Fundamental J. Thermal Science and Engineering, Vol. 2, Issue 2, 2012, Pages 63-87 Published online at http://www.frdint.com/
:esphras and Keywords axial wall conduction, conjugate heat transfer, laminar flow, step
change, numerical study, P-1 approximation, participating medium.
*Corresponding author
Received December 14, 2012
© 2012 Fundamental Research and Development International
EFFECT OF AXIAL WALL CONDUCTION ON HEAT
TRANSFER PARAMETERS FOR PARALLEL-PLATE
CHANNEL HAVING RADIATION PARTICIPATING MEDIUM
AND STEP CHANGE IN BOUNDARY CONDITIONS
M. A. HASSABa, M. KHAMIS MANSOUR
b,* and M. SHAWKY ISMAIL
a
aDepartment of Mechanical Engineering
Faculty of Engineering
Alexandria University
Alexandria, Egypt
bDepartment of Mechanical Engineering
Faculty of Engineering
Beirut Arab University
Beirut, Lebanon
e-mail: [email protected]
Abstract
This paper presents a theoretical model to investigate the effect of axial
wall conduction on the heat transfer characteristics for laminar flow
through a thick-walled channel, taking into account the radiation effect in
gases as a participating medium in the radiation exchange. A partition
separates the channel of interest into two regions (upstream and
downstream region) in which two different coolant fluids bound the
channel in those regions. A numerical model has been developed in this
study using a finite-difference technique to solve the conjugate heat
transfer problem by creating a self-made computer code. The modified
P-1 approximation is employed to solve the radiation part of the problem.
HASSAB, MANSOUR and ISMAIL
64
The calculated results from the computer program have been compared
with a commercial code “ANSYS FLUENT 12” and a good agreement has
been obtained. The effects of the optical thickness ,oτ the conduction-to
radiation parameter N, single scattering Albedo coefficient ω and wall
emissivity ζ on the heat transfer characteristics are presented in this
study. The heat transfer characteristics are represented by local Nusslet
number Nu, fluid temperature distribution, and solid temperature
distribution.
The study shows that the significant change in those parameters could lead
to an enhancement of 24% for the average Nusselt number (Nu) at certain
conditions.
List of Symbols
iB Biot number, w
i K
thB ∞=
pC Specific heat at constant pressure, ( )K.kgJ °
1∞h Outside heat transfer coefficient for the upstream fluid,
( )K.mW 2 °
2∞h Outside heat transfer coefficient for the downstream fluid,
( )K.mW 2 °
H Vertical distance from center line to the inner surface of the solid
domain, m
fK Fluid domain thermal conductivity, ( )K.mW °
wK Solid domain thermal conductivity, ( )K.mW °
rK Thermal conductivity ratio, fw KK
pK Planck mean coefficient
RK Rosseland mean coefficient
L Domain length parallel to the fluid flow, m
N Conduction to radiation parameter, 34
*
i
f
T
KN
σ
β=
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
65
Nu Local Nusselt number, k
Hhi=Nu
P pressure, kPa
P* Dimensionless pressure, 2inletu
PP*
ρ=
Pe Peclet number, Pr*RePe =
Pr Prendtl number, f
p
k
cPr
µ=
inconvQ Dimensionless convective heat transfer from inner fluid, *
*
yx
∂
θ∂∆
outconvQ Dimensionless convective heat transfer to the outer fluid,
( )∞θ−θ∆ *xKB ri
axialQ Dimensionless axial heat transfer through the solid domain,
*xtK rr
∂
θ∂
radQ Radiative heat flux, 2mW
*rQ Dimensionless radiative heat flux, ( )
−σ ∞1
3rad 4 TTTQ ii
Re Reynolds number, µ
ρ=
UHRe
t Thickness of the solid domain, m
rt Thickness ratio of the solid domain, Httr =
T Temperature, K°
iT Inlet fluid temperature, K°
meanT Local bulk temperature of the fluid domain, K°
surfaceT Interface temperature between the solid and fluid domains, K°
1∞T Outside temperature for the left side (upstream), K°
2∞T Outside temperature for the right side (downstream), K°
inletu Inlet fluid velocity in X-direction, sm
HASSAB, MANSOUR and ISMAIL
66
U Velocity in X-direction, sm
*U Dimensionless velocity in X-direction, inlet* uUU =
V Velocity in Y-direction, sm
*V Dimensionless velocity in Y-direction, inlet* uVV =
X Horizontal coordinate, m
*X Dimensionless horizontal coordinate, HXX =*
Y Vertical coordinate, m
*Y Dimensionless vertical coordinate, HYY =*
Greek Symbols
θ Dimensionless temperature, ( ) ( )11 ∞∞ −−=θ TTTT i
ρ Fluid density, 3mkg
