Empirical Equations
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A common equation for volume flow out of a tank (in the good old pre metric days) is
[math]\displaystyle{ Q = 4.90 A \sqrt h }[/math]
where [math]\displaystyle{ h }[/math] is the depth of the tank in feet and [math]\displaystyle{ A }[/math] is the pipe area.
This equation makes no sense if you try and do dimensional analysis since it gives dimension [math]\displaystyle{ [L^{5/2}] }[/math] for the right hand side and [math]\displaystyle{ [L^3T^{-1}] }[/math] for the left hand side and we might conclude: This cannot be correct
However many generations of engineers successfully used this for designing tanks without worrying about dimensional analysis. (Their main concern was minimizing floating point multiplications, which required getting out the slide rule)
Of course the equation makes perfect sense when you understand the 'constant' 4.90 has dimensions [math]\displaystyle{ L^{1/2}T^{-1} }[/math] which makes everything work out.
In the current enlightened day and age we would probably nondimensionalise everything and write
[math]\displaystyle{ Q = .61A \sqrt {2gh} }[/math]
which would apply in the any system of units, but that doesn't make the old equation invalid. And the old equation can be evaluated with just two floating points multiplies plus a square root.
- The word empirical denotes information gained by means of observation, experience, or experiment, as opposed to theoretical.