How to calculate average gas lifting force in water?

Question

Let say at you have 10 litres of some gas (air, hydrogen, oxygen..) in 100 meter under water it can lift about 10kg

when it goes 50m up it can expand to 20 litres so now you have 20 kg lifting force. When it is just 10m deep it is expand 500 litres and can lift 500kg. but how to calculate average lifting force? (example for power calculations)

Answer 1

The average lifting force will be the sum of the lifting force at the original starting depth, plus the lifting force at the surface, divided by a factor of two.

The trick is to calculate the lifting force at the surface, which cannot be infinity, which is where you are going with your explanation of how the lifting force is being multiplied as the gas ascends to the surface.

I believe you will have to know the volume of the gas at the surface in order to calculate its displacement, or lifting force at the surface.

Answer 2

It sounds like your refering to the buoyancy force (Fbuoy=rho*V*g).  This will be counterbalanced by the weight of the object.  

If your object is in water you can consider the fluid incompressible, in other liquids the density may be some function of height.  If the internal gas is compressible, volume can be solved using the ideal gas law where pressure and temperature are some function of the depth of the submerged body in the fluid.  

Pressure can be solved by integrating the object over the top and bottom fluid depth,  Pressure=rho*g*H (hydrostatic relation).  As far as temperature you'll either have to assume it constant, find a known equation, or derive its change through height by solving the systems heat trasnfer properties.

To find the average lifting force use the mean value theorm by integrating the buoyancy function over the two points and dividing by the distance between them, or finding the local rate of change between two points and applying the mean value theorm.