4* Analogy Between Heat and Momentum Transfer in Turbulent Flow
Tải bản đầy đủ - 0trang
67706_06_ch06_p350-419.qxd
5/14/10
12:45 PM
Page 383
6.4 Analogy Between Momentum and Heat Transfer in Turbulent Flow
383
Similarly, the shearing stress caused by the combined action of the viscous forces
and the turbulent momentum transfer is given by
m
t
du
= a
+ eM b
r
r
dy
(6.48)
According to the Reynolds analogy, heat and momentum are transferred by analogous processes in turbulent flow. Consequently, both q and vary with y, the distance from the surface, in the same manner. For fully developed turbulent flow in a
pipe, the local shearing stress increases linearly with the radial distance r. Hence, we
can write
y
t
r
=
= 1 ts
rs
rs
(6.49)
and
qc>A
(qc>A)s
=
y
r
= 1 rs
rs
(6.50)
where the subscript s denotes conditions at the inner surface of the pipe. Introducing
Eqs. (6.49) and (6.50) into Eqs. (6.47) and (6.48), respectively, yields
y
ts
m
du
a1 + eM b
b = a
rs
r
r
dy
(6.51)
and
qc,s
Asrcp
a1 -
y
k
dT
b = -a
+ eH b
rs
rcp
dy
(6.52)
If eH = eM, the expressions in parentheses on the right-hand sides of Eqs. (6.51) and
(6.52) are equal, provided the molecular diffusivity of momentum > equals the
molecular diffusivity of heat k> rcp, that is, the Prandtl number is unity. Dividing Eq.
(6.52) by Eq. (6.51) yields, under these restrictions,
qc,s
Ascpts
du = - dT
(6.53)
Integration of Eq. (6.53) between the wall, where u ϭ 0 and T ϭ Ts, and the bulk of
the fluid, where u = Uq and T ϭ Tb, yields
qsUq
= Ts - Tb
Ascpts
which can also be written in the form
ts
qs
hqc
1
=
=
2
As(Ts - Tb) cprUq
cprUq
rUq
(6.54)
since qhc is by definition equal to qs>As(Ts - Tb). Multiplying the numerator and the
denominator of the right-hand side by DHk and regrouping yields
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
384
5/14/10
12:45 PM
Page 384
Chapter 6 Forced Convection Inside Tubes and Ducts
hqc
qhcDH
DHmk
m
k
Nu
= a
ba
ba
b =
= St
cpm UqDHr
k
RePr
cprUq DHmk
where St is the Stanton number.
To bring the left-hand side of Eq. (6.54) into a more convenient form, we use
Eqs. (6.13) and (6.14):
ts = f
rUq 2
8
Substituting Eq. (6.14) for s in Eq. (6.54) finally yields a relation between the
Stanton number St and the friction factor
St =
f
Nu
=
RePr
8
(6.55)
known as the Reynolds analogy for flow in a tube. It agrees fairly well with experimental data for heat transfer in gases whose Prandtl number is nearly unity.
According to experimental data for fluids flowing in smooth tubes in the range
of Reynolds numbers from 10,000 to 1,000,000, the friction factor is given by the
empirical relation [17]
-0.2
f = 0.184ReD
(6.56)
Using this relation, Eq. (6.55) can be written as
St =
Nu
-0.2
= 0.023ReD
RePr
(6.57)
Since Pr was assumed unity,
Nu = 0.023Re0.8
D
(6.58)
m -0.8
qhc = 0.023Uq 0.8D-0.2k a b
r
(6.59)
or
Note that in fully established turbulent flow, the heat transfer coefficient is
directly proportional to the velocity raised to the 0.8 power, but inversely proportional to the tube diameter raised to the 0.2 power. For a given flow rate, an increase
in the tube diameter reduces the velocity and thereby causes a decrease in hqc proportional to 1> D1.8. The use of small tubes and high velocities is therefore conducive to
large heat transfer coefficients, but at the same time, the power required to overcome
the frictional resistance is increased. In the design of heat exchange equipment, it is
therefore necessary to strike a balance between the gain in heat transfer rates
achieved by the use of ducts having small cross-sectional areas and the accompanying increase in pumping requirements.
