## Large-Cell Model for Radiation Heat Transfer in Multiphase Systems

* Following from: *Computational models for radiative transfer in disperse systems

* Leading to: *Thermal radiation modeling in melt-coolant interaction

The radiation model discussed in this section has been recently developed by Dombrovsky (2007a) for multiphase flows typical of the so-called fuel-coolant interaction (FCI) when a high-temperature core melt (80%UO_{2}+20%ZrO_{2}) falls into a water pool. Various aspects of FCI have been widely investigated during last two decades because of the possibility of severe accidents with regard to light-water nuclear reactors. The complexity of different stages of high-temperature core-melt interaction with water is one of the reasons for the present-day state of the art; however, some important physical processes still have not been considered in detail. The efforts of many researchers have focused on hydrodynamic simulation of melt jet breakup (Dinh et al., 1999; Bürger, 2006; Pohlner et al., 2006) and the specific problems of steam explosions (Fletcher and Anderson, 1990; Theofanous, 1995; Fletcher, 1995; Berthoud, 2000). At the same time, the radiation heat transfer in the multiphase medium containing polydisperse corium particles of temperature of about 2500-3000 K has not been a subject of detailed analysis. The papers by T.-N. Dinh et al. (1999), Fletcher (1999), and Dombrovsky (1999a, 2000a) were probably the first publications where the important role of radiation heat transfer was discussed. It was noted that a part of thermal radiation emitted by particles can be absorbed far from the radiation sources because of the semitransparency of water in the short-wave range.

The general problem of radiation heat transfer between corium particles and ambient water can be divided into the following problems of different scales: thermal radiation from a single particle through a steam blanket to ambient water and radiation heat transfer in a large-scale volume containing numerous corium particles, steam bubbles, and water droplets. One can show that solutions to these problems can be incorporated in a general physical and computational model, as was done for similar problems of radiation heat transfer in other disperse systems (Dombrovsky, 1996). The single-particle problem has been analyzed in some detail by Dombrovsky (1999a, 2000a). The main focus was given to the significant contribution of electromagnetic wave effects in the case of very thin steam layers. The effect of semitransparency of nonisothermal oxide particles on the thermal radiation has also been studied by Dombrovsky (1999b, 2000b, 2002). The resulting physical features of particle solidification have been reported recently by Dombrovsky (2007b) and Dombrovsky and Dinh (2008). To the best of our knowledge, the first attempt to calculate radiation heat transfer in water containing corium particles was reported by Yuen (2004). It was assumed that there is no radiation scattering in the medium. The spectral radiative properties of the melt particles of various temperatures and sizes were ignored in this paper, and all the particles were considered as the sources of black-body radiation. The calculations by Yuen (2004) were based on the formal zonal method, which seems to be a poor choice for the problem considered. A more sophisticated model for radiation heat-transfer calculation in water containing numerous polydisperse corium particles of different temperatures and polydisperse steam bubbles has been suggested by Dombrovsky (2007a). This model, called the large-cell radiation model (LCRM), is sufficiently simple to be easily implemented into computational fluid dynamics (CFD) codes for multiphase flow calculations (Dombrovsky et al., 2009). The computational results for realistic conditions are considered in the article Thermal radiation modeling in melt-coolant interaction. In the present article, we focus on the radiation model.

### Two-Band Model with Conventional Semitransparency and Opacity Regions

To suggest an adequate model of radiation heat transfer in water containing hot corium particles and steam bubbles, one should take into account specific optical properties of water in the visible and near-infrared spectral ranges. It is well known that water is semitransparent in a short-wave range, and there is a strong absorption band at the wavelength λ = 3 μm (Hale and Querry, 1973). To estimate the role of nonlocal radiation effects, one can introduce the characteristic penetration depth of the collimated radiation in water: *l*_{λ} = 1/α_{w}. The spectral dependence of *l*_{λ} in the most interesting intermediate range of 0.8 < λ < 1.4 μm is illustrated in Fig. 1. One can see that *l*_{λ} decreases from about 0.5 m at the visible range boundary λ = 0.8 μm to *l*_{λ} = 1 mm at the wavelength λ = 1.38 μm.

