The paper deals with Simultaneous heat transfer by convection and radiation in a channel flow between two infinite black parallel plates is investigated. The effect of radiation on the heat transfer and the full thermal development of the flow is studied. The effect of scattering albedo, conduction-radiation parameter and the optical thickness are examined.

1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 3, March (2015), pp. 14-20© IAEME
14
EFFECT ON HEAT TRANSFER AND THERMAL
DEVELOPMENT OF A RADIATIVELY PARTICIPATING
FLUID IN A CHANNEL FLOW
NDZANA Benoît
Senior Lecturer, National Advanced School of Engineering,
University of Yaounde I, Cameroon
BIYA MOTTO
Frederic, Senior Lecturer, Faculty of Sciences,
University of Yaounde I, Cameroon
LEKINI NKODO Claude Bernard
P.H.D. Student; National Advanced School of Engineering,
University of Yaounde I, Cameroon
ABSTRACT
The paper deals with Simultaneous heat transfer by convection and radiation in a channel flow
between two infinite black parallel plates is investigated. The effect of radiation on the heat transfer
and the full thermal development of the flow is studied. The effect of scattering albedo, conduction-
radiation parameter and the optical thickness are examined. The radiation is shown to substantially
alter the heat transfer downstream before the thermally fully developed conditions. The full thermal
development is shown to exist for the constant wall temperature case, while it is pushed further
downstream and could not be seen for the constant wall heat flux case. While the radiation greatly
affects the heat transfer when the fluid is heated, for the cooling case radiation effect decreases along
the stream wise direction and vanishes at the fully developed conditions.
Keyword: Heat Transfer; Thermal Development; Channel Flow; Albedo; Radiation.
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND
TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 6, Issue 3, March (2015), pp. 14-20
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© I A E M E

2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 3, March (2015), pp. 14-20© IAEME
15
NOMENCLATURE
‫ܥ‬ = ‫ ݂ܿ݅݅ܿ݁݌ݏ‬ℎ݁ܽ‫ݐ݂݋ݐ‬ℎ݂݈݁‫݀݅ݑ‬
‫ܫ‬ = ‫ݕݐ݅ݏ݊݁ݐ݊݅ ݐ݊ܽ݅݀ܽݎ‬
‫ܫ‬௕ = ݈ܲܽ݊ܿ݇ ݂‫݊݋݅ݐܿ݊ݑ‬
݇ = ‫ݐ‬ℎ݁‫ݐ ݂݋ ݕݐ݅ݒ݅ݐܿݑ݀݊݋ܿ ݈ܽ݉ݎ‬ℎ݁ ݂݈‫݀݅ݑ‬
‫ܮ‬ = ܿℎ݈ܽ݊݊݁ ‫ݐ݀݅ݓ‬ℎ
ܰ௖௥ = ܿ‫݊݋݅ݐܿݑ݀݊݋‬ − ‫ݎ݁ݐ݁݉ܽݎܽ݌ ݊݋݅ݐܽ݅݀ܽݎ‬ =
݇ߚ
4ߪܶ௥௘௙
ଷ
ܲሺߤ, ߤ′ሻ = ‫݌ ݃݊݅ݎ݁ݐݐܽܿݏ‬ℎܽ‫݊݋݅ݐܿ݊ݑ݂ ݁ݏ‬
‫ݍ‬ሺ‫ݕ‬ሻ = ℎ݁ܽ‫ݔݑ݈݂ ݐ‬
‫ݏ‬ = ‫ݐܽ݌‬ℎ ݅݊ ܽ݊‫݊݋݅ݐܿ݁ݎ݅݀ ݕ‬
ܶ = ‫݁ݎݑݐܽݎ݁݌݉݁ݐ‬
‫ݑ‬ = ‫݀݅ݑ݈݂ ݂݋ ݕݐ݅ܿ݋݈݁ݒ‬
ߚ = ݁‫ݐ݂݂݊݁݅ܿ݅݁݋ܿ ݊݋݅ݐܿ݊݅ݐݔ‬
ℵ = ‫݁ݐܽ݊݅݀ݎ݋݋ܿ ݈ܽܿ݅ݐ݌݋‬ = ߚ‫ݕ‬
ℵ଴ = ‫ݐ ݈ܽܿ݅ݐ݌݋ ݈ܽݐ݋ݐ‬ℎ݅ܿ݇݊݁‫ݏݏ‬
ߤ = ܿ‫݈݁݃݊ܽ ݎ݈ܽ݋݌ ݂݋ ݁݊݅ݏ݋‬
ߦ = ݀݅݉݁݊‫݁ݐܽ݊݅݀ݎ݋݋ܿ ݁ݏ݅ݓ݉ܽ݁ݎݐݏ ݏݏ݈݁݊݋݅ݏ‬
ߪ = ܵ‫݂݊ܽ݁ݐ‬ − ‫ݐ݊ܽݐݏ݊݋ܿ ݊݊ܽ݉ݖݐ݈݋ܤ‬
߱ = ‫݋ܾ݈݀݁ܽ ݃݊݅ݎ݁ݐݐܽܿݏ ݈݁݃݊݅ݏ‬
= ܽ݊݃‫ݕݐ݅ݏ݊݁ݐ݊݅ ݐ݊ܽ݅݀ܽݎ ݂݋ ݊݋݅ݐܿ݁ݎ݅݀ ݎ݈ܽݑ‬
INTRODUCTION
The analysis of combined modes of heat transfer has been the subject of many investigators.
Different numerical techniques, iterative in nature and differing from each other in the way the
radiation part is handled, have been employed to solve the combined radiation convection problems.
The present work focuses on the effect of radiation on the heat transfer and the full thermal
development of an absorbing, emitting and anisotropically scattering fluid between two infinite black
parallel plates, as it combines with convection. The problem is to solve the energy equation, a
parabolic partial differential equation, which couples with the equation of radiative transfer, an
integro-differential equation. A multilayer technique is employed to obtain the radiation contribution.
Scaling laws derived in [1] and [2] have been utilized to treat the general case of an absorbing,
emitting and anisotropically scattering fluid by modelling it as an equivalent non-scattering medium.
The effect of some parameters such as the scattering albedo, the optical thickness and the conduction
radiation parameter are considered.
GOVERNING EQUATIONS
The governing equation to be solved, with the assumptions of constant properties, no internal
heat generation and no heat production due to viscous dissipation, is written as:
ߩܿ‫ݑ‬
డ்
డ௫
= ݇
డమ
డ௬మ −
డ௤ೝ
డ௬
(1)
Where the second term in the right hand side is the radiation contribution

