O H. Evaporation/condensation of water. Unresolved issues. Kazys Almenas. I. Phase change at low pressures, laminar conditions

Transcription

1 Vytautas Magnus University is proud to present exclusive books bearing the Libri Universitatis Vytauti Magni brand. The Acta Humana, Acta Natura and Acta Musica series present the Kazys Almenas papers, thoughts and insights of Vytautas Magnus University s LI BRI UNIVERSI TAT IS VY TAU T I MAGNI most renowned scientists and authors, Honorary Doctors and Professors. Dr. Kazys Almenas is an Emeritus Professor of Nuclear Engineering at the University of Maryland, MD, USA. He is a member of the Lithuanian Academy of Sciences, Dr. Honoris causa Vytautas Magnus University. He has authored Nuclear Engineering text books, and published extensively in the areas of neutron transport theory and nuclear safety related thermal hydraulics. His recent interest has been condensation/ evaporations phenomena, he participated in a series of experimental test programs investigating rapid condensation phenomena conducted at the Lithuanian Energy Institute, Kaunas, Lithuania. Evaporation/condensation of water. Unresolved issues I. Phase change at low pressures, laminar conditions б б O H H + б ISBN VYTAUTAS MAGNUS UNIVERSITY V E R S US AU R EUS

2 Evaporation/condensation of water. Unresolved issues

3 Kazys Almenas Evaporation/condensation of water. Unresolved issues I. Phase change at low pressures, laminar conditions VYTAUTAS MAGNUS UNIVERSITY VERSUS AUREUS

4 UDK 556 Al-146 This monograph has been reviewed by: Prof. Dr. Bal Raj Sehgal, Emeritus Professor of Nuclear Power Safety, Swedish Royal Institute of Technology, Sweden Prof. Dr. Jurgis Vilemas, Vytautas Magnus University, Kaunas, Lithuania Dr. Tony Pollman, Chair for Research, Naval Postgraduate School, Monterey, CA, USA Approved and recommended for printing by the Council of the Faculty of Natural Sciences of Vytautas Magnus University on 20 October 2014 (Protocol No ). ISBN (Print) ISBN (Print) ISBN (Online) ISBN (Online) Kazys Almenas, 2014 Vytautas Magnus University, 2015 Versus aureus Publishers, 2015

5 1. Introduction This study considers very familiar, but still not completely understood phenomena the evaporation and condensation of water. Mankind has used and studied these processes since antiquity; from a practical point of view these studies have been successful. Empirical and semi-empirical relationships have been developed which make it possible to predict the rate of condensation/evaporation processes for various applications over specified ranges of conditions. The properties of water and its phase changes have been analyzed by meteorologists, biologists, physicists and practically all categories of engineers. This has led to a fragmentation of the accumulated information, the development of discipline specific nomenclatures and empirical relationships which makes the transfer of information between disciplines more difficult. To summarize this brief overview in most of its numerous applications the phase changes of water can be predicted and controlled successfully, and yet, a fundamental understanding of the mechanism of these processes is lacking. To be specific: The kinetic energy spectrum of evaporating and condensing molecules is not known. Very high condensation rates can lead to the disruption of the liquid-vapor interface and sudden decreases of pressure. The mechanisms which generate these condensation shock or condensation implosion events are not understood. Water is the most abundant substance on the surface of our planet. It is a deceptively simple molecule, just an atom of oxygen and two of hydrogen, yet its 5