Figure 6.17 shows the effect of surface roughness on the friction coefficient.
We observe that the friction coefficient increases appreciably with the relative
roughness, defined as ratio of the average asperity height to the diameter D.
According to Eq. (6.55), one would expect that roughening the surface, which
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
5/14/10
12:45 PM
Page 385
6.4 Analogy Between Momentum and Heat Transfer in Turbulent Flow
385
0.1
0.09
0.08
0.05
0.04
0.07
0.06
0.03
0.05
f, friction factor
0.04
0.01
0.008
0.006
0.03
0.004
0.025
0.02
Laminar flow
f = 64
ReD
Equation 6.56
Laminar
flow
Transition
zone
0.002
0.001
0.0008
0.0006
0.0004
0.015
0.01
0.009
0.008
0.0002
0.0001
Complete turbulence, rough pipes
0.00005
Critical
zone
103
2
Relative roughness ε
D
0.02
0.015
3 4 5 6789104
2
3 4 5 6789105 2 3 4 5 6789106 2
Reynolds number ReD = ρuD/µ
0.0001
2 3 4 5 6789108
ε = 0.000005
D
ε = 0.000001
D
3 4 5 6789107
FIGURE 6.17 Friction factor versus Reynolds number for laminar and turbulent flow in tubes with various
surface roughnesses.
Source: Courtesy of L. F. Moody, “Friction Factor for Pipe Flow,” Trans. ASME, vol. 66, 1944.
increases the friction coefficient, also increases the convection conductance.
Experiments performed by Cope [28] and Nunner [29] are qualitatively in agreement with this prediction, but a considerable increase in surface roughness is
required to improve the rate of heat transfer appreciably. Since an increase in the
surface roughness causes a substantial increase in the frictional resistance, for the
same pressure drop, the rate of heat transfer obtained from a smooth tube is larger
than from a rough one in turbulent flow.
Measurements by Dipprey and Sabersky [30] in tubes artificially roughened
with sand grains are summarized in Fig. 6.18 on the next page. Where the
Stanton number is plotted against the Reynolds number for various values of the
roughness ratio e> D. The lower straight line is for smooth tubes. At small
Reynolds numbers, St has the same value for rough and smooth tube surfaces.
The larger the value e> D, the smaller the value of Re at which the heat transfer
begins to improve with increase in Reynolds number. But for each value of e> D,
the Stanton number reaches a maximum and, with a further increase in Reynolds
number, begins to decrease.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
386
5/14/10
12:45 PM
Page 386
Chapter 6 Forced Convection Inside Tubes and Ducts
4 × 10–3
St = NuD / ReD Pr
3
0.08
ε /D = 0.04
0.02
2
0.01
ε /D = 0.005
0.002
10–4
5 × 103 8 104
0.0005
Smooth pipe
6
5×
ε /D = 0.001
10–3
8
2
4
6 8 105
ReD
2
4
6 8 × 105
FIGURE 6.18 Heat transfer in artificially roughened tubes, St versus
Re for various values of e/D according to Dipprey and Sabersky [30].
Source: Courtesy of T. von. Karman, “The Analogy between Fluid Friction and Heat
Transfer,” Trans. ASME, vol. 61, p. 705, 1939.
6.5
Empirical Correlations for Turbulent Forced Convection
The Reynolds analogy presented in the preceding section was extended semi-analytically to fluids with Prandtl numbers larger than unity in [31–34] and to liquid metals with very small Prandtl numbers in [31], but the phenomena of turbulent forced
convection are so complex that empirical correlations are used in practice for engineering design.
