Heat transfer engineering an international journal, tập 32, số 6, 2011

Heat Transfer Engineering, 32(6):439–454, 2011
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457632.2010.506166

Correlations for Natural Convection
in Vertical Convergent Channels With
Conductive Walls and Radiative
Effects
LUIGI LANGELLOTTO1 and ORONZIO MANCA2
1

Centro Sviluppo Materiali S.p.A., Rome, Italy
Dipartimento di Ingegneria Aerospaziale e Meccanica Seconda Universit`a degli studi di Napoli, Naples, Italy

2

Natural convection in air, in vertical convergent channels, is analyzed to carry out thermal design and optimization criteria.
A scale analysis is developed to estimate the optimal geometrical configuration in terms of total volume and average wall
temperature. The best geometrical configuration obtained by this analysis is the parallel-plates channel. New correlations
for mass flow rate, radiative heat flux, and dimensionless maximum wall temperature are proposed in the emissivity range
from 0.10 to 0.90, convergence angle ranging from 0◦ to 10◦ , ratio between minimum and maximum channel spacing in the
range from 0.048 to 1.0, aspect ratio, the ratio between wall length and minimum channel spacing, in the range from 10 to
58, and average channel Rayleigh number in the range from 5.0 to 2.3 × 105. For the same convergence angle and ratio
between minimum and maximum channel spacing ranges, new average Nusselt number correlations are also given. These
correlations are evaluated for emissivity value equal to 0.90, for aspect ratio, referred to the minimum channel spacing,
ranging from 10 to 80 and average channel Rayleigh number ranging from 2.5 × 10−2 to 2.3 × 105.

INTRODUCTION
Cooling technology for electronic equipments and components requires a deep knowledge of heat transfer phenomena.

The main aim is to maintain a relatively constant component
temperature equal to or lower than the manufacturer’s maximum specified service temperature, in order to ensure system performance and reliability [1–3]. The design of natural
convection thermal control systems using simple relations is
certainly appealing. Particular interest has been devoted to the
channel configuration and several contributions have dealt with
this geometry [4]. An interesting problem is the heat transfer
in a convergent channel with two uniformly heated flat plates
[5–11].
The first numerical and experimental study of natural convection in water, in a convergent vertical channel, was carried
This work was supported by MIUR with Articolo 12 D. M. 593/2000 Grandi
Laboratori “EliosLab.” A special acknowledgment is given to the reviewers; their
suggestions have improved the article.
Address corrrespondence to Professor Oronzio Manca, Dipartimento di
Ingegneria Aerospaziale e Meccanica Seconda Universit`a degli studi di Napoli,
Via Roma 29–81031, Aversa (CE), Italy. E-mail:

out in [5]. The converging walls were maintained at the same
uniform temperature. Natural convection in air in a uniformwall-temperature convergent channel was investigated experimentally in [6], numerically in [7], and both numerically and
experimentally in [8]. A numerical study on natural convection,
in vertical convergent channels, with uniform wall temperature, for different convergence angles was carried out in [9]. A
Nusselt number composite correlation was proposed for convergence angle in the range 0–60◦ . Recently, a numerical simulation and optimization for a vertically diverging and converging
channel with laminar natural convection was accomplished in
[10]. For convergent channels, the results showed that the optimal angle between the two walls was approximately zero when
the Rayleigh number was large. The configuration of a vertical
convergent channel was numerically studied in [11]. The two
principal flat plates, at uniform heat flux, were considered with
finite thickness and thermal conductivity. An experimental investigation on natural convection in air, in vertical convergent
channels, with uniform wall heat flux was presented in [12].
For the lowest spacing, maximum wall temperature decreased
significantly, passing from the configurations of the parallel vertical plate to the configurations with convergence angles δ ≥ 2◦ .

439


heat transfer engineering

Experimental

Numerical and
experimental
Numerical

Bianco et al. [13,
14]
Bianco et al. [15]

vol. 32 no. 6 2011

0◦ to 60◦ bmin
0◦ to 10◦ bmin

UWHF + CW + 0◦ to 10◦ bmax
RE
10–58

4.4 to 2.9 × 108

4.4 to 2.9 × 108

10–58

θ∗ = f5

bmin
, ε, Ra ∗bmax
bmax

Nu = f3

1/
p
p
p ;Re = f6
= θ∗ 0 + θ∗ ∞

bmin b
,
, Ra bmax
bmax Lw

1
bav
p
p
, Ra ∗bmax ;θmax = f4 Ra = θ0 + θ∞ /p
bmin

bav
, Ra bmax
bmin

bmin b
qr
,
, Ra ∗bmax ;
= f5 ε, qref , Ra ∗bmax
bmax Lw
qr + q c

bmin
, ε, Ra ∗bmax
bmax

1
p
p
Nu = f1 Ra bmax = Nu0 + Nu∞ /p ;Nu = f2

Re = f4

1
p
θ∗max = f2 Ra ∗ = θ∗ 0 + θ∗ ∞ p /p ;θ∗ = f3

1
p
Nu∗ = f1 Ra ∗ = Nu∗ 0 + Nu∗ ∞ p /p

Nub = f1 Ra b ; Nu∗ b = f2 Ra ∗b

4.4 to 2.9 × 108

10–58

1
p
p
Nu = f Ra , δ = Nu0 + Nu∞ /p
Nub = f Ra b

Nu = f Ra

Correlation

Nub = f1 Ra b ; θmax = f2 Ra b

1 to 1 × 106
4.4 to 2.9 × 108

6.4 to 4.8 × 104

4 × 103 to 7 × 104 Nubmax = f bmax /Lw , Rabmax

Ra

4.4 to 2.9 × 108

10–58

0.5–50

10–58

6, 8.5, 12

bmin /2

0◦ to 8◦

L/b
11.4, 22.9

Reference
Length

0◦ to 15◦ bmax

δ

0◦ to 10◦ bmin and
bmax
UWHF + CW + 2◦ , 10◦
bmax
RE(4)
UWHF + CW + 0◦ to 10◦ bmax
RE

UWT
UWHF(2) +
CW(3)
UWHF + CW

UWT

UWT(1)

B. C. on channel
walls

(1) Uniform Wall Temperature; (2) Uniform Wall Heat Flux; (3) Conductive Wall; (4) Radiative Effects.

Present work

Numerical

Numerical
Numerical

Kaiser et al. [9]
Bianco and
Nardini [11]
Bianco et al. [12]

Shalash et al. [8]

Numerical and
experimental
Numerical and
experimental

Type of

investigation

Sparrow et al. [5]

Reference

Table 1 Comparison among different studies of natural convection in convergent channels

440
L. LANGELLOTTO AND O. MANCA


L. LANGELLOTTO AND O. MANCA

Radiative effects are particularly interesting in convergent channels, due to the large view factor toward the ambient [13, 14]. In
the first study, a numerical analysis was carried out in laminar,
two-dimensional steady-state conditions, with the two principal
flat plates at uniform heat flux and taking into account wall conductivity and emissivity. Average Nusselt numbers were evaluated and simple monomial correlations for average Nusselt
numbers, in terms of channel Rayleigh numbers, were proposed. In the second study, an experimental investigation on
natural convection in air, in a convergent channel, with uniform
heat flux at the walls, was carried out. Average Nusselt numbers
were evaluated and simple monomial correlations for dimensionless maximum wall temperatures and average Nusselt numbers were proposed in terms of channel Rayleigh numbers in the
same ranges given in [13]. Numerical results, obtained in [13],
were in very good agreement with experimental results given in
[14].
Design charts for the evaluation of thermal and geometrical
parameters, for natural convection in air, were proposed for natural convection in vertical convergent channels in [15]. Thermal
design and optimization of a channel in stack of convergent
channels were obtained employing the correlations among the
more significant dimensionless thermal and geometrical parameters.

Proposed correlations for natural convection in convergent channels, given in the already-mentioned papers, are
reported in Table 1. In the present paper, a scale analysis
is carried out following the procedure given in references
[16–20]. New correlations for convective heat transfer contribution in terms of Reynolds numbers, dimensionless wall
temperature, and global Nusselt numbers are proposed. More
accurate new correlations for the ratio between radiative and
global heat flux (radiative and convective heat fluxes) are
evaluated.
The new correlations extend the analysis presented in references [15] and [20–23]. They are obtained by enlarging the results given in [13] to large values of channel aspect ratio and low
Rayleigh numbers. This also allows evaluation of the thermal
behavior of the convergent channels in a possible fully developed flow. The analysis is proposed to evaluate the previously
mentioned variable for vertical convergent channel, with surface emissivity ranging from 0.10 to 0.90, for a single assigned
wall thickness and thermal conductivity, for convergence angle
from 0◦ to 10◦ , ratio between minimum and maximum channel
spacing, bmin /bmax , in the range from 0.048 to 1.0, aspect ratio,
Lw /bmin , in the range from 10 to 80, and global Rayleigh num∗
bers referred to bmin , Ra bmin , in the range from 2.5 × 10−2 to
2.3 × 105.
From a different point of view, the present study may be
conceived as an effort to estimate the right balance between
the control of the maximum wall temperature and an applied symmetrical wall heat flux. Moreover, this attempt can
also be viewed as the maximization of heat transfer for an
assigned available total volume that is constrained by space
heat transfer engineering

441

Figure 1 Sketch of the configuration: (a) physical domain; (b) computational
domain.

limitations. This goal has been studied in references [10],
[16], [19], and [24], and reviewed lately in [25] and more
recently in [26]. The present geometry is important in electronic cooling [9, 10, 27] and in solar energy components
[28, 29].

MODEL DESCRIPTION AND NUMERICAL
PROCEDURE
Model Description
The physical domain under investigation is shown in
Figure 1a. It consists of two nonparallel plates that form a vertical convergent channel. Both plates are thermally conductive,
gray, and heated at uniform heat flux. The imbalance between
the temperature of the ambient air, To , and the temperature of the
heated plates draws air into the channel. The flow in the channel
is assumed to be steady-state, two-dimensional, laminar, incompressible, with negligible viscous dissipation. All thermophysical properties of the fluid are assumed to be constant, except
for the dependence of density on the temperature (Boussinesq
approximation), which gives rise to the buoyancy forces. The
thermophysical properties of the fluid are evaluated at the ambient temperature, To , which is assumed to be 300 K in all
cases.
The ambient is assumed to be a black body at a temperature
of 300 K.
With the already mentioned assumptions, the governing
equations in the conservative form and primitive variables
vol. 32 no. 6 2011


442

L. LANGELLOTTO AND O. MANCA
Table 2

are:
∂u ∂v
+
=0
∂x ∂y

(1)

∂u
∂u
1 ∂p
∂ u ∂ u
+v
=−
+ν
+ 2
∂x
∂y
ρf ∂x
∂x2
∂y
2

u

2

− gβ (T − To )
(2)

Wall∗

T = To

HG

∂u
=0
∂x

∂v
=0
∂x

T = To

AB, EF, IL, OP

u=0

v=0

BC, DE, LM, NO

u=0

v=0

DN

u=0

v=0

u=0

v=0

∂u
=0
∂x
∂u
=0
∂y

∂v
=0
∂x
∂v
=0
∂y

(4)

CM

where the pressure is referred to the ambient pressure, po .
A two-dimensional conduction model is employed. The heat
conduction equations in the steady-state regime with constant
thermophysical properties is:

RQ

∂ T ∂ T
+ 2
∂x2
∂y
2

∂ Ts
∂ Ts
+
=0
∂x2
∂y2
2

(5)

(Tmax − To )k ∗
(T − To )k
; θmax = max
qc b
(qc + qr )b

Ra b = Gr Pr =
Re =

IR, QP
∗ The

∂T
=0
∂x
∂T
∂T
= ks
kf
∂x
∂x
∂T
∂T
= ks
+ q + qr
kf
∂n
∂n
∂T
∂T
kf
= ks
− q + qr
∂n
∂n
∂T
=0
∂x
∂T
=0
∂y

letters in the column are in reference to Figure 1b.

