Journal articles: 'Condensation; Condensate film'

1

Dong, Jing Lan, Wei Ping Yan, and Chao Hui Zhang. "Convective Condensation of Oxy-Coal Combustion Flue Gas of Laminar Fow in a Vertical Pipe." Applied Mechanics and Materials 325-326 (June 2013): 389–97. http://dx.doi.org/10.4028/www.scientific.net/amm.325-326.389.

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The problem of the oxy-fuel combustion flue gas condensation is the condensation of vapor in the presence of high concentration non-condensable gas. The vapor condensing at dew point temperature releases heat and diffuses on to the surface of the pipe through a non-condensable gas film. Thus it is treated as combined heat and mass transfer problem governed by mass, momentum and energy balance equations for the vaporgas mixture and diffusion equation for the vapor species. The flow of the falling condensate film is governed by the momentum and energy balance equations. The temperature at the gas-to-liquid interface, at which the condensation takes place, is estimated with the help of the heat balance and mass balance equations at the interface. The local values of the condensation Nusselt number, condensate Reynolds number, gasliquid interface temperature and pressure drop are estimated from the numerical results for different values of the system parameters at inlet, such as vapor component, temperature of vaporgas mixture, gas phase Reynolds number and total pressure. The thermodynamic calculations were made and analyzed using numerical calculation method under different conditions.

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Mitrovic, J. "Effects of Vapor Superheat and Condensate Subcooling on Laminar Film Condensation." Journal of Heat Transfer 122, no. 1 (August 9, 1999): 192–96. http://dx.doi.org/10.1115/1.521450.

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Nusselt’s model is employed to illustrate the effects of vapor superheat and condensate subcooling on laminar film condensation occurring under simultaneous actions of gravity and interfacial shear. The vapor superheat affects the condensation kinetics in cooperation with heat transfer in both phases. Under comparable conditions, the condensate film is thinner and the heat transfer coefficient larger for superheated than for saturated vapor. The heat flux on the cooling surface arising from the sensible heat of condensate increases as the critical point of the condensing substance is approached and, at this point, the Nusselt condensation model gives the single-phase boundary layer solutions. [S0022-1481(00)00701-5]

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Mosaad, El-Sayed. "Heat transfer performance of film condensation created by forced flow." Thermal Science 23, no. 3 Part B (2019): 2001–11. http://dx.doi.org/10.2298/tsci171218134m.

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In this work, condensate film on a vertical wall cooled on the external side by forced flow is analysed as a conjugate heat transfer problem. The treated case is that the condensate film and forced flow boundary-layer are in a parallel-flow arrangement. The mass, momentum and energy boundary-layer equations of the condensate film and forced flow are set in a dimensionless form to generalize the model. The parameters affecting the thermal communication between the condensate film and the forced flow are defined from the analysis. These parameters explain the relative impact of the three involved thermal resistances of solid wall, forced convection and film condensation on the local and mean Nusselt number. The study shows that the Nusselt number predicted by the present conjugate model is different from that predicted by a Nusselt-type model.

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Lin, Yan-Ting, and Sheng-An Yang. "Turbulent Film Condensation on a Nonisothermal Horizontal Tube." Journal of Mechanics 21, no. 4 (December 2005): 235–42. http://dx.doi.org/10.1017/s1727719100000678.

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AbstractA simple model has been developed for the study of turbulent film condensation from downward flowing vapors onto a horizontal circular tube with variable wall temperature. The interfacial shear at the vapor condensate film is evaluated with the help of Colburn analogy. The condensate film flow and local/or mean heat transfer characteristics from a horizontal tube with non-uniform temperature variation under the effect of Froude number, sub-cooling parameter and system pressure parameter has been conducted. Although the non-uniform wall temperature variation has an appreciable influence on the local film flow and heat transfer; however, the dependence of mean heat transfer on the non-uniform wall temperature variation is almost negligible.

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Nozu, S., and H. Honda. "Condensation of Refrigerants in Horizontal, Spirally Grooved Microfin Tubes: Numerical Analysis of Heat Transfer in the Annular Flow Regime." Journal of Heat Transfer 122, no. 1 (August 1, 1999): 80–91. http://dx.doi.org/10.1115/1.521439.

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A method is presented for estimating the condensation heat transfer coefficient in a horizontal, spirally grooved microfin tube. Based on the flow observation study performed by the authors, a laminar film condensation model in the annular flow regime is proposed. The model assumes that all the condensate flow occurs through the grooves. The condensate film is segmented into thin and thick film regions. In the thin film region formed on the fin surface, the condensate is assumed to be drained by the combined surface tension and vapor shear forces. In the thick film region formed in the groove, on the other hand, the condensate is assumed to be driven by the vapor shear force. The present and previous local heat transfer data including four fluids (CFC11, HCFC22, HCFC123, and HFC134a) and three microfin tubes are found to agree with the present predictions to a mean absolute deviation of 15.1 percent. [S0022-1481(00)01501-2]

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Galamba, D., and V. K. Dhir. "Transient Simultaneous Condensation and Melting of a Vertical Surface." Journal of Heat Transfer 107, no. 4 (November 1, 1985): 812–18. http://dx.doi.org/10.1115/1.3247508.

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Transient melting of a vertical wall as a result of condensation of saturated vapor is investigated analytically. Using a Nusselt-type analysis the equations governing the laminar melt and condensate films are solved by a combination of analytical and numerical finite-differencing techniques. The solutions in the axial distance-time plane fall into three regions: the piling region where the melt and condensate thicknesses are only functions of time, the Goursat region where the thicknesses are functions of both the axial distance and time, and the steady-state region where the film thicknesses are only functions of the axial distance. The time to achieve a steady-state condition is obtained analytically.

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Webb, R. L., T. M. Rudy, and M. A. Kedzierski. "Prediction of the Condensation Coefficient on Horizontal Integral-Fin Tubes." Journal of Heat Transfer 107, no. 2 (May 1, 1985): 369–76. http://dx.doi.org/10.1115/1.3247424.

