CC BY
8NUMERICAL STUDY OF MIXED CONVECTION COUPLED WITH RADIATION IN A VENTED PARTITIONED ENCLOSURE
A. Bahlaoui*, A. Raji*, M. Hasnaoui**, R. El Ayachi*, M. Naimi*,
T. Makayssi*, M. Lamsaadi*
*Faculty of Sciences and Technologies, Department of Physics, University Sultan Moulay Sliman, Team of Flows and Transfers Modeling (EMET), Laboratory of Physics and Mechanics of Materials, BP 523, Beni-Mellal, Morocco Tel.: 00212 23 48 51 12; Fax: 00212 23 48 52 01, E-mail: abderaji@fstbm.ac.ma; abderaji@yahoo.fr
"Faculty of Sciences Semlalia, Department of Physics, University Cadi Ayyad, UFR TMF, BP 2390 Marrakesh, Morocco Tel.: 00212 44 43 46 49; Fax: 00212 44 43 74 10, E-mail: hasnaoui@ucam.ac.ma
Received: 27 Sept 2007; accepted: 29 Oct 2007
The present work reports numerical results of mixed convection and surface radiation within a horizontal ventilated cavity, with an aspect ratio A = L'/H' = 2, heated from below and provided with an adiabatic partition, of a fine thickness, on the heated surface. Air, a radiatively transparent medium, is considered to be the cooling fluid. The effect of the governing parameters, which are the Reynolds number, 200 < Re < 5000, the partition position from the inlet, 0.25 < Lb < 1.75, and the emissivity of the walls, 0 < e < 0.85, on the fluid flow and heat transfer characteristics is studied in detail. The relative height of the partition, Hb = H'b/H', and the relative height of the openings, B = h'/H', are kept constant at 1/2 and 1/4 respectively.
Keywords: mixed convection, surface radiation, ventilated cavity, adiabatic partition
Organization: A teacher at the Faculty of Sciences and Technics in Beni Mellal (Morocco). Education: Doctorates in heat transfer (1994 and 2000) after attending Cadi Ayyad University. The title of the first doctorate: "Etude numérique des écoulements et des transferts de chaleur par convection dans des cavités en interaction et dans un canal de longueur finie". The title of the second thesis: "Etude numérique du phénomène de la convection mixte dans des cavités ventilées avec et sans effet de rayonnement".
Experience: Head of the physics department from 2000 to 2003.
Main range of scientific interests: Heat transfer by conduction, convection, radiation, porous media... Publications: from 2004: - Coupling between mixed convection and radiation in an inclined channel locally heated. Journal of Mechanical Engineering, 2004. - Interaction between natural convection and radiation in a square cavity heated from below. Numerical Heat Transfer, Part - A: Applications, 2004.
- Multiple steady state solutions resulting from coupling between mixed convection and radiation in an inclined. Heat and Mass Transfer, August 2005. - Combined effect of radiation and natural convection in a rectangular enclosure discreetly heated from one side. International Journal of Numerical Methods for Heat & Fluid Flow, 2006. - Multiplicité de solutions en convection naturelle couplée au rayonnement dans une cavité horizontale, Physical & Chemical News, 2006. - Mixed convection in a horizontal channel with emissive walls and partially heated from below. Numerical Heat Transfer, Part
- A: Applications, 2007. - Multiple steady state solutions for natural convection in a tilted rectangular slot containing non-Newtonian power-law fluids and subject to a transverse thermal gradient. Numerical Heat Transfer, Part - A: Applications, 2007. - Parallel flow convection in a shallow horizontal cavity filled with non-Newtonian power-law fuids and subject to horizontal and vertical uniform heat fluxes2. In press in Numerical Heat Transfer, Part - A: Applications, 2007. - Coupled natural convection and radiation in a horizontal rectangular enclosure discretely heated from below. Accepted in Numerical Heat Transfer, Part - A: Applications, 2007.
