A hot air balloon (assumed spherical) rests in equilibrium at an altitude where

Question

A hot air balloon (assumed spherical) rests in equilibrium at an altitude where the atmosphere pressure and temperature are P(infinity) and T(infinity), respectively. The mass of the basket, passengers, and other equipment below the ballon is Mb. The balloon's material is has surface mass density (mass per unit area) of w.
a) For a balloon of given radius r, find an expression for the internal temperature, Ti, of the air in the balloon to keep it aloft. (Hint: Balloon is open at the bottom so its internal pressure must + p infinity)
b) Would the required internal temperature be higher or lower if the same balloon were at higher elevation where p(infinity) was lower than used in (a)?

Explanation / Answer

a)Let the density of air at that altitude be x. accelration of gravity be g then from equating forces we have,buyoncy in UP direction, Weights of balloon and Mb in DOWN direction So, Mb + 4(pi)r2 = 4/3(pi)r3*x*g x = (Mb + 4(pi)r2)/(4/3(pi)r3*g) then by ideal gas equation we have P=x*T*R where R=0.287kJ/kg-k From this we get Ti = P(infinity)/x*R as given P=P(infinity) b)Given P(infinity) is lower so T at this level will be lower than that of Ti as all other parameters reamin constant

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