Open Journal of Applied Sciences, 2012, 2, 319-325
doi:10.4236/ojapps.2012.24046 Published Online December 2012 (
A 2D Finite Element Study on the Flow Pattern and
Temperature Distribution for an I sothermal Spherical
Furnace with the Aperture
Sudhakar Matle, Subbiah Sundar
Mathematics, IIT Madras, Chennai, India
Received October 11, 2012; revised November 12, 2012; accepted November 21, 2012
Calibration of radiation thermometers is one of the important research activities in the field of metrology. Many re-
searchers in recent times have conducted numerical simulations on the calibration furnace to understand and overcome
the experiment limitations. This paper presents a 2D numerical free convective study on the calibration furnace with the
aperture using finite element method. The focused issues here are: aspect ratio effect on the flow pattern and tempera-
ture fields, heat transfer mechanism in the aperture zone as well as in hump regime. It is concluded that flow and tem-
perature fields follow the same behavior in the hump regime as well as in the aperture zone. Also, it concluded that
penetrative convection is more dominant for the enclosure of high aspect ratio.
Keywords: Calibration Furnace; Thermal Penetration; Rayleigh Number; Aspect Ratio
1. Introduction
Temperature measurement of the very hot objects by
means of contact thermocouple is not always handy. For
this purpose, radiation thermometer is used to measure
temperature in terms of radiation emitted by the hot ob-
ject. Radiation thermometer is a temperature measure-
ment instrument and measures temperature of the very
hot object from a distance. For accuracy of the instru-
ment, it must be calibrated. Calibration is an act of ad-
justing instrument by comparison against any standard
surface. The Saturn (a hollow spherical cavity) is one of
such standards used in radiation thermometer calibration
[1]. For details on calibration furnace and the finite ele-
ment modeling to study experiment limitations, one can
refer authors’ previous work [2].
Many research efforts have been done finite element
modeling [3,4] of calibration furnaces, industrial fur-
naces, heated spheres [5,6] and solar cavity receivers for
quality equipment. Recently, Oluwole et al. [7] studied
the flow patterns in two salt bath furnaces using finite
element analysis. The implications of the heat flows on
long term stability of furnace performance were evalu-
ated. Khoei et al. [8] developed a finite element model
that is employed to simulate the furnace rotation and
analyze energy flows inside the furnace. Also, finite
element, stream function vorticity solutions for steady
state incompressible Navier Stokes equations are derived
in papers [9,10].
Natural convection in fluid filled enclosures [11,12]
with heating bottom and sides [13] have been studied
extensively. Sarris et al. [14] studied numerically natural
convection in a rectangular enclosure with heating from
the top wall with all others insulated by varying Rayleigh
number from 102 to 108 and the aspect ratio from 0.5 to 2.
Recently, Hartlep et al. [15] performed simulations of
Rayleigh-Benaurd convection in a large box of aspect
ratio 10 over a range of Rayleigh and Prandtl numbers
provide important insight into the choice of the aspect
ratio. In another study, Lee et al. [16] stressed the need
for very large aspect ratio domains by studying the natu-
ral convection in a horizontal fluid layer with a periodic
array of internal square cylinders.
Finite element mesh is a part of numerical simulation
study. Sensitivity analysis for parameter dependent opti-
mization problems is an active area of research in the
context of solution of partial differential equations. Beck-
er and Vexler [17,18] investigated sensitivity of the mesh
based on relative condition numbers which describe the
influence of small changes in measurements on the value
of interest functional.
This paper addresses a 2D numerical free convective
study on the calibration furnace with the aperture. In the
next section, the 2D problem, governing equations and
the corresponding boundary conditions are systematically
discussed. In Section 3, mathematical formulation of 2D
Copyright © 2012 SciRes. OJAppS
problem and then numerical simulation schematically
presented. Section 4 presents precise results on flow and
temperature fields near the proximity of the furnace sur-
face and in the aperture plane.
