Specific interface area and self-stirring in a two-liquid system experiencing intense interfacial boiling below the bulk boiling temperatures of both components
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b EPJ manuscript No. (will be inserted by the editor)
Specific interface area and self-stirring ina two-liquid system experiencing intenseinterfacial boiling below the bulk boilingtemperatures of both components
Denis S. Goldobin , , a and Anastasiya V. Pimenova , b Institute of Continuous Media Mechanics, UB RAS, Perm 614013, Russia Department of Theoretical Physics, Perm State University, Perm 614990, Russia
Abstract.
We present an approach to theoretical assessment of themean specific interface area ( δS/δV ) for a well-stirred system of twoimmiscible liquids experiencing interfacial boiling. The assessment isbased on the balance of transformations of mechanical energy and thelaws of the momentum and heat transfer in the turbulent boundarylayer. The theory yields relations between the specific interface areaand the characteristics of the system state. In particular, this allowsus to derive the equations of self-cooling dynamics of the system inthe absence of external heat supply. The results provide possibility forconstructing a self-contained mathematical description of the processof interfacial boiling. In this study, we assume the volume fractions oftwo components to be similar as well as the values of their kinematicviscosity and molecular heat diffusivity.
For a well-stirred multiphase fluid systems the mean interface area per unit volume, or the specific interface area, say S V ≡ ( δS/δV ), is a significant characteristic of thestate. It becomes even more important for the systems where the interface is activechemically or physically (for instance, interface of phase transition). The systems ofimmiscible liquids experiencing interfacial boiling [1,2,3,4,5,6,7,8,9,10,11,12,13,14] arean example of the systems where parameter S V has essential control on the evolvingprocess.It is well known that the boiling at the interface between two immiscible liquidscan occur bellow the bulk boiling temperatures of both components ( e.g. , see [1,2]).This phenomenon is widely reported in literature and finds many applications inindustry. Species from both liquid phases evaporate into the vapour layer formingbetween them. As a result, the condition for the vapour layer growth ( e.g. , see [1]) is n (0)1 ( T ) + n (0)2 ( T ) ≥ P k B T , a e-mail: [email protected] b e-mail: [email protected]
Will be inserted by the editor where n (0) j ( T ) is the number density of the saturated vapour of specie j at absolutetemperature T , P is the atmospheric pressure, and k B is the Boltzmann constant.The minimal interfacial boiling temperature T ∗ corresponds to equality in the lat-ter condition. Simultaneously, the bulk boiling of liquid component j requires n (0) j ( T )alone to exceed P / ( k B T ), meaning that interfacial boiling starts below the bulk boil-ing points of both components. The interfacial boiling process is essentially controlledby the specific interface area S V .The problem of calculation of S V cannot be addressed rigourously. Moreover, anydirect numerical simulation, being extremely challenging and CPU-time consuming,will provide results pertaining to specific system set-ups. Some general assessmentson S V can be highly beneficial. In this paper we perform these assessments for theprocess of direct contact boiling in a system of two immiscible liquids.The big picture of mechanical processes in the system is as follows. At the directcontact interface, a vapour layer grows and produces bubbles which breakaway fromthe interface and rise. The presence of vapour bubbles changes the fluid buoyancy andperforms a “stirring” of the system. This stirring enforces increase of the contact area S , while surface tension and gravitational segregation of two liquids tend to minimizethis contact area.For a two-liquid system experiencing direct contact boiling, the quantity of ourinterest depends on parameters of liquids and characteristics of the evaporation pro-cess, which are controlled by the mean temperature excess above T ∗ and the bubbleproduction rate [9]. In our study, we assume the