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Analysis of film break-up and dry patch stability de Rodriguez, Sara Gersberg 1975

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ANALYSIS OF FILM BREAK-UP AND DRY PATCH STABILITY by SARA GERSBERG de RODRIGUEZ L i e . i n Physics, Un i v e r s i t y of Buenos A i r e s , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1975 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th i s thesis for scho lar ly purposes may be granted by the Head of my Department or by h i s representat ives . It is understood that copying or pub l i ca t ion of th is thes is for f i nanc ia l gain sha l l not be allowed without my written permission. Department of /// <-"//. The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Sr/j l,;t / / /'} 7 3 -f — ABSTRACT The s t a b i l i t y of stationary dry patches i n a t h i n , heated l i q u i d f i l m was analyzed according to previous models and compared with recent experimental data. Previous analysis i n d i c a t e that dry patch s t a b i l i t y i s expressed i n terms of a balance of forces at the up- stream edge of the dry patch: a pressure force tends to rewet the dry patch, and the surface tension and thermocapillary forces causes the dry patch to spread. Roughly, the models are reduced to two types, the f i r s t evaluates pressure force applying a Bernoulli-type equation to the center-streamlines, the second uses the control-volume approach. The former method gives h a l f the pressure force predicted by the second model; i n both analyses the flow i s considered one-dimensional. In the present study the c o n t r a d i c t i o n was c l a r i f i e d by applying the con t r o l volume technique to a two- dimensional flow. Both methods give equivalent r e s u l t s f o r the l i m i t i n g case of c o n t r o l volume co i n c i d i n g with the center stream- l i n e . When experimental data are used the model that proposes a Bernoulli-type equation to f i n d pressure force best describes the balance of forces, s p e c i a l l y for low Reynolds numbers. For high Reynolds numbers pressure forces depart s i g n i f i c a n t l y from surface forces. i i i In the present study the force balance c r i t e r i o n f o r s t a b i l - i t y of dry patches was extended to the case of a wavy f i l m . Kapitza's analysis for surface waves on t h i n f i l m was used and the bi-dimensional character of the flow was considered through the i n t r o d u c t i o n of a c o e f f i c i e n t whose value was assumed equal to the steady case. Results show that a body force must be included together with pressure force to balance surface tension force. A better d e s c r i p t i o n of the flow f i e l d i s needed since Kapitza's analysis i s not v a l i d near the dry patch. A further model i s presented by means of which f i l m p r o f i l e s and pressure forces can be evaluated. The goal was to describe flow behaviour of a t h i n heated f i l m around a dry patch. Due to the complexity of the problem d i f f e r e n t assumptions at various stages were made. The problem was divided into two regions, s i m i l a r to a boundary layer method. In the outer region surface tension e f f e c t s were neglected and the patch acts l i k e a s o l i d object f o r the flow. Increases i n stagnation pressure are balanced by changes i n hydrostatic pressure. In the inner region surface tension e f f e c t s predominate over i n e r t i a l e f f e c t s . The free surface p r o f i l e s , v a l i d f o r a narrow range of low Reynolds numbers, are wedge-shaped and d i f f e r e n t from measured p r o f i l e s . In future work surface tension e f f e c t s i n the outer region must be included and the s o l u t i o n extended to a larger range of Reynolds numbers. i v TABLE OF CONTENTS Chapter Page 1. INTRODUCTION . 1 2. REVIEW OF LITERATURE 5 2.1 General 5 2.2 Thin Film Flow 5 2.2.1 Laminar Film Flow 6 2.2.2 Turbulent Film Flow 7 2.2.3 Wavy Film Flow . . . 8 2.3 S t a b i l i t y of Dry Patches . 11 2.3.1 Steady Case . 11 2.3.2 Unsteady Case 20 2.4 Contact Angle 24 3. COMPARISON WITH EXPERIMENTS OF PREVIOUS STABILITY MODELS AND SOME SIMPLE EXTENSIONS 28 3.1 Discussion of McAdam's Experiments 28 3.2 Relationship Between Various Models . . . . . . . 30 3.3 Comparison of Previous Models With Data 31 3.3.1 General 31 3.3.2 Low R e . . . . 34 3.3.3 High Re 37 3.4 Simple Extensions 39 3.4.1 Simple 2-D Extensions 39 V Chapter Page 3.4.2 Re-laminarization Effects 41 3.5 Discussion 42 3.5.1 Balance of Forces 42 3.5.2 Role of Thermocapillarity 44 3.6 Summary 46 4. ^ UNSTEADY EFFECTS 48 4.1 Evaluation of a 52 5. A RATIONAL TWO-DIMENSIONAL MODEL 56 5.1 General 56 5.2 Solution 58 5.2.1 Outer Region . 58 5.2.2 Inner Region 66 5.3 Discussion 69 6. SUMMARY AND CONCLUSIONS . . 73 FIGURES 76 REFERENCES 103 APPENDIX A - EVALUATION OF THE SHEAR FORCE AND BODY FORCE FOR A KAPITZA VELOCITY PROFILE 107 APPENDIX B - DEVELOPMENT OF THE SIMPLIFIED NAVIER-STOKES EQUATIONS USED IN CHAPTER 5 112 APPENDIX C - PROPERTIES OF CARBON DIOXIDE 120 v i ! LIST OF TABLES Table Page 3.1 Evaluation of Pressure Forces and Surface Forces according to Zuber & Staub at low Re 34 3.2 Evaluation of Pressure Forces According to Ponter et al. at low Re 35 3.3 Evaluation of Pressure Force, Body Force and Surface Tension Force According to Wilson 35 3.4 Evaluation of Pressure Forces and Surface Forces at High Re 38 3.5 Evaluation of Pressure Forces and Surface Forces When the 1/7 Power Law i s Used at Low Re . . . . . . . 42 4.1 Evaluation of Pressure Force and Surface Tension Force When the Control-Volume Method i s Applied to a Wavy Liquid Film 53 4.2 Evaluation of Pressure Forces and Surface Forces When the Thickness C r i t e r i o n i s Applied to a Wavy Film 54 5.1 Evaluation of Pressure Force According to Model Described i n Chapter 5 71 v i i LIST OF FIGURES Figure Page 1. Some mechanisms of f i l m break-up 76 2. Nuclear power plant . . . 77 3. Flow b o i l i n g regimes, upward flow 78 4. Dry patch on a s o l i d surface . . . 79 5. Forces acting at a dry patch (Ref. 22) 80 6. Annular flow with waves (Ref. 26) 81 7. T r i p l e front p o s i t i o n during two waves f o r d i f f e r e n t values of heat fluxes (Ref. 26) 82 8a. A l i q u i d drop on a h o r i z o n t a l plate 83 8b. Hysteresis of the contact angle 83 9. A drop on a t i l t e d p l ate 84 10. V a r i a t i o n of apparent contact angle due to surface roughness 85 11. Film p r o f i l e s for d i f f e r e n t times,T=2°C, Re=185,(Ref.1). . 86 12. Contact angle v a r i a t i o n with time, T=2°C, Re=185,(Ref.1). . 87 13. Film p r o f i l e s f o r d i f f e r e n t times,T=2°C, Re=310,(Ref.1). . 88 14. Contact angle v a r i a t i o n with time,T=2°C, Re=310,(Ref.1). . 89 15. Film p r o f i l e s f o r d i f f e r e n t times,T=2°C, Re=940,(Ref.1). . 90 16. Contact angle v a r i a t i o n with time,T=2°C, Re=940,(Ref.1). . 9 1 17. Average contact angle versus input Reynolds No. . . . . . 92 18. Average contact angle versus l o c a l Reynolds No 93 19. Scheme of a two dimensional flow around a dry patch. The dash l i n e s i n d i c a t e the c o n t r o l volume used 94 v i i i Figure Page 20. Pressure forces according to d i f f e r e n t models 95 21. Comparison of the t o t a l surface force (F + I r t j 1 ) according to Zuber and Staub with the surface force without thermal e f f e c t s (Fj) 96 22. Difference between pressure force and surface force according to Zuber and Staub model for low Re ( c i r c l e s ) . Difference between pressure force and surface forces when the 1/7 power law i s used as a v e l o c i t y p r o f i l e (squares) 97 23. Difference between pressure force and surface forces for high Re 98 24. Comparison of the pressure force according to the two dimensional model with Zuber and Staub c r i t e r i o n . I n l e t temperature = 2°C 99 25. Scheme of the f i l m surface p r o f i l e i n the v i c i n i t y of a dry patch on an i n c l i n e d plate 100 26. F i l m thickness p r o f i l e according to model described i n Chapter 5 1°1 27. Film thickness p r o f i l e according to model described i n Chapter 5 . 1 ° 2 28. Density of saturated l i q u i d versus temperature 120 29. Density of saturated CO2 vapor versus temperature . . . 121 30. Thermal conductivity CO2 versus temperature 122 31. Surface tension of l i q u i d (X^ versus temperature . . . 123 32. Dynamic v i s c o s i t y of CO2 versus temperature 124 33. Latent heat of CO2 versus temperature 125 34. Vapor pressure of C0 9 versus temperature 126 i x ACKNOWLEDGEMENTS I would l i k e to express my sincere thanks and appreciation to my supervisor, Dr. E.G. Hauptmann for h i s help and advice throughout the project. Also I would l i k e to thank Dr. D.W. McAdam for sharing with me hi s experimental data and his knowledge of the subject. This work was sponsored by the National Research Council of Canada. NOMENCLATURE a distance from o r i g i n to source and sink, (m) a c radius of c o l l a r , (m) A aspect r a t i o of body b width of body, (m) c thickness of a c o l l a r , (m) C phase v e l o c i t y , (m/s) d width of a c o l l a r , (m) FB body force, (N/m) F P pressure force, (N/m) F s shear force, (N/m) F c surface tension force, (N/m) F t h thermocapillary force, (N/m) Fr g Froude number, — — u _ (Pg) 1 2 2 a c c e l e r a t i o n due to gravity, (m/s ) h f i l m thickness in, p r e s e n c e of waves, (m) V latent heat of vaporization, (J/kg) average thickness of a wavy f i l m , (m) h p wave peak thickness, (m) k u 2TT . -1. wave number, — , (m ) k t thermal conductivity, (W/m ,0C) K t o t a l radius of curvature, (m) 1 heated plate length, (m) L length of body, (m) source and sink strength, (m /s) 2 pressure,(N/m ) 2 pressure at the bottom of the plate,(N/m ) heat f l u x , (W/m2) dimensional coordinate (m) dimensionless coordinate radius of curvature of dry patch, (m) Reynolds number, ^ ~ input Reynolds number time, (s) temperature, (°C) in t e r f a c e temperature, (°C) pla t e temperature, (°C) dimensional v e l o c i t i e s , (m/s) u/u X dimensionless v e l o c i t y /x Q/p , (m/s) dimensionless v e l o c i t i e s average v e l o c i t y , (m/s) reference v e l o c i t i e s , (m/s) i maximum v e l o c i t i e s at y = 6, (m/s) v e l o c i t y at average stream cross-section hp, (m uniform v e l o c i t y equal to the average v e l o c i t y a parabolic p r o f i l e , (m/s) uniform v e l o c i t y equal to the maximum v e l o c i t y a parabolic p r o f i l e , (m/s) X l l -2 We Weber number, pu 6/o" x,y,z dimensional coordinates, (m) x*,y*,z* dimensionless coordinates x g coordinate of stagnation point, (m) y + dimensionless coordinate, x v ZQ width of con t r o l volume, (m) Greek Symbols a angle of i n c l i n a t i o n of the p l a t e , (°) a(Chapter 4) r a t i o of t o t a l momentum f l u x i n the x d i r e c t i o n to input momentum f l u x 6 f i l m thickness, (m) 6 c r i t i c a l f i l m thickness, (m) c 6̂  maximum f i l m thickness, (m) 6 + dimensionless f i l m thickness, -^y- T flow rate, (kg/m«:s) I" c r i t i c a l flow rate, (kg/m's) I" flow rate that evaporates, (kg/m«s) X wavelength, (m) y dynamic viscosity(kg/m-s) V kinematic v i s c o s i t y , (m/̂ s) to wave c i r c u l a r frequency, ^ ~ , (s ̂ ) 3 p l i q u i d density, (kg/m ) 3 p^ vapour density, (kg/m ) a surface tension, (N/m) x i i i 8 contact angle, (°) 6 s t a t i c advancing contact angle, (°) 9̂  s t a t i c receding contact angle, (°) K 9' contact angle at the leading edge of a drop, (°) 9' contact angle at the rear, (°) R T wave period, (s) 2 T. shear stress at the i n t e r f a c e , (N/m ) i 2 Tn shear stress at the w a l l , (N/m ) Supra Index min minimum av average max maximum 1 1. INTRODUCTION In systems cooled with a f l u i d which can change phase, the w a l l - t o - l i q u i d heat transfer c o e f f i c i e n t i s high where the w a l l and l i q u i d are i n d i r e c t contact. Where t h i s contact i s not maintained the heat transfer c o e f f i c i e n t may be considerably reduced causing a r i s e i n the surface temperature. The r i s e i n temperature can lead to melting or rupture of the metal surfaces. As the w e t a b i l i t y of the surface by the l i q u i d i s of a c r i t i c a l importance, dry spot formation must be prevented. There i s l i t t l e information concerning the d e t a i l e d formation of dry spots or dry patches. One mechanism of dry patch formation i s when the free surface i s subject to a l o c a l temperature v a r i a t i o n (Figure 1). Surface tension, a, of the l i q u i d changes from point to point and tangential forces may be exerted producing a bulk movement of the l i q u i d from lower to higher surface tension regions (Marangoni e f f e c t s ) . For example when f i l m thickness i s not uniform(due to waves) the surface temperature i s higher i n thinner regions and lower i n thicker regions. Generally, for l i q u i d s surface tension decreases with temperature and there i s a net motion of l i q u i d from thinner regions (low o) to thicker regions (high a) and f i l m destruction can occur. Even i n case of evaporation at the i n t e r f a c e , some departure from the thermodynamic equilibrium i s p o s s i b l e , therefore, free surface temperature can be d i f f e r e n t from saturation temperature at that pressure, and Marangoni e f f e c t s can s t i l l be present. In the case of mass tra n s f e r , changes i n concentration produce v a r i a t i o n s of surface tension which i n turn can produce motion on the f l u i d leading to f i l m destruction. Nucleation (bubble formation) i s another possible cause of f i l m break-up. Film thickness underneath each bubble i s reduced and l o c a l l y destroyed. If the bubble stays on the wall f or a s u f f i c i e n t l y long time, the dry area at the base of the bubble spreads. Liquid f i l m break-up can be present i n b o i l i n g water-cooled reactors (Figure 2). This phenomenon i s c a l l e d dryout. When a sub- ^cooled f l u i d enters a heated tube, a flow regime c a l l e d annular flow can appear (Figure 3). In t h i s regime the w a l l of tube i s covered by a t h i n l i q u i d f i l m , while the centre of the tube has vapour moving at a higher v e l o c i t y . I f the f i l m breaks up, dryout occurs with the consequent decrease i n the heat transfer c o e f f i c i e n t . As most reactor systems operate with constant heat f l u x , the resultant temperature r i s e i s often s u f f i c i e n t to melt the f u e l rod or induce corrosion. McAdam [1] observed f i l m break-up i n a t h i n l i q u i d f i l m of flowing under gravity over a heated plate. Thermocapillary e f f e c t s i ( v a r i a t i o n of o" with T) and evaporation at the i n t e r f a c e were the main mechanisms for the formation of dry patches i n h i s t e s t s . Nucleation did not occur i n these experiments because the system was not s u f f i c - i e n t l y superheated to produce bubbles. In order to better understand how to prevent formation of dry patches, some workers have t r i e d to describe the flow behaviour 3 around those dry areas. Some mathematical models are a v a i l a b l e i n the l i t e r a t u r e describing the flow behaviour near the vertex of a dry patch (Figure 4). No consideration of the l a t e r a l flow, that i s , the flow that d i v e r t s around the dry patch i s taken into account. The " s t a t i c " s t a b i l i t