ARTICLES

Conjugate heat transfer in a two stage trapezoidal cavity stack

A.M. Aramayo^†, S. Esteban^‡ and L. Cardón^‡

† Departamento de Matemática, aaramayo@unsa.edu.ar
‡ Departamento de Física, cardon@unsa.edu.ar
Facultad de Ciencias Exactas, Universidad Nacional de Salta.
Av. Bolivia 5150, 4400, Salta, Argentina

Abstract —The flow pattern and heat transfer in a pair of cavities stacked one over the other, with a separating solid plate in between, are numerically studied. The configuration idealizes two trays in a multiple effect desalinator. The work focuses on the computational modeling of the two conjugate recirculations in the cavities, conductively coupled through the thin separating plate. The effect of the plate physical parameters, thickness and conductivity, on the heat transfer are also analyzed. Plates made of glass, steel and aluminum of 2, 3, 4 and 6 mm are studied. Cavity flow and temperature patterns, depicted by streamlines, velocity fields, and isotherms are analyzed for steady state. Attention was paid to the temperature variation over, and heat flow through the separating plate. Global Nusselt numbers are calculated for the bottom side of the lower cavity. Data are obtained for air as the working fluid (Pr=0.7) in the range 10³<Ra_H<10⁷.

Keywords —Cavity. Natural Convection. Conjugate Heat Transfer.

I. INTRODUCTION

The physics of basin type solar desalinators has attracted the interest of many researchers in the past, since the first one was built at Las Salinas, Chile, in 1872 (Duffie and Beckman., 1990). This type of desalinator consists of a closed cavity, of triangular or trapezoidal cross section; identified as the tray. In its bottom pan (the basin) is a small quantity of brine that is to be heated and evaporated in order to separate the water from the salts that may be present in it. Humid air in the interior of the tray is convected towards the cooler inclined cover of the cavity, where clean water is naturally removed by gravity, after condensation.

Natural convection in cavities related with the present and similar problems have been studied. Poulikakos and Bejan (1982), Asan and Namli (2000), Holtzman et al. (2000), Del Campo et al. (1988), numerically studied the convection in an attic space, a pitched roof, and other triangular cross sections respectively. Flack et al. (1979) and Flack (1980), experimentally addressed convection in triangular and trapezoidal cavities. Palacio and Fernández (1993) and Rheinländer (1982) studied the turbulent regime and mass transfer in solar still desalinators. Esteban et al. (2004) studied the heat and mass transfer in a trapezoidal cavity.

The two tray configuration posses a conjugate problem between the two recirculating flows coupled by heat conduction through the separating plates between trays. The recirculations are driven by the hot and cold plates of each cavity. The same problem was first addressed by Aramayo et al. (2004) in the case of a glass separating plate. Here we extend these results and we analyze the effect of the properties of the plate materials, considering glass, steel and aluminum. The materials cover a three order of magnitude range for the thermal diffusivity. We have also considered various commercial thicknesses, 2, 3, 4 and 6 mm.

Conjugate convection with conduction is an interesting problem in its own and it is receiving increasing attention since Anderson and Bejan (1980,1981) studied the problem of two fluid reservoirs separated by a vertical thin wall. Other conjugate natural convection problems have been addressed more recently. Polat and Bilgen (2003) studied the heat transfer from a wall to an open shallow cavity, Dong and Li (2004) studied the conjugate heat transfer between the top of a square cavity, a solid cylinder inside it, and the fluid in between. Aydin (2006) has studied double pane windows.

In this paper we address the conjugate heat transfer problem in a two stage trapezoidal cavity stack and the laminar convection of dry air inside both cavities.

In section II the mathematical description of the problem and the numerical method employed is presented. In section III the flow and temperature patterns, the effect of the Rayleigh number, the separating plate thickness and material are examined. Plate temperature and cavity temperature profiles are also presented. Global heat transfer data is discussed in section IV. Conclusions are presented in section V.

