Mechanistic Modelling of Water Vapour Condensation …/Theoretical Investigation... · Mechanistic Modelling of Water Vapour ... Figure 3.4 Condensation in presence of noncondensable

Mechanistic Modelling of Water Vapour

Condensation in Presence of Noncondensable Gases

Doctoral Thesis

by

Krzysztof Karkoszka

School of Engineering Sciences Department of Physics

Div. of Nuclear Reactor Technology Stockholm, 2007

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Abstract This thesis concerns the analytical and numerical analysis of the water vapour condensation from the multicomponent mixture of condensable and noncondensable gases in the area of the nuclear reactor thermal-hydraulic safety. Following an extensive literature review in this field three aspects of the condensation phenomenon have been taken into consideration: a surface condensation, a liquid condensate interaction with gaseous mixtures and a spontaneous condensation in supersaturated mixtures. In all these cases condensation heat and mass transfer rates are significantly dependent on the local mixture intensive parameters like for example the noncondensable species concentration. In order to analyze the multicomponent mixture distribution in the above-mentioned conditions, appropriate simplified physical and mathematical models have been formulated. Two mixture compositions have been taken into account: a binary mixture of water vapour with heavy noncondensable gas and a ternary mixture with two noncondensable gases with different molecular weights. For the binary mixture a special attention has been focused on the heavy gas accumulation in the near-interface region and the influence of liquid film instabilities on the interface heat and mass transfer phenomena. For the ternary mixture of gases a special attention has been paid to the influence of the light gas and induced buoyancy forces on the condensation heat and mass transfer processes. Both analytical and numerical methods have been used in order to find solutions to these problems. The analytical part has been performed applying the boundary layer approximation and the similarity method to the system of film and mixture conservation equations. The numerical analysis has been performed with the in-house developed code and commercial CFD software. Performing analytical and CFD calculations it has been found that most important processes which govern the multicomponent gas distribution and condensation heat transfer degradation are directly related to the interaction between interface mass balances and buoyancy forces. It has been observed that if the influence of the liquid film instabilities is taken into consideration the heat transfer enhancement due to the presence of different types of waves is directly related to the internal film hydrodynamics and shows up in the mixture-side heat transfer coefficient. The model developed for the dispersed phase growth shows that degradation of the condensation heat transfer rate, which is a consequence of degradation of the convective mass flux, should be taken into account for highly supersaturated gaseous mixtures and can be captured by combination with the mechanistic CFD surface condensation model. Keywords: condensation, noncondensable gases, CFD simulation, boundary-layer approximation, binary and ternary mixtures

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List of papers and publications Publications and papers included

I. Karkoszka K., Anglart H., 2006. CFD modeling of laminar film and spontaneous condensation in presence of noncondesable gas. Archives of Thermodynamics, Vol. 27, No. 2, 23 – 36.

II. Karkoszka K., Anglart H., 2005. Multidimensional multicomponent

model of consensation in presence of noncondensable gases. 11th International Topical Meeting on Nuclear Reactor Thermohydraulics, October 2-6, Avignon, France.

III. Karkoszka K., Anglart H., 2006. Numerical analysis of solitary wave

influence on the filmwise condensation in presence of noncondensable gases. 14th International Conference on Nuclear Engineering, July 17-20, Miami, Florida, USA.

IV. Karkoszka K., Anglart H., 2007. Laminar filmwise condensation of vapor

in presence of multi-component mixture of non-condensable gases. 12-th International Topical Meeting on Nuclear Reactor Thermohydraulics, September 30 – October 4, Pittsburgh, Pennsylvania, USA, to be presented.

V. Karkoszka K., Anglart H. Multidimensional effects in laminar filmwise

condensation of vapor in binary and ternary mixtures with non-condensable gases. Submited to the Nuclear Engineering and Design.

Publications and papers not included

VI. Karkoszka K., Anglart H., 2004. CFD Modelling of Direct-Contact Condensation in Presence of Non-Condensable Gases on Liquid Film Surface. 42nd European Two-Phase Flow Group Meeting, Genoa, Italy.

VII. Karkoszka K., Anglart H., 2005. CFD modelling of wall condensation

in presence of noncondensable gas. HEAT2005, Gdansk, Poland.

VIII. Karkoszka K., 2005. Theoretical Investigation of Water Vapour Condesation in Presence of Noncondesable Gases. Royal Institute of Technology, Stockholm, Sweden, licentiate thesis.

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Contribution to papers All papers, included and not included in the present thesis, have been written under supervision of Assoc. Prof. Henryk Anglart. All calculations, models and results have been developed, implemented and analyzed by the author. Summary of included papers Paper I describes the mechanistic modelling of forced convection water vapour condensation from the binary mixture with air. The filmwise condensation is investigated as well as effects of the direct-contact condensation and influence of the spontaneous water droplets nucleation. In all cases modelling is based on the resolution of the gaseous boundary layer in the vicinity of the liquid condensate. Paper II is mainly focused on the forced convection direct-contact and spontaneous condensation effects. It contains detailed discussion of the applied models as well as discussion about the noncondesable mass fraction distribution in the vicinity of the interface boundary layer. The most important conclusion from both Paper I and Paper II is that a proper mechanistic CFD model is able to predict the heat transfer degradation due to the presence of noncondensable gas. Paper III is focused on the liquid film structure influence on the heat transfer between gaseous and liquid phases. All calculations are performed with a two-dimensional in-house code which has been developed in order to give required flexibility in the film geometry modelling. This paper discuses how the film structure disturbances in forms of sinusoidal and soliton-shaped waves influence the local condensation heat transfer rate. It has been found that the internal vortex present inside the wave and interface boundary conditions are directly responsible for the enhancement of the heat transfer process. Paper IV presents the mechanistic modelling of water vapour free convection condensation from the ternary mixture of gases with the boundary layer approximation. Fully coupled, through interface balances, boundary layer equations for liquid and gaseous phases, are solved with the similarity method. Results show how resistance to the interface heat transfer process is influenced by the presence of noncondensable species with different molecular weights and what relations between those species are. Paper V is an extension of Paper IV, where also the free convection condensation from ternary mixture of gases has been investigated. As an extension to Paper IV both the boundary layer approximation and mechanistic modelling with the commercial CFD code have been applied. The main conclusion from this article is that mechanistic CFD modelling with carefully implemented interface balances shows local physical relations between noncondensable components and local

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noncondensable distribution fields. Both solutions of the boundary layer equations and mechanistic CFD model converge to each other for the binary mixture of gases.

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Contents

Abstract III

List of papers and publications V Contents IX

List of figures XI Nomenclature XV

Chapter 1 Introduction 1 1.1 Condensation process and its applications……………………………………....1 1.2 Review of research approaches….……………………………..……...………...3 1.3 Review of modelling approaches………………………………………..………3 1.3.1 Empirical correlations ………………………………………………………...3 1.3.2 Analogy between heat and mass transfer……………………………………...4 1.3.3 Diffusion layer model…………………………………………………………4 1.3.4 Boundary layer approximation and fully mechanistic model…………………4 Chapter 2 Literature review 7 2.1 Empirical and theoretical investigations………………………………….……..7 2.2 Thin film instability.…………………..……...…………………......................13 2.3 Spontaneous condensation.……………………………………....………….....16 Chapter 3 Physical model and assumptions 19 3.1 Free and forced convection condensation on vertical surfaces………………...20 3.1.1 Modelling of forced convection condensation on a vertical surface…...……20 3.1.2 Modelling of free convection, gravity driven condensation on a vertical surface………...……………………………………………………...22 3.2 Forced convection condensation on horizontal surfaces………………………23 3.2.1 Forced convection condensation on a horizontal, isothermal surface…...…..23 3.2.2 Direct-contact condensation on a horizontal, adiabatic surface…………..…23 3.3 Modelling of interactions between liquid film structures and gaseous boundary layer………………………………………………………..25 3.4 Model of spontaneous condensation in presence of noncondensable gases……………………………………….…………………..27 Chapter 4 Mathematical models 31

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4.1 Film and mixture conservation equations……………………………………...31 4.1.1 Liquid phase conservation equations………………………………………...32 4.1.2 Conservation equations for a ternary gaseous mixture….…………………...32 4.2 Balances at the liquid-gas interface………………………………………..…..33 4.3 Boundary layer approximation………………………………………………...35 4.3.1 Boundary layer conservation equations for a liquid film and a ternary mixture of gases…………………………….……………………….35 4.3.2 Interface balances with boundary layer approximation……………………...36 4.3.3 Modelling of buoyancy effects………………………………………………37 4.4 General formulation of ternary diffusivity coefficients………………………..38 4.5 Mathematical model of spontaneous condensation……………………………39 Chapter 5 Solution methods and tools 45 5.1 Modelling of film structure…………………………………………………….45 5.2 Solution of the boundary layer equations……………..……………………….46 5.2.1 Gravity driven condensation on a vertical surface………………….………..47 5.2.2 Application of similarity variables to the interface balance equations……………………………………………………...…………...48 5.3 Mechanistic modelling with a Computational Fluid Dynamics code…………49 Chapter 6 Results and discussion 53 6.1 Forced convection condensation on a vertical surface………………………...53 6.2 Forced convection direct-contact condensation on a horizontal, adiabatic surface…………..……………………………………………………….55 6.3 Influence of spontaneous condensation…...…………………………….……..56 6.4 Film structure influence on the heat and mass transport in the gaseous boundary layer..……………………………….………...………….57 6.4.1 Sinusoidal wave……………………………………………………………...58 6.4.1 Solitary-shaped wave………………………………………………………...59 6.5 Similarity solution of the boundary layer equations for free-convection gravity-driven condensation………………….……………….61 6.7 Mechanistic CFD analysis of free-convection, gravity-driven condensation.....67 Chapter 7 Concluding remarks 73 Acknowledgements 77

References 79 Appendix 1 87 Appendix 2 97 Papers I-V 103

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List of figures Figure 3.1 Physical model of free, gravity driven and forced convection film-wise condensation in presence of noncondensable gases on a vertical surface...…….....21 Figure 3.2 Physical model of forced convection condensation in presence of noncondensable gases on a horizontal, adiabatic surface…...………...………..24 Figure 3.3 Condensation in presence of noncondensable gas on a sinusoidal wave with prescribed geometry……...………………………...……..26 Figure 3.4 Condensation in presence of noncondensable gas on a solitary-shaped wave with prescribed geometry……...………...…….………….26 Figure 3.5 Physical model of spontaneous condensation in presence of noncondensable gase.……….……………………………………...28 Figure 4.1 Interface between liquid film and gaseous mixture……………………33 Figure 4.2 Change of free Gibbs energy as a function of droplet diameter for supersaturation ratio S > 1……………………………………………41 Figure 6.1 Comparison of the diffusion, mechanistic CFD model with commonly used correlations for inlet mixture velocity 8 m/s, K40∆T = ….…….54 Figure 6.2 Comparison of the direct-contact diffusion model with experimental data by Choi et al. (2002)……..………………………….…….55 Figure 6.3 Sensitivity study of the influence of spontaneous condensation during direct-contact condensation on a horizontally-stratified liquid film…...………..…56 Figure 6.4 Example of the calculated temperature field within sinusoidal wave with small amplitude (in K), 0.1air =

∞ϖ ………….………………………………….......58 Figure 6.5 Influence of the noncondensable gas on the gas-side heat transfer coefficient for the small amplitude sinusoidal wave, 0.1air =

∞ϖ , 20.air =∞ϖ and

30.air =∞ϖ , respectively…………………………………………………………….59

Figure 6.6 Example of the calculated temperature field within the liquid film for solitary-shaped wave, (in K), 0.1air =

∞ϖ …………………………...……………...60

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Figure 6.7 Influence of the noncondensable gas on the gas-side heat transfer coefficient for the solitary-shaped wave, 0.1air =

∞ϖ , 20.air =∞ϖ and 30.air =

∞ϖ , respectively……………………...…………………………………………………60 Figure 6.8 Influence of the addition of the light gas on the dimensionless mixture velocity profile……………………………...……………………………………...61 Figure 6.9 Comparison of the computed degradation in the condensation heat transfer with experimental data obtained by Al-Divany and Rose (1973),

5100.air =∞ϖ and 0240.air =

∞ϖ , respectively………………………………………62 Figure 6.10 Comparison of the computed degradation in the condensation heat transfer with experimental data obtained by Al-Divany and Rose (1973),

040.air =∞ϖ and 0680.air =

∞ϖ , respectively………………………………..………63 Figure 6.11 Comparison of the computed degradation in the condensation heat transfer with experimental data obtained by Al-Divany and Rose (1973),

120.air =∞ϖ and 190.air =

∞ϖ , respectively………...……………………...…..……63 Figure 6.12 Comparison of the computed degradation in the condensation heat transfer with experimental data obtained by Al-Divany and Rose (1973),

2540.air =∞ϖ …………………………………………………………….…..……...64

Figure 6.13 Influence of an additional light gas on the dimensionless air and helium mass fraction profiles, additional amount of helium 0.1%He =

∞ϖ …………………64 Figure 6.14 Influence of an additional light gas on the dimensionless air and helium mass fraction profiles, additional amount of helium %40.He =

∞ϖ …………………65 Figure 6.15 Influence of an additional light gas on the dimensionless air and helium mass fraction profiles, additional amount of helium %90.He =

∞ϖ …………………65 Figure 6.16 Influence of an additional amount of heavy or light gas on the degradation of the condensation heat transfer, 2.4%0.32,η 0airδ == ∞ϖ ………......66 Figure 6.17 Influence of an additional amount of heavy or light gas on the degradation of the condensation heat transfer, 1.5%0.34,η 0airδ == ∞ϖ . …………66 Figure 6.18 Comparison of the heat transfer coefficient calculated with the CFD mechanistic model and obtained from the Dehbi et al. (1991) correlation………...68

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Figure 6.19 Interface air mass fraction profile for an additional bulk amount of helium 0%He =

∞ϖ , 0.1%He =∞ϖ , 0.2%He =

∞ϖ and 0.3%He =∞ϖ , respectively,

4%air =∞ϖ ………………………………..………………………………….….…68

Figure 6.20 Interface helium mass fraction profile for an additional bulk amount of helium 0%He =

∞ϖ , 0.1%He =∞ϖ , 0.2%He =


4%air =∞ϖ …………………………………………………………………………69

Figure 6.21 Comparison of the average interface air mass fraction obtained by a solution of the boundary layer equations for vertical surface and by the mechanistic CFD model with the corner-type geometry, 4%air =

∞ϖ …………………..……...70 Figure 6.22 Comparison of the average interface helium mass fraction obtained by a solution of the boundary layer approximation for vertical surface and by the mechanistic CFD model with the corner-type geometry, 4%air =

∞ϖ ………...…. 70 Figure 6.23 Prediction of the heat transfer degradation due to addition of helium in bulk mixture for 4%air =

∞ϖ . ………………………………..……………………71 Figure A1.1 Transformation from the physical to the computational domain……89 Figure A1.2 Compass notation................................................................................92 Figure A2.1 The physical model of forced convection condensation from the binary mixture with noncondensable gas on a horizontal, isothermal surface……………97 Figure A2.2 Comparison of results obtained with the film thickness model given by Equation (A2.37) and results obtained by Sparrow et al. (1967) for the Schmidt number 0.55…………………………………………………................................102

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Nomenclature a heat diffusivity (m2/s)

Cp specific heat (J/kg/K)

d diameter (m)

D diffusivity coefficient (m2/s)

f dimensionless stream function

h specific enthalpy (J/kg)

hfg latent heat (J/kg)

g, g gravity acceleration and gravity vector (m/s2)

∆G change in Gibbs free energy (J)

j diffusive mass flux (kg/s/m2)

Ja Jakob number

k heat conductivity (W/m/K)

k Boltzmann constant (m2 kg s-2 K-1)

m mass flow rate (kg/s)

M molecular mass (kg/kmol) or mass (kg)

n mass flux (kg/s/m2)

N number of droplets

p pressure (Pa)

q, q heat flux and heat flux vector (W/m2)

Pr Prandtl number

R universal gas constant (kJ/K/kmol)

( ) 2/1MMLL µρµρR = in Appendix 2

S supersaturation ratio

Sc Schmidt number

t time (s)

T temperature (K)

u, v velocity components (m/s)

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u velocity vector

V volume (m3)

x coordinate (m) or molar fraction

y coordinate (m)

Greek symbols

α volume fraction

α, β expansion coefficients

Г mass transfer term (kg/s)

δ film thickness (m)

η dimensionless distance

η normal unit vector

θ dimensionless temperature

λ mean free path (m)

µ dynamic viscosity (kg/m/s)

ν kinematic viscosity (m2/s)

ξ constant in equation (2.81) (m3/4)

ρ density (kg/m3)

σ surface tension (N/m)

τ, τ shear stress and shear stress tensor (N/m2)

τ tangential unit vector

ω mass fraction

Ω dimensionless mass fraction

Subscripts

A noncondensable mixture component

B noncondensable mixture component

crit critical

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C condensable mixture component

He helium

L liquid

M mixture

w at wall

V water vapour

δ at the interface

τ tangential

Superscripts

∞ far-field location

sat saturation condition

out outlet location Other ∇ gradient vector

• scalar product

⊗ dyadic product

Chapter 1 Introduction

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Chapter 1 Introduction 1.1 Condensation process and its applications Condensation is a thermodynamic, egzoenergetic nonequilibrium process of phase transition from the gaseous into the liquid phase. Because of this nature its industrial applications have been investigated for many years. There exists a variety of heat exchanger designs whose function is based on this phenomenon. The most common one, present almost in every home, is the condenser in a refrigerator. Among its industrial applications, condensation is very often concerned in the energy sector. In conventional (coal) power plants, condensation of flue gases and spontaneous condensation of steam in the turbine condensers can be a source of very serious problems. In nuclear power plants condensation has important safety implications. For example during loss of coolant accident (LOCA) in the primary system of a water-cooled nuclear reactor large amount of water vapour can be released into the reactor containment. As a result integrity of the containment can be seriously threatened (SOAR report, 1999). There are several aspects which have to be taken into account during such events. One of these is the possibility of high over-pressurization of the reactor containment. In such condition it is very important to condense released steam as quickly as possible. There exist several types of systems which role is to provide fast condensation of water vapour. Containment spray systems and passive containment cooling systems (PCCS) can be considered as the examples. In both cases, efficiency of these systems depends not only on the type of condensation

Krzysztof Karkoszka – Mechanistic Modelling of Water Vapour Condensation…

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(from a thermodynamic point of view) but also on the composition of the gaseous phase, which usually contains a significant amount of noncondensable gas (Paladino 2004). Released water vapour during loss of coolant accident creates together with air or nitrogen (which are present for example in PWR reactor containments during normal operation conditions) mixture of gases. Additionally, if the reactor core during such event becomes uncovered, steam can react with very hot fuel pins cladding. Due to this reaction, a significant amount of hydrogen can be created. Another example where condensation from the mixture with noncondensable gases must be investigated is the hydrogen and oxygen accumulation in the nonvented parts of the boiling water reactor (BWR) pipeline system. Due to the radiolysis of water molecules in the reactor core, some amount of hydrogen and oxygen is always present in the coolant. It has been found that these two components can accumulate in the nonvented geometries due to the water vapour condensation on the imperfectly insulated internal structures. This is a long-term process, however recent investigations show (Stevanovic et al., 2005), that integrity of the pipeline system can be seriously threatened, because after many months, or even years, explosive mixture can be created. Even small amounts of noncondesable gases can strongly deteriorate the condensation heat and mass transfer process. Heavy, noncondensable gases tend to accumulate near the liquid-gas interface due to its impermeability to these components. Such behaviour creates an additional resistance to the mass transfer, because the condensable gas has to diffuse through the mixture gaseous layer. As results from nuclear safety analyses, the distribution of mixtures of light and heavy gases is also strongly coupled with the condensation process. As it is shown in this thesis the accumulation of noncondensable gases near the cold structures is driven by the interface impermeability for noncondensable components, as well as by the multicomponent diffusion and buoyancy effects. In such conditions, all these driving forces are strong functions of the local mixture composition. The condensation in presence of noncondensable gases is a very complex phenomenon. Justification which effects are important is rather difficult. However, as it has been deduced from the literature review presented in the next chapter, three different aspects can be considered because of theirs direct impact on the condensation heat and mass transfer process and applicability to the nuclear power safety analysis. These aspects are namely: the filmwise free and forced convection condensation from binary and ternary mixtures, the interaction between liquid film structure and diffusion mixture layer and the spontaneous condensation. All of these effects can play an important role during water vapour release into the reactor containment. They can also influence the performance of the passive containment cooling systems and the accumulation of explosive mixtures in the nonvented elements of BWR pipeline systems.


