Using gravity can be economical in some circumstances. It is also very reliable since it's pretty much constant. The place where gravity flow is most often used on a paper machine is the stuff box. The stuff box provides a constant head of pressure to the basis weight valve which meters in the appropriate amount of stock.
There are other places where gravity can be used to provide pressure. In order to utilize gravity one must utilize the Bernoulli equation. One of the few equations I learned in school that I was able to use in industry. Paper machines are constructed in such a way that there are various levels that can be used to provide pressure, from the basement to the mezzanine floor. If you go into enough paper mills you will also learn the term mezzaline, which is the proper way to say mezzanine in some mills. There is usually another floor above the mezzanine which is the deculator floor or some other floor. Not all paper machines have a deculator, but they could have a floor to provide access to dryer hood vents. Of course there is also the roof. I would not recommend putting anything on the roof unless it's relatively small.
I was able to utilize gravity flow in a project that required less fluctuation in a dye flow. The dye had a relatively small flow and the pump supplying the dye had been oversized. I was given a small budget of $20,000. The reason the flow was fluctuating was due to the small orifice size on the control valves and the oversized pump. The valves worked better when they were more open in the 50% range. I calculated the pressure I needed to open the valves in this range for the necessary flow. I was able to salvage a stainless tank and set it on the mezzanine floor, the control valves were in the basement. I used the pump to fill the tank to a level setpoint. I realized that the head pressure would change when the level in the tank changed, but it was within the operating range of the control valves.
The end result was a much happier stock tender. He didn't have to worry about keeping an eye on the dye flow. The valves were stroked regularly during breaks to ensure there was no build up at the orifice. In addition to the pressure reliability, the system did not go down if the pump went down. The tank could be filled manually with totes if necessary which added more reliability to the system. So when working with a limited budget and needing a reliable pressure source, look to gravity. The pressure is only limited by how high the tank or line can go.
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