II. MATHEMATICAL MODEL

A. The fluid model

The physical system under consideration, as shown schematically in Fig. 1, is a two-dimensional enclosure. Flow and heat transport in the fluid obeys the continuity, the Navier Stokes, and the thermal energy equations. The Boussinesq approximation is applied as usual: constant properties apart from the density in the body force term. Incompressible flow results. Being u and v the velocity components in the x and y directions, respectively, and T the temperature, the mathematical model is described by the following equations

Continuity

(1)

Momentum equation for the x direction

(2)

where v is the kinematic viscosity of the fluid.

Fig. 1. Calculation domain and boundary conditions.

Momentum equation for the y direction

(3)

where g is the acceleration of gravity, β_T volumetric expansion coefficient, T_m is the reference temperature at which all properties are to be evaluated.

Energy equation

(4)

where ρ is the density, k is the thermal conductivity and c_p is the thermal capacity of the fluid.

The above equations are non-dimensionalized applying the following transformations

where H is a properly chosen height scale. Moreover, T_m and ΔT are the mean temperature and the maximum temperature difference between T_H and T_C (Fig. 1). For the velocity scale, u_o, the characteristic velocity for a side differentially heated cavity (Bejan, 1995) is used

This scale is optimal for cavities with a lateral heated wall.

The selection of a proper length scale H is not straightforward due to the complexity of the configuration. It depends on the flow and heat transfer regime. For all calculations the length scale is taken as the height of the rectangular domain from which the actual configuration is carved off.

Introducing the above transformation, the Eqs. (1) to (4) become

	(5)
	(6)
	(7)
	(8)

where the Prandlt and Rayleigh number are defined as

and

B. The separating plate model.

Because the continuity, momentum and energy equations are to be solved for the whole domain, disregarding the material involved, previsions have to be taken to ensure that the solid part, in particular that corresponding to the separating plate, behaves as a rigid, not moving, and heat conducting material. To impose these constraints over the hydrodynamic problem, a very high viscosity is assigned to the solid parts. This produces a blocking effect that results in a null velocity field. Consequently, the convective term in the energy equation for the solid parts is set to zero, leaving us with the conduction equation as required

(9)

where the subscript s means solid properties.

C. Calculation Domain, Initial and Boundary Conditions

The calculation domain corresponding to a pair of trays is shown in Fig. 1. Boundary conditions and names are specified there. Initial conditions are zero velocity field and constant temperature in the whole domain. The initial temperature is set to be the average between the two boundary temperatures T_C and T_H.

A rectangular domain of L×H is discretized with a uniform structured grid of 100×315 control volumes. The aspect ratio of each control volume is set to match the inclination of the separating plate between the cavities. Only two cavities are considered and to clip them off from the rectangular domain, two triangular sections were blocked off on opposite corners, imposing a very high viscosity over them. Also, high conductivity is assigned to them in order to allow for the constant boundary temperatures, cold, T_C, and hot, T_H, imposed over the horizontal limits of the rectangle to be transferred to the upper and lower inclined plates of the stack (Aramayo et al., 2003). The vertical walls on the active part of the domain (excluding the blocked off triangular regions) are set to be adiabatic. The height of the rectangle is H=1, and the aspect ratio of the complete domain is B=H/L=1. Individual cavities have an aspect ratio A=h_R/L=0.65 and a right to left height ratio C=h_R/h_L= 0.65/0.02=32.5. Plates of dimensionless thickness of 0.002, 0.003, 0.004 and 0.006 are studied.

The resolution of the problem was done with Professor's Patankar program (Patankar, 1997) based on SIMPLER algorithm (Patankar, 1980). We use a relaxation parameter of 0.8, and a dimensionless time step of 0.1. As much as 5 internal iteration per time step were done. We integrate until steady sate is achieved.

