the pressure distribution of the lifting gas and the internal back pressure due to valving gas. To find maximum rates of ascent for various balloons would necessitate a complicated series of trial and error solution. In general, it has been more practical to determine a maximum rate of rise for normal operating conditions for any given size balloon by finding the maximum allowable rate for the balloon rising to its lowest normal operating level (i.e., we will find the maximum allowable rate for the worst normal operating conditions and consider it a maximum for all normal operating conditions.)
Let us take the case of a 20-foot diameter polyethylene balloon of .001" thickness. Lowest normal floating altitude is 20,000 ft. MSL. Let us assume that the balloon will be full and begin valving gas at 15,000 ft. MSL. Assume the appendix diameter to be $ \frac{1}{2} $ foot. Using equation (1) to find maximum allowable internal pressure and assuming the critical x-y plane to be that of maximum diameter $ \Delta Z=D/2 $ , we have:
$$
\Delta p _ {\mathrm {a l l .}} = \frac {4 S _ {\mathrm {f t}}}{D} = \frac {4 (9 0 0 / 2) \cdot . 0 0 1}{1 2 \cdot 2 0} = . 0 0 7 5 \mathrm {p s i}
$$
(Here we have introduced a factor of safety by saying $ S_{f}=9 0 0 / 2 $ instead of 900 psi, the ultimate strength in tension of polyethylene.) Pressure distribution:
$$
\Delta p _ {D / 2} = \Delta z \frac {d p}{d z} (1 - B) = \frac {\angle O}{2} \cdot 3. 3 8 \cdot 1 0 ^ {- 4} \cdot 8 6 2 = . 0 0 2 9 1 p s i
$$
Allowable back pressure:
$$
\Delta p _ {b p} = \Delta p _ {a l l} - \Delta p _ {0 / 2} = . 0 0 4 6 \mathrm {p s i}
$$
Maximum rate of rise using equation (13);
$$
\frac {d Z}{d t} = \sqrt {\frac {2 8 8 \Delta p _ {b p} g}{d _ {g}}} \left(\frac {2 7 8 0 0}{V} C _ {d} A _ {a}\right) f t / \sec
$$
$$
= 1 0 0. 7 \mathrm {f t / s e c}
$$
$$
= 6 0 0 0 \mathrm {f t / m i n}
$$
It is evident from this calculation that the rate of rise of the 20-ft. diameter polyethylene balloon is not a critical factor in bursting unless the open appendix becomes snarled and gas is not allowed to escape.
Rate of rise and appendix openings are important from the standpoint of balloon design. For operational reasons it is important to have a rapid rate of rise. In order to make most efficient use of weight, the balloon film should be thin. As mentioned