Lecture #2
Today’s Lecture
• Questions on Lecture #1 (Flight # 1)
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ASE 167M Lecture 2
The Atmosphere
Computer #1
Report # 1?
Flight Data # 1 ?
Theory ?
Curve Fitting ?
Etc ?
Auto-pilots?
• Atmosphere properties
• Atmosphere Toolbox (Matlab): Computer Project # 1
Revised by
Greg Holt
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Lecture #2
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Lecture #2
The Atmosphere
The Atmosphere
• Necessity for studying atmosphere conditions
– Performance of A/C completely dependent of the atmosphere conditions
• The atmosphere is the gaseous medium enveloping the
Earth
• The properties of the atmosphere are a function of time
and position (i.e. latitude, longitude, and altitude).
• These properties are nonlinear and deterministic but are
nonetheless unpredictable because of the scale and
interconnectivity of global meteorology (the Butterfly
effect).
– “The flapping of a butterfly's wings in China could cause tiny atmospheric
changes which over a period of time could effect weather patterns in New York.”
⇒ This models the unpredictability of local weather patterns
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Lecture #2
The Atmosphere (2)
The Atmosphere (3)
some information
Layers of the atmosphere
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50% of mass is within 5 km altitude
99% of mass is within 30 km altitude
Mass of atmosphere is 5x1015 tons
Mass of atmosphere ≈ 1 total mass of
300
water
• 78% nitrogen, 21% oxygen,
0.93% argon, 0.031% CO2
• Troposphere - sea level to 28,000/55,000 ft
(11 Km ) (most airplane/jet airlines)
– Contains 75% of our atmosphere’s mass
– Height varies from poles to equator
~30000 ft at poles
~36000 ft at equator
! Tropopause: top layer of the Troposphere. It is the boundary
region between Troposphere and Stratosphere
• Stratosphere - up to approx. 50 miles
– Contains ozone layer
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The Atmosphere (5)
The Atmosphere (4)
more cool information
• Mesosphere
• Troposphere:
– Middle Zone (out of 5) – “Upper Troposphere” regarding
temperature profile, but has a series of intense photochemical
reactions (dissociations and recombination)
tropos " Greek for turning
• Stratosphere:
• Ionosphere - up to approx. 650 miles
– Air moves only horizontally (jetstream)
– solar radiation strips electrons from oxygen and nitrogen
molecules
– Important for communications
• Ionosphere:
– Air particles are ionized by sun’s ultraviolet radiation
• Exosphere - out to “∞”
• Exosphere:
– Boundary region between the Earth’s atmosphere and the
interplanetary space
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– outside
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Lecture #2
Lecture #2
Temperature (T)
Temperature (T)
• Temperature varies linearly
with altitude. The rate of this
variation is called the
Temperature lapse rate
• What causes most of the planet’s thermal
effects?
Solar energy
– Dry adiabatic lapse rate = 10º C
per 1000 m (1 km)
– Wet adiabatic lapse rate = 6º C
per 1000 m (1 km)
– Difference is due to latent heat of
evaporation; due to
condensation, energy is
released which reheats the air
parcel to some extent # slower
cooling
Temperature Scales
Rankine: R = F + 459.68
Kelvin: K = C + 273.15
Celsius: C = 5/9 * ( F - 32 )
Fahrenheit: F = 9/5*C + 32
Celsius used freezing and boiling point of water
Fahrenheit ? (see http://www.straightdope.com/classics/a891215.html
http://www.crh.noaa.gov/pub/temp2.htm )
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315
250
1 km = 3281 ft
βi= lapse rate
β4
β3
186
Isothermal Regions
130
β2
82
65
36
300
Tropopause
β1
400
390
500
518
0
R
600
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Lecture #2
Pressure (P) and Density (ρ)
The Standard Atmosphere
• Pressure is the effect felt from the weight of the
atmosphere (force/area) and is continually
changing due to air movement (sloshing) and
temperature changes.
– Recall that altitude is measured by a pressure
reading.
• Density is the mass of air particles per unit
volume and is a function of T, P, and humidity.
– It is the most important quantity in A/C Performance
(Engine Thrust, Airspeed Indicator, CD and CL, etc)
– Water vapor may account for as much as 5% of
density by volume.
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h
(1000 ft)
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• It is impractical to consider all atmospheric
property variations for design and
performance of aircraft.
• The standard atmosphere is defined to
relate flight tests, wind tunnel tests,
general airplane design, and performance
to a common reference.
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Atmospheric Model Design
Considerations
The Standard Atmosphere (2)
• Many models available
• This definition represents average conditions
derived from math models and experimental
data:
– U.S. Standard Atmosphere (I & II)
– CIRA, MSIS, Harris-Priester, Jacchia-Roberts, …
– Weather balloons
– Sounding rockets
• For design and analysis we want P(h), T(h), ρ(h),
and a(h) ⇒ Functions of altitude only!
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• The atmospheric model is typically designed to
be representative of a certain operation point (i.e.
.78 M at 35,000 ft for an airliner).
• Designers must consider weather for
performance, flight planning, and hazard
assessment.
