Effect of surface radiation on RBC in cavities heated from below

<strong>Effect</strong> <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> on RBC in cavities heated from below☆
M. Ashish Gad 1 , C. Balaji ⁎
Department <strong>of</strong> Mechanical Engineering, Indian Institute <strong>of</strong> Technology Madras, Chennai, 600 036, India
article info
Available online 26 August 2010
Keywords:
Rayleigh–Benard convection
Natural convection
Surface <strong>radiation</strong>
Bottom heated enclosure
Onset <strong>of</strong> convection
1. Introduction
abstract
Rayleigh–Benard convection occurs in a fluid layer which is
confined between two thermally conducting plates, and is heated
from below to produce a fixed temperature difference. Rayleigh–
Benard convection (RBC) was first studied analytically by Lord
Rayleigh in 1916 in relation to the experiments made by Benard in
1900. Though Rayleigh–Benard convection has been widely studied in
the literature, studies on convection in a cavity heated from below
that simultaneously take into account effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> are
scarce. The study <strong>of</strong> bottom heated cavities is important for a variety
<strong>of</strong> engineering applications such as solar collector design, passive
energy storage and micro-manufacturing techniques such as
lithography.
2. Review <strong>of</strong> literature
Gille and Goody [5] conducted experimental studies on the effect
<strong>of</strong> radiative transfer on the onset <strong>of</strong> Rayleigh–Benard convection by
comparing data for dry air and NH 3 between parallel aluminum plates
maintained at different temperatures. Their experiments indicated
that the critical Rayleigh number in NH 3 is greatly increased
compared to that for air. Lan et al. [6], experimentally, determined
the onset <strong>of</strong> Rayleigh–Benard convection in horizontal layers <strong>of</strong>
varying mixtures <strong>of</strong> air and CO2. They found out a maximum increase
<strong>of</strong> 20% in the critical Rayleigh number. Lan et al. [7] used linear
stability analysis and weakly non-linear analysis to determine the
critical Rayleigh number for the onset <strong>of</strong> convection in the presence <strong>of</strong>
<strong>radiation</strong> in three-dimensional enclosure using spectral methods.
☆ Communicated by A.R. Balakrishnan.
⁎ Corresponding author.
E-mail address: balaji@iitm.ac.in (C. Balaji).
1 Graduate student.
0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.icheatmasstransfer.2010.08.003
International Communications in Heat and Mass Transfer 37 (2010) 1459–1464
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ichmt
Numerical investigations <strong>of</strong> combined free convection and <strong>surface</strong> <strong>radiation</strong> in enclosures for Rayleigh–
Benard configuration <strong>of</strong> air are carried out using FLUENT 6.3, with a view to determine the onset <strong>of</strong>
convection and to propose correlations for convection and <strong>radiation</strong> Nusselt numbers based on a detailed
parametric study. The onset <strong>of</strong> Rayleigh–Benard convection is delayed with an increase in the emissivity <strong>of</strong>
the sidewalls. The effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> on the onset <strong>of</strong> convection however diminishes with aspect ratio
(AR) and for AR=8, the effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> ceases. Post-onset, the effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> on the
convection heat transfer becomes insignificant beyond an aspect ratio <strong>of</strong> 5.
© 2010 Elsevier Ltd. All rights reserved.
Chunmei and Jayathi [3] carried out a numerical investigation <strong>of</strong> flow
transitions in deep three-dimensional cavities heated from below to
obtain the critical Rayleigh number for the onset <strong>of</strong> convection and the
transition to turbulence in tall cavities with aspect ratio varying from
1 to 5. To determine the onset <strong>of</strong> convection, steady simulations were
done starting with a Rayleigh number range across which the
transition occurs. Tong et al. [12] carried out numerical simulations
to determine the onset <strong>of</strong> natural convection in tall and shallow
rectangular enclosures for both bottom heated and differentially
heated configurations. The critical Rayleigh number was insensitive to
the aspect ratio for flat, wide enclosures but, on the contrary, the
critical value increased markedly with increasing aspect ratio for tall,
narrow enclosures.
