When I was working on problems related to entropy generation, there was this problem that asks if the process claimed in the problem is possible or not. The solution manual calculated the total entropy change and it was positive, therefore the total entropy generation is also positive, so it concluded that the process is possible (occurs spontaneously).
But I was thinking, the total entropy generation is the sum of the entropy generation of some systems, then if the total entropy generation is positive this doesn't mean that the entropy generation of each system is positive, and the process that was concluded possible may not be possible. Shouldn't we inspect for each system separately.
To make the question clearer, this is a derivation of the principle of the increase of entropy from my text book. At the end of the derivation (equation 6.39), you see that total entropy change equals to S(gen A) + S(gen B)+ S(gen C). Now if we inspect the total entropy change and it was positive, does this necessarily mean that S(gen A) is positive and S(gen B) is positive and S(gen C) is positive, shouldn't we inspect for each system separately.
Can any one help me with this? Thanks in advance.
Best Answer
It is not clear what you mean by "external entropy generation is due to heat transfer through finite temperature difference." This portion of the entropy generation is included either in the entropy generated in the system or the entropy generated in the surroundings (or both). However, the surroundings is usually modeled as an ideal reservoir (or reservoirs) in which case the entropy generation in the surroundings is zero. Therefore, this portion of the entropy generation is typically part of the entropy generated in the system.
With regard to the second part of your question, entropy generation is always positive, no matter where it occurs. However, entropy transfer as a result of heat flow from one entity to another can be either positive or negative, depending on whether the heat is flowing in or out. So, there are two ways that the entropy of an entity can change: entropy transfer from another entity or entropy generation (within the entity under consideration).
ADDENDUM
In their book Transport Phenomena by Bird, Stewart, and Lightfoot, they prove in Chapter 11, Example 11D.1., Equation of change for entropy, that the local rate of entropy generation per unit volume in a system experiencing an irreversible process is the sum of three contributions: one contribution proportional to the squares of local velocity gradients, a second contribution proportional to the square of the local temperature gradient, and a third contribution proportional to the squares of the local species concentration gradients. Since squares of the quantities are involved, they are all positive definite.