by Phil Lowell and Joe Reynolds

What, another article on the speed at which pibals
rise? Wasn't there recently an article by Mike Bien [*Balloon Life*,
page 18, July 95] that told more than we wanted to know about them? Well,
yes, but here is the new twist. We did some experimental work and by making
a force balance we derived a magic formula that only requires you to get
an average weight for your balloons and to measure the "diameter"
before you release them. You can then look up the balloon rise speed using
the figure or table in this article. An important point of this article
is that the balloon rise speeds given here are based on pibals that balloon
pilots use.

Just to give you a little background, engineers have been calculating the drop rate of spheres for decades - or is it centuries? All of this experimental work is boiled down to one number, a drag coefficient. Now a balloon is almost like a sphere, right? And gravity pushing a steel or polyethylene sphere down through air or water should be no different than gravity pushing a sphere up through air, right? Wrong!

Recently some learned folks have measured (instead of calculating the obviously already known solution) the rise rate of light spheres - like a pibal in air. They found the drag coefficients were greater by a factor of two or more [see D. G. Karamanev, et al, AIChEJ vol 42, 1789-92 (1996)]. The reason put forth for this discrepancy is that in the early experiments, spheres of steel, nylon, glass, etc. were used. These were heavy spheres and had a lot in inertia to resist twisting (a large angular moment of inertia for you science buffs). The pibals, on the other hand have a low moment of angular inertia. The drag forces exerted on the pibal cause it to twist and turn. Actually it will often corkscrew (spiral) up. This changes the effective drag coefficient. So much for talking about background, here is what we did.

We ran two sets of experiments. We first weighed the balloons empty. For the first set we filled them with welder's helium and tied them off. We clipped a binder clip on the tie-off and weighed the balloon. We put the balloon between two carpenter's squares and measured the diameter and "height" of the balloon. We call the balloon "tie-off" the bottom. The height is the distance between the bottom and the other end - which we call the top. We call the diameter the largest circle of balloon perpendicular to the bottom-top axis.

Now for the action. Inside a four story office building with a 69 foot atrium ceiling we released the balloons and timed them to hitting the glass ceiling. This was done after working hours when the AC was off.

The second set of experiments we ran with a mixture of helium and air - just for science. As an aside, welder's helium is very close to being pure helium. Party balloon helium is often an air/helium mixture. The air keeps the balloons from deflating and looking sick as the helium diffuses out. The party balloon helium filled balloons look OK, they just may not rise at a predictable speed. At any rate, we timed how fast these balloons hit the ceiling. For the sake of pure science, we filled some balloons with air and dropped them from the fourth floor and timed their descent (OK, so we ran out of helium before we ran out of balloons and were having too much fun wondering what folks would say Monday morning when they found the atrium ceiling covered with balloons).

For a sphere the lift would be the volume of the
sphere times the density difference between air and helium, **P**air
- **P**He** **minus the weight of the balloon itself. Since a balloon
is not a perfect sphere, we throw in a shape factor, k, for the volume of
the sphere.

Vballoon = k(/6)D3

Now for the heavy science. Every article like this needs some equations to make it look clever. So here are our equations. First we make a force balance.

Force up = Force down

Then we define these forces.

Force up = Lift force = (Volumeballoon)(densityair-helium) - empty weight of balloon

= k(/6)D3 (**P**air - **P**He) - Wballoon

Force down = drag force = Cd(/4)D2(**P**airV2/2gc)

We found that the shape factor, k, was 1.130, just a little greater than one, as expected. We found that the average drag coefficient was 0.989 - a little over twice the drag coefficient of a heavy spheres falling ( about 0.47).

We can solve these equations for the rise velocity of the pibal in terms of the measured diameter and the weight of the empty balloon. We solve these equations for "v" and get our magic formula.

v = 60{(2gc/Cd/**P**air)(2kD_p/3 - 4Wb/cD2)}1/2
ft/min

k = 1.130 Cd = 0.989 gc = 32.17 ft/sec2 **P**air**
= **0.0729 #/ft3 _p = 0.0628 #/ft3

We programmed the magic equation and solved it for several diameters and several empty balloon weights. We graphed these in Figure 1 and printed them in Table 1. Here is how you use Figure 1. Take a bunch of the balloons you intend to use as pibals and weigh them. Say eleven empty pibals weigh 30.8 grams. The average pibal weight is 30.8/11 = 2.8 grams (there are 28.4 grams in an ounce, if you have a scale that weighs in ounces). Inflate your pibal with helium. Measure the diameter. You can do this by using a carpenter's square as a caliper, measuring the pibals shadow on a surface perpendicular to the sun, etc. Suppose your pibal is 15 inches in diameter. To use the figure, go up the 15 inch line to about half way between the 2 and 4 gram pibal lines and read about 415 ft/min.

To use the table, enter the column marked 3 grams. Go down to the row marked 15 inches and read the value 416 ft/min. Our balloon only weighed 2.8 grams so we go a little toward the 2 gram column (423 ft/min) and estimate 418 ft/min. Don't waste your time trying to get closer than about 5 ft/min.

If you want to use the magic formula, put everything in consistent units. The factor of 60 in the magic formula converts from ft/sec to ft/min. If the temperature or pressure is markedly different from our standard conditions of 78· F and 750 mmHg you can change the density with the following formula.

**p**new** = **[(78
+ 460)/(tnew + 460)](Pnew/750) tnew in degrees F Pnew in mmHg

How close is Mike Bien's formula to this one? We can calculate the lift, which is his "L" for our 15 inch (15/12 = 1.25 ft) pibal. We need it in grams.

L = 1.130(3.14/6)1.253(0.0628)454 - 2.8 = 30.1 grams

From his formula:

v = 229x30.10.63/(30.1 + 2.8)0.42 = 451 fl/min

Not bad agreement for two completely independent
methods. Look at Mike Bien's article on page 14 of the August 1995 *Balloon
Life* for computing wind speed aloft.

Copyright © 1998 Balloon Life. All rights reserved.