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Prediction of dynamic contact angle histories of a bubble growing at a wall
International Journal of Heat and Fluid Flow 25 (2004) 74–80 www.elsevier.com/locate/ijhff
Prediction of dynamic contact angle histories of a bubble growing at a wall Cees W.M. van der Geld
*
Department of Mechanical Engineering, Division Thermo Fluids Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Received 17 July 2003; accepted 8 October 2003
Abstract 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 approximate expressions [Int. J. Heat Mass Transfer 36 (1993) 651, Int. J. Heat Mass Transfer 38 (1995) 2075]. In the present study an exact expression for the expansion force is derived for the case of a truncated sphere attached to a plane, infinite wall. The Lagrange–Thomson formalism is applied. Two Euler–Lagrange equations are derived, one governing the motion of the center of mass, the other governing expansion a kind of extended Rayleigh–Plesset equation. If a constitutive equation for the gas–vapor content of the bubble is given, initial conditions and these two differential equations determine the dynamics of the growing truncated sphere that has its foot on a plane, infinite wall. Simulations are carried out for a given expansion rate to predict the history of the dynamic contact angle. The simulations increase the understanding of mechanisms controlling detachment, and yield realistic times of detachment. 2003 Elsevier Inc. All rights reserved. Keywords: Contact angle; Boiling; Bubble growth; Added mass; Euler–Lagrange; Lift
1. Introduction Many criteria for bubble or droplet size at detachment are based on a force balance and on the assumption, often implicitly made, that detachment occurs when no shape can be found that satisfies the force balance. Often a bubble is modelled as part of a sphere with radius R, with actual details near the bubble foot neglected. Indeed the shape of boiling bubbles, rapidly growing in water at ambient pressure, or at higher pressures, is observed to be close to that of a truncated sphere (Stralen and Cole, 1979). The rapidity of the growth apparently allows the contact angle to deviate from the static value. In convective boiling, the equations governing bubble growth and motion have to comprise terms that account for hydrodynamics. It is obvious, that in order to derive an accurate detachment criterion from a force balance, *
Tel.: +31-40-2472923; fax: +31-40-2475399. E-mail address: [email protected] (C.W.M. van der Geld).
0142-727X/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2003.10.002
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., 1993; Helden et al., 1995). 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., 1993; Geld, 2000). 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.
C.W.M. van der Geld / Int. J. Heat and Fluid Flow 25 (2004) 74–80
In the present study, the liquid–vapor interface is taken to have the shape of a truncated sphere with its foot attached to a plane wall. Two parameters are required to specify a shape: radius R and height h of the center of the sphere above the wall. Note that prior to detachment, h differs from the height of the center of mass of the bubble above the wall, hCM . Shapes close to truncated spheres are important in practice, see above, and prediction of shape history corresponds to prediction of the history of the dynamic contact angle. It is noted that boiling conditions exist in which bubble growth is relatively slow and in which asymmetric interface distortion by pressure differences and shear in the direction of the liquid flow occurs. Bubbles may then even be found to slide over the heater surface. The analysis of the present paper does not apply to these conditions. The theoretical analysis of this paper can however be extended to account for more complex deformation, see also Geld and Kuerten (2001). In the case of a sliding bubble, the contact angle needs to vary along the contact line such, that the net force component in the plane of the bubble foot exactly balances the net fluid stress, and possibly gravitation, exerted in this plane. In boiling applications, radius R is usually a known function of time t. Often R is proportional to tn , with n 0:5. This paper will therefore consider situations when the bubble radius history is prescribed by such a time dependency. An equation governing h will be presented and used for simulations that may help our understanding of the mechanisms involved in bubble detachment. The bubble content is either pure vapor, non-condensing gas, or a mixture of both. The pressure inside, pb is taken to be homogeneous. Among the forces which act on a bubble attached to a wall along which a boundary layer flows are the following ones: • • • •
a force related to stochastic features of turbulence, one related to vorticity in the approaching liquid, one due to Marangoni convection and 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 computed exactly. It is noted that these forces, described with the aid of a so-called added mass tensor, retain their value in flows when vorticity does play a role. Whether vorticity would be generated at the body surface or would be present in the approaching rotational flow, it would not affect the added mass coefficients (Howe, 1995). In this sense, a complete description of 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 drag and lift by the vorticity in the main flow. However, the hydrodynamic forces that are computed without
75
vorticity already yield a notable lift force on a fastly 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, 2002; Geld, 2000). The analytical approach of this paper is an extension of the one used in previous studies (Geld, 2000; Geld, 2002).