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A theoretical model is developed for prediction of the condensation coefficient on horizontal integral-fin tubes for both high and low surface tension fluids. The model includes the effects of surface tension on film drainage and on condensate retention between the fins. First, the fraction of the tube circumference that is flooded with condensate is calculated. Typically, the condensation coefficient in the flooded region is negligible compared to that of the unflooded region. Then the condensation coefficient on the unflooded portion is calculated, assuming that surface tension force drains the condensate from the fins. The model is used to predict the R-11 condensation coefficient on horizontal, integral-fin tubes having 748, 1024, and 1378 fpm. The predicted values are within ±20 percent of the experimental values.

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Honda, H., S. Nozu, and Y. Takeda. "A Theoretical Model of Film Condensation in a Bundle of Horizontal Low Finned Tubes." Journal of Heat Transfer 111, no. 2 (May 1, 1989): 525–32. http://dx.doi.org/10.1115/1.3250709.

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The previous theoretical model of film condensation on a single horizontal low finned tube is extended to include the effect of condensate inundation. Based on the flow characteristics of condensate on a vertical column of horizontal low finned tubes, two major flow modes, the column mode and the sheet mode, are considered. In the column mode, the surface of the lower tubes is divided into the portion under the condensate column where the condensate flow is affected by the impinging condensate from the upper tubes, and the portion between the condensate columns where the condensate flow is not affected by the impinging condensate. In the sheet mode, the whole tube surface is assumed to be affected by the impinging condensate. Sample calculations for practical conditions show that the effects of the fin spacing and the number of vertical tube rows on the heat transfer performance is significant for R-12, while the effects are small for steam. The predicted value of the heat transfer coefficient for each tube row compares well with available experimental data, including four fluids and five tube bundles.

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Wang, Hua Sheng, and John W. Rose. "A Theory of Film Condensation in Horizontal Noncircular Section Microchannels." Journal of Heat Transfer 127, no. 10 (June 16, 2005): 1096–105. http://dx.doi.org/10.1115/1.2033905.

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The paper presents a theoretical model to predict film condensation heat transfer from a vapor flowing in horizontal square and equilateral triangular section minichannels or microchannels. The model is based on fundamental analysis which assumes laminar condensate flow on the channel walls and takes account of surface tension, interfacial shear stress, and gravity. Results are given for channel sizes (side of square or triangle) in the range of 0.5–5 mm and for refrigerants R134a, R22, and R410A. The cases considered here are where the channel wall temperature is uniform and the vapor is saturated at the inlet. The general behavior of the condensate flow pattern (spanwise and streamwise profiles of the condensate film), as well as streamwise variation of local mean (over section perimeter) heat-transfer coefficient and vapor mass quality, are qualitatively in accord with expectations on physical grounds. The magnitudes of the calculated heat-transfer coefficients are in general agreement with experimental data for similar, but nonidentical, channel geometry and flow parameters.

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10

Kazeminejad, H. "Effect of Vapour Drag on Laminar Film Condensation on a Vertical Rectangular Fin." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 207, no. 1 (January 1993): 63–69. http://dx.doi.org/10.1243/pime_proc_1993_207_099_02.

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A simple theory is presented for laminar film condensation of a pure vapour on a vertical rectangular fin which takes account of drag induced on the liquid film by the flow of the condensing vapour. Under these conditions, the governing conjugate differential equations for the fin and condensate flow are solved numerically to determine the fin temperature and condensate film thickness distributions. For the range of parameters investigated, it was found that the reduction in condensate thermal resistance due to vapour shear significantly enhances the heat-transfer rate to the fin and decreases the fin efficiency. The model also provides a clear picture of the relative effect of the gravity force, friction drag and momentum drag on the performance of the fin.

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11

Zhang, Jun Xia, Zeng Sheng Li, Bin Yao Gong, Hong Xing Zhao, and Yu Huai Zhao. "Analysis of Stagnant Non Condensable Gases Effects on Condensation Heat Transfer in a Vertical Tube." Applied Mechanics and Materials 148-149 (December 2011): 491–95. http://dx.doi.org/10.4028/www.scientific.net/amm.148-149.491.

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In a vertical condenser tube installed at the cold end of a non-vacuum separate type heat pipe, non condensable (NC) gases in the system is pushed by continuous vapor flowing from the hot end into the condenser tube at the cold end, gathering above condensate at the outlet of the condenser tube. Therefore, condensation heat transfer of vapor with the stagnant NC gases occurs in the condenser tube. It is necessary to comprehend the effects of stagnant NC gases on condensation heat transfer. A VOF method was adopted to analyze how stagnant NC gases affect condensation heat transfer, a mass fraction equation of NC gases was used to solve diffusion between NC gases and vapor, a Hertz-Knudsen-Schrage model was applied to deal with condensation rate of vapor on the surface of liquid film. Parameters, including volume fraction, velocity, pressure, mass fraction of NC gases and condensation heat transfer coefficients (HTC), were obtained. Results show that a lot of NC gases deposits in the condenser tube rear, leading a lot of vapor to condense at the condenser tube front. NC gases slightly affect condensation HTC of the tube front, and severely degrade condensation HTC of the tube rear. Furthermore, an increase in mass of NC gases causes a rise in pressure and velocity, improving condensation heat transfer.

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12

Chang, T. B. "Effect of Vapor Superheating on Mixed-Convection Film Condensation Along an Isothermal Vertical Cylinder." Journal of Mechanics 26, no. 1 (March 2010): N1—N7. http://dx.doi.org/10.1017/s1727719100003786.

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AbstractThis paper presents an analytical investigation into the effect of vapor superheating on the mixed-convection of a condensate layer flowing along the outside surface of an isothermal vertical cylinder. The governing system of partial differential equations is transformed into a dimensionless form using the nonsimilar transformation method. In investigating the heat transfer characteristics within the condensate layer and vapor phase, the analysis takes account of both the inertia effects and the convection effects within the condensate layer and the shear resistance at the liquid-vapor interface. The numerical results reveal that vapor superheating has a negligible effect on the temperature profile and local Nusselt number within the condensate layer. Moreover, it is found that a higher forced-flow intensity increases the temperature gradient in the vapor phase, but has a marginal effect on the temperature profile in the condensate layer. Finally, it is shown that the velocity at the liquid-vapor interface increases as the intensity of the forced-flow increases or as the ratio of the condensate layer viscosity to the vapor phase viscosity reduces.