2

The characteristic variables, for the investigated configuration in this paper, are the dimensionless maximum wall temperature, the channel Rayleigh number, the Reynolds number, and
the channel Nusselt number, defined as follows:
θmax =

T

∂v
=0
∂y

∂T
∂T
+v
= af
∂x
∂y

u

v

∂u
=0
∂y

(3)

2

u

AH and FG

∂v
∂ 2v ∂ 2v
∂v
1 ∂p
+v
=−
+ν
+ 2
∂x
∂y
ρf ∂y
∂x2
∂y

u

Boundary conditions for the fluid domain

(6)

gβqc b5
gβ(qc + qr )b5

Pr;
Ra
∗
=
Pr (7)
b
ν2 kLw
ν2 kLw

uav,bmin bmin
ν

Numerical Procedure
Since the two plates are placed in an infinite medium, from a
numerical point of view the problem has been solved with reference to a computational domain of finite extension, as depicted
in Figure 1b, by following the approach given in [11–13]. This
computational domain allows taking into account the diffusive
effects peculiar to the elliptic model. The imposed boundary
conditions are reported in Table 2 for the fluid domain and in
Table 3 for the solid domain. The pressure defect is equal to
zero at the inlet and outlet boundaries. The net radiative heat
flux from the surface is computed as a sum of the reflected
fraction of the incident and emitted radiative heat fluxes:

(8)

qr (xw ) = (1 − εw ) qin (xw ) + εw σT4w (xw )

(13)

and
Nub =

qc b
Tw,av − To k

Nu∗ b =

(qc + qr )b
Tw,av − To k

qin (xw ) =

(9)

Iin s · nd
s·n>0

where b is bmin or bav or bmax , and
1
qc =
Lw
1
qr =
Lw
1
Tav =
Lw

Table 3 Boundary conditions for the solid domain

Lw

qc,x (xw )dxw

0

(10)

Wall∗
DN

Lw

qr,x (xw )dxw

kf

(11)

Tw = Tf

0

CM

Lw

T(xw )dxw

kf

(12)
BL, OE

heat transfer engineering

∂T
∂T
= ks
− q + qr
∂n
∂n

Tw = Tf

0

It is worth noticing that an evaluation of qc separate from qr
is very difficult in practice. The value of (qc + qr ) is not equal
to the dissipated heat flux q due to the conductive heat losses
toward the ambient through the lower and upper edges of the
walls.

∂T
∂T
= ks
+ q + qr
∂n
∂n

BC, DE, LM, NO

∂T
=0
∂n
∂T
∂T
kf
= ks
+ qr
∂x
∂x
Tw = Tf

∗ The

letters in the column are in reference to Figure 1b.

vol. 32 no. 6 2011

(14)


L. LANGELLOTTO AND O. MANCA

SCALE ANALYSIS
For the convergent channel, the total volume (channel total
volume) is:
Vtot = Wbmax Lw cos δ

(15)

and it is greater than the channel volume as shown in Figure
1a. The geometrical optimization of the convergent channel, in
terms of maximum or average wall temperature, should take
into account the channel total volume. The heat transfer rate in
the channel is:
Q = 2hWLw Tw

(16)

In the optimization procedure, the channel total volume is
considered:
Qbmax = 2hWLw Tw bmax

(17)

60

Lw/bmin = 58.0

Tw - T o [K]

40 ε = 0.90
30
20
10
0

Experimental
Numerical

0

100

(18)

heat transfer engineering

200
xw [mm]

300

400

100

10

θ = 0°
θ = 2°
θ = 5°
θ = 10°

(b)

ε = 0.90

1
Present numerical data
Experimental data [14,31]
0.1
100

101

102

103

104

105

106

107

108

Ra'*bav
Figure 2 Comparison between numerical and experimental data: (a) wall temperature profiles, (b) average Nusselt number.

Combining Eq. (18) with the average channel Nusselt number, Eq. (9), we get:
Vtot Tw ∼

Qbmax b

2kNub

(19)

for an assigned heat transfer rate. The optimal channel configuration that minimizes the product Vtot Tw is the configuration
that maximizes the heat transfer as a function of the channel
total volume in terms of bmax .
In laminar, fully developed and two-dimensional natural convection between parallel plates, heated at uniform
heat flux, the maximum wall temperature is obtained at the
channel outlet section and the minimum Nusselt number is
[20]:
Nux=L =

For small convergent angles Eq. (17) becomes:
Qbmax ∼ 2hVtot Tw

(a)

50 δ = 10°
qΩ = 220 W/m2

Nu*bav

The computational fluid dynamics code FLUENT [30] was
employed to solve the present problem. The segregated method
was chosen to solve the governing equations, which were linearized implicitly with respect to the equation’s dependent variable. The second-order upwind scheme was chosen for the
energy and momentum equations. The Semi Implicit Method
for Pressure-Linked Equations (SIMPLE) scheme was chosen to
couple pressure and velocity. Similar considerations were made
for choice of the discrete transfer radiation model (DTRM),

which assumes all surfaces to be diffuse and grey. The convergence criteria of 10−6 for the residuals of the velocity components and of 10−8 for the residuals of the energy were assumed.
A grid dependence test is accomplished to realize the more
convenient grid size and radiative subdivisions by monitoring
the induced dimensionless mass flow rate and the average Nusselt number, referred to the minimum channel spacing for a
convergent channel system with Lw /bmin = 40.6, δ = 10◦ at
∗
Ra bmin = 30 and 220 and with Lw /bmin = 10.2, δ = 10◦ , and
∗
Ra bmin = 3.1 × 104 and 2.25 × 105 as reported in [13]. A more
detailed description on the numerical model is reported in [13].
A comparison between numerical and experimental [31] results is reported in Figure 2. In Figure 2a wall temperature
∗
profiles, obtained for Lw /bmin = 58.0, δ = 10◦ , and Ra bmin =
37, are shown. The comparison between the numerical and experimental data showed a good agreement with a maximum
percentage discrepancy of about 8%. In Figure 2b the comparison, in terms of average Nusselt number, is accomplished. A
very good accord between the numerical and experimental data
is observed.
Since the numerical results and experimental data are in good
agreement, the assumptions of steady-state, two-dimensional,
laminar, incompressible, with negligible viscous dissipation are
confirmed, as well as the Boussinesq approximation.

443

k
+
hb

48
Rab

−1

(20)

The average wall temperature is approximately equal to the
wall temperature at middle channel length and the average
vol. 32 no. 6 2011


444

L. LANGELLOTTO AND O. MANCA

Nusselt number is estimated by [20]:
Nux=L/2 =

k
+
hb

12
Rab

−1

(21)

The first term on the right-hand side of the Eqs. (20) and
(21) is negligible with respect to the square root as given in

[20]:
Nux=L

Rab
and Nux=L/2 ∼
48

Rab
12

Nub min xw =Lw

bav
bmin

Rab min
12

Nub min xw =Lw /2 ∼

3
2

3
2

(23)

For fully developed flow, the comparison between the
parallel-plate channel, Eqs. (20) and (21), and the convergent

channel, Eqs. (23), shows that, for the same bmin , the convergent
channel has a higher Nusselt number value; i.e., the convergent channel, with the minimum channel spacing, equal to the
parallel-plate channel spacing, presents lower maximum and
average wall temperature values.
For the Nusselt number referred to the average channel spacing it is:
Rabav bmin
and
48 bav

Nubav xw =Lw ∼

Nubav xw =Lw /2 ∼

Rabav bmin
12 bav

(24)

As shown in Eqs. (24), the Nusselt number for the convergent channel, referred to bav , is lower than the one for the
parallel-plate channel, i.e., the wall temperature in the convergent channel is greater than the one in the parallel-plate channel.
Further, the convergence angle limit is:
bav − Lw sin (δ) = 0 ⇒ δ = arcsin

bav
Lw

(25)

NuLw = 0.56 RaLw cos δ

Nub max xw =Lw ∼

Nub max xw =Lw /2 ∼

Rabav b2min b3av
12
b5max

(27)

0.2

(28)

For developing flow in convergent channels as limit condition, Eq. (28) is employed in terms of b/Lw with b equal to bmin
or bav or bmax :
b
Lw

(29)

As suggested in [20], a composite relation is obtained by
summing the two expressions, the equation for fully developed
limit, indicated with Nu0 [Eqs. (23)–(25)], and single-plate limit,
indicated with Nu∞ [Eq. (29)]. The binomial correlation is:
−p

Nu−p = Nu0 + Nu−p
∞

(30)

as a first approximation, the correlation exponent, p, is set equal
to 2.
The term Vtot Tw is evaluated by means of Eq. (30) using
bmin , bav , and bmax , respectively:
⎧
−2
Rab min bav 3/2
Qbmax bmin ⎨
(Vtot Tw )b min ∼
⎩
2k
12
bmin

+ 0.56 Rab min cos δ

(Vtot Tw )bav

⎫0.5
⎬
0.2 −2

+ 0.56 Rabav cos δ

(Vtot Tw )b max

(31)

⎭

⎧
Qbmax bav ⎨
∼
⎩
2k

For the Nusselt number referred to the maximum channel
spacing it is:
Rabav b2min b3av
and
48
b5max

bmax
2Lw

In laminar, developing, and two-dimensional natural convection along an inclined single plate, heated at uniform heat flux,
the average Nusselt number is [32]:

Nub ∼ 0.56 (RaLw cos δ)0.2

and

bav
bmin

bmax − 2Lw sin (δ) = 0 ⇒ δ = arcsin

(22)

In vertical channels, at uniform heat flux, with small convergence angle, the minimum and average Nusselt numbers,
referred to the minimum channel spacing, can be evaluated as
in [20]. It is obtained as:
Rab min
48

Also in this case, the Nusselt number for the convergent
channel, referred to bmax , is lower than the one for the parallelplate channel. The convergence angle limit is equal to:

Rabav bmin
12 bav

−2

⎫0.5
⎬
−2
0.2

(32)

⎭

⎧
Qbmax bmax ⎨
∼
⎩
2k

Rabav b2min b3av
12 b5max

−2

⎫0.5

(26)

heat transfer engineering

+ 0.56 Rab max cos δ

vol. 32 no. 6 2011

⎬
0.2 −2
⎭

(33)


L. LANGELLOTTO AND O. MANCA

445

In Figure 3c, the optimal configuration, in terms of channel
spacing, is given as a function of the channel convergence angle, for qc equal to 30 W m−2. The figure shows that the curves
tend to an asymptotic value equal to 9.9 × 10−3 m, which

represents the optimal configuration for the parallel-plate channel. Figure 3c shows that for (Vtot Tw )b min , increasing the
convergence angle, the minimum channel spacing decreases.
For fixed convergence angle, decreasing bmin , the total volume
decreases and the Nusselt number increases, Eq. (31), and then
the wall temperature decreases. In Vtot Tw referred to bav and
bmax , for fixed convergence angle, the Nusselt number and the
total volume increase as the reference channel spacing increases.

