The Carnot Cycle: Efficiency and Validity with Gases, Solids, and Liquids

[Lucw]

Hello

The Carnot cycle has 4 stages.
1. Isothermal relaxation. The gas provides mechanical energy to the piston and receives heat.
2. Adiabatic relaxation. The gas provides mechanical energy to the piston and cools.
3. Isothermal compression. The gas receives mechanical energy from the piston and gives up heat.
4. Adiabatic compression. The gas receives mechanical energy from the piston and heats up
Carnot has shown that the efficiency of such a cycle with a gas depends only on the temperatures of hot and cold sources.

If instead of using a gas, a solid is used. This one could push the piston when it is heated. And pull it when you cool it. Are Carnot's conclusions still valid?

Carnot's book is available here. https://www3.nd.edu/~powers/ame.20231/carnot1897.pdf. On page 107, Carnot gives his point of view.
Those that we need are the expansive force acquired by solids and liquids by a given increase of temperature, and the quantity of heat absorbed or relinquished in the changes of volume of these bodies).Thank you for your reactions.

Lucw

 

[DrClaude]

Are Carnot's conclusions still valid?

Depends what you mean by "Carnot's conclusions." Given two reservoirs at fixed temperatures $T_h > T_c$, then the maximum efficiency of any engine producing work by using that temperature difference is that of the Carnot engine working with the same reservoirs.

If that was not the case, if there existed an engine with a greater efficiency, then it would be possible, by coupling that engine with a Carnot engine working in reverse, to build a closed system without an external energy source whose net result would be to transfer heat from the reservoir at $T_c$ to the reservoir at $T_h$, violating the 2nd law.

 

[Lucw]

Hello DrClaude.

Thank for your reply.
I have not seen that from this angle.

So. If you heat a gas at a constant volume and then relax it by doing a job, it cools. It's the same for a solid.
(But if you relax the gas in a vacuum (Joule's experiment), it does not cool, so it must be the same for a solid ...)
And in the Carnot cycle, when we compress the gas, we "lose" mechanical energy. We spend it.
With a solid one could recover (not spend) mechanical energy after cooling it ...
I suppose that everything has been measured and checked.
Do you have references?

Have a good day.

Lucw

 

[russ_watters]

If instead of using a gas, a solid is used. This one could push the piston when it is heated. And pull it when you cool it. Are Carnot's conclusions still valid?

Carnot's conclusions are valid for what they apply to. It does not appear to me they apply here. Further/related, I don't see what your idea is even supposed to do. Please describe the steps in your process in a format similar to how you described the carnot cycle.

[edit]
Maybe I can figure it out:

Your setup could simply be a long cylinder with a rod and camshaft, but no piston.

Step 1: Hot air is circulated through the cylinder, heating the rod up and causing it to expand, applying a force/torque to the camshaft to start it turning.

Step 2: When the cam reaches bottom and the rod is fully extended, valves flip and cold air is circulated through to return it to starting position, applying a force/torque as the rod contracts.

I am guessing you think this will provide 100% energy conversion. I think what is probably missing is that energy has to be stored in the rod in order to provide the force and this energy is not recovered. The rod doesn't *just* expand; when the force is applied, that force restricts and reduces the expansion, storing energy like a spring.

Edit2...Some more:
There may also be a perception here that you have energy available in both strokes. That would mean x+x=2x = 200% efficiency. In reality if you are rejecting all of your heat it is x-x=0% efficiency.

 

[Lucw]

Hello Russ_Watters.

Carnot's cycle with a solid...
1 / The solid is heated at constant volume.(or one of its dimensions is kept constant ...)
2 / Then, it relaxes by supplying mechanical energy by pushing a piston. (How does his temperature evolve? ...)
3 / The solid is cooled at constant volume. Or by keeping one of its dimensions constant...
4 / Then it retracts by pulling the piston and thus providing mechanical energy. What can not make a gas; nor a liquid.

I suppose that Rudolf Clausius checked everything before universalizing the Carnot principle on gases.
Carnot who in his book of 1824 wrote ...:
I can't conclude on liquid and solids...

Lucw

 

[russ_watters]

2 / Then, it relaxes by supplying mechanical energy by pushing a piston. (How does his temperature evolve? ...)

Ok, it is what I described, but broken into four parts.

My understanding is that there is no spontaneous temperature change in step 2 and the amount of thermal energy converted to mechanical is so small it tends to be ignored and there is little or no spontaneous temperature change. You can look up if specific heat capacity varies with stress/strain.

 

[russ_watters]

I did a quick calc of thermal vs spring energy for a 10m x 1cm steel rod heated by 100C:

3,200,000 J is stored as thermal energy.
113 J is generated from the spring action, expanding 12mm at a peak force of 18,800N.

That's an efficiency of 0.0035%

Whether the rod cools at all when allowed to expand or not, it would be too small to noticeably affect the efficiency calculation.

 

[Lucw]

Hello

Euh.
274400 joule are store as thermal energy...
I will make the calculation for the generated mechanical energy.
But.
If we recover mechanical energy, we can not find it in the form of heat.
And if we let the piece of steel expand without constraint, then we must find it ...

Have a good day

Lucw