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13

Utaka, Yoshio, and Tetsuji Nishikawa. "MEASUREMENT OF CONDENSATE FILM THICKNESS FOR SOLUTAL MARANGONI CONDENSATION." Journal of Enhanced Heat Transfer 24, no. 1-6 (2017): 279–90. http://dx.doi.org/10.1615/jenhheattransf.v24.i1-6.200.

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14

Lu, Q., and N. V. Suryanarayana. "Condensation of a Vapor Flowing Inside a Horizontal Rectangular Duct." Journal of Heat Transfer 117, no. 2 (May 1, 1995): 418–24. http://dx.doi.org/10.1115/1.2822538.

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Condensation of a vapor flow inside a horizontal rectangular duct, using the bottom plate as the only condensing surface, was experimentally investigated. The experimental measurements included condensate film thickness and heat transfer coefficients with R-113 and FC-72. The condensate film thickness, measured with an ultrasonic transducer, was used to obtain the local heat transfer coefficient. The heat transfer coefficient increased with increasing inlet vapor velocity. The rate of increase was enhanced noticeably after the appearance of interfacial waves. Within the limited range of the experimental variables, a correlation between St and RegL was developed by a linear regression analysis. However, because of the effect of the interfacial waves, instead of a single correlation for the entire range of RegL, two separate equations (one for the wave-free regime and another for the regime with waves) were found. Analytical predictions of heat transfer rates in the annular condensation regime require the proper modeling of the interfacial shear stress. A properly validated interfacial shear stress model with condensation is not yet available. The measurement of condensate film thickness at several axial locations opens the door for determining the local interfacial stress and, hence, a model for the interfacial shear stress.

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Sapozhnikov, Sergey, Vladimir Mityakov, Alexandr Babich, and Elza Zainullina. "Study of condensation at the surfaces of tube with gradient heat flux measurement." MATEC Web of Conferences 245 (2018): 06010. http://dx.doi.org/10.1051/matecconf/201824506010.

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Gradient heat flux measurement is used for study of heat transfer during condensation of water steam at inner and outer surfaces of tube. Experimental setups allow producing experiments with minimal distortion of condensate film flow. Experiments were carried out for different directions of steam and cooling water flows and for different angles of tube inclination relative to the vertical. Heat transfer coefficients and their change along the length and perimeter of tube were measured. The obtained data allow to study formation of condensate film and parameters of film motion. The results are corresponding to classical ideas.

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Ait Hssain, Mustapha, Youness El Hammami, Rachid Mir, Sara Armou, and Kaoutar Zine-Dine. "Numerical Analysis of Laminar Convective Condensation with the Presence of Noncondensable Gas Flowing Downward in a Vertical Channel." Mathematical Problems in Engineering 2019 (June 20, 2019): 1–15. http://dx.doi.org/10.1155/2019/7941363.

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The purpose of this paper is to study and perform a numerical analysis of the simultaneous processes of mass and heat transfer during the condensation process of a steam in the existence of noncondensable gas (NCG) inside a descending vertical channel. In this study, the flow of the vapor-air mixture is laminar and the saturation conditions are prevailing at the inlet of the channel. The coupled control equations for liquid film, interfacial conditions, and mixture flow are solved together using the approach of finite volume. Detailed and valuable results are presented both in the liquid condensate film and in the mixing regions. These detailed results contain the dimensionless velocity and dimensionless temperature profiles in both phases, the dimensionless mass fraction of vapor, the axial variation of the dimensionless thickness of the film liquid δ⁎, and the accumulated condensate rate Mr as well the local Nusselt number Nuy. The relative humidity at the inlet varies from 60% to 100% and the inlet temperature from 40°C to 80°C. The results confirm that a decrease in the mass concentration of NCG by the increasing the inlet relative humidity has a direct influence on the liquid film layer, the local number of Nusselt, and the variation of condensation rate accumulated through the channel. The results also designate that an increase of the inlet relative humidity and the inlet temperature ameliorates the condensation process. The comparison made for the coefficient of heat transfer due to condensation process and the condensate liquid film thickness with the literature results is in good concordance which gives more credibility to our calculation model.

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Chang, T. B., and F. J. Wang. "Effects of channel width on film condensation in a top-closed vertical channel." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 222, no. 9 (September 1, 2008): 1763–71. http://dx.doi.org/10.1243/09544062jmes824.

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This article conducts a theoretical investigation into the condensation of a saturated vapour within a top-closed, vertical isothermal channel. The theoretical model takes account of the inertia and convection effects in a condensate layer and the shear stress at the liquid—vapour interface. Using a dimensionless transformation method, the complex partial differential governing equations are transformed into a corresponding system of ordinary differential equations, which are then solved using the forward Runge—Kutta shooting scheme. Numerical results indicate that the condensate flowrate decreases and the negative shear stress at the liquid—vapour interface increases as the channel width is reduced. Conversely, a larger channel width increases the shear stress at the channel wall, but has no apparent effect on the temperature profile within the condensate layer.

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Qu, Mengnan, Jia Liu, and Jinmei He. "Fabrication of copper-based ZnO nanopencil arrays with high-efficiency dropwise condensation heat transfer performance." RSC Advances 6, no. 64 (2016): 59405–9. http://dx.doi.org/10.1039/c6ra09699a.

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A copper-based zinc oxide nanopencil array film was reported. Compared with hydrophobic flat Cu surface, it exhibits condensate microdrop self-propelling function and maximal ∼140% enhancement in dropwise condensation heat transfer coefficient.

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Mahajan, R. L., T. Y. Chu, and D. A. Dickinson. "An Experimental Study of Laminar Film Condensation With Stefan Number Greater Than Unity." Journal of Heat Transfer 113, no. 2 (May 1, 1991): 472–78. http://dx.doi.org/10.1115/1.2910585.

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Experimental laminar condensation heat transfer data are reported for fluids with Stefan number up to 3.5. The fluid used is a member of a family of fluorinated fluids, which have been used extensively in the electronics industry for soldering, cooling, and testing applications. Experiments were performed by suddenly immersing cold copper spheres in the saturated vapor of this fluid, and heat transfer rates were calculated using the quasi-steady temperature response of the spheres. In these experiments, the difference between saturation and wall temperature varied from 0.5° C to 190°C. Over this range of temperature difference, the condensate properties vary significantly; viscosity of the condensate varies by a factor of nearly 50. Corrections for the temperature-dependent properties of the condensate therefore were incorporated in calculating the Nusselt number based on the average heat transfer coefficient. The results are discussed in light of past experimental data and theory for Stefan number less than unity. To the knowledge of the authors, this is the first reported study of condensation heat transfer examining the effects of Stefan number greater than unity.

