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High Temperature, Vol. 42, No. 1, 2004, pp. 133–139. Translated from Teplofizika Vysokikh Temperatur, Vol. 42, No. 1, 2004, pp. 143–150.
Original Russian Text Copyright © 2004 by Dombrovskii.
HEAT AND MASS TRANSFER
AND PHYSICAL GASDYNAMICS
The Propagation of Infrared Radiation in a Semitransparent
Liquid Containing Gas Bubbles
L. A. Dombrovskii
IVTAN (Institute of High Temperatures) Scientific Association, Russian Academy of Sciences, Moscow, 125412 Russia
Received May 13, 2003
Abstract—A theoretical model is suggested for the propagation of infrared radiation in a semitransparent liquid containing gas bubbles, which includes an approximate description of the radiation characteristics and radiation transfer in a disperse system. Calculations are performed for a layer of water containing vapor bubbles
illuminated by the thermal radiation of an external source. It is demonstrated that, for real values of the parameters, the scattering of radiation by bubbles may lead to the absorption of thermal radiation in a much thinner
layer of water. The possible application of the obtained results to the solution of a conjugate problem is
discussed.
INTRODUCTION
The problem treated in this paper arises in the case
of theoretical simulation of water cooling of burning
hot surfaces. A practical example is provided by the
delivery of water to the surface of the core melt in the
case of a serious failure of a nuclear reactor [1]. The
thermal radiation of a solid or melt with a temperature
of 2000–3000 K is largely associated with the near
infrared spectral region where water is semitransparent.
Therefore, a significant part of the radiation is not
absorbed in the surface layer but penetrates deep into
the water and leads to volumetric heat release [2]. The
forming vapor bubbles may, in turn, affect the propagation of the radiation. The complete physical formulation of the problem must take into account this feedback and the development of the process in time. At the
same time, in the first stage, it is of interest to investigate the variation of infrared radiation characteristics of
water in the presence of vapor bubbles and to solve the
problem on the propagation of thermal radiation in a
two-phase disperse system which both absorbs and
scatters the radiation.
The problem to be solved is rather general and
relates both to any semitransparent liquids and to solids
containing numerous bubbles or other spherical inclusions. Such structures are formed in the manufacture of
glass and are observed in some heat-insulating materials. The determination of the radiation characteristics
and the calculation of radiation transfer in such media
represent important elements of the calculation of the
thermal process conditions or of the heat-insulating
properties [3–5].
The following main assumptions are used in this
study to determine the radiation characteristics of a
medium containing bubbles: the absorption of radiation
occurs in a layer whose thickness significantly exceeds
the bubble size; all bubbles are spherical; the bubbles
are arranged at random; and the distance between bubbles significantly exceeds their sizes and the radiation
wavelength.
The first of these assumptions implies that only that
spectral range is treated in which the medium weakly
absorbs radiation (semitransparent region). The
assumptions of the random arrangement of bubbles and
of their not-too-high concentration lead one to believe
that the scattering of radiation by an individual bubble
does not depend on the presence of other bubbles [6, 7].
The restriction of our treatment to bubbles of spherical
shape significantly simplifies the determination of their
radiation characteristics.
The problem was analogously formulated in [8–10]
in calculating the radiation characteristics of glass containing gas bubbles. The relations used were valid only
in the Rayleigh–Gans approximation or for the region
of anomalous diffraction [11, 12]. In this study, the radiation characteristics of a medium containing bubbles are
analyzed using the rigorous theory of scattering.
RELATIONS FOR RADIATION
CHARACTERISTICS OF SPHERICAL
PARTICLES IN A SEMITRANSPARENT
ABSORBING MEDIUM
The classical Mie solution for the absorption and
scattering of radiation by a spherical particle relates to
the case when the particle is in a vacuum. According to
the Mie theory, the characteristics of absorption and
scattering depend on the diffraction parameter x =
2πa/λ (a is the particle radius, and λ is the radiation
wavelength) and on the complex refractive index of the
particle material m' = n' – iκ' (n' is the refractive index,
and κ' is the absorption index) [11–13]. The regular formulas for the efficiency factors of scattering Qs and
0018-151X/04/4201-0133 © 2004 MAIK “Nauka /Interperiodica”
134
DOMBROVSKII
extinction Qt , as well as for the factor of asymmetry of
scattering µ , have the following form [13]:
The following formulas are given in [8–10] for gas
bubbles in glass:
b
∞
2
2
2
Q s = -----2
( 2k + 1 ) ( a k + b k ),
x k=1
∑
(1)
∑
∑
(3)
2k + 1
+ -------------------- Re ( a k b k* ) .
k(k + 1)
Here, ak and bk are Mie coefficients expressed in
terms of the Riccati–Bessel functions; the asterisk indicates complex conjugate quantities.
