university of trieste school of doctorate in environmental and industrial fluid mechanics xxiii

University of Trieste

School of Doctorate in Environmental and Industrial Fluid

Mechanics

XXIII Cycle

NUMERICAL INVESTIGATION OF CONDENSATION

AND

EVAPORATION EFFECTS INSIDE A TUB

by

Andrea Petronio

April 2011

SUPERVISORS

Prof. V. Armenio

ASSISTANT SUPERVISORS

Ing. G. Buligan

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A Norah e Sarah,per tutte queste H!

Sommario

L’obiettivo 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 l’acqua per ilrisciacquo finale viene portata ad una temperatura di 70C che scalda le stoviglie per-mettendo a queste di immagazzinare energia termica. La lavastoviglie si raffreddadall’esterno, cosicche la vasca risulta essere piu fredda dei piatti posti all’interno. Inquesto contesto l’acqua 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 cosı comuni nelle applicazioni ingegneristiche, la lorocomprensione e lontana dall’essere 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 consideril’evoluzione del film liquido durante processi di asciugatura. Questo progetto vuolecontribuire allo sviluppo della ricerca in questo settore.

Il modello matematico del flusso d’aria in presenza di evaporazione e condensazionee stato implementato numericamente nell’ambiente 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 sottol’approssimazione di Boussinesq. Si assume inoltre l’approssimazione 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 d’aria, prescrivendo una condizione diDirichlet per la temperatura e per il vapore. Quest’ultimo all’interfaccia del liquidoe considerato in condizione di saturazione. Il calcolo della velocita di evaporazioneall’interfaccia, imposta anche come condizione al contorno per il campo di velocita, hapermesso la quantificazione del processo di evaporazione/condensazione consentendo il

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calcolo della massa d’acqua 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 estato evidenziato l’effetto sul flusso di evaporazione attorno al cilindro delle condizionialle pareti, considerate come bagnate o asciutte ed adiabatiche. Inoltre e stato valutatoanche l’effetto 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. Un’ulteriore analisi estata fatta simulando il processo fino all’asciugatura 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 dell’aria 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 l’instaurarsi del moto convettivo e l’effetto dell’asciugatura 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 l’interruzione del programma. Tale compor-tamento e stato spiegato dalla mancata dissipazione turbolenta nel flusso. L’attivazionedel modello LES di Smagorinsky con l’analogia di Reynolds per la determinazione dellediffusivita turbolente dei due scalari ha dato luogo ad una soluzione numericamentestabile. Tuttavia la eccessiva viscosita di sotto-griglia ha sovrastimato la diffusionedegli scalari, inficiando l’accuratezza della simulazione.

Per includere nel modello l’accoppiamento 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 l’implementazione delmodello di sotto-griglia LES dinamico lagrangiano che permettera di superare i limitiriscontrati nel utilizzo del modello di Smagorinsky.

Summary

The overall aim of the project is to get more insights regarding issues related to dish-washer’s 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. Thedishwasher 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 author’s 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 formulationof the Navier-Stokes equations, plus the transport equations for temperature and vaporconcentration, ω. 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 havebeen enlighten along with the dependence on the gap between the body and the wall.In Figure 5.4a an 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 thebuoyancy 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 simplifiedgeometry 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.

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Sponge region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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

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

2.3.1 Interaction with the substrate . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Evaporating drops physic . . . . . . . . . . . . . . . . . . . . . . 13

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

2.3.4 Packing ratio argument . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 E/C b.c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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

2.4.2 Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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

2.5 Thermal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.2 DDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.3 Film model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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

3 Numerical Implementation 31

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

3.1.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.2 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 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 in Left . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . 91

5.4 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

CONTENTS xi

Bibliography 107

xii CONTENTS

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. . . . . . . . . . . . . . . . 13

2.3 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 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 . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 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 θ → 0 . . . . . . . . . . . . . . . . . . . . . . . 63

4.10 Thin-film drop approximation . . . . . . . . . . . . . . . . . . . . . . . . 63

4.11 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 in Left . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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

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

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

4.19 Conductivity ratio r = 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. . . . . . . . . . . . . . . . . . . . . . . . 764.23 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 of Cd and Cl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.8 The geometry layout of the 2D dishwasher case. . . . . . . . . . . . . . . 875.9 Vapor concentration contour plot, for both tub’s walls conditions. . . . . 895.10 Evaporation velocity around the cup, for both tub’s 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

List of Tables

2.1 table table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 EC boundary condition summary . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Stability test parameters and results. . . . . . . . . . . . . . . . . . . . . 554.2 Results of the accuracy-Co test . . . . . . . . . . . . . . . . . . . . . . . 564.3 Drop evaporation rate comparison. . . . . . . . . . . . . . . . . . . . . . 62

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xvi LIST OF TABLES

Chapter 1

Introduction

1.1 Aim of the project

Nowadays 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 dishwasher’s 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 wateris drained out of the tub. Typically, the water for the final rinse cycle is heated up to

1

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 thedrying 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 author’s 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 ǫ, 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 takeplace in wall bounded flows. The relevant literature has been reviewed. This kind of

2

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 focusedon 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 almostall 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

ρ = ρ0 [1− βT∆T − βω∆ω]

where ρ0 is the reference density, βT and βω the volumetric expansion coefficientsdue to changes in temperature, ∆T , and vapor concentration content ∆ω 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 bymeans 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 in 2.3.

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

1the vapor flowing out the liquid interface

3

CHAPTER 1. INTRODUCTION

the thermal point of view. In particular both should incorporate the effect of the latentheat flux due to the change of phase and should correctly represent the film under dif-ferent air flow and substrate conditions, for example taking into consideration dishwarewith very different thermal diffusivity.

Finally the issue of turbulence in this kind of flows is considered. In real scale de-vices turbulence can appear both due to the forced inlet ventilation and in the wake ofbuoyant plumes above hot and wetted bodies. In the late stage of the PhD program thisissue arose during the study of the 3D full scale dishwasher. More turbulence modelhave been taken into account. As the evaporation takes place the dishware cools downand hence the buoyancy force is weakened along the drying process. This means thatthe flows under investigation must be considered as transient in nature. This limits thepossible choices among the turbulence models available mainly to the LES approach,since DNS is not affordable. In particular the Smagorinsky sub-grid closure model hasbeen applied along with the Reynolds analogy for the two scalars diffusivities. Unfor-tunately this model doesn’t fit well for the complex flows here considered, predictingover diffused fields. For all this reasons the Lagrangian Dynamic sub-grid scale modelappears the most appealing. Its implementation is in progress. In the present studythe implementation of the plane averaged dynamic model is included. The latter andthe Smagorinsky model available in the present release of OpenFoam have been testedon the turbulent channel flow case and the results compared against the data of Kim,Moin and Moser [24].

4

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 2.3.1 just 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. Ingeneral, for a binary mixture, the density and mass fraction are expressed by thefollowing constitutive relations

ρ = ρa + ρb (2.1)

1 = ωa + ωb

where the subscripts refer to species, a and b. The mass fraction is the mass concentra-tion ωa,b = ρa,b/ρ. At the liquid film interface the mass flux n equation for one speciesis

na = −ρDa,b∇ωa + ωa(na + nb) (2.2)

5

CHAPTER 2. FORMULATION

where na = ρaua, ua is the velocity of species a, Da,b is the diffusivity. There are twocontributions: the diffusion flux due to the concentration gradient ja = −ρDa,b∇ωa,usually referred as the Flick’s first law of diffusion; and ρua = ωa(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 +∂ρa∂t

− ra = 0 (2.3)

where ra is the rate of chemical production of a. Assuming that Da,b and ρ are constant,without production term, the mass flux expression along with the continuity one yieldto the transport equation for ωa

∂ωa

∂t+ u · ∇ωa = Da,b∇

2ωa (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

(dna

dn

)

i

= 0 (2.5)

(dnb

dn

)

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 = −ρDa,b

1− ωai

(dωa

dn

)

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 ωai is assumedto 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 T and the water vapor mass fractionω, 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 thegas are considered constant. They are the kinematic viscosity ν (m2/s), the thermal

6

2.2. GASEOUS PHASE

expansion coefficient βT (K−1), solutal expansion coefficient βω, the temperature dif-fusion coefficient DT (m2/s) and the mass diffusion coefficient Dω (m2/s). Densityvariations are retained as buoyancy term in the momentum equation along the gravitydirection 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

ρ

∂p

∂x+ ν∇2u

∂v

∂t+ u · ∇v = −

1

ρ

∂p

∂y+ ν∇2v (2.8)

∂w

∂t+ u · ∇w = −

1

ρ

∂p

∂z+ ν∇2w + g(βT∆T + βω∆ω)

Energy Equation∂T

∂t+ u ·∇T = DT∇

2T (2.9)

Concentration Equation

∂ω

∂t+ u · ∇ω = Dω∇

2ω (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 localizedjust 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 therecirculation 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 thevapor mass fraction equations respectively

7


2.2.2 LES turbulence model

In the home-appliances, as well as any other technical application, the flows are quiteoften 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 f is 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 fsgs is the sub-grid-scale contribution filteredout by the convolution 2.13. 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

∂ui∂xi

= 0 (2.15)

∂ui∂t

+∂uiuj∂xj

= −1

ρ

∂p

∂xi+ ν

∂2ui∂xj∂xj

−τij∂xj

(2.16)

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

τij = uiuj − uiuj (2.17)

which have to be modeled. In particular is possible to define the eddy viscosity suchthat

τaij = −2νtSij (2.18)

where τaij = τij−δijτkk/3 is the anisotropic part of the stress, and Sij =12

(∂ui

∂xj+

∂uj

∂xi

)

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

νsgs = (Cs∆)2|Sij | (2.19)

8

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

C ′s = Cs

(1− exp

(−y+/A+

))(2.20)

where y+ = yu∗

ν is the non-dimensional distance from the wall, and A+ is thedistance from the wall up to which the damping occurs, usually is set as A+ = 25.

The friction velocity, u∗, is defined as u∗ =√

τwρ , where τw 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 filter G of width ∆ is applied, then theobtained field is filtered again by the test filter G of width ∆, usually equal to2∆. The SGS-stress is found as

τij = uiuj − uiuj (2.21)

and then applying the test filter

Tij = uiuj − uiuj (2.22)

The so-called Germano identity holds

Lij = Tij − τij = uiuj − uiuj (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 τij and Tij then

τij = −2C2s∆

2|S|Sij (2.24)

Tij = −2C2s ∆

2|S|Sij (2.25)

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

C2s =

〈LijMij〉

〈MijMij〉(2.26)

where

Mij = 2∆2

|S|Sij −

(∆

∆

)2

|S|Sij

(2.27)

9


Averaging has been introduced because the constant Cs can 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 νsgs and eddydiffusivities for the temperature, DTsgs , and vapor concentration, Dωsgs . The two eddydiffusivities are evaluated by means of the Reynolds analogy

DTsgs =νsgsPr

Dωsgs =νsgsSc

(2.28)

Continuity Equation∇ ·u = 0 (2.29)

Momentum Equation

∂u

∂t+ u · ∇u = −

1

ρ

∂p

∂x+∇ · ((ν + νsgs)∇u)

∂v

∂t+ u · ∇v = −

1

ρ

∂p

∂y+∇ · ((ν + νsgs)∇v) (2.30)

∂w

∂t+ u · ∇w = −

1

ρ

∂p

∂z+∇ · ((ν + νsgs)∇w) + g(βT∆T + βω∆ω)

Energy Equation∂T

∂t+ u · ∇T = ∇ ·

((DT +DTsgs)∇T

)(2.31)

Concentration Equation

∂ω

∂t+ u ·∇ω = ∇ ·

((Dω +Dωsgs)∇ω

)(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 successfullyin slightly different research’s frames, as discussed in 1.2. A liquid film is properly

10

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 between hmand 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 doesn’tincorporate 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, it’s possible to obtain the geometry of the interface oncethe pressure term is prescribed.

∆p = σ

(1

R1+

1

R2

)

where ∆p is the pressure difference between the two phases, σ 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]

11


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 orthe 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 = ∆ρgL2/σ

where L is the characteristic length (equivalent diameter). Eo can be interpretedas the ratio of a inter-phase differential hydrostatic pressure ∆ρgL and capillarypressure σ/L. The effect of gravity on the interface should be negligible if Eo≪ 1,

or, defined a =(

2σ∆ρg

)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 holdsθ = π, the second is the partial wetting for which is 0 < θ < π and the third oneis the complete wetting condition with θ = 0. The contact angle θ between liquidphase and solid surface is the parameter that gives the degree of wetting. Atequilibrium it’s assumed that θ is relate to interfacial tensions between solid-gas,liquid-gas and solid-liquid by

σlg cos(θ) = σgs − σsl

called the Young’s 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 θ is increased because of the increased liquid-solid contact area

cos(θW ) = r cos(θY )

where r is the ratio between the actual to apparent surface area, θY is the Youngcontact angle and θW the one prescribed by Wenzel.

12

2.3. THIN-FILM

Figure 2.2: Young contact angle and interfacial tensions.

Moreover there are two more issues regarding θ

• 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 thechange of configuration from the one with three superficial tensions to one justwith γsg. 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 Π, called the disjoiningpressure, that for a polar liquid is

Π(h) =b

h3− e−h

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 Appendix A 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.

13


A constant base regime in which, as liquid evaporates, the contact line remains pinnedto the surface and hence the height of the drop and the contact angle reduce con-sequently; and a constant angle regime in which the drop base recedes as the totalvolume diminishes. The latter is applied for ideal surfaces mostly. If there are anydefects on the surface the drop remains pinned till the Young force can overcome theadditional adhesive force provided by roughness and suddenly the regime switches tothe constant angle mode. The switching can be enforced by liquid-air surface tensiongradient caused, for example, by variations of temperature or concentration of surfac-tant additives, see [38]. Water drops laying on real surfaces, i.e. with defects, havebeen found to be pinned for the largest part of the evaporation process, and to analyzejust this stage seems quite reasonable [4, 8, 9].

All the experimental and numerical work reported here has been done on a particulardrop class: these are surface tension-dominated drops, i.e. the maximum height h0 islower than the capillarity length lc, that for water is lc = 2, 72 mm; the evaporation inthe pinned regime covers the 90% of the overall process, [23]. For this kind of dropsthe spherical cap approximation holds so that the surface curve and the correspondingevaporation rate are given by

h(r, t) =

√(R

sin θ

)2

− r2 −R

tan θ(2.33)

−dV

dt=

2π

ρl

∫ R

0Jr

√(1 +

∂h

∂r

)2

dr (2.34)

being J the evaporative mass flux.

Theoretical results

In recent years some analytical results have been developed under some assumptions:no convective motion takes place inside the drop 5, in the gaseous region only diffusionis admitted, concentration above the drop has an assessment time tω = R2/Dω of theorder of ms, which is much less than the droplet life-time tD, typically of hundreds ofseconds. Among those the more stringent is the omission of the convective motion dueto buoyancy in the air region, which is responsible of an underestimation up to 30% ofthe overall evaporation rate as confirmed by [34]. The evaporating liquid volume rateis given as a function of the contact angle, where R is the radius of the circular dropbase and HR is the relative saturation far from the droplet

θ = pi/2 : −dV

dt=

2πRDω(1−HR)ωi

ρl(2.35)

0 < θ < pi/2 : −dV

dt= (0.27θ2 + 1.3)

πRDω(1−HR)ωi

ρl(2.36)

θ → 0 : −dV

dt=

4RDω(1−HR)ωi

ρl(2.37)

where the first and the third relations are found on analytical basis, the intermediateθ value relation is found via numerical computation by [14].

5In [23] it has been considered but was found to have a negligible effect for the most commonsituations and can be safely discarded [9].

14

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 Figure 2.3.

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 Figure 2.3a, the area undergoing phase change is multiplied by a

factordAi

dA, and consequently the liquid depletion

∆h =dAi

dA

ρg

ρl∆tV 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 ωi 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 in 5.3.

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

15


reasonable that below a certain pk 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 wholeresearch’s 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 in 2.5.If h 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 evaporationor condensation in air-vapor gas treated as an ideal gas mixture, so that the followingstandard relations applies. The vapor concentration is defined as ω = mv

mv+ma. At the

interface it is evaluated by

ωi =Mv

Ma

φi ps(Ti)

p− (1− Mv

Ma)φi ps(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 φ =mv

msis given by

φ =pv

ps(Ti)[kgv/kg

s] (2.41)

6at least from a macroscopic point of view.

16

2.5. THERMAL COUPLING

being pv and ps vapor partial pressures at actual and at saturation condition respec-tively. At the interface is assumed φi = 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

(dω

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 = −Dω

1− ωi

(dω

dn

)

i

(2.44)

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

normal to interface gradient,

(dω

dn

)

i

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

condition allows an inflow or outflow happen through wetted surfaces. As consequenceof 2.43, 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 =

ρaρlVe 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 ∼ 10−3.

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

17


liquid patch laying on a hotter body, and the surrounding air is far from saturation.The solid can conduct enough heat to maintain the water in favorable condition forevaporation despite the large latent heat flux leaving the liquid. If the solid cannotprovide enough heat and the concentration at interface is below the one in the gas,condensation occurs. For the same reasons also adiabatic substrates are found to dampevaporation, eventually promoting equilibrium configurations.

The same arguments apply with respect to the drying process of the dishware. Theheat excess of the solid bodies is released to water films and drops. If the evaporationlasts long enough eventually all the dishware completely dry. It is common experiencethat this is not always the case. Often some drops remains pinned inside a glass orsome portion of plastic bowls are still wetted after the dishwasher drying cycle. Themain reason behind is twofold: the first is probably related to the wetting propertiesof the dishware surface, the second lies in thermal properties of the specific object. Inthis study and in particular in the present section the problem of modeling the thermalcoupling is considered in order to take into account the different materials the tablewareare made of.

