Numerical Study of Entropy Generation for Natural Convection in Cylindrical Cavities

Natural convection flow into the cylindrical cavities could be used in energy-storage equipment such as the longitudinal cylindrical pipes installed in the back of the natural gas transmission pipeline terminals near cities filled by natural gases.

This storage method is useful, especially in the cold season when gas pressure falls. The integrated flat-plate solar collectors are the other examples related to the present study. In these collectors, the fluid as an absorber has flow into the longitudinal cylindrical pipe.

Heat-transfer processes and devices are inherently irreversible. Energy conservation requires the use of efficient thermodynamic heat-transfer processes, that is, minimization of irreversibility (entropy generation) due to temperature gradients and viscous dissipation [12-16].

There are a fair amount of studies about natural convection heat-transport problems in the literature. The optimal design criteria for thermal systems that minimize their entropy generation recently have been a topic of great interest, especially in the fields related to the geometry of a duct and natural convection in the enclosure. The present paper reports a numerical study of entropy generation in the incompressible natural convection flow within cylindrical cavities. Attempts are made to analyze the influence of the Rayleigh number, aspect ratio, and irreversibility distribution ratio on entropy production. Therefore, it is essential that an engineer focus on the irreversibility of heat-transfer and fluid-flow processes and try to establish the optimal design criteria for thermodynamic systems.

In the last numerical solutions of the heat transfer and fluid flow for enclosures, a code based on the enhanced SIMPLE method [17] couples the pressure with the velocity. In the current work, an artificial compressibility technique was applied to couple the continuity with the momentum equations. The idea of relaxing the incompressibility constraint by adding an artificial compressibility term has been known for a long time and has been used extensively in finite-volume as well as in finite-element approximations of the incompressible Navier–Stokes equations [18-20].

In a finite-volume method, mapping is not needed. Therefore, the scheme is applied directly in the physical domain. Among the various schemes proposed for the flux calculation in the finite-volume model, the Jameson’s numerical method is still of use because of its simplicity [21]. In this work, the Jameson’s flux-averaging finite-volume scheme for space discretization was developed.

One of the methods for solving incompressible flows is based on the artificial compressibility formulation of Chorin [22]. The artificial compressibility approach circumvents the difficulty of the pressure decoupling in the incompressible Navier–Stokes equations by adding a pseudo-time pressure derivative to the continuity equation. Then, the new system of equations can be iterated in pseudo-time until the divergence-free flow field is satisfied. The method can be used both for steady and unsteady flows, and there are a number of papers in the literature that describe implicit and explicit strategies for solving steady and unsteady flow problems in conjunction with artificial compressibility [23-31].

The artificial compressibility method leads to hyperbolic and hyperbolic–parabolic equations for inviscid and viscous incompressible (constant density) flows, respectively. The discretization schemes and solvers developed for artificial compressibility have many similarities with the methods developed for compressible flows. Therefore, numerical developments for compressible flows can be transferred to incompressible flows. Although artificial compressibility has been used extensively for constant-density flows, the development of numerical schemes in the framework of artificial compressibility for variable-density incompressible flows have received scant attention in the literature. Riedel [32] used an artificial compressibility formulation to construct an unstructured finite volume method for the solution of two-dimensional steady viscous, incompressible, reacting flows, while Lin and Jion [33] developed a surface-capturing total variation diminishing method with slope modification for a multi-fluid incompressible Navier–Stokes formulation.

To suppress the tendency for odd and even point decoupling, the artificial dissipation terms are added in the central differencing schemes. In this work, in order to overcome this problem, the discretization of the viscous and thermal conduction terms were simplified greatly using an enhanced scheme similar to the flux averaging in the convective term so that the artificial dissipation was not needed.

2. Governing equations

Fig. 1 shows the schematic view of the cylinder’s geometry and its thermal boundary conditions. According to this figure, the flow has two-dimensional characteristics. Therefore, the cylinder is assumed to have expanded adequately on the z axis. According to this figure, the cylinder cross section has a circle shape when . For other values of the aspect ratio, the cylinder cross section’s shape will be elliptic. All cylinder walls are assumed to be nonslipping. Therefore, under these assumptions, the nondimensionless variables at the cylinder walls are obtained as  (for all walls),  (on the left wall), and  (on the right wall).

                                     (1)

Fig. 1  Schematic view of the cylindrical cavity.

