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Multiplicity of solutions due to combined mixed convection and


									       Multiplicity of solutions due to combined mixed convection and
            radiation in a channel discreetly heated from below
                          A. BAHLAOUI*, A. RAJI* and M. HASNAOUI**
                                        Department of Physics
         * Faculty of Sciences and Techniques, Unit of Formation and Research of Chemistry
                           and Environment Sciences, B. P. 523, Beni-Mellal
                   ** Faculty of Sciences Semlalia, Unit of Formation and Research
                       of Thermics and Fluid Mechanics, B. P. 2390, Marrakech

Abstract: - In this work, we present a numerical study of the phenomenon of the mixed convection coupled
with radiation in a channel inclined with respect to the horizontal, discreetly heated by the bottom and subjected
to an external ventilation of fresh air. We study the emissivity effect, , of the walls of the channel on the
multiplicity of solutions. The dynamical and thermal flow structures is examined by presenting typical
streamlines and isotherms for Ra = 105, Re = 10 and  = 30°. The average convective, radiative and total
Nusselt numbers, at the levels of the various walls are calculated and presented for various values of  with an
aim of quantifying the contribution of the radiation to the overall heat transfer. The ratio of the heat quantities
leaving the channel through the cold top wall, Qtop, and through the exit, QE, is also presented in order to better
know the most favorable issue to the heat transfer.

Key-Words: - Mixed convection, radiation, discrete heating, numerical study, multiplicity of solutions.

Nomenclature                                                tot      total
B    aspect ratio of the channel, B  L/H                 T0       dimensionless reference temperature,
cv   convection                                                                    
                                                                      T0 = TC / (TH  TC ) 
Fij  view factor between Si and Sj elements                 u0      velocity of the entering flow, m/s
g    acceleration due to the gravity, m/s2                  u, v     dimensionless horizontal and vertical
 H  height of the channel, m                                        velocities, (u, v)  (u , v ) / u 
Ii                                           4
     dimensionless irradiation, I i  I / TC
                                       i                    x, y     dimensionless coordinates,
        dimensionless radiosity, J i  J  / TC                      (x, y)  (x , y) / H
Ji                                         i
L      length of the channel, m
Nr      convection-radiation interaction parameter,         Greek symbols
         N r   TC H  / (TH  TC )
                                                                thermal diffusivity of the fluid, m2/s
Nu      average Nusselt number                                    thermal expansion coefficient of the fluid,
 Nu     average normalised Nusselt number,                         1/K
         Nu = Nu/Nu( = 0)                                        emissivity of the walls of the channel
Pr      Prandtl number, Pr   /                                 thermal conductivity of fluid, W/mK
Qr      dimensionless radiative heat flux,                        kinematic viscosity of fluid, m2/s
         Q r  Q r / TC 4                                      dimensionless vorticity,    H / u 

Ra      Rayleigh number based on H ,                             dimensionless stream function,
                                                                        / u  H
                            
         Ra  g  (TH  TC )H  3 /                                           0
                                                                  Stéfan-Boltzman constant,
rd      radiation
                                                                     = 5,669 10-8 W/(m2 K4)
Re      Reynolds number, Re  u  H / 
                                                                  angle of inclination of the channel
t       dimensionless time, t  t  u  / H
T       dimensionless fluid temperature,                    Subscripts
                                 
         T  (T   TC ) / (TH  TC )                       b                bottom wall
TC      temperature of the cold wall, K                     CR               critical
                                                           H                heated surface
TH      temperature of the heated elements, K
                                                            I                inlet of the channel
E                exit of the channel
top              top wall

