Page 147 - Mechanical Engineers' Handbook (Volume 4)

136 Exergy Analysis, Entropy Generation Minimization, and Constructal Theory

How do we identify the geometric features that bring a flow architecture to the highest
level of global performance? There are many lessons of this type throughout engineering,
and, if remembered, they constitute strategy—they shorten dramatically the search for the
geometry in which all the features are ‘‘useful’’ in serving the global objective. Constructal
theory is about strategy, about compact lessons of optimal shape and structure, which are
fundamental and universally applicable. They are geometric relatives of truths such as the
universal observation that all things flow naturally from high to low (the second law of
thermodynamics).
Here are the classical lessons that abbreviate the search through the broad categories
listed as (1)–(6). Again, for simplicity assume that ducts are slender, and the flows are slow
so that in each cross section the regime is laminar and fully developed. Each lesson is
identified by the symbol of the geometric feature that it addresses: (1)–(3) a single duct with
large cross section offers a smaller flow resistance than two ducts with smaller cross sections
connected in parallel; (4) the lowest resistance belongs to the shortest duct, in this case the
straight duct between the two points; (5) the duct with cross-sectional geometry that does
not vary longitudinally has a lower resistance than the duct with variable cross section.
Summing up, out of the infinity of designs represented by (1)–(5) we have selected a
single straight duct with a cross-sectional shape that does not vary from one end of the duct
to the other. According to (6), however, there is still an infinite number of possible cross-
section shapes: symmetric vs. asymmetric, smooth vs. polygonal, etc. Which impedes the
flow the least? The answer becomes visible if we assume cross sections with polygonal
shapes. Start with an arbitrary cross-section shaped as a triangle. The area of the cross section
A is fixed because the total duct volume V and the duct length L are fixed, namely A
V/L. Triangular cross sections constrict the flow when one of the angles is much smaller
than the other two.
The least resistance is offered by the most ‘‘open’’ triangular cross section, which is
shaped as an equilateral triangle. Once again, if one very small angle and two larger ones
represent a nonuniform distribution of geometric features of imperfection (i.e., features that
impede the flow), then the equilateral triangle represents the constructal architecture, i.e., the
one with ‘‘optimal distribution of imperfection.’’
The same holds for any other polygonal shape. The least resistance is offered by a cross
section shaped as a regular polygon. In conclusion, out of the infinity of flow architectures
recognized in class (6) we have selected an infinite number of candidates. They are ordered
according to the number of sides (n) of the regular polygon, from the equilateral triangle (n
3) to the circle (n ). The flow resistance for Hagen-Poiseuille flow through a straight
duct with polygonal cross section can be written as (Ref. 5, pp. 127–128)
P LCp 2

˙ m 8V 2 A
where p is the perimeter of the cross section. As shown in Table 1, the dimensionless
perimeter p/A 1/2 is only a function of n. The same is true about C, which appears in the
solution for friction factor in Hagen-Poiseuille flow,
C
ƒ
Re
where Re U D / , D 4A/p, and U ˙m /( A). In conclusion, the group Cp /A depends
2
h
h
only on n, and accounts for how this last geometric degree of freedom influences global
performance. The group Cp /A is the dimensionless global flow resistance of the flow sys-
2
tem. The smallest Cp /A value is the best, and the best is the round cross section.
2