ANALYSIS AND PROCEDURES FOR CORRELATIONS
The results are obtained by the numerical procedure reported
in [13]. In this work, the analysis is focused on the radiative
effects on natural convection in air, in a convergent channel,
uniformly heated at the two principal walls. The wall thickness,
t, is 3.2 mm, with the ratio t/bmin varying in the range 0.080–0.64.
Its thermal conductivity is 0.198 W/m-K, with a solid-to-fluid
conductivity ratio ks /kf = 8.18. The input data are ranging from
∗
10 to 80 for aspect ratios, Lw /bmin ; Rayleigh numbers, Ra bmin ,
−2
5
ranging from 2.5 × 10 to 2.3 × 10 ; convergent angles, δ,
ranging from 0◦ to 10◦ ; and wall emissivities, ε, ranging from
0.1 to 0.9.
The percentage value of the conductive heat flux, qk , referred
∗
to the dissipated heat flux, q , for different Ra bmin values and
for the geometry here considered, are given in Table 4.

Figure 3 Wall temperature for channel total volume as a function of channel
spacing and convergence angle with reference channel spacing equal to: (a)

bmin ; (b) bav . (c) Optimal geometrical configurations, in terms of bmin , bav and
bmax values, as a function of convergence angle.

The values of Vtot Tw , for qc equal to 30 W m−2, as a
function of the convergence angle and the minimum and average channel spacing, are reported in Figures 3a and 3b. The
contours of Vtot Tw value in the (b,δ) plane are also given. It
is noted, in Figure 3a, that (Vtot Tw )b min , Eq. (31), is always
defined except for bmin equal to zero. The function shows that
the absolute minimum value is obtained for δ = 0◦ . For the
considered convective heat flux the optimal channel spacing is
bmin = 9.9 × 10−3 m. This value corresponds to the minimum
value of Vtot Tw , i.e., the minimum Tw with the minimum
compatible total volume Vtot . For (Vtot Tw )bav , Eq. (32), and
Figure 3b, for assigned bav value, the convergence angle limit,
δlimit , exists and a vertical asymptotic plane is detected for δ →
δlimit , according to Eq. (25). The optimal configuration in terms
of (Vtot Tw )bav , obtained by Eq. (32), is realized for δ = 0◦
and for the considered convective heat flux bav = 9.9 × 10−3 m.
The same results are obtained for (Vtot Tw )b max , Eq. (33), but
the results are not reported here.
heat transfer engineering

Mass Flow Rate
Mass flow rate, involved in the heat transfer, is an important parameter in design and control of electronic equipment
and solar energy in building. The following correlations for
mass flow rate, in a convergent uniformly heated vertical channel, as a function of thermal and geometrical parameters are
proposed. The mass flow rate for unit of width is defined as
follows:
˙ = ρuav,bmin bmin
m

(34)

where uav,bmin is the mean velocity at the minimum channel
section. From Eqs. (8) and (34):
˙ = Re µ
m

(35)

The Reynolds number, as a function of Ra bmax , is reported
in Figure 4. The figure shows that, when the Rayleigh number increases, for fixed aspect ratio and convergence angle, the
Reynolds number also increases. Decreasing the aspect ratio
(increasing the spacing), the Reynolds number increases significantly, whereas there is slight change in the mass flow rate in
the emissivity range 0.10–0.90. The maximum percent variation
vol. 32 no. 6 2011


446

L. LANGELLOTTO AND O. MANCA
Table 4 Conductive heat losses
∗

Ra bmax ≥ 30
∗
1.0 ≤ Ra bmax ≤ 30
Ra

∗

bmax

Lw /bmin ≤ 58
58 ≤ Lw /bmin ≤ 80
Lw /bmin = 80
Lw /bmin ≥ 80

≤ 1.0

0.048 ≤ bmin /bmax
0.048 ≤ bmin /bmax
0.048 ≤ bmin /bmax
0.048 ≤ bmin /bmax

between emissivity value equal to 0.90 and 0.10 is about 10%.
Figure 4 shows that the Reynolds number is highly dependent
on channel aspect ratio and convergence angle.
In order to reduce the Reynolds number scattering, the varimin
as a function of Ra bmax
able Re bbmax

Re

bmin
bmax

χ

bav

Lw

= f Ra bmax

η

is considered:

bav
Lw

≤ 1.0
≤ 1.0
≤ 0.7
≤ 1.0

qk /q ≈ 3%; qk /q ≤ 5%
qk /q ≈ 10%; qk /q ≤ 15%
qk /q ≥ 15%

A new correlation, in terms of Ra bmax , is evaluated by regression analysis:

Re = −4.62 · 103

bmin
Lw

3

+ 752

bmin
Lw

2

+ 18.6

bmin
Lw

η

(36)
104
(a)

A dependence on
is observed and the best correlations
Lw
for assigned bmin are carried out employing χ = −1 and η =
−5/2. The plot of Eq. (36) is reported in Figure 5a. In this
figure, a greater dispersion is observed for high values of
Ra bmax ( bLavw )η . The dispersion is due to the increase in relevance
of the aspect ratio in this zone. To obtain a monomial correlation
for the mass flow rate, in terms of geometrical and thermal
variables, the following relation is proposed in the form [15]:

bmax
Re

bmin

Lw
=α
bmin

Ra bmax

bav
Lw

−2.5 β

Re(bmax/bmin)

Lw
bmin

Lw
bmin

103

102

101 4
10

(37)

105

106

107
108
109
-2.5
Ra'bmax(bav/Lw)

1010

1011

100

200

300
400
500
Re (numerical)

600

700

700
(b)

600

Re

500

ε = 0.10
ε = 0.50
ε = 0.90

Re (Eq. 38)

1000

400
300

100
200

Lw/bmin = 58.0
Lw/bmin = 40.6

δ = 0°
δ = 2°
δ = 5°
δ = 10°

Lw/bmin = 20.3
Lw/bmin = 12.6

Lw/bmin = 10.2

10
100

101

102

103

104

105

106

107

108

109

Ra'bmax
Figure 4 Reynolds number versus Rayleigh number for various convergence
angles and wall emissivity values.

heat transfer engineering

100

0

0

min
Figure 5 (a) Re bbmax
as a function of Ra bmax ( bLavw )−2.5 . (b) Comparison
between numerical Reynolds numbers and Reynolds numbers by correlation
given by Eq. (38) with a percentage difference in a ±5% range.

vol. 32 no. 6 2011


L. LANGELLOTTO AND O. MANCA
−24.6 bLmin
w

bmin 2
×
; r = 0.996
bmax

Lw/bmin = 12.6
Lw/bmin = 10.2

c

(38)

and a simpler expression is also proposed:

Re = 52.9

−24.6
−2.50 0.249+0.233e

bav
Lw

bmin
Lw

0.01
100

3

(39)

bmin
+ 72.9
Lw

2

bmin
+ 54.6
Lw
−24.5 bLmin
w

Rab∗max

−0.429

Ra

bav
Lw

−2.50 0.242+0.230e

bmin 2
; r = 0.989
bmax

(40)

Moreover, a new correlation in simplified form, in terms of
∗
bmax , is proposed:
Re = 57.4

bmin
− 0.454
Lw
−24.5 bLmin
w

× Rab∗max

×

bav
Lw

bmin 2
r = 0.988
bmax

101

102

103

104 105
Ra'*bav

106

107

108

∗

bmin 2
r = 0.990
bmax

bmin
Re = −499
Lw

δ = 0°
δ = 2°
δ = 5°
δ = 10°

ε = 0.10
ε = 0.50
ε = 0.90

They present a better regression coefficients than the corre∗
lation presented in [15] in terms of Ra bmax :

×

0.1

bmin
− 0.436
Lw

× Rab max

×

Lw/bmin = 58.0
Lw/bmin = 40.6

Lw/bmin = 20.3

qr/(q +qr)

−0.0248] Rab max

1

−2.50 0.249+0.233e

bav
Lw

447

−2.50 0.242+0.230e

(41)

In Figure 5b, a comparison between the Reynolds number
value from numerical data and the Reynolds number values
calculated by the correlation in Eq. (38) are reported together
with an error level of ±5%.

r
Figure 6 Radiative heat flux ratio ( qcq+q
), as a function of Ra bav , for three
r
different wall emissivity values and different convergence angles.

is very useful when qc + qr is known, thereby allowing for
estimation of radiative heat losses.
The ratios between radiative and total heat fluxes, as a func∗
tion of Ra bav , are reported, for different angles and for three ε
values, in Figure 6. It is observed that for fixed Lw /bmin values,
∗
r
decreases when Ra bav increases, and the ratio
the ratio qcq+q
r
∗
variation decreases with increasing Ra bav and decreasing aspect
ratio. The percentage reduction of heat flux ratio decreases with
increasing convergence angle. Furthermore, the ratio values increase with decreasing aspect ratio value. Increase in the convergence angle produces a significant increase in heat flux ratio. For
δ = 10◦ , the radiative heat flux ranges between 20% and 40%
of the total heat flux. Figure 6 shows that, for fixed Rayleigh
number, the heat flux ratio decreases with decrease in the wall
∗
emissivity. The higher the Ra bav values, the higher is the vari∗
ation of the heat flux ratio. In fact, at Ra bav = 20, for ε = 0.10,
the heat flux ratio is 0.020, whereas for ε = 0.90, the ratio is
∗
0.032; at Ra bav = 2.4 × 107, the heat flux ratio is 0.11 and 0.27
for ε = 0.10 and ε = 0.90, respectively. The percentage varia∗
tions referred to ε = 0.90 are about 38% and 59% for Ra bav
7
equal to 20 and 2.4 × 10 , respectively.
For assigned wall length, Figure 6 allows to observe the dependence on the convergence angle and channel spacing when
the wall heat flux is fixed. Furthermore, the figure shows a data
scattering. In order to reduce the heat flux ratio scattering, the

c
is employed as suggested in [15]. A new comvariable qrq+q
ref
posite correlation for ε = 0.90, obtained by means of regression
analysis, is proposed:

Radiative Heat Flux

qr
=
qc + qr

Correlations to evaluate the ratio between radiative heat flux,
qr , and total heat flux, qc + qr , for a convergent channel and
r
,
surface emissivity are proposed. The heat flux ratio, qcq+q
r

×

heat transfer engineering

qr + qc
qref

1.12x10−3

0.153Ra ∗ bav
−0.26

vol. 32 no. 6 2011

4

+ 0.0481Ra ∗ bav

qref = 1 W/m2 ;

0.186

r2 = 0.991

4 1/4

(42)


448

L. LANGELLOTTO AND O. MANCA
Table 5 Coefficients and exponent of the Eq. (43)
ε
0.10
0.50

Table 6 Coefficients and exponent of the Eq. (44) with
Eqs. (45) and (46)

m

n

p

r2

0.0478
0.0525

0.135
0.173

−0.26
−0.26

0.981
0.990

Eq. (44)

αo
βo
α∞
β∞
α1
β1
n
m
γ

p
qref
a1
a2
a3
a4
r2

In Eq. (42), the value of reference heat flux is used to obtain
a dimensionless equation. A good accord is observed between
Eq. (42) and the numerical data set.
For ε = 0.10 and 0.50, at high Rayleigh numbers, a monomial
correlation is proposed:
qr
n
= m Ra ∗ bav
qc + qr

qr + qc
qref

p

(43)

where the reference heat flux is equal to the values given in
Eq. (42). The coefficient m and exponents n and p, as well as
r2 values, are reported in Table 5 for the different ε values. In
this case, simpler equations are proposed with respect to the one
given in [15]. Moreover, a new global correlation is evaluated

using all available data:
β

o
αo Ra ∗bav

qr
=
qc + qr

n

β

∞
+ α∞ Ra ∗bav

β

1
α1 Ra ∗bav
+ f

×

qr + qc
qref

n 1/n

bmin
bmax

εm
1 + γεm

p

(44)

f

bmin
bmax

= a1

bmin
bmax
bmin
bmax

+ a2

0.990

1.00

+ a2

3

0.1721
0.0237
0.0474
0.2968
0.0346
0.1743
4
1.075
7.916
−0.260
1.00
0.5062
−0.7256
0.2325
0.1250
0.991

Composite correlations between the dimensionless maximum wall temperatures and Rayleigh numbers, referred to the
maximum channel spacing bmax , are evaluated for ε = 0.1, 0.5,
and 0.9. The equations are obtained by means of the asymptotic
relations for the single tilted plates (large Rayleigh number,
Ra > 104) and for the fully developed limit (small Rayleigh
number, Ra < 102), following the procedure suggested in [20].

