Entropy generation in an inclined porous channel with suction/injection

Abstract The present work is concerned with an analytical study of entropy generation in viscous, incompressible fluid in an inclined channel with porous walls. The solution of the governing equations were obtained in closed form. The expression for the irreversibility ratio was also obtained and the results were presented graphically and extensively discussed for different values of the dimensionless parameters. The result indicates that wall inclination enhance entropy generation.


Introduction
Considering rst law of thermodynamics, temperature distribution of uid and heat transfer coe cient within the channel can be obtained, this law however does not account for the relative contribution of viscosity and heat convection for entropy generation in a system [1]. Second law analysis becomes suitable since it gives information regarding the relative viscosity and heat transfer in the system. This law plays a vital role in predicting the performance of the engineering processes, since entropy generation measures the destruction of the available work of the system. In order to improve its performance, it is important to determine factors responsible for entropy generation in the system. This process is made easy since by second law analysis, many di erent interactions in the process can be compared to identify the major sources of energy losses. This method was introduced by Bejan [2][3][4] and due to the importance of minimizing the entropy generation, many researchers in the last decade have developed interest in this area of study.
In particular, Makinde and Maserumule [5] investigated entropy generation in a Couette ow of viscous incompressible uid and concluded that viscous dissipation irreversibility dominates near the moving plate while heat transfer irreversibility dominates near the stationary plate. Makinde and Gbolagade [6] studied second law analysis of incompressible viscous ow through an inclined channel with isothermal walls. Makinde and Osalusi [7] investigated entropy generation in a liquid lm falling along an inclined porous heated plate. They found that heat transfer irreversibility dominates the entropy generation near the inclined heated porous plate while the uid friction irreversibility dominates near the liquid-free surface. Ajibade and Jha [8] studied entropy generation under the e ect of suction/injection and they concluded that an increase in suction on the cold porous plate increases dominance of heat transfer irreversibility, while an increase in injection on the cold porous plate decreases the dominance of heat transfer irreversibility over the uid friction irreversibility Hooman et al. [9] numerically studied entropy generation for forced convection in a porous channel with iso ux or isothermal walls and concluded that di erent arrangement of the parameter lead to completely di erent behaviour for both entropy generation number and Bejan number. Aziz [10] presented entropy generation in pressure gradient assisted Couette ow. Four di erent combinations of thermal boundary conditions are investigated. His results illustrate the e ect of pressure gradient, temperature asymmetry, heat ux, convection Biot numbers, and ambient temperatures and concluded that for certain combinations of thermal variables, the total entropy generated is minimized. Using local similarity solution technique and shooting quadrature, Makinde [11] numerically investigated second law analysis for variable viscosity hydro-magnetic boundary layer ow with thermal radiation and Newtonian heating while in conjunction with Aziz [12] presented a numerical study of Second law analysis for a variable viscosity plane Poiseuille ow with asymmetric convective cooling using an e cient numerical shooting technique with a fourth order Runge-Kutta algorithm. Vyas and Ranjan [13] analytically examined entropy analysis for MHD generalized Couette ow in a composite duct. In another work, using entropy generation analysis, the mixed convection ow between two vertical concentric pipes with constant heat ux at the boundaries and MHD ow e ects is considered by Mahian et al. [14]. It is concluded that with an increase in the Hartmann number, the energy cost increases.
Later on, thermo-economic analysis of a heat exchanger was performed by Mahian et al. [15] to estimate the total costs due to entropy generation for the ow of a refrigerant inside the heat exchanger. It was found that in the cases investigated, the parameter Gr/Re has no considerable e ect on entropy generation. The generalized axial Couette ow of an Ostwald-de Waele power law reactive uid between concentric cylindrical pipes has been numerically investigated by Makinde [16] using the shooting method coupled with the Runge-Kutta-Fehlberg integration technique.
In this present study, special attention has been given to the combined e ects of porosity and wall inclination with symmetric heating on entropy generation and irreversibility ratio. Problem is formulated, solved and discussed.