2. Statement of the Problem
The problem is addressed in the context of temperature
distribution and 2D flow pattern for the furnace in three
various types of enclosures, but, the method, ideas and
results can be compared to enclosure of any dimension
and 3D model. The qualitative behavior of flow and heat
transfer is the same for a general 2D model and 3D mo-
del due to bi-axial symmetry of the sphere. For this pur-
pose, a furnace slice, obtained by cutting along the equa-
torial line, kept in an enclosure of two dimensions is
considered as a computational domain for the present
study. The coordinates chosen for the enclosure compu-
tational domain of aspect ratio 1 are E (0, 0), F (0.3, 0),
G (0.3, 0.3) and H (0, 0.3). The furnace dimensions are
taken from the experiment and it is shown in Figure 1.
The enclosure is filled with air to keep maintain experi-
ment conditions.
Mathematical Model
The computational domain () is assumed to be union
of the two sub domains 1 (gas) and 2 (solid). The
schematic sketch of the computational domain is
shown in Figure 1. Concentric sub domains are estab-
lished with settings from the outermost to innermost
are: stainless steel (44.5 W/mK), ceramic SiO2 (1.4
W/mK), concrete (1.8 W/mK), copper (400 W/mK) and
ceramic (SiO2). The value in the parentheses represents
thermal conductivity of the corresponding material.
The innermost sub domain is filled with air at Boussi-
nesq. 20˚C.
Figure 1. Schematic sketch of the computational domain.
The flow is incompressible air and two dimensional,
there is no viscous dissipation, gravity acts in vertical
direction, air properties are constant and density varia-
tions are neglected except the buoyancy term. Continu-
ity and momentum equations are defined in the air do-
main 1 as follows:
uu p
uv u
xy x
uv vgTT
xy y
∂∂ ∂
∂∂ ∂
∂∂ ∂
+=−+Δ+ −
∂∂ ∂
The heat energy equation is defined in the entire com-
putational domain () to study the heat transport equa-
tion is as follows.
0 in
uv T
∇+= Ω
where everywhere except at the heat source re-
3. Mathematical Formulation
The steady state incompressible Navier-Stokes equations
in velcoty-vector potential formulation are presented as
⋅∇= Δ+−
with no slip boundary conditions
0, 0
while the velocity is given by
, ,u
=∇∇= ∂−∂ (5)
The stream function, and hence the velocity, is then
calculated from
Weak form of Equations (3)-(5) is as follows: find
such that
() ()
01 1
and HH
∈Ω ∈Ω
wugT TH
φνφωβφ φ
φψφω φ
∇=∇∇+ −∀∈
∇∇ =−∀∈ (6)
where , denotes the standard inner product on 1
for 2 norm. Let L2
be the standard con-
tinuous finite element space with the 2nd degree polyno-
mial on each element of a triangulation. Let 2
be the
subspace of 2
with zero boundary values. For the
Lagrangian finite element space,
() ()
dim and dim
hih ib
NXN==N+ where in i
Copyright © 2012 SciRes. OJAppS
the number of interior nodes ando N
b is number f
boundary nodes. Then the finite elem approximation
is as follows: find 22
hhh h
∈∈ such that
hhh h
Figure 2. In teand hsfer
phen an enclure, tile near toxi-
mie su is ste pronu-
assumption that the
is investigated based
from Rayleigh
order to betr underst
h hh
hhhh hh
ν φωφ
φψφω φ
∇∇ =−∀∈(7)
The velocity h
u is obtained from the stream
=∇ (8)
Clearly the velocity h
u sat
co t
isfies the divergence free
ndition everywhere andhe normal velocity h
un is
continuous across element boundary. The absolute value
of the extreme stream function is evaluated at the stag-
nant fluid, i.e.
