Bk Dutta Mass Transfer Solutions
Eliora Shopbell
The two-dimensional boundary layer flow of a non-Newtonian Casson fluid and heat transfer due to an exponentially permeable shrinking sheet with viscous dissipation is investigated. Using similarity transformations, the governing momentum and energy equations are transformed to self-similar nonlinear ODEs and then those are solved numerically by very efficient shooting method. The analysis explores many important aspects of flow and heat transfer of the aforesaid non-Newtonian fluid flow dynamics. For the steady flow of non-Newtonian Casson fluid, more amount of wall mass suction through the porous sheet is required in comparison to that of Newtonian fluid flow. Dual similarity solutions are obtained for velocity and temperature. The viscous dissipation effect has major impact on the heat transfer characteristic. In fact, heat absorption at the surface occurs and it increases due to viscous dissipation. For higher Prandtl number, the temperature inside the boundary layer reduces, but with larger Eckert number (viscous dissipation) it is enhanced.
On the other hand, Magyari and Keller [20] initiated a study of the boundary layer flow with heat transfer over an exponentially stretching sheet. The effect of wall mass suction on the boundary layer flow and heat transfer over an exponentially stretching sheet was studied by Elbashbeshy [21]. Al-Odat et al. [22] considered an exponential temperature distribution on the boundary layer flow towards an exponentially stretching surface. Later, Sajid and Hayat [23] obtained the series solutions for the boundary layer flow over an exponentially stretching sheet with thermal radiation using homotopy analysis method (HAM). Bidin and Nazar [24] and Ishak [25] numerically investigated the effect of radiation on the boundary layer flow and heat transfer over an exponentially stretching sheet. However, very limited attention has been given to study the boundary layer flow over an exponentially shrinking sheet though it is equally significant in many engineering processes as that of exponentially stretching sheet. The flow and heat transfer due to exponentially shrinking sheet were first discussed by Bhattacharyya [26] and the effect of magnetic field was illustrated by Bhattacharyya and Pop [27]. Rahman et al. [28] showed the effect of nanoparticles on the boundary layer flow past an exponentially shrinking sheet with second-order slip.
The analysis of the obtained numerical results using shooting method [16, 19] explores the condition for which the steady flow is possible for the non-Newtonian Casson fluid. According to Miklavčič and Wang [13] and Fang and Zhang [14], for Newtonian fluids, the steady two-dimensional flow due to a linearly shrinking sheet with wall mass transfer occurs only when the wall mass suction parameter is greater than or equal to 2. However in case of Newtonian fluid over an exponentially shrinking sheet the similarity solution is achieved when , which is consistent with the results obtained by Bhattacharyya [26]. On the other hand, it is quite different for non-Newtonian Casson fluid. For , the flow has dual similarity solutions for and consequently for no similarity solution exists. For, the dual similarity solutions exist if the ranges of are and hence no similarity solution exists for. Further, it is interesting to note that more increment in Casson parameter causes more reduction in the solution suction domain. For , the similarity solution exists when and thus no solution exists for . So, for the steady flow of Casson fluid (with decreasing values of ), more amount of wall mass suction is needed in comparison with that of Newtonian flow. This effect is physical realistic because when the Casson parameter decreases the yield stress becomes larger and for this more amount of vorticity generated due to shrinking and to suppressing the vorticity the requirement of mass suction is higher for the Casson fluid.
The variations of local skin friction coefficient and local heat transfer coefficient (which are proportional to the wall skin friction coefficient and the local Nusselt number or the rate of heat transfer, resp.) with suction for several values of Casson parameter are shown in Figures 2 and 3. From Figures 2 and 3, it is observed that the skin friction coefficient and the heat transfer rate increase with an increase in the values of for the first solution while for the second solution the values decrease. Hence, for non-Newtonian Casson fluid, the local skin friction coefficient is less compared to that of Newtonian fluid case for first solution and reverse result shows in case of second solution. It is also observed from these figures that the values of skin friction coefficient are always positive, which implies that the fluid exerts a drag force on the sheet and the heat transfer coefficient is negative which indicates the heat absorption at the sheet; that is, the heat flows from the ambient fluid to the sheet. Also, the values of temperature gradient at the sheetfor different values of the Eckert number Ec and Prandtl number Pr are plotted in Figures 4 and 5, respectively. From Figure 4 it is observed that for both the first and second solutions the heat transfer rate decreases with increasing values of Ec. Thus, more heat is generated in the boundary layer region due to the viscous dissipation and hence it reduces the heat transfer rate from the sheet; that is, it enhances the heat absorption, whereas, with the increase in Pr, the value of (Figure 5) increases for both solutions and for higher values of Pr it becomes positive, which implies that heat transfers from the hot sheet to the ambient fluid. In addition, to provide a clear view of the flow field the streamlines are plotted for both solutions for fixed values of suction parameter and Casson parameter in Figures 6 and 7.