Mathematical analysis
Steady ow of viscous incompressible uid is considered in an inclined heated channels formed by two in nite parallel porous walls. The x-direction is taken parallel to one of the channel porous wall while the y-direction is taken normal to it.
The geometry of the system under consideration in this present study is shown schematically in Fig. 1 consisting of uid owing steadily downstream in the x-direction. The basic equations governing the ow of viscous incompressible uid between two heated inclined parallel plates are; Conservation of mass Conservation of momentum and Conservation of energy Where ∇ = ∂ ∂x i + ∂ ∂y j + ∂ ∂z k. The physical quantities ρ, ν, ρ , C P , V , T and k are de ned in nomenclature. We consider a two dimensional ow so that V = (u, v, ) , where u and v are the horizontal and vertical components of velocity, respectively. Moreover, it is assumed that the ow is along x-axis which is steady fully developed and depending on y alone ( Fig. 1 Integrating Eq. (4), we have the vertical velocity as v = −v (a constant) which is the velocity of suction/injection. The dimensional governing momentum and energy equations for the steady ow of viscous incompressible uid with the associated boundary conditions are given by The non-dimensional quantities in the above equations are de ned as In dimensionless form, Eq. (6) to (9) are obtained as T (x, ± ) = Where Pr is the Prandtl number which is inversely proportional to the thermal di usivity of the working uid, s is the dimensionless suction/injection parameter. Positive values of s denotes injection at the porous wall y = h with a corresponding suction on the wall y = −h while negative values denote injection at the porous plate y = −h with a corresponding suction on the other plate. The physical quantities used in Eq. (10) are de ned in the nomenclature.
The solution to Eq. (11) and (12) with the boundary conditions (13) to (15) is given by

Entropy generation rate
When we consider the ow of a Newtonian incompressible uid under the in uence of Fourier law of heat conduction, the Cartesian coordinates of the volumetric rate of entropy generation is given by Bejan [9] as The above form of entropy generation shows that the irreversibility is due to two e ects; conduction (k) and viscosity (µ). As long as one of temperature or velocity gradient is present in a medium, entropy generation rate is nite and positive. In many fundamental convective heat transfer problem, Velocity and temperature distributions are simpli ed assuming that the ow is hydro-dynamically developed ∂V ∂x = and thermally developing ∂T ∂x ≠ or thermally Eq. is reduced to the following form: In dimensionless form, E G is given as the entropy generation number Ns which is, by de nition, equal to the ratio of the actual entropy generation rate to the characteristics entropy transfer rate E G,C the entropy transfer rate is given by White [18] as: Where T is the absolute reference temperature, q is the heat ux, ∆T is the temperature di erence and L is the characteristics length depending on the geometry of the channel and problem type. The expression at the center of the Eq. (20) is used for iso ux boundary condition while that on the right hand side is used for isothermal boundary condition.
In terms of the dimensionless velocity and temperature, the entropy generation number becomes and Ω = (T w − T )/T are the peclet number, Brinkman number and temperature di erence parameter respectively, the rst term Nx represents the entropy generation by heat transfer due to axial conduction, the second term Ny denotes the entropy generation due to heat transfer across di erent uid sections(transverse) within the channel and the last term is N f is the entropy generation due to uid friction.

Irreversibility analysis
Expression for entropy generation Eq. (21) gives the spatial distribution of entropy but fails to identify the relative contributions of each irreversibility to the total entropy generation. In order to have an idea whether it is the uid friction or heat transfer irreversibility that dominates the total entropy, Bejan [19] de ned the irreversibility distribution ratio (ϕ) as the ratio of the entropy generation due to viscous dissipation (N f ) to heat transfer (Nx + Ny). That is, For ≤ ϕ < , implies that heat transfer entropy generation dominates the irreversibility ratio while ϕ > , indicates that entropy generation due to uid friction dominates the ow and ϕ = , signi es that heat transfer and uid friction have equal contribution of entropy generation to irreversibility ratio.
In many engineering designs and energy optimization problem, the contribution of heat transfer entropy generation to the total entropy generation rate is required. Hence, Paoletti et al. [20] presented an alternative irreversibility distribution parameter in terms of Bejan number (Be) and de ned it as the ratio of the entropy generation due to heat transfer (N x + Ny) to the total entropy generation (Ns). The Bejan number is de ned mathematically as Obviously, the Bejan number ranges from 0 to 1. Be = , is the limit where the irreversibility is dominated by uid friction e ect while Be = , shows that entropy is generated solely by heat transfer irreversibility. The contribution of heat transfer and uid friction to entropy generation is equal when Be = . . According to Bejan [21], the viscous irreversibility term in eq. (21) may not be negligible, implying that it may be considered in entropy generation analysis, even in cases when the viscous dissipation term has been neglected in the energy equation which is the case in the present problem. Similar analysis has been presented in the work of Hooman and Gurgenci [22]. In many heat transfer problems, it is often possible and convenient to neglect the viscous dissipation term in the energy equation. This is particularly the case in heat transfer through gases at subsonic velocities (Bejan [23])

