# Film Evaporation from Wedge or Cone in a Porous Medium

Heat and mass transport to a liquid film falling along a vertical or inclined surface in a porous medium occur in many types of chemical and energy production processes. Some examples are geothermal energy production, packed-bed reactors and heat exchangers, oil recovery from petroleum reservoirs, heat transfer from buried vertical pipe lines, and gravity-assisted heat pipes. Flow and transport of heat and mass in a thin liquid film on the surface of a wedge or cone embedded in a porous medium during evaporation, gas absorption at the free surface, and heating and solid dissolution at the wall have been studied by Rahman and Faghri (1993a,b) and will be presented here.

Figure 9.14 Film evaporation on surface of wedge or cone embedded in a porous media.

The coordinate system used for the following analysis and numerical computation is shown in Fig. 9.14. The x-axis is directed along the length of the wedge or cone and the y-axis is perpendicular to the solid surface. The axis of symmetry of the wedge is oriented vertically along the direction of the gravitational force, while the solid wall is inclined by an angle θ. The continuity, momentum, and energy equations are

 $\frac{\partial ({{r}^{n}}u)}{\partial x}+\frac{\partial ({{r}^{n}}v)}{\partial y}=0$ (1)
 $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{{{\rho }_{\ell }}}\frac{\partial p}{\partial x}+\nu \left[ \frac{\partial }{\partial x}\left( \frac{1}{{{r}^{n}}}\frac{\partial ({{r}^{n}}u)}{\partial x} \right)+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}} \right]-\frac{\nu u}{K}+\frac{g({{\rho }_{\ell }}-{{\rho }_{v}})}{{{\rho }_{\ell }}}\cos \theta$ (2)
 $u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{{{\rho }_{\ell }}}\frac{\partial p}{\partial y}+\nu \left[ \frac{\partial }{\partial x}\left( \frac{1}{{{r}^{n}}}\frac{\partial ({{r}^{n}}v)}{\partial x} \right)+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}} \right]-\frac{\nu v}{K}+\frac{g({{\rho }_{\ell }}-{{\rho }_{v}})}{{{\rho }_{\ell }}}\sin \theta$ (3)
 $u\frac{\partial \phi }{\partial x}+v\frac{\partial \phi }{\partial y}={{\Gamma }_{eff}}\left[ \frac{\partial }{\partial x}\left( \frac{1}{{{r}^{n}}}\frac{\partial ({{r}^{n}}\phi )}{\partial x} \right)+\frac{{{\partial }^{2}}\phi }{\partial {{y}^{2}}} \right]$ (4)

Here, n = 0 or 1 denotes equations applicable to a wedge or a cone, respectively. The properties of the fluid and the medium are assumed to be constant. The terms containing permeability quantify the resistance to the flow due to the presence of the porous matrix; as $K\to \infty$, the flow equations approach those for a liquid medium with no solid matrix. The variable φ represents temperature or concentration, and Γeff is the corresponding diffusion coefficient for the different heat and mass transfer processes considered here. For heat transfer, Γeff is the effective thermal diffusivity, which is defined as

 ${{\Gamma }_{eff}}=\frac{\varepsilon {{k}_{\ell }}+(1-\varepsilon ){{k}_{s}}}{\varepsilon {{\rho }_{\ell }}{{c}_{p\ell }}}$ (5)

where $\varepsilon$ is the porosity of the medium. For mass transfer processes, mass diffusivity is

 ${{\Gamma }_{eff}}=\frac{{{\nu }_{\ell }}}{Sc}$ (6)

The porous matrix diffuses heat and therefore participates in the heat transfer process, but so far as mass diffusion is concerned, the fluid is the only active medium. An experimental or analytical correlation (if available) can be substituted for the linear averaging of thermal conductivity – as used in eq. (5) – without changing the analysis presented here.

Due to the small rate of evaporation, the flow is not affected by the loss of fluid at the free surface. In that situation, the latent cooling due to evaporation has a negligible influence on heat transfer from the wall. Mass transfer during absorption or dissolution neither affects the flow or heat transfer, since the rate of mass diffusion is very small compared to the fluid flow rate, and the heat of absorption or dissolution is negligible compared to the sensible heat transfer. Therefore, the momentum and transport equations are decoupled, and the flow field can be solved independently without considering any associated transport processes.

