Thermodynamic and kinetic considerations of nucleation and stabilization of acoustic cavitation bubbles in water

Ultrasonics Sonochemistry 15 (2008) 65–77 www.elsevier.com/locate/ultsonch

Thermodynamic and kinetic considerations of nucleation and stabilization of acoustic cavitation bubbles in water Pratap S. Bapat, Aniruddha B. Pandit

*

Chemical Engineering Department, University Institute of Chemical Technology, Matunga, Mumbai 400019, India Received 17 July 2006; received in revised form 11 January 2007; accepted 14 January 2007 Available online 3 February 2007

Abstract Qualitative explanation for a homogeneous nucleation of acoustic cavitation bubbles in the incompressible liquid water with simple phenomenological approach has been provided via the concept of the desorbtion of the dissolved gas and the vaporization of local liquid molecules. The liquid medium has been viewed as an ensemble of lattice structures. Validity of the lattice structure approach against the Brownian motion of molecules in the liquid state has been discussed. Criterion based on probability for nucleus formation has been defined for the vaporization of local liquid molecules. Energy need for the enthalpy of vaporization has been considered as an energy criterion for the formation of a vaporous nucleus. Sound energy, thermal energy of the liquid bulk (Joule–Thomson effect) and free energy of activation, which is associated with water molecules in the liquid state (Brownian motion) as per the modified Eyring’s kinetic theory of liquid are considered as possible sources for the enthalpy of vaporization of water molecules forming a single unit lattice. The classical nucleation theory has then been considered for expressing further growth of the vaporous nucleus against the surface energy barrier. Effect of liquid property (temperature), and effect of an acoustic parameter (frequency) on an acoustic cavitation threshold pressure have been discussed. Kinetics of nucleation has been considered.  2007 Elsevier B.V. All rights reserved. PACS: 82.60.Nh; 64.60.Qb; 05.60.Cd; 05.50.+q; 43.35.+d Keywords: Vaporous nucleus; Thermodynamics; Kinetics; Acoustic cavitation; Threshold pressure; Cubic lattice model

1. Introduction The possible sources of cavities considered so far in the acoustic cavitation literature to answer its dynamic behavior are pre-existing gas bubbles in the liquid and the formation of void in the liquid continuum due to the rupture in the liquid medium [1] and the local vaporization of liquid water [2,3]. The literature regarding the existence of these possibilities has been explained in brief as follows. The laboratory experiments are usually conducted in the standstill water. Epstein and Plesset [4] derived the mathematical expression for the time required for the complete dissolu-

*

Corresponding author. Tel.: +91 22 24145616; fax: +91 22 24145614. E-mail address: [email protected] (A.B. Pandit).

1350-4177/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ultsonch.2007.01.005

tion of the stand still gas bubbles in a pure liquid (saturated with the gas) due to the surface tension phenomenon. It requires few seconds (<10 s) for the complete dissolution of air bubbles of micrometer size in the pure water. The other possible source of the cavity is the formation of a void, which would be created in water when the molecules are separated by more than the van der Waals distance (Re = 0.4 nm). The Blake threshold pressure for such separation is about 1368 · 105 Pa [5]. However the maximum tensile strength of water found by an experimental method by Briggs [6] is much lower i.e. in the range of 200 · 105–300 · 105 Pa even though all precautions had been taken to purify the water to the maximum extent. In the case of liquid water, which is saturated with air, it is observed that negative pressure amplitude of 1 · 105– 2 · 105 Pa can form a cavity as reported in the review by

66

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

Brenner et al. [7]. The models are developed to explain the pre-existence of the stable gas bubbles having internal pressure either equal to or lower than the ambient pressure of the liquid so that these may be stabilized against gaseous diffusion due to the surface tension phenomenon. The liquid considered in these models is impure i.e. either containing solid crevices (crevice model) [1,8], or having dissolved contaminants exhibiting surfactant properties as described by Yount in the Varying Permeability model [9,10]. In the present work, the possibility of a gaseous nuclei formation by desorbtion of the dissolved gas molecules due to decrease in the liquid pressure has been considered. The most widely preferred theory for nucleation due to the vapor–liquid phase transition is the classical nucleation theory (hereafter referred as CNT) [2,11]. It accounts for the requirement of energy for the formation of new surface. During condensation, enthalpy would be given out from the initial vapor phase. However during vaporization, an external energy, equal to the latent heat of vaporization must be provided to the liquid to change its phase to a meta-stable state and then to a stable vapor phase. CNT does not account this for nucleation by vaporization. The latent heat of vaporization, DH at 20 C for water is 2454.3 kJ/kg [12]. The value of Gibbs free energy barrier gbar at 20 C for water is 30 kJ/kg as reported by Vortmann et al. [3]. In the present approach, we have considered the stable phase change and not the dynamic phase change (molecules in the liquid state in a cluster may momentarily change their phase to the vapor state and then may come back in the liquid state). Thus, DH which, we have used as the energy barrier in our model is much larger than the reported gbar based on the critical analysis of the graph by Vortmann et al. [3] for the variation of Gibbs free energy against specific volume during the phase change. Following three possible sources of energy for the enthalpy of vaporization have been considered in the present approach: a. Potential energy, which would be stored in the liquid bulk during the propagation of an acoustic sound energy. b. Thermal energy of the liquid water bulk, which would be available due to the Joule–Thomson effect. c. Free energy of activation, which is associated with a water molecule in the liquid state as per the Eyring’s kinetic theory of liquid. Referring to steam tables of water [12], one can conclude that about 95% of the total enthalpy change during the vaporization process of water is due to an increase in the internal energy alone in the temperature range of 283– 343 K. The properties like internal energy and specific volume are the state functions, and do not depend upon a path followed for the change of state. Hence the total energy change can be considered as an independent of a path to be followed i.e. either by the supply of heat or by the sup-

ply of sound (pressure) energy in its rarefaction region or by the surrounding liquid bulk. In the present approach, water has been viewed as a cubic lattice structure to estimate the enthalpy of vaporization per unit lattice. The cubic lattice approach to model the liquid state is well known [13]. It is used for successful prediction of transport phenomenon of liquid water, such as thermal conductivity, by Bridgman [14] and to explain the experimentally found relation between self-diffusion coefficient and viscosity over a temperature range of 0– 55 C [15]. Now onwards we follow a terminology, such as ‘nucleus’ means the presence of a gas/vapor phase in the liquid, which is not necessarily in mechanical equilibrium with the surrounding liquid and the ‘cavity’ means a nucleus in mechanical equilibrium with the surrounding liquid. Following assumptions have been made: a. Molecules have been considered to be spherical and rigid. Nucleus, which would be formed due to desorbtion/vaporization, has also been assumed to be of spherical shape. b. Intra-molecular interactions are considered in terms of thermodynamic properties like internal energy and enthalpy. c. Classical approach has been adopted which assumes that the properties of a liquid at macroscopic level equally hold true even at microscopic level. Here a term ‘void’ means an empty space available among the rigid molecules, which can only be occupied by dissolved gas or vaporizing liquid due to its volume expansion. 2. Lattice structure approach In the simple cubic structure particles are placed at the corners of cubes touching with each other as shown in Fig. 1. Thus eight water molecules form one cubic lattice. There is an empty space (void) enveloped by these eight water molecules. Thus each cube contains one void space in it. Each water molecule is participating in the formation of eight adjacent cubes. Hence an effective  number of particles (molecules) per unit cube is one ¼ 88 . Hence for the simple cubic structure, the total numbers of cubes are equal to the number of molecules present in the liquid bulk.

