thermal hydraulic characteristics of extended heated ...operation and safety in the natural...
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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:01 1
171701-9292-IJMME-IJENS © February 2017 IJENS I J E N S
Thermal Hydraulic Characteristics Of Extended
Heated Vertical Channels To Enhance Natural
Convection In The Core Of A Typical MTR Reactor
Said M. A. Ibrahim * Department of Mechanical Engineering, Faculty of Engineering, AL-Azhar University, Nasr City, Cairo 11371,Egypt
*E-mail: [email protected]
Professor of Mechanical Power Engineering & Energy
Abstract-- This research deals with natural convection heat
transfer from vertical heated cladded plates, which are
symmetrically placed in proposed chimneys of variable heights in
the core of a typical MTR reactor. The heated plates serve as
thermal pumps for pumping fluid of a symmetrical enclosure
beneath the chimney. The suggested chimneys are used for
increasing the length of the vertical heated channels of the
reactor core to give the chimney effect. In the thermal analysis of
natural convection in channel–chimney systems, the variables
that play an important role are heat flux, maximum wall
temperatures and geometrical parameters such as the height of
the heated channel, the channel spacing and the height and
spacing of unheated extensions. A simple numerical procedure to
obtain the thermal design charts, a thermal optimization of the
system and an uncertainty analysis due to the thermo- physical
properties is presented. The present results are obtained from a
real domain inside the reactor core data in the following
dimensionless parameter ranges: 5≤ Lh/ b≤ 20; 1:5≤ L/Lh ≤ 4;
1≤ B/b≤ 4; 102 ≤ Ra≤ 105 . This study results in enhancing the
reactor power in the free convection regime from a maximum of
400 kW up to 950 kW of thermal energy. This is quite significant
increase in reactor power in the natural convection regime which
adds to reactor safety. The results are of importance to reactor
operation and safety in the natural convection mode of
operation.
Keywords-- Thermal hydraulic- Natural convection- Chimney-
Vertical heated channel- MTR- Rayleigh number- Nusselt
number- Temperature profile- Aspect ratio- Expansion ratio-
Extension ratio.
1. INTRODUCTION
Nowadays more recent investigation trends in natural
convection heat transfer are oriented towards either seeking of
new configuration to enhance the heat transfer parameter or
the optimization of standard configurations. Natural
convection between heated vertical parallel
plates is a physical system frequently employed in
technological applications, such as thermal control in
electronic equipment, nuclear reactors, solar collectors and
chemical vapor deposition reactors and it has been extensively
studied both experimentally and numerically ( Gebhart, 1988
), ( Kimm and Lee, 1966 ), ( Manca et al, 2000 ). More recent
trends in natural convection research are to find new
configurations to improve heat transfer parameters or to
analyze standard configurations to carry out optimal
geometrical parameters for better heat transfer rates ( Manca
et al, 2000 ), ( Ledezma, 1977 ), ( Bejan et al, 2004 ).
Haaland and Sparrow ( 1983 ) were the first to show that
higher flow rate of fluid through a confined open-ended
enclosure can be induced by the chimney effect. They
introduced a numerical solution for natural convection flow in
a vertical channel with a point heat source or distributed heat
source situated at the channel inlet.
Oosthuizen ( 1984 ) studied numerically the heat transfer
enhancement caused by the addition of the straight adiabatic
extension at the exit of isothermal parallel-walled channel. He
solved the parabolic form of the governing equation by means
of a fully implicit forward marching procedure. The results
indicated that substantial increase (about 50 %) in the heat
transfer rate could be achieved, but very long adiabatic
sections were required.
Wirtz and Haag ( 1985 ) presented experimental results for
isothermal symmetrically heated plates with an unheated entry
channel portion. Their experiments were carried out over a
wide range of the Rayleigh number, from the single-plate limit
to the fully developed channel. They found that the flow is
quite insensitive to the presence of unheated entry section of
large channel spacing, while it is severely affected when the
gap spacing is small
Asako et al. ( 1990 ) examined numerically the heat transfer
increment due to an unheated chimney attached to a vertical
isothermal tube. The numerical results were obtained by a
control volume approach solving the full elliptic form of the
governing equation. They evaluated the optimum chimney
diameter where the maximum amount of heat is transferred
and found that for optimum chimney diameters the heat
transfer enhancement was up to 2.5 times for low Rayleigh
number and small chimney sizes.
Straatman et al. ( 1993 ) carried out a numerical and
experimental investigation of free convection in vertical
isothermal parallel walled channels, with adiabatic extension
of various sizes and shapes. They employed a finite element
discretization to solve the fully elliptic form of the governing
equation with the inlet boundary conditions based on Jeffrey-
Hamel flow. The experiments were performed with ambient
air, using a Mach-Zender interferometer. The increase in heat
transfer rates varied from 25 times at low Rayleigh number to
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1.5 times at high Rayleigh numbers. The authors proposed a
single correlation in terms of channel Rayleigh number and all
the geometric parameters, e.g. heated length ratio, expansion
ratio.
Lee ( 1994 ) investigated numerically the effect of the
unheated exit section for natural convection in vertical
channels with isotherm or isoflux walls. The results were
obtained by means of the boundary layer approximation. An
important finding was that an unheated exit determines larger
total heat transfer and flow rate than an unheated entry .
