A very large hot air balloon and its accompanying structure (ex: the basket, rig

Question

A very large hot air balloon and its accompanying structure (ex: the basket, rigging, equipment, etc.) have a mass of 400 kg. The air inside the balloon must be heated to a temperature of 200 degrees C for the balloon to life off the ground when the outside air temperature is 16.0 degrees C. To what temperature must the air be heated if the balloon is to lift off with the three 80.0 kg people aboard? (NOTE: this problem requires three pictures of the balloon (COLD, HOT, HOTTER) with the appropriate information labeled in each picture.) ANSWER IS : 492 Degrees C (need help figuring out how to get the answer)

Explanation / Answer

The heated air inside the envelope is at roughly the same pressure as the outside air. With this in mind we can calculate the density of the heated air at a given temperature, using the Ideal gas law, as follows:

P = RT

Where:

P is the absolute pressure of the gas, in Pa

is the density of the gas, in kg/m3

R is the gas constant, in Joules/kg.K

T is the absolute temperature of the gas, in Kelvins (K)


Now,

Normal atmospheric pressure is approximately 101,300 Pa

The gas constant for dry air is 287 Joules/kg.K

The air inside the envelope is typically heated to an average temperature of about 200 degrees Celsius, which is 473 K

Substituting the above three values into the Ideal gas law equation and solving for we get = 0.75 kg/m3. This is the density of the heated air inside the envelope. Compare this to normal (ambient) air density which is approximately 1.2 kg/m3.

because of buoyant force balloon starts to lift off

Vballon g +400= Vballonair g

Vballon=90.7 m3

the balloon is to lift off with the three 80.0 kg people aboard

Vballon g +400 +240= Vballonair g

=0.48 kg/m3

again

P = RT

T= 735 K

=462 degree c

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