1. No category

Columbia International Publishing
American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
doi:10.7726/ajhmt.2015.1003
Research Article
Heat Transfer and Flow-pattern Formation in a Cylindrical
Cell with Partially Immersed Heating Element
Neeraj Sharma1 and Gerardo Diaz1*
Received 5 January 2015; Published online 28 March 2015
© The author(s) 2015. Published with open access at www.uscip.us
Abstract
A non-Boussinesq weakly compressible- low numerical model is used to study the temperature and luid velocity distribution in a cylindrical tank with a heating element partially immersed in a temperature dependent
luid. This con iguration is commonly found in immersion heaters and electrolytic cells where the heating element does not extend along the entire axial length of the cylindrical vessel. A very particular vortex pattern
arises inside the cell which strati ies the domain dividing it into two regions of different temperatures. With
the increase in aspect-ratio, the Nusselt number at the surface of heating element increases, while the Nusselt
number at the inner surface of cylindrical tank decreases.
Keywords: Buoyancy- low; Strati ication; Immersed heating element
1. Introduction
Natural convection heat transfer and luid lows have been studied in various con igurations by many researchers with applications ranging from passive cooling of electronics to heat transfer in nuclear fuel rods. Bazylak et al.
(2006) studied natural convection in an enclosure with distributed heat sources. Their heat sources were located at the bottom wall and they found that a spacing equal to that of the source length provided the most
effective heat transfer. In addition, convective heat transfer from four in-line simulated electronic chips in a
vertical rectangular channel using water as the working luid was studied experimentally by Bhowmik et al.
(2005). They analyzed the effect of heat luxes, low rates and geometrical parameters such as chip number and
developed empirical correlations using Nusselt, Reynolds and Grashof numbers, based on channel hydraulic
diameter. Chadwick et al. (1991) studied both, experimentally and theoretically, the natural convection from
two-dimensional discrete heat sources in a rectangular enclosure, and found that the location of heat source
near the bottom of the enclosure maximizes the heat transfer in high Grashof number range. Buoyancy-driven
convection in the presence of mutually orthogonal heat generating baf les was studied by Hakeem et al. (2008)
in a square cavity with walls subjected to an isothermal temperature or outward heat lux. The problem was
formulated and solved in terms of vorticity-stream function for various con igurations of the baf les. Keyhani
et al. (1988) performed an experimental study of natural convection with discrete heat sources in an ethyl-glycol
*
Corresponding author: Tel: 1-209-228-7858, Email: gdiaz@ucmerced.edu
1 School of Engineering, University of California, 5200 North Lake Road, Merced, California 95343, U.S.A.
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
illed vertical cavity. With the help of heat transfer data and low visualization, they found that strati ication was
the primary factor in luencing the temperature of heated sections. Effective boundary conditions at the open
side of two and three dimensional open-ended structures was studied in Khanafer and Vafai (2000, 2002). Convective heat transfer in vertical or inclined annular enclosures with either constant heat lux at the inner wall or
constant temperature at inner wall have been studied both experimentally and theoretically by various authors
(Davis and Thomas, 1969; El-Shaarawl and Sarhan, 1981; El-Shaarawl and Al-Nimr, 1990; Lal and Kumar, 2013;
Mahfouz, 2012; Radhia et al., 2011; Shiniyan et al., 2013). Reddy and Narasimham (2008) performed a numerical study of conjugate natural convection in a vertical annulus with centrally located vertical heat generating
rod as the driving source using pressure-correction algorithm with the problem formulated in primitive form.
Their study has applications in spent nuclear casks and electronics equipment where temperature distribution
in the central vertical rod is of interest.
More recently, the numerical study of natural convection in a vertical annulus with localized heat source was
done in Sankar et al. (2012). Their study focused on the effect of size and location of a single iso lux heat source
on the buoyancy induced convection in a cylindrical annulus. They found that the rate of heat transfer is an
increasing function of radii ratio of the annulus. They also determined that the placement of the heater near the
center of the cylinder avoided hot spots and yielded maximum heat transfer.
Most of the studies found in literature related to convection in vertical annulus with either discrete or continuous heating source or constant wall temperature employ the Boussinesq approximation which limits the driving
temperature differences to relatively low values. A recent notable exception to this approach can be found in
Reddy et al. (2010), where the authors have studied non-Boussinesq conjugate natural convection in a vertical
annulus with axial heat generating rod which extends along the entire length of the outer cylinder.
The present study is aimed at formulating the steady-state natural convection analysis in a partial annulus geometry in the general framework of weakly compressible low without the use of the Boussinesq approximation.
