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Shape and structure morphing of systems with internal flows 47

2.3 Shape and structure
The fundamental problem of volume to point flow and the constructal
growth
The roots of the constructal law origins could be traced back to the performance optimi-
zation of engineered, artificial systems such as heat sinks, fluid systems, electrical windings,
etc.—systems of finite size, with purpose, with internal fluxes, under internal and external
constraints, whose structures and shapes are adapted and morphed to optimally function
(Bejan and Errera, 1997). The fundamental structural problem is the minimization of the
volume (territory) to point resistance of single flows (streams, currents, fluxes), which also
explains the morphology of natural flow systems, such as lungs, capillary beds, river basins,
etc. (Bejan, 2000a,b). The elemental system, or the unit cell out of which the growth (in
size) startsisthe seed forhigherorder tree ensemblesthatperform thesamefunction(vol-
ume-to-point discharge), optimally—with minimum flow resistance.
Fig. 2.1 (Morega and Proca, 2004) shows the conduction heat transfer implemen-
5
tation of the constructal principle (Bejan and Errera, 1997). The designer allots a small
quantity of material of higher conductivity to pave a better conduction path (dark) for
the heat generated in cell to reach a port on the boundary, which is otherwise insu-
lated. In the limit k 0 =k p ,, φ ,, 1, φ 5 D 0 =H 0 the analytic solution of the sta-
0 0
tionary diffusion problem may be used to calculate the resistance of M 0 , a design
parameter, defined as the ratio of the maximum internal temperature drop through
the heat current that leaves the cell through the port on the boundary

ΔT 0 1 H 0 k 0 L 0
5 3 1 3 ; ð2:1Þ
qwH 0 L 0 =k 0 8 L 0 2φ k p H 0
0
that shows off two optimization parameters: the cell aspect ratio, AR 0 5 H 0 =L 0 , which

is the cell shape factor, and the composition factor, k 0 =k p =φ , which presents the
0
material properties and the composition of the cell. The design resistance is minimum
for the slenderness factor

1=2

k 0 =k p
H 0
5 2 ;
ð2:2Þ
L 0 opt φ 0

that, interesting enough, spells out the optimal shape of M 0 .
The design may continue, with the first order ensemble, M 1 , Fig. 2.1b, which is in
fact a tree. An additional design parameter, assumed by the designer, intervenes here:
the size of the allotted high conductivity lane, D 1 . The design resistance of this

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The analog problem of volume-to-point current flow was addressed in the photovoltaic power genera-
tion structures (Morega and Bejan, 2005).