Re: Balloon lift calculations with different gases...

Balloon lift calculations with different gases...

How do you calculate the lift generated by hydrogen, helium, or
(theoretically) a vacuum in a balloon (probably mylar)?  How high
 could such balloons rise, given an ideal situation and a standard
mylar or similar balloon material?  What difference does a hot gas
make over a cold gas?  What other gases could provide lift in our
atmosphere (i.e. are lighter than air)?
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The answer:

Hi John,

The question you asked has to do with physics as a basic science, and with 
one of its specialized branches, which is fluid mechanics. The example 
you mentioned, the balloon, is an example for buoyancy and buoyant 
force. Perhaps you remember Archimedes  principle about floatation and 
buoyant force. That theory says that there is a force exerted on any 
body immersed or half-immersed  inside a fluid. That force is termed the 
buoyant force, and it is equal to the weight of the displaced volume from 
the fluid. The second point to discuss is some mechanics that will not 
exceed your secondary school information.

You are aware of Newton's laws, I am sure. Thus for a body moving with 
variable velocity (acceleration) like the balloon, we should be able 
to establish an equation of motion. There are forces that pull that 
balloon down. These forces are the weight of the people on board and 
the equipment (motor and fuel tanks). This is in addition to the weight of 
the balloon ( I mean the place where the gas is inside, whatever the gas 
name was. However, the balloon is moving upwards. Thus there should 
be forces that are pushing the balloon upwards.

This force is the one called the buoyant force. The buoyant force is equal 
to:

F (b) = (displaced fluid density)x(gravity acceleration)x(displaced volume)

Consider this fact that a lot do not consider, the balloon is a body 
immersed 
in a fluid, which is air. The displaced volume is the volume that the 
balloon replaces of air to occupy. Then the buoyant force for the 
balloon equals:

F (b) = (air density)x(9.81 m/sec^2)x(volume of the gas filled balloon)

In an equation
Net force = Upward pushing forces - Downward pulling forces
F = ma = F (up) - F (down)
Thus the complete equation

F = ma = F (b) - Weight of equipment- Weight of the gas container
(This equation ignores some effects; such as: wind direction, 
surrounding temperature and pressure)

Consider with me these facts; the weight of the balloon equipment is 
perfectly constant, and so is the buoyant force (in fact it changes 
somehow because as we ascend upward the density of air decreases, but this 
is not so significant because air density itself is 1.2 Kg/m^3 and 
variations will be in limits of 0.01 to 0.1 ). Thus if you are the driver 
of 
a balloon, how can you change the speed of the balloon? The only answer is 
to change the weight of the gas container. That container is full of a 
gas, probably hydrogen. By increasing the heat for hydrogen gas, its 
density decreases.  
There is one more equation that I would like to add:

Weight = (density)x(gravity acceleration)x(volume)

Thus if  return back to the equation
ma = F (b) - Weight of equipment- Weight of the gas container

m = mass of the balloon equipment, and the mass of the gas (total mass)
F (b) = (density of air)x(gravity acceleration)x( volume of the balloon) 
= nearly a constant value
Weight of the equipment = constant
Weight of the gas container = (gas density)x(gravity acceleration)x(volume)

For other gases, here are some with their densities in Kilogram per 
cubic meters:

air  = 1.29
Oxygen = 1.43
Hydrogen  = 0.0899
Helium = 0.179

You had a question about if a free vacuum. Okay, just create a vacuum, and 
we will discuss it later. But till that time, you should guess that 
the contribution of the gas weight will vanish. 
You asked also about the maximum height. I am not sure about the value nor 
the figure, because this all depends on the fuel you have. If you have 
enough containers to reach the moon, and you are able to resist variation 
in temperature, pressure, and oxygen % in air, you can make it I 
suppose.However, atmosphere does not extend to the moon, as you definitely 
know. 
 

That is all for now. Waiting for a reply or further explanation if you 
want. Thanks for your question, John. You really refreshed my mind.


Moataz Attallah
The American University in Cairo-Egypt
Mechanical Engineering undergraduate student
Email: mizoa@aucegypt.edu
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References:

Abbott, A.F. Ordinary Level Physics. Heinemann Educational Books, London. 
Fourth edition 1984.

Serway, Raymond. Physics for Scientists and Engineers. Saunders Collage
publishing, USA. Third updated edition 1990.