Campo et al. ( 1999 ) presented a numerical solution to the
wall temperature distribution and the thermal and the fluid
dynamic fields in a channel with partially isoflux heated
parallel plates. They found a reduction in the maximum wall
temperature when an insulated extension was placed
downstream of the heated part, the larger the Rayleigh number
the less relevant the reduction
Fisher et al. ( 1997 ) developed analytical solution for a
vertical parallel plate isothermal heat sink and chimney system
whereas Fisher and Torrance ( 1998 ) developed an analytical
solution for a pin-fin sink and chimney system. In the former
investigation a ridge of maximum total heat transfer was
observed with respect to the plate spacing and the heat sink
height, and the authors showed that smaller heat sinks can be
used together with a chimney without compromising the
thermal performance and without increasing the system size.
In the latter, the chimney effect was shown to enhance local
heat transfer rates in such a way that the minimum temperature
rise remains approximately constant while the height of the
heat sink relative to the total height is reduced.
Bianco et al. ( 1998 ) studied experimentally the free
convection in vertical isothermal parallel walled channels,
with adiabatic extension of various sizes and shapes with the
heated part at uniform wall heat flux. They presented a limited
investigation in terms of geometric parameters and Rayleigh
number. Auletta ( 2001 ) studied expermintally the effect of
adding adiabatic extensions for a vertical isoflux
symmetrically heated channel. They offered the best
configurations of their system to avoid the maximum wall
temperatures around the heated channels. This study was
useful for the present investigation.
Shahin and Floryan ( 1999 ) analyzed numerically the
chimney effect in a system of isothermal multiple vertical
channels. Each channel had an adiabatic extension. They
claimed that the interaction between multiple channels
increases the induced flow rate and that the associated
chimney effect is stronger than in a single channel with
adiabatic extension.
Fisher and Torrance ( 1999 ) carried out experiments on air
natural convection in a finned vertical parallel plate heater,
with an adiabatic downstream extension. The effect of fin
spacing and the channel length on the total heat transfer was
investigated and their results confirmed prior theoretical
predictions.
The present research is an applied one. It is based on studying
how to enhance the natural convection heat transfer around the
vertical heated channels in the core of a typical MTR reactor.
In doing so, chimneys were introduced to increase the heights
of the vertical channels in the reactor core in order to utilize
the chimney effect to do the job. The best system
configurations are based on a theoretical analysis which
includes all possible factors including heat transfer ones that
lead to our conclusions This type of applied thermal hydraulic
research in a complicated core of a real nuclear research
reactor is not readily available and is needed. The subsequent
increase in the reactor power in the natural convection mode is
rather important.
2. THE REACTOR CORE DATA
The reactor core is the main component concerned with the
performance of the neutronic and thermal hydraulic
calculations. MTR core is an array of aluminium cladded fuel
elements, absorber plates inside guide boxes, double wall core
chimney and irradiation boxes. Inside the core chimney there
are 30 grid positions with 6 x 5 configurations. It is divided
by two zones where two guide boxes (for absorber control
plate insertion) are placed. As a result, the core grid is divided
into a central area of 3x6 and two lateral areas of 1 x 6 each.
General data regarding the present MTR core and its fuel
elements is given in Table I.
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Table I
General Data of ETRR-2 Core
Reactor Type Open pool
Fuel Material U3O8-Al
Fuel enrichment (w % 235U) 19.7 %
Fuel elements dimension (cm x cm) 8.0 X 8.0
Shape of Fuel Plates Flat
Number of Fuel Plates 19
Active length (cm) 80
Fuel Plate dimension:
Thickness(cm) 0.150
Width (cm) 7.5
Fuel Meat dimension:
Width (cm) 6.4
Thickness (cm) 0.07
Water channel thickness between two fuel plates(cm) 0.270
Water channel thickness between two fuel elements (cm) 0.390
Weight of 235U (g) in fuel elements:
Standard fuel element ~404 g
Type one fuel element ~146 g
Type two fuel element ~209 g
Cladding Material Aluminium
Absorber Plates Material Ag-In-Cd
Moderator Light water
Coolant Light water
The physical model considered in the present work is a simple
design of a vertical channel with symmetrical heat generation
according to the fission of the fuel element. The channel
domain consists of entrance section, channel bundle section,
and exit section, as shown in Fig.1. The channel dimensions
are 80 mm length, 2.7 mm width and 800 mm in height.
Fig. 1. Drawing of the flow channel and its axes.
Y
X
g
Left Wall
Right Wall
Direction of Flow
Outlet
Inlet
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3. PROBLEM FORMULATION
The aim of this paper is to present a numerical analysis of
natural convection in single phase water in a symmetrically
heated vertical channel, considering the presence of two
downstream adiabatic extensions to enhance the “chimney
effect”. In the following, the heated part is indicated as
channel and the unheated part as chimney. The computational
domain for the heated vertical rectangular channels in the
cladding of the fuel assembly is depicted in Fig.2.
The domain is made up of a vertical channel with two parallel
plates, heated at uniform heat flux q; the height of the channel
plates is Lh whereas the distance between them is b. On top of
the channel, there is a chimney made up of two insulated
parallel and vertical plates; their height is (L-Lh) and the
distance between them is B. An enlarged computational
domain has been chosen. It is made up of the geometry
described previously and of two reservoirs of height Lx and
width Ly, which are placed upstream the channel and
downstream the chimney. The reservoirs are important
because they simulate the thermal and fluid dynamic behaviors
far away from the inflow and outflow regions.
Fig. 2. Computational domain of the problem.