2. Governing equations Two parameters describe the shape of the truncated sphere under consideration, see Section 1: 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. 1. Let U be defined as dh=dt, € def R ¼ d2 R=dt2 , 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 of ffi water-vapor. The radius of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bubble foot, R2 h2 , will be denoted as rfoot . 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, 2000; Geld, 2002). It turns out that energy T can be written in the form 2 T ¼ pR3 qL fa33 U 2 þ wx;3 U R_ þ traceðbÞR_ 2 g ð1Þ 3 where a33 , wx;3 , trace(b) are added mass coefficients, to be quantified below. With the Lagrange–Thomson
∆p
θ
Aw
Ab
R h
∆σ
Fig. 1. Schematic of bubble geometry, areas and contact angle.
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approach, expressions for the hydrodynamic work in terms of T are derived. The combining of the two corresponding expressions for the work in U yields Eq. (3), below. The work in R_ yields another governing equation, futher discussed in Section 4. The same approach, but with a different velocity potential, has been applied to a sphere growing near a plane wall in a previous study (Geld, 2002). The added mass coefficients are found to depend only on R=z. The method to compute them changes essentially at R=z ¼ 1. This is why Fig. 2 gives only values for shapes up to this point. Note that at R=z ¼ 0:5 the bubble is spherical. The Lagrange–Thomson method yields terms in €h € as well as the so-called hydrodynamic lift of the and R bubble, Lift . It is given by ! R R_ U oa33 R _ 2 1 otraceðbÞ def R Lift ¼ U z R z z oR=z oR=z z R_ 1 owx;3 _ a33 U þ wx;3 R=2 þ ð2Þ þ3 R 2 R oR=z The first term on the RHS of Eq. (2) is denoted term-1 in the following; similar for terms-2 and -3. The governing equation of h reads: 4 3 1 € € pR qL Lift a33 h wx;3 R 3 2 oVb oAb oAw ohCM ¼ Dp þr þ Dr qL gVb ð3Þ oh oh oh oh with Dp given by Dp ¼ pb pw þ qL ghCM . Here pw denotes the hydrostatic pressure at the wall and hCM denotes the height of the center of mass of the truncated sphere above the wall; Vb denotes the bubble volume, Aw the area of the bubble in contact with the plane wall, and Ab the remaining area, at the liquid–vapor interface. The contact angle, h, follows from h ¼ arccosðh=RÞ. The importance of the terms of Eq. (3) is studied in Section 3.
5 trace(β)
4 3 2
It can be shown that forces that have been used in the past, in particular the so-called surface tension force, Fr , and the so-called pressure correction force, Fcorr (Chesters, 1978), 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, 1 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 R0 , 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, h0 , and any predicted deviation of h from h0 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
oVb oAb oAw þr þ Dr ¼0 oh oh oh
ð4Þ
oVb oAb oAw þr þ Dr ¼0 ð5Þ oR oR oR Here Dp denotes the pressure drop across this interface. It is straightforward to show that the above equations yield both the Young equation for the static contact angle and the Laplace equation Dp ¼ 2r=R. Eq. (3) which governs h can be viewed as a direct extension of Eq. (4). Eq. (5) is related to an extended Rayleigh– Plesset equation that governs R and is discussed in Section 4. Dp
1
3. Simulations and analysis
0 –1 –2 0.5
α 33
ψ x,3 0.6
0.7
0.8 R/(2 h)
Fig. 2. Added mass coefficients.
0.9
Actual boiling bubbles contain a small quantity of inert gases, remnants from the cavity, and the bubble 1 1 Here only ðR=zÞ, but for example with deformation more parameters are required.
C.W.M. van der Geld / Int. J. Heat and Fluid Flow 25 (2004) 74–80
pressure, pb , is the sum of the partial pressures of both components: vapor and inert gas. Actual vapor bubble behavior therefore combines features of vapor bubble dynamics and gas bubbles dynamics, in a ratio determined by the concentration of gas. To understand the behavior of actual vapor bubbles, it is sufficient to study pure vapor and gas bubbles separately, as is done in the following. The behavior of a bubble filled with a constant quantity of inert gas will be examined first, because its slow growth behavior helps to understand the physics of the biggest force components active during growth of a vapor bubble. The constitutive equation of the gas content is taken to be pb Vbc ¼ c, c being either cp =cv or 1, dependent on the process, respectively, isentropic and isothermal; c is a constant. The higher the value of c, the faster the response on volume changes, but otherwise this parameter has little influence. If at time zero the bubble radius is increased at a rate of 0:5 R0 per s (c ¼ 1:4), the volume of the truncated sphere is increased, which has two consequences, see Fig. 3:
Fig. 3. The motion is easily sustained for many thousands of cycles in the absence of viscous damping. 2. At a longer timescale, i.e. at times ~t that satisfy ~t=T 1 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 dAb =dh being slightly more negative than the sum of all other forces. The dAb =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=R0 . 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
1. The initial jump in R_ causes an instability of height h. Oscillations at cycle time T result; T 2:2 ms in
200
m/s2
1.003
h0 = 0.42 mm R0 = 0.7 mm
1.002
0
– ∆ σ dA w /dh
1.001
0
1
2
3
4
–400
– σ dA b /dh 0
1
Velocity (mm/s)
Time (ms) 0.75
0
0.7
–2
0.65
–4
0.6
U 0
1
2
3
4
4
0.55
3
4
h/R0
0
1
2
Time (ms) 1.015
10
m/s 2
3
hCM /R0
Time (ms)
θ /θ0
1.01
g ∂ hCM / ∂ h 5
1.005
Other accelerations are negligible 0
2
Time (ms)
2
–6
∆ p dVb /dh
–200
R/R0 1
77
0
1
2
Time (ms)
3
rfoot /rfoot,0
1 4
0.995
0
1
2
Time (ms)
Fig. 3. Simulation of bubble growth with constant content of air.