For a thin film flow, the thickness of the film is usually much smaller than the physical dimension of the object over which flow is taking place. Similarly, the velocity normal to the wall is much smaller than the velocity component parallel to the wall. A scaling argument can be used to show that convection in the flow direction is dominant over diffusion in that direction, and therefore one can neglect diffusion in the flow direction. Moreover, in a gravity-driven flow through a porous matrix, small effective Reynolds numbers cause the momentum boundary layer to develop quickly and encompass the entire thickness after a short distance. Therefore, inertia can be neglected in most regions. The convection of heat or mass can still be significant, particularly if the Prandtl or Schmidt number was large, and thus was retained here. Keeping only the most significant terms, the governing equations (1) – (4) were transformed to

 $\mu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}-\frac{\mu u}{K}+g({{\rho }_{\ell }}-{{\rho }_{v}})\cos \theta =0$ (7)
 $p={{p}_{0}}+({{\rho }_{\ell }}-{{\rho }_{v}})g(\delta -y)\sin \theta$ (8)
 $u\frac{\partial \phi }{\partial x}+v\frac{\partial \phi }{\partial y}={{\Gamma }_{eff}}\frac{{{\partial }^{2}}\phi }{\partial {{y}^{2}}}$ (9)

where p0 is the ambient pressure. Equation (7) is a second-order non-homogeneous ordinary differential equation. Adding the homogeneous and particular solutions and applying boundary conditions at the wall (u = 0) and on the free surface (du/dy = 0), the following velocity distribution was obtained.

 $u={{u}_{D}}\left[ 1-\frac{\cosh {{\delta }^{*}}(1-{{y}^{+}})}{\cosh {{\delta }^{*}}} \right]$ (10)

where

 ${{u}_{D}}=\frac{K}{{{\mu }_{\ell }}}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g\cos \theta$ (11)

is the velocity obtained by direct application of Darcy’s law. The quantity δ * is the film height nondimensionalized with the permeability of the medium (${{\delta }^{*}}=\delta /\sqrt{K}$). It can be noticed that u increases with y+ and becomes maximum on the free surface. The average film velocity was obtained by integrating the local velocity across the film thickness.

 $\bar{u}={{u}_{D}}\left( 1-\frac{\tan {{\delta }^{*}}}{{{\delta }^{*}}} \right)$ (12)

It may be noted that eqs. (10) and (12) are applicable to both the wedge (n = 0) and the cone (n = 1), provided that for a cone, $\delta \ll r$ (i.e., away from the vertex).

The thickness of the film can be related to the velocity by satisfying the conservation of mass at any x-location. The volumetric flow rates for a wedge and a cone are respectively

 $Q=\bar{u}\delta$ (13)
 $Q=2\pi x\bar{u}\delta \sin \theta$ (14)

Substituting eq. (12) into eqs. (13) and (14), the following two equations for the film thickness are obtained.

 δ * − tanδ * = A for wedge (15)
 ${{\delta }^{*}}{{({{\delta }^{*}}-\tanh {{\delta }^{*}})}^{2}}=\frac{B}{{{x}^{+}}}\text{ for cone}$ (16)

where

 $A=\frac{Q}{{{u}_{D}}\sqrt{K}}$ (17)
 $B=\frac{Q{{\Gamma }_{eff}}}{2\pi \sin \theta {{K}^{3/2}}u_{D}^{2}}$ (18)
 ${{x}^{+}}=\frac{{{\Gamma }_{eff}}x}{\bar{u}{{\delta }^{2}}}$ (19)

