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Mike, interesting problem which can be readily solved through calculus,
or even more interestingly using numerical simulation on the computer
(programming required). The fundamentals as already mentioned is one of
buoyancy force v.s. drag force due to the diver moving through the
water column. When the buoyancy force equals the drag force, you have
achieved "terminal velocity." We are ignoring the initial start of
motion.
When buoyancy force = drag force => terminal velocity is achieved
The buoyancy force is:
(Displacement x water density) - (weight of the diver and kit in air)
The drag force is determined by:
Drag force = 0.5 * density * area * (velocity * velocity) * drag
coefficient
There are two primary forms of drag applicable here:
form (body) drag and frictional drag
Sometimes these two drag forces are simplified and combined into an
overall drag (overall drag coefficient). To estimate this coefficient
may I suggest an experiment.
Go to an inlet somewhere where there is a pronounced current and jump
in in your full diving kit, attached to a fixed point (anchored boat)
with a rope and a spring scale. Completely deflate your BC and use just
enough weight so that you will hang horizontal. I am assuming the
victim will ascend head up/feet down. Have your partner in the boat
read the force on the spring scale. Now completely fill your BC and
add enough weight to again hang horizontally completely in the flow.
Again have your partner read the spring scale. Try to do some
intermediate BC volumes and corresponding spring scale forces. Each
time you do this, throw a grapefruit in the water and time how long it
takes to travel a fixed distance. You now have enough information, drag
force v.s. BC volume and water velocity, to solve for a drag
coefficient v.s. BC volume curve. Look up the density of the water and
measure your maximum cross-sectional area at the various BC volumes.
Armed with this information, now go to a swimming pool and determine
the amount of buoyancy your BC has when fully inflated, and empty while
wearing it with the rest of your kit (can be negative). Just add weight
to maintain neutral buoyancy. Do this for various BC inflation
settings.
Now off to the computer. Remember Boyle's law which discussed the
change in volume v.s. pressure. Start your diver, in your program, at
depth and add some air and off he goes. Using programming loops and
small depth increments, you should be able to approximate the terminal
velocity curve along the path. The time to surface is the sum of the
depth increments divided by the calculated terminal velocity during
each increment. Is there error, most definitely. Would it be fun? Most
definitely, assuming you don't croak. Actually if this experiment
was conducted carefully and seriously, it could provide interesting
data regarding equipment streamlining efficiency. I've done this
experiment with ocean engineering students, but not with an object with
a changing buoyancy. I threatened them however with it. Maybe someday!
Doug Chapman
Mike, I'm from Cary originally. I noticed your company is in Raleigh.
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