µ Dynamic viscosity, Pa.s
ν Kinematic viscosity, ρµ=ν
ω Single scattering Albedo, ( )σ+
σ
K
ξ Identified parameter, ( )
( )ω−τε−
ε1
22 o
β Mean extinction coefficient, σ+pK
oτ Optically thickness, H∗β
σ Scattering coefficient for fluid
σ Stefan-Boltzmann constant
∈ Solid domain emissivity
∆ Difference
Subscripts
i inlet
r ratio
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
67
rad radiation
w wall
f fluid
1∞ Outside left fluid
2∞ Outside right fluid
1. Introduction
The interest in predicting of heat transfer characteristics in laminar forced flow
in pipes and channel flows is steadily under consideration particularly when the
thermal problem is out of ordinary form. For example, in some heat transfer devices
such a fired-heater exchanger and shell-and-tube heat exchanger, the surface wall is
exposed to significant step change in the bounding fluid temperature at the exchanger
baffles/partition. This significant change in the temperature intensifies the axial wall
conduction to participate in the heat transfer modes and changes the characteristics of
the convective heat transfer problem. In particular, when this channel has a
participating medium in radiation exchange, the heat transfer characteristics are
considerably changed. This study addresses this problem and highlights the
importance of the axial wall conduction’s influence on the heat transfer parameters
such as local Nusselt number, fluid temperature distribution, and solid temperature
distribution during the presence of radiative heat exchange. In addition, according to
authors’ knowledge and the open literature survey, no specific work has been
dedicated to address this problem including the radiation effect in the gas and this is
considered the main contribution in this research. The topic of combined conduction
and radiation heat transfer in the channel/tube having participating medium has been
studied extensively in the cited literature survey. A comprehensive review of
combined conduction-radiation heat transfer investigations has been given by Siegel
[1]. Also, numerous investigations of the cited review covered a wide range of
operating conditions for the combined convective and radiative heat transfer
problems either laminar or turbulent flow, developing or developed flow, gray or
nongray participating medium, most of those investigations are listed in Modest [2]
and Viskanta [3]. Some of those researches focus on the combined radiation and
convection inside channel such as in [4-14]. The foregoing review of the available
and recent literature review reveals that the effect of axial wall conduction in the
combined radiative and convective heat transfer problem has not been extensively
investigated yet. Therefore, this work is devoted to investigate this problem and this
HASSAB, MANSOUR and ISMAIL
68
could be considered the main contribution for this study. Nevertheless, Yang et al.
[15] pioneered a research to study the effect of axial fluid conduction on laminar
forced conduction of a participating gray, absorbing and emitting medium with a
fully developed velocity entering an isothermal semi-infinite long pipe. The objective
of their study is to investigate the effect of Peclet number, conduction-radiation
parameter, and optical thickness on heat transfer behavior in the thermal entrance
region of a circular pipe.
2. Mathematical Formulation
The geometrical configuration of the investigated problem is schematically
shown in Figure 1. A fluid with an average temperature inT and velocity inu is
flowing into parallel-plate channel with height 2H with a wall thickness of t.
The assumptions which have been adopted in this model are as follows:
[1] The fluid is Newtonian and incompressible with constant physical properties
in steady-state heat transfer problem.
[2] Negligible viscous heat dissipation and fluid axial conduction effects.
[3] The flow is laminar.
[4] Hydrodynamic and thermal developing flow.
[5] The partition is adiabatic so no heat flow across it from the coolant fluids.
[6] The participating gas is a gray, emitting, isotropically scattering, and
absorbing medium.
[7] The radiation model is a P-1 approximation with a one-dimensional radiation
equation.
Figure 1. Geometrical configuration of the computational domain.
