Some random thoughts pertaining to balloon ascent rate.
1. The lift force has to do with the difference between the density of the volume of gas inside the balloon and the ambient air density.
2. As the balloon rises, temperature changes - which changes balloon volume and hence density.
3. However ambient air density also drops.
4. Depending on the rate the density drops in the balloon, as well as ambient air, the lift force will change accordingly.
5. Other factors that come into play is the constant load, and the balloon drag.
6. Balloon drag, calculated off Reynold's numbers, also change with temperature.
7. Actual velocity is the integration of the state equations for the net force over time.
Air density can be read off atmospheric charts and extrapolated for inter-sample points. Temperature, too can be computed.
As for balloon volume, if we can assume that the pressure exerted by the gas is constant, then we can use Boyle's law to compute volume. The question is if that pressure is constant. A balloon at equilibrium should have the internal pressure equal to atmospheric pressure. This is what stops the balloon from expanding further. Therefore a drop in external pressure will cause the volume to increase until the pressure inside matches the pressure outside. This implies that as the volume increases, the pressure decreases. Actually, the external pressure plus the elastic force of the balloon is what is being matched. Thus when the external pressure decreases, the balloon will expand until the elastic pressure of the balloon makes up for the loss of external pressure.
Strictly speaking, we need to look beyond Boyle's, Charles' or the Gay-Lussac's laws to the combined gas law:-
pV/T = k
p = kT/V
If we state the external force on the gas inside the balloon as:-
CF + cfV
where F is the force exerted by atmospheric pressure, and f elastic force, cfV would be a force proportional to how stretched the balloon is, C and c are constants, we can re-state the state equation crudely as:-
CF + cfV = kT/V
CFV + cfVV = kT
As can be seen, the exact outcome of a reduction in F is dependent on the coefficients and the non-linear equation. In general it can be seen that a reduction in F needs to be compensated by a corresponding increase in V.
We can also see that a reduction in external pressure results in a corresponding reduction in internal pressure. In the steady state:-
CF + cfV = p
where p is the pressure inside the balloon. We further know that density changes with volume:-
r = m/V
where r is the density and m is the total mass of the gas inside the balloon, which can be assumed to be a constant in our case. Thus density decreases as volume increases. We also know from the ideal gas law that:-
r = Mp/RT
where M is the molar mass of the gas and R is the universal gas constant. This means that we can actually compute the density of the gas inside the balloon either from known the pressure, or from the volume of the balloon.
ref: http://aolanswers.com/questions/air_pre ... 7?#answers