Fig. 1. Cubic lattice structure approach for the liquid bulk.

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

Hence the total numbers of unit lattices (void) present in the one mol of water are equal to the Avogadro’s number of molecules. 3. Calculations and inferences 3.1. Formation of gaseous nucleus It is difficult to completely degass liquid water. Moholkar et al. [16] did one such attempt. They conducted an experiment with partially degassed demineralized water (total gas content was below 2 ppm) in an ultrasound bath (driving frequency: 35 kHz, root mean square pressure amplitude: 1.1 · 105 Pa) to measure the acoustic cavitation intensity at various locations. They found that the variation of the measured cavitation intensity with location, qualitatively matched well with the calculated variation of the amplitude of shock wave, which would be emitted by a gas bubble based on the numerical simulations. This underlines the importance of the consideration of desorbtion of dissolved gas and its effect in the cavitation phenomena. Formation of gaseous nucleus i.e. nucleus, which consisting of gas molecules, in the liquid water bulk has been considered. Consider a gas–liquid system in which two phases are separated by a plane surface. Let symbol Pg denotes the gas pressure. The saturation concentration of the dissolved gas x at Pg is given by the Henry’s law [17] and is as follows: x¼

Pg ; H

ð1Þ

where unit of x is mole fraction of dissolved gas in the liquid water, H is the Henry’s law constant at a given temperature having units of Pascal of gas pressure per mol fraction of dissolved gas in the liquid phase. With the propagation of the acoustic wave at a frequency f, having a pressure amplitude PA, the instantaneous local pressure Pl in the liquid bulk changes as a sinusoidal function of time t, and is as follows: P l ¼ P l0  P A  sinð2pftÞ;

ð2Þ

where Pl0 is the initial liquid pressure at t = 0. In the rarefaction region of the acoustic wave, if the local liquid pressure falls below the vapor pressure P Tvap then the local liquid molecules will transform to vapor phase (boiling). Hence at this condition the saturation solubility of the gas has been taken as zero. Thus the saturation solubility of the gas xl at a given temperature has been assumed to be a linear function of the local liquid pressure Pl and is as follows: xl ¼ x xl ¼

x0  ðP l  P Tvap Þ P l0  P vap

when P l ¼ P l0 ;

ð3aÞ

when P Tvap 6 P l 6 P l0

ð3bÞ

and xl ¼ 0

when P l < P Tvap :

ð3cÞ

67

Assume that the actual concentration of the dissolved gas is z percentage of its saturation concentration x at a given liquid temperature. Due to the decrease in the local liquid pressure Pl in the rarefaction region of the applied acoustic wave, solubility of gas (air in the present case) decreases linearly with decreasing pressure, according to Eq. (3b). The required local pressurehat which the liquid would bei z z come saturated with gas is 100 P l0 þ P Tvap 1  100 . Putting this expression for Pl in Eq. (2), the time instant tA at which the liquid water become super saturated has then been obtained with the use of Eq. (2) and is as follows:   z z 1 P l0 ½100P l0 þP vap ð1100Þ sin PA : ð4Þ tA ¼ 2pf Thus, in the first half of the rarefaction region, after time instant tA, desorbtion of the dissolved gas would take place. Due to the local desorbtion, there will exist a gas phase concentration gradient in the liquid phase. Hence the dissolved gas molecules will diffuse towards the desorbed gas phase (first nucleus). Thus, the dissolved gas will try to appear as a separate gas phase by the diffusion phenomenon until the saturation level at a new pressure is reestablished again. In the first half of the rarefaction region pressure decreases continuously to the minimum local pressure (Pl0  PA). The corresponding time instant tC is 4f1 . Knowing the instantaneous local pressure at these time instants i.e. at tA and at tC, we can calculate the average pressure A tC P tavg by using the following general formula: R tC ðP l0  P A  sinð2pftÞÞdt t tA tC P avg ¼ A tC  tA P A ½cosð2pftC Þ  cosð2pftA Þ : ð5Þ ¼ P l0 þ 2pf ðtC  tA Þ If this average pressure is below the vapor pressure then the average saturation concentration of the gas during this time period is to be taken as zero (near boiling condition). If the average pressure is above the vapor pressure then the average saturation concentration of the gas has been found by the following equation, corresponding to the instantaneous local pressure: A tC xtavg ¼

A tC P tavg  P vap  ðx  0Þ: P l0  P vap

ð6Þ

Let the symbol D indicates the diffusivity of the dissolved gas. Hence the characteristic length l in the liquid medium through which the desorbtion will take place is equal to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p DðtA  tC Þ. Thus gas molecules in the hypothetical sphere (or cubic lattice) of radius l will tend to desorb. Hence the magnitude of total number of gas molecules n, which would be desorbed in the given time interval of the rarefaction region can be calculated as follows:  z  4 A t C n¼  x  xtavg ð7Þ  pl3  qw  N A ; 100 3

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P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

where qw represents the density of liquid water and NA is the Avogadro’s number. As the size of the vaporous nucleus has been calculated at the vapor pressure, the gaseous nucleus size Rgas has also been calculated at the vapor pressure for convenience as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3nkT : ð8Þ Rgas ¼ 3 4pP Tvap

Recent experimental study of Lee et al. [19] using novel capillary system demonstrated that with an increase in the percentage-dissolved air in the liquid water, these exists an increase in the total bubble volume (gas volume occupied by uncollapsed or undissolved bubbles) due to the formation of bubbles and their subsequent coalescence. The coalescence of air nuclei has been quantified later (Section 5).