This con iguration inds applications in immersion heaters and in convective lows generated inside electrolytic
cells (Rao et al., 2014) with a partially immersed electrode.
2. Mathematical Formulation
The governing equations for buoyancy-driven low include mass, momentum, and energy conservation with
temperature dependent density, viscosity, thermal conductivity, and speci ic heat. In most of the literature, the
equations are simpli ied by invoking Boussinesq approximation which is valid only for small variation of density
with respect to temperature (∆ρ/ρ ≪ 1). In the present study, steady-state laminar luid low and heat transfer
is modeled without Boussinesq approximation, using equations for weakly compressible luid low at low Mach
number.
A schematic of the con iguration is presented in Fig. 1, where rw corresponds to the radius of the outer cylinder,
re the radius of the heating element, l is the submerged depth of the heating element, H is the height of the outer
cylinder, and R is the difference between the radius of the heating element and the cylindrical tank.
The governing equations for weakly compressible luid, as described in Bird et al. (2006), are written in nondimensional form as follows:
[
]
(
)
( )T ) 2¯
µA ( ¯ )
¯A ( ¯
ρ¯
¯ ⃗u
¯ · −pI + µ
¯ ⃗u
⃗g¯
−
ρ¯ ⃗u
¯·▽
¯ =▽
▽⃗u
¯+ ▽
¯
▽ · ⃗u
¯ I +
Re
3Re
Fr
( )
¯ · ρ¯⃗u
▽
¯ =0
(
)
(
)
1
¯ T¯ +
¯ · −k¯▽
¯ T¯ = 0
ρ¯C¯p ⃗u
¯·▽
▽
Pe
(1)
(2)
(3)
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
Fig. 1: Schematic of cylindrical container with partially immersed heating element.
The above set of non-dimensional equations are obtained using the following dimensionless variables and parameters:
ρ¯ = ρ/ρr ,
C¯p = Cp /Cp,r ,
¯
▽· = Lr ▽ ·,
µ
¯ = µ/µr ,
Ra =
gβTr L3r
,
νr αr
Re =
ρr ur R
,
µr
k¯ = k/kr ,
⃗u
⃗g¯ = ⃗g /g
¯ = ⃗u/ur ,
¯
p¯ = (p − p0 )/pr ,
T = (T − T0 )/Tr
ur
Fr = √
,
gLr
ρr Cp,r ur Lr
,
kr
µr Cp,r
R
Pr =
,
A=
kr
Lr
Pe =
(4)
(5)
where,
R = rw − re ,
Lr = l,
pr = ρr u2r ,
ρr = ρ (T0 ) ,
Tr = Te − T0 ,
µr = µ (T0 ) ,
ur =
νr = µr /ρr ,
kr = k (T0 ) ,
√
√
gβTr Lr / P r,
(6)
Cp,r = Cp (T0 ) ,
αr = kr / (ρr Cp,r )
Where ze is the vertical coordinate along the surface of the electrode measured from its lowest submerged end
and zw is the vertical coordinate along the cylindrical wall measured from the top surface. It is noted that the
length scale used to non-dimensionalize the governing equations is the submerged length of the heating element.
Choice of this length scale becomes apparent under the limit of very large partial annulus gap (R) for which
the problem reduces to that of natural convection around a heated vertical cylinder. Also, the representative
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
Reynolds number is calculated using R as the length scale which is the natural choice for de ining Reynolds
number for low inside annulus. The reference velocity is de ined so as to achieve dimensionless velocities of
the order of unity. The mathematical formulation has been extensively validated in (Dillon et al., 2009, 2010;
Reeve et al., 2004).
3. Results and Discussions
The numerical simulation of the governing equations for a partially submerged heating element described by
Eqs. 1 to 3 was performed using a weakly-compressible low formulation of the inite element method using
the commercial software COMSOL (2015). Estimation of thermal and outer boundary layer thicknesses, as explained in Bejan (1993), were taken into account in generating meshes so as to resolve the boundary layers
appropriately. Mesh independence was veri ied by comparing simulation results for a sequence of cases with
increasing mesh densities until mesh independence was achieved. A combination of triangular elements for
the bulk and quadrilateral elements for the boundary layer were used for all cases analyzed. Simulations were
carried out for different values of laminar Rayleigh number, Ra, and inner-cylinder aspect ratio, A. The values of parameters used in the simulations are given in Table 1 and the corresponding boundary conditions are
described in Table 2.