**Figure 1. The characteristic propagation depth of collimated near-infrared radiation in water.**

It is reasonable to separately consider the following conventional spectral regions:

- The short-wave semitransparency range λ < λ
_{*}= 1.2 μm, where*l*_{λ}*>*10 mm. There is a considerable radiation heat transfer between corium particles in this spectral range because the distance between neighboring millimeter-sized particles is usually less than 10 mm. One can use the traditional radiation transfer theory to calculate the volume distribution of radiation power. Both absorption and scattering of radiation by particles should be taken into account. - The opacity range λ
*>*λ_{*}, where*l*_{λ}< 10 mm. In this range one can neglect the radiation heat transfer between the particles. The radiative transfer problem degenerates because of strong absorption at distances comparable to both particle sizes and distances between the particles. One can assume that radiation emitted by the particle in this spectral range is totally absorbed in ambient water.

Of course, the above two-band radiation model should be treated as a simple approach, and the effect of a conventional value of the boundary wavelength λ_{*} may be a subject of further analysis.

In a multiphase flow typical of the FCI problem, numerous steam bubbles and core melt particles have a considerable effect on the radiative properties of the medium in the range of water semitransparency. Nevertheless, the above division of the spectrum into two bands according to the absorption spectrum of water remains acceptable (Dombrovsky, 2007a).

### P_{1} Approximation and Large-Cell Radiation Model for Semitransparency Range

The RTE for emitting, absorbing, refracting, and scattering medium containing *N* components of different temperatures can be written as follows (Dombrovsky, 1996; Siegel and Howell, 2002; Modest, 2003):

(1) |

where *n*_{λ} is the index of refraction of the host medium, and α_{λ,i} is the absorption coefficient of the composite medium component with temperature *T*_{i},

(2) |

By writing the last term on the right-hand side of RTE (1), we have assumed that every component of the medium is characterized by a definite temperature. This is not the case for large corium particles with considerable temperature difference in the particle. Nevertheless, the problem formulation should not be revised for opaque particles. It is sufficient to treat the value of *T*_{i} as a surface temperature of the particles of *i*th fraction. An essentially more complex problem should be considered for semitransparent particles when thermal radiation comes from the particle volume. It is a realistic situation for particles of aluminum oxide or other light oxides used as simulant substances in experimental studies of the core melt-coolant interaction. The problem of thermal radiation from semitransparent nonisothermal particles is considered in the articles Thermal radiation from nonisothermal spherical particles and Thermal radiation from nonisothermal particles in combined heat transfer problems. The solution obtained can be combined with the large-scale problem under consideration.

It is very difficult to use the complete description of the radiation heat transfer based on RTE (1) in the range of water semitransparency. Therefore, the simplified radiation models should be considered for engineering calculations. The integration of the RTE over all values of the solid angle yields the following equation of spectral energy balance:

(3) |

where *p*_{λ} is the spectral radiation power emitted in a unit volume of the medium. Note that Eq. (3) is a generalized form of Eq. (7) from the article The radiative transfer equation for the case of a multitemperature medium. The spectral balance equation (3) is considered as a starting point for simplified models for radiation heat transfer in multiphase disperse systems.

In the case of somewhat cool particles, the main part of thermal radiation is emitted in the range of water opacity. Thus it is reasonable to ignore the specific feature of the process in the short-wave range and assume water to be totally opaque over the whole spectrum. This approach can be called the opaque medium model (OMM). According to the OMM, thermal radiation emitted by single hot particles is absorbed in water at very small distances from the particle. In this case, the total power absorbed by water in a unit volume is equal to the power emitted by particles in this volume:

(4) |

where λ_{1} and λ_{2} are the boundaries of the spectral range of considerable thermal radiation. Obviously, this model overestimates the heat absorbed in water and cannot be employed to distinguish the radiation power absorbed at the steam/water interface near the particle and the power absorbed in the volume. The latter may be important for detailed analysis of heat transfer from corium particles to ambient water in calculations of water heating and evaporation. Simple estimates showed that contribution of short-wave radiation increases rapidly with the particle temperature, and one cannot ignore the spectral range of water semitransparency when corium particle temperature is greater than 2500 K. In other words, one can expect the OMM error to be considerable in this case.

The large-cell radiation model (LCRM) is based on the assumption of negligible radiation heat transfer between the computational cells. Note that the present-day computer codes for multiphase flows use computational cells of about 5-10 cm or greater and all parameters of the multiphase flow are assumed to be constant in every cell. In the range of water semitransparency, the local radiative balance in a single cell yields the following relation for radiation energy density instead of Eq. (3):

(5) |

As a result, the expressions for the integral radiation power absorbed in water can be written as

(6) |

where α_{λ,w} is the spectral absorption coefficient of water containing steam bubbles. The components *P*_{w}^{(1)} and *P*_{w}^{(2)} of the absorbed power correspond to the ranges of water semitransparency and opacity. One can assume that *P*_{w}^{(1)} causes the volume heating of water, whereas *P*_{w}^{(2)} causes the surface heating and evaporation of water near the hot particles. Obviously, the predicted contribution of the semitransparency range to the total absorbed power appears to be less than the corresponding value estimated by use of OMM. Note that LCRM does not include any characteristics of radiation scattering in the medium.

