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About this book

This book is an expanded form of the monograph, Dropwise Condensation on Inclined Textured Surfaces, Springer, 2013, published earlier by the authors, wherein a mathematical model for dropwise condensation of pure vapor over inclined textured surfaces was presented, followed by simulations and comparison with experiments. The model factored in several details of the overall quasi-cyclic process but approximated those at the scale of individual drops. In the last five years, drop level dynamics over hydrophobic surfaces have been extensively studied. These results can now be incorporated in the dropwise condensation model.

Dropwise condensation is an efficient route to heat transfer and is often encountered in major power generation applications. Drops are also formed during condensation in distillation devices that work with diverse fluids ranging from water to liquid metals. Design of such equipment requires careful understanding of the condensation cycle, starting from the birth of nuclei, followed by molecular clusters, direct growth of droplets, their coalescence, all the way to instability and fall-off of condensed drops. The model described here considers these individual steps of the condensation cycle. Additional discussions include drop shape determination under static conditions, a fundamental study of drop spreading in sessile and pendant configurations, and the details of the drop coalescence phenomena. These are subsequently incorporated in the condensation model and their consequences are examined. As the mathematical model is spread over multiple scales of length and time, a parallelization approach to simulation is presented. Special topics include three-phase contact line modeling, surface preparation techniques, fundamentals of evaporation and evaporation rates of a single liquid drop, and measurement of heat transfer coefficient during large-scale condensation of water vapor. We hope that this significantly expanded text meets the expectations of design engineers, analysts, and researchers working in areas related to phase-change phenomena and heat transfer.

Table of Contents

Frontmatter

Statics, Spreading, Coalescence

Frontmatter

Chapter 1. Droplet Statics

Abstract
Dropwise condensation (DWC) is accomplished in a two-phase (liquid-vapor) system at selected nucleation sites by transferring the latent heat release through the walls of the container. Heat release is at the liquid-vapor interface but heat transfer to the cooler substrate occurs through the footprint of the condensed liquid drop. Heat transfer rates determine the extent of condensation and in turn, the growth rate of the drop. Hence, the nature of contact of the condensed liquid phase with the solid wall strongly affects the energy and mass transfer rates in the condensation process. The solid-liquid contact area depends on the overall shape of the drop. In this context, a good understanding of the solid-liquid contact behavior in drops and full three-dimensional (3D) drop shapes is required. Drop motion is observed only during coalescence or instability. For a major portion of the condensation cycle, the drop is under near-static conditions. It is, therefore, important to study static drops for a detailed understanding of the condensation process. This chapter presents the fundamentals of wettability, followed by a discussion on contact angles, along with the mechanical and thermodynamic derivations of the Young’s equation. Physical and chemical methods to control wettability are outlined in the section on surface texturing. Hydrophobicity arising from the Cassie-Baxter and Wenzel models, along with superhydrophobicity and transition between the different wetting regimes is discussed next. The effect of plate inclination on pendant drops is presented in detail, highlighting contact angle hysteresis. The mathematical models used for the determination of equilibrium shapes of static pendant drops are discussed next, with a particular focus on substrate inclination. Mechanistic and energy approaches are presented. The dependence of drop shapes on various physical parameters, such as gravity, fluid density, interfacial tensions, drop volume, substrate texturing, and gravity, can be explored in this framework. An inverse method for the measurement of equilibrium contact angle is described and results obtained by the authors at the laboratory scale are presented. The notion of a dynamic contact angle for a drop moving on a textured surface is briefly described. The chapter concludes with recent trends in microscopic modeling of gas-liquid interfaces using molecular dynamic simulations.
Gaurav Bhutani, K. Muralidhar, Sameer Khandekar

