Page 19 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 5

The first principle, Eq. (1.3), written for an open control volume yields the following
(Bejan, 1988):
dE X X
_
5 Q 2 _ W 1 _ me 1 PvÞ 2 _ me 1 PvÞ; ð1:9Þ
ð
ð
dt
inlets outlets
where _m [kg/s] is the mass flow rate, e [J/kg] is the energy density, P [Pa] is the pres-
3
sure, v [m /kg] is the specific volume, and the symbolic sums account for the inlet and
outlet mass transfer through all permeable ports on the boundary.
For a system of volume V Σ , that is, bounded by the closed surface Σ, Eq. (1.9)
becomes
ð I I
@ ρeðÞ
dv 5 qUndA 2 _ W 1 ρhvUndA; ð1:10Þ
@t
V Σ Σ Σ
3 2
where ρ [kg/m ] is the mass density, q [W/m ] is the heat flux rate, v [m/s] is the
velocity, and h 5 u 1 Pv [J/kg] is the specific enthalpy. It is assumed that the only
form of energy storage is the specific internal energy, u [J/kg].
The closed surface integrals that replace the sums in Eq. (1.10) suggest that the
mass transport, the heat transfer, and the work transfer with the environment may
occur (may be distributed) everywhere all over the boundary. These surfaces or flux
integrals may be replaced with volume integrals by using Gauss integral (divergence)
theorem (Annex 1) to yield the following relation between the integrands

@ ρeðÞ
52rUq 2 ww 2 PrUv; ð1:11Þ
@t
@ρ
where mass conservation principle, 52 ρrÞUv, is used.
ð
@t
3
In Eq. (1.11) ww [W/m ] stands for a specific heat generation rate—the “work”
needed for the electrical current density to flow through the electroconductive
medium is converted into heat, through Joule Lentz effect. For instance, in an
electrokinetic problem ww 5 EJ,where E [V/m] is the electric field strength, and
2
J [A/m ] is the electrical conduction current density.
It is worth noting that the first principle, Eq. (1.11), or the energy equation, in a par-
ticular form or another, is the mathematical model skeleton of the heat transfer used
throughout this book.

Electromagnetic power transferred through the boundary
(at the electrical terminals)
_
Depending on the nature of the work interactions, W in Eq. (1.3) may have different
expressions. For instance, for a closed system situated in an external electromagnetic