The fluid and plate related physical quantities, such as α, ρ, k, c_p are evaluated at 80 C. Their values are shown in Table 1.

Table 1: Values of the physical properties.

All calculations are done with the same spatial and temporal discretization, and are advanced until the non-dimensional time reaches .

C. Separating plate discretization.

Implementation of the plate is done by means of assigning solid material properties, i.e. high viscosity, to a strip of control volumes with staircase like profile, as it is shown schematically in Fig. 2.

Fig. 2. Discretization of the separating plate.

Thus, the thickness of the plate depends on the number and height of the control volumes on the strip. The inclination of the plate matches the aspect ratio of the control volume. In this way the setting of the plate's thickness is very much grid dependent. To get around this problem and fix the discretization parameters for all the considered thickness cases, we adopt a unique number of vertical control volumes for the plate strip, which results in a unique virtual, computational or discrete thickness, e_D. Then, we adjust the properties of the plate, density, specific heat and conductivity, in a way that the heat capacity and thermal resistance of the virtual plate coincides with those of the real plate. This strategy was first used by Aramayo et al. (2004) in a study made for a glass separating plate. The discrete or virtual diffusivity is related with the discrete thickness, the diffusivity, and the thickness of the plate by

(10)

After non-dimensionalization of Eq. (9), the conduction equation for the plate stands as

(11)

The thermal resistance concept assumes a one-dimensional heat flow through the plate. On the contrary, the actual problem may have a small two-dimensionality depending on the inclination of the plate and on the characteristics of the fluid flow. On the conductive regime (low Rayleigh number) some heat may flow longitudinally along the separating plate. At high Rayleigh numbers, isotherms become parallel to the plate. Heat crosses the plate in its normal direction, then the resistance concept may be applied, as described, without error.

III. FLOW AND TEMPERATURE PATTERNS

A. Flow pattern

Figure 3 shows the streamlines and isotherms for Ra_H=10⁴ for a 2 mm thick separating plate, for the three studied materials. We focus on the lower tray for the flow features description.

Fig. 3. Flow patterns and isotherms for a 2 mm plate at Ra_H=10⁴, e=2 mm.

The first pair of plots are for the aluminum separating plate, the second for the steel one and the third for glass separating plate. In each tray, a unique counterclockwise recirculating flow over the hole cavity is observed for the three cases. The sense of the circulation can be observed in the vector field plots shown on Fig. 5. We interpret that a lateral heating regime (typical of enclosures heated from the side as opposed to heated from below (Bejan, 1995)) is taking place, driven by the vertical projection of inclined hot lower plate. The bottom tray recirculation is driven upwards by the lower hot inclined plate while the upper tray recirculation is driven downwards by the upper cold inclined plate.

Fig. 4. Flow patterns and isotherms for a 2 mm plate at Ra_T=10⁶.

Fig. 5. Velocity fields and isotherms for the glass e=2 mm case.

The isotherms, being almost parallel among themselves, show the same slope as that of the lower and upper plates of the stack. This means that the flow regime is dominated by conduction in the normal direction to both inclined driving plates.

Following a typical isotherm from bottom to top, the effect of the separating plate shows up as a small perturbation, (a small step), in the slope of the isotherm. Normals to the isotherms point to the direction of the heat flow, then, the perturbation on the isotherms shows the effect caused by the plate on the heat flow. For steel and aluminium plate an upward step is observed while for the case of the glass plate a downward step is observed.

For higher Rayleigh numbers, 10⁵ (Fig. 4) and beyond, many cells are observed. The size of each cell adapts itself to the local height of the cavity. There is a main pair of cells on the left and a stagnation zone on the right of the cavity. The big cell on the left is driven by the hot inclined lower plate. It flows counterclockwise and takes more than half of the horizontal length of the cavity, depending on the Rayleigh number. Its upward arm diffuses momentum to the second cell that runs in the clockwise direction. This cell flows, in the vicinity of the hot plate, in the opposite direction of the buoyancy force, which confirms that its movement is driven by diffusion of momentum from the first cell.

