Proceedings of CHT-12 ICHMT International Symposium on Advances in Computational Heat Transfer July 1-6, 2012, Bath, England CHT-12-LAD
THE USE OF TRANSPORT APPROXIMATION AND DIFFUSION-BASED MODELS IN RADIATIVE TRANSFER CALCULATIONS Leonid A. Dombrovsky Joint Institute for High Temperatures, NCHMT, Krasnokazarmennaya 17A, 111116, Moscow, Russia
E-mail:
[email protected]
ABSTRACT The paper presents a discussion of the use of both transport approximation for scattering phase function and diffusion-based models for radiative transfer in absorbing and anisotropically scattering media like many disperse systems in nature and engineering. The main attention is paid to heat transfer problems and traditional methods of identification of spectral radiative properties of dispersed materials when the details of angular distribution of the radiation intensity are not so important. The latter makes reasonable the use of the above mentioned approximations. In more complex applied problems, the diffusion approximation appears to be a good approach at the first step of a combined two-step solution. Some example problems solved recently by the author and his colleagues are used to illustrate the approach considered in the paper. NOMENCLATURE
a B c d D F g G I J k L n
particle radius Planck’s function heat capacity sample thickness radiation diffusion coefficient radiation source function function introduced by Eq. (19) radiation energy density radiation intensity diffusion component of the radiation intensity thermal conductivity latent heat of melting index of refraction
n q P r r R T u y, z
Greek symbols absorption coefficient extinction coefficient coefficient in boundary conditions emittance eigenvalue
0
dimensionless temperature, polar angle direction cosine
Φ
unit vector of external normal radiation flux divergence of integral radiation flux radial coordinate radius-vector reflectance temperature, transmittance velocity coordinates
asymmetry factor of scattering cosine of scattering angle reflectivity, density scattering coefficient optical thickness functions introduced by Eqs. (34) and (36) scattering phase function function introduced by Eq. (34)
Ψ
function introduced by Eq. (24) scattering albedo
Subscripts and superscripts absorption a critical, color c d - h directional-hemispherical external e gas g
m n r R tr w λ
melting normal radiative Rosseland transport wall spectral peculiar value
INTRODUCTION It is well known that a general mathematical formulation of the radiation transfer problem for a continuous absorbing and scattering medium is based on the integrodifferential radiative transfer equation (RTE) for the radiation intensity which is not only a function of three spatial coordinates but depends also on two angles. As a result, the exact numerical solution to multi-dimensional RTE is an extremely complicated problem even in the case of simple boundary conditions. The problem would be most easily solved if scattering is not taken into consideration and hence the integral term is equal to zero. Unfortunately, this assumption is unacceptable for many applications since scattering of radiation by particles, fibers, cracks, or pores is one of the main features of radiative transfer in disperse systems. The progress in numerical methods and modern computers is definitely insufficient to overcome the difficulties accompanied a formal approach based on direct solving the complete mathematical problems. Therefore, there is a remaining need in simple but reliable methods which are applicable to obtain approximate solutions to many practical problems, especially in typical situations in nature and engineering when radiation is only one of the interacting heat transfer modes. Fortunately, a specific of heat transfer problems (in contrast to some types of optical diagnostics and remote sensing) makes it possible to use relatively simple approximations because of predominant multiple scattering of radiation and not so important details of angular dependence of the resulting radiation intensity. This enables one to consider simplifications based on some assumptions on both scattering phase function and angular dependence of the radiation intensity. A simple presentation of the angular dependences of radiation intensity is a usual way to the so-called differential approximations or combined diffusion-based models considered in this paper. The topic on simplified radiation transfer models or various differential approximations is too wide for one paper. Therefore, this paper is focused mainly on the transport approximation and diffusionbased models. Moreover, the material presented summarizes only particular results of several studies reported by the author and his colleagues. Few selected theoretical models and computational data are reproduced below. Some more details can be found in recently published papers specified in the list of references. Of course, the resulting “picture” is not complete. Nevertheless, the author believes that this short overview will be useful for young researchers and engineers who are not so experienced in solving the problems under consideration.
RADIATIVE TRANSFER AND TRANSPORT APPROXIMATION In many studies, the authors employ a continuum approach to model the radiative transfer in a complex medium containing an absorbing host mediums and absorbing and scattering particles. The so-called radiative transfer equation (RTE) is considered in the traditional continuum theory. In the case of a one-temperature medium, the RTE can be written as follows [Modest 2003, Howell et al. 2010, Dombrovsky and Baillis 2010]:
I λ r , λ I λ r , λ 4
I r , Φ d B T r
λ
λ
λ
λ
(1)
4
The physical meaning of Eq. (1) is evident: variation of the spectral radiation intensity in direction takes place due to self-emission of thermal radiation (the last term), extinction by absorption and also by scattering in other directions, as well as due to scattering from other directions (the integral term). The absorption coefficient, λ , the scattering coefficient, λ , and scattering phase function, Φλ , depend on the coordinate r . For simplicity, Eq. (1) is written for the case of an isotropic medium when the coefficients of RTE do not depend on direction. Generally speaking, one can consider the most general form of the RTE, the so-called vector RTE (VRTE), which fully accounts for the polarization nature of light and is applicable to scattering media composed of arbitrary shaped and arbitrary oriented particles. The VRTE is formulated for the Stokes column vector (instead of the radiation intensity), and the scattering matrix is used instead of the scattering phase function in the VRTE integral term [Mishchenko et al. 2006]. The polarization effects are really important in remote sensing and in specific problems of microwave radiation. Fortunately, one can ignore polarization of the infrared radiation in most of applied problems for dispersed materials. Moreover, the details of scattering phase function are not important when hemispherical characteristics of the radiation field are considered. In this case, it appears to be sufficient to know the so-called asymmetry factor of scattering, , which is independent of polarization [Dombrovsky and Baillis 2010]. A complete and accurate solution to the RTE in scattering media is a very complicated task. One can find a lot of studies in the literature on specific numerical methods developed to obtain more and more accurate spatial and angular characteristics of the radiation intensity field. Several modifications of the discrete ordinates method (DOM) and statistical Monte Carlo (MC) methods are the most popular tools employed by many authors (see monographs [Modest 2003, Howell et al. 2010, Dombrovsky and Baillis 2010]). Fortunately, the well-known transport approximation appears to be highly successful method to solve many applied problems [Davison 1957, Dombrovsky 1996a,b, Dombrovsky and Baillis 2010]. According to this approximation, the scattering function is replaced by a sum of the isotropic component and the term describing the peak of forward scattering:
Φλ 0 1 λ 2λ 1 0
(2)
With the use of transport approximation, the RTE can be written in the same way as that for isotropic scattering, i.e., with λ 1 : (3) Gλ r I λ r , d I λ r , λtr I λ r , Gλ r λ Bλ T
4
where the “transport” scattering and extinction coefficients are defined as follows:
λtr λ 1 λ
λtr λ λtr λ λ λ
(4)
The transport approximation is widely used in neutron transport and radiative transfer calculations during many years. The quality of this approach has been analyzed in early papers by Pomraning [1965], Bell et al. [1967], Potter [1970], and Crosbie and Davidson [1985]. In the case of strong forward and backward scattering, one can introduce an additional delta-function in the backward direction [Sjöstrand 2001]. According to Williams [1966] this scattering function was used already by Fermi.
SIMPLE DIFFERENTIAL APPROXIMATIONS The complete mathematical formulation of radiative transfer problems is very complicated even for the simplest approximation of scattering function. The main difficulty is an angular dependence of the radiation intensity. At the same time, this angular dependence appears to be rather simple in many important applied problems. It enables one to use this property of solution to derive simple but fairly accurate differential approximations. The differential approximations have a long history. This is reflected in their well-known names: the Eddington method, the Schwarzschild–Schuster method, etc. The progress in computer engineering and numerical methods for boundary-value problems makes it possible to obtain more accurate solutions. Nevertheless, simple and physically clear differential approximations are widely used at present for solving the radiative transfer problems in scattering media, particularly in combined heat transfer problems [Özişik 1973, Viskanta 1982, 2005, Rubtsov 1984, Dombrovsky 1996a; Dombrovsky and Baillis 2010]. All the differential approximations for RTE are based on simple assumptions concerning the angular dependence of the spectral radiation intensity I . These assumptions enable us to deal with a limited number of functions I i r instead of function I r , and turn to the system of the ordinary differential equations by the use of integration of RTE. The same result can be obtained if the integral term in Eq. (1) is expressed in the form of the Gaussian quadrature [Case and Zweifel 1967]. Particularly, through the use of the DOM one can derive the same system of the ordinary differential equations as those by expansion of I on the spherical functions. Differential approximations are suitable for calculation of radiative transfer at arbitrary optical depth, but their possibilities in account for real scattering functions are very limited. For example, the mathematical formulations in the first approximation of the spherical harmonics method (P1) for linear and transport scattering functions are identical.