6.5.1 Ducts and Tubes
The Dittus-Boelter equation [35] extends the Reynolds analogy to fluids with
Prandtl numbers between 0.7 and 160 by multiplying the right-hand side of
Eq. (6.58) by a correction factor of the form Prn:
NuD =
hqcD
n
= 0.023Re0.8
D Pr
k
(6.60)
where
n = e
0.4 for heating (Ts 7 Tb)
0.3 for cooling (Ts 6 Tb)
With all properties in this correlation evaluated at the bulk temperature Tb,
Eq. (6.60) has been confirmed experimentally to within Ϯ25% for uniform wall
temperature as well as uniform heat-flux conditions within the following ranges of
parameters:
0.5 6 Pr 6 120
6000 6 ReD 6 107
60 6 (L> D)
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
5/14/10
12:45 PM
Page 387
6.5 Empirical Correlations for Turbulent Forced Convection
387
Since this correlation does not take into account variations in physical properties due
to the temperature gradient at a given cross section, it should be used only for situations with moderate temperature differences (Ts - Tb).
For situations in which significant property variations due to a large temperature difference (Ts - Tb) exist, a correlation developed by Sieder and Tate [16] is
recommended:
1/3
NuD = 0.027Re0.8
D Pr a
mb 0.14
b
ms
(6.61)
In Eq. (6.61), all properties except ms are evaluated at the bulk temperature. The viscosity ms is evaluated at the surface temperature. Equation (6.61) is appropriate for
uniform wall temperature and uniform heat flux in the following range of conditions:
0.7 6 Pr 6 10,000
6000 6 ReD 6 107
60 6 (L> D)
To account for the variation in physical properties due to the temperature gradient in
the flow direction, the surface and bulk temperatures should be the values halfway
between the inlet and the outlet of the duct. For ducts of other than circular crosssectional shapes, Eqs. (6.60) and (6.61) can be used if the diameter D is replaced by
the hydraulic diameter DH.
A correlation similar to Eq. (6.61) but restricted to gases was proposed by Kays
and London [17] for long ducts:
0.3
NuDH = CRe0.8
DH Pr a
Tb n
b
Ts
(6.62)
where all properties are based on the bulk temperature Tb. The constant C and the
exponent n are:
0.020 for uniform surface temperature Ts
C = e
0.020 for uniform heat flux q s–
n = e
0.020 for heating
0.150 for cooling
More complex empirical correlations have been proposed by Petukhov and
Popov [38] and by Sleicher and Rouse [37]. Their results are shown in Table 6.3 on
the next page, which presents four empirical correlation equations widely used by
engineers to predict the heat transfer coefficient for turbulent forced convection in
long, smooth, circular tubes. A careful experimental study with water heated in
smooth tubes at Prandtl numbers of 6.0 and 11.6 showed that the Petukhov-Popov
and the Sleicher-Rouse correlations argeed with the data over a Reynolds number
range between 10,000 and 100,000 to within Ϯ5%, while the Dittus-Boelter and
Sieder-Tate correlations, popular with heat transfer engineers, underpredicted the
data by 5 to 15% [38]. Figure 6.19 on the next page shows a comparison of these
equations with experimental data at Pr ϭ 6.0 (water at 26.7°C). The following
example illustrates the use of some of these empirical correlations.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
TABLE 6.3
5/14/10
12:45 PM
Page 388
Heat transfer correlations for liquids and gases in incompressible flow through tubes and pipes
Formulaa
Name (reference)
Dittus-Boelter [35]
NuD =
ne
Sieder-Tate [16]
Petukhov-Popov [36]
n
0.23Re0.8
D Pr
= 0.4 for heating
= 0.3 for cooling
mb 0.14
b
ms
where
0.5 6 Pr 6 120
(6.60)
6000 6 ReD 6 107
0.7 6 Pr 6 104
( f/8)ReDPr
K1 + K2( f/8)
Equation
6000 6 ReD 6 107
0.3
NuD = 0.027Re0.8
D Pr a
NuD =
Conditions
1/2
(Pr
2/3
0.5 6 Pr 6 2000
104 6 ReD 6 5 * 106
- 1)
(6.61)
(6.63)
f = (1.82 log10 ReD - 1.64)-2
K1 = 1 + 3.4f
1.8
K2 = 11.7 +
Sleicher-Rouse [37]
Pr 1/3
NuD = 5 + 0.015ReaDPrbs
where
a = 0.88 -
0.1 6 Pr 6 105
104 6 ReD 6 106
(6.64)
0.24
4 + Prs
b ϭ 1/3 ϩ 0.5eϪ0.6Prs
a
All properties are evaluated at the bulk fluid temperature except where noted. Subscripts b and s indicate bulk and surface temperatures,
respectively.
