Biomedical Engineering Reference
In-Depth Information
Likewise, a fully analogous analysis can be applied for systems
modeled in cylindrical and spherical coordinates. For cylindri-
cal geometry, the dimensionless temperature is given by
The following two sections present brief descriptions for how
the convective heat transfer coefficient,
h
, and the radiation heat
flux incident on a surface,
q
s
, can be computed to provide quan-
titative boundary conditions for conduction problems as may be
needed.
∞
∑
(
)
2
∗
−ζ
Fo
∗
θ=
Ce
J
ζ
r
(1. 27)
n
0
n
n
=
1
1.2.3 Convection Heat transfer
Convective boundary conditions occur when a solid substrate is
in contact with a fluid at a different temperature. The fluid may
be in either the liquid or vapor phase. The convective process
involves relative motion between the fluid and the substrate. The
magnitude of the heat exchange is described in terms of Newton's
law of cooling, for which the relevant constitutive property of
the system is the convective heat transfer coefficient,
h
(W/m
2
K).
The primary objective of convection analysis is to determine the
value of the convective coefficient,
h
, to apply in Newton's law of
cooling, which describes the convective flow at the surface,
Q
s
, in
terms of
h
, the interface area,
A
, between the fluid and solid, and
the substrate surface and bulk fluid temperatures, (
T
s
) and (
T
):
where
Fo
=
α
t
R
2
.
C
n
satisfies for each value of
n
ζ
ζζ+ζ
2( )
()
J
1
n
C
=
(1. 28)
n
2
2
J
J
()
n
0
n
1
n
and the eigenvalues
ζ
n
are defined as the positive roots of the
transcendental equation
()
()
J
J
ζ
ζ
1
0
n
n
ζ
=
Bi
(1. 29)
n
Bi
hR
k
.
For a spherical geometry, the dimensionless temperature is
given by
where
=
=
Q AT
(
−
T
).
(1. 33)
s
s
∞
There are four distinguishing characteristics of convective
flow that determine the nature and intensity of a convection heat
transfer process. It is necessary to evaluate each of these char-
acteristics to calculate the value for the convective heat transfer
coefficient,
h
. These characteristics and the various options they
may take are:
∞
∑
1
(
)
2
∗
−ζ
Fo
∗
(1. 30)
θ=
Ce
sin
ζ
r
n
n
∗
ζ
r
n
n
=
1
=
α
t
R
2
.
C
n
satisfies for each value of
n
where
Fo
1. The source of relative motion between the fluid and solid,
resulting in forced (pressure driven) or free (buoyancy
driven) convection.
() ()
(
4sin cos
2sin 2
ζ−ζ
ζ
n
n
n
(1. 31)
C
=
n
)
ζ+ ζ
n
n
2. The geometry and shape of the boundary layer region of
the fluid in which convection occurs, producing internal
or external flow. In addition, for free convection the orien-
tation of the fluid/solid interface in the gravitational field
is important.
and the eigenvalues
ζ
n
are defined as the positive roots of the
transcendental equation
1cot
n
−ζ ζ=
Bi
(1. 32)
n
3. The boundary layer flow domain, being laminar or
turbulent.
Bi
hR
k
.
Although the exact solution takes the form of an infinite
series, for many problems it is adequate to use only a limited
number of terms and still maintain an acceptable level of accu-
racy. If the analysis can be restricted to portions of the process
following the initial transient for which
Fo
> 0.2, then only the
first term is required. The closer the analysis must approach
the process beginning, the more terms must be included in the
calculation. In these cases the exact solution still can be com-
puted in a relatively straightforward manner (Diller 1990a, b),
although the detail that must be included increases with each
additional term. Unfortunately, in many classes of biomedical
processes, information concerning the initial transient behavior
is of greatest interest, and it is not possible to use the single term
approximation.
where
=
4. The chemical composition and thermodynamic state of
the fluid in the boundary layer that dictate numerical val-
ues for the constitutive properties relevant to the convec-
tive process.
The influence of each of the four principal characteristics must
be evaluated individually and collectively, and the value deter-
mined for
h
may vary over many orders of magnitude depending
on the combined effects of the characteristics. Table 1.1 presents
the range of typical values for
h
for various combinations of the
characteristics as most commonly encountered.
The relative motion between a fluid and solid may be caused
by differing kinds of energy sources. Perhaps most obviously, an
external force can be applied to the fluid or solid to produce the
motion (which is termed
forced convection
). This force is most