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Shen, Chaoqun, Lingbo Liu, Suchen Wu, Feng Yao, and Chengbin Zhang. "Lattice Boltzmann simulation of droplet condensation on a surface with wettability gradient." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 234, no. 7 (January 2, 2020): 1403–13. http://dx.doi.org/10.1177/0954406219898220.

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In this paper, a free energy lattice Boltzmann model of vapor condensation on a surface with a wettability gradient is developed and numerically analyzed to understand the microscopic behaviors of self-propelled droplet condensation. The effect of wettability gradient on droplet self-motion and coalescence as well as vapor condensation is examined and investigated. The condensation rate is presented during the whole droplet condensation process to analyze the role of wettability gradient on droplet condensation. The results indicate that the surface with wettability gradient is preferred for vapor condensation owing to the appearance of self-propelled droplet condensation. Condensate film is initially spread on the high surface energy region, and liquid nucleation sites form, grow and, subsequently, coalesce with other droplets on low surface energy regions. The condensation rate is higher on a surface with a larger wettability gradient due to the more effective removal of condensate. In addition, the condensation rate fluctuates with time at the quasi-steady-state stage. During the condensation process, the droplet coalescence triggers a sudden peak of condensation rate, and the generation of new nucleation results in a rapid increase in the condensation rate.

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Honda, Hiroshi, Huasheng Wang, and Shigeru Nozu. "A Theoretical Study of Film Condensation in Horizontal Microfin Tubes." Journal of Heat Transfer 124, no. 1 (August 14, 2001): 94–101. http://dx.doi.org/10.1115/1.1421048.

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A stratified flow model of film condensation in helically grooved, horizontal microfin tubes has been developed. The height of stratified condensate was estimated by extending the Taitel and Dukler model for a smooth tube to a microfin tube. For the upper part of the tube exposed to the vapor flow, laminar film condensation due to the combined effects of gravity and surface tension forces was assumed. For the lower part of the tube exposed to the stratified condensate flow, the heat transfer coefficient was estimated by an empirical equation for the internally finned tubes developed by Carnavos. The theoretical predictions of the circumferential average heat transfer coefficient by the present model and previously proposed annular flow model were compared with available experimental data for five tubes and five refrigerants. It was shown that the stratified flow model was applicable to wide ranges of mass velocity and quality as long as the vapor to liquid density ratio was larger than 0.05. Comparison was also made with the predictions of previously proposed empirical equations.

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Zhang, Jun Xia, Li Wang, and Jian Wen Wang. "Analysis of Annular and Stratified Flows for Vapor Condensation Heat Transfer in Horizontal Tubes." Advanced Materials Research 753-755 (August 2013): 2717–21. http://dx.doi.org/10.4028/www.scientific.net/amr.753-755.2717.

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The greater part of the horizontal condenser tube is occupied by the stratified and annular flows, which play important roles in condensation heat transfer coefficients. Both a volume of fluid (VOF) interface tracking method and k-ε two equations model was applied to analyze the characteristics of the stratified and annular flows for horizontal tubes, obtaining distribution of velocity, contours of both temperature and local condensation heat transfer coefficients. The computation shows that an annular flow having thinner liquid film attached on the inner surface of tubes appears at the inlet of the horizontal condenser tube because vapor exerts a higher interfacial shear stress on the gas-liquid interface, therefore, there are a higher local condensation heat transfer coefficients there. Next, as vapor quality decreases and condensate gravity increases along the condenser tube length, the condensation flow pattern transforms from the annular flow into the stratified flow in which local condensation heat transfer coefficients decreases and distributes unevenly along the circumference as liquid film at the bottom of the condenser tube become thicker. In addition, in the stratified flow, a wave structure is formed in the middle of the condensate pool at the bottom of the condenser tube because condensate on the top of the condenser tube slides into the bottom of the condenser tube and collapse each other. The heat transfer rate calculated by the present method is compared to those predicted from a Shah correlation, the agreement is found to be good, and both errors is within 7%.

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Brendle, Rainer. "Einstein Condensation in a Macroscopic Field." Zeitschrift für Naturforschung A 40, no. 12 (December 1, 1985): 1189–98. http://dx.doi.org/10.1515/zna-1985-1203.

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A new kind of thermodynamic limit is given for the model of an ideal Boson gas which scales the supports of the local Weyl operators into infinitesimal regions leaving the external potential fixed. A technical assumption in a paper of Davies on this subject is derived from geometrical arguments. The spatial distribution of the condensate density is calculated for an arbitrary potential being bounded from below. This is used for a simple qualitative explanation of the Helium film effect.

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Michael, A. G., J. W. Rose, and L. C. Daniels. "Forced Convection Condensation on a Horizontal Tube—Experiments With Vertical Downflow of Steam." Journal of Heat Transfer 111, no. 3 (August 1, 1989): 792–97. http://dx.doi.org/10.1115/1.3250753.

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The paper reports an experimental investigation into the condensation of steam flowing vertically downward over a single horizontal tube. A limited number of experiments with steam-nitrogen mixtures are also reported. For the case of “pure” steam, data have been obtained at near atmospheric pressure for vapor approach velocities in the range 5 to 81 m/s and with superheats in the range 2 to 40 K. The results are compared with theory and earlier experimental data for steam and other fluids. The vapor-side heat transfer coefficients were found to increase with velocity. For vapor velocities in excess of 30 m/s, the rate of increase of the vapor-side coefficient was greater than predicted by laminar condensate flow theory. This behavior has also been observed by earlier workers for refrigerant-113 and ethylene glycol and may indicate onset of turbulence in the condensate film. Superheat had insignificant effect on the heat transfer coefficient for the condensate film. The results for steam-nitrogen mixtures were generally in good agreement with existing equations for the “gas-layer” resistance.

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Kim, Chul Ju, Jae Seok Lee, Sung Hoon Kim, and Byung Ho Sung. "An Analysis of Liquid Film Condensation Occurring Inside Rotating Heat Pipes with a Trigonal Cross Section." Solid State Phenomena 120 (February 2007): 257–60. http://dx.doi.org/10.4028/www.scientific.net/ssp.120.257.

