Analytical Investigation Into Effects of Capillary Force on Condensate Film Flowing Over Horizontal Semicircular Tube in Porous Medium

Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 6693512, 10 pages https://doi.org/10.1155/2021/6693512

Research Article Analytical Investigation into Effects of Capillary Force on Condensate Film Flowing over Horizontal Semicircular Tube in Porous Medium

Tong-Bou Chang , Bai-Heng Shiue, Yi-Bin Ciou, and Wai-Io Lo

Department of Mechanical and Energy Engineering, National Chiayi University, Chiayi, Taiwan

Correspondence should be addressed to Tong-Bou Chang; [email protected]

Received 17 November 2020; Revised 21 February 2021; Accepted 6 March 2021; Published 18 March 2021

Academic Editor: Luca Chiapponi

Copyright © 2021 Tong-Bou Chang et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A theoretical investigation is performed into the problem of laminar filmwise condensation flow over a horizontal semicircular tube embedded in a porous medium and subject to capillary forces. *e effects of the capillary force and gravity force on the condensation heat transfer performance are analyzed using an energy balance approach method. For analytical convenience, several dimensionless parameters are introduced, including the Jakob number Ja, Rayleigh number Ra, and capillary force parameter Boc. *e resulting dimensionless governing equation is solved using the numerical shooting method to determine the effect of capillary forces on the condensate thickness. A capillary suction velocity can be obtained mathematically in the calculation process and indicates whether the gravity force is greater than the capillary force. It is shown that if the capillary force is greater than the condensate gravity force, the liquid condensate will be sucked into the two-phase zone. Under this condition, the condensate film thickness reduces and the heat transfer performance is correspondingly improved. Without considering the capillary force effects, the mean Nusselt number is also obtained in the present study 1/2 π 1/2 as Nu | ∗ � (2Ra Da/Ja) � (1 + cos θ) dθ. V2 �0 0

