ISSN 10637761, Journal of Experimental and Theoretical Physics, 2014, Vol. 119, No. 1, pp. 91–100. © Pleiades Publishing, Inc., 2014. Original Russian Text © A.V. Pimenova, D.S. Goldobin, 2014, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2014, Vol. 146, No. 1, pp. 105–115.
SOLIDS AND LIQUIDS
Boiling at the Boundary of Two Immiscible Liquids below the Bulk Boiling Temperature of Each Component A. V. Pimenovaa and D. S. Goldobina,b,* a
Institute of Continuum Mechanics, Ural Branch, Russian Academy of Sciences, ul. Akademika Koroleva 1, Perm, 614013 Russia b Perm National State Research University, Perm, 614990 Russia *email:
[email protected] Received December 23, 2013
Abstract—The problem of vapor formation in the system of two immiscible liquids is considered at the tem peratures that are lower than the bulk boiling temperature of each component and higher than the tempera ture at which the sum of the saturated vapor pressures of the components is equal to atmospheric pressure. In this situation, boiling occurs at the surface of direct contact between the two components. The kinetics of the vapor layer at the contact boundary is theoretically described, and a solution is obtained for the idealized case where the properties of the two liquids are close to each other. The relation between the solution for the vapor layer kinetics and the integral boiling characteristics of the system is considered, and the problem of cooling the system in the absence of a heat inflow is solved. DOI: 10.1134/S1063776114060053
distillation and affects the ignition of a liquid fuel [1]. In particular, this effect makes it possible to perform the distillation of the substances the boiling tempera ture of which at atmospheric pressure is higher than their decomposition temperature. For example, tetra ethyl lead, which is undissolved in water, boils at a temperature of 200°C and undergoes partial decom position. However, when it is mixed with water, it can be distilled at a temperature close to the boiling tem perature of water without decomposition. As for the combustion of a liquid fuel, we note kerosene as an example: its ignition requires a higher temperature than the ignition of a mixture of kerosene with water, since the kerosene vapor concentration over the liquid surface in the case of the mixture is higher than for “pure” kerosene. That is why water should not be used to extinguish oil and oil products, since this intensifies their boiling.
1. INTRODUCTION The system of immiscible liquids (i.e., the liquids the solubility of which in each other is low) is charac terized by the fact that its boiling temperature is lower than the boiling temperature of each component: the closer the boiling temperatures of the components to each other at a given pressure, the stronger the effect of decreasing the boiling temperature [1]. The mecha nism of this phenomenon consists in the fact that boil ing at the interface between the components rather than bulk boiling of the components takes place. In the vapor layer between the liquid phases, particles from both liquids evaporate; as a result, the pressure in this layer is equal to the sum of the saturated vapor pres sures of two liquids. For the vapor layer to grow, it is sufficient that the sum of these pressures exceeds atmospheric pressure, in contrast to the case of bulk boiling of a component, which requires the saturated vapor pressure of this component to exceed atmo spheric pressure. The mutual insolubility of the liquids is important, since a decrease in the main component concentra tion in the solution leads to a decrease in its saturated vapor pressure over the solution surface [1]. Thus, reduced saturated vapor pressures should be summed up in the case of mutual solubility of the liquids. In the case of complete mutual solubility, the effect can fully disappear or strongly weaken in time after the liquids are mixed and during their mutual dissolution at the interface. This effect is well known in engineering and sci ence. It is important in the technological processes of
Although this effect is well known, numerous experimental [2–5] and theoretical [6, 7] works are focused on studying the boiling of a system of immis cible liquids when one of the components is heated above its bulk boiling temperature. There is no ab ini tio theoretical description of the vapor layer kinetics for the situation where the system is below the bulk boiling temperature of each component. Boiling at the contact surface between immiscible liquids begins at temperature T∗, which is determined from the condition that the sum of the saturated vapor pressure of each liquid is equal to atmospheric pres 91
92
PIMENOVA, GOLDOBIN Liquid 2
Vapor mixture layer n1(L) = n1(0) (T10) n2(0) = n2(0) (T20) n2(z)
Liquid 1
n1(z) T1(z)
T2(z) p = p0
T2(0) = T20 κ2, χ2, ρ2, Λ2
0 0
z
T1(0) = T10 κ1, χ1, ρ1, Λ1 z
L z 0
Fig. 1. Growing vapor layer between immiscible liquids and the coordinate system. It is convenient to introduce transverse coordinate z for each of the three regions. (0)
sure. In terms of particle concentration n i urated vapors, we have
in the sat
p0 (0) (0) n 1 ( T ) + n 2 ( T ) = , * * kB T * where p0 is atmospheric pressure and kB is the Boltz mann constant. This phenomenon is of particular interest when the system is far from boiling of each component. Therefore, the purpose of this work is to consider the case where the temperature field in the system is slightly higher than T∗. The material of the work is presented as follows. In Sections 2 and 3, we use ab initio calculations to derive evolution equations for the vapor layer at the contact of two liquids and to find their solution to describe the layer growth. In Section 4, we consider the relation between the vapor layer kinetics and the integral char acteristics of the state of the system. In particular, we determine the average and maximum overheatings of the system at a given constant heat inflow and describe the cooling of the system overheated above T∗ in the absence of an external heat inflow. Section 5 summa rizes the discussion of the results. 2. VAPOR LAYER EVOLUTION Figure 1 shows the vapor layer between two immis cible liquids. At this stage, a theory is developed for the case where material parameters κi, χi, ρi, and Λi of two liquids are considered to be the same, and the system is taken to be reflection symmetric with respect to the median plane of the vapor layer. To analyze and describe the system dynamics, we use the following postulates for boiling. (1) The temperature in the liquid phase is nonuni form, which ensures a heat inflow due to heat conduc tion to the evaporation surface. (2) During evaporation, the liquid phase mass is lost and the surface moves toward the liquid phase.
