Basics of Fluid Mechanics, 2014a

4.3. PRESSURE AND DENSITY IN A GRAVITATIONAL FIELD 91
4.3.5.1 Ideal Gas in Varying Gravity
In physics, it was explained that the gravity is a function of the distance from the center
of the plant/body. Assuming that the pressure is affected by this gravity/body force.
The gravity force is reversely proportional to r 2 . The gravity force can be assumed that
for infinity, r →∞the pressure is about zero. Again, equation (4.11) can be used
(semi one directional situation) when r is used as direction and thus
∂P
∂r = −ρ G r 2 (4.69)
where G denotes the general gravity constant. The regular method of separation is
employed to obtain
∫ P
P b
dP
P
= − G RT
∫ r
r b
dr
r 2 (4.70)
where the subscript b denotes the conditions at the body surface. The integration of
equation (4.70) results in
ln P = − G ( 1
− 1 )
(4.71)
P b RT r b r
Or in a simplified form as
ρ
= P =
ρ b P b
e − G r−r b
RT rr b (4.72)
Equation (4.72) demonstrates that the pressure is reduced with the distance. It can be
noticed that for r → r b the pressure is approaching P → P b . This equation confirms
that the density in outer space is zero ρ(∞) =0. As before, equation (4.72) can be
expanded in Taylor series as
ρ
ρ b
= P P b
=
standard
{ }} {
1 − G (r − r b)
RT
−
correction factor
{
(
}}
)
{
2 GRT + G 2 r b (r − rb ) 2
2 r b (RT) 2 + ...
(4.73)
Notice that G isn’t our beloved and familiar g and also that Gr b /RT is a dimensionless
number (later in dimensionless chapter about it and its meaning).
4.3.5.2 Real Gas in Varying Gravity
The regular assumption of constant compressibility, Z, is employed. It has to remember
when this assumption isn’t accurate enough, numerical integration is a possible solution.
Thus, equation (4.70) is transformed into
∫ P
P b
dP
P = − G ∫ r
dr
ZRT r b
r 2 (4.74)