3
4
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C.W.M. van der Geld / Int. J. Heat and Fluid Flow 25 (2004) 74–80
forces, viz the hydrodynamic lift 43 pR3 qL Lift and the € even in the case of forces qL gVb ohohCM and 43 pR3 qL 12 wx;3 R, rapid bubble growth. Minor differences between the two surface area and the volume terms may account for main trends in h and correspond to differences between the instantaneous contact angle, h, and the static value, h0 . The oscillations on the smaller time scale hardly affect the detachment process, and would not occur if the bubble would be mainly vapor at constant saturation temperature. Detachment is therefore more conveniently studied by keeping the bubble pressure constant. This is the reason why in the simulations below the bubble pressure has been kept constant, with c ¼ 1 to model isothermal behavior. In order to facilitate the analysis of the effect of various terms in the governing equation, a Ôstandard’ simulation case is defined, to which other tests will be referred and compared. In the standard process, bubble radius R is growing steadily, at a rapidp rate shown in the ffiffiffiffiffiffiffiffiffiffiffi first plot of Fig. 4, according to R / t þ t0 . This case corresponds to actual vapor bubble growth at a pressure at the wall, pw , of 0.1 MPa. The three major accelerations, due to the volume and surface area changes are gathered in subplot (1,2) of Fig. 4. Only because of the rapid growth selected, the
acceleration due to lift has the same order of magnitude, see subplot (3,1). In all cases considered lift exceeds the remaining acceleration components, i.e. g ohohCM and € The last term, 1 wx;3 R, € is usually the smallest 12 wx;3 R. 2 in all the simulations of this paper. The initial bubble pressure can be chosen such that the volume change term, denoted with Dp dVb =dh in Fig. 4. However, in the standard case the bubble pressure is assumed to equal the hydrostatic pressure at the wall, pw , plus 2r=R0 . 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, pb;0 , 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 Dp0 þ gDqXz
at each height Xz 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.
200
1.3
h 0 = 0.42 mm
1.2
R 0 = 0.7 mm
m/s2
1.4
0
– ∆ σ dA w /dh
R/R0 0
0.5
1
1.5
2
– σ dAb /dh 2.5
3
–400
0
0.5
1
300
Velocity (mm/s)
1.5
2
2.5
3
2
2.5
3
2.5
3
Time (ms)
Time (ms) 1.5
200 1
U
100
hCM /R0 h/R0
0
0
0.5
1
1.5
2
2.5
3
0.5
0
0.5
1
Time (ms)
1.5
Time (ms)
50
1.5
g ∂ hCM / ∂ h m/s2
∆ p dVb /dh
–200
1.1 1
ð6Þ
rfoot /rfoot,0
1 0 2 2 – ψx3 d R/dt /2
–50
0
0.5
1
θ/θ0
0.5
L ift
1.5
Time (ms)
2
2.5
3
0
0
0.5
1
1.5
2
Time (ms)
Fig. 4. Simulation of rapid vapor bubble growth, isothermal and at constant bubble pressure.