volumes of both components to becommensurable; no phase can be considered as a medium hosting dilute inclusions ofthe other phase. Hence, the characteristic width of the neighborhood of the vapourlayer, beyond which the neighborhood of another vapour layer sheet lies, is H ∼ /S V . In this paper, as a first approach to the problem, we simplify our consideration,restricting ourselves to the case of liquids with similar values of those physical pa-rameters, which control the properties of the turbulent boundary layer and the heattransfer in it: kinematic viscosity and molecular heat diffusivity. The assumptions ofsimilar volumes of both fractions and similar physical parameters make our problemsymmetric with respect to interchange of two liquids and significantly simplify thestudy.The process of boiling of a mixture above the bulk boiling temperature of the morevolatile liquid is well-addressed in the literature [3,4,5,6,7,8,9]. Hydrodynamic aspectsof the process of boiling below the bulk boiling temperature have been theoreticallystudied in [13,14,15]. While in [13,14] the specific interface area S V was treated as acontrol parameter of the system dynamics, here we construct a theory allowing to findthe relationships between this parameter and the macroscopic characteristics of thestate of a well-stirred system. Specifically, in what follows, we perform an analyticalassessment of the dependence of S V on the evaporation rate (or heat influx) andmean temperature and derive equations of self-cooling in the absence of external heatsources for a well-stirred system.Our work heavily relies on the theory of turbulent boundary layer [16,17,18,19]which is employed for calculation of the rates of transfer of kinetic flow energy andheat from the bulk of phases towards the boiling interface. For liquids with viscosity properties similar to that of water, the viscous boundarylayer of flows turns out to be very thin compared to the realistic scales of folds of the ill be inserted by the editor 3 x u y ( ) y Fig. 1.
Sketch of the turbulent boundary layer near the flat rigid wall and notations interface , meaning the flow in the liquid bulk is either inviscid (practically improb-able) or turbulent (which is more plausible for the system under our consideration).Hence, one has to consider the turbulent currents near the interface; the descriptionof properties of these currents is available from the theory of a turbulent boundarylayer.The theory of a turbulent boundary layer has been developed by Karman [16] andPrandtl [17], and is also well presented in book [18], on which we heavily relay inthis section. For a turbulent boundary layer near a flat rigid wall beyond the ‘viscoussublayer’—a thin vicinity of the boundary where molecular and turbulent viscositiesare of the same order of magnitude—the ‘macroscopic’ (averaged over pulsations)flow u is a shear one (tangential to the boundary) and is controlled by the fluiddensity ρ , momentum flux to the boundary Π , and the distance from the boundary y . Specifically, dudy = u ∗ κ y , (1)where the reference turbulent pulsation velocity u ∗ is introduced as follows: Π = ρu ∗ , κ is the dimensionless Karman constant determined experimentally, κ ≈ . . Generally, u = u ∗ f ( ξ ) , ξ = yu ∗ /ν , (2)where ν is the kinematic viscosity. Beyond the viscous sublayer ( i.e. , for ξ & u = u ∗ κ ln yu ∗ ξ ν , (3)where constant ξ ≈ .
13 is determined from experiments. The average energy fluxto the boundary is q u = uΠ. (4)For the temperature field the average profile in the turbulent boundary layer isadditionally controlled by the Prandtl number Pr = ν/χ and is affected by the fact thetemperature field is a passive scalar, while the average velocity is a vector quantity,which is also correlated with the turbulent vortex cascade, T = β q T κ ρc P u ∗ h ln yu ∗ ν + f T (Pr) i + T , (5)where T is the boundary temperature, q T is the heat energy flux towards the bound-ary, c P is the specific heat at constant pressure; constant β = 0 . f T (Pr)(instead of ln(1 /ξ ), as compared to Eq. (3)) represent the mentioned differences be-tween u and T [18,19]. This statement will be underpinned with estimates below in the text. Will be inserted by the editor x u y Liquid 1Liquid 2 HH /2 Fig. 2.
