Heat Transfer Analysis of Viscous Incompressible Fluid by Combined Natural Convection and Radiation in an Open Cavity

The effect of radiation on natural convection of Newtonian fluid contained in an open cavity is investigated in this study. The governing partial differential equations are solved numerically using the Alternate Direct Implicit method together with the Successive Overrelaxation method. The study is focused on studying the flow pattern and the convective and radiative heat transfer rates are studied for different values of radiation parameters, namely, the optical thickness of the fluid, scattering albedo, and the Planck number. It was found that, in the optically thin limit, an increase in the optical thickness of the fluid raises the temperature and radiation heat transfer of the fluid. However, a further increase in the optical thickness decreases the radiative heat transfer rate due to increase in the energy level of the fluid, which ultimately reduces the total heat transfer rate within the fluid.


Introduction
There are many physical phenomena in which energy exchange due to radiation plays an important role, for instance, heat transfer in furnaces and combustion chambers, solar simulators and the utilization of solar energy, flow of the earth's mantle, the flow of oxide melts during crystal growth, processing of molten glass, and the solar air receivers.Heat leakage in evacuated spaces, energy dissipation in vacuum tubes, role of air and water as coolant in power plants, and cooling of electronic and optoelectronic devices also involve the energy exchange via radiation.One of the important features of radiation heat transfer is the nature of its dependency on temperature.Review of the radiation models that exist in the literature is given by Siegel and Howell [1].It should also be noted that the study of heat transfer in fluids which absorb and emit radiant energy becomes a complicated task due to the coupled, non-linear physical phenomena of internal radiation and natural convection.A detailed review of this coupling phenomenon of radiative and fluid transport can also be found in Modest [2].
In general, the combined mechanism of radiant and convective heat transfer in finite enclosures has received considerable attention.Larson and Viskanta [3] investigated the effects of free burning by taking a model of transient natural convection and radiation for gray diffuse walls with an arbitrary temperature in an enclosure.They showed that the difference between the temperatures of the hot and the cold walls reduces to about 1% of that of the hot wall, due to radiation effect.More interestingly, the air in the core region generally reached 33% of the hot wall temperature, whereas it obtains 13% of the hot wall temperature in the absence of radiation effect at the same time level.Lauriat [4] analyzed the effect of combined radiation and radiation phenomena for gray fluids.Bouallou and Sacadura [5] considered the porous media case of participating medium.Draoui et al. [6] further made an investigation of natural convection of participating fluids in a square enclosure.The study of flow properties for such kind of flows showed that there is a tendency of flow reduction and an increase in heat transfer for the process of combined conduction-radiation of fluids in 2 Mathematical Problems in Engineering enclosures.Moreover the centrosymmetric property of flow is perturbed due to the effect of radiation (see also Lauriat [4]).Salinger et al. [7] studied the effects of optical thickness and thermal gradients on the stability and structure of flows in a cylindrical container heated from below.Modest [2, page 466] indicated that there is a general misconception about lowest order differential approximation, that it may fail in optically thin limit.Rather there is some loss of accuracy only when small optically thin medium is sandwiched between hot and cold surfaces in the presence of Collimated irradiation.Derby et al. [8] examined the performance of the approximations for modeling a representative problem of heat transfer and buoyant flow in optically thick fluids.They remarked that although Rosseland diffusion had an advantage of less computational cost, an inaccuracy in the thermal boundary layer appears in it.However despite the fact that differential approximation is more costly in terms of computation due to an additional elliptic type differential equation, it gives more accurate results.Ridouane et al. [9] studied both transient and steady conditions for the effects of surface radiation on natural convection in a square enclosure heated from below and cooled from above.They showed that heat transfer across the cavity rises quickly due to active walls emissivity.Recent boundary layer study of combined mechanism of convection and radiation under different conditions can be seen in [10][11][12][13][14]. Liu et al. [15] used discrete ordinate method to investigate the combined mechanism of natural convection and radiation in a cavity and showed that radiative heat transfer was significantly reduced as the optical thickness was increased.
Most of the study of natural convection in enclosures has been devoted to the study of streamlines and isotherms.However, the investigation of heat function is also useful in studying the heat transfer characteristics of the phenomenon of natural convection.Kimura and Bejan [16] introduced heat function as the energy analog of stream function.Also, isotherms provide credible information about heat flow only in conduction dominated regime, whereas heatlines are locally parallel to the actual direction of energy flow in a domain.Thus heatlines study helps to comprehend the flow and temperature profile as the heat function formulation is based on identically satisfying the thermal energy equation (see also Bello-Ochende [17], Costa [18,19], and Deng and Tang [20]).The study of heat function profile becomes more important when the natural convection flow is combined with radiation, as the inclusion of radiation raises the energy level of the fluid in domain considerably (see Larson and Viskanta [3]).A very recent study of heatlines formulation for surface radiation case in enclosures has been made by Hossain et al. [21].Further, the problem of natural convection in open ended cavity is recently addressed by Saleem et al. [22,23].As far as the study of the effect of surface radiation in open cavity is concerned, the model was considered by Hinojosa et al. [24].They investigated the Nusselt number distribution for the natural convection and surface thermal radiation in a square tilted open cavity.However, more emphasis was laid on the effect of tilt angle on the Nusselt number distributions.Also the case of conjugate natural convection with conduction and surface radiation in open cavity was studied by Nouanegue et al. [25].
The present study aims to investigate the basic flow pattern and heat transfer characteristics in open ended domain.Since, to author's knowledge, the study of participating fluids in open ended cavities is not found in literature, such kind of models may have their applications in the situations where heat is rejected via spaces between reflecting surfaces, energy transfer in vacuum tubes, and flow of air and water as coolant in power plants, where heat is mainly rejected via radiation.Thus here we consider the case of combined natural convection and radiation of Newtonian fluid in an open ended cavity, whose left wall is maintained at a higher temperature.The density of fluid with temperature is considered to vary under Boussinesq approximation.Boussinesq approximation is a better choice for laminar case of such fluids.(See also [26].)Further, it is also worth mentioning here that modeling such flows requires that the temperature of the gas does not rise enough to a level where it causes generation of large number of ions in the gas.So high energy photons due to ionic exchange do not come into account and the frequency/energy level of radiation does not vary over a wide range (Modest [2]), and gray approximation remains valid therein.Air (Pr = 0.71) can be chosen for modeling such phenomenon (see also [3,4,6]) so that the ionic exchange does not take place even at high temperatures.Thus we consider a Boussinesq type fluid with gray radiant properties and consider the Rayleigh number only in the range of 2 × 10 5 ≤ Ra ≤ 7 × 10 5 .Attention is focused on investigating the effect of radiation parameters, namely, the optical thickness, scattering albedo, and the Planck number.