C.W.M. van der Geld / Int. J. Heat and Fluid Flow 25 (2004) 74–80
When the bubble is growing, its center of mass moves upward, which yields increasing value of h=R, as shown by subplot (2,2) in Fig. 4. The Dp dVb =dh-acceleration is proportional to R1 ð1 h2 =R2 Þ, and is therefore decreasing. At the same time, also the surface-area term labeled with rdAb =dh increases, since this term is negative and proportional to R2 . The other gravityrelated term increases since it is proportional to dhCM =dh. This derivative is positive for R0 ¼ 0:7 mm, and increasing with increasing R and with increasing h. Near detachment, the lift is negative, implying a hydrodynamic force directed towards the wall. The main contribution to the lift in this situation is given by term_ 1 of Eq. (2). The value of U =z R=R in it is positive near detachment. In the course of time, velocity U increases, whereas simultaneously Rz decreases towards 0.5, when the gradient oa33 =o Rz attains its most negative value. The gradient and term-1 are therefore negative. The main positive component of the hydrodynamic force is term-3 of Eq. (2) near detachment, but it is only about 20 % of the above negative term. This shows that lift counteracts motion away from the wall near detachment. The dependency of a33 on Rz is largely responsible for the essential dynamics near bubble detachment at constant bubble pressure. As a result of the negative lift, also the net acceleration dU =dt becomes negative near detachment. The velocity of the center of mass, however, is still positive and inertia is sufficiently large to cause detachment. Detachment actually occurs in the simulations if h ¼ R and rfoot ¼ 0. Prior to actual detachment, the ratio of the radius of the bubble foot rfoot to its initial value rapidly decreases, see subplot (3,2) of Fig. 4. Simultaneously, the instantaneous contact angle is predicted to deviate considerably from the static value, h0 , see Fig. 4. This is merely due to the imposing of the shape of a truncated sphere, whereas actually often some kind of a neck is formed to connect wall and main bubble volume. The actual shape of a detaching vapor bubble may therefore be different from that of a truncated sphere, but this will have little effect on the total predicted detachment time since the last stage of contraction of the bubble foot happens rapidly. The above standard case yields an acceleration and detachment history that is rather typical for boiling bubbles. Other test cases allow for the following conclusions. The total time of detachment increases if, for initial velocity U0 ¼ 0 and for the same growth rate and the same initial height h0 , initial radius R0 is increased. This trend is easily understood from the dependencies on R of the major acceleration contributions, as discussed above. The total time of detachment, td , hardly depends on the initial value h0 , since with increasing initial height the value of U_ decreases, while the distance the bubble needs to be elevated in order to reach lift-off decreases. Time td is decreased by increas-
79
ing the initial velocity U0 , by the time it would take to reach the value of U0 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 td is about doubled if the gravitation constant is chosen to be )9.82 rather than +9.82 m/s2 . 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 (1981). 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 Detachment of a vapor bubble from a plane, solid wall has been studied theoretically. The vapor–liquid interface shape has been approximated by that of a truncated sphere with radius R. A governing equation for height h above the wall has been presented. The forces related to gravity and surface energy densities are found to be major contributions. They manifest themselves via an oscillatory motion of h if gas residues fill up a substantial part of the bubble volume. The surface energy densities allow for the computation of the dynamic contact angle. The hydrodynamic forces computed comprise the lift force related to the added masses of a growing bubble moving away from the wall. The hydrodynamics are fully controlled by the dependencies of three added mass coefficients on a single parameter, R=ð2hÞ. Exact expressions for these coefficients have been derived and used for computation. There is no need anymore to adjust a fitting parameter of the so-called Ôbubble growth force’ (Klausner et al., 1993) to experimental data. The simulations presented have elucidated the mechanisms which are involved in the dynamics of bubble detachment. Apart from lift, the dependency of R a33 on 2h accounts for the main hydrodynamic action near detachment at constant bubble pressure. In the simulations of this paper, the time history of radius R has been prescribed in order to mimic the physics of actual vapor bubble growth and to highlight the effect of growth on detachment. However, the variational approach of this study also yields a governing equation for R, which can be considered as a kind of Rayleigh–Plesset equation for a truncated sphere at a plane wall.
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The approach followed in this paper, based on the use of generalized coordinates, can also be applied to predict deformation of a bubble in the vicinity of a plane wall (Geld and Kuerten, 2001). In many practical boiling situations deformation can be neglected, however.
References Chesters, A., 1978. Modes of bubble growth in the slow-formation regime of nucleate pool boiling. Int J. Multiphase Flow 4, 279–302. Geld, C.v.d., 2000. A note on the bubble growth force. Multiphase Sci. Technol. 12 (3,4), 215–232. Geld, C.v.d., 2002. On the motion of a spherical bubble deforming near a plane wall. J. Eng. Math. 42, 91–118.
Geld, C.v.d., Kuerten, J., 2001. Deformation and motion of a bubble near a plane wall. In: Michaelides, S. (Ed.), Fourth Int. Conf. on Multiphase Flow, New Orleans, pp. 1–12. CD-Rom. Helden, W.v., Geld, C.v.d., Boot, P., 1995. Forces on bubbles growing and detaching in flow along a vertical wall. Int. J. Heat Mass Transfer 38 (11), 2075–2088. Howe, M., 1995. On the force and moment of a body in an incompressible fluid, with application to rigid bodies and bubbles at low and high Reynolds numbers. Quart. J. Mech. Appl. Math 48, 401–426. Janssen, L., Stralen, S.v., 1981. Bubble behavior on and mass transfer to an oxygen evolving transparent nickel electrode in alkaline solution. Electrochim. Acta 26, 1011–1022. Klausner, J., Mei, R., Bernard, D., Zeng, L., 1993. Vapor bubble departure in forced convection boiling. Int. J. Heat Mass Transfer 36 (3), 651–661. Stralen, S.v., Cole, R., 1979. In: Boiling Phenomena, vols. 1 and 2. McGraw-Hill, Hemisphere, Washington.
