The Growth of Vapor Bubbles in the. Volume of Superheated Drops, Dispersed. in High-Boiling Liquid

Transcription

1 Appied Mathematica Sciences, Vo. 8, 2014, no. 151, HIKARI Ltd, The Growth of Vapor Bubbes in the Voume of Superheated Drops, Dispersed in High-Boiing Liquid Aexander K. Rozentsaig and Chesa S. Strashinskii Naberezhnye Cheny Institute of Kazan (Voga region) Federa Uniersity Tatarstan, Russian Federation Corresponding author: Aexander K. Rozentsaig , Naberezhnye Cheny, street Shami Usmano dom 76/39 (45/15/01) kartira 50, Russian Federation Copyright 2014 Aexander K. Rozentsaig and Chesa S. Strashinskii. This is an open access artice distributed under the Creatie Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, proided the origina work is propery cited. Abstract The growth of bubbes to the critica size, resuting in a oume of superheated dropets, which dispersed in high-boiing continuous medium iquid emusion. Based on the first aw of thermodynamics anaysis of the formation of a new iquid-apor interface surface and the nature of the phase transitions of the first kind in the cosed medium, which is imited to the surface of the dropets of the dispersed phase. Shows the presence of factors associated with the size of the bubbes of apor with a constant suppy of heat through the interface. Presents the dependencies that fundamentay different from those describe boiing in unimited oume fuid or near the surface of the heated wa. Keywords: iquid emusion, heat transfer, ow-boiing dropets, initiation of nuceation, apor bubbes, mechanisms of boiing

2 7520 Aexander K. Rozentsaig and Chesa S. Strashinskii 1 Introduction The formation of a continuous fim of apor near surfaces cooed was with hep homogeneous iquid can competey bock heat transfer. An effectie way to preent boiing crisis is the use of iquid emusions. The transfer process of aporization in the oume drops ow-boiing dispersed phase aows aoiding a sharp reduction in heat transfer [1, 2]. The fact that the high-boiing continuous medium of the emusion at a high enough concentration drops can inhibit the growth and merging of bubbes of apor. This is due to excessie capiary pressure inside drops and insuation of apor bubbes formed in the neighboring dropets. Thus preseres the homogeneity of the structure of non-equiibrium gas-iquid system in the dispersed phase of the emusion. Subcooed iquid continuous medium eiminates the appearance of fim apor on heating surfaces, which promotes uniform distribution of heat fow in the oume of iquid emusion. In genera, the unproductie expenses of heat required to moe a dispersed iquid in the apor state, wi be reatiey high, which creates conditions for increase efficiency of heat transfer to the emusions. Mechanisms of transfer of heat in emusion with ow-boiing of the dispersed phase, which caused to the efficiency of the process of heat exchange, significanty different from the cassica poo nuceate boiing - mechanism of boiing in an unimited oume of a homogeneous iquid. Dynamics of nuceate boiing in the dropets of bidistiate at soid wa of the heater was studied [3]. These experiments was carried out on the copper rough surface as we as on the poished soid surface. Marked important differences of aporization inside of oume dropets from conditions poo boiing. Intensie dropet boiing accompanied by continuous motion of a contact ine: the area of dropet base and its shape change, the number of bubbes aries continuousy as we as their aerage diameter and his time of ife. Shows that this boiing mechanism fundamentay differs from poo nuceate boiing. Resuts of the anaysis of the conditions of stabe nuceation in the ow-boiing disperse phase taking into account the turbuent mode of motion of iquid emusions presented [4]. Proposed cacuated reation for estimating the dependence of the minimum size of drops, whose boiing up can initiated by the resonance mechanism of their destruction by turbuent pusations on the superheating temperature. Take into account the effect of hydrodynamic factors on the time deay of boiing up of emusions with the ow-boiing dispersed phase. This gies broader understanding of the possibe modes of the bubbes boiing. More compex for the experimenta study is boiing inside the dropets of the dispersed phase, separated from the heating surface of the emusion high-boing continuous enironment. In this case, the mode representation based on the generaization of the resuts of studies of the physica processes inoed in the composition of the studied phenomenon.

