Expansion or compression -- which is more energy efficient?

[T C]

TL;DR

I am describing here a thought experiment where power consumption for expanding and compressing has been compared side by side to find out which one consumes more energy.

I want to share a thought experiment here. Suppose, we have a cylinder with a frictionless piston added to it. Enclosed in it is 1 gm-mole of a diatomic gas at 1 kg/cm^2 pressure and having volume of 0.0224 m^3. The temperature is 27°C i.e. 300°K.

Now, in the first scenario, the gas has been expanded adiabatically by pulling the piston outwards and the volume have been increased to 0.0672 m^3 i.e. three times to its initial volume. As the process is adiabatic, the temperature will fall to -80°C or 193.2°K. That's the temperature of the gas after expansion.

Now, in this scenario, without going into complex equations, we can assume that the cylinder is being pushed against atmospheric pressure i.e. 1 kg/cm^2. We know that the increase in volume and by multiplying both, the amount of energy consumed in the process has been found to be 448 J. and, kindly note that this is the highest amount of energy consumption. Actually the power consumption would be less as the pressure inside isn’t zero but will gradually decrease. But, for the sake of simplicity, let’s consider the power consumption to be this for now.

Now, in the second scenario, the same gas has been compressed to 1/3rd of its original volume. In that case, the temperature will rise to 465.54°K or 192.39°C. The power consumption is 3440 J in the whole process. Now, the hot has been cooled to 27°C i.e. 300°K and is being released to atmospheric pressure i.e. 1 kg/cm^2 pressure. The fall in temperature in that case would be the same as the first scenario as described before i.e. -80°C or 193.2°K. That's the temperature of the gas after being released.

In short, we get the same level of cooling with both the cases. But in the first case, the power consumption would be much less in comparison to the second. Want to know others opinions in this regards.

 

[russ_watters]

Without checking your math, please note that your first case doesn't consume/input mechanical work it outputs mechanical work. The gas pushing on the piston causes the expansion. The piston is not being pulled away from the gas (unless you are also doing work against the atmosphere...).

 

[T C]

(unless you are also doing work against the atmosphere...)

That's what I have suggested. The pressure inside is less while outside it's higher and that's atmospheric.

 

[jack action]

Now, the hot has been cooled to 27°C i.e. 300°K and is being released to atmospheric pressure i.e. 1 kg/cm^2 pressure.

It is not clear which processes (isochoric, isobaric, isentropic, isothermal) you are using to cool and depressurize the fluid in the second scenario. One could assume an isochoric cooling followed by an isothermal expansion, but you seem to talk about opening a valve between the cylinder and the atmosphere to equalize the pressure, and this would further cool the air coming out with an isentropic process, in addition to changing the mass of fluid present inside the cylinder.

Also, could you show your math so that we can see how you got your numbers, and we don't have to guess.

 

[T C]

It is not clear which processes (isochoric, isobaric, isentropic, isothermal) you are using to cool and depressurize the fluid in the second scenario.

Hope the first scenario is clear to you. The second scenario that I have described is like that. The compression is isenthalpic/adiabatic and you can use the adiabatic process equations here for 1 gm-mole of gas. Not very hard I think. After being compressed, the gas is being cooled in isobaric way. And, at the end, it has been released in an isenthalpic way. Hope it's clear now.

Also, could you show your math so that we can see how you got your numbers, and we don't have to guess.

For the first process of the second process? The first process isn't that complicated. Just think of an adiabatic/isenthalpic process where the gas is forcefully expanded to 3 times its initial volume. I think you can do the math by yourself.
The second process have been clearly described and I am sure you can do the math by yourself.

 

[jack action]

the gas is being cooled in isobaric way

That is also an isobaric compression, requiring more work.

And, at the end, it has been released in an isenthalpic way.

Do you mean like an isothermal expansion? Where both work is released and heat added?

The first process isn't that complicated.

Well, I'm glad it is, then it will be easy for you to do the work and show it to us. Just use the LaTeX notation to ease the reading of your equations.

A P-V diagram might be helpful as well.

I am sure you can do the math by yourself.

I'm sure I can too. It is just easier for me - or anyone else - to read your work and look for errors or misinterpretations than for me to make the whole work and hand it freely in a post, just to find out it is not what you meant because we misunderstood each other.

People want to help you after you did the work, not do it for you (even if you swear you already did it).

 

[T C]

First, I must admit my mistake. The calculated power for the first process that I have said before is wrong. It’s not 448 W but rather higher amount. And I am calculating it below:

Let’s calculate the process in this way:

The piston is moving against atmospheric pressure i.e. 101325 N/m^2 of force per unit area and the change in volume is (0.0672 m3 – 0.0224 m^3) i.e. 0.0448 m^3.

Therefore, the power consumption by the piston for moving against atmospheric pressure to increase the volume is (101325 X 0.0448) Joules i.e. 4539.38 Joules.

But, we also have to calculate the work done by the gas inside as its volume has increased to three times its initial volume and its temperature has fallen.

Considering the process to be adiabatic and using the adiabatic process formula of TVγ-1, the final temperature after the completion of the expansion process is T
₂
= T1(V1/V2)γ-1; the final temperature has been found to be 300°K (initial temperature) X (1/3)..4 (considering γ = 1.4).

Therefore, the final temperature is 193.31°K.

That means the fall in temperature is 106.68°K.

Therefore, the work done by gas inside the cylinder is 5 X 4.2 X 106.68 J i.e. 2240.28 Joules.

This amount has to be subtracted from the initial power consumption calculation for working against atmospheric pressure i.e. 4539.38 J.

At the end, the net power consumption has been found to be (4539.38 - 2240.28) Joules i.e. 2299.1 Joules.

That is sufficiently less than the 3440 Joules power consumption.

The reason behind such less power consumption during the expansion lies in one fact. And that is during the compression process, the work done on the gas increases its temperature and that heat has to be released for getting the necessary cooling.

But, during expansion, the work done by the piston have been stored as potential energy of the expanded gas and there no increase in temperature occurs during process and that means no wastage of energy as heat as part of the process itself.