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I'm playing around with the physics of hot air balloons, and I've come to two different estimates of the energy required to get a hot air balloon off the ground. The two estimates make wildly different predictions and I'm hoping to get some insight on which one is wrong.
Okay, so first from thermodynamics. Let's assume we're going to heat our balloon (volume [itex]V[/itex], basket mass [itex]m[/itex]) up to some temperature [itex]T_{\mathrm{in}}[/itex]. The ambient temperature is [itex]T_0[/itex], and the condition for liftoff is:
[tex]V(\rho_0 - \rho_{\mathrm{in}}) \ge m[/tex]
Some faffing around with the ideal gas law results in:
[tex]V \ge \frac{m \bar{R}}{p_0 (\frac{1}{T_0} - \frac{1}{T_{\mathrm{in}}})}[/tex]
where [itex]\bar{R}[/itex] is the specific gas constant ([itex]R / M_{r,\mathrm{air}}[/itex]).
So that gives us the volume of balloon we need given a specified temperature difference between the inside and outside of the balloon. Using this expression we find that we can lift off a basket of 300 kg with a 2500 m3 balloon heated to 60 deg C. That sounds reasonable.
Ok, so now, making the (extreme and unrealistic) assumption that we can do the heating adiabatically, and ignoring the gas lost out of the balloon on heating, we can guess the heat required from the burner:
[tex]\mathrm{d}Q = C \mathrm{d}T = \frac{7}{2}\frac{pV}{T} \mathrm{d}T \Rightarrow \Delta Q = \frac{7}{2} p V \ln{\frac{T_{\mathrm{in}}}{T_0}}[/tex]
This gives numbers of order 100 MJ.
Another way of guessing is to say that the "buoyancy energy" must balance the gravitational potential at the surface of the earth, or:
[tex]\Delta Q \ge m g r_{\mathrm{Earth}}[/tex]
This gives an estimate of order 20 GJ.
Which of these estimates is wrong? I think it must be the latter, not least because that sounds like an enormous amount of energy, but I can't work why it doesn't work.
Okay, so first from thermodynamics. Let's assume we're going to heat our balloon (volume [itex]V[/itex], basket mass [itex]m[/itex]) up to some temperature [itex]T_{\mathrm{in}}[/itex]. The ambient temperature is [itex]T_0[/itex], and the condition for liftoff is:
[tex]V(\rho_0 - \rho_{\mathrm{in}}) \ge m[/tex]
Some faffing around with the ideal gas law results in:
[tex]V \ge \frac{m \bar{R}}{p_0 (\frac{1}{T_0} - \frac{1}{T_{\mathrm{in}}})}[/tex]
where [itex]\bar{R}[/itex] is the specific gas constant ([itex]R / M_{r,\mathrm{air}}[/itex]).
So that gives us the volume of balloon we need given a specified temperature difference between the inside and outside of the balloon. Using this expression we find that we can lift off a basket of 300 kg with a 2500 m3 balloon heated to 60 deg C. That sounds reasonable.
Ok, so now, making the (extreme and unrealistic) assumption that we can do the heating adiabatically, and ignoring the gas lost out of the balloon on heating, we can guess the heat required from the burner:
[tex]\mathrm{d}Q = C \mathrm{d}T = \frac{7}{2}\frac{pV}{T} \mathrm{d}T \Rightarrow \Delta Q = \frac{7}{2} p V \ln{\frac{T_{\mathrm{in}}}{T_0}}[/tex]
This gives numbers of order 100 MJ.
Another way of guessing is to say that the "buoyancy energy" must balance the gravitational potential at the surface of the earth, or:
[tex]\Delta Q \ge m g r_{\mathrm{Earth}}[/tex]
This gives an estimate of order 20 GJ.
Which of these estimates is wrong? I think it must be the latter, not least because that sounds like an enormous amount of energy, but I can't work why it doesn't work.