Convective Flow of Two Immiscible Fluids and Heat Transfer With Porous Along an Inclined Channel with Pressure Gradient.

Transcription

1 Research Inventy: Internional Journal Of Engineering And Science Issn: -, Vol., Issue (February ), Pp - Convective Flow of Two Immiscible Fluids and He Transfer With Porous Along an Inclined Channel with Pressure Gradient., Simon Daniel,, Shagaiya. Daniel,, Department of Mhemics Kaduna Ste University, Kaduna, Nigeria Abstract: This paper presents a novel of Two Immiscible fluids flow and he transfer with porous along an inclined channel with pressure gradient. Two stionary ples held constant different temperures with one region filled with porous merial sured with a viscous fluid and another region with a clear viscous fluid. The differential equion is solved analytically. We found th the presence of porous structure, merial constants and different liquid can effectively control fluid flow. Results for a wide range of governing parameters such as Pressure parameter, Grashof number, angle of inclinion and rios of heights of the two layersare plotted on the velocity and temperure fields. The results obtained through this method are found to be in excellent agreement with earlier results obtained analytically. Keywords: Porous medium, convective flow, inclined channel Nomenclure specific he constant pressure accelerion due to gravity rio of the heights of the region height of the region I height of the region - II Grashof number Reynolds number permeability of the porous medium rio of the thermal conductivities effective thermal conductivity of the fluid sured porous medium in region- I thermal conductivity of the fluid in region-ii rio of the viscosities pressure nondimensional pressure gradient Prandtl number temperure, temperure of the boundaries velocity space coordines Greek Symbols rio of the coefficients of the thermal expansion porous parameter density of the fluid kinemic viscosity

2 Convective Flow Of Two Immiscible Fluids And He viscosity nondimensional temperure Subscripts & refers to the quantities for region I and II respectively wall conditions I Introduction Convective he transfer and fluid flow in a system containing simultaneously a reservoir and a porous medium sured with fluid is of gre mhemical and physical interest. More specifically the existence of a fluid layer adjacent to a layer of fluid sured porous medium is a common occurrence in both geophysical and engineering environments. Composite systems are part of numerous and other engineering applicions also, such as fibrous and granular insulion, porous insulion of ducts, ambient air he transfer from heir covered skin, grain storage, and drying paper. Freezing of soila and melting the ice frozen soils due to the change in weher conditions also require the knowledge of interaction mechanism between the fluid and porous layers. Two fluid flow and he transfer in an inclined channel containing porous and fluid layers was studied analytically by Malashetty et al []. Umavhi et al [,,,] analyzed steady and unsteady flow and he transfer of immiscible fluids in a horizontal channel. An extensive review of convective flow and he transfer between fluid and porous layers has been done by Prasad []. Srinivasan and Vafai [] have reported a theoretical study on two immiscible fluid systems in a porous medium, taking into account the non-darcian boundary and inertia effects. Chamkha [] analyzed the flow of two immiscible fluids in porous and non-porous channels. Masuoka [] has observed convective flow in a layer of fluid heed from below and divided by a horizontal porous wall suppresses the convection. Recently, he transfer in channels partially filled with porous media has received considerable tention and was the focus of several investigions Chikh et al [] and Vafai and Kim []In the present work, we study the convective flow of two immiscible fluids and he transfer with porous along an inclined channel with pressure gradient, containing porous layer sured with a fluid and a clear viscous fluid layer. Processes involving he and mass transfer are often encountered in the chemical industry, in reservoir engineering connections, with thermal recovery processes, and in the study of the dynamics of salty hot springs in the sea. II Mhemical Analysis We considered coupling of governing equions for the fluid region with the equions for the porous region with the equions for the porous region through an approprie set of mching conditions the fluid/porous medium interface, we assume the continuity of velocity, shear stress, temperure and he flux the interface. The region is occupied by clear fluid. The two walls of the channel are held different temperure and are inclined, making an angle with the horizontal. In the analysis, the porous medium is considered to be homogeneous and isotropic. The fluid within the porous medium sures the solid mrix and both are in local thermodynamic equilibrium. The fluids in the two regions are different with different properties. The flow is assumed to be steady, laminar and incompressible. Further, the fluid on both regions is assumed to be driven by a common constant pressure gradient and th existence of he transfer does not affect the pressure gradient with the assumptions mentioned aboved, the governing equions of motion and energy can be written as follows: Region I Region II The approprie boundary and mching conditions for the problem under considerion can be written as

3 Convective Flow Of Two Immiscible Fluids And He In order to non-dimensionalise the governing equions the following transformions are used The rios of different parameters th appears in governing equions are Substitution of the above nondimensional quantities in equions ()-(), results in the following equions. Region-I Region-II The transformed boundary and mching conditions are given by Solving the differential equions (), (), () and () sisfying the boundary and interface conditions () and () we obtain All the constants defined in the above equions are shown in the appendix

