CC BY
18NUMERICAL STUDY OF THE EFFECT OF THE INSULATION OF THE WALLS ON NATURAL CONVECTION IN A DIFFERENTIALLY HEATED
CAVITY
A. Raji*, M. Hasnaoui**, S. Goujon-Durand***, P. Vasseur****
*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, Béni-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
***Université Paris XII, France
****Ecole Polytechnique de Montréal, Canada
Received: 27 Sept 2007; accepted: 31 Oct 2007
In this work, we study numerically the natural convection in a square cavity using the Navier-Stokes equations with the Boussinesq approximation. A finite difference method based on the control volume notion is used to resolve, in nondimensional forms, the natural convection equations. The results are obtained for various combinations of the parameters of base, such as the Rayleigh number (103 < Ra < 5-106), the relative thermal conductivity (0.1 < K < < 100), the relative width of the walls (0 < e < 0.2) and the relative spacing (0.05 < espa < 0.3). The numerical simulations of natural convection in a cavity having walls of finite width have shown the possibility to reduce significantly the heat transfer by using appropriate isolation techniques.
Keywords: natural convection, numerical study, effect of the insulation
Organization: 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 was: "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 was: "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 fluids and subject to horizontal and vertical uniform heat fluxes. 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.
Abdelghani Raji
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
Nomenclature
A -aspect ration of the cavity, H IL '
e -relative width of the wall, lIL'
ens — relative width of the insulating layer, l 'n/L '
epa — spacing between the walls, l /!pa /L '
g — acceleration due to gravity, m/s2
h —mean convection heat transfer
K — relative conductivity, X/Xf
l — dimensional width of the wall, m
l ins — dimensional width of the insulating layer, m
l spa — dimensional spacing between the walls, m
P — dimensionless pressure, (P '+ p0 gy )/p0(a/L )2
Pr — Prandtl number, v/a
Ra — Rayleigh number, gPAT'L /3/(va)
rw —relative conductivity of the wall, Xw /Xf
rins — relative conductivity of the insulator, Xins/Xf
AT' —temperature difference, (TH - TC)
(u, v) —dimensionless horizontal and vertical velocities,
(u, v )/ (a/L )
(x, y) —dimensionless Cartesian coordinates, (x', y '/L')
Greek letters
a — thermal diffusivity (m2-s—') P —thermal expansion coefficient of the fluid (K—') r — diffusion coefficient X — thermal conductivity (W-m—'-K—') v —kinematic viscosity of fluid (m2-s—') y —dimensional stream function (m2-s—') T — dimensionless stream function, y/a 6 —dimensionless temperature, 6 = (T ' - T C)/ (T H - T C) p0 —fluid density at the temperature T0 (kg-m—3)
Subscripts and Superscripts
board — boarding C — cooled cond — conduction f—fluid H — heated ins — insulator max — maximum r —relative
s — solid spa — spacing w — wall — dimensional variables
Introduction
During the last decades, natural convection in closed cavities has received great efforts of research. This special attention was dictated by the importance of this phenomenon in the design of the physical models related to the field of solar engineering, the cooling of the printed circuit boards and the design of buildings. Hence in several practical applications, the use of materials with high thermal conductivity will lead to an undesirable increase of the temperature of the fluid inside the cavity. The maintenance of a desirable temperature can be achieved if an adequate insulation of the walls of the cavity is taken into account. Heat exchange between the walls of the cavity and the fluid can be controlled by recourse to thermally insulating materials. A problem of base well documented in the literature specialized in heat transfer is that of a rectangular enclosure without partition. An exhaustive review on this subject is presented in references [1-4]. Compared to the studies relating to rectangular cavities, those concerning partially or completely partitioned cavities are relatively less documented. It is obvious that the addition of one or more partitions has a considerable effect on the heat transfer and the fluid flow structure inside the cavity. Thereafter, the subject needs to be further explored and other works are necessary to improve our knowledge in this field by taking account of the influence of partitions; especially if it is known that their dimensions, their conductivities, their number and the way in which they are placed are among the parameters which control the importance of this influence in various technological aspects [5-10].
The main goal of this study consists in numerically simulating the two-dimensional natural convection in a differentially heated cavity whose vertical walls have a finite conductivity and a thickness. The effect of the insertion of an insulating material on the internal surfaces of the walls will be also investigated in parallel with the technique of the thermal boarding (use of a double wall separated by an insulator). The effects of these various techniques of insulation on the heat transfer and the fluid flow will be studied for various values of the Rayleigh number, Ra, the relative thermal conductivities rw and rins and for various combinations of the geometrical parameters.
