A 2D Finite Element Study on the Flow Pattern and Temperature Distribution for an Isothermal Spherical Furnace with the Aperture

doi:10.4236/ojapps.2012.24046

Paper Menu >>

Journal Menu >>

Open Journal of Applied Sciences, 2012, 2, 319-325

doi:10.4236/ojapps.2012.24046 Published Online December 2012 (http://www.SciRP.org/journal/ojapps)

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

Email: [email protected]m, [email protected]

Received October 11, 2012; revised November 12, 2012; accepted November 21, 2012

ABSTRACT

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

S. MATLE, S. SUNDAR

320

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 (Ω) is assumed to be union

of the two sub domains Ω1 (gas) and Ω2 (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.

H G

E F

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 Ω1 as follows:

()

amb

uu p

uv u

xy x

vvp

uv vgTT

xy y

νβ

∂∂

∂∂ ∂

+=−+Δ

∂∂ ∂

+=−+Δ+ −

∂∂ ∂

(1)

The heat energy equation is defined in the entire com-

putational domain (Ω) to study the heat transport equa-

tion is as follows.

0 in

uv T

∂∂

+=ΔΩ

∂∂

∇+= Ω

(2)

where everywhere except at the heat source re-

gion.

0Q=

3. Mathematical Formulation

The steady state incompressible Navier-Stokes equations

in velcoty-vector potential formulation are presented as

()

amb

ugT

ωνωβ

ψω

⋅∇= Δ+−

Δ=

(3)

with no slip boundary conditions

0, 0

∂

∂=

)

(4)

while the velocity is given by

(

, ,u

⊥⊥

=∇∇= ∂−∂ (5)

The stream function, and hence the velocity, is then

calculated from

Δ=.

Weak form of Equations (3)-(5) is as follows: find

such that

() ()

01 1

and HH

ψω

∈Ω ∈Ω

,,,

amb

wugT TH

φνφωβφ φ

φψφω φ

∇=∇∇+ −∀∈

∇∇ =−∀∈ (6)

where ⋅, denotes the standard inner product on 1

and

⋅ 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

0,h

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

S. MATLE, S. SUNDAR 321

the number of interior nodes ando N

ent

b is number f

boundary nodes. Then the finite elem approximation

is as follows: find 22

and

hhh h

ωψ

∈∈ such that

hhh h

φω

∇

Figure 2. In teand hsfer

phen an enclure, tile near toxi-

mie su is ste pronu-

assumption that the

is investigated based

from Rayleigh

hieved.

place

order to betr underst

,,,

h hh

hhhh hh

ν φωφ

φψφω φ

=∇∇∀∈

∇∇ =−∀∈(7)

The velocity h

u is obtained from the stream

thro

function

ugh

⊥

=∇ (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

ψω

=× (9)

The heat transfer equation defined on

fin

Ω1 as follows:

d hh

TY∈ such that

h h

τα

∇= ,

hhhh h

uTT Y

τ τ

∇∇ ∀∈ (10)

where is a finite element space of sec

ond degree

polynoml defined on 1

Ω.

On Ω2, heat transfer equation is defined as follows:

fin 2

TY∈

d hh

such that

∇= ,

h hh

TQk Y

ττ

∇ ∀∈ (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-

ment.

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

number.

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

S. MATLE, S. SUNDAR

322

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.

Temperatur

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.

S. MATLE, S. SUNDAR 323

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

increases.

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

while

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 =

0.435.

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-

in-

creases as the aspect ratio increases from 0.5 to 1.

the plane.

Figure 6 shows thermal penetration is 100% in the

aperture

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.

S. MATLE, S. SUNDAR

324

• Circulation strength has determined at the stagnant

fluid and concluded that strength of the circulation

increases with the Rayleigh number and the aspect ra-

tio.

• Penetrative convection is more at the high aspect ra-

tio.

• Enclosure of the high aspect ratio has been suggested

for stable solution.

• Solution has been optimized with the constraint T =

409.8˚C.

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.

REFERENCES

[1] D. Nutter and D. P. Dewitt, “Theory and Practice of Ra-

diation Thermometry,” Wiley-Intersci en ce, New York,

1989.

[2] S. Matle and S. Sundar, “Axi Symmetric 2D Simulation

and Numerical He at Transfer C haracteristics for the Cali-

bration Furnace in a Rectangular Enclosure,” Applied

Mathematical Modeling, Vol. 36, No. 3, 2012, pp. 878-

893. doi:10.1016/j.apm.2011.07.047

[3] S. J. Cogan, K. Ryan and G. J. Sheard, “The Effects of

Vortex Breakdown Bubbles in the Mixing Environment

inside a Base Driven Bioreactor,” Applied Mathematical

Modeling, Vol. 35, No. 4, 2011, pp. 1628-1637.