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1.2 Review of research approaches The literature review indicates that research on aspects related to the water vapour condensation in presence of noncondensable gases follows two main paths. These paths are, however, strongly coupled together. The first type of research is directly linked with industrial applications. Thus, the developed models are as simple as possible and easy to implement into engineering calculations. However they need to include enough information about the investigated phenomenon and especially about limitations of their applicability. Such models are usually based on the experimental investigations and are given in forms of correlations. The most important parameter, from the heat transfer point of view, is the heat transfer coefficient. This coefficient is usually obtained by the heat flux measurements and the knowledge of boundary conditions. Heat transfer coefficient usually determines further design of the given, engineering device. An extension of this approach is based on the inclusion of more fundamental knowledge, based on the physics, into the correlation development process. Application of the heat and mass transfer analogy can be given as an example. Further generalization of the research process is directed to the more fundamental studies. Besides of its industrial importance, this research also contributes to the fundamental knowledge about thermodynamics and multi-component diffusion effects in gaseous mixtures. There are two concepts which lie behind this approach. The first one is called the diffusion layer model (DLM) with its variation, which sometimes is called the thermal resistance model (TRM). The second approach is based on the solution of the fully coupled, liquid and mixture, conservation equations. These equations can also be simplified by the order of magnitude analysis and given in forms of so called boundary layer equations. With this approximation, different solution methods can be applied and even an analytical solution can be obtained. Application of numerical methods allows to solve two or even three-dimensional, fully coupled system of conservation equations for both liquid and gaseous phases. Both the boundary layer approximation and numerical methods permit to construct mechanistic models of the condensation phenomenon. 1.3 Review of modelling approaches 1.3.1 Empirical correlations Experimental data obtained from measurements are usually correlated and presented in forms of the average heat transfer coefficients or Nusslet numbers. These relations are usually functions of such parameters as the Reynolds number,


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the Prandtl number, the noncondensable mass fraction, etc. In the nuclear reactor containment safety analysis, where condensation from the mixture with multiple noncondensable gases must be taken into consideration, the most widely used correlations have been developed by Uchida, Tagami, Kataoka and Dehbi (Herranz et al., 1998). Unlike other correlations, the Dehbi’s correlation does not depend only on the noncondensable gas mass fraction, but also takes the effect of pressure into account. All these correlations are given for air-water vapour mixtures and are developed for reactor containment conditions during postulated sever accidents. Dehbi et al. (1991) performed a set of experiments where also the ternary mixture of water vapour, air and helium has been considered. The correlation for the average heat transfer coefficient has been given as a function of both noncondensable gases mass fractions and pressure of the system. Helium has been used as the simulant of hydrogen. Equivalence of helium and hydrogen in the process has been recently analysed by Peterson (2000), where based on the DLM, physical similarities between these two gases have been theoretically verified. 1.3.2 Analogy between heat and mass transfer The heat and mass transfer analogy is based on the similarity between heat and mass fraction transport equations. Two important dimensionless relations are characterized, namely the Nusselt number and the Sherwood number. Based on the analogy between transport processes, relation between them is given. Basically, knowledge of one dimensionless number allows determination of the other one. If also similarity between heat, mass fraction and momentum equation is taken into consideration, then the whole analysis can be based on so called Chilton-Colburn analogy (Bird et al., 2002). 1.3.3 Diffusion layer model The diffusion layer model is an extension of the heat and mass transfer analogy. In this approach a more physical description of the mass transfer process is considered, based on the solution of mass fraction transport equation at the liquid-gas interface. Another variation of this model is the thermal resistance model - TRM. This methodology is based on the similarity between heat transfer resistance network and mass transfer in the gaseous boundary layers (Herranz et al. 1998). 1.3.4 Boundary layer approximation and fully mechanistic model The next level of generalization is based on the solution of the system of conservation equations, for both liquid film and gaseous mixture phases. These


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equations describe conservation of mass, momentum, energy and additional species. Together with adequate balances at the liquid-gas interface, boundary conditions and equations of state, in principle, they can be solved in general applying different numerical methods. General conservation equations together with interface balances can also be simplified. Applying the order of magnitude analysis, they can be written in the form of, so called, boundary layer equations. Together with simplified interface conditions, such system of equations can be solved analytically, for example with the similarity method. This approach is very general and doesn’t need to rely on any empirical correlations. It can also be applied to different types of fluids, until they are considered to satisfy continuity requirements. Despite of its generality, this analysis is limited to simple geometries and definitions of applied similarity variables. Moreover, it is difficult to analyze complicated processes such as diffusion in the ternary mixture of gases, where significant amount of the light gas has been introduced. In such case strong buoyancy and multicomponent diffusion effects create complicated velocity fields. However this method can be applied to the binary mixtures of heavy gases and ternary mixtures with a small amount of light gas. Such analysis, as it is shown in the present thesis, can give interesting results and reasonable estimates of the fundamental, governing processes. Recent fast development of numerical methods and tools gives even more general capabilities of detailed investigation of governing process, which play important role during condensation from the multicomponent mixture of gases. In principle the fully coupled, unsteady, three dimensional system of conservation equations with adequate interface and boundary conditions can be solved, for example with the computational fluid dynamics (CFD) methods. These methods are based on discretization of conservation equations in time and space with finite difference, finite element or finite volume approximations. However, because of significantly long computational time, very often only steady-state and two-dimensional calculations are considered. Despite of these simplifications, complicated solutions can be obtained and even such effects as the ternary mass diffusion, interactions between wavy film and diffusion boundary layer as well as spontaneous condensation effects, can be investigated. With some experience, results can be properly analysed, understood and fundamental conclusions about physical transport phenomena can be drawn. Summary The above introduction to the applications and modelling approaches of the condensation phenomenon from the mixture of condensable and noncondensable gases gives a general overview of the industrial problems and research methods. Empirical correlations and solutions of the boundary layer equations for the binary


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mixture of gases have been applied for almost fifty years. However, recently developed more advanced techniques and tools, like for example CFD methods, give real possibility of detailed investigation of physical processes which govern the condensation phenomenon. The next chapter contains a literature review. It shows chronologically how research concepts of the condensation phenomenon modelling have been growing in time and what is the current state-of-the-art in this subject.

Chapter 2 Literature review

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Chapter 2

Literature review Introduction In this chapter a literature study of the condensation phenomenon from the multicomponent mixture of condensable and noncondensable gases has been presented. To start with, a general, chronological literature overview of the theoretical and experimental investigations is given. In the following, research on the specific aspects, such as the influence of the wavy flow hydrodynamics on the condensation heat and mass transfer, is discussed. This chapter also includes the literature review dealing with the spontaneous condensation from a binary mixture of gases. Finally, general summary and conclusions are presented. 2.1 Empirical and theoretical investigations Fundamentals of pure water vapour condensation as well as the basics of heat and mass transfer can be found for example in Chapman (1984) and Incropera and Dewitt (1996). The standard handbooks give an introduction to the field and present the Nusselt theory, which was the first attempt to study the laminar film-wise condensation. Basic discussion of the influence of noncondensable gases on the heat transfer is given in such books as Whalley (1996), Bird et al. (2002) and Baehr and Stephan (2006). One of the first articles, connected to the present subject, has been authored by Koh et al. (1961). It presents a solution of the gravity driven pure water vapour condensation boundary layer equations with the similarity method. Both liquid and

Krzysztof Karkoszka - Mechanistic Modelling of Water Vapour Condensation…

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gaseous phases are coupled through interface mass and momentum balances. In order to find an analytical solution to this problem, the so called similarity variables have been applied. These variables permit simplification of the system of partial differential equations to the system of ordinary differential ones. Minkowycz and Sparrow (1966) present solution of the gravity driven boundary layer equations for condensation from a binary mixture of gases. Again similarity variables have been applied, in order to solve the coupled system of equations for both, liquid and gaseous phases. Degradation of the heat transfer process due to the presence of noncondesable gas has been noticed and discussed. Sparrow et al. (1967) investigate forced-convection condensation on a horizontal surface from a binary mixture with noncondensable gas. Boundary layer equations are solved with the similarity method. Results show that heat transfer degradation due to the presence of noncondensable gas is much lower when the forced convection condensation is considered. This has been found to be in contrast to the gravity driven free convection condensation where reduction of the heat transfer rate in such situation is significant. Minkowycz and Sparrow (1969) show theoretical studies on the mixture superheating effect. It has been found that mixture superheating is more important for the gravity driven free convection condensation problems. Taitel and Tamir (1969) study direct contact condensation on the falling liquid sheet of water. The binary mixture of water vapour and air has been taken into consideration and theoretical model has been built based on the similarity method and integral solutions of the boundary layer equations. It has been noticed that the water sheet-mixture interface temperature decreases with the increase of the amount of the noncondensable gas, especially in the near leading edge region. Felicione and Seban (1973) solve boundary layer equations for the binary mixture of gases with the integral approach. Both the similarity solution and the integral method are found to give very similar results. Very important, experimental studies on the condensation phenomenon from the binary mixture of gases are performed by Al-Dwany and Rose (1973). Laminar free convection condensation from the steam-air mixture is investigated. Special attention is paid to preserve free convection conditions, because it has been found that the forced convection significantly increases the value of the heat transfer coefficient. Good agreement of experimental results with boundary layer solutions has been also reported. Sage and Erstin (1976) present similarity solution for the condensation from a ternary mixture of alcohol vapours. A limitation of these studies results from the fact that diffusivity coefficients are defined as constant values and the influence of noncondensable gas is not taken into consideration. However, overview of the ternary diffusion problem has been presented.


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Jackson (1977) theoretically investigates diffusion of ternary mixture components. In order to calculated ternary diffusivity coefficients, Stefan-Maxwell equation has been employed. Prosperetti (1979) studies in detail boundary conditions at the liquid-gas interface. A generic equation, which takes into account the interface balances of mass, momentum, energy and any number of additional species, has been presented with a clear, mathematical formalism. Garimella and Christensen (1990) build an experimental setup for investigations of the steam-air mixture behaviour in a small enclosure. It has been found that the heat transfer coefficient significantly depends on the initial amount of noncondensable gas only when this amount is small. For large initial values of noncondensable gas this dependence has been found to be less important. Chan and Yuen (1990) deduce the heat transfer coefficient from the measurements of the direct contact condensation from the air-steam mixtures on the horizontally stratified liquid water sheet. It has been noticed that for high condensation rates, the presence of noncondensable gas significantly reduces the heat transfer coefficient. Dehbi et al. (1991a, 1991b) present experimental investigations of the free convection condensation from steam-air and steam-air-helium mixtures. Based on the boundary layer approximation, correlations for the condensation heat transfer coefficients have been developed. These correlations are functions of the noncondensable mass (or molar) fractions and pressure value. While conducting the experiments, stratification and separation effects have been noticed and deliberately avoided by introduction of high amount of heavier gas. Siddique et al. (1993) investigate steam-air or steam-helium mixtures during condensation inside a vertical pipe. Influence on the heat transfer coefficient has been found to be dependent on the ratio between air-to-helium mass fractions. As the main conclusion it has been found that if the ratio between air and helium mass fractions is equal to unity, helium has the most dominant influence on the heat transfer degradation. However if molar fractions of these gases are equal to one, the dominance relies on the air side. Hassan and Banerjee (1996) show implementation of the six-equation model for condensation from a binary mixture of gases based on the heat and mass transfer analogy. It has been found that the heat and mass transfer analogy approach predicts value of the average condensation heat transfer coefficient more precisely than previously used correlations. Yao et al. (1996) discuss implementation of the transient, one-dimensional model based on the Couette flow, in the RELAP5/MODE3 code. One-dimensional transient system of equations has been closed by the correlations developed for prediction of the condensation heat transfer coefficients as well as by the heat and mass transfer analogy. Peterson (1996), based on the diffusion layer model, shows that commonly used Uchida correlation


10

can produce significant errors. Future, mechanistic modelling has been recommended. George and Singh (1996) implement a nine-equation model based on the Uchida and Gido-Koestel correlations into the GOTHIC code. This code is specially developed for aerosol behaviour investigations in the reactor containment geometry. From the numerical results, based on the average values of the heat transfer coefficients for binary mixtures, has been concluded that the Gido-Koestel correlation, which is based on the mechanistic approach, works better than the empirical Uchida correlation. Kuhn et al. (1997) perform number of experiments with a vertical tube. Pure steam, steam-air and steam-air-helium mixtures are considered. Based on the Couette flow model a more mechanistic correlation has been presented in form of the degradation factor. Such degradation factor has been split and written as a product of the steam-air and steam-helium factor. Both elements introduce degradation of the heat transfer rate due to the additional presence of particular gases. Karl and Weiss (1997) show an experimental approach for determination of the condensation heat transfer coefficients during condensation from binary mixtures, from the knowledge of nonconensable mass fraction profiles. This useful method can also be applied in order to determine temperature profile inside the liquid layer. Dehbi and Guentay (1997) show a numerical investigation of the condensation heat transfer to the secondary systems (for example coolants) based on the heat and mass transfer analogy. Such approach allows estimation of the condensation heat and mass transfer rates without knowledge of the condensation surface temperature, what normally is required by commonly used correlations. Also importance of the secondary system influence on the condensation heat transfer rate has been discussed. Anderson et al. (1998) investigate experimentally behaviour of the steam-air-helium mixture in the scaled AP600 containment geometry. During performance of this experiment, rolling waves and film structures on the vertical surface have been observed. It has been concluded that for a high amount of very light gas (like hydrogen or helium) stratification and separation of the mixture components can also occur. Such process can lead to the local accumulation of the light gas. Future, mechanistic modelling has been recommended. Chin et al. (1998) apply the finite volume numerical method in order to investigate the condensation heat transfer in the binary mixture of gases on inclined surfaces. Relation between an inclination angle and the heat transfer degradation factor has been discussed. Chen et al. (1998) employ the similarity method for investigation of the influence of noncondensable gas in the binary mixture on the condensation heat transfer rate to a vertical fin. Again it has been found that even a small amount of noncondensable gas significantly reduces the heat transfer rate between gaseous mixture and the fin surface. Herranz et al. (1998) present an improved diffusion layer model which is based on the analogy between heat and mass transfer for condensation from air-steam and air-steam-helium mixtures. Special application of this model addresses


11

the AP600 nuclear reactor containments and can be applied for prediction of the average heat transfer coefficient in this particular geometry. Park et al. (1999) investigate experimentally the influence of noncondensable gas in the Passive Containment Condenser (PCC) geometry. Experiments with single, vertical pipe have been performed and results correlated with the degradation factor approach. The degradation factor in this case is just a ratio between the total, experimental heat transfer coefficient and the film heat transfer coefficient deduced from the Nusselt theory. The final correlation has been given as the function of the Jakob number, the film Reynolds number and the air mass fraction. Dehbi et al. (1999) study experimentally mixtures of air, steam and helium, again for the PCC geometry. For this purpose, experimental facility with a vertical pipe has been built. Based on the experimental data and the boundary layer approximation, empirical results have been correlated. Srzic et al. (1999) investigate separation effects during condensation from a binary mixture of water vapour with a very light gas. Boundary layer equations are solved with the finite volume method. It has been found that light gases tend to separate. However the main conclusion is that light gases reduce condensation heat and mass transfer rates more significantly than heavy species. Cobo et al. (1999) present development of the simple model of condensation from the binary mixtures of gases that can be implemented into engineering codes. Condensation inside and outside of a vertical pipe has been investigated and adequate correlations have been written based on both, the degradation factor and the integral methods. Liu et al. (2000) perform set of experiments with condensation of steam-air-helium mixtures in a vertical tube. It has been noticed that the most notable correlations for containment applications are those by Uchida (1965), Gido and Koestel (1983) and Dehbi (1991). However the most important mechanistic approaches rely on the diffusion layer model by Herranz et al. (1995) and Peterson (1993). All these models, as a result, give average heat transfer coefficients. Experimental results obtained during these studies have been correlated and deduced heat transfer coefficients have been found to be in a good agreement with the diffusion layer model. Such conclusion can be treated as a proof that further modelling should be based on the mechanistic analysis. Peterson (2000) performs fundamental research on the diffusion mass transport in the multicomponent mixtures of gases. Extension and application of the diffusion layer model, based on the heat and mass transfer analogy, for multicomponent mixtures, has been discussed. This approach is based on the so called effective diffusivity coefficient. It has been noted that this coefficient for mixtures of light and heavy gases is mainly governed by the diffusivity of the heavy component. No and Park (2002) investigate condensation of water vapour from the binary mixture with air. Developed model has been based on the heat and mass transfer analogy and takes into account such effects as the entrance length and the waviness of the liquid film through empirical correlations. Siow et al. (2002) show numerical


