Basics of Fluid Mechanics, 2014a

4.3. PRESSURE AND DENSITY IN A GRAVITATIONAL FIELD 87
Separating the variables in equation (4.53) and changing the formal ∂ to the informal
d to obtain
dP
P = − gdh
R (T 0 − C x h)
(4.54)
Defining a new variable 7 as ξ =(T 0 − C x h) for which ξ 0 = T 0 − C x h 0 and d/dξ =
−C x d/dh. Using these definitions results in
dP
P = g dξ
RC x ξ
(4.55)
After the integration of equation (4.54) and reusing (the reverse definitions) the variables
transformed the result into
Or in a more convenient form as
ln P P 0
=
g ln T 0 − C x h
(4.56)
RC x T 0
Pressure in Atmosphere
( )
P T0 − C x h (
g
) RCx
=
P 0
T 0
(4.57)
It can be noticed that equation (4.57) is a monotonous function which decreases with
height because the term in the brackets is less than one. This situation is roughly
representing the pressure in the atmosphere and results in a temperature decrease.
It can be observed that C x has a “double role” which can change the pressure ratio.
Equation (4.57) can be approximated by two approaches/ideas. The first approximation
for a small distance, h, and the second approximation for a small temperature gradient.
It can be recalled that the following expansions are
(
P
= lim 1 − C ) g
RCx
x
h
P 0 h−>0 T 0
= 1 −
ghρ 0
P 0
{ }} {
gh
T 0 R −
correction factor
{ }} {
(
RgCx − g 2) h 2
2 T 0 2 R 2 − ... (4.58)
Equation (4.58) shows that the first two terms are the standard terms (negative sign is
as expected i.e. negative direction). The correction factor occurs only at the third term
which is important for larger heights. It is worth to point out that the above statement
has a qualitative meaning when additional parameter is added. However, this kind of
analysis will be presented in the dimensional analysis chapter 8 .
7 A colleague asked this author to insert this explanation for his students. If you feel that it is too
simple, please, just ignore it.
8 These concepts are very essential in all the thermo–fluid science. I am grateful to my adviser
E.R.G. Eckert who was the pioneer of the dimensional analysis in heat transfer and was kind to show
me some of his ideas.