1. Introduction gradient; (3) the shear force between the liquid and the steam was sufficiently small to be ignored; and (4) the inertial force *e problem of laminar film condensation in porous media in the liquid film could also be ignored. However, these has received considerable attention in the literature due to its assumptions do not necessarily hold in practical situations. importance in many engineering applications, including For example, the liquid film thickness is often nonconstant packed-bed heat exchangers, chemical reactors, petroleum in experimental tests, and hence a linear temperature gra- recovery, heat pipes, casting solidification, geothermal res- dient cannot be guaranteed. *us, the theoretical heat ervoirs, and oil recovery in reservoir engineering. In such transfer coefficient predicted by Nusselt is frequently slightly applications, the capillary force has a significant effect on the lower than that observed experimentally. Consequently, heat transfer performance and must therefore be taken into many scholars have attempted to modify and improve the account when designing and evaluating the system. original assumptions in [1] in order to achieve a better fit *e problem of laminar film condensation was first between the theoretical and experimental results. investigated by Nusselt [1], who considered a condensate Cheng [2] investigated the heat transfer coefficient of a film flowing down a vertical plate. In developing a theoretical condensate film on an inclined plate in a porous medium. solution for the heat transfer rate, Nusselt imposed four *e results showed that, for an inclination angle close to assumptions, namely, (1) the vertical wall had a constant zero, the flow condition at the edge of the plate could not be temperature; (2) the laminar film had a linear temperature found and the value can only be found when the inclined 2 Mathematical Problems in Engineering angle is larger than zero due to the ambiguity of the pressure used a nanoscale surface to increase the heat transfer rate in gradient. In a later study, this problem was resolved by Yang dropwise condensation. Xie et al. [19] investigated the and Chen [3] using an open channel hydraulics concept. condensation heat transfer performance of R245fa foam Chang [4] and Wang et al. [5] investigated the heat transfer insulation in tubes with and without lyophilic porous- performance of a condensate layer on a horizontal plate in a membrane-tube inserts, respectively. Wen et al. [20] in- porous medium in the absence of capillary forces. However, vestigated the condensation heat transfer performance by in many of these problems, the effective pore radii are small, using hydrophobic copper nanowires to induce a capillary and hence the influence of capillary forces on the heat force. *e results showed that a smaller hole diameter of the transfer performance is potentially significant and should be hydrophobic copper nanowires induces a larger capillary considered. Accordingly, Majumdar and Tien [6] studied the force and then enhances the condensation heat transfer effect of capillary forces on film condensate flow along a performance. Buleiko et al. [21] examined the effects of the vertical wall. *e results showed that capillary forces actually capillary force on the phase behavior of liquid and gaseous play a significantly effect on condensate flow. Udell [7, 8] propane and the dynamics of hydrate formation and dis- used a sand-water-steam system to perform a series of sociation in porous media. It was shown that the reactant theoretical and experimental investigations into the effect of with water and propane required more pressure to extract capillary forces on the heat transfer process in porous media than a simple material in a porous medium. saturated with both vapor and liquid phases. Fluid flow around a semicircular tube had been carried Heat transfer and fluid flow in porous media had been out by many researchers [22–27]. *is geometry is of par- studied by many researchers. EL-Hakiem and Rashad [9] ticular interest for chemical and food processing industries, investigated the effects of both radiation and the nonlinear heat pipe designs, solar energy applications, and the design Forchheimer terms on free convection from a vertical cyl- of heat exchangers. Moreover, since the capillary forces have inder embedded in a fluid-saturated porous medium nu- a significant effect on the heat transfer performance in merically by using the second-level local nonsimilarity porous media and must therefore be taken into consider- method. EL-Kabeir et al. [10] applied the group theoretic ation, accordingly, the present study investigates the in- method to solve problem of combined magnetohydrody- fluence of capillary forces on the laminar film condensation namic heat and mass transfer of non-Darcy natural con- flow on a finite-size horizontal semicircular tube embedded vection about an impermeable horizontal cylinder in a non- in a porous medium and maintained at a constant tem- Newtonian power law fluid embedded in porous medium perature. In order to determine the effects of the capillary under coupled thermal and mass diffusion, inertia resis- forces, the present study employs the concepts of capillary tance, magnetic field, and thermal radiation effects. Bakier pressure and a two-phase zone to analyze the liquid flow. *e et al. [11] developed a linear transformation group approach dimensionless heat transfer coefficient is then solved for method to simulate the problem of hydromagnetic heat various values of the Darcy number Da, Jakob number Ja, transfer by mixed convection along vertical plate in a liquid Rayleigh number Ra, Prandtl number Pre, and capillary saturated porous medium. *e steady two-dimensional parameter Boc. laminar forced flow and heat transfer of a viscous incom- pressible electrically conducting and heat-generating fluid past a permeable wedge embedded in non-Darcy high-po- 2. Analysis rosity ambient medium with uniform surface heat flux have Figure 1 shows a schematic illustration of the physical model been studied by Bakier and Rashad [12]. Rashad and EL- considered in the present study with curvilinear coordinates Kabeir [13] studied the coupled heat and mass transfer in (x, y) aligned along the tube surface and surface normal, transient flow by a mixed convection boundary layer past an respectively. *e tube is assumed to be horizontal, clean, and impermeable vertical stretching sheet embedded in a fluid- semicircular (with a radius R) and to be embedded in a saturated porous medium in the presence of a chemical porous medium. Pure quiescent vapor in a saturated state reaction effect. Mansour et al. [14] developed a thermal and with a uniform temperature T condenses on the tube nonequilibrium modeling of steady natural convection in- sat surface. If the wall temperature is lower than the vapor side wavy enclosures with the effect of thermal radiation. temperature, the resulting condensation forms a thin con- Mansour et al. [15] numerically investigated the problem of densate layer and wets the tube surface ideally. Under the double-diffusive convection in inclined triangular porous effects of gravity, the liquid film boundary layer thickness, δ, enclosures with sinusoidal variation of boundary conditions has a minimum value at the top of the tube and increases in the presence of heat source or sink. Sameh and Rashed gradually as the liquid flows downward over the surface of [16] investigated the effects of an electromagnetic force on the tube. Notably, the gravity force acting on the liquid film the heat transfer by natural convection flow of nanofluids in in the downward direction is opposed by the capillary force wavy enclosure filled by an isotropic porous medium. *ey induced by the porous medium, which pulls the film in the found that the increase in the Hartmann number and the upward direction with a velocity ] . decrease in Darcy number obstruct the fluid movement but 2 *e following assumptions are made for analyzing the they increase the rate of heat transfer. heat transfer characteristics of the condensate film: Liu et al. [17] found that the injection of copper bubbles into a porous medium increased the heat transfer rate under (1) *e flow is steady and laminar. Hence, the effects of joint boiling and condensation conditions. Lee et al. [18] inertia and convection in momentum equation are Mathematical Problems in Engineering 3

following energy balance exists for each element of the liquid ﬁlm with height δ and width dx: Tsat y2 d δ zΤ v2 �� ρu� h + C Τ − Τ �dy��dx + ρh ] dx � k dx, y g fg p sat fg 2 e x2 Porous dx 0 zy x δ media (5) θ