12

investigations of the condensation from binary mixtures of steam-air and refrigerant R134a-air, on a horizontal surface. The fully coupled numerical approach has been applied to the film and mixture boundary layer equations. Also main differences between these two mixtures have been reported. Simultaneously, Choi et al. (2002) present experimental data from the investigation of direct contact condensation in a horizontally stratified flow with and without noncondensable gas. It has been found that existing models in the RELAP5 code significantly over-predict experimental results when the presence of noncondensable gas is taken into consideration. Suggestions how to improve these models have been discussed. Shepel (2003) implements the Gido-Koestel correlation into a commercial CFD code. Results have been compared with TOSQAN experimental data. TOSQAN is a small-scale containment facility where condensation from binary and ternary mixtures of gases can be studied. Obtained numerical results have been found to be in good agreement for prediction of the total pressure transients during injections of the air-helium-water vapour mixtures. Maheshwari et al. (2004) study theoretically the heat transfer coefficient for a wide range of the film and gas Reynolds numbers in a vertical pipe. The analogy between heat and mass transfer has been applied in order to compute the condensation mass flux. It has been found that for low Reynolds numbers the mixture-side resistance to the heat transfer plays a dominant role. However, this situation is opposite for high Reynolds numbers, where the main resistance to the heat transfer processes relies on the liquid side. Slow et al. (2004) present numerical investigations of the heat transfer process during condensation from the binary mixture with air flowing between two, parallel vertical surfaces. It has been found that the heat transfer process and the liquid film thickness are strongly dependent on the inlet amount of noncondensable gas. Martin et al. (2005) investigate film condensation models for a large reactor containment geometry, implemented in a CFD code. These models rely on the degradation factor approach, available empirical correlations, the Hilton-Colburn heat-mass transfer analogy and the diffusion layer model. It has been concluded that correlations do not work properly behind their ranges and the heat-mass transfer analogy doesn’t work well for condensation on horizontal surfaces. Recently Stevanovic et al. (2005) develop a simplified mechanistic CFD model for the prediction of condensation from a ternary mixture of steam with hydrogen and oxygen in a closed at the top vertical pipe, where gaseous mixture has been introduced from the bottom side. Qualitative results are presented and transient mixture components propagation has been observed. However, physical relations between mixture components have not been discussed. Anyway, these investigations showed that fundamentals of the ternary mixture diffusion can be deeper understood through a fully mechanistic modelling. Stephan (2006) studies a forced-convection condensation from a binary mixture of condensable and


13

noncondensable gases. Based on the solution of boundary layer equations, a new correlation for the heat transfer coefficient has been proposed. Oh and Revankar (2006) investigate the condensation of water vapour from a binary mixture with air inside a vertical pipe. Relying on the heat and mass transfer analogy, an adequate correlation for the heat transfer coefficient has been developed. However it has been found that such correlation underestimates the value of heat transfer coefficient in the wavy region. Further studies of the wavy film flow region have been recommended. In the meantime Stevanovic et al. (2006) numerically investigate condensation of water vapour from the mixture with hydrogen and oxygen in complex pipeline geometry. Qualitative results of the noncondensable gas distribution have been shown and discussed. The main conclusion is that a mechanistic CFD approach can be used for future investigations of such fundamental effects like the ternary mixture diffusion and the stability of the thin liquid films. Siow et al. (2007) apply the finite volume method in order to solve the fully coupled system of boundary layer equations for condensation from binary mixtures of gases in an inclined, rectangular channel. It has been found that the most important parameter, affecting condensation heat transfer process, is the film Reynolds number. 2.2 Thin film instability As ones of the first, Banerjee et al. (1967) theoretically investigate influence of the wave hydrodynamics on the mass transport phenomena. It has been pointed out that increase of the interfacial area cannot explain the fact that mass transfer is enhanced by the presence of wavy structures. Sofrata et al. (1980) theoretically study film-wise condensation with wave initiation by extension of the basic Kapitza analysis. A new dimensional number, which has been proven to be a control parameter, based on the relation between Weber and Reynolds numbers, has been introduced. Alekseenko et al. (1985) derive, based on the integral method, one-dimensional unsteady wave equation. Maron et al. (1989) study thin liquid film flow patterns. They employ both the finite element and the finite difference methods in order to find solution to the momentum conservation and interface momentum balance equations. Relations between wave velocity, amplitude and internal hydrodynamics have been investigated. Patnail and Perez-Banco (1995) solve two-dimensional mass, momentum, energy and species conservation equations and show that the mass transfer enhancement is strongly dependent on the internal wave hydrodynamics. In the same year


14

Karapantsios and Karabelas (1995) study direct-contact condensation of water vapour-air mixtures on the periodically fed liquid films, falling on a vertical surface. It has been evidently observed that creation of the surface waves increases condensation heat and mass transfer rates. Karapantsios et al. (1995) measure local condensation rates during the direct-contact condensation of water vapour from a mixture with air. Influence of the film Reynolds number has been discussed. It has been noticed that increase of the film Reynolds number increases the condensation heat transfer coefficient. The opposite behaviour has been found with an increase of amount of the noncondensable gas. Further studies on the influence of liquid film structures on the diffusion layer region have been recommended. Park et al. (1996) investigate influence of the film Reynolds number and film surface structures on the heat transfer process during condensation of water vapour from a mixture with air. Vertical, rectangular channel is taken into consideration. It has been found that interaction between surface waves has very important influence on the mass transfer in the near-interface region. This interaction plays significant role for low film and mixture Reynolds numbers. It has been also speculated that the heat transfer process in the vicinity of wavy liquid film is enhanced due to the mixing phenomenon in the diffusion region. Yoshimura et al. (1996) study experimentally mass and heat transfer enhancements due to the presence of two-dimensional waves. Absorption of the additional gas into the liquid film has been investigated. The evidence of existence of the internal hydrodynamic structures inside the solitary wave has been presented. Additionally, Nosoko et al. (1996) investigate complex wave dynamics and waves coalescence. Park et al. (1997) study experimentally the influence of wavy interface on the water vapour-air condensation on the vertical surface. Based on the analogy between heat and mass transfer, a correlation for the heat transfer coefficient has been developed. It has also been noticed that the heat transfer process depends on both film and vapour Reynolds numbers. Enhancement of the heat transfer coefficient increases with increase of the film Reynolds number and decreases with increase of the mixture velocity. Jayanti and Hewitt (1997a) calculate with a commercial CFD code hydrodynamics and heat transfer processes in the wavy liquid films in the prescribed, wavy geometries. Obtained heat transfer coefficients and influence of the internal film hydrodynamics have been discussed. However, presence of the gaseous phase has not been considered during these studies. In another study, Jayanti and Hewitt (1997b) investigate wave hydrodynamics and the heat transfer applying again a CFD approach. Using the low Reynolds number k-ε turbulence model, they show that flow between waves can be considered to be laminar, however, inside wavy structures, effects of turbulence can appear. Karimi and Kawaji (1998) prove by the experimental observation of wavy falling liquid films that the simple Nusselt theory is not adequate for such problems, because empirically obtained velocity profiles highly deviate from the Nusslet description. Yang and Jou (1998) investigates theoretically wavy film influence on the heat and mass transfer process during condensation from the binary mixture of


15

air and water vapour. Correlations for the Nusselt and Sherwood numbers have been derived. Again, it has been noticed that wavy film structures increase condensation heat and mass transfer rates and presence of a noncondensable gas degrades this process. Miyara (2001) investigates with numerical methods the heat and mass transfer enhancement during condensation of pure water vapour due to the presence of two-dimensional waves. It has been concluded that main mechanisms that are responsible for this effect are film thinning and convection effects inside the waves. Kit et al. (2001) employ the integral method in order to solve two-dimensional Navier-Stokes equations with the interface momentum balance. Wavy dynamics inside the vertical tube is taken into consideration and some results, like for example two-dimensional velocity fields, have been presented. Gao et al. (2003) study numerically two-dimensional instabilities in free-falling liquid films. The direct numerical simulation of Navier-Stokes equations has been performed and the volume of fluid method has been employed for the liquid surface reconstruction. It has been found that low frequency inlet perturbations tend to create solitary-shaped wavy structures with circulation zones inside. However high frequency inlet disturbances develop low amplitude sinusoidal waves. It has been also noticed that these instabilities are dependent on the gravity, viscosity and surface tension effects. All these effects are included in the nondimensional relation, called the Kapitza number, which shows which effects dominate for given flow conditions. Park and Nosoko (2003) show experimental observations of transition from two-dimensional, regular waves to three-dimensional structures. It has been found that these structures significantly influence transport phenomena processes and further numerical investigations have been recommended. Drosos et al. (2004) investigate experimentally flow dynamics of the falling liquid films with high Kapitza numbers (such liquids as water, butanol, etc). It has been shown that for these fluids, flowing with the film Reynolds number over 200, the wave structure is only a weak function of the film Reynolds number. Kunugi and Kino (2005) investigate theoretically two and tree-dimensional waves. By performing the direct numerical simulation of Navier-Stokes equations, it has been found that there exist circulation zones between capillary and solitary waves, which significantly increase heat and mass transfer transport processes. Very similar structures to those investigated experimentally by Park and Nosoko (2003) have been obtained. Alekseenko et al. (2005) perform number of experimental observations in order to study in details wave hydrodynamics of falling liquid films with low Reynolds numbers. Aktreshev and Alekseenko (2005) deduce, from the linear stability theory, that stability of the free-falling liquid films in a long wave film region, during condensation of pure vapours, is improved by the mass transfer condensation effects.


16

2.3 Spontaneous condensation Young (1991, 1993) investigates theoretically, based on the irreversible thermodynamics analysis, process of spontaneous condensation and evaporation of the small liquid droplets in the supersaturated pure water vapour and mixture of water vapour with noncondensable gas. It has been shown how to calculate evolution of droplets diameters in time and what the most important parameters for this process are. Fox et al. (1997) perform qualitative studies on the fog formation in water vapour-air mixtures. Condensation on the walls of vertical enclosure has been taken into consideration and the whole process of fog formation has been investigated based on the kinetic theory of gases. Kaufmann and Hilfiker (1999) investigate theoretically a fog formation in mixtures with inert gases. The importance of heterogeneous condensation has been shown and methods of its prevention discussed. Kang and Kim (1999) investigate numerically and empirically the film water vapour condensation from the mixture with air. They found that for small temperature differences between the cooling surface and gaseous phase, mixture can be treated as a superheated gas. However for large temperature differences fog formation should not be neglected. Karl and Hein (1999) study effects of spontaneous condensation from binary mixtures of water vapour with air or nitrogen. Importance of the influence of fog formation on the heat and mass transfer rates has been discussed. It has been noticed that when intensive formation of dispersed phase is observed, the convective heat and mass transfer to the cooling surfaces is significantly reduced. In such conditions the heat transport process is governed mainly by the conduction through the noncondensable gas layer. Manthey and Schaber (2000) show experimentally and numerically that local effects during the spontaneous condensation of humid air cannot be neglected for the highly supersaturated mixtures. Karl (2000) performs experimental investigations on the spontaneous condensation in gaseous boundary layers. It has been found that the most important parameter is the interface temperature. It has also been noticed that effect of the additional, noncondensable gas cannot be neglected in the future investigations. Summary Literature studies on the problem of condensation from multicomponent mixtures of gases have been presented in this chapter. It can be noticed that most of research has been directed to the condensation of water vapour from the binary mixture with heavy, noncondensable gas. The main results are usually shown in the form of correlations for an average heat transfer coefficient.


17

Different approaches have been also employed in the literature: pure empirical correlations, the analogy between heat and mass transfer, the diffusion layer model (based on the Couette flow) and finally the fully mechanistic approach with mass, momentum and energy balances at the liquid-gas interface. Unfortunately, only in few cases, condensation from the ternary mixture has been considered. Also the mechanistic modelling has been applied very rarely in the citied investigations. However, many researchers are aware of the fact that only the fully mechanistic modelling can lead to the deeper understanding of fundamental relations governing the multicomponent mixture distribution and condensation heat and mass transfer processes. In the above literature review, the importance of such effects as wavy liquid film interaction with the gaseous boundary layer and spontaneous condensation appearance in the supersaturated mixtures has been discussed. All these effects, together with the surface condensation are expected to exist during severe accident- conditions mentioned in the introduction. Because of such formulation of the problem, in the next chapters the above effects have been investigated in details using the mechanistic approach and applying different solution methods.

Chapter 3 Physical models and assumptions

19

Chapter 3

Physical models and assumptions Introduction This chapter contains a description of physical models, together with most important assumptions and simplification applied in the present thesis. It begins with a physical description of the free and forced convection condensation on a vertical wall in presence of a multicomponent mixture of noncondensable gases. In the next section, the direct-contact condensation from a binary mixture of gases on a horizontally stratified liquid film has been presented. Further, a physical model of wavy film flow and its interaction with the gaseous boundary layer has been described. Chapter ends with a description of a model of spontaneous condensation from a binary mixture with noncondensable gas. The physical models have been graphically illustrated and important assumptions and simplifications have been clearly listed for each of them. Based on the literature review presented in Chapter 2, three aspects of the condensation process, which should be considered in the nuclear safety analysis, have been chosen as the object of present studies. These aspects can be classified as follows:

free and forced convection condensation on vertical and horizontal surfaces,

influence of liquid film structure on the condensation from multicomponent mixture of gases,

Krzysztof Karkoszkak - Mechanistic Modelling of Water Vapour Condensation…

20

influence of spontaneous condensation on the heat and mass transfer process in the supersaturated mixtures with noncondensable gas.

Additionally, two variants of the forced convection condensation on the horizontal surface have been distinguished:

condensation on the isothermal surface, condensation on the liquid film sheet flowing along adiabatic surface.

Further, the similarity method applied in this thesis uses the liquid film thickness as a boundary condition instead of the bulk mixture temperature. Main reason for this is the much shorter computational time. The model of condensation on the horizontal, isothermal surface has been used in this context in order to show that this concept is equivalent to the specification of the bulk mixture temperature as the boundary condition. It has also been compared with previous, theoretical investigations found in the literature (Chapter 5 and Appendix 2). 3.1 Free and forced convection condensation on vertical surfaces The first model described in this chapter is dealing with the film-wise condensation from a multi-component mixture of gases on a vertical, smooth surface. With regard to flow conditions, two situations have been considered: forced convection condensation and free convection, gravity driven condensation. 3.1.1 Modelling of forced convection condensation on a vertical surface The schematic of the model has been shown in Figure 3.1 (wavy region is not considered). Since a number of assumptions and simplification has been applied, they have been listed separately for the liquid film, the gaseous phase and geometry. Additionally, a method of solution has been pointed out for each model in order to clarify further analysis of the solution methodology performed in Chapter 5.


21

Figure 3.1 Physical model of free, gravity driven and forced convection film-wise condensation in presence of noncondensable gases on a vertical surface. Assumptions and simplifications with regard to the liquid film

flow of the liquid phase is laminar, interface velocity is set to the wall velocity and calculated from the Nusselt

model, there is no slip between liquid and gaseous phases, film temperature is equal to the wall temperature, heat of condensation is totally transferred through the liquid film, film interface is impermeable to noncondensable component, liquid film properties are assumed to be constant.

Assumptions and simplifications with regard to the gaseous phase

binary mixture of water vapour and air, mixture enters the physical domain with a fully developed velocity profile, mixture is in the saturation state at the liquid-gas interface,

1NBA =+++ ∞∞∞ ϖϖϖ K

Saturated mixture of condensable and noncondensable gases

x, u

y, v

Free convection

∞Mv

∞Mu

∞MT

Temperature profile

δT WT

∞Vϖ

δVϖ

g

δ

Forced convection

Water vapour mass fraction profile


22

flow of the mixture can be laminar or turbulent, only water vapour diffusion process drives condensation heat and mass

transfer, there is no spontaneous creation of the water droplets in the gaseous phase, bulk amount of noncondensable gas and bulk mixture temperature are

known and uniform, mixture components are in dynamic and thermal equilibrium.

Assumptions and simplifications with regard to geometry

vertical, smooth, isothermal wall.

Solution method

two-dimensional steady-state CFD computations with implemented

mechanistic condensation model, which is based on the resolution of the near-wall boundary layer.

3.1.2 Modelling of free convection, gravity driven condensation on a vertical surface Schematic of the model has been shown in Figure 3.1. Again a number of assumptions and simplification has been applied and those which have direct influence on the description of this model have been presented below. Assumptions and simplifications with regard to the liquid film

film flow is assumed to be laminar, film flow is driven by the gravity forces, film thickness is considered to be proportional to 4/1x based on the order

of magnitude analysis, for both the boundary layer solution and the CFD calculations,

there is no slip between the liquid film and the gaseous phases, heat of condensation is totally transferred through the liquid film, film interface is impermeable to noncondensable components, liquid film properties are assumed to be constant.

Assumptions and simplifications with regard to the gaseous phase

binary or ternary mixture of water vapour, air and helium, mixture is in the saturation state at the liquid-gas interface, flow of the mixture is laminar, mixture flow is driven by the interfacial shear, buoyancy forces and

condensation effects,


23

bulk amount of noncondensable gases and bulk mixture temperature are known and are uniform.

with the boundary layer approximation, ternary diffusivity coefficients have been calculated with respect to the bulk mixture composition (see chapter 4),

with CFD calcualtions, ternary diffusivity coefficients have been calculated as local functions of the mixture composition applying the Maxwell-Stefan equation,

mixture components are in dynamic and thermal equilibrium. Assumptions and simplifications with regard to geometry

smooth, vertical wall, additional, horizontal surface bounding the vertical wall from the top is

considered in CFD calculations in order to investigate the effect of geometry on the noncondensable gas accumulation.

Solution method

similarity solution of the fully coupled boundary layer equations for both

liquid and gaseous phases, two-dimensional, CFD, pseudo-transient calculations with implemented

mechanistic condensation model. 3.2 Forced convection condensation on horizontal surfaces 3.2.1 Forced convection condensation on a horizontal, isothermal surface Detailed description of the physical and mathematical models of forced convection condensation on a horizontal, isothermal surface can be found in Appendix 2. This model generally has been applied in order to prove the assumption that specification of the liquid film thickness (see Chapter 5) is equivalent to the specification of the wall temperature as a boundary condition. Obtained results have been found to be in very good agreement with the analysis performed by Sparrow et al. (1967). 3.2.2 Direct-contact condensation on a horizontal, adiabatic surface The model of direct-contact condensation on a horizontally flowing, subcooled liquid film sheet from the binary mixture of gases, has been drawn schematically in


24

Figure 3.2. Because of the given experimental database by Choi et al. (2002) this model has been mainly applied for empirical validation of the implemented mechanistic approach. As in the previous section, the most important assumptions and simplifications have been listed below.