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 3, March (2015), pp. 14-20© IAEME
16
The boundary and inlet conditions are:
ܶሺ‫ݕ‬ሻ = ܶ଴ at ‫ݔ‬ = 0
డ்
డ௬
= 0 at ‫ݕ‬ =
௅
ଶ
(2)
ܶሺ‫ݕ‬ሻ =ଵ or ‫ݍ‬ሺ‫ݕ‬ሻ = ‫ݍ‬ఠ at ‫ݕ‬ = 0
In a dimensionless form, the energy equation is written as:
‫ݑ‬∗ డ்∗
డξ
=
డమ்∗
డℵమ
−
ଵ
ே೎ೝ
డ௤ೝ
డℵ
(3)
The conduction – radiation parameter ܰ௖௥ represents the relative importance of conduction to
radiation in the transverse direction of the flow.
The divergence of the radiative heat flux is written from [3] as:
ௗ௤ೝ
ௗℵ
= ሺ1 − ‫ݓ‬ሻ[4ߪܶସ
− ‫ܩ‬ሺℵሻ] (4)
Where ‫ܩ‬ሺℵሻ us the average radiative intensity given by:
‫ܩ‬ሺℵሻ = 2ߨ ‫׬‬ ‫ܫ‬ሺℵ଴ߤ′ሻ݀ߤ′
ାଵ
ିଵ
(5)
The radiation intensity ‫ܫ‬ሺℵ, ߤሻ is governed by the equation of radiative transfer:
ଵ
ఉሺ௦ሻ
ௗூሺ௦, ሻ
ௗ௦
+ ‫ܫ‬ሺ‫,ݏ‬ ሻ = ሺ1 − ‫ݓ‬ሻ‫ܫ‬௕ሺܶሻ +
௪
ସగ
‫׬‬ ‫ܫ‬ሺ‫,ݏ‬ ௜ሻܲሺ , ௜ሻ݀ ௜ସగ
(6)
A solution of the equation of transfer is needed in the analysis of this problem. However, the
knowledge of the temperature is necessary. We see clearly that the equation of radiative transfer
couples with the energy equation. For an anisotropically scattering medium, even with the use of
approximate methods, the difficulty in solving the equation of transfer still remains. Thus, we seek a
solution for the anisotropically scattering medium by solving the equation of transfer for a simpler
problem. This is achieved by using the scaling laws which model an anisotropically scattering layer
by a non-scattering one.
DISCRETIZATION
The energy equation is discretized using the control volume approach [4] and a fully implicit
scheme. The generation term or the radiation contribution term is linearized using the tangent
method [4]. The discretized energy equation for a node (i,j) is written as:
ܽ௜௝ܶ௜,௝ିଵ + ܾ௜௝ܶ௜,௝ + ܿ௜௝ܶ௜,௝ାଵ = ݀௜௝ (7)
Where
ܽ௜௝ = −
∆ξ
௨ೕ∆ℵమ (8)