Chapter 2. Spreading of Sessile and Pendant Drops on Partially Wetting Surfaces

Abstract
Spreading dynamics of an initially spherical liquid drop over a partially wetting textured surface is analyzed by solving an integral form of the governing equations. The mathematical model extends Navier-Stokes equations by including surface tension at the gas-liquid boundary and a force distribution at the three-phase contact line. While interfacial tension scales with local surface curvature, the motion of the contact line depends on the departure of the instantaneous contact angle from its equilibrium value. The numerical solution is obtained by discretizing the spreading drop into equal volume disk elements. The Bond number range considered is 0.01–1. Results obtained for sessile and pendant drops are in conformity with limiting cases reported in the literature. They further reveal multiple oscillatory timescales that have also been reported in experiments. Spreading of water and glycerin drops over fully and partially wetting surfaces is studied in terms of excess pressure, wall shear stress, and the footprint radius. Water drops show oscillations during spreading while glycerin spreads uniformly over the surface. The mathematical model also predicts heat transfer rates from the liquid drop to a colder substrate. A pendant drop exhibits rich dynamical behavior including inertial oscillations and gravitational instability, given that gravity tries to detach the drop against wetting contributions. Thus, a unique aspect of a pendant drop is that it may detach from the surface at an intermediate stage of spreading and not achieve a stable equilibrium shape. Unlike commonly known hydrodynamic instabilities, inertial oscillations, however, play a stabilizing role in keeping the pendant drop attached to the surface.
Aashutosh Mistry, K. Muralidhar

Chapter 3. Coalescence Dynamics of Drops over a Hydrophobic Surface

Abstract
Fluid motion arising from the coalescence of two liquid drops is discussed. Experiments are described in which two small water drops are placed on a chemically textured hydrophobic surface (apparent contact angle ~150°), either in sessile or in pendant modes, respectively, just touching each other, under atmospheric conditions. Equal and unequal drop volumes have been studied. The Bond number of the combined drop falls within 0.01–0.04. The resulting coalescence process has been imaged by a high-speed camera, till the combined drop reaches equilibrium. The sequence of images arising from coalescence is analyzed. The position of the center of mass of the combined drop is determined, with displacement yielding the velocity components. The centroid displacement data show that two timescales describe the harmonic content of flow oscillations. These are related to the high initial flow velocities generated, followed by viscous relaxation of the fluid at later times. Scale analysis in terms of force pairs and energy components delineate experimental trends in velocity and wall shear stress. Shear stresses are momentarily developed at the wall at the short timescale, with a magnitude depending on the drop volumes. These are smaller in the pendant mode compared to the sessile. The possibility of including the coalescence details of individual droplet pairs in a complete dropwise condensation model is then discussed.
Praveen Somwanshi, K. Muralidhar, Sameer Khandekar

Chapter 4. Introduction to Evaporative Heat Transfer

Abstract
Evaporation refers to the change of phase occurring at the air or gas interface formed with a liquid medium. The discussion here is restricted to evaporation of a water body. The quantity of importance is the evaporation rate, or, the evaporative mass flux. In a continuum formulation, it is determined by requiring continuity of temperature and pressure at the interface, followed by visualizing a layer of air saturated with water vapor just above the water body. The gradient in humidity in the gas phase sets up mass fluxes of water vapor and hence decides the evaporative mass flux. Additional phenomena such as cooling of the water body and fluid convection will define the overall transport process. This description is referred to as the quasi-equilibrium model. In contrast, the non-equilibrium model, based on the kinetic theory of gases postulates the appearance of a Knudsen layer at the air-water interface. The extent of jump in temperature across the Knudsen layer can significantly affect the evaporation rate. These two models are described at length in this chapter.
Manish Bhendura, K. Muralidhar, Sameer Khandekar

Modeling Dropwise Condensation

Frontmatter

Chapter 5. Introduction to Condensation

Abstract
Condensation involves change of phase from the vapor state to the liquid. It is associated with mass transfer, during which vapor migrates towards the liquid-vapor interface and is converted into liquid. Condensation process is initiated by subcooling, a temperature difference between the bulk vapor and the solid surface. Subsequently, energy in the form of the latent heat must be removed from the interfacial region either by conduction and convection through the droplet and conduction through the substrate. This chapter introduces classification and significance of various physical processes in dropwise condensation, while comparing it with the filmwise form of condensation. The importance of surface wettability and equilibrium contact angle on the formation of drops is highlighted. The shape of the drop plays a central role in fixing conduction resistance, the onset of gravitational instability with respect to static equilibrium, as well as its motion over the substrate. Post instability, fresh nucleation ensures that the dropwise condensation process is intrinsically cyclic, with a characteristic timescale, area coverage, and drop size distribution.
Sameer Khandekar, K. Muralidhar