Education: A diploma of higher studies in Fluid Mechanics and Energetics (2000) of the Faculty of Sciences Semlalia, Marrakech, Morocco and a National Doctorate in Fluid Mechanics and Heat Transfer of the Faculty of Sciences and Technologies (F.S.T.), Béni-Mellal, Morocco (2006). Organization: A teacher in the F.S.T., Béni-Mellal, Morocco, during the period 2004-2007. Member of the Team of Flows and Transfers Modeling (EMET) in the F.S.T., Béni-Mellal, Morocco. The principal publications which I carried out are those cited above.
Abdelghani Raji
Ahmed Bahlaoui
HI
131
Nomenclature
A - aspect ratio of the cavity (= L' / H') B - relative height of the openings (= h'/ H') cv - convection
Fj - view factor from St surface to Sj one g - acceleration due to gravity, m/s2 h - height of the openings, m H - height of the cavity, m Hb - relative height of the partition (= H'b / H')
I - dimensionless irradiation (= I'/oT' 4) J - dimensionless radiosity (= J'/oT'4) L ' - length of the cavity, m
Lb - dimensionless x-direction distance of the partition from the inlet (= L'b / H')
Nr - convection-radiation interaction parameter (= oT'4 / q')
Nu - average Nusselt number
Pr - Prandtl number (= v /a)
q' - imposed wall heat flux, W/m2
Qr - dimensionless radiative heat flux (= Q'r /oT'4)
Ra - Rayleigh number (= g p q'H'4/avA,)
rd - radiation
Re - Reynolds number (= u'H' / v)
t - dimensionless time (= t'u'0 /H')
T - dimensionless fluid temperature (= A(T' - T')/ q' H')
T/- dimensional fluid temperature, K
T - dimensionless mean temperature
Tmax - dimensionless maximum temperature
T' - common temperature of the left vertical cold wall and
imposed flow, K
T0 - dimensionless reference temperature (= AT' / q' H') u0 - velocity of the imposed flow, m/s (u, v) - dimensionless horizontal and vertical velocities (=(uv')/ u0)
(x, y) - dimensionless coordinates (= (x', y') / H')
Greek symbols
a - thermal diffusivity of fluid, m2/s P - thermal expansion coefficient of fluid, 1/K At - dimensionless time step e - walls emissivity
A - thermal conductivity of fluid, W/(K-m) v - kinematic viscosity of fluid, m2/s Q - dimensionless vorticity (= fl' H' / u0) T - dimensionless stream function (= ^'/u'0 H') o - Stefan-Boltzman constant (= 5.67-10-8 W/m2-K4)
Subscripts and Superscripts
' - cold temperature H - heated wall max - maximum value min - minimum value ' - dimensional variable
Introduction
Mixed convection heat transfer in ventilated systems continues to be a fertile area of research, due to the interest of the phenomenon in many technological processes, such as the design of solar collectors, thermal design of buildings, air conditioning and recently the cooling of electronic circuit boards. In the literature, numerous analytical, numerical and experimental studies dealing with mixed convection in ventilated geometries have been reported without radiation effect. The effect of the latter can be neglected in the case of configurations with non emissive or weakly emissive boundaries which is not the case in general since the contribution of radiation to the overall heat transfer could be significant. In the absence of radiation, mixed convection in a square enclosure provided with a partially dividing partition was studied numerically by Hsu et al. [1], How and Hsu [2] and Hsu and Wang [3]. Results of the simulations indicate that the heat transfer and flow structure are strongly dependent on the height, the conductivity ratio and the location of the conducting baffles. Laminar mixed convection in a two-dimensional enclosure with assisting and opposing flows was studied numerically by Raji and Hasnaoui in the case of a cavity uniformly heated from one or two side walls [4-6]. The obtained results show that the Re-Ra plane can be divided in regions corresponding to the dominance of the forced convection or to the mixed convection regime where the heat transfer is maximum. Recently, mixed convection from a flush-mounted uniform heat sources in a rectangular enclosure with openings was numerically investigated by Bhoite et al [7]. and Saha et al. [8]. In the case of ventilated cavities, the numerical study, conducted by Raji and Hasnaoui [9] on combined mixed convection and radiation, showed that the contribution of radiation could be important even though the cooling fluid is transparent to radiation. The neglected effect of thermal radiation is mainly justified by the fact that the heat transfer is especially ensured by mixed or forced convection. However, moderate temperature differences give rise to significant radiation effects and the fact of neglecting their contribution becomes non realistic. The main objective of the present study consists of examining the effect of the Reynolds number, Re, the horizontal position of partition, Lb, and the emissivity of the walls, e, on flow and thermal fields. Variations, versus the main controlling parameters, of maximum and mean temperatures are also explored.