Figure 3.2 Physical model of forced convection condensation in presence of noncondensable gases on a horizontal, adiabatic surface. Assumptions and simplification with regard to the liquid film

entering liquid film sheet is fully developed and has thickness inδ and

given, known temperature, flow of the liquid film is laminar, during calculations thickness of the liquid film has been neglected and its

velocity has been approximated with the wall velocity, there is no slip between film and gaseous phases, wall is insulated and film temperature is assumed to be uniform in y

direction, condensation heat flux increases only the internal energy of the liquid film, film interface is impermeable to noncondensable components, liquid film properties are assumed to be constant.

Assumptions and simplification with regard to the gaseous phase

binary mixture of water vapour with atmospheric air, mixture is in the saturation state at the liquid-gas interface, flow of the mixture can be laminar or turbulent,

∞Mu

x, u

y, v

Temperature profile

Water vapour mass fraction profile

∞MT

δT

0y

TW

L =∂∂

∞Vϖ

δVϖ

inδ δ

1NBA =+++ ∞∞∞ ϖϖϖ K

Saturated mixture of condensable and noncondensable gases


25

mixture inflow velocity is uniform and equal to the film interface velocity, mixture flow is driven by the entrance velocity, the interfacial shear and

condensation effects, condensation is driven by the diffusive mass flux in the vicinity of the

liquid-gas interface boundary layer, bulk amount of noncondensable gas and bulk mixture temperature are

known and uniform, mixture components are in dynamic and thermal equilibrium.

Assumptions and simplification with regard to geometry

smooth, horizontal, adiabatic wall.

Solution method

two-dimensional, steady-state CFD calculations with implemented

mechanistic condensation model. 3.3 Modelling of interactions between liquid film structures and gaseous boundary layer A physical model of the wavy film interaction with the gaseous boundary layer is based on the free convection condensation model. The difference is that the liquid film enters the physical domain with a prescribed wavy shape. This shape is assumed to be sinusoidal or similar to the typical solitary wave profile. Based on the literature review in Chapter 2, both types of waves have been confirmed to exist in the laminar wavy flow region. Both sinusoidal and solitary-shaped film models are shown schematically in Figures 3.3 and 3.4. Also the most important assumptions and simplifications have been listed below.


26

0 1 2x 10

-4

0

0.005

0.01

0.015

0.02

y - direction [m]

x -

dir

ecti

on

[m

]g

Figure 3.3 Condensation in presence of noncondensable gas on a sinusoidal wave with prescribed geometry.

0 1 2 3 4x 10

-4

0

0.01

0.02

0.03

0.04

0.05

0.06

y - direction [m]

x -

dire

ctio

n [m

]

g

Figure 3.4 Condensation in presence of noncondensable gas on the solitary-shaped wave with prescribed geometry. Assumptions and simplification with regard to the liquid film

film flow is laminar-wavy, film flow is driven by buoyancy forces, film thickness it given by the prescribed geometry of the wave, there is no slip between film and gaseous phases, wall temperature is given and is uniform, film interface is impermeable to noncondensable components,

∞Vϖ

δVϖ

∞Vϖ

δVϖ

Binary, saturated mixture of gases

Binary, saturated mixture of gases


27

liquid film properties are assumed to be constant.

Assumptions and simplification with regard to the gaseous phase binary mixture of water vapour with atmospheric air, mixture is in the saturation state at the liquid-gas interface, flow of gaseous phase is laminar, mixture flow is driven by the interfacial shear, buoyancy and condensation

effects, bulk amount of noncondensable gas and bulk mixture temperature are

known and are uniform, mixture components are in dynamic and thermal equilibrium.

Assumptions and simplification with regard to geometry

smooth, vertical, isothermal wall.

Solution method

in-house developed, two-dimensional, pseudo-transient code with implemented, mechanistic condensation model (see Appendix 1 for detailed description of the model implementation).

3.4 Model of spontaneous condensation The spontaneous condensation model is based on the nucleation of small water droplets in the supersaturated pure vapours or mixtures. Formation of such dispersed phase can lead to a significant reduction in heat and mass transfer rates to cooling surfaces. These reductions are mainly governed by the degradation of the convection mass transfer rates due to the nucleation of the dispersed phase (as pointed out in the literature review). For very high supersaturation ratios this convective mass transfer can even vanish. Additionally, if some amount of noncondensable gas is present in the gaseous phase, the whole condensation process is governed by the water vapour diffusion through the region in the vicinity of the droplet interface. Physical model which takes this aspect into consideration has been schematically shown in Figure 3.5.


28

Figure 3.5 Physical model of spontaneous condensation in presence of noncondensable gase. Only qualitative studies have been performed on this model. For this purpose, previous concept of direct-contact condensation on the horizontally flowing liquid film sheet has been extended, in order to include spontaneous condensation effects. There are several assumptions and simplifications to this model which have been listed below. Assumptions and simplification with regard to the dispersed phase

critical droplet diameter is calculated from the equilibrium conditions for free Gibbs energy,

all water droplets are assumed to have the same dimensions, the size distribution of the water droplets is Gaussian, droplet surface pressure has been assumed to be equal to the water vapour

partial pressure .

Assumptions and simplification with regard to the gaseous phase

binary, supersaturated mixture of water vapour and air, mixture is in the saturation state at the liquid-gas interface, condensation heat and mass transfer is driven by the water vapour

diffusion through the gaseous boundary layer in the vicinity of the liquid droplet surface,

mixture components are in dynamic and thermal equilibrium.

d ∞Vϖ

SVϖ

Supersaturated mixture of condensable and noncondensable gases

Nucleated water droplets


29

Solution method

two-dimensional steady-state CFD calculations with implemented surface condensation model and additional volume mass sources.

Summary Physical models, based on the different aspects of condensation from the multicomponent mixture of gases, have been presented in this chapter. In order to give clear picture of applied models, the most important assumptions and simplifications have been listed. Also solution methodology has been pointed out for each model. All described above effects can naturally exist simultaneously. However, such, separate studies facilitate better understanding of the particular processes which are important for given, physical conditions. In the next chapter, the mathematical formulation of the physical models presented above has been described.

Chapter 4 Mathematical models

31

Chapter 4

Mathematical models Introduction In this chapter a general mathematical description of physical models, discussed in the previous section, is presented. First fundamental laws, which are governing transfer of mass, momentum and energy in both liquid and gaseous phases, are described. In the next section mass, momentum and energy balances at the liquid-gas interface have been written in the most general form. Further on, using the order of magnitude analysis, the system of conservation equations and interface balances are simplified to, so called, boundary layer equations. Next section gives a mathematical description of the buoyancy effects and the modelling of multicomponent diffusivities. Chapter ends with description of a mathematical model of the spontaneous condensation in the supersaturated, binary mixture of gases. 4.1 Film and mixture conservation equations Conservation equations for liquid and gaseous phases have been described in this section. These relations come from the general conservation laws for mass, momentum and energy with the assumptions that both fluids can be treated as Newtonian fluids and pressure tensor satisfies the Stokes hypothesis.


32

4.1.1 Liquid phase conservation equations Liquid phase conservation equations are described by the following relations: Conservation of mass of the liquid phase

( ) 0ρtρ

LLL =•∇+

∂∂ u . (4.1.1.1)

Conservation of momentum of the liquid phase

( ) LLLLLLLL pρρ

tρ τguuu

•∇+∇−=•∇+∂

∂ . (4.1.1.2)

Conservation of energy of the liquid phase

( ) ( )LLLL ht

h qu •∇=•∇+∂∂ . (4.1.1.3)

4.1.2 Conservation equations for a ternary gaseous

mixture The system of conservation equations for the gaseous phase has been formulated for the mixture which is a composition of three gases. All variables with subscript “M” are related to the mixture, “A” letter has been related to the heavy noncondensable gas (e.g. air), “B” letter to the light noncondensable gas (e.g. helium) and “C” letter to the condensable gas (water vapour). With the above notation, conservation equations for the gaseous phase are described by the following relations: Conservation of mass of gaseous phase

( ) 0ρtρ

MMM =•∇+

∂∂ u . (4.1.2.1)

Conservation of momentum of gaseous phase

( ) MMMLMMMM pρρ

tρ

τguuu

•∇+∇−=•∇+∂

∂ . (4.1.2.2)


33

Conservation of energy of gaseous phase

( ) ( )MMMM ht

h qu •∇=•∇+∂∂ . (4.1.2.3)

Conservation of A-component of gaseous phase

( ) ( ) ( )B12A11AMMAM DDρ

tρ

ϖϖϖϖ

∇•∇+∇•∇=•∇+∂

∂ u . (4.1.2.4)

Conservation of B-component of gaseous phase

( ) ( ) ( )B22A21BMMBM DDρ

tρ

ϖϖϖϖ

∇•∇+∇•∇=•∇+∂

∂ u . (4.1.2.5)

Mass fraction of C-component can be calculated from the following constrain:

BAC 1 ϖϖϖ −−= . (4.1.2.6) 4.2 Balances at the liquid-gas interface In order to close the system of conservation equations, in addition to boundary conditions and equations of state, the interface mass, momentum and energy balances must be specified. Figure 4.1 schematically shows a flux of any scalar or vector quantity through the liquid-gas interface. It is assumed that the interface can be described by the following equation:

( )tx,δy = . (4.2.1)

Figure 4.1 Interface between liquid film and gaseous mixture.

( )tx,δy =

Mixture - MLiquid - L

η−

flux vector

τ−

τ

η


34

Generally equation (4.2.1) can be written as a certain function “F”, which is dependent on time and position (in two-dimensional space). This function can be given in the following form: ( ) ( )tx,δyty,x,F −= . (4.2.2)

With the above definition tangential and normal vectors τ and η at any point of the interface can be calculated as:

1FxF,-

yF −∇

∂∂

∂∂

=τ , 1FyF,

xF −∇

∂∂

∂∂

=η . (4.2.3 – 4)

Using these vectors, the mass, momentum, energy and species interface balances are described by the following relations: Interface mass balance

( ) ( ) ( ) 0ρρ δLLδMM =−•−+•− ηuuηuu . (4.2.5) Interface momentum balance

( )( ) ( )( ) ( ) 0ρρ LδLLLMδMMM =−•+−⊗+•+−⊗ ηPuuuηPuuu . (4.2.6) Interface energy balance

( )( ) ( )( ) ( ) 0hρhρ LδLLLMδMMM =−•+−+•+− ηquuηquu . (4.2.7) Interface balance of species “A”

( )( ) ( )( ) ( ) 0ρρ ALδLALAMδMAM =−•+−+•+− ηjuuηjuu ϖϖ . (4.2.8) Interface balance of species “B”

( )( ) ( )( ) ( ) 0ρρ BLδLBLBMδMBM =−•+−+•+− ηjuuηjuu ϖϖ . (4.2.9) Because of the impermeability of the liquid film surface to noncondensable components, species balances are reduced to the following forms:

( )( ) 0ρ AMδMAM =•+− ηjuuϖ , (4.2.10)

( )( ) 0ρ BMδMBM =•+− ηjuuϖ . (4.2.11)


35

4.3 Boundary layer approximation Conservation equations given in section 4.1.1 can be simplified using the boundary layer approximation. Applying the order of magnitude analysis, it can be found that some terms in the boundary layer region can be neglected. This is very useful approach because it allows finding the analytical solution of the system of boundary layer equations. 4.3.1 Boundary layer conservation equations for a liquid film and a ternary mixture of gases With the assumption of constant properties, boundary layer equations for the liquid phase can be written as follows: Mass conservation

0y

vx

u LL =∂∂

+∂∂ . (4.3.1.1)

Momentum conservation

2L

2

LL

Mx

LL

LL y

uν

ρρ

1gy

uv

xu

u∂∂

+

−=

∂∂

+∂∂ ∞

, 0y

pL =∂∂ . (4.3.1.2)

Energy conservation

2L

2

LL

LL

L yT

ay

Tv

xT

u∂∂

=∂∂

+∂∂ . (4.3.1.3)

Boundary layer equations for the ternary gaseous mixture can be written as follows: Mass conservation

0y

vx

u MM =∂∂

+∂∂ . (4.3.1.4)

Momentum conservation

2M

2

MM

Mx

MM

MM y

uν

ρρ

1gy

uv

xu

u∂∂

+

−=

∂∂

+∂∂ ∞

, 0y

pM =∂∂ . (4.3.1.5)


36

Energy conservation

2M

2

MM

MM

M yTa

yTv

xTu

∂∂

=∂∂

+∂∂ . (4.3.1.6)

Conservation of species “A”

2B

2

122A

2

11A

MA

M yD

yD

yv

xu

∂∂

+∂∂

=∂∂

+∂∂ ∞∞ ϖϖϖϖ . (4.3.1.7)

Conservation of species “B”

2B

2

222A

2

21B

MB

M yD

yD

yv

xu

∂∂

+∂∂

=∂∂

+∂∂ ∞∞ ϖϖϖϖ . (4.3.1.8)

All ternary diffusivity coefficients in Equations (4.3.1.7) and (4.3.1.8), have been defined with respect to the bulk mixture composition. Mass fraction of condensable component “C” can be found from Equation (4.1.2.6). Pressure gradient terms in Equations (4.3.1.2) and (4.3.1.5) have been combined with gravity effect with the assumption that pressure field in both fluids is given by the value of the hydrostatic pressure. 4.3.2 Interface balances with boundary layer approximation Assuming a stationary shape of the liquid-gas interface:

( ) 0t

ty,x,F=

∂∂ , (4.3.2.1)

and performing again the order of magnitude analysis, it can be found that interface balances with the boundary layer approximation can be simplified to the following forms: Mass balance

δv

δ

MMM

δ

LLL ndxdδuvρ

dxdδuvρ =

−=

− . (4.3.2.2)


37

Momentum balance

δM

M

δL

L yu

µy

uµ

∂∂

=

∂∂ , δ

MδL pp = . (4.3.2.3 – 4)

Energy balance

δMM

Mδv

δV

δLL

Lδv

δL y

Txδ

xTknh

yT

xδ

xTknh

∂∂

−∂∂

∂∂

+=

∂∂

−∂∂

∂∂

+ . (4.3.2.5)

Impermeability condition imposed on the flux of component “A”

0y

Dy

Dρxδuvρ

δ

B12

A11MMMA =

∂∂

+∂∂

+

∂∂

− ∞∞ ϖϖ . (4.3.2.6)

Impermeability condition imposed on the flux of component “B”

0y

Dy

Dρxδuvρ

δ

B22

A21MMMB =

∂∂

+∂∂

+

∂∂

− ∞∞ ϖϖ . (4.3.2.7)

Relations (4.3.2.2) through (4.3.2.7) represent simplified form of the interface conservation equations. Additional relation can be written assuming that both phases have the same tangential velocity components at the liquid-gas interface. Assuming that the gradient of liquid film thickness is very small, this approximation simplifies just to the following relation:

δM

δL uu = . (4.3.2.8)

4.3.3 Modelling of buoyancy effects The first term on the right hand side of Equation (4.3.1.5) contains buoyancy forces and an approximation that pressure is equal to the hydrostatic one in both liquid and gaseous phases. It can be shown that this term for the ternary mixture of gases can be rewritten as the following sum (Karkoszka and Anglart, 2007):

( ) ( )∞∞∞

−+−=− BBAAM

M βαρρ

1 ϖϖϖϖ , (4.3.3.1)


38

where constants βα, are referenced as the ternary expansion coefficients of the mixture components “A” and “B”, respectively. Taking into account the ideal gas law, these coefficients can be written in the following forms:

( )( ) ( ) BABCABAACBBA

BACB

MMMMMMMMMMMMMM

α−−+−

−= ∞∞ ϖϖ

, (4.3.3.2)

( )( ) ( ) BABCABAACABA

BACA

MMMMMMMMMMMMMM

β−−+−

−= ∞∞ ϖϖ

. (4.3.3.3)

4.4 General formulation of ternary diffusivity coefficients Expressions describing ternary diffusivity coefficients ( 11D and 22D ) as functions of the local mixture composition can be found from the Maxwell-Stefan equation (Bird et al., 2002). This expression relates molar fraction gradients of any component “i” to the differences between diffusive mass fluxes of the mixture components:

( )∑=

−

−=∇

N

1i ij

ijji

MMSi D

jjxxρ1x . (4.4.1)

Both noncondensable diffusive fluxes in one dimension can be described by the following equations:

∂∂

+∂∂

=y

Dy

Dρj B12

A11MA

ϖϖ , (4.4.2)

∂∂

+∂∂

=y

Dy

Dρj B22

A21MB

ϖϖ . (4.4.3)

Molar fraction of “i” component of the mixture in terms of mass fractions can be defined as:

∑=

= N

1jjj

iii

/M

/Mx

ϖ

ϖ . (4.4.4)


39

Combining both expressions (4.4.1) and (4.4.4) and taking into account the following relations between mixture components mass fractions and diffusive fluxes:

∑=

=N

1ii 1ϖ and ∑

=

=N

1ii 0j , (4.4.5 – 6)

the ternary diffusivity coefficients can be calculated as functions of local mass fractions of both noncondensable mixture components:

( )BA11 ,fD ϖϖ= , ( )BA21 ,fD ϖϖ= , (4.4.7 – 8)

( )BA12 ,fD ϖϖ= , ( )BA22 ,fD ϖϖ= . (4.4.9 – 10) It has been noticed that cross-diffusivity effects described by diffusivity coefficients with subscripts “12” and “21” are around two orders of magnitude smaller than the main diffusivities. Thus the following relations can be written:

1211 DD >> , 2122 DD >> . (4.4.11 – 12) With the above, diffusive fluxes can be simplified to the following forms:

yDρj A

11MA ∂∂

=ϖ ,

yDρj B

22MB ∂∂

=ϖ . (4.4.13 – 14)

Full description of these coefficients can be found in Appendix attached to Paper IV. 4.5 Mathematical model of spontaneous condensation Spontaneous (or volume) condensation process can be described as the mass transfer from the vapour phase to the dispersed phase. Assuming a spherical shape of water droplets, mass of a single droplet Md is just a function of its diameter dd (Karkoszka and Anglart 2005): )f(dM dd = . (4.5.1) Thus transport of mass to a single droplet (i.e. its growth in time) due to the nucleation process can be described as follows (McCallum et al. 1999):


40

( )d2dd

dd

d ddtddπρ

21

dtdV

ρdt

dM== . (4.5.2)

In the above expression ρd and Vd denote density and volume of a single droplet, respectively. Taking relation (4.5.2) into account the total mass flow rate to the dispersed phase - assuming that dispersed water droplets have the same diameters - can be calculated as follows:

( )∑ ∑∑= ==

===

N

1j

N

1jjd,

2jd,d

jd,N

1jdjd,d

tot ddtddπρ

21

dtdV

ρVdtdρ

dtdM . (4.5.3)

As can be seen, the total condensation mass flow rate is a function of the number of droplets “N”, and droplet evolution:

=dt

)d(dN,fM d

total. (4.5.4)

In order to find value of the volumetric condensation rate, estimation of number of nucleated droplets and theirs growth in time is required. A critical diameter, above which water droplet is allowed to grow by condensation process, can be found from the Gibbs free energy defined as:

ln(S)nkTσπd∆G v2d −= , (4.5.5)

where σ, k, Tv, S are droplet surface tension, Boltzmann constant, local vapour temperature and supersaturation, respectively. Symbol “n” is the number of molecules within a single droplet and can be calculated from:

molecule1

d

VV

n = . (4.5.6)

In the above expression Vd and V1 molecule are volume of a single droplet and volume of a single droplet molecule, respectively. Gibbs free energy described by Equation (4.5.5) is just a difference between energy which is required to form a spherical surface and isothermal compression of vapour described by the supersaturation ratio (Kaufmann et al. 1998):

sat

v

pp

S = , (4.5.7)


41

where pv and psat denote a water vapour partial pressure and saturation pressure, respectively. There exists a critical value of the droplet diameter dcrit, referred to as the Kelvin-Helmholtz diameter, which will continue growing due to the condensation process. This value can be calculated from the following equilibrium condition for the Gibbs free energy:

( ) 0∆G)d(d

d

d

= . (4.5.8)

From the above relation, the critical droplet diameter can be calculated as:

kTln(S)σV4

d OH1crit

2= . (4.5.9)

Solution of Equation (4.5.8) has also been shown graphically in figure 4.2.