4. NUMERICAL STUDY The numerical calculations were performed for the velocity
and temperature fields inside the chimney and the box. The
conservation equations were solved numerically. The
governing equations solved by FLUENT ( 2014 ) are the
Navier-Stokes equations combined with the continuity
equation, the thermal transport equation, and constitutive
property relationships.
Continuity Equation ( 2014 )
𝜕𝑝
𝜕𝑡+
𝜕
𝜕𝑥(𝜌𝑣𝑥) +
𝜕
𝜕𝑟(𝜌𝑣𝑟) = 𝑆𝑚
(1)
Navier Stokes Equation ( 2014 )
Conservation of momentum in an inertial (non-accelerating)
reference frame
𝜕
𝜕𝑡(𝜌�̅�) + ∇. (𝜌�̅��̅�) = −∇𝑝 + ∇. (𝜏̅) + 𝜌�̅� + �̅�
(2)
where p is the static pressure, τ is the stress tensor (described
below), and ρg and F are the gravitational body force and
external body forces (e.g., that arises from the interaction with
the dispersed phase), respectively. F also contains other model
dependent source terms such as porous-media and user-
defined sources.
The stress tensor τ is given by
𝜏̿ = 𝜇[(∇�̅� + ∇�̅�𝑇) −2
3∇. �̅�𝐼] (3)
Where µ is the molecular viscosity, I is the unit tensor, and the
second term on the right hand side is the effect of volume
dilation.
For two dimensional axisymmetric geometries, the axial and
radial momentum conservation equations are given by
𝜕
𝜕(𝜌𝑣𝑥) +
1
𝑟
𝜕
𝜕𝑥(𝑟𝜌𝑣𝑥𝑣𝑥) +
1
𝑟
𝜕
𝜕𝑟(𝑟𝜌𝑣𝑟𝑣𝑥) = −
𝜕𝑝
𝜕𝑥+
1
𝑟
𝜕
𝜕𝑥[𝑟𝜇 (2
𝜕𝑣𝑥
𝜕𝑥−
2
3(∇. �̅�))] +
1
𝑟
𝜕
𝜕𝑟[𝑟𝜇 (
𝜕𝑣𝑥
𝜕𝑟+
𝜕𝑣𝑟
𝜕𝑥)] + 𝐹𝑥 (4)
and
L
Lh
B
b
x
y
Chim
ney
Vertical
channel
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𝜕
𝜕𝑡(𝜌𝑣𝑟𝑥) +
1
𝑟
𝜕
𝜕𝑥(𝑟𝜌𝑣𝑥𝑣𝑟) +
1
𝑟
𝜕
𝜕𝑟(𝑟𝜌𝑣𝑟𝑣𝑟) = −
𝜕𝑝
𝜕𝑟+
1
𝑟
𝜕
𝜕𝑥[𝑟𝜇 (2
𝜕𝑣𝑟
𝜕𝑟−
2
3(∇. �̅�))] − 2𝜇
𝑣𝑟
𝑟2 +2
3
𝜇
𝑟(∇�̅�) + 𝜌
𝑣𝑧2
𝑟 +
1
𝑟
𝜕
𝜕𝑥[𝑟𝜇 (
𝜕𝑣𝑟
𝜕𝑥+
𝜕𝑣𝑥
𝜕𝑟)] + 𝐹𝑟 (5)
Where
∇. 𝑣 =𝜕𝑣𝑥
𝜕𝑥+
𝜕𝑣𝑟
𝜕𝑟+
𝑣𝑧
𝑟 (6)
The tangential momentum equation for 2D forced may be written as: [15]
𝜕
𝜕(𝜌𝑣𝑧) +
1
𝑟
𝜕
𝜕𝑥(𝑟𝜌𝑣𝑥𝑣𝑧) +
1
𝑟
𝜕
𝜕𝑟(𝑟𝜌𝑣𝑟𝑣𝑧) =
1
𝑟
𝜕
𝜕𝑥[𝑟𝜇
𝜕𝑣𝑧
𝜕𝑥] −
1
𝑟2
𝜕
𝜕𝑟[𝑟3𝜇
𝜕
𝜕𝑟[
𝑣𝑧
𝑟]] − 𝜌
𝑣𝑟𝑣𝑧
𝑟 (7)
The boundary conditions for the energy equation are based on
the natural convection 2D analysis. The heat flux,
corresponding to the input power of, for instance, 100 W, has
been imposed on the plate. For a constant heat flux, the wall
temperature of the plate is uniform. Therefore, the plate was
defined in the simulations exactly as if it was built in reality;
it had the core which generates heat, and the external layers
defined as ‘‘conducting walls.’’ The thermal conductivity of
aluminum was taken as 180 W/m K. For the other boundaries,
FLUENT makes it possible to incorporate the heat transfer
coefficients of the walls and the outside temperatures in the
calculation of the inside temperature field. Thus, the
calculations were performed both for adiabatic walls and for
walls with heat-transfer coefficients in the real plate of the
reactor. The temperature of the surroundings is imposed at the
entrance opening. As for the exit opening, FLUENT ( 2014 )
adjusts the boundary condition there, extrapolating the
temperature values from the interior grid cells adjacent to the
exit.
5. RESULTS AND DISCUSSIONS
Results for the parametric analysis are carried out for water, Pr
= 0.71, in the Rayleigh number ranges from 102 to 105 and for
a channel aspect ratios of Lh/b = 5, 10 and 20. The expansion
ratio, B/b, is in the range 1 – 4 and the extension ratio, L/Lh,
ranges from 1.5 to 4. No local flow separation around the
entrance corner was found in all considered cases.