(45)
bmin
bmax

2

+ a3

bmin
bmax

+ a4
(46)

The coefficients and exponents of Eq. (44) are reported in
Table 6.
r
The comparison between the qcq+q
ratios obtained numerr
ically and from Eq. (44), taking into account Eq. (46), are
reported in Figure 7 together with an error level of ±5%. It
is observed that the best agreement among the data and pro∗
∗
posed correlation is obtained for high Ra bav value (Ra bav >
4
10 ).
The main advantage of Eq. (44), with respect to the correlations in Eqs. (42) and (43) and ones in [15], is that it is a
single equation for all emissivity values. Furthermore, Eq. (44)
r
ratio with respect to the
provides a better estimation of qcq+q
r
previous correlations.
heat transfer engineering

/(qr+qc) Eq. (44) with Eq. (46)

= a1ln

0.1840
0.0746
0.0408
0.03569
0.0357
0.2257
4
1.075
7.916
−0.260
1.00
0.0027
0.1501

0.10

qr

bmin
bmax

Eq. (46)

Dimensionless Maximum Wall Temperature

min
With two different equations for f ( bbmax
):

f

Eq. (45)

ε = 0.90
ε = 0.50
ε = 0.10
0.01
0.01

0.10
qr

1.00

/(qr+qc) (numerical)

Figure 7 Comparison between the qr (qr + qc ) ratio obtained numerically
and that from the correlation Eq. (44) and Eq. (46), with a percentage difference
in a ±5% range.

vol. 32 no. 6 2011


L. LANGELLOTTO AND O. MANCA

The composite correlation between θbmax and Ra bmax , for
ε = 0.9, is:
θb max =

−0.414

4

6.13Ra bmax

4 1/4

−0.197

+ 1.88Ra bmax

(47)

with r2 = 0.997, in the ranges: 4.4 ≤ Ra bmax ≤ 2.9 × 103,
10 ≤ Lw /bmin ≤ 80, 0◦ ≤ δ ≤10◦ (0.048 ≤ bmin /bmax ≤ 1.0).
Considering the variable
Ra ∗mod = Ra ∗bmax

bmin
bmax

ε0.10

(48)

a new correlation is proposed for ε = 0.1, 0.5, and 0.9:
θ∗b max =

−0.628

9.39Ra ∗ mod

2

−0.242

+ 2.48Ra ∗ mod

2 1/2

(49)

Nubmin =

449

0.408

0.475Ra bmin

−4

0.200

+ 0.660Ra bmin

−4 −0.25

(54)

for 1◦ < δ < 10◦ and 0.02 < Ra bmin < 1.9 × 105 with r2 = .999.
Numerical data and the two composite correlations are reported in Figure 8a. A good accord between the numerical data
and the correlations is observed. A slightly better agreement
is observed for higher Nusselt numbers corresponding to the
higher Rayleigh numbers.
In order to obtain a single composite correlation that takes
into account all convergence angles, a different monomial correlation for fully developed flow, Ra bmin < 50, is evaluated and
employed. The following monomial correlation is proposed in
terms of bav /bmin to take into account the convergence angles:
n

∗

−3

with r = 0.978, in the ranges 1.4 × 10 ≤ Ra ≤ 4.2 × 10 ,
10 ≤ Lw /bmin ≤ 80, 0◦ ≤ δ ≤ 10◦ (0.048 ≤ bmin /bmax ≤ 1.0).
A good agreement is observed in the comparison between the
numerical data and Eq. (49). The comparison shows that greater
∗
differences are found for the highest Ra values. Equation (49)
∗
is defined in a larger Ra bmax and aspect ratio range, extending
its validity in the fully developed region with respect to the
equation given in [15].

2

8

Nusselt Number Correlation

0
Nu0,bmin = m0 Ra bmin

m0 = 0.468 − 0.29

bav
bmin

n0 = 0.40 + 0.142

bav
bmin

The average convective Nusselt numbers, defined in the Eq.
(9), as a function of channel Rayleigh number referred to bmin ,
is given in the following for two asymptotic conditions: fully
developed flow Ra bmin < 50 and single plate limit Ra bmin >
800. The correlation for fully developed flow depends on the
convergence angle. The following correlations are obtained by
means of regression analysis:
0.589

0.408

Nu0,bmin = 0.475Ra bmin

(51)

for 1◦ < δ < 10◦ and Ra bmin < 50 with r2 = .991.
For the single plate limit, the monomial correlation is:
0.200

Nu∞,bmin = 0.660Ra bmin

(52)

for 0◦ < δ < 10◦ and Ra bmin > 800 with r2 = .997.
The composite correlations are obtained following the procedure suggested in [20]:
−p
−p
N u− p = N u0 + N u∞
For p = 4, the two composite correlations are evaluated:
0.589

0.182Ra bmin

−4

−3

bav
bmin

0.468 − 0.29

0.40−0.142

× Ra bmin

0.200

+ 0.660Ra bmin

bav
bmin

−3

−3

⎤−4
⎦

−4 −0.25

(56)

(50)

for δ = 0◦ and Ra bmin < 50 with r2 = .992; and

Nubmin =

(55)

A composite correlation is obtained from the Eqs. (52) and
(55), with p = 4:
Nubmin =

Nu0,bmin = 0.182Ra bmin

−3

0.200

+ 0.660Ra bmin

−4 −0.25

(53)

for δ = 0◦ and 0.2 < Ra bmin < 1.9 × 105 with r2 = .995; and
heat transfer engineering

for 0◦ < δ < 10◦ and 0.02 < Ra bmin < 1.9 × 105 with r2 = .998.
In this case, a more noticeable difference is observed for
Nubmin < 1.0, 0.0◦ < δ < 1.0◦ , and 0.02 < Ra bmin < 50. It is
interesting to observe that in Eq. (55), the coefficient m0 and
the exponent n0 are functions of bav /bmin , which confirm the
scale analysis results. These functions are reported in Figure 8b
and a horizontal asymptotic value is observed in both functions.
Moreover, the critical values correspond to the zone where the
functions change from vertical to horizontal asymptote.
A similar analysis is given for average total Nusselt number,

defined in Eq. (9), which takes into account both radiation and
convective heat fluxes. Average total Nusselt number, as function of total channel Rayleigh number, referred to bmin , is given
in the following for the two asymptotic conditions: fully de∗
∗
veloped flow, Ra bmin < 100, and single plate limit, Ra bmin >
1.0 × 103. The correlation for fully developed flow depends on
the convergence angle. In fact, by means of numerical data, the
following correlations are obtained employing the regression
vol. 32 no. 6 2011


450

L. LANGELLOTTO AND O. MANCA

for δ = 0◦ and Ra

10
(a)

∗

bmin

< 100 with r2 = .997; and

Nu∗0,bmin = 0.492Ra ∗bmin

0.392

(58)

∗

1
Nubmin

δ = 0.0°
δ = 1.0°
δ = 1.5°
δ = 2.0°
δ = 5.0°
δ = 10°

0.1

for 1◦ < δ < 10◦ and Ra bmin < 100 with r2 = .992.
For the single plate limit, the monomial correlation is:
Nu∗∞,bmin = 0.725Ra ∗bmin

100

101

102

103

104

105

106

Ra'bmin

∗

0.5

Nu∗bmin =

0.4
0.3

n0 Eq. (55)
m0 Eq. (55)

0.2

n0
m0
0

2

4

0.201Ra ∗bmin

0.545

−4

+ 0.725Ra ∗bmin

0.210

−4 −0.25

∗

(b)
m0 and n0 of Eq. (55)

Nu∗bmin =

for δ = 0◦ and 0.08 < Ra bmin < 2.2×105 with r2 = .988.
From Eqs. (58) and (63):

0.6

0.1

(59)

for 0◦ < δ < 10◦ and Ra bmin > 1.0×103 with r2 = .975.
Two composite correlations are obtained, with p = 4, from
Eqs. (57) and (59) resulting in:

Eq. (53)
Eq. (54)

0.01 -3
10 10-2 10-1

0.210

6
bav/bmin

8

10

12

0.492Ra ∗bmin

0.392

−4

+ 0.725Ra ∗bmin

0.210

(60)

−4 −0.25

(61)
is carried out for 1◦ < δ < 10◦ and 0.02 < Ra bmin < 2.2 × 105
with r2 = .992.
Numerical data and the two composite correlations are reported in Figure 8c. A good agreement between numerical data
and correlations is observed.
In the same way employed to obtain Eqs. (55), the following
monomial correlation is carried out, in terms of bav /bmin , to take
into account all convergence angles:
n∗

Nu∗0,bmin = m∗0 Ra ∗bmin 0

100
(c)

m∗0 = 0.520 − 0.322

bav
bmin

n∗0 = 0.288 + 0.340

bav
bmin

Nu*bmin

10
δ = 0.0°

δ = 1.0°
δ = 1.5°
δ = 2.0°
δ = 5.0°
δ = 10°

1

0.1

Eq. (60)
Eq. (61)

0.01
10-2

10-1

100

101

102 103
Ra'*bmin

104

105

106

Figure 8 (a) Nusselt number versus Rayleigh numbers and correlations given
by Eqs. (53) and (54). (b) Coefficient m0 and exponent n0 in the Eq. (55). (c)
Total Nusselt number vs total Rayleigh numbers and correlations given by Eqs.
(60) and (61).