From a modeling point of view the thermal coupling is called Conjugate Heat Trans-fer (CHT) problem between the gaseous flow and the solid with the interposition of thethin liquid film. Two more issues need to be addressed to fulfill the CHT task, namelythe choice of a Domain Decomposition method and a suitable energy equation for thethin film. In the following each subject is tackled.

2.5.2 Domain Decomposition Method

In the numerical approximation of the solution of P.D.Es it is often required to de-compose the domain in more sub-domains, mainly for three different reasons [39]: asdata structure decomposition used in parallel computing, as a preconditioning methodfor large problems or a technique to couple different kind of P.D.Es across interfacesof disjointed sub-domains, i.e. the subdivision of the physical domain in sub-regionsthat are modeled with different equations or even with different numerical methods.The latter procedure is also referred as heterogeneous domain decomposition [30]. Inliterature Domain Decomposition Method (DDM) refers to all mentioned subjects, butnow just the last one is considered.

Assume that an entire domain Ω is split in two sub-domains Ω1 and Ω2. The domaindecomposition technique allows to find the solution over Ω starting from solutions overthe smaller regions, provided suitable boundary conditions. The DDM methods can bedivided in two main groups based don how the splitting is performed: with overlappingor without overlapping of sub-domains as schematically represented in Figure 2.4. Tothe former belongs the popular Schwarz method which impose a Dirichlet conditionon both the boundaries Γ1 and Γ2 dividing the two regions as shown in Figure 2.4b.Among the latter there are the Dirichlet-Neumann (DN) and the Neumann-Neumannmethods (NN). In general non-overlapping methods require a simpler implementationand lower computational and communication costs. In particular DN is found to beeffective in many applications, as for examples in [27, 7]. For this reasons in this workthe DN technique has been preferred and then implemented.

18


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

Figure 2.4: Decomposition of domain.

Equivalence theorem

A non-overlapping decomposition requires that Ω1 ∪ Ω2 = Γ and Ω1 ∩ Ω2 = ∅ asdepicted in 2.4a, moreover ni is the outward normal of the each sub domain Ωi such thatn1 = −n2. The following theorem gives a mathematical basis to the non-overlappingDDM [31].

Given L as elliptic operator, u is the solution to following differential problem overthe domain Ω with Lipschitz boundary ∂Ω

Lu = f in Ωu = 0 on ∂Ω

(2.46)

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

Lui = f in Ωi

u = 0 on ∂Ωi/Γ(2.47)

provided the coupling condition to be applied at the interface Γ

u1|Γ = u2|Γ(∂u1∂n1

)

Γ

=

(∂u2∂n2

)

Γ

(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 Ω× (0, T )

B.C. on ∂Ω× (0, T )I.C. in Ω at t = 0

(2.49)

In the DDM for time dependent P.D.Es implicit temporal discretization schemesshould be preferred. The estimation of the values required at the interface often involves

19


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 natureof 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 Ω1

u(k)1 = u

(k−1)2 on Γ

u(k)1 = 0on Ω1/Γ

(2.50)

Lu(k)2 = f in Ω2

∂u(k)2

∂n2=∂u

(k)1

∂n1on Γ

u(k)2 = 0 on Ω2/Γ

(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 in 2.51 is a Neumann problemwith the derivative value at its boundary Γ is computed by the solution of 2.50 in 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 2.50 can be relaxed as

u(k)1 = θu

(k−1)2 + (1− θ)u

(k−1)1 on Γ

that lowers the error at each iteration given the relaxation parameter θ. The existenceof a θ 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

∂T s

∂t= DT s∇2T s (2.52)

where T s is the temperature in the solid body to avoid ambiguity, and consequently DT s

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 assumption 2.3

20


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

∂T l

∂n≫

∂T l

∂t1,2(2.53)

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

∂T l

∂t= DT l

∂2T l

∂n2(2.54)

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

T l = T l(y) with 0 ≤ y ≤ h(x) (2.55)

being y 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 totackle 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(∂T s

∂n

)

w

= −kl(∂T l

∂n

)

w

(2.56)

T s = T l (2.57)

Phase change process is associated to the latent heat, L, release. This is gener-ally large, for example L ≈ 2.26 106 [J/kg] for water evaporation. Thous the evapora-tion/condensation latent heat flux LJ is found to be the ruling term of the heat balanceat the liquid/gaseous boundary

LJ = −ρiLDω

1− ωi

(∂ω

∂n

)

i

[J/m2s] (2.58)

then at the interface the following applies

−kl(∂T l

∂n

)

i

= −kg(∂T g

∂n

)

i

− JL (2.59)

T l = T g

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).

21


water PP glass ceramic metal

D [m2/s] 1.4 10−7 10−7 3 10−7 1.1 10−6 10−5

h [mm] 0.5 2 2 2 2td [s] 2 40 13 3.6 0.4

Table 2.1: table table

Diffusion time scale

Tableware can have very different geometries but usually display high aspect ratios,and a characteristic length scale for their thickness ranging from millimeters to roughlyone centimeter. But they thermal properties can vary of orders of magnitudes, seefor example the data in Table 2.1. A scale analysis helps in the choice of the propersolution (2.55). The fundamental parameter is found to be the temperature diffusiontime scale, td, defined as

td = l2ρcpk

= l2/DT (2.60)

It can be interpreted as the characteristic settlement time in response to a temperaturechange.

Assuming a typical thickness of 0.5 mm for the water layer the corresponding set-tlement time is twd = 2 s. This time has to be compared with the diffusion times ofthe typical materials the dishware are made of. For example, to cover most of thepossibilities, polypropylene, PP, glass, ceramics and metal typical thermal propertieshave been considered. The td are reported in Table 2.1 for a typical thickness of 2 mm.The ratio between twd and the generic solid tsd is spread on two order of magnitude

0.05 <twdtsd< 5

In light of these considerations is possible to further simplify the FTF model. Forexample a liquid model can be based on a constant heat flux assumption, prescribinga linear variation of the temperature as

T l(y) = T sw −

y

kl

[kg(∂T g

∂n

)

i

+ JL

](2.61)

that is equivalent to assume an instant diffusion model for which DTl→ ∞. This model

can be a good approximation for cases in which tld ≪ tsd holds, and the temperaturein the liquid settles much faster than in the solid. Another choice is to consider atemperature profile constant at the same value of the underlaying wall

T l(y) = Tw

and parametrize the heat fluxes. A better approach is to apply an energy balance onthe infinitesimal liquid water control volume Vl, assuming heat flux only through theinterfaces

∂El

∂t=∂Ein

∂t−∂Eout

∂t(2.62)

22


∂

∂t

∫

Vl

ρlcpTl dV =

[ks

(∂T s

∂y

)

w

dSs + kg

(∂T g

∂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 of T and 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 temperatureis 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 Figure 2.5c in 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 Neumannboundary 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= α

∂2T

∂x2T (0, t) = a∂T (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

Bn sin

((2n+ 1)π

2lx

)exp

(−(2n+ 1)πα

2l2t

)

23


(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.

24


where the Fourier coefficients are computed asBn =

2

l

∫ l

0g(x) sin

((2n + 1)π

2lx

)dx

g(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

Bn exp

(−(2n+ 1)πα

2l2t

)cos

((2n + 1)π

2

)(2.66)

= b (2.67)

while the gradient at x = 0 is computed as

∂T

∂x(0, t) = b+

∞∑

0

Bn exp

(−(2n+ 1)πα

2l2t

)(2n + 1)π

2lcos (0) (2.68)

= b+∞∑

0

Bn exp

(−(2n+ 1)πα

2l2t

)(2n + 1)π

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. Usually T (x)is quite smooth so that N can 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 1mm thick liquid film is computed provided aDirichlet 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 of Tw(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 Figures 2.6 and 2.7 are 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. if tdw < 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 thereforecannot be effective in the thermal coupling algorithm.

25


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

(b) slow variation tds = 25 s, tdw ≪ tds thin plastic1mm, thick glass (3mm)

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

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

26


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

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

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

Figure 2.7: Film temperature distributions for different cases.

27


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

ν

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 =√

(βT∆T + βω∆ω)gL

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

Grω =gβω∆ωL

3

ν2GrT =

gβT∆TL3

ν2

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 Pr ≈ 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

Nut = Nus +Nul

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

Dω

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

28

2.7. MODEL SUMMARY

Equation Condition

Temperature assigned value Ti

Concentration ωi =Mv

Ma

ps(Ti)

p−(1−MvMa

) ps(Ti))

Momentum Ve = −Dω

1− ωi

(dω

dn

)

i

Film ThicknessdH

dt=ρaρwVe

Table 2.2: EC boundary condition summary

2.7 Model summary

In conclusion to this chapter the mathematical model is here summarized. The equa-tions that are solved in Chapter 5 are given along with the boundary conditions for thewetted surfaces in the table 2.2.

Continuity Equation∇ ·u = 0

Momentum Equation

∂u

∂t+ u ·∇u = −

1

ρ

∂p

∂x+ ν∇2u

∂v

∂t+ u · ∇v = −

1

ρ

∂p

∂y+ ν∇2v

∂w

∂t+ u · ∇w = −

1

ρ

∂p

∂z+ ν∇2w + g(βT∆T + βω∆ω)

Energy Equation∂T

∂t+ u ·∇T = DT∇

2T

Concentration Equation∂ω

∂t+ u · ∇ω = Dω∇

2ω

29


30

Chapter 3

Numerical Implementation

Open Source Field Operation and Manipulation, OpenFoam in short, is essentially acollection of C++ libraries that are used to create executable files, applications, whichare called solvers and 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 aredynamically 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. Anumber of different discretization schemes are available for each operation. These

31

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 is solvedin terms of fluxes at the cell faces, and then the velocity is obtained again at the cellcenter.

The discretization procedure based on the Finite Volume Method, FVM, adoptedin OpenFoam is discussed in the following. As reference the general transport equationfor a scalar φ, with a source term Sφ is considered

dρφ

dt︸︷︷︸Temporal derivative

+ ∇ · (ρUφ)︸︷︷︸Convection term

= ∇ · (ρΓ∇φ)︸︷︷︸Diffusion term

+ Sφ(φ)︸︷︷︸Source term

(3.1)

where Γ is the diffusivity coefficient.

3.1.1 Spatial discretization

The solution domain is split in a set of cells defined by a computational mesh overwhich the P.D.Es’ system must be discretized and then solved. In the FVM procedurethe cell are referred as Control Volumes, CV, of volume Vc, with center, named P ,located in xp such that ∫

Vc

(x− xp)dV = 0

in which all the dependent variables are stored. As depicted in Figure 3.1, each CV isbounded by a number of flat faces, denoted by the label f ; S is the face area vectordefined to be normal to the face itself and with magnitude equal to the face area. S

points outward into the neighboring cell with center N ; d denotes the vector betweenthe two centers. The mesh is said to be orthogonal if d is orthogonal to the face, i.e.if d is parallel to S. Since 3.1 is a second order partial differential equation, to ensurea good overall accuracy of the solution the numerical scheme must be at least of thesecond order too. OpenFoam achieves a second-order accurate method allowing thedependent variables to vary linearly within the CV around the point P and time t:

φ(x) = φ(xP ) + (x− xP ) · (∇φ)xP

φ(t+∆t) = φ(t) + ∆t

(∂φ

∂t

)

t

32

3.1. FVM DISCRETIZATION

Figure 3.1: A Control Volume and its parameters.

In deriving all the discretized operators the generalized Gauss’s Theorem will beextensively recalled by taking advantage of the following identities

∫

Vc

∇ ·a dV =

∫

∂Vc

dS · a

∫

Vc

∇φdV =

∫

∂Vc

dSφ (3.2)

∫

Vc

∇a dV =

∫

∂Vc

dSa

where ∂Vc is the surface enclosing the volume of CV, and dS the infinitesimal surfacewith its normal pointing outward of ∂Vc; a and φ are arbitrary vector and scalar fieldsrespectively.

Central Differencing Scheme In the discretized formulation, in order to take ad-vantage of computing surface integrals, the values at the cell face centers are required.For this task OpenFoam implements a large number of the most used interpolationschemes for CFD. In this study the Central Differencing Scheme, CDS, is adoptedexclusively, if not differently specified. The values at the face centers are computed by

φf = fxφP + (1− fx)φN

with fx the ratio

fx =fN

PN

CDS is known to be second order accurate even on unstructured mesh [11].

33


Gradient Operator In OpenFoam the gradient operator can be applied at the prim-itive variable at the cell center. The Gauss theorem simply yields to

∫

Vc

∇φdV =

∫∂V dSφ ≈

∑

f

Sφf (3.3)

Often is also required to obtain the surface normal gradient, ∇⊥f φ, at the cell faces. If

n is the face normal

∇⊥f φ = n · ∇fφ ≈

φN − φP|d|2

d (3.4)

This approximation keeps second-order accuracy if the mesh is orthogonal, i.e. S isparallel to d. Otherwise a non-orthogonal correction should be introduced.

Convection term The convection term ∇ · (ρUφ) appearing in (3.1) is integratedover the Control Volume replacing the volume integral with the surface one invokingthe Gauss theorem as follow

∫

Vc

∇ · (ρUφ)dV =

∫

∂Vc

dS · (ρUφ)

≈∑

f

S · (ρU)fφf (3.5)

=∑

f

Ffφf

where Ff = S · (ρU)f is the mass flux through the face cell. All the quantities at thefaces are evaluated by interpolation.

Diffusion term The diffusion term ∇ · (ρΓ∇φ) of (3.1) is treated in a similar manner

∫

Vc

∇ · (ρΓ∇φ) dV =

∫

∂Vc

dS · (ρΓ∇φ)

≈∑

f

S · (ρΓ∇φ)f (3.6)

=∑

f

(ρΓ)fS · (∇φ)f

the integral over Vc of the divergence operator becomes a summation over the faces ofthe cell of the surface normal gradient, as defined above in (3.4). If S is not parallel tod the non-orthogonal correction is required to properly treat S · (∇φ)f and retain thesecond order accuracy of the discretization scheme. The corrected gradient term reads

S · (∇φ)f = ∆ ·∇⊥f φ︸︷︷︸

orthogonal contribution

+ k · (∇φ)f︸︷︷︸non−orthogonal correction

(3.7)

with ∆, chosen to be parallel to d, and k. The two vector are determined by theparticular correction adopted, satisfying the condition

∆+ k = S

34

3.1. FVM DISCRETIZATION

Figure 3.2: The decomposition of the S in the “over-relaxed” correction approach.

as schematically represented in Figure 3.2. OpenFoam choses the “over-relaxed” ap-proach, as described in [17, 6], moreover in [17] it is proved that the “over-relaxed” isthe most robust, convergent and computationally efficient among the tested procedures.The orthogonal term is given by (3.4) multiplied by

∆ =d

d ·S|S|2

This is meant to increase the importance of the values at the CV center, φP and φN ,as the non-orthogonality increases. The correction term is evaluated by interpolatingcell centered gradient (3.3)

(∇φ)f = fx (∇φ)P + (1− fx) (∇φ)N

Although this methods is also second order accurate, it’s computed over a larger com-putational molecule, thous has larger truncation error than (3.4). The correction iscomputed explicitly for computational reasons as a consequence can lead to unboundeddiffusive terms and instability in particular on highly non-orthogonal meshes. To con-clude, the correction should be performed to keep the overall accuracy but a goodquality mesh is required, otherwise stability issues may necessitate to drop the non-orthogonal treatment lowering the order of the scheme.

3.1.2 Time discretization

In the following the numerical method for dealing with the temporal derivative and theintegration in time of (3.1) will be examined. Among the many implemented schemeswithin the software, the focus will be on the second order backward differencingscheme, BD in short, since it has been almost exclusively used in the present study.The time integral of (3.1) reads

∫ t+∆t

t

dρφ

dt+∇ · (ρUφ) dt =

∫ t+∆t

t∇ · (ρΓ∇φ) + Sφ(φ) dt (3.8)

and the spatial discretization form gives

∫ t+∆t

t

(dρφ

dt

)

P

Vc +∑

f

Ffφf

dt =

35


Figure 3.3: Boundary condition discretization scheme.

∫ t+∆t

t

∑

f

(ρΓ)fS · (∇φ)f + (Su + SpφP )Vc

dt (3.9)

in which φ is assumed to vary linearly in time. BD takes advantage of three time levels,denoted by the index n, n− 1 and n− 2, from the current to two previous time steps.The Taylor expansions of φn−2 and φn−1 can be combined together, [6], to find thesecond order approximation of the temporal derivative at the present time step

(dφ

dt

)n

=32φ

n − 2φn−1 + 12φ

n−2

∆t(3.10)

In this scheme the temporal variation of face fluxes are neglected, so that the fulldiscretized equation (3.1) is

(32ρφ

n − 2ρφn−1 + 12ρφ

n−2

∆t

)

P

Vc +∑

f

Ffφnf −

∑

f

(ρΓ)fS · (∇φ)nf = (Su + SpφnP )Vc

(3.11)The linear system must be solved for φnP .

3.1.3 Boundary condition

The discretization procedure requires the values at the faces of each cell, namely φf or∇fφ. When this computation involves faces laying on the boundary surface a numericaltreatment of each boundary face for all the variables is needed. In Figure 3.3 the vectord connects the cell center to the boundary face center, b, and dn is its componentparallel to S. The condition applied is assumed to be valid on the whole face. Thenon-orthogonality, for both corrected and non-corrected schemes, is always addressed

36

3.2. PISO ALGORITHM

by taking into account dn as discussed in [17, 6, 32]. The basic boundary conditionsimplemention is described in most of the OpenFoam available documentation, see forexample [6].