103
9
8
Range of experimental data
7
6
Petukhov-Popov
5
Nusselt number, NuD
Sleicher-Rouse
4
Sieder-Tate
3
Dittus-Boelter
2
102
3 × 104
388
105
Reynolds number, ReD
2 × 105
FIGURE 6.19 Comparison of predicted and measured Nusselt number for turbulent flow of water in
a tube (26.7°C; Pr ϭ 6.0).
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
5/14/10
12:45 PM
Page 389
6.5 Empirical Correlations for Turbulent Forced Convection
EXAMPLE 6.5
SOLUTION
389
Determine the Nusselt number for water flowing at an average velocity of 10 ft/s in
an annulus formed between a 1-in.-OD tube and a 1.5-in.-ID tube as shown in Fig.
6.20. The water is at 180°F and is being cooled. The temperature of the inner wall
is 100°F, and the outer wall of the annulus is insulated. Neglect entrance effects and
compare the results obtained from all four equations in Table 6.3. The properties of
water are given below in engineering units.
T
(°F)
m
(lbm /h ft)
k
(Btu/h ft °F)
r
(lbm/ft3)
c
(Btu/lbm °F)
100
140
180
1.67
1.14
0.75
0.36
0.38
0.39
62.0
61.3
60.8
1.0
1.0
1.0
The hydraulic diameter DH for this geometry is 0.5 in. The Reynolds number based
on the hydraulic diameter and the bulk temperature properties is
ReDH =
(10 ft/s)(0.5/12 ft)(60.8 lbm/ft3)(3600 s/h)
rUqDH
=
m
0.75 lbm/h ft
= 125,000
The Prandtl number is
Pr =
cpm
k
=
(1.0 Btu/lbm °F)(0.75 lbm/h ft)
= 1.92
(0.39 Btu/h ft °F)
The Nusselt number according to the Dittus-Boelter correlation [Eq. (6.60)] is
0.3
NuDH = 0.023 Re0.8
= (0.023)(11,954)(1.22) = 334
DH Pr
Using the Sieder-Tate correlation [Eq. (6.61)], we get
0.3
NuDH = 0.27Re0.8
DH Pr a
mb 0.14
b
ms
= (0.027)(11,954)(1.24) a
0.75 0.14
= 358
b
1.67
1.5 in.
1 in.
Water in
annulus
180°F
10 ft/s
Insulation
Inner wall temperature = 100°F
FIGURE 6.20 Schematic diagram of annulus for
cooling of water in Example 6.5.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
390
5/14/10
12:45 PM
Page 390
Chapter 6 Forced Convection Inside Tubes and Ducts
The Petukhov-Popov correlation [Eq. (6.63)] gives
f = (1.82 log10ReDH - 1.64)-2 = (9.276 - 1.64)-2 = 0.01715
K1 = 1 + 3.4f = 1.0583
K2 = 11.7 +
NuDH =
1.8
= 13.15
Pr0.33
f ReDHPr/8
K1 + K2(f/8)1/2( Pr 0.67 - 1)
(0.01715)(125,000)(1.92/8)
=
1.0583 + (13.15)(0.01715/8)1/2(0.548)
= 370
The Sleicher-Rouse correlation [Eq. (6.64)] yields
NuDH = 5 + 0.015ReaDPrbs
a = 0.88 b =
0.24
= 0.88 - 0.0278 = 0.852
4 + 4.64
1
0.5
0.5
+ 0.6 Pr = 0.333 +
= 0.364
s
3
16.17
e
ReD = 82,237
NuDH = 5 + (0.015)(82,237)0.852(4.64)0.364
= 5 + (0.015)(15,404)(1.748) = 409
Assuming that the correct answer is NuDH = 370, the first two correlations underpredict NuDH by about 10% and 3.5%, respectively, while the Sleicher-Rouse
method overpredicts by about 10.5%.