y of a dry patch i s expressed i n terms of a balance of forces at the upstream edge of the dry patch: a pressure or stagnation force tends to rewet the dry patch, and the surface tension and thermocapillary forces cause the dry patch to spread. The scope of the present study i s to c r i t i c a l l y examine e x i s t i n g models to see how they agree with recent experimental data recorded by McAdam, and make some suggestions f o r a better understanding of the flow behaviour i n order to prevent the formation of dry spots. As a f i r s t step, Chapter 2 includes a review of l i t e r a t u r e of those aspects re l a t e d to dry patch s t a b i l i t y . Chapter 3 consists of a s e l e c t i o n of c r i t e r i a for s t a b i l i t y of dry patches. McAdam's data are analyzed and the d i f f e r e n t c r i t e r i a are checked. The r e s u l t s and l i m i t a t i o n s of the previous models are discussed. As a f i r s t m odification to the previous models a new model i s proposed to evaluate the magnitude and consequence of the flow that diverts around the dry patch. The model proposes an i d e a l flow around a body whose shape i s s i m i l a r to the shape of a dry patch so that v e l o c i t i e s and pressure forces can be evaluated. McAdam's r e s u l t s show that the contact angle and f i l m t h i c k - ness change with time even though the dry patch i s stationary. Surface waves are responsible for th i s v a r i a t i o n and can produce a range of contact angles and f i l m thicknesses before the patch rewets or r e t r e a t s . This e f f e c t i s not accounted for i n previous models. The forces acting on the upstream edge of the dry patch i n the presence of waves are evaluated using the momentum theorem i n Chapter 4, assuming that the wave pattern can be described by Kapitza's a n a l y s i s . Chapter 5 presents an approximate s o l u t i o n for the increase i n thickness which occurs j u s t upstream of a dry patch. For s i m p l i c i t y the s i t u a t i o n considered i s a s l i g h t l y i n c l i n e d p l a t e , with stagnation pressure balancing the increase i n hydrostatic pressure a r i s i n g from the increase of f i l m thickness near the dry patch. For a v e r t i c a l p late, increase i n stagnation pressure i s balanced by changes i n curvature of the free surface. Although contact angles on an i n c l i n e d p late are probably lower than those for an equivalent flow on a v e r t i c a l p l ate the differences are expected to be small. Thus estimates of pressure forces from an i n c l i n e d model may be used as a rough check against McAdam's data. A c a l c u l a t i o n of type i s done and r e s u l t s are discussed. 5 2. REVIEW OF LITERATURE 2.1 General The s t a b i l i t y of a dry patch has been analyzed by several investigators to predict conditions under which the dry patch remains sta b l e . This c r i t e r i o n i s usually obtained by a force balance at the upstream stagnation point of a dry patch. The main forces are pressure force, surface tension force, and thermocapillary force developed as a r e s u l t of the v a r i a t i o n of surface tension with surface temperature. This approach takes intb account the contact angle, the flow behaviour far from the dry spot, the thickness of the f i l m , changes i n temperature of the free surface and the ph y s i c a l properties of the liquid-vapour system. D e t a i l s of the flow and temperature f i e l d s around the dry spot are not considered, except i n the v i c i n i t y of the "nose" of the dry patch. Some papers deal with experimental observations of dry patches and t h e i r s t a b i l i t y [2], [3], [4] others [5], [6] attempt t h e o r e t i c a l d e s c r i p t i o n s , while some workers [7] [8] have tested the c r i t e r i a against experimental data. In what follows, the f l u i d - f l o w regions are roughly c l a s s i f i e d for con- venience as t h i n f i l m , laminar, turbulent, and wavy flow. 2.2 Thin Film Flow To understand the previous models i t i s necessary to review flow i n thi n f i l m s . Because of t h e i r many i n d u s t r i a l a p p lications 6 they are a subject of continuous study. Various regimes of t h i n f i l m flow can be distinguished and because of the presence of a f r e e surface, the c l a s s i f i c a t i o n of the regimes must take i n t o account surface tension and viscous e f f e c t s . Thus Reynolds number Re, Weber number. We, and Froude number, Fr, are the most us e f u l dimensionless p h y s i c a l quantities for flow pattern c l a s s i f i c a t i o n . Film flow can be subdivided into laminar or turbulent regimes depending on whether Re i s smaller or larger than a c r i t i c a l Re. In addition, the free surface may be smooth or wavy depending on Fr and We. I t has been shown [9] that gravity waves f i r s t appear i n a water f i l m when Fr = 1-2 and c a p i l l a r y surface e f f e c t s become important i n the neighbourhood of We = 1. Thin f i l m flow can be c l a s s i f i e d as follows [10]: laminar without r i p p l i n g Re < 1 to 6 laminar with r i p p l i n g 6 < Re < 250 to 500 turbulent Re > 250 to 500 . Studies [11,12] that are s t i l l the subject of discussion, i n d i c a t e Re = 0 for inception of laminar i n s t a b i l i t y during flow down a v e r t i c a l surface. I t should be noted that the presence of waves does not necessarily mean the flow i s turbulent. 2.2.1 Laminar Film Flow When a l i q u i d f i l m flows under gravity on a v e r t i c a l p late shear forces at the i n t e r f a c e can be neglected. Nusselt [13] obtained the f u l l y developed v e l o c i t y p r o f i l e f o r a steady, viscous flow with no shear or wave motion at the free surface as 7 U = U T ( 6 " 2) ' U T = U ( Y = 6 ) • • • - t 2 - 1 ) 6 Under these approximations the flow r a t e r = • • • • ( 2 • 2 ) These v e l o c i t y p r o f i l e s were confirmed by Dukler [14], Cook and Clark [15] among others. If the width of the flow channel i s f i n i t e , a d d i t i o n a l terms a r i s i n g from viscous edge e f f e c t s , drag and c a p i l l a r y forces modify the former s i m p l i f i e d formulas. 2.2.2 Turbulent Film Flow Although no theories e x i s t on t h i n f i l m flow f o r the turbulent regime, Dukler and Bergelin [14] developed a s i m p l i f i e d r e l a t i o n between f i l m thickness and pressure drop, assuming that the u n i v e r s a l v e l o c i t y p r o f i l e developed by Von Karman for pipes applies to the l i q u i d phase i n two-phase f i l m flow. Their r e s u l t s are + + + u = y , 0 < y <• 5 (laminar sublayer) u + = -3.05 + 5.0 £ny +, 5 < y + < 30 (buffer layer) . . . (2.3) u + = 5.5 + 2.5 &ny +, 30 < y + < 6 + (turbulent zone) where + , x u = u/u + x . y = yu /V + x 6 = Su /v (dimensionless f i l m thickness) and x , , N l / 2 u = (T./p) ( f r i c t i o n v e l o c i t y ) , By i n t e g r a t i n g the dimensionless v e l o c i t i e s over the f i l m thickness Dukler and Bergelin found the r e l a t i o n s h i p between Re and 6 + to be Re = <5+(3 + 2.5 in 6 +) - 64 (2.4) For flow down a v e r t i c a l p l a i e without shear at the i n t e r f a c e [14] 1/2 .3/2 (2.5) An expression [16] for the v e l o c i t y p r o f i l e i n the turbulent zone that f i t s the experimental data w e l l and i s easier to handle i s u = u T(y/5) 1/7 . .(2.6) 2.2.3 Wavy Film Flow Between the smooth laminar and turbulent regimes the flow i s characterized as laminar flow with surface waves. This region has been studied by many workers [11], [17], [18]. Determining the Re for the inception of wavy flow i s the subject of study of many investigators [19]. For v e r t i c a l plates, the onset of waves occurs at very low Re. The Kapitza [17] treatment of wavy f i l m flow predicts an inception Re = 5.8 for water on a v e r t i c a l w a l l , but i s only v a l i d 9 for long wavelengths with respect to mean f i l m thickness. Benjamin [11] presented a d e t a i l e d treatment of the onset of a two-dimensional (2-D) i n s t a b i l i t y i n t h i n f i l m flow considering c a p i l l a r y e f f e c t s . He found that f o r v e r t i c a l p l a t e the inception Reynolds number i s equal to zero. Castellana and B o n i l l a [12] also found Reynolds number i s equal to zero f o r wave inception. For the case of a wavy f i l m flow on a v e r t i c a l wall,Levich [18] has shown that the complete Navier-Stokes equations can be reduced to the f a m i l i a r form of the boundary layer 2 ---*-+ v 9 + g (a) p dx „ 2 ° 9 y (b) . . . .(2.7) 8 x - + 8 y - = ° ' ( C ) At y = h(x) the boundary conditions at the free surface must be s a t i s f i e d . . . .(2.8) 3 u 9ju du 3 t 3 x 3 y ¥ = ° 3 y 9 u 3 y At the s o l i d w a l l y = 0 u = v = 0 .(2.9) (2.10) \ 10 As v ^ 3h at at the free surface the continuity equation can be expressed 3h 3t _3_ 3x (fudy) .(2.11) Kapitza was the f i r s t to attempt a s o l u t i o n of t h i s system. In h i s analysis the term v i n equation (2.7a) was omitted. I t was dy assumed that the v e l o c i t y d i s t r i b u t i o n i n the f i l m could be given by the usual parabolic expression of the form Here u(x,t) i s the average v e l o c i t y over the cross-section and i s a function of the p o s i t i o n along the f i l m and time. Kapitza assumed that f i l m thickness could be represented by h = h^ + h^ i|>, where h^ i s the average f i l m thickness and h ^ i s the d e v i a t i o n of the surface from that average. He also assumed that for undamped waves, qua n t i t i e s l i k e f i l m thickness and average l i q u i d v e l o c i t y are functions of the argument (x - Gt) where C i s the phase v e l o c i t y of the waves. With these conditions, by s u b s t i t u t i n g u i n Equations (2.7a) and (2.11) Kapitza obtained a f i r s t , undamped approximate s o l u t i o n for the thickness h^ c l o s e l y r e l a t e d to Nusselt's formula for laminar flow. In t h i s approximation the phase v e l o c i t y i s C = 3 u^ where UQ i s the v e l o c i t y at average stream cross section h^. In a second approximation Levich [18] followed Kapitza's method and c a r r i e d out an analysis of the wavy flow to determine the condition under which the energy supplied 2 u .(2.12) 11 to the f i l m by gravity forces was balanced by the d i s s i p a t i o n of energy by viscous forces. In t h i s context i t was found ti = h (1*46) h : thickness of a peak p U D 0 P 0 p C = 2.4 u h -!r- = 0.93 h_: mean f i l m thickness with waves o 0 6 : smooth f i l m thickness at the same flow rate. The expression f o r the v e l o c i t y f i e l d was found to be ? 2 u(x,y,t) = 3 u n [ l + 0.6 sin(kx-ajt)- 0.3 s i n (kx-wt) ] - ° h 2h 2 . . . .(2.13a) 2 3 v(x,y,t) = - 1.8 u„k cos(kx-wt)[l-sin(kx-wt)](^ — ^ ) • ° 2 h 6h 2 . . . .(2.13b) Both Kapitza's and Levich's r e s u l t s are v a l i d under the condition that • A > 13.7 hp. This corresponds to Re - 50 f o r a v e r t i c a l water or C0 2 f i l m . 2.3 S t a b i l i t y of Dry Patches 2.3.1 Steady Case Bankoff [20], postulated two stages of f i l m break-up. An i n i t i a l stage i n which t h i n spots are produced i n the f i l m by growth 12 of an unstable surface wave, and a second stage, "break-up stage," i n which l i q u i d i s displaced from the s o l i d surface. In the f i r s t stage w e t a b i l i t y properties of the system measured through the contact angle do not play a r o l e , while i n the second stage the contact angle i s the major factor i n f l u e n c i n g s t a b i l i t y of dry spots. Hartley and Murgatroyd [ 5 ] have considered the s p e c i f i c case of an isothermal f i l m flowing under gravity i n the presence of a dry patch. They considered a patch of the shape shown i n Figure 4 and assumed that the l i q u i d i n the c e n t r a l stream segment, AB, stagnates, while the remain- ing flow follows the stream l i n e s shown. A cross s e c t i o n of the l i q u i d near the dry patch i s shown i n Figure 4. They considered that the f r e e surface acts l i k e a membrane and a dry patch i s stable when the meniscus (curved region of the film) i s i n mechanical equilibrium. In t h e i r analysis they equated "upstream" surface tension force (due to the i n t e r a c t i o n of the liquid-vapour system with the r e s t of the l i q u i d and with the plate) to "downstream "pressure force (that force required to bring the l i q u i d at B to r e s t from i t s v e l o c i t y i n the undisturbed f i l m at A). The upstream f o r c e , assuming that curvature of the meniscus (Figure 4,plane x,y) i s larger than the curvature of the patch (Figure 4, plane x,z)(true f o r t h i n f i l m s ) , has two components; adz represents the force that the r e s t of the f l u i d applies to the membrane, and O" cos 6 dz represents the force that the p l a t e applies to the membrane (8 i s the contact angle, see Figure 4 ) . The t o t a l up- stream surface tension force for an isothermal flow i s O"(l-cos 9) dz. The downstream force was found by Hartley and Murgatroyd applying the B e r n o u l l i equation to the t h i n viscous l i q u i d flowing over a v e r t i c a l plate 13 P + I u 2 ( ) = ! B + I V B 2 , V = 0 . . . . ( 2 . 1 4 ) p 2 J p 2 r6 2 F = p ^ - dy , . . . .(2.15) P J 0 2 where F = pressure force per unit width. P No explanation i s given about the v a l i d i t y of a Bernoulli-type equation. Probably the workers assumed that the work done by the gravity force equals the work done by the shear force. This i s a good assumption far from the dry patch but not i n the v i c i n i t y near the dry patch where the f l u i d i s slowed down. Hartley and Murgatroyd evaluated the Equation (2.15) f o r an isothermal, steady laminar f i l m with no shear at the i n t e r f a c e . The v e l o c i t y p r o f i l e i s parabolic (Equation 2.1) and the equilibrium of forces can be expressed i n terms of thickness 6 as ^ P 3 l y « 5 = 0(1 - cos 9) . . . . .(2.16) They found a minimum, or c r i t i c a l f i l m thickness of (2.16) ~i o 1/5 r y n2/5 .„ . n. 5 c = x . 7 2 [ o - c 1 - ^ 6 ) ] [£l • • • • -(2.17) In terms of mass flow rate this becomes, r = 1 . 6 9 ^ ) 1 / 5 [ 0(1-cos 9) ] 3 / 5 c v pg y 14 According to t h i s formula i f the thickness of the f i l m i s larger than 6^, the patch rewets, while f o r f i l m thicknesses smaller than &c dry patches can be established. The existence of a dry patch i s an e s s e n t i a l condition i n the analysis as the f i l m could quite possibly be thinner without breakdown of the f i l m i f the surface i s already wetted. As another a p p l i c a t i o n of t h e i r c r i t e r i o n Hartley and Murgatroyd considered the case of a laminar or turbulent f i l m motivated by surface shear only. They also postulated that the sum of the k i n e t i c and surface energy of the unbroken f i l m was minimized at the c r i t i c a l thickness for s t a b i l i t y . This gives a d i f f e r e n t c r i t i c a l thickness 6^ c a l l e d "minimum thickness from power c r i t e r i o n " although (except for the contact angle) the same parameters are involved 6^ = 1.34 ( o 7 p ) 1 / 5 ( y / p g ) 2 / 5 Bankoff [21] assumed that the l i q u i d f i l m w i l l break up into r i v u l e t s when the t o t a l mechanical energy ( k i n e t i c plus surface) per unit area i s the same i n the two configurations. For low flow rates, energy considerations favour a break-up into p a r a l l e l r i v u l e t s . The minimum f i l m thickness 6 £ found i s a function of the contact angle formed between the r i v u l e t and the s o l i d surface. 