The simplest differential approximations, brought together in recent book by Dombrovsky and Baillis [2010], as in early monographs [Adrianov 1972, Dombrovsky 1996a], by the general term “diffusion approximation”, give the following representation of the spectral radiation flux: (5) qλ Dλ Gλ and differ only by the expression for the radiation diffusion coefficient Dλ . Sometimes, the term “diffusion approximation” is related only to the case when Dλ 1 3 λtr [Zeldovich and Raizer 1966], that corresponds to the Eddington approximation. It is known that Eq. (5) can be also derived based on some assumptions concerning the angular dependence of radiation intensity. Substituting Eq. (5) into the radiation energy balance qλ λ 4Bλ T Gλ r (6)
we obtain the nonhomogeneous modified Helmholtz equation for the spectral radiation energy density:
DλGλ λGλ λ 4Bλ T
(7)
For an internal region of the optically thick volume, one can use the equilibrium radiation intensity in Eq. (5) and the radiation diffusion coefficient from the Eddington approximation:
B T qλ 4Dλ λ T T Integration of Eq. (8) over the spectrum yields
Dλ
1 3 λtr
(8)
q kr T kr
16 T 3 3 trR
trR
(9a)
4T 3 1 B T λtr λT d
(9b)
Here k r is the so-called radiative conductivity and trR is the Rosseland mean transport extinction coefficient. Equations (9) are called the radiative conduction approximation or the Rosseland approximation. This approximate model is sometimes called the Rosseland diffusion approximation, but one should avoid confusion between this model and the above defined diffusion approximation (5). Obviously, the radiative conduction (Rosseland) approximation is applicable only inside optically dense media at large optical distances from the boundaries and from the regions with strong variation of temperature and medium properties. It is known that diffusion approximation is a good approach for radiative transfer in optically thick media with small absorption and relatively high scattering. The biological tissues obey these conditions in the near infrared spectral range. Therefore, the diffusion approximation (P1) is employed in many medical applications such as disease diagnostics and medical imaging using optical techniques. One can refer also to recent papers by Vera and Bayazitoglu [2009] and Dombrovsky et al. [2011e, 2012a] where two kinds of the diffusion approximation we employed as components of the combined heat transfer models for laser induced hyperthermia of tumors. Although media such a skin, bone, brain matter, and breast tissue satisfy the conditions of the diffusion approximation applicability, there also exist low-scattering almost clear regions in a human body. The diffusion approximation fails to accurately describe the radiation propagation in these regions [Arridge et al. 2000, Chen et al. 2001]. Moreover, the diffusion theory can give inaccurate description of transition zones between materials with significantly different absorption and scattering properties. The alternative combined techniques to overcome these difficulties have been developed in papers by Arridge et al. [2000], Ripoll et al. [2000], Aydin et al. [2002, 2004], Hayashi et al. [2003], and Koyama et al. [2005]. For instance, Hayashi et al. [2003] suggested a hybrid Monte Carlo – diffusion method. In this method, the time consuming Monte Carlo calculation is employed only for the thin low-scattering layer of cerebrospinal fluid, and hence, the computational time of the hybrid method is dramatically shorter than that of the Monte Carlo method. One can also recall a recent paper by Gorpas et al. [2010] where a coupled radiative transfer equation and diffusion approximation model for solving the forward problem in fluorescence imaging was applied.
COMBINED COMPUTATIONAL MODELS Radiative transfer is often characterized by complex angular dependence of the radiation intensity. In this case, the diffusion approximation can be incapable of accurately predicting the radiation field in real applications, especially for media that are characterized by strong spatial variation of radiative properties and large temperature gradients [Dombrovsky 1997]. At the same time, even for the most complicated cases, the field of the radiation energy density obtained by using the diffusion approximation can be successfully used as an initial guess in multi-step solution methods. Edwards and Bobcôo [1967] and Bobcôo [1967] have proposed an iteration method employing the diffusion approximation as an initial approach for isotropically scattering media. In the case of isotropic scattering, the spectral source function Fλ on the right-hand side of RTE is isotropic:
I λ r , λ I λ r , Fλ r
Fλ r λ Gλ r λ Bλ T 4
(10)
Therefore, it is reasonably first to obtain the spectral radiation energy density Gλ r by the diffusion approximation and then to solve the RTE with the known source function Fλ r . This idea can also be realized in the case of anisotropic scattering when the transport approximation for the scattering (phase) function is employed. Indeed, the RTE in transport approximation is similar to the RTE for isotropic scattering:
I λ r , λtr I λ r , Fλtr r
tr Fλtr r λ Gλ r λ Bλ T 4
(11)
and the diffusion approximation, as before, can be easily applied to the first stage of the iterative solution procedure. This modification of the method was employed by Adzerikho et al. [1979] to model the thermal radiation of finite cylinder. Dombrovsky and Barkova [1986] have used the FEM at the first stage of solution procedure and developed an algorithm applicable for volumes of complex form and substantially nonuniform properties of the medium. The following two-step solution was suggested. First, the variational problem for functional
Gλ2 4 λ Bλ T Gλ dV w Gλ2 2 4Bλ Tw Gλ dS (12) 2 2 V S in the medium volume is solved and the spectral radiation energy density Gλ r in the nodes of finite element mesh is determined. The spectral radiation intensity I λ r , is calculated at the second step of solution by direct integration of the RTE. This procedure can be easily realized in the case when no radiation illuminates the volume surface at s 0 and the formal solution along the ray s is as follows: D 2 G λ
2
λ
s s l I λ s Fλtr l exp λtr l dl λtr l dl dl 0 0 0
(13)
The error of the above described combined method is influenced by the transport approximation. It is also not obvious if the only iteration is sufficient to obtain high-accuracy results. The accuracy of the combined method for some model problems has been analyzed in some details [Dombrovsky and Lipiński 2010, Dombrovsky and Baillis 2010].
ANALYTICAL SOLUTION TO NORMAL EMITTANCE The material of this section is based on computational study reported recently by Dombrovsky et al. [2011b]. Particularly, a novel approximate analytical model for the normal emittance of a refracting and scattering medium layer is presented below. A plane layer of the model participating medium shown schematically in Fig. 1 is considered. The medium is isothermal, absorbing, emitting, scattering, and refracting. The radiative properties of both the disperse phase and host medium are constant across the layer. The absorption index of the host medium is assumed to be very small as typical for semitransparent materials in their semi-transparency ranges. The surroundings are radiatively nonparticipating and there is no external irradiation on the medium layer. The two-step solution method for the RTE was applied by Dombrovsky et al. [2011b] to the model problem shown in Fig. 1. In this method, the transport approximation for the scattering phase function is employed in the first step. For a plane parallel isothermal layer of an absorbing, emitting, refracting and scattering medium, the transport RTE is integrated over the azimuthal angle, leading to [Modest 2003, Howell et al. 2010]:
I I F tr
0 tr tr0
(14)
Figure 1. Plane-parallel layer of an isothermal, absorbing, emitting, scattering, and refracting medium.