The suction is very important to maintain the steady flow near the sheet by delaying the separation. Since the suction is necessary, the effects of suction parameter on the velocity and temperature profiles are important in analytical as well as practical point of view. Figures 12 and 13 demonstrate the velocity and temperature profile for different values of suction . For first solution, velocity boundary layer thickness decreases with increasing values of suction and it increases for second solution. The temperature profiles for different values of are illustrated in Figure 13 and from this figure it is clear that the temperature decreases with increasing for both solutions. It is also noted that in second solution heat absorption at the surface is found for and high heat transfer from the sheet is observed for and 2.7.
The steady boundary layer flow of Casson fluid and heat transfer over a permeable exponentially shrinking sheet with viscous dissipation are studied. The governing equations are transformed and solved numerically using shooting method. The study reveals that the steady flow of Casson fluid due to exponentially shrinking sheet requires some more amount of mass suction than the Newtonian fluid flow. In all cases, when similarity solution exists, it is found to be dual solutions for velocity and temperature distributions. For first solution, the skin friction coefficient and heat transfer rate decrease with decreasing values of Casson parameter and opposite behavior is observed for second solution. In many cases, heat absorption at the sheet occurs. Moreover, due to the viscous dissipation effect, the heat absorption increases. The temperature inside the boundary layer increases with Eckert number and also thermal overshoot is observed. For less amount of mass suction heat absorption is found for both solutions, but for large mass suction heat transfer from the sheet occurs in both cases.
In this research work, a coupled heat and mass transfer model was developed for salt recovery from concentrated brine water through an osmotic membrane distillation (OMD) process in a hollow fiber membrane contactor (HFMC).The model was built based on the resistance-in-series concept for water transport across the hydrophobic membrane. The model was adopted to incorporate the effects of polarization layers such as temperature and concentration polarization, as well as viscosity changes during concentration.
Different approaches are used to model the OMD process, however, the resistance-in-series (RIS) model approach is a widely applied and acceptable technique to predict water transport flux through the membrane [18,19,20,21]. Most of the models used in literature based on the RIS model have not considered temperature/concentration polarization effects. Similarly, the effect of physical properties was considered constant during the concentration process. Ahmad et al. [13] studied the RIS model for osmotic membrane evaporation of fruit juices and found that physical properties like viscosity and density variations during the process had a significant effect on water flux. Ni et al. [22] studied the effect of temperature polarization in direct contact membrane distillation for NaCl concentration and observed that temperature polarization was enhanced with the increase of feed temperature and the decrease of feed and permeate velocity. Similarly, Lou et al. [23] investigated coupled temperature and concentration polarization in the membrane distillation treatment of brines with the help of the CFD heat and mass transfer model and observed the effect of coupled temperature and concentration polarization. Salmon et al. [24] experimentally applied the concept of OMD for the crystallization of inorganic salts using two HFMC in series. In another similar study, Salmon et al. [25] described the experimental and theoretical analysis of mass and heat transport of inorganic species during the OMD process.
In the current research work, an RIS model coupling heat and mass transfer resistances was adapted to model HFMC for the recovery of inorganic salts from the brine solution via OMD operation. The model incorporates the effect of concentration polarization in the feed side as well as the osmotic solution side. An aqueous stream of Na2CO3 was taken as an inorganic salt. The model was successfully validated with experimental data under the same operating and module geometrical conditions from the literature [25]. The effects of temperature/concentration polarization of water flux were studied through the simulation of the developed model. Finally, the feasibility of large-scale OMD operation for recovery of Na2CO3 was studied by estimating the required membrane area for industrial conditions which provides an insight into the mass and heat transport phenomena and their implications in scale-up OMD-based HFMCs.