In the case of the steel separating plate the stagnation zone observed in the other cases leaves the way to an additional very weak recirculation.

The upward jet produced by the co-directional arms of the main pair of cells, gives rise to an easy to note a deflection on the isotherms, which reflects the upward convection of heat. A second downward plume is also noticed, and even a third, upward one, in some cases.

In all the studied cases, a noticeable temperature gradient is observed in the separating plate. As expected, for the same thickness, the greater gradient corresponds to glass plate, which is the more thermal resistant.

The effect of the thickness of the separating plate on the flow pattern and isotherm map was studied for the three materials. Isotherms were studied for the steel case at Ra_H=10⁵. The thickness of the plate increases the relative resistance to the cross through flow of heat and allows for less lateral conduction. The same is observed for all the studied Rayleigh numbers.

B. Effect of the Rayleigh number.

The effect of the Rayleigh number on the heat and fluid flow is shown in Fig. 5 for a 2 mm glass separating plate case. Velocity fields and isotherms are shown, from top to bottom, for Ra_H=10⁴, Ra_H=10⁵, Ra_H=2.5×10⁵ and Ra_H=10⁶. Isotherms show more clearly the transition from the conductive regime to the convective regime that occurs between Ra_H=10⁴ and Ra_H=10⁵. Velocity fields show an intensification of the flow in the vicinity of the cold and hot plates as the Rayleigh number increases. The heat plume that began to appear at Ra_H=10⁵ becomes stronger with higher Rayleigh numbers. Over the separating plate, small lateral heat transfer can be observed alongside the plate at Ra_H=10⁴, the small step on the isotherms. For higher Rayleigh numbers this becomes a long downward step along the length of the separating plate, leaving isotherms inside the plate almost parallel to the plate itself. This indicates that as the Rayleigh number increases the heat flows dominantly in the plates' normal direction. Consequently, the conductivity of plate material is increasingly less important.

With increasing Rayleigh number, the velocity field is stronger. A second weak recirculation pair of cells appears in the otherwise stagnation zone near the tip of the cavity. Cells keep the same sense of recirculation for all Rayleigh numbers.

C. Plate temperature profiles.

Figures 6 and 7 show the temperature profiles all along the separating plate length for 2 mm and 6 mm thick plates, respectively. Each figure shows the data for steel and glass plates for Ra_H=10⁴ and Ra_H=10⁶. Bottom (T_bottom) and top (T_top) plate surface temperatures are shown. The plate temperature drop (ΔT_plate) may be calculated from the data shown in these figures as ΔT_plate= T_bottom-T_top.

Fig. 6. Temperature distribution in the separating plate, e=2 mm.

Fig. 7. Temperature distribution in the separating plate, e=6 mm.

For Ra_H=10⁴ all the observed temperature profiles are linear. This is consistent with the evenly spaced isotherms described before. For 2 mm, Fig. 6, the temperature drop is not appreciable for the steel plate, while for the glass, the temperature drop is about 4% of ΔT. For 6 mm, Fig. 7, temperature drops of 3.5% and 22% are appreciated for steel and glass respectively. This behavior is partially and qualitatively explained by the contribution of conductivity and thickness on thermal resistance, but the actual values of the temperatures reached at the plates surfaces depend also on the cavity resistance. This behavior was described by Bejan (1995) as a "floating" temperature between the two convective layers.

For Ra_H=10⁶, the profiles show the effect of convection and they are no longer linear. For 2 mm steel glass plate, there is a barely appreciable temperature difference across the plate. In the center, the temperature is equal to the mean reference temperature. The profiles show, to the left of the plate, the heating effect of the bottom tray main recirculation. To the right, they show the cooling effect of the upper tray main recirculation.