A typical acoustic source of intensity 10 W/cm2, which radiates sound energy at a frequency of 20 kHz with a pressure amplitude PA of 5.42 · 105 Pa, has been considered for air–water system at initial pressure (Pl0) of 1.01325 · 105 Pa. The Henry’s law constant for the air– water system is 7.81 GPa of liquid pressure/mol fraction of dissolved air in an aqueous solution [17]. Diffusivity of the dissolved air in the liquid water has been taken as 2 · 109 m2/s. (Calculations are done for the estimation of Rair, but are not shown in the present article as they are straight forward). At z = 100% and x = 1.3 · 105 mole fraction of dissolved air, we have tA ¼ 0 s; A tC A t C ¼ 0; l ¼ tC ¼ 1:25  105 s; P tavg ¼ 2:44  105 Pa; xtavg 7 4 3 1:58  10 m; qw ¼ 5:56  10 mol=m ; n ¼ 7186; Rair ¼ 121 nm. Thus, at 100% saturation condition, Rair works out to be 121 nm. Fig. 2 represents the effect of the percentagedissolved air z on the radius of the gaseous nucleus Rair formed at the vapor pressure of water at 303 K. As shown in Fig. 2, Rair decreases with a decrease in the concentration of the dissolved air expressed as a function of percentage saturation at 303 K. If we consider the effect of surface tension, then there may be a possibility of coalescence of two or more gaseous nuclei, which are separated only by a single layer of water molecules to minimize the surface tension force. The experimental study of Pandit and Davidson [18] regarding hydrodynamics of the rupture of thin liquid film has revealed that below the critical film thickness of 0.07 lm, the liquid film gets rupture naturally without requirement of any external puncturing device. Hence referring to the experimental work of Pandit and Davidson [18], it is very likely that such two or more nuclei formed in adjacent cubic lattices may coalesce spontaneously to produce relatively larger bubble (stable nucleus).

3.2. Probability of nucleus formation in degassed water

Rair (nm)

150

In a real liquid, molecules are not stationary. There exists a continuous migration of liquid molecules, which is known as the Brownian motion. Vaporous nucleus formation is possible only in the rarefaction region where the molecules, sitting at the corners of a cubic lattice are pulled away simultaneously. Each lattice unit is surrounded by other lattices, there has to be a simultaneous motion of a set of lattices influenced by a sufficient magnitude of the rarefaction pressure, which results into a finite probability (significantly less than one) of the nucleus formation. In the cubic lattice structure, a molecule may vibrate in the following possible directions as shown in Fig. 3: a. Along three axes: Along any axis, a molecule can vibrate in the two directions i.e. positive direction (away from centre) and negative direction (towards the centre). Hence the total directions are 6 (=2 · 3). b. Along planes, which pass through two axes and those are perpendicular to the third axis. For e.g., in XY plane a molecule can vibrate in 4 diagonal directions. There are three such planes, which are X  Y, X  Z, and Y  Z. Hence the total possible directions are 12 (=4 · 3). c. Along the diagonals of a three dimensional cube. There are 8 such directions. Hence the total directions for a molecular motion in the cubic lattice structure are 26. However, only 7 directions allow a molecule to migrate away from the other seven molecules as shown in Fig. 3. Now, for the possible nucleus formation, all the eight molecules forming a single lattice unit must simultaneously move away from the center of the unit lattice. Thus, the probability of this situation is 78 out of 268 i.e. 1 in 36,224 vibrations of an individual

100

50 0 0

50 z

100

Fig. 2. Plot of Rair against z at 303 K. An acoustic source of intensity 10 W/cm2 having PA = 5.42 Pa and f = 20 kHz has been considered. Rair is at the vapor pressure of water at 303 K. i.e. 4.23 · 104 Pa.

Fig. 3. Possible directions for migration of a single liquid molecule in a cubic lattice structure.

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

molecule (or lattice). Thus one in 36,224 molecular movements may actually lead to the formation of a homogeneous nucleation by the evaporation of the water molecules provided the energy criterion is satisfied. 3.3. Energy criteria for the formation of vaporous nucleus Energy of vaporization of water molecules forming one unit lattice, DHlattice has now been calculated using thermodynamic data available in the standard steam tables [12]. Various possible energy sources, discussed earlier have been considered. 3.3.1. Enthalpy of vaporization per unit lattice In this calculation it has been assumed that every molecule is participating in the formation of eight adjacent lattice units. Therefore the enthalpy of vaporization per unit lattice, DHlattice at 303 K has been calculated as follows: DH lattice ¼

DH g mol ¼ 7:27  1020 J=void: NA

ð9Þ

Thus the enthalpy of vaporization per unit lattice in the liquid water at 303 K is 7.27 · 1020 J. The same calculation procedure has then been repeated to calculate DHlattice at different temperatures T in the temperature range of 283–343 K. DHlattice is found to decrease from 7.405 · 1020 J at 283–6.975 · 1020 J at 343 K as reported in Table 1. In the following discussion, the considered sources of energy have been quantitatively evaluated to satisfy the energy need for the vaporization per unit lattice: 3.3.2. Acoustic irradiation as a possible source The reported maximum threshold pressure by Briggs [6], P max th in the degassed water for the stable cavity inception is about 295 · 105 Pa. Hence the required intensity I of the acoustic source at this threshold pressure has been calculated as follows: I¼

2 ðP max ð295  105 Þ2 th Þ ¼ 2qw C 2  1000  1:500  103

 3  108 W=m2 ¼ 3  104 W=cm2 :

69

An instantaneous power supplied by the acoustic source is a function of time t and the location of the receiving point with respect to the source, y. For the time being the location has been considered to be adjacent to the vibrating surface of the source (y = 0) as shown in Fig. 4 to neglect the pressure wave damping effect due to the medium impedance on the acoustic energy as a function of distance. The actual receipt of the energy by one lattice, located at a distance from the source will be significantly less. The sound wave, which has been considered in the present approach, is a simple harmonic plane wave. Hence, one has to approximate the spherical nucleus, which would be formed to the cylindrical form, such as the axis of the cylinder is parallel to the direction of the propagation of the sound wave. Hence a cylinder whose length lcylinder and diameter dcylinder are same, and equal to the diameter of the initial nucleus has been considered. As discussed later (Section 3.4), the diameter of a vaporous nucleus is 12.34 nm at 303 K. Thus, at 303 K, the linear dimensions of the hypothetical cylinder are equal to 12.34 nm. Hence, the surface area of a flat side of the cylinder, Acylinder through which the sound energy would be transmitted or accepted is [=p · (dnucleus)2] 1.19 · 1016 m2. Expression for the total potential energy stored, in the cylinder during the rarefaction region i.e. from time t equal to zero to sð¼ 2f1 ¼ 2:5  105 sÞ has been obtained as follows [20]:  max 2 1 P th  4p2 f 2 P:E: ¼ lcylinder  Acylinder  qw 2 qw Cf  

   sin 4pf s  Cy þ sin 4pf Cy 1 sþ  : ð11Þ 2 4pf Hence, the calculated total potential energy stored in the cylinder (Eq. (11)) in the rarefaction region of the acoustic sound wave is about 1.65 · 1022 J, which is about two orders of magnitude less than that required to form one nucleus (7.27 · 1020 J/void) by evaporation. Thus, the sound energy alone cannot be the source for the vaporization and the growth of the cavity.

