petronio (2011) phd t - numerical investigation of condensation and evaporation effects inside a tub

8/12/2019 Petronio (2011) PhD T - Numerical Investigation of Condensation and Evaporation Effects Inside a Tub

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University of Trieste

School of Doctorate in Environmental and Industrial Fluid

Mechanics

XXIII Cycle

NUMERICAL INVESTIGATION OF CONDENSATION

ANDEVAPORATION EFFECTS INSIDE A TUB

by

Andrea Petronio

April 2011

SUPERVISORS

Prof. V. Armenio

ASSISTANT SUPERVISORS

Ing. G. Buligan


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iii

A Norah e Sarah,per tutte queste H!


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Sommario

Lobiettivo principale del progetto di ricerca sviluppato nella presente tesi di dottoratoe quello di comprendere meglio le problematiche riguardanti le prestazioni di asciu-gatura della lavastoviglie, con speciale riferimento alla modellazione dei fenomeni dievaporazione e condensazione che avvengono nella vasca. Tipicamente lacqua per il

risciacquo finale viene portata ad una temperatura di 70C che scalda le stoviglie per-mettendo a queste di immagazzinare energia termica. La lavastoviglie si raffreddadallesterno, cosicche la vasca risulta essere piu fredda dei piatti posti allinterno. Inquesto contesto lacqua puo evaporare dalle superfici delle stoviglie e condensare sullepareti della vasca stessa.

Il sistema fisico puo essere descritto come un flusso in presenza di cambiamenti difase. Tali tipologia di flussi ha un ruolo cruciale in molti processi naturali e tecnologici,in particolare in quelli in cui si hanno asciugatura o formazione di condensa sulle super-fici solide. Tuttavia, pur essendo cos comuni nelle applicazioni ingegneristiche, la lorocomprensione e lontana dallessere completa. Il complesso problema fisico puo esseresuddiviso in tre sotto-problemi: la trasmissione del calore tra il corpo bagnato ed illiquido sulla sua superficie; il trasferimento di calore e massa tra la fase liquida e quellagassosa; il flusso della fase gassosa che risulta essere molto influenzato dalle forze digalleggiamento dovute alle variazioni di densita causate dalla diffusione di temperaturae concentrazione di vapor acqueo.

Dallo studio della letteratura risulta che tale problema non sia stato ancora investi-gato completamente. In particolare non e mai stato proposto un modello adatto a scopiingegneristici, cioe per problemi di larga scala con geometrie complesse, che considerilevoluzione del film liquido durante processi di asciugatura. Questo progetto vuolecontribuire allo sviluppo della ricerca in questo settore.

Il modello matematico del flusso daria in presenza di evaporazione e condensazione

e stato implementato numericamente nellambiente open-source OpenFoam. Il modelloconsiste nella formulazione delle equazioni di Navier-Stokes per flussi incomprimibili piule equazioni del trasporto per la temperatura e la concentrazione di vapore. Entrambigli scalari sono considerati attivi e le variazioni di densita sono state incorporate sottolapprossimazione di Boussinesq. Si assume inoltre lapprossimazione a film sottile, percui si e inteso che film liquidi, gocce ed, in genere, le zone bagnate di un solido possanoessere considerate come un film liquido continuo. Tale film sottile e stato interpretatocome una condizione al contorno per il flusso daria, prescrivendo una condizione diDirichlet per la temperatura e per il vapore. Questultimo allinterfaccia del liquidoe considerato in condizione di saturazione. Il calcolo della velocita di evaporazioneallinterfaccia, imposta anche come condizione al contorno per il campo di velocita, ha

permesso la quantificazione del processo di evaporazione/condensazione consentendo il

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calcolo della massa dacqua evaporata/condensata.Il modello numerico e stato validato con i dati di letteratura per poi essere applicato

nello studio del flusso su un cilindro bagnato, tra due piani paralleli. In questo lavoro e

stato evidenziato leffetto sul flusso di evaporazione attorno al cilindro delle condizionialle pareti, considerate come bagnate o asciutte ed adiabatiche. Inoltre e stato valutatoanche leffetto della distanza del cilindro stesso dalle pareti.

Successivamente il modello e stato applicato ad una geometria 2D della lavastoviglie.I risultati mostrano che il flusso evolve secondo uno schema preciso: le forze di galleg-giamento danno luogo ad un moto convettivo che si alza dalle stoviglie piu calde edumide, che poi scende lungo alle pareti piu fredde e meno umide. Unulteriore analisi estata fatta simulando il processo fino allasciugatura completa di un film uniformementedistribuito su tutte le stoviglie. Negli stadi intermedi del processo e stato osservato che,attorno alle porzioni di stoviglie gia asciutte, il galleggiamento risulta essere ridotto ela velocita dellaria minore, per il mancato rilascio di vapor acqueo.

Un passo ulteriore verso la modellazione della lavastoviglie e stato condotto con-siderando una geometria 3D semplificata per testare il modello e verificare le caratter-istiche richieste alla griglia computazionale. Anche per questa configurazione e statoosservato linstaurarsi del moto convettivo e leffetto dellasciugatura sul flusso.

Infine si e iniziato a studiare il caso della lavastoviglie 3D. La simulazione e potutadurare pochi secondi fisici, nei quali hanno iniziato a svilupparsi sopra le stoviglie icaratteristici plume. Successivamente delle instabilita numeriche hanno dato luogo avalori di pressione non fisici determinando linterruzione del programma. Tale compor-tamento e stato spiegato dalla mancata dissipazione turbolenta nel flusso. Lattivazionedel modello LES di Smagorinsky con lanalogia di Reynolds per la determinazione dellediffusivita turbolente dei due scalari ha dato luogo ad una soluzione numericamente

stabile. Tuttavia la eccessiva viscosita di sotto-griglia ha sovrastimato la diffusionedegli scalari, inficiando laccuratezza della simulazione.

Per includere nel modello laccoppiamento termico che caratterizza il processo diasciugatura e stata scelta la tecnica di decomposizione di domini detta di Dirichlet-Neumann in quanto e risultata essere efficace e semplice da implementare. Essa imponela continuita della temperatura e il bilancio dei flussi di calore attraverso le interfacce.

Inoltre e stato proposto un modello opportuno per la distribuzione della temper-atura nel film liquido, per riprodurre nella maniera corretta il trasferimento di caloreattraverso il film stesso. Ciascuna di queste due parti e stata implementata e testataindividualmente cosicche la loro inclusione nel modello potra avvenire in un succes-sivo sviluppo della presente ricerca. Inoltre e in fase di sviluppo limplementazione delmodello di sotto-griglia LES dinamico lagrangiano che permettera di superare i limitiriscontrati nel utilizzo del modello di Smagorinsky.


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Summary

The overall aim of the project is to get more insights regarding issues related to dish-washers drying performances, with special regard on the modeling of condensationand evaporation process inside the tub. Typically, the water for the final rinse cycle isheated up to 70C to heat up the dishware, i.e. to store thermal energy into them. The

dishwasher cools down from the outside, hence the tub wall is colder than the dishes.Therefore, the water on the dishes is evaporated from the dishware and condensed onthe tub wall.

The physical system under consideration can be described as a mixed convectionflow in presence of phase change phenomena. This kind of systems have crucial rolein several natural and technological processes, in particular in those involving dryingand wetting of solid surfaces. Although such flows are quite common in engineeringapplications, their understanding is far from being complete. The complex physicsinvolved can be divided in three sub-problems: the exchange of heat between wettedsolid bodies and the thin liquid film or drops laying on their surface; the heat and masstransfer between liquid phase and the surrounding gas through change of phase; thegaseous flow which is greatly influenced by the buoyancy forces due to density variationsarising from the diffusion of temperature and vapor concentration.

To the best of authors knowledge, in literature such a problem has not been com-pletely faced yet. In particular a model suitable for engineering purposes, i.e. on largescales and complex geometries, of liquid film during drying process has never beenproposed yet. This PhD project has been meant to start the research on such topic.The numerical implementation has been carried out in the OpenFoam environment, anopen-source C+ + CFD tool.