where I I λ 2n 2 Bλ T is the dimensionless spectral intensity of radiation and the source function in the right-hand side of Eq. (14) is:
F tr
tr
1
G tr I tr , d
G tr 1 tr
2
(15)
1
Hereafter, the subscript is omitted for brevity. The spectral incident radiation G is proportional to the spectral radiation energy density. In the above relations, tr tr z is the actual transport optical thickness, tr0 tr d 2 is a half of the overall transport optical thickness of the plane layer, d is the geometrical thickness of the plane layer, tr tr tr is the transport albedo of the medium, and cos , where the angle is measured from the external normal direction. Note that Eqs. (14) and (15) are true in the case of an isotropic medium. A more complex formulation should be considered for anisotropic media when radiative properties of a small volume of the medium depend on the angle of the radiation incidence. The boundary conditions for RTE (14) are as follows:
I tr
I tr0 , ' I tr0 ,
0
0
(16)
tr 0
where ' is the directional-hemispherical reflectivity of the boundary given by Fresnel’s relations [Born and Wolf 1999, Modest 2003]. Normal emission is determined by finding I n tr I tr ,1 and I n tr I tr ,1 from:
dI n I n F d tr
I 0 I 0 n
n
0 tr
n n
dI n I n F d tr
(17a)
n 1 n ( 1) n 1
I
I
n
2
'
0 tr
(17b)
Note that the above expression for n is true in the case of a weakly absorbing medium with n . The following formal solution for the normal emittance is then obtained as:
n 1 n I n
0 tr
1 n 2 exp tr0 1 Ctr
tr0
F cosh d tr
0
tr
Ctr n exp 2 tr0
(18)
An analytical relation for the source function is needed to evaluate the integral in Eq. (18). For the problem under consideration, this relation can be found using the modified two-flux approximation developed by Dombrovsky et al. [2005, 2006]. According to this approximation, the dimensionless irradiation is expressed as: G
1 c g 2c 1 tr
c 1 1 n 2
1 tr c
(19)
where the function g is determined by the following boundary-value problem:
g 0 0
2
d 2g 2 g 2 2 d tr2
(20a)
1 c g tr0 2 g tr0
41 tr 1 c 2 1 tr c
(20b)
1 n 2n 2 1 n n 1
(20c)
Note that the use of normal reflectivity n in the expression for coefficient in Eq. (20c) is an additional approximation and it may lead to a considerable error for high values of the refractive index and the scattering albedo of the medium. In a more accurate approach, one can use a value of the reflectivity averaged over the angles, , as it was done by Siegel and Spuckler [1994]:
1 2
3n 1n 1 n2 n2 12 ln n 1 2n3 n2 2n 1 8n4 n4 1 ln n 2 6n 1 n2 13 n 1 n2 1n4 1 n2 1n4 12
(21)
The analytical solution to the above problem is as follows:
g 2 1
cosh tr 1 c c0 s0 2
s0 sinh tr0
c0 cosh tr0
(22)
The resulting relation for the normal emittance, derived on the basis of the modified two-flux approximation and by subsequent analytical integration of the RTE, is as follows:
1 c trΨ 1 n n 1 1 exp 2 tr0 1 Ctr 1 tr c c 1 c s 0 0 2
(23)
where
Ψ
sinh 1 tr0 sinh 1 tr0 1 2 sinh tr0 1 1
(24)
Obviously, these equations reduce to the well known exact formula for a non-scattering medium ( tr 0 ) [Modest 2003, Howell et al. 2010]: 1 n n 1 exp a0 a0 d (25) 1 n exp a0
One can see that spectral dependences of n , tr and tr0 determine the spectral normal emittance of a refracting, scattering, and absorbing medium.
Following [Dombrovsky 2011b] compare the above approximate analytical solution with reference numerical results. In the case of a nonrefracting medium, the boundary condition is significantly
simplified and the total internal reflection of the radiation at the medium boundaries is omitted from analysis. This allows for application of alternative solutions for the spectral irradiation based on the two-flux approximation or the P1 approximation [Dombrovsky and Baillis 2010]. These solutions can be written in the following form: trΨ 0 n 1 (26) 1 exp 2 tr 2 c s 0 0
The function Ψ remains the same as introduced by Eq. (24), but the coefficients and are modified as: 1 2 4 N appr 1 tr (27) 2 1 N appr N mod 4 where N appr 0 and N appr 1 correspond to the two-flux and P1 approximations, respectively.
N mod 0 and N mod 1 correspond to Marshak’s boundary condition and Pomraning’s boundary condition, respectively [Dombrovsky and Baillis 2010]. In the limiting case of an optically thick medium, Eq. (26) is simplified to: 2tr (28) n 1 1 2 A comparison of the normal emittance obtained using Eq. (28) and the reference numerical methods is presented in Table 1 from paper [Dombrovsky et al. 2011b]. Note that DOM and MC results shown in Table 1 are reported for the absorption optical thickness a0 10 . Table 1 The Normal Emittance of an Optically Thick Scattering Medium
tr 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95
n Two-flux (DP0) 0.9823 0.9621 0.9389 0.9116 0.8787 0.8377 0.7842 0.7082 0.5811 0.4635
P1 0.9792 0.9558 0.9290 0.8978 0.8606 0.8150 0.7564 0.6750 0.5435 0.4263
P1,mod 0.9804 0.9582 0.9328 0.9030 0.8674 0.8234 0.7664 0.6868 0.5563 0.4385
DOM 0.9836 0.9647 0.9427 0.9166 0.8847 0.8445 0.7912 0.7147 0.5850 0.4644
MC 0.9837 0.9648 0.9428 0.9167 0.8851 0.8448 0.7914 0.7148 0.5852 0.4645
Additional MC computations were performed for a0 100 , for which the MC results remained practically unchanged with the decimal accuracy from Table 1. Good agreement is observed between the analytical and numerical results. The analytical solution based on the two-flux approximation tends to be more accurate than those based on the P1 and P1,mod approximations. A comparison of the normal emittance obtained using the analytical solution and the reference numerical methods for selected values of the absorption optical thickness a0 21 tr tr0 is presented in Fig. 2. One can see that the analytical results are in excellent agreement with those obtained using the reference numerical methods. Note that presence of scattering can considerably increase the normal emittance for an optically thin medium. The latter trend was observed for the hemispherical emittance in [Dombrovsky 1996a].
1.0
n
4
0.8
3 0.6
2
0.4
0.2
1 0.0 0.0
0.2
0.4
0.6
0.8
tr
1.0
Figure 2. Normal emittance of a layer of scattering nonrefracting medium. Comparison of the derived analytical solution (solid lines) and numerical calculations (circle points – DOM, triangle points –MC): 1 – a0 0.1 , 2 – a0 0.5 , 3 – a0 1 , 4 – a0 1 .
A comparison of the normal emittance obtained using the analytical solution and the reference numerical methods for the refractive index of the host medium of n 2 is presented in Fig. 3.
1.0
n
4
0.8
3
0.6
2 0.4
0.2
0.0 0.0
1
0.2
0.4
0.6
0.8
tr
1.0
Figure 3. Normal emittance of a layer of refracting and scattering medium at n 2 . Comparison of the derived analytical solution (solid lines – with the use of the normal reflection coefficient in Eq. (23) and dashed lines – with the use of averaged reflection coefficient (21)) with exact numerical calculations (circle points – DOM, triangle points – MC) at n 2 : 1 – a0 0.1 , 2 – a0 0.5 , 3 – a0 1 , 4 – a0 1 .
The agreement between the analytical solution and exact numerical results is satisfactory, and it further improves when the average reflection coefficient given by Eq. (21) is used at the first step of the analytical solution. It is important that the effect of scattering on normal emittance is similar to that observed for the above analyzed case of n 1 . The increase of emittance due to scattering has been obtained for absorptance of porous ceramics by Dombrovsky et al. [2007b]. It was noted that a porous sample of a weakly absorbing substance exhibits significant absorption in the case when numerous pores lead to very long path of photons in the sample. One should recall that according to [Dombrovsky et al. 2007b] this effect can be used to determine a very low absorption coefficient of dense materials. In the above analysis, we considered the transport RTE assuming that it is sufficient to obtain satisfactory results for arbitrary scattering phase function of the medium. The error of the transport approximation for the problem under consideration has been also analysed in [Dombrovsky et al. 2011b]. It was shown that scattering phase function has only a minor effect on the normal emittance and the transport approximation is applicable to many practical problems.