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
69
The conjugate heat transfer problem consisting of the flowing fluid and the tube
wall is governed by the respective energy equations in 2D form. Due to the axial
symmetry of the problem, it is sufficient to solve the lower half of the model. The
outer surfaces of the channel are exposed to coolant fluids with different
temperatures of 1∞T and 2∞T “step change boundary conditions”. The coolant fluid
in the left side is called an upstream fluid while the right side is a downstream fluid.
It is assumed that both fluids have the same convective heat transfer coefficient
which is kept uniform in the numerical model at .∞h Doing so, the mathematical
formulation of the governing equations in its derivative and non-dimenstional form is
given below.
2.1. Governing equations for fluid domain
2.1.1. Continuity equation
.0=∂
∂+
∂
∂
Y
V
X
U (1)
2.1.2. Momentum equations
,1
2
2
2
2
∂
∂+
∂
∂ν+
∂
∂
ρ−=
∂
∂+
∂
∂
Y
U
X
U
X
P
Y
UV
X
UU (2)
.1
2
2
2
2
∂
∂+
∂
∂ν+
∂
∂
ρ−=
∂
∂+
∂
∂
Y
V
X
V
Y
P
Y
VV
X
VU (3)
2.1.3. Energy equation coupled with the radiation term
.rad2
2
Y
q
Y
T
*c
k
Y
TV
X
TU
p ∂
∂−
∂
∂
ρ=
∂
∂+
∂
∂ (4)
2.1.4. P-1 radiation model
( ) .*163 3rad2
rad2
Y
TTKqKK
Y
qPRP ∂
∂=σ+−
∂
∂ (5)
By introducing the following dimensionless quantities,
,,,,, *
1in
1in
*in
*HYYHtt
TT
TTuVVuUU r ==
−
−=θ==
∞
∞
( ),
4,
4,,
31
3rad*
2in
*
ii i
f
ii
rf
wr
T
KN
TTT
K
KK
u
PP
σ
β=
−σ==
ρ=
∞
HASSAB, MANSOUR and ISMAIL
70
( )( ),1*
22,,, ω−τ
ε−
ε=ξβ=τ
σ+
σ=ωσ+=β oo
pp *H
KK
.Nuand,,Pr,Rek
Hh
K
thB
k
cUH i
wi
p==
µ=
µ
ρ= ∞
Equations (1) to (5) can be renormalized as follows:
,0*
*
*
*=
∂
∂+
∂
∂
Y
V
X
U (6)
,Re1
*
*
2*
*2
2*
*2
*
**
*
**
X
P
Y
U
X
U
Y
UV
X
UU
∂
∂−
∂
∂+
∂
∂=
∂
∂+
∂
∂ (7)
,Re1
2*
*2
2*
*2
*
**
*
**
∂
∂+
∂
∂=
∂
∂+
∂
∂
Y
V
X
V
Y
VV
X
VU (8)
,1
PrRe1
*
*
2*
2
**
**
Y
Q
NpeYY
VX
U ro
∂
∂τ−
∂
θ∂=
∂
θ∂+
∂
θ∂ (9)
( ) ( ) .1413 22
*2
YY
Qoo
r
∂
θ∂τω−=τω−−
∂
∂ (10)
2.2. Governing equation for the solid domain
Assuming steady-state conditions and constant properties of the wall, the energy
equation for 1D is adopted (neglecting the temperature gradient in y-direction) and it
is written in dimensionless form as follows:
.0*
2=
∂
θ∂
X (11)
2.3. Boundary conditions
2.3.1. At the inlet section
At the inlet section for the fluid domain, the velocity and temperature are
considered as uniform while for the solid domain the velocity is zero and the wall is
adiabatic.