Mathematical Formulation
Consider two-dimensional flow of a viscous incompressible absorbing/emitting and scattering Boussinesq type fluid confined in an open rectangular cavity formed by the regions between two horizontal planes at  = 0 and  = , and the two vertical planes at  = 0 and the open end along  = , where  is the length and  is the height of the cavity.The left wall is assumed to be emissive, whereas its temperature is isothermally maintained at   .The temperature of the fluid that enters the cavity region is supposed to be at   at  = 0 (where   >   ).In order to represent the temperature at the open end, we use the subscripts "in" for incoming and "out" for outgoing fluid, respectively.Flow configuration and dynamical boundary conditions are shown in Figure 1.
Here,  and V are the components of velocity along the  and  axes, respectively,  is the fluid temperature, and  is the magnitude of acceleration due to gravity whereas   , ( = , 2, 3) is the emissivity of left, top, and bottom walls, respectively.The unsteady motion of incompressible fluid and the equations for conservation of mass, momentum, and energy in two-dimensional Cartesian coordinate system are given by where ] is the kinematic viscosity,  is the coefficient of volume expansion,  is the density,   is the molar specific heat at constant pressure,  is the coefficient of thermal conductivity,  is the time, and   is the radiation flux vector in two dimensions.We have considered semitransparent fluid with gray radiant properties.The equation of radiant energy for optically thick absorbing/emitting and scattering fluid is thus given by (see [4][5][6]) where   is mean extinction coefficient, Ω 0 is scattering albedo, and  is Stefan-Boltzmann constant.The boundary conditions are expressed as follows: ≥ 0. First we describe the boundary conditions at the solid walls.The velocity components at the solid walls are given by no slip condition The temperature of the left wall is given by whereas the temperature at the nonisothermal walls is given by Finally the boundary conditions at the solid walls for thermal radiation are Here,  is the direction normal to the wall and  = 1, 2, 3, respectively, represent the left, top, and bottom wall as labeled in Figure 1.(See [6].)To set up the open end boundary conditions, notice that the velocity component , V should satisfy the continuity equation at the open end.Moreover, V/ = 0 is a physically more realistic condition for vertical velocity at the open end as compared to V = 0, as the fluid may not necessarily leave the domain horizontally.Further, (/) out = 0 is found to be a suitable condition at the open end for temperature.Most recently, these conditions are also addressed by the same authors (Saleem et al. [22,23] and see the references therein).Further, since it is required to confine the effects of wall radiation within the cavity, this suggests that the divergence of radiation intensity at the open end for thermal radiation should be zero.Thus the boundary conditions at the open end of the cavity are summarized by the following equations In order to transform the problem into stream vorticity form, we define the stream function and vorticity by the following relations: Further introduce the nondimensional variables by using the following transformations: where ,  are the nondimensional coordinate axis, , V are the nondimensional velocity components,  and  are the nondimensional stream and vorticity functions,  is the time,  0 is the reference temperature,  is the nondimensional temperature,  0 is the mean temperature in nondimensional form,  is the dimensionless radiant energy, and  0 is the optical thickness.The treatment of radiative flux divergence in ( 2) is based on two-dimensional differential approximation.The nondimensional form of radiative flux and its divergence are given by the following relations (see also [1,2,[4][5][6]): where  , and  , are the radiative fluxes along the  and  components, respectively.Thus the governing equations in nondimensional form finally become with the following boundary conditions: where are, respectively, the Prandtl number, the Rayleigh number, and the Planck number (also known as conduction-radiation parameter).The temperature boundary conditions for horizontal walls and radiation energy conditions at the three solid walls in nondimensional form are now given by ([1, 2, 4-6]) Heatlines are a measure of the path followed by the heat function across the flow region.Following the formulation of heat function defined by Kimura and Bejan [16], we construct a Poisson-type heat function equation that identically satisfies the energy transport equation (2) given by (see also [15][16][17][18][19]) Now defining  = /Δ as the nondimensional heat function and making use of ( 12), we get the following definition of heat function: The wall boundary conditions for the heatlines are followed by the definition of heat function given by (17) (see also [15][16][17][18][19]): Finally we define the heat transfer from the left wall.When energy transfer takes place in combined mode, the net heat transfer rate is also defined as the combined mechanism of these modes (Lauriat [4]).Thus for combined mechanism of convection and radiation the net heat transfer for solid wall boundaries can be defined by the following relations (see [4][5][6]21]): From here it follows that the average Nusselt number is defined as where  is the element of length  along the wall.Equations ( 13) to ( 16) are solved with the boundary conditions given by ( 17) and ( 19) to solve the system.The heat lines are given by ( 21) whereas the average heat transfer is calculated from (24).