The solutions of eqs. (15) and (16) give the film thickness distribution for the wedge and cone, respectively. It may be noted that δ * remains constant for a wedge, whereas it decreases with x for the case of a cone because more area is available to the flow as the fluid moves downstream. The transport equation (9) can be nondimensionalized as

 \begin{align} & \text{For wedge: }{{u}^{+}}\frac{\partial \phi }{\partial {{x}^{+}}}=\frac{{{\partial }^{2}}\phi }{\partial {{y}^{+2}}} \\ & \text{For cone: }{{u}^{+}}\left( 3-\frac{1}{{{\delta }^{*}}} \right)\frac{\partial \phi }{\partial {{x}^{+}}}+\frac{{{u}^{+}}}{{{x}^{+}}}\left[ \frac{1}{{{\delta }^{*}}}-\frac{{{y}^{+}}}{1-\cosh ({{y}^{+}}{{\delta }^{*}})+\sinh ({{y}^{+}}{{\delta }^{*}})} \right]\frac{\partial \phi }{\partial {{y}^{+}}}=\frac{{{\partial }^{2}}\phi }{\partial {{y}^{+2}}} \\ \end{align} (20)

In addition to the simplified analytical solution for δ * , Rahman and Faghri (1993b) obtained a complete numerical solution of these equations using a two-dimensional finite-difference method. To appropriately handle the variation of film height with flow, a curvilinear body-fitted coordinate system was used. The free surface formed one of the boundaries of the computation domain, and grid cells between the wall and free surface were generated by an algebraic interpolation between boundary values. The cell faces, in general, were nonorthogonal to each other. The equations were solved in the physical domain using the co-variant components of velocity and force vectors.

Figure 9.15 Variation of film thickness along a wedge.
Figure 9.16 Variation of film thickness along a cone.

The distribution of dimensionless film thickness along the wedge is shown in Fig. 9.15. Ethyl alcohol at 20 ºC was used for fluid properties of the film. The porous matrix was made of glass beads 0.98 mm in diameter, which provide a porosity of $\varepsilon =0.35$ (for uniformly-packed spheres) and a permeability of $K=3.3\times {{10}^{-9}}\text{ }{{\text{m}}^{\text{2}}}$. The wedge angle was $\theta ={{15}^{\circ }}$. A flow rate of Q = 9.5x10-5 m2/s was used, which produced laminar flow through the pore structure as well as a laminar fluid film. These specifications resulted in a value of A = 250 for the system. The range of parameter A chosen for the analytical solution was established by preserving the validity of the volume-averaging technique used in the formulation as one limit and by keeping the flow as laminar as the other limit. It can be seen that the film thickness increased with the parameter A, which was dependent on the flow rate and the permeability of the medium. Comparison of the numerical and analytical solutions revealed that the film height distributions were practically coincident. Therefore, neglecting terms for a simplified analysis introduced a very small error in the film thickness.

For the different transport problems considered here, the film thicknesses corresponding to the thin film flow adjacent to a cone are shown in Fig. 9.16. Since the vertex of the cone is a singular point with zero area, the flow is expected to change very rapidly in the vicinity of the vertex. Moreover, the flow at that location will be determined by how the fluid is introduced to the cone (in the form of a liquid jet, etc.). Therefore, the analysis and computations presented here considered flow at some distance from that region so that a thin film could be attained. The most significant parameter for flow over a cone was B. Unlike the parameter A used in relation to the wedge, this parameter contains Γeff, which varied with the type of transport (heat transfer, gas absorption, or solid dissolution). For numerical computation, the cone angle was $\theta ={{15}^{\circ }}$. The same porous medium and fluid combination was used as that for the computation of flow over a wedge. A flow rate of Q = 5.9x10-6 m3/s was used for the numerical computation. The analytical and numerical methods had negligible difference in their predictions of film thickness. The results and heat and mass transfer can be found in Rahman and Faghri (1993b).

## References

Rahman, M.M., and Faghri, A., 1993a, “Transport in a Thin Liquid Film on the Outer Surface of a Wedge or Cone Embedded in a Porous Medium, Part I: Mathematical Analysis,” International Communications in Heat and Mass Transfer, Vol. 20, pp. 15-27.

Rahman, M.M., and Faghri, A., 1993b, “Transport in a Thin Liquid Film on the Outer Surface of a Wedge or Cone Embedded in a Porous Medium, Part II: Computation and Comparison of Results,” International Communications in Heat and Mass Transfer, Vol. 20, pp. 29-42.