purpose of the Standard Atmosphere has been defined by the World Metheorological Organization (WMO). The accepted standards are the COESA (Committee on Extension to the Standard Atmosphere) US Standard Atmosphere 1962, updated by US Standard Atmosphere 1976.

The basic variables representing the thermodynamics state of the gas are the Density, ρ, Temperature, T and Pressure, p.

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Earth Atmosphere

The Density, ρ, is defined as the mass, m, per unit volume, v, and has units of kg/m3.

v

mv

0lim

The Temperature, T, with units in degrees Kelvin ( : K). Is a measure of the average kinetic energy of gas particles.

The Pressure, p, exerted by a gas on a solid surface is defined as the rate of change of normal momentum of the gas particles striking per unit area.

It has units of N/m2. Other pressure units are millibar (mbar), Pascal (Pa), millimeter of mercury height (mHg)

S

fp n

S

0

lim

kPamNbar 100/101 25

mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 The Atmospheric Pressure at Sea Level is:

6

Earth Atmosphere

Physical Foundations of Atmospheric Model

gHd

AP

A

AdPP

gHdAg

gH

The Atmospheric Model contains the values of Density, Temperature and Pressure as function of Altitude.

Atmospheric Equilibrium (Barometric) Equation

In figure we see an atmospheric element under equilibrium under pressure and gravitational forces

0 APdPPHdAg g

or gg HdHgPd

In addition, we assume the atmosphere to be a thermodynamic fluid. At altitude bellow 100 km we assume the Equation of an Ideal Gas

where V – is the volume of the gas N – is the number of moles in the volume V m – the mass of gas in the volume VR* - Universal gas constant

TRNVP *

V

m

M

mN &

MTRP /*

Earth Atmosphere

mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2

We must make a distinction between:- Kinetic Temperature (T): measures the molecular kinetic energy and for all practical purposes is identical to thermometer measurements at low altitudes. - Molecular Temperature (TM): assumes (not true) that the Molecular Weight at any altitude (M) remains constant and is given by sea-level value (M0)

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Earth Atmosphere

TM

MTM 0

To simplify the computation let introduce:- Geopotential Altitude H- Geometric Altitude Hg

Newton Gravitational Law implies: 2

0

gE

Eg HR

RgHg

The Barometric Equation is gg HdHgPd

The Geopotential Equation is defined as HdgPd 0

This means thatg

gE

Eg Hd

HR

RHd

g

gHd

2

0

Integrating we obtaing

gE

E HHR

RH

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Earth Atmosphere

Atmospheric Constants

DefinitionSymbolValueUnitsSea-level pressureP01.013250 x 105N/m2

Sea-level temperatureT0288.15: K

Sea-level densityρ01.225kg/m3

Avogadro’s NumberNa6.0220978 x 1023/kg-moleUniversal Gas ConstantR*8.31432 x 103J/kg-mole -: KGas constant (air)Ra=R*/M0287.0J/kg--:K

Adiabatic polytropic constantγ1.405Sea-level molecular weightM028.96643

Sea-level gravity accelerationg09.80665m/s2

Radius of Earth (Equator)Re6.3781 x 106m

Thermal Constantβ1.458 x 10-6Kg/(m-s-: K1/2)

Sutherland’s ConstantS110.4: KCollision diameterσ3.65 x 10-10m

10

Earth Atmosphere

Physical Foundations of Atmospheric Model

Atmospheric Equilibrium Equation

HdgPd 0At altitude bellow 100 km we assume t6he Equation of an Ideal Gas

TRMTRP a

MRR

a

aa

/

**

/

HdTR

g

P

Pd

a

0

Combining those two equations we obtain

Hd

AP

A

AdPP

HdAg

H

Assume that T = T (H), i.e. function of Geopotential Altitude only. The Standard Model defines the variation of T with altitude based on experimental data. The 1976 Standard Model for altitudes between 0.0 to 86.0 km is divided in 7 layers. In each layer dT/d H = Lapse-rate is constant.

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Earth Atmosphere

Layer Index

GeopotentialAltitude Z,

km

GeometricAltitude Z;

km

MolecularTemperature T,

: K

0 0.0 0.0288.150

111.0 11.0102216.650

220.0 20.0631216.650

332.0 32.1619228.650

447.0 47.3501270.650

551.0 51.4125270.650

671.0 71.8020 214.650

7 84.8420 86.0186.946

1976 Standard Atmosphere : Seven-Layer Atmosphere

Lapse RateLh;

: K/km

-6.3

0.0

+1.0

+2.8

0.0

-2.8

-2.0

12

Earth Atmosphere

Physical Foundations of Atmospheric Model

KT

kmH

150.288

10

20

30

km11

km20

km32

650.216

K650.228

• Troposphere (0.0 km to 11.0 km). We have ρ (6.7 km)/ρ (0) = 1/e=0.3679, meaning that 63% of the atmosphere lies below an altitude of 6.7 km.