The two-dimensional incompressible viscous flow governing equations in nondimensional form and with artificial compressibility can be written as

where  are dimensionless Cartesian velocity components,  are dimensionless Cartesian axes,  is the dimensionless static pressure,  is the dimensionless temperature,  is the aspect cavity ratio,  is dimensionless time, Pr is the Prandtl number, Ra is the Raleigh number, and  is the artificial compressibility that is given as follows:

3. Formulation of heat-transfer characteristics

The volumetric entropy generation in the flow due to the heat transfer and fluid friction can be written as

The heat-transfer contribution of the volumetric entropy generation of the two-dimensional flow system is

The fluid-friction contribution of the volumetric entropy generation of the two-dimensional flow system is

The dimensionless quantity of these equations can be written as follows:

where  is the ratio between the viscous and thermal irreversibilities in the following equation:

where  is the bulk temperature. The dimensionless total entropy generation is the integral over the system volume of the dimensionless local entropy generation:

The Bejan nondimensional number (Be) is defined as follows:

When , the irreversibility due to the heat transfer dominates. For , the irreversibility due to the viscous effect dominates. For , heat-transfer and fluid-friction entropy generation are equal.

4. Nusselt number

The heat-transfer coefficient in terms of the local Nusselt number  is defined as

where n denotes the normal direction to a plan. The local Nusselt number at the heat wall-plane   is defined as

Therefore, the average Nusselt number is calculated as follows:

5. Grid features

Here, grids are generated by using coding program as an easy method which is used in this investigation. In order to describe this method, first consider a circle with Cartesian xy coordinates, as shown in Fig. 2. In the customary way, this circle could be approximated by quadrilateral ABCD. In this case, four cells in the form of triangles OAB, OBC, OCD, and ODA result from the grid-generation procedure. In the new current method, the circle is approximated by square 1234. In this case, four cells in the form of squares OA2B, OB3C, OC4D, and OD1A result from the grid-generation procedure. 

Fig. 2  Depiction of the current grid characteristics.

Fig. 3  Algebraic rectangular grids for a circular cavity.

Fig. 3 shows the grid-generation processes schematically for a circle cavity (i.e., A = 1) according to the following steps:

(1) As shown in Fig. 3a, the structured grid is generated algebraically for the square cavity. Clustering has been used near the square cavity walls.

(2)  As shown in Fig. 3b, the cells in the vicinity of the curvilinear boundaries are selected completely as the computational domain (i.e., where pieces of them lay out of the physical domain). Then, the remaining cells between the curvilinear corner and the square cavity walls are withdrawn. Finally, all cells are established in the form of a rectangle.  

(3) Now, the required boundary conditions can be imposed onto the extra cells’ boundaries (see Fig. 3b).

6. Finite-volume scheme for space discretization

By integrating from conservation Eqs. (1)–(4) over a control volume , which is bounded by surface , and applying the Gauss divergence theorem, the following is derived:

Fig. 4  Cell view in the finite-volume method.

Therefore, these equations can be arranged in the suitable vector form as follows:

where

The computational domain is divided into rectangular cells (see Fig. 4), and a system of ordinary differential equations is obtained by applying Eq. (23) to each cell separately.  

In order to find the convective terms in the boundaries of the cell (i,j), we choose the custom flux averaging scheme. In this scheme, each quantity, such as  (i.e., the value of  on the boundary denoted by number 1, as shown in Fig. 4), is evaluated as the average of the cells on either side of the face, that is .

Flux averaging is applied here for the viscous and conduction terms. In this method, we need to know the first-order derivatives, such as  and so on (at the cell centers). These functions are obtained from the Gauss divergence theorem. For example, we have . Therefore, these derivatives are saved for all cells as the functions that can be used in the next time step. Thus, each quantity, for example, , is found, such as  and so on.

After discretizing Eq. (23) around the cell (i,j) and considering the cell-centered assumption for the variables, the following are the results:

where  is the area of the cell (i,j).  

7. Numerical solution processes

An explicit classical fourth-order Runge–Kutta integration algorithm was applied to ordinary differential Eq. (25) to find the steady-state condition. To study the effect of the grid size on the solutions, a series of calculations was performed by varying the number of grid points in each direction. For , it was found that, at , the Bejan number changes less than 0.0867 percent as the grid point varies from to . Therefore,  grid points are considered optimal for this study. For the case , the grid points were adjusted according to the aspect-ratio  values. The convergence criterion was used for the mass conservation residue, established as . In this study, the selected fluid flow is air (with Pr = 0.7).

8. Results and discussions

     In order to show the validation of this study, the code was reproduced to solve a similar problem in [8] (i.e.,  and the square boundary cavity). Then, the mean Nusselt number values for different Ra numbers were compared between the present study and the literature, as seen in Table 1. It can be seen that the results obtained from the present model are as good as were expected.

     The curves in Fig. 5 show the histories of the mass conservation residues (Res) for different values of the artificial compressibility parameters ( ) in the cases of different aspect ratios and different Rayleigh numbers. One can see that, in all situations, good convergence speed has been obtained. Also, for each selected value of , in the case of the constant aspect ratio, the residues are converged with the constant number. This means that the value of   cannot change the exactness of the steady-state results. However, the convenience choice of   can be useful for the solution procedure.   

    The streamline is shown in Fig. 6, which depicts three different aspect ratios ( ) in the cases of two Rayleigh numbers ( ). In all figures, it can be observed that, with an increase in the Rayleigh number, the circulation of the fluid flow is affected by the thickness of the hydrodynamic boundary layer. This phenomenon is followed by an increase in the fluid acceleration due to the buoyant force action layers.

