Biomedical Engineering Reference
In-Depth Information
Conditions of validity: convection averaged across laminar
region,
L
, for flow over a flat plate, 0.6 ≤ Pr.
T
s
< T
∞
0.5 .33
(1. 56)
Nu
=
0.664 Re
Pr
L
L
Conditions of validity: local convection in turbulent region for
flow over a flat plate, Re
x
≤ 10
8
, 0.6 ≤ Pr ≤ 60.
0.8 .33
(1. 57)
Nu
=
0.296Re
Pr
x
x
ρ
∞
, T
∞
Conditions of validity: convection averaged across the combined
laminar and turbulent regions of total length,
L
, for flow over a
flat plate, 0.6 ≤ Pr.
g
0.8
0.33
(1. 58)
Nu
=
(0.037Re
−
871)Pr
L
L
Conditions of validity: convection averaged across the entire
surface around a cylinder of diameter,
D
, in perpendicular flow,
0.4 ≤ Re
D
≤ 4 × 10
5
, 0.7 ≤ Pr.
FIGURE 1.6
Growth of a free convection boundary layer on a vertical
cooled plate.
n
0.33
(1. 59)
Nu
=
C
Re
Pr
D
D
are some of the most commonly applied free convection correla-
tion relations. Many make use of a dimensionless constant, the
Rayleigh
number
,
Ra
=
Gr
·
Pr
.
Conditions of validity: free convection averaged over a vertical
plate of length
L
, including both the laminar and turbulent flow
regions over the entire range of
Ra.
where the values of
C
and
n
are functions of Re
D
as given
below:
Re
D
C
n
0.4-4
0.989
0.330
4-40
0.911
0.385
2
1/6
0.387
1 .492 /Pr
Ra
40-4,000
0.683
0.466
L
(1. 61)
Nu
=
0.825
+
L
8/27
4,000-40,000
0.193
0.618
(
)
9/16
+
40,000-400,000
0.027
0.805
Conditions of validity: free convection averaged over a vertical
plate of length
L
for laminar flow defined by 10
4
≤
Ra
L
≤ 10
9
.
Conditions of validity: convection averaged across the entire
surface around a sphere of diameter,
D
, properties based on
T
∞
,
3.5 ≤ Re
D
≤ 7. 6 × 10
4
, 0.71 ≤ Pr ≤ 380.
1/4
Nu
=
0.59
Ra
(1.62)
L
L
0.25
µ
µ
Conditions of validity: free convection averaged over a vertical
plate of length
L
for turbulent flow defined by 10
9
≤
Ra
L
≤ 10
13
.
(1.6 0)
0.5
0.67
0.4
Nu
=+
2 0.4Re .06Re Pr
+
D
D
D
s
1/3
Nu
=
0.1
Ra
(1.63)
L
L
1.2.3.3 Free Convection Correlations
Since free convection processes are driven by buoyant effects,
determination of the relevant correlation relations to deter-
mine the convection coefficient must start with analysis of the
shape and orientation of the fluid/solid interface. This effect is
illustrated with the free convection boundary layer adjacent to
a vertical cooled flat plate shown in Figure 1.6. Note that the
flow velocity is zero at both the inner and outer extremes of
the boundary layer, although the gradient is finite at the solid
interface owing to viscous drag of the fluid. The environment
is assumed to be quiescent so that there is no viscous shearing
action at the outer region of the boundary layer. The following
Conditions of validity: free convection averaged over the upper
surface of a heated plate or lower surface of a cooled plate having
a dimension
L
; 10
4
≤
Ra
L
≤ 10
7
.
1/4
Nu
=
0.5
Ra
(1.6 4)
L
L
Conditions of validity: free convection averaged over the upper
surface of a heated plate or lower surface of a cooled plate having
a dimension
L
; 10
7
≤
Ra
L
≤ 10
11
.
1/3
Nu
=
0.15
Ra
(1.65)
L
L