The analyzed configuration is applied to a nuclear research
reactor core chimney cooling. Typical geometrical dimensions
are referred to Lh = 0.8 m, with L / Lh = 3 m, b is in the range
of 5 - 40 mm and, consequently, B changes from 55 to 110
cm. Heat flux ranges between 3 and 500 W/m2. Two actual
limiting cases are b = 50 cm and a heat flux of about 50 W/m2
and the corresponding Rayleigh number is 100 and b = 60 cm
with the same heat flux distribution and Ra = 105. The highest
considered heat flux, 500 W/m2 related to Ra = 105, is
attained for b equal to about 50 cm. Figure 3 illustrates the
velocity contour for various chimney designs of the channel.
B/b=1
B/b=1.5
B/b=2
B/b=3
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L/Lh=1
L/Lh=2
L/Lh=3
Fig. 3. Velocity contours.
Wall temperature profiles for Ra = 102 and 105 and for L/Lh =
1.5 and 4 are shown in Figs. 4 – 7 with Lh/b = 5, 10, and 20
and for different expansion ratio values. In all cases the
highest value of maximum wall temperature is attained for the
simple channel configuration. These profiles allow the
evaluation of the different thermal behaviors of the channel–
chimney system in terms of the channel aspect ratio. In all
temperature profiles, the maximum wall temperature is not
attained at the channel outlet section but at a slightly lower
value of the axial coordinate due to the diffusive effects,
according to the experimental results given by Haaland and
Sparrow ( 1983 ). The value Xmax of the section at which the
maximum wall temperature is attained depends on the
geometrical parameters and Ra values. In fact, for the simple
channel configuration, the point Xmax is the lowest among the
various configurations for the assigned Ra and Lh/b, as shown
in Figs. 4 – 7; this effect is more evident for the lowest Ra, as
given in Figs. 4 – 5. The Xmax value, for the same channel
length, increases with increasing Lh/b value because of the
decreasing diffusive effects toward the external ambient.
Moreover, increasing the Rayleigh number, the Xmax value
increases because of the decreasing diffusive effects, as it is
noted from comparing Fig. 4 with Fig. 6 and Fig. 5 with Fig.
7. A sharp decrease of wall temperature in the outlet section
zone is present due to also the cold inflow inside the chimney,
which reaches the outlet section of the channel.
For the lowest Rayleigh number, Ra = 102, and L/Lh = 1.5,
Fig. 4 indicates that wall temperatures decrease with
increasing the expansion ratio up to B/b between 2 and 3 and,
for higher B/b values wall temperatures increase again, where
in Fig. 4 𝜃𝜔 is the dimensionless temperature, and X the
dimensionless distance. Moreover, the decrease in the wall
temperature at the outlet region for B/b ≤ 3 is lower than that
for the simple channel. For B/b = 4, this decrease is almost
equal to that for the simple channel due to a cold inflow from
the outlet section of the chimney. The cold inflow in the
chimney was observed by Haaland and Sparrow ( 1983 ) and a
fluid stream flows down along the adiabatic extensions. It
reaches the horizontal wall of the chimney, mixes with the hot
plume-jet and goes out of the channel. A consequence of the
cold inflow or down flow is a reduction of chimney effect,
which gets stronger with increasing the aspect ratio as
indicated in Fig.4 (b) and (c). It is possible to estimate the
position along the adiabatic wall of the chimney where the
vorticity goes to 0. In general, it is observed that the number
of configurations with a complete down flow increases with
increasing Ra value whereas the number of configurations
with a partial separation from the wall decreases. The
separation is present for B/b = 2 only when L/Lh is equal to
1.5 at Ra ≤ 104 whereas, for Ra = 105, only a complete down
flow is observed. Some possible guide lines to evaluate critical
conditions related to the beginning of flow separation and
complete down flow will be provided in Figs. 8–12. In fact,
after the optimal conditions, thermal and fluid dynamics trends
indicate a worsening of the chimney effect.
The difference between the maximum values of the wall
temperature for the simple channel and for B/b = 2 increases
with increasing Lh/b, just as the increasing difference between
the maximum values of the wall temperature for the simple
channel and for B/b = 4. It is possible to affirm that increasing
Lh/b allows to enhance the channel–chimney system heat
transfer with respect to the simple channel, particularly for the
configurations with B/b > 1.5.
Thus comparing the maximum wall temperatures for the
simple channel and the present suggested channel-chimney
system allowed to determine the best configuration for better
heat transfer and also to minimize the maximum wall
temperatures.
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(a) Lh/b = 5
(b) Lh/b = 10
Ra=102 L/Lh=1.5 Lh/b=5
Ra=102 L/Lh=1.5 Lh/b=10
θω
θω
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(c) Lh/b = 20
Fig. 4. Heated wall temperatures at Ra = 102 and L/Lh = 1.5 for different channel aspect ratios.
For L/Lh = 4, Fig. 5 depicts that the absolute differences
strongly increase with respect to the previous case (L/Lh = 1.5)
and this shows that the chimney effect is remarkably
improved. Moreover, these differences increase with
increasing the channel aspect ratio, Lh/b. The configuration
with B/b = 4 gives the lowest wall temperature values, but it
has to be underlined that the decrease of the maximum wall
temperature is significant even for B/b =1.5, whereas the
reduction from the configuration with B/b = 1.5 to the
configuration with B/b = 4 is reasonably lower. In fact, the
percentage variations of the maximum wall temperature
between the configuration with B/b = 1.5 and the simple
channel, in reference to the value pertinent to the configuration
with B/b = 1.5, is almost 60 %, whereas the variation between
the configuration with B/b = 4 and that with B/b = 1.5 is
almost 19 % of that for Lh/b = 5. Therefore increasing the
channel aspect ratio enhances the thermal behavior of the
channel–chimney system for both low L/Lh and large L/Lh
values, and for low Rayleigh number values.
