## Balloon Volumetric Calculations

Project to send a balloon into lower space with payload
Daniel Wee
Posts: 2442
Joined: Wed 25 Feb 25 2009 8:00 pm

### Balloon Volumetric Calculations

The tricky part with accurately calculating the balloon volume has to do with the elastic force. The other parameters we should be above to estimate - pressure and temperature. We also know the gas density's relationship with temperature. As such, using Boyle's Law, we can work out the volume - but we don't know the elasticity factor which acts to compress the volume. This may be assumed to be linear from some point onwards (that is once the balloon is fully inflated.)

The key performance parameters of interest are:-
1. balloon volume
2. lift force
3. rate of of ascent at a given altitude and temperature
4. rupture volume

http://www.thefreelibrary.com/Lift+off!-a0111897749

Daniel
Attachments
balloon-release-study.pdf
Ascent Rate Calculator.xls
Daniel Wee
Posts: 2442
Joined: Wed 25 Feb 25 2009 8:00 pm

### Re: Balloon Volumetric Calculation

Somewhat related is the calculation of the rate of ascent of the balloon. Some reference(s):-

http://showcase.netins.net/web/wallio/ASCENT.html

Daniel
Daniel Wee
Posts: 2442
Joined: Wed 25 Feb 25 2009 8:00 pm

### Re: Balloon Volumetric Calculation

Some random thoughts pertaining to balloon ascent rate.

1. The lift force has to do with the difference between the density of the volume of gas inside the balloon and the ambient air density.
2. As the balloon rises, temperature changes - which changes balloon volume and hence density.
3. However ambient air density also drops.
4. Depending on the rate the density drops in the balloon, as well as ambient air, the lift force will change accordingly.
5. Other factors that come into play is the constant load, and the balloon drag.
6. Balloon drag, calculated off Reynold's numbers, also change with temperature.
7. Actual velocity is the integration of the state equations for the net force over time.

Air density can be read off atmospheric charts and extrapolated for inter-sample points. Temperature, too can be computed.

As for balloon volume, if we can assume that the pressure exerted by the gas is constant, then we can use Boyle's law to compute volume. The question is if that pressure is constant. A balloon at equilibrium should have the internal pressure equal to atmospheric pressure. This is what stops the balloon from expanding further. Therefore a drop in external pressure will cause the volume to increase until the pressure inside matches the pressure outside. This implies that as the volume increases, the pressure decreases. Actually, the external pressure plus the elastic force of the balloon is what is being matched. Thus when the external pressure decreases, the balloon will expand until the elastic pressure of the balloon makes up for the loss of external pressure.

Strictly speaking, we need to look beyond Boyle's, Charles' or the Gay-Lussac's laws to the combined gas law:-

pV/T = k

or

p = kT/V

If we state the external force on the gas inside the balloon as:-

CF + cfV

where F is the force exerted by atmospheric pressure, and f elastic force, cfV would be a force proportional to how stretched the balloon is, C and c are constants, we can re-state the state equation crudely as:-

CF + cfV = kT/V
CFV + cfVV = kT

As can be seen, the exact outcome of a reduction in F is dependent on the coefficients and the non-linear equation. In general it can be seen that a reduction in F needs to be compensated by a corresponding increase in V.

We can also see that a reduction in external pressure results in a corresponding reduction in internal pressure. In the steady state:-

CF + cfV = p

where p is the pressure inside the balloon. We further know that density changes with volume:-

r = m/V

where r is the density and m is the total mass of the gas inside the balloon, which can be assumed to be a constant in our case. Thus density decreases as volume increases. We also know from the ideal gas law that:-

r = Mp/RT

where M is the molar mass of the gas and R is the universal gas constant. This means that we can actually compute the density of the gas inside the balloon either from known the pressure, or from the volume of the balloon.

Daniel

Daniel Wee
Posts: 2442
Joined: Wed 25 Feb 25 2009 8:00 pm

### Re: Balloon Volumetric Calculation

This now brings us to fluid mechanics, and the Archimedes' Principle in particular. This basically says that:-

F = rgV

where F is the buoyant force, g is gravitational acceleration, r is the atmospheric air density and V is the displacement volume.

What is immediately striking about this is the fact that F is not directly related to the density or pressure inside the balloon. It is the volume of the balloon that gives the upward force. The nett lift, however, needs to take the mass of gas inside the balloon, the weight of the balloon itself, and the weight of the payload into consideration as these are the downward force. Given that the downward force is constant, the lift force should increase as volume increases. Atmospheric density, however, decreases so if the rate of increase of volume is lower than the lapse rate of density, then you will end up with a reducing lift as the balloon ascends.

These complex inter-related equations would form the basis for writing the state equations that tell us the amount of lift, which when integrated over time, should yield the velocity, assuming unimpeded travel.

Daniel

ref: http://biotsavart.tripod.com/balloon.htm
Daniel Wee
Posts: 2442
Joined: Wed 25 Feb 25 2009 8:00 pm

### Re: Balloon Volumetric Calculation

Next we need to know the drag force on the spherical balloon. In order to do this, we will be looking at the drag coefficient and Reynold's numbers:-

http://www.scribd.com/doc/5174368/Drag- ... n-a-Sphere

CD = F / (0.5 r vv A)

where F is the drag force, r is the density of ambient air, v is velocity and A is the cross sectional area. We would need to work out A from V, where:-

V = 4/3 * pi * rrr

and

A = pi * rr

and

V = 4/3 * A * r
r = V/A * 3/4

we can therefore write, by isolation:-

rrr = V * 3/4 / pi
r = (V * 3/4 / pi) ^ (1/3)

and, by substitution:-

A = pi * (V * 3/4 / pi) ^ (2/3)
A = pi * 9/16 * VV/AA
A = (pi * 9/16 *VV) ^ (1/3)

Drag force can be described as:-

F = 1/2 * r * vv * A * f(R)

where R is the Reynold's number and f() is an experimentally determined function.

Daniel
Daniel Wee
Posts: 2442
Joined: Wed 25 Feb 25 2009 8:00 pm

### Re: Balloon Volumetric Calculation

Here's what looks like an approximation table, using Hydrogen:-

http://ukhas.org.uk/guides:balloon_data

Daniel
Daniel Wee