'                dimensional variable

1 Introduction
The study of heat transfer by mixed convection in
inclined channels has received a growing interest          The non-dimensional governing equations, written
during the last decades. This interest is dictated by      in - formulation are:
the role played by such configurations in the field of               1   2  2  Ra   T   T 
                                                               u     v   2  2   2 cos θ  sin            (1)
the habitat, the design of the solar heat collectors and    t    x    y Re  x
                                                                                   y  Re Pr 
                                                                                                x   y 
more recently the cooling of the electronic cards          T    T    T   1   2T  2T 
(because of the tendency to the miniaturization of            u    v                                         (2)
                                                           t    x    y RePr  x 2 y 2 
                                                                                          
the components). An exhaustive review of the
literature shows that the case of the mixed                 2Ψ  2Ψ
                                                                 2 Ω                                            (3)
convection in a rectangular channel was examined           x 2  y
by several authors and the references [1-4] are
quoted only by way of indication. However, the             The stream function and the vorticity are related to
effect of the radiation was often neglected in the         the velocity components by the following
majority of the available studies in spite of its          expressions:
significant contribution to the heat transfer in such          Ψ         Ψ                  v u
                                                            u      , v        and Ω =                   (4)
systems. Its effect was rather taken into account in           y         x                  x y
rectangular closed [5-6] or opened [7-8] geometries
in the case of natural convection. In comparison with
these studies, the case of opened systems with             2.1 Boundary conditions
radiant walls in mixed convection hardly starts to         u=v=0                     on the rigid walls
arouse interest [9].                                       T = v =  = 0, u = 1      at the inlet of the channel
                                                           and  = y
       The objective of this work consists in studying     T=0                       on the top cold wall
the coupling between mixed convection and                  T=1                       on the heated elements
radiation in a rectangular channel by examining the          T
                                                                 NrQr  0          on the adiabatic portions
effect of the emissivity, , of the walls on the             y
temperature distribution and the flow structure
within the channel. A detailed attention will be given     The vorticity on the rigid walls is calculated by
to the contribution of the radiation to overall heat       using the Woods formula [10]:
transfer for different combinations of the governing
parameters.                                                ΩW  
                                                                      Ω W 1 
                                                                                    Ψ W 1  Ψ W                 (5)
                                                                    2          Δη 2

2 Analysis and modelling                                   where, w, denotes the wall and  the space step of
The geometry considered is depicted in Fig. 1. It
                                                           space in the normal direction to the latter.
consists of a channel of finite length, with an aspect
ratio B  L/H = 10, inclined with respect to the         The boundary conditions at the exit of the channel
horizontal and discretely heated from below. The           are obtained by mean of extrapolation technique
top wall of the channel is maintained at a cold            similar to that used in references [1-4].
uniform temperature. The system is submitted to an
imposed flow of ambient air, parallel to the plates.
                                                           2.2 Radiation equations
The flow is considered two-dimensional and
                                                           The calculation of the radiative heat exchange
laminar. The fluid properties are considered
                                                           between the channel and its surrounding (through
constant and the assumption of an incompressible
                                                           the inlet and the exit) is based on the radiosity
fluid obeying the Boussinesq approximation is
                                                           method. In addition, the radiative heat transfer
                                                           between the surfaces is expressed by the following
  y’, v’                                           Exit
                                                           set of equations:
               Cold wall
                                                  x’, u’
                                       Heated section
Inlet                                 Adiabatic portion
                                              T     
                                                                4                                                   Equations (1) and (2) were discretized by using a
J i  (1   i )               Fij J j   i  i  1                                                        (6)   finite difference technique. Hence, all the diffusive
                          sj                   To                                                                 terms were approached by centred differences while
                                                                                                                    a second order upwind scheme was used for the
The radiative heat flux leaving a surface Si is                                                                     convective terms in order to avoid possible
evaluated by:                                                                                                       instabilities frequently encountered in mixed
                                                                                                                  convection problems. The integration of equations
Q r  J i  I i   i ( i  1) 4 
                       To                             F           ij   Jj
                                                                                                              (7)   (1) and (2) was ensured by the Alternate Direction
                                                                          