Figure 4.2 Change of free Gibbs energy as a function of droplet diameter for supersaturation ratio S > 1. If the nucleated droplet diameter is below the critical value it will evaporate immediately. Inserting this critical value into Equation (4.5.5) it is possible to calculate the critical change of the Gibbs free energy crit∆G required to create the droplet which can grow due to the condensation process. Following the classical theory of homogenous nucleation, the rate of formation of new droplets described by number of droplets “N”, can be calculated assuming the Gauss distribution of the dispersed phase:

critd

crit∆G

[ ]md

[ ]J∆G


42

( )∆G/kTexpNN 0 −= . (4.5.10)

In the above expression, N0 is a factor which can be calculated from the following equation:

21

3OH1d

2v

0

2πM

σ2ρρ

N

= . (4.5.11)

Number of droplets of the critical size can be calculated with consideration that change in the free Gibbs energy must be equal to the critical one:

( )/kT∆GexpNN crit0crit −= . (4.5.12) Due to the Kelvin effect, a higher partial pressure is required to maintain mass equilibrium for the curved surface as compared to the flat one. Using the Kelvin equation, the supersaturation ratio at the droplet surface can be calculated as follows:

[ ])RTd/(ρσM4expS ddVK = . (4.5.13) In the above equation “R” describes the universal gas constant. Because satdK /ppS = , it is now possible to calculate the steam partial pressure at the droplet surface:

[ ])RTd/(ρσM4exppp ddVsatd = . (4.5.14) Applying the diffusion layer theory and assuming that water droplet diameter is larger than the mean free path λ of vapour molecules given by:

21

v

v

8πRT

pµ3

λ

= , (4.5.15)

growth of the water droplet can be calculated from the following expression:

( ) ( )RTdρ

ppMD4d

dtd

dd

satvVAVd

−= , (4.5.16)

where DAV is the binary diffusivity coefficient.


43

With the above, it is possible to calculate the time interval which is required for a droplet to growth from diameter critd to diameter 2d :

( ) ⇒−

= ∫∫2

crit

d

ddd

satVAC

dt

0)d(dd

ppMD4RTρ

dt (4.5.17)

( ) ( ) ( )2crit

22

satVVAC

d

d

d

2

satVVAC

d ddppMD8

RTρ2d

ppMD4RTρt

2

crit

−−

=−

= . (4.5.18)

Having the above expression, it is possible to approximate the average growth of droplet in the following way:

( )3crit

32

dddddd dd

tρπ

61

tmm

t∆m

dtdMm crit2 −=

−=== . (4.5.19)

Defining value dm as the average condensation mass flow rate due to the droplet growth, the total mass flow rate in the unit volume can be calculated from the following relation:

∑=

=N

1iid,

totald, mV1

Vm . (4.5.20)

Summary In the present chapter a mathematical description of the previously developed physical models has been presented. Beginning with the general form of conservation equations for both liquid and gaseous phases, the system of conservation equations has been simplified with the boundary layer approximation. The same procedure has been applied to the general form of interface mass, momentum and energy balances. In the following sections the modelling of buoyancy effects with boundary layer approximation as well as general derivation of the ternary diffusivity coefficients has been described. Finally this chapter ended with derivation of the spontaneous condensation model, which lends itself to the multidimensional CFD analyses. Next chapter contains descriptions of solution methods (mentioned in Chapter 3) and short presentation of tools which have been applied in present calculations.

Chapter 5 Solution methods and tools

45

Chapter 5

Solution methods and tools Introduction This chapter contains description of solution methods and tools. Firstly the modelling of the interface interactions between wavy film and multicomponent gaseous mixtures is described. Next, the application of the similarity method to the coupled system of boundary layer equations - for the liquid and gaseous phases - is presented. Finally the application of a commercial CFD code to the mechanistic modelling of condensation is described. 5.1 Modelling of film structure A two-dimensional pseudo-transient CFD code has been developed in order to model the interactions of the wavy liquid film with binary mixtures. The interface temperature, which is the same for both liquid and gas domains, has been calculated from first principles with the application of the energy balance equation. This equation can be in general written as follows:

δM

δL qq = . (5.1.1)

It has been assumed that liquid-gas interface is impermeable to noncondensable component. Mathematically, this condition has the following form:


46

0δA =n . (5.1.2)

Assuming that λA << where A is the wave amplitude and λ the wave length, the energy balance can be simplified to the following form:

δM

MfgδV

δL

L yT

khny

Tk

∂∂

−=∂∂

− . (5.1.3)

Following the above assumption, the interface mass flux of the condensed vapour can be calculated as:

δV

δV

AVMδV y1

Dρn

∂∂

−=

ϖϖ

. (5.1.4)

Both liquid and gaseous domains are computed separately and linked through the above interface conditions by the iteration process. Within this process, firstly the initial temperature for the liquid-gas interface is guessed. Then its value is updated during iterations, until the interface balances for energy and mass fractions are satisfied within a given tolerance. More details about the implementation of the numerical scheme can be found in Appendix 1. 5.2 Solution of the boundary layer equations In order to solve analytically the coupled system of boundary layer equations for liquid and gaseous phases, the similarity method has been employed. Details concerning application of this method to the free convection gravity driven condensation problem have been presented below. The same method has been used in order to find a solution of the problem of forced convection condensation from the binary mixture on a horizontal, isothermal wall. The calculations have been performed in order to prove that method of specification of the liquid film thickness δ = ψ1x1/2 for the horizontal wall and δ = ψ2x1/4 for the vertical wall is equivalent to the specification of the wall temperature as a boundary condition. Therefore, description of the method of analytical solution of the forced convection condensation on the horizontal surface and prove of the above assumptions by comparison with previous theoretical studies found in the literature, have been included in the Appendix 2.


47

5.2.1 Gravity driven condensation on a vertical surface In order to find an analytical solution of the system of boundary layer equations, partial differential equations have to be transformed to the system of ordinary differential equations. Therefore, the following similarity variables have been introduced for the liquid phase:

4/1L

L xyc

η = , ( )LL4/3

LLL ηfxcνψ = , ∞

∞

−−

=TTTT

θ WL

L, (5.2.1.1 – 3)

where θf,,η are dimensionless distance, dimensionless stream function and dimensionless temperature, and constant c includes fluid properties. Applying the above relations, the liquid phase momentum equation can be now written in the following form:

04dηdf2

dηfdf3

dηfd4

2

L

L2L

L2

L3L

L3

=+

−+ . (5.2.1.4)

Mass conservation equation has been automatically satisfied by the definition of the stream function ψ and doesn’t need to be considered any longer. For the ternary mixture of gases, similarity variables have been defined as follows:

( )4/1

MM x

δycη

−= , ( )MM

4/3MMM ηfxνcψ = ,

∞

∞

−−

=TTTT

θWM

M, (5.2.1.5– 7)

∞=A

AAΩ

ϖϖ ,

∞=B

BBΩ

ϖϖ . (5.2.1.8 – 9)

With these relations the mixture momentum and “A” and “B” mass fraction transport equations can be transformed to the following relations:

( ) ( ) 01Ωβ41Ωα4dηdf

2dη

fdf3

dηfd

4 BBAA

2

M

M2M

M2

M3M

M3

=−+−+

−+ ∞∞ ϖϖ , (5.2.1.10)

0dηΩd

Sc4

dηdΩf3 2

M

A2

11M

AM =− ∞

, 0dηΩd

Sc4

dηdΩf3 2

M

B2

22M

BM =− ∞

, (5.2.1.11– 12)

where ∞Sc is the Schmidt number calculated with the bulk state the reference.


48

The boundary conditions expressed in terms of the similarity variables become as follows:

0dηdf

W

L

L = , 0f WL = , 1θW

L = , (5.2.1.13 – 15)

0dηdf

M

M =∞

, 0θδM = , 1ΩA =∞ , 1ΩB =∞ . (5.2.1.16 – 19)

5.2.2 Application of similarity variables to the interface balance equations The interface balance equations of mass, momentum, energy and concentrations of species “A” and “B” can be written as follows:

( ) ( ) δM

2/1MM

δL

2/1

L

ML2/1LL fµρf

ρρρµρ =

− , (5.2.2.1)

δ

M

Mδ

L

L2/1

L

ML

dηdf

dηdf

ρρρ

=

− , (5.2.2.2)

( ) ( )δ

2M

M2

2/1MM

δ

2L

L22/1

L

ML2/1LL dη

fdµρdη

fdρρρµρ =

− , (5.2.2.3)

JaCh

TT

dηdθ

ckck

dηdθ

fPr43

PLfg

W

δ

M

M

LL

MMδ

L

L

δLL

=−

=

−

∞ , (5.2.2.4)

δM

δ

M

A

11

δA f

43

dηdΩ

Sc1Ω ∞= , δ

M

δ

M

B

22

δB f

43

dηdΩ

Sc1Ω ∞= . (5.2.2.5)

In Equation (5.2.2.4) symbol “Ja” denotes the Jakob number. Calculating this number together with application of the ideal gas law, the temperature difference

wTT∆T −= ∞ can be obtained with the assumption that the whole condensation


49

heat flux in the liquid phase is transported to the cold wall by conduction. This is a reasonable approximation because the liquid film thickness, as considered in these studies, is very small. 5.3 Mechanistic modelling with a Computational Fluid Dynamics code Numerical methods employing the computational fluid dynamics (CFD) approach are not limited to the boundary layer approximation. Instead, general unsteady conservation equations, with proper boundary conditions, can be solved for a given geometry in the discrete space. In the present work both liquid and gaseous domains are modelled separately and are linked together through interface conditions. The basic assumption pointed out in Chapter 3 is that the mixture of gases is in the saturation state and can be treated as an ideal gas. For modelling of the liquid film and the mixture the separate domain approach can be applied. In this model the mass conservation equations for the liquid and gaseous phases are as follows:

( ) ( )[ ] Γ=•∇+∂∂

LLLLL ρρt

uαα , (5.3.1)

( ) ( )[ ] Γρρt MMMMM −=•∇+

∂∂ uαα , (5.3.2)

where Γ is the mass transfer term which is equal to zero everywhere except at the interface, and Lα , Mα are volume fractions of the liquid film and mixture, respectively. The volume fractions satisfy the following condition 1=+ ML αα . Because it has been assumed that only one phase can exist in every computational domain, 0,1 ML == αα for the liquid film domain and 1,0 ML == αα for the mixture domain, respectively. The main advantage of such formulation relies on parallel computations of both domains. Based on the order of magnitude analysis, it has been assumed that the liquid film thickness - for the gravity driven flow - can be described by the following expression: ( ) 4/1ψxxδ = , (5.3.3)

where ψ is related to the liquid properties. The same methodology has been applied with the analytical solution to the boundary layer equations, as shown in Appendix


50

2. With this assumption, the condensation mass flux can be calculated using a simple Nusselt solution of the following form:

( )dxdδ

δµ

ρρgρdxdm 2

L

MLL −==Γ . (5.3.4)

Effect of the interface shear stress in Equation (5.3.4) has been neglected because of low velocities of liquid film and vapour. Unsteady momentum equations for the liquid and mixture domains can be written in the following two-dimensional form:

( ) ( )

( ) δLLL

LLLLLLLLLL ρρρt

uτ

guuu

Γ+•∇+

++∇−=•∇+∂∂

α

αααα p , (5.3.5)

( ) ( )

( ) δMMM

MMMMMMMMMM ρρρt

uτ

guuu

Γ−•∇+

++∇−=•∇+∂∂

α

αααα p . (5.3.6)

Assuming that the gaseous phase behaves like the ideal gas, local mixture density can be calculated as a function of the local composition of the mixture:

=

∑=

N

1i i

isatM

MM

MRT

pρ

ϖ. (5.3.7)

The gaseous mixture has been also assumed to be in the saturated state. That means its temperature can be evaluated with the following relation:

( )satsatM pfT = , (5.3.8)

where the saturation pressure for the ternary mixture is described by the following expression:

( )( ) ( ) ( )CACBCBCABACBA

BACMBAsat

1MM1MMMMPMMp

ϖϖϖϖϖϖϖϖϖϖ

−+−+++

= . (5.3.9)

Neglecting cross-diffusivity effects, unsteady transport equations for the mixture components “A” and “B” can be written in the following forms:


51

( ) ( )[ ] 0Duραραt A11MAMMAMM =∇−•∇+

∂∂ ϖϖϖ , (5.3.10)

( ) ( )[ ] 0Dρρt B22MBMMBMM =∇−•∇+

∂∂ ϖϖαϖα u . (5.3.11)

The mass fraction of component C is then defined by the following constrain:

BAC 1 ϖϖϖ −−= . (5.3.12) Mixture density and diffusivity coefficients 11D and 22D depend on the local mixture composition and generally can be defined as (see also Chapter 4):

( )BAM ,fρ ϖϖ= , ( )BA11 ,fD ϖϖ= , ( )BA22 ,fD ϖϖ= . (5.3.13 – 15) Boundary conditions are given by no-slip conditions for the liquid film at the wall and composition of the mixture in the bulk:

0vu WL

WL == , ∞∞

BA ,ϖϖ . (5.3.16 – 18) Because mixture flow is driven by the falling liquid film as well as condensation effects and buoyancy forces, the von Neumann boundary conditions have been imposed on the gaseous domain. The amount of mixture flowing into and out of this domain is calculated from the overall mass balance given by the following relation:

outM

δM mmm +=∞ , (5.3.19)

where ∞

Mm , δm and outMm are respectively: mass flow rate from the bulk mixture

towards the cold wall, interface mass flow rate between gaseous and liquid domains due to the phase change and total mixture mass flow rate at the outlet from the mixture domain. The applied CFD code imposes for scalar variables the Dirichlet boundary conditions at the inlet and the von Neumann boundary conditions at the outlet. However, for the buoyant flow some of the mixture - due to the presence of very light gas - can also enter the mixture domain through the outlet and therefore all scalar gradients in such region are approximated with the assumption of fully developed flow (i.e. zero gradients). Summary Methods of solution of the boundary-layer equations and a mechanistic set of conservation equations have been described in this chapter. The results of prediction with the present models are presented and discussed in the next chapter.

Chapter 6 Results and discussion

53

Chapter 6

Results and discussion Introduction This chapter discuses computational results which have been obtained with models described in the previous chapters. It begins with solutions of the forced convection condensation problem on a vertical surface. Next, a comparison of the diffusion CFD model with experimental data for forced convection direct-contact condensation on an adiabatic horizontal surface is presented. Further, qualitative investigations of the influence of the spontaneous condensation on the condensation heat transfer coefficient have been included. In the following section the influence of the wavy film flow on the heat transfer coefficient for condensation from binary mixtures has been studied. The chapter ends with a discussion of the results obtained from the boundary layer modelling with the similarity method and the mechanistic modelling using the CFD framework. Finally the results of calculations are summarized and analysed. 6.1 Forced convection condensation on a vertical surface The physical model of forced convection condensation from the binary mixture of water vapour and air on a vertical, isothermal surface has been described in Chapter 3. A mixture of water vapour and air flows downwards and condenses on a cold surface. It has been assumed that whole condensation process is driven by the


54

diffusion of water vapour through the near-interface boundary layer. As it has been mentioned in the previous chapters, such boundary layer exists, because liquid-gas interface is almost impermeable to the noncondensable gas. Because only water vapour crosses the liquid-gas interface, noncodensable species tends to accumulate in this region. This process creates additional resistance to the condensation heat and mass transfer and results in the reduction in the heat transfer coefficient. Because in the experimental studies for PCCS water vapour condenses inside vertical pipes, a vertical wall can be viewed as its two-dimensional equivalent. This allows a comparison of the commonly used empirical correlations, developed for the pipe geometry, with present, numerically-obtained results. Condensation model based on the resolution of the boundary layer has been implemented into a commercial CFD code (AEA Technology, 2000) through user subroutines. It calculates condensation heat and mass transfer fluxes at the liquid-gas interface from the heat and mass balance equations. Comparison of calculations of the average heat transfer coefficient with correlations given by Uchida, Tagami, Kataoka and Dehbi is shown in Figure 6.1. As can be observed, the numerical result follows trends given by these expressions. It means that the diffusive mass flux calculated by the code in the mechanistic way corresponds to the physical results, if the average values are taken into consideration.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1000

2000

3000

4000

5000

6000

7000

inlet noncondensable mass fraction

heat

tra

nsfe

r co

effi

cien

t [W

/m2 /K

]

Uchida correlationTagami correlationDehbi correlationdiffusion model

Figure 6.1 Comparison of the diffusion, mechanistic CFD model with commonly used correlations for inlet mixture velocity 8 m/s, K40∆T = . Results depicted in Figure 6.1 show that, at least for this simple geometry, the heat transfer degradation processes due to the presence of noncondensable gas can be investigated and predicted with the purely mechanistic approach, without any correlations. Additional resistance created by the accumulation of the noncondensable gas along the cold surface is quite uniform (except the region near


55

to the leading edge). This brings about a possibility of the numerical prediction of the experimentally correlated heat transfer coefficients. 6.2 Forced convection direct-contact condensation on a horizontal, adiabatic surface Geometrical conditions create the necessity of investigation of behaviour of the mechanistic model for different geometrical configurations. Horizontal surface can be considered as one of the examples. Modelling of the condensation from the binary mixture of water vapour and air by solving boundary layer equations has been described in Appendix 2. In this section CFD studies are performed on the modelling of the water vapour condensation on a horizontally-stratified liquid-water sheet flowing along an adiabatic wall. Physical model of this phenomenon and important simplifications have been pointed out in Chapter 3. As it has been done in the previous section, a model based on the proper resolution of the near-interface boundary layer and interface heat and mass balances, have been implemented in the commercial CFD code CFX. The most important assumption is that the whole heat of condensation is converted into the internal energy of the flowing liquid phase. A simple energy balance (described in Paper I) and obtained numerically mass flux of condensing vapour at the interface can be applied, to produce results shown in Figure 6.2.

