International Journal of Heat and Mass Transfer 96 (2016) 199–209

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

A simplified model for the shielding of fire thermal radiation by watermists

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.01.0280017-9310/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +7 910 408 0186.E-mail address: ldombr@yandex.ru (L.A. Dombrovsky).

Leonid A. Dombrovsky a,⇑, Siaka Dembele b, Jennifer X. Wen c

a Joint Institute for High Temperatures, Krasnokazarmennaya Str. 17A, NCHMT, Moscow 111116, Russiab Fire, Explosion and Fluid Dynamics Research Group, School of Mechanical and Automotive Engineering, Kingston University, London SW15 3DW, UKcWarwick FIRE, School of Engineering, University of Warwick, Coventry CV4 7AL, UK

a r t i c l e i n f o

Article history:Received 22 December 2015Received in revised form 11 January 2016Accepted 11 January 2016

Keywords:Fire radiationWater mistRadiative transferHeat transferComputational model

a b s t r a c t

A solution for the complete problem of attenuation of fire radiation by water mist is presented. This solu-tion is based on simplified approaches for the spectral radiative properties of water droplets, the radiativetransfer in the absorbing and scattering mist, and transient heat transfer taking into account partial evap-oration of water mist. A computational study of the conventional model problem indicates the role of themain parameters and enables one to formulate some recommendations to optimize possible engineeringsolutions. The method developed is also applied to more realistic case study of a real fire. It is suggestedto decrease the size of supplied water droplets with the distance from the irradiated surface of the mistlayer. The advantage of this engineering solution is confirmed by numerical calculations. Potential pos-sibility of microwave monitoring of water mist parameters is analyzed on the basis of Mie theorycalculations.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Since the ratification of the Montreal protocol in 1987, phasingout halon agents due to their negative environmental impacts, theuse of water sprays and mists in fire protection has gained momen-tum. Water mist is defined according to the NFPA as sprays inwhich 99% of the volume is in droplets with diameters less than1000 microns. Water spray/mist systems can be used for the dilu-tion of toxic releases [1]. The scope of the present study is its fireapplication. There are two main strategies for using watersprays/mists in fire protection. In the first one, the intention is toextinguish or control the fire by applying the spray directly ontothe fire source. Such applications have been well reviewed in [2]and considered also in [3]. In the second application strategywhere there is no direct contact between the fire source and spray,the curtain of spray/mist is used as a radiation attenuation shieldto protect potential targets which could be equipments or humanbeings [3]. The present study is concerned with such radiationshielding applications of water mist curtains. In the process indus-tries, spray/mist curtains provide an effective mean to protectflammable targets (e.g. storage tanks) in the event of fires. Theycould also serve as protection against fire radiation for personnel

during evacuation on-board carrier and chemical ships duringmaritime transport [4]. In some countries, fire engines used byfirefighters to combat forest fires are fitted with water spraycurtains as emergency personnel protection. Water spray curtainscan also be employed as compartmentation to protect people infire events [5].

Research on water spray/mist shielding has received consider-able attention in the past two decades. Although the main mecha-nisms of radiation attenuation by a two-phase water spray havebeen identified as absorption and scattering by droplets andabsorption by the gas phase (mainly water vapor), a rigorousmodel that account for the coupled radiation, heat and mass trans-fers in the spray is complex to develop and is too involved compu-tationally. Such models are important to better design andoptimize water sprays and mists for reliable and cost-effectivesolutions.

The bulk of the literature on water spray/mist curtain shieldinghas been devoted to radiation modeling by uncoupling it fromother phenomena, in order to predict the transmittance and atten-uation of the curtain. Obviously, the Beer–Lambert law used in [6]is inapplicable for the transmittance calculations in the problemunder consideration. Therefore, the two-flux model was employedin more recent papers [7–10] for the transmittance calculations.These studies show that smaller droplets in high concentrationprovide better attenuation of the spray. However the important

Nomenclature

a radius of dropletA attenuation parameter introduced by Eq. (17)c specific heat capacityCD drag coefficientf v volume fraction of dropletsH height of the mist curtaink thermal conductivityL latent heat of evaporationm complex index of refractionn index of refractionq radiative fluxQ efficiency factor of absorption or scatteringR reflectancet timeT temperatureu velocityW absorbed radiation powerx diffraction parametery horizontal coordinatez vertical coordinate

Greek symbolsa absorption coefficiente dielectric constantg dynamic viscosity of gas

j index of absorptionk wavelengthl cosine of an angleq densityr scattering coefficient or electrical conductivityr0 Stefan–Boltzmann constantx scattering albedo

Subscripts and superscriptsa absorptiond droplete externalel electricalf flameg gash hemisphericaln-h normal-hemisphericalrel relaxations scattering or statict transmittedtot totaltr transportw waterk spectral

200 L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209

phenomena such as mass transfer and droplets evaporation werenot considered in these papers.

More detailed description of radiative transfer based on the Dis-crete Ordinates Method (DOM), Finite Volume Method and evenMonte Carlo simulation were also used in computational studiesof water mists [11–17]. Some of these studies have coupled theradiation, heat, mass and momentum transfer in sprays using thecombined Eulerian–Lagrangian approach for both dynamic andthermal non-equilibrium of droplets and ambient gas. Howeverthe complexity of this approach is an obstacle to their widespreaduse.