DIRECTIONAL-HEMISPHERICAL REFLECTANCE AND TRANSMITTANCE The measurements of directional-hemispherical transmittance and reflectance are widely used in present-day identification procedures for obtaining the information on the main radiative properties of semi-transparent dispersed materials. One can recommend review papers by Baillis and Sacadura [2000], Sacadura [2011], and Dombrovsky et al. [2011c] on this subject. There are many similar problems related with a propagation of diffuse or collimated external radiation through a layer or cloud of particles. One should recall attenuation of solar radiation by water mist and sprays analyzed in recent paper by Dombrovsky et al. [2011a]. Consider the problem of radiation transfer in plane-parallel layer of an absorbing, refracting and scattering medium. We will limit our consideration to the one-dimensional azimuthally symmetric problem when one surface of the layer is uniformly illuminated along the normal direction by randomly polarized radiation. In the case of homogeneous isotropic medium, the transport RTE and the associated boundary conditions can be written as follows:
I I tr Id tr 2 1
I 0, RI 0, 1 R 1
1
(29a)
I tr0 , RI tr0 ,
0 (29b)
where I I λ n 2 I e , I e is the incident spectral radiation intensity, R is the Fresnel’s reflection coefficient. Following the usual technique [Sobolev 1972, Dombrovsky 1996], present the radiation intensity I as a sum of the diffuse component J and the term which corresponds to the transmitted and reflected directional external radiation: I J
1 R1 exp tr 1 Ctr exp tr 1 1 R1Ctr
(30)
where Ctr R1 exp 2 tr0 , R1 R1 . The mathematical problem statement for the diffuse component of radiation intensity is as follows:
1 1 R1 J exp tr Ctr exp tr J tr Jd tr 2 1 1 R1C 0 0 J 0, R J 0, J tr , R J tr , 0
(31a)
(31b)
The directional-hemispherical reflectance and transmittance can be expressed through the diffuse component of the radiation intensity: 1
Rd h R
0 d h
1 R J 0, d
1 R J tr0 , d 1
Td h T
0 d h
0
(32)
0
where the first terms are given by the well-known equations [Modest 2003]: 0 d h
R
R1
2 1 R1 Ctr
0 d h
T
1 R1Ctr
2 1 R1 exp 0
1 R1Ctr
(33)
tr
According to the modified two-flux approximation suggested by Dombrovsky et al. [2005, 2006], the following presentation of the radiation intensity is considered: tr , 1 c J tr , tr , c c , 1 tr c
c 1 1 n 2
12
(34)
This approximation takes into account an important effect of the total internal reflection at n 1 . Note that the case c 0 (at n 1 ) corresponds to the usual two-flux model. The intermediate angle interval c c gives no contribution to the radiation flux and the words “two-flux” are applicable to the modified approximation too. We will not reproduce here further mathematical transformations which can be found in papers by Dombrovsky et al. [2005, 2006] and in the book by Dombrovsky and Baillis [2010]. The limited length of the paper does not allow also a detailed analysis of the resulting solution accuracy. The only simple example for the simplest case of a nonrefracting medium is given in Fig. 4, which illustrates rather high accuracy of the approximate solution. 70
Rd-h , %
Td-h , %
100
1 80
60
5 50
2 40
60
3
4
30
3
40
20
4
20
2 10
1
5 0 0.0
0.2
0.4
0.6
0.8
tr
1.0
0 0.0
0.2
0.4
0.6
0.8
tr
Figure 4. Directional-hemispherical transmittance and reflectance calculated using approximate analytical solution (dashed lines) and the exact numerical solution for the transport scattering function (solid lines): 1 – tr0 0.2 , 2 – 0.5, 3 – 1.0, 4 – 2.0, 5 – 5.0.
1.0
The modified two-flux approximation has been successfully employed in studies of spectral radiative properties of various dispersed materials [Dombrovsky et al. 2005, 2007b, 2010, 2011d, 2012b]. The same method was recently used in a combined transient thermal model for laser hyperthermia of superficial tumors [Dombrovsky et al. 2011e]. It is interesting that there is a very simple relation between the solution for Rd -h and Td -h obtained using the above described method and the two-step analytical solution for normal emittance n presented in the previous section of the present paper. This relation follows from the Kirchhoff law for thermal radiation and looks as follows:
n 1 Rd-h Td h
(35)
It means that the accuracy of this approach is rather high even for n 1 when Eq. (21) is employed for the reflectivity in the two-flux solution instead of the normal reflectivity R1 used in [Dombrovsky et al. 2005, 2006, 2007b].
THERMAL RADIATION FROM SEMI-TRANSPARENT SPHERICAL PARTICLES Thermal radiation of a nonisothermal particle is an especially interesting problem for particles of a semi-transparent material (in particular, for metal oxide particles). The point is that materials, which are semi-transparent in the infrared spectral range, are usually characterized by low thermal conductivity and, as a result, by a comparatively large temperature difference between the center and the surface of the particle. On the other hand, in the case of a small index of absorption, the solution to the problem is expected to be more complex because of possible considerable contribution of radiation emitted from the central core of the particle [Dombrovsky 2007a]. A rigorous statement of the problem must take into account effects of interference as is done in the Mie theory. In the paper, only the case of a large particle (as compared with the wavelength) is considered. The geometrical optics approximation and the RTE can be employed to determine thermal radiation emitted by such particles. Some interesting solutions for particles in the Mie scattering region can be found in papers by Mackowski et al. [1990], Tuntomo et al. [1991], Lai et al. [1991], Choudhury et al. [1992], Velesco et al. [1997], and Dombrovsky [1999a, 2000a]. Modified Differential Approximation for Spherical Particles An analysis of angular dependencies of
the radiation intensity in the range 0 n 0 [Dombrovsky, 1999a, 2000a] showed that the following approximation of a dimensionless radiation intensity may be rather good:
, 1 I , 1, - , 1
1 *
2
0 n
(36)
Where is the current optical thickness measured from the particle center. This approach was called MDP0 (modified DP0), and this is also a modification of the two-flux approximation. Further mathematical transformations and the final formulation of the boundary value problem can be found in papers by Dombrovsky [2000a, 2002] and also in the book by Dombrovsky and Baillis [2010]. The MDP0 approximation is much simpler than RTE. At the same time, the error in MDP0 is not large both for heat generation profiles and thermal radiation from the particle. This is illustrated in Figs. 5 and 6 for isothermal particles at n 2 . Similar estimates have been obtained for nonisothermal particles [Dombrovsky and Baillis 2010].
_
1.0
W 3
1
a b
a
0.8
2
b
0.6
2 1
1a,b 2 3 4
0.4
3 0 0.0
0.2
0.4
0.6
0.8
_
r
1.0
0.2 0.0
0.5
1.0
1.5
2.0
2.5
0
3.0
Figure 5. [Left panel] Heat generation profiles calculated by numerical solution of the RTE (a) and by the MDP0 approximation (b): 1 – 0 0.2 , 2 – 0 2 , 3 – 0 5 . Figure 6. Spectral emissivity of spherical particle: 1 – Mie theory calculations (1a – x 50 , 1b – x 300 ), 2 – RTE solution inside the particle, 3 – MDP0 approximation, 4 – DP0 approximation. Oxide Particles in Plasma Spraying Plasma spraying is widely used for coating deposition and
advanced material forming. To control the plasma spraying process, one should be able to evaluate on-line the key physical process variables, including the bulk temperature of particles [Fauchais and Montavon 2007, Fauchais and Vardelle 2010]. The particle temperature is usually determined experimentally from the ratio of the thermal radiation detected at two closely related wavelengths (two-color pyrometry). As a result, the so-called color temperature is obtained. It the case of a semitransparent particle, the color temperature is intermediate between the surface temperature and the center temperature of the particle, but it may not coincide with the bulk (volume averaged) temperature. The difference between the measured value of color temperature and the bulk temperature may be considerable because of great temperature difference inside the particles of oxides, which are characterized by a very small thermal conductivity. The temperature profiles inside the particles have been analyzed long ago [Fiszdon 1979, Yoshida and Akasi 1997] but the semi-transparency of oxide particles was ignored even in more recent paper by Wan et al. [1999]. The particle semi-transparency in thermal spraying was first taken into account by Dombrovsky and Ignatiev [2001, 2003]. Particle concentration in a plasma spray jet is usually small; therefore, particles do not interact with each other, and their influence on plasma parameters is not large. On the other hand, the optical thickness of the jet is small, and thermal radiation can be easily taken into account in the heat balance for a single particle. The resulting equations for a particle moving along the jet axis are as follows: du 3CD g ug u ug u u0 u0 u (37a) dy 8a T 1 2 T c L T Tm u (37b) r k P y, r y r 2 r r T T Nu 0 k kg Tg y T y, a T 0, r T0 (37c) r r a 2a r r 0
3200
3500
T, K
T, K
a
3000
3000
b
2800
y=6cm
2500
2600
c 2000
1500
1 2 0.00
0.02
0.04
0.06
0.08
y, m
2400
0.10
2200
1 2
y=3cm 0
5
10
15
20
r, m
25
Figure 7. [Left panel] Temperature of aluminum oxide particle in the plasma jet: 1 – without radiation, 2 – with radiation; a – surface, b – average (bulk) temperature, c – center. Figure 8. Temperature profiles in an aluminum oxide particle at two different cross sections of the plasma jet: 1 – without radiation, 2 – with radiation.