Sketch of the vapour layer between two immiscible liquids
For realistic situations, u ∗ is in range 0 . . / s and the argument of the loga-rithm in Eq. (3) is of the order of magnitude of 1 for y ∼ − m. On this scale the freeinterface is inflexible because of the surface tension. Interface deformations becomeplausible for the scale of 1 mm, where the logarithm argument ( yu ∗ ) / ( ξ ν ) ∼ and the further increase of the distance y by a factor 10 causes the relative changeof the logarithm only by 1 /
4. Hence, the dominant change of average fields of flowand temperature occurs on the scales, where the interface is nearly inflexible, andthe theory of the turbulent boundary layer can be relevant for our system with theliquid–liquid interface in the role of the boundary (Fig. 2). However, one should keepin mind, that on the rigid wall the velocity is strictly zero, while for a free inflexibleinterface the vortexes with currents along the interface are admitted. Therefore, thequantitative characteristics of the boundary layer profiles can be altered comparedto their values for the case of a rigid wall. Since constants κ and β are related tothe scale-independent properties of profiles, while the difference between the cases ofa rigid wall and a free inflexible interface influences the processes within the viscoussublayer, one can expect the change of ξ and f T (Pr), but κ and β must hold thesame. The latter remark is important for us, because of unfortunate lack of knowledgeon the turbulent boundary layer near a free inflexible surface.Employing the knowledge on the turbulent boundary layer properties, one canestimate the rate of viscous dissipation of the kinetic energy of the liquid flow andthe heat flux to the interface. The kinetic energy flux from the bulk of certain liquid component towards the inter-face can be assessed on the basis of relation (4). On the other hand, this flux is aviscous loss of the kinetic energy of macroscopic liquid flow in the bulk; q u ( y = H/ ≈ δ ˙ W visc δS , (6) ill be inserted by the editor 5 where δ ˙ W visc is the viscous energy loss per area δS of the interface. Hence, δ ˙ W visc δV ≈ (cid:18) δSδV (cid:19) q u ( y = H/ S V Πu ( y = H/ S V ρu ∗ κ ln Hu ∗ ξ ν . (7)The space-average kinetic energy of liquid flow, say W liq , k , is mainly contributedby average flow u , as the contribution of turbulent pulsation flow can be neglected [18];therefore, δW liq , k δV ≈ H H/ Z ρu y ≈ ρu ∗ κ ln Hu ∗ eξ ν , (8)where e is the Euler’s number.One can evaluate the average temperature from Eq. (5), finding the relation be-tween h T i and q T ; h T i = 2 H H/ Z T d y ≈ T + β q T κ ρc P u ∗ (cid:18) ln Hu ∗ ν + f T (Pr) − (cid:19) . (9)We assume the interface temperature to be equal to the minimal temperature requiredfor the growth of the vapour layer, T = T ∗ . Indeed, with the data provided in [14],one can estimate the typical difference ( T − T ∗ ) observed in experiments as ∼ − K.Hence, we can neglect it for our consideration in this paper. From Eq. (9), one canevaluate the dependence of heat flux on the average overheat ( or the temperatureexcess above T ∗ ) h Θ i = h T − T ∗ i ; q T ≈ κ ρc P u ∗ h Θ i β (cid:18) ln Hu ∗ ν + f T (Pr) − (cid:19) . (10) Let us derive relationships between the macroscopic parameter S V = ( δS/δV ), so-called specific interface area, of the system state and the heat influx rate per unitvolume ˙ Q V = δQ/ ( δV δt ) for a statistically stationary process of interfacial boiling.Statistical stationarity of the enforced stirring process implies as well balance of me-chanical energy fluxes.One should distinguish macroscopic degrees of freedom and internal (thermody-namic) degrees of freedom. It is important, because while the viscous dissipation ofthe macroscopic kinetic energy into the internal