(3) The substance from the liquid surfaces evapo rates into the vapor layer. On the surface contact of the vapor layer with the liquid phase of one phase, parti cles of the other substance do not pass to the liquid phase, since the liquids are almost insoluble in each other. The particle concentration of the gas compo nent over the surface of its liquid phase is equal to the particle concentration of the saturated vapor of this substance at local temperature T = T* + Θ (Fig. 1), (0)
n 2 ( z = 0 ) = n 2 ( Θ ),
(0)
n 1 ( z = L ) = n 1 ( Θ ).
It is known that, during evaporation from a free liq uid surface (e.g., in vacuum), the vapor concentration over the surface remains lower than the saturated vapor concentration due to a finite flow rate of the evaporating substance [8–10]. In the case under study, evaporation occurs into the vapor layer, at the bound aries of which the gas particle concentration drop tends toward zero at temperature T T∗ (i.e., the diffusion outflow can become asymptotically small), rather than into the open halfspace. Thus, at a suffi ciently weak overheating of the system, the accepted assumption regarding local thermodynamic equilib rium at the vapor–liquid boundary should remain valid in continuity at a sufficient accuracy. (4) Since the mechanical inertia of the system is negligibly small relative to the thermal and diffusion “inertia,” the pressure within the vapor layer is consid ered to be constant and equal to atmospheric pressure, p = p0. (5) Assuming that the vapor layer is an ideal gas, we can write the total volume concentration of particles as n 1 + n 2 = n 0 = p 0 /k B T, where T is the local temperature. During the vapor layer growth, the component particle concentration drop across it turns out to be related to the saturated vapor concentration, which experimentally depends on temperature [1, 8]. Total particle concentration n0 depends on temperature much weaker, as a power function. As a result, the change of n0 associated with the deviation of temperature from T∗ may be neglected against the background of the changes of n1, 2, and we can write n1 + n2 = n0
p0 = . * kB T*
(1)
The system of these statements is not a rough approximation: all assumptions are valid (at least) at the early stages of vapor layer growth. In Section 4, we comprehensively analyze the results obtained and the validity of these assumptions for the parameters of real substances. We now transform these statements into a mathematical formulation. It is convenient to choose a moving coordinate sys tem in the liquid phase regions so that point z = 0 cor
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responds to the surface of contact with the vapor layer (Fig. 1). Inside the vapor layer, the redistribution of gas component concentration occurs due to diffusion pro cesses, which are described by the Fick law J i = – D 12 ∇n i ,
(2)
where D12 is the interdiffusion coefficient. Coefficient D12 is independent of the component concentration and is a power function of temperature. Against the background of large relative changes of ni, small rela tive changes in temperature for a power D12(T) func tion give small relative changes of D12, and they may be neglected on the assumption of a constant value of D12 = D12(p0, T∗). Correspondingly, the change in the gas mixture component concentration obeys the law
n1
(3)
For convenience of solution, we introduce coordinate system (˜t , ζ) in which the vapor layer width is time independent, z = L (˜t ) ⎛ ζ – 1⎞ , ⎝ 2⎠
t = ˜t ,
κ ∂T 2 – 2 Λ 2 ∂z
2 · ∂n D ∂n n· i – ζ L i = 122 2.i L ∂ζ L ∂ζ
(5)
When one of the vapor region boundaries (e.g., boundary 2) travels δL2, the number of particles δN2 having passed through interface area S is related to the number of particles of type 2 in the volume SδL2 that is added to the vapor layer from the side of liquid 2 as follows: n2
z = 0 δL 2
∂n 2 = D 12 ∂z
∂n 2 = ⎛ D 12 ⎝ ∂z
δN δt + 2 S z=0
κ ∂T 2 – 2 Λ 2 ∂z z=0
(7)
z=0
,
(8)
z=0
(9) z=0
at the interface. The heat transfer in the liquid phases is described by the heat conduction equation 2
∂T ∂T ∂T i + v li i = χ i 2i , ∂t ∂z ∂z
(10)
where χi is the thermal diffusivity of the ith liquid and vli is the liquid velocity in the coordinate system related to the interface. The shift velocity at the inter face is proportional to the liquid volume passed to the gas phase, κ 1 ∂T 1 1 v l1 = – n l1 Λ 1 ∂z
