For an inclined plate with heated surface facing upward with approximately constant heat flux, the correlation obtained is of the form Nuq = 0.14[Gr · Pr1/3 − Grcr· Pr1/3]+ 0.56Grcr· Pr c
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Here NuL,q represents the Nusselt number atx = L and Gr∗ = gβqL4/kν2 In a later study, Vliet and Ross (1975) obtained a closer corroboration for data in air with the following relationships:
Nux,q = 0.55(Gr
∗
x · Pr)0.2 for laminar flow (7.79)
0.17(Gr∗
x · Pr)0.25 for turbulent flow (7.80)
Nuq can be obtained by computing the mean temperature difference and using the overall heat transfer rate provided by Vliet and Ross (1975)
7.7.2 Inclined and Horizontal Flat Surfaces
As discussed earlier, the results obtained for vertical surfaces may be employed for surfaces inclined at an angle γ up to about 45° with the vertical, by replacing g
withg cos γ in the Grashof number For inclined surfaces with constant heat flux,
Vliet and Ross (1975) have suggested the use of eq (7.74) for laminar flow, with the replacement of Gr∗xby Gr∗xcosγ for both upward- and downward-facing heated
inclined surfaces In the turbulent region also, eq (7.76) is suggested, with Gr∗x replaced by Gr∗xcosγ for an upward-facing heated surface and with Gr∗
x replaced
by Gr∗xcos2γ for a downward-facing surface
Several correlations for inclined surfaces, under various thermal conditions, were given by Fujii and Imura (1972) For an inclined plate with heated surface facing upward with approximately constant heat flux, the correlation obtained is of the form
Nuq = 0.14[(Gr · Pr)1/3 − (Grcr· Pr)1/3]+ 0.56(Grcr· Pr cos γ)1/4
for 105 < Gr · Pr cos γ < 1011and 15°< γ < 75° (7.81) where Grcris the critical Grashof number at which the Nusselt number starts deviating from the laminar relationship, which is the second expression on the right-hand side
of eq (7.81) This correlation applies for Gr > Grcr The value of Grcris also given for various inclination angles Forγ = 15, 30, 60, and 70°, Grcris given as 5× 109,
2× 109, 108, and 106, respectively For inclined heated surfaces facing downward, the expression given is
Nuq = 0.56(Gr · Pr cos γ)1/4 for 105< Gr · Pr cos γ < 1011, γ < 88° (7.82)
The fluid properties are evaluated atT w −0.25(T w −T∞), and β at T∞+0.25(T w −T∞).
For horizontal surfaces, several classical expressions exist For heated isothermal surfaces facing downward (lower surface of heated plate), or cooled ones facing upward (upper surface of cooled plate), the correlation given by McAdams (1954) which has been used extensively, is
Nu= 0.27Ra1/4 for 105 Ra 1010 (7.83) Fujii and Imura (1972) give the corresponding correlation as
Trang 2EMPIRICAL CORRELATIONS 563
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Nu= 0.58Ra1/5 for 106≤ Ra ≤ 1011 (7.84) Over the overlapping range of the two studies by Fujii and Imura (1972) and Rotem and Claassen (1969), the agreement between the two Nusselt numbers is fairly good
For the heated isothermal horizontal surface facing upward and cold surface facing downward, the correlations for heat transfer are given by McAdams (1954) as
Nu= 0.54Ra
1/4 for 104 Ra 107 (7.85) 0.15Ra1/3 for 107 Ra 1011 (7.86) The corresponding correlation given by Fujii and Imura (1972) for an approximately uniform heat flux condition is
Nuq = 0.14Ra1/3 for Ra> 2 × 108 (7.87)
7.7.3 Cylinders and Spheres
A considerable amount of information exists on natural convection heat transfer from cylinders For vertical cylinders of large diameter, ascertained from eq (7.47), the correlations for vertical flat surfaces may be employed For cylinders of small diameter, correlations for Nu are suggested in terms of the Rayleigh number Ra, where Ra and Nu are based on the diameterD of the cylinder.