Problem formulation
The configurations under study with the system of coordinates are sketched in Fig. 1. They consist of a differentially heated cavity whose vertical walls have a finite thickness without insulation (Fig. 1, a), with an internal insulation (Fig. 1, b) and with an insulation using the thermal boarding technique (Fig. 1, c). For these three types of configurations, the horizontal walls are adiabatic. The fluid flow is assumed to be laminar
and two-dimensional and the Boussinesq approximation is adopted for the variations of the density in the buoyancy term.
L'
H'
/
y, v A
1'
I8
x , ui —►
The parameters r and K are respectively the diffusion coefficient and the relative thermal conductivity. A value of unity is allotted to r in the fluid area while an "infinite" value (1015) is affected for it in the solid area. The relative thermal conductivity K is equal to unity in the fluid medium and its value is kr for the solid walls. The parameters r and K are easily implanted in the numerical code and their use satisfies automatically the boundary conditions at the solid-fluid interfaces. The thermal and dynamic boundary conditions associated to this problem are:
0 = 0 0 = 1 d0
T = 0
u = v = Y = 0
for x = 0; for x = 1;
for y = 0 and y = 1; on the solid walls.
(4)
ins, 1 ins )
,(K 1 )
The average heat quantity transferred through the active walls of the cavity is defined by mean of the following expression:
1 d0
Q = J^7 dy
Nu =■
Q
Qa
(5)
(6)
where Qcord is the heat flux by conduction evaluated by equation (5) for Ra = 0.
(K, 1 )
fyspa 1 spa )
/ (K 1' )
Fig. 1. Studied configurations
The equations governing the dynamic and thermal fields are those of Navier-Stokes coupled with the equation of energy transport. In stationary and non-dimensional form, they are written as:
du du dP r dx dy dx
í 32 32 \
d u d u
ax2"+ärT
(1)
dv + dv = dP+r
dx dy dy
ídv Pv}
dx2 + d .2
dy 2
+ RaPr0, (2)
d0 d0 „
u — + v— = K dx dy
íd0 afe^
dx2 + dy2
(3)
Numerics
The discretization of the governing equations, written in primitive variables, is carried out by using a finite difference method and the numerical integration is based on the control-volume approach with the SIMPLER algorithm and the hybrid scheme for interpolation [11]. The technique of the finite volume method is based on the principle of conservation of the various physical quantities (mass, momentum and energy). In order to prevent that physical incompatibilities of the pressure do not generate a convergence of the total solution towards a nonphysical solution, a uniform shifted grid was used for the velocity. The numerical code was validated by comparing the results obtained in the case of a differentially heated cavity with those of the Bench-Mark solution of De Vahl Davis [3]. As it can be seen in Table 1, the agreement observed is excellent and the maximum relative deviation remains less than 1.67 and 1.34 % respectively in terms of Tmax and Q. The numerical code was also validated by comparing our results with those of the references [12-14] in the case of a cavity containing an obstacle placed at its centre. The comparative results, presented in Table 2, show an excellent agreement with a maximum deviation less than 1 %. Also for all the computations, it was carefully verified that the heat provided by the heated wall to the fluid is equal to that leaving the cavity through the cold wall with 0.2 % as a maximum difference. It is interesting to announce also that in the pure conduction
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
T
T
1
a
T
b
T
T
c
mode, the heat flux through a cavity containing various solid layers, having each one a thermal conductivity, coincides perfectly with that calculated analytically by using the concept of thermal resistance. The agreement is excellent and the maximum deviation remains lower than 0.5 %. The sensitivity of the results with respect to grid size is summarized in Table 3 in terms of Tmax and Nu. It can be retained that a uniform grid of 80x80 can describe correctly the heat transfer and fluid flow. In fact, a refinement of the mesh to 120x120 involves relatively weak maximum deviations but induces consequently higher computing times. The parameter of relaxation varies between 0.4 for low values of Ra and 0.1 for high values of this parameter. Finally, the convergence criterion, based on the correction of the pressure, is checked when the corrected terms are sufficiently weak and the existing difference between the fields of pressure before and after each correction is nonsignificant.
