UNCORRECTED PROOF ARTICLE IN PRESS. HFF 6483 No. of Pages 7, DTD = November 2003 Disk used. 2 Prediction of dynamic contact angle histories

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1 2 November 23 Disk used 2 Prediction of dynamic contact angle histories 3 of a bubble growing at a wall 4 Cees W.M. van der Geld * 5 Department of Mechanical Engineering, Division Thermo Fluids Engineering, Eindhoven University of Technology, 6 P.O. Box 53, 56 MB Eindhoven, The Netherlands 7 Received 7 July 23; accepted 8 October 23 8 Abstract 9 A fast growing boiling bubble at the verge of detaching from a plane wall is usually shaped as a truncated sphere, and experiences various hydrodynamic forces due to its expansion and the motion of its center of mass. In a homogeneous flow field, one of the forces is the so-called bubble growth force that is essentially due to inertia. This force is usually evaluated with the aid of 2 approximate expressions [Int. J. Heat Mass Transfer 36 (993) 65, Int. J. Heat Mass Transfer 38 (995) 275]. In the present study 3 an exact expression for the expansion force is derived for the case of a truncated sphere attached to a plane, infinite wall. The 4 Lagrange Thomson formalism is applied. Two Euler Lagrange equations are derived, one governing the motion of the center of 5 mass, the other governing expansion a kind of extended Rayleigh Plesset equation. If a constitutive equation for the gas vapor 6 content of the bubble is given, initial conditions and these two differential equations determine the dynamics of the growing 7 truncated sphere that has its foot on a plane, infinite wall. Simulations are carried out for a given expansion rate to predict the 8 history of the dynamic contact angle. The simulations increase the understanding of mechanisms controlling detachment, and yield 9 realistic times of detachment. 2 Ó 23 Elsevier Inc. All rights reserved Introduction 23 Many criteria for bubble or droplet size at detach- 24 ment are based on a force balance and on the assump- 25 tion, often implicitly made, that detachment occurs 26 when no shape can be found that satisfies the force 27 balance. Often a bubble is modelled as part of a sphere 28 with radius R, with actual details near the bubble foot 29 neglected. Indeed the shape of boiling bubbles, rapidly 3 growing in water at ambient pressure, or at higher 3 pressures, is observed to be close to that of a truncated 32 sphere (Stralen and Cole, 979). The rapidity of the 33 growth apparently allows the contact angle to deviate 34 from the static value. 35 In convective boiling, the equations governing bubble 36 growth and motion have to comprise terms that account 37 for hydrodynamics. It is obvious, that in order to derive 38 an accurate detachment criterion from a force balance, 39 all forces should be accurately known. If a mechanism is acting that can not be described precisely, approximate expressions, one or two fitting parameters and comparison with experiments might offer a way out. Such fitting procedures have indeed been applied (Klausner et al., 993; Helden et al., 995). One of the forces that has been known only imprecisely, and has been fitted to experiments, is related to the added masses of a growing sphere near a plane wall. It is usually denoted with Ôbubble growth force (Klausner et al., 993; Geld, 2). This force is further discussed, and actually quantified, below. But what would happen if one force component would not have been identified, while only an approximate expression for another force is used, comprising a fitting parameter? This fitting parameter would probably be given an unrealistic value, and possess a high inaccuracy. The inaccuracy would increase with increasing range of controlling process conditions, such as the liquid velocity. It is therefore important to aim at a complete description of the hydrodynamics involved. In the present study, the liquid vapor interface is taken to have the shape of a truncated sphere with its * Tel.: ; fax: address: c.w.m.v.d.geld@tue.nl (C.W.M. van der Geld). International Journal of Heat and Fluid Flow xxx (23) xxx xxx X/$ - see front matter Ó 23 Elsevier Inc. All rights reserved. doi:.6/j.ijheatfluidflow.23..2