4 (y) Convective Flow Of Two Immiscible Fluids And He III Results and Discussion In the previous section, an analytical solution for the problem flow of two immiscible fluids through inclined channel was obtained. Equion ()-() are evalued numerically, fixing some of the parameters and varying the parameters. In this section, a representive set of results has been selected for presention in graphical form. We have written a MATLAB programme to compute and genere the graphs for the velocity and temperures. Some representive results are presented in the form of line graphs in fig. - to interpret the effects of these parameters. The effect of Pressure parameter on velocity and temperure profiles is shown in fig. and. For fixed porous parameter, the pressure parameter increases on Region-II as both velocity and temperure increases. Its increases more pronounced on temperure. The frictional drag resistance against the convective is very large and as a result, the velocity is very small in porous region while on the temperure it is more significant. The porous parameter decreases the fluid velocity thereby increases temperure. The effect of Pressure parameter is more on the temperure profile. In fig. and, we illustre how the Grashofnumber affects the velocity and temperure distributions in the inclined channel for fixed values of parameter mention above and varying. The explanion offered for fig. and displays the velocity and temperure profiles in the inclined channel for varying angle of inclinion with fixed parameter. As we can see, the effect of increasing increases the velocity. The driving force increases with the inclinion angle. In temperure profile shown on fig.; as temperure increases oscille which is more pronounced on region II toward to ple In fig. and, we illustre the effects of rios of the heights on velocity and temperure. We observe th the velocity and temperure increases with increases in the values of the rio of the heights. On the temperure profile, in region-i is fl and region-ii convective he transfer increases more toward to ple. IV Conclusions The problem of laminar flow in an inclined channel with an immiscible fluid was analyzed. The basic equions governing the flow were solved analytically. The effect of Pressure and other parameters on the flow was promoted for a clear fluid condition and suppressed for porous condition. Both the velocity and temperure increases on clear fluid condition. The convective he transfer increases the right wall and decreased the left wall for porous region. The angle of inclinion oscilled on both velocity and temperure in the inclined channel. The re of he transfer decreased the left wall and increased the right wall for both velocity and temperure for different parameters Fig. Velocity profiles showing effect of Pressure Parameter Fig. Temperure profiles showing effect of Pressure Parameter

5 (y) (y) Convective Flow Of Two Immiscible Fluids And He Fig.Velocity profiles showing effect of Grashof number Fig.Temperure profiles showing effect of Grashof number FigVelocity profiles showing effect of Angle of inclinion Fig.Temperure profiles showing effect of Angle of inclinion

6 (y). Convective Flow Of Two Immiscible Fluids And He Fig.Velocity profiles showing effect of Rio of heights Fig.Temperure profiles showing effect of Rio of heights..... Appendix

7 Convective Flow Of Two Immiscible Fluids And He V Reference [] Malashetty, M.S., Umavhi, J.C. and Prhap Kumar, J., Two fluid flow and he transfer in an inclined channel containingporous and fluid layer, He and Mass Transfer,, -.. [] Umavhi, J.C., Malashetty, M.S. and Abdul Meen, Fully Developed Flow and He Transfer in a Horizontal Channel Containing Electrically Conducting Fluid Sandwiched between Two Fluid Layers, Int. J. Appl. MechanicalEngineering,, -,. [] Umavhi, J.C., Chamkha, A.J., Abdul Meen and Al-Mudhaf, A., Unsteady Two Fluid Flow and He Transfer in a Horizontal Channel, He Mass Transfer,, -,. [] Umavhi, J.C., Chamkha, A.J., Manjula, M.H. and Al-Mudhaf, A., Flow and He Transfer of a Couple-Stress Fluid Sandwiched between Viscous Fluid Layers, Canadian J. of Physics,, -,. [] Umavhi, J.C., Prhap Kumar, J. and Chamkha, A.J., Flow and He Transfer of a Micropolar Fluid Sandwiched between Viscous Fluid Layers, Canadian J. of Physics,, -,. [] Prasad, V., Convection Flow Interaction and He Transfer between Fluid and Porous Layers, Proc. NATO Advanced Study Institute on Convective He and Mass Transfer in Porous Media, Turkey,. [] Srinivasan, V. and Vafai, K., Analysis of Linear Encroachment in Two-Immiscible Fluid Systems, ASME J. of Fluid Engineering,, -,. [] Chamkha, A.J., Flow of Two-Immiscible Fluids in Porous and Non-Porous Channels, ASME J. of Fluid Engineering,, -,. [] Masuoka, T.,. Convective currents in a horizontal layer divided by a permeable wall, Bull. Japan, Soc. Mech. Eng., Vol., pp. -. [] Chikh, S., Boumedian, A., Bouhadef, K. and Lauri, G., Analytical solution of non-darcian forced convection in an annular duct partially filled with porous medium, Int. J. He Mass Transfer, Vol., pp. -.. [] Kim, S.J. and Choi, C..,. Convection he transfer in porous and overlying layers heed from below, Int. J. He MassTransfer, Vol., pp. -. [] Kuznetsov, A.V.,. Fluid mechanics and he transfer in the interface region between a porous medium and a fluid layer: a Boundary layer solution, J. Porous Media, (), -.