How can the surface tension of a spherical balloon affect its radius over time?

[member 428835]

Homework Statement

Consider the process of blowing up a spherical balloon. Measurements indicate that the “surface tension” of the balloon material is $k$ (assumed constant here with units of force per unit length). Assuming that an air compressor used to blow up the balloon can deliver a constant mass flow rate of air of $\dot{m}$ , plot the balloon radius, $r_b$, as a function of time assuming $r_b(t) = 0$.

Homework Equations

Conservation of Mass

The Attempt at a Solution

I know the answer is $$\frac{4 \pi}{3} \left. r_b' \right.^2(2+r_b') = t'\\
r_b'=r_b\frac{\rho_{atm} r T}{k}\\
t' = t \frac{\dot{m}}{\rho_{atm}}\left(\frac{\rho_{atm} R T}{}k\right)^3$$

But I'm not sure how to arrive there. I was thinking $$m = \rho V = \rho \frac{4}{3}\pi r^3 \implies \dot{m} = \rho 4 \pi r^2 \frac{dr}{dt}$$. Any ideas?

 

[Simon Bridge]

You should describe the physics you plan to use ... note: you last string of maths appears to assume that the density (of air inside the balloon?) is a constant.
If so, you should probably revisit that assumption.

 

[Ssnow]

Hi, I think that

r′b=rbρatmrTk

is $r_{b}\frac{\rho_{atm}RT}{k}$ and that in the equation:

t′=t˙mρatm(ρatmRTk)3

the $k$ is in the denominator. I think you must remember also the relation between the pressure and the ray of the balloon $\Delta P=\frac{4k}{r}$, and a question I suppose that $t$ is the time and $r_{b}$ the radius but $t',r_{b}'$ what are in your notation?

 

[member 428835]

You should describe the physics you plan to use ... note: you last string of maths appears to assume that the density (of air inside the balloon?) is a constant.
If so, you should probably revisit that assumption.

How would I know (intuitively) that density is not constant? I thought volume would increase do to constant density. At any rate, we then have $$\dot{m} = \frac{d\rho}{dt} \frac{4}{3}\pi r^3+\rho 4 \pi r^2 \frac{dr}{dt}$$
But I'm not sure how to proceed from here.

Hi, I think thatis $r_{b}\frac{\rho_{atm}RT}{k}$ and that in the equation:
the $k$ is in the denominator.

Yes, you are right, I'm sorry for the typos! The correct answer is

$$r_b'=r_b\frac{\rho_{atm} r T}{k}\\
t' = t \frac{\dot{m}}{\rho_{atm}}\left(\frac{\rho_{atm} r T}{k}\right)^3$$

where $t'$ and $r_b'$ are non dimensional parameters for time and radius.

I think you must remember also the relation between the pressure and the ray of the balloon $\Delta P=\frac{4k}{r}$

Where did you get this formula? It definitely seems relevant.

For what it's worth, this question isn't homework (I just posted it here because this seemed appropriate). The question is from this site I found for practice problems:
https://engineering.purdue.edu/~wassgren/notes/COM_PracticeProblems.pdf
and it's problem 23.

Thanks so much for your responses!

 

[Ssnow]

yes, I think that $\frac{r_{b}'}{r_{b}}=\frac{\rho_{atm}rT}{k}$ is another way to write $\frac{1}{r}=\frac{\Delta P}{4k}=\frac{\rho_{atm}rT}{4k}$ using the law that I told you before ...

I am curious on these problems, your link doesn't work on my pc do you have another link ?

Hi

 

[Simon Bridge]

How would I know (intuitively) that density is not constant?

You know that gasses are compressible, that the balloon is elastic, and that you are adding air to the balloon. Is there any reason to assume that the air density inside the balloon is constant?
To handle it you need to know the relationship between particle number, density, and pressure the gas exerts.
That is why I said you should describe the physics you plan to use.

 

[member 428835]

yes, I think that $\frac{r_{b}'}{r_{b}}=\frac{\rho_{atm}rT}{k}$ is another way to write $\frac{1}{r}=\frac{\Delta P}{4k}=\frac{\rho_{atm}rT}{4k}$ using the law that I told you before ...

I am curious on these problems, your link doesn't work on my pc do you have another link ?

Hi

Thanks! And not sure why the link doesn't work? Anyways, here's the main website I found the problem from:
https://engineering.purdue.edu/~wassgren/notes/

You know that gasses are compressible, that the balloon is elastic, and that you are adding air to the balloon. Is there any reason to assume that the air density inside the balloon is constant?
To handle it you need to know the relationship between particle number, density, and pressure the gas exerts.
That is why I said you should describe the physics you plan to use.

Yea, I see what you're saying now about describing the physics.

I thought it would not compress because volume would expand before compressing the fluid. I'll try to apply what you both have said when I get a chance and see if I can get the solution.

 

[TSny]

I think you must remember also the relation between the pressure and the ray of the balloon $\Delta P=\frac{4k}{r}$

This equation is appropriate for a bubble which is modeled as a thin spherical shell which has both an inner spherical surface and an outer spherical surface. See here for a derivation:



You will need to modify this for the case of the balloon which is treated as having only one elastic spherical surface.