Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

Most recently, restraint is built into the fabric of the balloon to reduce the likelihood of bursting. Exterior netting has been employed.

There is a 1976 (PDF) report on this subject, BT-1044.09, "The First Year of the ATMOSAT Project", that may be downloaded from The Goddard Library describing such balloons. The abstract follows:

The pressure balloon will often be partially filled with lifting gas at ground level and then ascend until fully expanded and continue ascending until its overall density matches that of the surrounding atmosphere. In the later segment of the ascent the gas pressure will rise.

Such balloons have a limit to the pressure that they are able to withstand without bursting but within that upper limit the balloon will float at a more or less stable altitude. At times some gas may need to be vented to avoid the threat of bursting.

Having reached a stable altitude with a pressurized balloon it is possible, by venting a small amount of gas, to descend and float at stable altitudes that are above the altitude at which full expansion occurred.

Venting enough gas to drop below the full expansion altitude will return the balloon to earth unless enough ballast is released to allow the balloon to again rise and fully expand. Release of a smaller amount of ballast will slow the rate of descent.

The Web-based 2D Calculator version of the spreadsheet, available from the upper row of navigation tabs is used for the modeling.

The spreadsheet Parameter Table follows:

The partially filled balloon and payload had a density of 1.1 at its sea level launch site. This is less than the surrounding atmospheric density of 1.22... and thus the balloon rises into the air.

The expansion factor provided to the macro is 2.5. The balloon reaches full expansion to a radius of 30 metres about 32 minutes later at an altitude of ~20,784 metres.

It continues to rise for just over a minute, stopping at its float altitude of ~ 21,461 metres. At float altitude the balloon and payload have a density equal to that of the surrounding atmosphere, ~0.07039.

Graphs of the altitude and velocity of the balloon ascent versus time are shown next.

The velocity of ascent over time is the more interesting.

At launch, the balloon's rate of rise rapidly reaches ~9.0 metres per second, it then speeds up reaching ~ 14.0 metres per second at full expansion. Its rate then falls rapidly as float altitude is approached.

The velocity does not drop to zero on reaching float altitude. The kinetic energy of the balloon will force it to overshoot that altitude. Once above float altitude the balloon looses buoyancy to the point where gravity will accelerate it back down through float altitude, having lost a little of its energy to atmospheric friction.

The balloon will again become buoyant and soon reverse direction to move upward through float altitude again.

This up then down cycle repeats with the maximum excursion from float altitude slowly diminishing with each cycle.

Rather than continue calculating through a great many cycles looking for convergence to the float altitude, the macro is designed to stop calculating at the calculation step when float altitude is first exceeded.

Of course the calculated behaviour is not an accurate representation. A great many influences such as atmospheric temperature, radiant energy from the sun, stretch in the balloon fabric, condensing of residual moisture in the lift gas, etc., have been ignored. Nonetheless, the calculated behaviour should bear a resemblance to the actual behaviour of such a balloon.

To test the effect of floating, supply the spreadsheet with the platform altitude of 21,465 metres, 10 metres above float altitude. Set the balloon radius to 30 metres.

No expansion or contraction relative to this platform altitude is wanted, therefore set the balloon expansion factor to zero. The macro then ignores expansion.

The overall density of a pressurized balloon does not change at altitudes above the fully expanded altitude. Thus employ the density that was achieved at that altitude.

On running the macro the results are:

Note that the balloon falls to approximately its previous float altitude. The small difference is largely due to the finite step size of 1 second.

We could equally well have lowered the platform altitude to 10 metres below float altitude and the balloon would have ascended to approximately float altitude.

The effect of venting a little of the lifting gas will be to increase the overall density a little. One has to be careful about how much, else the balloon will descend to earth. This is illustrated next.

On the spreadsheet, increase overall density by 0.01. Since we expect with this change that the balloon will no longer be fully expanded, set the balloon expansion ratio to 0.4.

Set the target altitude to zero. This will cause calculation to stop at the first step at which the altitude is less than 0.0.

Change the velocity reporting altitude to 0.0 metres and run the macro.

That case is shown next:

From the volume seen in D13, the balloon deflated to just under its launch radius at an altitude of ~ 1091 metres.

From the step-by-step table, not shown, the velocity at that altitude was ~10.91 metres per second.

The balloon lands at a velocity of ~10.36 metres per second, e.g. the payload hits the earth at ~37 km. per hour or ~ 23 miles per hour!

An amount of ballast could have been cast out before reaching the ground to reduce this speed.

Less lift gas vented at the peak altitude would have resulted in a lower speed at earth impact.

Venting a still smaller amount of lift gas at the 21,455-metre altitude, an amount limited so as to retain full expansion, will result in a lower float altitude.

Try adding half as much, 0.005, to the overall density and set the expansion factor at 0.0 as we plan on remaining in the full expansion altitude range.

See the result next.

The balloon fell ~ 429 metres in a period of ~1 minute.

**
**

The balloon is reported to have an expanded to a diameter of about 122 metres, a radius of ~ 61 metres and be 11 to 12 metres wide at lift off. That suggests about a 10:1 expansion ratio. Kittinger is reported as saying that it took ~ 90 minutes to reach an altitude of 31,330 meters.

To model the Kittinger ascent, choose the balloon expansion factor and the initial radius r such that their product is the fully expanded radius of 61 metres.

The principle result of adjusting the expansion factor is to affect the float altitude.

Choose a platform altitude of 1500 metres for the launch site. Choose the total density as slightly less than the atmospheric density at the site altitude so that the balloon will rise.

The principle result of adjusting the overall density is to affect the time taken to reach the float altitude.

A few manual iterations adjusting both the expansion factor and the overall density one will arrive at a result such as that following.

Both the time taken to reach float altitude and the float altitude are very close to the values reported by Kittinger.

The model suggests that the value for the expansion factor is lower than that might be deduced from the stated 11 to 12 metre width of the balloon at launch. The model suggests a launch width of ~ 30 metres.

From the photo, the writer deduces that the 30-metre width may not be that far off the mark.

See
www.balloonfiesta.com
for an interesting article.

Modeling projectiles in and out of Earth's atmosphere.

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