Biomedical Engineering Reference
In-Depth Information
layer defines the region wherein viscous drag causes a velocity
gradient as the interface is approached. The region outside the
boundary layer where the viscous properties do not affect the
flow pattern is called the inviscid free stream. The fluid veloc-
ity outside the boundary layer is designated by
v
∞
(m/s), and the
boundary layer thickness by δ(m), which increases with distance
along the interface from the point of initial contact between the
fluid and solid. In this case, it is assumed that the interface is
planar and the fluid flow is parallel to the interface. In like man-
ner, a thermal boundary layer develops in the flowing fluid as
heat transfer occurs between a solid substrate and fluid that are
at dissimilar temperatures.
The velocity and temperature boundary layers have similar
features. Both define a layer in the fluid adjacent to a solid in
which a property gradient exists. The temperature boundary
layer develops because there is a temperature difference between
the fluid in the free stream
T
and the solid surface
T
s
. A tem-
perature gradient exists between the free stream and the surface,
with the maximum value at the surface, and which diminishes
to zero at the outer limit of the boundary layer at the free stream.
The temperature gradient at the surface defines the thermal
boundary condition for conduction in the solid substrate. The
boundary condition can be written by applying conservation of
energy at the interface. Since the interface has no thickness, it
has no mass and is therefore incapable of energy storage. Thus,
the conductive inflow is equal to the convective outflow as illus-
trated in Figure 1.5.
An important feature of the convection interface is that there
is continuity of both temperature and heat flow at the surface,
the latter of which is expressed in Equation 1.34:
over different regions of an interface as a function of the local
boundary layer characteristics. In some cases it is necessary to
determine these local variations, requiring more detailed calcu-
lations. Often it is sufficient to use a single average value over the
entire interface, thereby simplifying t
h
e analysis. The averaged
heat transfer coefficient is denoted by
h
L
, where the subscript
L
defines the convective interface dimension over which the aver-
aging occurs.
Values for the convective heat transfer coefficient appropri-
ate to a given physical system are usually calculated from cor-
relation equations written in terms of dimensionless groups of
system properties. The most commonly applied dimensionless
groups are defined as follows.
he
Nusselt number
is a dimensionless expression for the con-
vective heat transfer coefficient defined in Equation 1.35. It can
be written in terms of local or averaged (over an entire interface
surface) values:
hx
k
hL
k
L
Nu
=
or
Nu
=
.
(1. 35)
x
L
f
f
An important distinction to note is that
k
f
is the thermal con-
ductivity of the fluid, not of the solid substrate. The Nusselt
number describes the ratio of total convection effects to pure
thermal conduction in the fluid. In principle, its value should
always be greater than 1.0 since convection is a combination of
both conduction plus advection in the fluid.
Nu
represents the ratio of the temperature gradient in the
fluid at the interface with the solid to an overall reference tem-
perature gradient based on a physical dimension of the sys-
tem,
L
. This dimension has a different meaning, depending
on whether the flow geometry is internal or external. For an
internal flow in which the boundary layer occupies the volume
normal to the interface, the dimension represents the cross-
sectional size of the flow passage, such as the diameter,
D
. For
an external flow configuration in which the size of the boundary
layer can grow normal to the interface with no physical restric-
tion, the relevant dimension is the distance along the interface
from the leading edge at which the fluid initially encounters the
solid substrate, such as the length,
L
. It is important to identify
dT
dy
(
)
−
k
=
hT
−
T
.
(1. 34)
f
s
∞
y
=
0
The magnitude of convection heat transfer is directly depen-
dent on the size and flow characteristics within the boundary
layer. As a general rule, thicker boundary layers result in a larger
resistance to heat transfer and, thus, a smaller value for
h
. he
result is that there can be local variations in convective transport
T
∞
v
∞
y
T
s
, h
x
dT
dy
,
k
FIGURE 1.5
Convective boundary condition at the surface of a conducting tissue.