The mathematical model for gaseous flow with the evaporation and condensationat wetted surfaces has been implemented. It consists of the incompressible formulation

of the Navier-Stokes equations, plus the transport equations for temperature and vaporconcentration,. Both are treated as active scalars. The density variations are takeninto account by means of the Boussinesq approximation. A thin film approximation isassumed, meaning that all liquid films, liquid patches and drops on the wetted surfacesare considered as a continuous film. Moreover the thin liquid film is treated as aboundary condition for the gaseous flow. It prescribes a Dirichlet condition for thetemperature and for the vapor concentration that at the liquid interface is consideredin saturation. The evaporation/condensation process is evaluated by the evaporationvelocity, V e, at the liquid-gas interface by the relation (2.44) as explained in 2.4,providing the boundary condition for the velocity field, and the water mass transferrate.

This numerical model has been validated against literature results of[19] and then

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exploited firstly in the study of the archetypal case of the flow over a wetted cylinderconfined between two parallel walls [28]. In this work the effects of walls conditions,wetted or dried and adiabatic, on mean evaporative fluxes around the cylinder have

been enlighten along with the dependence on the gap between the body and the wall.In Figure5.4aan instantaneous vapor concentration contour plot is shown for the casewith wetted walls and cylinder placed at one forth of the channel width. It shows thecomplex pattern flow that typically arises in this problems.

Successively a first application of this model on a real configuration has been per-formed on a 2D dishwasher. In Figure 5.9b is shown an instant contour plot of thevapor concentration, that also reveals the complex flow pattern among the dishware.An interesting analysis has also been carried out taking retrieving the actual drying.An initial uniform film thickness has been prescribed on the tableware, and the evolu-tion in time has been followed up to the complete drying. In intermediate situationsthe surface portion already dried cannot release more vapor into the flow, lowering the

buoyancy near the body and slowing down the flow velocity in the plume.A further step towards the real-scale 3D dishwasher has been done on a simplified

geometry in order to check the model capabilities and the mesh requirements in a 3Dcase. The results shows again that the buoyancy force starts the convective cells andthe evolution of the liquid thickness takes place.

Finally the case of a real-scale 3D dishwasher, with the proper geometry has beenalso tackled. The simulation ran for few physical seconds and the characteristic plumesdevelop as can be seen in Figure 5.16, then a numerical instability appears leading tounphysical pressure values. This is thought to be due to turbulence missing dissipa-tion. The standard Smagorinsky LES model with the Reynolds analogy for the eddydiffusivity of the two scalars has been applied. It stabilized the solution, but the re-

sults appear to be to much diffused due to an overestimation of the sub-grid viscosity.In general the flow in the real-scale dishwasher appears to be turbulent. In order toproperly take into account the effects of turbulence in home-appliance applications theLES approach is preferable, being DNS computation not affordable and the flow tran-sient in nature. The geometries involved are very complex and the flows are stronglyanisotropic. Among the possible LES models known in literature the most appealingfor such a task is the Lagrangian dynamic sub-grid scale model proposed in [22]. Theimplementation of Lagrangian dynamic model is in progress.

The drying process is known to be strongly affected by the thermal coupling of theliquid phase either with the solid substrate and the air[9]. In order to incorporates suchan important features the Dirichlet-Neumann domain decomposition technique hasbeen chosen. It appears to be effective and straightforward to implement. It enforcesthe continuity of temperature and the balance of the heat fluxes across each interface.In order to properly take into account the heat transfer mechanism through the liquidlayer a suitable temperature film model has been proposed. As expected the rulingterm in the heat transfer process among the three media is the latent heat flux ofevaporation. Each of these parts have been already implemented and some test casehave been performed in order to validate the code.


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Contents

Sommario v

Summary vii

1 Introduction 1

1.1 Project aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Formulation 5

2.1 Mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Gaseous phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Sponge region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 LES turbulence model . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Thin-film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Interaction with the substrate. . . . . . . . . . . . . . . . . . . . 112.3.2 Evaporating drops physic . . . . . . . . . . . . . . . . . . . . . . 13

2.3.3 Modeling hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.4 Packing ratio argument . . . . . . . . . . . . . . . . . . . . . . . 152.4 E/C b.c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.4 Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Thermal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.2 DDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.3 Film model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Model summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Numerical Implementation 31

3.1 FVM discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.2 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.3 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 PISO algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Current model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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3.3.1 Current model algorithm . . . . . . . . . . . . . . . . . . . . . . 43

3.3.2 E/C implementation . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.3 Film implementation . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 LES models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.1 Smagorinsky model. . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.2 Plane averaged dynamic model . . . . . . . . . . . . . . . . . . . 50

4 Validation and Testing 53

4.1 PISO algorithm testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 Current model stability test . . . . . . . . . . . . . . . . . . . . . 53

4.1.2 Accuracy-Co number test . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Validation test case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 LES turbulence models test . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Drop test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.1 Drop as a thin film . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.5 DDM validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5.2 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5.3 Constant heat source inLeft . . . . . . . . . . . . . . . . . . . . 68

4.5.4 Solid-fluid case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5.5 Multi-dimensional cases . . . . . . . . . . . . . . . . . . . . . . . 71

4.5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Cases studied 79

5.1 Wetted cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1.2 Non-wetted walls . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1.3 Wetted walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2 2D DW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.2 Case set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2.3 Case discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.4 Drying process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3 Plate in a box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.1 experimental comparison . . . . . . . . . . . . . . . . . . . . . . 915.4 3D DW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Conclusions 97

A Dynamic film model 101

A.1 Film Thickness Evolution Equation. . . . . . . . . . . . . . . . . . . . . 101

A.1.1 Basic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.1.3 Long-wave Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 103

A.1.4 Film Thickness Evolution Equation. . . . . . . . . . . . . . . . . 104

A.1.5 Incorporating evaporation and condensation. . . . . . . . . . . . 105


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CONTENTS xi

Bibliography 107


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List of Figures

1.1 The complex geometry of a dishwasher tub. . . . . . . . . . . . . . . . . 1

2.1 Wetting sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Young contact angle and interfacial tensions. . . . . . . . . . . . . . . . 132.3 The surface curvature and fringe field effects. . . . . . . . . . . . . . . . 15

2.4 Decomposition of domain. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Models for the temperature prediction in the thin film. . . . . . . . . . . 24

2.6 Film temperature distribution 1. . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Film temperature distribution 1. . . . . . . . . . . . . . . . . . . . . . . 27

3.1 A Control Volume and its parameters. . . . . . . . . . . . . . . . . . . . 33

3.2 over-relaxed correction. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Boundary condition discretization scheme. . . . . . . . . . . . . . . . . . 36

3.4 The PISO algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 The current algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Ghost liquid cell balance. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.7 Finite film thickness model sketch. . . . . . . . . . . . . . . . . . . . . . 48

4.1 PISO test case geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Accuracy-Co test case scheme. . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Comparison of velocity profile . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Mixture density and mass flux . . . . . . . . . . . . . . . . . . . . . . . 584.5 Mean velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Rms data comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.7 Drop test mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.8 Evaporation velocity vectors. . . . . . . . . . . . . . . . . . . . . . . . . 62

4.9 Evaporation velocity for 0 . . . . . . . . . . . . . . . . . . . . . . . 634.10 Thin-film drop approximation . . . . . . . . . . . . . . . . . . . . . . . . 634.11 Reference geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.12 Analytical solution with sinusoidal source . . . . . . . . . . . . . . . . . 66

4.13 Temperature evolution in (0.25, 0.25) . . . . . . . . . . . . . . . . . . . . 67

4.14 Time evolution at interface with a cosinusoidal heat source. . . . . . . . 68

4.15 heat source only inLeft . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.16 Distribution of temperature . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.17 Constant heat source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.18 Larger time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.19 Conductivity ratior = 100. . . . . . . . . . . . . . . . . . . . . . . . . . 73