.0and0at*,,,0, 4radinlet HYXTQTTPPVuU iii ≤≤=σ===== (12)
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
71
.0andat0 LXHYVU ≤≤=== (13)
In dimensionless form as:
.10and0at0,,1 ***2inlet
*** ≤≤==ρ
===θ= YXVu
PPQU i
r (14)
.01at0 **** HLXYVU ≤≤=== (15)
2.3.2. At the center line
At the center line the boundary condition is the symmetrical axis
.0,0at0rad LXYY
Q
Y
T
Y
V
Y
U≤≤==
∂
∂=
∂
∂=
∂
∂=
∂
∂ (16)
In dimensionless form as:
.0,0at0 ***
*
**
*
*
*HLXY
Y
Q
YY
V
Y
U r ≤≤==∂
∂=
∂
θ∂=
∂
∂=
∂
∂ (17)
2.3.3. At the wall
( ) .20,at1rad2
2LXHYTThQ
Y
TK
X
TtK fw ≤≤=−=+
∂
∂−
∂
∂∞∞ (18)
( ) .2,at2rad2
2LXLHYTThQ
Y
TK
X
TtK fw ≤≤=−=+
∂
∂−
∂
∂∞∞ (19)
In dimensionless form as:
.20,1at** ****2*
2HLXYtBKQ
NYX
tK rirro
rr ≤≤=θ=τ
+∂
θ∂−
∂
θ∂ (20)
( ) .20,1at** **2
**2*
2HLXYtBKQ
NYX
tK rirro
rr ≤≤=θ−θ=τ
+∂
θ∂−
∂
θ∂∞ (21)
And Marshak’s boundary condition [16] for a cold surface, in a simplified manner is
expressed as:
( )
.0,at0*22 rad
rad LXHYQKY
Qp ≤≤==
ε−
ε+
∂
∂ (22)
And it could be written in dimensionless form as:
HASSAB, MANSOUR and ISMAIL
72
.0,1at0* ****
*
HLXYQY
Qr
r ≤≤==ξ+∂
∂ (23)
3. Numerical Procedure and Model Validation
The system set of the governing equations and boundary conditions are
discretized numerically using a finite-difference method. Pressure-correction scheme
has been adopted to couple the pressure and velocity in the continuity and momentum
equations and Gauss-Seidel technique has been implemented as an iterative method
to solve the system of finite-difference equations. Numerous grid sizes have been
examined to achieve the mesh-independent solutions as displayed in Figure 2. It can
be seen from this Figure that after 26000 grid number there is no significant change
in the numerical results, this number has been obtained through 221 cells in the flow
direction and 118 cells in the normal direction to the flow. A computer code has been
developed to solve the governing equations simultaneously. In order to guarantee the
reliability of our self-coded program it should be assessed first before its
implementation in the parametric study. The assessment will be accomplished
through comparison with a well-known and robust program code such as ANSYS
FLUENT 12. The self-coded program is run at predefined conditions ( ,250Re =
6.0,1.0,5.0,766,01.0,2.0,5.0,767.0Pr =τ==ω===−=θ∆= ∞ orir NkBt
and )15.0=ζ and the equivalent of those conditions physically have been given to
the ANSYS FLUENT 12 program for the sake of the comparison. As it is noticed
from Figure 3, most of the predicted results of the self-coded program are mainly
coincident with those of the commercial code with a deviation of 6%.
Figure 2. Mesh-independent history.
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
73
Figure 3. Comparison between the self-coded program and ANSYS FLUENT 12
program.
4. Results and Discussion
A case study has been selected to highlight the effect of the axial wall
conduction on the conjugate heat transfer problem taking into account the radiation in
the participating medium. The self-coded program has been catered by the following
selected parameter ,5.0,01.0,2.0,5.0,7.0Pr,250Re =ω==−=θ∆== ∞ ir Bt
0.1,05.0,1800 =τ== or Nk and .25.0=ζ
The output results are represented in Figures 4 to 8. Figure 4 is prepared to show
the effect of the axial wall conduction and the radiation heat flux on the local Nu
along the channel. As it can be obviously seen from this Figure, that both cases “with
and without radiation effects” have a turnover point in the Nu trend at the partition
position. The enhancement in the local Nu could reach to 50% in the case without
radiation while it could reach only 8% in the second case.
The main explanation of this turnover point has been addressed in details in
Hassab et al. [17]. However, in this paper, it focuses on the effect of the radiative
medium on the conjugate heat transfer problem. The slight enhancement in the Local
Nu could be explained as follows; the high temperature difference between the two
coolant fluids "upstream and downstream fluid" at the partition plate intensifies the
magnitude of the axial wall conduction through the channel wall. However, the
radiative heat flux warms up the wall surface and attempts to decrease the gradient of
the wall surface temperature as shown in Figure 6. This warming makes the energy to
be dissipated to the outer fluid more than non-participating medium case.