Table 1
Validation of the numerical code in terms of the maximum stream function and the average heat quantity
Present study Bench Mark [3]
Ra Т max Q Т A max Q
1G3 1.176 1.1G2 1.174 1.117
1G4 5.G7S 2.221 5.G71 2.23S
1G5 9.642 4.496 9.612 4.5G9
1G6 17.G31 S.SSG 16.75G S.S17
Table 2 Comparative results of the average heat quantity for a square enclosure containing a conducting square solid body at the centre
Ra kr Present study House et al. [12] Merrikh and Lage [13] Lee and Ha [14]
1G5 G.2 4.623 4.624 4.6G5 4.631
1G5 5 4.317 4.324 4.2SG 4.324
Table 3
Variations of the maximum stream function, T max, and the average heat quantity, Q, according to the grid size for Ra = 106
Mesh max Q CPU(s)
4Gx4G 17.55 9.14 191
SGx8G 17.G3 S.S6 iG3i
i2Gx12G 16.92 S.S2 6346
Results and discussion
1st configuration: walls without insulation
Typical results of the streamlines and the isotherms illustrating the effect of the relative conductivity of the walls are presented in Fig. 2 for Ra = 106, e = 0.05 and rw ranging between 0.1 and 50. For walls with a low thermal conductivity (rw = 0.1), Fig. 2, a shows that the fluid circulation is relatively weak. The corresponding isotherms indicate that the horizontal heat gradients in the solid walls are important indicating a great resistance to the heat flux between the extreme faces of each wall.
d
lO6, e
Fig. 2. Streamlines and isotherms for Ra and different values of rw:
a) rw = 0.1 (Tmax = 6.62, Nu = 1.46),
b) rw = 1 (Wmax = 11.42, Nu = 3.92),
c) rw = 10 (Wmax = 16.01, Nu = 7.13)
d) rw = 50 (Tmax = 16.85, Nu = 7.76)
- O.O5
a
b
c
This will lead to a limited interaction with the fluid inside the cavity. For rw = 1, Fig. 2, b shows an improvement of the fluid circulation characterized by higher values of Tmax and a spacing of the isotherms at the level of the walls as a consequence of the reduction in their thermal resistance. By increasing rw to 10, Fig. 2, c shows that the fluid circulation continues to be intensified. The corresponding isotherms show an uniformisation of the temperatures of the two walls and a better interaction of the latters with the fluid. For high values of relative conductivity (rw = 50), Fig. 2, d shows that the flow and temperature fields are qualitatively similar and quantitatively comparable to those described in the preceding figure corresponding to rw = 10.
109876-
Nu 5 -432-
e = 0 e = 0.05 e = 0.1 e = 0.2
r„ = 50
/
/
T m
103 2520 15 10
104 105 Ra
10°
--- e = 0 rw = 50
e = 0.05 /■//
— e = 0.1
e = 0.2 ' 'S'S y :
103
104
Ra
105
10°
b
Ra, the effect of the thickness becomes visible and the Nusselt number decreases when this last parameter increases. For Ra = 106 and e = 0.2, the reduction of Nu reaches approximately 48 % when compared to the reference case (e = 0). This tendency is a consequence of the increase in the conductive thermal resistance of the walls with the thickness.
The effect of the Rayleigh number on the Nusselt number and the maximum stream function is presented in Fig. 4 for e = 0.05 and various values of relative conductivity rw of the walls. For the lower considered value of rw (rw = 0.1), the Nusselt numbers remain generally close to unity, showing thus the resistant effect of the walls to the heat transfer and this even for high values of the Rayleigh number. For a given value of Ra, we can note an increase in the heat quantity restored to the fluid by the active surfaces. This increase of Nu with rw cease as soon as this last parameter becomes higher than 50 as it can be seen in Fig. 5 where the Nusselt number and the maximum stream function tend asymptotically to limiting values. For Ra = 106, it is interesting to note that the use of walls with a conductivity such rw = 10 would lead to a reduction of 9.8 % of Nu when compared to walls of relative conductivity rw = 50. This
reduction reaches walls is rw = 5.
