Heat Transfer Engineering, 32(6):506–513, 2011
C Taylor and Francis Group, LLC
Copyright ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457632.2010.506378
Laminar Natural Convection Heat
Transfer From Isothermal Cylinders
With Active Ends
MOHAMMAD ESLAMI and KHOSROW JAFARPUR
School of Mechanical Engineering, Shiraz University, Shiraz, Iran
Calculation of free convection heat transfer from isothermal bodies of different shapes is of fundamental importance in
science and engineering. In the present study, a new analytical model is developed to calculate laminar natural convection
heat transfer from finite isothermal cylinders of arbitrary aspect ratio and orientation (vertical, horizontal, and inclined)
with active ends in fluids of any Prandtl number. This method is based on the new concept of Dynamic Body Gravity
Function. Also, a new dimensionless parameter called Body Fluid Function is introduced and applied in this method. Results
of this dynamic model are presented for six different circular cylinders and compared with the available experimental data.
Excellent agreement is found between the developed model and experimental results in a wide range of Rayleigh number for
all cylinders discussed. This shows that the present method is a powerful tool for modeling laminar natural convection from
isothermal bodies.
INTRODUCTION
Because of its importance in engineering, natural convection heat transfer from isothermal cylinders has been studied
extensively in the past decades. Many correlations are available
for horizontal, vertical, and inclined isothermal cylinders in the
open literature. Unfortunately, most of the studies are dedicated
to very long cylinders where the effects of the ends are negligible or the case where ends are insulated. Very little attention
has been paid to finite cylinders with active ends, despite their
significance in industrial fields. Some of the specific applications may be mentioned as heat transfer from electronic devices
and short circular fins, heat treatment of die-cast disks, or mass
diffusion from short capsules.
Morgan [1] has reviewed 34 published experimental and 23
analytical and numerical studies of natural convection around
horizontal isothermal cylinders and discussed possible reasons
for the dispersion in the results. He compared the five most
commonly referenced correlations of Morgan [2], Kuehn and
Goldstein [3], Raithby and Hollands [4], McAdams [5], and
Churchill and Chu [6]. He also mentioned the close agreement
between the results of Clemes et al. [7], Kuehn and Goldstein
[8], and Farouk and Guceri [9].
Address correspondence to Professor Khosrow Jafarpur, School of Mechanical Engineering, Shiraz University, Shiraz, Iran. E-mail: kjafarme@shirazu.ac.ir
As natural convection from the outer surface of a vertical
cylinder is similar to that of a flat plate when the cylinder diameter is much larger than the boundary-layer thickness, most of
the studies are dedicated to slender cylinders or more complex
situations such as mixed convection. Minkowycz and Sparrow
[10] used a nonsimilar method to obtain boundary layer velocity and temperature profiles for isothermal vertical cylinders
placed in air. Sparrow and Gregg [11] and Fujii and Uehara
[12] have also studied natural convection from the outer surface
of a vertical circular cylinder. Recently, Popiel et al. [13] have
proposed empirical correlations for free convective heat transfer
from the outer surface of vertical slender cylinders. Among the
published research, Oosthuizen [14] has considered the effects
of active ends in free convection from vertical cylinders and reported the experiment for short vertical cylinders with one end
exposed and the other attached to an insulated plane. Jafarpur
and Yovanovich [15] also proposed models for natural convection from horizontal thin isothermal elliptic disks with both ends
active. A special case of this is a thin vertical circular cylinder.
Their model shows good agreement with experimental results.
Moreover, Kobus and Wedekind [16] conducted experiments on
horizontal isothermal circular disks, which are very thin vertical
circular cylinders with both ends active.
In addition, the problem of free convective mass transfer at
vertical cylinders of varying aspect ratio with active ends has
been studied experimentally by Krysa and Wragg [17]. They
506
M. ESLAMI AND K. JAFARPUR
have compared the measured value of mass transfer rate from
the entire surface of different vertical cylinders with the predictions obtained by summation of the mass transfer rate at the
individual surfaces. As the summation method overpredicted the
experimentally obtained mass transfer rates, they introduced an
interference factor to correct the predictions. This empirically
obtained correction factor was found to be a function of cylinder aspect ratio [17]. Later, Wragg and Krysa [18] reported the
value of interference factor for some body shapes and showed
that its functional dependency on aspect ratio is quite different
for each geometry.
