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.