From the information about thermal expansion on Wikipedia, I found that if I heat a steel rod that's 1-meter in length and 1-cm in diameter steel rod to 100 degrees Celsius, then it will expand by approximately 1 mm (based on linear expansion coefficient).
Figure 1: Heating a steel rod from $20°\:\rm C$ to $120°\:\rm C$ causes about $1\:\rm mm$ of linear expansion.
Question: How can the work produced by the expansion of the steel rod be calculated?
There are a lot of equations in internet about ideal gas, etc., but not much for solid bodies.
The idea is to calculate maximum possible work produced by rod. I assume simplest case, no heat loss, vacuum everywhere, linear thermal expansion only to one direction along rod axis (or we can neglect expansion on other axis, because it is much smaller).
The first law of thermodynamics, $$W = Q - U$$ where $W \equiv$ work, $Q \equiv$ amount of heat, and $U \equiv$ change of internal energy.
From this perspective the only question is to calculate internal energy. But could I find how to calculate internal energy for the metal (steel) rod given a specific temperature?
I did my homework and here is some more calculation. For steel rod 1 m length, 1 cm diameter with initial temperature around 0 °C, if I want to heat it from 0 to 100 °C I will need 28302 joules. Length increase will be 1.2 mm. This rod (based on ultimate strength) can hold max 3927 kg, which will give us work 47 joules. It is 0.167 % of energy which I spend for heating. If I take Kevlar, then efficiency will be more 0.7 %.
Now the question is, if I find some very strong, very light material, with very high thermal expansion, so that efficiency (like shown above) will be more than 100 %. It should not be possible, because of energy conservation law. In this case internal energy will be that main reason of efficiency less than 100 %. How to calculate this internal energy?
You can ask, am I inventing combustion engine with solid body as work body instead of gas. Answer is yes, why not :) Most probably it is not possible, otherwise it would be done already. I just want to understand why it is not possible. Maybe it was proven, that gas is most effective, then it would be good to have a link to the article where it was proven.