ð10Þ

Thus an acoustic source, which radiates sound energy with intensity of 300 MW/m2 at a frequency f of 20 kHz, has been considered in this typical case.

Acoustic horn Liquid water Spherical nucleus

Table 1 b coal and Pnucleation (CNT) for liquid water Calculated values of DH lattice ; R T (K)

P Tvap * ðPaÞ

DHlattice (1020 J)

R (nm)

Rcoal (nm)

b coal R ðnmÞ

Pnucleation (Pa)

283 293 303 323 343

1227 2337 4241 12,340 31,160

7.405 7.334 7.264 7.121 6.975

9.12 7.44 6.17 4.41 3.30

13.13 10.71 8.88 6.35 4.75

8.68 8.78 8.88 9.06 9.23

1125 2113 3779 10,690 26,249

*

[12].

Hypothetical Cylinder

d

d

d

Enlarged View Fig. 4. Schematics of the location of the nucleus and its approximation in the cylindrical form.

70

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

3.3.3. Thermal energy of a liquid bulk as a possible source If a fluid is expanded adiabatically at a constant enthalpy H then its temperature drops down. This is known as the Joule–Thomson effect; and aJT is defined as follows:   DT : ð12Þ aJT ¼ DP H The reported value of aJT for liquid water [21] is in the order of 108 K/Pa. In this section, the possibility of a surrounding liquid bulk as a source of energy supply for the vaporization has been explored. The temperature drop required between the unit lattice and the surrounding liquid bulk has been calculated as follows. Bridgman’s simple theory of energy transport in pure liquids has been taken into consideration. The theory has been explained in brief by Hirchfelder et al. [13] and by Bird et al. [14]. According to Bridgman’s theory, the expression for a thermal conductivity k 0 is as follows:  23 N k0 ¼ 3  K  C; ð13Þ V   where NV is the volume per molecule (cm3/molecule), K is the Boltzman’s constant (1.38 · 1023 J/molecule) and C represents the velocity of sound in the liquid medium (m/ s). The velocity of sound increases with an increase in the acoustic frequency [13]. At lower frequencies (that is lower compared to the characteristic frequency of liquid molecule) the velocity of sound is given by the expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi C P oP C¼ ; ð14Þ C V oq T

k0 ¼ 3



6:023  1023 1:808  105

23

 1:3805  1023  1:500  103

¼ 0:6434 W=m K:

Thus, the calculated thermal conductivity of liquid water at 303 K with the use of Eq. (13) is 0.6434 W/m K, which is in agreement with the measured value of 0.618 W/(m k) [23]. The calculations of thermal conductivity at various temperatures in the temperature range of 283–343 K have then been compared with the standard reported values [23]. The difference between the calculated value and the reported value is about 10%. This implies that lattice structure approach appears to be valid. As shown schematically, in Fig. 5 cross-shaded circles represent two-dimensional view of the selected unit lattice. Four molecules along a length (single shaded circle) are required to cover the unit lattice. Hence the total number of molecules required to cover the lattice in the threedimensional space are 64 (=43). It has been assumed that the instantaneous energy requirement is provided in the form of thermal energy. The thermal energy for vaporization is provided with the aid of instantaneous decrease in the temperature of these 64 molecules from bulk temperature T to temperature Tinst. Hence the temperature of the unit lattice and of the surrounding molecules will be at temperature Tinst at the time of vaporous nucleus formation. Due to the temperature difference between 64 molecules and the remaining liquid molecules; heat transfer will then take place. Temperature of the unit lattice at the time of vaporous nucleus formation has been assumed to be at 303 K. Specific heat of liquid water bulk at 303 K is 4.174 · 103 J/kg K. Hence specific heat of a single water molecule C Pmolecule in the liquid state is 1:247  1022

where the symbols CP and CV represents the specific heat at constant pressure and at constant volume respectively. The symbols P and q represent liquid pressure and liquid density respectively. h oq i The data of isothermal compressibility xW ¼ q1 oP T has been taken from Hodgman [22]. At temperature 303 K, the isothermal compressibility of the liquid water xW is 4.46 · 1010 m2/N. Its density q is 995.7 kg/m3. Hence   oP 1 m2 ¼ 2:25  106 2 : ¼ oq T ¼303 xw q s Hence the velocity of sound C according to Eq. (14) has been calculated as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ 1  2:25  106 ¼ 1:500  103 m=s: Mass of one mol of water is 18 g. Hence its molar volume V is equal to 1.808 · 105 m3/mol. The Avagadro number is 6.023 · 1023 molecules/mol. Hence the thermal conductivity of liquid water at 303 K has been calculated with the use of Eq. (13) as follows:

Fig. 5. Schematic of the energy transfer to the nuclei from the surrounding liquid bulk. Cross-shaded circle represents molecules of the nucleus. Single-shaded circle represents molecules covering the nucleus. Plain circle represents molecule of the liquid bulk.

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77



3

71



18 ¼ 4:17410 J=molecule K. Hence the equation for the 6:0231026

energy balance is as follows: DH lattice ¼ 64  C Pmolecule  ðT  T inst Þ:

ð15Þ

DHlattice at 303 K is 7.27 · 1020 J. Hence the required temperature difference (T  Tinst) is 9.1 K. Thus, if the initial liquid bulk temperature is at 312.1 K, then the vaporization of a unit lattice will take place at 303 K provided the said temperature drop occurs due to the Joule–Thomson effect. The required temperature drop at different vaporization temperatures (283–343 K) for the formation of a single nucleus is around 9 K. However as reported previously, the Joule–Thomson coefficient for liquid water is of the order of 108 K/Pa. Thus the theoretical pressure drop required to create a temperature differential of 9 K is of the order of 109 Pa (equivalent to an acoustic intensity of 3.33 · 107 W/cm2), which is very large as compared to the reported threshold pressure (200 · 105–300 · 105 Pa). Hence the thermal energy (liquid enthalpy) alone as a source energy for the enthalpy of vaporization of unit lattice is insufficient, though at an acoustic intensity of 3 W/ cm2, a temperature differential of 3 · 103 K will be created and may contribute to the overall vaporization process. 3.3.4. Free energy of collision among molecules in the liquid state as a possible source In the liquid state, molecules migrate continuously in the form of collisions with the surrounding molecules. Each collision is associated with certain energy DE of collision. There exist a finite number of collisions per second per molecule for a given DE. If it so happen that 8 molecules forming a unit lattice migrate away (Brownian motion) from the center of the unit lattice, then there exist a possibility of vaporization provided the total energy of collisions of 8 molecules (8 · DE) is greater than the earlier estimated heat of vaporization per unit void. To calculate the value of DE, kinetic theory of liquid developed by Eyring and coworkers [24] and later improved by Li and Chang [15] has been referred. Postulates of Eyring’s kinetic theory of liquid are as follows: a. In a pure liquid at rest, the motion of the individual molecule is largely confined to vibrations of each molecule within a cage formed by its nearest neighbors as shown in Fig. 6. The cage is represented by the energy barrier of height NDGA where DG is referred as molar ‘‘free energy of activation’’ and NA is the Avogadro’s number. b. Liquid bulk is formed by the formation of layers of liquid molecules. A distance of d separates these layers from each other. In the simple cubic lattice structure, d is equal to p1ffiffi2 times the diameter of a molecule. c. Further, Eyring had proposed that a liquid, at rest continually undergoes rearrangements in which one molecule at a time escapes from its ‘‘cage’’ into an

a

δ

Fig. 6. Eyring’s model for the cubic lattice structure.