Prediction of dynamic contact angle histories of a bubble growing at a wall Cees W.M. van der Geld
*
Department of Mechanical Engineering, Division Thermo Fluids Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Received 17 July 2003; accepted 8 October 2003
Abstract 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 approximate expressions [Int. J. Heat Mass Transfer 36 (1993) 651, Int. J. Heat Mass Transfer 38 (1995) 2075]. In the present study an exact expression for the expansion force is derived for the case of a truncated sphere attached to a plane, infinite wall. The Lagrange–Thomson formalism is applied. Two Euler–Lagrange equations are derived, one governing the motion of the center of mass, the other governing expansion a kind of extended Rayleigh–Plesset equation. If a constitutive equation for the gas–vapor content of the bubble is given, initial conditions and these two differential equations determine the dynamics of the growing truncated sphere that has its foot on a plane, infinite wall. Simulations are carried out for a given expansion rate to predict the history of the dynamic contact angle. The simulations increase the understanding of mechanisms controlling detachment, and yield realistic times of detachment. 2003 Elsevier Inc. All rights reserved. Keywords: Contact angle; Boiling; Bubble growth; Added mass; Euler–Lagrange; Lift
1. Introduction Many criteria for bubble or droplet size at detachment are based on a force balance and on the assumption, often implicitly made, that detachment occurs when no shape can be found that satisfies the force balance. Often a bubble is modelled as part of a sphere with radius R, with actual details near the bubble foot neglected. Indeed the shape of boiling bubbles, rapidly growing in water at ambient pressure, or at higher pressures, is observed to be close to that of a truncated sphere (Stralen and Cole, 1979). The rapidity of the growth apparently allows the contact angle to deviate from the static value. In convective boiling, the equations governing bubble growth and motion have to comprise terms that account for hydrodynamics. It is obvious, that in order to derive an accurate detachment criterion from a force balance, *
Tel.: +31-40-2472923; fax: +31-40-2475399. E-mail address: [email protected] (C.W.M. van der Geld).
0142-727X/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2003.10.002
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., 1993; Helden et al., 1995). 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., 1993; Geld, 2000). 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.
C.W.M. van der Geld / Int. J. Heat and Fluid Flow 25 (2004) 74–80
In the present study, the liquid–vapor interface is taken to have the shape of a truncated sphere with its foot attached to a plane wall. Two parameters are required to specify a shape: radius R and height h of the center of the sphere above the wall. Note that prior to detachment, h differs from the height of the center of mass of the bubble above the wall, hCM . Shapes close to truncated spheres are important in practice, see above, and prediction of shape history corresponds to prediction of the history of the dynamic contact angle. It is noted that boiling conditions exist in which bubble growth is relatively slow and in which asymmetric interface distortion by pressure differences and shear in the direction of the liquid flow occurs. Bubbles may then even be found to slide over the heater surface. The analysis of the present paper does not apply to these conditions. The theoretical analysis of this paper can however be extended to account for more complex deformation, see also Geld and Kuerten (2001). In the case of a sliding bubble, the contact angle needs to vary along the contact line such, that the net force component in the plane of the bubble foot exactly balances the net fluid stress, and possibly gravitation, exerted in this plane. In boiling applications, radius R is usually a known function of time t. Often R is proportional to tn , with n 0:5. This paper will therefore consider situations when the bubble radius history is prescribed by such a time dependency. An equation governing h will be presented and used for simulations that may help our understanding of the mechanisms involved in bubble detachment. The bubble content is either pure vapor, non-condensing gas, or a mixture of both. The pressure inside, pb is taken to be homogeneous. Among the forces which act on a bubble attached to a wall along which a boundary layer flows are the following ones: • • • •
a force related to stochastic features of turbulence, one related to vorticity in the approaching liquid, one due to Marangoni convection and 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 computed exactly. It is noted that these forces, described with the aid of a so-called added mass tensor, retain their value in flows when vorticity does play a role. Whether vorticity would be generated at the body surface or would be present in the approaching rotational flow, it would not affect the added mass coefficients (Howe, 1995). In this sense, a complete description of 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 drag and lift by the vorticity in the main flow. However, the hydrodynamic forces that are computed without
75
vorticity already yield a notable lift force on a fastly 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, 2002; Geld, 2000). The analytical approach of this paper is an extension of the one used in previous studies (Geld, 2000; Geld, 2002).
