Laminar free and forced convection cooling of a semi permeable open cavity

1 Mechanical Engineering Department, University of Taubaté, Rua Daniel Danelli, s/no Jardim Morumbí 12060-440 Taubaté SP Brazil. 2 Mechanical Engineering Department and Department of Basic and Environmental Sciences (School of Engineering at Lorena), University of Taubaté and University of Sao Paulo, Brazil. 3 Mechanical Engineering Department, University of Campinas Campinas SP Brazil.


INTRODUCTION
The heat transfer in enclosures has been studied for a variety of engineering applications.Results have been presented in research surveys (Bruchberg et al., 1976;Kakaç et al., 1987) and it has become a main topic in convective heat transfer textbooks (Bejan, 1984).Usually the enclosures are closed and natural convection is the single heat transfer mechanism.There are however, several applications in passive solar heating, energy conservation in building and cooling of electronic equipment, where open cavities are employed (Chan and Tien, 1985;Hess and Henze, 1984;Penot, 1982).
Frequently the devices employed for the cooling of electronic equipment are based on forced convection (Sparrow et al., 1985).Studies on Computational Fluid Dynamics (CFD) applied to analysis of electronics cooling had been developed in several works.Specially theory *Corresponding author email: carlos.chaves@unitau.br,Tel/Fax: +55 12 3621-8002.Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License on natural convection in enclosures (Ostrach, 1972) and electronics cooling enclosure used as part of a larger telecommunication radar system (Boukhanouf and Haddad, 2010).Also studies of flow characteristics are done numerically with CFD and compared to experimental approaches, that is, manifolds that are widely used in electronic cooling equipment (Gandhi et al., 2012), shielded heat sink also widely used in that (Shaalan et al., 2012), and analysis to eliminate fan used in electronic device cooling (Nickell, 1997).
Several studies on natural convection in open cavities had been done.Steady-state natural convection taking place in rectangular cavities filled with air was studied both experimental and numerically in Bairi et al. (2007).A steady buoyancy-driven flow of air in a partially open square 2D cavity with internal heat source, adiabatic bottom and top walls, and vertical walls maintained at different constant temperatures was investigated numerically in Fontana et al. (2011).The numerical results of heat transfer calculations in an open cavity considering natural convection and temperaturedependent fluid properties were presented by Juarez et al. (2011).The natural convection occurring from open cavities was analysed by Prakash et al. (2012) in three different cavity shapes namely cubical, spherical and hemispherical geometries having equal heat transfer area.The thermal behaviour of airborne electronic equipment submitted to natural convection in closed parallelogrammic air-filled cavities was examined by Bairi et al. (2012).Also mixed convection in open cavity had been reviewed through several works.Wong and Saeid (2009) investigated the opposing mixed convection arises from jet impingement cooling of a heated bottom surface of an open cavity in a horizontal channel filled with porous medium through a numerical study.Stiriba et al. (2010) had analyzed the effects of mixed convective flow over a three-dimensional cavity that lies at the bottom of a horizontal channel through a numerical study, in which was found that the flow becomes stable at moderate Grashof number and exhibit a threedimensional structure, while for both high Reynolds and Grashof numbers the mixed convection effects came into play.
A finite element analysis was performed on the conjugated effect of joule heating and magnetohydrodynamic on double-diffusive mixed convection in a horizontal channel with an open cavity (Rahman et al., 2011).
Laminar mixed convective flow over a threedimensional open cavity with heating from below at constant temperature was numerically simulated using direct numerical simulation and the most hydrodynamic and thermal aspects of the flow were presented by Stiriba et al. (2013).Magneto-hydrodynamic mixed convection in a lid driven cavity along with a heated circular hollow cylinder positioned at the centre of the cavity was studied numerically by Farid et al. (2013).Altemani and Chaves (1988) presented a numerical study of heat transfer inside a semi porous two-dimensional rectangular open cavity for both local and average Nusselt numbers at the heated wall and for the isotherms and streamlines of the fluid flowing inside the open cavity.Chaves et al. (2005) presented a numerical program in finite volumes applied to the transient natural convection heat transfer by double diffusion from a heated cylinder buried in a saturated porous medium.Chaves et al. (2008) presented a work where it was done a numerical analysis of the heat transfer inside a semi porous two-dimensional rectangular open cavity, where forced and natural convection where considered and the bottom and the opposite wall was heated.
This paper presents a continuation of work of Chaves et al. (2008) using the program development in Chaves et al. (2005) where numerical analysis of heat transfer was done inside a semi porous two-dimensional rectangular open cavity.It is constituted by two vertical parallel plates closed at the bottom by an adiabatic surf ace and open at the top, as indicated in Figure 1.
One of the vertical plates is porous and there is a fluid flow forced normal to it in order to cool the other vertical plate.This second plate transfer a uniform heat flux to the cavity.In addition to the forced convection, the analysis considered the influence of natural convection effects.Isotherms and streamlines are presented for the fluid flow inside the open cavity.Local and average Nusselt numbers are obtained for the uniformly heated plate for several values of aspect ratio to the parameters governing the heat transfer: Re p and Gr.

