How Hot Must the Air in a Balloon Be to Lift a Given Mass?

[erok81]

Homework Statement

I have a balloon with a volume of 500m³
Outside air temp of 300K
Mass to lift of 300kg
Molar mass of air is 28 g/mol (I didn't end up using this)

I am to find the temperature inside the balloon to barely lift the given mass. I have apparently forgotten everything buoyancy related.

Homework Equations

F_b=ρgV

The Attempt at a Solution

Okay...I know that in order for the balloon to lift F_b > F_g (subscript b = buoyancy and subscript g is gravity force)

Using F_b=ρgV I tried taking the differences of the the two air densities and setting it equal to my mass I need to lift. As seen here.

(ρ_air-ρ_ballon)Vg=mg

Since I need to find the temp required, I found this relationship.

ρ=P/R_specT where P=pressure, R_spec is a known value, and of course T=temp. R_spec = 287.058 J/kg*K for dry air

P_ballon and P_air are equal.

Subbing that stuff in I get...

$\left(\frac{P}{R_{spec}T_{air}}-\frac{P}{R_{spec}T_{balloon}}\right) Vg=m_{cargo}g$

I enter my values into the above equation...the units work out, but I am leaving them off for brevity.

$\left(\frac{1.013e5}{(287.058)(300)}-\frac{1.013e5}{287.058T}\right)(500)(9.81)=(300)(9.81)$

After some rearraging etc I end up with T=612K.

Looking up hot air balloon temps I see my answer is waaay off.

What I am not understanding here that is causing my incorrect answer?

 

[deep838]

R_spec is a known value, and of course T=temp. R_spec = 287.058 J/kg*K for dry air

Ok... but shouldn't it be different for dry air at 300K and air at a higher temperature? Consider that and try solving the problem once again. I don't see any other errors.

Could you give the answer as well?

 

[erok81]

Hmm...I'm not sure that it depends on temp.

R_spec=R/M where R is gas constant and M is molar mass.

With that said, I don't see how it can vary with temp.

 

[deep838]

Oh yeah... sorry! That said, I have no idea on how to help you. I really see no error in your working. You could go through your calculations once again! 600 K does seem to be too much to me as well.

 

[zeralda21]

1.
Using F_b=ρgV I tried taking the differences of the the two air densities and setting it equal to my mass I need to lift. As seen here.

(ρ_air-ρ_ballon)Vg=mg

I have a very similar problem so I'd rather just bump this thread.

I don't understand the step above. It seems that there are two upward forces(one outside of the balloon) but from my intuition I can only count one(the one inside). So can someone clarify this step?

 

[haruspex]

Your answer looks correct. It could be unreasonably high because the given conditions (volume, load) are unreasonable.

 

[zeralda21]

Your answer looks correct. It could be unreasonably high because the given conditions (volume, load) are unreasonable.

I think you have misunderstood my concern. This is not my thread but I have a very similar exercise which I solved using the formulas above(indeed correct), without understanding one of the steps. This one:

(ρair-ρballon)Vg=mg

Since ρVg=F_{b} I understand this as there are THREE acting forces, one inside the balloon and one outside the balloon(which is in the NEGATIVE y-direction) and mg. I find this strange since the balloon is not moving.

 

[haruspex]

I think you have misunderstood my concern.

I was responding to the OP. Didn't notice it was a bit old. That's a risk with piggy-backing on an existing thread.

I solved using the formulas above(indeed correct), without understanding one of the steps. This one:

(ρair-ρballon)Vg=mg

Since ρVg=F_{b} I understand this as there are THREE acting forces, one inside the balloon and one outside the balloon(which is in the NEGATIVE y-direction) and mg. I find this strange since the balloon is not moving.

I can't follow your reasoning. Which density are you using in ρVg=F_b, and what exactly is F_b?
There are forces all over the place, but you can group them into:
- gravity on the balloon contents
- net force from surrounding air
- suspended load
It's a statics question: what temperature will generate enough lift to balance the other forces. What do you find strange?