Inflation System: Calculating Pressure & Volume

[AlanJones15]

(We are designing a portable pontoon with inflatable base, however neither of us has much knowledge of inflation systems or fluid mechanics)

How can you calculate the internal pressure of an inflated bladder as it gets pushed under the water by the weight it supports? And how this pressure increases as the bladder bends and gets compressed reducing its volume?

 

[FredGarvin]

Well, it's going to have to reach equilibrium with it's surroundings. Therefore the pressure inside the bladder will have to be equal to the pressure outside of it.

The pressure due to depth is $$p=\rho g h$$ where

$$\rho$$ is the fluid density
$$g$$ is the gravitational constant
$$h$$ is depth below the water surface

 

[AlanJones15]

Thanks although I know pressure at depth calculations.
I see that there should be equilibrium but will there be no more pressure than the pressure at depth outside? Perhaps caused by the change in shape resulting in change of volume?There is the upward buoyancy force which will equal the weight pushing the bladder down (until it displaces the same volume of water equivalent to the weight)

just like a partly inflated football bladder, the pressure inside will change if its pushed down. how to analyse it though as the pontoon can move up and done unlike a ball on the ground.

 

[AlephZero]

Well, it's going to have to reach equilibrium with it's surroundings. Therefore the pressure inside the bladder will have to be equal to the pressure outside of it.

Er, not quite.

The external pressure due to depth is $$p=\rho g h$$. If the bladder contains gas, its internal pressure is constant.

The difference between the pressures is the stress in the membrane surrounding the bladder.

In general, the shape of the bladder is rather hard to calculate. However you can often make two assumptions:

1. The membrane doesn't stretch significantly.
2. The bladder is fully inflated, i.e. the internal pressure is greater than the maximum external pressure (at the deepest point).

With those assumptions, the shape and the volume will be approximately the same whether the bladder is in water or in air. The buoyancy force depends only on the volume of water displaced, not the shape, so that's all you need to know.

The extra weight of air in the bladder if it is slightly over-inflated is negligible compared with the weight of water it displaces. Obviously the strength of the membrane will limit the maximum working pressure

 

[FredGarvin]

I can't say one way or another if the assumptions you have made are valid or not. I assumed a general case in which the bladder will expand and contract with the surroundings as necessary. There is a limit to that volume change, but there is no mention to the materials used. If one goes with the rigid bladder then you are more in line with things. When I think of a bladder, I think of something much more flexible.

However, if the volume change is acceptable, then the changing volume will alter the pressure in the bladder as load is applied. As load is applied, the bladder will shrink in size, increasing it's internal pressure and sink farther down until the outside pressure equalizes.

 

[AlephZero]

There are two ways the bladder could expand and contract. One way is by stretching the material (like blowing up a toy balloon). The other way is by the material "folding up" (like a half-inflated airbed).

I was thinking of a fairly substantial "membrane" (e.g. like an inflatable boat) so the stretching would be small. There doesn't seem any advantage in making the pontoon out of a stretchy material, just for the sake of it.

Either way, I would be worried by significant changes of volume because of possible stability problems. If the pontoon sinks lower in the water, the water pressure increases and its volume reduces, therefore the buoyancy reduces and it sinks further.

Another design issue is that the only way a membrane (with no bending stiffness) can support a load is by having a curved shape. Presumably you want your pontoon to have a flat top, and flat bottom - otherwise it would be deeper to displace the same volume of water and give the same buoyancy. So you probably need to make it from a raft of cylinders not one big bladder. That would also be good for safety, since a local puncture would be less likely to sink it.

Personally I would design it to be pressurized so it was effectively rigid. The water pressure at the bottom of the structure is about 0.1 atm per meter depth. Assuming your pontoon has a depth of the order of 1m, an internal pressure of say 0.5 or 1.0 atm should be enough to do the job. You would still have the advantages of easy transportation etc.

 

[FredGarvin]

In practical terms, you are right. I think that you would have to make the bladder as rigid as is feasible.

 

[AlanJones15]

Thanks for all the input guys!

Have taken on your points about pressurising it so it is rigid, that would seem to eliminate problems. I guess the thing I was having trouble getting my head round was the fact of the weight acting down and the buoyancy force up, I was picturing these would cause the internal bladder pressure to rise even more though what you're saying is that the weight simply causes the pontoon to sink down into the water until the point where displacement (buoy force)= the weight.


Am contacting manufacturers for bladder materials. I take it for specifying the pressures its a case of the bladders need to be 0.5 - 1 bar for when submerged, then plus another 1 bar for the atmospheric pressure pushing down on the water? and can then calculate the stresses in the membrane.