Numerical Study of Thin Film Condensation in Forced Convection on an Inclined Wall Covered with a Porous Material

The present work presents a study of forced convection condensation of a laminar film of a pure and saturated vapor on a porous plate inclined to the vertical. The Darcy-Brinkman-Forchheimer model is used to write the flow in the porous medium, while the classical boundary layer equations have been exploited in the pure liquid and in the porous medium taking into account inertia and enthalpy convection terms. The problem has been solved numerically. The results are mainly presented in the form of velocity and temperature profiles. The obtained results have been compared with the numerical results of Chaynane et al. [1]. The effects of different influential parameters such as: inclination (ϕ), effective viscosity (Re K ), and dimensionless thermal conductivity λ * on the flow and heat transfers are outlined.


Introduction
The study of convection in a porous medium is the object of much covetousness both at the scientific and industrial levels. Indeed, many actors are interested in this field during these last years because of its importance in some important How to cite this paper: Ndiaye, G., Sambou, V., Ndiaye Chaynane R. et al. [1] presented an analytical and numerical study of film condensation on a wall inclined from the vertical and covered with a material.
The Darcy-Brinkman model is used to write the flow in the porous medium, Asbik M. et al. [2] have analytically studied the laminar thin film condensation of a pure and saturated vapor flowing in forced convection over a vertical porous plate. The transfers in the porous medium and the pure liquid are respectively described by the Darcy-Brinkman model and the classical boundary layer equations. Thermal dispersion is considered in the heat equation for heat transfer in the porous layer. The analysis of the velocity, temperature, and local Nusselt number profiles shows the influence of thermal dispersion on the condensate flow and heat transfer. The results show that the increase in the thermal dispersion coefficient leads to a considerable increase in heat exchange.
The comparison of the results deduced from the analytical expressions with those obtained by solving the conservation equations of momentum and heat using a numerical method leads to a quantitatively satisfactory agreement. The difference is less than 10%.
Asbik M. et al. [3] were interested in the analytical study of a condensation film deposited by forced convection on a vertical surface covered with a porous layer. A forced convection condensation problem in a porous thin film is considered. The flow in the region is described by the Darcy-Brinkman-Forchheimer (DBF) model, while the classical boundary layer equations without inertia and enthalpy terms are used in the pure condensate region. In order to solve this problem, an analytical method is proposed. Then, analytical solutions for the flow velocity, temperature distributions and local Nusselt number are obtained. The results are presented mainly in the form of velocity and temperature profiles in the porous layer. Renken K. J. et al. [4]- [10] were the first to show numerically via a finite-difference scheme the effect of the porous layer thickness on the transfers by studying laminar thin film condensation on a vertical surface with a porous coating. Their model simulates two-dimensional condensation inside a highly permeable, thinly conductive porous layer. The local volume averaging technique is used to establish the energy equation. The Darcy-Forchheimer model is employed to describe the flow field in the porous layer while the classical boundary layer equations are used in the pure condensate region. They also developed experiments on forced convection condensation in thin and porous coatings. This study presents the results of forced convection heat transfer experiments of condensation on plates with a thin porous coating. The composite system consists of a porous, thin, highly conductive, and permeable material bonded to a cold surface in isothermal condensation that runs parallel to the saturated vapor flow, were interested in studying forced convective condensation on a thin porous layer led plate. The system consists of a relatively thin permeable highly conductive composite placed parallel to the vapor flow. The pores have thicknesses ranging from 0 to 254 micrometers which are used as a passive technique for heat transfer enhancement. The use of a thin porous layer on isothermal surfaces under the condition of forced convection shows that the thinner the porous layer the more heat transfer is increased. Thus, the experiments demonstrated the advantages of using a thin porous coating in a forced convection heat transfer environment.
Ndiaye M. et al. [11] [12] [13] [14] [15] proposed a numerical model for the study of pure saturated vapor condensation of thin-film type in forced convection on a wall covered with a porous material. They analyzed the influences of Prandtl, Froude, Reynolds, and Jacob number, the dimensionless thickness of the porous layer, and the thermal conductivity ratio on the transfers in the porous medium and in the liquid phase. They had also retained that the effects of inertia could no longer be ignored in the porous medium as soon as the Reynolds number is higher than 7.
Louahlia et al. [16] have provided a numerical study of condensation by forced convection of R134a between two vertical plates. The physical model is based on the solution of the boundary layer equations taking into account the parameters often neglected in theoretical studies of condensation, namely the pressure gradient, the inertia and enthalpy convection terms, and the variation of the physical properties. Their computational results are compared with experimental values. The effects of turbulence, gravity, steam velocity, pressure forces, and tangential stresses are also analyzed.
Ma X. H. et al. [17] presented a numerical study of film condensation on a vertical porous plate coated with a porous/fluid composite system based on the dispersion effect.