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Rotational heat pipes (RHPs) with a trigonal cross section was recently proposed by Lee et al. Inside the tube of these RHPs the condensate liquid film flows down the wall in the direction vertical to the tube axis due to centrifugal force and then the liquid returns to the evaporator along the edges. The Nusselt analysis model of laminar film condensation was applied and the solutions were compared with the experimental data. Concerning about the Nu vs Re the same relation was obtained as that for vertical plate except the length scale. The thickness of condensate film was found to be uniform along the surface due to increasing centrifugal force with surface length for a given heat flux and rotational speed.

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Zaster, Svitlana, and Eric R. Bittner. "Role of dark excitations in the nonequilibrium condensation of exciton polaritons in optically-pumped organic single crystal microcavities." International Journal of Modern Physics B 29, no. 22 (September 7, 2015): 1550157. http://dx.doi.org/10.1142/s021797921550157x.

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We present a reaction/diffusion model for the formation of a lower polariton condensate in a microcavity containing an organic semiconducting molecular crystalline film. Our model–based upon an anthracene film sandwiched between two reflecting dielectric mirrors–consists of three coupled fields corresponding to a gas of excitons created by an external driving pulse, a reservoir of dark states formed by the nonemissive decay of excitons in to nearby electronic states, and a lower polariton condensate. We show that at finite temperature, the presence of the dark reservoir can augment the exciton population such that a lower critical pumping threshold is required to achieve the critical exciton densities required to sustain a stable condensate population. Using linear-stability analysis, we show that a variety of dynamical regimes can emerge depending upon the characteristics of the cavity and the lattice temperature.

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Chang, Tong-Bou, Bai-Heng Shiue, Yi-Bin Ciou, and Wai-Io Lo. "Analytical Investigation into Effects of Capillary Force on Condensate Film Flowing over Horizontal Semicircular Tube in Porous Medium." Mathematical Problems in Engineering 2021 (March 17, 2021): 1–10. http://dx.doi.org/10.1155/2021/6693512.

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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. The 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. The 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 as N u ¯ | V 2 ∗ = 0 = 2 R a D a / J a 1 / 2 ∫ 0 π 1 + cos θ 1 / 2 d θ .

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Wang, Chi-Chang, and Cha’o-Kuang Chen. "Combined Free and Forced Convection Film Condensation on a Finite-Size Horizontal Wavy Plate." Journal of Heat Transfer 124, no. 3 (May 10, 2002): 573–76. http://dx.doi.org/10.1115/1.1458019.

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Mixed-convection film condensation with downward flowing vapors onto a finite-size horizontal wavy plate is studied by a simple mathematical model and the spline alternating-direction implicit method. Effects of the wavy geometry, the interfacial vapor shear and the pressure gradient on the local condensate film thickness and the heat transfer characteristics have been studied independently. Results show that the pressure gradient tends to increase the heat transfer rate and to decrease the influence of the wavy amplitude. The appropriate wave number which can enhance the maximum condensation heat transfer rate is found in the neighborhood of lunder all circumstances.

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Salam, Bodius, David A. A. McNeil, Bryce M. Burnside, and Sumana Biswas. "FORCED CONVECTION FILM CONDENSATION ON A HORIZONTAL TUBE WITH VARIABLE PROPERTIES." Journal of Mechanical Engineering 40, no. 2 (June 28, 2010): 79–89. http://dx.doi.org/10.3329/jme.v40i2.5347.

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The paper presents the results of numerical calculations for filmwise condensation of downward flowing pure saturated steam on a horizontal tube. A tube of 19.05 mm diameter was used. Steam approach conditions used were 5000 N/m2 (Tsat = 32.9 oC) pressure and velocity 5 – 100 m/s. Tube wall temperatures were considered to be constant at 22.9 oC and 30.9 oC, giving condensate subcooling DT = 10 K and 2 K respectively. Earlier theoretical studies omitted the variations of physical properties with pressure arising from flow of vapor over the tube surface. The present work takes into account these property variations. The velocity and pressure distributions were taken from the potential flow theory. At low condensate subcooling, DT = 2 K, and high steam velocities, significant reduction of average heat transfer coefficient was predicted when property variations were taken into account, compared to constant property values. The mean heat flux predicted considering the variation of properties was up to 9% and 42% less than that obtained for DT = 10 K and 2 K respectively.Keywords: Laminar filmwise condensation; horizontal tube; boundary layer separation; numerical.DOI: 10.3329/jme.v40i2.5347Journal of Mechanical Engineering, Vol. ME 40, No. 2, December 2009 79-89

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30

Honda, H., B. Uchima, S. Nozu, E. Torigoe, and S. Imai. "Film Condensation of R-113 on Staggered Bundles of Horizontal Finned Tubes." Journal of Heat Transfer 114, no. 2 (May 1, 1992): 442–49. http://dx.doi.org/10.1115/1.2911293.

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Film condensation of R-113 on staggered bundles of horizontal finned tubes with vertical vapor downflow was experimentally investigated. Two tubes with flat-sided annular fins and four tubes with three-dimensional fins were tested. The condensate flow and heat transfer characteristics were compared with the previous results for in-line bundles of the same test tubes and a staggered bundle of smooth tubes. The decrease in heat transfer due to condensate inundation was most significant for the in-line bundles of the three-dimensional fin tubes, whereas the decrease was very slow for both the staggered and in-line bundles of the flat-sided fin tubes. The predictions of the previous theoretical model for a bundle of flat-sided fin tubes agreed fairly well with the measured data at a low vapor velocity. The highest heat transfer performance was provided by the staggered bundle of flat-sided fin tubes with fin dimensions close to the theoretically determined optimum values.

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31

Honda, H., N. Takata, H. Takamatsu, J. S. Kim, and K. Usami. "Effect of Fin Geometry on Condensation of R407C in a Staggered Bundle of Horizontal Finned Tubes." Journal of Heat Transfer 125, no. 4 (July 17, 2003): 653–60. http://dx.doi.org/10.1115/1.1560153.