energy is possible, the direct trans-formation of the internal energy into the macroscopic one is forbidden by generalprinciples of thermodynamics. The only mechanism of the energy supply for macro-scopic motion is the buoyancy of generated vapour bubbles. Let us now comparethe amount of energy in macroscopic and internal degrees of freedom. The energy ofthermal motion of atoms corresponds to characteristic atom velocities 10 − m / s.Particularly for water, the latent heat of evaporation is even significantly larger than Will be inserted by the editor the kinetic energy of thermal motion of its atoms at T = 300 K. For macroscopicdegrees of freedom the mechanical motion is enforced by potential forces of gravityand surface tension. Hence, the net energy associated with these degrees of freedomis of the same order of magnitude as the macroscopic kinetic energy. For the realisticsystems of our interest, the macroscopic stirring flow of velocity 1 m / s can be consid-ered as very strong, which means that the energy of macroscopic flow is typically bya factor 10 or more smaller than the internal energy. Thus, the macroscopic degreesof freedom make vanishingly small contributions into the balance of internal energy.Summarizing,(i) the supply of the macroscopic mechanical energy into system is associated solelywith the buoyancy of generated vapour bubbles,(ii) the amount of generated vapour bubbles is determined by the heat transfer to theboiling interface.The flow and consequent stirring in the system are enforced by the buoyancy ofthe vapour bubbles, while other mechanisms counteract the stirring of the system.These other mechanisms are gravitational stratification of two liquids, surface tensiontending to minimise the interface area and viscous dissipation of the flow energy. Asdiscussed above, all the heat inflow into the system can be considered to be spentfor the vapour generation; ˙ Q V V −→ ( Λ n (0)1 + Λ n (0)2 ) ˙ V vap , where V is the systemvolume, ˙ V vap is the volume of the vapour produced in the system per unit time, Λ j is the enthalpy of vaporization per one molecule of liquid j , and n (0) j is the saturatedvapour pressure of liquid j . Thus,˙ V vap = ˙ Q V VΛ n (0)1 + Λ n (0)2 . (11)The potential energy of buoyancy of rising vapour bubbles ρV vap gh/ h isthe linear size of the system, h ∼ V / , ρ is the average density of liquids, the vapourdensity is zero compared to the liquid density) is converted into the kinetic energy ofliquid flow, the potential energy of a stirred state of the two-liquid system, the surfacetension energy and dissipated by viscosity forces. In a statistically stationary state,the mechanical kinetic and potential energies do not change averagely over time andall the mechanical energy influx is to be dissipated by viscosity; ρV vap gh/ −→ ˙ W visc τ , where ˙ W visc is the rate of viscous dissipation of energy, τ is the time of generation ofthe vapour volume V vap , V vap = ˙ V vap τ . Hence, ρ ˙ V vap g h W visc . (12)In turn, the viscous dissipation of the kinetic energy of flow ˙ W visc is determined byEq. (7).Further, we have to establish the relationship between the flow kinetic energyand the mechanical potential energy in the system. Rising vapour bubbles “pump”the mechanical energy into the system, while its stochastic dynamics is governed byinterplay of its flow momentum and the forces of the gravity and the surface tension onthe interface. Hamiltonian systems with huge number of degrees of freedom experiencethermalization, or one can say, they tend to the state of thermodynamic equilibrium.In thermodynamic equilibrium, the total energy is strictly equally distributed betweenpotential and kinetic energies related to quadratic terms in Hamiltonian. (The latterstatement is frequently simplified