Tw where ke is the effective thermal conductivity of the porous medium with saturated liquid and subscript 2 R indicates that the referred properties belong to the two- phase zone. *e right-hand side of (5) represents the energy Two-phase Liquid Liquid Two transferred from the liquid film to the tube surface. zone film film phase Meanwhile, the first term on the left-hand side of the zone equation expresses the net energy flux across the liquid film (from x to x + dx), while the second term expresses Figure 1: Schematic diagram of physical model and curvilinear the net energy sucked into the two-phase zone by the coordinate system. capillary force. *e continuity equation at the interface between the negligible. Moreover, the condensate film travels two-phase zone and the liquid film has the form very slowly and thus has negligible kinetic energy. � � zu � zυ � (2) *e vapor temperature, wall temperature, and 2� + 2� � 0. (6) �y � �y � physical properties of the dry vapor, condensate, and zx 2 0 zx 2 0 porous medium, respectively, remain constant. Moreover, the no slip condition between the two-phase (3) *e liquid film flow in the porous medium is gov- zone and the liquid film is given by erned by Darcy’s law. ρgK u | � u| � sin θ. ( ) 2 y2�0 y�δ 7 Given these assumptions, from the concept of convec- μe tion in porous media [28], the governing equations for the liquid film can be formulated as follows: Applying the relation dx � Rdθ, (7) can be rewritten as � � Continuity equation is as follows: zu � zu� ρgK cos θ 2� � � � . (8) �y � �y� zu zv zx 2 0 zx δ μe R + � 0. (1) zx zy According to Udell [7], the suction velocity produced by the capillary force can be written as Momentum equation in x-direction is as follows: KK zP μ v � re � c − ρg cos θ�, (9) ρ − ρ �g sin θ − e u � 0. (2) 2 μ zy v K e 2 Energy equation is as follows: where z2Τ K � s3, (10a) α � 0, (3) re ezy2 σ Pc � √�� f(s), (10b) where μe and αe are the dynamic viscosity of the liquid K/ε condensate and effective thermal diffusivity, respectively; ρ 2 3 and ρv are the liquid density and vapor density, respectively; f(s) � 1.417(1 − s) − 2.120(1 − s) + 1.263(1 − s) , v and u and are the velocity components in the x- and y- (10c) directions, respectively. Assuming that the vapor density ρv is much less than the in which s is the dimensionless saturation and has a value of liquid density ρ, (2) can be rewritten as s � 1 at y2 � 0. Substituting (10a)–(10c) into (9), the suction velocity is ρgK obtained as u � sin θ. (4) μe K zs According to Fourier’s conduction law, the first law of 3 √��σ v2 � s � f(s)· − ρg cos θ�. (11) thermodynamics and Nusselt’s classical analysis method, the μe k/ε zy2 4 Mathematical Problems in Engineering

Substituting (11) and (8) into (6), the continuity equation at the interface between the two-phase zone and the liquid ﬁlm can be rewritten as ρgK K ⎧⎨ σ zs 2 zs 2 z2s zs ⎫⎬ cos θ + √�� ⎡⎣s3f″� � + 3s2f′� � + s3f′ − 3s2ρg cos θ ⎤⎦ � 0. (12) R ⎩ k zy zy 2 zy ⎭ μe μe /ε 2 2 zy2 2

2 2 According to Majumdar and Tien [6], the saturation (z s/zy2) � 0 at y2 � 0. Equation (12) can then be rewritten proﬁles near the interface of the two-phase zone and the as liquid ﬁlm is almost linear when y2 � 0. In other words,

�� 2 �� s2 g �gs2 � − ( (K ) )�s3f″ + s2f′�( g R) zs � 3 ρ cos θ ± 3ρ cos θ 4 σ/ /ε 3 ρ / cos θ � �� (13) � � 3 2 . zy2 � 2(σ/ (K/ε) )�s f″ + 3s f′� y2�0

*e suction velocity v2 can be calculated by substituting (13) into (11) and then rearranging to give