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Experimental results are presented that show the effect of fin geometry on condensation of downward flowing zeotropic refrigerant mixture R407C in a staggered bundle of horizontal finned tubes. Two types of conventional low-fin tubes and three types of three-dimensional-fin tubes were tested. The refrigerant mass velocity ranged from 4 to 23 kg/m2 s and the condensation temperature difference from 3 to 12 K. The measured condensation heat transfer coefficient was lower than the previous results for R134a, with the difference being more significant for smaller mass velocity. The effect of fin geometry on the condensation heat transfer coefficient was less significant for R407C than for R134a. The effect of condensate inundation was more significant for the three-dimensional-fin tubes than for the low-fin tubes. By using the dimensionless heat transfer correlation for the condensate film that was based on the experimental data for R134a, a superficial vapor-phase heat transfer coefficient was obtained for condensation of R407C. The vapor-phase heat transfer coefficient showed characteristics similar to the vapor-phase mass transfer coefficient that was obtained in the previous study for R123/R134a.

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32

Uma, B. "Effect of Wind Stress on the Dynamics and Stability of Nonisothermal Power-Law Film down an Inclined Plane." ISRN Mathematical Physics 2012 (January 26, 2012): 1–31. http://dx.doi.org/10.5402/2012/732675.

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Dynamics and stability of a nonisothermal power-law liquid film down an inclined plane is considered in the presence of interfacial shear. Linear stability characteristics of the power-law liquid film using normal mode approach reveal that isothermal and evaporating films are unstable for any value of power-law index while there exists a critical value of power-law index for the case of condensate film above which condensate film ow system is always stable. This critical value of power-law index increases with the increase in shear stress at the interface. Weakly nonlinear stability analysis using method of multiple scales divulges the existence of zones due to supercritical stability and subcritical instability. The nonlinear evolution equation is solved numerically in a periodic domain. The results reveal that (1) for an isothermal dilatant (pseudoplastic) liquids, the maximum wave amplitude is always smaller (larger) than that for a Newtonian liquid and the amplitude of permanent wave increases with the increase in interfacial shear; (2) condensation of pseudoplastic film happens for the earlier instant of time when the phase change parameter increases and the effect of interfacial shear makes the film more corrugated; (3) dilatant (pseudoplastic) evaporating liquid film attains rupture faster (slower) than that of Newtonian liquid film, and the interfacial shear does not influence the time at which rupture occurs.

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33

Honda, H., B. Uchima, S. Nozu, H. Nakata, and E. Torigoe. "Film Condensation of R-113 on In-Line Bundles of Horizontal Finned Tubes." Journal of Heat Transfer 113, no. 2 (May 1, 1991): 479–86. http://dx.doi.org/10.1115/1.2910586.

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Film condensation of R-113 on in-line bundles of horizontal finned tubes with vertical vapor downflow was experimentally investigated. Two tubes with flat-sided annular fins and four tubes with three-dimensional fins were tested. The test sections were 3×15 tube bundles with and without two rows of inundation tubes at the top. Heat transfer measurements were carried out on a row-by-row basis. The heat transfer enhancement due to vapor shear was much less for a finned tube bundle than for a smooth tube bundle. The decrease in heat transfer due to condensate inundation was more marked for a three-dimensional fin tube than for a flat-sided fin tube. The predictions of the previous theoretical model for a bundle of flat-sided fin tubes agreed well with the measured data for low vapor velocity and a small to medium condensate inundation rate. Among the six tubes tested, the highest heat transfer performance was provided by the flat-sided fin tube with fin dimensions close to the theoretically determined optimum values.

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34

Hijikata, K., Y. Fukasaku, and O. Nakabeppu. "Theoretical and Experimental Studies on the Pseudo-Dropwise Condensation of a Binary Vapor Mixture." Journal of Heat Transfer 118, no. 1 (February 1, 1996): 140–47. http://dx.doi.org/10.1115/1.2824027.

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When a water–ethanol binary mixture condenses on a flat plate, one observes that the liquid film condensate rises locally and eventually forms many droplets on the film. Usually, filmwise condensation is expected because both substances are completely soluble in each other and they wet a copper plate well. This paper presents the droplet growth mechanism during so-called pseudo-dropwise condensation. Instability analysis is used to determine the transition from filmwise condensation to pseudo-dropwise condensation theoretically. In a stress balance at the vapor–liquid interface, the analysis considers not only the surface tension itself, but also the surface tension variation due to changes in temperature and concentration, assuming saturation conditions at the interface. Numerical results indicate that the Marangoni effect plays a more important role than the absolute value of the surface tension in pseudo-dropwise condensation. The change in surface tension with temperature is not always negative; it becomes positive for certain mixtures due to the dependence on concentration. Pseudo-dropwise condensation is only realized when surface tension increases with temperature. This analysis qualitatively predicts the critical Marangoni number experimentally observed during water–ethanol mixture condensation.

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35

Honda, H., and S. Nozu. "A Prediction Method for Heat Transfer During Film Condensation on Horizontal Low Integral-Fin Tubes." Journal of Heat Transfer 109, no. 1 (February 1, 1987): 218–25. http://dx.doi.org/10.1115/1.3248046.

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A method for predicting the average heat transfer coefficient is presented for film condensation on horizontal low integral-fin tubes. Approximate equations based on the numerical analysis of surface tension drained condensate flow on the fin surface are developed for the heat transfer coefficients in the upper and lower portions of the flooding point below which the interfin space is flooded with condensate. For the unflooded region, the equation is modified to take account of the effect of gravity. These equations are used, along with the previously derived equation for the flooding point, to determine the wall temperature distribution, and in turn the average heat transfer coefficient. It is shown that the present model can predict the average heat transfer coefficient within ±20 percent for most of the available experimental data including 11 fluids and 22 tubes.

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36

Harley, C., and A. Faghri. "Two-Dimensional Rotating Heat Pipe Analysis." Journal of Heat Transfer 117, no. 1 (February 1, 1995): 202–8. http://dx.doi.org/10.1115/1.2822304.

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A detailed transient numerical simulation of rotating heat pipes is presented. This two-dimensional, axisymmetric formulation accounts for the thin liquid condensate film on the inner surface of the rotating pipe wall, the vapor flow in the vapor space, and the unsteady heat conduction in the pipe wall. The thin liquid film is coupled to the vapor velocity at the liquid–vapor interface, and the effects of the vapor pressure drop and the interfacial shear stress are included in the Nusselt-type condensation analysis.