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xiv LIST OF FIGURES

4.20 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.21 Solid-fluid case with heat source . . . . . . . . . . . . . . . . . . . . . . 754.22 Multi-dimensional case scheme. . . . . . . . . . . . . . . . . . . . . . . . 76

4.23 Multidimensional error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.24 Temperature error along the whole domain . . . . . . . . . . . . . . . . 78

5.1 Flow around a cylinder between two parallel walls . . . . . . . . . . . . 805.2 Accumulation of vapor concentration within the channel . . . . . . . . . 825.3 Accumulation of vapor concentration within the channel . . . . . . . . . 825.4 Vapor mass fraction distribution in the channel . . . . . . . . . . . . . . 835.5 Instantaneous evaporation velocity . . . . . . . . . . . . . . . . . . . . . 845.6 Time averaged evaporation velocity. . . . . . . . . . . . . . . . . . . . . 855.7 Spectra ofC dand C l . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.8 The geometry layout of the 2D dishwasher case.. . . . . . . . . . . . . . 87

5.9 Vapor concentration contour plot, for both tubs walls conditions.. . . . 895.10 Evaporation velocity around the cup, for both tubs walls conditions. . . 905.11 Intermediate phase of the dishes drying . . . . . . . . . . . . . . . . . . 915.12 Intermediate phase of the plate drying . . . . . . . . . . . . . . . . . . . 925.13 Mean evaporation mass flux during the drying process of the plate. . . . 925.14 Experimental data of the whole washing cycle from Bonn university. . . 935.15 The drying cycle experimental data from Bonn university. . . . . . . . . 945.16 3D Dishwasher case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.17 Smagorinsky 3D Dishwasher case . . . . . . . . . . . . . . . . . . . . . . 96


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xvi LIST OF TABLES


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Chapter 1

Introduction

1.1 Aim of the projectNowadays the customers are more and more demanding regarding the performance oftheir appliances: referring to dishwashers, the consumer expects dry dishes once thecycle is finished but is used putting dishes of several materials inside the tub. Manytimes it happens that dishes (the ones made by plastic, for example) are not fullydried. The effect is clear and was described using a zero-dimensional model, and,therefore, also the limitations of the present drying system are evident. Understandingand modeling of the drying process by means of a 3D CFD model in a dishwasher areneeded to improve and optimize the present and future drying systems of dish washers.

Figure 1.1: The complex geometry of a dishwasher tub.

The overall aim of the project is therefore to understand better the physics regardingthe dishwashers drying performances, with special focus on the modeling of conden-sation and evaporation inside the tub. A dishwasher includes a chamber containingthe disheware, the tub. The latter is filled by approximated 4-6 liters of tap water percycle. Depending on the level of dirtiness of the dishware, 2-4 water cycles are usedfor a full washing program. By the rotating spray arms the water is distributed in thetub and wets the dishes. A pump circulates the water. With a drain pump the water

is drained out of the tub. Typically, the water for the final rinse cycle is heated up to

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CHAPTER 1. INTRODUCTION

70 C to warm up the dishes storing thermal energy. The tub walls are colder than thedishes since the dishwasher cools down from the outside. Therefore the water dropletson the dishes are evaporated and condensed on the tub wall. In order to improve the

drying process in advanced drying systems, the air is circulated through an air duct andpasses inside a condenser unit. Other solutions are to vent the tub just by opening thedoor or to heat up the air and blow it into the tub. The study develops the modelingof evaporation and condensation effects in a dishwasher by means of ComputationalFluid Dynamic (CFD). It has to give the possibility to explore new drying solutions fordishwasher configurations for the improvement of the drying performances.

1.2 The approach to the problem

Mixed convection flows in presence of condensation and evaporation phenomena havecrucial role in several natural and technological processes, in particular in those in-volving drying and wetting of solid surfaces. Although such flows are quite commonin engineering applications, their knowledge is far from being complete. The complexphysics involved can be briefly sketched as follows: wetted solid bodies exchange heatwith the liquid laying on their surfaces; the liquid phase exchanges mass and heat withthe surrounding gas through the change of phase; the consequent diffusion of tem-perature and vapor concentration result in density variations that greatly impact thegaseous flow introducing buoyancy forces. In such a problem the way the surface iswetted has crucial importance. The liquid can be spread onto the solid as a continuousfilm but most likely as a distribution of liquid patches or sessile droplets or even in acombination of both. Such behavior depends on the complex dynamics arising from theinteractions among many factors, the most important being the surface tension of theliquid, the interfacial tensions at the contact line and gravity, not to mention the effectsof evaporation and condensation. A detailed description of the topic can be found in[26] and in [5] in which all the subject briefly discussed here have been extensivelyreviewed. The complete modeling of the dynamic of the liquid phase is beyond thepresent computational capabilities and not practical for many technical and industrialapplications as pointed out by many authors, for examples by [3]. In particular thepresence of a liquid phase on large surfaces still needs a conceptual interpretation. Inthis study the liquid is modeled in the limit of the thin liquid film approximation. Suchapproximation is invoked in a large part of the literature regarding this topic but, tothe authors knowledge, its systematic definition is still lacking. Nevertheless heuris-tics arguments can be provided, as done in the present thesis in 2.3, to justifies theapproximation even in different frames. The thin liquid film have been used to modelthe interface between porous media and the external flow, as in [20] in which the wallsurface is characterized just by the porosity parameter , and its impact on the flow isinvestigated: the higher the porosity the higher heat transfer. The same mathematicalmodel has been adopted to model the liquid/air interface to compute the mass exchangebetween a falling film on a vertical wall and the air flow in [ 10], or the vapor mass fluxreleased from an horizontal pool of water in [40].

In the present study the main focus firstly has been on the liquid-gas interaction,to better understand how evaporation and condensation over solid wetted surfaces take

place in wall bounded flows. The relevant literature has been reviewed. This kind of

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1.2. PROBLEM APPROACH

problem has been faced mostly through numerical modeling, see for example [13],[15],since uncertainties in experimental approaches may come from the difficulties to controlthe parameters ruling the process. Almost all the investigated cases have been focused

on the study of laminar flows moving within straight channels with different conditionsat wetted walls, considering only steady-state solutions. Moreover in the larger partof literature the mathematical models adopted parabolic equations neglecting axialdiffusion of the transported quantities, see for example [16] where the case of inclinedchannels is studied. When complex flow has to be predicted with flow reversal andwith vortical structures, full elliptic formulation must be considered as pointed outby[19]. The mass transfer process is modeled by means of the Stefan flow 1, imposedas a boundary condition at the wetted surfaces. Only in [19]the evaporation effect isimplemented considering the Stefan flow as a source term placed in the first cells nextto the wetted wall.

The physical properties of the humid air are assumed to be constant in almost

all the literature. The exact value is evaluated by the one-third rule, see [15], [13]or [20], or given as experimental data. The density is computed under the Boussinesqapproximation as

= 0[1 TT ]

where 0 is the reference density, T and the volumetric expansion coefficientsdue to changes in temperature, T, and vapor concentration content of the airparcel.

The mathematical model adopted in this work consists of the full incompressibleNavier-Stokes set of equations where the buoyancy forces are taken into account by

means of the Boussinesq approximation. At the wetted walls the thin liquid film ismodeled as semi-permeable boundary condition which prescribes a Dirichlet conditionfor temperature and consequently for the saturated vapor concentration. The evalua-tion of the Stefan flow permits the evaporation/condensation of the liquid/vapor phaseat the liquid-gas interface. Such an approach gets rid of important physical phenomenasuch as droplets nucleation or evaporation that, in most of the cases, are beyond thepresent simulation capability. The set of equations and the boundary conditions areimplemented within an unsteady incompressible Navier-Stokes solver developed usingthe OpenFoam library. The new solver has been validated against literature numericalresults of [19] in the case of laminar plane channel flow.