The latent heat of melting is treated here as an equivalent increase in specific heat capacity in the narrow temperature interval near the melting temperature ( is the Dirac function). The value of heat generation rate on the right-hand side of the energy equation is determined by MDP0 approximation. We will not give here semi-empirical relations for the drag coefficient CD and the Nusselt number Nu . The corresponding equations as well as approximations of thermal properties of argon plasma and typical profiles of the plasma velocity and temperature along the jet axis can be found in [Dombrovsky and Ignatiev 2001, 2003]. The numerical results for aluminum oxide particles are presented below. The following approximate relation for the absorption coefficient of molten aluminum oxide was used in the calculations:
λ 800 10 7 0.6 exp T 1000 2.95
(38)
where is expressed in microns and T is in Kelvin. The calculated temperature for particle of radius a 25 μm is shown in Figs. 7 and 8. One can see the great temperature difference inside the particle during melting. Note that radiation effect on the particle temperature is insignificant. It is much less than that predicted by opaque particle model. Consider the relationship between the calculated color temperature Tc and the bulk temperature T of aluminum oxide particle. The latter quantity is defined as a
T 3 T r r 2 dr a 3
(39)
0
The calculated values of Tc and T for a particle of radius a 25 μm moving along the plasma jet axis are shown in Fig. 9. It is interesting that variation of particle color temperature along the jet axis is not monotonic. This result is explained by a strong increase in absorption coefficient of aluminum oxide with temperature. The considerable difference between measured color temperature and bulk temperature of particles in plasma spraying should be taken into account in optical diagnostics of plasma spraying process. Some suggestions on estimates of bulk temperature from experimental data were reported by Dombrovsky and Ignatiev [2003].
2900
T,K 2800
2700
1 2
2600
2500 0.00
0.02
0.04
0.06
0.08
y, m 0.10
Figure 9. Comparison of color temperature Tc (1) and bulk temperature T (2) of an aluminum oxide particle in the plasma jet.
Cooling and Solidification of Core Melt Droplets The problem of melt–water interaction is especially
important for analysis of hypothetical severe accidents in light-water nuclear reactors. A severe accident involves melting of the core and possible subsequent interaction of the core melt (UO2ZrO2 composition) with water. The fuel–coolant interaction (FCI) may lead to steam explosion with a significant part of the melt thermal energy converted into the mechanical energy of the detonation wave [Berthoud 2000, Magallon 2009]. Physical processes that govern steam explosion energetics are multifaceted and complex [Fletcher and Theofanous 1997, Dinh et al. 1999b]. The probability of steam explosion is determined by the so-called premixing stage of FCI when the melt jets are fragmented to numerous small droplets. The fragmentation of the melt jets and large droplets is considered as very important process. Many experiments with core melt jets have failed to produce strong steam explosions. On the contrary, spontaneous explosions have been observed when Al2O3 melt jets are employed. It appears that explosivity of the core melt is extremely low. One can assume that this result is explained by different conditions of droplet fine fragmentation because of different character of cooling and solidification of core melt and alumina particles [Uršič et al. 2011]. Simple estimates show that thermal radiation is the main mode of heat transfer from a single particle to the ambient water [Dinh et al. 1999a, Fletcher 1999, Dombrovsky 1999b, 2000b]. For this reason it is interesting to focus on radiative cooling of these particles. The main part of the radiation emitted by particles is absorbed in the ambient water, at least in the case of not too high volume fraction of particles [Dombrovsky 2003]. It allows us to assume that radiation heat transfer between the particles is insignificant compared to local heat transfer to surrounding water and consider a model problem for single particles. There is no free space in this paper to present a mathematical statement of the radiative-conductive problem and discuss some interesting results on effects of thermal radiation on transient temperature field in single oxide particles. It is important that computational analysis was based on MDP0 approximation for radiative transfer both in alumina and core melt particles. It should be only noted that a very interesting effect of predominant internal cooling of semi-transparent alumina particles has been obtained at the first time. This effect is illustrated in Fig. 10, where typical dimensionless profiles in single particles of alumina and corium are presented. One can see quite different cooling and solidification dynamics of these particles (the value 1 corresponds to a conventional solidification temperature).
1.02
1.05
1.00
1.00
0.95
0.98 0.90
t
0.96
0.2s 0.4 0.6
0.94 0.0
0.2
0.4
0.6
0.8
t
0.85
0.2s 0.4 0.6 0.8
0.80
_
r
1.0
0.75 0.0
0.2
0.4
0.6
0.8
_
r
1.0
Figure 10. Temperature profiles in an aluminum oxide particle of radius a 1 mm (left panel) and in a corium particle of the same radius (right panel).
A reader interested in radiative cooling and solidification of oxide particles and radiative transfer in a multiphase system containing core melt particles can be addressed to recent papers by the author and his colleagues [Dombrovsky 2007a,b, 2009, 2010, Dombrovsky and Dinh 2008, Dombrovsky et al. 2009] Note that a combined method including approximate calculation of a two-dimensional field of the radiation energy density and subsequent ray-tracing procedure to solve transport RTE was implemented in a regular computer code for multiphase flow calculations and makes possible to predict thermal radiation from the zone of intense FCI interaction in the case of severe accident of nuclear reactors [Dombrovsky and Davydov 2010]. It should be noted that MDP0 approximation can be employed to determine transient temperature and thermal stress profiles in semi-transparent particles under high-flux irradiation typical for some thermochemical reactors which use a concentrated solar radiation. A solution to this particular problem has been derived by Dombrovsky and Lipiński [2007] and can be also found in the book by Dombrovsky and Baillis [2010].
NUMERICAL SOLUTION TO MULTI-DIMENSIONAL PROBLEMS Following paper [Dombrovsky et al. 1991], consider first two model problems involving radiative transfer in a cylindrical volume of a homogeneous and isothermal absorbing and scattering medium as described below. The choice of the model problems is motivated by engineering applications concerned with both near-field thermal effects and remote sensing of thermal radiation of two-phase jets containing highly scattering micron-size particles of alumina [Laredo and Netzer 1993, Surzhikov 2004]. In contrast to the real jets, the model problems assume uniform radiative properties. However, the selected values of the radial optical thickness tr 5 , the scattering albedo tr 0.95 , and the asymmetry factor of scattering 0.5 are close to those encountered in the real problems. The length-to-radius ratio of the cylindrical volume is assumed to be equal to 4. The results presented below are obtained by using the collision-based MC method [Farmer and Howell 1998] in the second step of the combined method as described by Dombrovsky and Lipiński [2010]. For each problem, the distribution of the dimensionless radiative flux leaving the cylindrical volume is computed as
q 2q
1
q d
1
(40)
where q is the radiative flux intercepted by a large black and cold sphere of radius much greater than the cylinder length and the center coincident with the center of the cylindrical volume, = cos, where is the polar angle measured from the axis of the cylindrical volume. The following methods are employed: (1) the combined method incorporating the P1 approximation, FEM and the transport approximation in the first step to obtain the radiation source function, and MC to compute the radiative flux in the second step; (2) the Monte Carlo method (referred further to as “transport MC”) incorporating the transport approximation, and (3) the reference MC method providing the complete radiative transfer solution in a single step and using the Henyey–Greenstein scattering phase function.
Problem 1 The own thermal radiation of the medium is considered and no radiation sources at boundaries are present. The numerical results for this symmetric problem are presented in Fig. 11. One can see that the combined method agrees very well with both the transport MC method and the reference method. It is interesting that the variation of the radiation flux with the asymmetry factor of scattering is negligible. Beside the small error of the transport approximation, the combined method is expected to be acceptable for multi-dimensional radiative transfer problems with predominant contribution to the source function by emission, especially those problems where hemispherical characteristics are of interest. The computational time reduction for the transport MC method is 46% and 94% for 0.5 and 0.95, respectively, as compared to the reference method [Dombrovsky and Lipiński 2010]. The combined method is even faster, by more than 1 and 2 orders of magnitude for 0.5 and 0.95, respectively, than the reference method. The computational time savings for both methods become more pronounced with an increasing fraction of forward scattered radiation. This is explained by the fact that the computational time required by the reference method increases with the increasing value of the asymmetry factor of scattering.
_
1.2
q
q
1.5
_
1.0
1.0 0.8
1 2 3 4
0.6
0.4 -1.0
-0.5
0.0
0.5
1 2 3 4
0.5
1.0
0.0 -1.0
-0.5
0.0
0.5
Figure 11. Angular variation of the dimensionless radiation flux for problem 1 (left panel) and problem 2 (right panel): 1 – combined method; 2 – transport MC, 3 – reference MC at 0.5 , 4 – reference MC at 0.95 .