6 = c 1.72 ( ^ ) 1 / 5 ( y / p g ) 2 / 5 f ( 9 ) .(2.18) 15 The function f(9) i s a function of contact angle and geometry of the of forces as contact angle at the stagnation point. It i s i n t e r e s t i n g to note that contact angle was not experi- mentally measured i n these works, so Hartley and Murgatroyd [5] could only work with the power c r i t e r i o n . They used other workers' experi- mental r e s u l t s to evaluate t h e i r a n a l y t i c a l c r i t e r i o n , with varying degrees of success. Hewitt and Lacey [4] designed an experiment to s p e c i f i c a l l y test dry patch s t a b i l i t y . They found that the upstream surface tension force was about eight times the pressure force using a s e s s i l e drop value of 8. Ponter et at. [7] presented a model to predict l i q u i d f i l m breakdown i n the presence of mass transfer. They assumed a dry patch of the same shape as Hartley and Murgatroyd (Figure 4). If the dry patch i s stable surface force along AB must balance the f l u i d force brought about by the loss i n momentum i n bringing the l i q u i d to r e s t at B. According to Ponter the balance of forces i s represented by For a steady, laminar flow without waves or shear at the i n t e r f a c e r i v u l e t s . This contact angle i s not subject to the same conditions u dy = c ( l - c o s 8) J 0 f 5 2 .(2.19) u (2.1) Substitution into Equation 2.19 gives 16 9 o ^ p • S- 2 L- = a ( l - cos 6) . . . . . ( 2 . 2 0 ) Note that the l e f t hand side of t h i s equation i s two times that evaluated by Hartley and Murgatroyd (see 2.16). C r i t i c a l thickness i s then given by * = L 4 9 5 [g ( 1 - C n ° s 9 ) ] 1 / 5 [ U / P g ] 2 / 5 . . . . . ( 2 . 2 1 ) c p or i n terms of the mass flow rate (Equation 2.2) r c/y = 1.116 ( i - cos e ) ° - 6 ( £ | V / 5 . c y g N r / u defines the minimum l i q u i d flow rate or minimum wetting rate to c sustain a stable dry patch. Ponter [7] measured mass flow rate, contact angle and surface tension during absorption to te s t the v a l i d i t y of the model. Close agreement was found between contact angles measured under flow conditions and those measured from a s e s s i l e drop under the same conditions. Data for water films i n the presence of a saturated a l c o h o l - a i r mixture are reported and show good agreement between the observed and predicted minimum wetting rates. Zuber and Staub [ 6 ] extended the Hartley and Murgatroyd analysis [ 5 ] to the case with heat trans f e r and proposed two a d d i t i o n a l forces due to thermal e f f e c t s ; thermocapillary force, due to a non- uniform temperature at the free surface and vapour thrust, due to change i n momentum fl u x experienced by the f l u i d p a r t i c l e s when they pass from l i q u i d to vapour phase. They 17 assumed a l i n e a r temperature p r o f i l e and approximated the meniscus by a wedge shape to s i m p l i f y c a l c u l a t i o n s . - Their conditions f o r a stable dry shape to s i m p l i f y c a l c u l a t i o n s . Their conditions for a stable dry patch i s _p_ ,gAp,2 .5. _ n „ ~ v . da . Q_ 0 , n , Q ,2 Ap 2 ». x 15 W 6 c " ^ C 1 ^ 0 3 9 ) + dT 6 c k C O s 9 + P v ( p ^ r } p C ° S 9 6 c ' t V x,g where Ap = p - p . ' ' ' ' ( 2 - 2 2 ) The minimum f i l m thickness 6^ w i l l permit the wetting of the e n t i r e surface. In t h i s analysis as w e l l as the previous ones i t appears that only one value of 6 i s possible for each value of 8. McPherson [22] deals with a h o r i z o n t a l f i l m with a shear stress at the i n t e r f a c e . He also presents the p i c t u r e of a meniscus as a p h y s i c a l surface, and equilibrium of forces at the vertex of the dry patch implies that the patch may be "quasi-stable". Unbalance of forces corresponds to a s i t u a t i o n where the patch grows either upstream or downstream. He includes the forces considered by Zuber and Staub [6], as w e l l as vapour shear at the f i l m vapour i n t e r f a c e , hydrostatic head from the l i q u i d f i l m , and drag at the small step i n the f i l m . The heated length upstream of the dry patch edge (see Figure 5) i s divided into a region(1-m) over which the f i l m i s being decelerated and diverted with no change i n thickness and a region (m) over which evaporation reduces the thickness to zero. The v e l o c i t y p r o f i l e i n the evapora- t i o n length (m) does not change and i s i d e n t i c a l to that at m. To evaluate the vapour thrust force he considered a non-constant heat fl u x due to conduction i n the wall from the dry to the wet region. Thermocapillary e f f e c t i s considered together with surface tension force i n the expression 18 F a = 0 ( T i ) _ a ( T p ) c o s 0 * ' " - ( 2* 2 3 ) where a(T^) i s the surface tension evaluated at the i n t e r f a c e and 0(T^) i s the surface tension evaluated at the point where the l i q u i d , vapour and plate are i n contact. The resultant force acting on the vertex of the dry patch i s the sum of the stagnation force, vapour thrust, shear force, surface force, body force, and drag force. McPherson applied h i s analysis to a known case of dryout taken from experiments conducted at Harwell [22]. He concluded that f o r a steam-water system the most important forces are upstream surface tension force F_, (thermal e f f e c t s included), downstream deceleration F and inter- a p f a c i a l shear forces F s F = F + F + F a p s If F = 0 stable dry patches F > 0 f i l m rewets F < 0 dry patch grows upstream. Using a l i n e a r v e l o c i t y p r o f i l e F = o-(T.)-a(T ) cos0+ | p 6 ( u 2 - u T 2 ) + T . ( ^ ) ( l - — ) . .'.(2.24) l p j L°° Le l z uL°° McPherson also proposed a mechanism for maintaining a "quasi-stable" dry patch. According to McPherson when a dry patch i s f i r s t formed the upstream surface force w i l l decelerate the approaching f i l m causing a 19 downstream force. I f F i s n e g l i g i b l e , F must balance F . As no P s a measurements of contact angle were recorded at Harwell, McPherson shows that for every possible contact angle there i s some shear force which w i l l balance the surface force (or shear plus pressure force i n case the l a t t e r i s not n e g l i g i b l e ) . He suggested that a s e l f - a d j u s t i n g process w i l l occur, but i f upstream forces are larger than downstream forces f o r a l l degrees of deceleration, the patch w i l l move upstream. On the other hand, once a "quasi-stable" dry patch i s formed, rewetting i s only possible through some perturbation of the system such as an increased f i l m flow or droplet deposition r a t e . f i l m flow down a v e r t i c a l plate where a dry patch has already formed. The flow i s assumed to be undisturbed by the dry patch except i n a t h i n region around i t s boundary c a l l e d t h e " c o l l a r T h e thickness of the three times the thickness of the l i q u i d f a r from the patch. I t i s proposed that the c o l l a r resembles a boundary layer i n which surface It i s i n t e r e s t i n g to point out that the pressure balance at the apex of dry patch i s In a recent paper, Wilson [23] considers an isothermal l i q u i d c o l l a r can be predicted and according to the theory developed i s almost tension forces replace the f a m i l i a r viscous drag. 1 s i n 0 o(~ R—) a c • R + 2 pgd , .(2.25) where a c = radius of c o l l a r c .= thickness of c o l l a r R = radius of curvature of dry patch boundary d = width of c o l l a r 20 Wilson assumed a c o l l a r whose cross s e c t i o n i s an arc of a c i r c l e making the appropriate angle with the s o l i d boundary so that d, a , and c can c be r e l a t e d . Equation 2.25 states that pressure inside the meniscus, o i s represented by two terms. The f i r s t , ( r i g ht hand side Equation 2.25) represents stagnation pressure assuming a laminar v e l o c i t y p r o f i l e (in terms of a force i s equal to Ponter's pressure force) and the second term, represents the pressure due to a body force. Wilson's paper i s the f i r s t to formally analyze an increase i n thickness near the dry spot. This bulge or c o l l a r was observed experimentally by Hewitt [4], Ponter [7] and McAdam [1]. i 2.3.2 Unsteady Case As was explained i n the review of t h i n l i q u i d f i l m s , surface waves are present f o r v e r t i c a l flows at a l l Re. The e f f e c t of waves on the process of rewetting has only recently been considered. Hsu and Simon [24] stressed the importance of waves i n producing temperature differences at the free surface. These i n turn are capable of sus t a i n - ing steady surface tension force differences which d i s t o r t the f i l m . and make i t thinner, leading to formation of a dry patch. Simon and Hsu i n another paper [25] conducted an experimental and a n a l y t i c a l i n v e s t i g a t i o n of the breakdown due to heating of a f a l l i n g subcooled l i q u i d f i l m . Two flow regimes, the c a p i l l a r y wave and r o l l wave regimes were found. C a p i l l a r y waves are weak so that i f a dry patch i s formed the surface remains dry and i s not rewetted because pressure force i s not large enough to overcome surface forces. They found a 21 constant value of We during experiments with water and water-glycerol flow down a heated v e r t i c a l p l a t e . Simon and Hsu [25] assumed that a constant We i s the quantity c o n t r o l l i n g s t a b i l i t y of a dry patch and found a c r i t i c a l thickness a f t e r rearrangement of We: 6 = 0.66 [ - ] 1 / 5 [ u / p g ] 2 / 5 . . . . .(2.26) C Q This formula does not include contact angle or heating e f f e c t s . I t i s consistent with the assumptions made by Simon and Hsu [25] that minimum f i l m thickness when breakdown occurs i s independent of the process through which t h i s f i l m thickness i s reached and i s the same as the isothermal case. The r o l l wave regime occurs at a higher flow r a t e and i n t h i s regime the f i l m breakdown o s c i l l a t e s between a dry and a wetted surface condition. Dryout occurs at wave troughs, however, wave crests have enough momentum to rewet a dry patch. According to Simon and Hsu, the rewetting process i s determined by heat f l u x and other parameters not involved i n the capillary-wave regime. Thompson [8] t r i e d to fi n d the e f f e c t of surface K a v e s i n t n e presence of a dry patch by evaluating the pressure force using l i q u i d f i l m thickness at the wave hollow, average, base of surface waves, and wave peak. He considered a water f i l m driven by steam flow and evaluated the surface, thermo- c a p i l l a r y and vapour thrust forces with Zuber and Staub's [6] formulae for three dryout conditions. 22 The i n t e r f a c i a l shear term was obtained from McPherson's equation by assuming that stagnation occurs over a distance 1 equal to t o t a l heated length (see Figure 5). Pressure force i s found by adopting e i t h e r a logarithmic or l i n e a r v e l o c i t y p r o f i l e . Thompson's re s u l t s show that vapour thrust, body and drag forces are several orders of magnitude smaller than the pressure and surface tension forces, while i n t e r f a c i a l shear and thermocapillary forces are only one order of magnitude smaller. He concludes that s t a b i l i t y of a dry spot depends on the r e l a t i v e magnitude of the pressure and surface tension forces (as would be the case f o r an isothermal flow). Although no stable dry patches were observed, i n at l e a s t two of three cases examined his c a l c u l a t i o n s give F^ > F^ (evaluated for an average thickness). However, Thompson [8] found that F^ using the peak thickness i s an order of magnitude larger than F . In t h i s case the dry patch i s unstable, as was observed experimentally. The main conclusions of h i s paper are: 1. s t a b i l i t y of a dry patch w i l l be governed by F^ and F , 2. e f f e c t i v e k i n e t i c energy of surface waves i s much greater than the average k i n e t i c energy of the f i l m . Stable dry patches cannot e x i s t when surface waves are present. 3. contact angle of a stationary i n t e r f a c e under flow con- d i t i o n s i s nearly the same as the s t a t i c angle of a s e s s i l e drop, and 4. nucleating i s the main thermal i n s t a b i l i t y . 23 McAdam [1] makes some observations about the r o l e of waves i n the r e - wetting process: although waves were present i n most of h i s experiments, t h e i r e f f e c t was not always to rewet the dry patch. He presented a sequence of photographs showing that waves caused d i s t o r t i o n of the stagnation point and narrowing of the dry patch but did not r e s u l t i n complete rewetting. Mariy et al. [26] also stress the importance of waves i n the rewetting process i n a study of the motion of the 3-phase front (point where the l i q u i d , the vapour and the s o l i d are in,contact). They analyzed flow i n a h o r i z o n t a l tube, with a t h i n l i q u i d f i l m driven by i t s own vapour, with the following assumptions: the dry patch behaves l i k e the edge of an annular l i q u i d f i l m at the end of the 2-phase flow; the f i l m front i n the absence of waves i s stable and of thickness 6 , c the heat supplied being j u s t s u f f i c i e n t to evaporate the incoming l i q u i d at the t r i p l e phase fron t ; for each contact angle there i s a unique value of the c r i t i c a l thickness &c, (Equation 2.15) and rewetting occurs i f the f l u i d becomes thicker than 6^. They considered the flow per- turbed with waves c a l l e d "harmonics". The flow model consists of a mass of l i q u i d moving at the average v e l o c i t y , plus an excess of l i q u i d moving as a r i g i d body at the phase v e l o c i t y (Figure 6). Mariy et al. [26] suggest f i l m thickness increases at the 3-phase front when waves approach , so downstream forces overcome up- stream forces and the t r i p l e phase front moves forward. The subsequent advance and r e t r e a t of the front depends on whether the excess l i q u i d i s evaporated before the next wave a r r i v e s , i n proper circumstances 24 the front w i l l move forward and rewet the surface. Continuity and energy equations based on the model of "waves" are solved to determine movement of the stagnation point. Dryout data recorded by Barnet et at. [27] and Thompson and Macbeth [28] were used to show the dependence of positions of the stagnation point on parameters such as flow rate and heat f l u x . Their r e s u l t s are shown i n Figure 7. When surface heat f l u x i s less than or equal to the dryout value excess l i q u i d due to the "waves" i s not evaporated and the 3-phase front moves forward. If heat f l u x i s high enough to evaporate more than the incoming f l u i d the 3-phase front recedes. No general c r i t e r i o n can be extracted from the i r paper, as movement of the 3-phase front i s strongly dependent on the model of waves used. Mariy's [26] work w i l l not be considered elsewhere i n the present study. 2.4 Contact Angle A free l i q u i d drop or a drop i n contact with a s o l i d w i l l assume a shape which minimizes the free energy (Helmholtz function) of . the system. In absence of gravity t h i s i s equivalent to minimizing the surface area of the drop [29]. For a two-dimensional system (Figure 8a), the minimization of the free energy i s represented by Young's equation: C 0 cos 0 = O - o „ . . . . ( 2 . 