Nu∗0,bmin = 0.201Ra ∗bmin

0.545

(57)

heat transfer engineering

(62)
−2

For p = 4, a composite correlation is obtained from Eqs. (59)
and (64):
Nu∗bmin =

0.520 − 0.322

∗
× Rabmin

bav
bmin

−2

bav
0.288 − 0.340
bmin

+ 0.725Ra bmin 0.210
∗

analysis:

−2

−4 −0.25

−2

−4

(63)

for 0◦ < δ < 10◦ and 0.02 < Ra bmin < 2.2 × 105 with r2 =
.991.
In Figure 9a, all proposed correlations are reported and they
are compared for ε = 0.10, with the correlation given in [12],
vol. 32 no. 6 2011


L. LANGELLOTTO AND O. MANCA

451

10

Nubmin

(a)

1

0.1
10-1

ε = 0.10 [12]
ε = 0.50
ε = 0.90
ε = 0.90 [13]
ε = 0.90 δ = 0°
ε = 0.90 1° < δ < 10°

100

101

102
103
Ra'bmin

104

105

106

10

Nu*bmin

(b)

1

0.1
10-1

ε = 0.10 [12]
ε = 0.50
ε = 0.90
ε = 0.90 [13]
ε = 0.90 δ = 0°
ε = 0.90 1° < δ < 10°

100

101

102
103
Ra'*bmin

104

105

106

Figure 9 Nusselt number versus Rayleigh numbers for several wall emissivity
values.

and for ε = 0.90, with the correlation proposed in [13]. In Figure
9a, the correlations for convective average Nusselt number, as
a function of convective channel Rayleigh number, referred to
the minimum channel spacing, are compared. The figure shows
that the greatest differences among the correlations are detected
for ε = 0.1.
In Figure 9b, the correlation for total average Nusselt number,
as a function of total channel Rayleigh number, referred to the
minimum channel spacing, for several wall emissivities, are
reported. In this case as well, the greatest differences among the
correlations are detected for ε = 0.1.
Channel Optimization
In the same manner of scale analysis, Vtot Tw is obtained
by means of Eq. (56):
(Vtot Tw )b min =

Qbmax bmin
2k

0.468 − 0.29

bav

bmin

−3

heat transfer engineering

Figure 10 (a) Wall temperature for channel total volume as function of minimum wall spacing, bmin , and channel convergence angle, δ. (b) Optimal geometrical configurations in terms of channel spacing as a function of convergence
angle. (c) Optimal channel spacing as function of convective wall heat flux.

0.40−0.142

×Ra bmin

bav
bmin

−3

⎤−4
⎦

0.200

+ 0.660Ra bmin

⎫0.25
⎪
−4 ⎬
⎪
⎭

(64)

In Figure 10a, the term Vtot Tw , Eq. (64), evaluated
for qc equal to 30 W m−2, as a function of bmin and δ, is
reported. In this figure the contours in the (b,δ) plane are also
given. It is observed that the minimum value is obtained for
δ = 0◦ and bmin = 0.0104 m, which represents the optimal
configuration for convergent channel. These values are almost
equal to the ones estimated by the scale analysis. Moreover, the
two diagrams in Figures 10a and 3a seem similar. This confirms that the scale analysis provides a good estimation for this
configuration.
vol. 32 no. 6 2011


452

L. LANGELLOTTO AND O. MANCA

In Figure 10b, the optimal channel spacing, evaluated by
means of Eqs. (64), (31), (32), and (33), is given as a function of
the convergence angle. The figure shows that the curves present
asymptotic values, corresponding to the optimal configuration,
the parallel-plate channel. For all curves, the asymptotic values
are almost equal. For the dimensionless quantities referred to
bmin and for small convergence angle, the figure shows that the
optimal configurations obtained by the numerical solution, Eq.
(64), and scale analysis, Eq. (31), have a similar trend. In Figure
10b, it is observed, for Eq. (64), that the minimum value of the
optimal minimum channel spacing value is about 8.1 × 10−3 m

attained for δ = 0.44◦ .
By increasing the heat flux the optimal minimum channel
spacing decreases as shown in Figure 10c. This result is in
agreement with the results reported in [15] and [33].

CONCLUSIONS
Natural convection in air, in convergent channels, symmetrically heated at uniform heat flux, in a steady-state regime, was
studied. A scale analysis allowed estimation of optimal geometrical configurations from Nusselt number correlations, for
single plate and fully developed flow, in terms of the channel
Rayleigh numbers, the ratio bmin /bmax , and the convergence angle. A new optimization procedure was obtained in terms of the
minimum value of the product of average wall temperature and
total volume, Vtot Tw , as a function of convergence angle and
minimum, average, and maximum wall spacing. It was observed
that in all cases, the best configuration is δ = 0◦ . These results
are different from the ones given in previous papers [10, 12, 13,
15].
Monomial and composite correlations were estimated by the
numerical results obtained by the numerical model proposed in
[13]. The correlation equations were accomplished for mass flow
rate, radiative heat flux, and dimensionless maximum wall temperature in the emissivity range from 0.10 to 0.90, convergence
angle from 0◦ to 10◦ , ratio between minimum and maximum
channel spacing, bmin /bmax , in the range of 0.048 to 1.0, 10 ≤
∗
L/bmin ≤ 58, and 5.0 ≤ Ra bmin ≤ 2.3 × 105. Average Nusselt
number correlation was proposed for the emissivity value of
0.90, convergence angle ranging from 0◦ to 10◦ , bmin /bmax in
the range of 0.048 to 1.0, 10 ≤ L/bmin ≤ 80, and 2.5 × 10−2 ≤
∗
Ra bmin ≤ 2.3 × 105. It was observed that all correlations were
in very good accord with the numerical data.

The analysis of Nusselt number, in fully developed flow,
in the channel, shows that the asymptotic value for δ = 0◦ is
different from the ones with δ > 0◦ . Moreover, for δ ≥ 1◦ ,
all values were along a single asymptotic curve, Eqs. (56) and
(63) for Nu and Nu∗ , respectively. This asymptotic curve was
considered the border line between the fully developed flow and
the developing flow for low Ra values.
The new proposed optimization procedure was applied employing the evaluated Nusselt number correlation. The optimal
heat transfer engineering

configuration was obtained for δ = 0◦ . This confirms the result
obtained by means of the scale analysis. The optimal value of
minimum channel spacing decreases with increase in the wall
heat flux as shown in [15] and [33].

NOMENCLATURE
a
b
g
Gr
Gr’
h
I
k
L
Lx , Ly
˙
m
n
Nu

p
Pr
q
Q
r
Ra
Ra
Re
s
t
T
u
v
V
W
x, y

thermal diffusivity (m2/s)
channel spacing (m)
acceleration of gravity (m/s2)
Grashof number
Gr b / Lw , Eq. (7)
Convective heat transfer coefficient (W/m2-K)
radiation intensity (W/m2)
thermal conductivity (W/m-K)
channel length (m)
reservoir dimensions (m)
mass flow rate for width unit (kg/s-m)
normal to the wall
average Nusselt number, Eq. (9)

reduced pressure referred to ambient pressure (N/m2)
Prandtl number
heat flux (W/m2)
heat transfer rate (W)
regression coefficient
average Rayleigh number
Ra b/Lw , Eq. (7)
average Reynolds number, Eq. (8)
ray direction vector
wall thickness
temperature (K)
velocity component along x axis (m/s)
velocity component along y axis (m/s)
volume (m3)
plate weight (m)
coordinates (m)

Greek Symbols
β
δ
ε
µ
ν
θ
ρ
σ

volumetric coefficient of expansion (1/K)
half angle from the vertical (degrees)
emissivity

dynamic viscosity (kg/m-s)
kinematic viscosity (m2/s)
dimensionless temperature, Eq. (6)
density (kg/m3)
Stefan–Boltzmann constant (W/m2-K4)
hemispherical solid angle (sr)

Subscripts
av
b

average
referred to the channel spacing
vol. 32 no. 6 2011


L. LANGELLOTTO AND O. MANCA

c
ch
f
in
k
max
min
mod
o
r
ref
tot

s
w
0
∞

convective
channel
fluid
incident
conductive
refers to a maximum value
refers to a minimum value
modified, see Eq. (48)
ambient air
radiative
reference value
total
solid
wall
Ohmic dissipation
referred to the fully developed flow channel
referred to the single plate limit

Superscripts
∗

convective and radiative

REFERENCES
[1] Sathe, S., and Sammakia, B., A Review of Recent Developments in Some Practical Aspects of Air-Cooled Electronic

Packages, Journal of Heat Transfer, vol. 120, pp. 830–839,
1998.
[2] Bar-Cohen, A., and Kraus, A. D., Advances in Thermal
Modelling of Electronic Components and Systems, vol. 4,
ASME Press, New York, 1998.
[3] Yeh, L. T., and Chu, C., Thermal Management of Microelectronic Equipment, ASME Press, New York, 2002.
[4] Manca, O., Morrone, B., Nardini, S., and Naso, V., Natural
Convection in Open Channels, in Computational Analysis of Convection Heat Transfer, eds. B. Sund´en and G.
Comini, WIT Press, Southampton, UK, 2000.
[5] Sparrow, E. M., Ruiz, R., and Azevedo, L. F. A., Experiments and Numerical Investigation of Natural Convection
in Convergent Vertical Channels, International Journal of
Heat and Mass Transfer, vol. 31, pp. 907–915, 1988.
[6] Kihm, K. D., Kim, J. H., and Fletcher, L., Investigation of
Natural Convection Heat Transfer in Vertical Converging
Channel Flows Using a Specklegram Technique, Journal
of Heat Transfer, vol. 115, pp. 140–148, 1993.
[7] Said, S. A., Investigation of Natural Convection in Convergent Vertical Channels, International Journal of Energy
Research, vol. 20, pp. 559–567, 1996.
[8] Shalash, J. S., Tarasuk, J. D., and Naylor, D., Experimental
and Numerical Studies of Natural Convection Heat Transfer in Vertical Converging Channel Flows, Proc. 4th International Conf. on Experimental Heat Transfer, Fluid
Mechanics and Thermodynamics, vol. 4, pp. 2167–2174,
1997.
heat transfer engineering

453

[9] Kaiser, A. S., Zamora, B., and Viedma, A., Correlations
for Nusselt Number in Natural Convection in Vertical Converging Channels at Uniform Wall Temperature by Numerical Investigation, International Journal of Heat and Fluid
Flow, vol. 25, pp. 671–682, 2004.
[10] Bejan, A., da Silva, A. K., and Lorente, S., Maximal Heat

Transfer Density in Vertical Morphing Channels With Natural Convection, Numerical Heat Transfer Part A, vol. 45,
pp. 135–152, 2004.
[11] Bianco, N., and Nardini, S., Numerical Analysis of Natural Convection in Air in a Vertical Convergent Channel
With Uniformly Heated Conductive Walls, International
Communications in Heat and Mass Transfer, vol. 32, no.
6, pp. 758–769, 2005.
[12] Bianco, N., Manca, O., and Nardini, S., Experimental Investigation on Natural Convection in a Convergent Channel With Uniformly Heated Plates, International Journal
of Heat and Mass Transfer, vol. 50, pp. 2772–2786, 2007.
[13] Bianco, N., Langellotto, L., Manca, O., and Naso, V.,
Numerical Analysis of Radiative Effects on Natural Convection in Vertical Convergent and Symmetrically Heated
Channels, Numerical Heat Transfer Part A, vol. 49, pp.
369–391, 2006
[14] Bianco, N., Langellotto, L., Manca, O., and Nardini, S., An
Experimental Study of Radiative Effects On Natural Convection in Air in Convergent Channels, International Journal of Heat and Mass Transfer, vol. 53, pp. 3513–3524,
2010.
[15] Bianco, N., Langellotto, L., Manca, O., and Nardini, S.,
Thermal Design and Optimization of Vertical Convergent
Channels in Natural Convection, Applied Thermal Engineering, vol. 26, pp. 170–177, 2006.
[16] Bejan, A., Shape and Structure, From Engineering to Nature, Cambridge University Press, Cambrige, UK, 2000
[17] Bejan, A., Convection Heat Transfer, 3rd ed., Wiley, New
York, 2004.
[18] Bejan, A., Optimal Internal Structure of Volumes Cooled
by Single-Phase Force and Natural Convection, Journal of
Electronic Packaging, vol. 125, pp. 200–207, 2003.
[19] da Silva, A. K., and Gosselin, L., Optimal Geometry of L
and C-Shaped Channels for Maximum Heat Transfer Rate
in Natural Convection, International Journal of Heat and
Mass Transfer, vol. 48, pp. 609–620, 2005.
[20] Bar-Cohen, A., and Rohsenow, W. W., Thermally Optimum Spacing of Vertical, Natural Convection Cooled,
Parallel Plates, Journal of Heat Transfer, vol. 106, pp.