�� ⎧⎪ 2 �� ⎫⎪ ⎪ s2 g �gs2 � − ( (K ) )�s3f″ + s2f′�( g R) ⎪ K ⎨ σ 3 ρ cos θ ± 3ρ cos θ 4 σ/ /ε 3 ρ / cos θ ⎬ v � s3 √�� f(s) �� − ρg cos θ . (14) 2 ⎪ K 3 2 ⎪ μe ⎩⎪ /ε 2(σ/ (K/ε) )�s f″ + 3s f′� ⎭⎪

√� For analytical convenience, let the following dimen- where Boc � (δ ε /ρgK) is a capillary parameter repre- sionless parameters be introduced: senting the ratio of the capillary force to the gravitational √� Bo δ ε force and Da is the Darcy number. If c is large (i.e., a small Bo � , Bond number), the capillary force dominates the gravita- c gK ρ tional force and hence the contribution of the two-phase (15) K zone to the total liquid flow is significant. Da � , By differencing (10c), it can be derived that f′ � − 1.417 R2 and f″ � − 4.24 when y2 � 0 and s � 1. Substituting these values into (14) gives

�� ρgK √�� ] | � �0.2503 cos2 θ + 3.774Bo Da cos θ − 0.7497 cos θ�. ( ) 2 y2�0 c 16a μe

As described above, v2 is the suction velocity produced the downward direction and no liquid condensate is sucked by the capillary force. As a result, it cannot have a negative into the two-phase zone (i.e., v | � 0 physically). 2 y2�0 value physically. However, a value of v | < 0 can be For analytical convenience, let the suction velocity 2 y2�0 obtained mathematically in the calculation process and produced by the capillary force be expressed in the following implies that the gravity force is greater than the capillary dimensionless form: force. In other words, the condensate ﬁlm ﬂows entirely in

] | ��√�� ∗ 2 y2�0 2 V2 � � 0.2503 cos θ + 3.774Boc Da cos θ − 0.7497 cos θ, (16b) ρgK/μe � Mathematical Problems in Engineering 5

where (ρgK/μe) is the characteristic velocity (i.e., the Darcy Substituting (4) and (17) into (5) yields velocity, ud � (ρgK/μe). Equation 16(a) can then be rewritten as follows:

ρgK ∗ ] |y � � · V | 2 2 0 μ 2 e (17) or ] | � u · V∗. 2 y2�0 d 2

d hfg ∗ ΔTμeke δ (δ sin θ) + δ · V2 � 2 . (18) dx hfg +(1/2)CpΔT ρ gK� hfg +(1/2)CpΔT�

Let the following dimensionless parameters be addi- Fortunately, for the particular case where the capillary ∗ tionally defined: force is neglected (i.e., V2 � 0), the analytical solution of the dimensionless local liquid film thickness can be derived dx � Rdθ, using the separation of variables method as follows: 1/2 CpΔT ∗ 2Ja 1 Ja � , δ | ∗ �� . (21) V2 �0 �hfg +(1/2)CpΔT� RaDa 1 + cos θ In order to solve δ ∗ from (20) (i.e., for the case where the ρ2gP R3 Ra � re , capillary force cannot be neglected), the following effective 2 (19) μe capillary suction function, F, is introduced to represent the effect of the capillary force on the thickness of the con- μeCp densate layer: Pre � , k ∗ e δ ∗ � 1 − F, (22) δ δ |V∗�0 δ∗ � . 2 R or With these definitions, (18) can be rewritten as 2Ja 1 1/2 ∗ (23) d 1 Ja δ �(1 − F)�� . δ∗ δ∗ sin θ � +�1 − Ja �δ∗V∗ � . (20) RaDa 1 + cos θ dθ 2 2 RaDa Substituting (23) into (20) yields However, even with the corresponding boundary conditions, i.e., (dδ/dθ) � 0 at θ � 0, it is difficult to solve δ∗ from (20).