3.2 PISO algorithm

OpenFoam standard solvers for the incompressible Navier-Stokes equations adopt mainlytwo algorithms, the SIMPLE, for steady-state problems, and PISO for transient ones.Both lag the non-linearity in the convective term and require iterative procedures toenforce the continuity condition.

The PISO (Pressure Implicit with Splitting of Operators) algorithm originally pre-sented by Issa in 1986, is a pressure-velocity calculation procedure. Its implementationin OpenFoam is described and schematically represented in the diagram in Figure 3.4:

1. initial condition for pressure, and velocity must be provided.

2. set up the coefficient matrix, UEqn, for the momentum equation without thepressure contribution. All terms are treated implicitly except for the volumetricsource S

fvVectorMatrix UEqn

(

fvm::ddt(U)

+ fvm::div(phi, U)

- fvm::laplacian(nu, U) - S

);

In a tensorial notation this can be written as

Cu− S

The non-linearity of convection is linearized by freezing the convective flux phiat the previous time step as

∑

f

S · (ρun−1)funf

3. momentum predictor solves velocity using the last known value of pressure,i.e. the gradient is treated explicitly.

solve(UEqn == -fvc::grad(p));

In general the FVM method will bring to a coefficient matrix for the actual timen, C, in which the convective term has been linearized with the value at n− 1, ifr collects all the explicit terms then

Cu = r−∇ · pn−1

The predictor step will solve this equation for u, a non-divergence free velocityfield.

37


4. the PISO loop takes care of the momentum-pressure coupling leading possi-bly also to a proper momentum solution. The number of iterations is prescribedand the momentum balance is not a priori guaranteed. The OpenFoam refer-ences [17], [6] and [32] assures that in most of the cases 2 or 3 PISO loops areenough. Although in complex geometries or in presence of buoyancy forces addi-tional loops are required as proved by the present work in Section 4.1.1. Moreoverit has been proven that the number of loops affects just the stability of the solu-tion but not its accuracy. The UEqn matrix can be split in the diagonal terms,A, and the off-diagonal terms, H ′

C = A+H′

so that the current corrector step solution, denoted by ∗∗, can be computed fromthe last correction or from the previous time step firstly entering the loop, withthe ∗

Au∗∗ +H′u∗ = r −∇p∗

replacing H = r −H′u∗, this yields to

u∗∗ = A−1H−A−1∇p∗

enforcing now the continuity of the velocity field, ∇ ·u∗∗ = 0, the pressure equa-tion is found

∇2(A−1∇p∗) = ∇ ·A−1H

In the code this steps can be recognized: discarding the predictor result, from thelast time step solution of velocity, the diagonal coefficients are extracted and thereciprocal values are stored and used to compute the velocity lacking the pressureinfluence:

volScalarField rUA = 1.0/UEqn.A();

U = rUA*UEqn.H();

and then reconstructing the flux at the cell faces using this U

phi = (fvc::interpolate(U) & mesh.Sf())

+ fvc::ddtPhiCorr(rUA, U, phi);

we solve the pressure equation

fvm::laplacian(rUA, p) == fvc::div(phi)

The pressure is solved iteratively until the prescribed tolerance is reached, as-suring a divergence-free velocity. At this stage the non-orthogonal correction isperformed for a prefixed iteration number on the pressure gradient as describedin 3.1.1

5. in the explicit velocity correction step the velocity field is then corrected inan explicit manner as a consequence of the new pressure distribution

U -= rUA*fvc::grad(p);

38

3.2. PISO ALGORITHM

Figure 3.4: The PISO algorithm.

meaning that at the last corrected velocity without pressure influence is takenout by the pressure contribution that satisfies the continuity

u∗∗ = u∗ −A−1∇p∗

6. points 4 to 5, are computed for a fixed number of iterations, each time updatingthe values of H by the most recent u available

To summarize the PISO loop consists of an implicit momentum predictor followed bya series of pressure solutions and explicit velocity corrections, as shown in Figure 3.4.

Rhie-Chow interpolation This method gives an oscillation-free velocity in line withRhie-Chow interpolation, even though the grid isn’t staggered. This can be understoodanalyzing the diffusion term in the Laplacian equation for p

∫

V∇ · (Γ∇p) dV =

∫

AΓ∇p · dA ≈

∑

f

ΓfSf · (∇p)f

this discretization uses the value of the gradient of p on the cell face, once it is calculatedusing neighboring center cell values, as described in (3.4). Then in the correction of uthe explicit gradient term of p is calculated from the cell face values of p

∫

V∇p dV =

∫

SdS p ≈

∑

f

Sfpf

In the following the commented main code of the incompressible flow solver, calledIcoFoam:

39


int main(int argc, char *argv[])

// these header files contain source code for common tasks

// that are used in numerous applications.

# include "setRootCase.H"

# include "createTime.H"

# include "createMesh.H"

# include "createFields.H"

# include "initContinuityErrs.H"

// * * * * * * * * * * * * * * * * * * * * * * * * //

Info<< "\nStarting time loop\n" << endl;

// use the runTime object to control time stepping

for (runTime++; !runTime.end(); runTime++)

Info<< "Time = " << runTime.timeName() << nl << endl;

# include "readPISOControls.H"

# include "CourantNo.H"

// set up the linear algebra for the momentum equation.

// The flux of U, phi, is treated explicity

// using the last known value of U.

fvVectorMatrix UEqn

(

fvm::ddt(U)

+ fvm::div(phi, U)

- fvm::laplacian(nu, U)

);

// solve using the last known value of p on the RHS.+

// This gives us a velocity field that is

// not divergence free, but approximately satisfies momentum.

// See Eqn. 7.31 of Ferziger & Peric

solve(UEqn == -fvc::grad(p));

// --- PISO loop---- take nCorr corrector steps

for (int corr=0; corr<nCorr; corr++)

// from the last solution of velocity, extract the diag. term from

// the matrix and store the reciprocal

// note that the matrix coefficients are functions of U due to

40

3.2. PISO ALGORITHM

// the non-linearity of convection.

volScalarField rUA = 1.0/UEqn.A();

// take a Jacobi pass and update U. See Hrv Jasak’s thesis eqn. 3.137

// and Henrik Rusche’s thesis, eqn. 2.43

// UEqn.H is the right-hand side of the UEqn minus the product of

// (the off-diagonal terms and U).

// Note that since the pressure gradient is not included in the UEqn.

// above, this gives us U without the pressure gradient.

// Also note that UEqn.H() is a function of U.

U = rUA*UEqn.H();

// calculate the fluxes by dotting the interpolated velocity

// (to cell faces) with face normals

// The ddtPhiCorr term accounts for the divergence of the face

// velocity field by taking out the

// difference between the interpolated velocity and the flux.

phi = (fvc::interpolate(U) & mesh.Sf())

+ fvc::ddtPhiCorr(rUA, U, phi);

// adjusts the inlet and outlet fluxes to obey continuity,

// which is necessary for creating a well-posed

// problem where a solution for pressure exists.

adjustPhi(phi, U, p);

// iteratively correct for non-orthogonality.

// The non-orthogonal part of the Laplacian is calculated from

// the most recent solution for pressure,

// using a deferred-correction approach.

for (int nonOrth=0; nonOrth<=nNonOrthCorr; nonOrth++)

// set up the pressure equation

fvScalarMatrix pEqn

(

fvm::laplacian(rUA, p) == fvc::div(phi)

);

// in incompressible flow, only relative pressure matters.

// Unless there is a pressure BC present, one cell’s pressure

// can be set arbitrarily to produce a unique pressure solution

pEqn.setReference(pRefCell, pRefValue);

pEqn.solve();

// on the last non-orthogonality correction, correct the flux

// using the most up-to-date pressure

if (nonOrth == nNonOrthCorr)

//The .flux method includes contributions from all implicit terms

// of the pEqn (the Laplacian)

phi -= pEqn.flux();

41


// end of non-orthogonality looping

# include "continuityErrs.H"

// add pressure gradient to interior velocity and BC’s.

// Note that this pressure is not just a small

// correction to a previous pressure, but is the entire pressure field.

// Contrast this to the use of p’

// in Ferziger & Peric, Eqn. 7.37.

U -= rUA*fvc::grad(p);

U.correctBoundaryConditions();

// end of the PISO loop

runTime.write();

Info<< "ExecutionTime = " << runTime.elapsedCpuTime() << " s"

<< " ClockTime = " << runTime.elapsedClockTime() << " s"

<< nl << endl;

// end of the time step loop

Info<< "End\n" << endl;

return(0);

// ********************************************* //

3.3 Current model implementation

The mathematical model described in Chapter 2 has been implemented within theOpenFoam environment, taking advantage of the PISO algorithm structure discussedin 3.2. In particular the new code takes care of the two active scalars, temperatureand vapor concentration, and adds the buoyancy term in the momentum equation con-sistently to the equation set (2.7), (2.8), (2.9), (2.10). The boundary condition foreach variable has been developed to match the requirements of the evaporation andcondensation model given in 2.4. Moreover two features are made available based onthe specific case needs: the adjustable time stepping procedure is able to vary theadvancing time step adapting to the flow conditions, and a sponge region can be over-lapped to the computational domain near the outlet section in order to preserve fromnumerical instability issues, as explained in 2.2.1 . Finally the thin-film model sketchedthrough 2.5 is coded. It acts as a property the boundary retrieving the temperatureprofile for each boundary face with non zero film thickness and the temperature gra-dients at the film-air and solid-film interfaces. Also the Dirichlet-Neumann couplingalgorithm has been coded and tested. Both the FTF model and the DNM have still tobe included into the main solver.

42

3.3. CURRENT MODEL

Adjustable time step The adjustable time stepping procedure is meant to keepmaximum Courant number constant. This value is computed each time step consideringthe flux through each cell face, S ·u, and the distance between the two neighboroughcell center, |d|

CoM =S ·u

|S| |d|∆t (3.12)

The reduction of time-step applies whenever the CoM exceed the prescribed value,while when it’s below the threshold the increment is damped in order to avoid unstableoscillations. The maximum increment allowed is of the 20% of the previous time step.

Sponge region In general the definition of the sponge region is case dependent.From a coding point of view it can be set up allowing the diffusion and the expansioncoefficients to vary in the domain. In the present codes it is achieved by creating ascalar field for each parameter from some functions of the type given by (2.11). Asany other field it requires also to be prescribed consequently onto the boundaries.

3.3.1 Current model algorithm

At the beginning of each time step loop the transport equation for temperature andvapor concentration, named T and C in the code, are solved retaining the last velocityfield available for the adventive term. The coefficient matrices are built up and thelinear systems solved

solve

(

fvm::ddt(T)

+ fvm::div(phi, T)

- fvm::laplacian(DT, T)

);

solve

(

fvm::ddt(C)

+ fvm::div(phi, C)

- fvm::laplacian(DC, C)

);

while the velocity and pressure are treated as already described in 3.2 except thatexplicit buoyancy source term is added. The local density variation is multiplied bygravity acceleration vector g.

fvVectorMatrix UEqn

(

fvm::ddt(U)

+ fvm::div(phi, U)

- fvm::laplacian(nuDamp, U) + g*(betat*(T-T0) + betac*(C-C0))

);

The diffusion and expansion coefficients, named with custom fashion DT, DC, nu, betat,betac, are prescribed as scalars in a separated file, or as fields in the first time stepdirectory if a sponge region is needed. The algorithm is summarized by the diagram inFigure 3.5.

43


Figure 3.5: The current algorithm.

44

3.3. CURRENT MODEL

3.3.2 Evaporation/Condensation boundary condition implementation

In general the model of the evaporation and condensation at the film interface, asdescribed in 2.4, requires Dirichlet boundary conditions for T, C, H and U in such away that each of them is a function of the other. Again these couplings are solved bysegregated approach. Each field is evaluated retaining the most recent values availableof the other fields. This implementation is general, meaning that works for a spatial andtime varying interface values, so that it can be applied straightforwardly to the interfacemodel also in the conjugate heat transfer problem. In this case the temperature field ofthe film is calculated instead of being prescribed. From here on each boundary variableis considered in details

Temperature The temperature of the film interface is the only variable to prescribe.The Dirichlet condition can be assigned by fixedValue boundary condition typeas for the field T, for a generic boundary patch

type fixedValue;

value uniform 336.15;

Vapor concentration At the interface the saturation condition hypothesis holds forthe field C. The vapor concentration value is enforced at the interface by settinga generic boundary patch

type wettedWall;

The wettedWall patch type takes care of computing and updating the relationgiven by (2.40)

void wettedWallFvPatchScalarField::updateCoeffs()

forAll(*this, i)

if (H_interface[i] == 0.0) ci[i] = cinternal[i]; // zeroGradient condition

else

ps = 611.85*Foam::exp((17.502*(T_interface[i]-273.15))/(240.9+ (T_interface[i]-273.15)));

ci[i] = coeff*ps/(Patm - 0.378*ps);

scalarField::operator=(ci);

Moreover if the surface is completely dried, i.e. the film thickness Hinterface = 0,the check allows the switch to a locally zero gradient boundary condition.

Evaporation velocity The evaporation/condensation process is ruled by the evapo-ration velocity. It is computed and assigned as boundary condition for U consis-tently to (2.44), by setting the generic patch type

type evaporationVelocity;

Again there is a check, if Hinterface = 0, then the velocity is set to be zero.

45


Figure 3.6: Ghost liquid cell balance.

void evaporationVelocityFvPatchVectorField::updateCoeffs()

scalarField nGradC(C_interface.snGrad());

forAll(*this, i)

if (H_interface[i] == 0.0) evaporationVelocity[i] = 0;

else

evaporationVelocity[i] = -( DC_interface[i]/1-C_interface[i])*nGradC[i];

vectorField::operator=(evaporationVelocity * n);

Film thickness The film thickness, H in the code, is modeled as a feature of eachboundary face representing the interface. Its patch type is set by

dish1

type filmThickness;

value uniform 1e-4;

where the value is updated each time step cell by cell following the mass balancegiven in (2.45)

∆h =ρgρl∆tV e (3.13)

clipping the value to zero, if the current iteration produces negative values.

void filmThicknessFvPatchScalarField::updateCoeffs()

scalarField dh = ratio* ( n & U_interface) *mesh.time().deltaT().value();

scalarField Hupdate = *this + dh;

forAll(Hupdate, i)

if (Hupdate[i] < 0.0) Hupdate[i] = 0.0;

scalarField::operator=(Hupdate);

Before moving to the new time step, the film thickness variable must be explicitlyupdated at all the boundaries, since it doesn’t undergo the solve method

forAll(H.boundaryField(), j) H.boundaryField()[j].updateCoeffs();

46

3.3. CURRENT MODEL

It worth to note that this boundary condition can be retained also if a dynamicmodel for the film thickness is embedded in the present solver.

The consistency of the present film computation in the FVM frame can be sup-ported by considering one ghost cell layer, as in Figure 3.6, placed beyond the com-putational domains, below the wetted surface. It stores the liquid amount. Eachghost cell, GC, is therefore discretized consequently, not allowing flow amongadjacent GC. The balance of mass requires the GC volume to reduce or increasecorrespondingly reducing or increasing the height of the cell, H. The only fluxesadmitted are those through the interface face, if , namely diffusion and advection

∫

GCρl∂H

∂tdS =

∫

ifρg

(Dω

∂ω

∂n− ωVe

)· dS (3.14)

=

∫

if−Ve ·dS (3.15)

≈ −Ve ·Si (3.16)

A remark about the drying process: the current modeling can predict the evolutionof the film thickness to complete drying in effective way but in roughly manner. Thelast step leading to Hinterface = 0 permits an evaporative flux higher than what isat disposal at the surface. Considering once again the length scale of the flows underconsideration, this kind of error is found to be altogether negligible, compared to theunder-resolution near the contact lines if present. It has been observed that the currentdrying process is affected by the local condition of the flow as expected, but present alsoa mesh size dependence. It happens that a boundary face has a higher evaporation ratethan the surrounding just due to numerical errors. In the end it will dry sooner andat that point the model will switches to the non-wetted boundary condition yielding tothe fringe effect described in 2.3.3. This may be unphysical but such an issues can beovercome only by a proper dynamic model for the film that will allow film rupture anddrop formation or by considering a droplet distribution instead of a film. Embed anyof these two approach seems exceedingly beyond the present computational capability.

3.3.3 Finite thickness film implementation

The temperature distribution inside the finite thickness film is implemented followingthe description given in 2.5. The liquid film is still treated as a boundary condition buta temperature profile corresponds to each cell face along its thickness, that will act asa buffer layer between the air and the solid body temperature. The structure of thesolver has to be modified slightly in order to include this new feature, along with theDirichlet-Neumann algorithm to couple either the film and air or solid, film and air.

A new data structure has been define within OpenFoam frame, a N -dimensionedarray field, named NVectorField, meant to extend the already implemented vectors andmatrices. The number of elements can be prescribed by the user, based on the actualneeds.

The NVectorField represents the temperature inside the film. The boundary typefilmT implements the model discussed in 2.5.