                                                            j                                                       Implicit method (ADI). At each time step, the
                                                                                                                    Poisson equation was treated by using the point
Since the working fluid (air) is transparent to the                                                                 successive over-relaxation method (PSOR) with an
radiation, the contribution of the latter appears only                                                              optimum over-relaxation coefficient equal to 1.74
in the thermal boundary conditions. Thus, the heat                                                                  for the considered grid. The set of equations
received by radiation on the adiabatic portions is                                                                  described by Eq. (6), representing the radiative
restored to the fluid by conduction according to the                                                                transfer between the different elementary surfaces of
relation:                                                                                                           the channel is solved by using the Gauss-Seidel
    T                                                                                                              method. The numerical code performed was
       Nr Qr  0                                                                                            (8)   validated by comparing the results obtained with
                                                                                                                    those of Yücel et al. [2] in the absence of the
                                                                                                                    radiation effect. The relative difference observed in
2.3 Heat transfer                                                                                                   terms of the Nusselt numbers is lower than 1.6% for
The average Nusselt numbers characterising the                                                                      various combinations of the governing parameters
contributions of the mixed convection and radiation                                                                 (Ra, Re and ).
through the heated wall (bottom), the cooled one
(top), the inlet and the exit of the channel are
respectively defined as:                                                                                            3 Results and discussion
                                                                                                                    For this problem, multiple steady solutions have
                                                                                                                    been obtained. Hence, in Figs. 2a-2c and 3a-3c,
                     1 T
                          B                                                  B
Nu b (cv) = -             
                     B 0 y
                                         dx ; Nu b (rd) =
                                                                       N rQr                       dx        (9)   streamlines and isotherms illustrating two different
                                  y0                                                        y0                    solutions are presented for various values of , Ra =
                                                                                                                    105, Re = 10 and            = 30°. The solution
                     1 T
                                                                                                                    corresponding to Figs. 2 was obtained using a pure
Nu top (cv ) = -
                     B y              dx ; Nu top ( rd ) =
                                                                  NrQr                          dx          (10)   mixed convection solution ( = 0) corresponding to
                            0     y 1                                   0               y 1
                                                                                                                    lower value of Ra as initial condition. Hereafter, this
                                                                                                                    kind of solution will be called as S1 type. The
                      1                                         1

Nu I (cv) = -                          dy ; Nu I (rd) =  N r Q r                           dy              (11)
                                                                                                                    solution presented in Figs. 3, known as S2 and S3
                            x   x 0                           0                    x 0
                                                                                                                    type, was obtained by using the pseudo-conductive
                                                                                                                    regime. Other kinds of initial conditions were also
                1                                                                1                                  tried, but the final state was either one of those
               T             
Nu E (cv ) = 
               
                   + Re Pr u T  dy ; Nu E (rd )  N r Q r
                               x B                                                               dy       (12)   presented in Figs. 2 and 3. Fig. 2a, obtained for  =
            0                                     0                                          x B                   0, shows a complex multicellular structure
                                                                                                                    characterised by the presence of a large cell
For each of the precedent equations, the total                                                                      occupying, from the inlet, a good part of the space
Nusselt number is evaluated as being the sum of the                                                                 offered inside the channel. The effect of radiation
corresponding convective and radiative Nusselt                                                                      leads to the split of this cell to three cells of
numbers.                                                                                                            different sizes when the emissivity reaches a critical
                                                                                                                    value CR1 = 0.32. This behaviour is illustrated in
The heat quantities leaving the channel through the                                                                 Figs. 2b and 2c for two values of  beyond this
cold wall (top), Qtop, and through the exit, QE, are                                                                threshold. The corresponding isotherms are well
respectively defined as:                                                                                            marked by the multicellular behaviour of the flow.
            B                              1
                T                              T                                                                The effect of the emissivity is also important in the
Q top        y
                          dx ; Q E          x  Re Pr u T  N Q              r   r              dy   (13)   case of S2 and S3 solutions presented in Fig.3. In
            0        y 1                  0                                                     x B
                                                                                                                    fact, significant changes are observed in the flow
                                                                                                                    structure when  is increased. Hence, in Fig. 3a,
2.4 Method of solution
                                                                                                                    obtained for  = 0, the flow structure is characterised
by the presence of a closed cell occupying almost
the totality of the space offered in the channel (S3
solution). The corresponding isotherms indicate an
important heat exchange between the cell and the
end part of the cold plate, at the vicinity of the exit.
However, this behaviour changes when the effect of
the radiation is considered (Figs. 3b and 3c). The
closed cell (S3 solution) is replaced by a reversal
flow (S2 solution) more favourable to the evacuation
of heat through the exit of the channel (for >0.1).
The isotherms examination shows a very limited
heat transfer through the cold wall. In fact, the flow
admitted through the exit, being at the ambient
temperature, enters by the upper part of the channel
and skirts the cold wall of the latter. It is clear to
deduce from the flow structure that the S1 solution is
more favourable than the S2 and S3 solutions to the
heat evacuation through the cold surface while the
opposite is true concerning the heat evacuation
through the exit.