Subsequently the important feature of the thermal coupling among solid bodies,

liquid film and air flow is addressed. A renewed interest in this field appeared inthe recent years. Important works have been done in enlightened the crucial role ofthe heat supplied by solid substrate to the liquid film or drops, as for examples [9]and in [38] both experimentally and numerically. Moreover these studies pointed outhow subtle can be the evaporation process of a single drop and warned about oversimplified approach. In addition they point out the present model limits, as discusseddeeper in2.3.

The thermal problem presents two main issues: finding a suitable method to guar-antee the continuity of the temperature and of the heat flux across each interface,namely solid/liquid and liquid/vapor, and properly modeling the thin liquid film from

1

the vapor flowing out the liquid interface

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Chapter 2

Problem formulation

The problem under consideration is in general a mixed convection flow. The fluidmotion is recognized to occur for both natural convection and, if present, by forcingaction. The fluid of interest is essentially moist air confined by wetted surfaces. Thephysical main sketch is given by the gaseous phase, an ideal gas mixture of dry air andwater vapor, and the liquid phase, namely water, laying on the wetted solid surfaces.At the same time the model is quite general and can be applied to a wide range ofbinary mixture. Evaporation or condensation will occur at the interface separatingthe gas from the liquid species. In particular the model assumes that the liquid has acharacteristic thickness much smaller than the characteristic length scale of the flow andalso of the solid substrate itself. These assumptions justify the thin film formulation,in which the liquid phase can be taken into account as proper boundary condition.

Moreover a more realistic approach that includes the liquid phase as a distribution ofdroplets has to face the extremely complex physics that rules each single sessile drop.These aspects are briefly introduced in 2.3.1just for sake of completeness.

A complete overview of the mathematical modeling adopted is presented throughoutthis chapter.

2.1 Mass transfer for air-water mixture

Here the case of humid air is considered. The water content in the gas region is treatedas a concentration. The vapor can condense/evaporate near colder/hotter surfacesrespectively, while in the gaseous bulk the air-water mixture is in a stable state. In

general, for a binary mixture, the density and mass fraction are expressed by thefollowing constitutive relations

= a+ b (2.1)

1 =a+ b

where the subscripts refer to species,a and b. The mass fraction is the mass concentra-tion a,b= a,b/. At the liquid film interface the mass fluxnequation for one speciesis

na= Da,ba+ a(na+ nb) (2.2)

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CHAPTER 2. FORMULATION

where na =aua, ua is the velocity of species a, Da,b is the diffusivity. There are twocontributions: the diffusion flux due to the concentration gradient ja = Da,ba,usually referred as the Flicks first law of diffusion; and ua = a(na+ nb), called the

bulk motion contribution. This, also referred as the diffusion-advection term, occurswhen one species diffuses through a stationary second species causing the overall phaseto have a net motion due to the movement of the first component.

The continuity equation must hold for any species, from a mass balance on a dif-ferential control volume

na+ a

t ra= 0 (2.3)

whererais the rate of chemical production ofa. Assuming thatDa,band are constant,without production term, the mass flux expression along with the continuity one yieldto the transport equation for a

at

+ u a= Da,b2a (2.4)

Evaporation velocity Thanks to the mass transfer theory a diffusive-like modelfor evaporation and condensation process is found. It can be applied at the interfacebetween a pure substance liquid film and a binary mixture gas prescribing a massflux through the interface. In a steady-state problem, neglecting diffusion along theinterface, we have the following boundary conditions

dnadn

i

= 0 (2.5)

dnbdn

i

= 0

The species b is insoluble into the liquid phase of species a, as the case of air andwater, so that nb = 0 at interface and hence in remaining of the domain. For thisreason sometimes b is referred as stagnant gas. Conversely for a it is found, with somealgebra, that:

nai = Da,b1 ai

dadn

i

(2.6)

where the physical quantities are considered at the interface: at a given temperatureTi and at given relative pressure pai of the soluble gas, its mass fraction ai is assumed

to be in saturated condition and can be computed. Such interface model is calledsemi-impermeable.

2.2 Gaseous phase model

In the present study the gaseous flow is considered incompressible and affected by thetransport of two active scalars: the temperature Tand the water vapor mass fraction, i.e. the percentage of water vapor in the moist air (kgv/kgm). The thermal andsolutal gradients yield buoyancy forces by density variations. The usually adopted as-sumptions are retained developing the model. The thermo-physical properties of the

gas are considered constant. They are the kinematic viscosity (m2

/s), the thermal

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2.2. GASEOUS PHASE

expansion coefficient T (K1), solutal expansion coefficient , the temperature dif-

fusion coefficient DT (m2/s) and the mass diffusion coefficient D (m

2/s). Densityvariations are retained as buoyancy term in the momentum equation along the gravity

direction under the Boussinesq approximation. Viscous dissipation and compressibilityeffect are neglected in the energy transport equation and the Dofour and Soret effects 1

are considered negligible. Therefore the governing equations for this physical systemare the following set of coupled P.D.Es

Continuity Equation

u= 0 (2.7)

Momentum Equation

u

t + u u= 1

p

x + 2u

v

t + u v=

1

p

y+ 2v (2.8)

w

t + u w=

1

p

z+ 2w+ g(TT+ )

Energy EquationT

t + u T =DT

2T (2.9)

Concentration Equation

t + u = D

2 (2.10)

2.2.1 Sponge region

In some complex flows, as for example stratified ones, the outflow condition must benon-reflective in order to preserve the accuracy of the solution preventing the distur-bances in form of internal waves to propagate upward. We use a sponge region localized

just before the outlet section. This is found to be effective in this task as explainedin [1] and moreover it has a stabilizing effect on the numerical simulation damping the

recirculation across the outlet section. In the case herein investigated, a zero gradientcondition for velocity and the scalars is imposed at the outlet together with the spongeregion acting along the 10% of the channel length, in which fluid viscosity, thermal andconcentration diffusivity are artificially increased according to an exponential law like

A(x) A ex (2.11)

with A any of the parameters. Moreover within the sponge region the buoyancy forceis neglected, i.e. the concentration and temperature are treated as passive scalars.

1the diffusion due to concentration (Dofour) and thermal (Soret) gradients in the energy and the

vapor mass fraction equations respectively

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2.2.2 LES turbulence model

In the home-appliances, as well as any other technical application, the flows are quite

often turbulent. Specific models have to be included in order to properly reproducethe flow characteristics. The large eddy simulation, LES, approach is preferable, beingDNS computation not affordable, because of the computational cost required by full-scale simulation. The LES technique aims to directly solve the large-scale structures ofthe flow and to model the effect of the small-scale eddies. Here on only a brief overviewof LES is given, whereas a more detailed description can be found in [35, 29]. Thescale separation is formally done by applying an appropriate low-pass filter to the flowvariables. For a monodimensional case

f=f+ fsgs (2.12)

where fis the variable to be evaluated, the resolved part is obtained as

f=

f(x)G(x, x)dx (2.13)

where G is the convolution kernel of the filter. The cut-off length determines theseparation of scales: roughly speaking eddies of size larger than are large eddies whilethose smaller than are small eddies and fsgsis the sub-grid-scale contribution filteredout by the convolution2.13. There are different filtering operators, in the present workit has been adopted the so-called top-hat filter defined, in the physical space, as

G(x, x) =

1/ if |x x| /2

0 otherwise (2.14)

where the filter width is assumed to be the cube root of the cell volume = V1/3c .Filtering the continuity and the Navier-Stokes equations leads to

uixi

= 0 (2.15)

uit

+ uiuj

xj=

1

p

xi+

2uixjxj

ijxj

(2.16)

for the resolved quantities. The effects of the small structures are represented by thesub-grid scale stresses

ij =uiuj uiuj (2.17)

which have to be modeled. In particular is possible to define the eddy viscosity suchthataij =2tSij (2.18)

where aij =ij ijkk/3 is the anisotropic part of the stress, and Sij = 12

uixj

+ ujxi

is the resolved stress tensor. The following two SGS models have been considered inthe present study:

Smagorinsky model in which the eddy viscosity is obtained from a scale analysisassuming local equilibrium between turbulent kinetic energy production and theviscous dissipation at the smallest scales

sgs = (Cs)

2

|Sij | (2.19)

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2.2. GASEOUS PHASE

where Cs is a constant that as to be tuned on the specific case.

