An infrared scattering by evaporating droplets at the initial stage of a pool fire suppression by water sprays

The computational analysis of downward motion and evaporation of water droplets used to suppress a typical transient pool fire shows local regions of a high volume fraction of relatively small droplets. These droplets are comparable in size with the infrared wavelength in the range of intense flame radiation. The estimated scattering of the radiation by these droplets is considerable throughout the entire spectrum except for a narrow region in the vicinity of the main absorption peak of water where the anomalous refraction takes place. The calculations of infrared radiation field in the model pool fire indicate the strong effect of scattering which can be observed experimentally to validate the fire computational model.

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droplets is considerable throughout the entire spectrum except for a narrow region in the vicinity of the main absorption peak of water where the anomalous refraction takes place.
The calculations of infrared radiation field in the model pool fire indicate the strong effect of scattering which can be observed experimentally to validate the fire computational model.

Introduction
The complex behaviour of water sprays used in suppression of pool fires is an important engineering problem [1][2][3][4][5]. Strictly speaking, one should take into account the effect of water sprays on flow field parameters. It is really important in the regular regime of fire suppression. Following [6], a preliminary probe stage of the fire suppression with a very small flow rate of water is considered. Obviously, the effect of a low flow rate water spray on the fire parameters at the probe stage is negligible. It means that one can consider the motion, heating, and evaporation of single water droplets in the flame without taking into account any feedback effects. It is also assumed that a relatively thin water spray moves down parallel to the flame axis. This approach is convenient to focus on the most important special features of the droplet behaviour at the probe stage. Of course, a simplified model for droplet motion and evaporation is insufficient to obtain accurate results for the interaction of a fast developing turbulent fire and water droplets. The latter is especially important when the characteristic times of the flow field changes and displacement of the droplets are comparable with each other.
In the limit of a relatively slow variation of the flow field, the evaporation of droplets accompanied by a decrease in their velocity may lead to a significant volume fraction of small droplets at several specific local areas. In the reality, both the position and parameters of these local areas in the developing flame are changed rather rapidly.
It is known that small water droplets are characterized by a strong scattering of light at the wavelength comparable with the droplet size [7]. This optical effect, which is commonly observed in the visible for the natural mists, has been recently considered as a promising way to improve shielding of fire radiation by multi-layered water sprays [8,9]. It is important that the gaseous combustion products in the fire do not scatter the radiation. The radiation scattering by soot aggregates in the infrared range is also negligible (a considerable scattering by soot aggregates can be observed in the visible range only) [10][11][12][13]. As a result, one can observe local regions of small water droplets near the flame axis because of their infrared scattering. This may be a scattering of the flame self-emission, but one can also use an external irradiation.
The objective of this paper is two-fold: (1) to study the behaviour of moving water droplets which can be collected in some local areas of the flame before their total evaporation and (2) to estimate the effect of scattering of the flame infrared radiation by these small droplets on the radiation source function, which is responsible for the observed flame emission.
It seems natural to consider first the typical trajectories and evaporation dynamics of water droplets to find the local regions where the volume fraction of highly scattering small droplets is expected to be significant.

Motion and evaporation of water droplets
The interaction of water sprays with fires has been modelled computationally in many papers especially during the last two decades [14][15][16][17][18]. However, there is no need to discuss here the state-of-the-art in this field because the present paper is focused on another particular problem. At the probe stage of the fire suppression, one can supply only the droplets falling down not far from the flame axis. For simplicity, the spherical water droplets are assumed to have the same radius at the initial cross section of the spray. It is sufficient to consider a few selected trajectories of droplets falling initially along the flame axis without any interactions between the droplets. According to [16,19], the following sets of equations are derived for the droplet motion: where the subscripts "g", "d" and "w" refer to the gaseous medium, droplet and water, u is the velocity, a is the droplet radius, D C is the drag coefficient, Re is the Reynolds number.
It is assumed that water droplets are first heated up to the saturation temperature without evaporation (at where w L is the latent heat of evaporation of water. A similar model has been recently used in analysis of water mist curtains [8,9]. This simplified evaporation model is also sufficient for qualitative estimates of the present paper because of very fast heating of moving droplets and large molar fraction of water vapour in the flame. It is not necessary at the moment to consider sophisticated evaporation models like that developed in paper [20]. A transfer from the droplet heating to its evaporation is given by equation: In contrast to recent papers [3,4,21], the effects of thermal radiation are neglected in the above model as compared with convective heat transfer from ambient hot gases. This assumption can be revised on the basis of the radiation field calculations. Obviously, the local relative volume fraction of water droplets can be calculated as follows: Equation (5) indicates that the volume fraction of water droplets decreases with their evaporation, but a strong increase in the volume fraction can be observed in the regions of slowly moving droplets.