The idea is to discretized the temperature profile as follows: each boundary face hasits own film thickness Hinterface = Hb that is uniformly divided in N points, placed at

47


Figure 3.7: Finite film thickness model sketch.

hi =Hb

N i for i = 1, . . . N . The initial distribution is stored as T (hi) for i = 0, . . . N − 1in the NVectorField. The model is schematically represented in Figure 3.7

The filmT boundary condition reads the temperature of the interface, Tinterface andof the corresponding surface below the film, computes the heat flux summing the latentheat and the conduction contribution

scalarField HF_interface( (- ka_.value() * T_interface.snGrad() + // conduction contribution

rhoa_.value() * L_.value() * ( n & U_interface) // latent heat contribution

) //

/ kl_.value()

);

then the analytical solution derived in 2.5 is computed and stored for each cell face.Eventually the error is evaluated as the difference between the given Tinterface and thecomputed T (hN ) for the film-air coupling

TErr = mag( film[c][(nx_-1)] - T_interface[c] ) ;

In the conjugate heat transfer solver this error dictates if the solution accuracy isreached, i.e. the continuity of the temperature between liquid and gas 1. Again thefilm temperature boundary condition requires an explicit call to update the field.

int heatFluxErrCorrCnt = 1;

for(int heatFluxErrCorrNr=0; heatFluxErrCorrNr < heatFluxErrCorrCnt; heatFluxErrCorrNr++)

bool checkAll = true;

for (int nonOrth=0; nonOrth<=nNonOrthCorr; nonOrth++)

solve ( fvm::ddt(Tint) + fvm::div(phi, Tint) - fvm::laplacian(DT, Tint) );

forAll(film.boundaryField(), j)

if(film.boundaryField()[j].type()=="filmT")

film.boundaryField()[j].updateCoeffs();

if(!refCast<const filmTFvPatchNVectorField>(film.boundaryField()[j]).myCheck())

checkAll = false;

if (!checkAll) heatFluxErrCorrCnt++; // add loop untill convergence

1the heat flux balance is assumed to be exactly matched applying the analytical solution

48

3.4. LES MODELS

To prescribe the full thermal coupling among solid, film and gas, filmT returns also thetemperature at the solid wall Tw and the corresponding heat flux, HFwall, simply by

HFwall = klT (h2)− T (h1)

h2 − h1(3.17)

Setting the solution parameters The analytical solution as given in 2.5.3 needssome physical constants to be prescribed by the user in the in physicalConstant dictio-nary:

• L the latent heat of evaporation for the liquid

• kl the heat conductivity of the liquid, kl

• ka the heat conductivity of the gas, ka

• rhoa the reference density of the gas, ρa

• rhol the reference density of the liquid, ρl

• alpha the temperature diffusivity coefficient of the liquid, α

• molarRatio the molar ratio between gaseous mixture and the liquid species

• Patm the reference atmospheric pressure

along with two numerical solution parameters to be fixed in the HeatSolverParameters

dictionary:

• nx the number of discretization points for film thickness

• nf number of odd Fourier coefficients, prescribing up to which harmonic thenumerical solution will be computed

both to be chosen based on the evolution of the temperature in the film: if largevariations and rapid oscillations are expected to happen then nx and nf should beincreased to capture the smallest feature, otherwise they can be relaxed in order tospeed up the computation. The right choice of the parameter relay both on the physicalconsideration exploited in 2.5.3 and some experience with the solver.

3.4 LES Turbulence model implementation

The implementation of LES models in OpenFoam is based on a common interfaceprocedure that takes care of computing the anisotropic part of the total stress, theresolved one, that for incompressible fluids is simply 2νS, plus the SGS contribution2νsgsS. Setting νeff = ν + νsgs as the effective viscosity the two stress tensor can besummed up

2νeffS = νeff(∇u+∇uT

)(3.18)

The divergence of the total stress tensor is then split in two terms

∇ ·[νeff

(∇u+∇uT

)]= ∇ · (νeff∇u) +∇ ·

(νeff∇uT

)(3.19)

In laminar flows the second term is identically zero because of continuity. This totalstress tensor is discretized by

49


tmp<fvVectorMatrix> GenEddyVisc::divDevBeff(volVectorField& U) const

return

(

- fvm::laplacian(nuEff(), U) - fvc::div(nuEff()*dev(fvc::grad(U)().T()))

);

where the viscous term is treated implicitly, while the second term is fully explicit.It must be noted that only the anisotropic part of νeff∇uT is used, enforcing thecontinuity equation

dev(uT)= uT −

1

3uT I (3.20)

uT I = ∇ ·u = 0 (3.21)

Such an interface that discretizes directly the laminar stress and the SGS contributionrequires the knowledge of the latest available velocity field and of the νsgs. The latter isprovided by the specific SGS model implementation. In the following, the Smagorinskymodel available in OpenFoam is introduced and then the implementation of the planeaveraged dynamic model is discussed.

3.4.1 Smagorinsky model

OpenFoam implementation of the standard Smagorinsky model is unusual. Lack ofreferences makes hard to distinguish the classical formulation of νsgs as given in 2.19.Two constants are required, ck and ce, set by default to 0.094 and 1.048 2. Then twoquantities are computed

k =ckce∆2|S|2 (3.22)

νsgs = ck k∆ = ck

√ckce∆2|S| (3.23)

in which it is possible to recognize C2s = ck

√ckce. The default value of the constant is

Cs = 0.1677.

3.4.2 Plane averaged dynamic model

With reference to the mathematical description given in 2.2.2, the following parts areadded or modified in the code in order to obtain a dynamic model with plane averaging.The deviatoric part of the resolved stress tensor is computed as

void classicDynSmagorinsky::correct(const tmp<volTensorField>& gradU)

LESModel::correct(gradU);

volSymmTensorField D = dev(symm(gradU));

updateSubGridScaleFields(D);

2the values can be set in the proper input file

50

3.4. LES MODELS

and then passed to compute the constant Cl = C2s

volScalarField classicDynSmagorinsky::Clilly(const volSymmTensorField& D)

volSymmTensorField Lij = dev(filter_(sqr(U())) - (sqr(filter_(U()))));

volSymmTensorField Mij =

filter_( sqrt(scalar(2.0)) * mag(D)*(D) ) -

sqr(filter_to_mesh_ratio_) * sqrt(scalar(2.0)) * mag(filter_(D)) * filter_(D);

volScalarField LijMij = Lij && Mij;

volScalarField MijMij = Mij && Mij;

ieFLfieldAverage(LijMij); // averaging...

ieFLfieldAverage(MijMij);

volScalarField rtrn = LijMij;

forAll(rtrn,iT)

if (MijMij[iT] != 0)

rtrn[iT] /= MijMij[iT];

if(rtrn[iT] < 0)

rtrn[iT]=0;

rtrn.dimensions() /= MijMij.dimensions();

return rtrn;

where the for-cycle is used to clamp to zero the negative values at the numerator of 2.26.filtertomeshratio is the ratio between the two filter width, and it is assumed to be 2.The averaging is then performed by

void classicDynSmagorinsky::ieFLfieldAverage(volScalarField &toAverage)

const fvMesh& mesh= averagingIndex.mesh();

List<scalar> totalValues;

List<scalar> weightSizes;

totalValues.setSize(averagingIndex.dimensions()[dimensionSet::LENGTH]);

weightSizes.setSize(averagingIndex.dimensions()[dimensionSet::LENGTH]);

totalValues=scalar(0);

weightSizes=scalar(0);

// calculation

forAll(toAverage,nuI)

totalValues[averagingIndex[nuI]] += toAverage[nuI] * mesh.V()[nuI];

weightSizes[averagingIndex[nuI]] += mesh.V()[nuI];

reduce(totalValues,sumOp<List<scalar> >() ); // MPI part

51


reduce(weightSizes,sumOp<List<scalar> >() );

forAll(totalValues,sgsI)

totalValues[sgsI] = totalValues[sgsI] / weightSizes[sgsI];

if(totalValues[sgsI] < 0 )

totalValues[sgsI] = 0;

// setting new field

forAll(toAverage,nuI)

toAverage[nuI] = totalValues[averagingIndex[nuI]];

in which it has been necessary to identify each plane by an index over which the loopscan perform the plane averaging. Finally the νsgs is computed

nuSgs_= Cl * (sqrt(scalar(2.0)) * mag(D)) / scalar(2.0);

Throughout the implementation, where possible, ∆ and constant values have beencanceled out.

52

Chapter 4

Validation and Testing

4.1 PISO algorithm testing

Some tests have been performed in order to better understand the PISO algorithm be-havior and reliability in particular for flows with evaporation and condensation process.The main result is that the number of momentum corrector loops affects the stabilityof the solution but just slightly its accuracy. The loop number that assures stability isflow dependent. Experience about flows which are to be simulated is necessary. Thetime advancement scheme adopted in the present work is implicit in order to take ad-vantage of larger time stepping. A second test is reported in the following to properlyset an upper limit to the maximum Courant number which allows not compromisingthe accuracy of the solution.

4.1.1 Current model stability test

The PISO algorithm’s main assumption states that the nonlinear term of Navier-Stoke’sequation can be linearized. For a time step n the convection term is replaced byun−1 · ∇un, so that the matrix coefficients are based on the previous time step solution.Then the Poisson equation for pressure is iteratively solved, correcting and updatingthe velocity at each loop for a prescribed number of cycles, as explained in muchmore detail though out (3.2). OpenFoam developers suggest to never use less then 2correction steps, and claim that 2 correctors should be enough for most of the cases.Actually with just one loop the solution diverges in few time steps. Complex flows withevaporation and condensation have been proven in this study to require more than 2loops. The exact number depends on the problem under examination.

A test has been performed on a box of size comparable to a dishwasher tub with asimplified cup inside, as shown in Figure 4.1. The goal was to retain all the importantphysical features of interest but on a simpler configuration. The standard SmagorinskyLES model has been applied with the Reynolds analogy taking care of the eddy diffu-sivity of the two scalars as described in 2.2.2. Even if the results are to much diffused, itstabilized the solution and the test can be carried out enlighten just the characteristicof the PISO algorithm.

The cup is a semi-sphere shell with an external diameter D = 0.2m and a thicknessS = 0.005m. The bottom cuts the shell at Rc = 0.08944 from its center. The box edgemeasures L = 0.5m, while the outflow pipe placed at top face’s box has a diameter

53

CHAPTER 4. VALIDATION AND TESTING

Figure 4.1: PISO test case geometry.

Do = 0.1m and a length Lo = 0.2m. Just natural convection is considered. E/Cboundary condition are enforced on the cup and on the box: the former has prescribedvalues of temperature Tc = 333K and vapor concentration Cc = 0.132, while on thelatter Tb = 328K and vapor concentration Cb = 0.102. The initial velocity is zeroeverywhere, while the two scalars are prescribed as T0 = 333K and C0 = 0.105. Atthe outlet section a zero gradient condition is applied for all fields except pressure thatis fixed to be zero. Moreover the pipe is treated as sponge region. Evaporation takesplace from the wetted cup, condensation occurs on the box’s walls.

As the simulation evolves, the plumes of both scalars rise above the cup and aconvective motion starts, reaching eventually a steady state. The results of the PISOstability test are summarized in Table 4.1 for three different meshes, numbered 1, 2and 3 from the coarsest to the finest. All meshes are unstructured with tetrahedralelements. The columns provide the following data: number of the cells, maximum andaverage non-orthogonality of the mesh, maximum skewness of the cells, the number oforthogonality correction, and PISO loops, the physical time reached by the solution,and if the solution reached the prescribed time, i.e. proved to be stable.

As main achievement, the test has proved that the number of iteration impacts justthe stability of the solution but just slightly its accuracy, i.e. adding more iterations toan already stable setting doesn’t improve the overall accuracy1. As a side consideration,it is to be mentioned how the orthogonality correction impacts on the solution. It canbe seen from the Table 4.1 that increasing only the number of orthogonality loopsimproves the simulation but are not enough to guarantee its numerical stability.

1an analysis on a DNS turbulent channel at Reτ = 180 with different numbers of loops confirmedthis behavior

54

4.1. PISO ALGORITHM TESTING

test n of non-o. non-o. skewn. orth. PISO time stable

cells max av. corr. corr.

test 1 111709 68,3 23,4 0,865 0 2 12,98 no1 2 20 no2 2 39 no2 5 60 yes2 3 60 yes4 2 30 no

test 2 159575 69,5 23,5 1,005 1 2 6,9 no0 2 7 no4 2 22 no2 3 60 yes

test 3 1261362 70,5 19 1,1 1 2 9,5 no4 2 18,5 no2 3 30 yes

Table 4.1: Stability test parameters and results.

4.1.2 Accuracy-Co number test

Among the time discretization scheme available in OpenFoam, mainly two have beenadopted in the cases run so far: the fully implicit Crank-Nicholson, C-N., and thebackward scheme. Both are second order accurate and implicit. The time steppingvaries during simulations. It is selected in order to match the maximum Courantnumber, Co, in the volume. These schemes are applied for the momentum equationsand the both scalar transport equation. Their clear advantage is that Co limit can berelaxed to values exceeding 1 allowing longer time steps. In order to check the accuracyof the solution with increasing maximum Co some test are performed. The unsteadyflow around a cylinder between parallel walls has been considered. A schematic pictureof the case is given in Figure 4.2. The inlet velocity profile is parabolic with a meanvalue Um, and the Reynolds number can be defined as Re = UmH/ν based on thechannel width H. In particular we consider the case of Re = 500 that yields to avortex shedding regime. The body is placed in the centerline of the channel and theblockage ratio is D/H = 0.2, where D is the cylinder diameter.

The results have been checked with respect to the mean drag and lift coefficients,Cd, Cl respectively

Cd =2F x

ρU2mHD

Cl =2F y

ρU2mHD

where F x and F y are the longitudinal and transversal time-averaged force componentacting on the body respectively. They have been checked also with respect to the nondimensional period of the main oscillations, TCd, TCl, i.e. the inverse of the Strouhalnumber St−1

St−1Cd =

UmtCd

HSt−1

Cl =UmtCl

H

where tCd and tCl are the periods of the main harmonic component.The reference line in 4.2 refers to the data provided by [43], in which only Cd and

TCd were provided. The first set of tests were performed adopting Crank-Nicholson

55


Figure 4.2: Accuracy-Co test case scheme.

Time Scheme Co P O Cd Cl TCd TCl

Reference 0.29 0.73

Crank-Nic. 4 2 0 0.31225 −1.6495e−3 0.3593 0.73143Crank-Nic. 0.8 2 0 0.30663 −1.5810e−4 0.3531 0.70621Crank-Nic. 0.4 2 0 0.30605 −1.1272e−4 0.3531 0.70621

Backward 4 2 0 0.31208 −1.62030e−3 0.3593 0.73143Backward 0.8 2 0 0.30623 −1.98155e−5 0.3531 0.70621Backward 0.8 4 0 0.30623 −1.97060e−5 0.3531 0.70621Backward 0.8 2 2 0.30623 −1.98160e−5 0.3531 0.70621Backward 0.8 4 2 0.30623 −1.98150e−5 0.3531 0.70621Backward 0.4 2 0 0.30562 −1.80360e−5 0.3531 0.70621

Table 4.2: Results of the accuracy-Co test

time scheme, for different Co numbers with 2 pressure correction loops, P, and 0 non-orthogonality corrector loops, O. The second set used the backward scheme. It was alsochecked the influence of higher P or O on the solution. Table 4.2 summarizes the testfindings: there are no effects of P and O on the results; at the same Co the C-N. givesin general lower values of Cd with respect to the backward, while the other quantitiesare identical; the main differences are due to the different Co choice: the lower theCo the lower the Cd while the other quantities below Co = 0.8 remains unchanged.The difference in Cd between the solution with backward scheme at the extrema valuestested , i.e. Co = 4 and Co = 0.4, is of the 2%. This result justify the use of the implicitscheme allowing larger time steps without loosing accuracy. This can be important insimulating long processes respect to the time scale of the flow.

4.2 Validation test case

A steady, developing counter gravity flow of a vapor-air mixture in a vertical channelwith evaporation from wetted walls is analyzed. The geometry and mesh used by [19]

56

4.2. VALIDATION TEST CASE

0

0.05

0.1

0.15

0.2

0.25

0 0.005 0.01 0.015 0.02

u (

m/s

)

cross-wise section (m)

Velocity profile at for cross section at x=1, 0.2, 0.01m

OF Laaroussi 2009

Figure 4.3: Comparison of velocity profile in three section along the channel betweenthe present model and the solution proposed in [19]

are reproduced along with the boundary conditions. The channel length (L = 2 m) andwidth (H = 0.02 m), giving an aspect ratio γ = L/H = 100 and hydraulic diameterDh = 2H = 0.04 is discretized with a (600 × 70) cells grid, with stretching parametersξx = 1.007 and ξy = 1.02. Among the various cases analyzed in the reference, wefocused on the case of air-water vapor mixture with the following boundary conditions:at inlet the velocity is U0 = 0.13 m/s (Re = 300), vapor mass fraction ω0 = 0 (dry air)and temperature, T0 = 327.50 K; at interface the vapor concentration is ωi = 0.1 andtemperature Ti = 327.50 K. The constant physical properties, evaluated with the 1/3-rule, are the kinematic viscosity ν = 1.74 · 10−5 m2/s for dry air, thermal expansioncoefficient βT = 0.00305 K−1, solutal expansion coefficient βω = 0.584, temperaturediffusion coefficient DT = 2.580 · 10−5 m2/s for dry air and mass diffusion coefficientDω = 3.290 · 10−5 m2/s. For this configuration the buoyancy force is only due to thesolutal contribution since ∆T = Ti−T0 = 0. Hence GrT = 0, while the solutal Grashofnumber is Grω = 1.212 · 105, given ∆ω = ωi−ω0. The corresponding solutal Richardsonnumber is also evaluated Riω = 1.346.