1.0
Problem 2 The cylindrical medium is non-emitting (cold) but the bottom wall of the cylinder (the wall observed from the cylinder center at 1 ) is diffusely emitting and perfectly absorbing. The results of calculations for this problem are also presented in Fig. 11. One can see that the error of the combined method is significant. These results were anticipated based on the early analysis by Dombrovsky et al. [1991], revealing the large error of the two-step combined method for the searchlight effect. A more detailed numerical study by Dombrovsky and Lipiński [2010] showed that the relative error of the combined method with respect to the transport MC method decreases with the optical thickness, especially for higher values of the scattering albedo (at least in the range of tr 0.5 ). This result is explained by a smoother angular distribution of the radiative intensity in the case of predominant scattering and higher optical thickness of the medium. The behavior of the combined method can be explained as suggested by Dombrovsky [1996a]: The “correction” of the strong (or even step-wise) angular dependence of radiative intensity in the P1 approximation was treated as an additional fictitious scattering or “scattering by approximation”. As a result, the P1 based results are similar to those obtained for media with more intense scattering. In our case, one can see that the two-step combined method overestimates the scattering in the backward hemisphere and in the directions close to the normal, whereas the remaining radiation into the directions close to the forward hemisphere appears to be considerably underestimated. The variation of the radiation flux with the asymmetry factor of scattering in Problem 2 is more pronounced as compared to that in problem 1. Nevertheless, the error of the transport MC method is not large. The directional flux distribution obtained by the transport approximation is physically correct, i.e. the location of the flux maximum, and the flux trends are predicted correctly. In contrast, the results of the combined method are far from the reference solution. Note that the engineering estimates at the realistic values of the problem parameters for rocket boosters can still be based on the combined two-step method [Dombrovsky 1996a,b]. In many engineering applications, only the hemispherical characteristics are of interest. A comparison of the computational results shows that both the combined and transport MC methods can provide satisfactory results. It is important that both the transport MC and combined methods enable for large reduction in the computational time also for problem 2. Interestingly, the computational time is practically the same as that for problem 1 and the effect of the asymmetry factor of scattering on the computational time of the reference method is very similar for both problems. The analysis of the results obtained for problems 1 and 2 let us conclude that the combined method can be used for radiative transfer problems in scattering media with medium emission being the dominant source of radiation as compared to the contribution by wall emission and by incident external fluxes. In the case of high optical thickness, the combined method can be employed even for more complicated problems including those characterized by a considerable contribution of the searchlight effect to the total radiation of the medium. The combined method becomes more appropriate when hemispherical fluxes (or equivalent directional-hemispherical transmittance and reflectance) are of interest. Examples include inverse radiative transfer analysis for hemispherical flux measurements with isothermal emitting samples. In such studies, properties of the samples are varied in a numerical simulation while repeating the simulations until good agreement between the experimental and numerical data is obtained. The reduction of the computational time in a single simulation will lead to significant reduction of the overall computational time of the inverse algorithm. Consider now propagation of a collimated external radiation. It was shown in previous section of the paper that diffusion approximation (DP0 or P1) can be employed to determine hemispherical reflectance and transmittance of a scattering medium even in the case of collimated incident
radiation. The analysis presented was concerned with one-dimensional problems typical of laboratory studies of the radiative properties of thin samples (see book by Dombrovsky and Baillis [2010] for further details). In some other applied problems, it is also important to consider a twodimensional axisymmetric problem when the diameter of the normally incident beam is comparable with the thickness of the medium layer. As earlier, the spectral radiation intensity should be presented as a sum of the diffuse component J and the term that corresponds to the nonscattered external radiation:
I λ r, z, J λ r , z, I λe r exp ztr nz
z
λtr dz tr z
(41)
0
where I e is the spectral radiation intensity of the incident beam. In transport approximation, the resulting RTE has the following form: tr J λ λtr J λ λ Gλ r , z I λe r exp ztr 4
(42)
The presentation of the radiation intensity in the form (41) enables one to consider an equivalent problem for the diffuse component of the radiation field. The latter problem can be solved by the diffusion-based approximate methods. The applicability of the diffusion approximation to the problems of this type was studied by Kolpakov et al. [1990] and by Dombrovsky et al. [1991]. The exact solution obtained by Nelson and Satish [1987] for a Gaussian beam
I λe r ~ exp r r0
2
(43)
illuminating a plane-parallel layer of a nonabsorbing, isotropically scattering medium was used to estimate the diffusion approximation error. In a more general case, the Monte Carlo numerical solution by using the computer code developed by Surzhikov [1987] was used to obtain the benchmark solution. We do not reproduce here the numerical data of papers [Kolpakov et al. 1990, Dombrovsky et al. 1991]. The main conclusions obtained in these papers are summarized as follows: (1) The diffusion approximation gives fairly good results at average and large optical distances from the conventional boundary of the beam (in the region of essential multiple scattering contribution). The maximum error in the total radiation intensity computed by using the diffusion approximation is observed in the vicinity of the beam boundary for the optically thin medium layer. (2) The radial profile of the radiation flux at the bottom of the medium layer is accurately predicted by the diffusion approximation when the layer optical thickness is large. This result is independent of the incident beam diameter. (3) The calculations for various scattering functions at typical parameters of the erosional plume (the characteristic optical thickness is about unity) showed that transport approximation is fairly accurate for both transmitted and reflected radiation.
CONCLUSIONS A systematic presentation of significant simplifications of the radiative transfer problem in absorbing and anisotropically scattering media based on the use of both transport approximation for scattering phase function and diffusion-based models for radiative transfer was presented in the paper. Several particular problems discussed in the paper showed that some modifications of the diffusion approximation and a combination of the simplified approach with ray-tracing procedures or Monte Carlo calculations appeared to be very effective tool to solve many problems of presentday interest, including combined heat transfer problems in multiphase disperse systems and identification of spectral radiative properties of advanced materials at room and elevated temperatures. The examples taken from recent publications of the author and his colleagues do not
exhaust a set of diverse engineering problems characterized by considerably role of radiation. To the author’s mind, similar combined methods are expected to be fruitful in solving of many other applied problems of different nature.
ACKNOWLEDGEMENTS The author is grateful to the Russian Foundation for Basic Research for the financial support of this work (grant no. 10-08-00218a). The author is also grateful to Wojciech Lipiński and Jaona Harifidy Randrianalisoa for their cooperation and contribution to particular computational studies.
REFERENCES Adrianov, V.N. [1972], Fundamentals of Radiative and Combined Heat Transfer, Energiya, Moscow (in Russian). Adzerikho, K.S., Antsulevich, V.I., Lapko, Ya.K., and Nekrasov, V.P. [1979], Luminescence of Two-Phase Inhomogeneous Media of Cylindrical Geometry, Int. J. Heat Mass Transfer, Vol. 22, No. 1, pp 131-136. Arridge, S.R., Dehghani, H., Schweiger, M., and Okada, E. [2000], The Finite Element Model for the Propagation of Light in Scattering Media: A Direct Method for Domains with Nonscattering Regions, Med. Phys., Vol. 27, No. 1, pp 252-264. Aydin, E.D., de Oliveira, C.R.E., and Goddard, A.J.H. [2002], A Comparison Between Transport and Diffusion Calculations Using a Finite Element – Spherical Harmonics Radiation Transport Method, Med. Phys., Vol. 29, No. 9, pp 2013-2023. Aydin, E.D., de Oliveira, C.R.E., and Goddard, A.J.H. [2004], A Finite Element-Spherical Harmonics Radiation Transport Model for Photon Migration in Turbid Media, J. Quant. Spectr. Radiat. Transfer, Vol. 84, No. 3, pp 247-260. Baillis, D. and Sacadura, J.-F. [2000], Thermal Radiation Properties of Dispersed Media: Theoretical Prediction and Experimental Characterization, J. Quant. Spectr. Radiat. Transfer, Vol. 67, No. 5, pp 327363. Bell, G.I., Hansen, G.E., and Sandmeier, H.A. [1967], Multitable Treatment of Anisotropic Scattering in SN Multigroup Transport Calculations, Nucl. Sci. Eng., Vol. 28, pp 376-383. Berthoud, G. [2000], Vapor Explosions, Ann. Rev. Fluid Mech., Vol. 32, pp 573-611. Bobcôo, R.P. [1967], Directional Emissivities from a Two-Dimensional, Absorbing-Scattering Medium: The Semi-Infinite Slab, ASME J. Heat Transfer, Vol. 89, No. 4, pp 313-320. Born, M. and Wolf, E. [1999], Principles of Optics, Seventh (expanded) edition, Cambridge Univ. Press, New York. Case, K.M. and Zweifel, P.F. [1967], Linear Transport Theory, Addison-Wesley, Reading, MA. Chen, B., Stamnes, K., and Stamnes, J.J. [2001], Validity of the Diffusion Approximation in Bio-Optical Imaging, Appl. Optics, Vol. 40, No. 34, pp 6356-6366. Crosbie, A.L. and Davidson, G.W. [1985], Dirac-Delta Function Approximation to the Scattering Phase Function, J. Quant. Spectr. Radiat. Transfer, Vol. 33, No. 4, pp 391-409. Choudhury, D.Q., Barber, P.W., and Hill, S.C. [1992], Energy-Density Distribution Inside Large Nonabsorbing Spheres by Using Mie Theory and Geometric Optics, Appl. Optics, Vol. 31, No. 18, pp 35183523. Davison, B. [1957], Neutron Transport Theory, Oxford Univ. Press, London. Dinh, T.N., Dinh, A.T., Nourgaliev, R.R., and Sehgal, B.R. [1999a], Investigation of Film Boiling Thermal Hydraulics under FCI Conditions: Results of Analyses and Numerical Study, Nucl. Eng. Design, Vol. 189, No. 1-3, pp 251-272.