The horizontal cylinder has been of interest to several investigators McAdams (1954) gave correlation for isothermal cylinders as
Nu= 0.53Ra
1/4 for 104< Ra < 109 (7.88) 0.13Ra1/3 for 109< Ra < 1012 (7.89) For smaller values of Ra, graphs are presented by McAdams (1954) A general expression of the form Nu = C · Ra n is given by Morgan (1975), with C and
n presented in tabular form Churchill and Chu (1975b) have given a correlation
covering a wide range of Ra, Ra≤ 1012, for isothermal cylinders as
Nu=
[1+ (0.559/Pr)9/16]16/9
1/62
(7.90)
This correlation is recommended for horizontal cylinders since it is convenient to use and agrees closely with experimental results
For natural convection from spheres, too, several experimental studies have pro-vided heat transfer correlations Amato and Tien (1972) have listed the correlations for Nu obtained from various investigations of heat and mass transfer In a review pa-per, Yuge (1960) suggested the following correlation for heat transfer from isothermal spheres in air and gases over a Grashof number range 1< Gr < 105, where Gr and
Nu are based on the diameterD:
Trang 3TABLE 7.2 Summary of Natural Convection Correlations for External Flows over Isothermal Surfaces
Vertical flat surfaces Nu= 0.825 + 0.387Ra1/6
[1+ (0.492/Pr)9/16]8/27
2
10−1< Ra < 1012 Churchill and Chu (1975a) Inclined flat surfaces Above equation withg replaced by g cos γ γ ≤ 60°
Heated, facing downward Nu= 0.27Ra1/4 3× 105≤ Ra ≤ 3 × 1010 McAdams (1954) Horizontal cylinders Nu=
0.60 + 0.387 Ra
[1+ (0.559/Pr)9/16]16/9
1/62
10−5≤ Ra ≤ 1012 Churchill and Chu (1975b)
Trang 4EMPIRICAL CORRELATIONS 565
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Nu= 2 + 0.43Ra1/4 for Pr= 1 and 1 < Ra < 105 (7.91) For heat transfer in water, Amato and Tien (1972) obtained the correlation for isother-mal spheres as
Nu= 2 + C · Ra1/4 for 3× 105≤ Ra ≤ 8 × 108 (7.92) withC = 0.500 ± 0.009, which gave a mean deviation of less than 11% A general
correlation applicable for Pr≥ 0.7 and Ra 1011is given by Churchill (1983) as
Nu= 2 + 0.589Ra1/4
1+ (0.469/Pr)9/164/9 (7.93)
Several of the important correlations presented earlier are summarized in Table 7.2
Correlations for various other geometries are given by Churchill (1983) and Raithby and Hollands (1985) Nu and Ra are based on the heightL for a vertical plate, length
L for inclined and horizontal surfaces, and diameter D for horizontal cylinders and
spheres All fluid properties are evaluated at the film temperatureT f = (T w +T∞)/2.