3 The growth of apor bubbes in the oume of superheated drops Thermodynamic Mode the Growth of Vapor Bubbes Within the Voume of Dispersed Drops Consider the process of formation and growth of the radius R bubbes of apor in the oume of superheated drops with radius R dispersed in high-boiing continuous medium iquid emusion (R > R). Limited to a uniform size distribution, the number of drops of the dispersed phase N in the oume of emusion Ve determined by the ratio: 4 R 3 N WdVe (1) 3 where Ve = Vd + Vc, Wd = Vd /( Vd + Vc), and Vd and Vc are the oumes dispersed and continuum phases of iquid emusion. Per unit oume of the emusion, the number of ow-boiing dropets dispersed phase is defined as 3 N 3W d /( 4R ) (2) They correspond to the tota area S-, representing the surface of the 2 iquid-iquid interface of the continuous and disperse phases S- = 4 R N = 3 Wd / R and excess surface energy d, associated with the dispersion of dropets in the oume of the continuous medium 3 Wd d S (3) 2 R R Surface forces that imit the ocation of the bubbes within the oume drops and inhibit their growth, cause, on the other hand, aggregate instabiity of the emusion, the tendency to coaescence of the dispersed phase [5]. Therefore, consideration is imited to the conditions when the impact can assumed negigibe. The increase in the temperature of the emusion creates conditions for aporization in the oume of ow-boiing iquid inside the dropets of the dispersed phase. Interfacia surface, separating it from the high-boiing continuous medium, becomes ike a soid surface during heating of a homogeneous iquid. As the warm aboe the temperature of saturated apor of the iquid drops inside the dropet at the interface conditions for the formation of a critica apor bubbe, abe to exist and grow in size. The temperature of the apor in the bubbes assumed equa to the saturation temperature. It wi proided by the constant suppy of heat through the interface surface S-, that separates steam phase in oume of the dropets of the dispersed phase. Depending on the concentration of the dispersed phase and therma conditions boiing drops may fai not ony compete or partia eaporation of them, but aso the reerse process of condensation when ack a energy from outside.

4 7522 Aexander K. Rozentsaig and Chesa S. Strashinskii Heat spent for eaporation, formation of new iquid-apor interface surface within oume of drops, and the change of the surface area of the drops themsees. It is beieed, that the source of heat for the drops of the dispersed phase is surrounding their continuous medium of the emusion, the amount of which is to simpify the modeed compex migration process reies unimited. Further consideration is aso imited to changing the size of the apor bubbe radius for a fixed oume of the dropets of the dispersed phase Interfacia surface, imiting ow-boiing iquid Bubbes of the apor in boiing dispersed dropet Low-boiing dropets of the dispersed phase High-boing continuous medium Soid wa of the heater Figure 1: The scheme for nuceate boiing in oume ow-boiing dispersed phase imited by the interfacia surface We write the first aw of thermodynamics for apor bubbes formed in the oume of each dropet of the dispersed phase iquid emusion with radius R: dq di V dp (4) where dq is the heat input from the continuum phase to the drops of the dispersed phase, di is the change of enthapy of the phase transition, V = 4 R 3 n /3 - tota oume of bubbes of apor phase, the number of which n depends on the ee of heat energy input, dp is the pressure change inside of the bubbes. Express the amount of the energy required for the formation of bubbes of steam in each of the dropets of the dispersed phase, as dq qs dt (5) where q is the specific heat fow from the superheated ayer interfacia surface bubbes with radius R, Sg- = 4 R 2 n is the tota interfacia area of a apor bubbes in a singe drop of the dispersed phase.

5 The growth of apor bubbes in the oume of superheated drops 7523 The enthapy change is: di = ρl dv, (6) where ρ - density bubbes formed pair, L is the specific heat of aporization. The pressure inside the apor bubbes, which are formed inside of oume the dropets of the dispersed phase, is formed of the foowing components: p p 0, (7) R R where p0 is the pressure in the continuous medium of the emusion, σ- - coefficient of surface tension at the interface of apor and iquid phases in oume drops dispersed phase, σ- is the coefficient of surface tension at the interface of the dispersed phase and the continuous medium of the emusion. Substituting (5) - (7) in (4) and assuming spherica symmetry of the process of eaporation in a drop of ow-boiing iquid, get qs dt = ρldv - V dp, dv = 4 R 2 n dr, dp dr dr 2 2, R R R qdt LdR ( dr dr ). (8) R R R To determine the reationship between R and R use the aw of conseration of mass of separate drops during eaporation: V 0 V (V V ), (9) where ρ - density iquid phase drops, V0 and V - initia and current oume drops. The condition of conseration of mass of each dropet with initia radius R0, after the formation of n apor bubbes, haing a spherica shape R0 nr ( R nr ) (10) Away from the critica point wi ρ << ρ. Then due to the change in oume of the dispersed phase due to the formation of bubbes of apor phase inside the ow-boiing dropets are R R nr (11) Taking into account expressions for R and dr obtained from (11), equation (8) can written as foows:

6 7524 Aexander K. Rozentsaig and Chesa S. Strashinskii 1 n qdt LdR ] R [ dr (12) n ( R0 / R ) The differentia equation (12) reatie to the size of the apor bubbes R is noninear. It is rather greaty simpifies the rea situation, to sere as a basis for exact or numerica soutions, ike a deeper understanding situations [6]. Equay important is the quaitatie anaysis of the mechanisms of heat transfer, which considered faorabe therma conditions [7]. Further mechanisms eaporating ike steam exposion [8, 9], excuded consideration of the reatiey ow and constant ee of heat fux density in the oume of the continuous medium iquid emusion. In a first approximation, the number of bubbes n in each indiidua drop is considered equa to one, assuming the heat fux is insufficient for growth of other iabe embryos apor phase, which potentiay aso can grow under more faorabe conditions. 3 Anaysis of the Mechanisms of Growth of Bubbes Within the Voume of Drops Limited Interfacia Surface Let us consider some imiting cases, when the equation (12) reduced to the inear mind and integrates without the use of numerica methods. So of coarse emusions, when apor bubbes notaby smaer dropet size of the fuid in which they are formed, i.e. by R0 >> R, equation (12) is substantiay simpified: qdt 1 LdR R dr With sufficienty arge size bubbes of steam, when the increase in their inear dimension sma compared with the increase in the apor phase and the second summand in the right hand side can negected, equation (13) takes we known from the works of other authors form [10-12]: (13) qdt LdR, dr / dt q /( L). (14) Obiousy, in this case, the growth of apor bubbes imited ony by the suppy of heat, required for the process of aporization, and does not expicity depend on their size. This mode corresponds to the conditions boiing in unimited oume of stationary fuid, where there is no need to consider the aaiabiity of the deeoped interfacia surface. Howeer, in the case of the ery sma bubbes, rapidy increasing radius with the increase oume of the apor phase, on the contrary, the first term of the right side of equation (13) wi preai oer the atter. In this imiting case, the rising steam bubbes wi be described another equation:

7 The growth of apor bubbes in the oume of superheated drops 7525 qdt dr (15) R Equation (15) corresponds to a fundamentay different mode iew, when the growth of fine bubbes within the oume of the dropets of the dispersed phase iquid emusions imited ony by the energy of surface tension forces. Moreoer, under these conditions, the roe of surface forces with decreasing bubbe radius becomes a the more important that, accordingy, there aaiabe physica concepts [11]. There are many theoretica and experimenta studies [10], that are connected with the initia stage of the phase transition of the iquid - apor. It is estabished, that in the superheated iquid can eoe oer time, ony apor bubbes, the radius of which exceeds the critica aue Rcr. For the apor bubbes of smaer size exist a ery high probabiity of returning them to the iquid phase. This a priori information corresponds to the initia condition R tt 0 R cr, which must satisfy the soution of the mode equation (15) for the case of aporization in the imited interfacia surface oume of the superheated fuid. The mode equation (15) is integrated in an anaytica form for the stationary fow of heat q(t) = q0, its soution is non-inear: R (t ) R кр q exp( 0 t) (16) It is true to a certain bubbe size, R*, when the rate of growth is dependent on increasing the oume of the apor phase. The dependence of the growth of the steam bubbes exceeding this imit, takes inear: * q0 R ( t ) R t (17) L