2 2 November 23 Disk used 2 C.W.M. van der Geld / Int. J. Heat and Fluid Flow xxx (23) xxx xxx 62 foot attached to a plane wall. Two parameters are re- 63 quired to specify a shape: radius R and height h of the 64 center of the sphere above the wall. Note that prior to 65 detachment, h differs from the height of the center of 66 mass of the bubble above the wall, h CM. Shapes close to 67 truncated spheres are important in practice, see above, 68 and prediction of shape history corresponds to predic- 69 tion of the history of the dynamic contact angle. It is 7 noted that boiling conditions exist in which bubble 7 growth is relatively slow and in which asymmetric 72 interface distortion by pressure differences and shear in 73 the direction of the liquid flow occurs. Bubbles may then 74 even be found to slide over the heater surface. The 75 analysis of the present paper does not apply to these 76 conditions. The theoretical analysis of this paper can 77 however be extended to account for more complex 78 deformation, see also Geld and Kuerten (2). In the 79 case of a sliding bubble, the contact angle needs to vary 8 along the contact line such, that the net force compo- 8 nent in the plane of the bubble foot exactly balances the 82 net fluid stress, and possibly gravitation, exerted in this 83 plane. In boiling applications, radius R is usually a 84 known function of time t. Often R is proportional to t n, 85 with n :5. This paper will therefore consider situa- 86 tions when the bubble radius history is prescribed by 87 such a time dependency. An equation governing h will 88 be presented and used for simulations that may help our 89 understanding of the mechanisms involved in bubble 9 detachment. The bubble content is either pure vapor, 9 non-condensing gas, or a mixture of both. The pressure 92 inside, p b is taken to be homogeneous. 93 Among the forces which act on a bubble attached to a 94 wall along which a boundary layer flows are the fol- 95 lowing ones: 96 a force related to stochastic features of turbulence, 97 one related to vorticity in the approaching liquid, 98 one due to Marangoni convection and 99 one mass transfer across the liquid vapor interface. These forces are neglected in the present study, but all forces which are not related to vorticity in the flow are 2 computed exactly. It is noted that these forces, described 3 with the aid of a so-called added mass tensor, retain 4 their value in flows when vorticity does play a role. 5 Whether vorticity would be generated at the body sur- 6 face or would be present in the approaching rotational 7 flow, it would not affect the added mass coefficients 8 (Howe, 995). In this sense, a complete description of 9 bubble dynamics near detachment is aimed at. The only agencies that might be important in practice and have been omitted from the analysis are the contributions to 2 drag and lift by the vorticity in the main flow. However, 3 the hydrodynamic forces that are computed without 4 vorticity already yield a notable lift force on a fastly 5 growing truncated sphere. In the present study, gravity is supposed to act perpendicular to the wall. Capillary forces related to motion of the contact line and expansion of the liquid vapor interface are accounted for. The drag force is not computed in the simulations of this paper since in lowviscous liquids it is usually negligible (Geld, 22; Geld, 2). The analytical approach of this paper is an extension of the one used in previous studies (Geld, 2; Geld, 22). 2. Governing equations Two parameters describe the shape of the truncated sphere under consideration, see Section : radius R and center height h ¼ z=2. These parameters are taken to be independent. The independent variation of height h and radius R allows for non-equilibrium values of the contact angle h, see Fig.. Let U be defined as dh=dt, R def ¼ d 2 R=dt 2, g denotes the gravity constant. Dr denotes the difference in surface energy densities of wall area in contact with vapor and with water, and r is the surface tension coefficient pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of water-vapor. The radius of the bubble foot, R 2 h 2, will be denoted as r foot. The derivation of the equations governing R and h starts with a computation of the total energy dissipation rate in the liquid, by integration of the mechanical energy balance. Next, the work terms related to _R and U, that are independent, are separated, yielding two equations. The velocity field in the liquid and its kinetic energy, T, are assessed with the aid of a velocity potential in a similar way as done in previous work (Geld, 2; Geld, 22). It turns out that energy T can be written in the form T ¼ 2 3 pr3 q L fa 33 U 2 þ w x;3 U _R þ traceðbþ _R 2 g ðþ where a 33, w x;3, trace(b) are added mass coefficients, to be quantified below. With the Lagrange Thomson approach, expressions for the hydrodynamic work in terms of T are derived. The combining of the two correθ p A w R h σ Fig.. Schematic of bubble geometry, areas and contact angle. A b