The current literature clearly shows the advances in modelingand improving the understanding of water spray/mist curtain infire radiation mitigation. However these models are mainlyemployed by the research community and are rarely used in thewater mist industry which is currently developing fast and needssuch tools. The complexity and computing cost are clearly twomajor obstacles for the application of current methods. There is aneed nowadays to develop engineering models for water mist cur-tain, which retains the physics of the problem and at the same timeoffers acceptable computing cost. The present study aims toachieve such a goal.

The objectives of the present paper are as follows: (1) todevelop a simplified but complete model for the combined heattransfer processes in a semi-transparent layer of water dropletsused as a shield for infrared radiation of fires, (2) to study compu-tationally the role of the main parameters of water mist and to givepreliminary recommendation on possible optimization of engi-neering solutions for fire protection, (3) to present a numericalsolution for realistic case study, (4) to suggest possible principalapproach to the microwave monitoring the mist parameters takinginto account the effect of both the size and temperature of waterdroplets on absorption and scattering of the microwave radiationby the mist layer.

The methodology of the present paper is based on a combina-tion of a set of approximate 1-D solutions for the radiative transfer

problem and a simplified heat transfer model for heating and evap-oration of water droplets. A relatively small absorption of radiationby water vapor, which is generated by partially evaporating waterdroplets, is neglected in a simplified model. The analysis of theanalytical solution for radiative heat transfer through the mistlayer makes it possible to suggest a decrease in the size of suppliedwater droplets with the distance from the irradiated surface of themist layer. Subsequent numerical calculations for more realisticcase study of fire protection confirm the advantages of the sug-gested engineering solution.

An analysis of possible microwave monitoring of water mistparameters is also given in the paper. The calculations based onthe rigorous Mie solution for single water droplets showed thatimportant information on average values of both the size and tem-perature of water droplets can be obtained using the measure-ments of directional-hemispherical reflectance of sub-millimeterradiation from the mist layer.

2. Spectral properties of water droplets

The spectral optical constants, n and j, of pure water are wellknown [18,19]. For convenience of subsequent analysis, spectraldependences of these quantities in the most important part ofthe infrared range are presented in Fig. 1. The spectral characteris-tics of absorption and scattering of spherical water droplets can becalculated using the Mie theory [20–22]. Because of a simplifiedradiative transfer model used in the present paper, we will focuson two dimensionless far-field characteristics which can beobtained from the analytical Mie solution: the efficiency factor ofabsorption, Q a, and the transport efficiency factor of scattering,Q tr

s . According to Mie theory, the values of Qa and Q trs depend on

both the complex index of refractionm ¼ n� ij and the diffraction(size) parameter x ¼ 2pa=k, where a is the droplet radius. The exactMie calculations are time-consuming especially for large dropletswith x >> 1. Fortunately, one can use the following analytical

1.1

1.2

1.3

1.4

1.5

λ, μm

n

a

2 4 6 8 10 12 14

2 4 6 8 10 12 141E-5

1E-4

1E-3

0.01

0.1

λ, μm

κ

b

Fig. 1. Spectral indices of refraction (a) and absorption (b) of pure water [18,19].

10 20 30 40 50

0.1

1

2

2

Qstr

Qa

Qa, Q

str

a, μm

1

Fig. 2. Efficiency factor of absorption and transport efficiency factor of scatteringfor water droplets of various radius: 1 – k ¼ 3 lm, 2 – k ¼ 5 lm; solid lines – Mietheory, dashed lines – approximation (1a–c).

L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209 201

approximation suggested in paper [23] for semi-transparent parti-cles (see also [24]):

Q a ¼4n

ðnþ 1Þ2½1� expð�4jxÞ� ð1aÞ

Q trs ¼ C

q=5 when q 6 5ð5=qÞc when q > 5

�ð1bÞ

where

C ¼ 1:5nðn� 1Þ expð�15jÞ c ¼ 1:4� expð�80jÞ ð1cÞA comparison of approximation (1a)-(1c) with Mie theory cal-

culations for water droplets at two typical wavelengths is givenin Fig. 2. As one can expect, approximate relations give sufficientlyaccurate results both for Qa and Q tr

s in the spectral range of a weakabsorption. As to water absorption band (at wavelength k ¼ 3 lm),only the approximation for the absorption efficiency factor isimportant because Q tr

s << Q a in this spectral range.The real mists contain water droplets of very different size in

every small volume of the mist. Therefore, the calculations of boththe absorption coefficient and transport scattering coefficient ofthe mist at every wavelength should take into account the localsize distribution of water droplets. This can be done using the fol-lowing relations for absorption coefficient and transport scatteringcoefficient of the mist [21,24] (hereafter the subscript k is omittedfor brevity):

fa;rtrg ¼ 0:75f va30

Z 1

0fQ a;Q

trs ga2FðaÞda ð2Þ

where FðaÞ is the size distribution function. It is convenient to usethe following traditional notation:

aij¼Z 1

0aiFðaÞda

�Z 1

0ajFðaÞda ð3Þ

In the case of i ¼ jþ 1, the values of aij are the average radii ofdroplets.

The integration according to Eq. (2) would strongly increase thecomputational time. Fortunately, the so-called monodisperseapproximation when all the particles are assumed to have thesame Sauter radius, a32, is often applicable [24,25]:

a ¼ 0:75f va32

Q a rtr ¼ 0:75f va32

Q trs ð4Þ

It was shown in [24] that this simplification may lead to consider-able errors in the case when particles of different size have differentvelocities and/or temperatures. Of course, more detailed calcula-tions should be made to estimate these errors in the problem underconsideration. It should be recalled that monodisperse approxima-tion is inapplicable for thermal radiation calculation in the case ofa strong difference in temperatures of particles of different size.The known examples are: the thermal radiation of particles inplasma spraying [26,27], the radiation from two-phase combustionproducts in exhaust plumes of aluminized-propellant rocket engi-nes [28], and the radiative cooling of core melt droplets in nuclearfuel–coolant interaction [29,30]. Nevertheless, the monodisperseapproximation is used in the present paper to simplify the mathe-matical procedure. A verification of this assumption might be a sub-ject of a separate study.

