On a sunny morning in a ?eld, the atmospheric pressure is 105 N/m2 and the tempe

Question

On a sunny morning in a ?eld, the atmospheric pressure is 105 N/m2 and the
temperature is 279 K. The ?lm crew has ?lled a balloon with 1000m3 of preheated
air at atmospheric pressure and a temperature of 300 K. The mass of the empty
balloon and its load is 240 Kg. Under these conditions, the balloon rises above its
burden, but the airship remains grounded. To begin the ascent, the air within the
balloon is brought to a higher temperature. It expands but remains at the pressure
of 105 N/m2 . No air enters or leaves the balloon. How much heat must be added to
the air in the balloon to initiate the ?ight?

Explanation / Answer

Density of air is 1.293 on the given temperature.
So we have to find out the temperature which will cause so much expansion as the buoyant force will be equal to the whole weight.
The weight of balloon = 240 * 9.81 = 2354.4 N.

at 300 degree celcius and 1 atm pressure the density of air is 0.616 Kg/m3

The air in the balloon weighs 1000 *0.616*9.81 N = 6042.96 N

The buoyant force on the balloon should equal 6042.96+2354.4 N = 8397.36 N

The final volume of the air should be equal to (8397.36 / 1.293) m3 = 6494.5 m3.

Lets say the final temperature of the air in the balloon is T2

P2 V2/T2=P1V1/T1

or, T2= V2*T1*P2/V1

or, T2= 6494.5*(300+273)/1000 = 3721 K

required temperature rise = 3721-573 K = 3148 K

Required Heat = 3362 kJ

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