The factor of absorption efficiency Qa and the transport factor of extinction efficiency Qtr, which are of interest to us (these quantities are required to perform an
approximate calculation of radiation transfer [13, 14]),
are determined by the formulas
Qa = Qt – Qs ,
Q tr = Q t – Q s µ.
(4)
It is also convenient to use the transport factor of
scattering efficiency,
tr
Q s = Q tr – Q a = Q s ( 1 – µ ).
2
(7)
Obviously, in the case of particles, cavities, or gas
bubbles which do not absorb radiation, the absorption
efficiency factor Qa in an absorbing medium is negatr
tive, and the transport factor of scattering efficiency Q s
is positive.
b
g
the complex refractive index for gas mb, Q a was calculated by the complex refractive index for gas mg, and
the diffraction parameter was taken to be the same in
both cases, x = 2πa/λ. No validation of formulas (8) and
(9) is given in [8]. One can demonstrate both theoretically and using direct calculations that formula (8) is
valid only in the limiting case of |mg – 1| 1, κgx 1
(the inequalities |mb – 1| 1, κbx 1 for gas are a priori
valid) when the Rayleigh–Gans approximation is valid
[11]. As to formula (9), it is erroneous, because the scattering depends on the ratio mb/mg rather than on the
value of mb.
THE EFFECT OF BUBBLES
ON THE RADIATION CHARACTERISTICS
OF A SEMITRANSPARENT MEDIUM
The coefficient of absorption of elementary volume
of a medium containing polydisperse particles or bubbles with the size distribution function F(a) is determined by the formula [13]
∞
f
4πκ
2
Σ a = ---------- + 0.75 ------v- Q a a F ( a ) da,
a 30
λ
∫
(10)
0
where fv is the volume particle density,
∞
a ij =
∞
∫ a F ( a ) da ∫ a F ( a ) da.
i
j
0
(11)
0
Similarly, the transport coefficient of scattering has
the form
(6)
where r ≥ a is the distance to the particle center. For the
semitransparent region treated in this study, κx 1. In
this case, the coefficient C is independent of the distance r and is determined by the simple formula
2
-.
C = --------2 2
n x
(9)
where, according to [8], Q a and Q s were calculated by
(5)
Mundy et al. [15] demonstrated that the formulas of
the Mie theory are also valid for particles in a refracting
and absorbing medium with an arbitrary complex
refractive index m = n – iκ. In so doing, the complex
quantities m̃ = m'/m (for cavities or gas bubbles, m̃ =
1/m) and x̃ = mx must be substituted for m' and x as
independent variables in calculating the Mie coefficients, and the coefficient 2/x2 in formulas (1)–(3) must
be replaced by
4κ exp [ – 2κx ( r/a ) ]
-,
C = ----------------------------------------------------------------------------------2
2
( n + κ ) [ 1 + ( 2κx – 1 ) exp ( 2κx ) ]
b
b
(2)
∞
k(k + 2)
4
-------------------- Re ( a k a *k + 1 + b k b *k + 1 )
µQ s = -----2
x k=1 k+1
(8)
Qs = Qs ,
∞
2
Q t = -----2
( 2k + 1 )Re ( a k + b k ),
x k=1
g
Qa = Qa – Qa ,
∞
tr
Σs
f
tr 2
= 0.75 ------v- Q s a F ( a ) da,
a 30
∫
(12)
0
The transport coefficient of extinction is Σtr = Σa =
tr
Σs .
In a monodisperse approximation, formulas (10)
and (12) are written as
Q
4πκ
Σ a = ---------- + 0.75 f v ------a ,
a
λ
(13)
tr
Qs
tr
Σ s = 0.75 f v ------.
a
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THE PROPAGATION OF INFRARED RADIATION
α
1.3
135
Qstr
0.5
1
2
3
1.2
1.1
0.4
4
0.3
3
0.2
2
0.1
1
I
II
1.0
0
10
20
30
x
0
5
10
15
20
25
30
x
Fig. 1. The relative factor of absorption efficiency for bubbles in media with different optical properties: (1) n = 1.2,
(2) 1.3, and (3) 1.5; I – κ = 10–3, II – κ = 10–4.