0 0.5 1 1.51000

1500

2000

2500

3000

3500

4000

4500

5000

channel length [m]

heat

tra

nsfe

r co

effi

cien

t [W

/m2 /K

]

Choi et al. 2002DC diffusion model

Figure 6.2 Comparison of the direct-contact diffusion model with experimental data by Choi et al. (2002). Average, measured heat transfer coefficient by Choi et al. (2002) has been compared with computational results. Even if the calculated heat transfer coefficient is slightly under-predicted, still application of the mechanistic model seems to give reasonable results. This is an important conclusion since - as it has


56

been pointed out in the literature review - existing correlations in the engineering codes as well as models based on the heat and mass transfer analogy tend to give inadequate results in such conditions. 6.3 Influence of spontaneous condensation A physical and mathematical model of the spontaneous condensation from a binary mixture of gases has been described in Chapters 3 and 4. This model, in analogy to the surface condensation, is also based on the resolution to the diffusion boundary layer near the liquid drop-gas interface. Based on the kinetic theory of gases and the equilibrium condition for existence of the dispersed water droplets, condensation rates have been described as sinks to the equation of the mass conservation. Experimental data which could be compared with computational results could not be found. However, a sensitivity study has been performed to verify the over-all performance of the model. Figure 6.3 presents the degradation of the heat transfer coefficient along a cold surface where a large temperature difference between the bulk mixture and the cold surface has been applied.

0 0.1 0.2 0.3 0.4 0.51000

1500

2000

2500

3000

channel length [m]

heat

tra

nsfe

r co

effi

cien

t [W

/m2 /K

]

DC, ωair∞ = 0.1

DC + SC, ωair∞ = 0.1

DC, ωair∞ = 0.2

DC + SC, ωair∞ = 0.2

Figure 6.3 Sensitivity study of the influence of spontaneous condensation in direct-contact condensation on a horizontally-stratified liquid film. It can be concluded that inclusion of the spontaneous condensation model can lead to a slight reduction of the heat transfer coefficient degradation for highly supersaturated binary mixtures. As it has been found from the literature review in Chapter 2, degradation of the condensation heat transfer relies on the decrease of the convection mass flux towards the cold surface. The predicted degradation of the heat transfer coefficient shown in Figure 6.3 is a result of this effect.


57

6.4 Film structure influence on the heat and mass transport in the gaseous boundary layer Literature review indicates that the heat and mass transfer process associated with the surface condensation is highly dependent on the flow conditions. If only liquid film phase is taken into consideration, it has been found by many researchers that its structure is highly influencing the condensation heat and mass transfer rates. This effect is especially pronounced for very thin liquid films, low mixture flows and low noncondensable mass fractions in the multicomponent gaseous mixture. From the literature review and the present studies it has been found that there exist three processes which are responsible for the heat and mass transfer increase. The first one is strongly coupled with the liquid film hydrodynamics. Internal velocity fields, which are dependent on the wave structure, determines enhancement of the mass transfer within the wave. This process has been also observed experimentally several times. The other process is coupled with the existence of the gaseous boundary layer. Film irregularities create nonuniform accumulation of the noncondensable species at the liquid-gas interface. Third mechanism is connected with the liquid film thinning due to the creation of the wavy structures. Physical model of the film structure influence on the condensation process from binary mixture of gases has been described in Chapter 3. In order to model wavy film shapes in a convenient way, an in-house CFD code has been developed. Details about the implementation of the mathematical model can be found in Appendix 1. Shapes of the waves have been prescribed in such a way that both types of structures, discussed in the literature review, have been investigated. Unfortunately there is a lack of detailed experimental database with which the results presented below could be compared. Such experimental measurements could be based for example on the Raman Spectroscopy Method (Karl and Weiss, 1997) which could be employed for the film temperature field measurements. However, even if the present studies are kept only on a qualitative level, they tend to confirm general empirical observations found in the literature. From the experimental observation of the thin wavy film flows it has been found that two types of film structures can be distinguished. The first type of waves has a simple sinusoidal shape. These waves are characterised by small amplitudes and short wavelengths. The second type of waves sometimes is called a roll wave. They are similar to the solitary structures and characterised by large amplitudes and long wavelengths.


58

6.4.1 Sinusoidal wave Computed temperature field within the sinusoidal wave has been presented in Figure 6.4. On both sides of the computational domain (low and high x), periodic boundary conditions have been specified. On the one hand the velocity and the coupled temperature fields are distorted because the falling wave has larger velocity than the rest of the liquid film. Higher temperature on the trailing edge of the wave has been observed. Because of the existence of the internal disturbances of the film velocity field, the thermal energy resulting from the water vapour condensation is evidently faster convected towards the cold surface in the upper part of the wavy domain (red arrow in Figure 6.4 shows schematically this process). On the other hand, the velocity field observed in the lower region tends to convect cold liquid from the wall towards the gas-liquid interface (blue arrow). The net effect of the mixing is an enhanced convective heat transfer within the wave are.

y - direction [m]

x - d

irec

tion

[m]

0 1 2x 10

-4

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

333.5

334

334.5

335

335.5

336

336.5

g

Figure 6.4 Example of the calculated temperature field within sinusoidal wave with small amplitude (in K), 0.1air =

∞ϖ . In Figure 6.5 the gas-side heat transfer coefficient, defined in Paper 3, has been plotted along the wave surface. Because of the higher temperature and the fact that gaseous mixture exists in the saturation state, the value of the heat transfer coefficient is higher at the trailing edge (red ellipse) of the wave as compared with the leading edge (blue ellipse). Because the interface conditions are given by the saturated state of the gaseous mixture, the higher temperature is responsible for the lower amounts of noncondensable gas at the liquid-gas interface. In this way, the effect of noncondensable gas accumulation, which creates the main resistance to the mass transfer process, is lower at the trailing edge as compared with the leading edge of the wave. From both Figures 6.4 and 6.5 it can be concluded that the heat transfer process caused by the wavy film hydrodynamics and presence of noncondensable gas is more enhanced in the trailing region of the wave.


59

0 0.005 0.01 0.015 0.02100

150

200

250

300

350

400

450

500

550

x - direction [m]

gas

- sid

e

he

at tr

ansf

er c

oeffi

cien

t [W

/m2 /K

]wave profile

ωair∞ = 0.1

ωair∞ = 0.2

ωair∞ = 0.3

g

Figure 6.5 Influence of the noncondensable gas on the gas-side heat transfer coefficient for the small amplitude sinusoidal wave, 0.1air =

∞ϖ , 20.air =∞ϖ and

30.air =∞ϖ , respectively.

6.4.2 Solitary-shaped wave All conclusions from the previous section hold for the large amplitude solitary-shaped wave modelling. However, in this condition the existence of both higher and lower temperature convective fluxes is more evident. Temperature filed within such wave is shown in Figure 6.6. It can be noticed that - in a similar way as for the sinusoidal wave - the higher temperature exists in the trailing part of the wave. The distortion of the film velocity field is much more evident for large-amplitude waves and an internal vortex can be observed. This vortex is responsible for an enhancement of the heat transfer flux in the trailing edge of the wave. In Figure 6.7 the gas-side heat transfer coefficient has been plotted. Its behaviour can again be explained by higher amounts of the water vapour in the higher-temperature region, which decreases the noncondensable gas degradation effect.


60

y - direction [m]

x -

dire

ctio

n [m

]

0 1 2 3x 10

-4

0.01

0.02

0.03

0.04

0.05

333.5

334

334.5

335

335.5

336

336.5

337

337.5

338

g

Figure 6.6 Example of the calculated temperature field within the liquid film for solitary-shaped wave, (in K), 0.1air =

∞ϖ .

0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

500

600

700

x - direction [m]

gas

- si

de h

eat

tr

ansf

er c

oeffi

cien

t [W

/m2 /K

]

ωair∞ = 0.1

ωair∞ = 0.2

wave profile

ωair∞ = 0.3

g

Figure 6.7 Influence of the noncondensable gas on the gas-side heat transfer coefficient for the solitary-shaped wave, 0.1air =

∞ϖ , 20.air =∞ϖ and 30.air =

∞ϖ , respectively.


61

6.5 Similarity solution of the boundary layer equations for free-convection gravity-driven condensation Physical and mathematical models of the free-convection, gravity-driven condensation from the multicomponent mixture of condensable and noncondnesbale gases have been described in detail in Chapter 3 and 4. In Chapter 5 two methods, which can be applied to obtain a solution of this problem, have been presented. The first one rests on the boundary layer approximation for which an analytical solution can be obtained using the similarity method. This method has been applied by many researchers and can lead, as it is shown further, to very interesting conclusions. In these calculations it has been assumed that the thickness of the liquid film can be approximated with the relation given by Equation (5.3.3). With such approximation the wall temperature has been calculated and this approach has been proved to be equivalent to the specification of the wall temperature as the boundary condition (see Appendix 2). In the following calculations the condensation from binary and ternary mixtures has been considered. In Figure 6.8 dimensionless mixture velocity profiles have been plotted as function of dimensionless distance from the liquid-gas interface, for 1.5% of air mass fraction and 0.1% to 0.9% of helium mass fraction, respectively. It can be observed how an addition of the light gas influences buoyancy forces. In particular, for mass fractions of helium higher than 0.7 %, some amount of the mixture moves against the gravity forces.

0 1 2 3 4 5

-0.02

0

0.02

0.04

0.06

0.08

ηM

f' M(η

)

ωair0∞ = 1.5%, ηδ = 0.34

ωHe∞ = 0, 0.1- 0.9%

g

Figure 6.8 Influence of the addition of the light gas on the dimensionless mixture velocity profile.


62

In Figures 6.9 through 6.12 the calculated degradation of the condensation heat flux has been compared with experimental data obtained by Al-Diwany and Rose (1973) for a binary mixture of water vapour and air. It can be noticed that analytical results obtained with the boundary layer approximation fits well to the experimental data. Adequate explanation of such behaviour is that buoyancy forces in the mixture of water vapour with heavy gas like air acts in such a way that both phases move downwards. In these conditions the boundary layer approximation with mechanistically specified interface balances seems to be a very good approximation of the physical problem. In Figures 6.9 through 6.12 the interface heat flux has been normalised to the heat flux obtained from the Nusselt analysis. This approach shows how the condensation heat transfer flux is degraded by the presence of noncondensable gas with respect to the pure water vapour condensation for the same temperature difference. Another conclusion which can be drawn from the graphs is that the degradation of the condensation heat flux is more pronounced for small amounts of noncondensables. For high noncondensable mass fractions the heat transfer degradation effect becomes much smaller. Such behaviour has also been observed by several researchers as indicated in the literature review.

0 10 20 30 40 500

0.5

1

T∞ - T W

q cal/q

Nu

ωair∞ = 1.5 %

0 10 20 30 40 50 60 700

0.5

1

T∞ - T W

q cal/q

Nu

ωair∞ = 2.4 %

boundary layer approx. with film model

Al - Diwany and Rose 1973

boundary layer approx. with film modelAl - Diwany and Rose 1973

Figure 6.9 Comparison of the computed degradation in the condensation heat transfer with experimental data obtained by Al-Divany and Rose (1973),

5100.air =∞ϖ and 0240.air =



63

0 10 20 30 40 50 60 700

0.2

0.4

T∞ - T W

q cal/q

Nu

ωair∞ = 4 %

10 20 30 40 50 60 700

0.1

0.2

T∞ - T W

q cal/q

Nu

ωair∞ = 6.8 %

boundary layer approx. with film modelAl -Diwany and Rose 1973


Al -Diwany and Rose 1973


040.air =∞ϖ and 0680.air =


10 20 30 40 50 60 700

0.1

0.2

T∞ - T W

q cal/q

Nu

ωair∞ = 12 %

10 20 30 40 50 60 700

0.05

0.1

0.15

T∞ - T W

q cal/q

Nu

ωair∞ = 19 %



Al -Diwany and Rose 1973


120.air =∞ϖ and 190.air =



64

10 20 30 40 50 60 700

0.02

0.04

0.06

0.08

0.1

0.12

T∞ - T W

q cal/q

Nu

ωair∞ = 25.4 %



2540.air =∞ϖ .

Dimensionless profiles shown in Figures 6.13 through 6.15 allow studying relations between heavy and light noncondensable gases. Dimensionless air and mass fraction profiles have been shown in such manner that the influence of the light gas on the heavy one can be compared with the situation in which only the heavy gas is present. It can be noted that addition of the light gas results in the higher interface accumulation of the heavy gas. This process is governed by two main forces which are strongly coupled. The first one is the buoyancy force, which pushes the mixture upstream with reference to the liquid phase. The second effect is coupled to the impermeability of the liquid-gas interface, which forces the noncondensable mass fractions and the corresponding gradients to satisfy the interface mass balances.

0 1 2 3 4 50

10

20

30

40

50

ηM

Ωai

r, Ω

He

ωair0∞ = 1.5 %, ηδ = 0.34

Ωair ω

air0

∞, ω

He

∞=0.1 %

ΩHe ω

air0

∞, ω

He

∞=0.1 %

Ωair ω

air0

∞ + 0.1 %, ω

He

∞=0

Figure 6.13 Influence of an additional light gas on the dimensionless air and helium mass fraction profiles, additional amount of helium 0.1%He =

∞ϖ .


65

0 1 2 3 4 50

10

20

30

40

50

ηM

Ωai

r, Ω

He

ωair0∞ = 1.5 %, ηδ = 0.34

Ωair ω

air0

∞, ω

He

∞=0.4 %

ΩHe ω

air0

∞, ω

He

∞=0.4 %

Ωair ω

air0

∞ + 0.4 %, ω

He

∞=0

Figure 6.14 Influence of an additional light gas on the dimensionless air and helium mass fraction profiles, additional amount of helium %40.He =

∞ϖ .

0 1 2 3 4 50

10

20

30

40

50

60

70

ηM

Ωai

r, Ω

He

ωair0∞ = 1.5 %, ηδ = 0.34

Ωair ω

air0

∞, ω

He

∞=0.9 %

ΩHe ω

air0

∞, ω

He

∞=0.9 %

Ωair ω

air0

∞ + 0.9 %, ω

He

∞=0

Figure 6.15 Influence of an additional light gas on the dimensionless air and helium mass fraction profiles, additional amount of helium %90.He =

∞ϖ . In Figures 6.16 and 6.17 the degradation of the condensation heat transfer process has been plotted. Figure 6.16 shows the relationship between the additional amount of air or helium, which is introduced to the mixture, and the corresponding increase of the temperature difference between the bulk and the wall. Initially the mixture consists of water vapour and air with %4.20air =∞ϖ . Introduction of 1% of helium causes an increase of temperature difference 0∆T∆T− by 20 K. If instead 1% of air


66

is introduced, the increase of the temperature difference will be around 9 K. Thus, the heat transfer degradation is smaller in the latter case.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

∆T - ∆T0

ωai

r∞

, ωH

e∞

, [%

]

ηδ = 0.32, ωair0∞ = 2.4 [%]

ωair∞

ωHe∞

Figure 6.16 Influence of an additional amount of heavy or light gas on the degradation of the condensation heat transfer, 2.4%0.32,η 0airδ == ∞ϖ . From both Figures 6.16 and 6.17 can be noticed that this behaviour is not linear and the difference between bulk and cold wall temperature due to addition of air or helium increases almost exponentially. This nonlinear behaviour can be explained by previously mentioned interface impermeability conditions and interactions with the buoyancy forces. This behaviour has also been noticed by several researchers, as pointed out in the literature review. Experimental investigations show that light gases act more destructively on the condensation heat flux than the heavier ones.

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

∆T - ∆T0

ωai

r∞

, ωH

e∞

[%]

ηδ = 0.34, ωair0∞ = 1.5 [%]

ωair∞

ωHe∞

Figure 6.17 Influence of an additional amount of heavy or light gas on the degradation of the condensation heat transfer, 1.5%0.34,η 0airδ == ∞ϖ .


67

Figures 6.16 and 6.17 have been plotted for two different dimensionless film thicknesses given by the constant δη . It can be noticed that reduction of the condensation heat flux is more significant if high condensation mass fluxes are taken into consideration.

6.7 Mechanistic CFD analysis of free-convection gravity-driven condensation The free-convection gravity-driven condensation has also been investigated with the computational fluid dynamics approach. A ternary mixture of water vapour, air and helium has been taken into consideration. Water vapour condenses on a vertical, cold surface, which additionally has been bounded on the upper side by a horizontal wall. It has been assumed that there is no condensation on this surface. Flow of the liquid film is totally driven by the buoyancy forces. Gaseous phase is driven by the momentum transfer from the liquid phase and by condensation effects. Mechanistic model, which rely on the mass, momentum and energy balances at the liquid-gas interface, has been implemented into a commercial CFD code (see Chapter 5). Both liquid and gaseous phases are driven by free convection condensation effects. The film thickness is calculated from Equation (5.3.3), (by analogy with the boundary layer analysis). Mixture of gases has been assumed to be in the saturated state. With an assumption that it satisfies the ideal gas law, the mixture temperature can be calculated with the knowledge of the partial pressure of the water vapour. Example of the solution is shown in Figure 6.18. The heat transfer coefficient has been calculated with the assumption that the condensation heat is transferred through the liquid phase by the conduction. This approximation is valid for thin films, where the convection effects are negligible. The heat transfer coefficient shown in Figure 6.18 has been obtained for bulk mixture composition of 4% of air and 0.1% of helium. The temperature difference WTT −∞ has been obtained by variation of the liquid film thickness. Qualitative comparison with the experimental correlation obtained by Dehbi et al. (1991) has also been presented. The adjective “qualitative” is used here since the Dehbi correlation has been developed for mixtures with larger amounts of noncondensable gas. Unfortunately, as it results from the literature study, this is the only correlation which has been found for ternary mixtures of gases. Even if Dehbi’s experimental data have been obtained for reactor containment conditions, free convection conditions have been preserved and therefore these measurements can be used as a qualitative reference for validations of the mechanistic CFD calculations.