So, the channel aspect ratio is important in upgrading the
channel-chimney system effect, for low Rayleigh numbers.
(a) Lh/b = 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
Ra=102 L/Lh=1.5 Lh/b=20
Ra=102 L/Lh=4 Lh/b=5
θω
θω
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(b) Lh/b = 10
(c) Lh/b = 20
Fig. 5. Heated wall temperatures at Ra = 102 and L/Lh = 4.0 for different channel aspect ratios.
For the largest Rayleigh number value considered, Ra = 105,
Figs. 6 and 7 reveal that the wall temperatures are lower than
those for the configurations pertinent to Ra = 102. For
L/Lh=1.5, Fig. 6, illustrates that the configuration with B/b =
1.5 shows the lowest maximum wall temperature values,
whereas the configuration with B/b = 4 has wall temperature
values similar to the ones pertinent to the simple channel, for
all the analyzed Lh/b values. In this configuration, the down
flow is already present for B/b = 2. This is due to the larger
velocity of the hot jet coming out of the channel, which
determines the fluid separation from the adiabatic chimney
wall. Also in this case, the Lh/b increase produces an
enhancement of the channel–chimney system with respect to
the simple channel, as observed in Fig. 6.
Here again, for the largest Rayleigh number values considered
the aspect ratio is an important factor in showing the
superiority of the channel-chimney system over the simple
channel system.
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
3.5
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
Ra=102 L/Lh=4 Lh/b=01
Ra=102 L/Lh=4 Lh/b=01
θω
θω
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(a) Lh/b = 5
(b) Lh/b = 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.05
0.1
0.15
0.2
0.25
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
0 1 2 3 4 5 6 7 8 9 100.05
0.1
0.15
0.2
0.25
0.3
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
Ra=105 L/Lh=1.5 Lh/b=5
Ra=105 L/Lh=1.5 Lh/b=10
θω
θω
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(c) Lh/b = 20.
Fig. 6. Heated wall temperatures at Ra = 105 and L/Lh = 1.5 for different channel aspect ratios.
For Ra = 105 and L/Lh = 4, Fig. 7 shows that the lowest wall
temperatures are obtained for B/b = 2. This indicates that, by
also increasing the chimney height remarkably, the cold
inflow will be present, causing a decrease in the chimney
effect. In fact, for Lh/b = 5, Fig. 7(a), it is observed that the
wall temperature decreases up to B/b = 2 and then it increases
again for B/b≤ 3.0. For the highest analyzed Lh/b values, Figs.
7 (b) and (c), it is observed that the difference between the
wall temperature values for B/b = 2 and the ones for the
simple channel increases. An increase in the chimney effect,
when the channel aspect ratio increases, for the highest
Rayleigh number for all the analyzed L/Lh values, is also
present. For Ra = 105 the cold inflow determines optimal
configurations for B/b ≥ 2 for the highest extension ratio.
(a) Lh/b = 5
0 2 4 6 8 10 12 14 16 18 200.05
0.1
0.15
0.2
0.25
0.3
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
Ra=105 L/Lh=1.5 Lh/b=20
Ra=105 L/Lh=4 Lh/b=5
θω
θω
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(b) Lh/b = 10
(c) Lh/b = 20.
Fig. 7. Heated wall temperatures at Ra = 105 and L/Lh = 4 for different channel aspect ratios.
To obtain quantitative values and furnish a better analysis of
the thermal behavior of the present system, the values of the
ratio 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 ( ratio of the the maximum temperature
of the channel–chimney system and the one of the simple
channel ) as a function of the expansion ratio are reported in
Figs. 8 and 9, for L/Lh from 1.5 to 4 where 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 is
defined as the maximum normalized temperature in the
channel walls. For Ra = 102, Fig. 8 indicates that the ratio
𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 is less than 1 for all the analyzed
configurations. In agreement with the wall temperature
profiles, the ratio decreases, attaining a minimum value, and
then it increases for L/Lh = 1.5 for all Lh/b values, whereas for
Lh/b = 5 the ratio 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 attains a minimum value as
well as for the configuration with L/Lh = 2, as observed in Fig.
8(a). For other analyzed L/Lh values the profile of the ratio
𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 does not show a minimum or a maximum
value in the considered interval. It is interesting to observe that
the difference between the ratio 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 for L/Lh = 1.5
and that for L/Lh = 4 increases when the expansion ratio
increases, for a fixed B/b value. For Lh/b = 10, Fig. 8(b)
depicts that the values of the ratio 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 are always
lower than the ones corresponding to the configuration with
Lh/b = 5. Moreover, the differences between the values at L/Lh
= 1.5 and at L/Lh = 4 still increase and for B/b = 1 the value is
about 0.105, whereas it is about 0.345 for B/b = 4. For Lh/b =
20 the values are very close to those for Lh/b = 10 and the
differences are the same.
