Calculation of the amount of pumps needed
As previously mentioned the additional amount of pumps needed per length is proportion to the force of gravity. We can neglect centripetal acceleration because it is to small to make much difference. I claim this can be written mathematically as:
dN/dh=(1/k)*/(P*g/rho)/ (P_o*g_o/rho_o)^2
Where:
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rho is the density of the gas on the high pressure side of the pump
P is the pressure of the gas on the high pressure side of the pump
g is the force of gravity at the pump
G is the universal gravitational constant
M_E is the mass of the earth
R is the radius of the earth
r is the distance from the center of the earth
h is altitude
N is the number of pumps.
k is the number of attenuation constants between pumps. The fraction of gas remaining is given by e^(-k)
k=1 gives 0.3679, k=5 gives 0.0067, k=10 gives 4.5400e-005
k=1 seems the most practical.
From: http://www.elmhurst.edu/~chm/vchembook/123Adensitygas.html
Methane Data
Here are some densities:
Densities of Common Elements and Compounds
(Substance Density kg/m^3)
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Hydrogen gas 0.000089e3
Helium gas 0.00018e3
Air 0.00128e3
Carbon Dioxide 0.001977e3
Water 1.00e3
Methane 0.0006557e3
The calculations will be done for air. Notice that methane is lighter then air.
To find the number of pumps needed we integrate the above expression from the radius of the Earth to GEO.
N=(1/k)int(P*g/rho),/(P_o*g_o/rho_o)^2, h=0…36e9)
Not that (P_o*g_o/rho_o) is the distance over which the pressure drops by 1/e.
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=1000 Pa * 9.8 m/s^2/0.00128e3 kg/m^2=7.6563e+003 m for air.
In the calculation below we will use 6.92105e3 instead of 7.6563e+003 so are integral agrees exatly at the first pump which will be at 7.6563e+003.
Which is about 7.7 km. If the pressure at the high pressure side of each side is the same the expression for N becomes
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N=(1/(k*6.92105e+003))int(g/g_o, h=0…36e6)= (1/k)int(G M_E/(h+R)^2/g_o, h=0…36e6) /0.3013
= (1/k*6.92105e+003)*int(6.67e-11 * 5.98e24/h^2/9.9, h=6e6…42e6)
=(1/k)* (-6.67e-11 * 5.98e24)/(9.9*6.92105e+003)*((1/42e6)-(1/6.38e6))=1000/k
Now to get it for others we can do this trick
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For hydrogen:
1000*(density of hydrogen/density of air)
= 1000*0.000089e3/0.00128e3=69.5313
For methane:
773.8850*0.0006557e3/0.00128e3=512.66
So that is 70 pumps for hydrogen, 513 pumps for methane and 1000 pumps for air. The pumps start out being spaced 7 km apart and get further apart as the altitude increases. Also note that when the pumps get further apart the tubes must get wider to keep the viscous forces the same.