Experimental works are also reported for inclined isothermal cylinders by Oosthuizen [19], Al-Arabi and Khamis [20],
and Oosthuizen and Mansingh [21]. Effects of ends are not
considered again in these experiments. However, Stewart [22]
and Raithby and Hollands [23] have presented analytical solutions to natural convection from the outer surface of an isothermal inclined cylinder. For the case of natural convection heat
transfer from thin inclined isothermal circular disks, Kobus and
Wedekind [24] presented experimental data and proposed an
empirical correlation. This investigation was also extended to
the case of isoflux circular thin disks at arbitrary angles by Kobus
[25]. Flow visualizations of free convection mass transfer from
an inclined cylinder are also reported by Wragg [26]. Moreover, free convective mass transfer from thin circular disks with
arbitrary inclination has been studied experimentally by Krýsa
et al. [27].
Considering all the studies just mentioned, the lack of a general method to calculate natural convection heat transfer from
isothermal circular cylinders of any aspect ratio and orientation
is evident. Moreover, the role of active ends in free convection
heat transfer is not clearly described in the few published results.
Therefore, the objective of the present work is to provide a general method to calculate laminar natural convection heat transfer
from isothermal circular cylinders of arbitrary aspect ratio (L/D)
and orientation (horizontal, vertical and inclined) with active
ends over a wide range of Rayleigh number (0 < Ra√ A < 108 ).
The present analytical method is based on models proposed
by Yovanovich [28] and Jafarpur [29]. Two new parameters,
dynamic body gravity function and body fluid function, are introduced as discussed in detail by Eslami [30]. Results of the
present dynamic model are compared with the available experimental data for six different isothermal cylinders of various
aspect ratio and orientations.
507
√
Lee et al. [31] have shown that the characteristic length A
in the above equation is a superior choice for bodies of arbitrary
shape. N u 0√ A is the conduction limit (value of Nu as Ra → 0),
which can be found by either analytical or numerical methods
as discussed by Yovanovich [28], Jafarpur [29], and Bigdely
[32]. F(Pr) is the Prandtl function proposed by Churchill and
Churchill [33]:
F(Pr) =
0.670
9/16 4/9
1 + 0.5 Pr
(2)
Jafarpur [29] has reviewed an extensive number of previously
published investigations on the Prandtl function. Comparing various expressions proposed for different geometries, he showed
that Eq. (2) can be used as a universal Prandtl function for any
body shape.
The body gravity function G √ A in Eq. (1) accounts for the
body shape and orientation with respect to the gravity vector.
For two-dimensional and axisymmetric bodies, Lee et al. [31]
derived the following relation for body gravity function (BGF):
G√
A
=
1
A
A
P × sinθ
√
A
3/4
1/3
dA
(3)
where P is the local perimeter of the body and θ is the angle
between the normal to the surface and the gravity vector. For
three-dimensional bodies of arbitrary shape, BGF may be obtained from the results of Stewart [22] and Raithby and Hollands
[23]:
G√
A
=
1
7
A /8
x2
ω2
ω1
x1
4 2
h x/3 h ω/3 dω
3/4
dx
(4)
where x and ω are surface coordinate lines and hx and hω are
scale factors described in detail by Raithby and Hollands [23]
and Stewart [22].
As evaluation of the integrals in Eqs. (3) and (4) may be
difficult for bodies of complex geometry, Lee et al. [31] proposed
the following technique to calculate BGF of different bodies.
When a body consists of two or more distinct body shapes in
combination, these relations can be applied. For the case where
surfaces are placed in parallel with respect to gravity vector
(they transfer heat independently), BGF is obtained from
N
7/
G√A =
G √ Ai Ãi 8
(5)
i=1
DYNAMIC MODEL
The following general expression for laminar natural convection heat transfer from three dimensional isothermal convex
bodies was first developed by Yovanovich [28]:
1/
N u √ A = N u 0√ A + F(Pr)G √ A Ra√4A
and when surfaces are considered to be in series, the expression
to calculate BGF of the N combined surfaces is
G√
(1)
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A
N
=
i=1
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3/4
4/
G √3A
i
7/
Ãi 6
(6)
508
M. ESLAMI AND K. JAFARPUR
In the preceding equations, Ãi is the fraction of each body area
relative to the total surface area:
Ãi =
Ai
A
is the modified lower bound for BGF as defined by
where G low
Eslami [30]. This is found to be a function of a new dimensionless parameter called the body fluid function (BFF) [30]:
(7)
Using Eqs. (5)–(7) the BGF of different body shapes is easily obtained. This includes bodies containing horizontal flat surfaces as discussed by Yovanovich and Jafarpur [34] and Jafarpur
[29]. At this point two important questions arise:
1. How should one consider different surfaces in series or parallel? For example, in the case of a vertical circular cylinder,
is the top surface in series with respect to the cylinder body
or in parallel? Does this question have a definite answer for
all values of Ra?