It should be noted that the above consideration is based on thewidely used hypothesis of independent scattering [31]. It meansthat each droplet is assumed to absorb and scatter the radiationin exactly the same manner as if other droplets did not exist. Inaddition, there is no systematic phase relation between partialwaves scattered by individual droplets during the observation timeinterval, so that the intensities of the partial waves can be addedwithout regard to phase. In other words, each particle is in thefar-field zones of all other particles, and scattering by individualparticles is incoherent.

3. Radiative transfer model

To choose relatively simple but physically sound radiativetransfer model, consider the main characteristics of the real

202 L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209

problem. First of all, the optical thickness of the mist layer shouldnot be too small to reach a significant attenuation of the incidentflame radiation. Therefore, the problem under consideration ischaracterized by multiple scattering at least in the range of watersemi-transparency where the scattering cannot be neglected. Inthe case of multiple scattering, the details of scattering phase func-tion are not important and one can use the simplest transportapproximation [24,32].

It would be also good to use a set of local 1-D problems insteadof considerably more complicated 2-D radiative transfer problem.The use of 1-D solutions is not only the obvious way to simplifythe mathematics. It is important that 1-D problems can be solvedwith sufficiently accuracy using simple differential approximationwithout any additional iteration. Note that a similar approachbased on a set of 1-D solutions in the case of relatively small 2-Deffects has been recently used in radiative transfer calculations[33].

The schematic presentation of the problem in Fig. 3 makes clearsome other assumptions of the computational model: (1) the mistof water droplets is generated by a set of small nozzles at the top ofthe mist layer; (2) the flat mist layer of constant thickness is con-sidered in the model; (3) one surface of the mist layer is diffuselyirradiated by the flame which is also flat but a variation of radiativeflux with the height is included in the model.

It is assumed that a protected wall exposed by the transmittedradiation is relatively cold and reflection of the radiation from thewall is negligible. An approximate radiative transfer model used inthe present paper is based on a set of 1-D problems for several hor-izontal layers of the mist curtain. It is assumed that all the mistparameters in every layer are constant and there is no radiativetransfer between the neighboring layers (in z-direction). As aresult, the radiation model is the z-direction is similar to theLarge-Cell Model suggested for multiphase flows in paper [29].Of course, the polarization effects due to scattering of radiationby water droplets are negligible, and one can use the scalar radia-tive transfer theory. With the use of transport approximation, the1-D radiative transfer equation (RTE) across the mist layer can bewritten as follows [24,34,35]:

l @I@y

þ btrI ¼rtr

2

Z 1

�1Iðy;lÞdl l ¼ cos h 0 < y < d ð5Þ

where Iðy;lÞ is the spectral radiation intensity at point y in direc-tion l (after the integration over the azimuth angles), btr ¼ aþ rtr

is the transport extinction coefficient. The boundary conditions attwo surfaces of the mist layer are written as follows:

Ið0;lÞ ¼ 2pef IbðT fÞ Iðd;�lÞ ¼ 0 l > 0 ð6Þ

layer N

layer 2

0

Hd0 y

prot

ecte

d su

rfac

e

diff

use

radi

atio

n fr

om fi

re

mist layer

water droplets

z

layer 1

Fig. 3. Scheme of the problem for diffuse irradiation of water mist layer.

where Ib is the Planck function. The above boundary condition atthe irradiated surface of the mist denotes that we use the simplestassumption of an optically gray fire radiation in the model problem.In other words, the external spectral radiative flux is assumed to bedirectly proportional to the blackbody radiation at temperature T f .The coefficient ef is the conventional constant hemispherical emis-sivity of the flame. It means that integral radiative flux from unitsurface area of the flame is expressed as follows:

qf ¼ efZ 1

0pIbðT fÞdk ¼ efr0T

4f ð7Þ

where r0 is the Stefan–Boltzmann constant. This approach is oftenused in engineering calculations of fire radiation [36].

The diffuse irradiation of the mist makes the problem underconsideration much simpler than the problem of shielding of solarradiation by water mist considered in paper [37] because there isno need in a separate consideration of directed and diffuse radia-tion and the two-flux method can be employed immediately (notonly to the diffuse component of the radiation field). Accordingto the two-flux method (Schuster–Schwarzschild approximation[35]) the radiation intensity is presented as a combination of twoangular independent components in backward and forwardhemispheres:

Iðy;lÞ ¼ 2pef IbðT f Þ �J�ðyÞ;l < 0JþðyÞ;l > 0

�ð8Þ

After integration of the RTE over two hemispheres, one canobtain the following boundary-value problem for the dimension-less function g ¼ J� þ Jþ [24]:

� ddy

Ddgdy

� �þ ag ¼ 0 D ¼ 1=ð4btrÞ ð9aÞ

y ¼ 0; Ddgdy

¼ ðg � 2Þ=2 y ¼ d; Ddgdy

¼ �g=2 ð9bÞ

where D ¼ 1=ð4btrÞ is the spectral radiation diffusion coefficient.The dimensionless spectral radiative flux from the shadow side ofthe mist layer is

�q ¼ qðdÞ2pef IbðT f Þ ¼

gðdÞ2

ð10Þ

The normalized profile of integral (over the spectrum) radiationpower absorbed in the mist is determined as follows:

�WðyÞ ¼Z 1

0�wðyÞdk �wðyÞ ¼ aðyÞgðyÞ ð11Þ

Note that the value of �W is expressed in m�1. It should berecalled that �w is the spectral value.