Table 1: Values of various parameters used in simulations
Parameter
Cp,r
g
kr
µr
ρr
ρa
T0
Unit
J kg−1 K−1
m s−2
W m−1 K−1
Pa s
kg m−3
kg m−3
K
Values
4180
9.81
0.6
8.55 x 10−4
997
1.204
300
Table 2: Boundary Conditions
Boundary
axis
electrode surface
cylinder wall
top
bottom
Thermal
axisymmetric
T =(1
)
¯ T¯ = Nu T¯
n
ˆ · (−k¯▽
)
¯ T¯ = Nu T¯
n
ˆ · −k¯▽
insulation
Fluids
axisymmetric
wall, no-slip
wall, no-slip
wall, slip
wall, no-slip
Figures 2(a) and 2(b) show the results of the simulations in terms of streamlines and temperature distribution
for two different Rayleigh numbers. It is observed that for a low value of Rayleigh number (Ra = 0.14 × 105 ),
the low pattern shown in Fig. 2(a) is markedly different with only one vortex formed within the entire domain
as opposed to the multiple vortices that are formed for a higher Rayleigh number (Ra = 84.2 × 105 ), as shown
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
in Fig. 2(b). It is seen that the temperature distribution is also quite different between the two cases, with high
temperatures being mainly located near the surface of the heating element (inner cylinder) for Ra = 0.14 × 105
in contrast to the two-zone temperature distribution seen for Ra = 84.2 × 105 , where a relatively uniform
temperature pro ile is seen in the upper section of the domain (in between the two cylinders) that is located on
top of a much cooler region at the bottom of the tank.
(a) Ra = 0.14 × 105
(b) Ra = 84.2 × 105
Fig. 2: Temperature distribution and streamlines for low with different Rayleigh numbers and same
aspect ratio (A = 2). (a) Ra = 0.14 × 105 , (b) Ra = 84.2 × 105 .
A typical variation of vertical velocity along the radial direction at half depth of the heating element, at z e = 0.5,
is shown in Fig. 3 for Ra = 10.5 × 105 and A = 4. The plot shows the formation of two boundary layers,
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
one at the surface of heating element and the other at the surface of the outer cylinder. Velocities inside the
boundary layer near the heating element are nearly ive times higher because of the stronger buoyancy effects
in that region. Figure 4 shows a typical variation of temperature in the radial direction at two different vertical
0.3
v
0.2
0.1
0
−0.1
0
1
2
3
4
5
r
Fig. 3: Variation of vertical velocity with respect to radial direction at a location where z e = 0.5 for
Ra = 10.5 × 105 and A = 4.
locations, i.e. z w = 0.5 and z w = 3.0, for a case of multiple vortex formation with Ra = 10.5 × 105 and
A = 4, where the dotted line denotes the location of the surface of the heating element. The igure shows a
clear thermal strati ication with higher temperatures in the region adjacent to the submerged heating element.
Similar thermal strati ication patterns have also been reported in a experimental study performed by the authors
(Sharma et al., 2013) for the case where the heating element is replaced by an electrode inside an electrolytic
cell. It is seen that the temperature pro iles at both vertical locations remain relatively uniform for most of the
domain but it varies inside the boundary layers. The local Nusselt numbers at the surface of heating element
1
z w = 0.5
z w = 3.0
0.9
T
0.8
0.7
0.6
0.5
0.4
0
1
2
3
4
5
r
Fig. 4: Variation of temperature with respect to radial direction at depths of z w = 0.5 and z w = 3 for
Ra = 10.5 × 105 and A = 4. Dotted line indicates the location of the cylindrical surface of the heating
element.
and on the internal wall of outer cylinder have been de ined as,
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
N ue
=
N uw
=
( ¯)
1
∂T
− ¯
|∆T |max ∂ r¯ e
( ¯)
¯
H
∂T
− ¯
|∆T |max ∂ r¯ w
(7)
(8)
where, |∆T¯|max is the maximum driving temperature difference (de ined as |T¯e − T¯w |z¯w =0 ) as suggested for
thermally strati ied luid reservoirs in Bejan (1993).
Average Nusselt numbers at the two surfaces are de ined as,
N ue
1
= − ¯
|∆T |max
N uw
= −
1
|∆T¯|max
( ¯)
∂T
d¯
z
∂ r¯ e
0
∫ H¯ ( ¯ )
∂T
d¯
z
∂ r¯ w
0
∫
1
(9)
(10)
Figure 5 shows the variation of local Nusselt number at the surface of the heating element with respect to the
distance from the lower end, z e , as the boundary layer develops along this surface with velocities in the upward
direction. As the luid rises, it absorbs the thermal energy from the heating element and gains temperature.