0 2 4 6 8 10 12 14 16 18 200.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
0 2 4 6 8 10 12 14 16 18 200.05
0.1
0.15
0.2
0.25
0.3
0.35
X
Simple Channel B/b=1 B/b=1.5 B/b=2 B/b=3 B/b=4
Ra=105 L/Lh=4 Lh/b=10
Ra=105 L/Lh=4 Lh/b=20
θω
θω
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(a) Lh/b = 5
(b) Lh/b = 10
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=3
Ra = 102 Lh/b = 5𝜃𝜔𝑚𝑎𝑥
/𝜃𝜔𝑚𝑎𝑥
0
B/b
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
Ra = 102 Lh/b = 10
𝜃𝜔𝑚𝑎𝑥
/𝜃𝜔𝑚𝑎𝑥
0
B/b
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(c) Lh/b = 20 Fig. 8. Ratio of the maximum wall temperature to the simple channel one vs.
B/b for different extension ratio values at Ra = 102 .
At Ra = 105, Fig. 9 indicates that the optimal configurations,
such as that for which the 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 value is minimum,
are those with the expansion ratio value, B/b, between 1.5 and
2 for all the considered extension ratios. Moreover, for L/Lh =
1.5 and 2, the configuration with B/b = 4 shows a channel
thermal behavior equal to the simple channel one for all the
analyzed channel aspect ratio values, the 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 ratio
being equal to 1. This is due to the downflow which is present
for these configurations. Moreover, the 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 values
decrease with increasing Lh/b whereas the differences between
the values at L/Lh = 1.5 and L/Lh = 4 increase.
The above results lead to determine the best configurations for
the channel-chimney system in order to avoid or rather
mitigate the maximum wall temperatures around the heated
vertical channels.
(a) Lh/b = 5
0.4
0.5
0.6
0.7
0.8
0.9
1
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
Ra = 102 Lh/b = 20
𝜃𝜔𝑚𝑎𝑥
/𝜃𝜔𝑚𝑎𝑥
0
B/b
0.8
0.85
0.9
0.95
1
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
𝜃𝜔𝑚𝑎𝑥
/𝜃𝜔𝑚𝑎𝑥
B/b
Ra=105 Lh/b=5
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(b) Lh/b =10
(c) Lh/b = 20.
Fig. 9. Ratio of the maximum wall temperature to the simple channel one vs.
B/b for different extension ratio values at Ra = 105 .
The values of the normalized mass flow rate ratio ∆ψ ∆ψ0⁄ (
the ratio of mass flow rate of the channel–chimney system to
that of the simple channel system ), as a function of the
expansion ratio are reported in Figs. 10 and 11, for L/Lh
ranging from 1.5 to 4, and for Ra = 102 and Ra = 105,
respectively. The values of the ratio ∆ψ ∆ψ0⁄ are always
greater than 1, showing that the mass flow rate in the channel–
chimney system is always greater than that in the simple
channel, except for Ra = 105 at the lower L/Lh values. In fact,
for these configurations for B/b = 4, ∆ψ ∆ψ0⁄ is almost equal
to 1 for all Lh/b values. For Ra = 102, Fig. 10, it is observed
that the mass flow rate, pertinent to the channel– chimney
system, is about two and half times that of the simple channel
when B/b ≤ 3 for L/Lh = 4 and for all values ofLh/b. The
differences between the ∆ψ ∆ψ0⁄ ratios for B/b = 1 for the
different analyzed extension ratios are far lower than the same
differences for B/b = 4. This means that, for a fixed and low
extension ratio, the increase in the expansion ratio produces
variations significantly lower than those pertinent to the higher
L/Lh values. In most configurations with a fixed L/Lh value,
the maximum values of ∆ψ ∆ψ0⁄ are present for B/b in the
range 1.5 – 4.
0.75
0.8
0.85
0.9
0.95
1
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
𝜃𝜔𝑚𝑎𝑥
/𝜃𝜔𝑚𝑎𝑥
B/b
Ra=105 Lh/b=10
0.75
0.8
0.85
0.9
0.95
1
1.05
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
Ra=105 Lh/b=20
𝜃𝜔𝑚𝑎𝑥
/𝜃𝜔𝑚𝑎𝑥
0
B/b
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(a) Lh/b = 5
(b) Lh/b = 10
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
B/b
∆ψ
∕∆ψ
Ra=105 Lh/b=5
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
Ra=105 Lh/b=10
B/b
∆ψ
∕∆ψ
0
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(c) Lh/b = 20.
Fig. 10. Ratio of the dimensionless mass flow rate to the simple channel one vs.
B/b for different extension ratio values at Ra = 102 .
For Ra = 105,as shown in Fig. 11, the maximum values of the
normalized mass flow rate ∆ψ ∆ψ0 ⁄ ratio are always obtained
for B/b≤ 2 and they depend more significantly on Lh/b rather
than for the case of Ra = 102, especially for L/Lh = 3 and 4.
These results confirm that the chimney effects are worsened
for the channel–chimney system when the down flow is
present in the chimney and they allow for a quantitative
evaluation of the decrease in the mass flow rate. Moreover,
comparing the configurations for Ra = 102, Fig. 10, with those
for Ra = 105, Fig. 11, it is observed that, for B/b = 1, an
increase in L/Lh leads to a larger increase in ∆ψ ∆ψ0⁄ ratio for
Ra = 105 for all the analyzed Lh/b values.
These results determine the extreme importance of the coolant
mass flow rate which must not be allowed to decrease in the
reactor core.
(a) Lh/b = 5
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
∆ψ
∕∆ψ
0
B/b
Ra=105 Lh/b=20
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
Ra=105 Lh/b=5
∆ψ
∕∆ψ
0
B/b
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(b) Lh/b = 10
(c) Lh/b = 20.
