of the surrounding atmosphere. Upon reaching the altitude at which it is full it will still have an unbalance in the direction of increase of altitude due to the excess buoyancy causing ascent. This unbalance is gradually decreased as the balloon rises (with a fixed volume) into less dense air. Meanwhile excess gas pressure is relieved by valving gas through the appendix until the balloon system is in a condition of equilibrium. The portion of the ascent after the balloon has become full is known as the "leveling-off" period.
The lifting gas which is valved out through the appendix will cause a "back pressure" inside of the balloon which must be transferred to the balloon fabric or film. In other words, there must be a pressure difference across the appendix opening during this period to force the excess lifting gas out of the balloon. Let us analyze this back pressure by the method used by Picard. Using the rules of subsonic aerodynamics, Picard suggests that air at sea level escaping at 1333 ft/sec. produces a back pressure of 1 atmosphere and that back pressure induced is proportional to the square of escape velocity of the gas and inversely proportional to the density of the gas escaping. Volume of gas lost in ascent through 1 foot is, within a reasonable degree of accuracy:
(6)
$$
\frac{\Delta V}{\Delta Z} = \frac{V}{P} \frac{dp}{dz} \cdot \frac{T + \Delta T}{T}
$$
$\frac{\Delta V}{\Delta Z}$ = volume lost per foot of ascent (ft.³/ft.)
V = balloon volume (ft.³)
P = pressure of free air (psi)
$\frac{dp}{dz}$ = pressure change with increase of Z (psi/ft)
T = temperature of air (°C abs.)
$\Delta T$ = change in air temperature during rise (°C)
For ascent in the troposphere this relationship will reduce to:
(7)
$$
\frac{\Delta V}{\Delta Z} = \frac{V}{27,800} \frac{\text{FT}^3}{\text{FT}}
$$
The velocity of escape of gas, then:
(8)
$$
V = \frac{dz}{dt} \cdot \frac{V}{27,800} \cdot \frac{1}{A_0}
$$
V = velocity of escape of lifting gas (ft./sec.)
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