# Second Law Analysis in Convective Heat and Mass Transfer

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{1/3}for mass transfer. A general expression for the Sherwood number as a function of the main parameters of the problem is also proposed over a very wide range. At second step, heat transfer and flow strucure are investigated [4]. They concluded that the thermosolutale convective flows may be classified in four different regimes: (i) unicellular regime, where the flow is dominated by the thermal effect; (ii) multicellular regime , where the thermal and the solutal effects are comparable in the central part of the cavity; (iii) flow globally driven by the solutal buoyancy force, with a persisting thermal cell in the centre; (iv) unicellular regime, where the solutal force is dominating. Bergman and Hyun [5] investigated the simulation of two-dimensional thermosolutal convection in liquid metals induced by horizontal temperature and species gradients. Results show two distinct regimes of behavior, in which large solutal buoyancy forces lead to enhanced mass transfer rates. The regimes are separated by a transition region where thermal and solutal buoyancy forces can become balanced, resulting in a velocity reduction throughout the cavity and small mass transfer rates. Double diffusive steady natural convection in a vertical stack of square enclosures, with heat and mass diffusive walls was studied numerically by Costa [6]. It has been established that changes on the buoyancy ratio are shown to affect seriously the temperature and concentration fields, the path followed by the heat and mass flows, and also the heat and mass transfer parameters. Mamou and Vasseur [7] investigated the onset of double diffusive convective flows in an inclined fluid layer, when constants fluxes of heat and mass are applied on the two opposing boundary of the layer. It has been demonstrated, on the basis of the parallel flow approximation, the existence of a subcritical Rayleigh number, for the onset of finite amplitude convection. Transient double diffusive natural convection in a horizontal enclosure was investigated numerically and analytically by Bennacer and al. [8]. It has been found that there are three distinct regimes, for lower buoyancy ratio (N) value the convective cell is essentially due to thermal forces, for high N value the transfer is diffusive and the stabilizing solutal stratification suppresses the flow and intermediate domain (moderate N value) the transfer decreases with N.Linear nonequilibrium thermodynamics (LNET) theory for coupled heat and mass transport was studied by Demirel and Sandler [9]. It has been demonstrated that the theory of LNET can play crucial role in the proper definition of the coupled heat and mass flows. Also, it has been suggested the use of the resistance type of phenomenological coefficients in the phenomenological equations in which the conjugate forces and flows are identified by the dissipation function. Modeling Soret effect coefficient measurements in porous media considering thermal and solutal convection was investigated by Benano-Melly and al. [10]. It has been found that multiple convection-roll flow patterns can develop when solutal and thermal buoyancy forces oppose each other, depending on the Soret number value. The effects of concentration and temperature on the coupled heat and mass transport in liquid mixtures are studied by Demirel and Sandler [11]. Using published experimental data on the thermal conductivity, mutual diffusivity and heats of transport, the degree of coupling between heat and mass flows has been calculated for binary and ternary non ideal liquid mixtures. The extent of coupling and the thermal buoyancy ratio are expressed in terms of the transport coefficients to obtain a better understanting of the interactions between heat and mass flows in liquid mixtures. It was found that the composition of the heavy component bromobenzene changes the direction and magnitude of the two-flow coupling in ternary mixture. The influence of Grashof number on the double diffusive natural convection in a rectangular enclosures was investigated by Benissad and Afrid [12]. Results allowed to observe complex and varied flow patterns with different conditions of numerical experimentations. In the traditional approach in numerical computation of double diffusive convection problems, the quantities to be computed are usually temperature, pressure, concentration, mass and heat flow rates, but infrequently involving entropy properties.

## 2. Entropy generation for convective heat and mass transfer

_{e}s and d

_{i}s. The first is the entropy change due to exchange of matter and energy with the exterior, the second is the entropy due to “uncompensated transformations”, the entropy produced by the irreversible processes in the interior of the system [28]:

_{e}s + d

_{i}s

- d
_{i}s = 0 for reversible processes - d
_{i}s > 0 for irreversible processes

_{α}is given by the ideal gas equation state:

## 3. Mathematical modeling

#### 3.1. Flow and governing equations

_{1}and W

_{2}are at different but uniform temperature and concentration (T

_{1}, C

_{1}) and (T

_{2}, C

_{2}) respectively, while the two other walls are impermeable and adiabatic. The fluid is modeled as a Newtonian, Boussinesq incompressible fluid whose properties are described by it’s kinematic viscosity ν , its thermal and solutal diffusivities, a and D respectively, and its thermal and solutal volumetric expansion coefficients β

_{T}and β

_{S}respectively.

