Liquid film condensation along a vertical surface in a thin porous medium with large anisotropic permeability
 Arthur S O Sanya^{1}Email author,
 Christian Akowanou^{1},
 Emile A Sanya^{1} and
 Gerard Degan^{1}
Received: 18 July 2014
Accepted: 20 October 2014
Published: 6 November 2014
Abstract
The problems of steady film condensation on a vertical surface embedded in a thin porous medium with anisotropic permeability filled with pure saturated vapour are studied analytically by using the BrinkmanDarcy flow model. The principal axes of anisotropic permeability are oriented in a direction that noncoincident with the gravity force. On the basis of the flow permeability tensor due to the anisotropic properties and the BrinkmanDarcy flow model adopted by considering negligible macroscopic and microscopic inertial terms, boundarylayer approximations in the porous liquid film momentum equation is solved analytically. Scale analysis is applied to predict the orderofmagnitudes involved in the boundary layer regime. The first novel contribution in the mathematics consists in the use of the anisotropic permeability tensor inside the expression of the mathematical formulation of the film condensation problem along a vertical surface embedded in a porous medium. The present analytical study reveals that the anisotropic permeability properties have a strong influence on the liquid film thickness, condensate mass flow rate and surface heat transfer rate. The comparison between thin and thick porous media is also presented.
Keywords
Introduction
Many investigations have been directed recently to attack film condensation in a porous medium in both steady (Cheng 1981; Chui et al. 1983; Nakayama and Koyama 1987; Vovos and Poulikakos 1987; Reeken et al. 1994) and transient problems (Cheng and Chui 1984; Ebinuma and Nakayama 1990; Masoud et al. 2000). Since the basic understanding of the transfer mechanisms of heat, momentum and mass in porous media during phase change have still developed, the two phase change flows have possible applications in geophysics and engineering problems such as geothermal energy, steam stimulation of oil fields, food drying and heat pipes problems (Ebinuma and Nakayama 1990). AlNimr and Alkam (1997) have studied steady film condensation on a vertical plate imbedded in a porous medium and have considered the BrinkmanDarcy model by neglecting the macroscopic and microscopic inertial terms to obtain analytical solutions for the three following requests: liquid film thickness, condensate mass flow rate and convective heat transfer coefficient. In the same way, Masoud et al. (2000) have adopted the Brinkmanextended Darcy model to found analytical solutions that describe the transient behavior of the three latter requests. The current results show the effect of the permeability of the porous material on several issues including the velocity profiles, the film thickness and the time required to reach steady state conditions.
Moreover, in all the above studies, the porous medium is assumed to be isotropic whereas, in the several applications, the porous materials are anisotropic (Degan et al. 1995). Despite this fact, film condensation in such anisotropic porous media has not received any attention.
Nevertheless, the study of natural convection in anisotropic porous medium is now available in the heat transfer literature (Kimura et al. 1993; Ni and Beckermann 1993; Zhang 1993; Degan and Vasseur 1996). Degan et al. (1995) have found analytical and numerical solutions for natural convection in a fluidsaturated porous medium filled in a rectangular cavity. In their work, the porous medium is assumed to be both hydrodynamically and thermally anisotropic. The principal directions of the permeability are oriented in a direction that is oblique to the gravity vector, while those of thermal conductivity coincide with the horizontal and vertical coordinate axes. The results showed that the analytical solutions can faithfully predict the flow structure and heat transfer for a wide range of the governing parameters.
In this paper, the problem of steady laminar film condensation on a vertical surface embedded in a thin porous medium with anisotropic permeability, with its principal axes oriented in a direction that is oblique to the gravity vector, based on the classical analysis by Nusselt (1916) is presented using the BrinkmanDarcy flow model (Tong and Subramanian 1985, Chan et al. 1970). Analogous to the work of Degan et al. (1995) who have studied the anisotropic behavior of the porous materials in the case of the natural convection in the neighborhood of vertical plate embedded in porous medium, the film condensation problem over a vertical plate embedded in a porous medium studied by AlNimr and Alkam (1997) is extended to cope with the reality which shows that the natural or industrial materials are anisotropic in their properties. With the consideration of the anisotropic permeability, the flow permeability tensor adopted is the same with previous report given by Degan et al (1995). The boundarylayer approximations are used only for the momentum equation in the porous liquid film region. The analysis aims to obtain closedform expressions for the liquid film thickness, the condensate mass flow rate and the surface heat transfer rate.
Mathematical formulation

