Archimedes principle - special case

[Lucky7]

Homework Statement

We have a chinese lantern (balloon) made of paper, cylinder shaped with the following sizes:
base diameter - 45 cm, height - 70 cm. Mass of the balloon is 57 g (21 g from this is mass of "fuel" - the fuel is wax paper!). Fuel is then ignited at the centre of the base, which is open. Therefore the balloon will soar. The question is, how much additional weight can be hung to the balloon and the balloon still takes off. Outside temperature is 5 °C (Let us denote it as cold air).

Homework Equations

Just see the text.

The Attempt at a Solution

First I can compute volume of the balloon, which is of course V = π*r²*h = π*22.5²*70 cm³ = 111330.189662 cm³≈0.11133m³.
Now the gravitational force of air pressed up by the balloon gives the magnitude of buoyancy:
G_{cold_air}=ρ_{cold_air}*V*g = 1.2697*0.11133*g≈1.38723 N.
Now we should determine the volume of the fuel, which is
V_fuel= m_fuel/ρ_fuel=0.021/650 = 0.00003 m³.
Now we will compute G_{hot_air}=ρ_{hot_air}*(V-V_fuel)*g = 1.1277*0.11130*9.871373 ≈ 1.23175 N. Now we can use Archimedes principle:
G_{cold_air} = G_{hot_air} + m*g + M*g, and we want to solve this equation for M, which is our burden, that can be carried:
M = (G_{cold_air}-G_{hot_air}-m*g)/g, which gives us M≈-0.04116 kg, which is rubbish...

There are just too many determinants, that were omitted:
1) temperature of hot air in the ballon, I chose 40 °C, but how can I know? What if it is 60 °C, then the density would be higher, but how can I know?
2) It would be best if all fuel would just burn out, then the temperature would be the highest and the mass the lowest, so should I wait till the fuel is burn out and only after that I bind the burden?
3) I omitted the air pressed up by the wax paper and the burden.
4) ... inf)
There are just too many...
I'd be so much grateful for your help!

 

[CWatters]

Do you know the energy content of the fuel and the specific heat capacity of air? That may allow you to calculate the temperature of the air in the balloon after the fuel is burnt. Some assumptions may have to be made!

 

[Lucky7]

This is all I know, we should probably make some of our own assumptions, yeah...

 

[haruspex]

Since you only need it to take off eventually, yes, you can restrict attention to the time at which the fuel is almost gone. I suspect that assuming all the heat stays in the balloon will give a significant overestimate of the temperature. You could calculate the volume of gas from the combustion products, assume that mixes uniformly with the original air, find the resulting expansion, and hence find how much heat is lost by spillage out of the envelope. (There'll also be radiative losses, but they're probably a lot less.)
Of course, the gases in the balloon will no longer be just hot air - there'll be a lot of CO2, affecting the density.

 

[HalfLostAndSearching]

Hello,

I would like to provide some additional insights and considerations to your solution attempt.

1) Temperature of hot air: You are correct in stating that the temperature of the hot air inside the balloon will affect its density and therefore its buoyancy. However, in this case, we can assume that the hot air will be at a relatively constant temperature throughout the duration of the flight. This is because the fuel (wax paper) will continue to burn and produce hot air, maintaining a fairly consistent temperature inside the balloon. Therefore, for the purpose of this calculation, we can assume a constant temperature of the hot air.

2) Burning of fuel: It is true that the mass of the balloon will decrease as the fuel burns, which will affect its buoyancy. However, since we are only interested in determining the maximum additional weight that can be hung to the balloon, we can assume that the fuel will burn out completely before the balloon reaches its maximum carrying capacity. This means that we can use the initial mass of the balloon (57 g) and the mass of the fuel (21 g) in our calculations.

3) Air pressed up by the wax paper and burden: These factors can be taken into account by considering the volume of the balloon that is occupied by the fuel and the additional weight. This will slightly decrease the overall volume of the balloon and therefore affect its buoyancy.

4) Other factors: As you mentioned, there are many other factors that can affect the maximum carrying capacity of the balloon, such as wind, air pressure, and the shape and material of the burden. These factors can be difficult to account for in a theoretical calculation and may require experimental testing to determine the exact maximum carrying capacity.

In conclusion, while your solution attempt is a good starting point, there are several other factors that need to be taken into consideration in order to accurately determine the maximum additional weight that can be hung to the balloon. It may be helpful to conduct experiments with different weights and conditions to get a more precise answer.