The radiation balance equation (3) can also be employed without ignoring the radiation flux divergence. To realize such a possibility, one should find a relation between the spectral radiation flux and radiation energy density. In P_{1} approximation, the known representation of the radiation flux is assumed to make the problem statement complete:

(7) |

and the spectral radiation energy density can be determined by solving the following boundary-value problem:

(8a) |

(8b) |

where _{n} is the unit vector of external normal to the boundary surface of the computational region. The boundary condition (8b) corresponds to the case of zero external radiation and no reflection from the boundary surface. The angular dependence of radiation intensity in the region of intensive FCI is expected to be smooth. Therefore, P_{1} can be used instead of the RTE to analyze the quality of LCRM. Note that boundary-value problem (8) is formulated for the complete computational region (not for single cells). After solving this problem for several wavelengths in the range of λ_{1} < λ < λ_{*}, one can find the radiation power absorbed in water:

(9) |

The total radiative heat loss from corium particles is

(10) |

where α_{λ,c} is the absorption coefficient of polydisperse corium particles. It is important that *P*_{c}^{(1)} ≠ *P*_{w}^{(1)} due to heat transfer by radiation in semitransparent medium:

(11) |

The P_{1} approximation takes into account the radiative transfer between all the computational cells. It is an important advantage of this model, especially in the case of semitransparent cells. A long-time experience in the use of P_{1} for solving various engineering problems has shown that the predicted field of radiation energy density is usually very close to the exact RTE solution. One can see that P_{1} also gives the radiation flux at the boundary of the computational region. In contrast to the radiation energy density, the radiation flux error may be significant (see the article, An estimate of P1 approximation error for optically inhomogeneous media). Therefore, a more sophisticated approach should be employed to determine the radiation coming from the FCI region.

The complete solution to the two-dimensional radiation heat-transfer problem in a multiphase flow typical of fuel-coolant interaction is too complicated even when the P_{1} approximation is employed. The main computational difficulty is related to the wide range of optical thickness of the medium at different wavelengths. One should consider not only the visible radiation when optical thickness of the medium is determined by numerous particles, but also a part of the near-infrared range characterized by the large absorption coefficient of water. As a result, the numerical solution of the boundary-value problem (8), generally speaking, cannot be obtained by using the same computational mesh at all wavelengths. There is no such difficulty in LCRM, which is simply an algebraic model and can be easily implemented into any multiphase CFD code.

In the Lagrangian calculations of the transient temperature of corium particles, the value of integral (over the spectrum) radiation heat flux from the unit surface of a single particle is used. In OMM, this value is determined as follows:

(12) |

In LCRM and P_{1}, we have the following expressions for the radiation flux:

(13) |

The complete formulation of the problem must include the relations for radiative characteristics of particles and steam bubbles. These relations have been derived in the papers of Dombrovsky et al. (2007a, 2009) (see also Radiative properties of gas bubbles in semi-transparent medium, Thermal radiation from spherical particle to absorbing medium through narrow concentric gap, and Thermal radiation modeling in melt-coolant interaction).

In Lagrangian modeling of motion and cooling of an isothermal particle of radius *a*_{i}, the following energy equation is usually employed:

(14) |

For simplicity, it is assumed here that the particle is totally opaque and optically gray (ε_{c} = *const*). Generally speaking, ψ_{i} ≠ 1 and the values of ψ_{i} can be determined from the large-cell model. To clarify the physical sense of coefficient ψ, consider the case of monodisperse corium particles when

(15) |

where α is the absorption coefficient of corium particles, and ζ_{0}(*T*) is the part of blackbody radiation at temperature *T* in the range of water semitransparency:

(16) |

Obviously, the coefficient ψ varies in the range between 1 - ζ_{0} and 1, where the lower limit corresponds to the high volume fraction of corium.