116–123, 1984.
[21] Kheireddine, A. S., Houla Sanda, M., Chaturvedi, S. K.,
and Mohieldin, T. O., Numerical Prediction of Pressure
Loss Coefficient and Induced Mass Flux for Laminar Natural Convective Flow in a Vertical Channel, Energy, vol.
22 no. 4, pp. 413–423, 1997.
[22] Manca, O., and Nardini, S., Composite Correlations for
Air Natural Convection in Tilted Channels, Heat Transfer
Engineering, vol. 20, no. 3, pp. 64–70, 1999.
vol. 32 no. 6 2011


454

L. LANGELLOTTO AND O. MANCA

[23] Olsson, C. O., Prediction of Nusselt Number and Flow
Rate of Buoyancy Driven Flow Between Vertical Parallel
Plates, Journal of Heat Transfer, vol. 126, pp. 97–104,
2004.
[24] da Silva, A. K., and Bejan, A., Constructal Multi-Scale
Structure for Maximal Heat Transfer Density in Natural
Convection, International Journal of Heat and Fluid Flow,
vol. 26, pp. 34–44, 2005.
[25] da Silva, A. K., Lorente, S., and Bejan, A., Constructal
Multi-Scale Structures for Maximal Heat Transfer Density,
Energy, vol. 31, pp. 620–635, 2006.
[26] Andreozzi, A., Campo, A., and Manca, O., Compounded
Natural Convection Enhancement in a Vertical ParallelPlate Channel, International Journal of Thermal Sciences,
vol. 47, no. 6, pp. 742–748, 2008.
[27] Incropera, F. P., Convection Heat Transfer in Electronic

Equipment Cooling, ASME Journal of Heat Transfer, vol.
110, pp. 1097–1110, 1988.
[28] Chena, Z. D., Bandopadhayay, P., Halldorsson, J.,
Byrjalsen, C., Heiselberg, P., and Li, Y., An Experimental Investigation of a Solar Chimney Model With Uniform
Wall Heat Flux, Building and Environment, vol. 38, pp.
893–906, 2003.
[29] Harris, D. J., and Helwig, N., Solar Chimney and Building Ventilation, Applied Energy, vol. 84, pp. 135–146,
2007.
[30] Fluent 6.2 User Manual, Fluent, Inc., Lebanon, NH, 2006.
[31] Bianco, N., Langellotto, L., Manca, O., Nardini, S., and
Naso, V., Converging on New Cooling Technology, Fluent
News, p. 28, summer, 2005.
[32] Fujii, T., and Imura, H., Natural Convection Heat Transfer from a Plate With Arbitrary Inclination, International
Journal of Heat and Mass Transfer, vol. 15, p. 752,
1972.

heat transfer engineering

[33] Manca, O., and Nardini, S., Thermal Design of Uniformly
Heated Inclined Channels in Natural Convection With and
Without Radiative Effects, Heat Transfer Engineering, vol.
22, no. 2, pp. 1–16, 2001.

Luigi Langellotto is a researcher at Centro Sviluppo
Materiali S.p.A. (CSM), Rome Italy. He received
his Ph.D. in mechanical engineering from Seconda
Universit`a degli Studi di Napoli (SUN). He is CSM
project leader in an RFCS project and several industrial projects with TenarisDalmine S.p.A. or ABS
S.p.A. His main scientific activities are on natural
convection in an open-ended cavity; thermal control

of electronic equipment; solar systems; analytical and
numerical solutions in material processing such as
seamless pipe rolling, strip rolling, and ingot casting; and numerical analysis
of austenite deformation and decomposition. He has co-authored more than 10
refereed journal and conference publications.
Oronzio Manca is a professor of mechanical engineering at Facolt`a di Ingegneria della Seconda Universit`a degli Studi di Napoli (SUN), Naples, Italy. He
has been coordinator of the Industrial Engineering
Area at SUN since January 2005. His main scientific activities are on active solar systems; passive
solar systems; refrigerant fluids; natural and mixed
convection in an open-ended cavity with and without
porous media; conduction in solids irradiated by moving heat sources; combined radiative and conductive
fields in multilayer thin films; analytical and numerical solutions in material
processing; thermal control of electronic equipment and solar systems; and heat
transfer augmentation by nanofluids. He is a member of the ATA Campania
Committee. He is a member of the American Society of Mechanical Engineering, and Unione Italiana di Termofluidodinamica UIT. He has co-authored
more than 270 refereed journal and conference publications. He is currently a
member of the editorial advisory boards for The Open Thermodynamics Journal, The Open Fuels & Energy Science Journal, and Advances in Mechanical
Engineering.

vol. 32 no. 6 2011


Heat Transfer Engineering, 32(6):455–466, 2011
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457632.2010.506167

An Inverse Analysis for Parameter
Estimation Applied to a Non-Fourier
Conduction–Radiation Problem

RANJAN DAS,1 SUBHASH C. MISHRA,1 T. B. PAVAN KUMAR,1
and RAMGOPAL UPPALURI2
1

Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India
Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati, India

2

Retrieval of parameters in a non-Fourier conduction and radiation heat transfer problem is reported. The direct problem is
formulated using the lattice Boltzmann method (LBM) and the finite-volume method (FVM). The divergence of radiative heat
flux is computed using the FVM, and the LBM formulation is employed to obtain the temperature field. In the inverse method,
this temperature field is taken as exact. Simultaneous estimation of parameters, namely, the extinction coefficient and the
conduction–radiation parameter, is done by minimizing the objective function. The genetic algorithm (GA) is used for this
purpose. The accuracies of the estimated parameters are studied for the effects of measurement errors and genetic parameters
such as the crossover and mutation probabilities, the population size, and the number of generations. The LBM-FVM in
combination with GA has been found to provide a correct estimate of parameters.

INTRODUCTION
Any differential equation governing a transient phenomenon
is subjected to initial and boundary conditions, and such a problem is mathematically well posed. In the area of heat transfer,
with medium and boundary properties known, the objective of
such a problem remains the determination of temperature and/or
heat flux distributions. Problems of this type belong to direct
problems. However, there are many situations where properties
and/or initial and/or boundary conditions remain unknown and
the temperature/heat flux histories are known from experiments.
The estimation of unknown quantities for this class of problems
falls under the purview of inverse problems. They are mathematically ill posed and accuracy of their solution depends on the
measured data. Inverse problems find applications in many areas of engineering such as material science [1], circuit analysis

[2], turbomachinery [3], manufacturing science [4], and design
of radiant enclosures [5].
In the area of heat transfer, inverse problems have been investigated by many researchers [5–16]. Erturk et al. [5] estimated

Address correspondence to Professor Subhash C. Mishra, Department of
Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati
781039, India. E-mail: scm

boundary conditions in a transient radiative enclosure. Reinhardt
and Hao [6], Alifanov and Nenarokomov [7], and Chen et al. [8]
have studied inverse heat conduction problems. Reinhardt and
Hao [6] proposed a method for solving noncharacteristic Cauchy
problems for parabolic equations. They used a conjugate gradient method in their analysis. Alifanov and Nenarokomov [7]
solved an inverse boundary heat conduction problem used to
investigate thermal processes between solids and environment.
They used an iterative regularization method based on minimizing the residual functional by means of gradient methods of
the first kind and spline approximation of unknown functions.
From the knowledge of temperature measurements within the
conducting solid, Chen et al. [8] retrieved temperature and heat
flux on a surface. A Kalman filter scheme was used in their
inverse analysis.
With known transient temperature data of a plate-finned tube
heat exchanger, Huang et al. [9] estimated thermal contact conductance. They used the conjugate gradient method for minimization. Franca and Howell [10] performed inverse design of
radiative enclosures for thermal processing of materials. Their
methodology was based on a truncated singular value decomposition method. They estimated heat input to a heater located at
the top of a three-dimensional enclosure that can satisfy a prescribed time-dependent temperature curve on a surface located
at the base of the enclosure.

455

456

R. DAS ET AL.

An inverse analysis to simultaneously estimate the effective thermal conductivity, the effective volumetric heat capacity, and the heat transfer coefficient between a porous medium
and a hot wire was performed by Znaidia et al. [11]. They used
the Levenberg–Marquardt method to solve the inverse problem.
Frankel and Arimilli [12] determined the convective and radiative loads from temperature measurements. They used Gauss
filter methods in their analysis. Most recently, Pereverzyev et al.
[13] carried out an inverse analysis for temperature reconstruction in a glass cooling process. They used the fast derivative-free
iterative method. Kim and Baek [14] performed an inverse analysis in a cylindrical enclosure involving conduction and radiation. They estimated heat flux distribution on the design surface.
The Levenberg–Marquardt method was used in their analysis.
They used the finite-volume method (FVM) for the energy equation. Das et al. [15, 16] recently did simultaneous estimation of
parameters for one-dimensional (1-D) [15] and two-dimensional
(2-D) [16] conduction–radiation problems. They used the lattice Boltzmann method (LBM) and the FVM [17] in conjunction
with genetic algorithm (GA) [2, 18]. All the past studies dealing
with inverse heat transfer problems involved consideration of
Fourier’s law of heat conduction.
In Fourier heat conduction, effects of any thermal disturbances in the system, in the form of a sudden rise in boundary
temperatures or a sudden appearance of a heat source at any
location in the medium, propagate with infinite speed and hence
establish instantaneously. This assumption is not universally applicable in all situations. For example, the validity of Fourier’s
law of heat conduction breaks down when we consider heat
transport through a processed meat/skin [19–22]. Further, at
a very low temperature, heat transport by conduction is not
governed by Fourier’s law [23–25]. When any material is subjected to a pulse radiation, at short time levels, a discontinuity in
the temperature profile is observed, which cannot be explained
through Fourier’s law of heat conduction [26]. In the area of
non-Fourier conduction and/or radiation heat transfer problems,

a few studies have been reported in the past [27–32].
Although literature reports dealing with parameter estimations in heat transfer problems involving Fourier’s law of heat
conduction are available to a good extent, no work has been
reported in the field of inverse non-Fourier heat transfer involving transient conduction–radiation. The present work is,
therefore, aimed at estimation of parameters in a non-Fourier
conduction–radiation problem.
In the present work, we simultaneously estimate parameters,
namely, the extinction coefficient and the conduction–radiation
parameter, in a non-Fourier conduction–radiation heat transfer
in a planar geometry. The LBM and FVM are used to solve
the direct problem. This is due to the fact that the LBM is an
efficient numerical method and, being mesoscopic in approach,
it presents a clear physical meaning. Most recently the usage of
the LBM has been explored for a wide variety of fluid flow and
heat transfer problems [33–37]. The FVM is also an efficient
method to compute the radiative information and is less prone
to the ray effect. With transient temperature distributions known
heat transfer engineering

from the direct method, in the inverse method, we use the LBM
and the FVM in conjunction with the GA to simultaneously
retrieve the extinction coefficient and the conduction–radiation
parameter. The GA is used as an optimization tool because the
probability of obtaining the solutions near to the global optimal
domain is expected to be higher because of its evolutionary
approach [38]. In both the direct and the inverse methods, the
FVM is employed to compute the radiative information and the
LBM is used to solve the energy equation. In the inverse method,
optimization is achieved using the GA.