z=0
κ 2 ∂T 2 1 = – n l2 Λ 2 ∂z
z=0
v l2
, (11) ,
where nl1 and nl2 are the bulk densities of the numbers of particles in the liquid phases. As noted above, we restrict ourselves to the sym metric case, where both liquids have the same quanti tative thermodynamic characteristics. In this case, the vapor layer uniformly expands toward both sides, δL1 = δL2 = δL/2. For the system to boil, the liquid must be overheated. In this case, the temperature field of the system differs from the minimum boiling tem perature T∗ by a small overheating Θ, T = T + Θ ( t, ζ ), Θ T . * * The temperature dependence of the particle concen tration of the saturated vapor can be linearized,
(6)
n (0) n 1, 2 = 0 + γΘ, 2
⎞ δt, ⎠
z=0
where κ2 is the thermal conductivity of liquid 2 and Λ2 is the enthalpy of vapor formation of substance 2 per particle. The first term in the righthand side of Eq. (6) corresponds to the diffusion flux of particles caused by the concentration gradient inside the vapor region, and the second term is related to the passage of parti cles from the liquid to the gas phase. Only the diffusion term
δt z=0
n 0 ∂n 1 = D 12 n 1 z = 0 ∂z
D 12 ∂n 1 L· 2 = n 1 z = 0 ∂z
(4)
where ζ ∈ [–1/2; 1/2] (vapor layer boundaries corre spond to ζ = –1/2 and 1/2, see Fig. 1) and L is the vapor layer thickness. In this case, Eq. (3) takes the form
∂n 1 = D 12 ∂z
is important for the other substance at this boundary. When solving Eqs. (6) and (7) simultaneously and tak ing into account dn1 + dn2 = 0, we can obtain the rela tionships
2
∂n ∂n i = ∇ ( D 12 ∇n i ) = D 12 2.i ∂t ∂z
z = 0 δL 2
93
(12)
(0)
where γ ≡ ( ∂n 2 /∂T ) T = T and correction γΘ is taken *
into account due to the strong exponential tempera ture dependence of the saturated vapor particle con centration (in the framework of this consideration, we omit the similar terms related to power dependences as small terms against the background of the terms related to the exponential dependences). The mathe matically strong exponential dependence means that
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PIMENOVA, GOLDOBIN (0)
γT∗/ n 2 (T∗) 1 (for polynomial dependences on the absolute temperature, analogous relationships have an order on about unity). For boundary conditions, Eq. (12) gives n2
ζ = – 1/2
n = 0 + γΘ, 2
n1
ζ = 1/2
n = 0 + γΘ. 2
(13)
Since the pressure inside the vapor layer remains con stant, we have ( n1 + n2 )
(14) + Θ ) = n0 T . * In combination, Eqs. (13) and (14) give additional boundary conditions n1
ζ = – 1/12
ζ = – 1/2 ( T *
n n n = 0 – ⎛ 0 – γ⎞ Θ ≈ 0 – γΘ, ⎠ 2 ⎝T 2 * n n 2 ζ = 1/2 ≈ 0 – γΘ. 2
α1 = –α2 .
θ
z=0
= 0,
∂θ ∂z
z=0
Λ = – 2 n l2 v l2 . κ2
(22)
It is convenient to search for the solution to Eq. (21) as a sum with the term Cn θ = θ 0 – l2 z + f ( z ), γ n0
(23)
v f = f 0 exp ⎛ l2z⎞ . ⎝ χ2 ⎠
(16)
The general solution to this equation has the form (18)
where C is a constant, which is a solution parameter. Since we consider a weak overheating of the system (γΘ n0/2), the solution to Eq. (9) can be written in the form 4D 12 L = Ct. n0
(21)
At t = 0, Eqs. (18) and (19) give θ(0) = 0 and Eq. (11) gives ( ∂θ/∂z ) z = 0 . Thus, the boundary conditions for Eq. (21) take the form
z = 0 ζ,
For coordinatelinear dependences, Eq. (5) takes the form L· (17) α· ζ – ζ α = 0. L z = 0,
2 2D 12 C ∂θ ∂θ + v l2 = χ 2 2 . ∂z γn 0 ∂z
which is linear in z. The substitution of Eq. (23) into Eq. (21) leads to the relation
z = 0 ζ.
α = CL = 2γΘ
2D 12 2 Θ 2 = C t + θ ( z ). γn 0
2
Boundary conditions (13) and (15) require
n n 2 = 0 – 2γΘ 2
(20)
with allowance for Eqs. (8) and (18). The solution to Eq. (10) can be represented in the form of the sum of terms, one of which depends only on the coordinates and the other is linear in time,
(15)
3. SOLUTION OF THE VAPOR LAYER EVOLUTION EQUATIONS Note that Eq. (5) permits solutions with a linear particle concentration profile,
n n n 1 = 0 + α 1 ζ = 0 + 2γΘ 2 2
2D 12 C v 12 = n l2
In this representation, Eq. (10) takes the form
Thus, the evolution of the vapor layer is fully deter mined by Eqs. (5) and (10), in which L· = 2 L· 2 and vli are determined by Eqs. (9) and (11), respectively, with boundary conditions (8), (13), and (15).