Dinh, T.N., Bui, V.A., Nourgaliev, R.R., Green, J.A., and Sehgal, B.R. [1999b], Experimental and Analytical Studies of Melt Jet – Coolant Interactions: A Synthesis, Nucl. Eng. Design, Vol. 189, No. 1-3, pp 299-327. Dombrovsky, L.A. [1996a], Radiation Heat Transfer in Disperse Systems, Begell House, New York. Dombrovsky, L.A. [1996b], Approximate Methods for Calculating Radiation Heat Transfer in Dispersed Systems, Therm. Eng., Vol. 43, No. 3, pp 235-243. Dombrovsky, L.A. [1997], Evaluation of the Error of the P1 Approximation in Calculations of Thermal Radiation Transfer in Optically Inhomogeneous Media, High Temp., Vol. 35, No. 4, pp 676-679. Dombrovsky, L.A. [1999a], Thermal Radiation of a Spherical Particle of Semitransparent Material, High Temp., Vol. 37, No. 2, pp. 260-269. Dombrovsky, L.A. [1999b], Radiation Heat Transfer from a Spherical Particle via Vapor Shell to the Surrounding Liquid, High Temp. 37 (6) (1999) 912-919. Dombrovsky, L.A. [2000a], Thermal Radiation from Nonisothermal Spherical Particles of a Semitransparent Material, Int. J. Heat Mass Transfer, Vol. 43, No. 9, pp 1661-1672. Dombrovsky, L.A. [2000b], Radiation Heat Transfer from a Hot Particle to Ambient Water through the Vapor Layer, Int. J. Heat Mass Transfer, Vol. 43, No. 13, pp 2405-2414. Dombrovsky, L.A. [2002], A Modified Differential Approximation for Thermal Radiation of Semitransparent Nonisothermal Particles: Application to Optical Diagnostics of Plasma Spraying, J. Quant. Spectr. Radiat. Transfer, Vol. 73, No. 2-5, pp 433-441. Dombrovsky, L.A. [2003], Radiation Transfer Through a Vapour Gap Under Conditions of Film Boiling of Liquid, High Temp., Vol. 41, No. 6, pp 819-824. Dombrovsky, L.A., 2007a. Thermal Radiation of Nonisothermal Particles in Combined Heat Transfer Problems, In: P.M. Mengüç and N. Selçuk, eds., “Fifth Int. Symp. Radiat. Transfer”, 17-22 June 2007, Bodrum, Turkey, (dedication lecture). Dombrovsky, L.A. [2007b], Large-Cell Model of Radiation Heat Transfer in Multiphase Flows Typical for Fuel–Coolant Interaction, Int. J. Heat Mass Transfer, Vol. 50, No. 17-18, pp 3401-3410. Dombrovsky, L.A. [2009] Thermal Radiation Modeling in Multiphase Flows of Melt-Coolant Interaction, Chapter 4 in the book “Advances in Multiphase Flow and Heat Transfer”, v. 1, ed. by L. Cheng and D. Mewes, Bentham Sci. Publ., pp 114-157. Dombrovsky, L.A. [2010], An Extension of the Large-Cell Radiation Model for the Case of SemiTransparent Nonisothermal Particles, ASME J. of Heat Transfer, Vol. 132, No. 2, paper 023502. Dombrovsky, L.A. and Barkova, L.G. [1986], Solving the Two-Dimensional Problem of Thermal-Radiation Transfer in an Anisotropically Scattering Medium Using the Finite Element Method, High Temp., Vol. 24, No. 4, pp 585-592. Dombrovsky, L.A. and Ignatiev, M.B. [2001], Inclusion of Nonisothermality of Particles in the Calculations and Diagnostics of Two-Phase Jets Used for Spray Deposition of Coatings, High Temp., Vol. 39, No. 1, pp. 134-141. Dombrovsky, L.A. and Ignatiev, M.B. [2003], An Estimate of the Temperature of Semitransparent Oxide Particles in Thermal Spraying, Heat Transfer Eng., Vol. 24, No. 2, pp 60-68. Dombrovsky, L.A. and Lipiński, W. [2007], Transient Temperature and Thermal Stress Profiles in SemiTransparent Particles under High-Flux Irradiation, Int. J. Heat Mass Transfer, Vol. 50, No. 11-12, pp 21172123. Dombrovsky, L.A. and Dinh, T.N. [2008], The Effect of Thermal Radiation on the Solidification Dynamics of Metal Oxide Melt Droplets, Nucl. Eng. Design, Vol. 238, No. 6, pp 1421-1429. Dombrovsky, L.A. and Baillis, D. [2010], Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, New York and Redding (CT). Dombrovsky, L.A. and Davydov, M.V. [2010], Thermal Radiation from the Zone of Melt-Water Interaction, Computational Thermal Sciences, Vol. 2, No. 6, pp 535-547.
Dombrovsky, L. and Lipiński, W. [2010], A Combined P1 and Monte Carlo Model for Multi-Dimensional Radiative Transfer Problems in Scattering Media, Comput. Thermal Sci., Vol. 2, No. 6, pp 549-560. Dombrovsky, L.A., Kolpakov, A.V., and Surzhikov, S.T. [1991], Transport Approximation in Calculating the Directed-Radiation Transfer in an Anisotropically Scattering Erosional Flare, High Temp., Vol. 29, No. 6, pp 954-960. Dombrovsky, L., Randrianalisoa, J., Baillis, D., and Pilon, L. [2005], Use of Mie Theory to Analyze Experimental Data to Identify Infrared Properties of Fused Quartz Containing Bubbles, Appl. Optics, Vol. 44, No. 33, pp 7021-7031. Dombrovsky, L., Randrianalisoa, J., and Baillis, D. [2006], Modified Two-Flux Approximation for Identification of Radiative Properties of Absorbing and Scattering Media from Directional-Hemispherical Measurements, J. Opt. Soc. Am. A, Vol. 23, No. 1, pp 91-98. Dombrovsky, L.A, Tagne, H.K, Baillis, D., and Gremillard, L. [2007b], Near-Infrared Radiative Properties of Porous Zirconia Ceramics, Infrared Phys. Tech., Vol. 51, No. 1, pp 44-53. Dombrovsky, L.A., Davydov, M.V., and Kudinov, P. [2009], Thermal Radiation Modeling in Numerical Simulation of Melt–Coolant Interaction, Comput. Thermal Sci., Vol. 1, No. 1, pp 1-35. Dombrovsky, L., Lallich, S., Enguehard, F., Baillis, D. [2010], An Effect of “Scattering by Absorption” Observed in Near-Infrared Properties of Nanoporous Silica, J. of Applied Physics, Vol. 107, No. 8, paper 083106. Dombrovsky, L.A., Solovjov, V.P., and Webb, B.W. [2011a], Attenuation of Solar Radiation by Water Mist and Sprays from the Ultraviolet to the Infrared Range, J. Quant. Spectr. Radiat. Transfer, Vol. 112, No. 7, pp 1182-1190. Dombrovsky, L.A., Randrianalisoa, J.H., Lipiński, W., and Baillis, D. [2011b], Approximate Analytical Solution to Normal Emittance of Semi-Transparent Layer of an Absorbing, Scattering, and Refracting Medium, J. Quant. Spectr. Radiat. Transfer, Vol. 112, No. 12, pp 1987-1999. Dombrovsky, L.A., Baillis, D., and Randrianalisoa, J.H. [2011c], Some Physical Models Used to Identify and Analyze Infrared Radiative Properties of Semi-Transparent Dispersed Materials, J. Spectr. Dynamics, Vol. 1, paper 7 (20pp.) Dombrovsky, L.A., Rousseau B., Echegut P., Randrianalisoa J.H., and Baillis, D. [2011d], High Temperature Infrared Properties of YSZ Elecrolyte Ceramics for SOFCs: Experimental Determination and Theoretical Modeling, J. Amer. Ceramic Soc., Vol. 94, No. 12, pp 4310-4316. Dombrovsky, L.A., Timchenko, V., Jackson, M., and Yeoh, G.H. [2011e], A Combined Transient Thermal Model for Laser Hyperthermia of Tumors with Embedded Gold Nanoparticles, Int. J. Heat Mass Transfer, Vol. 54, No. 25-26, pp 5459-5469. Dombrovsky, L.A., Timchenko, V., Jackson, M., and Lipiński, W. [2012a], An Advanced Thermal Model for Indirect Heating Strategy of Laser Induced Hyperthermia, Int. J. Heat Mass Transfer, submitted. Dombrovsky L., Ganesan K., and Lipiński W., 2012b, Combined Two-Flux Approximation and Monte Carlo Model for Identification of Radiative Properties of Highly Scattering Dispersed Materials, Proc. Int. Symp. Adv. Comput. Heat Transfer (CHT-12), July 1–6, 2012, Bath, England. Edwards, R.N. and Bobcôo, R.P. [1967], Radiant Heat Transfer from Isothermal Dispersion with Isotropic Scattering, ASME J. Heat Transfer, Vol. 89, No. 4, pp 300-308. Farmer, J.T. and Howell, J.R. [1998], Comparison of Monte Carlo Strategies for Radiative Transfer in Participating Media, Adv. Heat Transfer, Vol. 31, pp 333-429. Fauchais, P. and Montavon, G. [2007], Plasma Spraying: From Plasma Generation to Coating Structure, Advances in Heat Transfer, Vol. 40. pp 205-344. Fauchais, P. and Vardelle, M. [2010], Sensors in Spray Processes, J. Therm. Spray Tech., Vol. 19, No. 4, pp 668-694. Fiszdon, J.K. [1979], Melting of Powder Grains in a Plasma Flame, Int. J. Heat Mass Transfer, Vol. 22, No. 5, pp 749-761.
