Entropy generation in Marangoni convection flow of heated fluid in an open ended cavity

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Abstract

Numerical simulations have been conducted to study the influence of thermocapillary forces on the natural convection of a Newtonian fluid contained in an open cavity. The heated molecules of the fluid are allowed to enter the cavity region to cause the convection flow. The top horizontal surface of the cavity is assumed to be flat and free. The non dimensional partial differential equations that govern the flow and thermal fields are solved using alternate direct implicit (ADI) method together with successive over relaxation (SOR) scheme. The simulation results show that the active spot of maximum entropy generation depends on the magnitudes of Grashof number and Prandtl number, respectively. Entropy generation rate increases with the increase in the Marangoni number. The entropy generation number increases from 1 to 1.5 in the range 0  Ma  175 for Ec = 10−4, Gr = 2 × 103, Pr = 0.054. The Eckert number is considered in the range of 0  Ec  0.0002.

Introduction

Immense amount of study has been conducted for thermogravitational natural convection in rectangular enclosures. The basic examples of such studies are given in [1], [2], [3]. However, in the presence of a free top surface, the temperature gradient of the surface tension at the free surface may also induce the flow, called the thermocapillary flow or Marangoni convection. Thermocapillary influence is significant in small scale systems. Its application in industry includes crystal growth, liquid melting and resolidifying, glass manufacturing and welding. Since the flow velocity in such kind of flows generally remains less than the sonic velocity, it is reasonable to consider incompressible fluids in such kind of flows. Earlier investigations suggested that the convective flow is significantly modified by the presence of large Marangoni number. Zebib et al. [5] studied high Marangoni convection in a square cavity with free surface and showed that for sufficiently low surface tension Reynolds numbers, Re = Ma/Pr, a driven cavity type flow structure appears with concentration of streamlines near the cold wall for Pr < 1, and concentration of streamlines occurs near the hot wall for Pr > 1. Srivinasan and Basu [6] used ADI formulation to study thermocapillary flow in a cavity during laser melting. They used the assumption that the free surface is flat, which is valid when the capillary number, Ca=Δσ/σ¯<<1, where Δσ is maximum difference in surface tension and σ¯ is the mean value of surface tension. They showed that for low Marangoni number, conduction remains the main mechanism of the heat transfer. Burgman and Ramadhyani [7] studied the combined mechanism of buoyancy and thermocapillary driven flow in a square cavity. They established the result that both the buoyancy and surface tension effects assist the flow in the same direction and oppose the flow in the opposite direction. Moreover normalized average Nusselt number increase with the increase in Rayleigh number (Gr × Pr). Carpenter and Homsy [8], discussed the transition of flow in a cavity, from buoyancy dominated to thermocapillary dominated regime, by defining a parameter G = Ra/Ma, where 0  G  1 showed the dominant thermocapillary regime whereas at G = 10, buoyancy effects were dominant. Hadid and Roux [9] conducted a detailed study of the combined mechanism of buoyancy and thermocapillary driven flows in differentially heated cavities. They showed that at a given value of aspect ratio, the maximum number of vortex cells formed was a characteristic of Grashof number and surface tension Reynold number. Moreover, below certain value of Grashof number, the thermocapillary effects do not tend to destabilize the flow phenomenon. However, the counteracting mechanism of combined flow adds to the complexity of flow pattern. Chen and Hwu [10] suggested that thermocapillary flow may undergo oscillatory motion when the Marangoni number is larger than a certain critical value. This critical value largely depends upon the degree of surface deformation at the free surface and the convection between the interface and the ambient fluid. Rudraiah et al. [11] studied buoyancy and thermocapillary driven flow in the presence of a Magnetic field, and showed that heat transfer increases with the increase in Marangoni number, whereas it decreases on inclusion of Magnetic field. A comparison of pattern formation, between the experimental observations and numerically computed results of the transient thermocapillary flow in a rectangular cavity has been made by Sakurai et al. [12]. Hossain et al. [13] studied buoyancy and thermocapillary driven flow with heat generation in the presence of Magnetic field. However, the emphasis was laid on the study of the effect of magnetic field. It should be pointed out at this end that an excellent review paper on the appropriate way of viewing the Marangoni effect is presented with examples by Tadmor [14].

The heat transfer study for the case of open rectangular cavity, compared to the closed rectangular domains is comparatively more complex, due to the application of the boundary conditions at the open end. A steady state pattern in the open ended cavity results that consists of fluid entrained from one half of the open end and the expulsion of fluid from the other half of the open end. Thus open end aids in stabilizing the flow pattern in the sense that it not only permits the accelerated fluid to escape freely out of the flow region, but also aids in the entrainment of fluid in the vacant regions. Chan and Tien [15] considered both the cases of open rectangular cavity with extended and confined domains, and showed that the flow in the core region was insensitive to the way the approximate boundary conditions were set at the open end, particularly when the opening is sufficiently away from the core region. Moreover by confining the computation within the domain of the open cavity, they observed that the approximate boundary conditions of the opening made no significant impact on the heat transfer characteristics of the vertical wall. They also compared their numerical results with experimental data to check the validity of the temperature conditions applied at the open end. Vafai and Ettefagh [16] conducted a detailed study of instabilities in buoyancy driven flows in open cavity. They used combination of ADI method with second upwind differencing technique to solve the system of equations. A detailed description of the fluid flow, while it shifts from steady to the oscillatory, and then to turbulent regime, is given in their work. Angirasa et al. [17], [18] also confined the computational domain within the cavity itself, and used ADI method together with SOR scheme for the solution of stream-vorticity formulation. For vertical velocity at the open end, they used the condition v¯=0, whereas Chan and Tien [15] suggested that v¯/x¯=0, which is practically more realistic than the condition v¯=0, since it is not necessary that the accelerated fluid would leave the flow region horizontally. However comparing their results in [15], [17], one can see that both these conditions made no significant difference for large Grashof numbers. A discussion about open end boundary conditions has also been made by Vafai and Ettefagh [16].