n i = n 0 /2 + α i ζ,
the regions, e.g., region 2, due to the symmetry of the problem), and liquid velocity v12 in it (Eq. (11)) has the form
(19)
The state of the system in the liquid phase is described by Eq. (10) (it is sufficient to consider one of
(24)
The matching of the solution with boundary condi tions (22) yields constants of integration θ0 and f0, and the desired overheating temperature field can be writ ten in the form 2D 12 C n ᏸ n χ2 ⎞ Θ = t + l2 C z + ⎛ 2 – l2 ⎝ γn 0 n0 γ c p, l2 n 0 2D 12 γ⎠ 2
2
(25)
2D 12 C ⎞ z . × 1 – exp ⎛ – ⎝ χ 2 n l2 ⎠ Here, ᏸ2 and cp, l2 are the specific enthalpy of vapor formation and the specific heat of liquid 2. Hereafter, it is convenient to use the physical char acteristic of the process (e.g., L· ) through which con stant C can be expressed as n0 · C = L 4D 12
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95
rather than constant C. It is also convenient to intro duce the designation n L· (26) Z = 0 z n l2 2χ 2
Physical properties of water and nheptane at the bulk boil ing temperature and atmospheric pressure
for the dimensionless argument of the exponent in Eq. (25). We now consider the last term in Eq. (25), which is nonlinear in z . This term describes the temperature boundary layer in the liquid phase, the characteristic thickness of which is χ2nl2/2D12C. It is interesting that the amplitude of this term for real substances (table) is
ᏸ
2
2 = 5.4 × 10 K, c p, l2
2 n l2 χ 2
6
= 1.0 × 10 K 2n 0 D 12 γ
(27)
for water and
ᏸ
H2O
nHeptane
373.15
371.58
ᏸ2, J/kg
2.26 × 106
3.17 × 105
cp, l2, J/(kg K)
4.22 × 103
2.50 × 103
χ2, m2/s
1.70 × 10–7
0.66 × 10–7
ρl2, kg/m3
0.96 × 103
0.61 × 103
ρg, kg/m3
0.63
2.50
γ 2γ = , K–1 (0) n0 n
0.039
0.056
1.0 × 10–5
1.0 × 10–5
ηg, kg/(m s)
1.24 × 10–5
0.74 × 10–5
n0/nl2
0.66 × 10–3
4.1 × 10–3
Bulk boiling temperature, K
D12, m2/s (for pair) 2
2
2 = 1.3 × 10 K, c p, l2
n l2 χ 2 4 = 0.7 × 10 K 2n 0 D 12 γ
(28)
for nheptane (the values for many organic substances are close to each other). However, the theory in this work is constructed for an overheating temperature of about 1 K. Moreover, the effect is most important until bulk boiling of one component starts; that is, the over heating that is important for this effect cannot exceed several tens of kelvins. This means that the system should always be far from saturation of the term that describes the temperature boundary layer: for physical systems, only a very small initial segment of the boundary layer at Z 1 is important. However, a linear approximation turns out to be insufficient to describe the system. When writing Eq. (25) in the form 2 n 0 L· ᏸ –Z Θ = t + 2 ( 1 – e ) 8D 12 γ c p, l2 2
n l2 χ 2 –Z + ( Z – 1 + e ), 2n 0 D 12 γ we can see (estimates (27), (28)) that the amplitude of the second term in the last expression is three orders of magnitude lower than the amplitude of the third term, in which the constant and linear contributions vanish during expansion in terms of Z. For this difference between the amplitudes, the part of the third term that is quadratic in Z can contribute as strongly as the linear part of the second term. Thus, for physically real situ ations, Eq. (25) can be reduced to the simpler form 2 2 n 0 L· ᏸ2 1 n l2 χ 2 2 (29) Θ ≈ t + Z + Z 2 2n 0 D 12 γ 8D 12 γ c p, l2 almost without loss of accuracy. When analyzing the structure of the spatial part of Θ, θ(z) ~ 102 [K]Z + 105 [K]Z2, we can see that dimensionless complex is Z ~ 10–2, i.e., remains small, even for the maximum
overheating θ(z) ~ 10 K (bulk boiling of the compo nents begins at a much higher overheating) that can be achieved as the distance from the boiling surface increases. At an overheating θ higher than 0.1 K, the quadratic term gives the main contribution. 4. RELATION BETWEEN THE VAPOR LAYER KINETICS AND THE AVERAGE MACROSCOPIC PARAMETERS OF THE SYSTEM In contrast to the previous section (where the prob lem of the vapor layer growth was strictly solved), the consideration in this section has an estimating charac ter and aims to analyze the relation between the main results obtained in Section 3 and the characteristics and control parameters of the system, such as the heat inflow per unit mixture volume. A steady boiling regime is of interest. Under these conditions, we can assume that constant average heat supply takes place in the system and maintains a con stant average overheating of the system and a constant vapor formation rate. The geometric characteristics of the system are assumed to be statistically stable. 4.1. Separation of the Vapor Layer An important geometric parameter of a mixture of two liquids is the average area of their contact in unit mixture volume, δS/δV. This parameter depends on the liquid parameters and the process of vapor forma tion, which is determined by the average overheating of the system and the rate of vapor bubble formation [11]. We concentrate on the mesoscopic problem of the vapor layer growth, and the macroscopic parame ters of the system are taken to be specified (reader con
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PIMENOVA, GOLDOBIN y
The length of the corresponding section along contact boundary l is related to 2H as follows:
Vapor phase Liquid 1
dy = cos ϑdl,
Hcurv ~ 2H
B Liquid 2
ϑ
A
(a)
(b)
(c)
Fig. 2. Bubble formation in the growing vapor layer between immiscible liquids and the “nulling” of the vapor layer.