HdHLTR

gHd

TR

g

P

Pd

aa

0

00

kmKLHLTT /3.60

Integrating this equation we obtain

H

a

P

P

HdHLTR

g

P

PdS

S 0 0

0 1

0

0

00 lnln0

T

HLT

RL

g

P

P

aS

S

HenceaRL

g

SS HT

LPP

0

0

0

1

and

10

0

0g

RL

S

S

a

P

P

L

TH

13

Earth Atmosphere

Physical Foundations of Atmospheric Model

HdTR

g

P

Pd

Ta

*0

Integrating this equation we obtain

TTaS

S HHTR

g

P

P

T

*0ln

Hence T

Ta

T

HHTR

g

SS ePP

*

0

andS

STTaT P

P

g

TRHH ln

0

*

H

HTa

P

P T

S

TS

HdTR

g

P

Pd*

0

• Stratosphere Region (HT=11.0 km to 20.0 km). Temperature T = 216.65 : K = TT* is constant (isothermal layer), PST=22632 Pa

KT

kmH

150.288

10

20

30

km11

km20

km32

650.216

K650.228

14

Earth Atmosphere

Physical Foundations of Atmospheric Model

HdHHLTR

gHd

TR

g

P

Pd

SSTaa

*00

PaPHPkmKLHHLTT SSSSSST 5474.9,/0.1*

Integrating this equation we obtain

H

H SSTa

P

P S

S

SS

HdHHLTR

g

P

Pd*

0 1

*

*0 lnln

T

ST

aSSS

S

T

HHLT

RL

g

P

P

Hence aRL

g

S

T

SSSS HH

T

LPP

0

*1

and

10

* g

RL

SS

S

S

TS

aS

P

P

L

THH

Stratosphere Region (HS=20.0 km to 32.0 km).

KT

kmH

150.288

10

20

30

km11

km20

km32

650.216

K650.228

15

Earth Atmosphere

1962 Standard Atmosphere from 86 km to 700 km

Layer IndexGeometricAltitude

km

MolecularTemperature

K

KineticTemperature

K

MolecularWeight

LapseRateK/km

7 86.0 186.946 186.946 28.9644 +1.6481

8100.0 210.65 210.0228.88 +5.0

9110.0 260.65 257.0028.56+10.0

10120.0 360.65 349.4928.08+20.0

11150.0 960.65 892.7926.92+15.0

12160.01110.651022.2026.66+10.0

13170.01210.651103.4026.49 +7.0

14190.01350.651205.4025.85 +5.0

15230.01550.6513223024.70 +4.0

16300.01830.651432.1022.65 +3.3

17400.02160.651487.4019.94 +2.6

18500.02420.651506.1016.84 +1.7

19600.02590.651506.1016.84 +1.1

20700.02700.651507.6016.70

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Earth Atmosphere

1976 Standard Atmosphere from 86 km to 1000 kmGeometric Altitude Range: from 86.0 km to 91.0 km (index 7 – 8)

78

/0.0

TT

kmKZd

Td

Geometric Altitude Range: from 91.0 km to 110.0 km (index 8 – 9)

2/12

8

2

8

2/12

8

1

1

a

ZZ

a

ZZ

a

A

Zd

Td

a

ZZATT C

kma

KA

KTC

9429.19

3232.76

1902.263

Geometric Altitude Range: from 110.0 km to 120.0 km (index 9 – 10)

kmKZd

Td

ZZLTT Z

/0.12

99

Geometric Altitude Range: from 120.0 km to 1000.0 km (index 10 – 11)

ZR

ZRZZ

kmKZR

ZRTT

Zd

Td

TTTT

E

E

E

E

1010

1010

10

/

exp

KT

kmR

km

E

1000

10356766.6

/01875.03

17

Earth Atmosphere

18Central Air Data Computer

Earth Atmosphere

References

SOLO

19

S. Hermelin, “Air Vehicle in Spherical Earth Atmosphere”

Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993

R.P.G. Collinson, “Introduction to Avionics”, Chapman & Hall, Inc., 1996, 1997, 1998

Earth Atmosphere

John D. Anderson, “Flight”, 4th Ed., McGraw Hill, 2000, Ch. 3, “The Standard Atmosphere”

20

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

21

Earth AtmospherePolytropic Process

A Polytropic Process is a Thermodynamic Process that is reversible and obeys the relation:

where P is the pressure, V is the volume, n the Polytropic Index, and C is a constant.

CVP n

The equation is a valid characterization of a thermodynamic process assuming that the process is quasi-static and the values of the heat capacities, Cp and CV , are almost constant when n is not zero or infinity. (In reality, Cp and CV are actually functions of temperature and pressure, but are nearly constant within small changes of temperature).