Chapter 6. Modeling Dropwise Condensation: From Atomic Scale to Drop Instability

Abstract
The large body of literature available on the subject suggests the following three independent mechanisms of dropwise condensation: (1) The vapor condenses primarily between the droplets, i.e., the droplet-free area. This condensate layer gets transported to the droplets in their vicinity by surface diffusion. According to this model, the thin film between the droplets and the free surface of the droplets contribute to overall heat transfer. (2) While vapor condensation begins in a filmwise mode, the film reaches a critical thickness and ruptures due to surface tension-driven instability forming droplets. It is postulated that major part of the heat transfer takes place at this very thin condensate film, while the droplets mainly act as liquid collectors. (3) Droplets are only formed at individual nucleation sites, while the area between the droplets is regarded to be inactive with respect to condensation. In this model, heat transfer occurs only through the droplets and is primarily limited by their heat conduction resistance. Majority of the studies support this mechanism, in which the condensate is in the form of discrete drops located at the nucleation sites on or underneath a lyophobic substrate. A mathematical model based on the third mechanism is developed in this chapter. A comprehensive mathematical model of dropwise condensation underneath an inclined substrate, with and without a wettability gradient, is presented. The dropwise condensation process is hierarchical because it starts from the atomic scale and progresses on to the macroscale. The mathematical models of various sub-processes in dropwise condensation are described and correlated to experimental observations. Individual models of atomic level condensation, nucleation of a drop of minimum radius, growth by direct condensation, coalescence, instability, and fresh nucleation leading to a condensation cycle, are described. The model is then presented in the form of a numerical algorithm.
Sumeet Kumar, Smita Agrawal, Basant Singh Sikarwar, N. K. Battoo, K. Muralidhar, Sameer Khandekar

Chapter 7. Finite Time Coalescence in Dropwise Condensation

Abstract
Augmentation in the instantaneous wall shear stress and wall shear stress arising from coalescence of adjacent liquid drops within the dropwise condensation cycle is examined in this chapter. Instead of a first principles modeling of the coalescence process, a scale analysis that estimates velocity, length, and timescales is adopted. Results show that the time-averaged fluxes and stresses are barely impacted though the instantaneous values are large enough to be of consequence.
Praveen Somwanshi, K. Muralidhar, Sameer Khandekar

Chapter 8. Simulation in a Parallel Environment

Abstract
Condensation is either in filmwise or dropwise mode depending upon the wetting properties of the condensing liquid. Condensation occurs in the form of a film if the condensing liquid wets the substrate; otherwise, it is in the form of drops. Dropwise condensation is studied in this chapter. The number of drops to be tagged at a given time is the product of available nucleation sites and surface area. Numerical simulation with increasing surface area becomes computationally intensive. The simulator is developed using OpenMP and MPI architecture. The simulator can be used for a larger surface of size, e.g., 50 mm × 50 mm.
Praveen Somwanshi, K. Muralidhar

Chapter 9. Simulation: Dropwise Condensation of Water

Abstract
Simulations have been performed for water vapor condensation underneath a horizontal and an inclined textured substrate. A horizontal surface having unidirectional wettability gradient has also been considered. Here, the effect of thermophysical properties, physico-chemical properties of the substrate, promoter layer thickness, nucleation site density, saturation temperature, degree of subcooling, effect of wettability gradient and angle of inclination are parametrically explored. Quantities such as the contact angle and contact angle hysteresis play an important role. We have used a nucleation site density of 106 cm−2 in most of the simulations reported. The simulation data is presented in the form of condensation patterns, area coverage as a function of time, and area-averaged heat transfer coefficient as a function of time. The cycle time and the maximum drop size at instability are tabulated. Based on numerical data, heat transfer coefficients of water vapor condensation are expressed as correlations. In order, the horizontal, inclined, and the graded surface experience (a) larger to smaller drop sizes, (b) longer to shorter cycle times, and (c) lower to higher heat transfer coefficients.
Basant Singh Sikarwar, Praveen Somwanshi, K. Muralidhar, Sameer Khandekar