It is convenient to introduce dimensionless optical thickness ofmist layer measured from the irradiated surface:

strðyÞ ¼Z z

0btrdy str;0 ¼ strðdÞ ð12Þ

In dimensionless variables, Eqs. (9a) and (9b) can be written asfollows:

d2gds2tr

� n2g ¼ 0 n ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�xtr

pxtr ¼ rtr=btr ð13aÞ

str ¼ 0;dgdstr

¼ 2ðg � 2Þ str ¼ str;0;dgdstr

¼ �2g ð13bÞ

where xtr ¼ rtr=btr is the spectral transport albedo of the medium.In the case of xtr independent of str, one can obtain the followinganalytical solution to the boundary-value problem (13a,b) atxtr < 1 (n–0):

L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209 203

g ¼ 4ðE2tr;0=Etr � cEtrÞ

ð2þ nÞðE2tr;0 � c2Þ ð14aÞ

c ¼ 2� n2þ n

Etr ¼ expðnstrÞ Etr;0 ¼ expðnstr;0Þ ð14bÞ

The analytical expressions for the dimensionless spectral radia-tive flux at the shadow side of the mist layer and the profile ofabsorbed radiation power are as follows:

�q ¼ 2Etr;0ð1� cÞð2þ nÞðE2

tr;0 � c2Þ ð15Þ

WðzÞ ¼ 2pefZ 1

0

�wðzÞIbðT fÞdk �w ¼ 4aðE2tr;0=Etr � cEtrÞ

ð2þ nÞðE2tr;0 � c2Þ ð16Þ

In real situations, the transport albedo may be not constantacross the mist layer and one need a numerical solution to theboundary-value problem (9a,b). Of course, it is not difficult andcan be easily done. Nevertheless, it is interesting to consider thesimplest case of xtr ¼ const.

To illustrate the above derived analytical solution, consider acase study for the uniform layer of water mist with the averageSauter radius of droplets from a32 ¼ 3 lm to a32 ¼ 30 lm at vari-ous values of an ‘‘attenuation parameter”, A, of the mist layer.The latter dimensionless parameter is defined as follows:

A ¼ f vda32

ð17Þ

0.0

0.2

0.4

0.6

0.8

1.0

a = 3 μm 5 μm12

λ, μm

ωtr

a

1 2 3 4 5 6

1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

a = 10 μm 20 μm 50 μm12

λ, μm

ωtr

b

Fig. 4. Transport scattering albedo of monodisperse water mists: 1 – Mie theory, 2– approximation (1a–c).

The following characteristics of the flame radiation are considered:

T f ¼ 1500 K ef ¼ 0:9 ð18ÞConsidering first the calculated values of xtr in the most importantspectral range one can see in Fig. 4 that water mist containing dro-plets with radius a32 < 20 lm is a weakly-scattering medium onlyin the wavelength range of 2:7 < k < 3:2 lm, whereas the scatter-ing is considerable or even predominant outside this range. As tothe short-wave range of k < 2:5 lm, the scattering cannot beignored even in the case of a32 ¼ 50 lm (see Fig. 4b). It is also con-firmed by the data of Fig. 4 that analytical relations (1a)-(1c) give anacceptable approach which can be used instead of time-consumingMie theory calculations. Therefore, these approximate relations areused in subsequent calculations.

The main results of radiative transfer calculations are presentedin Fig. 5. The integral (over the spectrum) transmitted radiativeflux is determined as follows.

qt ¼ 2pefZ 1

0

�qIbðT f Þdk ð19Þ

According to Fig. 5, the value of qt decreases strongly with theeffective thickness of water mist layer. The curves of qtðAÞ in theimportant range of 3 < a32 < 30 lm are similar to each other andthe value of qt at a fixed value of A decreases slightly with the dro-plet size. So, the attenuation parameter A is the most importantquantity which determines the radiative flux transmitted throughthe mist layer.

It is also interesting to determine the radiative flux reflectedfrom the irradiated surface of the mist layer. The dimensionlessreflectance can be obtained as follows:

Rh ¼ gð0Þ � 1 ð20ÞWith the use of analytical solution (14a) we have:

Rh ¼ 42þ n

1� c=E2tr;0

1� c2=E2tr:0

� 1 ð21Þ

In the limiting case of purely absorbing (non-scattering) med-ium (n ¼ 2 and c ¼ 0) this leads to the physically obvious resultof Rh ¼ 0. The opposite case of a scattering but non-absorbing med-ium (n ¼ 0) cannot be considered using the above solution becauseof obvious degeneration of the Eq. (13a) [24]. At the same time, onecan use Eq. (21) to obtain the following important relation for thelimit of optically thick mist layer:

Rh ¼ 1� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�xtr

p

1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�xtr

p ð22Þ

1 2 3 4 5 6 720

40

60

80

100

120

140

a32

3 μm 5 μm 20 μm 30 μm

A

q t , kW

/m2

Fig. 5. Radiative flux transmitted through the mist layer.