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37

Sapozhnikov, Sergey, Vladimir Mityakov, Alexander Babich, and Elza Zainullina. "Gradient heat flux measurement during condensation at the surfaces of pipes." E3S Web of Conferences 140 (2019): 06006. http://dx.doi.org/10.1051/e3sconf/201914006006.

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Gradient heat flux measurement was used to study steam condensation at the inner and outer surfaces of pipes. Experimental setups were developed, manufactured and tested. The setups were able to incline the pipes for different angles relative to vertical and to rotate them around their main axes. Local heat transfer coefficients (HTC) along the pipe length and perimeter were determined. Formation and motion of condensate film were studied. The results are corresponding to classical ideas and give us some new information.

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38

Utaka, Yoshio, and Tetsuji Nishikawa. "Measurement of Condensate Film Thickness for Solutal Marangoni Condensation Applying Laser Extinction Method." Journal of Enhanced Heat Transfer 10, no. 2 (2003): 119–30. http://dx.doi.org/10.1615/jenhheattransf.v10.i2.10.

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39

Homescu, D., and P. K. Panday. "Forced Convection Condensation on a Horizontal Tube: Influence of Turbulence in the Vapor and Liquid Phases." Journal of Heat Transfer 121, no. 4 (November 1, 1999): 874–85. http://dx.doi.org/10.1115/1.2826078.

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An implicit finite difference scheme is used to solve the problem of condensation of pure vapors flowing vertically downwards around a horizontal tube. The incompressible flow equations coupled at the interface for the liquid and vapor phases are solved. The pressure gradient, inertia, and enthalpy convection terms are retained in this analysis, and the influence of turbulence in the two phases is considered. The calculated results for laminar flow and those from different mixing length turbulence models are compared with experimental results for condensation of steam and R113. The results presented show that the average condensation heat transfer coefficients obtained using Kato’s turbulence model in the condensate film and Pletcher’s model in the vapor phase, are in good agreement with the experimental data.

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40

Tsizh, B., and Z. Dziamski. "Technological Methods of Forming Thin Semiconductor Layers. Part 1." Scientific Messenger of LNU of Veterinary Medicine and Biotechnologies 21, no. 91 (April 23, 2019): 20–24. http://dx.doi.org/10.32718/nvlvet-f9104.

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The review and analysis of the basic technological methods of formation of thin layers of semiconductor materials is presented. The timeframe for the occurrence of thin film technologies and the main centers of their localization are specified. It is shown that nowadays structure, properties and basic methods of obtaining thin films sufficiently well studied for not only simple but also complex, multi-component inorganic semiconductor materials, new areas of application and increase of requirements to the operational characteristics of devices on their basis require improvement of existing technologies and development of new methods for their synthesis, which involves a detailed analysis of the known, and the search for new, progressive methods of preparation. Due to the fact that the main methods for obtaining thin films of inorganic semiconductor materials are vacuum condensation and chemical precipitation, the first part of the review describes the methods of their vacuum application, in particular, thermal spraying in an open vacuum. It is shown that the most common way of obtaining thin films is the thermal spraying under resistive heating of the evaporator with the source material.We analyze the special structural and technological changes and improvement of traditional methods and systems of thermal spraying, which allow to equalize the ratio of the chemical composition of thin films and the source material, improve the stoichiometry of condensates, and ensure their homogeneity.The designs of thermal evaporators with resistive heating of crucibles in an open vacuum with sublimation or evaporation of one and two substances are presented. It is shown how these types of evaporators exclude the transfer of solid particles into evaporating or sublimation into the vapor phase and eliminate direct vapor deposition on the condensation surface, which more or less protects against heterogeneous condensate inclusions.It is shown that the methods analyzed or their modifications are nowadays the necessary means for the creation of thin-film semiconductor structures with predetermined properties, while vacuum deposition, in particular, traditional and modified thermal spraying in a vacuum due to its simplicity(but at the same time its ability to effectively control a large the number of technological factors and create the necessary conditions for the growth of condensates) remains one of the most common ways of obtaining thin films, including inorganic semiconductors.

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41

Dharma Rao, V., V. Murali Krishna, K. V. Sharma, and P. K. Sarma. "A Theoretical Study on Convective Condensation of Water Vapor From Humid Air in Turbulent Flow in a Vertical Duct." Journal of Heat Transfer 129, no. 12 (April 1, 2007): 1627–37. http://dx.doi.org/10.1115/1.2767678.

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The problem of condensation of water vapor from humid air flowing in a duct in turbulent flow is formulated theoretically. Vapor condensing at the dew-point temperature of the vapor-air mixture diffuses to the wall of the duct through an air film. The flow of the condensate is laminar. The condensing vapor releases both convection and latent heats to the wall of the duct. Thus, it is treated as a combined heat and mass transfer problem. The mass, momentum, and energy balance equations for the vapor-air mixture flowing in the duct and the diffusion equation for the vapor species are considered. Ti, the temperature at gas-to-liquid interface, at which condensation takes place, is estimated with the help of the heat balance and mass balance equations at interface. The local and average values of the condensation Nusselt number, condensate Reynolds number, gas-liquid interface temperature, and pressure drop are estimated from the numerical results for different values of the system parameters, such as relative humidity and temperature of air at inlet, gas phase Reynolds number, and total pressure at inlet. The gas phase convection Nusselt and Sherwood numbers are also computed. A comparison of the present work with experimental data, for the case of in-tube condensation of vapor from humid air, shows satisfactory agreement.

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42

Narain, A. "Modeling of Interfacial Shear for Gas Liquid Flows in Annular Film Condensation." Journal of Applied Mechanics 63, no. 2 (June 1, 1996): 529–38. http://dx.doi.org/10.1115/1.2788900.

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Internal flow of pure vapor experiencing film condensation on the walls of a straight horizontal duct is studied. The commonly occurring annular case of turbulent (or laminar) vapor flow in the core and laminar flow of the liquid condensate—with or without waves on the interface—is emphasized. We present a new methodology which models interfacial shear with the help of theory, computations, and reliable experimental data on heat transfer rates. The theory—at the point of onset of condensation—deals with issues of asymptotic form of interfacial shear, nonuniqueness of solutions, and selection of the physically admissible solution by a stability type criteria. Other details of the flow are predicted with the help of the proposed modeling approach. These predictions are shown to be in agreement with relevant experimental data. The trends for film thickness, heat transfer rates, and pressure drops are also made available in the form of power-law correlations.