The results have been checked as follow. Firstly we compare the velocity profiles inthree different sections. A perfect matching with the reference’s results is found as canbe seen in figure 4.3. The bulk density and the accumulated vaporized mass along thechannel length, defined respectively as

ρm (x) =

∫H0 ρ (y) u (y) dy∫H0 u (y) dy

Φm,v (x) = 2

∫ L

0ρi (x) ui (x) dx

are shown in figure 4.4a and 4.4b. The agreement is satisfactory despite the off-sets

57


at the outlet that are of 0.2% and 3% for ρm and Φm,v respectively. This can beattributed to the different interpretation of the evaporation effect: the Stefan flowis added as source term placed in the cells next to the wetted wall instead of beingincluded in the boundary condition.

1.02

1.03

1.04

1.05

1.06

1.07

0 0.5 1 1.5 2

ρm

(kg

⋅ m

-3)

stream-wise direction (m)

Bulk mixture density

Laaroussi 2009 OF

(a) Bulk mixture density

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.5 1 1.5 2

Φm

,v (

g ⋅ s

-1)

stream-wise direction (m)

Vaporized mass flux rate

Laaroussi 2009 OF

(b) Vaporized mass flux

Figure 4.4: Comparison between results of the present simulation and reference one:a) Bulk mixture density b) Vaporized mass flux

58

4.3. LES TURBULENCE MODELS TEST

0

5

10

15

20

0.01 0.1 1 10 100 1000

U/u

*

y+

KMM Dyn OF

Smag OF

Figure 4.5: Comparisons of the mean velocities.

4.3 LES turbulence models test

The turbulent plane channel flow has been selected as a test case for the turbulenceLES models. The quantity are expressed in non-dimensional form. The dimensions inthe stream-wise, vertical and span-wise directions are: 2πδ, 2δ and πδ, where δ = 1 isthe half-channel width. The flow is driven by constant pressure gradient ∂p

∂x = 1 TheReynolds number based on wall shear velocity is set to be Reτ = 395, which determinesthe viscosity to be ν = 1/395 = 2.5316 10−3 . The mesh spacing in stream-wise directionis set to be ∆x+ = 50, in the span-wise direction ∆z+ = 15. As usual the wall unitsare normalized by ν

u∗

. The grid in the vertical direction is symmetric and stretchedin a way, such that the first cell is within y+ = 1, 8 cells below y+ = 11, with total80 cells between the walls. The case has been solved applying first the OpenFoamimplemented Smagorinsky model with Cs = 0.065, and with the Van Driest correction;then by applying the plane averaged dynamic model implemented as described in 3.4Both simulations started from an instantaneous field provided by a previous DNS run.

The results are compared with the data of Kim, Moin and Moser [24], labeled inthe figures legends as KMM. In blue solid line are shown the reference data, the valuesobtained with the Smagorinsky model are plotted in red, while the plane averegedmodel results are coloured in green. In Figure 4.5 the profiles of the mean velocity,U/u∗, are plotted. The y+ axes is given in logarithmic scale. Respect to the referencecase the Smagorinsky model under-predicts the mean velocity throughout the half-channel width. The mean velocity obtained by the dynamic approach is closer to thecorrect one, although it over-predicts velocity in the logarithmic layer. In the Figure 4.6rms of the three velocity components are plotted. Both models results fit quite wellwith the DNS case data, and their quality is in agreement with the dynamic modelsresults available in literature.

59


0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

rm

s(u)

y+

KMM Dyn OF

Smag OF

(a) rms of the stream-wise velocity

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

rm

s(v)

y+

KMM Dyn OF

Smag OF

(b) rms of the wall-normal velocity

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

rm

s(w

)

y+

KMM Dyn OF

Smag OF

(c) rms of the span-wise velocity

Figure 4.6: Rms data comparisons of the three velocity components plotted along y+.

60

4.4. DROP TEST

Figure 4.7: Mesh grid of the boundaries for the drop test domain.

4.4 Evaporating drop test

A further test has been performed considering the evaporation process of a single sessiledrop to check the performance of the current solver on fundamental problems. Thesessile drop geometry is approximated as a spherical cap, and is assumed to be pinnedfor all its life-time. Its contact radius and its surface temperature are fixed to R =0.00135m and TD = 295K respectively. Following the simulation procedure suggestedin [9], a far-field condition is supplied for the vapor concentration imposing a relativehumidity of 40% at gaseous domain boundaries, along with a zero gradient conditionfor temperature. To properly compare the numerical outputs to the analytical onesprovides by 2.35, the buoyancy force is switched off and the problem becomes purelydiffusive. The steady state solution is considered for three different contact angles,namely θ = π/2, θ = π/12 and θ → 0.

Despite the radial symmetry of the configuration the solver is based on Cartesiancoordinates hence it is possible to simplify the problem only taking into considerationjust a quarter of the drop volume applying symmetry boundaries but retaining a 3Dformulation. The computational domain around the drop is a quarter of a sphere, asdepicted in Figure 4.7 for θ = π/2, with a radius RD equal to 10R for θ = π/2, θ = π/12and equal to 20R for θ → 0. The resolution of the drop surface and near the contactline is prescribed to be around R/100 matching the requirements for the evaluation ofthe evaporation velocity.

The comparison result, summarized in Table 4.3, shows an overall good agreement.The evaporating rate is expressed as nL/s computed taking the drop surface integralof the mass flux. In [9] it is suggested to use RD as large as 320R, and that a domainof only 20R lead to an error of around 5%. Hence the error in the present comparisonis addressed mainly to the small computational volume around the drop.

61


θ RD OF [nl/s] theory [nl/s] ǫdVdt

dVdt

π/2 10R −2.61 −2.41 −8%π/12 10R −1.65 −1.59 −4%→ 0 20R −1.57 −1.53 −2%

Table 4.3: Drop evaporation rate comparison.

(a) θ = pi/2 (b) θ = pi/12 (c) θ → 0

Figure 4.8: Evaporation velocity vectors at the drop surfaces for three contact angles.

In the Figure 4.8 the evaporation velocity vectors are plotted over the drop surfaceto enlighten the different distribution depending on the contact angle. The smallerthe θ, the higher the value near the contact line. Moreover the distribution of the thevelocity along the radius is completely different, from a homogeneous distribution forθ = π/2, as predicted by the theory, to a curve that peak at the contact angle forsmaller angles. This typical evaporation velocity profile around drop surface is shownin Figure 4.9 for θ → 0.

4.4.1 Drop as a thin film

A further test has been performed to understand if the thin-film approximation can beapplied to directly model spherical cap drops during the evaporation process. Of coursethe approximation cannot provide the correct concentration diffusion in the volumesurrounding the drop, moreover, the vapor distribution cannot be modified by the dropchange of shape as evaporation takes place. In order to reduce the approximation errorthe flat surface is corrected for each θ to match the corresponding curved surface. Thephysical configuration is the same of the test discussed at the beginning of this section,the pinned drop has an initial contact angle θ = π/2. With the thin film approximationthe steady process gives an evaporation rate of 2.21 nL/s. The initial error is +8%. Asevaporation goes on, the thin film model cannot reproduce the theoretical predictionsunderestimating the depletion of liquid water, as showed by Figure 4.10. The red lineis the analytical solution and the blue one is the result of the simulation.

It clearly appears that such an approach is not effective in modeling a single evap-orative drop.

62

4.4. DROP TEST

Figure 4.9: Evaporation velocity along the drop radius for θ → 0.

Figure 4.10: Thin-film approximation of a single drop during evaporation.

63


4.5 Domain Decomposition validation

In the following the implementation of the Dirichlet-Neumann decomposition methodis validated in the frame of the conjugate heat transfer, and the comparison with themethod included in the OpenFoam releases is discussed. In particular the methods aretested for CHT problems with heat sources. The latter is interesting from an engineer-ing perspective and is found to be the most challenging for the coupling methods.

In general, CHT methods should solve the heat transport equation over more com-putational domain interconnected by common interfaces. The main role is to providethe correct coupling conditions at those boundaries in order to guarantee the continuityof the solution, namely

T1 = T2

k1∂T1∂n1

= k2∂T2∂n2

(4.1)

The purpose of CHT is to solve regions with different physical parameters or modeledby different equations, as for instance a fluid region bounded by a solid one. Is it foundthat the OpenFoam coupling condition,

solidWallMixedTemperatureCoupled

is not trustworthy being either not accurate nor robust. Although in many technical ap-plications the approximation requirements can be relaxed and the solidWallMixedTem-

peratureCoupled can be effective whenever the requirements are more stringent or theproblem has fast transients or has heat source the solidWallMixedTemperatureCoupled

fails. The implementation of the DNM overcomes all these drawbacks.

4.5.1 Methodology

In brief the methodology of the validation procedure is given below. A test series hasbeen done on mono-dimensional CHT problems with heat sources in one or more solidregions. Other test on two-dimensional case are further computed without heat source.The coupling problem between fluid and solid can be solved analogously because, de-spite the different transport mechanism, the interface conditions remains the same ifthe usual no-slip and no-penetration conditions hold.

For all the test the results have been compared to analytical solutions, if available,or to a numerical one computed over one domain. For sake of simplicity both referencesare called analytical solutions.

The performances of the transmission conditions are evaluated with respect to thesteady state solutions along the whole domain and to the time evolution of the tem-perature along the interface.

The reference geometry is two joint rectangles of 0.5× 0.5m each. Along the x-axiseach domain is discretized by 80 cells, and by 40 cells along y-axis. The left regions isnamed Left and the right one Right. The edges are identified as reported in Figure 4.11.The boundary conditions for temperature over 1Left and 1Right depend on the specifictest; over topLeft and bottomLeft and over topRight and bottomRight respectively havebeen applied cyclic, zero gradient boundary conditions. The interfaces are named 2Left

and 2Right, over which have been applied the solidWallMixedTemperatureCoupled andthe DN coupling conditions.

64

4.5. DDM VALIDATION

Figure 4.11: Reference geometry.

For sake of simplicity and without loss of generalization the physical parametershave been fixed to: specific heat cp = 1 J/kg m, thermal conductivity k = 1W/K mand density ρ = 1 kg/m3. The evolution time scale of the solution is then evaluated by

ts =l2

α= 1

where α is thermal diffusivity α = k/ρcpm2/s.

Analytic solution

The mono-dimensional heat equation for a solid region reads

∂T

∂t= α

∂2T

∂x2+ F (x)

where F (X) is the heat source, expressed in K/s, arbitrary distributed. Applied Dirich-let boundary conditions

T (0) = T0 T (l) = Tl

where l is the domain length and f(x) is the initial condition, the analytical solution is

T (x, t) = Tp(x) + Tic(x, t) + Ψ(x)

given

ψ(x) = −1

α

∫ ∫F (x)dx2

Ψ(x) = ψ(x) + c1x+ c2

65


-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

T (

C)

x (m)

Evoluzione temperatura

Figure 4.12: Analytical solution with sinusoidal source. The distribution of temperaturegiven by the blue lines for different time steps; the initial condition in green; the steady-state solution in red.

c2 = T0 − ψ(0)

c1 =Tl − T0 + ψ(0) − ψ(l)

lTp(x) = T0 − (Tl − T0)x

g(x) = f(x)− Tp(x)−Ψ(x)

Bn =2

l

∫ l

0g(x) sin

(nπlx)dx

Tic(x, t) =∞∑

0

Bn sin(nπlx)e−nπα

l

2

t

For the particular case in which T0 = 0, Tl = 0, and l = 1, given F (x) = sin(2πx),one sinusoidal period, the temperature evolves as shown in Figure 4.12. This solutionis compared with the numerical solution over one domain. The perfect matching forevery time step validates the numerical method with the source term. In this way thenumerical solution over one domain can be used as reference solution.

66

4.5. DDM VALIDATION

4.5.2 Test cases

Sinusoidal heat source

The above mentioned problem is here solved over the two regions by means of thecoupling condition under examination. The time step is constant and equal to 0.01ts,the steady-state temperature is reached after 0.2 ts. Both steady-state solutions con-verge to the analytical one, but a huge difference can be seen during the transitory,as showed by the plot in Figure 4.13, where the temperature evolution is remarked in(0.25m, 0.25m) correspondingly to the peak value of the source. The analytical solu-tion, represented by the red line, match the solution provided by the DN method, inblue. The solidWallMixedTemperatureCoupled cannot reproduce the correct behavior,in green.

0

0.005

0.01

0.015

0.02

0.025

0 0.1 0.2 0.3 0.4 0.5

T (

C)

t (s)

Transitorio interfaccia

IE-FluidsOF-1.6

Analytic

Figure 4.13: Temperature evolution in (0.25, 0.25) for the sinusoidal source: in red theanalytical solution; in blue the DN solution; in green the solution with the standardOpenFoam coupling.

Cosinusoidal heat source

The symmetry of the source distribution eases the coupling. To further test the per-formance of the methods a non symmetric source is considered

F (x) = cos(π

2x)

along with boundary conditions on 1Left fixed at T0 = 1 and on 2Left equal to Tl =0. The temperature evolution at interface, as can be seen in Figure 4.14, remarks

67


how the DN method matches the analytical profile while the OpenFoam conditionunderestimates the temperature for all the transitory period.

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

T (

C)

t (s)


IE-FluidsOF-1.6

Analytic

Figure 4.14: Time evolution at interface with a cosinusoidal heat source.

Sinusoidal heat source in Left

If the sinusoidal heat source is considered again but only in the Left region

F (x) = sin(2πx) 0 ≤ x ≤ 0.5

the solution provided by the implemented method match again the analytical profileand the solidWallMixedTemperatureCoupled behaves poorly as depicted in Figure 4.15for the temperature at the interface. To better notice the differences between the twosolutions in Figure 4.16 the temperature distribution along both regions is plotted att = ts; a discontinuity at x = 0.5m shows up in the curve obtained by the solid-

WallMixedTemperatureCoupled technique.

4.5.3 Constant heat source in Left

So far only Dirichlet boundary conditions have been applied. In the following caseswith zero gradient or cyclic boundary conditions on top and bottom boundaries arediscussed. On 1Left are applied either zero gradient or symmetric B.C.. Due to thelatter conditions the problem has a time scale three time larger than the previouscases, underlining the crucial role of a correct solution during the transitory period asthe Figure 4.17 shows. The results are qualitatively the same of the already discussed

68

4.5. DDM VALIDATION

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

T (

C)

t (s)


IE-FluidsOF-1.6

Analytic

Figure 4.15: Time evolution at interface with a sinusoidal heat source only in Left

region.

cases. The DN technique performs well and solves accurately the CHT problem withany tested boundary condition.

Influence of the time step

Further test show the influence of different time steps. In particular a time step of 0.1 tsis chosen and again a DN approach provides the correct solution while the OpenFoamone degrades further as described by Figure 4.18.

Influence of the physical parameters

In engineering applications different materials are modeled by different physical pa-rameters. The numerical methods under consideration are tested over the two regionswhich have a thermal conductivity ratio of r = kl/kr 6= 1. For instance a good conduc-tor, the Right region, is in contact with an insulator, the Left region and the ratio isfixed to r = 100/1 = 100, with a time step of 0.01 ts.

The temperature evolution in Figure 4.19 shows that solidWallMixedTemperature-

Coupled ’s solution are even worse when physical parameters differ between the tworegions. On the contrary DNM gives accurate result.

69


Figure 4.16: Distribution of temperature along both domains, expressed in C. x-axisx in m at t = ts.

4.5.4 Solid-fluid case

To include a test in which a fluid flow is bounded by solid walls the following case hasbeen tested and compared with the literature results provided in [33]. In particularthe CHT transitory period is considered for a laminar flow in a pipe. The flow is fullydeveloped and in thermal equilibrium with the solid: at time t = 0 holds Ts = Tl = Ti =0, where the subscripts refer to solid, liquid regions and initial condition respectively. .The external surface temperature of the pipe is increased to Tw = 100C for t > 0. Thetemperature evolution is monitored at the interface between solid an liquid. The radialsymmetry makes the problem mono-dimensional, nevertheless to take into account thecurvature effects in OpenFoam a complete section has to be modeled. In Figure 4.20the section is depicted; the structured mesh in the solid region is grey colored. It has 80cells along the circumference for 9 cell layers. Over the internal region the steady-statetemperature contour is plotted along with the mesh structure: near the walls the meshis structured analogously to the solid one in 80 × 9 cells, and the non-structured partthe mesh has 1382 triangular cells.

The velocity is imposed as a parabolic profile with a mean velocity of u = 25m/s.The radius of the fluid section is 0.5m while the pipe thickness is 0.1m. The physicalproperties are for the solid region αs = 10−4 m2/s; for the fluid region apply cp =1000 J/kgK, ν = 0.005m2/s, αl = 2.5 · 10−5 m2/s. The time step is fixed to 0.01 s. The

70

4.5. DDM VALIDATION

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

T (

C)

t (s)


IE-FluidsOF-1.6

Analytic

Figure 4.17: Time evolution of temperature at the interface with a constant heat sourcein region Left : time step of 0.01ts, symmetry condition on 1Left and cyclic on top andbottom boundaries.

data have been chosen to match the ones provided by [33].The equations that describe the heat transfer process are:

∂Ts∂t

= αs∇2Ts (4.2)

∂Tl∂t

= αl∇2Tl +

ν

cp(∇ · u)2 (4.3)

where the third term on the second equation is the viscous dissipation, that is treatedas a source term into the fluid. The simulation’s result with the DD method matchesthe reference’s data as the plot shows in Figure 4.21. Also OpenFoam’s solution fitsquite well although an error occurs at the steady-state value of about 0.12%, largerthan the DD error, 10−4%. Such behavior is not surprising since the source term isquite small and the transitory period quite long.

