Page 113 - Mechanical Engineers' Handbook (Volume 4)

102 Thermodynamics Fundamentals

The entire left-hand side in this inequality is by definition the entropy generated by the
process,
S S 2 Q
S gen 2 1
1 T
The entropy generation is a measure of the inequality sign in the second law and hence a
measure of the irreversibility of the process. As shown in the next chapter, the entropy
generation is proportional to the useful work destroyed during the process. 1,3,4 Note again
that any heat interaction ( Q) is accompanied by entropy transfer ( Q/T), whereas the work
transfer W is not.

4 ENERGY MINIMUM PRINCIPLE
Consider now a closed system that executes an infinitesimally small change of state, which
means that its state changes from (U, S, ...) to (U dU, S dS, ...). The first law and the
second law statements are
Q W dU
Q
dS 0
T
If the system is isolated from its environment, then W 0 and Q 0, and the two laws
dictate that during any such process the energy inventory stays constant (dU 0), and the
entropy inventory cannot decrease,
dS 0
Isolated systems undergo processes when they experience internal changes that do not require
intervention from the outside, e.g., the removal of one or more of the internal constraints
plotted qualitatively in the vertical direction in Fig. 2. When all the constraints are removed,
changes cease, and, according to dS 0, the entropy inventory reaches its highest possible
level. This entropy maximum principle is a consequence of the first and second laws. When
all the internal constraints have disappeared, the system has reached the unconstrained equi-
librium state.
Alternatively, if changes occur in the absence of work transfer and at constant S, the
first law and the second law require, respectively, dU Q and Q 0, hence
dU 0
The energy inventory cannot increase, and when the unconstrained equilibrium state is
reached the system energy inventory is minimum. This energy minimum principle is also a
consequence of the first and second laws for closed systems.
The interest in this classical formulation of the laws (e.g., Fig. 2) has been renewed by
the emergence of an analogous principle of performance increase (the constructal law) in
5
the search for optimal configurations in the design of open (flow) systems. This analogy is
based on the constructal law of maximization of flow access, 1,6 and is summarized in the
next chapter.

5 THE LAWS OF THERMODYNAMICS FOR OPEN SYSTEMS
If ˙m represents the mass flow rate through a port in the control surface, the principle of mass
conservation in the control volume is