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Derivation of Standard Atmosphere
Derivation of Standard Atmosphere
• Recall equation of state for a perfect gas
(neglecting intermolecular forces)
• Method:
– Use the hydrostatic equation (the force
balance on a fluid at rest) to get a free body
diagram
– Develop differential equation of pressure
w.r.t. altitude
– Solve this DE for differing temperature
profiles of the layers of the atmosphere
P = ρRT
– R is the specific gas constant. Value depends upon
the gas and system of units and is derived from
experimental data
• R= 287 m2/s2*K ⇒ for dry air
• R = 459.2 m2/s2*K ⇒ in the presence of water vapor
• R= 1716 ft*lb/(slug*R)
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The Hydrostatic Equation
Temperature Profile
h
(1000 ft)
W = ρVg = ρ (dh) g
P+dP
Ftop = ( P + dP )(1)(1)
315
Fbot = ( P )(1)(1)
1
Fbot = Ftop + W
1
β3
1
°R
=
slope ft
186
or
Isothermal Regions
W
β2
130
P = ( P + dP ) + ρdhg
82
dP = − ρgdh
65
P
dP −g
=
dh
P RT
36
•Solved by numerical integration
•Differing solutions for T(h)
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Tropopause
β1
400
390
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518
0
R
600
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Lecture #2
Isothermal Solution to D.E.
Varying Temperature Solution to D.E.
dP −g
=
dh
P RT
• Recall D.E.
dP
• T(h) = T1
dP −g h
∫ P = RT ∫ dh
P
h
1
−g
• Recall D.E. P = RT dh
• Temperature varies as T(h) = T1+β(h-h1)
P
dT
1
= const = β ⇒ dh = dT
β
dh
1
• Substitute T and dT into D.E.
 P  −g
ln  =
(h − h1 ) ⇒ P(h ) = P1e
 P1  RT
 g 
−
 (h− h1 )
 RT1 
 g 
dP −g dT
=
∫
P Rβ T T
P
• Solve D.E.
• Use the equation of state to get density
P
T
1
1
∫
 g 
ASE167M Lecture 2
 P  −g  T 
 T  −  R β 
ln  =
ln  ⇒ P(T ) = P1  
 P1  Rβ  T1 
 T1 
−g
 g

 T  −  Rβ +1
P
ρT  T  βR
=
=   ⇒ ρ(T ) = ρ1 
P1 ρ1T1  T1 
 T1 
−
 (h− h1 )
P
ρRT
RT
=
⇒ ρ (h) = ρ1e  1 
P1 ρ1 RT1
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=
250
dh
1 km = 3281 ft
βi= lapse rate
β4
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Lecture #2
Lecture #2
Speed of Sound and Viscosity
Initial Conditions for First Layer
• First layer is assumed to begin at sea level
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• Speed of the sound
P0 = 2116.2 lb/ft2
T0 = 518.69 0R (59 F)
ρ0 = 2.377x10-3 slug/ft3
Values repeated in lab manual
a (T ) = kRT
– k = 1.4 is the specific heat ratio for air
– Note that a(T) = a(T(h)) = a(h)
• We assume sea level conditions for our simulator
flights.
• Viscosity
µ = 2.27 ×10-8 T 3/2 (T + 198.6)-1
– Note again that dependence on T implies
dependence on h.
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Lecture #2
Atmospheric properties variation
Pressure, Density, and Temp. Altitudes
• Solutions of the
diff. equations for
the Standard
Atmosphere
• We can define pressure, temperature, and
density altitudes since each property in the
standard atmosphere corresponds to a
given h.
– Note that you must keep track of layer
changes since there is not a one to one
mapping between h and T.
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Lecture #2
Lecture #2
Computer Projects - Class
Administration
Pressure, Density, and Temp. Altitudes
• Behavior of aircraft depends on density
altitude.
• Altimeter depends on pressure altitude.
h
h(P1)
h(P1)
• Pick any programming language
– MATLAB
– Fortran, C, C++, Visual Basic, …
– Available at LRC
P1=const.
• Individual computer assignment
• There is no specific time for this “lab”
P2=const.
Same h reading in a/c
P3=const.
x
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Lecture #2
Computer Project 1 (1)
Computer Project 1 (2)
• 1) Write toolbox subroutine to calculate T(h),
P(h), ρ(h), a(h)
• Plot each property vs. h using your graphics
package of choice. Do NOT give reams of
numerical output or a single altitude reading per
sheet. A condensed table will suffice.
– Formula for this routine are in lab manual
– Make a function [Pres,Temp,dens,sos] = atmos(alt)
• Several if statements
Altitude (h)
Properties
T,P,sos,dens
• 2) Main Program – for test cases and problems
• Loop over altitude h in 1000 ft increments
• Display regions of interest (changing values) if
2000000 ft. is too much – You decide!!!!
– Call atmos(h,P,T,dens,sos)
• End loop
• Test cases from manual – COMPARE WITH TABLE
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Lecture #2
Lecture #2
Computer Project 1 (3)
•
•
Answer questions 3 and 4
Calculate drag on an orbiter at 160 N.M.
circular orbit
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•
Computer Project 1 (4)
µ = 1.40765x1016 ft3/s2
rc = re + h,
re = 2.09257x107 ft
1 N.M. = 1852/0.3048 ft
S = 5200 sq. ft. and 400 sq. ft.
CD = 2.0
Vc =
D=
• Reports:
µ
rc
1
ρV 2 SCD
2
– USE FORMAT FROM WEBPAGE…
– DUE IN TWO WEEKS
Calculate the drag on an aircraft flying at 400
knots at 30,000 ft with CD = .05 and S = 600 sq.
ft.
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Lecture #2
This Week
• Computer Program
• No Lab class
Next Week:
• Flight # 1 report due – 6pm
• Equations of motion / trim
• Flight # 2 Briefing
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