Hot Air Balloon: Find inside temperature given size, load

[cdeustice]

Homework Statement

Assuming normal pressure and temperature outside, and normal pressure inside, what should be the inside temperature (in Celsius) if the hot air balloon has diameter 17 meters and carries 446.3 kg load? (Except the mass of air inside, everything else is included in the "load")

Homework Equations

See attached picture

The Attempt at a Solution

See attached picture-Solved for temperature above the last equation because I ran out of room on the page

 

[SteamKing]

Homework Statement

Assuming normal pressure and temperature outside, and normal pressure inside, what should be the inside temperature (in Celsius) if the hot air balloon has diameter 17 meters and carries 446.3 kg load? (Except the mass of air inside, everything else is included in the "load")

Homework Equations

See attached picture

The Attempt at a Solution

See attached picture-Solved for temperature above the last equation because I ran out of room on the page

WOW! T = 1758 C! That's pretty toasty! Unfortunately, there are a few problems with your solution:

1. Hot air balloons are called that for a reason: they literally use hot air to provide lift, not helium, as is used in a blimp, for example.

2. A temperature of 1700 C would incinerate any reasonably light material with which to construct the balloon envelope. A balloon operating at this temperature would have to be constructed of asbestos or refractory brick in order to keep from bursting into flame.

3. What's the formula for the volume of a sphere? What you have written in your solution ain't it.

 

[cdeustice]

(Pi/6)d^3 is the volume of a sphere. Please give more specific advice.

WOW! T = 1758 C! That's pretty toasty! Unfortunately, there are a few problems with your solution:

1. Hot air balloons are called that for a reason: they literally use hot air to provide lift, not helium, as is used in a blimp, for example.

2. A temperature of 1700 C would incinerate any reasonably light material with which to construct the balloon envelope. A balloon operating at this temperature would have to be constructed of asbestos or refractory brick in order to keep from bursting into flame.

3. What's the formula for the volume of a sphere? What you have written in your solution ain't it.

 

[SteamKing]

(Pi/6)d^3 is the volume of a sphere. Please give more specific advice.

I see now what you did, and it's OK. Points 1 and 2 still apply, however.

 

[cdeustice]

I see now what you did, and it's OK. Points 1 and 2 still apply, however.

How does my work look now?

 

[SteamKing]

Well, I was wrong. I though when I told you your previous solution was too warm, you would take the hint.
Now, according to your calculations, the air inside the balloon is over twice the surface temperature of the sun,
you know, the big shiny ball which hangs in the sky during the day:

http://en.wikipedia.org/wiki/Sun

I don't know what the final answer to this problem is, but when you calculate that you must be hotter than the sun
for the balloon to work, you know you've made some serious errors somewhere. And that's an important part
of physics, too. You should know that water boils at 100 C, that lead melts at 327 C, that iron melts at 1538 C, etc.,
stuff which you can also look up.

I think your formula involving the SG is leading you astray in this problem. Dry air at standard conditions has a density
of about 1.2 kg / cu.m., so your balloon envelope should contain more than 61.74 kg of air before heating. I suggest
you manipulate the PV = nRT formula so that you can calculate the density of air at a given temperature.
Then, knowing how much lift you need to generate, you should be able to calculate the T required in a rather
straightforward manner.

 

[cdeustice]

Can you give an example of how to do the ideal gas calculation of density?

 

[SteamKing]

This is an example calculation:

http://chemistry.about.com/od/gaslawproblems/a/Density-Of-An-Ideal-Gas.htm

Note that the value of R is still selected to remain compatible with the units for P, V, and T.

 

[Simon Bridge]

Most familiar with the molar form of the ideal gas equation?
From buoyancy - get the mass-density of the air needed so the balloon will float with the required load.
From the mass-density, get the molar density n/V (what is the mass of one mole of air?)

From the ideal gas equation, get T for that molar density. T=P/[(n/V)R]
Note: it is easier to just type out the working than scan the paper, then upload the scanned image.
As already mentioned - be careful to watch your units for the Rydberg constant.

You can finesse things by modifying the ideal gas equation for the fact that air is not an ideal gas - but you may not have notes for that yet.

 

[SteamKing]

From the ideal gas equation, get T for that molar density. T=P/[(n/V)R]

As already mentioned - be careful to watch your units for the Rydberg constant.

Uhh, in the ideal gas equation, R is the universal gas constant, not the Rydberg constant.

 

[Simon Bridge]

Ugh: need... coffee...