Though the literature on free convection in enclosures is vast,
investigations which explore the effect <strong>of</strong> <strong>radiation</strong> on convection are
scarce. Balaji and Venkateshan [1,2] carried out numerical investigations
<strong>of</strong> free convection coupled with <strong>surface</strong> <strong>radiation</strong> in a
differentially heated square cavity. They showed that <strong>surface</strong>
<strong>radiation</strong> leads to a drop in the convective component but this
reduction tends to be compensated by the radiative transfer between
active walls. Ridouane et al. [10] discussed the effects <strong>of</strong> <strong>surface</strong>
<strong>radiation</strong> on convection in an air filled square cavity heated from
below cooled from above using a numerical model based on finite
differences. Ridouane and Hasnaoui [11], numerically, investigated
the influence <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> on the flow and temperature
patterns. The oscillatory behavior, characterizing the unsteady-state
solutions during the transitions from bicellular flows to the unicellular
flow were observed and discussed. Laboratory experiments exploring
the effects <strong>of</strong> <strong>radiation</strong> on convection are scarce. Among the few
studies are the one by Ramesh and Venkateshan [8] who conducted
interferometric studies on the problem undertaken numerically by
Balaji and Venkateshan. Later, Ramesh et al. [9] analyzed the effect <strong>of</strong>
boundary conditions on natural convection in a square enclosure with
<strong>surface</strong> <strong>radiation</strong>.

1460 M.A. Gad, C. Balaji / International Communications in Heat and Mass Transfer 37 (2010) 1459–1464
Nomenclature
English symbols
AR aspect ratio <strong>of</strong> cavity, L/H
Cp specific heat <strong>of</strong> fluid, kJ/kg K
g gravitational acceleration, 9.81 m/s 2
Gr Grash<strong>of</strong> number, gβΔTH 3 /ν 2
H characteristic height <strong>of</strong> the domain, m
L characteristic length <strong>of</strong> the domain, m
NRC <strong>radiation</strong> conduction interaction parameter, σ TH 4 d/
k(TH−T c)
Nu Nusselt number
NuC convection Nusselt number, q″ C L/kf ΔT
NuR <strong>radiation</strong> Nusselt number, q″R L/kf ΔT
Pr Prandtl number
Ra Rayleigh number, gβΔTH 3 /αν
T temperature <strong>of</strong> fluid, K
T∞ ambient temperature, K
TC temperature <strong>of</strong> cold plate, K
TH temperature <strong>of</strong> hot plate, K
To reference temperature, K
TR temperature ratio, TC/TH
U velocity in x direction, m/s
V velocity in y direction, m/s
x, y coordinate directions
Greek symbols
α thermal diffusivity, k
,m
ρCp 2 /s
β coefficient <strong>of</strong> thermal expansion, K −1
ΔT (TH−TC), K
ε emissivity
εH emissivity <strong>of</strong> the hot wall
εC emissivity <strong>of</strong> the cold wall
εB emissivity <strong>of</strong> the bottom wall
εT emissivity <strong>of</strong> the top wall
εR emissivity <strong>of</strong> the right wall
εL emissivity <strong>of</strong> the left wall
ρ density <strong>of</strong> fluid, kg/m 3
ρo density <strong>of</strong> fluid at reference temperature, kg/m 3
μ dynamic viscosity <strong>of</strong> the fluid, Ns/m 2
ν kinematic viscosity <strong>of</strong> fluid, m 2 /s
Though the preceding literature review shows that convection in a
cavity has received considerable attention, the problem <strong>of</strong> convection
in a cavity heated from below where <strong>surface</strong> <strong>radiation</strong> influences the
heat transfer has not received much attention. In consideration <strong>of</strong> this,
the objectives <strong>of</strong> the present study are to determine the effect <strong>of</strong>
<strong>surface</strong> <strong>radiation</strong> on the onset <strong>of</strong> Rayleigh–Benard convection for
various aspect ratios and to study the effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> on the
fluid flow and heat transfer characteristics for bottom heated cavities.
3. Methodology
3.1. Governing equations
All the problems considered for the present study involve twodimensional,
laminar steady convection with <strong>surface</strong> <strong>radiation</strong>. The
medium under consideration is air which is considered to be <strong>of</strong>
constant thermo-physical properties except for density. The density
changes are modeled via the Boussinesq approximation. Viscous heat
dissipation and compressibility effects are considered to be negligible.