Fletcher, D.F. [1999], Radiation Absorption During Premixing, Nucl. Eng. Design, Vol. 189, No. 1-3, pp 435-440. Fletcher, D.F. and Theofanous, T.G. [1997], Heat Transfer and Fluid Dynamic Aspects of Explosive MeltWater Iteractions, Adv. Heat Transfer, Vol. 29, pp 129-213. Gorpas, D., Yova, D., and Politopoulos, K. [2010], A Three-Dimensional Finite Elements Approach for the Coupled Radiative Transfer Equation and Diffusion Approximation Modeling in Fluorescence Imaging, J. Quant. Spectr. Radiat. Transfer, Vol. 111, No. 4, pp 553-568. Hayashi, T., Kashio, Y., and Okada, E. [2003], Hybrid Monte Carlo – Diffusion Method for Light Propagation in Tissue with a Low-Scattering Region, Appl. Optics, Vol. 42, No. 16, pp 2888-2896. Howell, J.R., Siegel, R., and Mengüç, M.P. [2010], Thermal Radiation Heat Transfer, CRC Press, New York. Kolpakov, A.V., Dombrovsky, L.A., and Surzhikov, S.T. [1990], Transfer of Directed Radiation in an Absorbing and Anisotropically Scattering Medium, High Temp., Vol. 28, No. 5, pp 753-757. Koyama, T., Iwasaki, A., Ogoshi, Y., and Okada, E. [2005], Practical and Adequate Approach to Modeling Light Propagation in an Adult Head with Low-Scattering Regions by Use of Diffusion Theory, Appl. Optics, Vol. 44, No. 11, pp 2094-2103. Lai, H.M., Leung, P.T., Poon, K.L., and Young, K. [1991], Characterization of the Internal Energy Density in Mie Scattering, J. Opt. Soc. Am. A, Vol. 8, No. 10, pp 1553-1558. Laredo, D. and Netzer, D.W. [1993], The Dominant Effect of Alumina on Nearfield Plume Radiation, J. Quant. Spectr. Radiat. Transfer, Vol. 50, No. 5, pp 511-530. Mackowski, D.W., Altenkirch, R.A., and Mengüç, M.P. [1990], Internal Absorption Cross Sections in a Stratified Sphere, Appl. Optics, Vol. 29, No. 10, pp 1551-1559. Magallon, D. [2009], Status and Prospects of Resolution of the Vapour Explosion Issue in Light Water Reactors, Nucl. Eng. Tech., Vol. 41, No. 5, pp 603-616. Mishchenko, M.I., Travis, L.D., and Lacis, A.A. [2006], Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering, Cambridge Univ. Press, New York. Modest, M.F. [2003], Radiative Heat Transfer, Second edition, Acad. Press, New York. Nelson, H.F. and Satish, B.V. [1987], Transmission of Laser Beam through Anisotropic Scattering Media, AIAA J. Thermophys. Heat Transfer, Vol. 1, No. 3, pp 233-239. Özişik, M.N. [1973], Radiative Transfer and Interaction with Conduction and Convection, Wiley, New York. Pomraning, G.C. [1965], A Flexible Numerical Approach to Linear Transport Problems, Trans. Amer. Nucl. Soc., Vol. 8, pp 488-489. Potter, J.F. [1970], The Delta Function Approximation in Radiative Transfer Theory, J. Atmos. Sci., Vol. 27, No. 6, pp 943-949. Ripoll, J., Nieto-Vesperinas, M., Arridge, S.R., and Dehghani, H. [2000], Boundary Conditions for Light Propagation in Diffusive Media with Nonscattering Regions, J. Opt. Soc. Am. A, Vol. 17, No. 9, pp 16711681. Rubtsov, N.A. [1984], Radiation Heat Transfer in Continuous Media, Nauka, Novosibirsk (in Russian). Sacadura, J.-F. [2011], Thermal Radiative Properties of Complex Media: Theoretical Prediction Versus Experimental Identification, Heat Transfer Eng., Vol. 32, No. 9, pp 754-770. Siegel, R. and Spuckler, C.M. [1994], Approximate Solution Methods for Spectral Radiative Transfer in High Refractive Index Layers, Int. J. Heat Mass Transfer, Vol. 37, Suppl. 1, pp 403-413. Sjöstrand, N.G. [2001], On an Alternative Way to Treat Highly Anisotropic Scattering, Annals Nucl. Energy, Vol. 28, No. 4, pp 351-355. Sobolev, V.V. [1975], Light Scattering in Planetary Atmospheres, Pergamon Press, Oxford (original Russian edition: Nauka, Moscow, 1972).
Surzhikov, S.T. [1987], On Directional Thermal Radiation Calculations for Light-Scattering Volumes by Use of the Monte Carlo Method, High Temp., Vol. 25, No. 4, pp 820-822 (in Russian). Surzhikov, S.T. [2004], Three-Dimensional Model of the Spectral Emissivity of Light-Scattering Exhaust Plumes, High Temp., Vol. 42, No. 5, pp 763-775. Tuntomo, A., Tien, C.L., and Park, S.H. [1991], Internal Distribution of Radiant Absorption in a Spherical Particle, ASME J. Heat Transfer, Vol. 113, No. 2, pp 407-412. Uršič, M., Leskovar, M., and Mavko, B. [2011], Improved Solidification Influence Modelling for Eulerian Fuel–Coolant Interaction Codes, Nucl. Eng. Design, Vol. 241, No. 4, pp 1206-1216. Velesco, N., Kaiser, T., and Schweiger, G. [1997], Computation of the Internal Field of a Large Spherical Particle by Use of the Geometrical-Optics Approximation, Appl. Optics, Vol. 36, No. 33, pp 8724-8728. Vera, J. and Bayazitoglu, Y. [2009], Gold Nanoshell Density Variation with Laser Power for Induced Hyperthermia, Int. J. Heat Mass Transfer, Vol. 52, No. 3-4, pp 564-573. Viskanta, R. [1982], Radiation Heat Transfer: Interaction with Conduction and Convection and Approximate Methods in Radiation, Proc. 7th Heat Transfer Conf., München, Vol. 1, pp 103-121. Viskanta, R. [2005], Radiative Transfer in Combustion Systems: Fundamentals and Applications, Begell House, New York. Wan, Y.P., Prasad, V., Wang, G.X., Sampath, S., and Fincke, G.R. [1999], Model of Power Particle Heating, Melting, Resolidification, and Evaporation in Plasma Spraying Processes, ASME J. Heat Transfer, Vol. 121, No. 3, pp 691-699. Williams, M.M.R. [1966], The Slowing Down and Thermalization of Neutrons, North-Holland Publ. Co., Amsterdam. Yoshida, T. and Akasi, K. [1997], Particle Heating in a Radio-Frequency Plasma Torch, J. Appl. Phys., Vol. 48, No. 6, pp 2252-2260. Zeldovich, Ya.B. and Raizer, Yu.P. [1967], Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Acad. Press, New York (also Dover Publ., NY, 2002; orig. Russian ed.: Nauka, Moscow, 1966).