to a less accurate statement, that energy is equally ill be inserted by the editor 7 distributed between kinetic and potential energies associated with each degree offreedom.) Although in turbulent systems the viscosity plays a crucial role, the kineticenergy is dissipated only on the edges of wavenumber spectrum, meaning the systemis weakly dissipative and its dynamics is nearly conservative. Being not exactly in thecase where one can rigorously speak of thermalization of the stochastic Hamiltoniansystem dynamics, we still may assess the kinetic energy of flow to be of the sameorder of magnitude as the mechanical potential energy of the system. Thus, W liq , k ∼ W liq , p g + W liq , p σ , (13)where W liq , p g and W liq , p σ are the gravitational potential energy and the surface ten-sion energy, respectively. We set the zero levels of these potential energies at thestratified state of the system with a flat horizontal interface.The gravitational potential energy of the well-stirred state with uniform distribu-tion of two phases over height is W liq , p g = ∆ρV g h , where ∆ρ is the component density difference. The surface tension energy is W liq , p σ ≈ ( σ + σ ) S V V , where we have neglected the interface area of the stratified state compared to the area S V V in a well-stirred state. Due to the presence of the vapour layer between liquidsthe effective surface tension coefficient of the interface is ( σ + σ ) but not σ as itwould be in the absence of the vapour layer. S V and balance of energy transfer S V as a function of average overheat h Θ i From Eqs. (7), (12), and (11), one can find for ˙ Q V :˙ Q V ≈ Λ n (0)1 + Λ n (0)2 gh S V u ∗ κ ln Hu ∗ ξ ν . (14)On the other hand, the heat inflow to the interface area within the volume δV ,˙ Q V δV , is contributed by heat flux q T from two sides of the interface, 2 q T δS , i.e. ,˙ Q V = 2 S V q T , and, with Eq. (10), one can evaluate˙ Q V = S V κ ρc P u ∗ h Θ i β (cid:18) ln Hu ∗ ν + f T (Pr) − (cid:19) ≈ S V κ ρc P u ∗ h Θ i β ln Hu ∗ ξ ν . (15)Matching heat consumption (14) for the vapour generation and turbulent heat transferto the interface (15), one obtains h Θ i ≈ Λ n (0)1 + Λ n (0)2 ρc P gh β u ∗ κ ln Hu ∗ ξ ν ≈ β ( Λ n (0)1 + Λ n (0)2 ) ρ c P gh δW liq , k δV . (16) Will be inserted by the editor
Eq. (13) provides the relation between W liq , k and potential energy, which is an explicitfunction of macroscopic parameter S V . Hence, h Θ i ≈ Θ g (cid:20) k h S V (cid:21) , (17)where Θ g = 2 β ( Λ n (0)1 + Λ n (0)2 ) ∆ρρ c P and the characteristic wavenumber for interfacial waves k = s ( ρ − ρ ) gσ + σ , where, as noted above, the effective surface tension coefficient of the contact interfacein the presence of the vapour layer is ( σ + σ ) [15]. Noteworthy, the relative impor-tance of the first and second terms in the brackets in Eq. (17) depends on the verticalsize of the system h ∼ V / .For the n -heptane–water system, l k ≡ /k ≈ . Θ g ≈ .
346 K [14]. Fora well-stirred system the distance between sheets of the folded interface H ∼ S − V ≪ h .One can distinguish two limiting cases;(1) h/S V ≪ l k , which corresponds to the case of a surface-tension dominated system,(2) h/S V ≫ l k , which corresponds to the case of a gravity-driven system.The temperature Θ g is remarkably small compared to the maximal overheat Θ ofthe n -heptane–water system 20 K, which can be attained before the bulk boilingof components can occur. Thus, according to Eq. (17), the boiling regime will betypically surface-tension dominated (since the second term in the brackets will betypically large compared to 1).An inverse form of Eq. (17) provides S V as a function of the average overheat h Θ i ; S V ≈ k h (cid:18) h Θ i Θ g − (cid:19) . (18) ˙ Q V Let us rewrite Eq. (14) in terms of the vapour generation