Based on the above assumptions, the governing equations for mass,
momentum and energy for a steady two-dimensional flow in the fluid
domain are given below in terms <strong>of</strong> primitive variables.
Continuity equation
∂U ∂V
þ ¼ 0 ð1Þ
∂X ∂Y
X momentum equation
U ∂U ∂U
þ V ¼
∂X ∂Y
1 ∂P
ρ ∂X þ ν ∂2U ∂X2 þ ∂2U ∂Y2 !
Y momentum equation
U ∂V ∂V
þ V ¼
∂X ∂Y
1 ∂P
ρ ∂Y þ ν ∂2V ∂X2 þ ∂2V ∂Y2 !
þ gβðT T∞Þ ð3Þ
Energy equation
U ∂T ∂T
þ V
∂X ∂Y ¼ α ∂2T ∂X2 þ ∂2T ∂Y2 !
For the problem under consideration, the performance <strong>of</strong> discrete
ordinates (DO) <strong>radiation</strong> heat transfer model in FLUENT was found to
be better than the <strong>surface</strong>-to-<strong>surface</strong> (S2S) <strong>radiation</strong> heat transfer
model for the range <strong>of</strong> emissivity under consideration. In the present
study, the optical thickness is set to 0, signifying a transparent
medium.
3.2. Physical model and boundary conditions
The geometry under consideration is shown in Fig. 1. The fluid
zone is initialized as being at rest. The bottom and top plates are
isothermal, with the bottom plate at a higher temperature and the
horizontal sidewalls are adiabatic. The emissivity <strong>of</strong> the top and
bottom walls is kept 0.85 throughout the study. The emissivity <strong>of</strong> the
horizontal sidewalls is a variable (ε L=ε R). These boundary conditions
are expressed mathematically in Eqs. (5)–(9) and are shown in Fig. 1.
The important non-dimensional numbers for the problem are:
aspect ratio—AR, temperature ratio—TR, Grash<strong>of</strong> number—Gr and the
<strong>radiation</strong> conduction interaction parameter, N RC.
For the horizontal sidewalls,
∑q =0 ð5Þ
ε R = ε L = ε ð6Þ
For the bottom wall,
T = T H; ε H =0:85 ð7Þ
Fig. 1. Geometry and boundary conditions.
ð2Þ
ð4Þ

For the top wall,
T = T C = 303; ε C =0:85 ð8Þ
For all the walls,
U = V =0 ð9Þ
3.3. Solution procedure
The governing Eqs. (1)–(4) and the radiative transfer equation are
discretized using the finite volume method and the resulting
equations are solved in an iterative procedure using the standard
implicit, second order upwind solver with velocity and pressure
coupling achieved by the SIMPLE algorithm. Computations have been
carried out using FLUENT 6.3 [4]. Segregated, implicit, 2-D, laminar
steady incompressible flow solver has been employed for the
numerical study. The convergence criterion for the continuity, X and
Y momentum quantities and discrete ordinate intensity residuals is
set as 10 −6 and for the energy equation it is 10 −12 .
3.4. Procedure to determine onset <strong>of</strong> convection
In the present study, steady state simulations, starting with a
Rayleigh number range across which the transition occurs, are carried
out and a bisection algorithm is applied to detect the critical Rayleigh
number. The onset <strong>of</strong> convection is obtained by observing the Nusselt
number. The point at which a sudden change in ∂Nu
occurs, is deemed
∂Ra
to herald to the onset <strong>of</strong> convection. It is necessary to mention that
there will be negligible yet non-zero convection currents along the
horizontal sidewalls due to conductive heat transfer from these walls
and so a linear stability analysis will not be useful in this case.
Furthermore, we are looking at the onset <strong>of</strong> Rayleigh–Benard
convection which occurs due to unstable stratification.
3.5. Validation <strong>of</strong> numerical procedure
The numerical procedure is validated with the correlations for
convection and <strong>radiation</strong> Nusselt numbers generated numerically for
differentially heated square cavity by Balaji and Venkateshan [2].
Table 1 provides a comparison <strong>of</strong> results with the correlations for
Ra=50,000. The numerical procedure is found to be satisfactory over
the entire range <strong>of</strong> emissivity and Rayleigh number.
