Aluminum liquid flow under pressure question

[thsiao]

Please look at the photo. In the photo, is a drawing of a casting furnace, with melted aluminum at the level of where the door starts. Assume that we have 2 different riser tubes inside this casting furnace, one with an inner diameter of 1.5" and the other one with an inner diameter of 2.5"... and assume we start to apply pressure into the casting furnace, causing the melted aluminum to start rising in the riser tube. At any given time, which riser tube will have a higher level of melted aluminum inside? In other words, which riser tube will start to fill up higher?

My common sense leads me to believe that the one with the smaller diameter since the applied pressure is the same (assume 14 psi).

Is there a formula/equation/law that can prove this?


Thanks!

 

[Mapes]

Hi thsiao, welcome to PF. Ignoring the small effect of surface tension, the level will be the same in both tubes. The pressure under the surface of a fluid is the product of its density, the acceleration of gravity, and the depth. In other words, the pressure is not a function of width, and so the height of a pressurized fluid, as specified in your schematic, will not be a function of tube width.

 

[thsiao]

Hi thsiao, welcome to PF. Ignoring the small effect of surface tension, the level will be the same in both tubes. The pressure under the surface of a fluid is the product of its density, the acceleration of gravity, and the depth. In other words, the pressure is not a function of width, and so the height of a pressurized fluid, as specified in your schematic, will not be a function of tube width.

But assuming that the same psi pressure is used, then isn't that equal to the same volume trying to be pushed out of the tubes? So if the inner diameter is smaller (smaller area) then the same volume should be causing it to rise faster since the area is smaller?

In other words, if the same volume of fluid is trying to be pushed out of these 2 different tubes, and given that one has a smaller inner area... wouldn't the one with the smaller inner area have the fluid higher since its trying to accommodate for the same volume?

Kinda like if you hook up a thin hose vs. a thicker hose to a water outlet, the thin hose will release the water faster (though slower than the thicker hose).

 

[Mapes]

It's not equivalent to the same volume. Only the total volume of aluminum is conserved, not the volume of aluminum per tube.

 

[thsiao]

It's not equivalent to the same volume. Only the total volume of aluminum is conserved, not the volume of aluminum per tube.

From what I've read online about the Bernoulli's Principle...

The volume traveling across these tubes would be the same as the applied pressure is the same. What will vary is the velocity in which the liquid travels through the tube. The thinner tube will have a higher velocity and its inner pressure will be lower, and the thicker tube will have a lower velocity but its inner pressure will be higher.

So though the final volume ejected will be the same, the thinner tube will have metal higher and the thicker tube initially and the velocity from the thinner tube will be higher.

Would this be a right assumption?
__________________

 

[FredGarvin]

Since this is a static condition, you are wrong. The pressure is all that matters. The cross sectional area of the tubes cancel out. Do a FBD of the tubes and you should see that the pressure of the aluminum bath has to be counteracted by the pressure [tex]p=(\rho)(g)(h)/tex

Look up how a manometer works and you'll see that the tube design means nothing in this case. In basic fluids classes, you'll see an apparatus along the lines of this to demonstrate this very fact:
http://www.kbescientific.com.sg/Image885.gif

 

[GT1]

The smaller diameter tube has higher hydraulic resistance - does it have any effect on the height?

 

[Mapes]

The smaller diameter tube has higher hydraulic resistance - does it have any effect on the height?

It would depend on how fast the liquid is rising. I assumed that the speed is <<1 m/s and that the system can be treated as one at steady state, but the original poster will have to confirm.