3 2 November 23 Disk used C.W.M. van der Geld / Int. J. Heat and Fluid Flow xxx (23) xxx xxx 3 53 sponding expressions for the work in U yields Eq. (3), 54 below. The work in _R yields another governing equa- 55 tion, futher discussed in Section The same approach, but with a different velocity 57 potential, has been applied to a sphere growing near a 58 plane wall in a previous study (Geld, 22). 59 The added mass coefficients are found to depend only 6 on R=z. The method to compute them changes essen- 6 tially at R=z ¼. This is why Fig. 2 gives only values for 62 shapes up to this point. Note that at R=z ¼ :5 the 63 bubble is spherical The Lagrange Thomson method yields terms in h and R as well as the so-called hydrodynamic lift of the 66 bubble, L ift. It is given by! L def ift ¼ R _R z R U U oa 33 z or=z R R z _ 2 otraceðbþ z or=z þ ow x;3 R þ 3 _ 2 R or=z R a 33U þ w x;3 _R=2 ð2þ 68 The first term on the RHS of Eq. (2) is denoted term- in 69 the following; similar for terms-2 and -3. The governing 7 equation of h reads: 4 3 pr3 q L L ift a 33 h 2 w x;3 R ¼ Dp ov b oh þ r oa b oh þ Dr oa w oh q oh CM LgV b oh ð3þ 72 with Dp given by Dp ¼ p b p w þ q L gh CM. Here p w de- 73 notes the hydrostatic pressure at the wall and h CM de- 74 notes the height of the center of mass of the truncated 75 sphere above the wall; V b denotes the bubble volume, A w 76 the area of the bubble in contact with the plane wall, and 77 A b the remaining area, at the liquid vapor interface. The 78 contact angle, h, follows from h ¼ arccosðh=rþ. The 79 importance of the terms of Eq. (3) is studied in Section 3. 8 It can be shown that forces that have been used in the 8 past, in particular the so-called surface tension force, F r, 82 and the so-called pressure correction force, F corr (Chesters, 978), are accounted for by Eq. (3). The demonstration of this, as well as the derivation of Eq. (3), will be presented in a subsequent paper. Added mass coefficients are a convenient tool for the fast computing of system dynamics. It might be tedious to quantify the tensor for all the parameters on which it depends, but prediction of dynamics is straightforward and rapid when this work is done, and the results are for example fitted by polynomial expressions. Prediction of system dynamics then only requires integration in time. In the simulations of Section 3 this is done numerically, using a combined Taylor and multi-step Adams Bashforth algorithm. Initial conditions are taken to be corresponding to the equilibrium shape of the truncated sphere with radius R, i.e. Dr and bubble pressure are initially such that the Young equation Dr ¼ r cosðhþ and the Laplace equation Dp ¼ 2r=R are satisfied. The initial contact angle is therefore the static one, h,and any predicted deviation of h from h is due to the hydrodynamics of a growing bubble with the shape of a truncated sphere. This implies that Eq. (3) should yield the static contact angle if no fluid motion would occur. That this indeed is the case is shown below. Thermodynamic equilibrium of a truncated sphere in the absence of gravity can be investigated by variation of the grand thermodynamic potential, also named grand canonical, with respect to the two parameters R and h. If the chemical potentials of both the liquid and vapor/gas phases are taken to be constant, this yields the following two equations: Dp ov b oh þ r oa b oh þ Dr oa w oh ¼ trace(β) ð4þ ¼ ð5þ Dp ov b or þ r oa b or þ Dr oa w or Here Dp denotes the pressure drop across this interface. 25 It is straightforward to show that the above equations 26 yield both the Young equation for the static contact 27 angle and the Laplace equation Dp ¼ 2r=R. Eq. (3) 28 which governs h can be viewed as a direct extension of 29 Eq. (4). Eq. (5) is related to an extended Rayleigh 22 Plesset equation that governs R and is discussed in 22 Section Simulations and analysis 223 α 33 ψ x, R/(2 h) Fig. 2. Added mass coefficients. Actual boiling bubbles contain a small quantity of inert gases, remnants from the cavity, and the bubble pressure, p b, is the sum of the partial pressures of both components: vapor and inert gas. Actual vapor bubble behavior therefore combines features of vapor bubble Here only ðr=zþ, but for example with deformation more parameters are required

4 2 November 23 Disk used 4 C.W.M. van der Geld / Int. J. Heat and Fluid Flow xxx (23) xxx xxx 229 dynamics and gas bubbles dynamics, in a ratio deter- 23 mined by the concentration of gas. To understand the 23 behavior of actual vapor bubbles, it is sufficient to study 232 pure vapor and gas bubbles separately, as is done in the 233 following. 234 The behavior of a bubble filled with a constant 235 quantity of inert gas will be examined first, because its 236 slow growth behavior helps to understand the physics of 237 the biggest force components active during growth of a 238 vapor bubble. The constitutive equation of the gas 239 content is taken to be p b V c b ¼ c, c being either c p=c v or, 24 dependent on the process, respectively, isentropic and 24 isothermal; c is a constant. The higher the value of c, the 242 faster the response on volume changes, but otherwise 243 this parameter has little influence. 