#### 3.2. Boundary and initial conditions

- U = V = 0 for all walls
- θ = ϕ = 0.5 on plane X = 0
- θ = ϕ = −0.5 on plane X = 1
- $\frac{\partial \theta}{\partial Y}=\frac{\partial \phi}{\partial Y}=0$ on planes Y = 1 and Y = 0

- At ζ = 0, U = V = 0, P = 0, ϕ = 0 and θ = 0.5 − X for whole space

#### 3.3. Dimensionless entropy generation

_{o}. According to Bejan [13], the characteristic entropy transfer rate is given by:

_{o}and ΔT are respectively, the thermal conductivity, the characteristic length of the enclosure, a reference temperature and a reference temperature difference.

_{o}and T

_{o}are respectively the reference concentration and temperature, which are in our case, the bulk concentration and the bulk temperature. Thus the dimensionless form of the local entropy generation rate can be obtained on using the system of the dimensionless variables defined in (18), after rearrangement we obtain:

_{i}(1 ≤ i ≤ 3), and called irreversibilities distribution ratios, are given by:

_{o}(1 − β

_{T}(T − T

_{o}) − β

_{S}(C − C

_{o}))), where ρ

_{o}is the reference density evaluated at C

_{o}and T

_{o}. The set of the dimensionless equations (9–17), show that the problem is governed by the dimensionless numbers of Pr, Sc, Gr

_{T}and N. The dimensionless thermal Grashof number, the buoyancy ratio and the inclination angle are the control parameters of the problem. On the other hand, if the temperature and the concentration are brought variables, the local entropy generation should be calculated in a dimensional form. As a consequence, the numerical simulation is easier carried out in dimensionless form due to the reduced number of parameters.

## 4. Numerical procedure

## 5. Results and discussions

_{T}, N, Pr and Sc) are used in the governing combined heat and mass transfer equations. In order to keep the number of simulations manageable, the ranges of some of these parameters were reduced. The thermal Grashof number is varied from 10

^{2}to 10

^{4}, only the cooperating situation is investigated, and the buoyancy ratio is kept positive and ranging between 0 and 10. The exploitation of the entropy generation equation limits the choices of the Prandtl and the Schmidt numbers to the case of a gaseous mixture only. For a gaseous mixture, the following definition of the Lewis number is used: ( Le = D/a ), both Le ≥ 1 and Le ≤ 1 are possible because (D) is of the same order of magnitude as (a). Furthermore the Prandtl and Schmidt numbers are fixed at 0.75 and 1.5 respectively. The parameter λ

_{1}is fixed at 10

^{−4}. In the case of nonzero buoyancy ratio the terms λ

_{2}and λ

_{3}are fixed at 0.5 and 10

^{−2}respectively. The local entropy generation rate is a function of temperature and velocity gradients in the x and y directions in the entire calculation domain. It is then a good indicator of grid dependence. Grid refinement tests have been performed for the case 10

^{2}≤ Gr

_{T}≤ 10

^{4}and 2 ≤ N ≤ 10. Results show that when we pass from a grid of 31x31 to a grid of 41x41, total entropy generation undergoes an increase of 3%. We conclude that the grid 31x31 is sufficient to carry out a numerical study of this flow. This grid is retained for all following investigations. The numerical simulations presented in this work has been conducted in order to study the effects of the inclination angle of the enclosure, the thermal Grashof number and the buoyancy ratio on entropy generation in steady state conditions. For comparison purposes, the presentation of results starts in Fig.2 with the influence of the inclination angle at different thermal Grashof numbers on the total entropy generation, in the case of no solute transfer (N = 0). In this case the solutal Grashof number is zero(Gr