Fig. 11. Ratio of the dimensionless mass flow rate to the simple channel one vs.
B/b for different extension ratio values at Ra = 105 .
Analogous trends are obtained for the Nu/Nu0 ratio, where Nu
is the average Nusselt number pertinent to the channel–
chimney system and Nu0 is the one corresponding to the
simple channel. The values of the ratio Nu/Nu0 as a function
of the expansion ratio are reported in Figs. 12 (a) and (b), for
L/Lh from 1.5 to 4 and Lh/b = 10, for Ra = 102 and Ra = 105,
respectively. The trends and the dependence on Lh/b are
qualitatively very similar to those shown in Figs. 10 and 11
whereas the differences are more pronounced between the
ratios given for Ra = 102 as in Fig. 12(a), and that for Ra = 105
as in Fig. 12(b). In fact, for Ra = 102, Nu/Nu0 reaches a
maximum value of about 1.8, whereas for Ra = 105 the
maximum value is slightly higher than 1.2. This indicates that,
for the lowest considered Ra value, the heat transfer enhances
more significantly in the channel–chimney system, whereas,
for the highest considered Ra value, the heat transfer
enhancement due to the employment of chimney is larger than
20% with respect to the simple channel. The maximum wall
temperature, average Nusselt number and mass flow rate ratio,
for smaller L/Lh, present their minimum and maximum value,
respectively, at B/b = 1.5 and, for higher B/b value, 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 increases and Nu/Nu0 and ∆ψ ∆ψ0 ⁄ decrease due to
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
Ra=105 Lh/b=10∆
ψ∕∆
ψ0
B/b
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/lh=3 L/Lh=4
B/b
∆ψ
∕∆ψ
0
Ra=105 Lh/b=20
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the presence of cold inflow, which determines a decrease of
the chimney effect. For L/Lh > 1.5, the cold inflow starts at
higher B/b values and, for B/b = 2, 𝜃𝜔𝑚𝑎𝑥/𝜃𝜔𝑚𝑎𝑥0 attains the
minimum value whereas Nu/Nu0 and ∆ψ ∆ψ0 ⁄ present the
maximum value.
(a) Ra = 102
(b) Ra = 105.
Fig. 12. Ratio of the average Nusselt number to the simple channel one vs. B/b
for different extension ratio values and Lh/b = 10.
This observation is more evident in Fig. 13, where the
maximum values of the ratio Nu/Nu0 are founded for different
L/Lh, Ra and Lh/b values. For Lh/b = 20, there is always an
enhancement of the thermal behavior of the system and the
maximum Nu/Nu0 ratio, (Nu/Nu0)max, increases when L/Lh
increases. For Ra = 102, with L/Lh passing from 1.5 to 4 , the
percentage increase of the ratio (Nu/Nu0) max is about 40 –
45 %, whereas for Ra = 105 it is only about 12 %.
1
1.2
1.4
1.6
1.8
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 l/Lh=3 L/Lh=4
Nu
/Nu
0
B/b
Ra=102 Lh/b=10
0.95
1
1.05
1.1
1.15
1.2
1.25
1 1.5 2 2.5 3 3.5 4
L/Lh=1.5 L/Lh=2 L/Lh=3 L/Lh=4
B/b
Nu
/Nu
0
Ra=105 Lh/b=10
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(a) Ra = 102
(b) Ra = 105.
Fig. 13. Maximum values of Nu/Nu0 vs. L/Lh for different Lh/b and Ra values.
6. CONCLUSIONS
The natural convection flow induced by a localized heat
source on the wall of a vertical channel in the core of MTR
reactors which uses plate type fuel elements with walls at
ambient temperature has been investigated numerically and
asymptotically. Numerical solutions have been computed for
an infinitely long channel and used to validate the asymptotic
scaling for large values of a Rayleigh number based on the
channel width. Simplified boundary layer equations have been
written on the basis of this scaling. The vertical extent of the
flow is found to be finite, and the limiting forms of the
solution around the upper and lower ends have been
computed.
Average Nusselt number, as a function of time, showed
minimum and maximum values and oscillations before the
steady state according to the temperature profiles. The profiles
show that, in terms of Nusselt number, for Ra = 102 the worst
configuration is B/ b = 1 and the best is for B/b = 4, whereas
for Ra = 105 the best configuration is B/b = 2 and the worst is
for B/b = 4. To conclude increasing the Ra value the optimum
B/b value, in terms of Nusselt number, decreases and the
worst configuration is obtained at higher B/b value.
Temperature wall profiles, as a function of axial coordinate,
enables the evaluation of thermal performances of the
channel–chimney system in terms of maximum wall
temperatures for different expansion ratios, as a function of the
channel aspect ratio. For the considered Rayleigh number
1.08
1.1
1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
1 2 3 4 5 6
Lh/b=5 lh/b=10 Lh/b=20
(Nu
/Nu
0) m
ax
L/Lh
Ra=105
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1 2 3 4 5 6
Lh/b=5 Lh/b=10 Lh/b=20
(Nu
/Nu
0) m
ax
L/Lh
Ra=102
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values, the difference between the highest and the lowest
maximum wall temperatures increased with increasing channel
aspect ratio. This behavior becomes greater as the extension
ratio goes up . These differences decreased significantly for
the highest Rayleigh number value. Optimal configurations for
assigned L/Lh and Lh/b were evaluated in terms of B/b
corresponding to the minimum value of maximum wall
temperatures. The optimal B/b values depend strongly on Ra
and L/Lh values and slightly on the channel aspect ratio. A
more significant increase of maximum average Nusselt
number referred to the simple channel value was obtained for
the lowest considered Ra value, Ra = 102, Lh/b = 20 and L/Lh
= 4 and it was about 80%, whereas for Ra = 105 this increase
was only about 24% for the same Lh/b and L/Lh values. This
mainly means that the reactor could be operated up to 950 kW,
in the free convection regime, instead of only a maximum
design value of 400 kW. Increasing the operating power of the
reactor in the natural convection mode of operation by about
2.38 times is of extreme importance as far as the reactor safety
and operation are concerned in this regime.