0, 0element area
ext uv
The heat transfer equation defined on
Ω1 as follows:
d hh
TY such that
h h
∇= ,
hhhh h
τ τ
∇∇ ∀∈ (10)
where is a finite element space of sec
ond degree
polynoml defined on 1
On 2, heat transfer equation is defined as follows:
fin 2
d hh
such that
∇= ,
h hh
∇ ∀∈ (11)
A quadratic triangular finite element m
ic, there is a rapid de-
ents on locally refined meshes.
esh of maxi-
m element size 0.02 is chosen for the current numeri-
cal study. Maximum element size is defined as the ratio
of maximum edge length to the unit length of the ele-
Sensitivity of the mesh is examined for two different
meshes composed of 9989 nodes and 10,667 nodes based
on relative condition numbers. From Table 1, measure-
ments at m
k (the point on the cavity surface exactly
opposite to the aperture) have more influenced than b
(point at the cavity bottom) and t
k (point at the cavity
top). It is reasonable to say that at m
k, the solution gra-
dient is very high due to curvature. Therefore, the mesh
is refined locally towards direction normal to m
4. Results and Discussion
As the heat passes through ceram
crease in temperature but not zero and it is shown in
Table 1. Relative condition numbers for point measure-
N b
k m
k t
9989 0.000 0.0001 0.00012 13433
eat tran
nomena i
ty of th
rface top
he prof
udied. Th
he pr
file is si
soidal in the interval range 0.3 < x < 0.5 indicates the
possibility of occurrence of Rayleigh-Benaurd convec-
tion in the region above surface top.
Three enclosures of aspect ratio 0.5 (narrow), 1 and
1.5 (wide) are chosen for current numerical investigation.
Also, it is important to notify the
mensionless temperature θ is 1 at the heating coil B
(0.26, 0.4) only when cavity attains the experiment tem-
perature Ts = 409.8˚C at C (0.29, 0.4). No radiation ef-
fects are considered in the current study. Typical flow
and temperature fields, mechanism of heat transfer and
thermal penetration in the aperture region are thoroughly
discussed by varying Rayleigh numbers from 104 to the
high Rayleigh number achieved.
4.1. Flow and Temperature Fields
For the aspect ratio 1, flow pattern
on vorticity values in a stagnant fluid
number 104 to the high Rayleigh number ac
At the Rayleigh number 104, the flow is not rotational
at the region where the boundary layer entrains into the
main stream. Therefore, stream function satisfies La
uation. Hence the strength of circulation at the low
Rayleigh number is 0.000064 (x = 0.4, y = 0.604). At the
Rayleigh number 105, strength of the circulation is
0.00068 (x = 0.485, y = 0.585). At this stage, convective
flow cell is formed rotating in an anti-clock wise direc-
tion. At the high Rayleigh number 8.2 × 105, strength of
the circulation is 0.00188 (x = 0.43, y = 0.601) and
therefore both clockwise and anti-clockwise flow pattern
predicted with high velocity. Thus, as expected circula-
tion strength increase with the Rayleigh number and the
coordinate shifts from close to the hot cell towards inside,
(in an anti-clockwise sense) i.e. the flow penetrates into
the enclosure more and more with an increasing Rayleigh
In the aperture plane, strength of circulation is
0.0000145 (x = 0.51, y = 0.4) at the Rayleigh number 104.
x in m
Figure 2. Temperature profile at the vertical midpoint along
horizontal layer of cross sec tion.
10,667 0.000146 0.000114 0.000114
Copyright © 2012 SciRes. OJAppS
Therefore, only weak celredicted in between x =
n strength is 0.00252 (x =
.4, 0.603) for Rayleigh numbers
creases from 105 to 4.9 × 105, strength
eases to
e boundary layer on the circumference of
ls are p
0.51 and x = 0.6. As the Rayleigh number increases to
105, the corresponding circulatio
0.51, y = 0.4). Thus, circulation strength increases with
the Rayleigh number and the cold fluid penetrates into
the aperture more and more with an increasing Rayleigh
number. Also, there is one more point (x = 0.58, y = 0.4)
at which fluid is stagnant. The corresponding circulation
strengths are 0.01 (Ra = 104), 0.064 (Ra = 105) and 0.42
(Ra = 8.2 × 105). It indicates there is a strong convective
cell around that point.