2. As Eq. (1) is based on boundary-layer assumptions, what is
the effect of thick boundary layers at low values of Rayleigh
number?
To answer the first question, Jafarpur [29] showed that the
two alternatives of series or parallel combination provide two
lower and upper bounds for the body gravity function of a single body shape. For example in the case of a vertical circular
cylinder, considering the top surface in series with the rest of
the body gives the lower bound for BGF (Glow ), and assuming it
in parallel, the upper bound for the body gravity function (Gup )
is obtained. In other words, when the change in geometry of
the body shape (in the direction of boundary layer growth) is
such that the boundary layer tends to separate on a surface, that
part should be once considered in series and the other time in
parallel. Therefore, two correlations can be found for a single
body shape by substituting Glow and Gup in Eq. (1), respectively:
1/
N u √ A = N u 0√ A + F(Pr)G low Ra√4A
1/
N u √ A = N u 0√ A + F(Pr)G up Ra√4A
(8)
(9)
Investigation of experimental results of Jafarpur [29] and
Hassani [35] reveals that N u √ A falls between these two bounds
when a wide range of Ra is considered. More accurately, Nu
is lower than Eq. (8) at Ra of ∼102 due to the effect of thick
boundary layers, and increases monotonically until it reaches the
upper bound Eq. (9) at Ra of ∼108. This phenomenon is named
dynamic behavior of body gravity function and is observed for
a variety of geometries as reported by Jafarpur [29]. This means
that the value of BGF varies at different values of Ra between
a lower and an upper bound. In other words, using a constant
value of G √ A in Eq. (1) is not accurate over a wide range of Ra.
Before proposing a new model to consider the dynamic behavior of body gravity function, a correction factor must be applied to the lower bound to consider the effects of thick boundary
layers at lower values of Ra:
= C × G low
G low
(10)
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BFF =
N u 0√ A
F(Pr) × G low
(11)
C = 0.34 + 0.046 × BFF
(12)
The conduction limit and body gravity function in Eq. (11)
are functions of body shape and F(Pr) is a function of fluid
properties. This is why the name body fluid function is chosen
for this parameter. C is the universal correction factor for thick
boundary-layer effects at small Ra. Finally the dynamic body
gravity function (Gdyn ) proposed by Eslami [30] is obtained
from the following expression:
G dyn =
BFF +
1/
G up
Ra√4A
G low
BFF + C ×
1/
Ra√4A
G low
(13)
Therefore, the desired dynamic model for natural convection
heat transfer from isothermal bodies becomes:
1/
N u √ A = N u 0√ A + F(Pr)G dyn Ra√4A
(14)
This new general method is next applied to six different
isothermal cylinders with active ends and results are compared
with the available experimental work.
RESULTS AND DISCUSSION
Figure 1 shows the geometry of six isothermal cylinders with
active ends, tested previously by Hassani [35] and Jafarpur [29].
Values of Nu from Eq. (14) are compared with their experimental
values. First, one has to find N u 0√ A .
Yovanovich [36] proposed the following expression to calcu
late the conduction limit of circular cylinders with 0 ≤ L D ≤
8:
0.76
8.00 + 6.95 L D
N u 0√ A = (15)
1/2
L
2π + 4π
D
For circular cylinders of larger aspect ratio the numerical
method of Bigdely [32] may be used. The conduction limit is
independent of orientation so the preceding equation is used for
all cylinders in this article.
Next, one must find the dynamic body gravity function (Gdyn )
in Eq. (14). To clarify the method of obtaining this function,
results are put into three categories of vertical, horizontal, and
inclined cylinders.
vol. 32 no. 6 2011
M. ESLAMI AND K. JAFARPUR
509
Table 1 Upper and lower bounds of BGF for cylinders illustrated
in Figure 1
Figure 1 Configuration of the six different cylinders considered in this study.
Vertical Cylinders
A circular cylinder is composed of three distinct surfaces:
the cylindrical side and the two end surfaces. Using Eq. (3), the
BGF of the side for a vertical cylinder is easily calculated:
1/4
1
P
D /8
= 1.154
(16)
G side−ver = √
L
A
Body shape
Glow
Gup
Vertical cylinder, L/D = 0.1
Vertical cylinder, L/D = 0.5
Vertical cylinder, L/D = 1.0
Horizontal cylinder, L/D = 0.1
Horizontal cylinder, L/D = 1.0
Inclined cylinder, L/D = 1.0, ϕ = 45◦
0.758
0.897
0.936
1.088
1.051
0.940
0.900
1.031
1.044
1.115
1.169
1.151
discussed and are reported in Table 1 for the purpose of quick
reference. Knowing the two bounds, one must now find BFF
and Gdyn using Eqs. (10)–(13).