8 7526 Aexander K. Rozentsaig and Chesa S. Strashinskii R, м t, с Figure 2: The dependence of the radius of the apor bubbe from time to time with a constant suppy of heat to the drop of water with a specific fow q0 = 10 W/m 2, corresponding to different mode iews: 1 - (13), 2 - (14), 3 - (15). The fig.2 presents the resuts of numerica soutions of the mode equations (13) - (15), thermophysica parameters correspond to conditions when the high-boiing continuous medium dispersed reatiey arge drops of water. Note that the cure (1), summarizing the growth of apor bubbes corresponds to the exponentia dependence on time, based on the experimenta data [12-14]. A comparison of the cure (1) with the cures (2) and (3) shows that the scope is rather narrow and restricted by the size of the bubbes cose to the size of critica apor bubbes under conditions of the fuctuation of formation apor phase [12]. Conditions initiated by the aporization meet a wider range of externa factors, which are not incuded in the mode equation (12). 4 Concusions 1. Thermodynamic mode of aporization in the imited oume of the dropets of the dispersed phase iquid emusion comprises in the range of thermophysica factors are characteristics of interfaces between immiscibe iquids forming the emusion, and between iquid and apor of the dispersed phase. 2. Mechanisms of heat transfer in the presence of we-deeoped interfacia surface significanty enhance the physica understanding of the processes of aporization in an unimited oume of stationary fuid. Therefore, for the radius of the apor bubbes cose to the critica aue obtained exponentia reationship between

9 The growth of apor bubbes in the oume of superheated drops 7527 them and the surface tension coefficient between the apor and the iquid phase within a oume of ow-boiing dropets of the emusion. 3. The proposed mode of growth of apor bubbes in superheated emusion (13) with a ow-boiing dispersed phase to justify additiona physica iew for expaining the deay of the boiing. References [1] N.V. Buano, B.M. Gasano, Pecuiarities of boiing of emusions with a ow-boiing disperse phase, Internationa Journa of Heat and Mass Transfer, 51 (2008), [2] N.V. Buano, B.G. Baidako, Bubbe Boiing of Emusions with a Low-Boiing Disperse Phase, Heat Transfer Research, 32(2001), [3] S.Ya. Misyura, Nuceate boiing in bidistiate dropets, Internationa Journa of Heat and Mass Transfer, 71(2014), [4] A.K. Rozentswaig, C.S. Strashinskii, Hydrodynamic aspects of boiing up of a disperse phase in a homogeneous turbuent fow of an emusion, High Temperature, 49(2011), [5] A.K. Rosenzweig, Ch.S. Straszynski, The coaescence of ow-boiing dispersed phase in a turbuent fow of the cooing emusion, Russian Journa of Appied Chemistry, 8(2008), [6] A.P. Soodo, Differentia mode of nuceate boiing, High Temperature, 45(2007), [7] A.K. Rozentsaig, Ch.S. Strashinskii, Mechanisms of boiing of an emusion with a ow-boiing disperse phase in a turbuent fow of a homogeneous emusion, Journa of Engineering Physics and Thermophysics, 83(2010), [8] Yu.A. Zeigarnik, Yu.P. Iochkin, V.S. Grigor e, A.A. Oksman, Notes Concerning Some Aspects of Vapor Exposion, High Temperature, 46(2008), [9] Yu.A. Zeigarnik, Yu.P. Iochkin, E.Z. Koro, Thermomechanica mechanism for fine fragmentation of iquid dropets under condition of apor exposion, High Temperature, 42(2004),

10 7528 Aexander K. Rozentsaig and Chesa S. Strashinskii [10] B.M. Dorofee, V.I. Vokoa, Dynamics of apor bubbes growth in boiing owing to the excess enthapy surrounding superheated iquid, High Temperature, 46(2008), [11] B.M. Dorofee, The ariation of the radius of the apor bubbe under condition boiing in the oume of subcooed iquid and the sound puses generated by bubbes, High Temperature, 40(2002), [12] B.M. Dorofee, V.I. Vokoa, An acoustic method of inestigation of the process of boiing, High Temperature, 43(2005), [13] Guhman A.A. Appication of simiarity theory to the study of the processes of heat-mass transfer. M.: Higher schoo, 1974 (in Russian). [14] Skripo, V.P., Metastabe Liquids, New York: Wiey, Receied: September 11, 2014; Pubished: October 28, 2014