[4] A. Riahi and J. H. Curvan, “Full 3D Finite Element Cos-

serat Formulation with Application in Layered Struc-

tures,” Applied Mathematical Modeling, Vol. 33, No. 8,

2009, pp. 3450-3464. doi : 10.1016/j.apm.2008.11.022

[5] A. Prhashanna and R. P. Chhabra, “Free Convection in

Power-La w Fluids from a Heated Sphere,” Chemical En-

gin eering Science, Vol. 65, No. 23, 2010, p p. 6190-6205 .

doi:10.1016 / j . c es . 2010. 09.003

[6] M. Z. Salleh, R. Nazar and I. Pop, “Modelling of Free

Convection Boundary Layer Flow on a Solid Sphere with

Newtonian Heating,” Acta Applicandae Mathematicae,

Vol. 112, No. 3, 2010, pp. 263-274.

doi:10.1007 / s10440-010-9567-5

[7] O. O. Oluwole, P. O. Atanda and B. I. Imas ogie, “Finite

Element Modeling of Heat Transfer in Salt Bath Fur-

naces,” Journal of Minerals and Materials Characteriza-

tion and Engineering, Vol. 8, No. 3, 2009, pp. 209-236.

[8] A. R. Khoei, I. Masters and D. T. Gethin, “Numerical

Modeling of the Rotary Furnace in Aluminium Recycling

Processes,” Journal of Materials Processing and Tech-

nology, Vol. 139, No. 1-3, 20 03, pp. 567-572.

doi:10.1016 / S 0924-0136(03)00538-7

[9] P. A. B. d e Sampaio , P. R. M. Lyr a, K . Morgan and N. P.

Weatherill , “Petr o v-Galerkin Solutions of the Incom-

pressible Navier-Stokes Equations in Primitive Variables

with Adaptive Remeshing,” Computer Methods in Ap-

plied Mechanics and Engineering, Vol . 106, No. 1-2,

1993, pp. 143-178. doi:10.1016/0045-7825(93)90 18 9-5

[10] W. N. R. Stevens, “Fini te Element Stream Function-Vor-

ticity Solution of Steady Laminar Natural Convection,”

International Journal for Numerical Methods in Fluids,

Vol. 2, No. 4, 1982, pp. 349-366.

doi:10.1002 / fld.16 500204 04

[11] I. Goldhirsch, R. B. Pelz and S. A. Orszang, “Numerical

Simulation of Thermal Con vection in a Two D imensional

Finite Box,” Journal of Fluid Mechanics, Vol . 199, 1989,

pp. 1-28. doi:10.1017/S0022112089000273

[12] D. E. Fitzjarrald, “An Experiment Study of Turbulent

Convectio n in Air,” Jou rnal of Fluid Mechanics, Vol. 73,

No. 4, 1976, pp. 337-353.

[13] E. Bilgen and R. B. Yedder, “Natual Convection in En-

closure with Heating and Cooling by Sinusoidal Tem-

perature Profiles on One Side, ” International Journal of

Heat and Mass Transfer, Vol. 50, No. 1-2, 2007, pp. 139-

150. doi:10.1016/j.ijheatmasstransfer.2006.06.027

[14] I. E. Sarris, I. Lekakis and N. S. Vlachos, “Natural Con-

vection in a 2D Enclosure with Sinusoidal Wall Temper-

ature,” Numerical Heat Transfer, Part A: Applications: An

International Journal of Computation and Methodology,

Vol. 42 , No. 5, 2002, pp. 513-530.

doi:10.1080 / 104077 80290059675

[15] T. Hartlep, A. Tilgner and F. H. Busse, “Large Scale

Stru ctu r es in Rayleigh-Benaurd Convection at High Ray-

leigh Numbers,” Ph ysical Review Letters, Vol. 91, No. 6,

2000. doi:10.1103/physRevLett.91.064501

[16] J. R. Lee, M. Y. Ha and S. Balachander, “Natural Con-

vection in a Horiz onta l Flui d Layer with a Periodic Array

of Internal Square Cylinders -Need for Very Large Aspect

Ratio Domains,” International Journal of Heat and Fluid

Flow, Vol. 28, No. 5, 2007, pp. 978-987.

doi:10.1016/j.ijheatfluidflow.2007.01.005

[17] R. Becker and B. Vexler, “A Posteriori Error Estimation

for Finite Element Discretizations of Parameter Identifi-

cation Problems,” Nu m erische Mathamatics, Vol. 96, No.

3, 2004, pp . 1-102.

[18] R. Becker and B. Vexler, “Mesh Refinement and Nu-

merical Sensitivity Analysis for Parameter Calibration of

Partial Differential Equations,” Journal of Computational

Physics, Vol. 206, No. 1, 2005, pp. 95-110.

doi:10.1016 / j . jcp.20 04.12 .0 18

S. MATLE, S. SUNDAR 325

Nomenclature

y Cartesian coordinates

,uv Velocity vectors

p Pressure

T Solution temperature, [˚C]

Ra Rayleigh number

Acceleration due to gravity, 2





Thermal conductivity

[

]

WmK

Pr Prandtl number = 0.71

Heat source





Greeks

Thermal diffusivity, 2









Coefficient of thermal expansion,

[

]

C Specific heat capacity at constant pressure,

[

]

KgK

Density of air, 3

kg m









Emissivity of the cavity=0.98

Dimensionless temperature

()

ref

TT T−Δ