adjoining ‘‘hole’’ and the molecules thus move in each of the Cartesian coordinate directions in jumps of length a at a frequency of k* per molecule where k* is given by the rate equation k ¼

KT RDGT e U h

ð16Þ

where h is the Plank’s constant (6.624 · 1034 Js), K is the Boltzmann constant (1.38 · 1023 J/molecule pffiffiffi K). For the cubic lattice structure, a is equal to 2 times the diameter of a molecule. Hence the ratio of d to a is 0.5. The free energy of activation can be calculated from the viscosity data against temperature as follows. For a Newtonian fluid, the mathematical expression for the viscosity in terms of ‘‘free energy of activation’’ after the correction suggested by Li and Chang [15] is as follows:  2   nxyz  nxy d hN A DG l¼ exp ; ð17Þ a RU T 2 nxyz Ve where Ve is the molar volume of the liquid phase, nxy represents the total number of (closest) surrounding molecules in a plane and nxyz represents those in space. For the cubic lattice structure nxy is equal to 4 and nxyz is equal to 6. Hence Eq. (17) for the cubic lattice structure becomes   1 hN A DG l¼ exp : ð18Þ 24 Ve RU T Thus, according to Eq. (18), viscosity changes exponentially with DG. After manipulation of the terms in Eq. (18) we have ! l Ve 1 DG 1 DG 1 ¼ 3:178 þ : ð19Þ ln ¼ ln þ hN A 24 RU T RU T Fig. 7 represents the variation in the left hand side term of Eq. (19) against the reciprocal of temperature for liquid water to obtain the value of DG. The data has been taken from Weast and Astle [25]. The of a linear fit  equation  le V for the liquid water is ln N A h ¼ 2:072  103 T 1  3:178. The regression coefficient of the linear fit is 0.98, which indicates that the assumption of the cubic lattice structure is indeed adequate. Hence, the numerical value of DG for the liquid water is 17.227 · 103(=2.072 ·

72

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

Hence the mathematical expression for the number of vibrations of a single molecule in this  time period  as per the Eyring’s theory is given by k   4f1  tnucleous . If these

6

vibrations are equal to or greater than the minimum number of vibrations required (36,224) for a possible formation of a homogeneous nucleation then there is a possibility for nucleus inception irrespective of the applied external pressure. Therefore the required condition is   1   tnucleus P 36; 224: k  ð20Þ 4f

4

~

ln

μV

N Ah 2

0 2

3 -1

-3

4 -1

T (10 K ) Fig. 7. Hollow circles represent the variation of left hand side term of Eq. (19) with reciprocal of temperature T for liquid water for cubic lattice structure in the temperature range of 273–363 K. Black line presents a linear trend.

103 · RU) J/mol as per the cubic lattice structure approach.  

Thus, free energy contained by a water molecule DG is NA 20 J/molecule. Hence the total free energy associ2.86 · 10 ated with molecules is 22.88 · 1020 J which is about three times that, required for the vaporization of a unit void (DHlattice = 7.21 · 1020 J at 303 K). DHlattice is about 7 · 1020 J in the temperature range 283–343 K as shown in Table 1. If we consider the energy conservation, then the free energy associated with eight molecules forming a unit lattice during the Brownian motion in the liquid state is sufficient for the enthalpy of vaporization of three adjacent unit lattices into voids. These three lattices may then coalesce to form a single vaporous nucleus. Thus the energy associated with Brownian motion of molecules in the liquid state is sufficient for the enthalpy of vaporization of cubic lattices, provided a. All the eight molecules migrate away from the center of the void and b. the local liquid pressure is below the vapor pressure at a given temperature. Let Pnucleus represents a local pressure at which a vaporous nucleus tends to form. In the following discussion, the reported effect of an acoustic frequency on Pnucleus [26] has been qualitatively explained based on the cubic lattice model.

At 303 K, the value of k* has been calculated with the use of Eq. (16) as follows: k ¼

1:38  1023  303 17:227103 e 8:314303 ¼ 6:77  109 Hz: 6:624  1034

Thus, at 303 K, the value of k* is 6.77 GHz for the liquid water. If we put tnucleus equal to zero in Eq. (20), then the value of f at 303 K is obtained as follows: f ¼

k 6:77  109 ¼ ¼ 46:7 kHz: 4  36; 224 4  36; 224

Thus, at 303 K the value of f is 46.7 kHz, which represents the upper limit of frequency up to which Pnucleus should be independent of the acoustic frequency. It has been reported [26] that the intensity for a vaporous nucleus formation is independent of the acoustic frequency below 10 kHz and then increases with an increase in the acoustic frequency. The reported intensity [in terms of pressure Pth using Eq. (10)] against the acoustic frequency has been shown in Fig. 8. Local applied negative pressure pulls all the molecules apart from each other, favoring the possibility of simultaneous movement of molecules away from each other. With an increase in the acoustic frequency, number of collisions per molecule in the first quarter will decrease. For example, at the acoustic frequency of 400 kHz, the number of vibrations of a  single molecule  in the first quarter of the acoustic 1 6:77109 wave are ¼ 4  400103 4230, which is less than the minimum required vibrations (36,224). In the absence of an external acoustic field, it has been assumed that molecules in the liquid state vibrate randomly. Thus, there is no preferential direction in which 1.E+08 1.E+07

Pth (Pa)

3.3.4.1. Effect of an acoustic frequency. The possibility of the formation of a homogeneous nucleus exists in the first quarter of the acoustic wave, when the local pressure continuously decreases below the vapor pressure. At the acoustic frequency f, let tnucleus represents the time instant at which the local instantaneous pressure decreases to Pnucleus at a given temperature T. Time instant at the end of the first quarter of the acoustic wave is 4f1 . Thus the total time period in the first quarter of the acoustic wavein which the  local liquid pressure remains below Pnucleus is 4f1  tnucleus .