Figures 2–4 compare the results of the present model,
Eq. (14), with experimental data of Hassani [35] and Jafarpur
[29] for three vertical cylinders of L/D =0.1, 0.5, and 1.0, respectively, as illustrated in Figures 1a–c. The experiments were
carried out with a heated body suspended inside a pressure vessel containing air at a wide pressure range (from 0.8 to 700 kPa).
Hence, Rayleigh number can be changed in the whole range of
laminar flow (from Ra of ∼10 to 108) [29, 37]. The uncertainty
in Ra and Nu is dominated by the uncertainty in pressure measurements (especially at lower pressures) and is less than 10%
as reported by Hassani and Hollands [37].
To show the dynamic behavior of body gravity function,
the modified lower bound and upper bound are also included
in Figure 2. The empirical correlation proposed by Kobus and
Wedekind [16] is also included. It is found that this correlation
underestimates the experimental data and the analytical model.
But excellent agreement between the present model and experimental data is observed for all three cylinders in the whole range
of 0 < Ra√ A < 108 . This shows that the dynamic BGF is a
powerful tool to correlate Nu number over a wide range of Ra.
Obviously, Eq. (3) cannot be used to calculate the BGF of
horizontal flat surfaces. But results of semi-empirical studies are
available and can be applied. Based on Yovanovich and Jafarpur
[34] and Jafarpur [29], body gravity functions for horizontal top
and bottom surfaces are obtained by the following equations:
G top = 0.952
(17)
1
G top = 0.476
(18)
2
Considering all three surfaces in series and using Eq. (6),
one can easily obtain the lower bound for body gravity function,
Glow . Assuming the top surface in parallel with the other two
combined in series results in Gup . Values of the two bounds for
the six circular cylinders shown in Figure 1 are calculated as
G bottom =
heat transfer engineering
Figure 2 Comparison of the results of present model with experiments of
Hassani [35] for a vertical cylinder of L/D = 0.1 in air (Pr = 0.72).
vol. 32 no. 6 2011
510
M. ESLAMI AND K. JAFARPUR
Figure 3 Comparison of the results of present model with experiments of
Jafarpur [29] for a vertical cylinder of L/D = 0.5 in air (Pr = 0.72).
Horizontal Cylinders
Figure 5 Comparison of the results of present model with experiments of
Hassani [35] for a horizontal cylinder of L/D = 0.1 in air (Pr = 0.72).
can be found using Eqs. (6) and (20):
The two ends of a horizontal cylinder are vertical circular
plates and their BGF is easily obtained from Eq. (3):
G end = 1.021
(19)
Applying Eq. (3) to a horizontal circular surface, one can
find its BGF as previously derived by Lee et al. [31]:
G side−hor = 0.891
L
D
1/8
(20)
Combining these three surfaces as parallel by using Eq. (5),
the lower bound for body gravity function, Glow , is obtained.
Also, one may consider top and bottom halves of the horizontal
circular side in parallel. This idea is supported by investigating
streamlines in the numerical solution of Kuehn and Goldstein
[8]. Therefore, the BGF for a half horizontal circular cylinder
Figure 4 Comparison of the results of present model with experiments of
Hassani [35] for a vertical cylinder of L/D = 1.0 in air (Pr = 0.72).
heat transfer engineering
G hal f −hor = 0.972
L
D
1/8
(21)
Combining the two ends along with the two half circular
cylinders all in parallel, Gup is found for a horizontal circular
cylinder with active ends. Figures 5 and 6 show the results of
the present model for two horizontal cylinders of L/D =0.1 and
1, respectively, as illustrated in Figures 1d and e, along with
experimental data of Hassani [35]. Accurate results for such a
large range of aspect ratio and Ra number are observed.
Inclined Cylinders
Stewart [22] and Raithby and Hollands [23] calculated the
body gravity function for the circular surface of an isothermal
inclined cylinder using equations similar to Eq. (4). They have
Figure 6 Comparison of the results of present model with experiments of
Hassani [35] for a horizontal cylinder of L/D = 1.0 in air (Pr = 0.72).
vol. 32 no. 6 2011
M. ESLAMI AND K. JAFARPUR
Figure 7 Comparison of the results of present model with experiments of
Hassani [35] for an inclined cylinder of L/D = 1.0 and ϕ = 45◦ in air (Pr =
0.72).
reported values that cover different inclination angles ϕ and
aspect √
ratiosL/D. Converting the results to the characteristic
length A, the value of BGF for the outer surface of an inclined
cylinder of ϕ = 45◦ and L/D = 1 (Figure 1f) is:
G side−inc = 1.085
(22)
Also, Eq. (3) gives the BGF for the two inclined ends:
1
G end = (sin ϕ) /4 × 1.021
(23)
Assuming the three surfaces in series, the lower bound is
obtained while the upper bound for BGF is calculated by considering these three surfaces in parallel. The resulting correlation
is compared with experimental work of Hassani [35] in Figure
7. Excellent agreement is again observed between the proposed
model and experimental data points.
