ae 1350 lecture #4 previously covered topics preliminary thoughts on aerospace design specifications...
TRANSCRIPT
AE 1350Lecture #4
PREVIOUSLY COVERED TOPICS
• 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.
• For example, from last lecture,
• 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, .
SV
LCL
2
21
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 orEquation of State
• Most gases satisfy the following relationship between density, temperature and pressure:
• p = RT– p = Pressure (in lb/ft2 or N/m2)– = “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:
• • where,
– a = speed of Sound (m/s or ft/s) = Ratio of Specific Heats = 1.4
– R = Gas Constant– T = temperature (in degrees K or degrees R)
RTa
Temperature vs. Altitude
Temperature, degrees K
Altit
ude,
km
288.16 K
11km216.66K
25 km
47 km, T= 282.66 K
53 km
79 km165.66 K
90 km
TroposphereStratosphere
Pressure varies with Height
The bottom layers have to carry more weight than those at the top
Consider a Column of Air of Height dhIts area of cross section is A
Let dp be the change in pressure between top and the bottom
Pressure at the top = (p+dp)
Pressure at the bottom = p
dh
Forces acting on this Column of Air
Force = Pressure times Area = (p+dp)A
Force = p A
Weight of air= gA dh
dh
Force Balance
Force = (p+dp)A
Force = p A
gA dh
Downward directed force= Upward force(p+dp)A + g A dh = pA
Simplify:
dp = - g dh
Variation of p with T
dp = - g dh
Use Ideal Gas Law (also called Equation of State):
p = R T = 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 Regionswhere 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.
Let us assume T = T1 +a (h-h1)
The slope ‘a’ is called a Lapse Rate.
h
h=h1
T=T1
Variation of p with T when T varies linearly (Continued..)
From previous slide, T = T1 +a (h-h1)An infinitesimal change in Temperature 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)
aR
g
T
T
p
p
11
loglog aR
g
T
T
p
p
11
Variation of with T when T varies linearly
From the previous slide, in regions where temperature varieslinearly, we get:
aR
g
T
T
p
p
11
Using p = RT and p1 = 1RT1, we can show that density varies as:
1
11
aR
g
T
T
Variation of p with altitude hin regions where T is constant
In some regions, for example between 11 km and 25 km, thetemperature 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 Equation 1
Integrate: log p = - g/(RT) h + C
Variation of p with altitude hin 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, 1
1
hhRT
g
ep
p
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.