I want to find the simplest method to determine whether a proposed heat transfer process violates the second law of thermodynamics. Specifically I am looking for a method that meets the following needs:
- A general method that can be used to analyse any heat transfer process.
- I want to do this without resorting to word definitions such as the
Clausius statement of the second law.
To explain my progress so far, a few days ago the "obvious answer" that I would have given was that $\Delta S_{Universe}$ for the process must be $>=0$.
However, on closer inspection $\Delta S_{Universe}>=0$ does not always mean a process is possible. An example of a process that satisfies this criterion, but is clearly impossible, is using a thermal reservoir to heat a body from $T_R-20K$ to $T_R + 10K$.
The reason that this process is impossible is that the part of the process that involves heating from $T_R$ to $T_R+10K$ involves heat transfer from a cooler body to a hotter body, which results in a decrease of $S_{Universe}$.
Based on this insight I came up with the following rule: Every stage of the process must result in a positive $\Delta S_{Universe}$.
The way we can test for this condition is as follows:
- Formulate an equation for $\Delta S_{Universe}$
- Differentiate the above equation with respect to the state variable of interest (in this case the temperature of the body)
- Set $ \displaystyle\frac{d\Delta S_{Universe}}{dT_{Body}} = 0$ in the above equation
- Solve for $T_{Body}$, which will henceforth be referred to as $T_{MaxS}$
- Check whether either of the following conditions are true: $T_{BodyFinal} < T_{MaxS} < T_{BodyInitial}$ or $T_{BodyInitial} < T_{MaxS} < T_{BodyFinal}$
- If either of the above conditions are true, then by the mean value theorem, some part of the process must have resulted in a negative $\Delta S_{Universe}$. If not, then the process obeys the second law of thermodynamics.
The above method appears to be applicable to all processes, however the issue is that even for a simple two body system this method is difficult and time consuming to carry out. For a worked example of this method see the previous thread that I started:
Can calculations find positive entropy change for heat transfer from cold reservoir to hot body?
Therefore my question is: Does anyone know of a simpler method to analytically determine whether a proposed process obeys the second law of thermodynamics? This must also meet the criteria mentioned above.
I appreciate anyone's time and thank you in advance.
Best Answer
The simplest way would be to measure efficiencies. Carnot's theorem says that the maximal efficiency of a heat engine (or, as you call it, a "heat transfer process") is:$$\eta = 1-\frac{Q_2}{Q_1}=1-\frac{T_2}{T_1}.$$Clausius's based his concept of entropy, that$$\oint\frac{dQ}{T}\le0,$$on Carnot's theorem, where $dQ>0$ means heat is absorbed by the system and "$=$" means an irreversible cycle.
Thus, if Carnot's theorem is violated, so is the 2nd Law.