Problem formulation
The configuration under study, together with the system of coordinates is depicted in Fig. 1. It consists of a ventilated rectangular cavity. The bottom wall is uniformly heated with a constant heat flux and provided with a vertical adiabatic baffle. The upper horizontal and right vertical walls are considered insulated, while the
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
left side of the cavity is cooled with a constant temperature. The system is submitted to an imposed flow of ambient air through an opening located on the lower part of left vertical wall. The forced flow leaves the cavity through an outflow opening placed on the higher part of the right vertical wall. The inner surfaces, in contact with the fluid, are assumed to be gray, diffuse emitters and reflectors of radiation with identical emissivities. The fluid properties are evaluated at a mean temperature and the airflow is assumed to be, two-dimensional, laminar, incompressible and obeying the Boussinesq approximation. Under these assumptions, the dimensionless governing equations, written in terms of vorticity and stream function formulation, are as follows:
ЭО дО dO 1 — + u—- + V—- = —
dt dx dy Re
д2О д2О
+
_ dx2 Эу2 _
Ra
Re2 Pr
dT
dx
1
dT dT dT "Т" + u^T + V^- = ——— dt dx dy Re Pr
d2T d2T dx2 + Эу2
"Г, (1)
(2)
Э2Т
Э2Т
dx2 dy
= -О.
(3)
The stream function and the vorticity are related to the velocity components by the following expressions:
ЭТ ЭТ dv du
u = —— , v = - —— and О = — - —. dy dx dx dy
(4)
Cold wall.
Air jet
/////////////////////////
y , v
и
h '
Adiabatic
Hb
, Heat flux q' Fig. 1. Schematic of the studied configuration
H
+
L
u
L
Boundary conditions The boundary conditions, associated to the problem are as follows: u = v = 0 on the rigid walls; T = v = Q = 0, u = 1 and T = y at the inlet of the cavity; T = 0 on the
dT
left vertical cold wall; -3"+ NrQr = 1on the lower
dy
horizontal heated
dT
wall; - — + NrQr = 0 on the dn
Ji = Ei
A4
T
-+1
iV
(1 -Ei ) X Fj Jj
(5)
j=i
The dimensionless net radiative heat flux leaving an element of surface Sj is evaluated by:
Qr = J, - I, =E,
( Th +1)4 -X Fj Jj
adiabatic walls; "n" being the normal direction to the considered adiabatic wall.
For this problem, the boundary conditions are unknown at the outflow opening. Values of u, v, T, T and Q are obtained at each time step by mean of an extrapolation technique [9-11].
Radiation equations
The calculation of the radiative heat exchange between the cavity and its surrounding (through the inlet and the exit) is based on the radiosity method. In addition, the radiative heat transfer between the system surfaces is expressed by the following set of equations in non-dimensional form:
j=1
(6)
Heat transfer
The average Nusselt numbers, characterizing the contributions of mixed convection and thermal radiation through the heated wall, are respectively defined as:
NuH (cv) = -
I f -1 i^T
A0 Tlay.
dx ;
y=o
1
Nuh (rd) = A f T [NrQr
■Í1 (Nr'
dx.
(7)
y=0
The total Nusselt number, Nu, is evaluated as being the sum of the corresponding convective and radiative Nusselt numbers, i.e. Nu = Nu(cv) + Nu(rd).