Chapter 10. Dropwise Condensation of Bismuth on Horizontal and Vertical Surfaces

Abstract
Simulation of dropwise condensation of bismuth vapor on a subcooled hydrophobic surface is discussed in the present chapter. The process starts from nucleation of drops, followed by their growth and coalescence, resulting in drop instability that removes them from the surface. Fresh nucleation occurs at the exposed nucleation sites, thus creating a cycle of vapor condensation and liquid removal. The drop size distribution over the surface determines the instantaneous surface averaged wall heat flux. Wall shear stresses are generated during coalescence process and also when large drops start moving due to instability, which is gravitational in origin; hence, the largest drop diameter achieved depends on the surface orientation. Near-horizontal and vertical surfaces have been studied in the present work. Drop instability affects the periodicity of the condensation process and the average drop size and thus, the wall heat flux and wall shear stress. Coalescence of adjacent drops is a momentary step with a timescale of milliseconds, but the velocities generated are substantial. Coalescence velocities and time intervals have been determined by scale analysis, and the sensitivity of wall heat flux and wall shear stress to these ensuing velocities are delineated. The multiscale model developed is computationally intensive and has been simulated on large condensing surface areas using MPI on a parallel architecture. Bismuth condensation properties have been compared with water. Large heat fluxes and shear stresses are seen to be attained sporadically during coalescence for short time instants and do not contribute significantly to the surface-averaged values. On the other hand, wall shear stresses are found to be large enough to damage the hydrophobic coatings and degrade surface wettability, thereby hindering dropwise mode of condensation.
Praveen Somwanshi, K. Muralidhar, Sameer Khandekar

Dropwise Condensation Experiments

Frontmatter

Chapter 11. Dropwise Condensation: Experiments

Abstract
Experimental determination of the heat transfer coefficient during dropwise condensation is a difficult task because of the many intricacies involved. The driving temperature difference is small, essentially resulting in a high heat transfer coefficient. Further, uncertainties associated with the microscale sub-structure of contact line shapes and motions, dynamic temperature variations below the condensing drops, effect of roughness and inhomogeneity of the substrate structure, control of true boundary conditions, microscale instrumentation, and transport dynamics of coalescence, merger, wipe-off, renucleation cycles, and the leaching rates of the promoter layer add to the difficulty in conducting repeatable experiments. Very high heat transfer rates (and therefore a very low temperature differential) coupled with the above factors also hinder generation of repeatable experimental data. Consequently, many conflicting experimental results have been published over the years, some results showing considerable scatter. In this chapter, experimental results of condensation patterns and the corresponding predictions of numerical simulation for water vapor are compared. The prediction of the model is in fair agreement with the experimental data for condensation of water vapor. Average heat flux as a function of degree of subcooling for water and mercury are compared. Although there is some discrepancy in the data obtained, major phenomena related to dropwise condensation underneath horizontal substrates are well simulated by the mathematical model.
Basant Singh Sikarwar, K. Muralidhar, Sameer Khandekar

Chapter 12. Surface Preparation: Some Techniques

Abstract
Non-wetting surfaces are surfaces that promote dropwise condensation. However, most raw surfaces have high free surface energy and exhibit high-wetting behavior. Accordingly, in most practical applications, lowering the surface energy becomes a prerequisite to achieving the dropwise mode of condensation. Five surface engineering techniques, namely silanization of glass, chemical texturing of copper, anodization of aluminum, laser micro-machining of the copper surface, and ion implantation, are discussed. Although each technique is distinct in nature, all of them work on modifying the surface texture and/or chemical properties to develop non-wettable surfaces. Chemical texturing of glass is performed by silanation using octyl-decyl-tri-chloro-silane (C18H37C13Si) in a chemical vapor deposition process. Copper surface is coated with a low surface energy material 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10-heptadecafluoro-1-decanethiol (CF3(CF2)7CH2CH2SH) to promote dropwise condensation. Physical texturing of aluminum is achieved by anodic oxidation of the material to generate the structured nanopores on the surface to lower the free surface energy. In the case of laser machining, material evaporation using laser ablation process is adopted to develop the desired physical structures on the surface, typically on the microscale length. Irradiation via Argon ion beams of 0.5 keV is executed on copper, aluminum, and gold surfaces to lower the surface energy by implanting the foreign molecules of very low dispersive energy inside the surface. It is demonstrated that different techniques can be adopted to fabricate the non-wetting surfaces, the surfaces which aid in dropwise condensation. Therefore, depending on the material of the surface in actual applications, one can choose a technique to envisage the dropwise condensation.
Mahesh Kumar Yadav, Praveen Somwanshi, Sameer Khandekar, Sanghamitro Chatterjee, Mohit Gonga, K. Muralidhar, Sudeep Bhattacharjee