Fig. 2. The transport factor of absorption efficiency for bubbles in nonabsorbing media with different refractive indices: (1) n = 1.2, (2) 1.3, (3) 1.4, and (4) 1.5.
We will examine the effect of monodisperse bubbles
on the radiation characteristics of a medium. For clarity, we will rewrite formula (13) as
For a polydisperse medium, approximate formulas
have the form
4πκ
Σ a = ---------- ( 1 – f v α ),
λ
α = – Q a / ( 8κx/3 ).
(15)
The results of calculation of the α(x) dependence by
the algorithm similar to that suggested in [13] for the
most interesting range of variation of the optical constants of the medium are given in Fig. 1. One can see
that, even with x > 10 and κx < 0.01, the parameter α
approaches the asymptotic value for large bubbles in a
weakly absorbing medium, α = 1. In the near-infrared
spectrum, the condition x > 10 is valid for all bubbles
with a radius a > 4 µm. Because fv 1, we have
αfv 1; therefore, according to Eqs. (15), the effect of
the bubbles on the absorption of radiation in the semitransparent region is negligibly low and may be ignored
on the assumption that Σa = 4πκ/λ.
A series of calculations using the Mie theory have
demonstrated that, in a medium that weakly absorbs
radiation, the absorption has almost no effect on the
scattering of radiation by bubbles, and it is sufficient to
tr
treat the Q s (x) dependences for κ = 0 given in Fig. 2.
The results of approximate calculations demonstrate
that, even for x > 10, the transport factor of scattering
efficiency may be regarded as a constant quantity,
which corresponds to the transition to the region of geotr
metrical optics. The respective values of Q s may be
estimated by the formula
tr
Q s = 0.9 ( n – 1 ),
(16)
whence
4πκ
Σ a = ---------- ,
λ
(18)
n–1
tr
Σ s = 0.675 f v -----------.
a 32
(19)
We will compare the transport scattering coefficient
with the absorption coefficient using the ratio between
them,
n–1
tr
Σ s /Σ a ≈ 0.34 f v -----------,
κx 32
2πa 32
-.
x 32 = ------------λ
(20)
It follows from (20) that, with a bubble concentration fv > 10κx32, radiation scattering may dominate over
absorption. With an invariable spectral absorption
index for liquid, the importance of scattering is defined
by the ratio of the volume density of the bubbles to their
average radius fv /a32.
THE METHOD OF CALCULATION
OF RADIATION TRANSFER IN AN ABSORBING
AND REFRACTING MEDIUM CONTAINING
SCATTERING PARTICLES
We will treat the problem on radiation transfer in a
semi-infinite layer of an absorbing and refracting
medium containing radiation-scattering particles. We
will assume that the surface of the medium is uniformly
illuminated by diffuse randomly polarized outer radiation. The equation of radiation transfer in the transport
approximation has the form [13]
tr 1
n–1
tr
Σ s = 0.675 f v -----------.
a
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Vol. 42
(17)
Σ
∂I
µ -------λ + Σ tr I λ = -----s- I λ dµ,
∂z
2
∫
–1
No. 1
2004
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136
DOMBROVSKII
and boundary condition for the spectral radiation intensity Iλ(z, µ),
2 e
I λ(0, µ) = RI λ(0, – µ) + ( 1 – R )2n q λ , 0 ≤ µ ≤ 1. (22)
Here, µ = cosθ, the angle θ is reckoned from a nore
mal directed into the medium, q λ is the spectral flow of
outer radiation, and R(µ) is the reflection coefficient,
1 µ – nµ' 2
nµ – µ' 2
R ( µ ) = ---  ------------------ +  ------------------ ,
 nµ + µ'
2  µ + nµ'
2
µ' =
2
1 – n (1 – µ ),
R ( µ ) = 1,
µ ≤ µc =
µ > µc ,
The differential approximation is provided by the
DP0-approximation of the method of dual spherical
harmonics, or two-flow approximation [13]. We
assume that
–
+
–
I λ ( z, µ ) = ϕ 0 ( z ) + [ ϕ 0 ( z ) – ϕ 0 ( z ) ]Θ ( µ )
(27)
and integrate Eq. (21) separately over the intervals
−1 < µ < 0 and 0 < µ < 1 to derive, after simple transformations, the following boundary-value problem for
–
+
the function g0 = ϕ 0 + ϕ 0 :
(23)
dg
d
– -----  D λ --------0 + Σ a g 0 = 0 ,
dz 
dz 
2
1 – 1/n .