2 7 ) &g sg si where 0\.. = i n t e r f a c i a l energies or surface tension ig = r e f e r to liquid-vapour 25 sg = r e f e r to solid-vapour si = r e f e r to s o l i d - l i q u i d i n t e r f a c e s 9 = angle of contact. Equation (2.24) i s s a t i s f i e d when surface tensions vary smoothly and the surface of the s o l i d i s smooth and r i g i d . In a r e a l system these conditions are d i f f i c u l t to s a t i s f y . for a given system. However, experimental evidence suggests a c h a r a c t e r i s t i c of wetting i s the a b i l i t y of a l i q u i d drop to have many d i f f e r e n t stable angles on a s o l i d surface (hysteresis e f f e c t ) . For example, i f a droplet of the same l i q u i d i s added to drop "a" to make a larger drop "b" (or withdrawn to form droplet "c") the base of the drop usually stays constant and the contact angle changes (Figure 8b). Two r e l a t i v e l y reproducible angles are the larger angle ( i n drop " b " ) , c a l l e d advancing s t a t i c angle and the smaller (drop i n " c " ) , c a l l e d the receding s t a t i c angle. The d i f f e r e n c e 9̂  - 9̂  i s c a l l e d contact angle h y s t e r e s i s . I t has been shown [29] that for a drop on an i n c l i n e d p late (Figure 9) equilibrium of forces i s represented by the equation Young's equation predicts one and only one stable contact angle mg s i n a a angle of i n c l i n a t i o n of the plate w width of drop 0 R contact angle at rear contact angle at the leading edge. 26 The angle a can be increased as long as 6' ± 8 and 6* ± 8 . However, R R A A when 8^ = 0 R (minimum angle) and 8^ = 8^ (maximum angle), equilibrium of forces cannot be s a t i s f i e d and the drop w i l l r o l l . If there were no hysteresis (unique value of contact angle) the drop would r o l l on the plate for any angle "a," [29]. In the case of a dry patch the l i q u i d meniscus can be assimilated to a large drop. However, stagnation of the flow increases the pressure in s i d e the drop and the system i s removed from the conditions of equilibrium f o r which Young's equation i s v a l i d . Therefore, because of hysteresis e f f e c t s stable dry patches can e x i s t . One of the e a r l i e s t explanations of hysteresis suggested that the receding angle would be smaller than the advancing because the surface had been wetted by the advancing l i q u i d . Other explanations for h ysteresis are based on roughness and heterogeneity of the s o l i d surface. Johnson and Dettre [29, 30], e x p l i c i t l y show the e f f e c t of roughness on wetting (see Figure 10). These observations on contact angle correspond to the case where the common l i n e (L V S) does not move. Ponter [31] and Thompson [8] measured contact angles under flow conditions as w e l l as contact angles of a s e s s i l e drop and claimed good agreement. Nevertheless, i t could be argued that while both values are close, they are not equal. These arguments can explain the discrepancy found by Hewitt [1] when 8 i s replaced by the equilibrium value i n the balance of forces for the s t a b i l i t y of a dry patch. A proper measurement of the contact angle under flow conditions i s necessary to test the c r i t e r i o n for dry patch s t a b i l i t y . 27 McPherson [22] suggested that sinousoidal roughness of the heater may be a factor a f f e c t i n g dry patch s t a b i l i t y . This r e s u l t s i n a larger force which tends to hold back an advancing f i l m and a smaller force which tends to reduce the force on a receding f i l m . Hysteresis e f f e c t s were not considered i n the previous models [5,6] i n the sense that the thickness c r i t e r i o n f o r rewetting i s no longer v a l i d . i The flow rate can be increased and the dry patch remains stationary. In the present study t h i s e f f e c t i s c a l l e d "anchoring". Observations made by McAdam [1] agree with t h i s concept. When the common l i n e moves along the s o l i d , the contact angle formed between the l i q u i d and the s o l i d i s c a l l e d "dynamic" contact angle. Movement of the common l i n e would appear to be a co n t r a d i c t i o n of no s l i p conditions at the s o l i d w a l l . Some papers [32, 33] try to explain how the 3-phase fro n t might a c t u a l l y move but these studies are beyond the scope of the present i n v e s t i g a t i o n . McAdam [1] measured dynamic contact angles and found the common l i n e o s c i l l a t e s even for stable dry patches due to the presence of waves. Ponter et at. [34] developed an experimental method to measure contact angle of a water drop on a smooth copper surface under ei t h e r isothermal or heat transfer conditions. For the isothermal case, the v a r i a t i o n of 0 with temperature i s almost l i n e a r and the slope of the curve i s negative. For the non-isothermal case they found only small differences i n the i n i t i a l contact angle and the equilibrium contact angle under isothermal conditions. Ponter -et at. [34] also reported r e s u l t s of water breakdown on a copper surface. They compared experimental values of minimum wetting rates with those predicted by Zub er and Staub and found good agreement for high temperature d i f f e r - ences between the surface and i n l e t water temperature. I 28 3. COMPARISON WITH EXPERIMENTS OF PREVIOUS STABILITY MODELS AND SOME SIMPLE EXTENSIONS 3.1 Discussion of McAdam's Experiments A l i q u i d CO2 f i l m under gravity flow over a heated glass plate was used by McAdam [1] to study f i l m break-up under b o i l i n g conditions. The t o t a l length of the heated plate was 163 mm and was designed so that the bottom 25 mm sec t i o n of the plate could be observed. Constant heat was supplied to the plate and dryout data were obtained by s e t t i n g the flow rate while r a i s i n g the heat f l u x u n t i l a dry patch was observed. Film break-up occurred when Re based on f i l m thickness ranged from 185 to 1000. System i n l e t pressures and temperatures correspond to saturation conditions and ranged from 2°C to 18°C. A s p e c i a l l y designed s c h l i e r e n system was developed to obtain quantitative measurements. Photographs of f i l m break-up were taken with a movie camera and the frames were i n d i v i d u a l l y analyzed. Once the data were processed, information on contact angle under flow conditions and l i q u i d f i l m p r o f i l e upstream of the t i p of dry patch was obtained. Some of the data from McAdam's thesis are plo t t e d i n Figures 11 to 16. The i n l e t temperature was 2°C and Re = 185,310,940. McAdam observed stationary patches (Figures 11 & 15), dry patches growing upstream or receding (Figure 13) and growing downstream or advancing (Figure 13). The representation of the f i l m thickness and of the contact angle (Figures 12, 14, 16) shows a v a r i a t i o n with time even i f the dry patch was c l a s s i f i e d as stationary. 29 In the present study a t t e n t i o n was focussed on stationary dry patches since t h i s case i s the most important one f o r s t a b i l i t y considerations. The receding and advancing cases should include new forces that were not considered i n any of the previous models, nor i n the present research. V a r i a t i o n with time of contact angle and thickness near the dry patch i s a t t r i b u t e d to perturbations or waves i n the flow. In.order to apply the previous c r i t e r i a an average stationary contact angle and an average thickness was used. In Figure 17, 9 (average contact angle) i s p l o t t e d versus Re input numbers for a l l the stationary dry patches analyzed i n the present study. Heat fluxes and temperatures are also i n d i c a t e d . Generally the heat f l u x necessary f o r formation of a dry patch increases when i n i t i a l flow rate increases at constant bulk temperatures (Figure 17). For almost constant Re (y 700) the heat f l u x i s constant 2 (y 24000 W/m ), however, for Re ^ 900 the required heat f l u x ranges 2 from 30000 to 63000 W/m . The contact angle decreases f o r most saturation temperatures when input Re and heat fluxes increase. This i s i n agreement with the fac t that generally contact angle decreases with temperature of the s o l i d surface [34]. Since Hartley and Murgatroyd predict that i n isothermal flow the contact angle increases with increasing Re, the r e s u l t s shown i n Figure 17 i n d i c a t e that thermal e f f e c t s might be very important i n determining dry patch s t a b i l i t y . 30 3.2 Relationship Between Various Models As a f i r s t step i n comparing the previously discussed models with experiments, i t can be shown that i n some cases these models are. rel a t e d to each other. For example, Hartley and Murgatroyd [5] and Zuber and Staub's [6] c r i t e r i a are equal when the dry patches are formed i n isothermal l i q u i d f i l m s flowing over unheated surfaces. McPherson's [22] analysis can e a s i l y be converted to the Zuber and Staub c r i t e r i o n for the case of a draining l i q u i d f i l m flowing over a heated p l a t e . In th i s case F g = 0 (Equation 2.24) and the c r i t e r i o n for dry patch s t a b i l i t y reduces to 0 7 6 2 2 | j ^ ( u L o o - u L e ) dy = a(T.) - a(T p ) cos 6 . . . .(3 where U j ^ = i s the v e l o c i t y upstream of the dry patch u = i s the v e l o c i t y at the edge of the dry region. Assuming a parabolic v e l o c i t y p r o f i l e (McPherson assumed a l i n e a r pro- f i l e ) and that the f l u i d stagnates completely u = 0, the i n e r t i a J_»£ term reduces to the same expression used by Zuber and Staub. Although McPherson presented the surface force as an a l t e r n a t i v e to the upstream force proposed by Zuber and Staub i t i s i n f a c t the same expression. If the term 0"(T\) cos 6 i s added and subtracted to the r i g h t hand side of Equation (3.1) then 31 a(T.) - a(T ) cos 9 + a(T.) cos 9 - a(T.) cos 9 i p x x = a(T ) [1-cos 0] + [a(T.)- a(T ) ] c o s 9 . X 1 p M u l t i p l y i n g and d i v i d i n g by AT = T^ - T^, t h i s becomes 0(T.)-a(T ) a(T.) [1-cos 9] + T - T ? (T - T.) cos 9 . P i As the v a r i a t i o n of C with T i s approximately l i n e a r a(T.) [1-cos 9]+ |^|| AT cos 9 . Assuming a l i n e a r temperature p r o f i l e as d i d Zuber and Staub, the up- stream forces can be expressed as C ( T . ) ( l - c o s 0 ) + | ^ | ̂  cos 9 . t This i s the same as r i g h t hand side of Equation 2.22 when vapour thrust e f f e c t s are n e g l i g i b l e . Ponter's Equation (2.20) for the pressure force i s twice the pressure force found by Zuber and Staub, this d i f f e r e n c e i s discussed l a t e r i n Section 3.4. 3.3 Comparison of Previous Models With Data 3.3.1 General After studying the d i f f e r e n t models that analyzed dry patch s t a b i l i t y through an equilibrium of forces, Zuber and Staub, Ponter 32 et al. , and Wilson's c r i t e r i a were selected to be tested with McAdam's data. From d i f f e r e n t approaches Bankoff [21] and Simon and Hsu [25] presented minimum f i l m thicknesses for the formation of dry patches. With these f i l m thicknesses, pressure forces are evaluated and compared to the surface forces proposed by Zuber and Staub. In order to check any of the previously discussed models the data needed are l o c a l data near the dry patch, but not perturbed by i t s presence. For example McAdam measured Re at the top of the heated plate, (input Reynolds numbers, Re^) but dry patches were observed and recorded i n any p o s i t i o n of the bottom 25 mm of the pla t e . Evaporation along the plate might a f f e c t the input Re and consequently the input thickness of the l i q u i d f i l m . Thus l o c a l Re and l o c a l f i l m thicknesses corrected for evaporation are considered i n order to check the previous c r i t e r i a . Under steady conditions, and as the temperature of the free surface i s at saturation temperature, i t i s assumed that the heat supplied to the plate i s equal to the heat f l u x at the i n t e r f a c e i and equal to the heat absorbed by vaporization. The thickness changes because of evaporation, and as the temperature of the i n t e r f a c e does not vary there i s a temperature gradient along the p l a t e . Under the conditions j u s t described T Ql e = x = where r . y R e h i i g Q = heat supplied to the plate T • = flow rate that evaporates 33 input flow rate latent heat of vaporization length of heated plate where evaporation occurs before a dry patch i s formed v i s c o s i t y . r = ( i - x) r . , r = l i q u i d flow r a t e a f t e r evaporation. Local Reynolds number, As the dry patches appear randomly i n the l a s t 25 mm of the plate i t 25 was considered that an average 1 = (163 m m ) • I t w a s assumed that the l i q u i d f i l m thickness w i l l not experience appreciable further changes by evaporation. Figure 18 shows average contact angles as a function of the l o c a l Re. The same general features described i n Figure 17 are observed. When Re increases the heat fluxes increase, while contact angle decreases when heat f l u x increases, except f o r a saturation temperature = 18°C. In the l a t t e r case the degree of evaporation i s very high (89%). This i s due to the f a c t that the latent heat at T = 18°C i s much less than at lower temperatures. To d i s s i p a t e a constant amount of heat 2 (65000 W/m ) more f l u i d must evaporate. The assumption that the heat supplied to the plate i s equal to the heat of vap o r i z a t i o n might be in c o r r e c t . Local Re less than 270 are c l a s s i f i e d as low Re (Re input = 185, 310, 420), while l o c a l Re la r g e r than 270 are c l a s s i f i e d as high Re. 34 3.3.2 Low Re Zuber and Staub's expression (Equation 2.22) f o r dry patch s t a b i l i t y reduces to _1_ 15 p 5 c g 2 V , 6 Q 0(1 - cos 0) + ~ ~ - cos dT kfc V P v \ .(3.2) when p « p v Table 3.1 shows the magnitude of the d i f f e r e n t forces. Vapour thrust e f f e c t s were found n e g l i g i b l e f or a l l cases. TABLE 3.1 Evaluation of Forces According to Zuber & Staub'at Low Re T(°C) Re.. l Re 6 xl0'5(m) c F xl0 5(N/m) P F axl0 5(N/m) F , xl0 5(N/m) th F +F . xl0 5(N/m) 0 th 2 185 144 7.8 158 146 43 189 2 310 221 9.0 319 109 121 230 9 420 269 9.2 382 88 175 263 As indicated previously, the pressure force proposed by Ponter [7] i s two times the pressure force presented by Zuber and Staub [6]. For a laminar v e l o c i t y p r o f i l e , using McAdam's data, the magnitude of the forces are shown i n Table 3.2. 