204 L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209

Eq. (22) describes correctly the increase of radiation reflectionwith the medium transport albedo and can be used in engineeringestimates.

Considering now a variation of the total radiation power, W tot,absorbed in the mist:

W tot ¼ 2pefZ 1

0�wtotIbðT fÞdk ð23Þ

This quantity is important because there is an obvious relationbetween the value of W tot and the required flow rate of water. Onecan use the following radiative balance equation for the spectralvalues:

�wtot ¼ 0:5ð1� RhÞ � �q ð24ÞHaving substituted Eqs. (15) and (21), the following analytical rela-tion can be obtained:

�wtot ¼ 1� 22þ n

1þ ð1� cÞ=Etr;0 � c=E2tr;0

1� c2=E2tr:0

ð25Þ

The results of calculations with the use of analytical solution(25) are presented in Fig. 6. One can see that the absorbedradiation power W tot increases monotonically with both theattenuation parameter and the droplet size but the effect of theseparameters is relatively insignificant in the ranges of A > 3 anda32 > 20 lm. This is an important qualitative result which shouldbe further examined on the basis of more detailed analysis of realwater mists.

It should be noted that the formal use of the results obtained inthe above radiative transfer analysis is insufficient to chose ashielding mist to protect an object wall from the intense fire

40

60

80

100

120

140

160

180

30 μm

20 μm

10 μm

5 μm

a32 = 3 μm

A

Wto

t , kW

/m2

a

1 2 3 4 5 6 7

5 10 15 20 25 3080

100

120

140

160

180

10

5

3

a32, μm

A = 2

Wto

t , kW

/m2

b

Fig. 6. Integral radiation power absorbed in the mist layer.

radiation. The mist containing relatively small droplets looks morepromising because of great value of the attenuation parameter butthe volumetric absorption of the incident radiation near the irradi-ated surface of the mist and small velocities of the falling dropletswill lead to high rate of the mist evaporation. In the opposite caseof very large droplets, the radiation is not practically reflected fromthe mist because of low scattering and also a considerable attenu-ation of the fire radiation can be reached only in the case of a geo-metrically thick mist layer with high flow rate of water. Of course,the latter variant is not the optimal one. Most likely, the droplets ofan average size may be a good choice.

One should recall that thermal conditions near the irradiatedsurface of the mist and the conditions at the opposite shadow sideof the mist are quite different. This makes interesting the use ofmore sophisticated engineering solution with a variable size ofsupplied water droplets across the mist layer. It is natural to have

relatively large droplets with average radius að1Þ32 at the irradiated

side and much smaller droplets with radius að2Þ32 << að1Þ32 at the sha-

dow side of the mist layer. This idea will be examined on the basisof the case study considered below.

4. Transient heat transfer model

Strictly speaking, the heat transfer model for water mistexposed by thermal radiation from fire should be based on CFDmodeling of the flow field and convective heat transfer in combina-tion with radiative heat transfer modeling. The general problem istoo complicated especially because of possible dynamic and ther-mal non-equilibrium of evaporating water droplets. On the otherhand, the practical sense of detailed modeling is not obvious atthe moment because of great uncertainty in many parameters ofparticular processes. In the present paper, a simplified problemstatement is considered without some details which can beignored at this stage of the research.

First of all, it is assumed that the shape of the main streamregion can be presented as a plane-parallel layer (see Fig. 3) andthe effects of viscous boundary interaction with ambient air canbe ignored. Following the above principal suggestion, we considerthe mist containing two separate layers. It is assumed that the firstlayer (from the flame side) contains large droplets and the secondlayer contains relatively small droplets. From the engineering pointof view, such approach will provide more stability to the curtain asthe larger droplets have higher momentum where as the smallerdroplets will achieve higher attenuation.

Generally speaking, the gas flow with suspended droplets ischaracterized by dynamic and thermal nonequilibrium of waterdroplets with respect to the gas flow. But this effect is degeneratedin the limits of both large and small droplets. In the first case, thedecrease in droplet velocity due to the drag force is relativelysmall, whereas the motion of small droplets is determined by thegas flow.

The specific effects on parameters of small water droplets in theentrance region of the flow characterized by and also the hydrody-namic effects in the vicinity of the ground surface that is impingedby a mist flow are not considered. This is a natural assumptionbecause the region of the mist formation may be positioned at agreater height that the fire and the part of fire characterized by asignificant thermal radiation is usually observed at some distancefrom the ground. In other words, we consider a middle part ofthe long mist layer.

The following modes of heat transfer seem to be the mostimportant and should be included in the computational model:the heating of water droplets by external radiation from fire, par-tial evaporation of these droplets, and the falling flow of the mist.One cannot exclude that turbulent heat transfer across the mist

L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209 205

layer should be also taken into account, but the liming volume ofjournal paper does not allow to analyze this effect.

The radiative heating of water droplets can be calculated usingthe field of radiation power absorbed in the mist. This part of thegeneral model was a subject of previous sections of the paper.The droplet size is not constant across the composite mist layerand also because of evaporation under the action of nonuniformabsorption of fire radiation. As a result, the medium transportalbedo is also not constant and one should use numerical solutionto the boundary-value problem (13a,b) instead of analytical solu-tion (14a,b).

It is assumed that water droplets are isothermal and their tem-perature is the same as that of ambient gas. It means that radiationpower is spent to heat both droplets and gas and also to evaporatethe droplets. The simplest equilibrium evaporation model is con-sidered and the evaporation at temperatures less than the satura-tion temperature at normal atmospheric conditions, Ts ¼ 373 K, isneglected. Possible overheating of water droplets is also not con-sidered in the model. The effects of surface tension are neglected.

