Page 153 - Mechanical Engineers' Handbook (Volume 4)

142 Exergy Analysis, Entropy Generation Minimization, and Constructal Theory

Physics requires that the first partial derivatives of R have opposite signs, Y 0 and Y
L
V
0, as noted earlier in this section. In general, when the flow architecture has not reached
equilibrium, R can be decreased by means 1, 2, and 3. Then the general version of the last
equation is
dR
YdL YdV
V
L
where the inequality sign refers to the time arrow of structural changes in a flow configuration
that, at least initially, was not of the equilibrium type. This inequality is a concise statement
of the three analytical formulations of the constructal law that we discussed so far:

R minimum at constant L and V
V minimum at constant R and L
L maximum at constant V and R
Another way to summarize the analytical formulation that we have just constructed is
by recognizing the analogy between the analytical constructal law and the analytical for-
mulation of classical thermodynamics (cf. the preceding chapter in this handbook). The
analogy is presented in Table 2. It is stressed further by Fig. 2 in Chapter 3, which is from
1
present-day thermodynamics. Figure 2 in Chapter 3 expresses the energy minimum princi-
ple, which states that as the internal constraints of a closed system are removed at constant
volume and entropy, the energy approaches a minimal value. Figure 2 in Chapter 3 is analo-
gous to Fig. 14a in this Chapter.
The analytical formulation of the constructal law presented in this section expresses a
universal phenomenon: figures such as Fig. 12 characterize the evolution toward equilibrium
configuration in any flow system with global objective, global constraints, and freedom to
morph. In Ref. 6, this was demonstrated through examples from three wide classes of flow
architectures: flow between two points, flow between a circle and its center, and flow between
one point and an area. Many other examples can be contemplated, and they will all reveal
the image of Fig. 12 on the road to equilibrium flow architectures.
At equilibrium the flow configuration achieves the most that its freedom to morph has
to offer. Equilibrium does not mean that the flow architecture (structure, geometry, config-
uration) stops changing. On the contrary, it is here at equilibrium that the flow geometry
enjoys most freedom to change. Equilibrium means that the global performance does not
change when changes occur in the flow architecture.

Table 2 The Concepts and Principles of Classical Thermodynamics and Constructal Theory 6
Thermodynamics Constructal Theory
State Flow architecture (geometry, configuration, structure)
Process, removal of internal constraints Morphing, change in flow configuration
Properties (U, S, Vol, . . .) Global objective and global constraints (R, L, V,. ..)
Equilibrium state Equilibrium flow architecture
Fundamental relation, U(S, Vol, . . .) Fundamental relation, R(L, V,. ..)
Constrained equilibrium states Nonequilibrium flow architectures
Removal of constraints Increased freedom to morph
Energy minimum principle: Constructal principle:
U minimum at constant S and Vol R minimum at constant L and V
Vol minimum at constant F and T V minimum at constant R and L
S maximum at constant U and Vol L maximum at constant V and R