8
16.6 % when the conductivity of the
Nu
7654321
-rw = 0.1
- — ■ rw = 1
- - - - rw = 5 ' /
---rw = 10
rw = 50
- ^ - ' ^ - "*
103
104
Ra
105
10°
Fig. 3. Variations of Nu and Wmax with Ra for e = 0.05, rw = 50 and different values of the relative width of the walls
The thickness of the walls was varied in the range 0 < e < 0.2 in order to analyze its effect on the fluid flow and heat transfer. The results obtained are presented in Fig. 3 in terms of variations of Nu and Tmax with Ra for rw = 50 and various values of e. The results of base obtained for e = 0 (absence of conduction in the walls) are also presented for a comparison purpose. For values of Ra < 1500, the heat transfer is done mainly by conduction and is slightly affected by the thickness of the walls. For higher values of
1 m
25
20
15
10
- rw = 0.1
— — - rw = 1
----rw = 5 ■/ ■
rw = 10
rw = 50
103
104
Ra
105
10°
b
Fig. 4. Variations of Nu and Vmax with Ra for e = 0.05 and different values of the relative conductivity rw of the walls
1
a
5
a
0
5
0
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
Nu
4х
1 m
9 8 76 5 4 32
1
0,1
28 24 20 16 12 8 4
0,1
- Ra = 103 e = 0.05
- - Ra = 104 ^ *
- - . Ra = 105 /
---Ra = 106 / /
/
/
/ . ■ "
/ _ ■ ' / - ' -----
10
100
Ra 3 103 e =0.05
- - Ra = 104 — - —----
Ra = 105
— -Ra = 106
- " - - - - - ~ - - - 1
10
100
b
contributes largely to the heat transfer. The increase of Ra supports the fluid circulation within the cavity and generates consequently an increase of Nu. Also, for a given value of Ra, the Nusselt number decreases when relative conductivity of the insulator decreases. A comparison with the results of the already studied configuration shows an appreciable reduction of the rate of heat transfer. The percentages of this reduction compared to walls of different relative conductivities are presented in Table 4 for Ra = 106 and for various values of the relative conductivity of the insulator.
Table 4
Percentage of reduction of Nu due to the addition of an insulation for Ra = 106 and various values of the conductivity of the insulator
Nu - Nuins rins
X 100 Nu 0.1 0.2 0.5 1.0 5.0
rw = 50 68.3 56.6 38.4 26.2 11.2
rw = 10 64.8 52.0 31.6 18.1 1.6
r = 5 ' w ^ 62.0 48.1 26.1 11.4 ---
Nu
Fig. 5. Effect of the relative thermal conductivity rw of the walls on Nu and for e = 0.05 and different values of the Rayleigh number
2n configuration: walls with insulation In this section, we will examine the effect of the increase of the thermal resistance of the walls by adding an insulating layer on the interior surfaces. The conductivity of this layer will be varied in order to obtain a rate of heat transfer comparable or lower than that obtained by using materials of relative conductivities of rw = 5 or 10. The thicknesses of the insulating layer and that of the wall are fixed respectively at eins = 0.0125 and e = 0.0375, in such a way that the total thickness remains equal to that of the already studied configuration (eins + e = 0.05). The effect of the addition of an insulating layer on internal surfaces of the walls will be examined for a relative conductivity of the walls fixed at rw = 50 and a thickness e = 0.0375. Hence, variations of Nu and Tmax with Ra are shown in Fig. 6 for eins = 0.0125 and various values of the relative conductivity of the insulator. It can be seen that for the low values of Ra, the Nusselt number is practically independent of the relative conductivity of the insulator. It is a foreseeable result since these values of Ra correspond to a mode for which conduction
rins = 0.1 rw = 50
rins °.2
' ' " ' rins = °.5 /
----r = 1 ' ins / ,
-----r = 5 ' ins , ' У
T m
103
25
20
15
10
104
Ra
105
106
rn = 0.1 rw = 50
---rins = °.2 /
- - rins = °.5
---r = 1 ' ins
----r- = 5 ' ins
103
104
Ra
105
106
b
Fig. 6. Effect of Ra on Nu and Wmax for rw = 50 and various values of the relative thermal conductivity rins of the insulating layer
1
г
a
0
1
r
1
a
5
0
3r configuration: case of thermal boarding Another technique of insulation consists in using two walls separated by a fluid or solid layer playing the role of an insulator. The thickness of the space between the two walls is taken equal to that of the insulator (espa = = 0.0125). The results obtained by using this technique of insulation are found identical to those of the second configuration. For the same values of rins, the maximum deviation compared to this situation remains lower than
0.45 %. Hence, the air was considered as insulator and its thickness was varied to study its impact on the heat transfer.
The effect of the spacing between the walls is examined in this section by producing streamlines and isotherms for Ra = 5-106, rw = 50 and different values of espa. The walls have each one a thickness equal to e = 0.025 so that the total thickness is equal to that of the wall of Fig.
