Page 45 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 31

Helomoltz problems, are depicted usually using field lines (streamlines in fluid
mechanics), and arrows. Further on the scalar fields (temperature, species, etc.) in cou-
pled diffusion convection (Morega and Nishimura, 1996) are visualized, in general,
through isosurfaces of those scalars and the field lines of their conjugated fluxes. For
instance, the solution to the stationary convection diffusion heat transfer problem for
incompressible flow, described by the following energy equation:

@T
ρc 1 uUrÞT 5 r krTÞ 1 qw; ð1:35Þ
ð
ð
@t
where qw is the local heat source, is presented through isothermal surfaces, T is a con-
stant, and heat flux 2krT vector field lines, as the solution of the regular conduction
(diffusion) problem.
In fact this diffusion-type visualization is less relevant and rather confusing here
because the convective transport of energy that adds to the conduction is not evi-
denced in this way. To solve this difficulty, Eq. (1.35) may be rewritten as follows:
ρcrU uT 2 krTÞ 5 qw ; ð1:36Þ
ð
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |{z}
heat source = sink
H f
where a new quantity, the heatfunction (Bejan, 1984) emerges

H f 5 uT 2 krT : ð1:37Þ
|{z} |ffl{zffl}
transport diffusion
A vast body of work is devoted to present the energy paths unveiled through this vector
fieldthatsumsupthe twounderlyingheattransfermechanisms(Morega, 1988; Morega and
Bejan, 1993, 1994). This new function is much more than a merely mathematical conse-
quence of diffusion convection. It has a strong physical meaning too by relating the trans-
port and diffusion mechanisms in a single quantity, which makes H f avaluable aid in
evidencing the enthalpy corridors that lead the convection diffusion heat transfer.
Similarly all convection diffusion scalar fields may be visualized using such companion
vector fields. For example, in convection diffusion mass transfer problems, the mass func-
3
tion, M f , may be introduced through M f 5 uc 2 Drc (Bejan, 1984), where c [mol/m ]is
2
the species concentration, and D [m /s] is the mass diffusivity of the species.
Consider the stationary forced convection mass transfer problem in a parallel plate channel,
in laminar, Newtonian, incompressible, fully developed (Hagen Poiseuille) flow. The channel
is 70-μm wide and the maximum velocity is 1 mm/s. The fluid is a methanol water mixture
3
with 0 90 wt.% CH 3 OH, ρ 5 800 1000 kg/m , η 5 0.891 1.033 poises. The species is
25 2
salicylic acid, and its diffusivity is D5 {1,0.5,1}3 10 cm /s (Chaaraoui et al., 2017). A
patch placed on the upper wall is the mass source @c 5 0. The horizontal walls are imperme-
@n
able, the inlet has a homogeneous Dirichlet condition, and the outlethomogeneous