The Propagation of Infrared Radiation in a Semitransparent Liquid Containing Gas Bubbles

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1 High Temperature, Vol. 4, No., 4, pp Translated from Teplofiika Vysokikh Temperatur, Vol. 4, No., 4, pp Original Russian Text Copyright 4 by Dombrovskii. HEAT AND MASS TRANSFER AND PHYSCAL GASDYNAMCS The Propagation of nfrared Radiation in a Semiansparent Liquid Containing Gas Bubbles L. A. Dombrovskii VTAN (nstitute of High Temperatures) Scientific Association, Russian Academy of Sciences, Moscow, 54 Russia Received May 3, 3 Absact A theoretical model is suggested for the propagation of infrared radiation in a semiansparent liquid containing gas bubbles, which includes an approximate description of the radiation characteristics and radiation ansfer in a disperse system. Calculations are performed for a layer of water containing vapor bubbles illuminated by the thermal radiation of an external source. t is demonsated 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. NTRODUCTON The problem eated 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 []. The thermal radiation of a solid or melt with a temperature of 3 K is largely associated with the near infrared specal region where water is semiansparent. Therefore, a significant part of the radiation is not absorbed in the surface layer but peneates deep into the water and leads to volumeic heat release []. 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 semiansparent liquids and to solids containing numerous bubbles or other spherical inclusions. Such suctures 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 ansfer 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 sie; all bubbles are spherical; the bubbles are arranged at random; and the distance between bubbles significantly exceeds their sies and the radiation wavelength. The first of these assumptions implies that only that specal range is eated in which the medium weakly absorbs radiation (semiansparent region). The assumptions of the random arrangement of bubbles and of their not-too-high concenation 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 resiction of our eatment to bubbles of spherical shape significantly simplifies the determination of their radiation characteristics. The problem was analogously formulated in [8 ] 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 [, ]. n this study, the radiation characteristics of a medium containing bubbles are analyed using the rigorous theory of scattering. RELATONS FOR RADATON CHARACTERSTCS OF SPHERCAL PARTCLES N A SEMTRANSPARENT ABSORBNG MEDUM 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 πa/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) [ 3]. The regular formulas for the efficiency factors of scattering Q s and 8-5X/4/ MAK Nauka /nterperiodica

2 34 DOMBROVSK extinction Q t, as well as for the factor of asymmey of scattering µ, have the following form [3]: Q s ---- ( k + ) ( a k + b k ), 4 µq s ---- x k Q t ---- ( k + )Re( a k + b k ), x k x k kk Re ( + ) ( k + a a * k k + + b k b k * + ) Re k + ( kk ( + ) a b* ). k k () () (3) Here, a k and b k are Mie coefficients expressed in terms of the Riccati Bessel functions; the asterisk indicates complex conjugate quantities. The factor of absorption efficiency and the ansport factor of extinction efficiency Q, which are of interest to us (these quantities are required to perform an approximate calculation of radiation ansfer [3, 4]), are determined by the formulas Q t Q s, Q Q t Q s µ. (4) t is also convenient to use the ansport factor of scattering efficiency, Q s Q s ( µ ). Q (5) Mundy et al. [5] demonsated that the formulas of the Mie theory are also valid for particles in a refracting and absorbing medium with an arbiary complex refractive index m n iκ. n so doing, the complex quantities m m'/m (for cavities or gas bubbles, m /m) and x mx must be substituted for m' and x as independent variables in calculating the Mie coefficients, and the coefficient /x in formulas () (3) must be replaced by C 4κ exp[ κx( r/a) ] , ( n + κ )[ + ( κx ) exp( κx) ] (6) where r a