Entropy of a thermodynamical system refers to the unavailability of useful work. Physically entropy generation is associated with thermodynamical irreversibility, which is a common phenomenon in all kinds of heat transfer designs. Greater rate of entropy generation in any thermal system destroys the useful work and greatly reduces the efficiency of the system. Earlier studies reveal that the entropy generation rate increases with the increase in buoyancy parameters. Magherbi et al. [19] first studied the transient behavior of entropy generation at the onset of convection, and compared the cases of low and high Rayleigh numbers. They considered the irreversibility coefficient, say λ in the range 10-5λ10-1. They showed that maximum entropy generation number increases with increase in Rayleigh number. Erbay et al. [20] studied the entropy generation in a square enclosure with two cases of completely and partially heated left vertical wall. For both partially and completely heated and cold side walls, the concentration of isolines of entropy generation along the walls were found to increase with increasing values of Rayleigh number. Moreover entropy generation because of viscous effects was found to be negligible as compared with that of thermal effects, whereas the viscous irreversibility coefficient was considered as low as 10−10. Entropy generation study in rectangular channel has been made in [21], [23]. Mahmud and Fraser [22] conducted Entropy generation analysis in Magnetohydrodynamic convection in porous cavity. Further studies were also made by Costa [24], Mourad et al. [25], Ilis et al. [26] and Oliveski et al. [27]. More recently, El Jery et al. [28] studied the effect of externally applied magnetic field on entropy generation in natural convection flow in a square cavity discussed different cases of irreversibility coefficients in the range 10-4λ10-2. They showed that in the absence of magnetic field, viscous irreversibility decreases whereas thermal irreversibility increases with the increase in Prandtl number. Boundary layer study of Marangoni convection includes the work of Pop [29] and Chamkha [30].

In modern thermal engineering, the optimal designs of thermal systems are subjected to the minimization of entropy generation in a system. One of the modern aspects of thermodynamics study is to focus on specifying the origin of entropy generation in a thermal system. This can be achieved by identifying the active spots of entropy generation in a particular geometry. In the case of closed rectangular enclosures, thermal irreversibility dominates the viscous irreversibility [20]. The motivation of present study arises in identifying the active spots of entropy generation, and investigating the effect of surface tension gradient on the irreversibility in the open rectangular cavity. As to authors knowledge, study of entropy generation rate and Marangoni convection, when the flow is induced due to the energy of the heated fluid, is not available in the literature. Thus we consider the case of a square cavity whose right wall is kept open for flow entrainment and exit, the top surface is free and we shall study the effect of flow parameters on bottom and left cold solid walls. Since the thermocapillary convection has its physical application in liquid melting, we consider the case of liquid melts. That is, the Prandtl number as low as 0.054, which is appropriate for liquid metals and semi conductor melts. In view of the study of Marangoni convection [6], [7], [8], [9], [10], [11], [12], [13], [14], we discern that the thermocapillary forces can significantly effect the temperature and flow field. Thus entropy generation, being a function of temperature and velocity, will also be affected by the presence of thermocapillary forces, as we shall see in the coming sections.

Section snippets

Mathematical formulation

Here we consider unsteady two-dimensional buoyancy and thermocapillary natural convection flow of a viscous incompressible fluid confined in an open square cavity formed by the regions between two horizontal planes at y¯=0 and y¯=H, and the two vertical planes at x¯=0 and the open end along x¯=H, where H is the height of the cavity. The temperature of left vertical wall and the bottom wall is isothermally maintained at T¯C whereas the fluid that enters the cavity region is at a relatively

Method of solution

For numerical simulation, here, successive over relaxation method with residual tolerance of order 10−5 is applied on the stream function Eq. (13). Considering H to be the reference height of the cavity, we take the uniform mesh size h = H/(jmax), where jmax is the maximum number of equi-spaced intervals along coordinate axes. The relaxation parameter, say ‘ε’ is obtained from the relation (see [4]):ε=21-(1-ξ)ξ,whereξ=cosπjmax-1+cosπjmax-122.From these calculated values of stream function, the

Results and discussion

We have considered the transient natural convection flow of viscous incompressible fluid confined in isothermally cold open cavity, with free top surface whereas the heated fluid that enters the cavity causes the convection to develop. The case of solid top surface is also taken into consideration for the sake of direct comparison. Numerical solution of the partial differential equations that govern the flow, represented by Eqs. (14), (15), (16) together with the initial and boundary conditions

Conclusion

A numerical study of combined thermogravitational and thermocapillary natural convection flow of a heated Newtonian fluid in a side open cavity has been conducted. The effect of Prandtl number, Marangoni number, Eckert number and Grashof number has been presented graphically. It is observed that due to low thermal conductivity and heat transfer, the active spot of entropy generation rests at the center of the opening of buoyancy cell when the values of Prandtl number and Grashof number are not

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