Page 150 - Mechanical Engineers' Handbook (Volume 4)

9 Constructal Theory 139

meable’’ the flow system is. The constant-V plane that cuts through Fig. 13a is the same as
the plane of Fig. 12.
The constructal law is the statement that summarizes the common observation that flow
structures that survive are those that morph in one direction in time: toward configurations
that make it easier for currents to flow. This holds for natural and engineered flow structures.
The first such statement was 1,5
For a finite-size system to persist in time (to live), it must evolve in such a way that it provides
easier access to the imposed currents that flow through it.
If the flow structures are free to change (free to approach the base plane in Fig. 13a),
they will move at constant L and constant V in the direction of progressively smaller R.If
the initial configuration is represented by point 1 in Fig.13b, then a more recent configuration
is represented by point 2. The relation between the two configurations is R
R (constant
2
1
L, V). If freedom to morph persists, then the flow structure will continue toward smaller R
values. Any such change is characterized by
dR
0 (constant L, V)
The end of this migration is the equilibrium flow structure (point e), where the geometry of
the flow enjoys total freedom. Equilibrium is characterized by minimal R at constant L and
V. In the vicinity of the equilibrium point we have
dR 0 and dR 0 (constant L, V)
2
The R(V) curve shown in Fig. 13b is the edge of the cloud of possible flow architectures
with the same global size L. The curve has negative slope because of the physics of flow:
the flow resistance always decreases when the flow channels open up, ( R/ V) 0.
L
The constant-R cut through the configuration space shows another way of expressing
the constructal law. If free to morph, the flow system will evolve from point 1 to point 2
at constant L and R. In the limit of total freedom, the geometry will reach another equilibrium
configuration, which is represented by point e . The alternative analytical statement of the
constructal law is
dV
0 (constant L, R)
For changes in structure in the immediate vicinity of the equilibrium structure, we note
2
dV 0 and dV 0 (constant L, R)
Paraphrasing the original statement of the constructal law, we may describe processes of
type 1–2 –e as follows:
For a system with fixed global size and global performance to persist in time (to live), it must
evolve in such a way that its flow structure occupies a smaller fraction of the available space.
The constant-V alternative to Fig. 13 is shown in Fig. 14. The lower drawing is the
projection of the space of possible flow architectures on the base plane R-L. The continuous
line is the locus of equilibrium flow structures at constant V, namely the curve R(V), where
( R/ L) 0. The fact that the slope is positive is flow physics: the flow resistance always
V
increases as the distance that must be overcome by the flow increases.
The constructal law statement can be read off Fig. 14b in two ways. One is the original
statement : at constant V and L, the evolution is from a suboptimal structure (point 1) to
1,5
one that has a lower global resistance (point 2). If the flow geometry continues to morph
freely, the structure approaches the equilibrium configuration (point e).