The main drawbacks of this model are: it predicts a non-physical eddy viscos-

ity close to the wall; it is not able to reproduce local re-laminarization and re-transition to turbulence; the constant depends on the numerical scheme adoptedto integrate the governing equations. The first drawback is overcome by the in-troduction of the Van-Driest damping function. It improves the SGS-viscositybehavior near the walls. The coefficient Cs is explicitly damped near the wall by

Cs = Cs

1 exp

y+/A+

(2.20)

where y+ = yu is the non-dimensional distance from the wall, and A+ is the

distance from the wall up to which the damping occurs, usually is set asA+ = 25.

The friction velocity,u, is defined asu=

w , wherew is the wall shear stress.

Dynamic models Germano at al. in [12] firstly proposed a dynamic procedure toobtain the Smagorinsky constant that depends on the resolved scales energy con-tent. Cs can adapt to flow condition locally and in time overcoming the mainissues of the basic Smagorinsky model. This approach takes advantage of filteringtwice the velocity field. Firstly the grid filterG of width is applied, then theobtained field is filtered again by the test filterG of width, usually equal to2. The SGS-stress is found as

ij =uiuj uiuj (2.21)

and then applying the test filter

Tij =uiujuiuj (2.22)The so-called Germano identity holds

Lij =Tijij =uiujuiuj (2.23)Lij are the resolved turbulent stresses that represent the intermediate scales ef-fects between the two filter widths. Notably it can be computed directly fromthe resolved velocity field. If the Smagorinsky model can be applied to determineboth ij and Tij then

ij

=2C2s

2|S|Sij

(2.24)

Tij =2C2s2|S|Sij (2.25)

In particular Lilly proposed in [21], originally without averaging procedure de-notes by , the following Cs formulation

C2s = LijMij

Mij Mij (2.26)

where

Mij = 22

|S|Sij

2

|

S|

Sij

(2.27)

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Averaging has been introduced because the constantCscan be extracted out fromthe filtering operation only if the function is smooth in space. From a practicalpoint of view this avoids numerical instabilities to rise in the flow field.

It should be noted that the averaging has to be performed along directions ofturbulence homogeneity over which Cs is assumed not to vary. The model issaid to be dynamic because the sub-grid stresses can adapt to the particular flowcondition, in particular they are zero in laminar flow or close to the solid walls.

It has been tested in the OpenFoam context the standard Smagorinsky the dynamicmodels. It has been found that OpenFoam dynamic model averages the constant in thethree-dimensional space. This is not correct when the flow is not homogeneous in thevolume. Therefore for the plane channel flow test case discussed in 4.3,a modificationto the OpenFoam code has been required to permit the average over the planes ofhomogeneity, i.e. parallel to the solid walls. The plane averaged dynamic model has tobe considered as a first step towards the Lagrangian dynamic model.

If the turbulence has to be included in the mathematical model given at the begin-ning of this chapter, the equations 2.8, 2.9, 2.10 have to be restated as follows. Theturbulence contribution has to be considered through the eddy viscosity sgs and eddydiffusivities for the temperature,DTsgs , and vapor concentration, Dsgs . The two eddydiffusivities are evaluated by means of the Reynolds analogy

DTsgs =sgs

P r Dsgs =

sgsSc

(2.28)

Continuity Equation u= 0 (2.29)

Momentum Equation

u

t + u u=

1

p

x+ ((+ sgs)u)

v

t + u v=

1

p

y+ ((+ sgs)v) (2.30)

w

t + u w=

1

p

z+ ((+ sgs)w) + g(TT+ )

Energy Equation

Tt

+ u T = (DT + DTsgs)T (2.31)Concentration Equation

t + u =

(D+ Dsgs)

(2.32)

2.3 Thin-film assumption

The liquid wetting the solid surfaces is present as a thin liquid film such that it can betreated as a boundary condition. This assumption has already been applied successfully

in slightly different researchs frames, as discussed in 1.2. A liquid film is properly

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2.3. THIN-FILM

define [41] as thin whenever the typical thickness hm of a free surface layer of liquid issmall compared to its length scale parallel to the substrate l

hm l

Liquid layers roughly range from drop to film based on the increasing ratio betweenhmand l. The hypothesis underlying the present study assumes that a continuous liquidfilm can effectively model an otherwise extremely complex physics system, extendingthe previous definition of thin film to any kind of liquid layers which thickness aremuch smaller then the characteristic length scale of the overall system. For example thetypical dimension of a dishwasher 2 is 1 m and while droplets have hm 1mm. Themodel should be applicable to a wide range of solid substrate materials, with differentsurface properties, i.e. all the range of dishware, over which a droplet distributionshould be considered, eventually taking into account the dynamic process of each drop.

On a large scale the inclusion of all these features may be not so crucial and a film witha suitable thickness can, at least, predict the overall effect of evaporation.

2.3.1 Interaction with the substrate

The film or drop evolution to a complete dryness is strongly dependent on several pa-rameters, and it is also affected by different forces and processes. These are mainlygravity force, free surface stress, surface tension, evaporation and condensation pro-cesses, and in particular cases also the disjoining pressure. The conditions at the solidsubstrate are also crucial in the de-wetting and wetting processes. Interactions betweenthe surface and the fluid are regulated by some micro-scale physical and morphologi-

cal features parameters as surface roughness and chemical heterogeneity

3

, interfacialtensions and contact angles. From an engineering perspective, i.e. on a macroscopicpoint of view, the overall effect can be summarized by the surface wettability. The lat-ter is a parameter known to be crucial in drop-wise condensation4, in capillary liquidfilm evaporation and for boiling heat transfer. In brief solid-liquid-gas interactions rulethe dynamic of the drops evaporative process. Even though the present model doesntincorporate any of these effects directly, it is important to be aware of them in orderto correctly take advantage of the thin-film assumption. In the following, the mostimportant effects and parameters are considered.

The curvature-radii effect. The Young-Laplace equation relates the curvature ofliquid-vapor interface and surface tension to the pressure difference, called capil-

lary pressure. From it, its possible to obtain the geometry of the interface oncethe pressure term is prescribed.

p=

1

R1+

1

R2

where p is the pressure difference between the two phases, is the interfacialtension,R1 and R2 are the curvature radii at a point.

2or better the distance between wetted surfaces3defects of the surface4actually, the heat transfer coefficient for surface with low wettability is much larger than that of

film condensation[18]

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Figure 2.1: Sketches of the three qualitatively different wetting situations for a simpleliquid on a smooth solid substrate.

The effect of gravity. The effect of gravity can be expressed through the definitionof a capillary length: when the system size, say the characteristic film thickness or

the drop diameter, is small enough then the gravitational effect is negligible. Tinyliquid drops are nearly spherical while larger drops present more deformations.These effects is ruled by the Bond or Eotvos number

Eo = gL2/

whereL is the characteristic length (equivalent diameter). Eo can be interpretedas the ratio of a inter-phase differential hydrostatic pressure gL and capillarypressure /L. The effect of gravity on the interface should be negligible ifEo 1,

or, defined a =

2g

the capillary length, L a

Contact angle and partial wettability effect. A liquid brought in contact with asolid ideal surface can spread until it becomes a spherical drop, a sessile drop thatlies on a limited area leaving dry the rest, or a film, as depicted in figures 2.1(a),(b) and (c) respectively. The first is the non-wetting condition for which holds= , the second is the partial wetting for which is 0 < < and the third oneis the complete wetting condition with = 0. The contact angle between liquidphase and solid surface is the parameter that gives the degree of wetting. Atequilibrium its assumed that is relate to interfacial tensions between solid-gas,liquid-gas and solid-liquid by

lgcos() = gs slcalled the Youngs equation, graphically represented in Figure 2.2,that gives theadditional condition at the contact line.

