Examination of radiative fraction of small-scale pool fires at reduced pressure environments

Journal Pre-proof Examination of radiative fraction of small-scale pool fires at reduced pressure environments Jiahao Liu, Zhihui Zhou PII:

S0379-7112(19)30418-7

DOI:

https://doi.org/10.1016/j.firesaf.2019.102894

Reference:

FISJ 102894

To appear in:

Fire Safety Journal

Received Date: 7 August 2019 Revised Date:

26 September 2019

Accepted Date: 16 October 2019

Please cite this article as: J. Liu, Z. Zhou, Examination of radiative fraction of small-scale pool fires at reduced pressure environments, Fire Safety Journal (2019), doi: https://doi.org/10.1016/ j.firesaf.2019.102894. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Examination of radiative fraction of small-scale pool fires at reduced pressure environments Jiahao Liua,*, Zhihui Zhoub,* a

College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 201306, China; b

China Waterborne Transport Research Institute, Beijing 100088, China;

Abstract: Determining flame radiative fraction is necessary in the estimation of radiation heat transfer of a fire, while the radiative fraction for a given fuel is usually regarded as constant, even at sub-atmospheric pressures. To ascertain the factors affecting this parameter, a series of small-scale n-heptane pool fires with diameters of 4~10 cm are performed at low pressures by means of an altitude chamber in this work. Based on a point source model, the radiative fractions of tested pool fires are determined, and coupled effects of pressure and pool diameter on this parameter are observed. A dimensional scaling theory is further proposed to interpret these effects, i.e.

~

.

.

.

, and the experimental results can be well correlated with this model. The

findings will be helpful in evaluating the radiative heat fluxes of pool fires at reduced pressures more accurately, further providing guidance on the detection of incipient fires at sub-atmospheric pressure environments. Keywords: Radiative fraction; pool fire; altitude chamber; sub-atmospheric pressure

1. Introduction The magnitude of the radiative transfer to targets external to the flame affects the hazard posed by a particular fire and influences the fire spread rate [1-4]. Radiative fraction (

), as an

important parameter representing the fraction of heat released in the form of radiation [3], can be expressed as

=

/ , where

is the energy radiated by the fire,

release rate of the fire. For a jet diffusion flame with constant

is the actual heat

, the radiative fraction is believed

to be affected mainly by the soot formation and flame volume, both of which are dependent on the ambient pressure ( ) [2, 5, 6]. Most et al. [7] investigated the effect of pressure on the diffusion

flames of ethane pool fire in the pressure range of 0.03~0.3 MPa, and they found that the radiative

~

fraction exhibited a very slightly decreasing tendency with the increasing pressure i.e.

/

.

Recent researchers also performed a series of experiments to examine the effect of low pressures on the radiative heat flux, and there seems to be a consensus that the radiative heat flux as well as radiative fraction displays a positive proportionality to pressure [5, 8]. Hu et al. [5] deduced a theoretical relationship of

~

/

, and verified it via propane jet fires at different altitudes.

Fang et al. [8] also found the weak dependence between them by employing methane fuel, empirically expressed in the forms of momentum-driven flames.

~

.

for buoyancy-driven flames and

~

.

for

The conclusions drawn above are mainly based on the gaseous fire tests with precisely controlled heat release rates, which are extremely different from the diffusive liquid pool fires, where the heat feedback from the flame to the fuel surface is critical to determine the heat release rate, and thereby affects the heat transfer from the flame in turn. For a buoyancy-controlled diffusion liquid pool fire, the heat release rate ( ) or burning intensity (

, termed as the burning

rate per unit burning area [9]) is also affected by the ambient pressure as well as the pool dimension [10-13], while the varied

will in turn change the soot yield and flame envelope.

These complex interactions will collectively impose effects on the radiative fraction, while the mechanisms for that still remain unclear. Comparative tests on n-heptane pool fires at two altitudes (Hefei city: 50 m, 101 kPa; Lhasa city: 3650 m, 64 kPa) have confirmed that the burning rate and radiation heat flux at higher altitude are lower [7, 12, 14]. Zhou et al. [14] developed a dimensional scaling theory to interpret the dependency of flame radiation fraction on pressure, i.e. the radiative fraction of a fire with fixed burner size and fuel flow rate can be theoretically expressed as

~

~

.

, where

is the flame temperature. As for the luminous n-heptane fuel at atmospheric pressure, its radiative fraction fluctuates within 0.33~0.4 , slightly depending on the pool dimension, when the pool diameter ( ) is less than 3 m [3, 15, 16]. Although previous researchers have indicated the pressure effect on radiative fraction according to the measured flame radiation and corresponding qualitative inference, their inherent physical relationship has not been revealed thoroughly and determined quantitatively. In effect, in their work, the radiative fraction is still believed to be weakly dependent on pressure, and is roughly regarded as a constant for simplification. Taking n-heptane as an example,

= 0.3 is

usually used for the simplified calculation regardless of the ambient pressure [12, 14]. In this work, small-scale n-heptane pool fires ( =4~10 cm) are conducted in the pressure range of 40~101 kPa by means of an altitude chamber. An attempt to identify the dependence of radiative fraction on relevant parameters is made through the measured radiative heat flux, mass loss, and flame height data. This is of great significance in obtaining accurate radiative fraction at different pressures and further estimating the accurate energy radiated from a fire.