In the lower tray, the upward jet between the two counterroating cells, impinges the plate approximately at x=0.6, conveying heat to both sides of this location and heating the plate from x≈0.2 to x≈0.7. At x≈0.8 the opposite effect of the second downward intercell jet is also noticed. At the right, from x≈0.9 to x=1.0 the flow is almost stagnant and the general flow regime may be considered as conductive. At the left, the flow also stagnates, as it turns around the upper corner of the tray. The plate there, from x=0.0 to x≈0.2, is conductively cooled from the above tray.

D. Vertical temperature profiles.

Cross cavity temperature profiles for a 6 mm steel plate are shown in Fig. 8 and 9 for various horizontal locations and Rayleigh numbers. In these figures, the flat line that frames the cavity profile corresponds to the blocked triangular parts of the domain that have been cut off, and they have no meaning.

Fig. 8. Temperature profiles for Ra_H≤10⁵

Fig. 9. Temperature profiles for Ra_H>10⁵.

For Ra_H=10⁴, and even for Ra_H=10⁵, the piecewise quasi-linear profile indicates that a low Rayleigh conduction regime takes place at both cavities as well as in the separating plate. For higher Rayleigh numbers, Ra_H=2.5×10⁵ to Ra_H=10⁶, the cross cavity temperature profiles are no longer linear.

IV. HEAT TRANSFER

A. Global Nusselt number.

Global heat transfer is calculated. In the adjacency of the hot (or cold) plate, heat transfer can be considered to be convective or conductive as

(12)

where is the convective heat transfer coefficient, L' is the length of the bottom plate, and η is the normal direction coordinate. Non-dimensionalizing with Q_k=k(T_H-T_C)L'/H, (the heat transfer due to conduction in the absence of movement), the dimensionless heat transfer is given by the Nusselt number, and can be calculated as the mean dimensionless temperature gradient as

(13)

The main features of the temporal evolution of the global Nusselt number towards steady state are studied. As a typical example, Fig. 10 shows this evolution for the lower plate of the bottom cavity, for the 4 mm glass and aluminum plates cases, for Rayleigh numbers from 10⁴ to 10⁶.

Fig. 10. Variation of the global Nusselt with the Rayleigh for e=4 mm.

The first instants of this evolution (from t=0 to t=10) have a similar trend for all Rayleigh regimes. A steep decrease of the Nusselt number is observed. This shows the response of an initially isothermal fluid to the impulsive imposition of a temperature boundary condition. After a minimum value is reached the subsequent behavior depends on the regime characterized by the Rayleigh number.

For low Rayleigh number (Ra_H=10⁴) the Nusselt number reaches a steady state value of order 1 immediately after the boundary condition is applied and the oscillation it caused in the boundary layer attenuates and reaches its final thickness. This order of magnitude indicates a conduction dominated regime.

For higher Rayleigh numbers, after reaching the minimum, a Nusselt number rebound is observed. A local maximum is reached to fall down again smoothly, toward the steady state value. The rebound and immediately subsequent evolution is related to the build up of a boundary layer that grows and contracts until it reaches its equilibrium thickness (Siegel, 1958). All the process has many features in common with other natural convection cavity flows and it has been studied by many authors (Nicolette et al., 1985; Cardón, 2003).

The Nusselt number is depend on the Rayleigh number in a way we describe in the next section. This dependency is affected by the plate`s thickness and conductivity. For Rayleigh (Ra_H=10⁶), the Nu number is 68% greater in the case of glass compared to aluminum one.

The effect of thickness is studied for Ra_H=2.5×10⁵ and the results are shown in Fig. 11 for the glass and aluminum plates cases.

Fig. 11. Global Nusselt number on the lower plate at Ra_H=2.5×10⁵.

As the thickness of the plate increases the Nusselt number diminishes, being the decrement much more important in the case of the glass plate. Taking the value for the 2 mm plate as a reference, the relative decrement of the Nusselt number for the 6 mm plate is 2% and 48% for the aluminum and glass plates respectively.