The nonuniform volumetric heating of large water droplets andmore accurate analysis of droplet evaporation in presence of ther-mal radiation are not considered in the present paper. The detailsof more comprehensive models can be found in the literature[38–42]. Note that the monodisperse approximation is employedfor both layers of the mist and the evaporation is treated as theonly effect resulting in change of the droplet size. In other words,possible effects of agglomeration or fragmentation of droplets arenot considered.

The falling flow of the mist is not calculated in the paper andeffects of steam generation and lift forces due to the gas heatingare not taken into account. Instead, two parallel uniform flowswith constant velocities are considered. Of course, this is a strongassumption, but this approach seems to be an important stage ofthe general study for possible gradual size variation of supplieddroplets on the basis of much more detailed CFD calculations.

An applicability of the assumption of local dynamic equilibriumof small water droplets and ambient air flow can be analyzed onthe basis of the following equation for the motion of a sphericalwater droplet in viscous gas in z-direction (see Fig. 3) [43]:

43pa3qw

dud

dt¼ 1

2CDpa2qðu� udÞ2 ð26Þ

where u and ud are the gas and droplet velocities, q and qw are thedensities of gas and water, CD is the drag coefficient. For slow rela-tive motion (when Reynolds number Red ¼ 2qju� udja=g << 1, thecase of small particles – the Stokes flow) the drag coefficient is asfollows:

CD ¼ 24=Red ð27Þand Eq. (26) can be written in the form:

treldud

dt¼ u� ud trel ¼ 2

9qwa

2

gð28Þ

where trel is the relaxation time. Note that it is sometimes conve-nient to introduce the Stokes number defined as Stk ¼ treljr~uj toestimate the dynamic nonequilibrium of inertial particles in a vis-cous medium [43,44]. In our case, it is sufficient to compare directlythe length of the droplet relaxation path and the height of a singlehorizontal layer of the computational region (see Fig. 3). Accordingto [9], the following value of an average initial velocity of water dro-plets is used in the estimates:

ud;0 ¼ 5 m s�1 ð29ÞThe calculations for water droplets with radius of a ¼ 30 lm

showed that ud;0trel � 54 mm << DH ¼ H=N at H ¼ 10 m and

N 6 100. It means that dynamic nonequilibrium of small waterdroplets can be neglected in the simplified computational model.It assumed also that there is a thermal equilibrium between thedroplets and ambient gas in every horizontal layer of small dro-plets. This approach is not good for the upper layer but it seemsnot so important for the simplified model problem. As a result,the only local temperature is used at every point of the secondlayer of the mist. Moreover, it is assumed that the gas mediumwith suspended particles can be treated as an equivalent continu-ous medium with average local dynamic and thermal parameters.The initial velocity of the medium in this mist layer is determinedas follows:

u2 � ud;0f v;inqw=qg ð30ÞOn the contrary, it is assumed that velocity of large water dro-

plets in the first layer of the mist is constant and the presence ofambient gas has no effect on the droplet motion. As to variationof the droplet temperature from the initial value to the saturationtemperature, it is determined by the local volumetric absorption ofthe fire radiation.

The computational model is based on dividing the mist volumeinto several horizontal layers for subsequent solution of a set ofcoupled 1-D heat transfer problems for every layer of the samethickness starting from the upper layer j ¼ 1 (see Fig. 3). The num-ber of layers, N, was varied in the methodological calculations toreach an acceptable computational error.

In the case of negligible turbulent heat transfer across the mistlayer, the approximate mathematical formulation of a heat transferproblem for every layer is as follows:

ðqcÞjuT jþ1 � ðyÞ � T jðyÞ

DH¼ W rad;jðyÞ T1ðyÞ ¼ T0 j ¼ 1; . . . ;N � 1

ð31aÞwhere

u ¼ u1 and ðqcÞj ¼ f v;jðqcÞw at 0 < y < y� ð31bÞ

u ¼ u2 and ðqcÞj � ðqcÞg þ f v;jðyÞðqcÞw at y� < y < d ð31cÞIt is assumed here that f v;j << 1 and variation of volumetric

heat capacity of gas mixture due to evaporation of water isinsignificant.

Eq. (31a) can be considered as a result of the use of an explicitfinite-difference scheme for the obvious differential equation. Inthe case of T�

jþ1ðyÞ 6 Ts, where Ts ¼ 373 K is the saturation temper-ature, the value of T�

jþ1 is the real temperature and T jþ1ðyÞ ¼ T�jþ1ðyÞ.

When the formal use of Eq. (30a) gives T�jþ1ðyÞ > Ts, the saturation

temperature is reached: T jþ1ðyÞ ¼ Ts. As to the current volume frac-tion of water droplets, it can be estimated using the followingrelation:

f v ;jþ1ðyÞ ¼ f v;jðyÞ 1� qgW rad;jðyÞ

LDHu

T�jþ1 � Ts

T�jþ1 � Tj

" #ð32Þ

Obviously, W rad;jðyÞ ¼ 0 at f v ;jðyÞ ¼ 0 and negative values off v ;jþ1ðyÞ should not appear in correct calculations.

According to the traditional classification of combined heattransfer problems [45], the above problem is an example of theso-called radiative heat transfer in moving media. The radiativeboundary layer (an outer part of the thermal boundary layer inthe case of optically thin viscous boundary layer) and the liquiddroplet radiator for space applications are two known examplesof the same class of heat transfer problems [21,24]. In ourparticular case, the problem is a bit more complicated because oftwo-layered medium and evaporation of water droplets.