Usually a real surface is not either perfectly smooth and defect-free hence aneffective contact angle may be used to replace the theoretical one. In particularthe Wenzel equation and its modification, see [18], takes into account effect ofroughness: the is increased because of the increased liquid-solid contact area

cos(W) = r cos(Y)

wherer is the ratio between the actual to apparent surface area, Y is the Young

contact angle and Wthe one prescribed by Wenzel.

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2.3. THIN-FILM

Figure 2.2: Young contact angle and interfacial tensions.

Moreover there are two more issues regarding

Contact line singularity.

The contact line, i.e. the region where the solid, liquid and vapor phasetouch, is a singular point in the macroscopic sense, since the no-slip conditionat the surface simply denies any movement. To model a moving front thiscondition must be relaxed. It can be done effectively by means of a numericalprecursor film, as suggested for example in [36].

Contact-angle hysteresis.In general the contact angle is found to be a function of the velocity of thecontact line. In particular the contact angle remembers its moving history,giving rise to hysteresis effect.

Disjoining pressure. The model of transition to a complete dryness should allow the

change of configuration from the one with three superficial tensions to one justwithsg. Below a certain film thickness an additional force appears between thetwo interfaces separating the liquid (from gas and solid), due to intermolecularinteractions. This can be expressed adding a pressure term , called the disjoiningpressure, that for a polar liquid is

(h) = b

h3 eh

This relation, with the right choice of constant b, ensures that the total surfaceenergy for a thin film correctly interpolates between the thick film and the drysubstrate.

2.3.2 Evaporating drops physic

Despite its apparent simplicity, the everyday life experience of a drop evaporation isstill an active research field, see for example [9]. Often it will be also necessary toconsider the dynamic behavior of the liquid interface, as, for example, in the study of asliding drop as in [42,36]or as in[25] to exploit evaporating drops with moving contactlines. In AppendixA a brief review about the modeling of the dynamic liquid film isreported, and the difficulties associated with this approach are addressed.

Nevertheless some basic facts about one single drop physic help dealing with theeven more challenging task of the large drop number system characterization. Accord-

ing to[23], two main regimes have been recognized during a sessile droplet evaporation.

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2.3. THIN-FILM

(a) sub-caption (b) sub-caption

Figure 2.3: The surface curvature and fringe field effects.

2.3.3 Modeling hints

From a modeling perspective the physic of the drops gives some useful hints to beacknowledged in any film/drop formulation. There are two main aspects to be awareof, namely the surface curvature effect and fringe field effect, sketched in Figure2.3.

Curvature effect Surface curvature is responsible for increasing the total interfacearea exposed to evaporation respect to a flattened layer. Considering just a simplelinear variation, as in Figure2.3a,the area undergoing phase change is multiplied by a

factor dA

idA, and consequently the liquid depletion

h= dAi

dA

g

ltV e (2.38)

Fringe effect Fringe field effect appears whenever there is a steep or discontinuouschange in boundary conditions. In the specific cases for the vapor concentration thereis a switch across the contact line: from a Dirichlet type prescribing i for the liquidinterface to a zero gradient condition representing the dry wall. Hence near the contactline the iso-concentration lines accumulate yielding to steeper concentration gradient,and eventually to larger evaporation velocity, see Figure 2.3b.

In this study the additional surface curvature term is present explicitly, i.e. thesurface is properly reproduced, in evaporating drops simulation reported in 4.4 and asa parametrized term as explained in 4.4.1, in the attempt of adopting the thin filmapproximation for the drop problem. In all the other cases the term is disregarded. Onthe other hand the fringe effect is implicitly embedded in the model, as suggested bythe results of the drying plate in5.3.

2.3.4 Packing ratio argument

In light of the physics above discussed the continuous thin film is thous found rep-resentative also for drops system. Defined the packing ratio pk as the mean dried

distance between two drops, ld, and the mean diameter of the drops, D, it appears

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reasonable that below a certainpk value vapor concentration fills up the holes amongthe liquid patches reducing the fringe effect and eventually leading to an homogeneousevaporation rate above drops, as provided by a continuous liquid film.

2.4 Evaporation/Condensation boundary condition

The evaporation/condensation process takes place over the interface between liquid andgaseous region. It is assumed to be a diffusion-limited process, i.e. the interface is inthermodynamic equilibrium, in contrast to the reaction-limited regime. Nearly wholeresearchs literature relies on the former even though some debate is still ongoing [25].

Moreover in the thin-film approximation the actual thickness of the film h is notdiscretized but its value is retained as a local, say cell-wise, property in order to trackthe evolution of the liquid depletion from a wetted to a completely dried 6 condition.

Eventually evaporation/condensation process is modeled through a set of boundaryconditions for the fields involved, plus the film thickness. Whenever any point of thedomain boundary completely dries out, i.e. reaches zero thickness, also a switch in theprescribed b.c. must occurs consequently, assuming that re-wetting phenomena are notpermitted. Actually, since the phase change occurs at the interface, they will be betterreferred as evaporation/condensation interface boundary conditions, EC b.c. in short.Each condition is discussed in the following along with the switching to the dry surfacecondition.

2.4.1 Temperature at interface

EC b.c. prescribes for the temperature at the interface a Dirichlet condition

Ti = Ti(x, t) (2.39)

In general Ti can be non uniform and unsteady, matching the requirements of thespecific film condition, as for example in thermally coupled problems as described in2.5.Ifh goes to zero then the condition should switch to match the temperature of the solidsubstrate.

2.4.2 Vapor concentration at interface

The diffusion-limited regime leads to assume the gaseous region at the interface to bein saturation condition. The focus in this study is mainly on liquid water evaporation

or condensation in air-vapor gas treated as an ideal gas mixture, so that the followingstandard relations applies. The vapor concentration is defined as = mvmv+ma . At theinterface it is evaluated by

i =MvMa

ips(Ti)

p (1 MvMa )ips(Ti)

(2.40)

whereMa= 28.97 g/mol,Mv = 18.02 g/mol are the molar mass of air and water vapor

respectively. The relative humidity =mvms

is given by

= pv

ps(Ti) [kgv/kg

s] (2.41)

6

at least from a macroscopic point of view.

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2.5. THERMAL COUPLING

being pv and ps vapor partial pressures at actual and at saturation condition respec-

tively. At the interface is assumed i = 1. To compute the saturation pressure anapproximated formula is adopted [2]

ps(Ti) = 611.85 exp

17.502 (Ti 273.15)

240.9 + (Ti 273.15)

[Pa] (2.42)

where pressure is set to be Patm = 101325 Pa. This is strictly valid for a water-airinterface, but it can be replaced by any other relation for the specific mixture underconsideration. On the dried region a Neumann condition applies

d

dn

i

= 0 (2.43)

allowing no more fluxes.

2.4.3 Evaporation velocity

From the mass flux relation 2.6 it is readily understood that the vapor leaves theinterface with a velocity specified by

Ve= D1 i

d

dn

i

(2.44)

that is the condition for the momentum equation. It is linked to field through the

normal to interface gradient,

d

dn

i

, and also by its value at the interface i. This

condition allows an inflow or outflow happen through wetted surfaces. As consequenceof2.43,on dried patch evaporation velocity is identically zero.

2.4.4 Film thickness

The phase change is ruled by Ve. In particular the intensity of the process is propor-tional to the modulus, Ve, while its sign defines if condensation or evaporation is takingplace at a specific time on a specific interface point. A simple film mass balance at eachpoint gives the variation rate of the film thickness h

dh

dtdSi =

al

Ve dSi (2.45)

It must be notice that the liquid water depletion rate is related to the ratio betweenair and liquid water densities. Typically values of this ratio are of the order 103.