                                                           Fig. 3: Streamlines and isotherms obtained for Re =
                                                                   10, Ra = 105,  = 30° and various values of
                                                                   the emissivity of the walls (using the
                                                                   pseudo-conductive regime): a)  = 0, b)  =
                                                                   0.5 and c)  = 0. 8.
                                                           Variations, with , of the normalised Nusselt
                                                           numbers (with respect to pure mixed convection)
                                                           evaluated on the bottom wall are presented in Fig. 4
                                                           for the two flow structures previously defined. A
                                                           close examination of this curve shows that the
                                                           emissivity effect on the mixed convection Nusselt
                          b)                               number is very limited while it is significant on the
                                                           radiation heat transfer rate which increases almost
                                                           linearly with . Consequently, the total Nusselt
                                                           number undergoes a linear increase with this
                                                           parameter. A comparison of the mixed convection
                                                           and radiation contributions to the overall heat
                                                           transfer shows that the contribution of the mixed
                          c)                               convection remains more important as long as  is
                                                           lower than a critical value which about 0.53/(0.43) in
Fig. 2: Streamlines and isotherms obtained for Re =        the case of S1/(S2 and S3). It is useful to note that, in
                                                           general, the reversal flow (S2) is more favourable to
        10, Ra = 105,  = 30° and various values of
                                                           the heat transfer in comparison with the multicellular
        the emissivity of the walls (using a pure
                                                           flow structure (S1). The variations of the average
        mixed convection solution): a)  = 0, b)  =
                                                           Nusselt numbers with , evaluated at the inlet, are
        0.5 and c)  = 1.
                                                           presented in Fig. 5. It is clear that the contribution of
                                                           the mixed convection to the overall heat transfer is
                                                           negligible and essentially due to the cells which
                                                           extend until the inlet of the channel. However, the
                                                           contribution of the radiation is important and its
                                                           importance increases by increasing the value of the
                                                           emissivity. Finally, it is to note that the overall heat
transfer at the inlet is independent of the type of      increases quickly when  is increased towards its
solution which is expected since the latter depends      critical value for which a maximum is reached
mainly on the radiation effect.                          followed by a linear decreasee, resulting from the
                                                         big change undergone by the flow structure (passage
      3.5                                                from monocellular (S3) to reversal flow (S2)). The
      3.0         S1 Solution                            radiative transfer varies in a monotonous way with
                  S2 and S3 Solutions                    the emissivity. A comparison between the
      2.5         *
                    Nub(tot)                             convective and radiative heat transfer quantities,
*     2.0   *
  Nub         Nub(rd)                                    evaluated at the level of the cold wall (Fig. 6) shows
                                                         that the contribution of radiation becomes greater
      1.0                                                than that of the mixed convection when the
      0.5                    *                           emissivity value is higher than 0.1/(0.58) in the case
      0.0                                                of a monocellular and reversal/(multicellular) flow.
        0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0      However, the contribution of the mixed convection
                                                        to the overall heat transfer is always higher than that
                                                         of the radiation at the exit of the channel (Fig. 7) for
Fig. 4 : Variation of the average normalized Nusselt     the three flow structures.
         number on the bottom wall with the                    3.5
         emissivity of the walls for Ra = 105, Re =                       S1 Solution                        total
         10 and  = 30°.                                                  S2 and S3 Solutions
                                                               2.5                                       radiation
     2.0                                                 *     2.0
               S1 Solution       NuI(tot)                  Nutop
               S2 and S3 Solutions                             1.5                            radiation
     1.5                                                                        total                   convection