is the distance to the particle center. For the semiansparent region eated in this study, κx. n this case, the coefficient C is independent of the distance r and is determined by the simple formula C n x (7) Obviously, in the case of particles, cavities, or gas bubbles which do not absorb radiation, the absorption efficiency factor in an absorbing medium is negative, and the ansport factor of scattering efficiency Q s is positive. The following formulas are given in [8 ] for gas bubbles in glass: b g, (8) (9) where, according to [8], and were calculated by g the complex refractive index for gas m b, was calculated by the complex refractive index for gas m g, and the diffraction parameter was taken to be the same in both cases, x πa/λ. No validation of formulas (8) and (9) is given in [8]. One can demonsate both theoretically and using direct calculations that formula (8) is valid only in the limiting case of m g, κ g x (the inequalities m b, κ b x for gas are a priori valid) when the Rayleigh Gans approximation is valid []. As to formula (9), it is erroneous, because the scattering depends on the ratio m b /m g rather than on the value of m b. THE EFFECT OF BUBBLES ON THE RADATON CHARACTERSTCS OF A SEMTRANSPARENT MEDUM The coefficient of absorption of elementary volume of a medium containing polydisperse particles or bubbles with the sie disibution function F(a) is determined by the formula [3] where f v is the volume particle density, a ij Q s Q b s, b () () Similarly, the ansport coefficient of scattering has the form () The ansport coefficient of extinction is Σ. n a monodisperse approximation, formulas () and () are written as Q s b 4πκ f v Q λ a a Fa ( ) da, a 3 a i Fa ( ) d a a j Fa ( ) da..75 f v Q s a Fa ( ) da, a 3 4πκ Q f a λ v -----, a Q.75 f s v a (3) (4) HGH TEMPERATURE Vol. 4 No. 4

3 THE PROPAGATON OF NFRARED RADATON 35 α.3 Qs x Fig.. The relative factor of absorption efficiency for bubbles in media with different optical properties: () n., ().3, and (3).5; κ 3, κ x Fig.. The ansport factor of absorption efficiency for bubbles in nonabsorbing media with different refractive indices: () n., ().3, (3).4, and (4).5. We will examine the effect of monodisperse bubbles on the radiation characteristics of a medium. For clarity, we will rewrite formula (3) as 4πκ ( f (5) λ v α), α /8κx/3 ( ). The results of calculation of the α(x) dependence by the algorithm similar to that suggested in [3] for the most interesting range of variation of the optical constants of the medium are given in Fig.. One can see that, even with x > and κx <., the parameter α approaches the asymptotic value for large bubbles in a weakly absorbing medium, α. n the near-infrared specum, the condition x > is valid for all bubbles with a radius a > 4 µm. Because f v, we have αf v ; therefore, according to Eqs. (5), the effect of the bubbles on the absorption of radiation in the semiansparent region is negligibly low and may be ignored on the assumption that 4πκ/λ. A series of calculations using the Mie theory have demonsated 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 eat the Q s (x) dependences for κ given in Fig.. The results of approximate calculations demonsate that, even for x >, the ansport factor of scattering efficiency may be regarded as a constant quantity, which corresponds to the ansition to the region of geomeical optics. The respective values of Q s may be estimated by the formula whence Q s.9( n ), n.675 f v a (6) (7) For a polydisperse medium, approximate formulas have the form 4πκ , λ n.675 f v a 3 (8) (9) We will compare the ansport scattering coefficient with the absorption coefficient using the ratio between them, Σ n πa s /.34 f v , x () κx λ t follows from () that, with a bubble concenation f v > κx 3, radiation scattering may dominate over absorption. With an invariable specal absorption index for liquid, the importance of scattering is defined by the ratio of the volume density of the bubbles to their average radius f v /a 3. THE METHOD OF CALCULATON OF RADATON TRANSFER N AN ABSORBNG AND REFRACTNG MEDUM CONTANNG SCATTERNG PARTCLES We will eat the problem on radiation ansfer 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 polaried outer radiation. The equation of radiation ansfer in the ansport approximation has the form [3] µ λ Σ λ Σ s λ dµ, () HGH TEMPERATURE Vol. 4 No. 4

