I have a question that relates to an arrangement of natural gas pipelines but for simplicity I will be describing a less complex problem.
Imagine that I have two large pressure vessels connected by a closed 6” diameter valve. The smaller pressure vessel is 119.23 cu-ft in volume and contains natural gas at 50 degrees Fahrenheit and a pressure of 450.1 psig. The other and much larger pressure vessel also contains natural gas also at 50 degrees Fahrenheit but has a pressure of 587.5 psig. The larger pressure vessel has an unknown volume. At time zero, the valve separating the pressures was cracked open slightly and the pressures begin to equalize. 12 minutes later, the pressure in the smaller vessel is 524.7 psig and the pressure in the larger vessel is 566.9 psig. Based on the data it appears that when the pressure difference across the valve was 137.4 (587.5-450.1) the rise in pressure in the smaller vessel was approximately 5.00 psig/min. After the 12 minutes, with only a pressure difference of 42.2 psig (566.9-524.7) it appeared that the rise in pressure in the smaller vessel was approximately 3.75 psig/min. We know that at if valve were left cracked open the pressure vessels would eventually equalize and with 0 pressure difference across the valve, there would be no flow of gas.
It certainly appears and would seem logical that the greater the difference in pressure across the valve would cause the pressure in the smaller vessel to rise more quickly. As the pressure difference across the valve decreased, so would the rate at which the pressure in the smaller vessel would rise. Similarly, I would assume that there would also be an asymptote that would be approached as the pressure difference were to grow very large. As greater and greater pressure differences across the valve were experienced, the amount of pressure increase per minute in the smaller vessel would be negligibly increasing.
My question is: is there enough information to generate a formula to estimate the amount of gas transferred based on pressure difference? Also, can the amount of gas transferred (cu-ft) in this 12 minute window be calculated?