206 L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209

5. Results for the case study

The following values of input parameters were used in the casestudy: H ¼ 10 m, d ¼ 1 m, T0 ¼ 300 K, qw ¼ 103 kg m�3, qg ¼1 kg m�3, f v;1 ¼ f v;in ¼ 10�4, að1Þ

32 ¼ 100 lm, að2Þ32 ¼ 30 lm,

u1 ¼ ud;0 ¼ 3 m s�1, cw ¼ 4:18 kJ kg�1 K�1, cg ¼ 1 kJ kg�1 K�1,L ¼ 2:26 MJ kg�1. Three variants of thickness of the front

0 2 4 6 8 1040

60

80

100

120

123

z, m

q t , kW

/m2

Fig. 7. Radiative flux transmitted though the mist layer of height 10 m at differentthickness of the front layer of supplied large water droplets: 1 – y� ¼ 0 (no largedroplets), 2 – y� ¼ 0:2 m, 3 – y� ¼ 0:5 m.

0

20

40

60

80

100

123

y, m

a 32, μ

m

a

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

123

y, m

fv

b

Fig. 8. Profiles of Sauter’s mean radius of water droplets (a) and local volumefraction of water (b) at the lower cross section of water mist layer: 1 – y� ¼ 0, 2 –y� ¼ 0:2 m, 3 – y� ¼ 0:5 m.

(irradiated) vertical layer of the mist are considered: y� ¼ 0,0:2 m, and 0:5 m. The first of these variants corresponds to the uni-form mist containing only small water droplets, two other variantsillustrate the effect of large droplets supplied from the side of fire.Note that numerical integration over the spectrum was conductedfor the wavelength interval from k1 ¼ 0:3 lm to k2 ¼ 6 lm. Theuniform step of Dk ¼ 0:02 lm was used in the spectral integration.

Some results of calculations for the case study are presented inFigs. 7 and 8. One can see in Fig. 7 that the presence of large dro-plets at the irradiated side of the mist layer may decrease thetransmitted radiative flux to the lower part of the protected object.This is explained by more favorable profile of the droplet size andthe resulting volume fraction of water droplets (see Fig. 8). Thisparticular result confirms potential advantages of the suggesteddecrease of the size supplied water droplets with the distance fromthe irradiated surface of the mist layer.

6. Possible microwave monitoring of water mist parameters

Microwave absorption and scattering by water droplets havebeen studied during many years because of developing applica-tions in remote sensing of the ocean ant atmosphere [46–48]. Inthis paper, possible microwave monitoring of the main parametersof a water mist is considered because this technique might be animportant part of the mist shielding properties. To analyze a possi-bility of such monitoring, consider first the microwave optical con-stants of water in the millimeter and centimeter spectral ranges.Following [49], the complex dielectric constant of water is deter-mined by the Debye relaxation model and the following modifiedequations:

2.0

2.5

3.0

3.5

4.0

4.5

5.0

3

2

n

λ, mm

1

a

0.5 1.0 1.5 2.0 2.5

0.5 1.0 1.5 2.0 2.50.5

1.0

1.5

2.0

2.5

3.0

2

3

1

κ

λ, mm

b

Fig. 9. Indices of refraction (a) and absorption (b) for water in sub-millimeter andmillimeter wavelength ranges: 1 – T ¼ 20 �C, 2 – 50 �C, 3 – 80 �C.

L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209 207

e0 ¼ e1 þ ðes � e1Þð1þ ð�ksÞ1�uSÞ1þ 2ð�ksÞ1�uSþ ð�ksÞ2�u

ð33aÞ

e00 ¼ es � e11þ 2ð�kSÞ1�uSþ ð�kSÞ2�u

ð�ksÞ1�uC þ relk18:8496

10�11 ð33bÞ

�ks ¼ ks=k S ¼ sinðup=2Þ C ¼ cosðup=2Þ ð33cÞwhere ks is the relaxation wavelength, e1 and es are the high- andlow-frequency dielectric constants. Hereafter, the values of k andks are expressed in millimeters. Eqs. (33a)-(33c) are reduced tothe classical Debye equations when the electrical conductivity rel

and the spread parameter u are equal to zero. In calculations forwater, the constant value of rel ¼ 12:5664 � 108 (Ohmmm)�1 wasused and other parameters were obtained using the followingrelations:

e1 ¼ 5:27137þ 0:0216474T � 0:00131198T2 ð34aÞ

u ¼ �16:8129=ðT þ 273Þ þ 0:0609265 ð34bÞ

ks ¼ 0:00033836exp½2513:98=ðT þ 273Þ� ð34cÞ

es ¼78:54ð1�4:579 �10�3~Tþ1:19 �10�5~T2�2:8 �10�8~T3Þ ~T ¼ T�25

ð34dÞNote that temperature T in the above equations is expressed incentigrade degrees.

0.1

1

10

Qa, Q

str

Qa

Qstr

2

a, μm

1a

24 26 28 30 32 34

25 30 35 40 45 50

0.01

0.1

1

10

Qa

Qstr

Qa, Q

str

2

a, μm

1b

Fig. 10. Efficiency factor of absorption and transport efficiency factor of scatteringfor water droplets in the sub-millimeter range at wavelengths 0.5 mm (a) and0.7 mm (b): 1 – T ¼ 20 �C, 2 – 80 �C.