4 36 DOMBROVSK and boundary condition for the specal radiation intensity λ (, µ), λ (, µ ) R λ (, µ ) + ( R)n q e λ, µ. () Here, µ cosθ, the angle θ is reckoned from a normal directed into the medium, q λ is the specal flow of e outer radiation, and R(µ) is the reflection coefficient, R ( µ ) -- µ nµ' nµ µ' , µ + nµ' nµ + µ' µ' n ( µ ), µ > µ c, R ( µ ), µ µ c /n. (3) n a particular case of a nonscattering medium, the problem given by Eqs. () (3) has an obvious analytical solution, λ (, µ ) ( R)n e q λ exp( /µ )Θ( µ µ c ), (4) where Θ is the Heaviside function. According to Eq. (4), the outer radiation, after entering a refracting medium, propagates within a cone with an apex angle of arccos( µ c ). The power of absorbed radiation P() was determined as P ( ) W λ ( ) dλ, where the specal density of radiation energy is n q λ e W λ ( ) λ (, µ ) dµ µ c ( R) exp ( /µ ) dµ. (5) (6) n the presence of scattering, the radiation field in the medium becomes significantly more complex. f 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. n this case, an approximate calculation of radiation ansfer 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 arbiary ratio between absorption and scattering, this may lead to a significant underestimation of the depth of radiation peneation. Therefore, follow [3, 4] and suggest a combined solution in which the differential approximation is used only to determine the integral term in the right-hand part of ansfer equation () and, in the second step of the solution, the equation of radiation ansfer with the known right-hand part is integrated. The differential approximation is provided by the DP -approximation of the method of dual spherical harmonics, or two-flow approximation [3]. We assume that λ (, µ ) ϕ ( ) + [ ϕ + ( ) ϕ ( ) ]Θ( µ ) (7) and integrate Eq. () separately over the intervals < µ < and < µ < to derive, after simple ansformations, the following boundary-value problem for + the function g ϕ + ϕ : d dg ---- D, (8) d λ Σ d a g dg, D λ d dg, d n g, (9) n 4n e ( q λ ) + (3) Here, D λ /(4Σ ) is the specal coefficient of radiation diffusion. Because, in the case of a numerical solution of the problem, boundary condition (3) is preassigned for some finite rather than for, a less rigid condition imposed on the derivative (instead of the possible condition g ) is preferred. The coefficient in the boundary condition on the liquid surface is determined, as in [6, 7], by the value of the reflection coefficient at µ, R( ) ( n ) /( n + ). (3) The solution of the boundary-value problem (8) (3) gives an approximate profile of the specal density of radiation energy entering the right-hand part of ansfer equation (), W λ ( ) λ (, µ ) dµ g ( ). (3) Note that, with constant coefficients and D λ, the problem given by Eqs. (8) (3) has a simple analytical solution, g ( ) n e q λ exp( Σ n a Σ ) Σ n a /Σ (33) Comparison of formulas (6) and (33) at Σ gives an idea of the error of the DP -approximation in a weakly scattering medium. HGH TEMPERATURE Vol. 4 No. 4

5 THE PROPAGATON OF NFRARED RADATON 37 Solutions to the equation of radiation ansfer () with the right-hand part calculated in the DP -approximation may be obtained using an integrating factor [8], λ (, µ ) λ (, µ ) λ (, µ ) exp( Σ /µ ) g µ () t t Σ exp dt, µ (34) -- t -----g (35) µ () t Σ exp t, µ µ d >. Relations (34) and (35) in combination with boundary condition () make it possible to calculate the specal radiation intensity in all directions at any point of the calculation region. First, λ (, µ) is determined by formula (35), and then the known quantity λ (, µ) from boundary condition () is used to calculate λ (, µ). After that, λ (, µ) is calculated by formula (34). However, it is not our objective to determine the angular dependence of the specal radiation intensity. t is sufficient to calculate the profile of the specal density of