35 TABLE 3.2 Evaluation of Pressure Forces According to Ponter et at. at Low Re Re F x 105(N/m) P 144 316 221 638 269 764 S i m i l a r l y , Wilson's [23] analysis (for unheated flow) was applied to Re^ = 185 at the lowest heat f l u x . The balance of pressure at the apex i s expressed by ,1 s i n 6 A ° (a " c _2_ p g 6_ 15 2 c y + 2 pgd (2.25) If R i s very large and c i s taken as the experimental t h i c k - ness equal to 15 x 10 ^ m, a^ and d can be evaluated through the c o l l a r model proposed by Wilson. The r e s u l t s are tabulated below. TABLE 3.3 Evaluation of Forces According to Wilson Re 6 x 105(m) c F x 10 5' (N/m) P F x 10 5 (N/m) F x 10 5 (N/m) a 144 7.8 316 180 140 Although heat transfer was present, the heat f l u x was only 5500 W/m̂ and Wilson's balance can be applied to McAdam's data as a f i r s t 36 approximation. Therefore, the increase of upstream forces due to heating effects would be insufficient to balance the pressure plus body forces. Bankoff [21] using a criterion of equating the total energy (kinetic plus surface) of a continuous film to the energy of a film that breaks up into rivulets, determines a minimum film thickness for the equilibrium situation. For the lower heat flux that corresponds to Re^ = 185, 6 = 1.73 x 10 5m c The pressure force i s far smaller than the surface forces when 6 c and a parabolic velocity profile i s used. The Simon and Hsu [25] criterion to evaluate the minimum film thickness is independent of the heat flux and only depends on the fl u i d properties: 6 c 0.666 ( f ) 1 / 5 ( ^ ) 2 / 5 • (2.26) For T = 2°C 6 = 3.5 x 10~5m c A pressure force evaluated with this film thickness and a parabolic velocity profile gives about 4 x 10 N/m. This value i s two orders of magnitude less than the upstream forces. Equation (2.26) was deduced by Simon and Hsu [25] taking a constant Weber number equal to 0.0145 for water and water-glycerol. As their criterion is independent 37 of heat f l u x i t i s i n t e r e s t i n g to compare t h e i r expression with Hartley and Murgatroyd. The Hartley and Murgatroyd c r i t e r i o n based on a We can be expressed as We = 1.67 (1 - cos 6) . This expression depends on the contact angle and i s equal to 0.55 when 8 equals 48°, the contact angle observed for Re = 185. Both c r i t e r i a [5] and [25] are d i f f e r e n t . A c t u a l l y , Simon and colleagues made measure- ments on a system of water and water-glycerol and extension of t h e i r conclusions to another system i s not j u s t i f i e d . In any case absence of the contact angle indicates that t h e i r analysis should be improved i n order to consider t h i s fundamental parameter i n the s t a b i l i t y of dry spots. 3.3.3 High Re For high Re v e l o c i t y p r o f i l e s are described by the u n i v e r s a l v e l o c i t y p r o f i l e (Equation 2.3) or by a power law such as seventh-root law (2.6). The Zuber and Staub c r i t e r i o n may be used for high Re by replacing the parabolic v e l o c i t y p r o f i l e with the 1/7 power law i n the 2 pressure force, F^ = p / -y- dy . The upstream forces (F^, F ^ ) can.be added and expressed as a unique force(see 3.2) equal to a(T.) - a(T ) cos 8 . i p i 38 When the increase in temperature of the plate i s evaluated near the dry patch using a linear temperature profile, the temperature at the point where the solid, liquid and vapour are in contact is larger than the c r i t i c a l temperature for CC^. This would mean a separating surface between liquid and vapour could not exist. Since dry patches weve observed, i t is believed that the temperature of the plate at 3-phase front must be less than the c r i t i c a l temperature. Therefore the linear temperature profile is not a good assumption even for thin film. When the temperature of the plate equals the c r i t i c a l temperature a(T^)=0, so that upstream forces have a maximum limit equal to d(T/) . Table 3.4 summarizes the different forces for 6 cases using the assumption above for maximum upstream force. TABLE 3.4 . Evaluation of Forces at High Re T(°C) Re. l Re 6 xl0 5(m) c F xl0 5(N/m) P Faxl0 5(N/m) 0? +F 1.j 1 A Xxl0 5(N/m) o th T (°C)* P 2 940 470 •13.0 860 80 425 78 9 700 453 11.9 700 58 305 41 13 750 477 12.1 679 89 250 44 855 531 : 12.5 817 61 250 55 18 750 450 10.8 463 43 170 48 1050 116(lam.) 6.5 76 50 170 67 The temperatures were evaluated through a linear p r o f i l e . Vapor thrust effects were found negligible relative to the magnitude of the other forces. As far as the previous models are concerned 39 (see Table 3.1 to 3.4) no stable dry patches can e x i s t , as i n most of the cases pressure force exceeds the surface force. 3.4 Simple Extensions 3.4.1 Simple 2-D Extensions The pressure force presented by Zuber and Staub i s hal f the pressure force used by Ponter et al. Zuber and Staub calculated the pressure force by applying the B e r n o u l l i equation to the center-stream l i n e . This increase i n pressure times the normal area y i e l d s the force. Actually t h i s i s not a proper procedure since only the center-stream- l i n e s stagnate. If a pressure force has to be found, the e n t i r e pressure d i s t r i b u t i o n along the meniscus should be known. Ponter applied the momentum equations to a cont r o l volume and considered the flow one- dimensional, although i t i s w e l l known that the flow i s at l e a s t 2-D i n the plane of the plate near the dry patch. The dif f e r e n c e between these two approaches i n the evaluation of the pressure force can be understood i f the momentum theorem i s applied to a co n t r o l volume shown i n Figure 19. Considering the flow bidimensional i n the v i c i n i t y of the dry patch, the complete steady expression f o r the x component force acting on the co n t r o l volume i s pufudydz + wdxdy] = F dz , 40 F = F + F + F + F , . t o t a l force per un i t width x B s 0 th F = body force per un i t width F = shear force at the wa l l per unit width, s In the approximation that F^ = -F , 2 p u dy dz + puwdxdy = ' ( F a + F t h ) d z • ' * * - ( 3 - 3 ) s o The f i r s t term of the pressure force i s the same as that used by Ponter. However, there i s a second term because the flow decelerates and d i v e r t s around the dry patch. This term has a sign opposite to the f i r s t , g iving a pressure force smaller than the one predicted by Ponter. A complete v e l o c i t y f i e l d must be known i n order to evaluate the r i g h t pressure force. As an attempt to f i n d the cont r i b u t i o n to the pressure force due to the l a t e r a l flow the v e l o c i t y f i e l d for an i n f i n i t e i d e a l flow around an obstacle i s proposed. The obstacle i s a Rankine h a l f body whose shape i s s i m i l a r to the inverted U-shape of the t h e o r e t i c a l dry patch, with v e l o c i t y components u = U + m — t , — k , w = m 2 2 ' w 2 2 X +Z X +z where U = average v e l o c i t y of the parabolic p r o f i l e m = source strength r e l a t e d to the width of the dry patch and the v e l o c i t y through the bU r e l a t i o n ——-2IT 41 b = width of dry patch measured on the movie f i l m x g = coordinate of the stagnation point = — Z Q = width of con t r o l volume. The width of the co n t r o l volume (Figure 19) i s chosen i n a way that the contour of the dry patch can be considered plane. This width i s much smaller than s i z e of the dry patch. The force per un i t width becomes x 2 F x = pU 6 - p6U — ( j - arctan (—) + (—- -) . . . .(3. 0 0 x + zn s 0 When the width of the con t r o l volume tends to zero, Equation (3.4) reduces to F x = (p/2)U 26 , which i s the same pressure force obtained by Zuber and Staub. In t h i s context i t can be seen that Zuber and Staub's pressure force represents a l i m i t i n g case, while Ponter's force does not take l a t e r a l flow into consideration. Some evaluations of the pressure force using Equation (3.4) are shown i n Figure 24. 3.4.2 Re-laminarization E f f e c t s The previous h i s t o r y of the l i q u i d f i l m probably determines the v e l o c i t y p r o f i l e near the dry patch. For example, i f a dry patch i s formed when a t h i n l i q u i d f i l m draining by gravity breaks-up, most 42 l i k e l y the v e l o c i t y p r o f i l e i n the v i c i n i t y of the patch i s parabolic. On the other hand, i f a dry patch i s formed i n a thick and turbulent flow, although the f i l m gets t h i n near the dry patch the flow might remain turbulent and not re-laminarize. The free surface can be smooth as McAdam [1] observed but the nature of the bulk f l u i d can be s t i l l turbulent. For these reasons, the 1/7 power law was used to evaluate the pressure force for l o c a l Re = 221 and 269, Rê ^ = 310 and 420 re s p e c t i v e l y . Table 3.5 shows the r e s u l t s of the c a l c u l a t i o n s . TABLE 3.5 Evaluation of Forces When the 1/7 Power Law i s Used at Low Re Re 6 x 105(m) c F x 10 5(N/m) p F a + F t h x 105(N/m) 221 9.1 265 230 269 9.55 308 263 The pressure forces evaluated through the 1/7 power law are lower than when the parabolic v e l o c i t y p r o f i l e i s used (Table 3.1). This could be an i n d i c a t i o n that although the f i l m i s very t h i n near the dry patch i t might not relaminarize. 3.5 Discussion 3.5.1 Balance of Forces Figure 20 shows the pressure force predicted according to Zuber and Staub [6], Ponter et al. [7] and Wilson [23], and includes 43 low Re as well as high Re cases. Figure 21 shows the t o t a l surface forces (surface force plus thermocapillary force) and the surface force alone ( i s o l a t e d from the thermal e f f e c t s ) . In general the estimate pressure forces are larger than surface forces and best r e s u l t s are obtained using Zuber and Staub's model. Figure 22 ( c i r c l e points) and Figure 23 show the di f f e r e n c e between the pressure and surface forces evaluated according to Zuber and Staub. Surface forces become larger than pressure forces f o r Re^ = 185, and Re^ = 1050. For the l a s t case the degree of evaporation was found to be very high (assuming that the heat supplied to the plate equals the heat absorbed by va p o r i z a t i o n ) . For low Re (Figure 22, c i r c l e s ) the agreement between the absolute value of the forces i s reasonable. The trend shows a departure of the balance of forces f o r increasing Re. At higher Re, (Figure 23) the pressure force exceeds the surface force by a factor of 3. This i s possibly due to the fac t that the assumptions made i n the development of the previous modesl are not v a l i d for high Re. The thicknesses estimated for low and high Re are smaller than the experimental thicknesses measured by McAdam [1]. If the experimental thicknesses are used to evaluate pressure force i t becomes far larger than the surface forces f o r any Re. McAdam measured f i l m thickness close to the dry patch and i t i s possible that the presence of a c o l l a r [23] r e s u l t s i n a larger measured thickness. Figure 24 shows the pressure forces according to Zuber and Staub c r i t e r i o n as w e l l as the 2-D model. A l l 44 forces are larger than the surface forces also shown in Figure 24, how- ever, the pressure force evaluated through the momentum theorem improves the agreement slightly. The simple 2-D model has some serious limitations. The flow decelerates and diverts around the dry patch implying a continuous increase in- pressure. As the liquid has a free surface the curvature and thickness must change near the dry patch which is not accounted for by the model. In addition, the assumption that the body force is equal and opposite to the shear force on the wall i s poor near the dry patch where the liquid i s slowed down appreciably. Nevertheless the model is useful to distinguish between Zuber and Staub, and Ponter's formalism, and shows why Wilson's model gives a larger pressure force. Wilson considered that the flow is disturbed by the dry patch only in a thin region around i t s boundary. fActually the disturbance is more extended, affecting the flow f i e l d and leading to a decrease in pressure force. The results of estimating the pressure force considering that the liquid might not re-laminarize are shown in Figure 22 (squares). The difference between the pressure and surface force diminishes but more data are needed. 3.5.2 Role of Thermocapillarity Figure 21 shows the total upstream force (surface plus thermocapillary forces). It also illustrates the surface force a ( l - cos 9) without thermocapillary effects. For a temperature, T = 2°C, three cases were analyzed: i Re. = 185, Re = 144; Re. = 310, Re = 221; Re. = 940, Re = 470. x x ' x 45 When the l o c a l Reynolds number increases the contact angle and there- fore the surface force decreases. At f i r s t sight t h i s behaviour, also observed by Ponter [7], i s i n con t r a d i c t i o n with the fac t that when Re increases, the contact angle must increase as Hartley and Murgatroyd p r e d i c t . This, however, ignores thermocapillary e f f e c t s , which are also important for dry patch s t a b i l i t y . When Reynolds number increases, heat f l u x and the t o t a l upstream force also increase (Figure 21). Therefore the thermocapillary force increases with Re to s t a b i l i z e the dry patches. Thompson [8] did not consider thermo- c a p i l l a r y e f f e c t s as important as i n the present research. The thickness estimated for the flow depends on the v e l o c i t y and temperature p r o f i l e . At high Reynolds number the 1/7 power law and l i n e a r temperature p r o f i l e give estimates of the temperature of the plate (T > 31°C) that do not agree with the observations of stationary dry patches. If a dry spot i s formed the temperature of the plate at the t r i p l e contact point cannot exceed the c r i t i c a l temperature of CC^T = 31°C). I f the dif f e r e n c e i n temperature between f l u i d and plate i s l i m i t e d by the c r i t i c a l temperature, the temperature p r o f i l e can not be l i n e a r . A l t e r n a t e l y , the thickness evaluated assuming a 1/7 power law v e l o c i t y p r o f i l e i s not the actual thickness of the f i l m . I t may also be possible that thermo- c a p i l l a r y e f f e c t s are present i n an extended region w e l l upstream of the dry patch, so that l o c a l thickness i s smaller than estimated, leading to a smaller value of the pressure force. This i s a very complex problem with flow and heat e f f e c t s combined, and a deeper analysis i s necessary i n order to understand the formation and s t a b i l - i t y of dry patches. 46 3.6 Summary Ponter's and Wilson's c r i t e r i a do not describe the s t a b i l i t y of dry patches measured by McAdam as w e l l as Zuber and Staub's. The c r i t e r i o n of equal Weber number [25] i s discarded i n view of i t s lack of g e n e r a l i t y . The c r i t e r i o n of equating the energy for a continuous f i l m to the energy of a configuration of r i v u l e t s gives a minimum f i l m thickness almost 4 orders of magnitude les s than the measured thickness (corrected for evaporation). The pressure force so evaluated i s 3 orders of magnitude less than surface forces. The energy c r i t e r i o n i s strongly dependent on the density and shape of r i v u l e t s and does not appear equivalent to a force balance c r i t e r i o n applied to the apex of an i s o l a t e d dry patch. For low Re,- Figure 22 ( c i r c l e s ) shows a r e l a t i v e l y good agreement of forces, while at high Reynolds number (Figure 23) the pressure force calculated through a 1/7 power v e l o c i t y law i s nearly three times the t o t a l upstream forces. The 1/7 power law v e l o c i t y p r o f i l e and l i n e a r temperature p r o f i l e do not.describe the f a c t that stationary dry patches were observed. The temperature of the p l a t e i s over-evaluated and as a l i m i t was assumed to be equal to the c r i t i c a l temperature of CC^. This implies, that no increase i n up- stream forces can be expected. Two simple extensions were proposed based on a more complete analysis of the problem, the 2-D extension and the r e - l a m i n a r i z a t i o n e f f e c t s [Figure 24 and Figure 22 (squares)]. 