B. Steady state Nusselt Correlations

Figures 12 and 13 show the Nu vs. Ra_H data for the two metallic plates cases and the glass case respectively. A range of thickness from 2 to 6 mm is covered. All plots show that the Nusselt number increase with Rayleigh number and decrease with thickness of the plate. In the metallic plate cases, the plate material or thickness make little difference on the heat transfer and all data lies almost in a unique line in the graphic. Instead, for the glass case, the plate thickness notoriously affects the heat transfer and the tendency lines spread over the graphic.

Fig. 12. Nu vs. Ra_H dependency for the metallic plate cases.

Fig. 13. Nu vs. Ra_H dependency for the glass plate cases.

The behavior is fitted to a Nu vs. Ra_H correlation of the power law form

with coefficients given in Table 2.

Table 2. Nu vs. Ra_H correlation coefficients.

For the metallic plate cases, the two correlations have almost the same exponent, 1/5. In the glass cases, the thickness of the plates affects both C and n coefficients. A unique correlation that takes into account the material and the thickness of the separating plate, as well as the aspect ratio of the cavity would be desirable, but this would require a complete theoretical framework that qualitatively explains the physics of the convection in each individual cavity in the tandem and a theory to explain the physics of the two cavities and the separating plate acting together. This theory is yet under development.

V. CONCLUSION

The conjugate heat transfer between two stacked trapezoidal cavities separated by an inclined plate is numerically studied. The problem is relevant to understand the physics of heat transfer and fluid flow in the interior of a stack of cavities.

The study is conducted in a Rayleigh number range that leads to conductive and convective flow regimes. Convective regime is reached for Ra_H=10⁶ for conductive plates, with maximum Nusselt number of 5 and 7 for steel and aluminum respectively. Conductive regime (Nusselt number of order one) is found for the glass separating plate.

The temperature profile over the separating plate explains lower heat transfer ratios encountered in individual tandem cavities, in comparison with stand alone cavity under constant upper hot and lower cold plate boundary conditions.

Considering each cavity alone, the temperature of the separating plate depends on the flow in the adjacent one. Then, the temperature of the separating plate may be considered as a complex boundary condition: hot at the right side end and cold at the left side end. In average, there is a smaller driving temperature difference. This results in lower heat transfer rate and also in flow stagnation in the short side part of the cavity.

The flow of fluid in the cavities shows a main recirculation cell and, at higher Rayleigh numbers, two minor cells. The multicellular pattern observed in the stand alone cavities, mostly related to horizontal layers of fluid heated from below, gives place to a unicellular convective flow. The last is typical of a laterally heated cavity. Here, the lateral heated wall can be interpreted as the projection of the inclined upper and lower plates. These aspects of the flow deserve further study.

The temperature difference across the thickness of the plate, ΔT_plate, is investigated and depends on the plate conductivity and thickness, and on a non-linear balance between the plate and the cavity thermal resistance. The cavity flow and its thermal regime depend on ΔT-ΔT_plate For the glass plate 6 mm thick ΔT_plate is 25% of the total temperature difference over the entire stack. Conducting plates, like aluminum and steel, less thermal resistant, do not affect significantly the heat transfer in the cavity.

At Ra_H=10⁶ the plate thickness accounts for a 2% Nusselt number difference for the metallic plates. For the glass plate, this difference grows to 48%. The result means that the thermal resistance of the plate cannot be ignored for relatively thick poor conducting plates like glass plates, although they may be ignored for high conducting materials like steel and aluminum.

Heat transfer rate data are correlated with a power law form for each separating plate. Further work on the conceptual model is required to obtain a unique theoretically based correlating formula that takes into account: the number of cavities in the stack, the aspect ratio of the cavities, and the material and thickness of the separating plates.

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Received: August 23, 2006.
Accepted: March 23, 2008.
Recommended by Subject Editor Rubén Piacentini.