The calculated spectral dependences of refraction andabsorption indices of water in the most promising parts of sub-millimeter and millimeter spectral ranges are presented in Fig 9.The index of refraction of water increases both with wavelengthand temperature. The latter is true at wavelength k > 0:85 mm.The spectral and temperature dependences of absorption indexin a long-wave part of the considered spectral range are morecomplex (Fig. 9b). It is clear that one cannot expect a significantabsorption of radiation by water droplets at k > 2 mm where theindex of refraction is too large [24]. At the same time, a combina-tion of moderate values of n and fast increasing values of j in thesub-millimeter wavelength range will result in unusual opticalproperties of single water droplets.

Consider the optical properties of small water droplets typicalof water mist. As above, we assume that droplets are spherical,their volume fraction is small and they are randomly positionedin space. The last assumptions are important but insufficient toneglect the so-called near-field dependent scattering effects. Oneshould recall the effect of coherent microwave scattering byclusters of water droplets is sometimes observed in cumulusclouds in turbulent atmosphere [50–52]. We assume that there isno such an effect in the problem under consideration and thehypothesis of independent scattering by single water droplets ofmists is true also in the microwave spectral range.

Some results of calculations for single droplets are presented inFigs. 10 and 11. The limiting volume of the paper makes possibleto illustrate only the most interesting effects observed insub-millimeter spectral range. The strong resonances of absorptionlike those studied in detail for gold nanoparticles in the so-called

0.0

0.2

0.4

0.6

0.8

ωtr

2

a, μm

1

a

5 10 15 20 25 30 35 40 45

10 15 20 25 30 35 40 45 500.0

0.2

0.4

0.6

0.8

ωtr

2

a, μm

1

b

Fig. 11. Transport albedo of a monodisperse water mist in the sub-millimeter rangeat wavelengths 0.5 mm (a) and 0.7 mm (b): 1 – T ¼ 20 �C, 2 – 80 �C.

208 L.A. Dombrovsky et al. / International Journal of Heat and Mass Transfer 96 (2016) 199–209

therapeutic window (0:6 < k < 1:4 lm) are observed. Thesenanoparticles are widely used in hyperthermia of tumors and othermedical applications related with targeted heating of biological tis-sues [53–55]. But it is not the same effect because of multiple res-onances and the accompanying resonances of scattering. The effectof water temperature appears to be also significant. Note that sim-ilar data for resonance absorption of sub-millimeter radiation bysingle water droplets have been recently reported in [56]. The com-putational data for the transport scattering albedo presented inFig. 11 are especially important because they indicate possiblemonitoring of the main parameters of water mists. Of course, itis a separate problem which is beyond the scope of the presentpaper. Therefore, we limit our consideration by computationalresults for the normal-hemispherical reflectance of a uniform opti-cally thick mist [57]:

Rn�h ¼ xtr

1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�xtr

p� �1þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�xtr

p� � ð35Þ

Fig. 12a indicates that the measurements of Rn�h at wavelengthk ¼ 0:5 mm can be used to estimate an average droplet tempera-ture in the case of 10 < a < 25 lm without more detailed informa-tion on size of water droplets. As to the average droplet size, it canbe obtained in the case of 40 < a < 50 lm from the measurementsat wavelength k ¼ 0:7 mm, and the result will be independent ofdroplet temperature (see Fig. 12b). Note that possible measure-ments of reflectance from the shadow side of the mist layer willgive us average values of both the droplet size and temperaturein this relatively cold and not evaporated region of the mist. Moredetailed information on variation of the retrieved values with

0

20

40

60

80

R n-h, %

2

a, μm

1

a

10 15 20 25 30 35 40 45 50

10 15 20 25 30 35 40 45 500

20

40

60

80

R n-h, %

2

a, μm

1

b

Fig. 12. Normal-hemispherical reflectance of an optically thick water mist layer inthe sub-millimeter range at wavelengths 0.5 mm (a) and 0.7 mm (b): 1 – T ¼ 20 �C,2 – 80 �C.

the distance from the shadow surface of the mist layer can beobtained from the measurements of directional–hemisphericalreflectance at various angles of incidence.

7. Conclusions

A simplified theoretical model for attenuation of fire radiationby water mist was developed. This spectral model is based on cal-culated absorption and scattering characteristics of water droplets,the local 1-D solutions for radiative heat transfer through the mistlayer, and transient heat transfer model taking into account heat-ing and evaporation of the droplets.

The computational data illustrate the main special features ofthe problem and enable one to estimate the effects of an averagedroplet size, the volume fraction of water, and the mist layer thick-ness on quality of the fire protection. The suggested analyticalsolution for radiative transfer and also the results of numericalanalysis of heat transfer in the mist are expected to be useful forphysically sound engineering estimates.

The case study for the fire radiation protection by water mist isconsidered. It was suggested to decrease the size of supplied waterdroplets with the distance from the irradiated surface of the mistlayer. The results of numerical calculations confirmed potentialadvantages of this engineering solution.

A possibility of microwave monitoring of water mist parame-ters was also considered in the paper. It was shown that importantinformation on average values of both the size and temperatureof water droplets can be obtained using the measurements ofdirectional–hemispherical reflectance of sub-millimeter radiationfrom the mist layer.

Conflict of interests

None declared.

Acknowledgments

This study was supported by the UK Royal Academy of Engi-neering (Grant DVF1415/2/22) and the Russian Foundation forBasic Research (Grants Nos. 13-08-00022a and 16-08-00157a).

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