47 However, agreement between pressure force and surface forces improved only s l i g h t l y with these extensions. Although l i m i t a t i o n s on the two-dimensional model are severe (for example, i t does not consider that the l i q u i d has a free surface), the d e s c r i p t i o n of some aspects of the flow behaviour i s more r e a l i s t i c than Zuber and Staub's. The thicknesses measured by McAdam exceed the thicknesses estimated by any previous models. I t i s f e l t h i s measurements apply to the thickness of a " c o l l a r " , an e f f e c t not described by the simple extensions. 48 4. UNSTEADY EFFECTS McAdam's [1] measurements of the contact angle show a dependence with time even though dry patches were stationary. The f a c t that dry patches stand s t i l l and the contact angle changes with time i n a rather p e r i o d i c way suggests the idea that waves of f a i r l y regular shape are responsible for the v a r i a t i o n i n the contact angle. When these waves change to a more i r r e g u l a r pattern (such as r o l l waves), increases i n amplitude and rewetting might be pos s i b l e . Hysteresis plays an important r o l e by allowing larger contact angles before the dry patch becomes unstable. Kapitza [17] and Levich [18] presented comprehensive studies of wave motion i n t h i n l i q u i d isothermal l a y e r s . Their studies enable the wavelength, frequency and amplitude of wave motion to be calculated. The treatment i s l i m i t e d to wavelengths greater than 14 h^ (h^ = mean f i l m thickness), which means Re numbers less than 50. As an attempt to analyze the unsteady e f f e c t s the force c r i t e r i o n balance for s t a b i l i t y of dry patches can be extended using Kapitza's expression for the v e l o c i t y f i e l d . A c o n t r o l volume i s chosen i n the same way as for the steady case and an i n t e g r a l equation of motion i s proposed. The forces acting on the volume are pressure force, surface tension force, viscous force and gravity and are now a l l functions of time. The c o n t r o l volume i s fixed to the p l a t e and the dry patch i s assumed to be a s o l i d object i n the flow (Figure 19). The flow stagnates, producing an o s c i l l a t o r y pressure force over the r i g i d surface that 49 represents the dry patch. In t h i s approximation the d e t a i l s of the flow near the leading edge of the f i l m are not considered, the contact angle does not enter the a n a l y s i s . The model i s the following: when a f l u i d with waves stagnates on the object, i t exerts a force that changes with time. I f the dry patch stands s t i l l the balance between pressure and upstream forces i s s a t i s f i e d for each instant. For example, the maximum pressure force must be equal to the maximum upstream force and the minimum pressure force i s balanced by the minimum upstream f o r c e . The minimum and the maximum surface forces determine a range of contact angles. The patch w i l l rewet when the pressure force i s high (or low) enough to overcome the anchoring. Perturbations can a l t e r the regular wave pattern described by Kapitza [17] and rewetting might be po s s i b l e . Taking a control volume from i n f i n i t y to the edge of the patch, the balance of forces per unit width i n the d i r e c t i o n of the main flow i s expressed as mArh g u rh ^ rnXrh eb. rhrnX p -r— dxdy + pu dy - puwdydx -I puvdx = F + pgdxdy Jo Jo 3 t Jo Jo Jo J o ° JQJO 0 pi A + T dx . . . .(4.1) 0 nA with n integer represents a distance f ar enough i n order to assume that the flow i s not perturbed from i t s condition at i n f i n i t y . The components of v e l o c i t y are ? u(x,y,t) = 3U Q[1 + 0.6 s i n (kx-OJt)- 0.3 s i n (kx-wt)]* 2 *(£ - -Z-yj . (2.13) h 2h 2 50 2 3 v(x,y,t) = -1.8 u k cos (kx-tot) [1 - sin(kx-oot) ] (^r- (2.13) where h = h + h n 0.46 s i n (kx-wt ) Formula (4.1) can be s i m p l i f i e d . The non-steady term i s equal to zero when the i n t e g r a t i o n i s performed over an integer number of wavelengths. The 4th term of the l e f t hand side i s zero because no flow crosses the top face of the c o n t r o l volume, the evaporation i s assumed to be n e g l i g i b l e . Kapitza's [17] treatment i s for undamped waves: the energy d i s s i p a t e d by wave motion must be balanced by the work done by gravity, which means that viscous shear and gravity forces cancel each other over one wave- length. A d e t a i l e d c a l c u l a t i o n of the shear and gravity force i s c a r r i e d out i n Appendix A. The balance of forces without heat a d d i t i o n i s , a f t e r s i m p l i f i c a t i o n : The second term of the l e f t hand side represents the contribution of the l a t e r a l flow to the pressure force. This term i s unknown beforehand because w, that i s the component of v e l o c i t y that appears due to the presence of a dry patch, i s zero because Kapitza's model i s one- dimensional. The second term i s evaluated through an approximation: The balance of forces i s expressed as (4.2) 5 1 w h e r e f h 2 a p u d y = a ( l - c o s 6) . . . . ( 4 . Jo a i s d e f i n e d a s [ 1 - ^ P u w d x d y j / p u d y s t e a d y c a s e T h e t e r m a r e p r e s e n t s a r e l a t i o n s h i p b e t w e e n t h e l a t e r a l m o m e n t u m f l u x a n d t h e i n c i d e n t m o m e n t u m f l u x . I t i s a s s u m e d t h a t t h i s r e l a t i o n s h i p i s a p p r o x i m a t e l y e q u a l t o t h e s t e a d y c a s e . T h e r e f o r e a i s e v a l u a t e d f o r a n i d e a l f l o w , s t e a d y , d i v e r g i n g a n d s t a g n a t i n g a r o u n d a n o b j e c t w h o s e s h a p e i s : s i m i l a r t o t h e s h a p e o f t h e d r y p a t c h . S u b s t i t u t i n g ( 2 . 1 3 ) i n f o r m u l a ( 4 . 3 ) 1 ft 2 9 9 F (t) = a . y ^ p u 0 h Q [ l + 0 . 6 S i n ( k x - a ) t ) - 0 . 3 s i n (kx-Wt)] z f o r x - 0 F ( t ) = a v l un P h n [ l + 0 . 3 6 s i n 2 w t + 0 . 0 9 s i n ^ w t - 1 . 2 s i n w t p 1 5 U U + 0 . 3 6 s i n 3 w t - 0 . 6 s i n 2 w t ] - a - j | 0 . 4 6 ^ ^ [ s i n w t 3 5 2 4 + 0 . 3 6 s i n w t + 0 . 0 9 s i n w t - 1 . 2 s i n u ) t + 0 . 3 6 s i n a ) t 3 - 0 . 3 6 s i n t o t ] F p ( t = 0 ) = 1 . 2 a p u 2 h Q F p ( t = T / 4 ) = 0 . 0 0 6 a p u 2 h Q F p ( t = T/2) ' 2 = 1.2 apu 0h 0 52 F (t = 3/4 x) P 3 apu 0h 0 The average value of the pressure force i s F r J0Vt)dt = - j a 1.2 p u Q h 0 [ l + 0.36 s i n ait + 0.09 s i n cat 2 1 - 0.6 s i n cat] dt + — f T 2 2 -a 1.2 0.46 pu h n[-1.2 s i n ait 0 U U + 0.36 s i n cot] dt F = a 1.33 pu 2h A P u u 4.1 Evaluation of a A v e l o c i t y p r o f i l e that can describe a flow around an object whose shape i s s i m i l a r to a dry patch was proposed i n Chapter 3. _ x 2 1 p<5U m (-̂  - arctan g—) - 6p y - ( — y-) a = i 3Q _ I s + z 0 pu 26 m» >̂ U } 2^ same as i n Chapter 3. 53 In order to check the criterion for st a b i l i t y of dry patches, (Equation 4.3) experimental data for local Reynolds number equal to 144 (Re^ = 185) were used. A Reynolds number of 150 is three times larger than the maximum Reynolds number permitted for the complete validity of Kapitza's treatment, but measurements below Re = 50 were not available. In any case this analysis is intended only as a starting point for the con- siderations of the st a b i l i t y of dry patches when unsteady effects are important. For this case (low heatflux) heating effects can be neglected. For dry patch s t a b i l i t y ' Fmin a p av = F av p a _max m̂ax F = F P a Table 4.1 shows the results of the calculations. TABLE 4.1 : Evaluation of Forces For a Wavy Liquid Film T(°C) Re. l Re a F x 10 5(N/m) P F 0 x 10 5(N/m) e (°) experimental 2 185 144 0.47 min av- max 0.79 159 396 104 141 181 41 48 55 Fmm P 54 The average and maximum pressure force are l a r g e r than the average and maximum surface forces, while the minimum pressure force i s two orders of magnitude smaller than the surface force. The fa c t that the body force compensates with the shear force over a distance equal to a wavelength does not hold near the dry patch. Kapitza's flow i s d i s t o r t e d , the pressure increases and the sinousoidal pressure described by Kapitza transforms into an asymmetric pattern. The flow slows down and the balance between shear and body force does not hold any more. As a r e s u l t of t h i s a net body force which also changes with time can be present i n the equilibrium of forces. The influence of the l a t e r a l flow evaluated through the c o e f f i c i e n t a has the same l i m i t a t i o n s as was pointed out i n Chapter 3. At t h i s stage i t i s i n t e r e s t i n g to compare t h i s simple model with some suggestions to treat unsteady e f f e c t s made by Thompson [8]. I f the pressure force i s evaluated as Thompson did using a thickness equal to the average, minimum and maximum, but with a parabolic v e l o c i t y p r o f i l e (Thompson used a l i n e a r and a logarithmic v e l o c i t y p r o f i l e ) the re s u l t s are rather d i f f e r e n t (Table 4.2). TABLE 4.2 Evaluation of Forces (Thickness C r i t e r i o n ) For A Wavy Film F x lO^N/m) P F a x 10^(N/m) min. 5 104 av. 109 . 141 max. 728 181 55 The r e s u l t s of Table 4.1 and Table 4.2 in d i c a t e that the evaluation of forces through the momentum theorem and through the thickness c r i t e r i o n are not equivalent. The maximum pressure force evaluated through the thickness c r i t e r i o n i s s i g n i f i c a n t l y larger than the surface tension force. Thompson [8] found the same r e l a t i o n s h i p for unstable dry patches, but i n the present analyzed case the dry patch i s stationary. Neither of these c r i t e r i a give a good agreement with experimental data. Nevertheless, the co n t r o l volume method of findi n g the pressure force i s w e l l founded while the thickness c r i t e r i o n to evaluate the force i n the presence of waves i s not j u s t i f i e d . The disagreement between the downstream force evaluated through the momentum theorem and the upstream force for stationary dry patches i s due to the fact that near the dry patch the flow i s two dimensional and the i n t e r n a l pressure increases. These properties are not described by Kapitza's a n a l y s i s . 56 5. A RATIONAL TWO-DIMENSIONAL MODEL 5.1 General An exact s o l u t i o n of the fundamental equations of motion and energy equation would be desirable for dry patch s t a b i l i t y studies. However, since these equations are non-linear and coupled, not only d i r e c t l y but through boundary conditions, i t seems u n l i k e l y that complete solutions could be found for the general case. The f i r s t u s e ful approximation i n order to get an i n d i c a t i o n of expected behaviour would be to consider an isothermal flow, the dry patch a r i s i n g from some change i n surface condition. However, the r e s u l t i n g Navier-Stokes equations for a viscous, free-surface flow around a dry patch of unknown shape are s t i l l unsoluble. This chapter suggests two further approximations i n order to get the simplest model from which useful information can be obtained. F i r s t the flow i s assumed steady, so that only " s t a t i c " s t a b i l i t y of the dry patch i s determined. I t i s e n t i r e l y p ossible that a "dynamic" s t a b i l i t y a n a l y s i s including surface waves might y i e l d d i f f e r e n t r e s u l t s . However, the s t a t i c model indicates some general features of the flow. Second, a procedure i s employed analogous to that of using combinations of p o t e n t i a l flow and boundary layer analysis to describe flow over a body. The dry patch i s assumed to act as a s o l i d body i n the flow, and the flow patterns around i t are determined by ignoring the boundary condition of prescribed contact angle at the edge of the patch (outer 57 region). This flow i s "patched" to a second type which properly accounts for boundary conditions, but does not involve d e t a i l s of the flow away from the dry patch (inner region). In the approximation of considering the dry patch as an obstacle f o r the flow, the idea i s to f i n d the flow behaviour of a l i q u i d f i l m with a free surface i n the presence of an object. Lamb [36] showed (see also S c h l i c h t i n g [37]) that when a l i q u i d between two plates i s driven by a pressure gradient past a c y l i n d r i c a l closed body of a r b i t r a r y cross section placed between the plate s , the r e s u l t i n g pattern of streamlines i s i d e n t i c a l with that i n p o t e n t i a l flow about the same shape. Hele-Shaw [38] used t h i s method to obtain experimental patterns of streamlines i n p o t e n t i a l flow about a r b i t r a r y bodies. The present model i s the following: a l i q u i d f i l m flowing along an i n c l i n e d plate i s forced by gravity past a s o l i d body with the shape of a dry patch. The pressure increases and i s taken into account h y d r o s t a t i c a l l y by increasing the f i l m thickness. The v a r i a t i o n of thickness i s gradual and curvature of the free surface i s considered s l i g h t enough to neglect surface tension e f f e c t s . The flow i s perturbed as i f an object of a si z e approximately equal to the s i z e of a dry patch i s placed between the free surface and the p l a t e . When the f l u i d stagnates i n front of the object the thickness of the l i q u i d f i l m i s a maximum. In the second part of the model, surface tension e f f e c t s are predominant and the thickness goes to zero and forms the contact angle 0 with the pl a t e (Figure 25). In t h i s region i n e r t i a l e f f e c t s are neglected and the increase i n pressure i s hydrostatic. The d i f f e r e n c e between the inner 58 and external pressure at the free surface i s equal to — K (K = radius of curvature). 5.2 Solution 5.2.1 Outer Region In the case of a slow l i q u i d f i l m flowing along an i n c l i n e d p l ate forced by gravity past an obstacle the s i m p l i f i e d equations of motion and continuity are (See Appendix B). 2 1 '3 u /r. „ x "p Ix = v ~1 g s i n a . . . .(5.9) 3y 1 9p /r. . p 3^ = - g c o s a . . . . (-5.11) i | £ - v . . . .(5.12) p 9 z 8y 2 with the assumptions that 600/L « 1 , tan a ^ a ^ 6ob/L . 59 The boundary conditions for this region are: u(y = 0) = 0 ' (fe) • = 0 w(y - 0) - 0 (fe) = 0 J y=o p(x, <S(x,z), z) = 0 As the component v of the velocity appears only in the equation of continuity as a f i r s t approximation i t is considered negligible. The set of equations is then P ax" = v 7 2 + 8 s i n a • • • ' ( 5 - 9 ) 9y 1 9p P 9y g cos a „ • . . . (5.11) ± f e = . . . -(5.12) 9u 9w - /r !/\ Tr- + v- = 0 . . . . . (5.14) 9x 9 z Integrating Equation 5.11, p (x,y,z) - p„(x,y = 0,z) = - pg cos a y , p (x,y,z) = p B(x,z) - pg cos a y . . . .(5.15) 60 Replacing the expression of the pressure i n Equations 5.9 and 5.12 1 „ 9 2u ., — g^- = V —j + g sxn a . . . . (5.16) 1 ^ 5 . = v ^ . . . . .(5.17) p 3z 3 y 2 If i n Equation 5.16 i t i s defined Pg = P B - Pg s i n a x , then 1 ^ - 2 - a - 5 - - V . . . .(5.16b) P dx ~ 2 1 3 P B „, v £ v . . . . .(5.17b) p 9z v „ 2 * 9y Equations 5.14, 5.16b and 5.17b form a set of equations s i m i l a r to the equations describing the Hele-Shaw experiments (Lamb [36], S c h i l c h t i n g [37]). A v e l o c i t y f i e l d can then be defined 2 2 W . = V X ' Z ) ( 5 O 6 T ~ TzT—) 6 (x,z) According to the boundary condition p(x,6(x,z), z) = 0 , I 61 P B = pg cos a 6 from Equation (5.15). Then repla c i n g u, w and p g i n Equations 5.16b, and 5.17b, 1 9 2 — g^- (pg cos a6 - pg s i n a x ) = - v — j u T , fe = ^ U r r + tan a . . . .