Page 63 - Computational Modeling in Biomedical Engineering and Medical Physics

50 Computational Modeling in Biomedical Engineering and Medical Physics

Figure 2.2 Volume-to-point pressure driven flow. The boundary is impermeable, except for the
output port.

The analysis is much simplified for K ,, K 0 and D 0 ,, H 0 , when the flows in
the K and K 0 regions are unidirectional, yielding

K @P K 0 @P
v 5 2 ; u 5 2 ;
μ @y μ @x ð2:3Þ
where P is pressure and μ is the kinematic viscosity. The peak pressure drop on the
cell (left upper and lower corners of the cell), P peak,1 divided through the volumetric
flow rate that crosses the port M, _mw, yields the design flow resistance of the cell
1
P peak;0 H 0 K 0 L 0
5 3 1 3 ;
_ mw 0 A 0 v 8 2φ K ð2:4Þ
L 0
0 H 0
where φ 5 D 0 =H 0 . As for the conduction problem discussed above, two factors are
0
evidenced as optimization parameters: the aspect ratio of the cell, AR 0 5 H 0 =L 0 , and
the composition factor φ 0 . The optimization (minimization) of Eq. (2.4) yields

H 0 21=2 1 21=2
; Δ ~ P 0 5 ;
0
5 2 φ ~ K 0 φ ~ K 0 ð2:5Þ
0
L 0 opt 2

~
~

where the definitions H i ; L i 5 ð H i ;L i Þ ~ K i D i are used. As expected, this
1=2 , K i 5 , φ 5
A K i H i
0
flux-gradient flow is analogous to the heat conduction problem above, hence the heat
transfer results Eqs. (2.1) and (2.2) may translated into their Darcy flow counterparts.
The constructal design sequence, toward higher order ensembles and its outcomes are
the same.
Moreover, for channels with clear fluid Hagen Poiseuille flow (Chapter 1:
Physical, Mathematical, and Numerical Modeling), if D i is small enough then
2
K i 5 D =12 (i 5 1,2,...)(Leopold et al., 1964). This analysis may be repeated
i
sequentially, noting that the permeabilities are no longer independent. The
results of this constructal growth as well as a comprehensive discussion on more
complex, three-dimensional flow structures may be found in Bejan (2000a,b).