Chapter 13. Measurement of Condensation Heat Transfer

Abstract
Optimum design of any thermal system requires detailed and experimentally verifiable knowledge of spatio-temporal heat transfer rates. Several techniques are available to estimate the heat flux and hence the heat transfer coefficient, including surface mountable sensors and thermocouple-based methods. Commercially available thermopile heat flux sensors are found to be unsuitable for measurement of condensation fluxes, mainly due to: (a) intrusion of the sensor with the ensuing condensation process, and (b) inherent measurement lag when the sensor is mounted away from the surface of interest. Thus, a noninvasive measurement system becomes highly desirable for surface textured-dependent phenomena such as condensation heat transfer. Use of inverse heat transfer technique, where temperature measurement at internal location(s) in the substrate is utilized to estimate the conditions prevailing at its boundary, is quite attractive for such heat transfer processes. Two case studies are presented in this chapter for measurement of filmwise and dropwise mode of condensation heat transfer to demonstrate its efficiency. Use of inverse heat transfer technique for well-designed experiments is found to be a very attractive and versatile technique to measure heat transfer rate for both modes of the condensation. The experiments on dropwise condensation of pure steam are also performed at different inclination angles, including limiting bounds of sessile and pendant mode, and high-quality experimental data are reported.
Mahesh K. Yadav, Maneesh Punetha, Abhinav Bhanawat, Sameer Khandekar, K. Muralidhar

Chapter 14. Measurement of Heat Transfer Rates under a Liquid Drop During Dropwise Condensation

Abstract
The quasi-periodic and statistical nature of drop formation, along with very high heat transfer coefficients at low operating temperature difeerentials, makes the experimental determintaion of transfer coefficients quite challenging. We demonstrate the use of high resolution Liquid Crystal Thermography (LCT) coupled with digital videography to measure the spatial distribution of temperature during dropwise condensation in the pendant mode over a polyethylene substrate. Using a one-dimensional heat transfer model, heat flux profiles through individual condensing droplets have been obtained. The measured heat flux as a function of drop diameter matches published data for large drop sizes but fails for small drops (< 0.4 mm) where the thermal resistance of the LCT sheet is a limiting factor. To a first approximation, the present work shows that drop size can be correlated to the local heat flux. It is demonstrated that the average condensation heat flux over a surface can be obtained entirely from the drop-size distribution.
Gagan Bansal, S. Khandekar, K. Muralidhar

Chapter 15. Evaporation Dynamics of a Sessile Droplet on a Hydrophobic Surface

Abstract
Contrasting dropwise condensation, evaporation of a water droplet on a hydrophobic but conducting surface under ambient conditions is discussed in the present chapter. Interfacial evaporation rates and evaporative flux are often predicted using the vapor mass diffusion model. It accounts for differences in the humidity ratio between the interface region and the far-field, the humidity flux being determined by Fickian diffusion. Two other factors that are relevant are evaporative cooling of the air-liquid interface and heat conduction in the solid-liquid-gas domain. Thus, thermophysical properties of the substrate become relevant, limiting droplet evaporation rates. In this chapter, an extended vapor diffusion model is built in COMSOL® that incorporates these additional transport mechanisms and predicts evaporation rates, interfacial evaporative flux, and temperature distribution at the base of the sessile droplet. The model is validated against controlled laboratory experiments for sessile droplet evaporation of water on copper and glass substrates that offer distinct wettability. The study reveals the limitations of the vapor diffusion model and the importance of additional mechanisms relevant to droplet evaporation.
Sachin K. Singh, Mohit Gogna, Sameer Khandekar, K. Muralidhar

Chapter 16. Closing Remarks and Prospects

Abstract
An overview of the dropwise condensation process is presented in the preceding chapters. It involves vapor-to-liquid phase change in the form of discrete drops on or underneath horizontal and inclined substrates. The process is hierarchical, in the sense that it occurs over a wide range of length and timescales. A mathematical model of dropwise condensation underneath textured surfaces, horizontal and inclined, is discussed in the text. The model starts from the formation of drops at the atomic scale at randomized nucleation sites, and follows its growth by direct condensation and coalescence, till individual drops are large enough to fall off or slide away. Fresh nucleation commences on the exposed surface devoid of drops and the cycle continues in a quasi-steady state. Predictions of numerical simulation are compared against experimentally derived condensation patterns and heat fluxes. The significance of droplet level details including statics, spreading, and coalescence is revealed in the text. Additional discussion related to surface preparation and measurement of heat fluxes on the drop-scale and the device-scale are presented. This final chapter closes with newer directions that are likely to be explored in the future.
Sameer Khandekar, K. Muralidhar

Backmatter

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