HASSAB, MANSOUR and ISMAIL
74
As expected, the heat transferred from the inner fluid is clearly increased during
presence of the radiation effect as shown in Figure 7. This increase is approximately
400% compared with that in the case of non-participating medium. This is an
indicator for the importance of the radiation calculation in practice such as in boilers
and fired water heaters.
This small temperature difference between the two coolant fluids around the
partition stimulates some amount of heat to be extracted through the axial conduction
in the channel wall. This amount makes small cooling for the wall surface. This is
translated into an increase in the heat extracted from the inner fluid. Hence, the ratio
between the radiation heat flux and the total heat extracted from inner hot fluid
decreases around the partition as shown in Figure 8.
It may be interesting to observe from Figure 6 that at a location in the front
region (from-0.14 to 0.0) the wall temperature is colder than the coolant fluid itself.
In case of participating medium in the channel, the release of thermal stresses takes
place with the outer convective heat flux. So, at fouling cases and relatively low Bi
number cases, the outer convective heat flux will be limited and may cause a failure
of the tube walls.
Figure 4. Local Nu for domain of interest for 0.1,05.0,5.0 =τ==ω oN and
.0.1=ζ
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
75
Figure 5. The ratio between _conv_in,and_conv_out QQ ratio between delta
inconvandaxial _Q_Q_ for 0.1,05.0,5.0 =τ==ω oN and .0.1=ζ
Figure 6. Temperature surface and its corresponding outside temperature for
0.1,05.0,5.0 =τ==ω oN and .0.1=ζ
Figure 7. _conv_inQ from the inner fluid for 0.1,05.0,5.0 =τ==ω oN and
.0.1=ζ
HASSAB, MANSOUR and ISMAIL
76
Figure 8. The ratio between _rad_inQ and the total heat in from inner fluid for
0.1,05.0,5.0 =τ==ω oN and .0.1=ζ
4.1. Parametric study
The relevant parameters to the radiative heat transfer and their impacts are
discussed in this section. Those parameters are optical thickness ,oτ conduction- to-
radiation N, wall emissivity ζ and Albedo scattering factor .ω The effect of those
parameters is represented by the variation in the local Nu number; surface
temperature distribution, and heat flux. It should be noted the other parameters are
kept constant in the parametric study. Those parameters are ,250Re =
1800,01.0,2.0,7.0Pr ==== rir kBt and .5.0−=θ∆ ∞
4.4.1. The effect of the optical thickness ( )oτ
The output results are displayed from Figure 9 to Figure 14. Figure 9 is
presented to show the effect of axial conduction and radiation heat flux on the local
Nu number at different optical thickness. It has been shown that the increase in the
optical thickness is followed by an increase in the local Nu number.
This increase starts in order of unity to order of hundreds, according to the range
of the optical thickness ( )oτ as shown in Figure 9, at prescribed ( ),oτ the local Nu
remains constant with the positive direction of the flow until reaches the partition.
Around the partition, it makes turnover point as a result of axial wall conduction
impact.
This turnover has a major effect at higher optical thickness values as shown in
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
77
Figure 9. It is interesting to mention that the magnitude of the axial conduction
through the walls gets improved at higher values of optical thickness as shown in
Figure 10. As the optical thickness increases the heat extracted from the inner hot
fluid increases as shown in Figure 12 until the optical thickness reaches unity. After
this value, the heat extracted from the hot fluid decreases as long as the optical
thickness is getting higher. This finding is emphasized in Figure 11. The increase in
the optical thickness is followed by an increase in wall surface temperature up to
,0.1=τo then the increase in the optical thickness is accompanied by the decrease in
the wall surface temperature as shown in Figure 11. At lower optical thicknesses, the
increase in the optical thickness intensifies the magnitude of the emissive power of
the medium. Hence, the combined radiative and convective heat transfer is getting
higher at this condition and accordingly, this causes a warming in the wall surface
temperature as illustrated in Figure 11. Alternatively, at higher optical thickness
,0.1>τo the increase in the optical thickness causes a significant attenuation in the
emissive power (see Figure 13), therefore, the total heat transfer is getting smaller at
this condition, making a decrease in the wall surface temperature. Figure 14 is
presented to underline this result by showing the effect of the optical thickness
variation on the convective outer heat transfer. At the beginning, the increase in the
optical thickness leads to an augmentation in the convective heat transfer as a result
of an increase in the difference between the outer coolant fluid and wall surface
temperature. Then, this difference is getting smaller and consequently the convective
outer heat transfer decreases as shown in Figure 14.