Result and discussion
The role of suction/injection on entropy generation in ow of viscous incompressible uid in an inclined porous channel is presented. The dimensionless parameters governing the ow are Prandtl number (Pr), which is inversely proportional to the thermal di usivity of the working uid, Brinkman number (Br), Peclet number (Pe) and the suc-  [24]). The numerical calculation for the dimensionless velocity (U), temperature (T), irreversibility ratio (ϕ) and entropy generation (Ns) have been carried out with the aid of MATLAB and the e ect of the dimensionless parameter on them are shown graphically in Figs. 2-11. Fig. 2 shows the variation of velocity for di erent values of suction/injection parameter. It can be observed that the uid velocity is parabolic in nature for di erent values of s. Considering s > , velocity decreases with increase in injection on the inclined heated wall s(y = ), while an increase in suction on the wall y = − has no signi cant e ect on the velocity. However, velocity is seen to decrease as we approach the heated wall. Fig. 3 and 4 shows the temperature variation for di erent values of s and Pr respectively. In Fig. 3, uid temperature is observed to increase towards the wall with maximum temperature on the heated walls. Fig. 4 revealed that uid temperature increases as Pr increases. In reality, as Pr increases, the thermal di usivity of the uid decreases leading to decrease in the di usion of the heat generated by uid friction within the channel. As a result, we have accumulation of heat within the channel leading to increase in uid temperature.
The ratio of entropy generation due to uid viscous dissipation to entropy generation due to heat transfer is presented in Figs. 5-9. Fig. 5, 6 and 7 shows the graph of irreversibility ratio for di erent values of group parameter (BrΩ − ), axial distance (x) and (Pe) respectively. The group parameter is an important parameter which helps in the analysis of the irreversibility, since it shows the ratio of the e ect of viscosity to temperature di erence in the system. We observed from these gures that irreversibility ratio increases from the middle of the inclined channel to its maximum values, then decrease towards the heated channel walls with entropy generation due to heat transfer dominating for all values of x and Pe (Fig. 6, 7). However, entropy generation due to uid friction dominates the irreversibility ratio for values BrΩ − ≥ , in the interval − . ≤ y ≤ − . and . ≤ y ≤ . (Fig. 5).
The in uence of di erent values of Prandtl number (Pr) on the irreversibility ratio is shown in Fig. 8. The irreversibility ratio is observed to increase transversely from the centerline towards the heated wall y = , with a decrease in Pr, a change in trend is observed close to the wall as irreversibility ratio increase with Pr for all Pr ≥ . A critical look into the graph shows that the irreversibility ratio increases from the centerline of the channel with Pr toward the wall y = − , till it reaches its maximum and then eventually decreases towards the heated wall y = − . It is interesting to note that entropy generation due to heat transfer greatly dominates the uid friction in the irreversibility for di erent values of Pr. The irreversibility ratio for di erent values of suction/injection parameter s is presented in Fig. 9. We observed two region of maximum value of ∅, for all values of s and a minimum point close to the centerline of the channel. It will be interesting to note that heat transfer irreversibility dominates throughout the channel. Its dominance can however be further enhanced by increasing injection close to the heated wall y = .
Entropy generation for di erent dimensionless parameter is presented in Fig. 10 and 11. Fig. 10 shows the entropy generation for di erent values of group parameter (BrΩ − ). We observed that entropy generation increases on both walls with increase in (BrΩ − ), while the minimum value of entropy is found within the channel (y = ). i.e., available energy in transverse direction at the middle of the channel is maximum. As expected, entropy generation on the wall with suction (y = − ) is higher when compared to the entropy on the wall with injection (y = ). This trend is however reversed by injecting through the wall (y = − ) and sucking the uid from the wall (y = ). Variation of entropy generation with suction/injection parameter is presented in Fig. 11. The graph shows that increase in suction on the wall (y = − ) increases entropy generation while increase in injection on the wall (y = ) decreases the entropy, indicating that introduction of suction/injection parameter encourages entropy generation in the inclined channel.

Conclusion
This paper presents the study of entropy generation in an inclined channel with porous walls. The velocity and tem- perature pro le are obtained and used to compute the entropy generation and irreversibility ratio for di erent values of suction/injection parameter (s), Prandtl number (Pr), group parameter (BrΩ − ) and Peclet number (Pe). Generally, we observed that; 1. Entropy generation on the suction wall is higher when compared to the entropy on the wall with injection, this indicates that suction of uid on the system encourages entropy generation. 2. Entropy generation near the two porous walls is higher than entropy generation within the channel, indicating that friction due to surface of the two walls increases entropy. 3. Entropy generation due to heat transfer dominates the irreversibility for di erent values of dimensionless suction/injection parameter (s), Prandtl number (Pr), and Peclet number (Pe). 4. Entropy generation due to viscous dissipation as a result of uid friction dominates the irreversibility ratio for values of group parameter (BrΩ − ≥ ).