radiation energy, W λ ( ) W λ ( ) + W + λ ( ), (36) + where W λ () λ (, µ)dµ and W () (, λ λ µ)dµ. We use Eq. (35) to find W λ ( ) -- Σ s g ()E t [ Σ ( t ) ] dt. (37) Relation (34) and boundary condition () yield the approximate integral relation W + λ ( ) W + λ ( ) + -- Σ s g ()E t [ Σ ( t) ] dt, (38) W + λ ( ) R( )W λ ( )E ( Σ ) (39) + [ R( ) ]n q e λ Ẽ ( Σ ). Formulas (36) (39) enable one to calculate the specal density of radiation energy. The integroexponential functions appearing in Eqs. (37) (39) are defined as E k ( y) µ k exp ( y/µ ) dµ, k ;, Ẽ ( y) exp ( y/µ ) dµ. µ c (4) (4) Formula (4) corresponds to the regular definition [8, 9], and Eq. (4) is distinguished by a nonero lower integration limit. After determining W λ (), the power of the absorbed radiation (P() is found by integration over the specum (by formula (5)). Note that the function P() satisfies the relation for energy balance, which is written for a semi-infinite layer of a medium as P ( ) d q, q q λ dλ, (4) where q is the integral flux of thermal radiation on the surface. We will follow [] and, along with the differential characteristic of absorption P(), use the function corresponding to the fraction of total integral flux of thermal radiation absorbed in a layer of liquid (, ), Q ( ) P ( ) d P ( ) d -- P ( ) d. q (43) CALCULATONS FOR WATER CONTANNG VAPOR BUBBLES We will eat the model problem on the propagation of thermal radiation in a thick layer of water containing vapor bubbles. The specum of radiation incident on the water layer is taken to be similar to the specum of blackbody radiation for some temperature T e, i.e., it is e assumed that q λ B λ (T e ). The concenation of vapor bubbles and their sie are assumed to be constant over the entire layer of water. The coefficients,, and Σ are independent of coordinate. t is obvious that the real problem is conjugate, because the profiles of volume concenation f v () and of the average bubble radius a 3 () affect the thermal radiation ansfer, and the volumeic heat release causes a variation of the f v () and a 3 () 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 f v and a 3 are eated as preassigned parameters. n the calculations of the specal radiation characteristics of water containing vapor bubbles, the absorption index for water in the infrared specum was determined by way of interpolation of the tabular data of [], and the refractive index was taken to be n.33 (in the semiansparent region, it varies from n.335 at λ.5 µm to n.34 at λ. µm). The vapor in the bubbles was assumed to be fully ansparent to thermal radiation. The specal coefficient of absorption HGH TEMPERATURE Vol. 4 No. 4

6 38, m. DOMBROVSK Q (a) λ, µm.8. (b) 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 specal ansport coefficient of scattering were determined by formulas (8) and (9). For better illusation of the ansparent region of water, Fig. 3 gives the values of water layer thickness at which the radiation on a given wavelength is almost fully absorbed. The value of was determined by the formula i.e., from the condition of the equality of the specal optical thickness of the water layer to ten. Curve (λ) in Fig. 3 defines the region in which the problem of radiation ansfer does not degenerate. The Q() dependences were calculated as follows. First, the analytical solution of (3) was used to determine the function g () which was a first approximation for the specal density of radiation energy W λ (). Then, formulas (36) (39) were used to find the next approximation for W λ (). The resultant function was substituted into Eq. (5) for the power of absorbed radiation P() and, finally, the function Q() was determined by formula (43). Figure 4 gives the calculated dependence Q() for different values of f v /a 3, because this particular ratio is the only parameter allowing for the effect of vapor bubbles on radiation ansfer. Also given for comparison is the Q() curve for f v. The calculations were performed for two temperatures of the thermal radiation source, namely, T e K and T e 3 K. One series of calculations was performed in the DP - approximation (without