Method of Solution
The flow is developed by coupling of buoyancy term in (14), which is updated by ( 15) and ( 16), stream function is obtained by coupling of ( 13) and ( 14), and finally velocity profile is updated using the nondimensional form of (10).The solution is complemented with the implementation of boundary conditions given in (17) and (19).The stream function equation ( 13) is solved using Successive Over Relaxation (SOR) method with residual tolerance of order 10 −5 .With  as the reference height of the cavity, we have considered a uniform grid of size ℎ = /( − 1), where  is the maximum number of grids along coordinate axes.Throughout the computation we take  = 1.The time dependent vorticity transport equation ( 14) and thermal conduction-convection equation ( 15) are solved using the Alternate Direct Implicit method.The details of the proposed method are also given in [21][22][23]27].A simple discretization procedure given by Bello-Ochende [17] is adopted for the solution of radiation equation ( 16) and heat function equation given by (21).As a simple case, all computations are performed for square cavity.That is,  = 1.Now in order to meet the criteria of convergence to reach the steady state, we define the error bound in the computed value of a variable , by the relation where  , is previously calculated value of a variable for ( × ) grid points.As a demonstration, Figure 2 shows the percentage error in the values obtained from the difference between the computed values of the Average Nusselt number, for different choices of mesh points.It can be seen that the maximum error between mesh sizes 61 × 61 and 71 × 71, in terms of Nu, drops to less than 1%.Thus a mesh of 61 × 61 is considered sufficient for the entire computation.The reduction in relative error justifies the grid independence of the solution.In order to further check the validity for the solution method of the proposed model, we have also revisited the work of Draoui et al. [6] using the solid wall conditions and the temperature and radiation condition given by [6] for the right wall.The comparison of streamlines, isotherms, and heatlines, for Ra = 10 5 , Pr = 0.7,  0 = 1, Ω 0 = 0.2, and  CR = 0.1, is shown in Figure 3. From this figure we assert that the numerical solution we obtained agrees well with the existing literature of heatlines.Intel 1.83 GHz processing machine is used for the entire computation.

