Mass Transport and Heat Transfer in the Microcirculation - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 237

which can be also be related to the viscous forces present within the media or fluid by

1
κ

- 2 - 2 r - 5

2

μr 0

7.6 Heat transfer is an integral component to many biological systems. Three basic mechanisms

for heat transfer include conduction, convection, and radiation. Most heat transfer formula-
tions begin with the Conservation of Energy principle, which states that

ð

E fluid 5 mc p T m 5

ρc p TðrÞuðrÞdA

A

From this equation, the temperature difference within a fully developed, two-dimensional
flow scenario can be solved from

2 2

ρc p u @ T

@x 1 v @ T

5 k @

@x 2 1 @

T T

@y 2 @y

or

u @ T

ρrc p dr

k

r @T
@r @z 5

if the viscous forces are negligible. This leads us to two specialized problems, one in which
there is a constant surface heat flux and a second where there is a constant surface tempera-
ture. In the case where we can assume that

q s is a constant, then the rate of heat of transfer

can be expressed as

Q 5 q s A s 5 mc p ðT o 2 T i Þ

and the mean fluid outflow temperature and the mean surface temperature become
T o 5 T i 1 q s A s
mc p

T s 5 T m 1 q s
h
If the conditions are such that we can make the assumption of a constant surface tempera-
ture, then the heat transfer and temperature conditions will be modified as

Q 5 hA s ΔT average

hAs
mcp

T o 5 T s 2 ðT s 2 T i Þe 2

Using the log mean temperature difference, the heat transfer rate can also be calculated from

Q 5 hA s ΔT lm

T i 2 T o
ln T s 2 T o
T s 2 T i

h i

ΔT lm 5