It should be noted that in general, the surface and film temperatures are not
known and therefore the use of Eq. (6.64) requires iteration for large temperature
differences. The main difficulty in applying Eq. (6.63) for conditions with varying
properties is that the friction factor f may be affected by heating or cooling to an
unknown extent. Thus, to account for variable property effects in the flow cross section due to a significant temperature difference between the tube surface and bulk
fluid, a correction factor is commonly employed. This is usually in the form of a
bulk-to-surface viscosity ratio or temperature ratio raised to some power, depending
on whether the fluid is heated or cooled in the tube; two examples are given in
Eqs. (6.61) and (6.62)
For gases and liquids flowing in short circular tubes (2 Ͻ L> D Ͻ 60) with abrupt
contraction entrances, the entrance configuration of greatest interest in heat exchanger
design, the entrance effect for Reynolds numbers corresponding to turbulent flow
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
5/14/10
12:45 PM
Page 391
6.5 Empirical Correlations for Turbulent Forced Convection
391
becomes important [40]. An extensive theoretical analysis of the heat transfer and
the pressure drop in the entrance regions of smooth passages is given in [41], and
a complete survey of experimental results for various types of inlet conditions is
given in [40].
The most commonly used and widely accepted correlation in current practice
for turbulent flows in circular tubes, however, and one that accounts for both variable property and entrance length effects is the Gnielinski correlation [42]. It is a
modification of the Petukhov and Popov [36] equation, is valid for the transition
flow and fully developed turbulent flow regimes (2300 … ReD … 5 * 106) as well
as a broad spectrum of fluids (0.5 6 Pr … 200), and is expressed as follows:
NuD =
(f> 8)(ReD - 1000) Pr
C 1 + (D> L)2/3 D K
1 + 12.7(f> 8)1/2( Pr 2/3 - 1)
(6.65)
where
(Prb> Pr s)0.11 for liquids
(Tb> Ts)0.45 for gases
and the friction factor f is calculated from the same expression used in the PetukhovPopov correlation of Eq. (6.65), as listed in Table 6.3. Note that instead of a viscosity ratio, the ratio of Prandtl number at bulk fluid and tube surface temperatures has
been used to account for variable property effects. This same correction factor can
be used as a multiplier to calculate f as well.
K = e
6.5.2 Ducts of Noncircular Shape
In many heat exchangers, rectangular, oval, trapezoidal, and concentric annular flow
passages, among others, are often employed. Some examples include plate-fin, ovaltube-fin, and double-pipe heat exchangers. The generally accepted practice in most
such cases, to a fair degree of accuracy as verified with experimental data [43], is to
use the circular-tube correlations with all dimensionless variables based on the
hydraulic diameter to estimate both the convective heat transfer coefficient and friction factor in turbulent flows. Thus, any of the correlations listed in Table 6.3 could
be employed, although the more popular recommendation in many handbooks is for
the Gnielinski correlation of Eq. (6.65)
The exception to this rule is the case of turbulent flows in concentric annuli
where the curvatures of the inner and outer diameters, or Di and Do, tend to have an
effect on the convective behavior, particularly when the ratio (Di> Do) is small [44,
45]. Based on experimental data and an extended analysis [44], the following correlation has been proposed:
NuDH = Nuc C1 + {0.8(Di> Do)-0.16}15 D
1/15
(6.66)
where Nuc is calculated from Eq. (6.65), again by using the hydraulic diameter of
the annular cross section, DH = (Do - Di), as the length scale. The duct-wall curvature effect, represented by the diameter ratio used in Eq. (6.66) is a modified
form of the correction factor considered by Petukhov and Roizen [45].
Furthermore, if effects of temperature-dependent fluid property variations in the
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
392
5/14/10
12:45 PM
Page 392
Chapter 6 Forced Convection Inside Tubes and Ducts
flow cross section have to be included in the analysis, then the same correction
factor K recommended in Eq. (6.65) may be employed for liquids or gases, as the
case may be.
6.5.3 Liquid Metals
Liquid metals have been employed as heat transfer media because they have certain advantages over other common liquids used for heat transfer purposes.