2.5 Full thermal coupling

2.5.1 Motivation

The influence of the thermal properties of the solid substrate on evaporation processesis now widely recognized [9]. The thermal coupling among the three media involved isresponsible either of supplying enough energy to sustain evaporation or to stop and start

the condensation whenever the liquid cools down. The former case is representative of

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(a) Non-overlapping method (b) Overlapping method

Figure 2.4: Decomposition of domain.

Equivalence theorem

A non-overlapping decomposition requires that 1 2 = and 1 2 = asdepicted in2.4a,moreoverniis the outward normal of the each sub domain isuch thatn1 = n2. The following theorem gives a mathematical basis to the non-overlappingDDM[31].

Given Las elliptic operator, u is the solution to following differential problem overthe domain with Lipschitz boundary

Lu =f in u = 0 on

(2.46)

such that u|i =ui, with i= 1, 2, where ui is the solution to Lui =f in i

u = 0 on i/ (2.47)

provided the coupling condition to be applied at the interface u1| = u2|

u1n1

=

u2n2

(2.48)

These are labeled also as transmission conditions. In particular this result allows tosolve the P.D.E. via an iterative procedure based on these interface conditions.

Generalization of DDM

The DDM is a suitable technique also for any time evolving arbitrary problem with Llinear or non-linear arbitrary differential operator

u

t + Lu(k) =f in (0, T)

B.C. on (0, T)I.C. in at t= 0

(2.49)

In the DDM for time dependent P.D.Es implicit temporal discretization schemes

should be preferred. The estimation of the values required at the interface often involves

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an explicit calculation followed by an implicit scheme to obtain information in theinterior of each sub-domain independently. Such an algorithm is often referred asan explicit/implicit domain decomposition algorithm. Because of the explicit nature

of computation of the interface values, the explicit/implicit algorithm is often justconditionally stable, but requires less severe conditions than for a fully explicit scheme.

Dirichlet-Neumann Method

The Dirichlet-Neumann method is an iterative algorithm for the Domain Decomposi-tion. For sake of simplicity the problem given in (2.46) is considered again in order to

explain how the DNM can be applied. Given u(0)2 , where the up-script is the number

of iteration, it is possible to solve the following two problems for k 1

Lu

(k)1 =f in 1

u(k)1 =u(k1)2 on u(k)1 = 0on 1/

(2.50)

Lu

(k)2 =f in 2

u(k)2

n2=

u(k)1

n1on

u(k)2 = 0 on 2/

(2.51)

One can recognizes that the set in 2.50 is a Dirichlet problem with the value at theinterface provided by the previous iteration, and the set in2.51is a Neumann problemwith the derivative value at its boundary is computed by the solution of2.50in the

current iteration. The equivalence theorem assures that if the sequences {u(k)1 },{u(k)2 }converge then both converge to the solution of (2.46). Even if the DNM is consistent,the convergence is not guaranteed. To overcome such a drawback in some cases isuseful to introduce a relaxation procedure in the calculation of one of the transmissionboundary value. For example for the Dirichlet condition in 2.50can be relaxed as

u(k)1 =u

(k1)2 + (1 )u

(k1)1 on

that lowers the error at each iteration given the relaxation parameter . The existenceof a for which the convergence is also guaranteed is proven just in the one and twodimensional case.

2.5.3 Finite thickness film model

The energy equation has to be solved in the solid, liquid film and gaseous region: theyhave different thermal properties and different heat transport mechanisms. In the solidthe process is assumed to be diffusive

Ts

t =DTs

2Ts (2.52)

where Ts is the temperature in the solid body to avoid ambiguity, and consequently DTs

is its diffusion coefficient. In the air flow the advection-diffusion equation (2.9) holds.

For the liquid phase some model options are available. The thin-film assumption2.3

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is retained and further extended assuming that also the solid bodies are much thickerthan hm. The velocity inside the liquid film u is then negligible and the convectionterm can be discarded. Moreover considering large wetted patches, hm/l 1, the

temperature variations along the surface tangential direction are considered negligiblecompared to the wall normal gradient, i.e.

Tl

n

Tl

t1,2(2.53)

being Tl the temperature in the liquid, so that its distribution inside the film reduceto an evolution equation

Tl

t =DTl

2Tl

n2 (2.54)

The solution of the mono-dimensional problem (2.54) at a specific point x along the

wetted surface isTl =Tl(y) with 0 y h(x) (2.55)

beingy the normal to wall axis in local frame of reference. This will be referred as thefinite thickness film, FTF, model for the energy equation in the liquid.

FTF boundary conditions

The liquid layer is interposed between the solid and air. The temperature is requiredto be continuous through the solid/liquid boundary, namely the wall, and through theliquid/gas interface. Across both the heat fluxes must be balanced too. For this purposecoupling boundary conditions should be enforced. In this frame it appears natural to

tackle the problem with the DD approach: each media is solved by its own given itsproper mathematical model, so that the transmission conditions as 2.48 take care ofthe temperature coupling. The following holds at the solid/liquid wall

ks

Ts

n

w

=kl

Tl

n

w

(2.56)

Ts =Tl (2.57)

Phase change process is associated to the latent heat, L, release. This is gener-ally large, for example L 2.26106 [J/kg] for water evaporation. Thous the evapora-tion/condensation latent heat fluxLJis found to be the ruling term of the heat balance

at the liquid/gaseous boundary

LJ=iL D1 i

n

i

[J/m2s] (2.58)

then at the interface the following applies

kl

Tl

n

i

=kg

Tg

n

i

JL (2.59)

Tl =Tg

In conclusion this can be addressed as a heterogeneous DD problem in which the solu-

tion can be provided by the DN method described in (2.5.2) provided the solution (2.55).

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t

Vl

lcpTl dV =

ks

Ts

y

w

dSs+ kg

Tg

y

i

dSi+ JLdSi

(2.63)

and compute the mean temperature of the liquid volume. As matter of fact this proce-dure disregards the continuity ofTand relaxes the coupling by assuming that the heatfluxes are those imposed from outside the liquid layer.

All the methods mentioned lack of generality, and can lead to substantial errors asdiscussed in 2.5.3. The proper approach is then to apply an analytical solution andsolve it numerically. All the issues of the simplified models are altogether solved. Thetemperature at wall and at interface are retrieved with their gradients permitting thecoupling among the media. The different behaviors are sketched in Figure 2.5 for aparticular situation. The solid body is assumed to release heat quite fast, for exampledue to the condition on the hidden boundary on the left of the Figure 2.5 , while onthe liquid interface a constant heat flux is applied. As response the film temperature

is expected to decrease. If the drop happens on a time period comparable to that ofthe diffusion time scale of the thin liquid film or even smaller, the temperature profileshould adjust in the transitory as in the Figure2.5cin which the liquid is giving backheat to the solid. This effect cannot be reproduced properly by the other simplifiedmodels, in particular the linear one would predict the opposite flux sign, Figure 2.5a.

Time dependent 1D energy equation

The mono-dimensional energy equation (2.54) can be solved analytically by means ofthe standard separation of variables method in terms of Fourier series, provided both theinitial and the boundary conditions. In particular prescribing Dirichlet and Neumann

boundary condition on the two boundaries permits the straightforward application ofthe model within the Dirichlet-Neumann coupling strategy. Moreover such boundaryconditions appear quite natural by considering that on the liquid-air interface the heattransfer is ruled mostly by the latent heat flux given by evaporation/condensation. Thesolution to the general problem (2.64)

T

t =

2T

x2T(0, t) = aT(l, t)

t = b

T(x, 0) = f(x)

(2.64)

is given by

T(x) =Tss(x) + Tic(x, t)

as the sum of the solution of the particular steady-state solution

Tss(x) =b x + a

and of an evolution equation satisfying initial conditions

Tic(x, t) =

0 Bnsin(2n + 1)

2l

x exp(2n + 1)

2l2

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(a) Instant diffusion model constant heat flux

(b) Energy balance model

(c) Analytical solution

Figure 2.5: Models for the temperature prediction in the thin film.

