8.5 Film Condensation in Porous Media

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1 y x T w δl Tsat T δl δ lv y2 Figure 8.29 Film condensation in a porous medium. g Liqui TwoVapor d film Phase region regio regio n n Figure 8.19 Film condensation in a porous medium. 1

2 The dominant forces in the condensation process are gravitational and capillary forces, and the latter dictates the thickness of the twophase region, δlv. The ratio of gravity and capillary forces is measured by Bond number: g (ρ l ρ v )K / ε Bo = σ (8.354) where K and ε are, respectively, permeability and porosity. When Bo ; 1, condensation in a porous medium is dominated by both gravitational and capillary force. The condensation is dominated by capillary force when Bo < 1. When gravity dominates, Bond number will be greater than 1 and there will be no two-phase region, in which case the analysis will be significantly simplified. In the following two subsections, an analysis of gravity-dominated condensation will be discussed first, followed by a discussion of the effect of surface tension on the condensation process. 2

3 8.5.2 Gravity-Dominated Film Condensation on an Inclined Wall It is assumed that the condensation is gravity-dominated and therefore the liquid and vapor are separated by a sharp interface, not a two-phase region. In addition, the following assumptions are made: The condensate film is very thin compared to the length of the inclined wall (δ = L ) so that boundary layer assumption is valid. The properties for the porous medium, liquid and vapor are independent from temperature. The inclination angle, φ, is small enough for the gravity component in the normal direction of the surface to be negligible. Darcy s law is valid for both liquid and vapor phases. 3

4 y x Vapor region Tsat φ T Liquid g Tw Figure 8.30 Gravity dominated film-condensation on an inclined wall in a porous medium. 4

5 Under these assumptions, the governing equations for the liquid layer are ul vl (8.355) = 0 x y K (8.356) ul = ( ρ l ρ v ) g cos φ µl Tl Tl 2Tl (8.357) ul vl =αl 2 x y y The boundary conditions are vl = 0, y= 0 (8.358) Tl = Tw, y= 0 (8.359) 5

6 At the interface, the boundary conditions are Tl = Tsat, y = δ l dδ m& = ρ l ul l vl, y = δ l dx T m& hlv = kml, y= δl y (8.361) (8.362) Combination of eqs. (8.361) and (8.362) T dδ l ρ l hlv ul vl = k ml, dx y (8.360) y= δl Introducing stream function ψ ψ ul =, vl = y x (8.363) (8.364) 6

7 and the following similarity variables η = ψ =α l θ (η ) = where y Ralx x (8.365) Ralx f (η ) (8.366) Tl Tsat Tw Tsat (8.367) ( ρ l ρ v ) g cos φ Kx Ralx = µ lα l (8.368) 7

8 The governing equations and the corresponding boundary conditions become f = 1 2θ f θ = 0 f (0) = 0 θ (0) = 1 θ (η δ ) = 0 1 Ja lθ (η δ ) = f (η δ ) 2 (8.369) (8.370) (8.371) 8

9 ηδ = Ralx δl x is the dimensionless liquid film thickness and Ja l = (8.375) c pl (Tsat Tw ) (8.376) is Jakob number that measures the degree of subcooling at the wall. Integrating eq. (8.369) and considering eq (8.371) f =η (8.377) which can be substituted into eqs. (8.370) and (8.374) (8.378) 2θ η θ = 0 hlv 1 Ja lθ (η δ ) = η δ 2 (8.379) erf (η / 2) θ (η ) = 1 erf(η δ / 2) (8.380) The solution of eq. (8.378) with eqs. (8.372) and (8.373) as boundary condition is 9

10 The dimensionless film thickness can be obtained by substituting eq. (8.380) into eq. (8.379) η δ2 πηδ ηδ (8.381) Ja = exp erf l 2 The heat flux at the wall is 4 2 kml (Tsat Tw ) Ra lx Tl qw = kml = θ l (0) x y y= 0 and the local Nusselt number is qw x Nu x = = kml 2 Ra1/ lx (8.382) (8.383) π erf(η δ / 2) Cheng recommended that eq. (8.383) be approximated 1/ 2 1 (8.384) 1 1/ 2 Nu x = Ralx 2 Jal π Average Nusselt number is obtained by integrating eq. (8.384) 1/ 2 (8.385) 1 2 Nu L = Jal π Ral1/L2 10