Polytropic Index

RelationEffects

n = 0P V0 =ConstEquivalent to an isobaric process (constant pressure)

n = 1P V = N k T(constant)

Equivalent to an isothermal process (constant temperature)

1 < n < γ-A quasi-adiabatic process such as internal combustion engine during expansion. Or in vapor compression refrigeration during compression.

n = γ-γ = Cp/CV is the adiabatic index, yielding an adiabatic process(no heat transferred)

n = ∞-Equivalent to an isochoric process (constant volume)

Earth Atmosphere

Explanation of the Thermal Gradient in the Atmospheric Layers by the Polytropic Process

A Polytropic Process is a Thermodynamic Process that is reversible and obeys the relation:

where P is the pressure, V is the volume, n the Polytropic Index, and C is a constant.

nn

nn CV

MCPMCCVP 000

Equation of an Ideal Gas MTRP /*

Take the Logarithmic Differentiation of those two equations

d

P

Pd

n

1

T

Tdd

P

Pd

By eliminating d ρ/ρ we obtain T

Td

n

n

P

Pd

1

We have

gggg Hd

Td

T

P

n

n

Hd

P

T

Td

n

n

Hd

P

P

Pd

Hd

Pd

11

Earth Atmosphere

Explanation of the Thermal Gradient in the Atmospheric Layers by the Polytropic Process (continue – 1)

gg

HgHd

Pd Barometric Equation

gg Hd

Td

T

P

n

n

Hd

Pd

1

*

1

RT

MP

Hd

Pd

Hg gg

Molecular Temperature (TM) versus Kinetic Temperature (T) TM

MTM 0

*

000 11

R

MHg

n

n

Hd

Pd

P

T

M

M

n

n

Hd

Td

M

M

Hd

Td g

ggg

M

*

01

R

MHg

n

n

Hd

Td g

g

M

This equation gives a relation between dTM/dHg , the Polytropic Exponent n and g (Hg).

Earth Atmosphere

Explanation of the Thermal Gradient in the Atmospheric Layers by the Polytropic Process (continue – 2)

*

01

R

MHg

n

n

Hd

Td g

g

M

The Temperature Gradient dTM/dHg , determine the Stability of the Stratification in the Stationary Atmosphere.

The Stratification is more stable when the temperature decrease with increasing height become smaller.

For dTM/dHg = 0 when n = 1, the Atmosphere is Isothermal and has a very stable stratification.

For n = γ = 1.405, the stratification is Adiabatic (Isentropic) with dTM/dHg =-0.98 :K per 100 m [-9.8 : K per km]. This stratification is indiferent because an air volume moving upward for a certain distance cools off through expansion at just the same rate as the temperature drops with height. This air volume maintains the temperature of the ambient air and is, therefore, in an indifferent equilibrium at every altitude.

Earth Atmosphere

Explanation of the Thermal Gradient in the Atmospheric Layers by the Polytropic Process (continue – 3)

Geopotential AltitudeH, km

0,0

11.0

20,0

32.0

47,0

51.0

71.0

84.8520

Thermal Lapse-rateL, 9K/km

-6.5

+0.0

+1.0

+2.8

+0.0

-2.8

-2.0

Polytropic Exponentn

1.2350

1.0000

0.9716

0.9242

1.0000

1.0893

1.0622

Variation of the Polytropic Exponent with Altitude

Earth Atmosphere

Explanation of the Thermal Gradient in the Atmospheric Layers by the Polytropic Process (continue – 4)

*

01

R

MHg

n

n

Hd

Td g

g

M

gg

HgHd

Pd Barometric Equation

Equation of an Ideal Gas MTRP /*

MT

R

HdHg

P

Pd gg

*

gM

ggM

RR

gM

gg

TM

MT

HTR

HdHg

HTR

HdHgM

P

PdM

0

*

0

*

0

gg HdHgHdg 0

Relation between Geopotential Altitude H, Geometric Altitude Hg.

gMgM

gg

HTR

Hdg

HTR

HdHg

P

Pd

0

Earth Atmosphere

Explanation of the Thermal Gradient in the Atmospheric Layers by the Polytropic Process (continue – 5)

Let carry the integration in two cases:

gMgM

gg

HTR

Hdg

HTR

HdHg

P

Pd

0

mxb

HbgRHgRHgHg gEgEgg

/10139.3:

1/21/17

002

0

1. The Non-isothermal layer with LZ:=dT/dHg ≠ 0

2. The Isothermal layer with LZ:=dT/dHg = 0

g

ig iiii

H

H ggZM

ggP

P HHLT

HdHb

R

g

P

Pd 10

iZMM

iZ

ZL

Tb

LR

g

ggM

Z

ii

ZZLTT

ZZLR

bgHH

TR

L

P

P

ii

i

iiZ

iM

iZ

i

i

i

0

1

exp1

0

1

ii

i

MMZiM

i

ii

TTLZZb

TR

ZZg

P

P

&02

1exp 0

2

Earth Atmosphere

29

LayerLevelName

BaseGeopotentialHeighth (in km)