68

10 20 30 40 50 60500

1000

1500

2000

2500

3000

T∞ - T w

Hea

t tra

nsfe

r co

effic

ient

[W/m

2 /K]

ωair∞ = 4%, ω

He∞ = 0.1%

Dehbi et al., 1991mechanistic CFD model

Figure 6.18 Comparison of the heat transfer coefficient calculated with the CFD mechanistic model and obtained from the Dehbi et al. (1991) correlation. In Figures 6.19 and 6.20 the interface mass fraction profile has been plotted for air and helium components along the liquid-gas interface. It can be noticed that this profile is nonuniform and high accumulation of the noncondensable gases occurs in the upper part of the vertical wall. This is caused by the additional, horizontal surface which bounds the vertical wall from the top. In this way corner-type geometry has been created and the geometry influence on the noncondensable gas accumulation has been captured.

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

Air mass fraction

Dis

tanc

e al

ong

film

inte

rfac

e

fro

m o

utle

t (0)

, ups

trea

m [m

]

He 0%

He 0.1%

He 0.2%

He 0.3%

High concentrationregion

Figure 6.19 Interface air mass fraction profile for an additional bulk amount of helium 0%He =

∞ϖ , 0.1%He =∞ϖ , 0.2%He =


4%air =∞ϖ .


69

0 0.02 0.04 0.06 0.080

0.02

0.04

0.06

0.08

0.1

Helium mass fraction

Dis

tanc

e al

ong

film

inte

rfac

e

from

out

let (

0), u

pstr

eam

[m

] He 0%He 0.1%He 0.2%He 0.3%

Highconcentrationregion

Figure 6.20 Interface helium mass fraction profile for an additional bulk amount of helium 0%He =

∞ϖ , 0.1%He =∞ϖ , 0.2%He =


4%air =∞ϖ .

Figures 6.21 and 6.22 show the average interface mass fraction profiles of air and helium. Comparison between results obtained from the boundary layer analysis and CFD calculations has been shown. Both methods converge to each other when the amount of additional light gas in the bulk mixture decreases. The explanation of this behaviour is as follows. The presence of only heavy gas creates more uniform noncondensable mass fraction profiles, and velocity fields calculated by both models are similar. However, with an increasing amount of the light gas, the geometry effect and buoyancy forces together with previously-mentioned impermeability conditions cause large differences in prediction of the interface air and mass fraction values. Unfortunately both methods have not been compared within the same geometry because of the difficulties with an adequate specification of the von Neumann boundary conditions in CFD calculations, which require fully developed flow conditions.


70

0 0.1 0.2 0.3 0.40.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

He [%]

Ave

rage

air

mas

s fr

actio

n 1

/NΣω

Airi

boundary layer approximation

mech. CFD model

Figure 6.21 Comparison of the average interface air mass fraction obtained by a solution of the boundary layer equations for the vertical surface and by the mechanistic CFD model with corner-type geometry, 4%air =

∞ϖ .

0 0.1 0.2 0.3 0.40

0.01

0.02

0.03

0.04

0.05

He [%]

Ave

rage

hel

ium

mas

s fr

actio

n 1

/N Σ

ωH

ei

boundary layer approximationmech. CFD model

Figure 6.22 Comparison of the average interface helium mass fraction obtained by a solution of the boundary layer approximation for the vertical surface and by the mechanistic CFD model with corner-type geometry, 4%air =

∞ϖ . In Figure 6.23 the CFD prediction of the condensation heat transfer degradation due to the additional presence of the light gas has been shown. The obtained results can be explained as follows. As it has been mentioned in the previous section, there exists strong relation between accumulation of both noncondensable gases at the liquid-gas interface. This relation is given by the interaction between interface, noncondensable mass balances and buoyancy effects. As it was observed with the


71

similarity method and also with the CFD approach (Figures 6.19 and 6.20), addition of the light gas to the binary mixture is responsible for the increase of amount of heavy gas at the liquid-gas interface. In this way, addition of the light gas to the binary mixture with heavy one has indirect influence on the degradation of the interface condensation heat flux.

10 20 30 40 50 600.1

0.15

0.2

0.25

T∞ - T W

q cal/q

Nu

Mech. CFD ωair∞ = 4%, ω

He∞ = 0.1%

Mech. CFD ωair∞ = 4%, ω

He∞ = 0%

Figure 6.23 Prediction of the heat transfer degradation due to addition of helium in bulk mixture for 4%air =

∞ϖ . Summary This chapter describes the major results obtained in this thesis as well as summarises and extends the conclusions presented in the attached papers. It has been shown that the fully mechanistic approach (without correlations) can be applied for the fundamental research into the condensation from the multicomponent mixtures of condensable and noncondensable gases. It has also been shown that mechanistic approach in combination with the numerical methods can be used for fundamental studies of the influence of the wavy film structure on the condensation heat transfer enhancement. Similarity solutions of the boundary layer equations, fully coupled through the interface balances, give possibility to study and understand the physical processes which govern noncondensable components accumulation in the ternary mixtures of gases. Fundamental relations between both species have been noticed. They are given by the impermeability conditions at the liquid-gas interface for the noncondensable components and by buoyancy effects. The buoyancy effects are also responsible for the counter-current flows of the mixtures if a proper amount of very light gas is introduced.


72

Mechanistic modelling of the gravity-driven condensation from the ternary mixture of gases with the CFD approach gives better understanding of relations between noncondensable species. Local effects can be studied in details. For example, the influence of the ternary diffusivity coefficients and given geometry on the noncondensable gas accumulation can be investigated. The most important conclusions from the boundary layer analysis and CFD studies are that both approaches converge to each other for the binary mixtures, however for the ternary mixture the CFD approach allows to predict local phenomena such as the nonuniform accumulation of the noncondensable gas at the interface. It has also been found that both methods predict the influence of an addition of the light gas on the behaviour of the heavy one. Because the interface balances must be satisfied the forces created by the addition of the light gas cause higher accumulation of the heavy gas. In this way the addition of the light species is indirectly responsible for the further degradation of the condensation heat flux. Other theoretical and experimental observations, found in the literature review, have been depicted in the present studies. It has been found that the addition of the light gas has a more destructive influence on the condensation heat flux than the heavy gas. The analysis of the influence of the wavy film structure on the condensation heat transfer in binary mixtures confirms previous investigations that enhancement of the heat transfer coefficient is determined by the wave hydrodynamics. However, it has also been found that this process is more significant for the gas-side heat transfer coefficient. Qualitative studies have also been performed on the influence of the spontaneous condensation on the heat and mass transfer process in the binary mixture of air and water vapour. By implementation of a model for the dispersed phase growth, the phenomenon of the reduction of the convective mass flux has been predicted. These results show that such process should be taken into consideration if very high supersaturation conditions are expected to exist in the gaseous mixture. The above calculations also show the ability of combination of the mechanistic surface condensation models with models predicting the dispersed phase growth.

Chapter 7 Concluding remarks

73

Chapter 7

Concluding remarks The research presented in this thesis has been concerned with studying various aspects of water vapour condensation in presence of noncondensable gases. Such phenomena have important implications in the safe performance of nuclear power plants and both better understanding and improved modelling capabilities are required. The present research has contributed to both above-mentioned areas. In particular, the influence of the spontaneous condensation, the wavy film structure and the multi-component mixtures on the condensation rates has been studied in detail and the underlying physical processes have been elucidated. To this end mechanistic models have been developed and implemented in the commercial CFD code CFX, as well as in CFD software developed for the purpose of the present research. To study the forced-convection condensation, a mechanistic diffusion model has been developed and implemented into the CFX code. The model employs the low Reynolds k-ε model for turbulence in order to resolve the near-interface region and to allow for detailed modelling of the interfacial mass, momentum and energy transfer. The model has been applied to predict the condensation rates from binary mixtures on vertical and horizontal walls, which are either isothermal or adiabatic. It has been demonstrated that the mechanistic model is able to capture the heat transfer degradation phenomenon and the mean calculated heat transfer coefficient is in good agreement with existing correlations and available experimental data. The spontaneous condensation model has been implemented into the commercial CFD code and combined with the model for the forced-convection condensation. The objective has been to study the influence of the spontaneous condensation on heat and mass transfer rates. The calculations indicate that the heat transfer


74

coefficient at the cold wall is slightly lower (that is the degradation level is higher) when the spontaneous condensation occurs. This effect is due to the reduction of the convective mass flux towards the cold wall surface. Even though the obtained results are qualitatively correct, further validation of the model is required once proper experimental data are available. The influence of the wavy film structure on condensation has been investigated with CFD software developed for that purpose. Using prescribed wave shapes (both sinusoidal and soliton-shaped waves have been investigated), detailed distributions of velocities, temperatures and concentrations have been obtained in the liquid film and the mixture regions. The results reveal several mechanisms of the enhancement of heat transfer. On the mixture side the heat transfer coefficient is influenced by the local concentration of noncondensable gas and it increases on the trailing part of the wave. In the liquid film region a vortex is developed inside the wave and contributes to the convective heat transfer between the cold wall and the interface. Finally, in the valley region between two wave crests the conductive heat transfer is increased due to thinner water film. The influence of multi-component mixtures on condensation heat transfer has been investigated both analytically and numerically. In the analytical approach the conservation equations have been cast into the boundary layer approximation form and solved using the similarity method. For that purpose proper linearization assumptions have been necessary. In the numerical approach, generally formulated conservation equations and balance conditions at the interface have been implemented and solved with the commercial CFD code. The validity of both models has been confirmed against experimental data obtained by Al-Divany and Rose (1973). The analytical model has been used for parametric studies of the influence of light gas in a ternary mixture on the heat transfer rates. It has been observed that even small amount of the light gas causes higher accumulation of the heavy gas at the interface. The process is influenced by both the buoyancy effects as well as by the impermeability of the interface to noncondensable gases. The parametric study indicates that heat transfer degradation increases exponentially with the increase of the gas concentration, and that the increase is much faster in case of lighter gases. It has been also observed that if the concentration of the light gas is higher than a certain threshold value, the gas moves upwards due to buoyancy forces and separates from the mixture. For binary mixtures and ternary mixtures with low gas concentrations both the analitycal and the numerical models coincide and are in good agreement with experimental data. However, with increasing concentration of gases in ternary mixtures a discrepancy between the two solutions is observed. The reason for the discrepancy is the neglect of the dependence of diffusion coefficients on concentration distributions. This dependence is fully included in the mechanistic CFD model. Based on the numerical studies, it is concluded that the boundary layer

Chapter 7 Concluding remarks

75

approximation approach tends to under-predict the effect of the heat transfer degradation in ternary mixtures with significant concentrations of noncondensable gases. Present research indicates that mechanistic modelling of condensation in presence of noncondensable gases is feasible and lead to better understanding of the governing phenomena. Future research should focus on extension of present models to turbulent and transient flows, and on extensive validation of the models against detailed experimental data. Since there is a lack of high quality experimental database, it is recommended that more experiments are performed with focus on measurement of local parameters in the vicinity of the interface.

77

Acknowledgements First of all I would like to thank my supervisor Professor Henryk Anglart for the technical support during this project. I would also like to acknowledge Svenskt Kärntekniskt Centrum (SKC) and my Reference Group for the financial support and technical suggestions. A special thanks to my fiancée, family and friends for so important spiritual support.

References

79

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Karimi G., Kawaji M., 1998. An experimental study of freely falling films in a vertical tube. Chemical Engineering Science 55 – 20, 3501 – 3512. Karl J., Weiss T., 1997. Measurement of condensation heat transfer coefficients at stratified flow using Linear Raman Spectroscopy. Proceedings of The 1st Pacific Symposium on Flow Visualization and Image Processing, Honolulu, 479 – 484. Karl J., Hein D., 1999. Effect of spontaneous condensation on condensation heat transfer in the presence of noncondensable gases. Proceedings of the 5th ASME/JSME Joint Thermal Engineering Confernce. Karl J., 2000. Spontaneous condensation in boundary layers. Heat and Mass Transfer 36, 37 – 44. Kaufmann S., Hilfiker K., 1999. Prevention of fog in the condensation of vapour from mixtures with inert as, by a regenerative thermal process. Int. J. Therm. Sci. 38, 209 – 219. Kil S. H., Kim S. S., Lee S. K., 2001. Wave characteristics of falling liquid film on a vertical circular tube. Int. J. Refrigeration 24, 500 – 509. Koh J. C. Y., Sparrow E. M., Hartnett J. P., 1961. The two phase boundary layer in laminar film condensation. Int. J. Heat Mass Transfer 2, 69 – 82. Kuhn S. Z., Schrock V. E., Peterson P. F., 1997. An investigation of condensation from steam-gas mixtures flowing downward inside a vertical tube. Nuclear Engineering and Design 177, 53 – 69. Kunugi T., Kino C., 2005. DNS of falling film structure and heat transfer via MARS method. Computers and Structures 83, 455 – 462. Liu H., Todreas N. E., Driscoll M. J., 2000. An experimental investigation of a passive cooling unit for nuclear power plant containment. Nuclear Technology and Design 199, 243 – 255. Maheshwari N. K., Saha D., Sinha R. K., Aritomi M., 2004. Investigation on condensation in presence of a noncondensable gas for a wide range of Reynolds number. Nucear Engieering and Design 227, 219 – 238. Manthey A., Schaber K., 2000. The formation and behaviour of fog in a tube bundle condenser. Int. J. Therm. Sci. 39, 1004 – 1014. Maron D. M., Brauner N., Hewitt G.F., 1989. Flow patterns in wavy thin films: numerical simulation. Int. Comm. Heat Mass Transfer 16, 655 – 666.

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Oh S., Revankar S. T., 2006. Experimental and theoretical investigation of film condensation with noncondensable gas. Int. J. Heat Mass Transfer, Article in press. Paladino D., 2004. Investigation on Passive Safety Systems in LWRs. Royal Institute of Technology, Stockholm, PhD thesis. Patnaik V., Perez – Banco H., 1995. A study of absorption enhancement by wavy film flows. Int. J. Heat Fluid Flow 17 – 1, 71 – 77. Park S. K., Kim M. H., Yoo K. J., 1996. Condensation of pure steam and steam-air mixture with surface waves of condensate film on a vertical wall. Int. J. Multiphase Flow 22, 893 – 908. Park S. K., Kim M. H., Yoo K. J., 1997. Effects of wavy interface on steam-air condensation on a vertical surface. Int. J. Multiphase Flow 23 – 6, 1031 – 1042. Park H. S., No H. C., 1999. A condensation experiment in the presence of noncondensables in a vertical tube of a passive containment cooling system and its assessment with RELAP5/MOD3.2. Nuclear Technology 127, 160 – 168. Park C. D., Nosoko T., 2003. Three-dimensional wave dynamics on a falling film and associated mass transfer. AIChE 49 – 11, 2715 – 2727. Perrin A., Hu H. H., 2006. An explicit finite-difference scheme for simulation of moving particle. Journal of Computational Physics 212, 166 – 187. Peterson P. F., 1996. Theoretical basis for the Uchida correlation for condensation in reactor containments. Nuclear Engineering and Design 162, 301 – 306. Peterson P. F., 2000. Diffusion layer modelling for condensation with multicomponent noncondensable gases. Journal of Heat Transfer 122, 716 – 720. Prosperetti A., 1979. Boundary conditions at a liquid-vapour interface. Meccanica, 34 – 47. Sage F. E., Estrin J., 1976. Film condensation from a ternary mixture of vapours upon a vertical surface. Int. J. Heat Mass Transfer 19, 323 – 333. Shepel S. V., 2003. Report on the first phase of the ISP – 47 project (TOSQAN). Siddique M., Golay M. W., Kazimi M. S., 1993. Local heat transfer coefficients for forced-convection condensation of steam in a vertical tube in the presence of a noncondensable gas. Nuclear Technology 102, 386 – 402.

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Yao G. F., Ghiaasiaan S. M., Eghbali D. A., 1996. Semi-implicit modelling of condensation in the presence of non-condensables in the RELAP5/MOD3 computer code. Nuclear Engineering and Design 166, 277 – 291. Yoshimura P. N., Nosoko T., Nagata T., 1996. Enhancement of mass transfer into falling laminar liquid film by two-dimensional surface waves – some experimental observations and modelling. Chemical Engineering Science 51 – 8, 1231 – 1240. Young J. B., 1991. The condensation an evaporation of liquid droplets in a pure vapour at arbitrary Knudsen number. Int. J. Heat Mass Transfer 34 – 7, 1649 – 1661. Young J. B., 1993. The condensation and evaporation of liquid droplets at arbitrary Knudsen number in the presence of an inert gas. Int. J. Heat Mass Transfer 36 – 11, 2941 – 2956. Zaleski S., Computation of multiphase flow by volume of fluid and high-order front tracking methods. Lecture notes.