4.5.5 Multi-dimensional cases

For multi-dimensional cases it can also be stated that the DD technique is more accuraterespect to the standard OpenFoam one. The results of the case described by theFigure 4.22: a square of dimensions 0.04m × 0.04m decomposed in three domains: onthe left two regions of one quarter of the square each and on the right one domain of half

71


0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

T (

C)

t (s)


IE-FluidsOF-1.6

Analytic

Figure 4.18: Time evolution at interface with a constant heat source only in Left region:time step 0.1ts

72

4.5. DDM VALIDATION

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20

T (

C)

t (s)


IE-FluidsOF-1.6

Analytic

Figure 4.19: Time evolution at interface with a constant heat source only in Left regionwith r = 100 and time step 0.01 ts

73


Figure 4.20: Cross section of the solid-liquid regions. In grey the solid part, the colorcontour shows the temperature inside the liquid and the geometry discretization.

of the square. Each domain has the same physical parameters, namely cp = 434 J/kgK,k = 100W/Km, ρ = 7854 kg/m3, and the same initial temperature T0 = 300K. Thetop border of the top-left region has a fixed temperature of Tin = 350K; the bottomborder of the right region has Tout = 250K Dirichlet boundary condition. All the otherborders are adiabatic. The temporal step is given, 0.01 s, the geometry is discretizedby an uniform grid of 40 × 40 cells. To test the performance of the methods errors,expressed in K, have been checked respect to the analytical solution. The Figure 4.23shows the error distribution all over the squared domain.

Dynamic test case

The last analysis considers a dynamic case over the same geometry in Figure 4.11.Oscillating temperature values have been applied at the boundaries 1Left and 1Right,a sinusoidal and cosinusoidal laws respectively

Tl = 325 + 25 cos(t2π/681) (4.4)

Tr = 325 + 25 sin(t2π/681) (4.5)

The physical parameter are for each domain cp = 434 J/kgK, ρ = 7854 kg/m3, k =100W/Km, the initial temperature is Ti = 325K. The steady-state error is evaluated

74

4.5. DDM VALIDATION

0

20

40

60

80

100

0 50 100 150 200 250 300 350 400

T (

C)

t (s)


OF-1.6IE-FluidsAnalytic

Figure 4.21: Evolution of temperature. Solid-fluid case with heat source in the fluidregion.

respect to the analytical solution and it is shown in Figure 4.24: both methods errorspeak at the interface although the DN coupling technique performs better.

4.5.6 Conclusions

In all the cases the error of the DN method has been checked, and at the interfaceis always found to be less than 10−5, the tolerance set for the simulations. On thecontrary the solidWallMixedTemperatureCoupled method is found to be quite ineffec-tive to transmit the heat across the interface, slowing the heat transfer process. Thisstandard OpenFoam coupling condition applies a non physical buffer that inhibits theheat conduction at the interface. This trend is worsened by larger time step and byhigh conductivity ratio between the domains.

75


Figure 4.22: Multi-dimensional case scheme.

76

4.5. DDM VALIDATION

Figure 4.23: Errors, expressed in K, of the temperature distribution over the threedomains. The DD method error is plotted in red crosses, in green crosses the solid-

WallMixedTemperatureCoupled.

77


Figure 4.24: Temperature error along the whole domain. In blue the DN method, thesolidWallMixedTemperatureCoupled in green.

78

Chapter 5

Cases studied

In the following chapter the cases studied by means of the evaporation/condensationmodel are presented and discussed in details. The boundary conditions adopted for thesolid wall in all the following cases are here restated for sake of clarity.

The adiabatic and impermeable condition prescribes the following boundary con-dition for temperature and vapor concentration respectively

∂ω

∂n= 0 (5.1)

∂T

∂n= 0 (5.2)

(5.3)

being n the normal to the solid walls. For velocity the non-slip condition holds.

Evaporation condensation condition or wetted wall condition, assigned an inter-face temperature Ti, prescribes the condition on vapor concentration and velocityas

ωi =Mv

Ma

ps(Ti)

p− (1− Mv

Ma) ps(Ti)

) (5.4)

Ve = −Dω

1− ωi

(dω

dn

)

i

(5.5)

see 2.4 for more details. If the drying process is activated the evolution of thefilm thickness is evaluated by

dH

dt=ρaρwVe

5.1 Evaporation/condensation around a wetted cylinder

confined between two parallel walls

5.1.1 Introduction

The archetypal problem of the flow around a wetted cylinder confined between twoparallel walls is here presented. The liquid phase is considered in the limit of the thin

79

CHAPTER 5. CASES STUDIED

film approximation and its effects are treated as boundary conditions. The walls canbe either impermeable and adiabatic or wetted. The effects of two different positions ofthe cylinder are investigated: the cylinder is placed at the centerline or at one quarterof the channel width. The main characteristics of each case are then discussed and thedifferences among them are enlightened.

Figure 5.1: Flow around a cylinder between two parallel walls. The study considers twocases in which evaporation takes place from the cylinder surface in two configuration:a) the channel walls are impermeable and adiabatic, b) in presence of condensation onthe liquid film wetting the walls

In the case of impermeable and adiabatic walls the evaporation process appearssubstantially steady and the fluxes around the surface are quite homogeneous with adefect in the rear of the body. The vapor flows in the upper part of the domain and tworegions of different concentration level are clearly visible. In the case of wetted wallscondition the buoyancy effect is increased together with the velocity inside the channel,the heat and mass transfer process from the solid body is enhanced. The evaporation isnon-homogeneous and unsteady. On average the whole surface of the cylinder permitshigher Stefan flow rates. The effects of distance from the wall is also investigated. Twodifferent position of the cylinder are considered, for the body placed at half or at onequarter of the channel width. The distance from the wall influences the Stefan flow, inparticular the case with wetted walls, determining preferential directions of the vaporflow around the cylinder.

As depicted in Figure 5.1 the flow develops in a straight channel of length L andwidth H, with aspect ratio λ = L/H = 5. A cylinder of diameter D = 0.2H is placedinside the channel at a distance equal to L/10 from the inflow section. In order toenlighten the different interaction mechanism between the walls and the bluff body twodifferent distances of the cylinder from the bottom wall are investigated, respectivelyd = H/2 and d = H/4. At the inlet uniform temperature T0 = 327.5 K and vaporconcentration ω0 = 0.05 are prescribed along with a parabolic velocity profile with amean velocity Um such that the Reynolds number is Re = UmH/ν = 500. The liquid

80

5.1. WETTED CYLINDER

film interface on the wetted cylinder has temperature Tc = 327.5 K and concentrationωc = 0.1 such to allow liquid evaporation. The channel walls can be either adiabaticand impermeable or wetted with fixed temperature Tw = 307.5K and concentrationωw = 0.034 allowing vapor condensation.

In some complex flows, as for example stratified ones, the outflow condition must benon-reflective in order preserve the accuracy of the solution preventing the disturbancesin form of internal waves to propagate upward. We use a sponge region localized justbefore the outlet section. This is found to be effective in this task as explained in [1] andmoreover has a stabilizing effect on the numerical simulation damping the recirculationacross the outlet section.

In the investigated case a zero gradient condition for velocity and the scalars is im-posed at the outlet along with the sponge region located as depicted in Figure 5.1 actingalong the 10% of the channel length, in which fluid viscosity, thermal and concentrationdiffusivity are artificially increased according to an exponential law like

A = A · e12(x−4.5)

with A being any of the parameters. Moreover within the sponge the buoyancy forceis neglected, i.e. the concentration and temperature are treated as passive scalars.A zero velocity field is imposed as initial condition along with homogeneous field formass fraction ωinit = 0.05 and temperature Tinit. We remark that T0 = Tc = Tinit,so that with non-wetted walls the temperature field will not affect the flow. Thephysical parameter are set to ν = 1.74 · 10−5, DT = 2.58 · 10−5, Dω = 3.29 · 10−5,βT = 3.05 · 10−3 and βω = 6.08 · 10−1.

For all the cases we compute drag and lift coefficients as

Cd =2F x

ρU2mHD

Cl =2F y

ρU2mHD

where F x and F y are the mean forces acting on the cylinder in the stream-wise andcrosswise directions respectively. Also their spectra are computed to evaluate the peri-odic feature of the flow.

5.1.2 Non-wetted walls

First we consider the case with adiabatic and impermeable walls. The Grashof andRichardson numbers are computed based on the difference between the values of Tand ω at the cylinder interface and at the inlet: Grω = 1.714 · 104, GrT = 0, Riω =6.856 · 10−2 and RiT = 0. The different cylinder positions yield to small differences inthe Stefan flow in the overall evaporation process. In particular the total amount of thevapor mass evaporated from the cylinder flowing in the channel is almost indistinguish-able from case to case as showed in Figure 5.2. The mass flow is quite homogeneousaround the surface with a defect in the rear of the body as it is showed in Figures 5.5aand 5.5b, the position of the defect is oscillating with a characteristic period. Whenthe accumulated vapor mass fraction ω reaches equilibrium, a frequency analysis is per-formed on Cd and Cl. From the spectra shown in Figures 5.7a and 5.7b we find a mainfrequency for the lift coefficient fl = 0.03027 Hz for both position of the cylinder. Thebuoyancy effects increases the frequency of the oscillations compared to the analogous

81


0.025

0.026

0.027

0.028

0.029

0.03

0 500 1000 1500 2000

ω

Time (s)

Total vapor mass fraction, non wetted walls

d = H/2d = H/4

Figure 5.2: Accumulation of vapor concentration within the channel

case without stratification. The drag coefficient presents spectral components at higherfrequencies, roughly twice fl. They are fd1 = 0.07129 Hz and fd2 = 0.05273 Hz forthe case with d = H/2 and d = H/4 respectively. For the case d = H/4 an additionalmain frequency of oscillation appears at f ′d2 = 0.0332. Note that this value is veryclose to fl. The vapor concentration accumulates in the upper part due the favorablebuoyancy plume induced by the solutal gradients of the domain and two regions ofdifferent concentration level are clearly visible in 5.4a. The velocity field develops inunsteady vortices without a clear dominant size in the downstream region.

0.021

0.022

0.023

0.024

0 200 400 600 800 1000 1200

ω

Time (s)

Total vapor mass fraction, wetted walls

d = H/2d = H/4

Figure 5.3: Accumulation of vapor concentration within the channel

5.1.3 Wetted walls

The system with wetted and colder walls presents remarkably differences from the pre-vious situation. Condensation, that occurs at the interface, greatly impacts the flowand the heat and mass transfer process. The buoyancy force and hence the charac-teristic velocity is increased. Unlike the adiabatic wall case, in the wetted wall casecondensation extracts vapor from the gaseous phase. Consequently the amount of va-por mass present initially in the channel decreases in time as illustrated in figure 5.3.An equilibrium state is reached roughly after 400s when the vapor that flows in thechannel from the inlet and evaporates from the cylinder is balanced by the vapor that

82

5.1. WETTED CYLINDER

(a) Non wetted walls, ω ranges from 0.05 to 0.01. The two different concen-tration regions are clearly visible.

(b) Wetted walls, ω ranges from 0.033 to 0.01.

Figure 5.4: Vapor mass fraction distribution in the channel, an instantaneous picture.The value of ω is given by the contour plot.

flows out at the outlet and condensates on the walls. As in the previous case the dif-ferent positions of the cylinder slightly affects this dynamics. Downstream the bodythe velocity field develops in well defined vortical structures of the size of the channelwidth. Near the cylinder the flow is greatly unsteady and poorly organized making thevapor well mixed, as can be seen in figure 5.4b. The analysis of the Cd and Cl showsscattered spectra as can be seen in Figures 5.7c and 5.7d reflecting the complexity ofthe flow. Nevertheless the main frequencies of the lift coefficients are still close to thenon wetted case ones: fl1 = 0.03085 Hz and fl2 = 0.03043 Hz for the case with d = H/2and d = H/4 respectively. The drag coefficient exhibits one clear peak for d = H/2at roughly twice the fl1 , fd = 0.06538 Hz. For d = H/4 we recognize two peaks, oneat higher frequency at fd = 0.08483 Hz and the second that is aligned to fl2 , similarlyto the adiabatic walls case. On average the whole surface of the cylinder permits Ste-fan flow rates higher than in the previous case. In this condition evaporation processis non homogeneous and unsteady around the body. It shows preferential directionsfor the mass flow depending on the cylinder’s position in the channel as depicted inFigure 5.5d for two plots of the instantaneous velocity vector pointing out the cylindersurface. This behavior is confirmed by the time averaged evaporation velocity scatterplot shown in Figure 5.6. Figure on top shows the cases with adiabatic-impermeablewall, on the bottom the wetted wall cases. In particular at d = H/2 the vapor flowsmostly from the front surface, but when the body is displaced at d = H/4 the vaporflows in the upper and from the bottom of the body. This preferential direction rotatesin time with some periodicity but in order to quantify this effect an average in time

83


(a) Non-wetted walls, cylinderat d = H/2

(b) Non-wetted walls, cylinderat d = H/4

(c) Wetted walls, cylinder at d = H/2(d) Wetted walls, cylinder at d = H/4

Figure 5.5: Instantaneous evaporation velocity vector plots for the different cases.

will be provide in a following work.

5.1.4 Conclusions

The archetypal case of the evaporation from a wetted cylinder confined between wettedor impermeable and adiabatic walls has been faced. Two body positions are considered:the cylinder placed at 1/2 or at 1/4 of the channel width. The main features of each floware discussed. The wetted walls greatly impact the heat and mass transfer by increasingthe concentration gradient and the buoyancy force. Moreover the body location affectsthe distribution of the Stefan flow around the surface. This preliminary study showsthe feasibility of the proposed method to deal with complex geometries in unsteadycases and encourages further developments in order to include into the model morephysical effects important in drying process, in particular the coupling of the energyequation with the solid substrate and the liquid thin film dynamics.

5.2 Two-dimensional dishwasher

5.2.1 Introduction

A first application of this model on a real configuration has been performed on a 2Ddishwasher model, obtained as a cross section os the actual machine. Figure 5.9b showsan instant contour plot of the vapor concentration, that also reveals the complex flow

84

5.2. 2D DW

(a) non-wetted wall, d = H/2 and d = H/4 from left to right

(b) wetted wall, d = H/2 and d = H/4 from left to right

Figure 5.6: Time averaged evaporation velocity vector scatter plots for the differentcases.

85


00.10.20.30.40.5

0 0.05 0.1 0.15 0.2 0.25

Cd,Cl

frequency (Hz)

Spectra of normalized Cd and Cl

CdCl

(a) Non wetted walls: cylinder at d = H/2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25

Cd,Cl

frequency (Hz)

CdCl

(b) Non wetted walls: cylinder at d = H/4

0

0.1

0.2

0 0.05 0.1 0.15 0.2 0.25

Cd,Cl

frequency (Hz)

CdCl

(c) Wetted walls: cylinder at d = H/2

0

0.1

0.2

0 0.05 0.1 0.15 0.2 0.25

Cd,Cl

frequency (Hz)

CdCl

(d) Wetted walls: cylinder at d = H/4

Figure 5.7: Spectra of the Cd and Cl for both wall conditions and both cylinderpositions.

86

5.2. 2D DW

Figure 5.8: The geometry layout of the 2D dishwasher case.

pattern among the dishware. The vapor plumes leaving from the different objects havedifferent vapor concentration values, i.e. different color intensity, because each objecthas a different temperature and hence a different vapor saturation condition. Evenfrom this first attempt some interesting features can be recognized: cup and glassin the upper part of the tub are filled up by vapor and hence cannot dry inside byjust evaporation, the inflow in the bottom left side of the tub has a nearly negligiblecontribution to the flow that is ruled by buoyancy force. The latter determines a risingflow in the core of the volume caused by the hot and humid air leaving regions nearthe dishware, and a downward flow near the colder and less humid air close the walls,over which condensation occurs. The general picture gives a convective motion risingfrom the center of the tub closing near the tub’s surface.

5.2.2 Case set-up

The rough dimension of the tub is 0.7 × 0.5m, its geometry is a simplified Diva2Electrolux configuration. Within the tub dishware are placed in a realistic fashion ascan be seen in Figure 5.8: two dishes on the bottom left region, further on the rightsix forks or spoons, some packed quite close one to each other; on the upper part thereare one glass and one cup. The dishwasher is not yet fully charged in order to speedup the testing of the numerical model in such complex configuration. The domain hasbeen discretized by about fifty thousand triangular cells. The linear dimension of thecells ranges from 2mm at the boundaries up to 5mm in the volume core. The meshhas an overall good quality, it appears quite homogeneous, and it has low values of thenon-orthogonality and the skewness parameters.

There are two inlet sections of 16×10mm each, on the bottom of the left tub’s wall

87


pointing one upward and the other downward, providing an air flow rate of 1.1L/s, at25C and 75% of relative humidity, the same of the initial condition in the volume.

The outlet section of 13.5×10mm is placed on the roof of the tub. The two bound-aries have been idealized in order to effectively model a far more complex ventilationsystem, the Diva1 dynamic drying. The dishware are considered wetted surface, so thatthe E/C boundary condition is applied on each body. The different temperature areapplied for different dishware, to roughly consider the different temperature diffusionof the different materials. For the following simulations the temperature of the disheshave been fixed at 63C, the forks at 43C and the glass and the cup at 53C. Boththe latter are placed upside down to prevent them to be filled up during the washingphase.

This particular choice of dishware temperatures is based on the following idea. Atthe beggining of the real drying cycle all the dishware are heated up to roughly 70C,then each object, dish, fork, glass or cup will cool down at its own rate, based on itsthermal properties and the air flow condition around it, eventually to a final tempera-ture around 40C. In absence of the thermal coupling method is found reasonable toprovide different temperatures based on the characteristic temperature diffusion timescales, defined in 2.60. So it appears clear that, for example, forks will have lowertemperature and for a longer period than the dishes, but still the exact values cannotbe computed since they depend on the flow.

