The Roswell Report: Fact vs. Fiction in the New Mexico Desert

This relationship is expressed as:

$$
\Delta p = \Delta z \left(\frac {d p}{d z}\right) _ {\mathrm {a i r}} (1 - B)
$$

where $ B=\frac{M_{g}}{M_{q}} $ $ M_{g}\& M_{q} $ are molecular weights of lifting gas and air, respectively.

Since, for a full balloon, $ \Delta z $ is constant at any altitude, and B (for our discussion) is a constant:

$$
Q \propto \sqrt {\frac {\left(\frac {d p}{d z}\right) _ {\mathrm {a i r}}}{d _ {g}}}
$$

The mass rate of flow is equal to the density of the lifting gas multiplied by the volumetric rate of flow:

$$
L = Q d _ {g} \propto \sqrt {\left(\frac {d p}{d z}\right) _ {\mathrm {a i r}} d _ {g}}
$$

Since the number of openings will not change with altitude, equation (7) expresses the relationship for mass rate of flow from a full balloon for any altitude. The leakage at any altitude may be expressed as a function of leakage at sea level:

$$
\frac {L _ {z}}{L _ {0}} = \left(\frac {\left(\frac {d p}{d z}\right) _ {\mathrm {a i r} - z}}{\left(\frac {d p}{d z}\right) _ {\mathrm {a i r} - \theta}} \frac {d g _ {z}}{d g _ {0}}\right) ^ {\frac {1}{2}}
$$

As an example, let us compare the leakage rates of a lifting gas through a full balloon at sea level, at 40,000 feet and at 100,000 feet.

Altitude	(dp/dz)air	dg
0	$\frac{1}{27}$	$\frac{1013}{288R}$
40,000	$\frac{1}{112}$	$\frac{188}{218R}$
100,000	$\frac{1}{1880}$	$\frac{10.9}{218R}$