(41) Bulk Cylindrical Tank

[Thomas Klemm]

(10-13-2018 01:41 PM)David Hayden Wrote:  I understand the formulas. It's the HP41C program at the end of the paper that I'm having trouble with.

There's a typo in formula (25-11): the square on \(v_0\) is missing.
Instead it should be:

\(V_{seg\ sphere} = \pi v_0^2(3R - v_0)/693\)

Cf. Spherical Cap

This formula is then used twice. Once with \(h=v_0\) and then again with \(h=v_0-v\). The difference between them is formula (25-12).
In both cases the radius \(R\) of the sphere is used.

But we can do without: Spherical Segment

\(V=\tfrac{1}{6}\pi h(3a^2+3b^2+h^2)\)

We know \(a=r\) and \(h=v\) but we have to calculate \(b\).

Quote:The hatch depth is the distance from the lower lip of the hatch to the edge of the tank.

I assume that this is the horizontal distance from the hatch to the circumference of the cylinder.

In lines 44-48 the distance \(v\) from the upper border of the cylinder to the juice level is calculated.
In lines 49-53 this value is then multiplied by the hatch depth and divided by the hatch height.
We could interpret this as an approximation of the difference \(\Delta r = r - b\).
At least something like this is calculated in lines 56-58.

However there are these strange X≠0? commands and I assume that the X<>Y command in line 55 is a bug.
Or then it should rather be X=0? instead of X≠0? in the line prior to that.

Lines 59-68 calculate \(r^3-b^3=(r-b)(r^2+rb+b^2)=\Delta r(r^2+rb+b^2)\).
In line 69-70 this is multiplied by the hatch height and then in lines 75-77 it's divided again by the hatch depth.

Thus we have a somewhat convoluted way to calculate \(v(r^2+rb+b^2)\).
The third term \(\tfrac{1}{3}v^2\) is missing but we might consider \(v\) rather small compared to \(r\) and \(b\) and thus neglect the term.
But there's still the mixed term \(rb\). What should we do with that?

Lines 80-98 just calculate the rest of the tank.

Maybe I could shed some light on that program and the formulas that were used but I must admit that I still don't fully understand it.

Kind regards
Thomas