BaseGeometricHeightz (in km)

LapseRate(in °C/km)

BaseTemperatureT (in °C)

BaseAtmosphericPressurep (in Pa)

0Troposphere0.00.0-6.5+15.0101325

1Tropopause11.00011.019+0.0-56.522632

2Stratosphere20.00020.063+1.0-56.55474.9

3Stratosphere32.00032.162+2.8-44.5868.02

4Stratopause47.00047.350+0.0-2.5110.91

5Mesosphere51.00051.413-2.8-2.566.939

6Mesosphere71.00071.802-2.0-58.53.9564

7Mesopause84.85286.000—-86.20.3734

Earth Atmosphere

ICAO_Standard_Atmosphere

30

LayerLevelName

BaseGeopotentialHeighth (in km)

BaseGeometricHeightz (in km)

LapseRate(in °C/km)

BaseTemperatureT (in °C)

BaseAtmosphericPressurep (in Pa)

0Troposphere0.00.0-6.5+15.0101325

1Tropopause11.00011.019+0.0-56.522632

2Stratosphere20.00020.063+1.0-56.55474.9

3Stratosphere32.00032.162+2.8-44.5868.02

4Stratopause47.00047.350+0.0-2.5110.91

5Mesosphere51.00051.413-2.8-2.566.939

6Mesosphere71.00071.802-2.0-58.53.9564

7Mesopause84.85286.000—-86.20.3734

Earth Atmosphere

ICAO_Standard_Atmosphere

31

Earth Atmosphere

There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 feet). The first equation is used when the value of Standard Temperature Lapse Rate is not equal to zero; the second equation is used when standard temperature lapse rate equals zero.

where     = Static pressure (pascals)    = Standard temperature (K)     = Standard temperature lapse rate -0.0065 (K/m) in ISA   = Height above sea level (meters)    = Height at bottom of layer b (meters; e.g.,    = 11,000 meters)     =Universal gas constant for air: 8.31432 N·m /(mol·K)    = Gravitational acceleration (9.80665 m/s2)     = Molar mass of Earth's air (0.0289644 kg/mol)

bLR

Mg

bbb

bb hhLT

TPP

*0

b

bb TR

hhMgPP

*0exp

Equation 1:

Equation 2:

32

Earth Atmosphere

There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 feet).

Subscript b

Height above sea levelhb

Static pressureStandard

temperatureTb

(K)

Temperature lapse rateLb

(m)(ft)(Pascals)(inHg)(K/m)(K/ft)

000101325.0029.92126288.15-0.0065-0.0019812

111,00036,08922632.106.683245216.650.00.0

220,00065,6175474.891.616734216.650.0010.0003048

332,000104,987868.020.2563258228.650.00280.00085344

447,000154,199110.910.0327506270.650.00.0

551,000167,32366.940.01976704270.65-0.0028-0.00085344

671,000232,9403.960.00116833214.65-0.002-0.0006096

33

Earth Atmosphere

34

Earth Atmosphere

Troposphere

Stratosphere

Termosphere

Mesoosphere

Tropopause

Stratopause

Mesopause

35

SOLO

Anders Celsius1701-1744

373

F C R K

672100212 Water Steam Point

Ice Point273492032

Absolute Zero0016.273460

15.273KC

THERMODYNAMICS

Temperature Scales

1. Fahrenheit

2. Celsius

3. Kelvin

4. Rankine

Daniel Gabriel Fahrenheit1686-1736

Mercury-in-GlassThermometer

1714

William John MacquornRankine

(1820-1872)

William Thomson Lord Kelvin(1824-1907)

Absolute Temperature1848

KR

5

9

67.459RF

325

9 CF

SOLO THERMODYNAMICS

Temperature Scales

from Celsiusto Celsius

Fahrenheit[°F] = [°C] × 9⁄5 + 32[°C] = ([°F] − 32) × 5⁄9

Kelvin[K] = [°C] + 273.15[°C] = [K] − 273.15

Rankine[°R] = ([°C] + 273.15) × 9⁄5[°C] = ([°R] − 491.67) × 5⁄9

Delisle[°De] = (100 − [°C]) × 3⁄2[°C] = 100 − [°De] × 2⁄3

Newton[°N] = [°C] × 33⁄100[°C] = [°N] × 100⁄33

Réaumur[°Ré] = [°C] × 4⁄5[°C] = [°Ré] × 5⁄4

Rømer[°Rø] = [°C] × 21⁄40 + 7.5[°C] = ([°Rø] − 7.5) × 40⁄21