2. Governing equations Two parameters describe the shape of the truncated sphere under consideration, see Section 1: 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. 1. Let U be defined as dh=dt, € def R ¼ d2 R=dt2 , 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 of ffi water-vapor. The radius of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bubble foot, R2 h2 , will be denoted as rfoot . 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, 2000; Geld, 2002). It turns out that energy T can be written in the form 2 T ¼ pR3 qL fa33 U 2 þ wx;3 U R_ þ traceðbÞR_ 2 g ð1Þ 3 where a33 , wx;3 , trace(b) are added mass coefficients, to be quantified below. With the Lagrange–Thomson
∆p
θ
Aw
Ab
R h
∆σ
Fig. 1. Schematic of bubble geometry, areas and contact angle.
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approach, expressions for the hydrodynamic work in terms of T are derived. The combining of the two corresponding expressions for the work in U yields Eq. (3), below. The work in R_ yields another governing equation, futher discussed in Section 4. The same approach, but with a different velocity potential, has been applied to a sphere growing near a plane wall in a previous study (Geld, 2002). The added mass coefficients are found to depend only on R=z. The method to compute them changes essentially at R=z ¼ 1. This is why Fig. 2 gives only values for shapes up to this point. Note that at R=z ¼ 0:5 the bubble is spherical. The Lagrange–Thomson method yields terms in €h € as well as the so-called hydrodynamic lift of the and R bubble, Lift . It is given by ! R R_ U oa33 R _ 2 1 otraceðbÞ def R Lift ¼ U z R z z oR=z oR=z z R_ 1 owx;3 _ a33 U þ wx;3 R=2 þ ð2Þ þ3 R 2 R oR=z The first term on the RHS of Eq. (2) is denoted term-1 in the following; similar for terms-2 and -3. The governing equation of h reads: 4 3 1 € € pR qL Lift a33 h wx;3 R 3 2 oVb oAb oAw ohCM ¼ Dp þr þ Dr qL gVb ð3Þ oh oh oh oh with Dp given by Dp ¼ pb pw þ qL ghCM . Here pw denotes the hydrostatic pressure at the wall and hCM denotes the height of the center of mass of the truncated sphere above the wall; Vb denotes the bubble volume, Aw the area of the bubble in contact with the plane wall, and Ab the remaining area, at the liquid–vapor interface. The contact angle, h, follows from h ¼ arccosðh=RÞ. The importance of the terms of Eq. (3) is studied in Section 3.
5 trace(β)
4 3 2
It can be shown that forces that have been used in the past, in particular the so-called surface tension force, Fr , and the so-called pressure correction force, Fcorr (Chesters, 1978), 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, 1 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 R0 , 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, h0 , and any predicted deviation of h from h0 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
oVb oAb oAw þr þ Dr ¼0 oh oh oh
ð4Þ
oVb oAb oAw þr þ Dr ¼0 ð5Þ oR oR oR Here Dp denotes the pressure drop across this interface. It is straightforward to show that the above equations yield both the Young equation for the static contact angle and the Laplace equation Dp ¼ 2r=R. Eq. (3) which governs h can be viewed as a direct extension of Eq. (4). Eq. (5) is related to an extended Rayleigh– Plesset equation that governs R and is discussed in Section 4. Dp
1
3. Simulations and analysis
0 –1 –2 0.5
α 33
ψ x,3 0.6
0.7
0.8 R/(2 h)
Fig. 2. Added mass coefficients.
0.9
Actual boiling bubbles contain a small quantity of inert gases, remnants from the cavity, and the bubble 1 1 Here only ðR=zÞ, but for example with deformation more parameters are required.
C.W.M. van der Geld / Int. J. Heat and Fluid Flow 25 (2004) 74–80
pressure, pb , is the sum of the partial pressures of both components: vapor and inert gas. Actual vapor bubble behavior therefore combines features of vapor bubble dynamics and gas bubbles dynamics, in a ratio determined by the concentration of gas. To understand the behavior of actual vapor bubbles, it is sufficient to study pure vapor and gas bubbles separately, as is done in the following. The behavior of a bubble filled with a constant quantity of inert gas will be examined first, because its slow growth behavior helps to understand the physics of the biggest force components active during growth of a vapor bubble. The constitutive equation of the gas content is taken to be pb Vbc ¼ c, c being either cp =cv or 1, dependent on the process, respectively, isentropic and isothermal; c is a constant. The higher the value of c, the faster the response on volume changes, but otherwise this parameter has little influence. If at time zero the bubble radius is increased at a rate of 0:5 R0 per s (c ¼ 1:4), the volume of the truncated sphere is increased, which has two consequences, see Fig. 3:
Fig. 3. The motion is easily sustained for many thousands of cycles in the absence of viscous damping. 2. At a longer timescale, i.e. at times ~t that satisfy ~t=T 1 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 dAb =dh being slightly more negative than the sum of all other forces. The dAb =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=R0 . 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
1. The initial jump in R_ causes an instability of height h. Oscillations at cycle time T result; T 2:2 ms in
200
m/s2
1.003
h0 = 0.42 mm R0 = 0.7 mm
1.002
0
– ∆ σ dA w /dh
1.001
0
1
2
3
4
–400
– σ dA b /dh 0
1
Velocity (mm/s)
Time (ms) 0.75
0
0.7
–2
0.65
–4
0.6
U 0
1
2
3
4
4
0.55
3
4
h/R0
0
1
2
Time (ms) 1.015
10
m/s 2
3
hCM /R0
Time (ms)
θ /θ0
1.01
g ∂ hCM / ∂ h 5
1.005
Other accelerations are negligible 0
2
Time (ms)
2
–6
∆ p dVb /dh
–200
R/R0 1
77
0
1
2
Time (ms)
3
rfoot /rfoot,0
1 4
0.995
0
1
2
Time (ms)
Fig. 3. Simulation of bubble growth with constant content of air.