Appendix 1

87

Appendix 1 Numerical implementation of the model of wavy film flow interaction with binary mixture of gases

A1.1 Numerical scheme Two-dimensional conservation equations, for both liquid and gaseous phases, in the general coordinate system can be written in the following, vector form:

JJγ

ηJβ

ηξJα

ξ

ηξt

nn2

2

nn2

2

nn

nnn

SqqqΠΓ

GFq

+

∂∂

+

∂∂

∂∂

+

∂∂

=

=∂∂

+∂∂

+∂∂ ∗∗∗∗

∗

(A1.1)

where n = L, M, denotes liquid or gaseous phase, respectively. Symbol nS in the above equation can be understood as a contribution vector to the transport equations and is a source term. In this case Sn contains buoyancy forces, which result from the nonuniformity in the noncondensable mass fraction field:

( )[ ]TAAn 000gα0 ∞−= ϖϖS , (A1.2) where α is given by the following relation:

( )( ) ∞

∞

−−−

=AVAA

AVA

MMMMM

αϖ

ϖ . (A1.3)

Further in the text, bold notation for vectors and matrices has been omitted. Vectors ∗

nq , ∗∗nF , ∗∗

nG are given by the following relations:

Jq

q nn =∗ ,

∇ΠΓ+

∂∂

+∂∂

=∗∗n

2

nn

nn

n qJξ

J

GyξF

xξ

F , (A1.4)


88

∇ΠΓ+

∂∂

+∂∂

=∗∗n

2

nn

nn

n qJη

J

GyηF

xη

G . (A1.5)

Vectors nnn G,F,q and Γ contain unknown variables and fluid properties. Including matrix Πn, vectors nnn G,F,q can be written as follows:

nA

2

2

n

nA

2

2

n

nA

n

vvT

p/ρvuv

va

G,

uuTuv

p/ρuua

F,Tvup

q

+=

+=

=

ϖϖϖ

(A1.6 - 8)

n

n

nAC

n

1000001000001000001000000

Π,

Daνν0

Γ

=

=. (A1.9 – 10)

Because the nonstaggered grid has been applied, checkerboard problem has been avoided by the so-called artificial compressibility method. This method has been used successfully in many fluid dynamics computations (Perrin and Hu 2006) and is based on the pseudo-transient formulation of the mass conservation equation. Therefore, the continuity equation has been written in the following form:

0yv

xua

tp 2 =

∂∂

+∂∂

+∂∂ . (A1.11)

With such expression, solution of the system of conservation equations doesn’t have physical meaning until the steady-state is reached. Coefficient “a” can be interpreted as the speed of sound in the medium if it was compressible. In order to perform such pseudo-transient calculations, Euler implicit scheme has been applied:

( ) ∆tq,tfqq 1m1m

m1m ++

+ += , (A1.12) where m is the m-th time step. Schematically the transformation from the physical to the computational domain is shown in the Figure A.1:

Appendix 1

89

Figure A1.1 Transformation from the physical to the computational domain. Between these two domains the following transformation coefficients are given:

1kj,1k2,j

1kj,1k2,j

ξξxx

ξx

+++

+++

−

−≈

∂∂ ,

k1,j2k1,j

k1,j2k1,j

ξηxx

ηx

+++

+++

−

−≈

∂∂ , (A1.13 – 14)

1kj,1k2,j

1kj,1k2,j

ξξyy

ξy

+++

+++

−

−≈

∂∂ ,

k1,j2k1,j

k1,j2k1,j

ξηyy

ηy

+++

+++

−

−≈

∂∂ . (A1.15 – 16)

With the above relations, the Jacobian of the transformation can be calculated from the following expresion:

∂∂

∂∂

−∂∂

∂∂

=ξy

ηx

ηy

ξxJ . (A1.17)

Coefficients γβ,α, , in two-dimensional space, have the following forms:

22

yξ

xξα

∂∂

+

∂∂

= ,

∂∂

∂∂

+∂∂

∂∂

=yη

yξ

xη

xξ2β , (A1.18 – 19)

22

yη

xηγ

∂∂

+

∂∂

= , (A1.20)

where

x

y ηphysical domain

computational domain

transformation

ξ


90

1Jηy

xξ

−

∂∂

=∂∂ ,

1Jηx

yξ

−

∂∂

−=

∂∂ ,

1Jξy

xη

−

∂∂

−=

∂∂ ,

1Jξx

yη

−

∂∂

=∂∂ . (A1.21 – 24)

Laplacians of ηξ, can be expressed as:

( ) ( )1

2

2

2

2

1

1

2

2

2

2

2

Jη

xηy

ηy

ηx

Jγ

J

xηξη

yyηξη

xJβ

Jξ

xηy

ξy

ηx

Jα

Jξ

−−

−

∂∂

∂∂

−∂∂

∂∂

+

∂∂∂

∂∂

−∂∂∂

∂∂

+

+

∂∂

∂∂

−∂∂

∂∂

=

∇

, (A1.25)

( ) ( )1

2

2

2

2

1

1

2

2

2

2

2

Jη

yξx

ηx

ξy

Jγ

J

yηξξ

xxηξξ

yJβ

Jξ

yξx

ξx

ξy

Jα

Jη

−−

−

∂∂

∂∂

−∂∂

∂∂

+

∂∂∂

∂∂

−∂∂∂

∂∂

+

+

∂∂

∂∂

−∂∂

∂∂

=

∇

. (A1.26)

Because of the structure of vectors Fn and Gn, the following relations can be written:

( ) ( )nqvqqGG,quq

qFF n

nnnn

nn Π−

∂∂

=Π−

∂∂

= , (A1.27 – 28)

where nq

F

∂∂ and

nqG

∂∂ are so called flux Jacobian matrices given as:

Appendix 1

91

nA

2

n

u0000u0T000uv0000u2/ρ1000a0

qF

ϖ

=

∂∂ , (A1.29)

nA

2

n

v0000vT0000v20/ρ100uv000a00

qG

ϖ

=

∂∂ . (A1.30)

Expanding into Taylor series about the m-th time level, 1mF + and 1mG + can be approximated by the following expression:

1mm

m1m ∆qqFFF ++

∂∂

+= , 1mm

m1m ∆qqGGG ++

∂∂

+= , (A1.31 – 32)

1mm

m1m ∆qqqqq ++

∂∂

+= , (A1.33)

where:

m1m1m qq∆q −= ++ . (A1.34) In order to compute derivatives of the transformation coefficients, the following, second order approximations have been applied:

( )21k2,j1k1,j1kj,

2

2

∆ξxx2x

ξx +++++ +−≈

∂∂ , (A1.35)

( )∆ξ∆η4

xxxxx

ηξk2,jkj,2kj,2k2,j ++++ −+−

≈∂∂

∂∂ , (A1.36)

( )22k1,j1k1,jk1,j

2

2

∆η

xx2xηx +++++ +−≈

∂∂ , (A1.37)


92

( )21k2,j1k1,j1kj,

2

2

∆ξyy2y

ξy +++++ +−≈

∂∂ , (A1.38)

( )∆ξ∆η4

yyyyy

ηξk2,jkj,2kj,2k2,j ++++ −+−

≈∂∂

∂∂ , (A1.39)

( )22k1,j1k1,jk1,j

2

2

∆ηyy2y

ηy +++++ +−≈

∂∂ . (A1.40)

Approximations of mixed derivatives with the five-point formula have been performed by employing additional coefficients in the following way:

( )

( ) ( )

( )nPES3

PWN2PEN1

nSWNWSENE

n

ψψψa

ψψψaψψψa∆ξ∆η4

1

ψψψψ∆ξ∆η4

1

ψηξ

−+−

+−+−−+=

=+−−=

=∂∂

∂∂

, (A1.41)

where: nn q

Jβψ

= .

Symbols NE, NS, etc. represent the so-called compass notation and gives addresses of the grid points (Figure A.2):

Figure A1.2 Compass notation. An example of approximation of coefficients mq

la and 1m∆qla

+ is presented below:

W

N

E

P S

NE

NW SW

SE

Appendix 1

93

−

+

=

P

m

E

m

N

mq1

NE

m qJβq

Jβq

Jβaq

Jβ m , (A1.42)

−

+

=

++++ +

P

1m

E

1m

N

1m∆q1

NE

1m ∆qJβ∆q

Jβ∆q

Jβa∆q

Jβ 1m , (A1.43)

P

m

E

m

N

m

NE

m

q1

qJβq

Jβq

Jβ

qJβ

am

−

+

= , (A1.44)

P

1n

E

1n

N

1n

NE

1n

∆q1

∆qJβ∆q

Jβ∆q

Jβ

∆qJβ

a1n

−

+

=+++

+

+ , (A1.45)

where: 1mm1m qq∆q ++ −≈ is used only in order to calculate mixed derivatives coefficients and should not be mixed with the time derivative. A1.2 Solution method In order to write a discrete form of the generic conservation equation (Equation A.1), the five-point formula with compass notation has been applied. The whole system of equations has been written in the following form:

ijij Q∆φA =⋅ , (A1.46) where ijA is the matrix of coefficients, j∆φ is the vector of unknowns and iQ is the vector of remaining coefficients. In a discrete form the previous equation becomes:

nlk,

1n1lk,S

1n1lk,N

1nl1,kW

1nl1,kE

1nlk,P Q∆φA∆φA∆φA∆φA∆φA =++++ +

−++

+−

++

+ . (A1.47) Such system of equations has been solved with the Matlab code by inversion:

i1

ijj Q)(A∆φ −= . (A1.48)


94

Coefficients of matrix ijA , with the compass notation, are given by the following expressions: 1. W (west)

( )( )

−

ΓΠ

+

ΓΠ

+

+

∇ΓΠ++−

∗∗242

2yx

P

aaJβ

∆ξ∆η4Jα

∆ξ

JξB

Jξ

AJξ

∆ξ21∆tJ

, (A1.49)

2. S (south)

( )( )

−

ΓΠ

+

ΓΠ

+

+

∇ΓΠ++−

∗∗342

2yx

P

aaJβ

∆ξ∆η4Jγ

∆η

JηB

Jη

AJη

∆η21∆tJ

, (A1.50)

3. P (central)

( ) ( )( )∗∗∗∗ −−+

ΓΠ−

+Γ+ 413222ij aaaa

∆ξ∆η4β∆t

∆ηγ

∆ξα∆t2δ1 , (A1.51)

4. N (north)

( )( )

−

ΓΠ

−

ΓΠ

−

+

∇ΓΠ++

∗∗212

2yx

P

aaJβ

∆ξ∆η4Jγ

∆η

JηB

Jη

AJη

∆η21∆tJ

, (A1.52)

5. E (east)

( )( )

−

ΓΠ

−

ΓΠ

−

+

∇ΓΠ++

∗∗312

2yx

P

aaJβ

∆ξ∆η4Jα

∆ξ

JξB

Jξ

AJξ

∆ξ21∆tJ

, (A1.53)

Appendix 1

95

with qFA

∂∂

= and qGB∂∂

= .

A1.3 Modeling of the boundary conditions at the liquid-gas interface Based on the previously presented numerical scheme, liquid and mixture domains have been computed separately and linked by interface boundary conditions. The interface temperature, which is the same for both liquid and gaseous domains, is calculated with the interface energy balance as follows:

δM

MfgδV

δL

L yT

khny

Tk

∂∂

−=∂∂

− . (A1.54)

In the above expression it has been assumed that the wave amplitude is much smaller than the wave length. Because the system of conservation equations is solved in the general coordinate system, this relation has been finally implemented as:

δ

M

MM

M

MMMfg

δV

δ

L

LL

L

LLL

ηT

yη

ζT

yζkhn

ηT

yη

ζT

yζk

∂∂

∂∂

+∂∂

∂∂

−=

=

∂∂

∂∂

+∂∂

∂∂

−. (A1.55)

In the same manner, the mixture mass transfer into the liquid domain can be implemented as follows:

δ

M

VM

M

VMδV

AVMδV ηy

ηζy

ζ1

Dρn

∂∂

∂∂

+∂∂

∂∂

−−

=ϖϖ

ϖ. (A1.56)

Relation between the interface noncondensable mass fraction and the interface temperature is given by the saturation conditions. Gradients at the liquid-gas interface are calculated with the second-order approximation, what is consistent with the applied numerical scheme:

∆ζφφ

ζφ PN −≈

∂∂ . (A1.57)


96

The relation (A1.57), even if it looks as the first order approximation, is in reality the second order one. Such result can be easily obtained by calculation of the unknown value of φ at point S with a linear extrapolation using points P and N.

Appendix 2

97

Appendix 2 Boundary layer approximation A2.1 Physical model

Figure A2.1 The physical model of forced convection condensation from the binary mixture with noncondensable gas on a horizontal, isothermal surface. The model of forced convection condensation from the binary mixture of condensable and noncondensable gas has been schematically shown in Figure A2.1. This model is specially constructed in order to confirm assumption that both, specification of the liquid film thickness based on the order of magnitude analysis, and the wall temperature are equivalent. There are several assumptions and simplifications used in this model. Most important ones have been listed below. Assumptions and simplifications with regard to the liquid film

flow of the liquid film is laminar, film thickness is considered to be proportional to the 2/1x based on the

order of magnitude analysis, there is no slip between film and gaseous phases, film interface is impermeable for the noncondensable component, liquid film properties are assumed to be constant,

∞Mu

x, u

y, v

Temperature profile Water vapour

mass fraction. profile

Saturated binary mixture of condensable and noncondensable gas

1CA =+ ∞∞ ϖϖ

∞MT

δT

WT

∞Cϖ

δCϖ δ


98

boundary layer approximation is valid for the liquid film phase. Assumptions and simplifications with regard to the gaseous phase

binary mixture of any condensable and noncondensable gas, mixture is in the saturation state at the liquid-gas interface, flow of the mixture is laminar, mixture inflow velocity and temperature profiles are uniform, bulk amount of noncondensable gases and the bulk mixture temperature

are known and are uniform, boundary layer approximation is valid for the gaseous phase, binary mixture components are in dynamic and thermal equilibrium.

Assumptions and simplifications with regard to geometry

smooth, horizontal, isothermal wall.

Solution method

boundary layer approximation with the similarity method.

A2.2 Liquid conservation equations In order to find the analytical solution of the liquid-side boundary layer, partial differential equations have been simplified to the system ordinary differential equations. Liquid phase velocity components, u and v, can be expressed with a definition of the stream function as:

yψ

u LL ∂

∂= ,

xψ

v LL ∂

∂−= . (A2.1 – 2)

Next, the following similarity variables can be introduced:

2/1L

L xyc

η = , ( )( ) 2/1

L

LLL

xνU

ψηf

∞= ,

δW

δL

L TTTT

θ−−

= . (A2.3 – 5)

Because the mass conservation equation is automatically satisfied by the definitions of the stream functions, the liquid-side momentum and energy equations become:

0dn

fdf

21

dηfd

2L

L2

L3L

L3

=+ , 0dηdθ

fPr21

dηθd

L

LLL2

L

L2

=+ . (A2.6 – 7)

Appendix 2

99

where Pr is the Prandtl number. For an isothermal surface with no-slip conditions, the above equations are subject to the following boundary conditions:

0dηdf

W

L

L = , 0f WL = , 1θW = . (A2.8 – 10)

A2.3 Mixture conservation equations Gaseous phase velocity components, u and v, can be expressed with a definition of the stream function as:

yψ

u MM ∂

∂= ,

xψ

v MM ∂

∂−= . (A2.11 – 12)

Introducing the following parameters, representing the dimensionless distance, the dimensionless stream function, the dimensionless temperature and the noncondensable mass fraction, respectively:

2/1M

M xδ)(yc

η−

= , ( )( ) 2/1

M

MMM

xνU

ψηf

∞= , (A2.13 – 14)

∞

∞

−−

=TTTT

θ WM

M,

∞=

A

AΩϖϖ , (A2.15 – 16)

the mixture momentum, energy and mass fraction conservation equations can be written as follows:

0dn

fdf

21

dηfd

2M

M2

M3M

M3

=+ , 0dηdθ

fPr21

dηθd

M

MMM2

M

M2

=+ ∞ , (A2.17 – 18)

0dηdΩfSc

21

dηΩd

MMM2

M

2

=+ ∞ . (A2.19)

In the above relations the Prandtl and the Schmidt numbers have been defined for the ambient conditions. Mixture side conservation equations are subject to the following boundary conditions:


100

1dηdf

M

M =∞

, 0θM =∞ , ( ) 1, =∞Ω x . (A2.20 – 22)

A2.4 Interface conditions Interface conditions are based on the balances of mass, momentum, energy and interface impermeability assumption for the noncondensable mass flux into the liquid phase. They can be written as follows:

δV

δ

MMM

δ

LLL ndxdδuvρ

dxdδuvρ =

−=

− (A2.23)

δM

δL uu = ,

δM

M

δL

L yu

µy

uµ

∂∂

=∂∂ , (A2.24 – 25)

δ

MMfg

δV

δL

L yT

khny

Tk

∂∂

−=

∂∂

− , (A2.26)

0y

Dρxδuvρ A

ACMMMA =∂∂

−

∂∂

−ϖ . (A2.27)

Applying similarity variables defined in the previous sections, the mass and momentum interface balances become:

δM

δL

2/1

MM

LL ffµρµρ

=

, δ

M

Mδ

L

L

dηdf

dηdf

= , (A2.28 – 29)

δ

2M

M2δ

2L

L22/1

MM

LL

dηfd

dηfd

µρµρ

=

. (A2.30)

Definitions of the heat conductivity of the liquid phase and the Jakob number as:

LLLL ρCpak = , ( )fg

WL

hTTCp

Ja−

=∞

, (A2.31 – 32)

permit to write another form of the Jakob number:

Appendix 2

101

δ

M

M

LL

MM

L

LδLL dη

dθckck

dηdθ

fPr21Ja

−−= . (A2.33)

Hence the interface nondimensional temperature is given by:

( ) δLLL

fg

δ

M

M

LL

MMδ

L

L

Mδδ

fPrCp0.5

hdηdθ

ckck

dηdθ

TTθ

−

−= ∞ , (A2.34)

where the latent heat is approximated to be a function of the interface temperature, only:

( )δfg Tfh = . (A2.35) With definition of the Schmidt number, the interface impermeability condition for the noncondensable gas allows to write the interface dimensionless mass fraction in the following form:

( ) 1δM

δ

M

δ

A

Aδ ScfdηdΩ2Ω −

∞ −==ϖϖ . (A2.36)

Method of finding the wall temperature is based on the assumption that liquid film thickness is given by the following relation:

2/1ψxδ = , (A2.37) where ψ in Equation (A2.37) is a constant expressed by the fluid properties. As a result, temperature difference W

M TT −∞ which satisfies the above statement is found. This can be proved to be equivalent to the specification of the wall temperature and finding the film thickness δ. The main reason of such treatment is the fast solution of the system of ordinary, nonlinear differential equations. This system is solved with the shooting method, where the two-sided boundary value problem is converted to the initial value one. Comparison of this method with solutions to the same problem obtained by Sparrow et al. (1967) is presented in Figure A.2.2:


102

0.5 1 1.5 2

0.4

0.5

0.6

0.7

0.8

0.9

1

1/ηδ

ωai

r∞

/ωai

rδ

0 0.5 1 1.5

0.4

0.5

0.6

0.7

0.8

0.9

1

RCp(Tδ-T W)/(h

fgPr)

ωai

r∞

/ωai

rδ

Sparrow et al. 1967

boundary layer approx.with film model

Sparrow et al. 1967boundary layer approx.with film model

Figure A2.2 Comparison of results obtained with the film thickness model given by Equation (A2.37) and results obtained by Sparrow et al. (1967) for the Schmidt number 0.55.

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