AE 1350
Lecture #4
PREVIOUSLY COVERED
TOPICS
•
•
•
•
•
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Preliminary Thoughts on Aerospace Design
Specifications (“Specs”) and Standards
System Integration
Forces acting on an Aircraft
The Nature of Aerodynamic Forces
Lift and Drag Coefficients
TOPICS TO BE COVERED
• Why should we study properties of
atmosphere?
• Ideal Gas Law
• Variation of Temperature with Altitude
• Variation of Pressure with Altitude
• Variation of Density with Altitude
• Tables of Standard Atmosphere
Why should we study
Atmospheric Properties
• Engineers design flight vehicles, turbine engines and
rockets that will operate at various altitudes.
• They can not design these unless the atmospheric
characteristics are not known. C  L
L
1
• For example, from last lecture,
rV2 S
2
• We can not design a vehicle that will operate
satisfactorily and generate the required lift
coefficient CL until we know the density of the
atmosphere, r.
What is a standard atmosphere?
• Weather conditions vary around the globe, from
day to day.
• Taking all these variations into design is
impractical.
• A standard atmosphere is therefore defined, that
relates fight tests, wind tunnel tests and general
airplane design to a common reference.
• This common reference is called a “standard”
atmosphere.
International Standard Atmosphere
Standard Sea Level Conditions
Pressure
101325 Pa
2116.7 lbf/ft2
Density
1,225 Kg/m3
0.002378 slug/ft3
Temperature
15 oC or 288 K
59 oF or 518.4 oR
Ideal Gas Law or
Equation of State
• Most gases satisfy the following relationship
between density, temperature and pressure:
• p = rRT
–
–
–
–
–
p = Pressure (in lb/ft2 or N/m2)
r = “Rho” , density (in slugs/ft3 or kg/m3)
T = Temperature (in Degrees R or degrees K)
R = Gas Constant, varies from one gas to another.
Equals 287.1 J/Kg/K or 1715.7 ft lbf/slug/oR for air
Speed of Sound
• From thermodynamics, and compressible
flow theory you will study later in your
career, sound travels at the following speed:
•
a  gRT
• where,
– a = speed of Sound (m/s or ft/s)
g = Ratio of Specific Heats = 1.4
– R = Gas Constant
– T = temperature (in degrees K or degrees R)
Temperature vs. Altitude
90 km
79 km
165.66 K
Altitude, km
53 km
47 km, T= 282.66 K
25 km
11km
216.66K
Stratosphere
Troposphere
288.16 K
Temperature, degrees K
Pressure varies with Height
The bottom layers have to carry more weight than
those at the top
Consider a Column of Air of Height dh
Its area of cross section is A
Let dp be the change in pressure between top
and the bottom
Pressure at the top = (p+dp)
dh
Pressure at the bottom = p
Forces acting on this
Column of Air
Force = Pressure times Area = (p+dp)A
dh
Weight of air= r g A dh
Force = p A
Force Balance
Force = (p+dp)A
Downward directed force= Upward force
(p+dp)A + r g A dh = pA
r gA dh
Simplify:
dp = - r g dh
Force = p A
Variation of p with T
dp = - r g dh
Use Ideal Gas Law (also called Equation of State):
p=rRT
r = p/(RT)
dp = - p / (RT) g dh
dp/p = - g/(RT) dh
Equation 1
This equation holds both in regions where temperature varies,
and in regions where temperature is constant.
Variation of p with T in Regions
where T varies linearly with height
From the previous slide,
dp/p = - g/(RT) dh
Equation 1
Because T is a discontinuous function of h (i.e. has breaks in its shape),
we can not integrate the above equation for the entire atmosphere.
We will have to do it one region at a time.
In the regions (troposphere, stratosphere), T varies with h linearly.
h
Let us assume T = T1 +a (h-h1)
The slope ‘a’ is called a Lapse Rate.
h=h1
T=T1
Variation of p with T when T varies linearly
(Continued..)
From previous slide,
An infinitesimal change in Temperature
T = T1 +a (h-h1)
dT = a dh
Use this in equation 1 :
dp/p = - g/(RT) dh
We get:
dp/p = -g/(aR)dT/T
Integrate. Use integral of dx/x = log x.
Log p = -(g/aR) log T + C
Equation 2
where C is a constant of integration.
Somewhere on the region, let h = h1 , p=p1 and T = T1
Log p1 = -(g/aR) log T1 + C
Equation 3
Variation of p with T when T varies linearly
(Continued..)
Subtract equation (3) from Equation (2):
log p - log p1 = - g/(aR) [log T - log T1]
log (p/p1)
= - g / (aR) log ( T/T1)
Use m log x
= log (xm)

log 

g



aR
T  
p

  log  
 T1  
p1 


 p  T 
    
 p1   T1 

g
aR
Variation of r with T when T varies linearly
From the previous slide, in regions where temperature varies
linearly, we get:
 p  T 
    
 p1   T1 

g
aR
Using p = rRT and p1 = r1RT1, we can show that density varies as:
 r  T 
    
 r1   T1 
 g


1 
 aR 
Variation of p with altitude h
in regions where T is constant
In some regions, for example between 11 km and 25 km, the
temperature of standard atmosphere is constant.
How can we find the variation of p with h in this region?
We start again with equation 1.
dp/p = - g/(RT) dh
Integrate: log p = - g/(RT) h + C
Equation 1
Variation of p with altitude h
in regions where T is constant (Continued..)
From the previous slide, in these regions p varies with h as:
log p = -g /(RT) h + C
At some height h1, we assume p is known and his given by p1.
Log p1 = - g/(RT) h1 + C
Subtract the above two relations from one another:
log (p/p1) = -g/(RT) (h-h1)
Or,
g

 h  h1 
p
RT
e
p1
Concluding Remarks
• Variation of temperature, density and
pressure with altitude can be computed for a
standard atmosphere.
• These properties may be tabulated.
• Short programs called applets exist on the
world wide web for computing atmospheric
properties.
• Study worked out examples to be done in
the class.