Studying the Internal Heat Transfer By Natural Convection Through an Inclined and a Modified Square Enclosure with a Triangular Top Wall .

A numerical analysis used to simulate laminar natural convective heat transfer in a two-dimensional inclined and a modified square enclosure having a triangular top wall has been performed. The model is applied to analyze natural convection in this enclosure where the left side wall is subjected to a constant heat flux while the other opposite wall is kept at isothermal cold temperature. The other enclosure walls are considered adiabatic. The enclosure under study is filled with air with Prandtl number is taken as 0.71. The two-dimensional flow is solved numerically by using the Finite Volume Method (FVM).The computations are carried out at different heights of triangular top wall. The Rayleigh numbers considered ranging from 10 to 10. The effect of inclination angle of the enclosure on the resulting convection process is investigated for inclination angles ranging from 0 to 60. The results of the present investigation show that the convective phenomenon is greatly influenced by the inclined top wall height and the dominance of the convective currents with the increase of Rayleigh number and inclination angle. The streamline and isotherm plots and the variation of the average Nusselt number at the hot left side wall are also presented. The results showed a good agreement with another published results.

The length of the heat source was varied from 20 to 80% of the total length of the bottom wall.The non-heated parts of the bottom wall were considered adiabatic.The Grashof number was varied from 10 3 to 10  al. [12] and the present work is that the present work deals with the inclined modified square enclosure using finite volume method rather than the vertical modified square enclosure using finite element method as in Saha et al. [12] work.

PROBLEM DESCRIPTION AND THE MATHEMATICAL ANALYSIS
The present model consists of a two-dimensional inclined modified square 2.The fluid properties are assumed constant except for the density variation in the buoyancy term which is treated according to Boussinesq approximation.
3.The fluid inside the enclosure is assumed Newtonian and incompressible while viscous dissipation effects are considered negligible.

4.
The model is assumed to be not subjected to any external flow; hence the internal natural convection phenomenon is studied for the present model.

5.
All the solid boundaries are assumed to be rigid with no-slip condition.
The laminar internal two-dimensional flow and the temperature distribution inside

BOUNDARY CONDITIONS
The boundary conditions which are used in the present study can be arranged as

NUMERICAL TECHNIQUE
In this section numerical method adopted in the present study is discussed.

GRID GENERATION
The grid generation calculation is based on the curvilinear coordinate system

DISCRETIZATION TECHNIQUE AND METHOD OF SOLUTION
The coupled governing equations (3 to 6) are transformed into sets of algebraic equations using finite volume method.The above equations are integrated over each control volume to obtain a set of discretized linear algebraic equations of the form: The convective terms were approximated by the power scheme and the diffusive terms with the central differencing scheme.The SIMPLE algorithm of Patankar [15] was used to couple continuity and momentum equations.The computational procedure is based on a collocated mesh, and the details can be found in the text by Ferziger and Peric [16].
The systems of algebraic equations are solved using the strongly implicit procedure (SIP) of Stone [17].The computation is terminated when the residuals for the continuity and momentum equations get below 10 -5 and the residual for the energy equation gets below 10 -8 .

GRID SENSITIVITY TEST
The numerical scheme used to solve the governing equations for the present work is a finite volume approach.It provides a smooth solutions at the interior domain including the corners.The enclosure is meshed with a non-uniform rectangular grid with a very fine spacing near the walls.As shown in Fig. 2, the 2-D computational grids are clustered towards the walls.The location of the nodes is calculated using a stretching function so that the node density is higher near the walls of the enclosure.Solutions are assumed to converge when the following convergence criteria is satisfied at every point in the solution domain :-   respectively.From this figure, the flow field is characterized by a primary recirculating air vortex near the left-hand wall where the temperature is high while its density is low due to the heat flux effect.This temperature and density differences between the left and right side walls causes this mentioned air vortex which filling most of the enclosure.When the range of Rayleigh numbers are low, the buoyancy force effect is small so for this case the convection heat transfer contribution is small.
From the other hand when the Rayleigh number range increases, the buoyancy force effect becomes greater so the vortices shape becomes irregular due to the active fluid motion causing from the higher temperatures near the left-hand wall and as a result behavior, is that the x-component of buoyancy effect due to external heating resulting from the temperature difference is small when the effect of inclination angle is small .
From the other hand, the effect of x-component of buoyancy forces becomes more strong, as the inclination angle increases.The results show also that when the Rayleigh number is low, there is a primary recirculating vortex which covers most of the enclosure size.At Ra = 10 6 , a two non-uniform recirculating vortices of different shape and opposite directions can be noticed.This irregularity in the vortices shape due to the flow confusion which causes a good convection heat transfer performance.