cerned can address himself to the extensive literature where the related problems are well described [12– 15]). The volumes of both components are assumed to be comparable; that is, none of the phases can be dis tinguished as a substantially main phase. The system is supposed to be well mixed (stirred) upon boiling, so that it can be treated as a statistically isotropic and homogeneous system [11]. The characteristic width of the vicinity of the vapor layer outside which the vicin ity of another layer begins is δS 2H ∼ ⎛ ⎞ . ⎝ δV⎠ –1
The characteristic distance of a change in the orienta tion of the contact surface in a well mixed system has the order of magnitude of 2H. The difference between the case of boiling below the bulk boiling temperature of each component and the case where a more volatile component is over heated above the temperature of its bulk boiling is reduced to the growth of the vapor layer and the heat transfer in its vicinity. On a macroscopic level, the mechanical processes in the liquid phases for these two cases should be similar. For the last case, the qualitative picture of boiling is well known from experiments [2–7, 11]: vapor bubbles form and leave the contact surface (Fig. 2). A viscous flow (Poi seuille flow) can appear because of the pressure gra dient in the vapor layer that is caused by the action of gravity on the liquid. Since the Poiseuille flow strongly depends on the vapor layer width, it can be insignificant for a sufficiently thin layer. When the growing layer reaches a certain rather large width, a vapor mass moves along the layer and forms a bub ble, which leaves the boiling boundary and floats up (Fig. 2). As a result, the vapor layer undergoes “null ing” and begins to grow again. Let us estimate the characteristic vapor layer renewal time. We consider a growing section of the vapor layer of height 2H: on the scale of this height, the slope of the contact surface changes substantially (Fig. 2a, section AB). We are interested in the nucle ation of separation of the vapor layer in this section.
where y is the vertical coordinate and ϑ is the angle of deviation of the contact surface from the vertical. For a uniform distribution of the orientation of the normal to the layer over a sphere (the system is assumed to be statistically isotropic), the mean value is 〈 cos ϑ 〉 = 1/2 and 2H ≈ 〈 cos ϑ 〉 l = l/2. Let the Poiseuille flow make contribution L· to the P
rate of change of the vapor layer thickness. For contri bution L· P to exist in a section of length l along the Poi seuille flow, the flow at the end of section AB should give L/2
∫
v l ( x n ) B dx n = – lL· P ,
– L/2
where vl is the vapor flow velocity along the vapor layer and xn is the coordinate that is measured from the cen ter of the layer and is normal to the layer. Since the thickness of the vapor layer is small as compared to its radius of curvature in the situation under study (before the formation of a vapor bubble), the Poiseuille flow profile has the shape corresponding to a steady flow between parallel planes, 2
2 1 ⎛ – dp v l ( x n ) = ⎞ ⎛ L – x n⎞ , ⎝ ⎠ ⎝ ⎠ 2η g dl 4
where ηg is the dynamic vapor viscosity, which depends on the mole fraction of a vapor component. However, for a weak overheating of the vapor region, this viscosity is assumed to be equal to the viscosity of the vapor mixture composition corresponding to infi nitely slow boiling at temperature T∗. For the pressure gradient, we have 1 dp ≈ 〈 cos ϑ 〉 dp ≈ dp . 2 dl dy dy When the system is mixed, the dynamic part of the pressure gradient related to the acceleration of liquid particles is nonzero. Since the mixing flow is induced by gravity, this part should be commensurable with the hydrostatic contribution to the pressure gradient: we can assume that dp/dy ~ ρlg, where g is the accelera tion of gravity. As a result, due to the effect of vapor penetration along the vapor layer, the contribution to the rate of change of the layer thickness is 3