(5.18) dx r.2 T go cos a 96 2v / c i n v 97 " -~^2 W T • • ' - ( 5 ' 1 9 ) go cos a fe + fe = 0 . . . . .(5.20) 9x dz Equations (5.18 to 5.20) form a set of non-linear p a r t i a l d i f f e r e n t i a l equations. If only a small increase i n thickness i s 2 2 expected 6 can be put approximately equal to 6^ i n the r i g h t hand side of Equations (5.18) and (5.19). Equation (5.20) becomes 3u 9w TT± + = 0 . . . . .(5.21) 9x dz 62 If this is done i t can be shown that u ̂, and ŵ, are velocity components of a potential flow and that the velocity potential is * 2 cos a g o„ CO $ = : ( 6 (x,z) - (tan a) x ) , 2V For example, i f 6 = 6m in the right hand side of Equations 5.18 and 5.19, the equations become |A = ^ u + t a n a , . . . .(5.22) OX r.̂  I gOoo cos a 86 _ 2v w„ . . . .(5.23) 9z ~ 2 g600 cos a Differentiating Equations (5.22) respect to x and Equation (5.23) respect to z and applying Equation (5.21), 2 V 6 (x,z) In addition, when Equation (5.22) is differentiated with respect to z and Equation (5.23) is differentiated with respect to x , 63 Then because of Equations 5.21 and 5.24, u T and ŵ , are harmonic functions. If i t i s assumed that the object has the shape of a complete Rankine body whose s i z e i s approximately the s i z e of the dry patch, u^ and w . can be expressed as tt i r x 4- a x - a u = IL + m i x 2 ~ 2 2 ' T T (x+a) + z ( x - a ) Z + z R Z Z -i w = m { ^ 2 2 2 i ' 1 (x+a) + tT (x-a) + z where m = source a = distance from o r i g i n to source and sink 2 U = pg6OT s i n a/2y v e l o c i t y f or x = 0 0 . Replacing u T and w T i n Equations 5.22 and 5.23 | 5 = _ Z 2 V {u T + m( X + I ? - X ~ * 2)} + t a n a , . . . .(5.25) 9 x  g 6 2 cos a 1 (x+a) Z+ z Z (x-a) Z+ z ° CO f = I 2 V m{ V-2 V~2} ' . . . . ( 5 . 2 6 ) 9 z g6 cosa (x+a) Z+ z Z (x-a) Z+ z Z The boundary condition i s that 6(-°°,z) = 5, 64 Substituting the value of U T i n Equation 5.25, the f i r s t term and t h i r d term of the r i g h t hand side cancel Then 8 6 - 2v r (x + a) x - a , / r _ m { } > . . .(5.27) gboo cos a (x+a) + z (x-a) + z Integrating 5.27 2 2 x - ~2v m 1 r (x+a) + z , , N o = —2 2 l n [ 2 2 -1 ( z ) g6oo cos a (x-a) + z Integrating 5.26 * - ~2v m .. , (x+a) 2 + z 2 , , ... . 6 - ~ 2 2 l n [ T — T ~ T ] + f ( x ) gOoo cos a (x-a) + z f(x) = f ( z ) = C , 2 2 \ 2v m i r (x+a) + z -, , <5(x,z) = 2 2 l n t 2 2 ] C ' g6oo cos a (x-a) + z «(-,« ) - 6 m • • c = 6 , <S(x,z) _ V n r (x+a) 2 + z 2 , . . gOoo cos a (x-a) + z 65 For z = 0 6(x,0) _ 6_ 2v i x + a . i m l n + 1 g&oo cos a x - a From the geometry of the Ranklne body [35] m = bg s i n a 6 o 2 arctan (a/b) 2 V 2 arctan a/b ' where Then a/b arctan (a/b) A = aspect r a t i o of the body = L/b L = length b = width . 5(x,0) 6oo = tan a 6 2 arctan (a/b) l n x/b - a/b x/b + a/b + 1 The maximum thickness occurs at x = -L y-L , o ) = tan a 26 o oarctan (a/b) - l n -A - a/b -A + a/b + 1 . (5.28) The r a t i o of the maximum thickness to the thickness at i n f i n i t y i s proportional to Re 66 5.2.2 Inner Region In t h i s region the flow i s considered at r e s t and the Navier-Stokes equations are expressed as 1 3p ' ~ TT~ = g s i n a P dx 1 9 P — = -g cos a p dy ^ - ° P dz Integrating p(x,y) = pg s i n a x - pg cos a y + C . The system of co-ordinates i s located now at the surface of the imaginary body. I f the boundary con d i t i o n , p(x = 0, y = 6 M) = 0 . . . .(5.29) i s chosen, then P(x,y) = pg s i n a x + pg cos a(6 - y) .(5.30) 67 The diff e r e n c e between the i n t e r n a l pressure ( l i q u i d pressure) and external pressure being the surface tension divided by the curvature P = P ± " P e = CT/K . The curvature i s expressed by . A i dx 2 j2 I = ~ _ ±JL K . 3/2 ,2 [ 1 + C g ) 2 l d x Equation (5.29) can be written as d2 pg s i n a x + pg cos a (6 - y) = -a — M dx The boundary conditions are y(x=0) = 6 M . , f ^ x = d > = " t a n If the v a r i a b l e r i s introduced equal to 5 -y 2 d r 2 pg(sina) x , — 2 - n r = K 6 g — — , n = J pg cos a .... .(5.31) dx a A nondimensional form of the Equation (5.31) can be formed by choosing the dimensionless variables 68 so that dx *2 2 2 * 2 2 d * (n d )r = (n d ) (-5—) (tan a)x 6M (5.32) The boundary conditions i n terms of the dimensionless v a r i a b l e s are r (x = 0) = 0 , — ( x = 1) = dx tan 6M The s o l u t i o n of (5.32) i s & & ^ d it r = A cosh(ndx )+ B sinh (ndx ; - - 5 — (tan a)x M The f i r s t boundary condition implies that the c o e f f i c i e n t A i s equal to zero, while the second boundary condition determines B. Thus and ndB cosh nd - tan a = tan 0 -S- , 6M 6M g = ( d ) tan 8 + tan a _ (tan 8 + tan a) 6,, (nd) cosh nd 6,, n cosh nd ' M M 69 * d ( t a n 9 + tan a) . , , , . N d . r = 6^ (dn) cosh nd s i n h ( n d x*> " S~ ( t a n a> x* . . . .(5.33) When x* = 1, r * = 1, these conditions define an equation for d: 1 = -jsp (tan 0 + tan a) tanh nd - tan a M M In terms of y, Equation (5.32) can be expressed as, y _ , • d (tan^ 0 + tan a) . , . , a , . ̂ ~ ^ - 1 ~ 6^ (dn) cosh nd S i n h ( n d x A ) " 6^ ( t a n a ) x * ' This model also predicts a pressure force i 2 F = Pg cos a(6 -y)dy = — ^- . . . .(5.34) F JQ 2 5.3 Discussion Figure (2 6) and (27) show the thickness p r o f i l e f or two Reynolds numbers when the plate i n c l i n a t i o n a = 2 ° . The contact angle, the width of the patch and the aspect r a t i o were chosen a r b i t r a r i l y equal to 35°, 100 x 10 ~*m and 2 r e s p e c t i v e l y . The model gives an increase i n thickness near the dry patch as was expected. The representation of the thickness for the inner region i s almost 70 a s t r a i g h t l i n e . This i s due to the low pressure generated r e l a t i v e to the surface tensions according to th i s model. At each point of the curve the radius of curvature i s much larger than the thickness. The matching of the solutions i s not smooth because the e f f e c t of surface tension was neglected i n the outer region. I t should be mentioned at t h i s point that i f instead of the boundary conditions (5.29) a d i f f e r e n t constant i s chosen d i f f e r e n t curvatures at the maximum thickness w i l l r e s u l t . The model permits the evaluation of a pressure force (Equation 5.34). Experimental data a v a i l a b l e correspond to observations of dry patches formed on a v e r t i c a l p l a t e . However, when the hysteresis i n contact angle i s small i t i s expected that surface force for a patch formed on an i n c l i n e d p late w i l l be si m i l a r to that on a v e r t i c a l p l a t e . Thus t h e i r pressure forces w i l l be s i m i l a r . For t h i s reason McAdam's data were used to evaluate a pressure force according to the present model. The model i s v a l i d for (Re 1- McAdam's lowest Reynolds number corresponds to 185. I f the length of the patch i s chosen ten times larger than the width, the product (Re i s of the order of 15, Li a c t u a l l y out from the range of v a l i d i t y of the model. Nevertheless, the pressure force was calculated to be 44 x 10 ^ N/m. The surface force was evaluated according to previous models, (F = O"(l-cos6)] . Table 5.1 shows the r e s u l t s of the c a l c u l a t i o n s . i 71 TABLE 5.1 Evaluation of Pressure Force According to Model Described i n Chapter 5 Re. l 6(°) a(°) A bxl0 5(m) SooxlO^m) 6 Mxl0 5(m) F xl0 5(N/m) P F^xlO 5(N/m) 185 t 48 2 10 100 27.4 31.64 44 146 If the surface force i s correct, the requirement of equilibrium means the model underestimates the pressure force. The s i m p l i f i c a t i o n s of the present approach introduce important l i m i t a t i o n s to the model. For example, the dependence of thickness on Reynolds number: i f the patch dimensions don't depend strongly on Re, when Re increases the r e l a t i v e increase of thickness decreases; when Reynolds number decreases, the r e l a t i v e increase of thickness increases. This gives a contradic- tory r e s u l t i n the l i m i t of zero Reynolds number since the increase i n thickness should go to zero. Perhaps the main r e s t r i c t i o n of t h i s analysis was equating thickness with thickness at i n f i n i t y i n the r i g h t hand side of Equations (5.18 and 5.19). This r e s t r i c t i o n means that the absolute increase i n thickness has to be considerably smaller than the i n i t i a l thickness. With that condition i t was found that absolute increase does not depend on i n i t i a l thickness, but only on the geometry of 3 3 the patch. As Re = C6 , then Re » CA5 (where C = constant, and AS i s increase i n thickness), while on the other hand (Re — - ) « 1. Thus there must be a range of Re where the r e s u l t s of the model describe the actual flow behaviour w e l l . For example i f a = 2°, A = 2, b = 100 x 10~5m then 0 « Re « 16. 72 Further measurements of patches i n the range where the present model i s v a l i d are needed to compare the f i l m p r o f i l e s and pressure force with the actual behaviour and r e a l forces. 73 6. SUMMARY AND CONCLUSIONS A study has been c a r r i e d out on the break-up of t h i n l i q u i d films and the s t a b i l i t y of dry patches formed on a heated p l a t e . Recent experimental data [1] have been used to check previous analysis of dry patch s t a b i l i t y . The d i f f e r e n t s t a b i l i t y c r i t e r i a are usually obtained by a force balance at the upstream stagnation point of a dry patch. They consider the p r i n c i p a l forces to be pressure, surface tension and thermocapillary forces. The previous models d i f f e r mainly i n evaluation of the pressure force: some workers [4][6] follow Hartley and Murgatroyd [5] and apply a Bernoulli-type equation to the center streamlines to f i n d the pressure force; others [23] follow Ponter et al. [7] and use a control-volume approach. In both analysis the flow i s considered one-dimensional and both methods give d i f f e r e n t r e s u l t s . In the present study the contradiction i s c l a r i f i e d by using a control-volume technique applied to a two-dimensional flow. In such a case both methods give equivalent r e s u l t s i n the l i m i t , when the control volume coincides with the center streamline. The experimental parameters used to check the previous models were heat f l u x , contact angle and Reynolds number. McAdam's thickness data were not used as they probably r e l a t e to a " c o l l a r " region [23], where the thickness i s larger than the thickness further upstream. When these thicknesses are used i n the previous c r i t e r i a , the forces which tend to rewet the dry patch are much, much larger 74 than those which tend to spread the patch. For t h i s reason upstream thickness corrected f o r evaporation was used to evaluate the forces. Zuber and Staub's [6] c r i t e r i o n of force balance best describes the s t a b i l i t y of "stationary" dry patches observed and measured by McAdam. At low Re, the balance between the pressure or stagnation force and surface forces i s reasonable. At high Re the pressure forces depart s i g n i f i c a n t l y from surface forces. Therefore, according to previous models no stable dry patches can e x i s t ! For a l l Re, the thermocapillary forces have an important r o l e since the measured contact angles decrease with Re for each saturation temperature. The main f a i l u r e of previous models i s that they are one- dimensional. As a f i r s t (and simplest) extension the present study considered an i d e a l flow around a body whose shape was s i m i l a r to the shape of a dry patch. The pressure force evaluated by t h i s model i s s t i l l larger than the surface force, although smaller than the force evaluated for the one-dimensional case. The model was us e f u l i n c l a r i f y i n g discrepancies between previous c r i t e r i a . A second model to account f o r unsteady e f f e c t s was proposed using the wave d e s c r i p t i o n for t h i n films presented by Kapitza [17]. Kapitza analyzed a one-dimensional flow i n the plane of the pl a t e , but i n the present study the bi-dimensional character of the flow was considered. A c o e f f i c i e n t evaluated f o r the steady two- dimensional case was applied using a c o n t r o l volume technique. 75 Pressure forces evaluated by the control-volume technique are compared with forces estimated when the thickness c r i t e r i o n [8] i s applied to the same waves. Although r e s u l t s are not very d i f f e r e n t , the present model can be improved since Kapitza's analysis i s not v a l i d near the dry patch. The most important l i m i t a t i o n of the simple two-dimensional model i s that changes i n curvature and thickness of the free surface near the dry patch are not considered. The model presented i n Chapter 5, describes the behaviour of a t h i n l i q u i d f i l m flowing by gravity on an i n c l i n e d p late where a patch has formed. The problem was divided i n t o two regions, "outer" arid "inner". In the outer region increases i n hydrostatic pressure a r i s i n g from the increase of f i l m thickness near the dry patch balance stagnation pressure. The s o l u t i o n obtained i s v a l i d for a narrow range of low Reynolds numbers. In the inner region the flow was assumed at r e s t , and the increase i n pressure i s compensated for by changes i n curvature of the free surface (surface f o r c e s ) . The estimated p r o f i l e s are wedge-shaped and d i f f e r e n t from measured p r o f i l e s . However, the flow behaviour should not be compared d i r e c t l y with experimental f i l m p r o f i l e s since they correspond to l i q u i d films at higher Re. In future work, assumptions made i n the development of the present model such as neglecting surface tension e f f e c t s i n the outer region must be a l l e v i a t e d . Thermal e f f e c t s should also be included, so that at each step a better d e s c r i p t i o n of the actual problem i s obtained. THERMOCAPILLARITY EVAPORATION cr(T) Figure 1 Some mechanisms of f i l m break-up NUCLEAR POWER PLANT HEAVY WATER COOLANT' TRANSFERS HEAT FROV URANIUM FUEL TO ORDINARY WATER IN BOILER (STEAM GENERATOR) I / • HEAT PRODUCED8Y / FISSIONING URANIUM FUEL (URANIUM) (NUCLEAR REACTION* Figure 2 Nuclear power plant 78 D R Y O U T 0 £ " ° SINGLE P H A S E S T E A M L IQUID D E F I C I E N T S P R A Y F L O W A N N U L A R F L O W S L U G F L O W B U B B L E F L O W S I N G L E P H A S E W A T E R F i g u r e 3 F l o w B o i l i n g R e g i m e s , Upward F l o w 79 Figure 4 Dry patch on a s o l i d surface 80 F e = FORCE DUE TO VAPOUR T H R U S T F d = DRAG F O R C E OVER S T E P IN FILM Fp s P R E S S U R E FORCE Fa = SURFACE F O R C E <r(Tf) - <r(Tp)cos 9 Figure 5 Forces acting at a dry patch (Ref. 22) Figure 6 Annular flow with waves (Ref. 26) 16 00 1400 12 00 I4C0 24 0 0 i - 20 OOl- 16 00 12 OOl- • 4 0 0 ' - Timo of two ptriods * 10000, f o o O - 8 001- 4 o o h g o oo «- -4 o o t - - 8 0 0 -I2 0 0 J - - I 8 0 O 1 - 14 00 Timi of two periods x 10000, t Figure 7 T r i p l e front p o s i t i o n during two waves for d i f f e r e n t values of heat fluxes (Ref. 26) 00 ho 8 • CONTACT ANGLE IN EQUILIBRIUM d A : STATIC ADVANCING ANGLE 0 R : STATIC RECEDING ANGLE Figure 8 (a) A l i q u i d drop on a h o r i z o n t a l plate (b) Hysteresis of the contact angle Figure 