A hot air balloon has a heated-air capacity of 102,000 ft 3 . Propane heaters ar

Question

A hot air balloon has a heated-air capacity of 102,000 ft 3 . Propane heaters are used to heat the air inside the balloon to 112°C. On a nice ballooning day near Portland, the ambient air temperature is 22°C. The atmospheric pressure is about 1.0 atm.

a. What is the density of the ambient air (outside the balloon)? Please show a calculation using the ideal gas law, do not just right down a number from your notes.

b. What is the density of the heated air inside the balloon? You can safely assume that the pressure of the heated air is essentially the same as the ambient air. Again show your calculation.

c. What is the lift capacity of the balloon? I.e., how many kg of mass can the volume of heated air lift? Show your calculation. Is this enough to carry some passengers and a bottle of champagne?

d. If instead of ballooning in Oregon, we are in Phoenix, Arizona and the ambient temperature is 43°C and the air pressure is still about 1.0 atm. Now, what is the lift capacity of the balloon at the warmer conditions? By what percentage did the lift capacity change from Oregon to Arizona?

Explanation / Answer

Ideal Gas la

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)

Part-a

Given is T= 22 degree celcius or 295 Kelvin

P = 1atm or 101325 Pa

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

by putting all the values in the ideal gas law equation, we get

101325 = * 287 * 295

= 1.19 kg/m3

Part -b

similarly for the density inside the baloon is

P = RT

101325 = * 287 * 385

= 0.917 kg/m3

Part-c

Volume of the heated air inside the baloon is V= 102,000 ft3 or 2888.31 m3

The lift capacity or The net buoyant force is defined here as the difference in density between the surrounding air and the heated air, multiplied by the envelope volume.

Thus

FB,net = (1.19-0.917) * 2888.31 = 788.5 Kg

This is the net buoyant force pushing upwards on the heated air inside the baloon. The hot air balloon components (such as envelope, gondola, burner, fuel tanks, and passengers) can at most weigh 788.5 kg in order for the buoyant force to be able to completely lift the hot air balloon off the ground.

Part-d

To find the lift capacity in Phoenix, first of all we have to find the air density outside the baloon

therefore applying again ideal gas law, we get

101325 = * 287 * 316

= 1.117 kg/m3

now the lift capacity of hot air baloon in Phoenix is

FB,net = (1.1170.917)×2888.31 = 577.66 kg

%change in lift capacity is = (788.5-577.66)*100/788.5 = 26.73%

thus the lift capacity drops by 26.73% from Oregon to Arizona

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