Thermal Hydraulic Characteristics Of Extended Heated ...1≤ B/b≤ 4; 102 ≤ Ra≤ 105. This study results in enhancing the reactor power in the free convection regime from a maximum

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.



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



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



𝜕

𝜕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



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.



(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



(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



(b) Lh/b = 10

(c) Lh/b = 20


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


0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

X


Ra=102 L/Lh=4 Lh/b=01

Ra=102 L/Lh=4 Lh/b=01

θω

θω



(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


0 1 2 3 4 5 6 7 8 9 100.05

0.1

0.15

0.2

0.25

0.3

X


Ra=105 L/Lh=1.5 Lh/b=5

Ra=105 L/Lh=1.5 Lh/b=10

θω

θω



(c) Lh/b = 20.


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


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


Ra=105 L/Lh=1.5 Lh/b=20

Ra=105 L/Lh=4 Lh/b=5

θω

θω



(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


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


Ra=105 L/Lh=4 Lh/b=10

Ra=105 L/Lh=4 Lh/b=20

θω

θω



(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


Ra = 102 Lh/b = 10

𝜃𝜔𝑚𝑎𝑥


0

B/b



(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


Ra = 102 Lh/b = 20



0

B/b

0.8

0.85

0.9

0.95

1

1 1.5 2 2.5 3 3.5 4




B/b

Ra=105 Lh/b=5



(b) Lh/b =10

(c) Lh/b = 20.

Fig. 9. Ratio of the maximum wall temperature to the simple channel one vs.


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




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


Ra=105 Lh/b=20



0

B/b



(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


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


Ra=105 Lh/b=10

B/b

∆ψ

∕∆ψ

0



(c) Lh/b = 20.

Fig. 10. Ratio of the dimensionless mass flow rate to the simple channel one vs.


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


∆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


Ra=105 Lh/b=5

∆ψ

∕∆ψ

0

B/b



(b) Lh/b = 10

(c) Lh/b = 20.

Fig. 11. Ratio of the dimensionless mass flow rate to the simple channel one vs.


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


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



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


B/b

Nu

/Nu

0

Ra=105 Lh/b=10



(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



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,



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

Documents

Thermal Hydraulic Characteristics Of Extended Heated ...1≤ B/b≤ 4; 102 ≤ Ra≤ 105. This study results in enhancing the reactor power in the free convection regime from a maximum