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43

Nakayama, A., and H. Koyama. "An Integral Treatment of Laminar and Turbulent Film Condensation on Bodies of Arbitrary Geometrical Configuration." Journal of Heat Transfer 107, no. 2 (May 1, 1985): 417–23. http://dx.doi.org/10.1115/1.3247431.

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A general solution procedure has been developed for laminar and turbulent film condensation problems. The procedure is designed to deal with both plane and axisymmetric isothermal bodies of arbitrary geometrical configuration. Inertia effects are fully considered by introducing a new parameter associated with the flow acceleration. A closed-form expression for the local Nusselt number is obtained for both laminar and turbulent flows. Calculations are carried out for laminar and turbulent condensate layers developed on flat plates, horizontal circular cylinders, and spheres. The results are compared with available predictions and measurements.

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44

Wang, Ying-Xin, Ling Zheng, Joel L. Plawsky, and Peter C. Wayner,. "Optical Evaluation of the Effect of Curvature and Apparent Contact Angle in Droplet Condensate Removal." Journal of Heat Transfer 124, no. 4 (July 16, 2002): 729–38. http://dx.doi.org/10.1115/1.1466460.

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The microscale transport processes in droplet condensation and removal due to interfacial phenomena were studied. In particular, this paper concerns the movement of a condensed ethanol sessile drop into a concave liquid film in the corner. An improved image analyzing procedure was used to evaluate the curvatures and contact angles for both the drop and the concave corner meniscus at different condensation rates. The experimental results demonstrated that the condensate removal rate was a function of the curvature and contact angle, which self-adjust to give the necessary force field. The use of a dimensionless, shape dependent, force balance was demonstrated. For small drops, the intermolecular force was found to be much larger than the gravitational force and dominated droplet removal. Microscale pressure fields can be experimentally measured whereas interfacial temperature differences cannot.

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45

Gstoehl, D., and J. R. Thome. "Film Condensation of R-134a on Tube Arrays With Plain and Enhanced Surfaces: Part II—Empirical Prediction of Inundation Effects." Journal of Heat Transfer 128, no. 1 (July 12, 2005): 33–43. http://dx.doi.org/10.1115/1.2130401.

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New predictive methods for R-134a condensing on vertical arrays of horizontal tubes are proposed based on visual observations revealing that condensate is slung off the array of tubes sideways and significantly affects condensate inundation and thus the heat transfer process. For two types of three-dimensional (3D) enhanced tubes, the Turbo-CSL and the Gewa-C, the local heat flux is correlated as a function of condensation temperature difference, the film Reynolds number, the tube spacing, and liquid slinging effect. The measured heat transfer data of the plain tube were well described by an existing asymptotic model based on heat transfer coefficients for the laminar wavy flow and turbulent flow regimes or, alternatively, by a new model proposed here based on liquid slinging. For the 26fpi low finned tube, the effect of inundation was found to be negligible and single-tube methods were found to be adequate.

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46

Idem, S. A., A. M. Jacobi, and V. W. Goldschmidt. "Heat Transfer Characterization of a Finned-Tube Heat Exchanger (With and Without Condensation)." Journal of Heat Transfer 112, no. 1 (February 1, 1990): 64–70. http://dx.doi.org/10.1115/1.2910366.

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The effects upon the performance of an air-to-water copper finned-tube crossflow heat exchanger due to condensation on the outer surface are considered. A four-tube, two-pass heat exchanger was tested over a Reynolds number range (based on hydraulic diameter) from 400 to 1500. The coil was operated both in overall parallel and overall counterflow configurations. Convective heat and mass transfer coefficients are presented as plots of Colburn j-factor versus Reynolds number. Pressure losses are, similarly, presented as plots of the friction factor versus Reynolds number. Enhancement of sensible heat transfer due to the presence of a condensate film is also considered.

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47

UTAKA, Yoshio, and Tetsuji NISHIKAWA. "611 Unsteady Measurement of Condensate Film Thickness for Marangoni Condensation by Using Laser Absorption Method." Proceedings of Yamanashi District Conference 2000 (2000): 185–86. http://dx.doi.org/10.1299/jsmeyamanashi.2000.185.

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48

KATOH, Yasuo, Ryuichi NAGATA, Eko SISWANTO, and Hiroshi KATSURAYAMA. "J132 Study on the condensate film behavior during condensation on flat plate in porous media." Proceedings of the Thermal Engineering Conference 2012 (2012): 309–10. http://dx.doi.org/10.1299/jsmeted.2012.309.

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49

Mahgoub, Mohamed. "Effect of Nature of The Surface of The Condensate Film on Condensation Heat Transfer.(Dept.M)." MEJ. Mansoura Engineering Journal 12, no. 1 (June 2, 2021): 77–97. http://dx.doi.org/10.21608/bfemu.2021.174534.

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50

Marto, P. J., D. Zebrowski, A. S. Wanniarachchi, and J. W. Rose. "An Experimental Study of R-113 Film Condensation on Horizontal Integral-Fin Tubes." Journal of Heat Transfer 112, no. 3 (August 1, 1990): 758–67. http://dx.doi.org/10.1115/1.2910451.

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Heat transfer measurements were made at near-atmospheric pressure on a smooth tube, on 24 integral-fin tubes having machined, rectangular-shaped fins, and on a commercial integral-fin tube. All tubes were made of copper. The vapor flowed vertically downward with a nominal velocity of 0.4 m/s. Vapor-side heat transfer coefficients were determined with a typical uncertainty of ± 7 percent using a “modified Wilson plot” technique. The vapor-side heat transfer coefficient of the integral-fin tubes (based upon the outside surface area of the smooth tube) was enhanced considerably more than the surface area enhancement provided by the fins. Heat transfer enhancements (for the same vapor-to-wall temperature difference) up to around 7 were measured for a corresponding area enhancement of only 3.9. The optimum fin spacing was found to lie between 0.2 and 0.5 mm, depending upon fin thickness and height. The data were compared with those of other investigations and with several existing theoretical models. Visual observations of condensate drainage patterns from the finned tubes were also made.

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Journal articles on the topic 'Condensation; Condensate film'

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