244 If at time zero the bubble radius is increased at a rate 245 of :5 R per s (c ¼ :4), the volume of the truncated 246 sphere is increased, which has two consequences, see 247 Fig. 3: 248. The initial jump in _R causes an instability of height h. 249 Oscillations at cycle time T result; T 2:2 ms in Fig The motion is easily sustained for many thousands 25 of cycles in the absence of viscous damping R/R Velocity (mm/s) At a longer timescale, i.e. at times ~t that satisfy ~t=t the value of h changes gradually such, that the volume growth is counteracted. For this, the oscillation-mean of the velocity U is negative, i.e. towards the wall, which for constant radius R would reduce the volume. The hydrodynamic lift is negligible in this case, because of the slow growth; only the forces related to volume and surface area changes contribute. The gradual change of h, at the longer timescale, is caused by the term comprising da b =dh being slightly more negative than the sum of all other forces. The da b =dh-term counteracts increase of bubble surface area by causing a decrease of height. Similarly, height h is found to increase if the bubble radius is decreased at the same rate. The initial estimate for the bubble pressure does not affect the gradual change of h if it is close to the hydrostatic pressure at the wall plus 2r=R. This case shows how important the surface area and volume terms in the force balance are for bubble dynamics. Each of them is usually at least one order of magnitude bigger than either one of the remaining forces, viz the hydrodynamic lift 4 3 pr3 q L L ift and the forces σ da b /dh U h =.42 mm R =.7 mm g h CM / h Other accelerations are negligible h CM /R p dv b /dh σ da w /dh θ/θ h/r r foot /r foot, Fig. 3. Simulation of bubble growth with constant content of air.

5 2 November 23 Disk used C.W.M. van der Geld / Int. J. Heat and Fluid Flow xxx (23) xxx xxx 5 oh 275 q L gv CM b oh and 4 3 pr3 q L 2 w x;3 R, even in the case of rapid 276 bubble growth. Minor differences between the two sur- 277 face area and the volume terms may account for main 278 trends in h and correspond to differences between the 279 instantaneous contact angle, h, and the static value, h. 28 The oscillations on the smaller time scale hardly affect 28 the detachment process, and would not occur if the 282 bubble would be mainly vapor at constant saturation 283 temperature. Detachment is therefore more conveniently 284 studied by keeping the bubble pressure constant. This is 285 the reason why in the simulations below the bubble 286 pressure has been kept constant, with c ¼ to model 287 isothermal behavior. 288 In order to facilitate the analysis of the effect of 289 various terms in the governing equation, a Ôstandard 29 simulation case is defined, to which other tests will be 29 referred and compared. In the standard process, bubble 292 radius R is growing steadily, at a rapid p rate shown in the 293 first plot of Fig. 4, according to R / ffiffiffiffiffiffiffiffiffiffiffi t þ t. This case 294 corresponds to actual vapor bubble growth at a pressure 295 at the wall, p w, of. MPa. 296 The three major accelerations, due to the volume and 297 surface area changes are gathered in subplot (,2) of Fig Only because of the rapid growth selected, the 299 acceleration due to lift has the same order of magnitude, Velocity (mm/s) h =.42 mm R =.7 mm R/R see subplot (3,). In all cases considered lift exceeds the remaining acceleration components, i.e. g oh CM oh and 2 w x;3 R. The last term, 2 w x;3 R, is usually the smallest in all the simulations of this paper. The initial bubble pressure can be chosen such that the volume change term, denoted with Dp dv b =dh in Fig. 4. However, in the standard case the bubble pressure is assumed to equal the hydrostatic pressure at the wall, p w, plus 2r=R. This standard case therefore corresponds to thermodynamic equilibrium at the inception of growth of a nucleus. This arbitrariness in the selection of the initial bubble pressure, p b;, is a consequence of the assumption of a homogeneous bubble pressure and a constant mean curvature of the interface, j. If evaporation and hydrodynamic stresses could be neglected, the actual local mean curvature should satisfy the Young Laplace equation jr ¼ Dp þ gdqx z ð6þ at each height X z of the liquid vapor interface. This equation can obviously never be satisfied with a constant bubble radius. Fortunately, the trends of the accelerations in simulations like the one shown in Fig. 4 are not affected by a slight change of initial bubble pressure. σ da b /dh U g h CM / h ψ x3 d 2 R/dt 2 /2 L ift h CM /R p dv b /dh h/r σ da w /dh θ/θ r foot /r foot, Fig. 4. Simulation of rapid vapor bubble growth, isothermal and at constant bubble pressure.