3
ρ l gL 1 ρ l gL δS L· P ≈ – . ≈ – 96η g H 48 η g δV
(30)
The property of L· P noted above, namely, a strong dependence on the layer thickness ( L· ~ L3), is again
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observed, whereas the evaporationinduced growth proceeds at a constant rate. Thus, for estimation, we can neglect penetration at the early stages of layer growth and assume that a bubble leaves the surface at the time when L· P becomes equal to the contribution of evaporation to the vapor layer growth rate. Thus, it follows from Eq. (30) that the limiting vapor layer thickness L∗ reached before the separation of a bubble can be related to the rate of vapor bubble growth due to evaporation, 1/3 48η g L ≈ ⎛ L· ⎞ . * ⎝ ρ l g ( δS/δV ) ⎠
48η g ⎞ 1/3 · –2/3 t ≈L L . ·* ≈ ⎛ * L ⎝ ρ l g ( δS/δV )⎠
n L· Z max = Z ( z = H ) = 0 n l2 4χ 2 ( δS/δV )
2 2 n 0 L· ᏸ 1 n l2 χ 2 2 Θ max ≈ t + 2Z max + Z max 2 2n 0 D 12 γ 8D 12 γ * c p, l2
(34)
and at the space and timeaverage overheating 2 2 2 n 0 L· t ᏸ Z max 1 n l2 χ 2 Z max 〈 Θ〉 ≈ * + 2 + 8D 12 γ 2 c p, l2 2 2 2n 0 D 12 γ 3
1 2χ 2 n l2 ⎛ 3η g χ 2⎞ 1/3 δS 4/3 = Z max 2 D 12 γ ⎝ ρ g g ⎠ δV
(35)
n l2 χ 2 2 ᏸ + 1 2Z max + 1 Z max . 2 c p, l2 6 2n 0 D 12 γ 2
For real substances, we have (H O)
4/3
Θ max2 ≈ 0.27 [ K/m ] ( δS/δV )Z max 2
6
2
+ 5.4 × 10 [ K ]Z max + 1.0 × 10 [ K ]Z max for the case of water and
(32)
We have to take into account that, when the vapor layer is renewed, vapor escapes and a certain overheat ing of the liquid, which is related to the timelinear term in Eq. (29), is retained in the vicinity of vapor. We first perform estimations using the assumption that the quantity of heat associated with this residual excess overheating may be neglected against the background of the other contributions to the balance of heat trans formations in the system and, then, make sure that this assumption is valid for real substances with a good accuracy. If this assumption holds true, the residual excess overheating after the renewal of the layer can result in the formation of only a small vapor layer dur ing relaxation; that is, the system can pass to a state that is close to solution (29) at the early stage of vapor layer growth, where L L∗. With Eqs. (29) and (32), we can calculate the max imum overheating of the system (temperature at the maximum distance between vapor layers at the time of layer renewal),
2χ 2 n l2 ⎛ 3η g χ 2⎞ 1/3 δS 4/3 = Z max D 12 γ ⎝ ρ g g ⎠ δV
where
(31)
An exact expression for L∗ should contain an addi tional dimensionless factor of about unity in parenthe ses, which takes into account the fine specific features of the system geometry and the processes occurring in the system. However, here we restrict ourselves to esti mation; moreover, it should be noted that, when the cube root is taken, the related correction due to this factor decreases strongly. After renewal, the vapor layer grows to this size in the time
97
( nC H 16 )
Θ max 7
4/3
≈ 0.0045 [ K/m ] ( δS/δV )Z max 2
4
2
+ 1.3 × 10 [ K ]Z max + 0.7 × 10 [ K ]Z max for nheptane (see table). We now can return to the problem of neglecting the residual excess overheating immediately after the renewal of the vapor layer. In the absence of a vapor layer, a local overheating above the minimum boiling temperature of the mixture T∗ means a local break in thermodynamic equilibrium, to which the system returns rapidly due to the loss of the overheating heat for the evaporation of the substance into the vapor layer. To an accuracy of an order of magnitude, we can assume that the state with a uniform contribution to 2 temperature (n0 L· t∗/8D12γ) and an absent vapor layer (L = 0) passes to state (29) of a new growing layer, which corresponds to certain time t0 instead of t = 0, as would be in the case of no overheating. The stage of layer growth t0 can be estimated from the balance (since the coordinate part of Eq. (29) is the same at different stages of growth) of the heat of over heating in the H vicinity of the layer and the heat of evaporation into the layer, n 0 L· t * ⋅ 2HδS + 0 c p, l2 ρ l2 8D 12 γ 2 n 0 L· t 0 = c p, l2 ρ l2 ⋅ 2HδS + Λ 2 n 0 L· t 0 δS, 8D 12 γ 2