2. Experimental setup A schematic diagram of the fire test in current study is shown in Fig. 1. The altitude chamber used here is identical to that in Refs. [17, 18], and its appearance, pressure controlling system and auxiliary systems are elaborated in Ref. [17]. Its internal dimension is 3(L)×2(W)×2(H) m with adjustable pressure ranges ranging from atmospheric pressure 101 to 40 kPa. The spatial arrangement of the experiment is also provided in Fig. 1. A SONY CCD camera with sampling frequency of 25 fps is 1 m horizontally away from the fire source to videotape the whole burning process. Four circular pans with diameters of 4, 6, 8, 10 cm and a depth of 2 cm are tested at four static pressures, i.e. 40, 60, 80 and 101 kPa. N-heptane with industrial purity above 99% is employed as the tested fuel, whose density is 683-685 kg/m3; boiling point is 98.5 oC.

Fig. 1. The schematic diagram of the altitude chamber tests The experimental arrangement is totally same to that in our previous work [17], except that

an array of three radiometers is vertically positioned at a horizontal distance of 0.6 m away from the pan center to measure the radiative heat fluxes. Their vertical interval is 10 cm, with the lowest one flush with the top of the pool rim, labeled as R1, R2 and R3 from the top down. The radiometers are manufactured by Medtherm Corporation (USA), which are a type of Gardon gauge, primarily intending for the measurement of radiation flux density from a field of view of 180°. The fuel thickness for all the tests is initially set as 1.5 cm, which is assigned to be thin-layer pool fire. Its combustion process can be partitioned into four typical stages, i.e. pre-burning, quasi-steady, boiling and decay stages [14, 17, 18], as shown in Fig. 2, where the measured radiative heat fluxes by the three radiometers exhibit the similar variation tendency as mass loss rate. For the closed altitude chamber, the effect of continuously deceasing oxygen concentration on the burning process should be considered. Assuming complete combustion, a general expression of residual oxygen mole fraction ( been derived in our previous work [18], i.e. = where

−

) for n-heptane pool fires in current chamber has 2.75 +

0.121 & ' + 1

is the initial oxygen mole fraction,

(1)

is the total mass of the fuel consumed over time.

For the extreme case of 10 cm pool fire at 40 kPa, the quantitative oxygen depletion over time can be estimated by Eq. (1), that is, 3.78% and 11.84% of the total for the combustion at the ends of quasi-steady and boiling stages, respectively [17]. Thus, considering the oxygen depletion in the chamber, only the data in quasi-steady stage with stable burning intensity, flame envelop as well as quasi-equilibrium heat feedback are intercepted to conduct the following analysis.

Fig. 2. The mass loss rate and measured radiative heat fluxes of 8 cm pool fire at 80 kPa In this work, each experimental configuration is repeated for at least three times to guarantee the reproducibility of the test. The method proposed by Moffat [19] is employed to estimate the

relative uncertainty of measurements, expressed as 2)/*̅ with the standard deviation ) = ,

0 ∑1 /23(./ .̅ )

4

, where *5 is the measured values of 6 set of repeated tests, *̅ is the average value.

As 95% of the data deviates within ±2) from the mean value, with the confidence of 95%, the

mean value must lie with the range of ±2) [19]. Take the critical parameter, burning intensity in

quasi-steady stage, as an example to show the analysis of experimental uncertainty in this work, as presented in Table 1. It is found that the relative uncertainty (2)/*̅ ) is less than 5%, reflecting

good repeatability of this work. The average values of repetitions for burning intensity, flame

height, radiative heat flux will be used for the following analysis. The relative uncertainties of flame height and radiative heat flux are less than 5% and 4%, respectively. Table 1. Measured burning intensities in quasi-steady stage of repeated tests and the estimated uncertainties. (cm)

(kPa)

Test #1

Test #2

Test #3

*̅ (g/s ∙ m )

2) (g/s ∙ m )