+
0
Method of solution
The non linear partial differential governing equations, Eq. (1)-(3), were discretized using a finite difference technique. The first and second derivatives of the diffusive terms were approached by central differences while a second order upwind scheme was used for the convective terms to avoid possible instabilities frequently encountered in mixed convection problems. The integration of equations (1) and (2) was ensured by the Alternating Direction Implicit method (ADI). At each time step, the Poisson equation, Eq. (3), was treated by using the Point Successive Over-Relaxation method
Effect of Ra and e on the mean convective Nuss of a square cavity for TH =
(PSOR) with an optimum over-relaxation coefficient equal to 1.88 for the grid (81x41) adopted in the present study. The set of Eq. (5), representing the radiative heat transfer between the different elementary surfaces of the cavity, was solved by using the Gauss-Seidel method. The numerical code was validated against the results of Akiyama and Chong [12] obtained in the case of a square cavity differentially heated. Comparisons, made in terms of convective Nusselt numbers, evaluated at the heated wall, showed a fairly good agreement with relative maximum deviations limited to 1.07 %/(1.36 %) for e = 0 / (1) for Ra varying in the range 103 < Ra < 106 (Table 1).
Table 1
elt number, Nucv, evaluated on the heating wall 298.5 K and TC = 288.5 K
e = 0 e = 1
Ra 103 104 105 106 103 104 105 106
Present work 1.118 2.257 4.627 9.475 1.250 2.242 4.192 8.100
Akiyama and Chong [12] 1.125 2.250 4.625 9.375 1.250 2.250 4.250 8.125
Results and discussion The effect of radiation on the flow structure and
temperature distribution inside the cavity is illustrated in
In the present study, the value of Rayleigh number, Ra, Fig. 2 for Re = 250 and various values of e. is fixed at 5-106.
a (fbiin = -0.25, fbiax = 0.28)
b) (Vmin = -0.22, lFmax = 0.27)
c) (Tmin = -0.17, Tmax = 0.27) Fig. 2. Streamlines and isotherms obtained for Re = 250, Lb = 1 and different values of e: a) e = 0, b) e = 0.15 and c) e= 0.85
The analysis of the streamlines in Fig. 2, a, obtained for e = 0, reveals the existence of open lines surmounted by a trigonometric cell whose formation is rather due to the shearing effect and a clockwise natural convection cell (its direction of rotation is imposed by the forced flow) under the open lines, located on the right of the baffle. The corresponding isotherms are tightened at the level of the heating wall indicating a good convective heat exchange between this wall and the open lines/(closed cell) on the left/(right) side of the baffle. An important heat exchange can also be seen between the lower cell and the open lines while this interaction is quasi absent between the forced flow and the upper cell. Consequently, the cold zone occupies a non negligible part of the available space on the left part of the fin (in the entrance region of the cavity), visibly reduced by increasing the emissivity. The reduction of the cold zone space, following the increase of e, indicates that the wall's radiation plays an important role in the homogenization of the fluid temperature inside the cavity. In addition, the increase of the emissivity reduces the importance of the upper cell (size and intensity) in favor of the open lines (Fig. 2, b, c) and leads to an increasing spacing of the isotherms in regions where the thermal interaction is important in the absence of radiation. Moreover, when the inner surfaces are radiatively participating, a heating of the adiabatic upper wall is observed and presents increasingly significant thermal gradients as the emissivity increases. Variations, versus Re, of the average Nusselt numbers, resulting from contributions of convection and radiation and the total Nusselt number, evaluated along the heated wall, are presented in Fig. 3, a-c for various values of e. As expected, Fig. 3, a shows a monotonous increase of Nuh(cv) with Re either with or without radiation effect. The rate of this increase becomes more important from Re ~ 1000 (an increase in the slope of the curves is observed from this threshold). This tendency is justified by the flow intensification with the inertia effect, promoted by the increase of Re. For a fixed value of this parameter, the increase of the emissivity of the walls leads to a noticeable decrease of the convection effect. The negative role of radiation on the natural convection is well known and it is confirmed here in the case of mixed convection. The effect of the emissivity of the walls on the radiative heat transfer component is presented in Fig. 3, b in terms of NuH(rd) variations with Re for e = 0.15, 0.5 and 0.85. Globally, it can be deduced that, for a given value of Re, the effect of radiation is important for e = 0.5 and 0.85 and it is characterized by an important increase of NuH(rd) with e. Also, the effect of Re on NuH(rd) is limited in the case of e = 0.15 but becomes increasingly positive by increasing e. The variations of the total Nusselt number with Re, presented in Fig. 3, c, show increasing tendencies of NuH with Re and e which means that the positive impact of radiation on the radiative Nusselt number is more important than its negative effect on the convective Nusselt number.