3
4
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forces, viz the hydrodynamic lift 43 pR3 qL Lift and the € even in the case of forces qL gVb ohohCM and 43 pR3 qL 12 wx;3 R, rapid bubble growth. Minor differences between the two surface area and the volume terms may account for main trends in h and correspond to differences between the instantaneous contact angle, h, and the static value, h0 . The oscillations on the smaller time scale hardly affect the detachment process, and would not occur if the bubble would be mainly vapor at constant saturation temperature. Detachment is therefore more conveniently studied by keeping the bubble pressure constant. This is the reason why in the simulations below the bubble pressure has been kept constant, with c ¼ 1 to model isothermal behavior. In order to facilitate the analysis of the effect of various terms in the governing equation, a Ôstandard’ simulation case is defined, to which other tests will be referred and compared. In the standard process, bubble radius R is growing steadily, at a rapidp rate shown in the ffiffiffiffiffiffiffiffiffiffiffi first plot of Fig. 4, according to R / t þ t0 . This case corresponds to actual vapor bubble growth at a pressure at the wall, pw , of 0.1 MPa. The three major accelerations, due to the volume and surface area changes are gathered in subplot (1,2) of Fig. 4. Only because of the rapid growth selected, the
acceleration due to lift has the same order of magnitude, see subplot (3,1). In all cases considered lift exceeds the remaining acceleration components, i.e. g ohohCM and € The last term, 1 wx;3 R, € is usually the smallest 12 wx;3 R. 2 in all the simulations of this paper. The initial bubble pressure can be chosen such that the volume change term, denoted with Dp dVb =dh in Fig. 4. However, in the standard case the bubble pressure is assumed to equal the hydrostatic pressure at the wall, pw , plus 2r=R0 . 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, pb;0 , 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 Dp0 þ gDqXz
at each height Xz 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.
200
1.3
h 0 = 0.42 mm
1.2
R 0 = 0.7 mm
m/s2
1.4
0
– ∆ σ dA w /dh
R/R0 0
0.5
1
1.5
2
– σ dAb /dh 2.5
3
–400
0
0.5
1
300
Velocity (mm/s)
1.5
2
2.5
3
2
2.5
3
2.5
3
Time (ms)
Time (ms) 1.5
200 1
U
100
hCM /R0 h/R0
0
0
0.5
1
1.5
2
2.5
3
0.5
0
0.5
1
Time (ms)
1.5
Time (ms)
50
1.5
g ∂ hCM / ∂ h m/s2
∆ p dVb /dh
–200
1.1 1
ð6Þ
rfoot /rfoot,0
1 0 2 2 – ψx3 d R/dt /2
–50
0
0.5
1
θ/θ0
0.5
L ift
1.5
Time (ms)
2
2.5
3
0
0
0.5
1
1.5
2
Time (ms)
Fig. 4. Simulation of rapid vapor bubble growth, isothermal and at constant bubble pressure.
