Physics behind the flow of gas coming out of a balloon?

Question

I'm working with stratospheric balloons (latex ones) and I want to put a valve on it so it can float for a longer time. I'm trying to define which valve I should use, which demands I estimate the flow os gas it can address. Also, it would be great to be able to model the problem mathematically, so I can design an altitude control system. Thinking about this, I got some doubts related to the physics of the latex balloon membrane and the gas inside of it. I'm trying to understand what exactly makes the gas inside a balloon come out when we open a valve placed on its "mouth". I know the pressure inside the ballon must be greater than the external in order to the gas flow. But when the system is in equilibrium, the internal pressure must be equal to the external, so the ballon doesn't expand/contract. Or ALMOST that. I think most people thinking about this just ignore or deem negligible the force the membrane of the balloon exerts in the gas inside of it. I came to the following conclusion: $$p_i = p_o + p_b$$ where: $p_o$ = external pressure $p_i$ = internal pressure $p_b$ = pressure exerted by the balloon membrane into the gas in its interior ---------- This justifies why the internal pressure would be greater than the external and the system would, still, remain in equilibrium. If this is correct, I can conclude that what will make the gas come out is pb, at any altitude. So, if I have pb, I could use Bernoulli's principle to calculate the velocity with which the gas come out and, from this, find which is the flow rate I need to model the control problem and guide me on the valve specification. I have two concerns with this approach: 1. Is it correct? Physically speaking, this is what really happens? Can someone give me more insights on the dynamics of mechanical tension on the membrane and gas pressure inside the balloon? 2. I can't find a way to calculate the pressure the membrane would exert on the gas (pb). I know that I can calculate the tensions that appears in the membrane surface accordingly with the internal pressure (with pressure vessels theory). I have a feeling that I can derive the force with which the membrane compress the gas from this tensions, and it seems to me it would be only a geometrical/calculus problem, but I'm struggling getting it forward. I have idealized a simplified model in which I apply Hook's Law, but I'm stuck with infinitesimal areas. Any one can guide me here?

Answer 1

The pressure differential (inside minus outside pressure), pb, is related to the surface tension in the balloon, s, and its radius, r, as: pb = 2s/r. If the latex is acting elastically, s increases linearly with r, and pb is essentially independent of r. If you want to prolong the life of the balloon as it enters the stratosphere, why not make the valve of the pressure-relief kind? That way, some gas in the balloon will be released whenever the differential pressure gets too high (due to decreasing exterior pressure). That will allow the balloon to reach a certain elevation and stay there, rather than bursting. Different relief pressure settings would correspond to different ceiling heights.