METHODOLOGY
The conservation equation of mass, momentum and energy, as well as their boundary conditions, will be expressed for the system indicated in Figure 1.Due to the low velocities usually associated with permeable walls, the natural convection will be considered in the analysis.It is assumed that the flow is laminar and occurs under steady state conditions.The natural convection will be treated via the Boussinesq approximation, that is, density variations are accounted for only when they contribute to buoyancy forces.In this problem, the buoyancy term is obtained from the y momentum equation terms representing the pressure and body forces: The density is related to temperature according to the Boussinesq approximation (Patankar, 1980;Kundu and Cohen, 2008): In Equation ( 2 2), (3), and (1) can be expressed by: The second term in this equation relates the buoyancy forces to temperature differences (T -Tp).From this formulation, the density will be assumed constant and equals to p  in all the equations, so that the subscript p may be deleted.It is also assumed that all the other properties of the fluid are constant.Viscous dissipation and compression work are not considered in the analysis, according to the low velocities, moderate temperature differences and laminar flow conditions assumed.In order to obtain the conservation equations in dimensionless form, the following variables were defined (Patankar, 1980;Kundu and Cohen, 2008): The equations expressing conservation of mass, x and y momentum and energy then become: In Equations ( 7) to (9) 2  is the Laplace operator in Cartesian coordinates.These equations are coupled and present two independent parameters, Gr and Pr.The first is the modified Grashof number, defined by Patankar (1980) and Sharma (2005): and the second is the Prandtl number of the fluid (Patankar, 1980;Sharma, 2005).
At the three solid boundaries of the open cavity, the velocity components are null, except the velocity of injection of the fluid (Up) at the porous wall.The thermal boundary conditions comprise a uniform (reference) temperature at the porous wall and a specified heat flux at the heated vertical wall.Expressed in dimensionless terms, the boundary conditions become: The dimensionless velocity component normal to the permeable is one parameter of this problem and it will be denoted the porous wall Reynolds number, Rep.The outflow boundary of the open cavity, at Y equal to H/D, is just a virtual boundary defining the calculation domain.In order to obtain a solution, two conditions must be satisfied at this boundary.First, there must be no backflow of fluid and second, there must be no diffusion from outside into the calculation domain.The first condition was verified checking the velocity profiles of each result obtained and discarding those results when a backflow was observed.The second was satisfied imposing artificially negligible partial derivatives of  and U in the vertical direction at the outflow boundary.The velocity component V was corrected at the outflow boundary in order to satisfy the conservation of mass in the whole domain.
The problem presents four independent parameters: H/D, Pr, Rep and Gr.For a fixed particular fluid, there are still three parameters governing the heat transfer: H/D, Rep and Gr.In the present work, a single value, equal to 0.72, was assigned to the Prandtl number.
The differential Equations ( 6) to (9) together with their boundary conditions, Equation (11), comprise a coupled system involving the four variables U, V, P and  .The equations were discretized using the control volume formulation described in Patankar (1980) and the solution was obtained employing the SIMPLE scheme.The convergence of the results was accepted when the relative change of the dependent variables was under 10 -3 .From the velocity field solutions, a steam function defined as: was evaluated along lines X = constant, with  = 0 at X = Y = O.
From the solution of the temperature field, the local heat transfer coefficient at the heated wall and a corresponding Nusselt number were expressed as Patankar (1980) and Sharma (2005): Where TW indicates the local temperature of heated wall.With the definition of the dimensionless temperature, Equation (5c), the Nusselt number becomes An average Nusselt number for the heated wall was obtained from In Equation ( 15  The adequacy of the grid fineness employed in the results is presented in Figure 2, related to the average Nusselt number defined in Equation ( 16).In either the absence or the presence of natural convection effects, a grid of 30x30 was adequate for the aspect ratio of 2 used in most of our results.Increasing the number of grid points from 900 to 1,600 would change the average Nusselt number by 0.2% (Figure 2).