the second step of solution), and the other series was performed using a combined computational model. t follows from the data in Fig. 4 that the DP -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 DP -approxima , (44) , mm Fig. 4. The effect of vapor bubbles on the absorption of radiation in a layer of water: (a) T e K, (b) T e 3 K; DP -approximation, combined computational model; () f v /a 3 () f v /a 3 m, (3) m, (4) m. tion somewhat decreases with increasing conibution 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 f v /a 3 m. Figure 5 illusates the effect of relatively low values of the parameter f v /a 3 on the thickness of the water layer in which the bulk of the power of thermal radiation is absorbed at T e 3 K. The values of given in Fig. 5 were determined using the equalities Q( ).7 and Q( ).8. n order to understand how real the preassigned values of f v /a 3 are, one can turn to the experimental data of [, ], according to which the value of f v /a 3 in the majority of cases varies from 5 to. n this range, the eated 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 K), note the presence of positive feedback: an increase in the concenation and sie of bubbles leads to an ever songer absorption of HGH TEMPERATURE Vol. 4 No. 4

7 THE PROPAGATON OF NFRARED RADATON 39, mm CONCLUSONS A theoretical model is suggested of for propagation of infrared radiation in a semiansparent liquid containing gas bubbles, which includes an approximate description of the specal radiation characteristics and radiation ansfer 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 demonsated 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 ansfer in a liquid containing gas bubbles; in this model, the DP -approximation is used in the first step of the solution, and then the equation of radiation ansfer is integrated. t is demonsated 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. t has been demonsated 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 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 ƒ v /a 3, m Fig. 5. The effect of vapor bubbles on the thickness of the water layer in which () 7% and () 8% of the power of thermal radiation is absorbed at T e 3 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. REFERENCES. Beshta, S.V., Vitol, S.A., Krushinov, E.V., et al., Teploenergetika, 998, no., p... Dombrovskii, L.A., Teplofi. Vys. Temp., 3, vol. 4, no. 6, p. 9 (High Temp. (Engl. ansl.), vol. 4, no. 6). 3. Reiss, H., High Temp. High Pressures, 99, vol., no. 5, p Baillis, D. and Sacadura, J.-F., J. Quant. Specosc. Radiat. Transfer,, vol. 67, p Fedorov, A.V. and Pilon, L., J. Non Cryst. Solids,, vol. 3, p Tien, C.L. and Drolen, B.L., Annu. Rev. Numerical Fluid Mech. Heat Transfer, 987, vol., p.. 7. Tien, C.L., ASME J. Heat Transfer, 988, vol., no. 4, p Fedorov, A.V. and Viskanta, R., Phys. Chem. Glasses,, vol. 4, no. 3, p Fedorov, A.V. and Viskanta, R., J. Am. Ceram. Soc.,, vol. 83, no., p Pilon, L. and Viskanta, R., Apparent Radiation Characteristics of Semiansparent Media Containing Gas Bubbles, Proc. th nt. Heat Transfer Conf., Grenoble,, vol., p Hulst, van de, H.C., Light Scattering by Small Particles, New York: Wiley, Bohren, C.F. and Huffman, D.R., Absorption and Scattering of Light by Small Particles, New York: Wiley, Dombrovsky, L.A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, Dombrovskii, L.A., Teploenergetika, 996, no. 3, p Mundy, W.C., Roux, J.A., and Smith, A.M., J. Opt. Soc. Am., 974, vol. 64, no., p Dombrovskii, L.A., Teplofi. Vys. Temp., 999, vol. 37, no., p. 84 (High Temp. (Engl. ansl.), vol. 37, no., p. 6). 7. Dombrovsky, L.A., nt. J. Heat Mass Transfer,, vol. 43, no. 9, p Siegel, R. and Howell, J.R., Thermal Radiation Heat Transfer, Washington: Hemisphere, Oisik, M.N., Radiative Transfer and nteractions with Conduction and Convection, New York: Wiley, Hale, G.M. and Querry, M.P., Appl. Opt., 973, vol., no. 3, p Hibiki, T. and shii, M., nt. J. Heat Mass Transfer,, vol. 45, no., p Hibiki, T., Situ, R., Mi, Y., and shii, M., nt. J. Heat Mass Transfer, 3, vol. 46, no. 8, p. 49. HGH TEMPERATURE Vol. 4 No. 4