9 A drop on a t i l t e d p l ate 85 SOLID SURFACE AT A, APPARENT CONTACT ANGLE <f> = Q + \ff AT B, APPARENT C O N T A C T A N G L E <f>=Q-\jr Figure 10 V a r i a t i o n of apparent contact angle due to surface roughness 0 3 0-2 E £ CO CO UJ z 0 1 S T A T I O N A R Y DRY P A T C H Q = 5 5 0 0 W / m 2 T = 2 ° C P =3 62 MPa (525 PSIA) 0 I 7 6 S E C D I R E C T I O N OF F L O W • 0 0 9 4 S E C 0 0 4 8 S E C 0 3 0 6 D I S T A N C E F R O M E D G E , mm Figure 11 Film p r o f i l e s for d i f f e r e n t times, (Ref. 1) 0 9 oo ON 60 ui UJ ac. o ui a ui - i e> z < o u 4 0 20 STATIONARY DRY PATCH Re= 185 T=2 °C P= 3.62 MPa (525psia) Q = 5 5 0 0 W / m 2 8 12 TIME , s X I 0 3 16 Figure 12 Contact angle v a r i a t i o n with time, (Ref. 1) I 2 Re = 3IO T = 2 ° C P=3-62 MPa (525 PSI A) Q = I 2 0 0 0 W / m 2 • R E C E D I N G * A D V A N C I N G 0 -3 0 6 D I S T A N C E F R O M E D G E , mm Figure 13 Film p r o f i l e s for d i f f e r e n t times, (Ref. 1) oo oo C O N T A C T A N G L E , ( D E G R E E S ) ro O -fe O o CD o 2 m y> X o ro • • 6 CO > JO o m » < o m z o O o z z J> z © JO Q o II ro o o o 30 0) II II 01 ro " • o OJ °* o — ro o Z ° 3 _ ro cn "6 CO ro S T A T I O N A R Y DRY P A T C H Re = 9 4 0 T = 2 ° C P = 3-62 M P a 1525 P S I A ) Q = 6 3 6 0 0 W/m z 0 0 0 0 5 S E C 0 0 0 I 6 S E C 0 0 0 2 6 S E C 0 0-3 0 6 0 9 D I S T A N C E F R O M E D G E , mm Figure 15 Film p r o f i l e s for d i f f e r e n t times, (Ref. 1) S T A T I O N A R Y DRY P A T C H Re * 9 4 0 T = 2 ° C P = 3-62 MPa (525 P S I A ) Q = 6 3 6 0 0 W / r n 2 T I M E , S E C X I 0 3 Contact angle v a r i a t i o n with time, (Ref. 1)  o O £6 94 Figure 19 Scheme of a two dimensional flow around a dry patch. The dash lines indicate the control volume used 95 9 0 0 100 3 0 0 5 0 0 7 0 0 REYNOLDS NO. Figure 20 Pressure forces according to d i f f e r e n t models 96 Figure 21 Comparison of the t o t a l surface force (F a+F t h) according to Zuber and Staub with the surface force without thermal e f f e c t s (F ) 97 150 E \ Z X u. < CO* Ui o or o ui o < u. tc •D 10 to Z3 Z Ul o cc o u. Ul CO CO Ul cc Q. 100 50 - 5 0 - 1 0 0 - 1 5 0 • LAMINAR V E L O C I T Y P R O F I L E • 1/7 P O W E R L A W V E L O C I T Y P R O F I L E 0 0 2 0 0 3 0 0 4 0 0 R E Y N O L D S NO. 5 0 0 Figure 22 Difference between pressure force and surface force according to Zuber and Staub model for low Re ( c i r c l e s ) . Difference between pressure force and surface forces when the 1/7 power law i s used as a v e l o c i t y p r o f i l e (squares) 6 0 0 £ v. m o x 500 u. <3 CO UJ o or o UJ o < u. DC (0 CO => Z UJ o cc o u. Ul cc •=> CO CO Ul cc 0_ 4 0 0 300 2 0 0 100 roo 3 0 0 5 0 0 R E Y N O L D S N O . 7 0 0 Figure 23 Difference between pressure force and surface forces fo high Re 99 Figure 24 Comparison of the pressure force according to the two dimensional model with Zuber and Staub c r i t e r i o n . Inlet temperature = 2°C Figure 25 Scheme of the f i l m surface p r o f i l e i n the v i c i n i t y of a dry patch on an i n c l i n e d plate o o Figure 26 Film thickness p r o f i l e according to model described i n Chapter 5 R e - 10 A = 2 o = 2 ° 0=35° 8 / S o o Figure 27 Film thickness p r o f i l e according to model described Chapter 5 103 REFERENCES 1. McAdam, D.W. "An Experimental Investigation of Dry Patch Formation and S t a b i l i t y i n Thin Liquid Films," Ph.D. Thesis, The University of B r i t i s h Columbia, 1975. 2. Norman, W.S., Mclntyre, V. "Heat Transfer to a Li q u i d Film on a V e r t i c a l Surface," Trans. Inst. Chem. Eng., 38, 301, 1960. 3. Norman, W.S., Binns, D.T. "The E f f e c t of Surface Tension Changes on the Minimum Wetting Rates i n a Wetted-Rod D i s t i l l a t i o n Column," Trans. Inst. Chem. Eng., 38, 294, 1960. 4. Hewitt, G.F., Lacey, B.M.C. "The Breakdown of the Liquid F i l m i n Annular Two Phase Flow," Int. J . Heat and Mass Transfer, 8̂, 781, 1965. 5. Hartley, D.E., Murgatroyd, W. " C r i t e r i a f o r the Breakup of Thin L i q u i d Films Flowing Isothermally Over S o l i d Surfaces," Int. J . Heat and Mass Transfer, 7_, 1003, 1964. 6. Zuber, N., Staub, F.W. " S t a b i l i t y of the Dry Patch Formed i n Liqui d Films on Heated Surfaces," Int. J . Heat and Mass Transfer, 9, 897, 1966. 7. Ponter, A.B., Davies, G.A., Ross, T.K., and Thornley, P.G. "The Influence of Mass Transfer on L i q u i d Film Breakdown," Int. J . Heat and Mass Transfer, 10_, 349, 1967. 8. Thompson, T.S., Murgatroyd, W. " S t a b i l i t y and Breakdown of Li q u i d Films i n Steam Flow with Heat Transfer," Internal Report, Queen Mary College, London, 1970. 9. Fu l f o r d , G.D. "Advances i n Chemical Engineering," Academic Press, 5, 151, 1964. B i r d , R.B., Stewart, W.E., Lightfoot, E.N. Transport Phenomenon, John Wiley and Sons, Inc., 1960. Benjamin, T. Brooks. "Wave Formation i n Laminar Flow Down an Inclined Plane," J . F l u i d Mech., 2, 554, 1957. Castellana, F.S., B o n i l l a , C.F. "Ve l o c i t y Measurements and the C r i t i c a l Reynolds Number for Wave I n i t i a t i o n i n F a l l i n g Film Flow," ASME, 70-HT-32, 1970. Nusselt, W. Z. ver Deut. Inf., _60, 541, 1916. Dukler, A.E., Bergelin, O.P. " C h a r a c t e r i s t i c s of Flow i n F a l l i n g Liquid Films," Chem. Eng. Progress, _43, 557, 1952. Cook, R.A., Clark, R.H. "The<Experimental Determination of Ve l o c i t y P r o f i l e s i n Smooth F a l l i n g L i q u i d Films," Canadian Journal of Chem. Eng., _49, 412, 1971., Reynolds, A.J. Turbulent Flows i n Engineering. John Wiley and Sons, 1974. Kapitza, P.L. J. Exp. Theor. Phys. U.S.S.R., 18, 3, 1948. Levich, V.G. Physicochemical Hydrodynamics. P r e n t i c e - H a l l , Inc., 1962. Tail b y , S.R., P o r t a l s k i , S. "The Hydrodynamics of Liquid Films Flowing on V e r t i c a l Surfaces," Trans. Inst. Chem. Eng., 38, 324, 1960. Bankoff, G.S. " S t a b i l i t y of Liqu i d Flow Down and Heated Inclined P l a t e , " Int. J . Heat and Mass Transfer, 14^ 377, 1971. Bankoff, G.S. "Minimum Thickness of a Draining L i q u i d Film," Int. J . Heat and Mass Transfer, 3A, 2143, 1971. McPherson, G.D. " A x i a l S t a b i l i t y of the Dry Patch Formed i n Dryout of a Two-Phased Annular Flow," Int. J . Heat and Mass Transfer, 13^ 1133, 1970. 105 23. Wilson, S.D.R. "The S t a b i l i t y of a Dry Patch on a Wetted Wall," Int. J . Heat and Mass Transfer, 17, 1607, 1974. 24. Hsu, Y.Y., Simon, F.F., Lad, J.F. "Destruction of a Thin L i q u i d Film Flowing Over A Heated Surface," Chem. Eng. Symposium Series, 61 (No. 57), 138, 1965. 25. Simon, F.F., Hsu, Y.Y. "Thermocapillary Induced Breakdown of a F a l l i n g Film," NASA TN D-5624, 1970. 26. Mariy, A.H., E l - S h i r b i n i , A.A., Murgatroyd, W. "The E f f e c t of Waves on the Motion of the Triple-Phase Front of a Dry Patch Formed i n a Thin Motivated L i q u i d Film," Int. J . Heat and Mass Transfer, 17, 1141, 1974. 27. Barnett, P.G. "An Experimental Investigation to Determine the Scaling Laws of Forced Convection B o i l i n g Heat Transfer," AEEW R-443, 1965. 28. Thompson, R. and Macbeth, R.V. " B o i l i n g Water Heat Transfer Burnout i n a Uniformly Round Tube," A.E.E.W. - R356 , 1964 . 29. Johnson, R.E., J r . and Dettre, R.H. "Wettability and Contact Angles," Surface & C o l l o i d a l Science, 2, 1969. 30. Johnson, R.E., J r . and Dettre, R.M. "Contact Angle Hysteresis," Advances i n Chemistry Series, 43. 31. Ponter, A.B., et al. "The Measurement of Contact Angles Under Equilibrium and Mass-Transfer Conditions," Int. J . Heat and Mass Transfer, 10^ 733, 1967. 32. Huh, Chun and Scriven, L.F. "Hydrodynamic Model of Steady Movement of a S o l i d - L i q u i d - F l u i d Contact Line," Journal of C o l l o i d and Interface Science, 35, 85, 1971. 33. Dussan, V.E.B. and Davis, S.M. "On the Notion of a F l u i d - F l u i d Interface Along a S o l i d Surface," J . F l u i d Mech., 65, 71, 1974. 106 34. Ponter, A.B., et al. "The Measurement of Contact Angles Under Conditions of Heat Transfer When a Liquid Breaks on a Vertical Surface," Int. J. Heat and Mass Transfer, 10_, 1633, 1967. 35. Vallentine, H.R. Applied Hydrodynamics, Butterworths, 1967. 36. Lamb, M. Hydrodynamics, Dover, 1945. 37. Schlichting, M. Boundary Layer Theory, McGraw-Hill, 1968. 38. Hele-Shaw, M.S. Trans. Inst. Nav. Archit, 11, 25, 1898. 107 APPENDIX A EVALUATION OF THE SHEAR FORCE AND BODY FORCE FOR A KAPITZA VELOCITY PROFILE The body force per unit width applied over a control volume of • length nA and thickness h Is "B fnX fnA pgdxdy = pg 0 J 0 h^ + 0.46 hp sin(kw-cot) dx For t = 0 FB = P g h Q n A The shear force per unit width applied on a control volume whose length is nA is rn. FS " ,3u , 3v. .' 0 ^ 3 xy=0 For thin films 3u > ; > _9_v 9y 3x For t = 0 108 •nA yGp) dx J 0 9 y y=0 mA 3 u„ 1+0.6 s i n kx-0. 3 s i n kx V h o 0.46 s i n kx dx 3 u 0 rnA 1 + 0.46 s i n kx dx + nA 0.6 1 + 0.46 sin kx dx 0.3 sin^kx 1 + 0.46 s i n kx dx (1) Each i n t e g r a l of Equation 1 i s solved below. F i r s t i n t e g r a l nA 1 + 0.46 s i n kx dx = _A_ 2TT r2iTn 1 + 0.46 s i n z dz where 2TT z = — x . Then 2TT r2nTT 1 + 0.46 s i n z dz = 2TT arctan tan nTf + 0.46 1 - 0.46 arctan 0.46 V' 2 1-0.46 1 - 0.46 109 _A_ 277 L *1- 0 . 4 6 nTT + a r c t a n — 2 \ / 1- 0.46 / 0.46 arctan / 1 - 0.46 2 /1 - 0.46 2 j nA J 2 1-0.46 The second i n t e g r a l can be s p l i t i n two known i n t e g r a l s , that i s 0.6 _A_ 2TT r2nlT s i n z 1 + 0.46 s i n z dz = 0.6 _A_ 0.46 2TT r 2n7T dz 2n7T 1 + 0.46 s i n z dz 0.6 0.46 nA 1-0.46 The t h i r d i n t e g r a l can be expressed as 0.3 _A_ 0.46 2TT (•2n77 r s i n z - s m z 1 + 0.46 s i n z dz 110 One integrated i s equal to 0.3 0.46' nX 1 - ' l - 0.46 2 ] 3u QX Adding the three i n t e g r a l s and multip l y i n g them by ny the h 0 f i n a l r e s u l t i s equal to 3 u o ny - r — X • 0.783 . . . . .(2) 0 The f a c t o r 0.783 represents the value of the function $ defined by Kapitza [17] and i s considered close enough to 0.8 used i n his a n a l y s i s . $ was determined by Kapitza through a graphical method and the numerical constants that appear i n the v e l o c i t y p r o f i l e are estimated with some small error. M u l t i p l y i n g and d i v i d i n g by U Q , the RHS of Equation (2)is: •x 2 ny — - — $ — ho uo equal to , Xn p S h 0 u 0 ~ Q from Kapitza's analysis and equal to pgh^nX I l l Then F = F and the assumption of undamped flow i s equivalent to the equality of shear force and body force, both forces integrated over a distance = to a wavelength. 1 1 2 APPENDIX B DEVELOPMENT OF THE SIMPLIFIED NAVIER-STOKES EQUATIONS USED IN CHAPTER 5 In the case of a three dimensional steady incompressible flow flowing along an i n c l i n e d p late that forms an angle a with the h o r i z o n t a l Navier-Stokes equation and the continuity equation are expressed as follows: 2 2 2 3 u , 3 u , 3 u 1 3 p . 3 u , 3 u . 3 u, . . u-^— + V ^ — + W T — = -r*- + v( ~ + — ~ + — T T ) + g s i n a . . ( 1 ) 3 x 3 y 3 z p 3 x .̂ 2 . 2 . 2 ° 3 o x 8 y dz 3 v 3 v , 3 v 1 3 P x . 3 2 v 3 2 v , 3 2 v , U 3 ^ + V 3 y - + W 3 i " ~ p 3 y (T~2 7~~2 7~2 ) " 8 C O S a ' " ( 2 ) 3 x 3 y 3 z u 9 w + v 9 w + w 9 w = _ i 9 £ + v ( 9 ? | + 9 i + ^ | ) . . ( 3 ) 3 x 3 y 3 z p 3 z 3 x 3 y 3 z The system of coordinates has i t s o r i g i n i n the bottom of the plate, the x, z plane i s p a r a l l e l to the pl a t e and the y-axis i s perpendicular to them. To know the r e l a t i v e importance of the d i f f e r e n t terms i n 113 the l a t t e r equations some s u i t a b l e c h a r a c t e r i s t i c magnitudes are selected as un i t s . Let u^, v^, ŵ , L, S^, and pg cos aSm denote these c h a r a c t e r i s t i c reference magnitudes, where u^, and ŵ  are the reference v e l o c i t i e s i n x, y and z d i r e c t i o n ; L represents the s i z e of the object i n z, x di r e c t i o n s and 6^ i s the thickness at i n f i n i t y , then * u * v + w u* = — , v* = — , w* = — U l V l W l x* = - , y* = J- , z* = ^ ' P * , 5 L pg cos aS The reference v e l o c i t y u^, can be taken as the average v e l o c i t y u = g s i n aS 2 3~V that corresponds to the one-dimensional problem. These dimensionless r a t i o s are of the order of one. If they are introduced into the continuity Equation (4) u 9 u * L 9 x * 6 9 y * W + _1 9w* L 9 z * . . . .(5) If w. i s considered of the same order as u 114 H i i H * V l 3 y * 3w* ) _ L I 3 x * u<5 3 y * 3 z * f - 0 . . . . . ( 6 ) Since the term between brackets i s of the order of one, has to be of the order of uS oo Introducing the dimensionless r a t i o s into the x component Navier-Stokes equations ~L~ I Sx*' V 3 y * W 3 7 * f = " 8 L " 8 ^ + 2 - 2 - 2 1 u 9 u* , u 3 u * • u 3 u* I , • . • + v < =• + —r + —~ y ) + g s i n a . . . . (7) L 3 x * 6 Z 3 y * L 3 z * ' v L M u v M 3 x * V 3 y * W 3 z * / g L ~ v d x * , i / 6»x 3 u * , 3 u * . 3 u * I . . , 6 ~ . + ) (.-K) o + o + (— 7 > + g sxn a (—) . L 3 x * 3 y * l 3 z * u v But u can be expressed as a function of the thickness of the f i l m and of the angle, 115 g s i n a6 3V then ) u * - ^ + v* - ^ + w * J- = -3 cotan a ( - ) ^ V L I 3x* + V 3y* + w 3z* • C O t a n L 3x* As the thickness of the f i l m i s much smaller than a character- i s t i c s i z e of the obstacle 6 00 u6 (5 If a modified Reynolds number _ , ™, , 0 0 i s defined such that i s ^ V L < < 1, and i f 6 00 cotan a • -r- i s assumed of order of one, (true for small a) Equation 8 reduces to 116 Converting back to dimensional units 3L 3p ± 3V 3 2u , _ n - + : — - —=• + 3 g s i n a = 0 n « „ 3x g s i n a - 2 Pg6 cos a Sy- Afte r rearrangements 2 - 7T fe + v + g s i n a 0 . . . . . (9) 3y Equation (9) i s v a l i d with the conditions <5 00 (Re — ) < < 1 CO a ^ — Li If the dimensionless r a t i o s are replaced i n the y component of the Navier-Stokes equations. 6 ( 3 0 ,u 2 N J . 3v* , . 3v* ̂  . 3v* I 3p* T V U 3 ^ + V* 3 ^ + W* 3 ^ = " 8 C ° S a 3 ^ u 6 c o w i [6l 3 2v* . 3 2v* , 6 ~ 3 2v* Dividing by g cos a 117 6 -2 , ( °°w u N ) * 9v* , . 9v* , 9v* f 3 D * u6 , ' i S 2 2 2 ^ v f — ) 1 ) 0 0 3v* 9v* . 9v* L 2 ) T 2 9 —7~ 9 <S„ g cos a ( 9x* 9v* L 9 7* + " T. ' .2 J T 9x* 9y* - 1 (10) The f i r s t factor i n the l e f t hand side of Equation 10, can be regarded as L L g cos a L 6 -gToTa" " ( T } o~ t a n a 3^ ' oo 6oo Replacing tan a by i t s approximate value = — , then L /"w^w 1 , „ /» 2 / 6 c o 6oo Likewise the f i r s t factor i n the second term of the r i g h t hand side of Equation 10, û co n g s i n a <52 v6 1 , CO . , oo . ]_ V — ^2 • ( — 3 ^ ><—>-2 — 6 r og cos a 6 g cos a ^ tan a — < < 1 . 6 CO F i n a l l y for — « 1 Equation 10 transforms i n t o 0 - - i ay* 1 Converting back to dimensional units 1 3 p P 37 = - g c o s a with the r e s t r i c t i o n s 6 00 (Re) (-£")<< 1 6 o tan a ^ a ^ — The z component of the Navier-Stokes equations transforms into -2 u . 3 w * . . 3 w * , . 3 w * i * + v * 7 r ~ - + w * 3 x * 3y" 3 z * gS cos a 3 £ * 3 z * + V 2 3 w * 3 x * 2 ^ V « 2 ^ 3 w * , °° 3 w * 2 ^ L ; „ .2 3 y * 3 z * M u l t i p l y i n g l e f t hand side and r i g h t hand side by 6 2 Vu 119 v u I Vu 3 x * 3 z * 9 y * Z A f t e r rearrangements and with the assumptions that 6 <5 (Re — ) < < 1 , tan a ^ — , , 9p_*_ 3 w * 3z* _ .2 3 y * Converting to dimension units 1 3 p 3 2w p 3 i = V T T ' • • • • ( " ) 3 y APPENDIX C PROPERTIES OF CARBON DIOXIDE  F i g u r e 29 Density of Saturated C0 2 Vap. F i g u r e 30 Thermal C o n d u c t i v i t y of CO K3    Temperature , °C F i g u r e 3 4 Vapor Pressure of C0 2 S3 ON

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