I have a case where I have an inner tube within an outer tube. The tubes can be anything from concentric to fully eccentric with the inner tube laying on the bottom of the outer tube. I want to calculate the height of fluid in the annulus knowing the volume. I know can calculate the area (and volume by extension) for each part using the following equations for the fully eccentric case:
$$A_o = cos^{-1}((r_o-h)/r_0)\cdot r_o^2 - (r_o-h)\sqrt{2r_oh-h^2}$$ $$A_i = cos^{-1}((r_i-h)/r_i)\cdot r_i^2 - (r_i-h)\sqrt{2r_ih-h^2}$$ $$V = (A_o-A_i)\cdot L$$
Where:
$r_o$ is the radius larger cylinder internal diameter
$r_i$ is the radius if the inner cylinder
$h$ is the height of the fluid
$A_o$ is the area outer
$A_i$ is the area inner
$L$ is the length of the cylinder
$V$ is the volume of fluid in the annulus
What I would like to do is solve the equation for h so I can directly calculate it.
Currently, I am solving each equation separately in excel and taking the difference. I build a table for incremental heights and then use the table to lookup the height for a given volume. It works but it seems like solving the equations would quicker. The math is beyond me though. Any ideas would be helpful.
Thanks!
Hogwildwa