In a particular case of a nonscattering medium, the
problem given by Eqs. (21)–(23) has an obvious analytical solution,
z = 0,
(28)
dg
n
2 e
- ( g 0 – 4n q λ ) ,
D λ --------0 = ------------2
dz
n +1
(29)
dg 0
-------- = 0.
dz
(30)
2 e
I λ ( z, µ ) = ( 1 – R )2n q λ exp ( – Σ a z/µ )Θ ( µ – µ c ), (24)
where Θ is the Heaviside function. According to
Eq. (24), the outer radiation, after entering a refracting
medium, propagates within a cone with an apex angle
of 2 arccos ( µ c ) . The power of absorbed radiation P(z)
was determined as
∞
P(z) =
∫ Σ W ( z ) dλ,
λ
a
(25)
0
where the spectral density of radiation energy is
1
W λ(z) =
∫ I ( z, µ ) dµ
1
2 e
Here, Dλ = 1/(4Σtr) is the spectral coefficient of radiation diffusion. Because, in the case of a numerical
solution of the problem, boundary condition (30) is preassigned for some finite z = ∞ rather than for z = z0, a
less rigid condition imposed on the derivative (instead
of the possible condition g0 = 0) is preferred. The coefficient in the boundary condition on the liquid surface
is determined, as in [16, 17], by the value of the reflection coefficient at µ = 1,
2
λ
–1
z = ∞,
2
R(1) = (n – 1) /(n + 1) .
(26)
∫
= 2n q λ ( 1 – R ) exp ( – Σ a z/µ ) dµ.
µc
In the presence of scattering, the radiation field in
the medium becomes significantly more complex. If
scattering prevails over absorption, the angular dependence of the radiation intensity does not exhibit the feature mentioned above and characteristic of the solution
in a nonscattering medium. In this case, an approximate
calculation of radiation transfer may be performed
using some differential approximation, as has been
done, for example, by Fedorov and Viskanta [8]. However, in the general case of an arbitrary ratio between
absorption and scattering, this may lead to a significant
underestimation of the depth of radiation penetration.
Therefore, I follow [13, 14] and suggest a combined
solution in which the differential approximation is used
only to determine the integral term in the right-hand
part of transfer equation (21) and, in the second step of
the solution, the equation of radiation transfer with the
known right-hand part is integrated.
(31)
The solution of the boundary-value problem (28)–
(30) gives an approximate profile of the spectral density
of radiation energy entering the right-hand part of
transfer equation (21),
1
W λ(z) =
∫ I ( z, µ ) dµ = g ( z ).
λ
0
(32)
–1
Note that, with constant coefficients Σa and Dλ, the
problem given by Eqs. (28)–(30) has a simple analytical solution,
2 e
2n q λ
- exp ( – 2 Σ a Σ tr z ).
g 0 ( z ) = ----------------------------------------2
n +1
1 + -------------- Σ a /Σ tr
2n
(33)
Comparison of formulas (26) and (33) at Σtr = Σa
gives an idea of the error of the DP0-approximation in a
weakly scattering medium.
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THE PROPAGATION OF INFRARED RADIATION
Solutions to the equation of radiation transfer (21)
with the right-hand part calculated in the DP0-approximation may be obtained using an integrating factor [18],
I λ ( z, µ ) = I λ ( 0, µ ) exp ( – Σ tr z/µ )
z
tr
1 Σ
z–t
+ --- -----s- g 0 ( t ) exp  – Σ tr --------- dt,

2 µ
µ 
∫
(34)
0
∞
Formula (40) corresponds to the regular definition
[18, 19], and Eq. (41) is distinguished by a nonzero
lower integration limit.