- e = 0 • Periodic solutions
- - e = 0.15 '
e = 0.5
e = 0.85 _____\ .
a)
200 500 1000 2000 5000
Re
--e = 0.15
■...--■"" -----e = 0.5
......... e = 0.85
b)
200 500 1000 2000 5000
Re
Re
Fig. 3. Variations, with Re, of the average Nusselt numbers on the heated wall for Lb = 1 and various values of e: a) NuH(cv), b) NuH(rd) and c) NuH
For practical applications, it is of great importance to know the impact of the governing parameters on mean and maximum temperatures of the fluid inside the cavity. Thus, the variations of these quantities with Re are presented in Figs. 4, a, b for different values of e. For all considered values of e, Fig. 4, a shows that the evolution
of T is characterized by a continuous decrease by increasing Re. In addition, for a given Re, the increase of e is accompanied by a decrease of the average temperature since the part of energy provided by the hot wall and leaving directly the cavity through the openings, without being transported by the fluid, increases with e. The evolution of the maximum temperature, presented in Fig. 4, b and generally located on the heated wall, is marked also by a monotonous decrease with Re and e which means that the overheating phenomenon can be avoided by increasing the emissivity of the walls and the velocity of the external imposed flow. Quantitatively, for Re = 200, the increase of the walls emissivity from zero to 0.85 generates a decrease in mean and maximum dimensional temperatures by
about 7.25 °C and 60.83 °C, respectively but this decrease drops to about 1.37 °C (case of e = 0) and 35.28 °C (case of e = 0.85) for Re = 5000.
Periodic solutions
0.00 200
500
1000 Re
2000
5000
5000
b
Fig. 4. Variations, with Re, of the temperature for Lb = 1 and various values of e: a) mean temperature T , b) maximum temperature Tmax.
Variations of the Nusselt numbers, with the position Lb of the baffle, are presented in Fig. 5 for Re = 300 and different values of e. Fig. 5, a shows that the convective component of heat transfer increases by moving away the partition from the cold wall but the rate of this increase becomes limited beyond Lb = 1.5; critical value from which the reduction of the space between the partition and the adiabatic vertical wall leads to a limited interaction between the closed cell and the open lines. Fig. 5, b shows that the increase of Lb is characterized by a limited decrease of NuH(rd) what means that the latter is almost independent of Lb. However, the figure shows clearly an increasing positive effect of radiation when e is increased leading to an enhancement of the total heat transfer (Fig. 5, c) and the latter is favored also by the increase of Lb. Improvements in terms of the total heat transfer of about 28.2 % and 18.1 % are obtained respectively for e = 0 and 0.85, when the parameter Lb is varied from 0.25 to 1.75. It should be mentioned that the solution is unsteady for Lb = 1.5. The corresponding results, presented by full circles in Figs. 5, a-c, were obtained as averaged values during flow cycles.
Nu„(cv)
1.75
Nu„(rd) 4
e = 0.15 e = 0.5 ......... e = 0.85
0.25
0.50
0.75
1.00 Lb
1.25
1.50
1.75
Fig. 5. Variations, with Lb, of the average Nusselt numbers on the heated wall for Re = 300 and various values of e: a) NuH(cv), b) NuH(rd) and c) NuH
Variations, with Lb, of mean and maximum temperatures are presented in Figs. 6 for Re = 300 and different values
of e. It is seen from Fig. 6, a that T decreases by increasing e and Lb. In fact, the increase of the overall heat transfer with both e and Lb (Fig. 5, c) and the mixed convection heat transfer component with Lb (Fig. 5, a), contributes to a better cooling within the cavity. It should be noted that for Lb > 1.5, the effect of the emissivity e
on T becomes limited. The variations of the maximum temperature, Tmax, reported in Fig. 6, b, show a decrease of this parameter by increasing e or Lb with a drastic decrease from Lb = 1.25, due to the competition between natural and forced convections, towards a minimum reached at Lb = 1.5, position for which the maximum of the overall heat transfer is reached.