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where the Fourier coefficients are computed as

Bn = 2

l l

0

g(x)sin(2n+ 1)

2l

x dxg(x) =f(x) bx a

(2.65)

The gradient value at x = l, is the imposed one so from the solution it reduces to b

T

x(l, t) =b+

0

Bnexp

(2n + 1)

2l2 t

cos

(2n+ 1)

2

(2.66)

=b (2.67)

while the gradient at x = 0 is computed as

T

x(0, t) =b +

0Bnexp

(2n + 1)

2l2 t

(2n+ 1)

2l cos (0) (2.68)

=b +0

Bnexp

(2n + 1)

2l2 t

(2n+ 1)

2l (2.69)

This analytical formulation can be effectively approximated by a finite series of Nsummations, to be chosen based on the harmonic content of the solution. UsuallyT(x)is quite smooth so that Ncan be relatively small.

A simple test on the FTF model

To test the performances of the FTF models the following case is described hereon.The evolution of temperature across a 1 mm thick liquid film is computed provided a

Dirichlet condition on the wall side, Tw(t), on the left, and a Neumann condition on theinterface, HFi, on the right, meant as evaporative heat flux. Moreover the former isprescribed to decrease linearly in time from 30 C to 29 C and then to remain clippedto the latest value, as it shown in Figure 2.6a. The linear variation ofTw(t) is chosento be tsd of the specific material underlaying the liquid. In this way the effect of thethermal properties of the solid body is roughly included. The heat flux is imposedbased on a typical evaporation velocity Ve = 1.6 10

4m/s for the water so that

HFi = 376.6 J/m2s

and the initial condition is prescribed to be

T(x, 0) =Tw(0) + HFi x

The test is meant to compare the analytical and the instant diffusion model for differenttsd representing the different usual dishware materials. The green line is the initialcondition, then the temperature on the left boundary decreases and eventually reachesthe final value. The two models evolve in time and the solutions of each are plotted atthe same simulation time.

In Figures2.6and2.7are plotted the results of the test. Different substrate thermaldiffusion properties are considered, as described in each caption. The instant diffusionmodel can be applied only if the substrate temperature varies slowly, i.e. iftdw < tds

holds, otherwise the FTF model should be preferred. The main drawback of the linearmodel is that it cannot reproduce the correct gradients at the interfaces and therefore

cannot be effective in the thermal coupling algorithm.

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(a) Wall side boundary condition for tdm = 10 s

(b) slow variation tds = 25 s, tdw tds thin plastic1 mm, thick glass (3 mm)

(c) moderate variation tds = 10 s, tdw < tds example:ceramic 3 mm

Figure 2.6: Wall side boundary condition example and film temperature distributionfor two different cases.

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(a) moderate variation tds = 8 s, tdw < tds exampleglass 2 mm

(b) same order variation tds = 5 s, tdw tds example:metal 2 mm

(c) fast variation tds = 0.2 s, tdw tds example:metal < 2 mm

Figure 2.7: Film temperature distributions for different cases.

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2.6 Non-dimensional parameters

Every fluid mechanics problem is usually characterized by some non-dimensional num-

bers based on the parameters of the flow under consideration. These numbers enlightenwhich are the ruling physical forces and can quantify them. For mixed convectionproblem in presence of evaporation and condensation the non dimensional parametersgoverning the flows are described here on. The Reynolds number,Re, quantifies therelative importance of the inertia force respect to the viscous one

Re=UcL

being Uc and L the characteristic velocity and length respectively. In the case of thechannel in 5.1 these are the inlet velocity and channel width. For bounded flows,

as in the dishwasher, in which the flow is sustained primary by buoyancy forces thecharacteristic velocity can be estimated by the following relation

Uc=

(TT+ )gL

where L is the height of the tub. The solutal Gr Grashof number and the thermalGrTGrashof number are defined as

Gr =gL

3

2 GrT =

gTT L3

2

These numbers quantify the relative strength between viscous force and buoyancy forcedue to humidity or temperature variations respectively. The Richardson number is thendefined as

Ri= Gr/Re2

which quantifies the relevance of buoyancy effect over the inertial one.

The characteristic molecular Prandtl and Schmidt numbers representing the impor-tance of the momentum diffusivity compared to diffusivities of the temperature or ofthe species respectively. For the physical system composed by air and water they areapproximately P r 0.7 and Sc 0.6. The Nusselt number is defined as the ratiobetween the contribution of the convection to the heat transfer over the conduction.

At the liquid interface the heat flux is the sum of the sensible heat (qs) and latent heat(ql) components, therefore the total Nusselt number may be written as

N ut= N us + N ul

providing a relative measure of the heat transfer mechanisms. The mass transfer processbetween the wetted surface and the humid air is evaluated by the Sherwood number,Sh

Sh = VeL

D

This compares the overall contribution, namely the Stefan flux, to the diffusion one.

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2.7. MODEL SUMMARY

Equation Condition

Temperature assigned valueTi

Concentration i = MvMaps(Ti)

p(1MvMa

)ps(Ti))

Momentum Ve= D

1 i

d

dn

i

Film Thickness dH

dt =

aw

Ve

Table 2.2: EC boundary condition summary

2.7 Model summaryIn conclusion to this chapter the mathematical model is here summarized. The equa-tions that are solved in Chapter5 are given along with the boundary conditions for thewetted surfaces in the table2.2.

Continuity Equation u= 0

Momentum Equation

u

t + u u=

1

p

x+ 2u

v

t + u v=

1

p

y+ 2v

w

t + u w=

1

p

z+ 2w+ g(TT+ )

Energy EquationT

t + u T =DT

2T

Concentration Equation

t + u = D2

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Chapter 3

Numerical Implementation

Open Source Field Operation and Manipulation, OpenFoam in short, is essentially acollection ofC++libraries that are used to create executable files, applications, whichare called solversand utilities:

solvers programs designed to solve a specific problem in computational continuummechanics

utilities programs that perform simple pre- and post-processing tasks, mainly involv-ing data manipulation and algebraic calculations

Is worth to point out that the software project is written in C++ taking full advantageof the capabilities of the Object Oriented Programming, OOP. From a practical pointof view every release of OpenFoam provides a set of pre-compiled libraries that are

dynamically linked during runtime. This allows the user to take advantage of its highmodular character, building more suitable application to his particular case. Amongother remarkable features, OpenFoam permits an easy and friendly high level code thatallows writing the P.D.Es in an intuitive way, miming the mathematical convention.For example the discretization of the Navier-Stokes equation is performed simply by

fvVectorMatrix UEqn(

fvm::ddt(U)

+ fvm::div(phi, U)

==

- fvc::grad(p)

+ fvm::laplacian(nu, U)

);

3.1 FVM discretization in OpenFoam

Among many numerical schemes available for solving P.D.Es equations, OpenFoamchooses the Finite Volume Method. More precisely OpenFoam subdivides its finitevolume method into two main branches (namespaces):

fvm finite volume method for implicit equations. It produces an fvMatrix objectfrom an operator. This object can simply be solved using the solve method. A

number of different discretization schemes are available for each operation. These

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CHAPTER 3. NUMERICAL IMPLEMENTATION

are loaded at run time using runTimeSelection, based on the schemes defined infvSchemes dictionary.

fvc finite volume calculus for explicit calculations. Given a field, a fvc performs acalculation and returns another field.

OpenFoam uses a collocated arrangement, i.e. all the quantities are stored at a singlepoint within a control volume, this point is at the control volume centroid. OpenFoamcan handle arbitrary polyhedral meshes without restriction to the number of points perface, nor to the number of faces enclosing a control volume. The faces of the first celllayer inside the computational domain conform to the boundaries. The clear advantageof a collocated grid is to minimize the number of coefficients to calculate and in dealingwith complex domains,[32].

In order to get rid of the oscillations due to the pressure-velocity decoupling, whichis the main drawback of this arrangement, a Rhie-Chow like interpolation procedureis adopted in the incompressible flow solvers: the Poisson pressure equation

petronio (2011) phd t - numerical investigation of condensation and evaporation effects inside a tub

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