Page 16 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

1.8 Principles of heat and fluid flow 13

include deposition, freezing and solid-to-solid transformation in solid, boiling/evap-
oration, recombination/deionization, and sublimation in gas and condensation and
melting/fusion in liquid. Heat transfer through living biological tissue involves heat
conduction in solid tissue matrix and blood vessels, blood perfusion (convective heat
transfer between tissue and blood), cooling of human body by radiation as well as
metabolic heat generation. Heat transfer processes are classified into three types:
conduction, convection, and radiation.
1.8.2.1 Conduction heat transfer
Conduction heat transfer is the transfer of heat through matter (i.e., solids, liquids, or
gases) without bulk motion of the matter. In another ward, conduction is the transfer
of energy from the more energetic to less energetic particles of a substance due to
interaction between the particles. Conduction heat transfer in gases and liquids is due
to the collisions and diffusion of the molecules during their random motion. On the
other hand, heat transfer in solids is due to the combination of lattice vibrations of the
molecules and the energy transport by free electrons. For example, heat conduction
can occur through wall of a vein in human body. The inside surface, which is exposed
to blood, is at a higher temperature than the outside surface.
To examine conduction heat transfer, it is necessary to relate the heat transfer to
mechanical, thermal, or geometrical properties. Consider steady-state heat transfer
through the wall of an aorta with thickness ∆x where the wall inside the aorta is at

higher temperature (T ) compare to the outside wall (T ). Heat transfer Q(W), is in Q˙ (W)
c
h
direction of x and perpendicular to plane of temperature difference. Heat transfer is
function of aorta wall higher and lower temperature, the aorta geometry and proper-
ties and is given by [51]:
A
Q ∝ ()( ∆T ) (1.17)
∆x Q˙∇(A)(∆T)∆x
or
h
c
Q = kA T ( h − T ) =−kA T ( c − T ) =−kA ∆T (1.18)
∆x ∆x ∆x Q˙=kA(Th−Tc)∆x=−kA(Tc−Th)∆x=
)
In Eq. (1.18), thermal conductivity (k,W/m K is transport property. Parameter A k,W/m K −kA∆T∆x
2
is the cross-sectional area (m ) of the aorta and ∆x is the aorta wall thickness (m). In ∆x
the limiting case of ∆ →x 0 Eq. (1.18) reduces to Fourier’s law of conduction: ∆x→0
Q =−kA dT (1.19)
dx Q˙=−kAdTdx
where dT is the temperature gradient and must be negative based on second law of dTdx
dx
thermodynamics.
A more useful quantity to work with is heat flux, ′′(W/m ), the heat transfer per q″ (W/m )
2
2
q
unit area:
q ′′ = Q
A (1.20) q″=Q˙A