Constant properties are assumed for the liquid film;

The liquid and solid porous domains are in local equilibrium;

The temperature distribution is in steady state and lower than the saturation temperature of the liquid film;

The gas is assumed to be a pure vapor at a uniform temperature equal to T_{ S } and no conduction of heat is assumed to take place at the liquidvapor interface.

The local film thickness is much larger than the pore or particle size. This makes it possible to use the local volumeaveraged treatment of the current heterogeneous fluidsolid system (Kaviany 1995).

The problem is gravity dominated and capillarity effects are then neglected. As a result, the twophase region, which contains a solid matrix saturated with vapor and liquid, is absent.

The shear stress at the liquidvapor interface is assumed to be negligible $\left(\frac{\partial {u}_{L}\left({\delta}_{L}\right)}{\partial y}=0\right)$; there is no need to consider the vapor velocity or thermal boundary layers.

The heat transfer across the film occurs only by conduction, in which case the liquid’s temperature distribution is assumed to be linear.

The boundarylayer approximation is valid in the momentum equation for a porous liquid film. However, both macroscopic and microscopic inertial terms are assumed to be negligible (Kaviany 1995).
Dimensional analysis
These conditions (712) arise because of the using of the vertical boundary layer hypothesis which impose that the equation (4) much be only dependent from the motion in x direction, that is to say the terms in y direction must be negligible with respect to the terms in the x direction to have the thin boundary layer along the vertical surface. Some examples are given in Bejan (1984).
Where the following nondimensional terms of the porous medium are in used:

$\mathrm{\text{The}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{Darcy}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{number}}:\phantom{\rule{1em}{0ex}}\mathit{Da}=\frac{{K}_{1}}{{L}^{2}}$(19)

$\mathrm{\text{The}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{Rayleigh}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{number}}:\phantom{\rule{1em}{0ex}}R{a}_{L}=\frac{{K}_{1}gL}{{\mu}_{L}{\alpha}_{L}}\left({\rho}_{L}{\rho}_{v}\right)$(20)

$\mathrm{\text{The}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{porosity}}:\phantom{\rule{1.25em}{0ex}}\u03f5=\frac{{\mu}_{L}}{{\mu}_{L,e}}$(21)
Resolution
The resolution of equation (13) can permit to determine the velocity profile, the condensate mass flow rate, the liquid film thickness and the convection heat transfer coefficient.
When K* = 1, i.e., a = 1, equation (28) becomes identical to solution obtained by AlNimr and Alkam (1997) in the case of condensation with negligible microscopic inertial term.
Where B is the liquid film width.
The energy balance which states that the rate of energy release due to condensation at the vaporliquid interface must equal the rate of heat transfer from the film to the wall can be considered to determine the film thickness δ_{ L }(x).
Where the following dimensionless parameters of the porous liquid film are defined as:

$\mathrm{\text{The}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{Jacob}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{number}}:\phantom{\rule{0.5em}{0ex}}Ja=\frac{C{p}_{L}\left({T}_{S}{T}_{W}\right)}{\phantom{\rule{0.25em}{0ex}}{h}_{Lv}\phantom{\rule{0.25em}{0ex}}}$(42a)

