A tank is a cylinder lying on its side with spherical caps on the ends. I need to find the volume of liquid, in gals, as the tank fills (depth gauge installed). The int dimensions are: Cyl dia = 78", cyl len = 226", depth end cap = 13" (total tank len = 252"). Method for finding the liq vol in the cyl portion is known and not a problem. Also, the formula for computing the volume capacity of a spherical cap is known. What I can't find is a formula giving the vol of liq in the cap as it fills. The formula must be coded in a PLC, which leaves out complex code such as Simpson integral code, etc.

Question

anonymous · Answer

$$V(h) = \frac{ \pi }{ 2}(rh ^{2}-\frac{ h ^{3} }{ 3 }) $$
for a cap

anonymous · Answer

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anonymous · Answer

logging off soon...

mccannon01 · Answer

Thank you, Algebraic!. However, the formula you offer gives 1/2 the volume of a cap. As I mentioned in my post finding the volume of a spherical cap is already known to me. Here are two formulas:
$$V = \frac{ 1 }{ 6 } \pi h(3a ^{2}+h ^{2})$$  where h = cap height and a = cap disk radius
$$V = \pi(3h ^{2}r-h ^{3})/3$$  where h = cap height and r = radius of the cap's parent sphere
What I'm trying to find is a formula for this volume:
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