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 3, March (2015), pp. 14-20© IAEME
17
ܾ௜௝ = 1 + 2
∆క೔
௨ೕ∆ℵమ + 4ܶ௜,௝೚೗೏
∆క೔
௨ೕே೎ೝ
(9)
ܿ௜௝ = −
∆క೔
௨ೕ∆ℵమ (10)
݀௜௝ = ܶ௜ିଵ,௝ + ቀ3ܶ௜,௝೚೗೏
ସ
+
ଵ
ସ
‫ܩ‬௜,௝೚೗೏
ቁ
∆క೔
௨ೕே೎ೝ
(11)
The system of equations obtained is solved by a tridiagonal matrix algorithm. A the first step, the
ܶ௜ିଵ,௝ s the inlet temperature.
COMPUTATION CONSIDERATION
The full radiation solution coupled with the convection problem is so computationnaly
intensive. Although the scaling greatly simplifies the radiation solution and reduces the calculation
time required to obtain the radiation contribution, the sudden change in the boundary conditions
causes the problem to be very stiff and presents numerical difficulties regardless of the computation
technique used for the radiation portion of the solution. Because of the stiffness of the problem at the
entrance region, very small axial steps are necessary to ensure acceptable accuracy. The present
work focuses also on reducing the computation time by generating a non-uniform grid in the axial
direction, which is keyed to the pure convection Nusselt number development. This allows for very
small steps in the beginning where all the large changes are taking place. Then, the larger steps
downstream still yield accurate results and significantly reduce the computation time. Using uniform
axial steps, up to 243 seconds of CPU time on a single processor Cray 2 is necessary for stable
solutions. With a non-uniform axial grid and taking advantage of the problem symmetry, the
computation time is reduced to 80 to 100 seconds with comparable accuracy. Some numbers of study
case for the grid generation is shown in table 1.
TABLE 1:
߱ = 1.0 ܰ௖௥ = 0.01 ℵ଴ = 2.0
Uniform grid Non-uniform grid
Non-uniform ½
channel
Entrance step size 0.0001 0.0001 0.0001
Largest step 0.0001 0.0288 0.0288
Downstream position 1.0000 0.9877 0.9877
Number of steps 10000 711 711
Nusselt number 7.5405 7.5401 7.5500
CPU time (cray 2) 243 sec 56 sec 21 sec
RESULTS
The effect of radiation on the heat transfer is shown through a study of the effect of various
parameters. Figure 1 shows the development of the bulk temperature which increases slowly in the
entrance region and then drastically goes to a maximum constant value downstream where the fully
developed condition is reached. The bulk temperature is seen to be higher in the entrance region for
lower albedo values since the medium is then more absorbing. The albedo is seen to have no effect in
the fully developed flow. A plot of the total (convection + radiation) Nusselt number along the axial
direction figure 2 shows first the usual decrease at the entrance region, but then unlike the pure

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 3, March (2015), pp. 14-20© IAEME
18
convection case which reaches a constant limiting value, it passes by a minimum and then sharply
increases to come finally to a limiting value. These results are in agreement with the results
presented by Chawla and Chan [5] for the entry region. The radiation effect dominates past the
minimum point. The minimum occurs earlier and is higher for a less scattering medium. Figure 3
shows the effect of radiation being stronger for smaller conduction-radiation parameter ie conduction
dominating radiation in the transverse direction, and also as the medium is optically thicker.
FIGURE 1: Effect of albedo on bulk temperature
FIGURE 2: Effect of albedo on total Nusselt number

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 3, March (2015), pp. 14-20© IAEME
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FIGURE 3: Effect of cond-rad parameter on total Nu
If a hot fluid enters the channel and is cooled, the scattering is shown to affect the heat transfer in the
entrance region. The effect of radiation vanishes downstream as the fluid is cooled down to finally
reach the fully developed condition. The same limiting solution as the pure convection case is then
reached. This is illustrated in figure 4.
FIGURE 4: Effect of albedo on total Nu (Cooling case)

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 3, March (2015), pp. 14-20© IAEME
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The same behaviour, that is the Nusselt number drastically increases beyond a certain axial
location, and the same effects of the albedo, optical thickness and the conduction-radiation
parameter, is seen in the heat transfer when the walls are a source of heat flux to the fluid which
enters cold. But the thermally fully developed conditions seem to be pushed further downstream.
CONCLUSION
The effect of radiation as it combines with convection is determined for this channel flow.
Unlike a pure convection problem, with a radiatively participating fluide the heat transfer coefficient
is substancially altered downstream before the full thermal development. The effect of radiation is
stronger for lower values of ܰ௖௥ and for higher optical thicknesses. If the walls are cold and the fluid
enters with a uniform hot temperature, the radiation effect vanishes along the axial direction as the
fluid is cooled down. The same trend of the Nusselt number is seen for constant heat flux boundaries,
going by a minimum then dramatically increasing. However the thermally fully developed conditions
are pushed futher downstream and could not be seen.
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