_{S}= 0), the concentration difference between the walls W

_{1}and W

_{2}is zero(ΔC = 0), and consequently the parameters λ

_{2}and λ

_{3}are zero (Eqs. (27,28)). Therefore the expression of the total entropy generation is reduced to the case of pure convective heat transfer that involves heat transfer and fluid friction irreversibilities given by the first and the second terms on the right hand side of equation (21) (σ

_{n}= σ

_{n,h}+ σ

_{n,f}). As can be seen in Fig. 2, for a thermal Grashof number Gr

_{T}= 10

^{2}and inclination angle ranging between 0° and 180°, the total entropy generation is practically unity (σ

_{n,T}= 1 ). This value corresponds to the entropy generation of a system at rest (characterized by a conduction regime). This is due to the fact that for small thermal Grashof number, there is practically no convection and the entropy generation due to fluid friction is zero, consequently the total entropy generation is reduced to the entropy generation due to heat transfer. At a fixed value of inclination angle the total entropy generation increases with the thermal Grashof number. This is because at higher Grashof number heat transfer due to convection begins to play a significant role increasing the flow velocity and in turn the entropy generation due to the viscous effects. Also the isotherms are deformed increasing the temperature gradient and consequently the entropy generation due to heat transfer.

_{T}≥ 10

^{4}, the total entropy generation increases and reaches a maximum at the inclination angle α ≈ 45°, then decreases and tends towards the value σ

_{n,T}= 1 for inclination angles near α ≈ 180°. Indeed, for inclination angle near α ≈ 45° buoyancy acts along both the active and adiabatic walls, there is more work done on the fluid by buoyancy thus increasing the total entropy generation via the augmentation of the convective heat transfer. As the inclination angle tends towards the value 180°, the velocity of the fluid diminishes because buoyancy and pressure oppose each other in the intrusion layer. This decreases entropy generation due to viscous effects. On the other hand, convective heat transfer decreases and the isotherms become nearly parallel to the active walls causing a decrease in the magnitude of the thermal gradient and consequently of the entropy generation due to heat transfer. Results concerning the entropy generation evolution versus inclination angle are in good agreement with those of Baytas [37], who investigated the entropy generation for natural convection in an inclined porous cavity.

^{4}corresponding for our notation to an inclination angle α ≈ 50°, which is close to the value found in Fig.2 (α ≈ 45°). Fig.3 illustrates the effect of the buoyancy ratio (N) on the total entropy generation for inclination angles α = 60°, 90° and 120° and thermal Grashof numbres Gr

_{T}= 10

^{2}, 10

^{3}and 10

^{4}.

_{T}= 10

^{2}and for buoyancy ratio 0 ≤ N ≤ 2.5. Fig.3 shows that curves of total entropy generation are identical for α = 60°, 90° and 120°, which indicates that the isotherms and isoconcentrations are nearly similar for the three considered inclination angles. It is important to note the linear behavior of the total entropy generation for relatively high thermal Grashof numbers (Gr

_{T}= 10

^{4}). To understand why the total entropy generation increases with the buoyancy ratio, we have plotted in Fig.4 the variation of the average Nusselt number on the heated wall and the total entropy generation versus the inclination angle. In this case, the viscous and diffusive irreversibilities are neglected (λ

_{1}, λ

_{2}and λ

_{3}<< 1) thus reducing the entropy generation to the heat transfer irreversibility only. Each pair of curves corresponds to a given value of the buoyancy ratio ranging from 0 to 10.

_{T}= 10

^{4}and for three angles of inclination of the cavity set to be equal 30°, 90° and 150° respectively as an illustrative example, the heat transfer irreversibilities and the diffusive irreversibilities are found similar and mainly confined to the lower and the upper corners of the heated and the cooled walls respectively except for α = 150° . As the buoyancy ratio increases, the indicated irreversibilities increase for α = 30° and 90°, while at fixed buoancy ratio, they decrease gradually for increasing the inclination angle. Entropy generation due to viscous effect increases with the buoyancy ratio, its maximum is localized in the middle of the active and adiabatic walls for α = 30° and in the middle of active walls for α = 90°. At 150°, irreversibility due to viscous effect decreases considerably indicating that the flow velocity diminishes and convection becomes insignificant. Total entropy generation covers the whole domain except for the center of the cavity at α = 30° and 90°. For α = 150°, the total entropy generation takes on a small value even for moderately high value of the buoyancy ratio and is localized in the entire domain indicating that viscous irreversibility covers the top and the bottom of the cavity, while heat and diffusif irreversibilities cover the center. Fig. 5 summarizes the local irreversibilities for a buoyancy ratio N = 10 as an example.