Liquid metals, such as sodium, mercury, lead, and lead-bismuth alloys, have relatively low melting points and combine high densities with low vapor pressures
at high temperatures as well as with large thermal conductivities, which range
from 10 to 100 W/m K. These metals can be used over wide ranges of temperatures, they have a large heat capacity per unit volume, and they have large convection heat transfer coefficients. They are especially suitable for use in nuclear
power plants, where large amounts of heat are liberated and must be removed in
a small volume. Liquid metals pose some safety difficulties in handling and
pumping. The development of electromagnetic pumps has eliminated some of
these problems.
Even in a highly turbulent stream, the effect of eddying in liquid metals is of
secondary importance compared to conduction. The temperature profile is established much more rapidly than the velocity profile. For typical applications, the
assumption of a uniform velocity profile (called “slug flow”) may give satisfactory
results, although experimental evidence is insufficient for a quantitative evaluation
of the possible deviation from the analytic solution for slug flow. The empirical
equations for gases and liquids therefore do not apply. Several theoretical analyses
for the evaluation of the Nusselt number are available, but there are still some unexplained discrepancies between many of the experimental data and the analytic
results. Such discrepancies can be seen in Fig. 6.21, where experimentally measured
Nusselt numbers for heating of mercury in long tubes are compared with the analysis of Martinelli [2].
Lubarsky and Kaufman [46] found that the relation
NuD = 0.625(ReDPr)0.4
(6.67)
empirically correlated most of the data in Fig. 6.21, but the error band was substantial. Those points in Fig. 6.21 that fall far below the average are believed to
have been obtained in systems where the liquid metal did not wet the surface.
However, no final conclusions regarding the effect of wetting have been reached
to date.
According to Skupinski, et al. [47], the Nusselt number for liquid metals flowing in smooth tubes can be obtained from
NuD = 4.82 + 0.0185(ReDPr )0.827
(6.68)
if the heat flux is uniform in the range ReDPr 7 100 and L> D 7 30, with all properties evaluated at the bulk temperature.
According to an investigation of the thermal entry region for turbulent flow of
a liquid metal in a pipe with uniform heat flux, the Nusselt number depends only on
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
67706_06_ch06_p350-419.qxd
5/14/10
12:45 PM
Page 393
6.5 Empirical Correlations for Turbulent Forced Convection
Lyon (theoretical)
Trefethen (mercury)
Johnson, Harinett, and Clabaugh (mercury
and lead-bismuth: laminar and transition)
Johnson, Clabaugh, and Hartnett (mercury)
Stromquist (mercury)
English and Barrett (mercury)
Untermeyer (lead-bismuth)
Untermeyer (lead-bismuth plus magnesium)
Nusselt number, NuD = hcD/k
102
393
Seban (lead-bismuth)
Isakoff and Drew (mercury: inside wall temperatures
calculated from fluid temperature profiles)
Isakoff and Drew (mercury: inside wall temperatures
calculated from outside wall temperature)
Johnson, Hartnett, and Clabaugh (lead-bismuth)
Styrikovich and Semenovker (mercury)
MacDonald and Quittenton (sodium)
Elser (mercury)
10
1
10
102
103
Peclet number, Pe = ReDPr
104
105
FIGURE 6.21 Comparison of measured and predicted Nusselt numbers for liquid metals heated in long tubes
with uniform heat flux.
Source: Courtesy of the National Advisory Committee for Aeronautics, NACA TN 3363.
the Reynolds number when ReDPr 6 100. For these conditions, Lee [48] found that
the equation
NuD = 3.0Re0.0833
D
(6.69)
fits data and analysis well. Convection in the entrance regions for fluids with
small Prandtl numbers has also been investigated analytically by Deissler [41],
and experimental data supporting the analysis are summarized in [49] and
[50]. In turbulent flow, the thermal entry length (L> DH)entry is approximately
10 equivalent diameters when the velocity profile is already developed and 30
equivalent diameters when it develops simultaneously with the temperature
profile.
For a constant surface temperature the data are correlated, according to Seban
and Shimazaki [51], by the equation
NuD = 5.0 + 0.025(ReDPr)0.8
in the range RePr 7 100, L> D 7 30.
Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
(6.70)