Exploring the Ideal Gas Law: A Balloon Problem

[Taulant Sholla]

Homework Statement

Homework Equations

Ideal gas law

The Attempt at a Solution

The solution to this problem assumes the pressure inside the balloon is the same as the outside pressure, i.e. atmospheric pressure. Is this a valid assumption? I would guess otherwise.

 

[BvU]

Is this a valid assumption? I would guess otherwise

Don't hot-air balloons have a big opening at the lower end ?

 

[Taulant Sholla]

Does this opening mean the inside pressure is always the same as the outside pressure? During it ascent wouldn't the pressure inside differ from the outside pressure?

 

[CWatters]

It's a big hole.

Let's work out how fast the air must escape to keep the pressure constant... These are only ball park figures. If you disagree post your own sums!...

Atmospheric pressure halves between sea level and about 18,000ft. So the volume of the balloon would double if no air could escape during such a climb. A typical balloon has a volume of 77,000 cubic feet and climbs are typically limited to 500ft/min. So roughly...

77,000* 500/18,000 = 2100 cubic feet/min

Must escape.

If the opening is say 10ft in diameter (a guess) then it has an area of 78 square foot. So the flow rate is..
2100/78= 27 feet per minute or about 6 inches per second. That shouldn't be a problem.

Just for info... At 500ft/min the air flowing down around the outside of the balloon is 500/60=8 feet per second. So the occupants won't notice the air coming out of the balloon.(Although I guess the might when the burner is running.)

 

[Taulant Sholla]

This is *very* clever - thank you! Yes, I suspect that enough air does indeed escape to keep the inside pressure the same as the outside pressure. Thank you again!

 

[jbriggs444]

This is *very* clever - thank you! Yes, I suspect that enough air does indeed escape to keep the inside pressure the same as the outside pressure. Thank you again!

Of course, there is pressure variation with altitude both inside and outside the balloon. And a [normally plugged] vent hole at the top as well as the one at the bottom. https://www.quora.com/Do-hot-air-balloons-have-a-hole-at-the-top

At the top of the balloon, internal pressure is greater than external pressure. This creates a net upward and outward force on the balloon material near the top. At the bottom of the balloon, internal and external pressure are equalized due to the hole. There is little net downward/inward force.

This pressure variation is the essence of buoyancy. "buoyancy" is nothing more than a word for local fluid pressure integrated appropriately over the surface area of an object. [Which is more conveniently calculated as a volume integral of fluid weight density: the weight of the displaced fluid]