For the aspect ratio 0.5, circulation strengths near the
hump regime are 0.00282 (0.435, 0.6), 0.0069 (0.49,
0.582) and 0.05775 (0
4, 105 and 4.9 × 105 respectively. At the Rayleigh num-
ber 104, penetration of the flow is not vertical and hence
the low flow velocity is predicted in the regime. As the
Rayleigh number increases, the flow penetration gradual-
ly comes to vertical and therefore, horizontal and vertical
penetration of the flow almost the same. At the low
Rayleigh number, the movement of the flow at the right
corner of the enclosure is very high when compared to
the movement of the flow at the left corner. As the
Rayleigh number increases, movement of the flow at the
left corner and at the right corner almost the same and
therefore, flow penetration gradually increases with the
Rayleigh number. Another observation is that the abso-
lute values of vorticity increases as the Rayleigh number
increases and hence the rotation of convective cell in-
creases with the Rayleigh number. One vortex is ob-
served for the Rayleigh number 105 while the two coun-
ter rotating large vortices are predicted at the high
Rayleigh number.
In the aperture plane, strength of circulation is 0.0098
(x = 0.58, y = 0.4) at the Rayleigh number 104. As the
Rayleigh number in
circulation increases from 0.05925 (x = 0.58, y = 0.4)
to 0.271 (x = 0.5835, y = 0.4) correspondingly.
For the aspect ratio 1.5, at the Rayleigh number 104,
circulation strength is weak with |ψext| = 0.00085 at x =
0.401, y = 0.603. As the Rayleigh number incr
5, correspondingly the location changes to x = 0.401, y =
0.604 with strength 0.0091 and penetration into the hump
region increases. Further increase in the Rayleigh number
results gradually full penetration into the hump with cir-
culation strength 0.1068 (Ra = 6.6 × 105) at the location x
= 0.05 and y = 0.603. Also, it is observed that the tem-
perature gradient is a decreasing function of the Rayleigh
number along the symmetry plane. The reason being (1)
formation of the vortex a little far away from the surface
involved in resisting heat transfer performance (2) pene-
trations from both horizontal and vertical directions are
almost the same. Therefore, length of the symmetry plane
participating in heat transfer to the right cold cell decreases.
In the aperture plane, circulation strength 0.0097 (x =
0.579, y = 0.4) at the Rayleigh number 104. As the
Rayleigh number increases to 105, location is movin
wards inside with strength 0.055 and level of penetra-
tion of the fluid increases. At the high Rayleigh number
achieved, circulation with maximum strength 0.289 at the
coordinate x = 0.58, y = 0.4. By comparison of isotherms
at the high Rayleigh number in terms of aspect ratio in-
dicates that the heat transfer rate is more at the high as-
pect ratio.
Figure 3(b) shows temperature contours at the high
Rayleigh number achieved for aspect ratios 0.5, 1 and 2.
ter layer surface is studied. The surface temperature
moves along the boundary layer in clockwise direction
and entrained into the main stream at the position where
in it encounters the temperature moves along the boun-
dary layer in an anti-clockwise direction. The mechanism
of heat transfer is by a single cell rotating in clockwise
direction in the left half driven by hot surface in between
0.36 and 0.43. The heat is transferred to the right half
along the symmetry line at x = 0.435, as a result of which
a counter-clockwise rotating cell is formed. Inner cell to
the left of the symmetry plane receives heat from the
outer convective cell and therefore records highest tem-
perature among hot convective cells while innermost cell
to the right of symmetry plane losses heat to outer con-
vective cells and hence records lowest temperature among
cold convective cells.
In case of the aspect ratio 1, temperature gradient along
the symmetry plane x = 0.435 increases as the Rayleigh
number increases.
Therefore, thermal penetration is more intensive with
higher temperature gradient along the symmetry plane,
(a) ar = 1
Figure 3. (a) Stream line plot for the aspect ratio 1; (b) Iso-
thermal plots at high Rayleigh numbers.