Results and Discussion
We have considered combined natural convection and radiation phenomena for participating fluid confined in an open square cavity.The left wall is considered at a temperature higher than that of the fluid entering from the ambient region.The effect of relevant radiation parameters, namely, the optical thickness, scattering albedo, and Planck number, on the flow profile and heat transfer rate has been numerically studied.The result is presented graphically in terms of streamlines, isotherms, heatlines, and heat transfer rate for different values of these governing physical parameters.a force from the incoming fluid, whereas the recirculation is higher in the half open vortex close to the opening.This counteracting mechanism develops a multicellular pattern as shown in Figure 4(c).Figure 5 shows the isotherms at these values of optical thickness.Figure 5(a) shows that, at  0 = 0.0, the isotherms are clustered close to the heated wall.It is due to the dominance of convection in the boundary layer region.However with the increase in optical thickness, the boundary layer thickness increases.This effect is qualitatively in agreement with the case of completely confined enclosures.The radiation energy adds to the total energy of isotherms and thus the high energy isotherms shift towards the core region as shown in Figure 5(b).At  0 = 1.0, the whole flow domain has the isotherms nearly equal to unity due to the dominance of radiant energy, shown in Figure 5(c).This might well be understood also from the heatlines pattern of Figure 6.

Effect of Optical Thickness
Figure 6 shows the heatlines while Ra = 7 × 10 5 , Ω 0 = 0.2, and  CR = 0.1 for (a)  0 = 0.0, (b)  0 = 0.5, and (c)  0 = 1.0, respectively.Obviously, at  0 = 0.0, there is no contribution of radiation in heat transfer and fluid flow.All fluid entering the flow domain absorbs heat from the solid wall and heatlines are negative throughout the flow region with higher values close to the solid wall as shown in Figure 6(a).The flow region, in which the recirculation is weaker (as in Figure 4(b)), is the region which absorbs heat at  0 = 0.5, given in Figure 6(b).The positive part of the heatlines pattern is the region of fluid with higher energy level.Figure 6(c) shows the heatlines pattern at  0 = 1.0.This clearly indicates the high energy level of fluid due to dominance of radiation effect.The heatlines are positive throughout the region, which indicates that the fluid in the region is at high energy level and emitting heat.The effect of optical thickness on heat transfer is described in Figure 7.It can be discerned that the overall heat transfer Nu pattern is dominated by the radiation heat transfer Nu  .In particular, overall heat transfer is almost the same as radiative heat transfer beyond  0 > 0.5.Heat transfer is maximum at  0 = 0.5, when the flow in the core region is very weak.The contribution of convective heat transfer asymptotically vanishes with the increase in the contribution of optical thickness.At  0 = 1.25, both the radiative and convective heat transfer rates coincide with "0, " beyond which the negative heat transfer signifies the energy level of fluid due to the fact that contribution of radiant energy is so high that it serves to reject heat to the solid wall.However, the smooth curve, for  0 > 1.75, shows that the flow pattern and transfer rates are not much affected beyond  0 = 1.75.