1.E+06 1.E+05 1.E+04 1.E+01

1.E+04 f (Hz)

1.E+07

Fig. 8. Reported Variation of the threshold pressure Pth with an ultrasonic frequency f [26].

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

molecules will vibrate. However if the external acoustic field is applied then the applied pressure will force the molecules to migrate in a particular fashion i.e. molecules will be forced to migrate away from each other in the rarefaction region. Hence the possibility of formation of nucleus will increase. Thus the applied negative pressure amplitude should increase the possibility of simultaneous migration of molecules away from each other with an increasing frequency. Hence as shown in Fig. 8, the reported trend [26] of pressure, Pth increases with an increase in the applied acoustic frequency above 10 kHz. 3.4. Estimation of an initial nucleus size of acoustic cavitation bubbles in the degassed water In the preceding section, an energy criterion for the enthalpy of vaporization of a single cubic lattice (void) has been satisfied with the cubic lattice model, hence we now proceed to estimate its size. Formation of a nucleus has been defined in this section followed by the estimation of its size. Effect of the temperature of the bulk liquid on the initial nucleus size has then been explored. It has been found from the steam table that the vapor state volume of one mol of water at 303 K is 0.59274 m3 at its vapor pressure (4.24 · 103 N/m2). Liquid state volume of one mol of water at 303 K is 1.8 · 105 m3 at its vapor pressure. The difference between these two volumes is the increase in the volume among the 6.023 · 1023 molecules as a result of the vaporization process, which occurs at/below the vapor pressure and at a given temperature. It has been obtained as 0.59272 m3. As discussed in Section 2, the number of unit lattice (void) in the 6.023 · 1023 molecules for the simple cubic structure is equal to the Avogadro’s number NA. Hence the volume of each unit lattice after flashing (vaporization) of water molecules is simply equal to 0:59274 i.e. 9.84 · 1025 m3. This expanded single NA lattice has been treated as an initial nucleus as shown in Fig. 9. Hence an initial radius of the spherical nucleus is 6.17 nm at 303 K and at the vapor pressure (4.24 · 103 N/m2) in the liquid water bulk, which is about 20 times less than the size of the gaseous nucleus (121 nm at 303 K) formed by desorbtion of dissolved gas molecules at initial 100% saturation condition. If we consider the energy analysis, then the radius of the vaporous nucleus after coalescence of three adjacent nuclei as discussed in Section p ffiffiffi 3 3.3.4 i.e. Rcoal at 303 K is 3R ¼ 1:44  6:17 109 Þ8:88 nm.

3.4.1. Effect of temperature of liquid water bulk With an increase in the temperature, vapor pressure increases and there is a decrease in the enthalpy of vaporization and the specific vapor volume [12]. Calculations have been repeated for different temperatures of liquid water bulk, T to estimate the corresponding nucleus size Rcoal formed by the evaporation of three unit lattices in the pure degassed water at the corresponding vapor pressures P Tvap . Rcoal decreases from 13.13 nm at 283 K ðP Tvap ¼ 1227 N=m2 Þ to 4.75 nm at 343 K ðP Tvap ¼ 31160 N=m2 Þ as shown in Table 1. Hence, if one wants to evaluate the effect of temperature on the nucleus size then there should be a common reference pressure. In the present case the vapor 3 2 pressure of water at 303 K ðP 303 vap ¼ 4:241  10 N=m Þ has been considered as the reference pressure. Thus the nuclei sizes Rcoal obtained at different temperatures T have been corrected as follows. An ideal gas law has been considered. Hence, the volume of the system (nucleus) is inversely proportional to the local pressure. Hence the mathematical b coal of the spherical expression for the corrected radius, R nucleus is as follows: sffiffiffiffiffiffiffiffiffi PT b coal ¼ 3 vap  Rcoal : R ð21Þ P 303 vap b coal corresponding to difThe corrected new values of new R ferent pressures at corresponding temperatures have been reported in Table 1. With an increase in the liquid temperb coal increases marginally from ature from 283 to 343 K, R 8.68 to 9.23 nm. 3.5. Calculation of the threshold pressure based on the lattice model So far we have considered the formation of a vaporous nucleus i.e. a vapor phase, which do not have a mechanical stability with the surrounding liquid. Now as already defined, a cavity means a presence of vapor phase in a spherical shape, which is in mechanical equilibrium with the surrounding liquid. Hence the pressure inside the cavity has to overcome the local pressure of the liquid and the pressure due to the surface tension (r). The value of the surface tension at 303 K is 0.072 N/m. Thus for the stable b coal in the pure degassed cavitation inception of radius R water, threshold pressure PT has been calculated at 303 K using Blake formula as follows: PT ¼

0:77  r 0:77  0:072 ¼ N=m2 b coal 8:88  109 R

¼ 62:13  105 N=m2 :

Unit lattice in liquid state

Unit lattice in vapor state

Fig. 9. Increase in volume of unit lattice due to vaporization.

73

ð22Þ

b coal In the above calculation, corrected value of Rcoal i.e. R has been used as per Eq. (14) to account for common reference vapor pressure at 303 K. Thus, the estimated theoretical Blake threshold pressure has been found to be approximately 62 · 105 Pa. With an increase in the temperature, surface tension decreases. Following formula, which is mentioned by

74

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

5

P T (10 Pa)

300

Briggs [6]

200 Cubic lattice model

100 0 270

300 330 T (K)

360

Fig. 10. Plot of PT against T for the degassed liquid water.

Carey [27] has been used to calculate the surface tension r at different liquid temperature T.  1:256   T T r ¼ 235:8 1  1  0:625 1  ; ð23Þ T cr T cr where Tcr is the critical temperature of water in Kelvin (647.3 K). Calculations have been done at different temperature values in the range of 283–343 K to predict the threshold pressures for the cavitation inception. Fig. 10 shows the trend of the predicted threshold pressure PT for the cavitation inception in the degassed incompressible pure water against the temperature of the liquid water T using (Eqs. (21)–(23)). With an increase in the temperature (283–343 K), there is b coal and decrease in the surface tension. an increase in R Hence, with an increase in the temperature, the predicted threshold pressure to form a single stable cavity also decreases. This prediction is qualitatively consistent with the reported experimental observation [6] over the temperature range (283–343 K) as shown in Fig. 10. The experimentally observed threshold pressure is in the range of 200 · 105–300 · 105 Pa as reported by Briggs [6]. Thus the estimated value is much closer to the experimentally observed threshold pressure than the earlier estimation of the cavitation threshold pressure (about 1367 · 105 Pa) based on the separation of water molecules by a distance greater than the van der Waals distance. However the rate of decrease of the observed threshold pressure is greater in magnitude than that has been predicted in the present mathematical model, which can be explained as follows. With an increase in the temperature, number of collisions of a molecule in its Brownian motion in the liquid state per second i.e. k* for a given free energy of activation DG increases exponentially as per Eq. (16). Hence there exists a possibility of simultaneous vaporization of three unit lattices at two or more locations with an increase in the temperature. If only a single layer of water molecules separates these locations then these units may combine to form one vaporous nucleus. This effect will further lower the predicted threshold pressure with an increase in the temperature. However in the present investigation this probable event has not been considered. So far we have considered the probability consideration and the criterion of the external energy requirement for the enthalpy of vaporization for the formation of the vaporous