$\mathrm{\text{The}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{local}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{Rayleigh}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{number}}:\phantom{\rule{0.5em}{0ex}}R{a}_{x}=\frac{{K}_{1}gx}{{\mu}_{L}{\alpha}_{L}}\left({\rho}_{L}{\rho}_{v}\right)$(42b)
Results and discussion
In what follows, the main results of our study related to the effects of anisotropic permeability parameters based on the BrinkmanDarcy flow model are compared essentially with Cheng’s steady solution (Cheng 1981) based on the Darcy flow model and Ebinuma’s steady solution (Ebinuma and Nakayama 1990) based on the Ergun model (Ergun 1952) and Sanya’s steady solution (Sanya et al. 2014) for thick porous medium with anisotropic permeability based also on the BrinkmanDarcy flow model.
In Figure 2, it is observed that, when the anisotropic parameters are held constant, for example, for θ = 30° and K* = 2.5, the film thickness and the condensate mass flow rate per unit width are both increased when the parameter ω is increased while the Nusselt number is decreased.
Conclusion
In this paper, the analyses by AlNimr and Alkam (1997) have been extended to take account the anisotropic properties of the porous medium, using the flow permeability tensor given by Degan et al. (1995). The BrinkmanDarcy flow model has been developed to investigate the steady film condensation on a vertical surface embedded in a thin porous medium with anisotropic permeability whose principal axes are noncoincident with the gravity vector. Within the boundary layer approximations, dimensionlessform expressions have been obtained that describe the steady behavior of the liquid film thickness, condensate mass flow rate and surface heat transfer rate. The relative significance of the anisotropic properties may be accounted, in general manner, by introducing the two anisotropic parameters named as the permeability ratio K* and the orientation angle θ of the principal axes. Both the permeability ratio and orientation angle of the principal axes have a strong influence on the liquid film condensation in the anisotropic porous media. As is can be noticed in this study, isotropic properties of the porous medium come with a = 1, that is for θ = 0 (or K* = 1.0), and the dimensionless thickness of the liquid film and the Nusselt number are found to be independent of K* (or θ). As the two anisotropic parameters increase, when θ varies from zero towards 90°, the surface heat transfer rate decreases while the liquid film thickness and the condensate mass flow rate grow, for both the thin and the thick porous media and this behavior is more drastic in the case of the thick porous medium.
Nomenclature
a, b, c, Constants, equations (3a,b,c)
B, Liquid film width (m)
C_{ P }, Specific heat capacity of fluid at constant pressure (J. kg^{ 1}. K^{ 1})
Da, Darcy number, equation (19)
g, Gravitational acceleration (m. s^{ 2})
h_{ Lv }, Latent heat of condensation (J. kg^{ 1})
h_{ x }, Local convective heat transfer coefficient (W. m^{ 2}K^{ 1})
Ja, Jacob number, equation (42a)
$\overline{\overline{K}}$, Flow permeability tensor, equation (2)
K_{1}, K_{2}, Flow permeability along the principal x and y axes, respectively (m^{2})
K*, Anisotropic permeability ratio, equation (3d)
k, Thermal conductivity (W. m^{ 1}K^{ 1})
L, Height of the surface (m)
Nu_{ x }, Local Nusselt number, equation (45)
Ra_{ L }, Rayleigh number, equation (20)
Ra_{ x }, Local Rayleigh number, equation (42b)
T, Temperature (K)
V, Velocity of the liquid film in the porous medium (m. s^{ 1})
u, v, Velocity components in x, y directions (m. s^{ 1})
P, Pressure (Pa)
x, y, Cartesian coordinates (m)
Greek symbols
α, Thermal diffusivity (m^{2}. s^{ 1})
δ_{ L }, Liquid film thickness (m)
ϵ, Porosity of the porous medium
μ, Dynamic viscosity (kg. m^{ 1}. s^{ 1})
Γ, Condensation mass flow rate per unit length (kg. s^{ 1}. m^{ 1})
ω, Positive parameter defined in equation (42e)
ψ, Stream function (m^{2}. s^{ 1})
ρ, Density of the fluid (kg. m^{ 3})
σ, Dimensional parameter defined in equation (42d)
θ, Orientation angle of the principal axes (°)
Superscripts
*, Dimensional quantities
Subscripts
e, Effective properties due to the presence of the porous domain.
L, Liquid region
s, Saturated properties
v, Vapor region
w, Refers to the vertical surface.
Declarations
Authors’ Affiliations
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