After determining Wλ(z), the power of the absorbed
radiation (P(z) is found by integration over the spectrum (by formula (25)). Note that the function P(z) satisfies the relation for energy balance, which is written
for a semi-infinite layer of a medium as
∞
tr
Σs
1
t–z
I λ ( z, – µ ) = --- ------ g 0 ( t ) exp  – Σ tr --------- dt,

2 µ
µ 
∫
137
µ > 0. (35)
∞
∫ P ( z ) dz = q,
0
q =
∫ q dλ,
λ
(42)
0
z
Relations (34) and (35) in combination with boundary condition (22) make it possible to calculate the
spectral radiation intensity in all directions at any point
of the calculation region. First, Iλ(z, –µ) is determined
by formula (35), and then the known quantity Iλ(0, –µ)
from boundary condition (22) is used to calculate Iλ(0, µ).
After that, Iλ(z, µ) is calculated by formula (34).
However, it is not our objective to determine the
angular dependence of the spectral radiation intensity.
It is sufficient to calculate the profile of the spectral
density of radiation energy,
–
+
W λ ( z ) = W λ ( z ) + W λ ( z ),
∫
1
–
(36)
+
where W λ (z) = 0 I λ (z, –µ)dµ and W λ (z) =
µ)dµ. We use Eq. (35) to find
∫
1
I (z,
0 λ
∞
–
W λ(z)
∫
(37)
Relation (34) and boundary condition (22) yield the
approximate integral relation
z
1 tr
+
+
W λ ( z ) = W λ ( 0 ) + --- Σ s g 0 ( t )E 1 [ Σ tr ( z – t ) ] dt, (38)
2
∫
0
+
–
W λ ( 0 ) = R ( 1 )W λ ( 0 )E 2 ( Σ tr z )
(39)
2 e
+ [ 1 – R ( 1 ) ]2n q λ Ẽ 2 ( Σ tr z ).
Formulas (36)–(39) enable one to calculate the
spectral density of radiation energy. The integroexponential functions appearing in Eqs. (37)–(39) are
defined as
1
∫µ
k–2
∞
z
Q(z) =
∫ P ( z ) dz ∫
0
0
exp ( – y/µ ) dµ,
k = 1, 2;
(40)
0
1
Ẽ 2 ( y ) =
∫ exp ( – y/µ ) dµ.
(41)
µc
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No. 1
2004
z
1
P ( z ) dz = --- P ( z ) dz.
q
∫
(43)
0
CALCULATIONS FOR WATER
CONTAINING VAPOR BUBBLES
We will treat the model problem on the propagation
of thermal radiation in a thick layer of water containing
vapor bubbles. The spectrum of radiation incident on
the water layer is taken to be similar to the spectrum of
blackbody radiation for some temperature Te, i.e., it is
e
1 tr
= --- Σ s g 0 ( t )E 1 [ Σ tr ( t – z ) ] dt.
2
z
Ek ( y ) =
where q is the integral flux of thermal radiation on the
z = 0 surface. We will follow [2] and, along with the differential characteristic of absorption P(z), use the function corresponding to the fraction of total integral flux
of thermal radiation absorbed in a layer of liquid (0, z),
assumed that q λ = Bλ(Te). The concentration of vapor
bubbles and their size are assumed to be constant over
tr
the entire layer of water. The coefficients Σa, Σ s , and Σtr
are independent of coordinate z.
It is obvious that the real problem is conjugate,
because the profiles of volume concentration fv(z) and
of the average bubble radius a32(z) affect the thermal
radiation transfer, and the volumetric heat release
causes a variation of the fv(z) and a32(z) profiles as a
result of absorption of radiation. However, in order to
solve the problem, one needs some kinetic model
describing the nucleation and growth of vapor bubbles
with due regard for the absorption of thermal radiation.
Such a general problem is beyond the scope of this
study; therefore, in the model calculations performed,
the constant values of fv and a32 are treated as preassigned parameters.
In the calculations of the spectral radiation characteristics of water containing vapor bubbles, the absorption index for water in the infrared spectrum was determined by way of interpolation of the tabular data of
[20], and the refractive index was taken to be n = 1.33
(in the semitransparent region, it varies from n = 1.335
at λ = 0.5 µm to n = 1.324 at λ = 1.2 µm). The vapor in
the bubbles was assumed to be fully transparent to thermal radiation. The spectral coefficient of absorption
138
DOMBROVSKII
Q
1.00
z0, m
100
10
(a)
0.95
4
1
3
1
0.90
0.1
0.85
0.01
0.001
0.5
1.0
1.5
2.0
2.5
λ, µm
Fig. 3. The thickness of the water layer in which the outer
radiation is almost fully absorbed. Estimation was made by
formula (44).
and the spectral transport coefficient of scattering were
determined by formulas (18) and (19).
