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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BOILING AT THE BOUNDARY OF TWO IMMISCIBLE LIQUIDS
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).
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Translated by K. Shakhlevich
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