Effect of Scattering Albedo.
Scattering albedo is a measure of radiative participation of the fluid.0 ≤ Ω 0 ≤ 1 shifts the fluid regime from perfectly participating to nonparticipating. Figure 8 shows the average total heat transfer against time at Ra = 2 × 10 5 ,  0 = 0.5, and  CR = 0.2 for different values of scattering albedo Ω 0 .The right hand side of (16) signifies that the fluid is perfectly transmitting at Ω 0 = 0.This justifies that with the increase in the value of Ω 0 , the scattering increases, which enhances the effect of convective heat transfer; ultimately Nu  increases whereas the radiative heat transfer decreases and the decrease seems more prominent because Ω 0 = 1 decouples the radiation equation from the thermal energy.Thus, at Ω 0 = 1, the radiative heat transfer is zero.Figure 8 signifies that Ω 0 has no significant effect on the overall heat transfer 0 ≤ Ω 0 ≤ 0.75; rather, there is a considerable decrease only between Ω 0 = 0.75 and Ω 0 = 1 due to the reason that at Ω 0 = 1 the fluid surface is perfectly scattering, and the contribution of Nu  vanishes at this value of Ω 0 .

Effect of Planck Number.
Planck number represents the ratio of conduction to radiation effects on the fluid.The greater the value of  CR is, the lesser the effect of radiation on the fluid will be.Figure 9 shows the streamlines at Ra core region absorbs heat by conduction due to decrease in the energy level of the fluid in the core region.
Finally Figure 11 shows the convective, radiative, and total heat transfer as a function of  CR while  0 = 1, Ω 0 = 0.2, and Ra = 5 × 10 5 .It can be observed that, for sufficiently low value of Planck number ( CR ≤ 0.5), both the radiation and convective heat transfer increase.The reason might be that, for low value of Planck number, radiation is assisted by natural convection.Looking up at (15) and (24), it becomes obvious that the contribution of radiation in total energy and total heat transfer should fade away in the limit  CR → ∞, and for sufficiently large  CR , the energy equation (15) gets decoupled from the radiant energy, whereas natural convection becomes the main mechanism to drive the flow.This is now evident from Figure 11 that, beyond  CR 0.5, there is a decrease in the radiation heat transfer with the increasing values of  CR , and it asymptotically reaches to zero in the limit  CR > 20 in this case, whereas the convective heat transfer rate increases and approaches total heat transfer curve for  CR > 20.Thus in this case for  CR > 20, the overall heat transfer rate is represented by convective heat transfer rate and the contribution of radiative heat transfer vanishes.Therefore we can say that there is a smooth transition of flow from radiation dominated regime to convection dominated regime.This is also physically evident from the definition of Planck number that, for  CR ≫ 1, the conduction effects would become dominant.However, the convective and total heat transfer attain almost a steady value beyond  CR > 15, due to the fact that the contribution of radiation in the total thermal energy becomes less and less as we increase the value of  CR .

Conclusion
An investigation of the effect of radiation and natural convection of viscous incompressible fluid in a square open cavity has been carried out.The main focus was the study of flow pattern and heat transfer rates for different values of radiation parameters 0 ≤  0 ≤ 2.5, 0 ≤ Ω 0 ≤ 1, and 0.05 ≤  CR ≤ 20.
The following conclusions are obtained.
(1) It was seen that with the increase in the optical thickness the strength of flow and energy level of the fluid increases, which ultimately results in the negation of heat transfer rate.Convective heat transfer Nu  decreases asymptotically to zero, whereas radiative heat transfer Nu  first increases in the range 0 ≤  0 ≤ 0.5 and then decreases due to rapid attenuation along propagation in the optically thick media.(2) The total heat transfer is not significantly affected in the range 0 ≤ Ω 0 ≤ 0.75, whereas due to rapid decrease in radiative heat transfer, overall heat transfer decreases near Ω 0 = 1.Prandtl number Ra: Raleigh number : Dimensional temperature (K)   ,   : Maximum and minimum temperature (K)  0 : Average/reference temperature (K) : Dimensional time (s) : Nondimensional time ,V: Velocitycomponents(ms −1 ) , V: Nondimensional velocity components , : Lengthandheightofthecavity(m) , : Dimensional coordinate axis (m) , : Nondimensional coordinate axis.

Figure 2
Figure2shows the result of grid dependence study at Ra = 5 × 10 5 ,  0 = Ω 0 =  CR = 0.5 for the choice of appropriate mesh size ℎ.For any variable , we define the relative error % between the computed values taking different grid points as given by