Biomedical Engineering Reference
In-Depth Information
evaluated. The following are some of the most commonly applied
convection correlation equations with the conditions noted for
which they are valid.
Conditions of validity: fully developed, laminar, uniform
temperature of wall surface, T s .
1.2.3.1 Interior Forced Convection Correlations
For the sake of simplicity, assume the flow to be through a cir-
cular conduit of diameter, D , and length, L . The conduit length
is assumed to be greater than the entrance region at the inlet
over which the boundary layers on opposing surfaces grow until
they meet at the centerline. Downstream of this point the entire
volume of the conduit is filled with boundary layer flow and is
termed fully developed. Fluid properties are evaluated at a mean
temperature, T m , which is an integrated average value for fluid
flowing in the boundary layer through the conduit. T m depends
on the velocity and temperature profiles within the flowing fluid,
which are quite different for laminar and turbulent boundary
layers. In addition, T m will change along the conduit from the
inlet to the outlet as heat is exchanged between the fluid and the
wall. Overall, the temperature at which the properties are evalu-
ated should reflect the average value for all of the fluid contained
in the conduit at any given time. If v m is the mean flow velocity
over the cross-sectional area, A c , of a conduit, then the mass flow
rate, m , is given by
(1.49)
Nu
=
3.66
D
Conditions of validity: fully developed, laminar, uniform heat
flux at the wall surface, q s .
Nu
=
4.36
(1. 50)
D
Conditions of validity: fully developed, turbulent, Re D ≥ 10 4 ,
L / D ≥ 10, 0.6 ≤ Pr ≤ 160, T s > T m .
0.8 .4
(1. 51)
Nu
=
0.23Re
Pr
D
Conditions of validity: fully developed, turbulent, Re D ≥ 10 4 ,
L / D ≥ 10, 0.6 ≤ Pr ≤ 160, T s < T m .
= mAv
.
cm
(1.45)
The net convective heat exchange between the fluid and con-
duit over the entire length equals the change in enthalpy of the
fluid between the inlet and outlet:
0.8 .3
(1. 52)
Nu
=
0.23Re
Pr
D
Conditions of validity: fully developed, turbulent, 3 × 10 3 ≤ Re D
≤ 5 × 10 6 , L / D ≥ 10, 0.5 ≤ Pr ≤ 2000.
QH
=−=
Hmhh mc
(
)
=
(
TT
).
(1.4 6)
out
in
out
in
pmout
,
min
,
(
)
Re
1000 Pr
1
80.79lnRe .64
D
Nu
=
(1.53)
At any cross section along the length of the conduit the rate of
energy flow associated with movement of the fluid (which is the
advection rate) is obtained by integrating across the boundary
layer:
(
)
D
2
23
(
)
12.7Pr 1
8 .79lnRe .64
D
1
+
(
)
12
D
As illustrated by Equations 1.51, 1.52, and 1.53, in some cases
alternative correlation relations are available to calculate a value
for h under the same conditions.
mc Tpvc TdA .
pm
=
(1.47)
p
c
c
1.2.3.2 Exterior Forced Convection Correlations
The properties of the fluid are determined for a state defined by
the temperature T f where
The velocity change with radius over the cross-sectional area in
the above integral is substantially different for laminar and tur-
bulent boundary layers. Eliminating the mass flow rate between
Equations 1.45 and 1.47 yields an expression for the mean tem-
perature over a circular cross-sectional area of outer radius r o ,
for constant density and specific heat:
TT
2
+
s
T
=
(1. 5 4)
f
which is the average of the wall and free stream fluid tempera-
tures. The length of the fluid/substrate interface is L . The fol-
lowing are some of the most commonly applied convection
correlation equations with the conditions noted for which they
are valid.
Conditions of validity: local convection in laminar region for
flow over a flat plate, 0.6 ≤ Pr.
r
o
2
(1.4 8)
T
=
vrTrrdr
() ()
.
m
2
vr
mo
0
The functions v ( r ) and T ( r ) are determined by the profiles of the
velocity and temperature boundary layers specific to the flow
conditions of interest. They provide a basis for determining
the mean temperature for defining the state at which the fluid
properties in the following convection correlation relations are
0.5 .33
Nu
=
0.332Re
Pr
(1. 55)
x
x
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