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Partially filled head volume

Partially filled head volume

Partially filled head volume

I need an equation for calculating the volume of a partially filled ASME F&D head on a vertical tank.  If anyone can help me out I would appreciate it.  Thanks in advance.


RE: Partially filled head volume

I went through my reference books and even managed to dig out (after finding it) some old engineering documents I hoped might have it with no success.  Horizontal, yes.  Vertical, no.

However, the GPSA data book has a tabular set of data for the fractional volume of a vertical head at different liquid depths.  You might want to try curve fitting a set of data points depending on the accuracy you need.

RE: Partially filled head volume

Checkout the thread
Thread770-12414 "Eliptical spheres" in the spreadsheet forum.

This is mostly about filling horizontal vessels, but prex' answer posted 22/10 includes a formula for filling the head of a vertical tank.


RE: Partially filled head volume

Thanks for the replies.  For anyone who may need similar information, I got the following formulas from Chicago Bridge & Iron Co bulletin #594, page 7:

Volume or contents of partially filled hemi-ellipsoidal heads with major axis vertical

Q= Partially filled volume or contents in ft^3
V= Total volume of one head in ft^3
R= Radius of cylinder in ft
delta= a/(KR) where
   a= liquid height
   KR= head height

Upper head   Q=1.5*V*(delta)*(1-(1/3*(delta)^2))
Lower head   Q=1.5*V*(delta)^2*(1-(1/3*(delta))

It probably gives you the same result as the one refrenced above by mgp.

Unfortunately I was unable to find a formula for an ASME F&D head but I was able to fit a equation to curvature of tank in question.  Using this, I can find the volume of a solid of revolution bound between the curve and a horizontal line at the liquid level.  Kind of long but I think it will work close enough.

Once again thanks for effort TD2K and mgp.


RE: Partially filled head volume

Check the reference
"Calculating Tank Volume" by Tim Broyles - Chemical Processing 2000 Fluid Flow Annual p.32

RE: Partially filled head volume

Gone are the days of everyone having the time to use second year university calculus and deriving the formulas from scratch by adding a couple of relatively simple integrals and solving for equation... I must be just too sentimental ...  

RE: Partially filled head volume

Hey Delta, I don't know if those days are gone, or not.  Last year I had to calculate the volume of a weir mounted up on the wall inside a vertical vessel.  The weir was open at the top and trapeziodally-shaped in section.  I had to calculate the volume (this was a skim trough in an IGF flotation cell) because the dump valve worked on a timer and we didn't want to blow a bunch of gas thru the dump valve and into the system.  So I set up a double integral, roated it thru 2pi, and it worked.  My old calc prof would have been proud.  I even did it in the field, in the control room, in the rain, during a startup!  How 'bout that!  

P. J. (Pete) Chandler, PE
Principal Engineer
Mechanical, Piping, Thermal, Hydraulics
Processes Unlimited International, Inc.
Bakersfield, California USA

RE: Partially filled head volume

Ahh, yes!  

RE: Partially filled head volume

The software cerebrotank give the exact parcial volume of vertical and horizontal tanks with any head commonly used.
Get a demo version in http://www.cerebromix.com.

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