5.2.3 Case discussion

Two simulations have been run: one considering the tub’s walls adiabatic and imper-meable, the second considering the tub wetted at 20C. For all the wetted surfaces theconcentration at interface is consequently computed assuming the saturation conditionwhile the depletion of the liquid film is not allowed if not differently specified as in 5.2.4.

The main flow is driven by buoyancy due to the warmer and higher vapor concen-tration on the dishware. In particular the dishes’ condition drives the main plume.The consequences of the different wall’s condition are qualitatively the same as for thewetted cylinder case discussed in 5.1: the flow velocity is higher if the wall’s are wettedpermitting condensation, i.e. colder and at lower vapor concentration. Moreover themixing of the scalars is completely different: the impermeable condition leads to a strat-ification of the flow. This is an important point since this situation must be avoided tolet the evaporation process continue. The wetted condition prevents the stratificationand allows drying process to take place. This notable difference is enlighten by theinstantaneous plots in Figure 5.9 once the steady-state is reached.

In general the vapor plume arises towards the top wall, hence the dishware placedabove the dishes are rapidly surrounded by a humid air that inhibits evaporation.Moreover the cups and glasses are filled up with warmer and more humid air, that atsome point stops the drying and can switch to condensation. This analysis is confirmedby experimental data or simply by common life experience. Often some drops can befound hanging at the bottom surface of glasses.

The effect on the drying process is more evident taking into consideration the evap-oration velocity around each single dishware. Here the evaporation around the cup isbriefly discussed. In Figure 5.10 an instantaneous evaporation velocity plot is shownfor both the configurations. For the stratified case the velocity is pointing inward the

88

5.2. 2D DW

(a) adiabatic and impermeable tub’s walls.

(b) wetted tub’s walls.

Figure 5.9: Vapor concentration contour plot, for both tub’s walls conditions.

89


(a) adiabatic and impermeable tub’s walls. (b) wetted tub’s walls.

Figure 5.10: Evaporation velocity around the cup, for both tub’s walls conditions.

liquid film and condensation occurs all over the cup’s surface. In such situation theinitial liquid water amount will increase in time. With wetted walls the drying processcontinues at least over the external surface, while the internal surface is exposed tohigh enough humidity fraction that the evaporation is inhibited.

5.2.4 Drying process

An interesting analysis has also been carried out retrieving the actual drying for the casewith wetted tub’s walls. An initial uniform film thickness of H = 5 · 10−5 m has beenprescribed on the dishwasher. This initial condition can be assumed as the final stageof the drying. The depletion in time can be tracked to complete drying of the dishware.In particular the dishes undergo drying faster. As the process takes place intermediatesituations develop, as the one shown in Figure 5.11. Once a computational surfaceface is dried, i.e. its film thickness reaches zero, it changes its boundary conditionfor the vapor concentration field to a zero wall normal gradient, i.e. a dried surface.This change, as dried surface enlarges, immediately affects the flow. As can be seen inFigure 5.11, the buoyancy is lowered and hence the flow velocity in the plume is sloweddown. As soon as the dishes dry, they cannot release any more humidity into thevolume. The condition for evaporation on all the other dishware improves in particularfor those just above the dishes.

5.3 Evaporation/Condensation from a plate in a box

A further step towards the real-scale 3D dishwasher has been done on a simplifiedgeometry in order to check the model capabilities and the mesh and computationalrequirements in a 3D case. Moreover the result of this case has been used to make afirst comparison with some experimental data at disposal in Electrolux CTI.

The tub has been reduced to a squared box with edges of 15 cm, with a wettedplate in its middle of dimension 10 × 10 × 1cm, as shown in Figure 5.12. Two pipesare connected to a lateral box’s face and to the top one in order to provide an inletand outlet respectively to the domain. Actually in the present study the inlet pipe is

90

5.3. PLATE IN A BOX

Figure 5.11: Vapor concentration contour in an intermediate phase of the dishes drying.

considered closed while along the outlet one is required in order to conserve the massas in all the other cases. Moreover in the outlet pipe a sponge region is applied.

The mesh consists of around one thousand tetrahedral cells, the dimensional re-quirements are the same adopted for the 2D dishwasher case’s mesh. The quality alsoin this case is good.

The plate is assumed wetted at a temperature of 63 C while the box’s surface isalso wetted at temperature of 20 C. The initial condition in the box volume is uniformat 25 C with a concentration of vapor of 0.015.

Natural convection starts the flow: the buoyancy force makes the convective cellsto develop. Once the steady-state is reached the depletion of the liquid thickness isswitched on. At this stage all the surfaces are covered by an uniform film thickness of2 · 10−5 m. The drying process, as for the dishes in the 2D dishwasher, begins from thelower part of the solid body, where the vapor concentration gradients at the surface arehigher. Once a first dried patch appears, as expected, it is possible to recognize theincreased evaporation rate near the contact line.

The mean evaporation mass flux, mw, from the plate is computed and its evolutionin time is plotted in Figure 5.13. At the beginning a steady-state can be recognized forwhich mw = 1.12 · 10−3 kg/m2. When the first portion of the plate surface dries outmw decreases until the complete drying is reached and finally the evaporation stops.

5.3.1 1st comparison with experimental data

The result of the evaporation process on the plate has been used to make a first com-parison with experimental data available at Electrolux CTI. This data were collectedduring a previous collaboration between Electrolux and University of Bonn. The ex-periment provided measurements on the upper basket of an actual dishwasher, of thedifferential weight and of the temperature during all the washing cycle. The differentialweight in particular measures the weight of the water laying on the tableware. Theresults are presented in the Figure 5.14. In Figure 5.15 the last part of the cycle, i.e.the drying phase, is considered only. In the first two-three minutes approximatively,

91


Figure 5.12: Vapor concentration and the film thickness contour in an intermediatephase of the plate drying.

Figure 5.13: Mean evaporation mass flux during the drying process of the plate.

92

5.4. 3D DW

Figure 5.14: Experimental data of the whole washing cycle from Bonn university.

the largest part of the weight loss is due by dripping. Afterwards the remaining waterdries through evaporation.

To compare these results the data of the simulation are extrapolated consideringthe steady-state value mw = 1.12 · 10−3 kg/m2, and can be considered representative ofthe first stage of the drying phase. The total evaporation rate on the total area of thedishware, At, used in the experiment is computed

Mw = mw ·At = 15.6 gmin

and the corresponding value is depicted by the magenta point in Figure 5.15.

Although the two systems are quite different, the purpose of the comparison wasto check at least the order of magnitude given by the simulations and to verify theeffectiveness of the modeling approach, both crucial issues for the industrial partner.

5.4 Three-dimensional dishwasher

Finally the case of the real-scale 3D dishwasher has been faced. The complex geometry,depicted in Figure 1.1, requires a preparatory stage in order to smooth the very smallgeometrical details of the original design and avoid extremely small cell’s volumes andto permit reasonable simulation time steps. The rough dimension of the tub is 0.7 ×0.5× 0.5m. On the roof of the tub a circular pipe 20 cm long is connected to the mainvolume. It is treated as a sponge region in order to prevent numerical instabilities. Theoutlet section has a diameter of 44mm. In this case there is no inlet section, flow issustained by natural convection driven by temperature an humidity conditions at thetub’s walls and at the dishware which are three cups on the upper part of the volume,seven dishes of three different dimension and shape in the lower region, sorted as inFigure 1.1.

The mesh is composed by about 8 · 105 tetrahedral cells. The linear dimension of thecells at the surfaces varies to follow the geometry of the cups and dishes from 1mm to8mm on tub’s walls. The cells volume ranges from 1mm3 to 1 cm3 from near boundaries

93


Figure 5.15: The drying cycle experimental data from Bonn university.

regions to bulk volume ones. The mesh quality check shows that the resulting meshhas some flaws, few cells are quite non-orthogonal and skewed. To preserve the overallsecond order accuracy of the numerical schemes the non-orthogonality correction mustbe switched on.

A temperature of 60 C and a vapor concentration of 0.105 are prescribed as initialcondition for air within the tub. The dishware are supposed wetted, the temperatureof the dishes is 60 C and 59 C for the cups. The walls are fixed at 55 C.

The simulation runs for few physical seconds and the characteristic plumes developas can be seen in Figure 5.16. Figure 5.16a shows the vapor concentration contoursin two vertical planes and in Figure 5.16b the corresponding velocity field. From thelatter is possible to recognize the usual convective motion.

Successively two kind of numerical instabilities appear both leading to unphysicalpressure values. The first kind of instability appears in the bulk volume above thedishware, i.e. where the maximum velocity magnitude is reached. This is thought tobe due to turbulence missing dissipation. The application of the Smagorinsky LESmodel stabilized the solution, but the results appear to be too much diffused dueto an over estimation of the sub-grid viscosity, see for example the istantaneus vapourconcentration and velocity fields in Figure 5.17. The second kind of instability shows uplater during the simulation, even with the Smagorinsky model. The pressure peaks closeto the dishware surfaces because the momentum is not well conserved. This problemhas been solved by tuning the number of momentum correction loops as discussedin 4.1.1.

In order to properly take into account the effects of turbulence in home-applianceapplications the LES approach is preferable, being DNS computation not affordableand the flow transient in nature. Since the geometries involved are very complex andthe flows strongly anisotropic, among the possible LES models known in literature themost appealing is the Lagrangian dynamic sub-grid scale model proposed in [22]. Suchmodel is not provided within the current OpenFoam release.

94

5.4. 3D DW

(a) Vapor contours in two vertical planes.

(b) Velocity vectors in two vertical planes.

Figure 5.16: Vapor concentration and velocity vectors in the 3D Dishwasher case.

95


(a) Vapor contours in two vertical planes.

(b) Velocity vectors in two vertical planes.

Figure 5.17: Vapor concentration and velocity vectors in the 3D Dishwasher case withSmagorinsky LES modelling.

96

Chapter 6

Conclusions and future work

The main goal of the PhD project has been to understand the physical issues related todishwasher’s drying performances, and to propose and implement a mathematical modelsuitable for such a task. The main focus has been on the modeling of condensationand evaporation process that affects the wetted surfaces, i.e. dishware, inside thetub, during the the drying phase. Before this phase the final rinse cycle heat-up thedishware. Then the thermal energy stored permits evaporation to take place. As theprocess continues the solid bodies cool down, diminishing the evaporation rates. Onthe tub’s walls condensation occurs.

The physical system has been considered as a mixed convection flow in presenceof phase change phenomena. In general this kind of systems with drying and wettingof solid surfaces are of great importance in many natural and technological processes.Nevertheless their knowledge is far from being complete. The complex physics involvedhas been divided in three sub-problems: the exchange of heat between wetted solidbodies 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.

In literature such a problem has not been completely faced yet. This study proposeda model for the drying process of liquid films suitable for engineering purposes, i.e. onlarge scales and complex geometries. The numerical implementation has been carriedout in the OpenFoam environment, an open-source C++ CFD tool. The mathematicalmodel consists of the incompressible formulation of the Navier-Stokes equations, plusthe transport equations for temperature and vapor concentration. Both are treated asactive scalars. The density variations are taken into account by means of the Boussi-nesq approximation. The thin liquid film is treated as a boundary condition for thegaseous flow. It prescribes a Dirichlet condition for the temperature and for the vaporconcentration that, at the liquid interface, is considered in saturation. The evapora-tion/condensation process is evaluated by the evaporation velocity at the liquid-gasinterface by the relation (2.44) as explained in 2.4, providing the boundary conditionfor the velocity field, and the water mass transfer rate.

The thin film approximation is assumed, meaning that all liquid films, liquid patchesand drops on the wetted surfaces are considered as a continuous film. In home appli-ances as the dishwasher, some surfaces, as example tub walls, have been found to belarge enough respect to the typical length scale of the droplets distributed over them.

97

CHAPTER 6. CONCLUSIONS

In such cases separation of scale can be properly invoked.

In order to provide a theoretical foundation to the thin film approach a futureinvestigation is needed. In particular the author proposes to include all the sensibleparameters of the droplet distribution, namely the contact angle, drop diameter andthe distance among droplets in a simplified model by means of a boundary homoge-nization technique. This aims to parametrize the wetted surface taking into accountthe geometrical features of the droplets 1.

The numerical model has been validated against literature results of [19], then astudy of the archetypal case of the flow over a wetted cylinder confined between twoparallel walls [28] has been carried out.

Successively a real configuration has been considered: the case of the 2D dishwasher.The analysis revealed how the vapor plumes leaving from the different bodies give riseto the complex flow pattern inside the volume. Some interesting features have beenrecognized such the filling up of the cup and glass in the upper part of the tub by vaporpreventing the evaporation process; or that the inflow at the bottom left side of thetub has a nearly negligible contribution to the flow. The buoyancy force rules the fluidmotion. The flow in the core of the volume is caused by the hot and humid air plumesabove the dishware, while close to the walls over which condensation occurs, the colderand less humid air sinks to the bottom. The general picture gives a convective motionrising from the center of the tub and closing near the tub’s surface. An initial uniformfilm thickness has been prescribed on the dishware, and the evolution in time has beenfollowed up to the complete drying. In intermediate situations the surface area that isalready dried cannot release more vapor into the flow, lowering the buoyancy near thebody and slowing down the flow velocity in the plume. The drying is found to beginfrom lower edges of the solid bodies.

A simplified 3D geometry has been consider as a successive step towards the caseof full scale 3D dishwasher. The results have shown again that the buoyancy forcestarts the convective cells and the evolution of the liquid thickness takes place. Thedrying process begins from the lower edge of the plate, where the vapor concentrationgradients at the surface are higher. As the first dried patch appears on the surface itis possible to recognize the increased evaporation rate near the contact line.

Finally the case of a real scale 3D dishwasher, with the proper geometry has beentackled. The fundamental issue of the meshing required some work in smoothing verysmall geometrical details of the original design in order to avoid extremely small cell’svolumes. The simulation can run for few physical seconds before the turbulence miss-ing dissipation leads to a numerical instability. In order to properly take into accountthe effects of turbulence in home appliance applications and because of the transientnature of the flow, the LES approach has been preferred. As a first choice, the stan-dard Smagorinsky LES model with the Reynolds analogy for the eddy diffusivity ofthe two scalars has been applied. The results appear to be to much diffused due toan overestimation of the sub-grid viscosity. The geometries involved are very complexand the flows strongly anisotropic. Therefore the most appealing model for such ap-plication would be the Lagrangian dynamic sub-grid scale model proposed in [22]. Itsimplementation is currently under development since OpenFoam doesn’t provide thismodel in the present release. Moreover the OpenFoam dynamic models are found to

1private communication with prof. V. Armenio

98

average the constant in the whole three-dimensional space, which is clearly not correct.As a first step towards the Lagrangian approach the dynamic model with plane aver-aging has been implemented. It has been tested on the turbulent channel flow at Reτand the results compared with the data of Kim, Moin and Moser [24], showing a goodagreement.

A second branch of the modelling required the thermal coupling among the mediaof the liquid phase either with the solid substrate and the air. The coupling methodcurrently provided in OpenFoam is found to be not accurate enough or even faultyfor some specific case. Hence in the present work the “Dirichlet-Neumann” domaindecomposition technique has been implemented. It enforces the continuity of temper-ature and the balance of the heat fluxes across each interface. The performed testsproved the effectiveness of the DN method also in presence of an arbitrary heat sourceor sink. In order to properly take into account the heat transfer mechanism throughthe liquid layer a suitable temperature film model has been proposed. As expected theruling term in the heat transfer process among the three media is the latent heat fluxof evaporation. The DNM and the film model have been independently implementedand tested in order to validate each part of the code. The inclusion of both parts in afinal version of the model is in progress.

In conclusion, during the PhD project the problem of the drying process has beenexploited from an engineering point of view. A mathematical model for the flow in thegaseous region has been proposed and implemented. The evaporation and condensationeffects are considered as boundary condition at the wetted surfaces. The current modelhas been applied to test cases and to real application cases proving its effectiveness inpredicting evaporation rates. Moreover it has been further developed to incorporatethe important effects of the thermal coupling of the liquid layer with the solid bodiesand the air flow.

99

CHAPTER 6. CONCLUSIONS

100

Appendix A

Dynamic thin film model

There are many physical phenomena and technological application involving liquidfilms; these can be unbounded free jet, as water dropping from the tap, a flow boundedbetween two walls, as in lubrication applications, and liquid on solid substrates. Thelatter is the main topic of this review. It has to be remarked that the adjective ”thin”referring to film thickness is relative to typical length-scales parallel to the substrate.

The interesting processes associated to thin film are the initial films rupture, thegrowth of individual holes, the evolution of the resulting hole pattern or drops forma-tion, and their associated evolution of the individual de-wetting/wettingfront. To facethis problems one has to keep in mind that the free interfaces are, in part, free bound-aries whose configuration must be determined as part of the solution. The numericalsolution of this problem modeled with full Navier-Stokes eq. with moving boundariesis beyond the present computational capabilities for most of the practical applicationcases. However it is possible to derive a reduced model, the so called ”Long-scaleevolution equation” [26]. The motivation supporting this model is based on the exper-imental evidence that most rupture and instability processes in thin films do occurson long-scales, i.e. the variations along the film are much more gradual than thosenormal to it, and slow in time. As in shallow-water theory or in lubrication theory,one can separate the variables and simplify the analysis, and finally write a single par-tial differential equation for the evolution of the film thickness profile, while the restof the unknowns, as temperature and velocity, are determined by algebraic functions.The main drawbacks of this approach are related to the strong non-linearity and thehigher-order spatial derivatives present in the governing equation.