(33)
ᏸ 1 n l2 χ 2 2 + 2Z max + Z max , 2 2n 0 D 12 γ c p, l2 2
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from whence, we have t ᏸ 8D 12 γ δS/δV⎞ –1 0 = ⎛ 1 + 2 (36) . ⎝ t c p, l2 n l2 L· ⎠ * The importance of the residual excess overheating is determined by the value of the timelinear contribu tion to the overheating at the stage of growth t0 as com pared to the characteristic overheating of the system Θmax, 2 n 0 L· 1 r = t 0 Θ max 8D 12 γ
and Eq. (38) takes the form · c p, l2 θ Z max = – . ᏸ 2 4χ 2 ( δS/δV ) 2
(37)
1 2χ 2 n l2 ⎛ 3η g χ 2⎞ 1/3 δS 4/3 t 0 = Z max . Θ max D 12 γ ⎝ ρ g g ⎠ δV t * From Eq. (33), we can calculate Zmax for a given max imum overheating Θmax. Then, using Eq. (34), we can determine the ratio L· /(δS/δV). To calculate r (Eq. (37)), we have to determine parameter δS/δV. When (δS/δV)–1 changes in the range from 1 to 10 mm, ratio r is always lower than 0.01 for water and lower than 0.002 for nheptane at Θmax = 0–10 K. Since r is always small, we can use Eq. (35) to deter mine the average overheating in the system. 4.2. Vapor Formation during a Constant Heat Inflow We now find the relation between the vapor layer growth parameters and a certain macroscopic charac teristic of the system, namely, the heat inflow per unit mixture volume per unit time, δQ q V = . δVδt Correspondingly, heat inflow qS = (δS/δV)–1qV per unit contact surface area takes place. Since a steady state boiling process is considered, the statistical char acteristics of the temperature field in the system remain unchanged in time, and the entire incoming heat should be consumed for vapor formation, qS = Λ2n0 L· 2 . Then, instead of Eq. (34) we have the equa tion qV Z max = 2 , 4ρ 12 ᏸ 2 χ 2 ( δS/δV )
of two liquids after their mixing in the absence of heat sources. The nontrivial contribution to heat transfer due to vapor formation in the situation where the tem perature of the forming system is higher than T∗ is of interest. Only the heat related to the overheating of the liquid phase (θ ≡ 〈 Θ〉 ) serves as an energy source for vapor formation. In this case, we have · q V = – ρ l2 c p, l2 θ ,
With allowance for Eq. (39), Eq. (35) takes the form of a nonlinear equation for the derivative of the average overheating temperature, 4/3 · δS θ θ = A 4/3 ⎛ – 2⎞ δV ⎝ ( δS/δV ) ⎠ (40) 2 · · –θ –θ ⎞ ⎛ + A 1 2 + A 2 2 , ⎝ ( δS/δV ) ⎠ ( δS/δV ) where the expressions for coefficients Ai are obviously determined from a comparison of the last equation with Eqs. (39) and (35). To describe the real system dynamics exactly, it is important to take into account the dependence of δS/δV on the overheating tempera ture. (Obviously, the more intense the vapor forma tion, the more intense the mixing of the system and the higher the characteristic δS/δV.) · For a large overheating, the term quadratic in θ in the righthand side of Eq. (40) is predominant. Assuming that 2 · θ θ ≈ A 2 ⎛ – 2⎞ (41) ⎝ ( δS/δV ) ⎠ at this stage and neglecting the change of δS/δV with time, we can find the decay time of θ to low values (in terms of Eq. (41), θ vanishes in a finite time), 2 A 2 〈 Θ〉 t = 0 τ 2 = 2 ( δS/δV )
(38)
which can be used to calculate the maximum (Eq. (33)) and average (Eq. (35)) overheatings in the system.
(42)
〈 Θ〉 t = 0 n l2 c p, l2 = 2 . 4 ᏸ 2 ( δS/δV ) 3χ 2 n 0 D 12 γ For water and nheptane, this time is ( H2 O )
4.3. System Cooling Dynamics in the Absence of Heat Inflow Using the results obtained for the relation between the average overheating in the system and the vapor layer kinetics, we can estimate the cooling of a system
(39)
τ2
( nC 7 H 16 )
τ2
–2
6
≈ 2.3 × 10 ( s m K 6
–2
– 1/2
≈ 2.0 × 10 ( s m K
–2
1/2
) ( δS/δV ) 〈 Θ〉 t = 0 ,
– 1/2
–2
1/2
) ( δS/δV ) 〈 Θ〉 t = 0 .
At (δS/δV)–1 ~ 1 mm to 1 cm, the time of the phase of rapid cooling is τ2 ~ 1 s to 1 min.