rate;˙ V vap V ≈ S V gh u ∗ κ ln Hu ∗ ξ ν = 4 S V gh u ∗ s δW liq , k /δV ) ρ = 4 S V gh u ∗ s ∆ρgh + 2( σ + σ ) S V ρ , (19)where we have employed relations (8) and (13) for W liq , k . From Eqs. (8) and (13), ∆ρg h σ + σ ) S V ≈ ρu ∗ κ ln u ∗ eξ νS V . (20)Eqs. (19)–(20) determine relation between S V and ( ˙ V vap /V ) with u ∗ as a parameterof this relation. ill be inserted by the editor 9 Unfortunately, the analytical calculation of an explicit dependence of S V on thegoverning parameter ( ˙ V vap /V ) from Eqs. (19)–(20) is problematic even though theargument of the logarithm function is a large number. However, at the end of theprevious section the system has been shown to be surface-tension dominated and onecan typically neglect the gravitational potential energy against the background of asurface tension energy. Then Eqs. (19)–(20) read˙ V vap V ≈ S V gh u ∗ s σ + σ ) S V ρ , (21)( σ + σ ) S V ≈ ρu ∗ κ ln u ∗ eξ νS V . (22)From Eq. (21), u ∗ ≈ gh r ρ σ + σ ) ( ˙ V vap /V ) S / V ;therefore, Eq. (22) can be rewritten as( σ + σ ) S V ≈ ρgh κ r ρ σ + σ ) ( ˙ V vap /V ) S / V × ln √ gh eξ ν (cid:18) ρ σ + σ ) (cid:19) / ( ˙ V vap /V ) / S / V ! . (23)Taking a square root of the later equation, one can recast it in the form S / V = α ( ˙ V vap /V ) / ln α ( ˙ V vap /V ) / S / V , (24)where α ≡ √ gh κ (cid:18) ρ σ + σ ) (cid:19) / , (25) α ≡ √ gh eξ ν (cid:18) ρ σ + σ ) (cid:19) / . (26)The dependence of S V on ( ˙ V vap /V ) has to be derived from Eq. (24).Notice, the argument of the logarithm function in Eq. (24) is of the same orderof magnitude as in Eq. (7), which can be estimated for the n -heptane–water systemwith material parameters from [14] and turns out to be not smaller than 10 –10 . Onecan exploit this to solve Eq. (24) iteratively. Indeed, multiplying a large argumentof the logarithm function by factor which can be non-small compared to 1 but smallcompared to the argument, one changes the logarithm only slightly. At the 0-thiteration, ( S (0) V ) / = α ( ˙ V vap /V ) / , and S (0) V = α / ( ˙ V vap /V ) / . At the 1-st iteration,( S (1) V ) / = α ( ˙ V vap /V ) / ln α ( ˙ V vap /V ) / ( S (0) V ) / , and S (1) V = α / ( ˙ V vap /V ) / ln / α α / ( ˙ V vap /V ) / . At the 2-nd iteration,( S (2) V ) / = α ( ˙ V vap /V ) / ln α ( ˙ V vap /V ) / ( S (1) V ) / , and S (2) V = α ( ˙ V vap /V ) ln α ˙ Q V ( ˙ V vap /V ) ln − α ˙ Q V ( ˙ V vap /V ) ! , (27)where α ˙ Q V ≡ α α / . Hence, S V = α ( ˙ V vap /V ) ln α ˙ Q V ( ˙ V vap /V ) ln − α ˙ Q V ( ˙ V vap /V ) × ln − α ˙ Q V ( ˙ V vap /V ) ln − α ˙ Q V ( ˙ V vap /V ) . . . !!!!! ≡ α ( ˙ V vap /V ) F , ∞ α ˙ Q V ( ˙ V vap /V ) ! , (28)where F ,n ( z ) ≡ ln (cid:16) z ln − (cid:16) . . . z ln − (cid:16)| {z } ( n −
1) times z (cid:17) . . . (cid:17)(cid:17) for n = 1 , , , ... . Function F / , ∞ ( z ) is plotted in Fig. 3(a).For different tasks, one may employ Eq. (28) to calculate S V for either given( ˙ V vap /V ) or given heat inflow ˙ Q . For the latter case, one has to recall the relation(11) between the vapour production and heat inflow,˙ V vap V = ˙ Q V Λ n (0)1 + Λ n (0)2 . Let us consider the dynamics of cooling-down of a system experiencing the interfacialboiling without external heat supply. The heat consumed for the vapour generationis provided from decrease of the mean system temperature;˙ Q V = ρc P ( −h ˙ Θ i ) . (29)Eq. (29) expresses how the temperature decrease rate controls the heat inflow ˙ Q V tothe interface and thus the vapour production rate ˙ V vap (see Eq. (11)). Then Eq. (19)yields ρc P ( −h ˙ Θ i ) Λ n (0)1 + Λ n (0)2 = 4 S V gh u ∗ s ∆ρgh + 2( σ + σ ) S V ρ , (30) ill be inserted by the editor 11 (a) z F / , i n f ( z ) (b) z F / , app ( z ) / F / , i n f ( z ) Fig. 3. (a): function F / , ∞ ( z ), (b): the ratio of approximate and exact functions F / , app ( z ) /F / , ∞ ( z ) (a) z F , i n f ( z ) (b) z F , app ( z ) / F , i n f ( z ) Fig. 4. (a): function F , ∞ ( z ), (b): the ratio of approximate and exact functions F , app ( z ) /F , ∞ ( z ) Eqs. (30) and (20) form a self-contained equation system which determines the rela-tion between S V and ( −h ˙ Θ i ) with u ∗ as a parameter of this relation.Until the latest stages of self-cooling the mean temperature excess h Θ i stays largecompared to Θ g , and one can consider the system to be surface-tension dominated.Then, similarly to Eqs. (21)–(22), one finds from Eqs. (30) and (20): ρc P ( −h ˙ Θ i ) Λ n (0)1 + Λ n (0)2 ≈ gh s σ + σ ) ρ u ∗ S / V , (31)( σ + σ ) S V ≈ ρu ∗ κ ln u ∗ eξ νS V . (32)From Eq. (31), u ∗ ≈ c P ( −h ˙ Θ i ) gh Λ n (0)1 + Λ n (0)2 ) 1 p σ + σ ) (cid:18) ρS V (cid:19) / ; and Eq. (32) yields S / V = e α q −h ˙ Θ i ln e α q −h ˙ Θ i S / V , (33)where e α ≡ κ s ρc P ghΛ n (0)1 + Λ n (0)2 (cid:18) ρ σ + σ ) (cid:19) / , (34) e α ≡ eξ ν s ρc P ghΛ n (0)1 + Λ n (0)2 (cid:18) ρ σ + σ ) (cid:19) / . (35)Employing relation (18), one can find˙ S V = k h Θ g h ˙ Θ i and recast Eq. (35) in the forms S / V = γ p − ˙ S V ln γ p − ˙ S V S / V ! , (36)where γ , ≡ Θ g k h e α , . (37)One can transform Eq. (36) into the equation of dynamics similarly to the trans-formation of Eq. (24) into Eq. (28); − ˙ S V = S / V γ F , ∞ γ /γ S / V ! , where F ,n ( z ) ≡ ln − (cid:16) z ln − (cid:16) . . . z ln − (cid:16)| {z } ( n −
1) times z (cid:17) . . . (cid:17)(cid:17) . The latter continuous fraction converges well when the argument z is large. In dimen-sionless terms, one can write d η d τ = F , ∞ (cid:0) η (cid:1) , (38)where dimensionless time τ = t γ r γ γ and η = 1 S / V r γ γ . ill be inserted by the editor 13 F / , inf ( z ) and F , inf ( z )Functions F / , inf ( z ) and F , inf ( z ) can be approximated with F / , app ( z ) = (cid:0) .
083 + ( F / , ( z )) . (cid:1) / . , (39) F , app ( z ) = 0 . F , ( z )) . , (40)respectively. In Figures 3 and 4, one can see that relative error of these approximationsin the relevant range of values of z is about 1%. For the system of two immiscible liquids experiencing interfacial boiling we have as-sessed the value of the mean specific interface area S V ≡ ( δS/δV ) as a function ofmacroscopic characteristics of the system state: • the mean overheat h Θ i above the minimal temperature of interfacial boiling T ∗ (Eq. (18)), • the vapour volume production per the unit volume of the system ( ˙ V vap /V ) (Eq. (28)), or • the heat inflow to the system ˙ Q V (Eqs. (28) and (11)).The calculations are based on the mechanical energy flux balance in the system andthe assumption of system stochatization. The system is assumed to be well-stirred byboiling. The heat and momentum transfer towards the contact interface is consideredto be turbulent one and obeys the theory of the turbulent boundary layer.With the dependencies between macroscopic characteristics, which we have ob-tained in this paper, one can construct a comprehensive self-contained mathematicalmodel of the process of interfacial boiling [13,14]. This model will be valid for thecase of the system of two immiscible liquids below the bulk boiling temperature ofboth components.For the case of no heat supply, within the framework of this approach, we havederived the equation of self-cooling dynamics of the system (38).The results are provided in the form of continued fraction–logarithm, which pos-sess slow convergence properties even though the arguments are as large as 10 –10 .For faster calculations we can suggest the approximations (39) and (40), the relativeerror of which is about 1%.The work has been financially supported by the Russian Science Foundation (grantno. 14-21-00090). References
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