Application Case Study
A pool fire is defined as a turbulent diffusion flame burning above a horizontal pool of vaporising fuel (e.g. liquid) where the fuel has zero or low initial momentum. Such fires are dominated by buoyancy since the velocity and momentum of the fuel vapour leaving the pool surface are negligible. In order to reproduce pool fires in laboratory conditions, most fire researchers used gas burners constructed of porous refractory material, where the gas fuel is supplied at very low velocity under controlled burning. Controlling the mass burning rate is a way to quantify with certainty the heat release rate which is an important quantity in pool fire studies. Moreover, bearing in mind that flaming combustion only occurs in the gas phase, this case also mimics the realistic scenario of a pool of liquid fuel with a low momentum vapour fuel initially formed at the surface of the pool before combusting with oxygen. In the present study, a methane gas burner is employed to replicate typical experimental condition of a burning liquid pool fire. This is common practice in CFD simulation of pool fires.
The flame is burning in the open quiescent environment with a heat release rate of 53 kW.
The numerical results of a CFD simulation from paper [6] for We have considered the developing flame at several time moments, but the selected flame appeared to be the most interesting because of two sub-regions with quite different local temperatures and axial velocities shown in Fig. 1. This specific flame is convenient to analyze the behavior of water droplets at very different conditions.
The temperature 300 0  T K was taken for water droplets at the initial cross section 5 . 0 0  z m of the water spray. As to the droplet size and initial velocity, a set of various realistic values of these quantities was considered. Some computational results including those obtained in [6] are presented in Fig. 2. The behaviour of water droplets moving along two parallel lines appeared to be quite different. However, the trajectories of evaporating droplets of different initial size are focused in small local areas. In the case of droplets moving along the axis, their initial velocity makes no difference because of a relatively small relaxation time (Fig. 2a). The evaporation of these droplets will be completed due to absorption of thermal radiation from a surrounding flame. On the contrary, the large droplets One should recall that the above result was obtained by ignoring the effects of turbulence and also time variation of the Reynolds-averaged flow field of developing flames. Obviously, the turbulence cannot affect the motion of inertial particles at high values of the Stokes number, which is proportional to the square of the droplet radius [19]. It is also interesting that turbulence can not only mitigate the particle collection. On the contrary, the transient effect of the so-called "preferential concentration" of particles with Stokes number about unity may be significant even in the homogeneous and isotropic turbulence [22][23][24].
The effect of a fast time variation of the developing flame parameters is expected to be much stronger than the effect of turbulence. Additional CFD calculations have shown that a characteristic time of considerable variations of the flow field shown in Fig. 1 is about 0.1-0.2 s. It means that the shape of the local areas with high volume fraction of small water droplets may be deformed and shifted at the probe suppression stage. More definite conclusion about the behavior of these local areas depends on the increase of water flow rate while transfer to the regular fire suppression.
In all cases, the moving local areas containing small evaporating droplets will lead to the infrared scattering by these small water droplets.

Infrared properties of water droplets
The spectral optical constants, n and  , of pure water are well known from early papers [25,26]. For convenience of subsequent analysis, spectral dependences of these quantities in the most important part of the infrared range are presented in Fig. 3. The spectral characteristics of absorption and scattering of spherical water droplets can be calculated using the Mie theory [27][28][29]. Two dimensionless far-field characteristics which can be obtained . Therefore, as it was done in recent studies [8,9], we use the following analytical approximation suggested in [30]  A comparison of approximation (6 a, b) with Mie theory calculations for water droplets at two typical infrared wavelengths can be found in paper [8]. As one can expect, the approximate relations are sufficiently accurate for both a Q and tr s Q in the spectral range of a weak absorption. As to water absorption band at wavelength of 3   µm, only the approximation for the absorption efficiency factor is important because a tr s Q Q  . The latter is typical of the narrow region of anomalous refraction [31,32].
The so-called monodisperse approximation, when all the droplets are assumed to have the same radius in the initial cross section of the spray, is used. The spectral values of absorption coefficient,  , and transport scattering coefficient, tr  , in a small volume containing monodisperse droplets are related with the efficiency factors as follows (the subscript λ is hereafter omitted for brevity) [7]: where v f is the local volume fraction of droplets. It is convenient also to introduce the corresponding normalized coefficients spraying [33][34][35][36], the radiation from two-phase combustion products in exhaust plumes of aluminized-propellant rocket engines [37,38], and the radiative cooling of core melt droplets in nuclear fuel-coolant interaction [39,40]. Fortunately, the subsequent calculations showed that it is not the case for the problem under consideration and the monodisperse approximation is really applicable.
The calculated values of a E and tr s E in the most important range of the problem parameters are presented in Fig. 4. It is clear that the radiation absorption by water droplets is significant in the absorption band of water. This absorption contributes to water evaporation and flame suppression. On the contrary, the scattering of the flame radiation by water droplets is more important outside the absorption band, where this effect is expected to be observed. It is interesting that spectral behaviour of both the absorption and transport scattering coefficients of droplets is similar for droplets of different sizes, but the scattering is much greater for relatively small droplets.
The present paper is focused on the radiation scattering by small water droplets. Therefore, the wavelength about 4 µm is the most interesting: the strong scattering of radiation by water droplets and relatively small absorption are typical for this spectral range (compare Figs. 4a and 4b).
The hypothesis of independent scattering does not work in the above obtained local areas of strong evaporation where the distance between the neighbouring droplets may be small as compared with both the droplet size and the radiation wavelength. According to [41][42][43][44][45], the dependent scattering may lead to significant changes in absorption and scattering characteristics of the turbulent gas flow with suspended clusters of closely spaced particles or droplets. One should recall papers [46][47][48], where this effect was observed and analysed as applied to cumulus clouds in the atmosphere. Of course, a study of this effect is a separate complex problem of both the physical optics and the turbulence of gas flows with suspended inertial particles and it is beyond the scope of the present paper. The ordinary relations for independent single droplets are used in subsequent estimates.