nucleus in the liquid medium. Now we have considered the energy required for the formation of a new surface during vaporization via classical nucleation theory approach. 4. Classical nucleation theory Classical nucleation theory accounts for the requirement of energy for the formation of a new surface of radius R during the phase change due to the surface tension r [2,11]. To fulfill this energy need partially; the initial phase (i.e. vapor phase in case of condensation or liquid phase in case of vaporization) would first shift to meta-stable state (super saturation) where excess free energy would be available. Let DF represents excess free energy per unit volume. Hence the net gain in the free energy of the system DX can be represented as 4 DXðRÞ ¼  pR3 DF þ 4pR2 r: 3

ð24Þ

Thus, the variation of DX with R attains maximum value at R ¼

2r  ðLaplace EquationÞ: DF

ð25Þ

A nucleus of critical size R* ensures that further growth of the nucleus will further decrease the difference between the free energy available for a given volume of nucleus and the energy consumed in the formation of a surface. From the lattice model approach, the coalescence nucleus size at 303 K is 8.88 nm. Surface tension is 0.0717 N/m. The value of DF has now been calculated according to Eq. (26) to ensure its further growth DF ¼

2r 2  0:0717 ¼ ¼ 16148648 J=m3 : Rcoal 8:88  109

ð26Þ

Let the degree of super saturation S is defined as S¼

P vap P nucleation

ð27Þ

;

where Pvap is the co-existing pressure (vapor pressure) and Pnucleation is the actual local pressure in the vapor phase. Considering the vapor phase as an ideal gas, the excess free energy available per unit volume due to the given degree of super saturation in the vapor phase at a given temperature T can be expressed as follows: DF  RU T ql lnðSÞ;

ð28Þ

where RU is the universal gas constant 8.314 J/mol K and ql represents molar density of liquid. The density of water at 303 K is 103 kg/m3 i.e. 106/18 g mol/m3. The degree of super saturation has then been calculated as follows (Eq. (28)): lnðSÞ ¼

DF 16148648 ¼ ¼ 0:1153; RU T ql 8:314  303  106 18

S ¼ 1:122: P

4241 Hence P nucleation ¼ vap ¼ 1:122 ¼ 3779 N=m2 . S Thus, in conclusion, when the actual local pressure, Pnucleation falls to 3779 Pa at 303 K, the inception of vapor-

P.S. Bapat, A.B. Pandit / Ultrasonics Sonochemistry 15 (2008) 65–77

ous nucleus of radius of 8.88 nm will take place according to the energy balance for enthalpy of vaporization as per lattice model and the nucleus will further grow as further growth ensures the decrease in the difference between the energy required for the formation of a new surface during its growth and the free energy available of the vaporous nucleus due to the super saturation in the vapor state, according to CNT. Table 1 reports the variation of local pressure Pnucleation below the vapor pressure required for the vaporous nucleus inception with its further growth with liquid temperature T. 5. Kinetics of homogeneous nucleation of acoustic cavitation bubbles In the following, the kinetics of nuclei formation based on the cubic lattice model and the kinetics of nuclei stabilization based on CNT have been discussed. In the cubic lattice model, out of the possible 36,224 vibrations of molecules, one could favorably result into the homogeneous nucleation (Section 3.2). Thus it can be concluded that out of 36,224 molecules, one molecule can undergo nucleation. One mol of water occupies 18 cm3. Hence in 1 cm3 of water, the total numbers of water mole 23

cules are ¼ 6:02310 18

3:346  1022 . As discussed in Section

3.3, the energy associated with the Brownian motion of molecules in the liquid state is sufficient for the enthalpy of vaporization of three cubic lattices. The symbol k* stands for total number of vibrations of a molecule per second during its Brownian motion. Hence the rate of formation of vaporous nucleus per second per cm3 of liquid water r_ has been expressed as follows. r_ ¼

k  3:346  1022 : 36; 224

ð29Þ

At 303 K, the value of k* is 6.77 GHz for the liquid water (Section 3.3.4). Hence the rate of nucleation at 303 K is 6.25 · 1027 events per cm3 per second (Cubic Lattice Model). The rate of stabilization of nucleus according to CNT is as follows [2]:   DX J ¼ J 0 exp ; ð30Þ RU T where J0 is the pre-exponential factor, RU is the Universal gas constant [8.314 J/(mol K)], and DX* denotes the maximum free energy barrier to nucleation. The expressions for J0 and DX* are given by Eqs. (31) and (32) respectively. rffiffiffiffiffiffiffi 2 2r ðqgas Þ J0  ; ð31Þ pm ql where r stands for surface tension, m represents mass of a single molecule, qgas denotes density of gas phase, and ql represents density of the liquid phase. The expression for the free energy barrier, DX* is as follows:

DX ¼

16pr3 ½3ðRT ql ln SÞ2 

:

75

ð32Þ

The molar density of the vapor phase has been calculated according to an ideal gas law, i.e. qgas ¼ P nucleation . RT Hence at a nucleation pressure of 3779 N/m2 at 303 K, the molar density of the vapor phase is qgas ¼ 3779 ¼ 1:5 mol=m3 . The molar density of liquid water 8:314303 at 303 K is 5.56 · 104 mol/m3. The mass of a water molecule is 2.98 · 1026 kg. Hence J0, and DX* have been calculated with the use of Eqs. (31) and (32) respectively as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  0:0717 ð1:5Þ2 ¼ 5  107 =ðm3 sÞ;  p  2:98  1026 5:56  104 16  p  ð0:0717Þ3 ¼ 2:368  014 J=mol: DX ¼ ½3ð8:314  303  5:56  104  0:1153Þ2  J0 