2Ja 1 dF Ja 1 2Ja 1 1/2 − (1 − F)� � ��sin θ +� − 2 − �1 − Ja �� · V∗�F RaDa 1 + cos θ dθ Ra 2 RaDa 1 + cos θ 2 (24) Ja 1 2Ja 1 1/2 + F2 +�1 − Ja �V∗� � �� 0. RaDa 2 2 RaDa 1 + cos θ Ja A � , Equation (24) is a ﬁrst-order diﬀerential equation of F RaDa with respect to θ. By setting θ � 0, the following polynomial 1/2 equation with respect to the initial boundary condition, Ja 1 Ja ∗ B �� − 2 − �1 − Ja �� · V2 �, (26) F|θ�0(i.e., F(0)), can be derived: Ra 2 RaDa AF|2 + BF| + C � 0, (25) θ�0 θ�0 1 Ja 1/2 C ��1 − Ja �V∗� � . where 2 2 RaDa 6 Mathematical Problems in Engineering

Substituting the derived value of F (0) into (17) and using number decreases. Furthermore, the dimensionless film the forward difference shooting method [29], the variation of thickness has a minimum value at the top of the tube (θ � F in the θ direction can be calculated. 0°) and increases with increasing θ. *is result is intuitive Let the local Nusselt number be defined as since the analyses consider the case of falling film condensation, and thus the effects of gravity minimize the film hθD thickness on the upper surface of the tube but cause the film Nuθ � , (27) ke thickness to increase toward an infinite value as the liquid flows over the tube and collects at the lower surface. *e where minimum film thickness on the upper surface of the tube k results in the maximum Nusselt number. However, as the h � e. (28) θ δ film thickness increases, the heat transfer performance reduces due to the lower temperature gradient. Consequently, Substituting hθ into (27), the local Nusselt number can the minimum Nusselt number occurs at the lower surface of be rewritten as the tube. Figure 4 shows the variation of the effective capillary D 2 1 2RaDa 1/2 Nuθ � � ∗ � �� (1 + cos θ)� . suction function F with θ as a function of the Jacob δ δ (1 − F) Ja number, Ja. As shown, F has a positive value for all values (29) of Ja and θ but decreases as Ja and θ increase. Figure 5 *e mean Nusselt number can then be obtained as shows the distribution of the dimensionless suction velocity, V∗ , along the surface of the tube as a function of 1/2 1/2/2 2 1 2RaDa π (1 + cos θ) Bo . It is seen that V∗ increases significantly with in- Nu � � � � dθ. (30) c 2 π Ja 0 (1 − F) creasing Boc, particularly at smaller values of θ. *is finding is reasonable since a higher value of Boc implies a By substituting F � 0 into (29) and (30), respectively, the stronger capillary force, which implies in turn a greater local Nusselt number and mean Nusselt number for the dimensionless suction velocity. It is additionally seen that ∗ special case of V2 � 0 can be derived as the dimensionless suction velocity has a maximum value at the top of the tube and decreases with increasing θ. *is 2RaDa 1/2 finding is again intuitive since the current analyses Nuθ|V∗�0 �� (1 + cos θ)� , 2 Ja consider the case of falling film condensation, and hence (31) the suction effect is the greatest at the upper surface of the 1/2 π tube and causes a spreading of the liquid in the horizontal 2RaDa 1/2 Nu |V∗�0 �� (1 + cos θ) dθ. direction. However, as the sucked liquid gradually falls in 2 Ja 0 the downward direction under the effects of gravity, the capillary ability of the interface reduces and hence the suction velocity also reduces. 3. Results and Discussions Figure 6 shows the variation of the mean Nusselt number, Nu, with the capillary parameter, Boc, as a function *e physical parameters used in (20) (i.e., R, Ja, Pre, Ra, of Ja, Ra, and Da. As shown, Nu increases with increasing Da, and Boc) must all be assigned reasonable values for Boc once Boc increases beyond a certain critical value (where simulating practical engineering problems. *e following this value depends on the values assigned to Ja, Ra, and Da). analyses take water-vapor as the working liquid and use *e turning points in the figure indicate at which the the dimensional and dimensionless parameter values cited capillary force effect should be taken into consideration. To in Udell [7,8] and Bridge et al. [30], as summarized in the left of the turning point, the capillary force has only a Table 1. small effect and can be neglected. In other words, the Figures 2 and 3 show the distributions of the dimen- condensate flow is dominated by the gravitational force, and sionless film thickness and local Nusselt number, respec- hence no liquid condensate is sucked into the two-phase tively, along the surface of the tube as a function of the Jacob zone. As a result, the value of Nu is close to that calculated ∗ number, Ja. (note that the remaining parameters are for Nu using V2 � 0 in (31). However, to the right of the assigned the characteristic values shown in Table 1, i.e., turning point, the value of Nu increases significantly with − 11 11 5 Da � 6.4 ×10 , Ra � 2 ×10 , and Boc � 6.1 × 10 ). In gen- increasing Boc. *is result is to be expected since a higher eral, the results show that as Ja increases, the dimensionless value of Boc implies a stronger capillary force, and hence a liquid film thickness increases, while the local Nusselt greater amount of liquid is sucked into the two-phase zone. Mathematical Problems in Engineering 7