## 6. Conclusion

## Nomenclature

a | thermal diffusivity (m^{2} s^{−1}) | Greek symbols | |

Be | Bejan number | α | inclination angle of the cavity |

C | concentration | β_{T} | coefficient of thermal expansion, (K^{−1}) |

C_{0} | bulk concentration, C_{0} = (C_{1} + C_{2})/2 | θ | dimensionless temperature |

D | mass diffusivity | μ | dynamic viscosity, (Kg m^{−1} s^{−1}) |

g | acceleration due to gravity (m s^{−2}) | ν | kinematic viscosity, (m^{2} s^{−1}) |

grad | gradient operator | ζ | dimensionless time |

Gr_{S} | solutal Grashof number | σ | entropy generation rate (J m^{−3} s^{−1} K^{−1}) |

Gr_{T} | thermal Grashof number | ϕ | dimensionless concentration |

J | flux density vector | λ_{i} | irreversibility distribution ratio, (i = 1,2,3) |

k | conductivity (J m^{−1} s^{−1} K^{−1}) | Ω | system volume |

L | cavity length (m) | Subscripts | |

p | pressure (N m^{−2}) | 1 | hot wall |

P | dimensionless pressure | 2 | cold wall |

Pr | Prandtl number | d | diffusion |

Sc | Schmidt number | f | friction |

t | time (s) | h | heat |

T | temperature (K) | n | dimensionless |

T_{0} | bulk temperature, T_{0} = (T_{1} + T_{2})/2 | p | steady state |

ΔT | temperature difference, ΔT = T_{1} − T_{2} | s | solutal |

ΔC | concentration difference, ΔC = C_{1} − C_{2} | T | total, thermal |

v | velocity vector | α | for specie α |

V | dimensionless velocity vector | Superscript | |

u, v | velocity components in x, y directions (m s^{−1}) | T | temperature |

x, y | Cartesian coordinates (m) | C | concentration |

X, Y | dimensionless Cartesian coordinates |

## References

- Platten, J.K.; Chavepeyer, G. Influence de la thermodiffusion sur la naissance de la convection libre en configuration de Rayleigh-Bénard. Entropie
**1994**, 184/185, 23–26. [Google Scholar] - Traore, Ph.; Mojtabi, A. Analyse de l’effet Soret en convection thermosolutale. Entropie
**1994**, 184/185, 32–37. [Google Scholar] - Bennacer, R.; Gobin, D. Cooperating thermosolutal convection in enclosures-I- Scale analysis and mass transfer. Int. J. Heat Mass Transfer
**1996**, 39, 2671–2681. [Google Scholar] [CrossRef] - Gobin, D.; Bennacer, R. Cooperating thermosolutal convection in enclosures-II- Heat structure and flow structure. Int. J. Heat Mass Transfer
**1996**, 39, 2683–2697. [Google Scholar] [CrossRef] - Bergman, T.L.; Hyun, M.T. Simulation of two-dimensional thermosolutal convection in liquid metals induced by horizontal temperature and species gradients. Int. J. Heat Mass Transfer
**1996**, 39, 2883–2894. [Google Scholar] [CrossRef] - Costa, V.A.F. Double diffusive natural convection in a square enclosure with heat and mass diffusive walls. Int. J. Heat Mass Transfer
**1997**, 40, 4061–4071. [Google Scholar] [CrossRef] - Mamou, M.; Vasseur, P. Hysterisis effect on thermosolutal convection with opposed buoyancy forces in inclined enclosures. Int. Comm. Heat Mass Transfer
**1999**, 26, 421–430. [Google Scholar] [CrossRef] - Bennacer, R.; Mohamed, A.A.; Akrour, D. Transient natural convection in an enclosure with horizontal temperature and vertical solutal gradients. International Journal of Thermal Sciences
**2001**, 40, 899–910. [Google Scholar] [CrossRef] - Demirel, Y.; Sandler, S.I. Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport. Int. J. Heat Mass Transfer
**2001**, 44, 2439–2451. [Google Scholar] [CrossRef] - Benano Melly, L.B.; Caltagirone, J.-P.; Faissat, B.; Montel, F.; Costeseque, P. Modeling Soret coefficient measurements experiments in porous media considering thermal and solutal convection. Int. J. Heat Mass Transfer
**2001**, 44, 1285–1297. [Google Scholar] [CrossRef] - Demirel, Y.; Sandler, S.I. Effect of concentration and temperature on the coupled heat and mass transport in liquid mixtures. Int. J. Heat Mass Transfer
**2001**, 45, 75–86. [Google Scholar] [CrossRef] - Benissad, S.; Afrid, M. Influence of the Grashof number on the natural convection double diffusion in a rectangular enclosure with a weak aspect ratio. Entropie
**2002**, 242, 44–55. [Google Scholar] - Bejan, A. Entropy generation minimization. CRC Press: New york, 1996. [Google Scholar]
- Nag, P.K.; Kumar, N. Second Law optimization of convective heat transfer through a duct with constant heat flux. Int. J. Energy Res.