Approximate method for radiative transfer calculations
We use the known P 1 approximation of the spherical harmonics method [8,49,50] to estimate the effect of radiation scattering by water droplets on thermal radiation field in the flame. In our case, this approach leads to the following boundary-value problem for the modified Helmholtz equation: is the irradiation function, D is the so-called radiation diffusion coefficient, is the transport extinction coefficient, g  and d  are the absorption coefficients of a gas medium and suspended water droplets, n  is the external normal to the boundary of the computational region. Of course, the spectral irradiation G , the gas The P 1 approximation is simple and very convenient for radiative transfer calculations especially in combined heat transfer problems. Moreover, this approach appears to be sufficiently accurate in the case when one needs only the divergence of radiative flux, , which is an important term in the energy equation [51][52][53][54]. At the same time, the radiative flux calculations at the boundaries of the computational region using the P 1 may lead to significant errors. A physical analysis of these errors has been made in early paper [55] and can be found also in monograph [7]. This drawback of P 1 can be compensated with the use of the combined two-step method and transport approximation for the scattering phase function, when the P 1 is employed in CFD calculations only, whereas a ray-tracing procedure is used to solve the spectral radiative transfer equation at the second step of the complete solution [7,[56][57][58][59]: where S is the so-called source function. However, the ray-tracing calculations are not necessary in the present paper because we are just going to compare several variants to estimate an effect of infrared scattering on the radiation field in the flame. According to [7,58,59], the finite-element method [60] with the simplest linear presentation of function G in triangular finite elements is used to solve a variational formulation of the P 1 boundary-value problem.

Effect of infrared scattering by evaporating water droplets
There is no need for complete calculations over the spectrum to estimate qualitatively the effect of infrared scattering by evaporating water droplets in the above described example pool fire. It is sufficient to consider spectral results at the representative wavelength of The non-uniform triangulation of the computational region with 1600 elements is shown in Fig. 5. The smaller areas of finite elements were made in the region of more optically dense medium. As usually, the radial coordinates of all the nodes were increased by the first radial interval of the mesh to avoid the formal difficulties in numerical calculations near the axis [7]. The calculated fields of function S in the most interesting part of the fire cross section are presented in Fig. 6. The circular regions (11) containing fast evaporating and almost immovable water droplets can be simply identified near the right-hand boundary and at the lower panel of Fig. 6b because of a considerable contribution of scattering to the radiative source function. Of course, there are no sharp edges of these areas in reality. One should use more accurate data for evaporating droplets to obtain smooth boundaries of the scattering region in the radiation calculations. However, the latter is not necessary for the qualitative estimates of the present paper. It is important that strong infrared scattering in local areas of small evaporating water droplets is physically sound and it is confirmed by direct calculations. To the best of our knowledge, this effect is computationally obtained for the first time.

Conclusion
The scattering of infrared radiation of a developing pool fire by evaporating water droplets in fire suppression by water sprays was predicted on the basis of theoretical and computational estimates. It was shown for the first time that small evaporating droplets with different initial size can be focused in local areas of the flame before their total evaporation.
The calculations for real developing flame showed that the effect of infrared radiation scattering by almost immovable and fast evaporating water droplets on the radiation field in the flame and also on the source function responsible for the infrared radiation from the flame is considerable. The latter can be observed at the initial stage of fire suppression using the              An initial stage of a pool fire suppression is considered.
 The motion, heating, and evaporation of water droplets are calculated.
 Focusing of evaporating water droplets in local regions is obtained.
 A strong infrared scattering by small water droplets is analyzed.
 The spectral region of a significant infrared scattering is determined.
 The use of the infrared scattering in flame observations is discussed.