C.W.M. van der Geld / Int. J. Heat and Fluid Flow 25 (2004) 74–80
When the bubble is growing, its center of mass moves upward, which yields increasing value of h=R, as shown by subplot (2,2) in Fig. 4. The Dp dVb =dh-acceleration is proportional to R1 ð1 h2 =R2 Þ, and is therefore decreasing. At the same time, also the surface-area term labeled with rdAb =dh increases, since this term is negative and proportional to R2 . The other gravityrelated term increases since it is proportional to dhCM =dh. This derivative is positive for R0 ¼ 0:7 mm, and increasing with increasing R and with increasing h. Near detachment, the lift is negative, implying a hydrodynamic force directed towards the wall. The main contribution to the lift in this situation is given by term_ 1 of Eq. (2). The value of U =z R=R in it is positive near detachment. In the course of time, velocity U increases, whereas simultaneously Rz decreases towards 0.5, when the gradient oa33 =o Rz attains its most negative value. The gradient and term-1 are therefore negative. The main positive component of the hydrodynamic force is term-3 of Eq. (2) near detachment, but it is only about 20 % of the above negative term. This shows that lift counteracts motion away from the wall near detachment. The dependency of a33 on Rz is largely responsible for the essential dynamics near bubble detachment at constant bubble pressure. As a result of the negative lift, also the net acceleration dU =dt becomes negative near detachment. The velocity of the center of mass, however, is still positive and inertia is sufficiently large to cause detachment. Detachment actually occurs in the simulations if h ¼ R and rfoot ¼ 0. Prior to actual detachment, the ratio of the radius of the bubble foot rfoot to its initial value rapidly decreases, see subplot (3,2) of Fig. 4. Simultaneously, the instantaneous contact angle is predicted to deviate considerably from the static value, h0 , see Fig. 4. This is merely due to the imposing of the shape of a truncated sphere, whereas actually often some kind of a neck is formed to connect wall and main bubble volume. The actual shape of a detaching vapor bubble may therefore be different from that of a truncated sphere, but this will have little effect on the total predicted detachment time since the last stage of contraction of the bubble foot happens rapidly. The above standard case yields an acceleration and detachment history that is rather typical for boiling bubbles. Other test cases allow for the following conclusions. The total time of detachment increases if, for initial velocity U0 ¼ 0 and for the same growth rate and the same initial height h0 , initial radius R0 is increased. This trend is easily understood from the dependencies on R of the major acceleration contributions, as discussed above. The total time of detachment, td , hardly depends on the initial value h0 , since with increasing initial height the value of U_ decreases, while the distance the bubble needs to be elevated in order to reach lift-off decreases. Time td is decreased by increas-
79
ing the initial velocity U0 , by the time it would take to reach the value of U0 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 td is about doubled if the gravitation constant is chosen to be )9.82 rather than +9.82 m/s2 . 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 (1981). 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 Detachment of a vapor bubble from a plane, solid wall has been studied theoretically. The vapor–liquid interface shape has been approximated by that of a truncated sphere with radius R. A governing equation for height h above the wall has been presented. The forces related to gravity and surface energy densities are found to be major contributions. They manifest themselves via an oscillatory motion of h if gas residues fill up a substantial part of the bubble volume. The surface energy densities allow for the computation of the dynamic contact angle. The hydrodynamic forces computed comprise the lift force related to the added masses of a growing bubble moving away from the wall. The hydrodynamics are fully controlled by the dependencies of three added mass coefficients on a single parameter, R=ð2hÞ. Exact expressions for these coefficients have been derived and used for computation. There is no need anymore to adjust a fitting parameter of the so-called Ôbubble growth force’ (Klausner et al., 1993) to experimental data. The simulations presented have elucidated the mechanisms which are involved in the dynamics of bubble detachment. Apart from lift, the dependency of R a33 on 2h accounts for the main hydrodynamic action near detachment at constant bubble pressure. In the simulations of this paper, the time history of radius R has been prescribed in order to mimic the physics of actual vapor bubble growth and to highlight the effect of growth on detachment. However, the variational approach of this study also yields a governing equation for R, which can be considered as a kind of Rayleigh–Plesset equation for a truncated sphere at a plane wall.
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The approach followed in this paper, based on the use of generalized coordinates, can also be applied to predict deformation of a bubble in the vicinity of a plane wall (Geld and Kuerten, 2001). In many practical boiling situations deformation can be neglected, however.
References Chesters, A., 1978. Modes of bubble growth in the slow-formation regime of nucleate pool boiling. Int J. Multiphase Flow 4, 279–302. Geld, C.v.d., 2000. A note on the bubble growth force. Multiphase Sci. Technol. 12 (3,4), 215–232. Geld, C.v.d., 2002. On the motion of a spherical bubble deforming near a plane wall. J. Eng. Math. 42, 91–118.
Geld, C.v.d., Kuerten, J., 2001. Deformation and motion of a bubble near a plane wall. In: Michaelides, S. (Ed.), Fourth Int. Conf. on Multiphase Flow, New Orleans, pp. 1–12. CD-Rom. Helden, W.v., Geld, C.v.d., Boot, P., 1995. Forces on bubbles growing and detaching in flow along a vertical wall. Int. J. Heat Mass Transfer 38 (11), 2075–2088. Howe, M., 1995. On the force and moment of a body in an incompressible fluid, with application to rigid bodies and bubbles at low and high Reynolds numbers. Quart. J. Mech. Appl. Math 48, 401–426. Janssen, L., Stralen, S.v., 1981. Bubble behavior on and mass transfer to an oxygen evolving transparent nickel electrode in alkaline solution. Electrochim. Acta 26, 1011–1022. Klausner, J., Mei, R., Bernard, D., Zeng, L., 1993. Vapor bubble departure in forced convection boiling. Int. J. Heat Mass Transfer 36 (3), 651–661. Stralen, S.v., Cole, R., 1979. In: Boiling Phenomena, vols. 1 and 2. McGraw-Hill, Hemisphere, Washington.