CONCLUSIONS
The following conclusions can be drawn from the results of the present work: 1-When the inclination angle is 0 o and the Rayleigh numbers are low, the buoyancy force effect is small and the convection heat transfer contribution is small.
2-When the Rayleigh number increases, the vortices shape becomes irregular and making a large convection heat transfer contribution.
3-When the inclination angle is 0 o the isotherm contours refer that the thermal field lines are almost linear and symmetrical at the upper part of the enclosure, so the diffusion heat transfer is dominated .While the convection heat transfer is dominated at the lower part of the enclosure.

4-When the Rayleigh number increases, the isotherms become non-symmetry
,uniform horizontally and linear vertically at the upper part of the enclosure.

5-
When the inclination angle is increased, the effect of inclination angle is small when the Rayleigh number is low.But when the Rayleigh number increases the fluid flow nature has a clear different behaviour.

6-
When the Rayleigh number is low, there is a primary recirculating vortex which covers most of the enclosure size.At Ra = 10 6 , a two non-uniform recirculating vortices of different shape and opposite directions can be noticed.
of the cold surface, (°C) U Dimensionless velocity component in x-direction u Velocity component in x-direction, (m/s) V Dimensionless velocity component in y-direction v Velocity component in y-direction, (m/s) X Dimensionless Coordinate in horizontal direction x Cartesian coordinate in horizontal direction, (m) Y Dimensionless Coordinate in vertical direction y Cartesian coordinate in vertical direction, (m) Thermal diffusivity, (m 2 /s) Volumetric coefficient of thermal expansion, (K -1 ) Dimensionless temperature Variable vector Enclosure inclination angle with horizontal Direction, (degree) Kinematic viscosity of the fluid, (m 2 /s) Density of the fluid, (kg/m 3 ) , Dimensionless body-fitted coordinates 1. INTRODUCTION Natural convection is observed as a result of the motion of the fluid due to density changes arising from heating process.The movement of the fluid in natural convection results from the buoyancy forces imposed on the fluid when its density of the heat transfer surface is described as a result of thermal expansion of the fluid in a non-uniform temperature distribution.Convection heat transfer is dependent on the movement of the fluid and the development of the flow of the fluid is also affected by the shape of the heat transfer surfaces.Both numerical and experimental methods have been used to obtain the solution of heat transfer and fluid flow problems.In most cases, the experimental methods are costly and time consuming due to necessary of expensive prototypes and instrumentations.On the other hand, numerical methods can offer considerable savings in design time and cost Saha [1], 2007.The problem of convective heat transfer in an enclosure has been studied extensively because of the wide application of such process.The flow inside an enclosure is much more complicated to investigate, as the boundary zone and the middle core zone never have the same effect for a certain boundary condition considered.Ali and Husain [2],1993, investigated the effect of corrugation frequencies on natural convective heat transfer and flow characteristics in a square enclosure of vee-corrugated vertical walls.This investigation showed that the overall heat transfer through the enclosure increased with the increase of corrugation for low Grashof number; but the effect was reversed for high Grashof number.Ganzarolli and Milanez [3] ,1995, performed a numerical study of steady natural convection in rectangular enclosures heated from below and symmetrically cooled from the sides.The size of the cavity was varied from square to shallow where the cavity width was varied from 1-10 times of the height.The heat source, which spanned the entire bottom wall, was either isothermal or at constant heat flux condition.They observed that, for the square cavity, the flow and thermal fields are not strongly affected by the isothermal or constant heat flux boundary condition at the bottom heat source.Aydin and Yang [4] , 2000 , investigated numerically the natural convection of air in a vertical square cavity with localized isothermal heating from below and symmetrical cooling from the sidewalls.The top wall as well as the non-heated parts of the bottom wall was considered adiabatic.The length of the symmetrically placed isothermal heat source at the bottom was varied.Their results showed that there a two counter rotating vortices were formed in the flow domain due to natural convection.Sharif and Mohammad [5] , 2005 , studied numerically natural convection in rectangular cavities by using a finite volume method.Their study was based on a configuration where a constant flux heat source was symmetrically embedded at the bottom wall.