7.7.4 Enclosures
As mentioned earlier, the heat transfer across a vertical rectangular cavity is largely
by conduction for Ra 103, which implies a Nusselt number Nu of 1.0 For larger
Ra, Catton (1978) has given the following correlation for the aspect ratioH/d in the
range 2 to 10 and Pr< 105:
Nu= 0.22
Pr
0.2 + PrRa
0.28
H d
−1/4
(7.94)
where the Nusselt and Rayleigh numbers are based on the distanced between the
vertical walls and the temperature difference between them For an aspect ratio be-tween 1 and 2, the coefficient in this expression was changed from 0.22 to 0.18 and the exponent from 0.28 to 0.29, with the aspect ratio dependence dropped Similarly, correlations are given for higher aspect ratios in the literature
For horizontal cavities heated from below, the Nusselt number Nu is 1.0 for Rayleigh number Ra 1708, as discussed earlier Globe and Dropkin (1959) gave
the following correlation for such cavities at larger Ra, 3× 105< Ra < 7 × 109:
Nu= 0.069Ra1/3· Pr0.074 (7.95)
For inclined cavities, Hollands et al (1976) gave the following correlation for air as the fluid withH/d 12 and γ < γ∗on the basis of several experimental studies:
Nu= 1 + 1.44
1− 1708
Ra cos γ 1−
1708(sin 1.8γ)1.6
Ra cos γ
+
Ra cos γ
5830
1/3
− 1
(7.96)
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whereγ is the inclination with the horizontal, γ∗ is a critical angle tabulated by Hollands et al (1976), and the term in the square brackets is set equal to zero if the quantity within these brackets is negative This equation uses the stability limit
of Ra= 1708 for a horizontal layer, given earlier For a horizontal enclosure heated
from below, with air as the fluid, Hollands et al (1975) gave the correlation
Nu= 1 + 1.44
1−1708
Ra
+
Ra 5830
1/3
− 1
(7.97)
Similarly, correlations for other fluids, geometries, and thermal conditions are given
in the literature
In this chapter we discuss the basic considerations relevant to natural convection flows External and internal buoyancy-induced flows are considered, and the govern-ing equations are obtained The approximations generally employed in the analysis
of these flows are outlined The important dimensionless parameters are derived in order to discuss the importance of the basic processes that govern these flows Lami-nar flows for various surfaces and thermal conditions are discussed, and the solutions obtained are presented, particularly those derived from similarity analysis The heat transfer results and the characteristics of the resulting velocity and temperature fields are discussed Also considered are transient and turbulent flows The governing equa-tions for turbulent flow are given, and experimental results for various flow configura-tions are presented The frequently employed empirical correlaconfigura-tions for heat transfer
by natural convection from various surfaces and enclosures are also included Thus, this chapter presents the basic aspects that underlie natural convection and the heat transfer correlations that may be employed for practical applications
NOMENCLATURE
Roman Letter Symbols
c p specific heat at constant pressure, J/kg· K
D diameter of cylinder or sphere, m
f stream function, dimensionless
F body force per unit volume, N/m3
g gravitational acceleration, m/s2
Gr Grashof number, dimensionless
Grx local Grashof number, dimensionless
Gr* heat flux Grashof number, dimensionless
h x local heat transfer coefficient, W/m2· K
¯h average heat transfer coefficient, W/m2· K
hφ heat transfer coefficient at angular positionφ, W/m2· K
Trang 6NOMENCLATURE 567
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k thermal conductivity, W/m· K
L characteristic length, height of vertical plate, m
m, n exponents in exponential and power law distributions,
dimensionless
M, N constants employed for exponential and power law
distributions of surface temperature, dimensionless
Nux local Nusselt number, dimensionless [= h x x/k]
Nu average Nusselt number for an isothermal surface,
dimensionless
Nuq average Nusselt number for a uniform heat flux surface,
dimensionless
Nuφ local Nusselt number at angular positionφ, dimensionless
Pr Prandtl number, dimensionless
q total heat transfer, W
q
x local heat flux, W/m2
q constant surface heat flux, W/m2
q volumetric heat source, W/m3
Ra Rayleigh number, dimensionless [= Gr · Pr]
Rax local Rayleigh number, dimensionless [= Grx· Pr]
Sr Strouhal number, dimensionless
t c characteristic time, s
∆T temperature difference, K [= T w − T∞]
T w wall temperature, K
plume centerline temperature, K
T∞ ambient temperature, K
u, v, w velocity components inx, y, and z directions, respectively,
m/s
V c convection velocity, m/s
x, y, z coordinate distances, m
Greek Letter Symbols
α thermal diffusivity, m2/s
β coefficient of thermal expansion, K−1
γ inclination with the vertical, degrees or radians
δ velocity boundary layer thickness, m
δT thermal boundary layer thickness, m
εH eddy diffusivity, m2/s
εM eddy viscosity, m2/s
η similarity variable, dimensionless
θ temperature, dimensionless [= (T − T∞)/(T w − T∞)]
µ dynamic viscosity, Pa· s
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ν kinematic viscosity, m2/s
Φv viscous dissipation, s−2
ψ stream function, m2/s
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