6 2 November 23 Disk used 6 C.W.M. van der Geld / Int. J. Heat and Fluid Flow xxx (23) xxx xxx 324 When the bubble is growing, its center of mass moves 325 upward, which yields increasing value of h=r, as shown 326 by subplot (2,2) in Fig. 4. The Dp dv b =dh-acceleration 327 is proportional to R ð h 2 =R 2 Þ, and is therefore 328 decreasing. At the same time, also the surface-area term 329 labeled with rda b =dh increases, since this term is 33 negative and proportional to R 2. The other gravity- 33 related term increases since it is proportional to 332 dh CM =dh. This derivative is positive for R ¼ :7 mm, 333 and increasing with increasing R and with increasing h. 334 Near detachment, the lift is negative, implying a 335 hydrodynamic force directed towards the wall. The main 336 contribution to the lift in this situation is given by term- 337 of Eq. (2). The value of U=z _R=R in it is positive near 338 detachment. In the course of time, velocity U increases, whereas simultaneously R z decreases towards.5, when the gradient oa 33 =o R z attains its most negative value. The 34 gradient and term- are therefore negative. The main 342 positive component of the hydrodynamic force is term of Eq. (2) near detachment, but it is only about 2 % of 344 the above negative term. This shows that lift counteracts 345 motion away from the wall near detachment. The 346 dependency of a 33 on R z is largely responsible for the 347 essential dynamics near bubble detachment at constant 348 bubble pressure. 349 As a result of the negative lift, also the net accelera- 35 tion du =dt becomes negative near detachment. The 35 velocity of the center of mass, however, is still positive 352 and inertia is sufficiently large to cause detachment. 353 Detachment actually occurs in the simulations if h ¼ R 354 and r foot ¼. Prior to actual detachment, the ratio of the 355 radius of the bubble foot r foot to its initial value rapidly 356 decreases, see subplot (3,2) of Fig. 4. Simultaneously, 357 the instantaneous contact angle is predicted to deviate 358 considerably from the static value, h, see Fig. 4. This is 359 merely due to the imposing of the shape of a truncated 36 sphere, whereas actually often some kind of a neck is 36 formed to connect wall and main bubble volume. The 362 actual shape of a detaching vapor bubble may therefore 363 be different from that of a truncated sphere, but this will 364 have little effect on the total predicted detachment time 365 since the last stage of contraction of the bubble foot 366 happens rapidly. 367 The above standard case yields an acceleration and 368 detachment history that is rather typical for boiling 369 bubbles. Other test cases allow for the following con- 37 clusions. The total time of detachment increases if, for 37 initial velocity U ¼ and for the same growth rate and 372 the same initial height h, initial radius R is increased. 373 This trend is easily understood from the dependencies 374 on R of the major acceleration contributions, as dis- 375 cussed above. The total time of detachment, t d, hardly 376 depends on the initial value h, since with increasing 377 initial height the value of _U decreases, while the distance 378 the bubble needs to be elevated in order to reach lift-off 379 decreases. Time t d is decreased by increasing the initial velocity U, by the time it would take to reach the value of U if the initial velocity would have been zero. As expected, the total time of detachment depends on the growth rate history and on gravity. This time t d is about doubled if the gravitation constant is chosen to be )9.82 rather than However, with gravitation directed upward rather than downward, detachment still occurs. This shows that detachment against gravity and subsequent downward motion of the bubble away from the solid wall are possible in practical circumstances. Rapid growth detachment against gravity was experimentally observed, see for example Janssen and Stralen (98). It is interesting to observe that the hydrodynamic lift force and gravity are relatively small near detachment, but still affect detachment profoundly. This will be further investigated in a subsequent paper. 