The present results highlighted the important significant
factors to enhance the reactor’s free convection heat transfer
for the channel-chimney system, such as the Rayleigh and
Nusselt numbers, aspect ratio, expansion ratio, and extension
ratio.
The present results allow to choose the favorite configurations
of the suggested channel-chimney system in the core of the
typical considered MTR reactor which avoid attaining the
maximum wall temperatures around the vertical heated
channels in the core and to improve the natural convection
heat transfer of the system. Conditions for keeping the coolant
mass flow rate in the core within the desired values are
lavished.
7. REFERENCES [1] Asako Y, Nakamura H, and Faghri M (1990). Natural convection
in vertical heated tube attached to thermally insulated chimney of a different diameter. ASME J. Heat Transfer. 112: 790-793.
[2] Auletta A, Manca O, Morrone B, Naso V (2001). Heat transfer
enhancement by the chimney effect in a vertical isoflux channel. Int. . of Heat and Mass Transfer 44: 4345-4357.
[3] Bejan A, da Silva AK, and Lorente S (2004). Maximal heat
transfer density in vertical morphing channels with natural convection. Numer. Heat Transfer. A 45: 135-152.
[4] Bianco N, Manca O, Morrone B, Naso V (1998). Experimental
analysis of chimney effect for vertical isoflux symmetricaaly heated parallel plates. Proceedings of the Eurotherm Seminar No.
85 on Thermal Management of Electronic Systems. III: 73-79.
[5] Campo A, Manca O, and Morrone B (1999). Numerical analysis of partially heated vertical parallel plate in natural convection
cooling. Numer. Heat Transfer. Part A 36: 129-151.
[6] Fisher TS, Torrance KE, and Sikka KK (1997). Analysis and optimization of a natural draft heat sink system. IEEE Tras. On
Component, Packaging Manufacturing Technol. Part A 20: 11-119.
[7] Fisher TS, and Torrance KE (1998). Free convection limits for pin fin cooling. ASME J. Heat Transfer. 120: 633-640.
[8] Fisher TS, and Torrance KE (1999). Experiments on chimney
enhanced free convection. ASME J. Heat Transfer. 121: 603-609. [9] Gebhart B, Jaluria Y, Mahajan RM, Sammaka B (1988).
Buoyancy-Induced Flows and Transport. Hemisphere Publ. Corp.,
New York. [10] Haaland SE, nd Sparrow (1983). Solutions for the channel plume
and the parallel-walled chimney. Numer. Heat Transfer. 6: 155-
172. [11] Kim SJ, nd Lee SW (1966). Air Cooling Technology for Electronic
Equipment. CRC Press, Boca Raton, FL.
[12] Ledezma GA, and Bejan A (1977). Optimal geometric arrangement of staggered vertical plates in natural convection.
ASME J. Heat Transfer. 119: 700-708. [13] Lee KT (1994). Natural convection in vertical parallel plates with
an unheated entry or unheated exit. Numer. Heat Transfer. Part A
25: 477-493. [14] Manca O, Morrone,B, Nardini S, Naso V (2000). Natural
convection in open channels. In Computational Analysis of
Convection Heat Transfer, Eds. Suden B, and Comini G, WIT Press, Southampton, UK, pp. 235-278.
[15] Oosthuizen PH (1984). A numerical study of laminar free
convection flow through a vertical open partially heated plane duct. ASME HTD. 32: 41-48.
[16] Shahin GA, and Floryan JM (1999). Heat transfer enhancement
generated by the chimney effect in systems of vertical channels. ASME J. heat Transfer. 121: 230-232.
[17] Straatman AG, Tarasuk JD, and Floryan JM (1993). Heat transfer
enhancement from a vertical isothermal channel generated by the chimney effect. ASME J. Heat Transfer. 115: 395-402.
[18] Wirtz RA, and Haag T (1985), Effects of an unheated entry on natural convection between heated vertical parallel plates. ASME
Paper 85-WA/HT-14.
8. NOMENCLATURE:
a thermal diffusivity m2/s
b channel gap M
B chimney gap M
g acceleration due to the gravity m/s2
Gr Grashof number
h (x) local convective coefficient W/m2k
K thermal conductivity W/m2k
L channel–chimney height m
Lh channel height m
Lx height of the reservoir m
Ly width of the reservoir m
Nu (x) local Nusselt number
Nu average Nusselt number
q heat flux w/m2
Ra Rayleigh number
Ra* channel Rayleigh number,
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171701-9292-IJMME-IJENS © February 2017 IJENS I J E N S
u , v velocity components along x m/s
U , V dimensionless components
x , y Cartesian coordinates M
X , Y dimensionless coordinates,
Pr Prandtl number
𝜃𝜔 Dimensionless temperature
Nu0 Normalized Nusselt number
Ψ stream function m2/s