Table 1: Physical parameters used in present analyses. Symbol Interpretation Typical value K Permeability 6.4 ×10− 13 m2 ρ Liquid density 957.9 kg/m3 R Radius of tube 0.1 m Cp Speciﬁc heat at constant pressure 4217 J/kg-°C − 4 μe Liquid viscosity 2.79 ×10 kg/m-s σ Surface tension 5.94 ×10− 2 N/m hfg Heat of vaporization 2257 kJ/kg ° ° ° ΔT Saturation temperature minus wall temperature (Tsat-Tw) 5 C, 10 C, 25 C ε Porosity 0.38 Ja (CpΔT/hfg + (1/2)CpΔT) 0.01, 0.02, 0.05 Pre (μeCp/ke) 1.76 Da (K/R2) 6.4 ×10− 11 2 3 2 11 Ra Ra � (ρ gPreR /μe ) 2 ×10 √� 5 Boc (σ ε /ρgK) 6.1 × 10

0.3 100

0.25 80

0.2 60 θ δ∗ Nu 0.15 40

0.1 20

Ra = 2 × 1011, Da = 6.4 × 10–11, Bo = 6.1 × 105 11 –11 5 c Ra = 2 × 10 , Da = 6.4 × 10 , Boc = 6.1 × 10 0.05 0 0204010 30 5060 70 80 90 0204010 30 5060 70 80 90 θ θ Ja = 0.05 Ja = 0.01 Ja = 0.02 Ja = 0.02 Ja = 0.01 Ja = 0.05 Figure ∗ Figure 2: Variation of dimensionless ﬁlm thickness δ with radial 3: Variation of local Nusselt number Nuθ with radial position θ as function of Ja. position θ as function of Ja. 8 Mathematical Problems in Engineering

0.8 1 × 103

0.6 1 × 102

0.4

F Nu

1 × 101

0.2

= 2 × 1011, = 6.4 × 10–11, = 6.1 × 105 Ra Da Boc 1 × 100 0 1 × 104 1 × 105 1 × 106 0204010 30 5060 70 80 90 Bo θ c 11 –10 Ja = 0.02, Ra = 2 × 10 , Da = 6.4 × 10 Ja = 0.01 11 –10 Ja = 0.05, Ra = 2 × 10 , Da = 6.4 × 10 Ja = 0.02 10 –11 Ja = 0.02, Ra = 2 × 10 , Da = 6.4 × 10 Ja = 0.05 10 –11 Ja = 0.05, Ra = 2 × 10 , Da = 6.4 × 10 Figure 4: Variation of eﬀective capillary suction function F with Turning points radial position θ as function of Ja. Figure 6: Variation of mean Nusselt number Nu with Boc as function of Ja, Ra, and Da.

1.2 Consequently, the ﬁlm thickness reduces and the heat 11 –11 Ja = 0.02, Ra = 2 × 10 , Da = 6.4 × 10 transfer rate increases due to the steeper temperature gradient.

4. Conclusion 0.8 *is study has analyzed the problem of laminar ﬁlm con-

∗ densation on a semicircular tube embedded in porous V2 medium and subjected to a capillary force. An effective capillary suction function, F, has been introduced to model 0.4 the effect of the capillary force on the thickness of the condensation film, thereby allowing both the local condensate film thickness and the local Nusselt number to be derived using a simple numerical method. *e main con- clusions of this study can be summarized as follows: 0 (1) If the capillary force effect is simply ignored, the local 0204060 80 liquid film thickness can be expressed analytically as ∗ 1/2 θ δ | ∗ �{ (2Ja/Ra Da)(1/1 + cos θ)} , while the V2 �0 = 6.1 × 105 Boc corresponding mean Nusselt number can be derived 5 1/2 π 1/2 Bo = 1.0 × 10 as Nu | ∗ � (2Ra Da/Ja) � (1 + cos θ) dθ. c V2 �0 0 Figure 5: Variation of dimensionless capillary suction velocity V∗ (2) When the capillary effect is considered, but the 2 capillary force is much less than the gravitational with radial position θ as function of Boc. Mathematical Problems in Engineering 9