Nusselt number distributions
Initially, just the effects of forced convection on the Nusselt number distributions will be considered to many aspect ratio H/D.The results presented in Figure 3 show that the Nusselt numbers increase with Re p and that they attain a uniform value within the cavity.The increase of Nusselt number is due to both the larger fluid flow through the cavity and the effect of inertia forces, causing the streamlines to come relatively closer to the heated wall.
The comparison of the streamlines indicated in Figure 4 shows that a larger fraction of fluid flow occurs closer to the heated wall when Re increases.In view of the definitions of Nu in Equations ( 13) and ( 14), Figure 3 also indicates that h and  attain uniform values.This thermal behaviour is analogous to that of the two-dimensional stagnation flow (Kays, 1966), mainly when Re p is small in Figure 3.The increase of fluid flow along the height of the heated wall, due to injection at the permeable wall, causes a smaller thermal penetration into the fluid in the upper portion of the cavity, as indicated in Figure 5.It is evident in Figure 5 the larger thermal resistance near bottom of the heated wall.
The effects of natural convection on the Nusselt number distributions will now be considered.The results obtained for a relatively low Re p are show in Figure 6.The curves of Nusselt number distributions are parametrized with the modified Grashof number.For the smallest values of Gr (10 and 100), the distributions are very similar to those shown in Figure 3.In this range of Gr the mechanism of forced convection is still dominant.The isotherms in the fluid are similar to those of Figure 5 and the heated wall thermal resistance is higher near the bottom of the cavity.As the Grashof number increases, there is an enhancement of heat transfer so that, as indicated in Figure 6, the entire heated plate becomes colder.The effects of the buoyancy forces, acting mainly close to the heated wall, share the dominant role on heat transfer with the forced convection effects.The Nusselt number profiles for Gr larger than 10 3 in Figure 6 indicate that the wall temperature initially decreases slightly from the bottom of the cavity and attains downstream a minimum value.The behaviour is distracted by forced convection effects.The minimum wall temperature occurs however, on the lower portion of the cavity instead of on the upper portion, as was the case in Figure 3. Downstream of this minimum, the wall temperature now increases monotonically to the upper end of the cavity.This behaviour is imposed by natural convection effects.The values of the Gr in Figure 6 are limited in the upper range (Gr < 10 4 ) due to the condition of no backflow of fluid at the outflow boundary.In this respect, the effects of natural convection are somewhat restrained for the relatively low value of Re p in this figure.
The streamlines and isotherms for Re p = 5 and Gr = 3,000 are showed in Figure 7. Due to the buoyancy induced flow, the streamlines, compared to the case for Gr = 0 in Figure 4, penetrate deeper into the cavity before  The Nusselt number distributions for a relatively high Re p are now shown in Figure 8. Again, for small values of Gr, there is almost no distinction from the case in the absence of natural convection effects.With this Re p , the parameter Gr can be increased to a much higher value than before without any backflow of fluid at the outflow boundary.For the largest values of Gr in Figure 8, the heated wall temperature is a minimum right at the bottom and increases monotonically along its length.In this case, the natural convection effects control the heat transfer.The streamlines and isotherms shown in Figure 9 correspond to Re p equal to 30 and Gr = 10 6 .The streamlines indicate that now half of the flow leaves the cavity within only 10% of its width, near the heated wall.The streamlines at the bottom go almost straight to the heated wall.The induced buoyant flow causes a bending of these lines slightly downward before turning upward.
The isotherms indicate a natural convection boundary layer development form the bottom of the heated wall.There is no increase in the thermal resistance of wall near the bottom of the cavity, as shown by the isotherms closest to the heated wall.