For better illustration of the transparent region of
water, Fig. 3 gives the values of water layer thickness z0
at which the radiation on a given wavelength is almost
fully absorbed. The value of z0 was determined by the
formula
10
z 0 = ------ ,
Σa
(44)
i.e., from the condition of the equality of the spectral
optical thickness of the water layer to ten. Curve z0(λ)
in Fig. 3 defines the region in which the problem of
radiation transfer does not degenerate.
The Q(z) dependences were calculated as follows.
First, the analytical solution of (32) was used to determine the function g0(z) which was a first approximation
for the spectral density of radiation energy Wλ(z). Then,
formulas (36)–(39) were used to find the next approximation for Wλ(z). The resultant function was substituted into Eq. (25) for the power of absorbed radiation
P(z) and, finally, the function Q(z) was determined by
formula (43).
Figure 4 gives the calculated dependence Q(z) for
different values of fv /a32, because this particular ratio is
the only parameter allowing for the effect of vapor bubbles on radiation transfer. Also given for comparison is
the Q(z) curve for fv = 0. The calculations were performed for two temperatures of the thermal radiation
source, namely, Te = 2000 K and Te = 3000 K. One
series of calculations was performed in the DP0approximation (without the second step of solution),
and the other series was performed using a combined
computational model. It follows from the data in Fig. 4
that the DP0-approximation gives qualitatively correct
results but underestimates the thickness of the water
layer in which the thermal radiation is absorbed by a
factor of almost two. The error of the DP0-approxima-
I
II
0.80
1.0
(b)
4
0.9
3
2
0.8
1
0.7
I
II
0.6
0.5
1
10
100
z, mm
Fig. 4. The effect of vapor bubbles on the absorption of radiation in a layer of water: (a) Te = 2000 K, (b) Te = 3000 K;
I—DP0-approximation, II—combined computational
model; (1) fv /a32 = 0 (2) fv /a32 = 10 m–1, (3) 100 m–1,
(4) 1000 m–1.
tion somewhat decreases with increasing contribution
by scattering. As was to be expected, the scattering of
radiation by vapor bubbles leads to the absorption of
radiation in a thinner layer of water. This effect
becomes significant even at fv /a32 ≥ 10 m–1.
Figure 5 illustrates the effect of relatively low values
of the parameter fv /a32 on the thickness ∆ of the water
layer in which the bulk of the power of thermal radiation is absorbed at Te = 3000 K. The values of ∆ given
in Fig. 5 were determined using the equalities Q(∆) =
0.07 and Q(∆) = 0.8. In order to understand how real the
preassigned values of fv /a32 are, one can turn to the
experimental data of [21, 22], according to which the
value of fv /a32 in the majority of cases varies from 5 to
100. In this range, the treated effect shows up quite
clearly.
Turning back to the discussion of the conjugate
problem in view of the effect of the absorption of radiation on the nucleation and growth of vapor bubbles
(bearing in mind, for example, film boiling on a surface
with a temperature above 2000 K), note the presence of
positive feedback: an increase in the concentration and
size of bubbles leads to an ever stronger absorption of
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THE PROPAGATION OF INFRARED RADIATION
∆, mm
35
tering of radiation by bubbles may lead to the absorption of thermal radiation in a much thinner layer of
water than in the case of water without bubbles.
The suggested theoretical model may be used in
solving a conjugate problem in view of the effect made
by volume absorption of radiation on the nucleation
and growth of vapor bubbles.
1
2
30
139
25
20
15
REFERENCES
10
5
0
5
10
15
20
25
30
ƒv /a32, m–1
Fig. 5. The effect of vapor bubbles on the thickness of the
water layer in which (1) 70% and (2) 80% of the power of
thermal radiation is absorbed at Te = 3000 K. The calculation was performed by the combined computational model.
radiation in a thin layer of water in the vicinity of the
surface. As a result, one can expect a periodic explosion-like boiling of liquid in this surface layer with a
splashing of fine droplets.