4. Conclusions a on R accounts for the main hydrodynamic action Detachment of a vapor bubble from a plane, solid 398 wall has been studied theoretically. The vapor liquid 399 interface shape has been approximated by that of a 4 truncated sphere with radius R. A governing equation 4 for height h above the wall has been presented. The 42 forces related to gravity and surface energy densities are 43 found to be major contributions. They manifest them- 44 selves via an oscillatory motion of h if gas residues fill up 45 a substantial part of the bubble volume. The surface 46 energy densities allow for the computation of the dy- 47 namic contact angle. 48 The hydrodynamic forces computed comprise the lift 49 force related to the added masses of a growing bubble 4 moving away from the wall. The hydrodynamics are 4 fully controlled by the dependencies of three added mass 42 coefficients on a single parameter, R=ð2hÞ. Exact 43 expressions for these coefficients have been derived and 44 used for computation. There is no need anymore to 45 adjust a fitting parameter of the so-called Ôbubble 46 growth force (Klausner et al., 993) to experimental 47 data. 48 The simulations presented have elucidated the 49 mechanisms which are involved in the dynamics of 42 bubble detachment. Apart from lift, the dependency of h near detachment at constant bubble pressure. 423 In the simulations of this paper, the time history of 424 radius R has been prescribed in order to mimic the 425 physics of actual vapor bubble growth and to highlight 426 the effect of growth on detachment. However, the vari- 427 ational approach of this study also yields a governing 428 equation for R, which can be considered as a kind of 429 Rayleigh Plesset equation for a truncated sphere at a 43 plane wall. 43

7 2 November 23 Disk used C.W.M. van der Geld / Int. J. Heat and Fluid Flow xxx (23) xxx xxx The approach followed in this paper, based on the use 433 of generalized coordinates, can also be applied to predict 434 deformation of a bubble in the vicinity of a plane wall 435 (Geld and Kuerten, 2). In many practical boiling 436 situations deformation can be neglected, however. 437 References 438 Chesters, A., 978. Modes of bubble growth in the slow-formation 439 regime of nucleate pool boiling. Int J. Multiphase Flow 4, Geld, C.v.d., 2. A note on the bubble growth force. Multiphase Sci. 44 Technol. 2 (3,4), Geld, C.v.d., 22. On the motion of a spherical bubble deforming 443 near a plane wall. J. Eng. Math. 42, 9 8. Geld, C.v.d., Kuerten, J., 2. Deformation and motion of a bubble near a plane wall. In: Michaelides, S. (Ed.), Fourth Int. Conf. on Multiphase Flow, New Orleans, pp. 2. CD-Rom. 446 Helden, W.v., Geld, C.v.d., Boot, P., 995. Forces on bubbles growing 447 and detaching in flow along a vertical wall. Int. J. Heat Mass 448 Transfer 38 (), Howe, M., 995. On the force and moment of a body in an incompressible fluid, with application to rigid bodies and bubbles 45 at low and high Reynolds numbers. Quart. J. Mech. Appl. Math , Janssen, L., Stralen, S.v., 98. Bubble behavior on and mass transfer to an oxygen evolving transparent nickel electrode in alkaline 455 solution. Electrochim. Acta 26, Klausner, J., Mei, R., Bernard, D., Zeng, L., 993. Vapor bubble 457 departure in forced convection boiling. Int. J. Heat Mass Transfer (3), Stralen, S.v., Cole, R., 979. In: Boiling Phenomena, vols. and 2. McGraw-Hill, Hemisphere, Washington