FORMULATION
Let us consider a planar participating medium (Figure 1). Its
thermophysical and optical properties are constant. The initial
temperature of the system is Tref . For time t > 0, the west and
the east boundaries of the medium are maintained at temperature TW (> Tref ) and TE (= Tref ) , respectively. The boundaries
of the medium are considered black. In the absence of convection and heat generation, the governing energy equation for the
problem under consideration is given by
∂T
∂qC
∂q R
=−
−
(1)
∂t
∂x
∂x
where ρ is the density, c p is the specific heat, qC is the
conductive heat flux, and q R is the radiative heat flux. The
medium considered is such that the conduction wavefront qC
takes some finite time to establish itself in the medium, and
thus the assumption of the infinite propagation speed of the
conduction wave in Fourier heat conduction does not hold true.
The radiation wave front q R propagates with the speed of light,
and thus for the time scale considered in the present study,
radiative transfer is an instantaneous process.
With finite propagation speed of the conduction wavefront,
the non-Fourier heat conduction equation is given by [23, 24]
ρc p

∂T

∂qC
+ qC = −k
∂t
∂x

(2)

where
= Cα2 is the thermal relaxation time, α is the thermal
diffusivity, and C is the speed of the conduction wavefront.
From Eqs. (1) and (2), we obtain the following expression:
ρc p

∂T
∂2T
+
∂t 2
∂t

=k

∂2T
∂q R
−
−
∂x2
∂x

∂
∂t

∂q R
∂x
(3)

In Eq. (3), the divergence of radiative heat flux
by [33]
∂q R
σT 4
= β (1 − ω) 4π
− G∗
∂x
π

∂q R
∂x

is given

(4)

where β is the extinction coefficient, ω is the scattering albedo,
and G ∗ is the incident radiation.
vol. 32 no. 6 2011


R. DAS ET AL.

457

Figure 1 Schematic of the 1-D planar geometry under consideration along with D1Q2 lattice of the LBM and control volume of the FVM, (b) Intensity I j in
m.
the direction j in the center of the elemental sub-solid angle

In dimensionless form, we define time ξ, distance η, temperature θ, conductive heat flux C , radiative heat flux R ,
conduction–radiation parameter N , and incident radiation G
in the following way:
ξ=
R

Ct
X

=

η=
qR
4
σTref

x
X

N=

θ=

T
Tref

kC
3
4ασTref

C

G=

qC
4
σTref
G∗
4
σTref
π

=

(5)

In dimensionless form, Eq. (3) is written as
1 ∂ 2 θ ∂θ
1 ∂ 2θ
1
=
+
−
A ∂ξ2
∂ξ
A ∂ξ2

4N

∂ R
1 ∂
+
∂η
A ∂ξ

∂ R
∂η

(6)

where A =
Based on the formulation given by Glass et al.
[31], in the present work, the value of A has been taken as 2.0.
Equation (4) in dimensionless form is written as
XC
.
α

∂ R
=
∂η

X β (1 − ω)
π

4πθ − G
4

S∗

I
is the dimensionless intensity and S =
4
σTref
/π
is the dimensionless source term, which for a linear

4
σTref
π
anisotropic scattering phase function is given by

ωβ
(9)
[G + πa cos δ R ]
4π
where a is the anisotropy factor and δ is the polar angle.
In Eq. (9), in case of a planar medium, incident radiation G
and net radiative heat flux
R are given by and numerically
computed from [33]:
S = β (1 − ω) θ4 +

π

M

I sin δ dδ ≈ 4π

G = 2π

=2

M

I cos δ sin δ dδ ≈ 2

(11)
where M is the number of rays considered over the complete
span of the polar angle δ (0 ≤ δ ≤ π).
Writing Eq. (8) for a discrete direction having index m and
m
integrating it over elemental solid angle
[33], we get

FINITE-VOLUME METHOD
where Dm and

(8)

heat transfer engineering

I sin δm cos δm sin δm
m=1

δ=0

Dm

dI
S
= −I +
d (ˆs β)
β

δm
2
(10)

π
R

The radiative transfer equation in any direction sˆ is given by
[33]

I m sin δm sin
m=1

δ=0

(7)

In the present work, we compute ∂∂ηR using the FVM [17,
33], and the LBM is employed to compute the temperature field
in both the direct and the inverse methods. In the inverse method,
the optimization is achieved using the GA. Next in this article
we provide a brief formulation of the FVM, the LBM, and the

GA.

∗

where I =

dIm
= X (−βI m + S m )
dη
m

D = 2π sin δm cos δm sin
= 4π sin δm sin

vol. 32 no. 6 2011

(12)

δm

(13)

are given by [33]

m

m

m

δm
2

(14)


458

R. DAS ET AL.

Table 1 Effect of the number of lattices in the LBM and control volumes in
the FVM and number of rays M on temperature distribution at three different
locations: ξ = 0.60, β = 0.5, ω = 0.8, N = 0.01, and θE = 0.5 and
θW = 1.0., for the non-Fourier conduction–radiation problem
Control volumes/lattices

x/ X = 0.20
LBM-FVM

Number of
directions M

Effect of control volumes/lattices
25
12
50
12
100
12
200

12
Effect of number of rays M
100
6
100
12
100
24

x/ X = 0.40
LBM-FVM

x/ X = 0.80
LBM-FVM

0.8434
0.8451
0.8465
0.8473

0.7191
0.7164
0.7154
0.7150

0.8706
0.8812
0.8862
0.8886
0.8682

0.8862
0.8913

0.8286
0.8465
0.8500

0.7041
0.7154
0.7174

Integrating Eq. (12) over the one-dimensional (1-D) control
volume, we get
m
D m = −X βI Pm dη
IEm − IW

m

+ X Sm
P dη

m

m
IEm + IW
2

Ib =

εb θ4b

+

1 − εb
π

M/
2
2π

I m sin δm cos δm sin δm

Once the intensity distributions are known, radiative information ∂∂ηR required for the energy equation is computed from
Eq. (7).

LATTICE BOLTZMANN METHOD
In the LBM, the discrete Boltzmann equation with
Bhatanagar–Gross–Krook (BGK) approximation in dimensionless form is written as [34, 35]
f i (η + ei ξ, ξ +

(16)

From Eqs. (15) and (16), the unknown cell-center intensity
I Pm in terms of known upstream intensity and the source term
computed from previous time level is calculated from the following [33]:
⎧
m
m
2D m IW

+ X Sm
dη
⎪
P
⎪
,
Dm > 0
⎨
m
m
2D + X β
dη
m
IP =
(17)
m
2 |D m | IEm + X S m
dη
⎪
P
m
⎪
⎩
,
D
<
0
m dη
2 |D m | + X β

(η, ξ) ,

ξ) = f i (η, ξ) −

ξ
(eq)
f i (η, ξ) − f i
τ

i = 1, 2, . . . b

(19)

where f i is the particle distribution function, ei =

(0.01, 0.50)

τ=

1
η
ξ

2

+

ξ
2

E

Pc

Pm

Estimated value (N , β)

Percent error (N , β)

0.0

0.3
0.8
0.8
0.3
0.3
0.8
0.8
0.3
0.3
0.8
0.8
0.3
0.3
0.8
0.8
0.3

0.03

0.03
0.3
0.3
0.03
0.03
0.3
0.3
0.03
0.03
0.3
0.3
0.03
0.03
0.3
0.3

(0.0111, 0.4777)
(0.0097, 0.5175)
(0.0107, 0.5703)
(0.0092, 0.4289)
(0.0106, 0.5611)
(0.0103, 0.4794)
(0.0109, 0.5240)
(0.0079, 0.4032)
(0.0119, 0.3982)
(0.0106, 0.4749)
(0.0111, 0.4148)
(0.0075, 0.4207)
(0.0142, 0.6311)
(0.0114, 0.5264)

(0.0074, 0.4113)
(0.0153, 0.6193)

(11.0, –4.46)
(3.0, 3.50)
(7.0, 14.06)
(–8.0, –14.22)
(6.0, 12.22)
(3.0, –4.12)
(9.0, 4.80)
(–21.0, –19.36)
(19.0, –20.36)
(6.0, –5.02)
(11.0, –17.04)
(–25.0, –15.86)
(42.0, 26.22)
(–14.0, 5.28)
(–26.0, –17.74)
(53.0, 23.86)

0.5

1.0

2.0

heat transfer engineering

η
ξ

is the
(eq)

velocity, τ is the dimensionless relaxation time, and f i
is the equilibrium distribution function. For a planar medium
problem under consideration, with D1Q2 lattice, b = 2 and the
relaxation time τ is computed from [35]

Table 2 Comparison of exact and estimated value of the parameters for different combinations of Pc and Pm for the non-Fourier
conduction–radiation problem
Exact value (N , β)

(18)

m=1

(15)

In the FVM, normally, the cell-center intensity I Pm is related
to cell-surface intensities using the diamond scheme:
I Pm =

In Eq. (17), for D m > 0, marching is from the west boundary
and in this case, for any control volume, the cell surface intensity
m
is a known quantity. When marching
at the west boundary IW
is done from the east boundary, D m < 0 and in this case, IEm is
known.

For a boundary having temperature θb and emissivity εb (in
the present work black boundary is assumed so, εb = 1.0), the
boundary intensity Ib is computed from [33]

vol. 32 no. 6 2011

(20)


R. DAS ET AL.

459

In the present case with D1Q2 lattice, temperature and heat
flux
are computed from the following [35, 36]:

1

Lines: Present work

2

Markers: Chu et al.

0.8

θ=

fi

(21)

ei f i

(22)

Temperature

i=1

0.6
2

=
0.4

i=1

ξ = 0.6

ξ = 0.3

To solve Eq. (19), an equilibrium distribution function is required, which for the non-Fourier conduction–radiation problem
is given by [36]

β = 1.0
ω = 0.5

0.2

Ν = 2.5

(eq)

0

0

0.2

0.4

0.6

Distance
(a)

0.8

1

1

2

Temperature

2
(eq)

fi

Markers: Chu et al.