It is also interesting to investigate the effect of inclination angle on natural convection heat transfer from isothermal circular
cylinders with active ends. Figure 8 compares Nusselt number
Figure 9 Experimental results of Hassani [35] for three horizontal, inclined
and vertical cylinders of L/D = 1.0 in air (Pr = 0.72)
from the present analytical method, Eq. (14), for three cylinders
of L/D = 1 at ϕ = 0, 45, and 90 degrees, respectively. It is
shown that the horizontal cylinder has the highest rate of heat
transfer, but with increasing inclination from horizontal orientation to 45◦ , Nu does not change significantly. In other words,
the rate of heat transfer decreases mostly when the cylinder is
in the near vertical orientation. This effect is clear from the lower
and upper bounds of body gravity function for the three cylinders as shown in Table 1. It is also confirmed by the available
experimental data (Figure 9).
Finally, the accurate results of this simple model show that the
present method of modeling body gravity function is a powerful
tool for calculation of natural convection heat transfer from
isothermal bodies of arbitrary shape over a wide range of Ra.
CONCLUSIONS
Evaluation of natural convection heat transfer from isothermal cylinders of arbitrary aspect ratio and inclination with active
ends using a new analytical model is presented. The method is
based on the concept of the dynamic body gravity function,
which itself is a function of a new dimensionless parameter the
body fluid function. Excellent agreement between the results of
the present model and available experimental results is observed
in the whole laminar flow range (0 < Ra√ A < 108 ) for all the
geometries discussed. This shows that no accurate result can
be obtained unless this dynamic BGF is employed. The present
model also shows that the rate of heat transfer decreases with
increasing inclination angle from horizontal to vertical orientation. Due to the presence of F(Pr), the proposed model can be
used for fluids of any Prandtl number.
NOMENCLATURE
Figure 8 Comparison of the results of present model for three horizontal,
inclined, and vertical cylinders of L/D = 1.0 in air (Pr = 0.72).
heat transfer engineering
511
A
Ã
total surface area, m2
fraction of sectional area defined by Eq. (7)
vol. 32 no. 6 2011
512
M. ESLAMI AND K. JAFARPUR
BFF
BGF
C
D
F(Pr)
g
G√A
G low
hx , hω
L
N
N u√ A
N u 0√ A
P
Pr
Ra√ A
x
body fluid function
body gravity function
universal correction factor
diameter of circular cylinder, m
Prandtl number function defined by Eq. (2)
gravitational acceleration, m/s2
body gravity function based on characteristic length
√
A
modified lower bound
scale factors
length of cylinder, m
number of distinct surfaces of a body shape √
Nusselt number based on characteristic length A
√
conduction limit based on characteristic length A
local perimeter of body with respect to gravity
vector, m
Prandtl number
√
Rayleigh number based on characteristic length A
surface coordinate line
[5]
[6]
[7]
[8]
[9]
[10]
Greek Symbols
ω
θ
ϕ
surface coordinate line
angle between normal to the surface and gravity vector,
radians
inclination angle, radians
[11]
[12]
Subscripts
cyl
dyn
hor
i
inc
low
up
ver
cylindrical surface
dynamic
horizontal
surface number i
inclined
lower bound
upper bound
vertical
[13]
[14]
[15]
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Mohammad Eslami is a Ph.D. student in mechanical engineering at Shiraz University, Shiraz, Iran. He
worked on analytical modeling of natural convection
heat transfer for his master’s thesis and received his
M.S. at Shiraz University in June 2008. His research
interests have included convection and conduction
heat transfer, numerical modeling of heat and fluid
flow, electrokinetic flow in microchannels, and solar
energy measurements and applications.
Khosrow Jafarpur is an associate professor of mechanical engineering at Shiraz University, Shiraz,
Iran. He received his Ph.D. at the University of Waterloo, Canada, in 1992 and joined Shiraz University
in the same year. His research includes free convection heat transfer, solar energy measurement, and solar stills, as well as heat transfer (and optimization)
in welding, porous media, and nanosystems. He is
the author or co-author of about 70 papers on these
topics.
vol. 32 no. 6 2011
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