[ 11 ] 3 to 10 6 . 6 .
, 2008 presented a numerical study of natural convection by using a penalty finite element method in a tilted square cavity with heated horizontal base and cold upper wall.Their study was based on a configuration where the top triangular wall of two different shapes was maintained at a constant low temperature.A constant heat flux source whose length is 20% of the total length of the cavity was discretely embedded at the left corner of the bottom wall while the remaining part of the bottom wall and the two sidewalls were considered to be adiabatic.Their study included computations for inclination angle of 45 o , while the Grashof number varies from 10 Their results explained that a significant effect of different triangular top surface on the heat transfer characteristics at the higher Grashof number and inclination angle.Saha et al. [12], 2008, carried out a numerical study for laminar natural convection heat transfer within a two-dimensional modified square enclosure having a triangular roof.The vertical sidewalls are differentially heated considering a constant flux heat source strip which was mounted with the left wall.The opposite wall is considered isothermal having a temperature of the surrounding fluid.The rest of the walls are adiabatic.Air is considered as the fluid inside the enclosure.The solution has been carried out on the basis of finite element analysis with different heights of the triangular roof.Fluid flow fields and isotherm patterns and the average Nusselt number were presented for the Rayleigh numbers ranging from 10 3 to 10 They concluded that the convective phenomenon is greatly influenced by the inclined roof height.From the above literature survey, it was found that there are some limited studies for non-linear top or bottom walled enclosure.The main purpose of present study is to investigate the effect of the inclination angle on the heat transfer and fluid flow for natural convection inside a modified square inclined enclosure with a triangular top wall by considering the constant heat flux at one side of the vertical walls for different Rayleigh number ranging from 10 3 to 10 6 .The other vertical wall is kept at a constant temperature and the rest of the walls, i.e. ,the bottom wall and triangular inclined top walls are at adiabatic condition.The present study is based on the configuration of Saha et al. [12], 2008.The main difference between Saha et

1 .
enclosure having a triangular top wall.The vertical side left wall is subjected to a constant heat flux while the opposite right wall is considered isothermal which having a temperature of the surrounding fluid.The rest of the walls are considered adiabatic.Air is considered as the fluid inside the enclosure with Pr = 0.71.The enclosure has the same height and width ( H = L ) and the top inclined triangular wall has the height E which is considered to have a length of 0.1 to 0.5 times L in the present study.The geometry under investigation is shown schematically in Fig. (1).Different heights of the triangular top wall have been considered for the present analysis.Fluid flow fields and isotherm patterns and the average Nusselt number are presented for the Rayleigh numbers ranging from 10 3 to 10 6 and the enclosure inclination angle, , is varied as 0 o , 20 o , 40 o and 60 o respectively.The numerical solution is obtained by using a finite volume scheme and the following assumptions are considered in the present analysis:-The flow is considered laminar , steady and two-dimensional.
the enclosure are described by the Navier-Stokes and the energy equations, respectively.The governing equations are transformed into a dimensionless forms by using the following dimensionless variables [12]:-The dimensionless forms of the governing equations of steady two-dimensional internal laminar natural convection are expressed in the following forms [12]:-Where Pr is the Prandtl number and Ra is the Rayleigh number.The previous dimensionless numbers are defined by [12]:-The average Nusselt number, ,at the hot left side wall is given by [12] :where is the dimensionless temperature at the hot left side wall.

follows:- 1 . 2 . 3 .
The left side wall (i.e.surface I) is subjected to a constant heat flux q , so:-The right side wall ( i.e. surface III ) is kept at a uniform cold temperature (T c ) The other surfaces of enclosure ( i.e. surfaces II , IV and V ) are thermally Figure 1: Schematic Physical Configuration and Boundary Conditions of the Modified and Inclined Square Enclosure Figure 2: A Typical 2D -Grid Distribution (100 x 100) with Non-uniform and Non-orthogonal Distributions for Inclination Cavity .

) 6 .x 10 3 to 4 x10 4 , 5 4 .
where φ represents a dependent variable U, V, P, and .In order to obtain grid independent solution, a grid refinement study is performed for Ra = 10 4 , Pr = 0.71, and = 0°.In the present work, eight combinations ( 80 x 80, 90 x 90, 95 x 95, 100 x 100, 150 x 150, 175 x 175, 200 x 200 and 225 x 225) of control volumes are used to test the effect of grid size on the accuracy of the predicted results.Fig. 3 shows the convergence of the average Nusselt number, Nu av , at the heated surface with grid refinement.It is observed that grid independence is achieved with combination of (100x100) control volumes where there is insignificant change in Nu av .The agreement is found to be excellent which validates the present computations indirectly.NUMERICAL RESULTS VERIFICATION For the purpose of the present numerical algorithm verification, a laminar natural convection problem inside the same tested model as obtained by Saha et al. [12] considering a constant heat flux condition is performed.The comparison is made using the following parameters: Pr = 0.71 , Ra = 6 and = 0°.Comparisons of the average Nusselt numbers at a heated left side wall are shown in Table 1.A general agreement between the present computation values and the previous values by Saha et al. [12] is seen to be very well with a maximum deviation of about 4.42%.Further validation is performed by using the present numerical algorithm to investigate the same problem considered by Saha et al. [12] using the same flow conditions, geometry, and the boundary conditions but the numerical scheme is different .The comparison is made using the following parameters: Pr = 0.71, Ra = 10 3 , E/L =0.1, 0.3, 0.5 and = 0°.Good agreement is achieved between Saha et al.[12] and the present numerical scheme for both the streamlines and temperature contours as shown in Fig.These validations make a good confidence in the present numerical model to deal with the same enclosure configuration problem but it is considered inclined to calculate the flow and thermal fields in the present work.