3.6. Grid independence study
The computational mesh is generated using GAMBIT 2.3. For the
present study, non-uniform mesh and boundary layer meshes are
considered Table 2 shows the results <strong>of</strong> a grid independence study for
a square cavity for Ra=3×10 4 and ε=0.85 with non-uniform mesh.
When the change in both the convection and <strong>radiation</strong> Nusselt
numbers are found to be less than 0.5% between two successive grids,
the grid is accepted and the former is chosen for subsequent
computations. A sudden change in the convection and <strong>radiation</strong>
Nusselt numbers is observed for the grid with 3600 cells as courser
meshes are not able to resolve the bicellular flow structure.
Table 1
Comparison <strong>of</strong> Nu C and Nu R obtained from simulations and correlations.
Case Nuc Nuc
(Balaji and Venkateshan)
M.A. Gad, C. Balaji / International Communications in Heat and Mass Transfer 37 (2010) 1459–1464
Table 2
Grid independence test for the square cavity (Ra=3×10 4 , ε=0.85).
Number <strong>of</strong>
cells
Convection Nusselt
number, NuC
Accordingly, a grid consisting <strong>of</strong> 5625 cells is selected, as highlighted
in Table 2.
4. Results and discussion
%
difference
4.1. <strong>Effect</strong> <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> on the onset <strong>of</strong> convection
Numerical simulations are carried out for laminar Rayleigh–
Benard convection with <strong>surface</strong> <strong>radiation</strong> spanning 6 different aspect
ratios (1, 2, 3, 5, 8, and 10) to determine the critical Rayleigh number
for the onset <strong>of</strong> convection for different emissivities <strong>of</strong> the sidewalls.
The procedure employed to determine the onset can be clearly seen in
Fig. 2.
The problem under consideration involves two parameters namely
aspect ratio and emissivity and hence the onset <strong>of</strong> convection is
obtained for 3 different emissivities <strong>of</strong> the sidewalls varied simultaneously
for both the sidewalls for every aspect ratio. The results are
shown in Fig. 3. The onset <strong>of</strong> Rayleigh–Benard convection is obtained
by running steady state simulations starting with the Rayleigh
number range across which transition occurs and applying the
bisection algorithm to detect the critical Rayleigh numbers by
observing the changes in the Nusselt number.
In general, the horizontal side wall creates a resistance to the
convection flow as boundary layers are formed near the horizontal
side walls. The no-slip condition near the side wall causes a hindrance
to the flow and the viscous forces increase, resulting in a higher
critical Rayleigh number for the onset <strong>of</strong> convection. Surface <strong>radiation</strong>
results in heat transfer from the horizontal sidewalls into the fluid
domain. This results in a reduction in the temperature gradient inside
the cavity which drives the Rayleigh–Benard convection. Thus, the
onset <strong>of</strong> convection is delayed due to <strong>surface</strong> <strong>radiation</strong>. It is clear from
the study, as the aspect ratio increases, the effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong>
reduces and it becomes negligible as the aspect ratio approaches 8. For
an aspect ratio <strong>of</strong> 10 or more, the critical Rayleigh number approaches
the value <strong>of</strong> 1708 which matches with the critical Rayleigh number for
the onset <strong>of</strong> convection in a fluid layer which is confined between two
horizontal plates and is heated from below and cooled from top
isothermally. Thus, for an aspect ratio 10 or more, the cavity mimics
an infinitely long cavity, regardless <strong>of</strong> the emissivity.
4.2. Fluid flow and heat transfer characteristics
Radiation Nusselt
number, NuR
4.2.1. Heat transfer in a square cavity (AR=1)
The present study addresses the problem <strong>of</strong> heat transfer in
cavities when the fluid is initially at rest and this leads to unique
solutions. The study does not address the problem <strong>of</strong> multiplicity <strong>of</strong>
solution reported by Ridouane et al. [10,11]. Fig. 4 (a) shows the
% error NuR NuR
(Balaji and Venkateshan)
1 3.425 3.645 6.0 4.308 4.208 2.4
2 3.577 3.834 6.7 0.395 0.359 9.9
3 3.602 3.683 2.2 4.250 3.872 9.8
%
difference
900 2.621 – 4.073 –
2025 2.653 1.22 4.071 −0.05
3600 2.899 9.27 3.781 −7.12
5625 2.898 −0.03 3.781 0
8100 2.898 0.00 3.782 0.03
1461
% error

1462 M.A. Gad, C. Balaji / International Communications in Heat and Mass Transfer 37 (2010) 1459–1464
Fig. 2. Change in Nusselt number with respect to Rayleigh number for a square cavity.