The Physical Model
We consider a saturated porous medium confined on a vertical plate, of thickness H, permeability K and porosity ε ( Figure 1). This flat plate tilted by an angle ϕ of length L is placed in a pure, saturated vapor flow of longitudinal velocity For the rest of our study, we have taken the following simplifying assumptions: 1) The porous substrate is isotropic and homogeneous.
2) The fluid saturating the porous medium is Newtonian and incompressible.
3) The flow generated is laminar and two-dimensional.
4) The work, induced by viscous and pressure forces, is negligible. 10) The porous matrix is in local equilibrium with the condensate.
11) The liquid-vapor interface is in thermodynamic equilibrium and the shear stress is assumed to be negligible.
12) The vapor and the film are separated by a distinct boundary.
13) The flow is considered of Bernoulli type in the pure vapor phase.

Equations
The equations characterizing our model are defined as follows: In the porous medium.
Porous layer 0 1 η < < The system of equations defining the motion is then written in the dimensionless form.

Continuity equation
Equation of the following momentum balance X.

Boundary and Interface Conditions
The basic equations described above are solved taking into account the boundary conditions specific to our problem. These are the following: At the wall 0 η = .

Mass and Heat Balance Equations
Since the dimensionless velocity and temperature depend on the thickness of the liquid film *, the dimensionless heat and mass balances are expressed by the following relationship respectively: With the mass flow equation which is given by: Represents the number of Jacob that compares the sensible and latent heat.
The Froude number which characterizes the ratio between the forces of inertia and gravity.  The Prandtl number calculated on the basis of the effective viscosity.

Numerical Resolution
The boundary layer equations are solved according to the implicit scheme. In this scheme, the partial derivative at nodes (i, j of cell i, j). The uniform mesh is To validate our model, we compared the results of R. Chaynane et al. [1] with those of our computational code in which, for the liquid medium, the inertial and enthalpic convection terms were considered and the tilt angle Φ with respect to the vertical is zero. We observe that the correspondence is acceptable as we will see later.
We show in this study the influence of parameters such as: tilt (ϕ), effective viscosity Re K and dimensionless thermal conductivity λ * on the flow and heat transfers. ε = 0.4; Pr = 2; υ * = 1; H * = 2.10 −2 ; Fr K = 10-2; Ja = 10 −5 The mesh sensitivity study led us to choose ΔX = 0.05 and ∆η = 0.02. The convergence criterion in the iterative process is set to 10-6.  The longitudinal velocity increases when the angle of inclination with respect to the vertical is low. Their profiles are similar to those of R. Chaynane et al. [1].
The velocity in the porous substance is linear and undergoes a slight variation at the porous medium/pure liquid interface due to the increase in buoyancy forces.
Contrary to R. Chaynane et al. [1] we note that the tilt angle does not influence the temperature profile. This means that the dynamic diffusion effects are negligible compared to the thermal diffusion effects.           Contrary to the work of R. Chaynane et al. [1], we note that the longitudinal velocity increases with the adimensional conductivity and is almost linear in both liquid/porous media if its value is between 2.5 and 5. This is due to the predominance of the liquid conductivity over the effective conductivity of the porous medium. However, when the adimensional conductivity is low (λ * ≤ 0.5 liquid medium with low conductivity) we note a significant variation at the interface of the two media.
The conductivity ratio λ * has a significant influence on the temperature distribution in the porous medium because it increases with it. At the same time, it favors heat transfer in the liquid phase due to the fact that large values of the dimensionless conductivity correspond to a highly conductive liquid phase. We note a strong variation of the temperature at the interface porous medium/liquid when the conductivity ratio becomes low (λ * ≤ 0.5) then in this case most of the transfers are made in the liquid.

Conclusion
We have presented a numerical study of thin-film condensation in forced convection on a wall inclined to the vertical and covered with a porous material. The equations were solved using the off-center implicit finite difference method. We showed the hydrodynamic and thermal influences of parameters such as the tilt angle Φ, the effective viscosity (Re K ), and the dimensionless thermal conductivity λ * . Taking into account our simplifying assumptions, we compared our results with those of R. Chaynane et al. [1]. Thus, we were able to show that the angle influences the velocity profile of both media (liquid and porous) but not the temperature profile. We showed that small values of the effective viscosity will increase the flow velocity in both media and give linear appearances in the porous and liquid media with little variation at the interface. We have shown that when the liquid is too conductive (λ * high), this favors an increase in velocity and temperature profiles.