Page 152 - Mechanical Engineers' Handbook (Volume 4)

9 Constructal Theory 141

The alternative is when structural changes are made such that R remains constant while
V is also fixed. Then the evolution in Fig. 14b is from point 1 to point 2 . Such changes
mean that
dL 0 (constant R, V)
and that the constructal law statement becomes
For a flow system with fixed global resistance (R) and internal size (V) to persist in time, the
architecture must evolve in such a way that it covers a progressively larger territory.
Equilibrium is reached at point e . The changes in flow structures in the immediate vicinity
of the equilibrium structure are such that the global external dimension at equilibrium is
maximal,

2
dL 0 dL 0 (constant R, V)
Accordingly, the constructal law states that the ultimate flow structure with specified global
resistance (R) and internal size (V) is the largest. A flow architecture with specified R and
V has a maximum size, and this global size belongs to the equilibrium architecture. A flow
structure larger than this does not exist. This formulation of the constructal law has impli-
cations in natural design, e.g., the spreading of species and river deltas without access to the
sea.
The original statement of the constructal law was about the maximization of flow access
under global size constraints (external L, internal V). This behavior is illustrated by the
structural changes 1–2–e in Figs. 13b and 14b. This means survival by increasing
efficiency—survival of the fittest. This is the physics principle behind Darwin’s observations,
the principle that rules not only the animate natural flow systems, but also the inanimate
natural flow systems and the engineered flow systems. Engineered systems are diverse species
of ‘‘man machine’’ beings.
The alternative shown by the changes 1–2 –e in Fig. 14b is survival by spreading:
growth as the mechanism for being able to persist in time. The limit to growth is set by the
specified constraints, in this case the fixed global flow resistance R and the global internal
size V. A given living species (river delta, animal population) will spread over a certain,
maximal territory.
An equivalent interpretation of the constructal principle is based on processes of type
1–2 –e , Fig. 13b. Flow architectures with the same performance (R) and size (L) evolve
toward compactness—smaller volumes dedicated to the internal ducts, i.e., larger volumes
dedicated to the working volume elements, which are the interstices. This is survival based
on the maximization of the use of the available space.
In summary, changes in performance (R) can be achieved through changes of three
types:
1. Flow configuration
2. Global external size, or covered territory, L.
3. Global internal size, or duct volume, V.
The examples discussed so far showed that changes may occur in one category, or simul-
taneously in two or three. The simplest illustration is possible for the case of equilibrium
flow architectures. For them the solid curves shown in Figs. 13b and 14b proclaim the
existence of the fundamental relation R(L, V), the differential of which is
dR YdL YdV (equilibrium)
V
L