Figure 9. Local Nu for domain of interest for 01.0,5.0,5.0 ==ω−=θ∆ ∞ N and
oτ=ζ ,1 is variable.
HASSAB, MANSOUR and ISMAIL
78
Figure 10. The ratio between delta _conv_inand_axial QQ for ,5.0=ω ,01.0=N
0.1=τo and oτ=ζ ,1 is variable.
Figure 11. Temperature surface and its corresponding outside temperature for
,5.0=ω 0.1,01.0 =τ= oN and oτ=ζ ,1 is variable.
Figure 12. _conv_inQ from the inner fluid for 01.0,5.0 ==ω N and oτ=ζ ,1 is
variable.
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
79
Figure 13. _in_radiationQ from inner fluid for 01.0,5.0 ==ω N and oτ=ζ ,1 is
variable.
Figure 14. _conv_outQ to the outer fluid for 01.0,5.0 ==ω N and oτ=ζ ,1 is
variable.
4.4.2. The effect of conduction to radiation parameter (N)
The local Nu number trend is almost flat in the front and behind regions at lower
values of N parameter as shown in Figure 15. Additionally, the axial conduction has
an insignificant effect interrupted by the lower value of _conv_in_axial QQ at
lower value of N parameter as shown in Figure 17. As the N parameter is small the
radiation heat flux increases and warming the wall surface subsequently the gradient
of the wall temperature can be neglected as shown in Figures 16 and 18.
HASSAB, MANSOUR and ISMAIL
80
The reverse is true for higher values of the N parameter the axial heat flux
generated and transfer from the front region to the backward region making a
valuable turnover in local Nu around the partition. Figures 19 and 20 illustrate this
effect through the variation in the heat extracted from the hot fluid and the convective
outer heat transfer with respect to the variation in N parameter. As shown in those
Figures, the increase in N parameter results in a decrease in the heat extracted and
also in the convective outer heat transfer and vice versa.
Figure 15. Local Nu for domain of interest for 1.0,5.0 =τ=ω o and 0.1=ζ and
N is variable.
Figure 16. Ratio between _conv_inand_in_radiation QQ for 1.0,5.0 =τ=ω o
and 0.1=ζ and N is variable.
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
81
Figure 17. The ratio between delta _conv_inand_axial QQ for 1.0,5.0 =τ=ω o
and 0.1=ζ and N is variable.
Figure 18. Temperature surface and its corresponding outside temperature for
1.0,5.0 =τ=ω o and 0.1=ζ and N is variable.
Figure 19. _conv_inQ from the inner fluid dimensionless for 1.0,5.0 =τ=ω o and
0.1=ζ and N is variable.
HASSAB, MANSOUR and ISMAIL
82
Figure 20. _conv_outQ to the outer fluid dimensionless for 1.0,5.0 =τ=ω o and
0.1=ζ and N is variable.
4.4.3. The effect of emissivity of the wall parameter ( )ζ
Figure 21 is presented to show the effect of axial wall conduction and the
radiation heat flux on the local Nu number at different wall emissivity ( ).ζ It has
been shown that the increase in wall emissivity, ε results in an increase in the local
Nu number as shown in Figure 21. As expected, the increase in the wall emissivity
according to Kirchoff’s law for gray opaque body is coupled by an increase in the
wall absorptivity property. In turn, the radiative heat flux is increased as the wall
emissivity increases and enhances the local Nu number as shown in Figure 21. In
addition, the increase in the wall emissivity/absorptivity wanes the effect of the axial
wall conduction, as seen in Figure 22, as a result of warming the wall surface up at
expense of an increase in the radiative heat flux towards the wall surface. This can be
evidenced by Figure 23; the hotter surface temperature is associated with the higher
wall emissivity.