force, the capillary effect can be ignored. However, if Subscripts the capillary force is greater than the gravitational force, the liquid condensate is sucked into the two- 2: Properties in the two-phase zone phase zone. Consequently, the condensate film o: Quantity at top of tube thickness reduces and the Nusselt number, Nu, c: Capillary increases. sat: Saturation property w: Quantity at wall (3) A dimensionless parameter, Boc, has been introduced to characterize the liquid flow induced by the e: Effective property. capillary force. *e results have shown that when Boc has a higher value (i.e., the capillary force is greater Data Availability than the gravitational force), the liquid film thickness *e data used to support the findings of this study are reduces, and the heat transfer rate is enhanced as a available from the corresponding author upon request. result of the higher temperature gradient. Conflicts of Interest Abbreviations *e authors declare that they have no conflicts of interest. Boc: Ratio of surface tension and gravity forces defined in (15) Acknowledgments Cp: Specific heat at constant pressure Da: Darcy number defined in (15) *e authors gratefully acknowledge the financial support F: Effective capillary suction function defined in (22) provided to this study by the Ministry of Science and f: Saturation parameter defined in (10c) Technology (MOST) of Taiwan under Grant nos. MOST g: Acceleration of gravity 106-2813-C-415-043-E and MOST 109-2221-E-415-006- h: Heat transfer coefficient MY2. hfg: Heat of vaporization Ja: Jakob number defined in (19) References k: *ermal conductivity [1] W. Nusselt, “Die Oberflachen Kondensation des Wasser- K: Permeability of the porous medium dampes,” Zeitsehrift des Vereines Deutscher Ingenieure, K : Saturation parameter defined in (10a) • re vol. 60, pp. 541–546, 1916. m: Condensate mass flux [2] P. Cheng, “Film condensation along an inclined surface in a Nu: Nusselt number defined in (27) porous medium,” International Journal of Heat and Mass Pc: Capillary pressure Transfer, vol. 24, pp. 983–990, 1981. Pre: Effective Prandtl number [3] S.-A. Yang and C. K. Chen, “Laminar film condensation on a R: Radius of circular tube finite-size horizontal plate with suction at the wall,” Applied Ra: Rayleigh number Mathematical Modelling, vol. 16, no. 6, pp. 325–329, 1992. s: Dimensionless saturation [4] T. B. Chang, “Laminar film condensation on A finite-size T: Temperature horizontal plate in A porous medium,” Journal of Chung T Cheng Institute of Technology, vol. 33, no. 1, pp. 69–76, 2004. Δ : Saturation temperature minus wall temperature [5] S.-C. Wang, C. o.-K. Chen, and Y.-T. Yang, “Film conden- u, v: Velocity component in x and y direction sation on a finite-size horizontal wavy plate bounded by a ud: Darcy velocity defined in (17) homogenous porous layer,” Applied 6ermal Engineering, V: Dimensionless capillary suction velocity defined in vol. 25, no. 4, pp. 577–590, 2005. (16b). [6] A. Majumdar and C. L. Tien, “Effects of surface tension on film condensation in a porous medium,” Journal of Heat Transfer, vol. 112, no. 3, pp. 751–757, 1990. Greek symbols [7] K. S. Udell, “Heat transfer in porous media heated from above with evaporation, condensation, and capillary effects,” Journal δ: Condensate film thickness of Heat Transfer, vol. 105, no. 3, pp. 485–492, 1983. δ0: Condensate film thickness at top of tube [8] K. S. Udell, “Heat transfer in porous media considering phase θ: Angle measured from top of tube change and capillarity-the heat pipe effect,” International μ: Liquid viscosity Journal of Heat and Mass Transfer, vol. 28, no. 2, pp. 485–495, ρ: Liquid density 1985. α: *ermal diffusivity [9] M. A. EL-Hakiem and A. M. Rashad, “Effect of radiation on σ: Surface tension non-Darcy free convection from A vertical cylinder embed- ε: Porosity. ded in a fluid-saturated porous medium with a temperature- dependent viscosity,” Journal of Porous Media, vol. 10, pp. 209–218, 2007. Superscripts [10] S. M. M. EL-Kabeir, M. A. EL-Hakiem, and A. M. Rashad, “Group method analysis of combined heat and mass transfer : Average quantity by MHD non-Darcy non-Newtonian natural convection ∗: Dimensionless variable adjacent to horizontal cylinder in a saturated porous 10 Mathematical Problems in Engineering

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