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After the stage of rapid cooling, the system is in the state with a small overheating and the predominant · term linear in θ in the righthand side of Eq. (40), · –θ θ ≈ A 1 2 . (43) ( δS/δV ) If the degree of mixing of the system could not trace the vapor formation intensity and δS/δV remained unchanged in time, the decay of the overheating would be exponential, θ ~ exp(–t/τ1), with the characteristic time A1 1 τ 1 = 2 = . 2 ( δS/δV ) 8χ 2 ( δS/δV )
(44)
For water and nheptane at (δS/δV)–1 ~ 1 mm to 1 cm, time τ1 ~ 1 s to 1 min is comparable with time τ2. The cooling of the system at the linear stage turns out to be infinite, the vapor formation intensity decreases with time, and the degree of mixing of the system should inevitably decrease (parameter δS/δV decreases). During infinitely slow vapor formation, δS/δV reaches its minimum ((δS/δV)min) rather than vanishing: during the major portion of time, the sys tem is in a stratified state with an approximately plane horizontal interface between light and heavy liquids. At this stage, the overheating temperature decays exponentially with the characteristic time 1 τ 1, ∞ = . 2 8χ 2 ( δS/δV ) min
(45)
Until the degree of mixing of the system is rather high (δS/δV (δS/δV)min), the dependence of δS/δV · on θ is assumed to have an empirical power character, · r δS/δV ∝ – ( θ ) , which does not take into account the lower bound set on δS/δV. The exponent is 0 < r < 1/2, and the condition r < 1/2 is associated with the fact · that Eq. (43) should give positive values of d(– θ )/dθ for physically realistic situations. In this case, Eq. (43) yields θ ∼ ( t – t0 )
– [ 1/2r – 1 ]
,
that is, the process of cooling is polynomially slow. Thus, in the absence of a heat inflow, the cooling of the overheated system due to vapor formation occurs at the following stages: (i) rapid cooling according to Eq. (41) in time τ2 (Eq. (42)), which depends on the initial overheating of the system; (ii) polynomially slow cooling according to Eq. (43) with characteristic time τ1 (Eq. (44)); and (iii) slow exponential cooling so that the system is stratified during the major portion of time and the interface remains approximately planar and horizon
99
tal; the characteristic time of cooling of the system at this stage is τ1, ∞ (Eq. (45)). 5. CONCLUSIONS We consider the boiling of a system of immiscible liquids at a temperature below the bulk boiling tem perature of each component, when boiling occurs at the contact boundary. Using ab initio calculations, we described the process and derived evolution equations for the vapor layer and the liquid temperature field in the vicinity of this layer (Eqs. (5) and (10) with bound ary conditions (8), (13), (15)). For the case of close characteristics of the liquids, we obtained the solution to these equations that corresponds to a growing vapor layer (Eq. (29)). The problem of the separation of vapor bubbles from a growing vapor layer was analyzed. Based on the results of this analysis, we found the characteristic growth time of the vapor layer (Eq. (32)) before its “nulling” because of the separation of bubbles and estimated the maximum (Eq. (33)) and average (Eq. (35)) overheatings of the system. The problem of cooling of the system in the absence of an external heat inflow was considered, and three characteristic stages of the process were distinguished. We calculated the values of the found characteris tics for the material parameters of water (liquid with the maximum heat capacity and the maximum spe cific heat of vapor formation) and nheptane (which is an example of saturated hydrocarbons characterized by low values of heat capacity and specific heat of vapor formation). It is interesting that, in the case of bulk boiling, the problem of the nucleation rate becomes a challenging problem for a theoretical description of the process kinetics. The solution of this problem requires the apparatus of the statistical physics of nonequilibrium thermodynamic processes [16] and the theory of hydrodynamic fluctuations [17]. In contrast, the pro cess of boiling at the boundary of direct contact between two immiscible liquids should be described in terms of macroscopic hydrodynamics of multiphase systems, and such a description was developed in this work. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 1401 31380mol_a) and the government of the Perm Terri tory (project no. S26/212). REFERENCES 1. V. T. Zharov and L. A. Serafimov, PhysicoChemical Fundamentals of Distillation and Rectification (Khimiya, Leningrad, 1975) [in Russian].
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2. H. C. Simpson, G. C. Beggs, and M. Nazir, Desalina tion 15, 11 (1974). 3. G. P. Celata, M. Cumo, F. D’Annibale, F. Guglier metti, and G. Ingui, Int. J. Heat and Mass Transfer 38, 1495 (1995). 4. M. L. Roesle and F. A. Kulacki, Int. J. Heat and Mass Transfer 55, 2160 (2012). 5. M. L. Roesle and F. A. Kulacki, Int. J. Heat and Mass Transfer 55, 2166 (2012). 6. S. Sideman and J. Isenberg, Desalination 2, 207 (1967). 7. A. A. Kendoush, Desalination 169, 33 (2004). 8. O. Knacke and I. N. Stranski, Prog. Met. Phys. 6, 181 (1956). 9. R. Ya. Kucherov and L. E. Rikenglaz, Sov. Phys. JETP 10, 88 (1959). 10. S. I. Anisimov and A. Kh. Rakhmatulina, Sov. Phys. JETP 37 (3), 441 (1973). 11. G. Filipczak, L. Troniewski, and S. Witczak, in Evapo ration, Condensation, and Heat Transfer, Ed. by A. Ahsan (InTech, Rijeka, Croatia, 2011), p. 123.
12. K. F. Gordon, T. Singh, and E. Y. Weissman, Int. J. Heat and Mass Transfer 3, 90 (1961). 13. C. B. Prakash and K. L. Pinder, Can. J. Chem. Eng. 45, 207 (1967). 14. C. B. Prakash and K. L. Pinder, Can. J. Chem. Eng. 45, 215 (1967). 15. T. G. Somer, M. Bora, O. Kaymakcalan, S. Ozmen, and Y. Arikan, Desalination 13, 231 (1973). 16. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 10: Physical Kinetics (Fizmatlit, Moscow, 2001; Butterworth–Heinemann, Oxford, 2002). 17. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 9: E. M. Lifshitz and L. P. Pitaevskii, Sta tistical Physics: Part 2. Theory of the Condensed State (Fizmatlit, Moscow, 2002; Butterworth–Heinemann, Oxford, 2002).
Translated by K. Shakhlevich
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