Copyright © 2012 SciRes. OJAppS
i.e., increasing heat transfer from the hot convective cell
to the cold convective cell. From the plot, as the Ray-
leigh number increases, thermal penetration shift towards
right while symmetry line shift towards left. Though
weak cells are generated beyond penetration limit at low
Rayleigh numbers, disappears as the Rayleigh number
In the aperture zone, the fluid flow is stagnant in be-
tween x = 0.5 and x = 0.51 while the fluid flow penetrat-
ing through the aperture. Therefore, high temperature
field is predicted in between x = 0.5 and x = 0.51. As the
Rayleigh number increases, the size of stagnant flow is
slightly reduced. From the temperature contours plot,
isotherms bend like inverted parabolas leave the aperture
toward inner surface of the cavity.
.2 × 105 for the aspect
r close
4.2. Free Convection in the Hump Regime
Local Nusselt number distribution along the horizontal
layer of cross section near the proximity of surface top
for various values of the aspect ratio 0.5, 1 and 1.5 and
for various values of Rayleigh number studied. For the
aspect ratio 1, maximum of local Nusselt number in-
creases as one moves from surface top to the environ-
ment. It indicates that heat transfer from the surface to
the ambient medium decreases. As the Rayleigh number
increases to 8.2 × 105, maximum of local Nusselt number
decreases as one move away from the surface.
Therefore, heat transfer from surface to ambient in-
creases. Also, heat transfer behavior is same qualitatively
for the aspect ratio 0.5 and 1.
Figure 4 shows the local Nusselt number along the
symmetry line and the line passes through hot convective
cells at the high Rayleigh number 8
ratio 1. From the plot, it clears that Nusselt numbe
to zero at two instances along the symmetry plane
it is almost zero at five positions along the line passes
through the hot cells. One important observation is that
number of positions at which Nusselt number proximity
Figure 4. Local Nusselt number along the symmetry line
and the line touches hot convective cell at x = 0.4 and x =
to zero represent the number of cells the line touches.
4.3. Free Convection in the Aperture Zone
In the aperture zone, local Nusselt number distribution
along the horizontal layer of cross section is studied for
various values of the aspect ratio 0.5, 1 and 1.5. Figure 5
shows the graph of local Nusselt number against Ray-
leigh number in the aperture plane for the aspect ratio 1.
From the plot, it clears that convection is gradually
dominated and therefore penetration of cold fluid into the
aperture increases as the Rayleigh number increases to
8.2 × 105. Another important observation is that many
weak convective cells are observed at Rayleigh number
104. As the Rayleigh number increases, weak cells vanish
and converted to very few strong convective cells inside
plane at the high aspect ratio 1.5. Another ob-
creases as the aspect ratio increases from 0.5 to 1.
the plane.
Figure 6 shows thermal penetration is 100% in the
servation is that number of strong convective cells
5. Conclusions
In view of results and discussions presented, the follow-
ing main conclusions are drawn.
Figure 5. Local Nusselt number distribution along hori-
zontal layer of cross section in the aperture zone for Ray-
leigh numbers 104, 105 and 8.2 × 105.
Figure 6. Local Nusselt number distribution along horizon-
tal layer of cross section in the aperture for the aspect ratios
0.5, 1 and 1.5.
Copyright © 2012 SciRes. OJAppS
Copyright © 2012 SciRes. OJAppS
Circulation strength has determined at the stagnant
fluid and concluded that strength of the circulation
increases with the Rayleigh number and the aspect ra-
Penetrative convection is more at the high aspect ra-
Enclosure of the high aspect ratio has been suggested
for stable solution.
Solution has been optimized with the constraint T =
6. Acknowled ge ments
The corre spond ing autho r is indebted to Ministry of Uni-
versity and Research, Cassino, Italy for their financial
support and thankful to Prof. Marco dell’Isola for tho-
rough discus sions on the calibration furnace.
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doi:10.1016 / j . jcp.20 04.12 .0 18
y Cartesian coordinates
,uv Velocity vectors
p Pressure
T Solution temperature, [˚C]
Ra Rayleigh number
Acceleration due to gravity, 2
Thermal conductivity
Pr Prandtl number = 0.71
Heat source
Thermal diffusivity, 2
Coefficient of thermal expansion,
C Specific heat capacity at constant pressure,
Density of air, 3
kg m
Emissivity of the cavity=0.98
Dimensionless temperature
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