**1989**, 13, 537–543. [Google Scholar] [CrossRef] - Shuja, S.Z.; Yilbas, B.S.; Budair, M.O. Local entropy generation in an impinging jet : minimum entropy concept evaluating various turbulence models. Computer Methods in Applied Mechanics and Engineering
**2001**, 190, 3623–3644. [Google Scholar] [CrossRef] - Shuja, S.Z.; Yilbas, B.S. A laminar swirling jet impingement on to an adiabatic wall effect of inlet velocity profiles. Int. J. Numerical Methods for Heat and Fluid Flow
**2001**, 11, 237–254. [Google Scholar] [CrossRef] - Shuja, S.Z.; Yilbas, B.S.; Budair, M.O. Investigation into a confined laminar swirling jet and entropy production. Int. J. Numerical Methods for Heat and Fluid Flow
**2002**, 12, 870–887. [Google Scholar] [CrossRef] - Shuja, S.Z.; Yilbas, B.S.; Rashid, M. Confined swirling jet impingement onto an adiabatic wall. Int. J. Heat Mass Transfer
**2003**, 46, 2947–2955. [Google Scholar] [CrossRef] - Al-Zaharnah, I.T.; Yilbas, B.S. Thermal analysis in pipe flow: influence of variable viscosity on entropy generation. Entropy
**2004**, 6, 344–363. [Google Scholar] [CrossRef] - Sahin, Z.A. The effect of variable viscosity on the entropy generation and pumping power in a laminar fluid flow through a duct subjected to constant heat flux. Heat Mass Transfer
**1999**, 35, 499–506. [Google Scholar] - Narusawa, U. The second Law analysis of mixed convection in rectangular ducts. Heat Mass Transfer
**2001**, 37, 197–203. [Google Scholar] [CrossRef] - Yapici, H.; Kayataş, N.; Kahraman, N.; Baştűrk, G. Numerical study on local entropy generation in compressible flow through a suddenly expanding pipe. Entropy
**2005**, 7 [1], 38–67. [Google Scholar] [CrossRef] - Hyder, S.J.; Yilbas, B.S. Entropy analysis of conjugate heating in a pipe flow. Int. J. Energy Res.
**2002**, 26, 253–262. [Google Scholar] [CrossRef] - Abassi, H.; Magherbi, M.; Ben Brahim, A. Entropy generation in Poiseuille-Benard channel flow. Int. J. Thermal Sciences
**2003**, 42, 1081–1088. [Google Scholar] [CrossRef] - Sahin, Z.A. Entropy generation in turbulent liquid flow through a smooth duct subjected to constant wall temperature. Int. J. Heat Mass Transfer
**2000**, 43, 1469–1478. [Google Scholar] [CrossRef] - Sahin, Z.A. Entropy generation and pumping power in a turbulent fluid flow through a smooth pipe subjected to constant heat flux. Exergy, an International Journal
**2002**, 2, 314–321. [Google Scholar] [CrossRef] - De Groot, S.R. Thermodynamics of irreversible processes; North-Holland, Amesterdam, 1966. [Google Scholar]
- Prigogine, I. Etude thermodynamique des processus irréversibles, 4th edition; Liège: Desoer, 1967. [Google Scholar]
- Hirschfelder, J.O.; Curtis, C.F.; Bird, R.B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. [Google Scholar]
- Shohel, M.; Roydon, A.F. Analysis of mixed convection-radiation interaction in a vertical channel: entropy generation. Exergy
**2002**, 2, 330–339, an International Journal. [Google Scholar] - Tasnim, H.S.; Shohel, M. Entropy generation in a vertical concentric channel with temperature dependent viscosity. Int. Comm. Heat Mass Transfer
**2002**, 29 (7), 907–918. [Google Scholar] [CrossRef] - Saabas, H.J.; Baliga, B.R. Co-located equal-order control-volume finite-element method for multidimensional, incompressible, fluid flow. Numerical Heat Transfer
**1994**, 26 (part B), 381–407. [Google Scholar] [CrossRef] - Prakash, C. An improved control volume finite-element method for heat and mass transfer, and for fluid flow using equal order velocity-pressure interpolation. Numerical Heat Transfer
**1986**, 9, 253–276. [Google Scholar] [CrossRef] - Hookey, N.A. A CVFEM for two-dimensional viscous compressible fluid flow. PhD thesis, McGill University, Montreal, Quebec, 1989. [Google Scholar]
- Elkaim, D.; Reggio, M.; Camarero, R. Numerical solution of reactive laminar flow by a control-volume based finite-element method and the vorticity-stream function formulation. Numerical Heat Transfer
**1991**, 20 (part B), 223–240. [Google Scholar] [CrossRef] - Abbassi, H.; Turki, S.; Ben Nasrallah, S. Mixed convection in a plane channel with a built-in triangular prism. Numerical Heat Transfer
**2001**, 39 (Part A), 307–320. [Google Scholar] - Baytas, A.C. entropy generation for natural convection in an inclined porous cavity. Int. J. Heat Mass Transfer
**2000**, 43, 2089–2099. [Google Scholar] [CrossRef] - Trevisan, O.V.; Bejan, A. Combined heat and mass transfer by natural convection in a vertical enclosure. J. Heat Transfer
**1987**, 109, 104–112. [Google Scholar] [CrossRef] - Viskanta, R.; Ranganathan, P. Natural convection in a square cavity due to combined driving forces. Numer. Heat Transfer
**1988**, 14, 35–59. [Google Scholar]