CONCLUSIONS
A theoretical model is suggested of for propagation
of infrared radiation in a semitransparent liquid containing gas bubbles, which includes an approximate
description of the spectral radiation characteristics and
radiation transfer in view of the scattering of radiation
by the bubbles.
The Mie theory was employed to calculate the effect
of bubbles on the absorption and scattering of radiation.
The calculation results have demonstrated that, in the
real range of variation of parameters, the bubbles have
almost no effect on the absorption coefficient and the
main effect is associated with the scattering of radiation. A simple formula is suggested for the scattering
coefficient, which approximates the results of exact calculations.
A combined computational model is suggested for
the calculation of radiation transfer in a liquid containing gas bubbles; in this model, the DP0-approximation
is used in the first step of the solution, and then the
equation of radiation transfer is integrated. It is demonstrated that the second step of solution significantly
refines the calculated profiles of absorption in a layer of
water.
Calculations were performed for a thick layer of
water containing vapor bubbles illuminated by the thermal radiation of an external source. It has been demonstrated that, for real values of the parameters, the scat-
HIGH TEMPERATURE
Vol. 42
No. 1
2004
1. Beshta, S.V., Vitol’, S.A., Krushinov, E.V., et al., Teploenergetika, 1998, no. 11, p. 20.
2. Dombrovskii, L.A., Teplofiz. Vys. Temp., 2003, vol. 41,
no. 6, p. 920 (High Temp. (Engl. transl.), vol. 41, no. 6).
3. Reiss, H., High Temp. High Pressures, 1990, vol. 22,
no. 5, p. 481.
4. Baillis, D. and Sacadura, J.-F., J. Quant. Spectrosc.
Radiat. Transfer, 2000, vol. 67, p. 327.
5. Fedorov, A.V. and Pilon, L., J. Non Cryst. Solids, 2002,
vol. 311, p. 154.
6. Tien, C.L. and Drolen, B.L., Annu. Rev. Numerical Fluid
Mech. Heat Transfer, 1987, vol. 1, p. 1.
7. Tien, C.L., ASME J. Heat Transfer, 1988, vol. 110, no. 4,
p. 1230.
8. Fedorov, A.V. and Viskanta, R., Phys. Chem. Glasses,
2000, vol. 41, no. 3, p. 127.
9. Fedorov, A.V. and Viskanta, R., J. Am. Ceram. Soc.,
2000, vol. 83, no. 11, p. 2769.
10. Pilon, L. and Viskanta, R., Apparent Radiation Characteristics of Semitransparent Media Containing Gas Bubbles, Proc. 12th Int. Heat Transfer Conf., Grenoble,
2002, vol. 1, p. 645.
11. Hulst, van de, H.C., Light Scattering by Small Particles,
New York: Wiley, 1957.
12. Bohren, C.F. and Huffman, D.R., Absorption and Scattering of Light by Small Particles, New York: Wiley,
1983.
13. Dombrovsky, L.A., Radiation Heat Transfer in Disperse
Systems, New York: Begell House, 1996.
14. Dombrovskii, L.A., Teploenergetika, 1996, no. 3, p. 50.
15. Mundy, W.C., Roux, J.A., and Smith, A.M., J. Opt. Soc.
Am., 1974, vol. 64, no. 12, p. 1593.
16. Dombrovskii, L.A., Teplofiz. Vys. Temp., 1999, vol. 37,
no. 2, p. 284 (High Temp. (Engl. transl.), vol. 37, no. 2,
p. 260).
17. Dombrovsky, L.A., Int. J. Heat Mass Transfer, 2000,
vol. 43, no. 9, p. 1661.
18. Siegel, R. and Howell, J.R., Thermal Radiation Heat
Transfer, Washington: Hemisphere, 1992.
19. Ozisik, M.N., Radiative Transfer and Interactions with
Conduction and Convection, New York: Wiley, 1973.
20. Hale, G.M. and Querry, M.P., Appl. Opt., 1973, vol. 12,
no. 3, p. 555.
21. Hibiki, T. and Ishii, M., Int. J. Heat Mass Transfer, 2002,
vol. 45, no. 11, p. 2351.
22. Hibiki, T., Situ, R., Mi, Y., and Ishii, M., Int. J. Heat
Mass Transfer, 2003, vol. 46, no. 8, p. 1409.