6 properties are so complex that no completely satisfactory theoretical model of the water molecule is available. It is a polar molecule, and has an exceptionally high dielectric constant. Because it is so small and has a high H to O atom ratio, it forms strong cohesive bonds, has a high surface tension, and an exceptional capacity for dissolving ionic compounds. As a result, all of the liquid water that we encounter is in fact a more or less concentrated ionic solution, sea water being the most abundant example. Other prominent examples include the various fluids found in the bodies of all living organisms including our own. Tap water, as well as distilled water in glass or plastic bottles, all contain sufficient contaminants to influence some of its properties. That is one of the reasons why in spite of an abundance of experimental studies, relatively few experimental studies of truly pure water are available. The first part of this study assesses empirical data and theoretical approaches relevant for condensation/evaporation processes occurring at low vapor pressures. Information available in several disciplines is considered. Classical theoretical and experimental studies based on kinetic theory are augmented by information from the fields of thermodynamics, physics, chemistry and potentially relevant computational approaches developed in neutron transport theory and molecular dynamics. It is an interdisciplinary study, therefore definitions and nomenclature which are self-evident for specialists of a given field have to be defined and to some degree explained. The presentation of an inherently complex phenomenon can be overwhelmed and eventually confused by detail, this is especially true if one attempts to deal with all of the complexities simultaneously. The phase-change of water is a complex issue, in order to keep the review manageable, first the simplest theoretical model is considered, additional phenomena and corresponding analytical approaches are introduced sequentialy. For this purpose a sequence of four control volumes, identified as CV-1 to CV-4, is employed. The schematics of these control volumes depict one dimensional representations of the flow geometry and define the assumptions and boundary conditions used by the associated theoretical models. An advantage of such an approach is that it approximately parallels the historical development of the analytical models. The literature concerned with these issues is very voluminous, some reviews of condensation/evaporation processes encompass well over 100 positions. In this study only selected papers, illustrating key developments, are referenced. 6

7 2. CV-1 Schematic Fig 1 represents the classical 1-D control volume employed for evaluating liquid-vapor phase transfer on the basis of kinetic theory. Boundary conditions are the bulk temperatures and pressures in the liquid and vapor regions. The interface is a zero thickness surface at which an abrupt change from liquid to vapor density occurs. For the conditions considered in this study the bulk vapor velocity resulting from condensation or evaporation is much smaller then average molecular velocities, bulk flow is laminar, therefore energy transport in both phases occurs by thermal diffusion. This implies that for non-equilibrium conditions, a step temperature difference must exist at the interface. As shown in the schematic, the evaluation of net mass and energy fluxes across the interface requires the specification of T fi, P fi on the liquid side of the interface and T gi, P gi on the vapor side. P g T g Cond. Evap. T gi P fgi T fi, P fi Tf FIG 1. DV-1 Idealized phase-change control volume 7

8 The specification of these parameters is the major cause of uncertainty. In the liquid phase the intermolecular distance is on the order of 0.1 nano-m. A direct measurement of the diffusive temperature gradient in the immediate vicinity of the interface is thus not possible, and T fi has to be assumed or inferred from other measurements. The pioneering work in this area harks back to the beginning of the previous century; Knudsen (1) showed that the evaporation rate of Mercury into a low pressure environment could be evaluated successfully by using kinetic theory and the assumption that the kinetic energy spectrum of the evaporating molecules has a Maxwellian distribution characterized by T gi (2,3,4,5). Numerous investigators applied this conclusion to other volatile materials including water. It has remained the dominant starting point for the development of analytical approaches for a very wide range of phase change problems. The use of this distribution is questioned in this study, therefore the reason why it has been employed is presented in the words of previous authors. In an extensive study of condensation published in 1970 Shankar (4) writes: We shall assume that the liquid emits vapor molecules in a Maxwellian distribution corresponding to the liquid temperature Tl In a thesis evaluating alternative analytical approaches for non-equilibrium condensation evaporation problems Bond (8) in 2000 provides the following justification: This is justified by assuming that at equilibrium (no net phase change conditions), the mass flux is zero, the temperature T across the surface is constant, and the vapor pressure is equal to the saturation pressure Psat(T). We then see that in equilibrium Pevap = Psat(T) This should be true for small perturbations from equilibrium, we can then replace Pevap with Psat(T) It is standard practice to assume the liquid phase is never far from equilibrium; this allows the molecules leaving the interface to be described by the Maxwellian. Applying these assumptions the absolute rate at which water molecules evaporate or condense is determined by integration over a half-range Maxwellian distribution at T fi and T gi. Net phase change rate is the difference of these two integrals. Knudsen assumed that all vapor molecules impacting the interface condense and all escaping liquid molecules remain in the vapor phase, the net phase change mass flux is then given by: 8