Average Nusselt numbers
The average Nusselt numbers are shown in Figure 10 as function of Re p and parametrized with the modified Grashof number.The relative enhancement of heat transfer due the effects of natural convection decreases with Re p .For the largest value of Gr in Figure 10, equals to 10 7 , the effects of natural convection are so dominant that the value of Nu is almost independent of Re p for the range investigated.
Another view is presented in Figure 11, where Nu is shown as a function of Gr and parametrized with respect to Re p .It is clear that the enhancement of heat transfer by natural convection occurs when Gr attains a minimum value.This Gr seems to increase slightly with Re p .As noticed before, Nu attains a limit value practically independent of Re p .

Aspect ratio
The results presented so far were obtained for an aspect ratio (H/D) of the open cavity equal to 2. In Figure 12, the Nusselt number distributions for the aspect ratios of 2, 4 and 8, obtained for a pair of values of Re p and Gr, are compared.The distribution for (H/D) equal to 2 is the same as that included in Figure 7, but in a much enlarged vertical scale.As the aspect ratio increases, the minimum temperature of the heated wall moves closer to the bottom of the cavity.As discussed before, this temperature characterizes the start up of a natural convection boundary layer along the heated wall.Thus, these profiles indicate a stronger effect of natural convection as the aspect ratio of the cavity increases.It is also noticed that the Nusselt number distributions in the region controlled by natural convection match each other.
In this region the heated wall temperature distributions seem to be independent of the aspect ratio of the cavity.

Conclusions
The study of laminar free and forced convection cooling

Figure 1 .
Figure 1.Coordinate system and thermal boundary conditions of the open cavity.
)w T indicates the average heated wall temperature.Expressed in dimensionless variables, the average Nusselt number becomes.

Figure 4 .
Figure 2. Adequacy of the grid fineness employed.

Figure 10 .
Figure 10.Average Nusselt numbers as a function of Rep.

Figure 11 .
Figure 11.Average Nusselt numbers as a function of Gr.

Figure 12 .
Figure 12.Influence of the aspect ratio.
of a semi permeable open cavity was investigated numerically.The equations describing the problem were expressed in Cartesian coordinates according to stream function formulation and numerically solved by the method of control volume.The program implemented allowed to achieve satisfactory results and enabled a better understanding of the influence of Reynolds and Grashof numbers on flows driven by heat.The results obtained show that the forced convection inside the semiporous open cavity studied may be greatly enhanced by natural convection effects.When Gr is small enough, just forced convection controls the heat transfer.In this case, the upper portion of the heated plate becomes the most convenient region for cooling purposes.When Gr increases, natural convection effects may become dominant and then the lower portion of the heated plate constitutes the coldest region.When the aspect ratio of the open cavity increases there seems to be an increase of the role played by natural convection effects.The investigation carried out was however, limited because the problem presents four parameters (H/D, Pr, Re p and Gr) and some choices had to be made about the range to be analysed.of gravity [m/s 2 ]; Gr, Modified Grashof number [-]; h ,Convective heat transfer coefficient [W/m 2 .°C];H, Height of the open cavity [m]; Nu, Local Nusselt number [-]; u N ,Average Nusselt number [-]; p, Pressure [Pa]; p * , Modified pressure [Pa]; P, Dimensionless pressure [Pa]; Pr, Prandtl number of the fluid [-]; q, Surface heat flux [W/m]; Re p , Porous wall Reynolds number [-]; T, Temperature [°C]; T p , Temperature of the porous wall [°C]; T w , Heated wall temperature [°C]; w T , Average temperature of the heated wall [°C]; u p , Injection velocity of fluid at the porous wall [m/s]; U, V, Dimensionless velocities [m/s]; x, y, Cartesian coordinates [m]; X, Y, Dimensionless Cartesian coordinates [m]; , Coefficient of thermal expansion [1/°C]; , Stream function [-]; , Kinematic viscosity [m 2 /s]; , Dimensionless temperature [°C]; w  , Dimensionless heated wall temperature [°C]; w  , Dimensionless average heated wall temperature [°C]; , Density [kg/m]; , Number of grid points in the domain [-]; , Viscosity [Pa.s]; , Thermal conductivity [W/m.°C].