variation <strong>of</strong> convection and <strong>radiation</strong> Nusselt numbers with Rayleigh
number for different values <strong>of</strong> the emissivity <strong>of</strong> side wall. Fig. 4 (a)
shows that <strong>radiation</strong> results in a reduction in the convection
component. With an increase in the emissivity <strong>of</strong> the sidewalls, the
convection component decreases except for Rayleigh number in the
range 10,000–50,000. However, in general <strong>surface</strong> <strong>radiation</strong> leads to
an overall increase in the heat transfer across the cavity. Three
different regimes are observed in the range <strong>of</strong> the Rayleigh number
(Ra) from 5×10 3 to 10 5 . The emissivity <strong>of</strong> the side walls has a
negligible effect on the convective heat transfer in the range <strong>of</strong>
Rayleigh number 4000bRab10,000. It is observed that flow structure
changes from unicellular to bicellular for 10,000bRab50,000 and a
corresponding increase in the convective heat transfer occurs for a
higher emissivity <strong>of</strong> the horizontal sidewall. For RaN50,000, the
convection is suppressed significantly with an increase in the
emissivity <strong>of</strong> the side walls. For the case where <strong>surface</strong> <strong>radiation</strong> is
not considered, unicellular solutions are found to exist. So, Rayleigh–
Benard convection has a tendency to produce unicellular flow
structure. When <strong>surface</strong> <strong>radiation</strong> is considered, for Rayleigh number
10,000 to 50,000 a bicellular flow structure is obtained. Figs. 5 and 6
show the streamlines and isotherms for Rayleigh number 30,000 and
90,000 when ε=0.85 (the maximum value <strong>of</strong> the stream function is
mentioned at the appropriate streamline in the figure).
4.2.2. Heat transfer for higher aspect ratio cavities
Fig. 7 (a) and (b) show the variation <strong>of</strong> the convection and
<strong>radiation</strong> Nusselt numbers for a cavity <strong>of</strong> aspect ratio 2, for different
Rayleigh number, Ra
4400
4100
3800
3500
3200
2900
2600
2300
2000
1700 1 2 3 4 5 6 7 8 9 10
Aspect ratio
= 0
= 0.1
= 0.5
= 0.85
Fig. 3. <strong>Effect</strong> <strong>of</strong> the emissivity <strong>of</strong> the sidewalls on the onset <strong>of</strong> convection for cavities <strong>of</strong>
different aspect ratios.
Convection Nusselt number, NuC
b
Radiation Nusselt number, NuR
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5
Rayleigh Number, Ra x10-4 5.00
4.75
4.50
4.25
4.00
3.75
3.50
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5
Rayleigh Number, Ra x10-4 Rayleigh numbers and emissivities <strong>of</strong> the sidewalls. For the case <strong>of</strong> a
cavity with AR=2, only bicellular solution is found to exist
irrespective <strong>of</strong> the emissivity <strong>of</strong> the sidewalls. The observations
made in the case <strong>of</strong> bicellular flow structure for a square cavity are
also valid for this case i.e. an increase in the convective heat transfer
and a decrease in the radiative heat transfer with an increase in the
emissivity <strong>of</strong> the sidewalls. Fig. 8 (a) and (b) show the stream function
and isotherms respectively when the emissivity <strong>of</strong> the sidewall is 0.85
and the Rayleigh number is 75,000. For the case <strong>of</strong> aspect ratio 3, for
the pure convection case, tri-cellular flow structure is found to exist
while for the case where <strong>surface</strong> <strong>radiation</strong> is considered for Rayleigh
number greater than 10,000, tetra-cellular flow structure is found to
= 0
= 0.10
= 0.50
= 0.85
= 0.10
= 0.50
= 0.85
Fig. 4. Variation <strong>of</strong> (a) convection and (b) <strong>radiation</strong> Nusselt number with Rayleigh
number for different values <strong>of</strong> the emissivity <strong>of</strong> the horizontal side walls (AR=1).
