balloon at any point $z$ may be expressed as:
$$
d_z = 2 \left[ \left( \frac{D}{2} \right)^2 - \left( \Delta z - \frac{D}{2} \right)^2 \right]^{1/2}
$$
(4)
$$
= 2 \left[ D \Delta z - \Delta z^2 \right]^{1/2}
$$
Therefore, maximum allowable balloon pressure at any plane $z - z$ will be:
(5)
$$
\Delta p_z = \frac{4 S_f t}{2 (D \Delta z - \Delta z^2)^{1/2}} \text{ psi}
$$
Equation (5) may be plotted in terms of bursting pressure and $\Delta z$ for any given diameter balloon. A straight line through the origin and tangent to the plot of Equation (5) will indicate the maximum allowable $(dp/dz)_{air} (1-B)$ for any given diameter balloon. Comparing the maximum allowable $(dp/dz)_{air}$ with a chart of altitude vs. pressure in the atmosphere will indicate the minimum altitude at which the balloon can be allowed to be full. From an altitude-buoyancy table for any given diameter balloon, the maximum allowable buoyancy, or maximum allowable gas inflation can be obtained.
Figure 26 is a plot of equations (3) and (5) for .001" polyethylene ( $S_f = \frac{900}{2} \text{ psi}$ ) balloons of 20', 30' and 70' diameters.
Fig. 25.
Relationship $d/\Delta Z$ , for balloon.
Fig. 26.
Graph of equations (3) and (5).
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