11 8.5.3 Effect of Surface Tension on Condensation in Porous Media The analysis in the preceding subsection is valid for gravitydominated condensation in porous media (Bo? 1 ). When the condensation is gravity-capillary forces dominated (Bo : 1 ) or capillary force dominated (Bo = 1), there will be a two-phase region that is saturated by a mixture of liquid and vapor, as shown in Fig The fraction of liquid in the pore space is defined as saturation γl= ϕl ε (8.386) where ϕ is volume fraction of the liquid in the porous media. 11

12 The continuity equation for the two-phase region ul vl ρl x y2 uv vv ρv = 0 x y2 (8.387) Mass fluxes for liquid and vapor are governed by Darcy s law KK rl m& l = pl νl KK rv m& v = pv νv (8.388) (8.389) Vapor flow is negligible compared to the liquid flow so that eq. ul vl (8.387) is reduced to = 0 x y2 (8.390) The velocity components in the x- and y- directions KK rl ( ρ l ρ v ) g (8.391) u = K u = l rl D µl 12

13 KK rl pl KK rl pc vl = = µ l y2 µ l y2 where ud is Daecian velocity. The capillary pressure is pc = (8.392) σ f (s) K /ε where f(s) is a Leverett s function f ( s ) = 1.417(1 s ) 2.120(1 s ) (1 s )3 and s is dimensionless saturation as γ defined γ s= l li 1 γ li The relative permeability in eqs. (8.391) and (8.392) is 3 K rl = s (8.393) (8.394) (8.395) (8.396) Substituting eqs. ( ) into eq. (8.390) s 3 x σ / K /ε ( ρ ρ ) g v l 2 s 2s (3 f sf ) sf 2 = 0 y2 x (8.397) 13

14 Subjected to the following boundary conditions s = 0, x = 0 s = 1, y2 = 0 s = 0, y2 Introducing the following similarity2 variable (ρ l ρ v )g η = y2 ( σ / K / ε ) x Eqs. (8.397)-(8.400) are transformed to 3η s 2(3 f sf ) s 2 s = 2sf s = 1, η = 0 s = 0, η (8.398) (8.399) (8.400) (8.401) (8.402) (8.403) (8.404) 14

15 The governing equations for the liquid film in dimensionless form ul vl = 0 (8.405) x y l 2 u 1 ul = 0 y 2 (8.406) 2θ l = 0 2 y where the dimensionless variables are defined as ul = (8.407) ul v, vl = l, x = ud ud x, y = K T Tw (8.408) δ y, δ l = l, θ = Tsat Tw K K The boundary conditions at the wall are l l u = v = θ = 0, y = 0 (8.409) 15

16 The boundary conditions at the interface between the liquid film and the two-phase region require that the velocity and shear stress in these two regions match, which makes the solution of the condensation problem very challenging. Majumdar and Tien (1990) proposed three models to handle the boundary condition at the interface between the liquid and the two-phase region and two of them are discussed below. Model 1. At the interface between the liquid and the twophase region, the shear stress is zero, i.e., u%l / y% = 0 at, which is the same as in classical Nusselt y% = δ l analysis. The velocity profile in the liquid layer is ul = 1 cosh y tanh δ l sinh y (8.410) 16

17 The liquid layer thickness can be obtained from an energy balance at the interface d δ Ja l 2 l (1 sech δ l ) (1 sechδ l ) = dx% Box% δ l Ra K where K 3/ 2 ( ρ l ρ v ) g Ra K = µ lα e (8.411) (8.412) Analytical solution of eq. (8.411) is not possible and it must be solved numerically. Model 2. This model also employs eq. (8.406) to obtain the velocity in the liquid layer except that the boundary condition at y = δ l is changed to ul = 1. Although it is not as rigorous as Model 1, it is an improvement over Cheng (1981) because it uses non-slip condition at the wall. The velocity profile in the liquid layer is ul = 1 cosh y coth δ l sinh y (8.413) 17

18 the overall energy balance at the interface d δ l Ja l 1 (8.414) 1 = 1 cosh δ dx% l Box% the local Nusselt number Nu x = δ l Ra K x% δ l (8.415) The parameter R in the figure is defined as Ra K σ K ε R= = Bo µ lα e (8.416) 18

19 Figure 8.31 Comparison of results of Model 1 and 2 with experiments: (a) aluminum foam metal, (b) polyurethane foam 19

LAMINAR FILM CONDENSATION ON A HORIZONTAL PLATE IN A POROUS MEDIUM WITH SURFACE TENSION EFFECTS

Journal of Marine Science and Technology, Vol. 13, No. 4, pp. 57-64 (5) 57 LAMINAR FILM CONDENSATION ON A HORIZONTAL PLATE IN A POROUS MEDIUM WITH SURFACE TENSION EFFECTS Tong-Bou Chang Key words: surface