1.5 x10 -4
303 K
315 K
Fig. 5. (a) Streamlines and (b) isotherms for Ra =30,000 and ε=0.85.

e present when the flow is at rest initially. It is pertinent to mention
that both flow structures exist in the entire range <strong>of</strong> Rayleigh numbers
considered. With an increase in the aspect ratio, the effect <strong>of</strong> <strong>surface</strong>
<strong>radiation</strong> on the convection heat transfer keeps reducing.
4.3. Correlations
Parameters identified by Balaji and Venkateshan [2] are used to
obtain useful correlations for the convection and <strong>radiation</strong> Nusselt
numbers for the post-onset region by using non-linear regression.
Convection Nusselt number, NuC
5.97 x10 -4
303 K
Fig. 6. (a) Streamlines and (b) isotherms for Ra=90,000 and ε=0.85.
b
Radiation Nusselt number, NuR
5
4.5
4
3.5
3
2.5
2
1.5
Rayleigh number, Ra x10-4 0 2 4 6 8 10
5.5
5
4.5
M.A. Gad, C. Balaji / International Communications in Heat and Mass Transfer 37 (2010) 1459–1464
= 0
= 0.1
= 0.85
= 0.1
= 0.85
Rayleigh number, Ra x10-4 4
0 2 4 6 8 10
339 K
Fig. 7. Variation <strong>of</strong> (a) convection and (b) <strong>radiation</strong> Nusselt number with Rayleigh
number (AR=2).
b
These are
6.5x10-4
Nuc =0:103 Ra 0:333 ð1+ εHÞ −0:023 −0:169 + 0:153 AR−0:024AR2
ð1+εÞ ð Þ
ð10Þ
ðNRC = ðNRC +1ÞÞ
4:401 AR 0:134
NuR =0:039 Ra 0:25 1:304 −0:258 + 0:175 AR−0:026AR
εH ð1+εÞ 2
ð Þ 0:882
NRC 1−T 4
R
0:618
AR 0:131
ð11Þ
The above correlations are valid for the following range <strong>of</strong> parameters:
5×10 3 ≤Ra≤105, 1≤AR≤5, 0.1≤ε H≤0.85, 0.1≤ε≤0.85. A quadratic in
the aspect ratio, AR is proposed for accommodating the non-linear effect
<strong>of</strong> the emissivity <strong>of</strong> the sidewalls on the convection and <strong>radiation</strong> Nusselt
numbers for different aspect ratios. The correlation coefficient <strong>of</strong> both
Eqs. (10) and (11) is 0.99.
5. Conclusions
303 K
333 K
Fig. 8. (a) Streamlines and (b) isotherms for Ra=75,000 and ε=0.85 (AR=2).
1463
Two-dimensional steady Rayleigh–Benard convection with <strong>surface</strong>
<strong>radiation</strong> in cavities <strong>of</strong> different aspect ratios was investigated
numerically using FLUENT 6.3. The onset <strong>of</strong> Rayleigh–Benard
convection is seen to be delayed with an increase in the emissivity
<strong>of</strong> the sidewalls. The effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> on the onset <strong>of</strong>
convection however diminishes with an increase in the aspect ratio
and at an aspect ratio <strong>of</strong> 8, the effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> ceases. For an
aspect ratio <strong>of</strong> 10 or more, the critical Rayleigh number approaches
the value <strong>of</strong> 1708 which matches with the critical Rayleigh number for
onset <strong>of</strong> convection for Rayleigh–Benard convection between two
infinitely long horizontal plates. Thus, for an aspect ratio 10 or more,
the cavity mimics an infinitely long cavity. Due to heat transfer
between the horizontal sidewalls and fluid zone, the flow characteristics
change significantly which lead to subsequent changes in the
heat transfer across the cavity. The effect <strong>of</strong> <strong>surface</strong> <strong>radiation</strong> on the
convection heat transfer becomes insignificant and thus convection–

1464 M.A. Gad, C. Balaji / International Communications in Heat and Mass Transfer 37 (2010) 1459–1464
<strong>radiation</strong> can be decoupled for cavities beyond an aspect ratio <strong>of</strong> 5.
Based on numerical data, useful correlations have been developed for
the convection and <strong>radiation</strong> Nusselt numbers.
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