Analytical Investigation Into Effects of Capillary Force on Condensate Film Flowing Over Horizontal Semicircular Tube in Porous Medium
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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 cal- culation 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 j ∗ � (2Ra Da/Ja) R (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.