In my book I discuss technical issues like the lift of helium,

descent speeds of parachutes, cosmic rays, etc. The next topic

I'm tackling is the ascent speed of a balloon. Here's what I have for

my notes so far. I'm looking for input, because I don't have the

complete picture yet. Please spend a little time looking these

notes over and making suggestions. Thanks.

A. Intro

We know the balloon will expand as it rises due to decreasing air

pressure. As a gas expands, as in a balloon, it cools. This

cooling decreases the volume of a gas. We can see that during a

flight that the balloon expands a lot more due to reduced air

pressure than it contracts due to cooling air temperatures (both

inside and outside the balloon). So cooling has a minor effect

compared to reducing air pressure. The forces acting on a balloon

are three, payload weight (W) pulling down, lift (L) pulling up, and

drag (D) pulling in the direction opposite of the balloon's motion (so

drag pulls down, just like the weight). Looking at flight data, we

see that the balloon ascends at a constant rate for the entire flight

[1]. As long as there is no net force acting on a balloon (or any

object for that manner), it moves with zero acceleration, or

constant velocity. Since the balloon climbs at a constant

veleocity, the force of lift of the balloon is balanced by drag and

weight, D + W = L. Lift and weight are constant on a balloon flight

[2], therefore the force of drag must be a constant throughout the

flight also.

B. Drag

The equation for drag is given as,

D = Cd * Ra * sq(V)/2 * rho

where,

Cd is a coefficient of drag

Ra is the reference area

sq(V)/2 is the square of the velocity, divided by 2

rho is the density of the air.

Cd is a dimensionaless constant. This means it's simply a

number and does not have any units (like pounds or feet) attached

to it. For a sphere, Cd is about 0.5

Ra is an area you choose to use. If you select a different area as

your Ra, then the Cd for that Ra is different, but Cd is still just a

number

I only what to determine how ascent speed changes as the

variables change, so I'm going to treat this equation as a

relationship of proportionality and drop the constants for the rest of

this discussion.

When you look at the US Standard Atmosphere (a model of how

air pressure, density, and temperature change as a function of

altitude), you see that the change in air density closely tracks the

change in air pressure. So I'm going to replace air density with air

pressure. The reason I do this is that we can see the change in air

pressure changes the balloon's volume and therefore it's reference

area. As a Ra, I'm going to select the frontal area occupied by the

balloon. This Ra is proportional to the two-thirds power of the

volume. The balloon's volume, ideally, is proportional to the inverse

of the air pressure. Observations show that the balloon's vertical

velocity, its ascent rate, is constant throughout a flight. So sq(V)/2

is also a constant. When I combine like terms and move the

constants out, I end up with the following,

D is proportional to (volume)^2/3 * pressure

Which simplifies to

D pro to 1/(pressure)^2/3 * pressure

Which simplified to

D pro to (pressure)^1/3

However, D must be a constant, whereas pressure is not a

constant! So what's wrong here?

The change in temperature can't be responsible for adjusting the

change in volume (that is due to changing air pressure) because

we see the temperature change in opposite direction in the tropo

and stratosphere. And yet, the ascent rates in both the tropo and

stratosphere are both constant.

So it appears to me that Cd is not a constant throughout a flight.

C. Cd

Cd models drag's depenedencies on things like shape, air

viscosity, and air compressibility. The balloon's shape does not

change significantly during a flight. The air compressiblity is not

siginificant belows speeds of 200 mph. This leaves only air

viscosity. Is the change in the air's viscosity proportional to the

cube root of the air's pressure? If so, the change in Cd would

oppose the chnage in drag caused by the change in air pressure

(or more exactly, air density). Does the Cd of an aircraft change

as the aircraft climbs?

Note 1

There is a knee in the ascent speed of a balloon as you approach

the stratosphere. In the startosphere, the air temperature

increases as the balloon climbs. Perhaps the slight extra increase

in the balloon's diameter due to increasing stratospheric

temperatures increases the balloon's drag slightly, causing it to

slow down a bit. As a future experiment, does a black balloon rise

faster than an identical white balloon or slower?

Note 2

The lifting ability of helium does decrease with decreasing

pressure. But this amounts to only something like a percent or

two. This is not enough to significantly effect our numbers.

Paul