a
a
b
c
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
0.07
0.05
0.03
0.01
■ • Periodic solutions - e = 0
e = 0.15
e - 0.5
e = 0.85
........
a)
0.25 0.50 0.75 1.00 1.25
Lb
1.50
1.75
0.30 0.25 Tm,s 0.20 0.15 0.10
- e = 0 b)
e = 0.15
e = 0.5
------ -- . e = 0.85
•---"
0.25 0.50 0.75 1.00 1.25 1.50 1.75
Lb
b
Fig. 6. Variations, with Lb, of the temperature for Re = 300 and various values of e: a) mean temperature T , b) maximum temperature Tmax
Concluding remarks
The problem of combined mixed convection and radiation inside a partitioned ventilated cavity has been studied numerically. Results of the study show that the radiation effect leads to a better homogenization of the temperature inside the cavity by reducing the cold zone space in the entrance region. It is found that the radiation effect reduces the convective Nusselt number component and the latter is favored by the Reynolds number, Re, and the displacement of the partition away from the inlet, Lb. All the parameters Re, Lb and e are found to have a positive effect on the total heat transfer. The better cooling of the cavity, expressed by the decrease of mean and maximum temperatures of the fluid, is obtained by the increase of the parameters Re, Lb and e. Also, the contribution of radiation to the overall heat transfer is generally not negligible even for the weaker value (e = = 0.15) considered for e.
References
1. Hsu T.H., Hsu P.T., How S.P. Mixed convection in a partially divided rectangular enclosure // Num. Heat Transfer. Part A. 1997. Vol. 31. P. 655-683.
2. How S.P., Hsu T.H. Transient mixed convection in a partially divided enclosure // Comput. Math. Appl. 1998. Vol. 36. P. 95-115.
3. Hsu T.H., Wang S.G. Mixed convection in a rectangular enclosure with discrete heat sources // Num. Heat Transfer. Part A. 2000. Vol. 38. P. 627-652.
4. Raji A., Hasnaoui M. Mixed convection heat transfer in a rectangular cavity ventilated and heated from the side // Num. Heat Transfer. Part A. 1998. Vol. 33. P. 533-548.
5. Raji A., Hasnaoui M. Corrélations en convection mixte dans des cavités ventilées // Revue Générale de Thermique. 1998. Vol. 37. P. 874-884.
6. Raji A., Hasnaoui M. Mixed convection heat transfer in ventilated cavities with opposing and assisting flows // Engineering Computations. Int. J. for Computer-Aided Engineering and Software. 2000. Vol. 17. P. 556-572.
7. Bhoite M.T., Narasimham G.S. V.L., Krishna Murthy M.V. Mixed convection in a shallow enclosure with a series of heat generating components // Int. J. Thermal Sciences. 2005. Vol. 44. P. 121-135.
8. Saha S., Saha G., Ali M., Quamrul Islam M. Combined free and forced convection inside a two-dimensional multiple ventilated rectangular enclosure // ARPN Journal of Engineering and Applied Sciences. 2006. Vol. 1, No. 3. P. 23-35.
9. Raji A., Hasnaoui M. Combined mixed convection and radiation in ventilated cavities // Engineering Computations. Int. J. for Computer-Aided Engineering and Software. 2001. Vol. 18. P. 922-949.
10. Bahlaoui A., Raji A., Hasnaoui M. Coupling between mixed convection and radiation in an inclined channel locally heated // Journal of Mechanical Engineering. 2004. Vol. 55. P. 45-57.
11. Bahlaoui A., Raji A., Hasnaoui M. Multiple steady state solutions resulting from coupling between mixed convection and radiation in an inclined channel // Heat and Mass Transfer. 2005. Vol. 41. P. 899-908.
12. Akiyama M., Chong Q.P. Numerical analysis of natural convection with surface radiation in a square enclosure // Num. Heat Transfer. Part A. 1997. Vol. 31. P. 419-433.
oo
-TATA —
oo
a