### Comparison of Diffusion and Large-Cell Models for Typical Problem Parameters

Following the paper by Dombrovsky (2007a), consider a one-dimensional axisymmetric problem of radiation heat transfer in water containing polydisperse steam bubbles and steam-mantled corium particles. In our sample problem we use the following similar profiles of the volume fractions of corium and steam:

(17) |

The following fixed values of parameters are considered: *R* = 0.5 m, *f*_{v0} = 0.5%. The function φ(*r*) and its “cell” approximation are shown in Fig. 2. The ordinates of the cell approximation for the number of cells *N* = 10 are calculated as follows:

(18) |

**Figure 2. Dimensionless profile of volume fraction of steam and corium considered in the model problem: 1 - smooth profile and 2 - stepwise approximation.**

The average radius of bubbles is assumed to equal 3 mm. The corium particles are treated as opaque ones. The emissivity of bulk corium was assumed to be independent of wavelength and temperature and equal to ε_{c} = 0.85. Because of the complexity of the general problem, two variants of the sample problem are considered below: one for monodisperse corium particles and one model for polydisperse corium characterized by different temperatures of small and large particles.

### Monodisperse Particles

Consider the case of monodisperse corium particles of radius *a*_{2} = 2.5mm and temperature *T* = 3000K. The results of calculations based on P_{1} approximation are presented in Figs. 3 and 4. One can see in Fig. 3 that there is a considerable difference between the radiation power emitted by corium particles in the semitransparency range and the power absorbed in water. It is explained by considerable radiation flux from the region in this spectral range (see Fig. 4). The difference between the calculations for a smooth profile of the particle volume fraction and a stepwise profile typical of cell approximation of the flow parameters is insignificant, especially for radiation power absorbed in water and spectral radiation flux at the boundary region. One can see in Fig. 4 that thermal radiation from the multiphase medium can be observed only in the visible range, and the corresponding radiative heat loss is negligible in the medium heat balance.

**Figure 3. Radiative heat loss from corium particles (a) and radiation power absorbed in water (b): 1 - in the range of water semitransparency, 2 - in the range of water capacity; I - smooth profile, II - stepwise approximation.**

**Figure 4. Spectral radiative flux at the boundary of the computational region: calculations for smooth profile ( I) and stepwise approximation (II) of the medium parameters.**

It follows from Eq. (3) that the large-cell model gives the only profile of radiation power. This profile is intermediate between the profiles obtained for corium and water in P_{1} approximation. One can see in Fig. 5 that the relative error of the large-cell model in total radiation power is not large (about 5-10%) because of the decisive contribution of the opacity range. It is important that this error can be estimated by comparison of the large-cell solution with the upper limit of radiative heat loss from corium particles:

(19) |

**Figure 5. Total radiative heat loss from corium particles (a) and radiation power absorbed in water (b): 1 - P _{1} approximation, 2 - large-cell model, 3 - maximum estimate (19).**

The latter statement is illustrated by curve *P*_{w}^{max}(*r*) plotted in Fig. 5.

### Polydisperse Particles

For simplicity, the following two-mode size distribution of particles is considered:

(20) |

with *a*_{1} = 0.5 mm, *a*_{2} = 3 mm, *T*(*a*_{1}) = *T*_{1} = 2000 K, *T*(*a*_{2}) = *T*_{2} = 3000 K. Obviously, the integral characteristics of size-distribution (20) are

(21) |

Note that ξ is the relative number of small particles, whereas the more representative volume fraction of these particles is given by ν = ξ*a*_{1}^{3}/*a*_{30}. The effect of polydisperse corium particles can be analyzed on the basis of the large-cell model. The local character of this model allows us to consider a single cell of the medium. One can write the following expressions for radiative cooling rate coefficients:

(22) |

Note that calculations showed the predominant role of visible radiation in direct heat transfer between the particles of different temperatures. For this reason, it is sufficient to use the “red” boundary of the visible spectral range λ_{red} = 0.8 μm instead of λ_{*} in approximate calculations of the function ζ(*T* ). The results of calculations presented in Fig. 6 showed that thermal radiation from relatively hot large particles of corium in the visible spectral range can lead to significant decrease in the radiative cooling rate of small particles. This effect should be taken into account in calculations of the fuel-coolant interaction. A more representative analysis of the LCRM error in realistic FCI problems can be found in the paper by Dombrovsky et al. (2009) and in the article Thermal radiation modeling in melt-coolant interaction.

**Figure 6. The coefficients of radiative cooling rate for corium particles of two fractions as functions of relative volume fraction of small particles: 1 - ψ _{1}, for small particles; 2 - ψ_{2}, for large particles.**

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