Increase in Entropy Principle

1. Non-flow Processes

We previously found from considerations of the Clausius Inequality that the following cyclic integral is always less than or equal to zero, where the equality occurred for a reversible cycle.

This lead to the definition of the property Entropy (S). Consider now an irreversible cycle in which process (1) -> (2) follows an irreversible path, and process (2) -> (1) a reversible path, as shown:

Thus the entropy change of an adiabatic process is always greater than or equal to zero, where the equality applies to reversible processes. However not all processes are adiabatic. Nevertheless we can always enclose a system in a surrounding environment which is adiabatic, thus considering the total entropy change of both the system and surroundings we obtain:

Thus the Increase in Entropy Principle states that for any process the total change in entropy of a system together with its enclosing adiabatic surroundings is always greater than or equal to zero. This total change of entropy is denoted the Entropy Generated during the process (Sgen [kJ/K] or sgen [kJ/kg.K]).

2. Flow Processes (Steady Flow)

We now consider the entropy generated during a steady flow process through a single-input/single-output Control Volume (CV) enclosed in an adiabatic surroundings as shown:

Notice that for a steady flow system there can be no change of any of its property values with time, thus the rate of increase of entropy can only be associated with the surroundings. Notice also that at station (2) we are also dumping entropy from the control volume into the surroundings, and at station (1) we are sucking entropy out of the surroundings, leading to:

Surprisingly the form of the specific entropy generated function sgen for a control volume is identical to that for a system.

For multiple-input, multiple-output control volumes under steady flow conditions, the entropy generated function is extended to:

where the summations (
) are taken over all the exit ports (e) and inlet ports (i).

Evaluating Entropy (the Tds relations)

Back to the Clausius Inequality


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Engineering Thermodynamics by Israel Urieli is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License