1, a. For an air layer of thickness espa = 0.05, Fig. 7, a shows a structure close to that obtained for a differentially heated cavity. The corresponding isotherms show a clear stratification in the central part of the cavity. An increase of espa to 0.1 supports the formation of convective cells of low intensities in the insulating layers (Fig. 7, b). The examination of the isotherms between the walls shows that conduction is the heat transfer dominating mode in these areas. From Fig. 7, c, it can be seen that an increase of espa to 0.15 is accompanied by an improvement of the fluid circulation in the insulating fluid layers and of a reduction of the general circulation of the fluid within the cavity which will lead to a reduction of the Nusselt number. The numerical tests carried out showed that espa = 0.15 is the optimum value which tolerates a weaker thermal interaction between the active walls of the enclosure for the considered value of Ra. A spacing beyond this critical value supports the role of natural convection in the insulating layers and generates a better interaction between the active walls of the cavity and the fluid. This results in a noticeable improvement of the general circulation of the fluid and an increase of Nu. The Fig. 7, d corresponding to espa = 0.2, shows a distortion of the isotherms between the walls resulting from the intensification of the flow in the insulating layers and a stratification of the temperature better than that observed in the case of Fig. 7, c.
The effect Ra on the Nusselt number and the maximum stream function is presented in Fig. 8 for rw = 50 and various values of espa. It can be seen that, for a fixed value of Ra, the Nusselt number decreases with the increase of the thickness of separation. Compared to the case of a wall without insulation, the technique of the boarding appears effective to reduce the rate of heat transfer through the cavity. This technique leads to performances better than those of materials having conductivities of rw = 5 and 10. Quantitatively, the percentage of this reduction is presented in Table 5.
Table 5
Percentage of reduction of Nu in comparison with a cavity without insulation for rw = 50, Ra = 106 and different values of the spacing
- Nuboard espa
A IUU Nu 0.05 0.1 0.2 0.3
rw = 50 39.88 62.03 74.22 70.36
d
Fig. 7. Streamlines and isotherms for Ra = 5-106, rw = 50, e = 0.025 and different values of the spacing between the walls: a) espa = 0.05 (Wmax = 18.78, Nu = 6.82),
b) epa = 0.1 (Wmax = 14.96, Nu = 4.17),
c) espa = 0.15 (Wmaax = 12.82, Nu = 3.90)
d) espa = 0.2 (Wmca = 15.28, Nu = 4.13)
a
b
c
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
Nu
— espa 1 rw = 50
- espa = 0.083
- - " eSpa = 0066
— e = 0.05
.<-* - ~
103
25
104
105
106
Ra
4х
T m
20
15
10
- -- - epa = О.1 rw = 50
- e = 0.083
с spa
— - - e = 0.066
spa
-e = 0.05
spa
í--"" *
103
104
Ra
105
10"
b
is important to underline that the performances of this technique are achieved to the detriment of the space allowed to the fluid inside the cavity.
Nu
7
64 5 4 3 2
Ra = 10" Ra = 5 x 106
rw =50
0,05
0,10
0,15
0,20
0,25
0,30
Fig. 8. Effect of the Rayleigh number on Nu and Wmax for rw = = 50 and various values of the relative spacing between the walls
From Table 5, it can be seen that the percentage of reduction of the heat transfer increases with espa as long as this parameter is lower or equal to 0.2. Beyond this value a decrease is observed for espa = 0.3. This testifies of the existence of a critical value of the spacing between the walls which would engender the maximum reduction of the heat transfer. In Fig. 9, the variations of the Nusselt number, with espa, are presented for rw = 50, e = 0.025 and different values of Ra. The examination of this curve confirms well that a spacing of about 0.2 induces a minimum heat transfer for Ra = 106. Beyond this critical value, any increase of espa would lead to an increase in the intensity of the flows in the space between the walls and consequently to an increase in the resulting heat transfer. For Ra = 5-106, it can be seen that the value of the critical spacing inducing a minimum heat transfer is of about 0.15. This critical value, lower than that corresponding to Ra = 106, is due to the early development of the effect of the natural convection in the fluid layers intended to play an insulating role. It appears thus obvious that the technique of the thermal boarding is effective to ensure a good insulation. But, it
Fig. 9. Variations of the Nusselt number with the spacing between the walls for various Ra
Conclusion
In this work, a numerical study was carried of on the natural convection in a square cavity with conducting vertical walls by using two techniques of heat insulation. The obtained results show that the thermal performances of walls with high relative conductivities can be improved considerably by using an insulating layer. The technique of the thermal boarding was found interesting to reduce in an appreciable way the heat transfer in comparison to the cases of the walls with or without insulation. In this technique, the existence of a range of the spacing between the walls which generates weak interactions with fluid medium was observed.
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