the original thread was more to do with flow of 3phase fluids, however you seem to ask about only the static conditions of the question concerning the diagram. so ...
from the original thread.
Two identical pipes (same ID and length), same elevation, each of them has two phases in them. They are both static, and they are both pressurized. The gas and oil content (by volume) are not the same in each pipeline, but the gas density is the same in pipe one as pipe two and the oil density is the same in pipe one as in pipe two. The temperatures are also the same.
Summary,
Two containers are indentical in all respects.
They are both static (no motion of container or fluids within).
They have pressure.
Oil and gas volumes in each are different.
Gas and oil densities in each pipe are equal, as well as their temperature.
Is the pressure the same in each pipe (pressure is a function of the pipe surface area) or different (pipe pressure a function of the surface area of the gas)?
Answers:
Is the pressure the same in each pipe?
Yes. The pressure is the same.
From
Avogadro’s law, equal volumes of gases at the same temperature and pressure contain equal number of moles or molecules. If the molar amount is doubled, the volume is doubled.
The pressure of the gas, from the ideal gas law, is
P = nRT/V
Density of a substance is calculated from its mass/volume, n is the number of moles, in the ideal gas law equation, where it also appears the volume, hence, the all the relationships are directly proportional to density.
We now can see that, if density of each gas is equal, the temperature of each gas is equal, then the ratio of pressures to volume is constant, so if the volume of gas in each pipeline is different, then the mass of the gas within each must also vary accordingly, the amount of which varies directly with the volume of the gas in each pipeline, which is the amount needed to quarantee that the pressures are equal. It is much the same for the oil, except the volume of oil does not very much with pressure, so we will ignore that. It can vary with temperature, but since temperatures are the same, the oil density is the same, so the amount of oil is directly proportional to the volume of oil, which we just said isn't much to do with pressure at all. The amount of oil is irrelevant.
([Is] pressure is a function of the pipe surface area) or different (pipe pressure a function of the surface area of the gas)?
Surface areas of pipe, gas or oil have nothing to do with the pressure, except as how surface areas might define the volume of the gas. Really to determine the pressure we only care about the volume of gas, the number of moles of gas and the temperature of the gas.
“What I told you was true ... from a certain point of view.” - Obi-Wan Kenobi, "Return of the Jedi"