**Figure 4.**Total entropy generation and the average Nusselt number versus inclined angle for Gr

_{T}= 10

^{4}: a) N=0, b) N=3, c) N=6, d) N=10

**Figure 5.**Local entropy generation for Gr

_{T}= 10

^{4}and N = 10 at α = 30°, 90° and 150°: 1) thermal irreversibility maps, 2) irreversibility maps due to concentration gradient, 3) irreversibility maps due to viscous effects, 4) total irreversibility maps.

© 2006 by MDPI. (http://www.mdpi.org) Reproduction for noncommercial purposes permitted.

## Share and Cite

**MDPI and ACS Style**

Magherbi, M.; Abbassi, H.; Hidouri, N.; Brahim, A.B.
Second Law Analysis in Convective Heat and Mass Transfer. *Entropy* **2006**, *8*, 1-17.
https://doi.org/10.3390/e8010001

**AMA Style**

Magherbi M, Abbassi H, Hidouri N, Brahim AB.
Second Law Analysis in Convective Heat and Mass Transfer. *Entropy*. 2006; 8(1):1-17.
https://doi.org/10.3390/e8010001

**Chicago/Turabian Style**

Magherbi, M., H. Abbassi, N. Hidouri, and A. Ben Brahim.
2006. "Second Law Analysis in Convective Heat and Mass Transfer" *Entropy* 8, no. 1: 1-17.
https://doi.org/10.3390/e8010001