                                                                    S3 S2
NuI 1.0                                                        0.5                          convection
                       NuI(rd)                                 0.0
                                                                 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
     0.5               NuI(cv)
     0.0                                                 Fig. 6: Variation of the average normalized Nusselt
       0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0               number on the top wall with the emissivity
                                                                of the walls for Ra = 105, Re = 10 and  =
Fig. 5: Variation of the average Nusselt number at
        the inlet of the channel with the emissivity                S3 S2                               S1 Solution
        of the walls for Ra = 105, Re = 10 and  =             3.0                                      S2 and S3
                                                               2.5           convection                 Solutions
                                                           NuE 2.0                      total
Variations, with , of the normalised Nusselt                  1.5
numbers on the top wall and the exit of the channel            1.0                    convection
are presented in Figs. 6 and 7. In the case of S1
solution (solid line), Fig. 6 shows that the amount of         0.5 radiation                  radiation
heat evacuated by convection through the cold                  0.0
                                                                 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
surface is insensible to the variations of . However,                                   
in the case of the dashed line, the increase of the
emissivity of the walls leads first to the decrease of   Fig. 7: Variation of the average normalized Nusselt
the convection effect due to the change of the flow              number at the outlet of the channel with the
structure which passes progressively from a                      emissivity of the walls for Ra = 105, Re =
monocellular flow (S3 solution) to a reversal one (S2            10 and  = 30°.
solution) when  reaches the value with CR2  0.1.
Above this critical value of , the convection effect    In such problems, it is useful to attach a special
becomes constant and practically negligible (Nut(cv)     importance to the quantities of heat leaving the
 0.25). On the other hand, the examination of Fig.      system through the cold surface and the exit in order
7, shows that, for the same solution, NuE(cv)            to quantify how the energy provided by the heating
                                                         elements is distributed between these two issues.
The quantification of the relative amount of heat          and structure fields. The radiative and total heat
evacuated by each one of these issues to the total         quantities evacuated through the cold wall and the
heat transfer is presented in Fig. 8 in terms of the       exit of the channel increase by considering the effect
variations, with the emissivity, of the ratio Qtop/QE.     of radiation, while the latter contributes to weaken
In the absence of the radiation effect ( = 0), the        the mixed convection at these issues.
flow structure is characterised by the presence of a
big cell. The corresponding flow structure is              References:
denoted as a solution of S3 type (dashed line). An         [1] T. Tomimura, and M. Fujii, Laminar Mixed
increase of the emissivity supports the appearance             Convection Heat Transfer between Parallel
of the return flow (S2 solution) which generates a             Plates with Localized Heat Sources, Proc. Int.
reduction in the quantity of the heat received by the          Symp. on Cooling Technology for Electronic
cold wall in favour of the exit (Qtop/QE decreases).           Equipement, Honolulu, Aung, W. ed, 1988, pp.
For  > 0.1, the flow structure remains qualitatively          233-247.
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only from the increase of the radiative heat transfer          Bilgen, Mixed Convection Heat Transfer in
at the level of the top wall. In the case of S1 solution       Open Ended Inclined Channels with Discrete
(solid line), the tendency of variation of the ratio           Isothermal Heating, Num. Heat Transfer, Vol.
Qtop/QE is very different. It shows that the heat              24, 1993, pp. 109-126.
generated by the heated elements is mainly                 [3] A. Raji, M. Hasnaoui and Z. Zrikem, Convection
evacuated through the cold top wall because of the             Mixte dans un Canal Incliné de Longueur Finie
significant effect of the mixed convection which is            contenant des Obstacles et Chauffé de Manière
supported by the multicellular flow structure. Also,           Isotherme et Discrète, Revue Générale de
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Fig. 8: Variation of the ratio Qtop/QE with the                Natural Convection-Conduction and Radiation
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       10 and  = 30°.                                         Cavity, ASME Transactions, Vol. 118, 1996, pp.
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4 Conclusion                                                   Convection and Radiation in Ventilated
In this work, the problem of mixed convection                  Cavities, Engineering Computations, Int. J. for
coupled with radiation in a channel discretely heated          Computer-Aided Engineering and Software,
from below has been investigated numerically. The              Vol. 18, No. 7, 2001, pp. 922-949.
taking into account of the effect of radiation             [10] L. C. Woods, A Note on the Numerical
contributes to the multiplicity of the solutions. The          Solution of Fourth Order Differential Equations,
obtained results show the existence of three                   Aero. Quart, Vol. 5, 1954, pp. 176-184.
solutions (multicellular S1, reversal flow S2 and
monocellular S3). The radiation heat exchange was
found to have a significant influence on the thermal

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