Figure 21. Local Nu for domain of interest for 1.0,05.0,5.0 =τ==ω oN and ζ
is variable.
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
83
Figure 22. The ratio between delta _conv_inand_axial QQ for ,05.0,5.0 ==ω N
1.0=τo and ζ is variable.
Figure 23. Temperature surface and its corresponding outside temperature for
1.0,05.0,5.0 =τ==ω oN and is variable.
4.4.4. The effect of Albedo parameter ( )ω
Figure 24 is introduced to show the effect of axial conduction and the radiation
heat flux on the local Nu at different Albedo scattering factors. As the Albedo
parameter ( )ω increases the local Nu number decreases. This can be explained by
that an increase in scattering coefficient relative to the absorbing coefficient leads to
vanish/disperse the radiative intensity and therefore, the radiation heat flux decreases
too. In this case the axial wall conduction reinforces his situation with decay in the
effect of the radiation as shown in Figure 25. Furthermore, as previously explained,
HASSAB, MANSOUR and ISMAIL
84
this makes the wall surface to be colder than usual in presence of weakness of the
radiation heat transfer as shown in Figure 26.
Figure 24. Local Nu for domain of interest for 0.1,05.0 =τ= oN and ζω, are
variables.
Figure 25. The ratio between delta _conv_inand_axial QQ for 0.1,05.0 =τ= oN
and ζω, are variables.
Figure 26. Temperature surface and its corresponding outside temperature for
0.1,05.0 =τ= oN and ω and ζ are variables.
EFFECT OF AXIAL WALL CONDUCTION ON HEAT …
85
5. Conclusions
The laminar forced convection inside channels with two parallel-plates has been
studied taking into account the axial wall conduction and radiative heat transfer in
participating medium. The combined convective and radiative conjugate heat transfer
problem has been solved numerically by a self-made computer code coupling the
solid and fluid domains by reasonable boundary conditions and assumptions. The
effect of some radiation parameters on the axial wall conduction and accordingly the
heat transfer characteristics (Nu number, wall mean temperature, and heat flux
distribution) has been investigated. Those parameters are optical thickness ,oτ
conduction to radiation N, wall emissivity ζ and Albedo scattering factor .ω The
following major conclusions can be drawn:
[1] Generally, an increase in the radiation heat flux results in a decrease in the
impact of the axial conduction as the radiation warms up the solid domain.
[2] The increase in the optical thickness produces attenuation to the impact of the
axial wall conduction particularly when the optical thickness is thinner “less
than unity”. However, at optical thickness larger than unity the radiative heat
transfer decreases allowing the axial wall conduction to be significant and take
a role to enhance the Local Nu number. At 10=τo the increase in local Nu
around the partition is 14 %.
[3] As it is expected the increase in the radiative heat transfer causes an increase
in the local Nu as a result to the increase in the total heat flux received by the
wall. This is clarified by the effect of conduction to radiation parameter, N; the
increase in this parameter has two competing adverse effects; an increase in
axial wall conduction and a decrease in average Nu number. For example, the
increase in N parameter from 0.01 to 1.0 makes the Nu to drop from 100 to 3.
It was noticed that the local Nu around the partition is increased responding to
the enhancement in the axial wall conduction at higher N parameter values.
[4] In presence of the participating medium in the channel, the wall emissivity and
scattering coefficient have adverse effects on the heat transfer parameters and
axial wall conduction. The channel wall which has higher emissivity
“unpolished/oxidized material” is characterized by high efficient thermal
performance but it is much exposed to the thermal stress around the partition
as consequence of weakness of the axial wall conduction contribution to cool
HASSAB, MANSOUR and ISMAIL
86
down the wall surface. On the other hand, the increase in the scattering
coefficient results in a decrease in the thermal effectiveness but it gives an
opportunity to the axial wall conduction to protect the channel wall from the
severe thermal stress.
The computer code has been assessed via the comparison with the commercial
software “ANSYS FLUENT 12” with a good agreement between the results obtained
from the self-coded program and commercial one.
Acknowledgements
The authors would like to acknowledge Arab Academy University for their
support by licensed ANSYS FLUENT 12 software to be used to carry out the
numerical calculation for comparison.
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