Hence with the use of Eq. (30), J has been calculated as follows:   2:368  1014 J ¼ 5  107  exp ¼ 5  107 =ðm3 sÞ: 8:314  303 Thus the rate of stabilization of vaporous nucleus according to CNT is 5 · 107/(m3 s) i.e. 50 events per cm3 per second. Similar calculations have been done to obtain the rate of stabilization of gas nuclei of size 121 nm generated at 303 K according to CNT. The mass of an air molecule (pseudo element) = 4.79 · 1026 kg, z = 100%, r = 0.0717 N/m. Then the calculated values are as follows. DF = 2.81 · 105 J/m3, S = 1.002, Pnucleation = 1.01123 · 105 Pa, qair = 40.14 mol/m3, J0 = 9.76 · 1011/(m3 s), 19 DX* = 2.783 · 10 J/mol and J = 2.84 · 104/(cm3 S). Thus, the rate of stabilization of gas nuclei of size 121 nm at 303 K is 2.84 · 104/(cm3 s) according to CNT. However this rate is lower by three orders of magnitude when compared to that of the experimental findings of Naidu et al. [28] which is equal to 2.64 · 107/(cm3 s) for a gas bubble of 2 lm. The radius 2 lm was selected [28] based on the mathematical analysis of size that grows explosively at the minimum pressure of liquid (Pl0–PA) as described by Young [1]. Thus CNT is not able to predict the rate of stabilization of nuclei for a cavitation phenomenon. In the following paragraph, a link between the present approach and the model developed by Naidu et al. [28] has been established via mass balance of dissolved air molecules per cm3. The likely coalescence rate of gas nuclei for their stabilization has also been quantitatively considered. At 100% saturation, the mole fraction of dissolved air in water is 1.3 · 105 at 303 K. The density of liquid water is 3.349 · 1022 molecules/cm3. Hence the total number of dissolved air per cm3 are (=1.3 · 105 · 3.349 · 1022) 4.34 · 1017. Air molecules per nucleus are 7185 (Section 3 3.1). Hence   the total number of gas nuclei per cm are 17 13 ¼ 4:3410 6:04  10 =cm3 . The initial partial pressure 7185 of vapor in the cavity [28] was considered to be equal to

76

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the vapor pressure (3168 Pa) at an operating temperature of 298 K. For the mechanical equilibrium, the pressure inside the bubble has to be 1.73 · 105 Pa. The volume of the bubble is 3.35 · 1017 m3 (for radius of 2 lm). Hence with the use of ideal gas law, the total number of air molecules are 1.38 · 109 per gaseous nuclei. Thus, total air molecules present in the bubbles per cm3 of liquid are (=1.38 · 109 · 2.64 · 107) 3.64 · 1016. This value is close to the total dissolved air molecules (4.34 · 1017) per cm3 at 100% saturation. Thus it is confirmed that the desorbtion of dissolved gas molecules is responsible for the formation of gas nuclei. Hence by comparing gas molecules per nucleus (7185) based on lattice model with gas molecules per bubble of size 2 lm [28]we obtain  the rate of coalescence 9 of gas nuclei equal to ¼ 1:3810 1:9  105 =s. Thus 1.9 · 7185 5 10 nuclei of radius 121 nm are likely to coalesce per second to form a stable bubble of radius 2 lm at 303 K when water is saturated with air. Similarly, in the case of vaporous nuclei formation in acoustic phenomenon, CNT may not be able to predict the rate of stabilization correctly. Hence we can say that the rate of formation and stabilization of vaporous nuclei via coalescence may vary from 50 to 6.25 · 1025 per cm3 per second at 303 K, depending on the quality of the water and other conditions. 6. Summary and conclusions In the present investigation a phenomenological approach to define a nucleus inception of acoustic cavitation bubbles in the liquid water by either the desorbtion of dissolved gas molecules or the vaporization of local liquid molecules has been made. The energy criterion for the enthalpy of local vaporization of water molecules and the probability consideration in the Brownian motion in the incompressible liquid bulk has been considered. A simple approach of cubic lattice structure has been made to quantify the local vaporization process in terms of unit lattice and the molecular movement. The vaporous nucleus has been said to form below the vapor pressure for a given liquid temperature by the vaporization of eight water molecules, which form a single unit lattice (void). The classical nucleation theory has then been considered to calculate the degree of super saturation, which would be required in the vapor phase for the growth of the vaporous nucleus against the energy requirement for the formation of a new surface. The conclusions have been summarized as follow: 1. At 100% saturation condition, radius of a gaseous nucleus, Rair works out to be 121 nm at 303 K, at an acoustic frequency of 20 kHz with a pressure amplitude of 5.42 · 105 Pa. There exist 6.04 · 1013 nuclei of size 121 nm per cm3 of water at 303 K base on the mass balance of dissolved air. It is likely that about 1.9 · 105 nuclei per second per cm3 coalesce to form a stable

2.

3. 4.

5.

6.

7.

8.

9.

bubble of 2 lm based on the findings of Naidu et al. [28]. The total free energy content by eight molecules of water forming a unit lattice (void) in their Brownian motion (22.88 · 1020 J) in the incompressible liquid state is sufficient to overcome the enthalpy required for the vaporization of three unit lattices (22 · 1020 J). The predicted nucleus size, which formed by vaporization of three unit lattices is in the range of 10 nm. It has been found that the potential energy stored in the unit lattice by the acoustic source in its rarefaction region (1022 J) at an acoustic power of 3 · 104 W/ cm2 (at 20 kHz) is insufficient for the enthalpy of vaporization of local water molecules. It has been found that the thermal energy of the surrounding liquid bulk of thickness of a single layer of water molecules, which would be available due to temperature drop during the pressure change in the rarefaction cycle in the Joule–Thomson effect, is in short supply of the required enthalpy of vaporization. The predicted threshold pressure based on the nucleus size required for a stable cavitation inception using the Blake formula is of the order of 60 · 105 Pa, which is close to the experimental observations [6]. The present approach qualitatively explains the decrease in the threshold pressure with an increase in the liquid temperature. The explanation for the observed dependency of the pressure, which is required for nucleus inception on the acoustic frequency has also been provided. The CNT is not able to predict the rate of stabilization correctly. The rate of formation and stabilization of vaporous nuclei via coalescence may vary from 50 to 6.25 · 1025 per cm3 per second at 303 K depending upon the quality of water and condition of cavitation.

Acknowledgements One of the authors (PSB) would like to acknowledge the financial support provided by the Department of Science and Technology, India. References [1] F.R. Young, Cavitation, McGraw-Hill Book Company, UK, 1989. [2] D.W. Oxtoby, Nucleation of first order phase transitions, Acc. Chem. Res. 37 (1998) 91–97. [3] C. Vortmann, G.H. Schnerr, S. Seelecke, Thermodynamic modeling and simulation of cavitating nozzle flow, Int. J. Heat Fluid Flow 24 (2003) 774–783. [4] P.S. Epstein, M.S. Plesset, On the stability of gas bubbles in liquid– gas solutions, J. Chem. Phys. 18 (1952) 1505–1509. [5] T.J. Mason, J.P. Lorimer, Applied Sonochemistry: The Uses of Power Ultrasound in Chemistry and Processing, Wiley VCH Verlag GmbH, Weinhein, 2002. [6] L. Briggs, Limiting negative pressure of water, J. Appl. Phys. 21 (1950) 721–722.

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