A.1 Film Thickness Evolution Equation

A.1.1 Basic Equation

Starting from the full set of the Navier-Stokes equation for a two dimensional case

• Continuity equation∂xu+ ∂zw = 0

• Momentum equation

ρ(∂tu+ u∂xu+ w∂zu) = −∂xp+ µ∇2u− ∂xφ

101

APPENDIX A. DYNAMIC FILM MODEL

ρ(∂tw + u∂xw + w∂zw) = −∂zp+ µ∇2w − ∂zφ

where φ is conservative body force potential, i.e. gravity.

• Energy equation

∂tT + u∂xT + w∂zT = α∇2T

A.1.2 Boundary conditions

• at solid substrate:

– no-slip and no-penetration conditions

us = 0 ws = 0

– slip condition at the contact line and no-penetration condition

us = β∂zu ws = 0

the slip is proportional to the shear stress through β, the slip coefficient, thatmust be chosen numerically small in order to let slip just near the contactline.

• at free surface, z = h(t, x), interface follows the flow field and undergoes toexternal stress

– kinematic condition

w = ∂th+ u∂xh

– force equilibrium

T ·n = −kσn+ ∂sσt+ f

where the forces on the surface is represented by the liquid stress tensor Tas the sum of the Young pressure pL = −kσ, with k the interface curva-ture, acting in the normal to surface direction, and by solutal or thermalMarangoni effect acting in tangential direction1, plus other forces such grav-ity and external shear stress represented by f. The two scalar condition alongthe tangential and normal directions are:

t : µ(∂zu+ ∂xw)(1 − (∂xh)2) + 2(∂zw − ∂xu)∂xh = τ + ∂sσ(1− (∂xh)

2)

n : p+2µ[−∂xu(∂xh)

2 − ∂zw + ∂xh(∂zu+ ∂xw)]

1 + (∂xh)2= −π −

σ∂2xh

(1 + (∂xh)2)3/2(A.1)

1as in case of dish-washing liquid and sparkling aid residuals in water

102

A.1. FILM THICKNESS EVOLUTION EQUATION

A.1.3 Long-wave Scaling

To find a suitable evolution equation a scale analysis is performed here on: as alreadysaid, in thin films the characteristic thickness is normally much smaller than the char-acteristic length parallel to the wetted surface, i.e. l ≪ L. If ǫ = l

L ≪ 1 holds, itfollows that

Z =z

lX =

ǫ

lx

From continuity it follows that, given the characteristic velocity U0:

U = u/U0 W = w/ǫU0

where U0 =µ

ρlif is assumed 1/Re → 1, where Re = U0lρ

µ . Time is scaled as:

T =ǫU0

lt

At this point one has to specify the length scale, usually l = h0 the initial film thickness,and L perturbation wave period in stability analysis.

Stresses and body-force and pressure are scaled as

(τ∗,Π∗) =h0µU0

(τ,Π) (Φ, P ) =ǫh0µU0

(φ, p)

and the slip coefficient β∗ = β/h0.

After some algebra the scaled system is

ǫRe(∂TU + U∂XU +W∂ZU) = −∂XP + ǫ2∂2XU + ∂2ZU − ∂XΦ

ǫ3Re(∂TU + U∂XW +W∂ZW ) = −∂ZP + ǫ2(ǫ2∂2XW + ∂2ZW )− ∂ZΦ

∂XU + ∂ZW = 0

with the boundary condition at Z = 0

W = 0 U = β∗∂ZU

setting H = h/h0, Re =U0h0

ν and Ca = U0µσ , at Z = H

t : (∂ZU + ǫ2∂XW )(1− ǫ2(∂XH)2)− 4ǫ2∂XU∂XH =

τ∗(1 + ǫ2(∂XH)2) + ∂XΣ(1 + (∂XH)2)1/2

n :2ǫ2

1 + ǫ2(∂XH)2[∂XU(ǫ2(∂XH)2 − 1)− ∂XH(∂ZU + ǫ2∂XW )] =

P +Π+C−1a ǫ3∂2XH

(1 + (∂XH)2)3/2

where Σ = ǫσµU0

. It’s essential that surface-tension effect is retained at leading order

in a correct model. Hence, with Ca = ǫ−3Ca, one possible approximation is found byletting

Re = Ca = O(1)as ǫ→ 0

103


Finally the equations at leading order are

∂2ZU = ∂XP + ∂XΦ

∂ZP = −∂ZΦ

∂TH + ∂X

∫ H

0UdZ = 0

with the boundary conditions at Z = 0

U = β∗∂ZU

and at Z = H

t : ∂Z = τ∗ + ∂XΣ (A.2)

n : 0 = P +Π+ Ca−1∂2XH (A.3)

Solving for U

U = (τ∗ + ∂XΣ)(Z + β∗) + ∂x(P +Φ)

(Z2

2−HZ − β∗H

)

that substituted into the mass conservation condition leads to the evolution equationfor the interface.

A.1.4 Film Thickness Evolution Equation

The resulting equations of the film thickness evolution are written in non-dimensionalform for the two dimensional cases:

∂TH +∇ ·

[(−→τ ∗ +∇Σ)

(H2

2+ β∗H

)]

−∇ ·

[(H3

3+ β∗H2

)∇

(Φi −

∂2XH

Ca−Π

)]= 0 (A.4)

Written in dimensional variables:

µ∂th + ∇ ·

[(−→τ +∇σ)

(h2

2+ βh

)]− ∇ ·

[(h3

3+ βh2

)∇(φi − σ∇2h−Π

)]= 0

(A.5)

From a mathematical point of view the model is a fourth order non-linear P.D.E,where the highest order terms are given by the presence of the surface tension andthe gravity force, with also the highest non-linearity given by h3; moreover gravityand surface tension are responsible of the coupling of the pressure term to H or h.In the literature, authors mainly focus on the stability issues for different physicalconfigurations, see among the others: film and drop placed above or below a flat plate,or on a inclined surface and so on, with or without the addition of other effects as heattransfer, variable viscosity and heterogeneity of the substrate.

Sometimes is possible to further simplify the model neglecting some of the effects.The most important cases are, here in one dimension for sake of simplicity:

104

A.1. FILM THICKNESS EVOLUTION EQUATION

constant shear stress and surface tension only

In this case β = φi = Π = 0 so the equation A.5 simplifies to

µ∂th+ τh∂xh+σ

3∂x(h

3∂3xh) = 0

It is proven that no instability occurs under the long wave regime.

constant surface tension and gravity only

In this case β = τ = Π = 0 and the gravity term is given by ρgh. The resultingequation is

µ∂th+1

3σ∂x(h

3∂3xh)−1

3ρg∂x(h

3∂3xh) = 0

in non dimensional form

∂TH +1

3Ca

−1∂X(H3∂3XH)−

1

3G∂X(H3∂3XH) = 0

whereG =ρgh2

0

µU0is the gravity number. In this case two configurations are possible:

a film placed above a solid surface, and a film below the surface.For the formerG > 0 and no instability occurs, for the latter G < 0 all wave numbers below acritical one are linearly unstable

k2 < kc = Bo

that happens to be given by the Bond number, that measure the relative impor-tance of gravity to capillary effects. The higher the surface tension (or lower thedensity difference between fluids) the shorter the instability region.

A.1.5 Incorporating evaporation and condensation

If the interface is at a temperature θ below or above the saturation one at given vaporpressure, condensation or evaporation occurs. To take into account this effects intothe model one has to add some terms to the boundary conditions. A mass balance atinterface gives

j = ρg(ug − ui) ·n = ρf (uf − ui) ·n

Since ρg ≪ ρf , sayρgρl

∼ 10−3, the normal velocity of the vapor is much greater thanthat the fluid at interface, meaning that phase change creates large acceleration of thegas and smaller back reaction, called the vapor thrust, that actually may be importantfor very high heat fluxes; comparing the dynamic pressures

ρgu2g,e =

j2

ρg≫ ρfu

2f,e =

j2

ρf

Similar approximation can be done for energy equation noting that even viscosity andthermal conductivity of the liquid are much larger that those of the gas. So one shouldadd at the right-hand side to the normal stress conditions expressed in (A.3)

j2

ρg

105


further an energy balance is needed

−k∂zθ = ifgj

meaning that all the heat conducted through the film to the interface is converted tolatent heat. One remark: if the film is heated or cooled by the solid surface, say it’s atfixed temperature, the interface temperature and j will be a function of h.

To conclude, it’s important to note that such models have already been used forengineering application, as reported in [37], to identify the main structures of a initiallyuniform film under evaporation, that are mainly drops and film patches with holes, withgood agreement with experimental results.

106

Bibliography

[1] V. Armenio. Mathematical modeling of stratified flows. In V. Armenio andS. Sarkar, editors, Environmental Stratified Flows. Springer, 2006.

[2] G. Comini. Lezioni di Termodinamica Applicata. SGE Editoriali, Padova, 2000.

[3] G. Comini, C. Nonino, and S. Savino. Modeling of coupled conduction and con-vection in moist air cooling. Numerical Heat Transfer-Part A-Applications, 51(1-2):23–38, 2007.

[4] S. David, K. Sefiane, and L. Tadrist. Experimental investigation of the effect ofthermal properties of the substrate in the wetting and evaporation of sessile drops.Colloids and Surfaces A: Physicochemical and Engineering Aspects, 298(1-2):108–114, Apr. 2007.

[5] P. G. de Gennes. Wetting: statics and dynamics. Reviews of Modern Physics,57(3):827, July 1985. Copyright (C) 2009 The American Physical Society; Pleasereport any problems to [email protected].

[6] E. de Villiers. The Potential of Large Eddy Simulation for the Modeling of Wall

Bounded Flows. PhD thesis, Thermofluids Section Department of MechanicalEngineering Imperial College of Science, Technology and Medicine, July 2006.

[7] E. Divo, F. Rodriguez, E. Steinthorsson, A. J. Kassab, and J. S. Kapat. Glenn-ht/bem conjugate heat transfer solver for large-scale turbomachinery models. Tech-nical Report NASA/CR-2003-212195, NASA Glenn Research Center, April 2003.

[8] G. Dunn, S. Wilson, B. Duffy, S. David, and K. Sefiane. A mathematical model forthe evaporation of a thin sessile liquid droplet: Comparison between experimentand theory. Colloids and Surfaces A: Physicochemical and Engineering Aspects,323(1-3):50–55, June 2008.

[9] G. J. Dunn, S. K. Wilson, B. R. Duffy, S. David, and K. Sefiane. The strong influ-ence of substrate conductivity on droplet evaporation. Journal of Fluid Mechanics,623:329–351, 2009.

[10] M. Feddaoui, H. Meftah, and A. Mir. The numerical computation of the evapo-rative cooling of falling water film in turbulent mixed convection inside a verticaltube. International Communications in Heat and Mass Transfer, 33(7):917–927,Aug. 2006.

107

BIBLIOGRAPHY

[11] J. H. Ferziger and M. Peric. Computational methods for fluid dynamics. Springer-Verlag, Berlin-New York, 1996.

[12] M. Germano, U. Piomelli, P. Moin, and W. H. Cabot. A dynamic subgrid-scaleeddy viscosity model. Physics of Fluids, 3(7), July 1991.

[13] Z. A. Hammou, B. Benhamou, N. Galanis, and J. Orfi. Laminar mixed convectionof humid air in a vertical channel with evaporation or condensation at the wall.International Journal of Thermal Sciences, 43(6):531–539, June 2004.

[14] H. Hu and R. G. Larson. Evaporation of a sessile droplet on a substrate. Journalof Chemical Physics B, 106:1334–1344, 2002.

[15] C. Huang, W. Yan, and J. Jang. Laminar mixed convection heat and mass transferin vertical rectangular ducts with film evaporation and condensation. InternationalJournal of Heat and Mass Transfer, 48(9):1772–1784, Apr. 2005.

[16] J. Jang, W. Yan, and C. Huang. Mixed convection heat transfer enhancementthrough film evaporation in inclined square ducts. International Journal of Heat

and Mass Transfer, 48(11):2117–2125, May 2005.

[17] H. Jasak. Error Analysis and Estimation for the Finite Volume Method with Ap-

plications to Fluid Flows. PhD thesis, Department of Mechanical EngineeringImperial College of Science, Technology and Medicine, June 1996.

[18] S. G. Kandlikar, M. Shoji, and V. K. Dhir. Handbook of Phase Change: Conden-

sation and Evaporation. Taylor & Francis, 1999.

[19] N. Laaroussi, G. Lauriat, and G. Desrayaud. Effects of variable density for filmevaporation on laminar mixed convection in a vertical channel. International Jour-nal of Heat and Mass Transfer, 52(1-2):151–164, 2009.

[20] J. Leu, J. Jang, and Y. Chou. Heat and mass transfer for liquid film evaporationalong a vertical plate covered with a thin porous layer. International Journal of

Heat and Mass Transfer, 49(11-12):1937–1945, June 2006.

[21] D. K. Lilly. A proposed modification of the germano subgrid-scale closure method.Physics of Fluids, 4(3), March 1992.

[22] C. Meneveau, T. S. Lund, and W. H. Cabot. A lagrangian dynamic subgrid-scalemodel of turbulence. Journal of Fluid Mechanics, 319:353–385, 1996.

[23] R. Mollaret, K. Sefiane, J. R. Christy, and D. Veyret. Experimental and numericalinvestigation of the evaporation into air of a drop on a heated surface. Chemical

Engineering Research and Design, 82(4):471–480, Apr. 2004.

[24] R. Moser, J. Kim, and P. Moin. Turbulence statistics in fully developed channelflow at reynolds number. Journal of Fluid Mechanics, 177:133–166, 1987.

[25] N. Murisic and L. Kondic. Modeling evaporation of sessile drops with movingcontact lines. Physical Review E, 78, 2008.

108

BIBLIOGRAPHY

[26] A. Oron, S. H. Davis, and S. G. Bankoff. Long-scale evolution of thin liquidfilms. Reviews of Modern Physics, 69(3):931, July 1997. Copyright (C) 2009 TheAmerican Physical Society; Please report any problems to [email protected].

[27] D. Panara and B. Noll. A coupled solver for the solution of the unsteady conjugateheat transfer problem. In Computational Methods for Coupled Problems in Science

and Engineering 2, Coupled Problems, Eccomas Thematic Conference, 2007.

[28] A. Petronio, V. Armenio, G. Buligan, and L. Padovan. Evaporation/condensationaround a wetted cylinder confined between two parallel walls. In proceedings of

the “12th Workshop on two phase flow predictions”, March 2010.

[29] U. Piomelli. Large-eddy and direct simulation of turbulent flows. Lecture series on”Introduction to Turbulence Modeling”. 2004.

[30] A. Quarteroni, F. Pasquarelli, and A. Valli. Heterogeneous domain decomposition:principles, algorithms, applications. In D. E. Keyes, T. F. Chan, and G. Meurant,editors, Fifth International Symposium on Domain Decomposition Methods for

Partial Differential Equations, SIAM Proceedings in Applied Mathematics. SIAM,1992.

[31] A. Quarteroni and A. Valli. Domain decomposition methods for partial differential

equations. Numerical Mathematics and Scientific Computation. Oxford UniversityPress, 1999.

[32] H. Rusche. Computational Fluid Dynamics of Dispersed Two-Phase Flows at High

Phase Fractions. PhD thesis, Imperial College of Science, Technology & MedicineDepartment of Mechanical Engineering, December 2002.

[33] O. S., E. E., W. E., and K. S. Unsteady conjugated heat transfer in laminar pipeflow. International Journal of Heat and Mass Transfer, 34(6):1443–1450, 1991.

[34] M. A. Saada, S. Chikh, and L. Tadrist. Numerical investigation of heat and masstransfer of an evaporating sessile drop on a horizontal surface. Physics of Fluids,22(112115), 2010.

[35] P. Sagaut. Large Eddy Simulation for Incompressible Flow, An Introduction, vol-ume Scientific Computation. Springer, 2001.

[36] L. W. Schwartz, D. Roux, and J. J. Cooper-White. On the shapes of droplets thatare sliding on a vertical wall. Physica D, 209:236–244, 2005.

[37] L. W. Schwartz, R. V. Roy, R. R. Eley, and S. Petrash. Dewetting patterns in adrying liquid film. Journal of Colloid and Interface Science, 234:363–374, 2001.

[38] K. Sefiane and L. Tadrist. Experimental investigation of the de-pinning phe-nomenon on rough surfaces of volatile drops. International Communications in

Heat and Mass Transfer, 33(4):482–490, Apr. 2006.

[39] B. F. Smith, P. Bjorstad, and W. Gropp. Domain Decomposition. CambridgeUniversity Press, Mar. 2004.

109

BIBLIOGRAPHY

[40] P. Talukdar, C. R. Iskra, and C. J. Simonson. Combined heat and mass transfer forlaminar flow of moist air in a 3D rectangular duct: CFD simulation and validationwith experimental data. International Journal of Heat and Mass Transfer, 51(11-12):3091–3102, June 2008.

[41] U. Thiele. Structure formation in thin liquid films. In Thin Films of Soft Matter,pages 25–93. 2007.

[42] U. Thiele, K. Neuffer, M. Bestehorn, Y. Pomeau, and M. G. Velarde. Sliding dropson an inclined plane. Colloids and Surfaces A: Physicochemical and Engineering

Aspects, 206:87–104, 2002.

[43] L. Zovatto and G. Pedrizzetti. Flow about a circular cylinder between parallelwalls. Journal of Fluid Mechanics, 440(-1):1–25, 2001.

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