9 J M 2 Rg Pgi Tgi In the literature this is usually referred to as the Hertz-Knudsen equation. Pfi Tfi (1) 3. CV-2 Schematic Practically from the start it was recognized that when Eq. 1 is used to determine phase change rates of fluids other then monatomic Hg, the theoretical mass fluxes are considerably larger than experimentally measured values. This is especially true for water; in 1931, Alty (2) showed that for evaporation at low pressures the discrepancy can reach two orders of magnitude. Alty reasoned that Eq. 1 correctly predicts the number of molecules impacting or escaping from an interface, but not all of these molecules actually condense or evaporate. He proposed that a fraction of the vapor molecules impacting the interface are reflected. At equilibrium conditions the condensing and evaporating mass fluxes are equal; therefore, a similar mechanism must exist for the molecules moving across the interface from the liquid side. These assumptions were widely accepted, the correction factors required to force agreement between theory and experiment are usually referred to as condensation and evaporation coefficients, in some publications as accommodation coefficients. The CV-2 schematic is an example of the figures used to illustrate these processes (6, 8, 10, 11). It shows two distinct correction factors, the condensation coefficient αc and the evaporation coefficient αe. In a number of publications both coefficient are assumed to have the same value. Numerous experimental studies seeking to measure these correction factors were conducted; this included several approaches which explored more detailed reflection mechanisms. For example, momentum preserving and specular 9

10 collisions were considered, escaping molecules were assumed to be reflected back to the liquid with and without loss of kinetic energy. All of these studies share common features the condensation/evaporation mechanisms are assumed to be symmetric and the correction factors are applied to the entire evaporation or condensation spectrum In 1953 Schrage (3) proposed a correction which accounts for the effect of bulk vapor velocity when net condensation or evaporation is present. The resulting widely used relationship for evaluating the phase-change mass flux is usually referred to as the Hertz-Knudsen-Schrage equation: (2) Eq. 2 became the starting point for further development of analytical models and is still employed for evaluating condensation/evaporation mass fluxes. Given a circumscribed range of liquid/vapor conditions and accommodation coefficients which were determined for these conditions, this approach can yield useful results. P g T y (1-α c ) Q c α c Q c α ε Q ε (1-α ε ) Q ε Tf FIG 2. V-2 Condensation/Evaporation Rates with Corrections Factors 10

11 Contents 1. Introduction 5 2. CV-1 Schematic 7 3. CV-2 Schematic 9 4. CV-3 Schematic Theory vs. experiment CV-4 Schematic Molecular characteristics of water at low pressures Agitational and total energy The hydrogen bond and its impact on the phase change of water The limits of kinetic theory Alternate computational approaches Insights from neutron transport theory Experimental verification Inferred spectrum of evaporating molecules Inferred spectrum of condensing molecules Conclusions, proposals and recommendations 47 Appendix A 53 Appendix B 61 69

12 Al-146 Almenas, Kazys Evaporation/condensation of water. Unresolved issues. I. Phase change at low pressures, laminar conditions. Monograph / Kazys Almenas. Kaunas: Vytauto Didžiojo universitetas; Vilnius: Versus aureus, p. ISBN (Print) ISBN (Print) ISBN (Online) ISBN (Online) This study assesses empirical data and theoretical approaches relevant for condensation/evaporation processes at low vapor pressures. Classical theoretical and experimental studies based on kinetic theory are augmented by information from the fields of thermodynamics, physics, chemistry and potentially relevant computational approaches developed in neutron transport theory and molecular dynamics. In the course of the overview a number of conclusions are reached, inferred distributions of evaporating and condensing molecular spectra are proposed and recommendation regarding further development of analytical approaches are presented. UDK 556 Kazys Almenas Evaporation/condensation of water. Unresolved issues I. Phase change at low pressures, laminar conditions Designer Kornelija BUOŽYTĖ Issuance by Order No. K Published by: Vytautas Magnus University K. Donelaičio g. 58, LT Kaunas leidyba@bibl.vdu.lt Versus aureus Publishers Rūdninkų g. 10, LT Vilnius info@versus.lt Printed by: BALTO print Utenos g. 41A, LT Vilnius