This in turn decreases the gradient of the temperature in the radial direction and, thus, lowers the magnitude
of the Nusselt number. N ue decreases from a value around 15 at the tip of the heating element to a value near
3 at the free surface of the liquid. As expected, Nusselt number increases with the increase in Rayleigh number.
Figure 6 shows the variation of the local Nusselt number at the outer-cylinder surface with respect to the dis-
Ra = 3.5 × 10 5
Ra = 7.0 × 10 5
Ra = 10.5 × 10 5
N ue
15
10
5
0
0.2
0.4
z¯e
0.6
0.8
1
Fig. 5: Variation of local Nusselt number along the vertical surface of the immersed heating element
for different Rayleigh numbers for A = 4.
tance from the top, z¯w , as the boundary layer develops from top to bottom and for a heating element with an
aspect ratio A = 4. Due to the thermal strati ication and formation of two vortices, the Nusselt number is, in
general, higher at the upper part of the cylinder compared to the lower section. Fluid that was in contact with
the heating element lows radially near the free surface of the domain and gets in contact with the upper section
of the outer-cylinder wall. Most of the heat transfer occurs due to the presence of the vortex located between
the heated element and the tank wall, therefore, increasing the value of N uw in the section of the tank wall in
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
3
Ra = 3.5 × 10 5
Ra = 7.0 × 10 5
Ra = 10.5 × 10 5
N uw
2.5
2
1.5
1
0
1
2
z¯w
3
4
5
Fig. 6: Variation of local Nusselt number along the surface of cylindrical tank for different Rayleigh
numbers and A = 4. Dotted line indicates the location of the bottom tip of the heating element.
close proximity to this vortex. The rest of the luid domain remains at a lower temperature and does not play a
signi icant role in transferring heat from the heating element to the tank wall, so N uw decreases considerably
near the bottom of the tank. The vertical dotted line in the igure shows the position of the tip of the submerged
heating element. It is clear that the submerged length determines the interface of thermal strati ication. It is
also seen that Nusselt number increases with respect to Rayleigh number. The effect of Rayleigh number on
14
12
A=4
A=3
A=2
N ue
10
8
6
4
2
4
6
8
Ra(×10 5 )
10
12
Fig. 7: Variation of average Nusselt number along the vertical surface of immersed heating element
with respect to Rayleigh number for different aspect ratios.
average Nusselt number at the heating element and outer-cylinder wall for different aspect ratios is presented
in Figs. 7 and 8, respectively. An increase in aspect ratio indicates an increase in the difference between the
radii of the heating element and cylindrical tank for a ixed height of the heating element. For a higher value of
aspect ratio (A=4), the heating element is able to dissipate thermal energy more effectively when compared to a
lower value aspect ratio, i.e. A=3 and A=2, because the vortex located in between the heating element and tank
wall contains a larger volume of luid so its bulk temperature tends to be lower than for smaller aspect ratios,
therefore increasing the potential of heat transfer at the heating-element surface.
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
7
6
A=4
A=3
A=2
N uw
5
4
3
2
1
2
4
6
8
R a(×10 5 )
10
12
Fig. 8: Variation of average Nusselt number along the wall of the cylindrical tank with respect to
Rayleigh number for different aspect ratios.
The trend is reversed for average Nusselt number at the tank wall, as seen in Fig. 8, where N uw decreases with
the increase in aspect ratio. The higher aspect ratio implies a larger volume of luid in between the heating element and the tank wall, which translates into lower bulk temperatures compared to cases with smaller aspect
ratios. The temperature difference between the bulk of the luid and the ambient temperature is smaller, therefore, heat transfer by convection is reduced. It is observed that higher Nusselt numbers are obtained with higher
Rayleigh numbers.
4. Conclusions
Buoyancy-driven low in a cylindrical tank with a partially immersed heating element has been studied numerically without using the Boussinesq approximation. This type of con iguration is commonly found in immersion
heaters and electrolytic cells. Thermal strati ication of the low inside the domain is reported and explained
based on the results of the model. The results indicate that in laminar conditions and when boiling does not
take place, the partial immersion of a heating element results in a much higher temperature value near the free
surface of the luid which remains almost uniform along the radial direction. A much lower temperature is observed near the bottom of the tank, generating a sort of two-temperature distribution over the entire domain
with the interface located at the depth where the tip of the heating element is located. The pattern occurs due
to the formation of vortices that appear inside the domain that depend on the value of the Rayleigh number and
aspect ratio.
Acknowledgements
This work has been partially supported by the California Energy Commission contract number PIR-08-036.
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Neeraj Sharma and Gerardo Diaz / American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 31-41
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