What Happens to Energy When a Balloon Deflates?

[XJL488Hax]

Homework Statement

Hi, I'm currently trying out a practical problem where I'm trying to calculate the energy present in an inflated balloon, as well as what happens to that energy once the air is let out from inside the balloon. The balloon is attached to a car, so that as the air escapes it propels the car forward. Unfortunately, I don't really have a problem statement, as this is a problem which I encountered while doing an external project, although a sample problem statement could be "Analyse and determine the energy changes within the system(the system being composed of the air, the balloon, and the car)."

Homework Equations

Work done by an expanding gas(considered adiabatic here) = area under a pV diagram
Kinetic energy of car = 1/2 mv^2
Energy of a Gas: E = 3/2 nRT = 3/2 PV

The Attempt at a Solution

First, I came up with a basic diagram to try to show the energy changes inside the car(shown below).

Basically, in its inflated state, energy in the balloon comes from two sources: The energy from the pressurised air inside the balloon, and the elastic potential energy of the balloon. I attempted to calculate the energy of the pressurised air by using the equation to calculate the kinetic energy of the gas molecules(which is the internal energy of the gas):
E = 3/2nRT = 3/2 PV.
For this experiment, I especially plotted a PV diagram for this purpose, which was obtained by measuring the internal pressure of the balloon at various volumes.

Using values which I had obtained from the PV diagram, I managed to calculate the energy to be around 394.36J. The work done by an expanding gas is the area under the curve of a PV diagram, and I assumed that the work done is on the balloon, meaning that as the gas expands, it transfers its energy to the balloon. Hence the work done should be the balloon's change in elastic potential energy. From the diagram, I calculated this amount to be about 263.48J. Hence the total energy should be around 657.84J.

Likewise, as the balloon deflates, it does work on the gas, transferring back its energy to the gas. However this energy is slightly less than the energy that was given to the balloon, because some energy was used to deform the balloon, a phenomenon known as the Mullins' Effect. The work done on the gas was calculated to be around 261.77J. As the balloon deflates, the air rushes out, causing its internal energy to be converted to kinetic energy. In addition, the energy transferred to it by the balloon is also converted to kinetic energy. This kinetic energy, in turn, is transferred to the car and is used to propel it. (Note that when the air is being expelled out, some energy will be lost due to internal viscosity)
The kinetic energy of the car increases steadily, maxing out at a certain velocity, before decreasing again due to various friction forces doing work against the car. Because it was difficult to measure the velocity of the air, I measured the velocity of the car using a motion analysis program, and use it to indirectly calculate the kinetic energy of the car and thus the air.

From the graph, the maximum velocity of the car is 93.2cm/s, and considering its mass is about 91.2g, its maximum kinetic energy is only about 0.04J. Hence I'm a bit confused here. This energy, is, hypothetically saying, supposed to be the maximum kinetic energy of the air, but the energy difference between this and its initial state(657.74J) is very large. Surely the energy losses aren't that much? Did I miscalculate anything or had some conceptual errors? Many thanks for your guidance! (also apologies for the long entry)

 

[mfb]

You have to put your car "into the tube" to use the full kinetic energy of the air to propel the car.

Your car is a rocket, and rockets are horribly inefficient if the exhaust velocity is much larger than the speed of the object. With the exhaust velocity and mass flow rate, you can calculate a force that pushes the car forwards, and translate this to an acceleration.

 

[Chestermiller]

XJL488hax:

This is a fairly complicated system analyze. It usually isn't a good idea to model the entire system all at one time. The three most important things about doing good modelling are

1. start simple
2. start simple
3. start simple

Try to break this problem down into simpler bite-size chunks and obtain solutions to these more manageable problems. Why? If you can't solve the simpler (sub-model) versions of the problem, then you will never be able to solve the entire problem in all its complexity. Plus, once you have analyzed a sub-model, you will already have some results under your belt.

I suggest starting with a model in which the balloon is replaced by a rigid container. I would have the container nailed down to the floor so that it isn't even being used to accelerate a model car. The deformability of the balloon just adds complexity to the situation. You can add that back in later. For a given starting pressure in the rigid container, I would just try to determine the force that the exiting gas exerts on the container as a function of time.

Regarding what you have done so far, the work under the PV curve is not the elastic energy stored in the balloon. The elastic energy stored in the balloon is equal to the inside pressure minus the outside pressure, integrated with respect to the volume change. Maybe that is one reason why your estimated stored elastic energy came out so high.

Chet