0.8

=

i=1

i=1

0.4

(24)

2
eq
f i ei

ξ = 0.6

ξ = 0.3

fi = θ
i=1

2

0.6

(23)

where wi and yi are the weights corresponding to the ith
direction, and they are chosen in such a way that the following
hold true:

Lines: Present work.

=

f i ei =

(25)

i=1

From Eqs. (23), (24), and (25), we get
β = 1.0
Ν = 0.25
0

0.2

1
(26)
2
By considering the effect of radiation, and non-Fourier heat
conduction, the equivalent form of Eq. (6) in the LBM is given

as
ξ
f i η, ξ − f i(0) η, ξ
f i η+ei ξ, ξ+ ξ = f i η, ξ −
τ
w1 = w2 = y1 = y2 =

ω = 0.5

0.2

0

= wi θ + yi ei

fi

0.4

0.6

Distance
(b)

0.8

1

1

0.8

−2 ξyi ei

ξ = 0.3

−

ξwi
4N

∂ R
∂η

(27)

Temperature

ξ = 0.6
0.6

Lines: Present work.
Markers: Chu et al.

GENETIC ALGORITHM

0.4
β = 1.0
ω = 0.5

0.2

0

Ν = 0.025
0

0.2

0.4

0.6

0.8

1

Distance
(c)
Figure 2 Comparison of temperature θ distributions in the direct method with
that of Chu et al. [28].

heat transfer engineering

Genetic algorithm [38] is an iterative optimization tool,
which, unlike deterministic methods, works with a group of
solutions collectively known as the population. Population undergoes gradual refinements in successive generations. This is
done by evaluating the fitness value of each individual in the
population, which is governed by a desired objective function.
The good individuals are retained and the bad ones are eliminated from the population, which in turn is subjected to reproduction, crossover, and mutation. Thus, in this manner the

refinement of the population takes place. The process of GA
is analogous to biological evolutions of any species in which
successive generations are conceived, born, and raised until
vol. 32 no. 6 2011


460

R. DAS ET AL.
108

10

5

10

3

10

1

Pc = 0.30, Pm = 0.03
Pc = 0.80, Pm = 0.03
Pc = 0.30, Pm = 0.30
Pc = 0.80, Pm = 0.30

104
102

100

10

-1

10

-3

10-2

10

-5

10-4

10

-7

10-6

0

20

40

60

80

Pc = 0.30, Pm = 0.03
Pc = 0.80, Pm = 0.03
Pc = 0.30, Pm = 0.30
Pc = 0.80, Pm = 0.30

6

10

Best fitness

Best fitness

107

0

100

20

40

60

80

100

Generations
(b)

Generations
(a)
108
10

8

Pc = 0.30, Pm = 0.03
Pc = 0.80, Pm = 0.30
Pc = 0.30, Pm = 0.30
Pc = 0.80, Pm = 0.03

106
104

104
10

2

10

0

10

-2

10

-4

Best fitness

Best fitness

106

Pc = 0.30, Pm = 0.03
Pc = 0.80, Pm = 0.03
Pc = 0.30, Pm = 0.30
Pc = 0.80, Pm = 0.30

102
100
10-2

0

20

40

60

80

100

10-4

0

20

Generations
(c)

40

60

80

100

Generations
(d)

Figure 3 Effect of different crossover probability and mutation probability on the best fitness: (a) E = 0.0, (b) E = 0.5, (c) E = 1.0, and (d) E = 2.0.

they themselves become ready to reproduce. Reproduction,
crossover, and mutation are the three main steps involved in
the GA. After generation of an initial population and evaluation

of its fitness, the process of reproduction starts. The generations
having good fitness values are replicated in the next population. Next, the crossover operation starts. In this process, pairs
from new strings mate to produce new offspring. The crossover
operation is governed by the crossover probability Pc , and it represents the number of individuals in the population undergoing
the crossover operation. The parents are replaced by the newly
produced offspring. Finally, through an assigned probability
termed as the mutation probability Pm , the mutation operator
randomly changes the genes in the string. The process continues until a satisfactory fitness value of the objective function is
attained.
In the present work, in the inverse analysis, the objective
function is defined as the summation of the squares of the differences between guessed temperature field θi and exact temheat transfer engineering

perature field θ˜ i [15],
n

J =

θ˜ i − θi

2

(28)

i =1

To account for the effect of measurement errors, biased errors
to the exact temperature field are added. Thus, the temperature
θmeasured when an error is included is expressed as [15]
θmeasured = θ˜ + E

(29)

where E is a biased error between 0 and 2.0. When there
˜ For
is no measurement error (E = 0.0) , θmeasured = θ.
estimation of unknown parameters, the minimization of the objective function [Eq. (28)] is required.
vol. 32 no. 6 2011


R. DAS ET AL.

RESULTS AND DISCUSSION
In the following, we present results of the inverse analysis.
The boundaries of the planar medium (Figure 1) are assumed to
be black. Initially the entire system is at a finite temperature TE.
With this, the dimensionless initial temperature is θ (η, 0) = θE .
For time t > 0, the west boundary is maintained at a higher
temperature T ref = TW = 2TE . In dimensionless form at time
ξ > 0.0, the east and the west boundaries are at temperatures
θE = 0.5 and θW = 2θE = 1.0, respectively. In the LBM,
the dimensionless velocity ei = ηξ is considered to be unity.
Therefore the non dimensional time step ξ = η is taken
for the analysis.
In Table 1, we present results of grid and ray independency
tests. For this, at time ξ > 0.0, we set the dimensionless temperatures of the west and the east boundaries at θW = 1.0
and θE = 0.5, respectively. We test the temperature θ distributions at time ξ = 0.60 and at three different locations,

5

10

namely, η = 0.20, 0.40, and 0.80. It is observed from Table
1 that beyond 100 control volumes and 12 rays, there is no significant change in the temperature θ distributions. Therefore, in
the present work we have provided the results considering 100
control volumes and 12 rays.
In order to verify the results of the direct problem (LBMFVM), in Figure 2, we compare the temperature θ distributions
with that of Chu et al. [28]. In this case, for time ξ > 0.0, the
dimensionless temperatures of the west and the east boundaries
are kept at θW = 1.0 and θE = 0.0, respectively. For extinction
coefficient β = 1.0 and scattering albedo ω = 0.50,
this comparison has been carried out at two different time
steps, e.g., ξ = 0.30 and 0.60. Three different cases for
conduction–radiation parameter, e.g., N = 2.5, 0.25, and
0.025, have been investigated. It is observed from Figure 2
that the temperature θ distributions obtained from the direct
method (LBM-FVM) compare very well with those given
by Chu et al. [28], who have solved the same problem using

Population size = 25
Population size = 50
Population size = 100

3

10

461

10

8

10

6

10

4

10

2

Population size = 25
Population size = 50
Population size = 100

10

Best fitness

Best fitness

1

10-1
10-3

100

10

-2

10-5

10-4

10-7

10
0

20

40

60

80

-6

0

100

Generations
(a)

20

40

60

80

100

Generations
(b)
8

10
8

10

Population size = 25
Population size = 50
Population size = 100

6

104

104

Best fitness

Best fitness

10

Population size = 25
Population size = 50
Population size = 100

106

102
100

2

10

100

10-2

10-2

10-4
-4

0

20

40

60

80

100

10

0

20

Generations
(c)

40

60

80

Generations
(d)

Figure 4 Effect of different population sizes on the best fitness: (a) E = 0.0, (b) E = 0.5, (c) E = 1.0, and (d) E = 2.0.

heat transfer engineering

vol. 32 no. 6 2011

100


462

R. DAS ET AL.
Table 3 Comparison of the exact and the estimated values of the parameters for different population size for the non-Fourier conduction–radiation
problem
Exact value (N , β)
(0.01, 0.50)

E

Population size

Estimated value (N , β)

Percent error (N , β)

0.0

25
50
100
25
50
100

25
50
100
25
50
100

(0.0072, 0.6681)
(0.0118, 0.4459)
(0.0097, 0.5175)
(0.0146, 0.7408)
(0.0071, 0.5657)
(0.0103, 0.4794)
(0.0046, 0.3470)
(0.0068, 0.6019)
(0.0106, 0.4749)
(0.0031, 0.3605)
(0.0135, 0.3513)
(0.0114, 0.5264)

(–27.0, 33.62)
(18.0, –10.82)
(3.0, 3.50)
(46.0, 48.16)
(–29.0, 13.14)
(3.0, –4.12)
(–54.0, –30.6)
(–32.0, 20.38)
(6.0, –5.02)
(–69.0, –27.90)

(35.0, –29.74)
(–14.0, 5.28)

0.5

1.0

2.0

the McCormack explicit predictor-corrector scheme and P3
approximation.
In the following pages we present results for the inverse analysis using the LBM-FVM in conjunction with the GA. For this
analysis, at time ξ > 0.0, we maintain the dimensionless temperatures of the west and the east boundaries at θW = 1.0 and
θE = 0.5, respectively. In the direct method (LBM-FVM), for
extinction coefficient β = 0.50, scattering albedo ω = 0.80,
and conduction–radiation parameter N = 0.01, we obtain the
temperature θ distributions at two different time levels, namley, ξ = 0.30 and 0.60. With the temperature θ distributions
available from the direct method (LBM-FVM), in the inverse
method (LBM-FVM-GA) two parameters, namely, the extinction coefficient β and the conduction–radiation parameter N ,
have been simultaneously estimated. The ranges for the extinction coefficient β and the conduction–radiation parameter N
are assumed to lie in the range (N , β) = (0.0 − 5.0).
To demonstrate the workability of the LBM-FVM-GA in the
inverse method, for a population size of 100, in Table 2, effects

of the crossover probability Pc and the mutation probability
Pm have been investigated. For scattering albedo ω = 0.80,
this study has been done for four different cases of Pc and
Pm , e.g., (Pc , Pm ) = (0.3, 0.03), (0.8, 0.03), (0.8, 0.3), and
(0.3, 0.3). The exact values of the parameters that were simultaneously estimated in the inverse method are β = 0.50 and
N = 0.01. This investigation is done for four different measurement errors, namely, E = 0.0, 0.5, 1.0, and 2.0. It can be

observed that for all measurement errors, a crossover probability Pc = 0.8 and a mutation probability Pm = 0.03 yield a
minimum error in the estimated value of the parameters.
For parameters considered in Table 2, in order to study the
effects of the crossover probability Pc and the mutation probability Pm on the convergence rate of the best fitness and their
effects on the number of generations required for the convergence, we present a comparison in Figure 3. It can be observed
that a crossover probability Pc = 0.8 and a mutation probability Pm = 0.03 provide a minimum value of the best fitness
as compared to other combinations of the crossover and the
1

1

Exact field, E = 0.0
Measured field, E = 2.0

0.9

0.9

ξ=0.3, 0.6

Temperature, θ

Temperature, θ

LBM-FVM
LBM-FVM-GA

0.8

ξ=0.3, 0.6

0.7

0.8

0.7

ξ = 0.3, 0.6
0.6

0.6

0.5

0

0.2

0.4

0.6

0.8

1

0.5

0

0.2

Distance, η
Figure 5 Comparison of exact and measured temperature profiles for different
measurement errors on temperature distribution E = 2.0.

heat transfer engineering

0.4

0.6

0.8

1

Distance, η
Figure 6 Comparison of exact and estimated temperature profiles on temperature distribution E = 2.0.

vol. 32 no. 6 2011