    Isotherms have been plotted in Fig. 7. These figures have been sketched for three different aspect ratios (  ) in the cases of two Rayleigh numbers (  ). We know that the temperature gradient causes these phenomena. It can be seen that the variation of the aspect ratio affects the temperature distribution within the cylinder. For a small Rayleigh number (i.e.,  ), the isolines are approximately smooth and vertical. In a high Rayleigh number (i.e.,   ), the isolines are horizontal at the cavity center and vertically clustered near the isotherm’s boundary walls. In the natural convection process, the biggest velocity gradients are found near the walls that have larger thermal gradients due to the action of the buoyant force.

Table. 1  Verifying for average Nusselt number.

Fig. 5  Histories of the mass conservation residues.

Fig. 6  The velocity field for  and  in different aspect ratios ( ).

The effects of the irreversibility ratio and the Rayleigh number on the total entropy generation ( ) are shown in Fig. 9. These figures have been drawn for different aspect ratios ( ). According to these figures, a higher aspect ratio has a greater entropy generation. Also, it can be seen that, with a decrease in the irreversibility ratio, the total entropy generation decreases. In addition, according to these figures, for a higher Rayleigh number, the values of the total entropy generation limit the constant value.

Fig. 7  Isotherms for  and  in different aspect ratios ( ).

 Fig. 8 shows isobars for different aspect ratios and Rayleigh numbers. The figure clearly illustrates how a stratified pressure field is formed. In the case of both a higher aspect ratio and Rayleigh number, the isobars are horizontal.

Fig. 8  Isobars for  and in different aspect ratios ( ).

     The effects of the aspect ratio, irreversibility ratio, and the Rayleigh number on the Bejan nondimensional number (Be) are shown in Fig. 10. Different behaviors are shown for the variation of the Bejan nondimensional number when the irreversibility ratio changes. It can be seen that, with a decrease in the irreversibility ratio, the Bejan number increases. This phenomenon obviously can be seen with a high Rayleigh number. As mentioned previously, when , the irreversibility due to the heat transfer dominates. For , the irreversibility due to the viscous effect dominates. For  , heat-transfer and fluid-friction entropy generation are equal.

Fig. 9  The effects of the irreversibility ratio ( ), aspect ratio ( ), and the Rayleigh number on the total entropy generation ( ).

Fig. 10  The effects of the irreversibility ratio ( ), aspect ratio ( ), and the Rayleigh number on the Bejan number ( ).

Fig. 11  The effects of the aspect ratio ( ), irreversibility ratio ( ), and the Rayleigh number on the total entropy generation ( ).

     The effects of the aspect ratio and the Rayleigh number on the total entropy generation ( ) are shown in Fig. 11. These figures have been drawn for different irreversibility ratios  . According to these figures, a higher irreversibility ratio has a bigger entropy generation. Also, it can be seen that, with an increase in the aspect ratio, the total entropy generation increases. In addition, according to these figures, with a higher Rayleigh number, the values of the total entropy generation increase exponentially as a function, and with a lower Rayleigh number, the values of the total entropy generation limit the constant value.

The effects of the aspect ratio, irreversibility ratio, and the Rayleigh number on the nondimensional Nusselt number (Nu) are shown in Fig. 12. Different behaviors are shown for variations of the Nusselt number when the aspect ratio changes. Due to the low temperature gradient, the Nusselt number is not high when there is a low Rayleigh number. When the temperature gradient increases, the Rayleigh number also increases, and as a result, the Nusselt number enhances exponentially as a function. When the Rayleigh number gradually increases, the buoyancy force intensifies, and finally, the dominant force and the viscosity of the fluid will begin to move. Increasing the Rayleigh number in all cases will increase the amount of the buoyancy force, or the turbulence of the flow within the chamber.

Also, one can try to correlate functions for the Nusselt number through the curves shown in Fig. 12. For example, the following equation results under the related conditions:

Fig. 12  The effects of the aspect ratio ( ) and the Rayleigh number on the Nusselt number.

 for , and ;

  for , and ; and

 for , and .

9.  Conclusions

     In this paper, an explicit finite-volume model was developed to study incompressible natural convection incylindrical cavities. An artificial compressibility technique was applied to couple continuity with momentum equations. In order to find the steady-state solutions, an explicit classical fourth-order Runge–Kutta integration algorithm was applied to the ordinary differential, obtained from a cell-centered finite-volume discretization scheme (see Eq. (25)). In this work, the aspect ratio, the Rayleigh number, and the irreversibility distribution ratio were limited as , , and , respectively. It was seen that, with a decrease in the irreversibility ratio, the total entropy generation decreases. Different behaviors are shown for variations of the Bejan nondimensional number when the irreversibility ratio changes. It was seen that, with a decrease in the irreversibility ratio, the Bejan number increases.

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