Fig. 3 .
Fig. 3. Variation of average Nusselt number along the hot left side wall as a function of control volume for Ra = 10 4 , Pr= 0.71, and = 0°.
making a large convection heat transfer contribution.The figure also shows that the stream line contours are collect morly near the bottom wall than the top one due to the triangular shape of the top wall.In the same figure, the isotherm contours refer that the thermal field lines are almost linear and symmetrical at the upper part of the enclosure, so the diffusion heat transfer is dominated because the generated buoyancy force is not strong enough to initiate fluid convection.While the convection heat transfer is dominated at the lower part of the enclosure.When the Rayleigh number increases, the isotherms become non-symmetry ,uniform horizontally and linear vertically at the upper part of the enclosure.The results indicate also that the thermal boundary layers near the hot and cold side walls increases and concentrated as the effect of the Rayleigh number increases.Figures (6,7 and 8) explain the streamlines and isotherms contours in a modified square inclined enclosure which having a triangular top wall when the range of Rayleigh numbers ( Ra ) are 10 3 , 10 4 ,10 5 and angle, , is varied as 20 o , 40 o and 60 o respectively.From these figures, the effect of inclination angle is small when the Rayleigh number is low.But when the Rayleigh number increases the fluid flow nature has a different behaviour where the fluid moving upwards in the interior of the enclosure and moving downwards in both the hot and cold side walls.The results indicate that the fluid internal motion near the left hot wall is more strong than that the opposite right cold wall.The reason of this

Furthermore, the thermal
field in the previous mentioned figures indicate that the isotherms lines are almost linear and symmetrical when the Rayleigh number is low, so the diffusion heat transfer is dominated.When the Rayleigh number increases, the isotherms become non-symmetry ,uniform horizontally and linear vertically at the upper part of the enclosure.Also, a thin thermal boundary layer is observed at the heated left side wall when the Rayleigh number and the inclination angle increases .Also, these figures indicate that when E / L ratio increases together with the increase in the Rayleigh number and the inclination angle, the spacing near the triangular inclined top walls decreases and the flow field enlarge to cover all the enclosure zone.The reason of this behaviour is that due to the increase in the flow velocity and a large surface area due to the increase of E/L ratio.The variation of the average Nusselt number along the hot left side wall is explained in Figure 9 when the range of Rayleigh numbers ( Ra ) are 10 3 , 10 4 ,10 5 and 10 6 respectively, E / L ratio equals 0.1 ,0.3 and 0.5 while the enclosure inclination angle, , is varied as 0 o , 20 o , 40 o and 60 o respectively.The figure explains that when the E/L ratio is high, the average Nusselt number increases together with the increase in the Rayleigh number and the inclination angle.This is due to the increase in the intensity of convection currents which causes a clear change in the heat transfer rate.

Sathiyamoorthy et al. [8] , 2007 ,
6.The effects of aspect ratio, inclination angles, and heat source length on the convection and heat transfer process in the

10 3 to 10 5 and for different aspect
ratios and position of heated plate.They found that when Grashof number increased ,the heat transfer rate increased in both vertical and horizontal position of the plate while, when aspect ratio of heated thin plate was decreased the heat transfer [10] decreased.Natarajan et al.[10], 2008, used a penalty finite element analysis to investigate the influence of uniform and non-uniform heating of bottom wall on natural convection flows in a trapezoidal cavity.The bottom wall was uniformly and non-uniformly heated while two vertical walls were maintained at constant cold temperature and the top wall was well insulated.They observed a symmetry while representing the flow patterns in terms of stream functions.Non-uniform heating of the bottom wall produced greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number showed overall lower heat transfer rate for non-uniform heating case.Sultana et al.