2)/*̅

101

6.59

6.66

6.79

6.68

0.20

2.99%

80

6.74

6.9

6.88

6.84

0.18

2.63%

4

60

7.16

7.14

7.45

7.25

0.34

4.69%

40

7.68

7.47

7.32

7.49

0.36

4.81%

101

9.27

9.11

9.07

9.15

0.22

2.40%

80

8.89

8.78

8.58

8.75

0.32

3.66%

60

7.96

8.11

8.2

8.09

0.24

2.97%

40

7.66

7.41

7.31

7.46

0.36

4.83%

101

9.19

9.1

9.4

9.23

0.30

3.25%

80

8.38

8.68

8.62

8.56

0.32

3.74%

60

7.25

7.37

7.61

7.41

0.36

4.86%

40

6.45

6.34

6.65

6.48

0.32

4.94%

101

10.4

10.12

10.29

10.27

0.28

2.73%

80

8.97

8.83

8.75

8.85

0.22

2.49%

60

7.5

7.66

7.55

7.57

0.16

2.11%

40

6.53

6.51

6.64

6.56

0.14

2.13%

6

8

10

3. Results and discussion 3.1 Basic data of burning intensity and flame height

Fig. 3. Mass burning intensity of n-heptane pool fire at atmospheric pressure in various literature To confirm the credibility of the current study, the burning intensities in quasi-steady burning stage at atmospheric pressure are plotted in Fig. 3 together with the results of n-heptane pool fires reported in various literature [6, 14, 16, 20-23]. It is perplexing to observe that the burning intensity for the same pool diameter is sometimes reported more than two values. This may be mainly attributed to the different range of mass loss curve that is intercepted for analysis. For example, Raj et al. [16] defined two plateaus to delineate the constant burning rate in quasi-steady and boiling stages, respectively, and thus, two distinct sets of data can be seen in Fig. 3. Similarly, different definitions of boiling stage by Zhou [14] and Kang et al. [20] also presents contrasting results. The burning intensities obtained by Fang et al. [6] are intercepted from the fully developed combustion period, which is determined as the region where the burning intensity approaches the maxima. Previous researchers tend to examine the burning behaviors in the fully developed period, while the uncertainty caused by the presence of boiling phenomena also imposes restrictions on that. By contrast, the burning intensities in quasi-steady burning stage are quite stable. The current results are well consistent with that obtained by Raj and Prabhu [16], Kang [20], Zhou et al. [14] during the quasi-steady stage, as the hollow symbols show in Fig. 3.

Fig. 4. Mass burning intensity versus pressure based on

∝

The average burning intensities in quasi-steady stage under different pressures are plotted in double logarithmic coordinate shown in Fig. 3, where the relative uncertainties are reflected as the error bars. Previous studies have indicated that the burning intensity has a power law dependence on the ambient pressure [6, 13, 14, 18], i.e.

∝

(2)

where the exponential factor A varies from 2 to negative values, depending on the dominating

heat feedback mechanism for a certain pool dimension [24]. The fitted results in Fig. 4 show that

A values are -0.12, 0.22, 0.39, 0.48 for 4, 6, 8, 10 cm pool fires, respectively, increasing with the

pool diameter. Current A values versus pool diameters are further presented in Fig. 5 together

with the results of n-heptane pool fires available in the literature for comparison [6, 13, 14, 25]. It

can be seen that A values in this work are quite approximate to that by Zhou et al. [14], but partly

differ from that by Fang et al. [6], especially for 10 cm pool. It is worthwhile to note that only the data by Wieser et al. [13] is obtained from the multi-pressure tests (fitted as that in Fig. 3), whereas the rest of the data are simply determined by the tests at two altitudes, Hefei and Lhasa, based on the deformation of Eq. (2), i.e.

BCDED /

FG G5

=H

BCDED / FG G5 I

. Practically, altitude

tests with at least three ambient pressures are necessary to determine A value. However, due to

the difficulties of conducting experiments at various altitudes, only limited data are available to

investigate the pressure effect on burning intensity. In Fig. 5, although A values scatter for pool

diameter larger than 10 cm, that for smaller ones are relatively stable, varying from negative to about 0.5. Based on the results, Fang et al. made an inference that a diameter of 10 cm may be the

turning point for A = 1, and they simply attribute the exceptional variations in A for diameters

of 10~15 cm to the vigorous, fully developed combustion at normal pressure [6].

Fig. 5. The variation of A value against pool diameter

With respect to small-scale pool fires, Drysdale [24] claimed that the dominated heat feedback mechanism is conduction as the burner diameter is less than 7 cm, and then the transition stage follows within 7~10 cm, and convection dominates within 10~20 cm. For pool diameter larger than 20 cm, the radiation heat feedback prevails, and the fuel regression rate tends to be independent of pool dimension [6, 12]. Through a unified analysis on n-heptane pool fires at two

altitudes, Zhou et al. [14] further suggested that apart from the radiative term (A > 1), dominated

heat feedback terms for small-scale pool fires can be partitioned into three parts with clear demarcation of A , i.e. conduction region ( A < 0 ), transition region ( 0 < A < 2/3 ), and

convection region (2/3 < A < 1). Thus, the dominated heat feedback mechanism is conduction for 4 cm pool fire with a negative A value [6, 14], while for 6~10 cm pool fires, they are

supposed to be in the transition region from conduction to convection, i.e. they may be influenced by both of them [17], and the convection part increasingly prevails with the increasing pool diameter, resulting in an increasing A value accordingly.

Fig. 6. Typical flame images and their corresponding binary ones of 4 and 10 cm pool fires at different pressures The flame height (L ) is determined by an image processing program based on MATLAB software, which has been successfully applied in previous work [18]. Specifically, the original video records are converted to grey-scale images and then binarized to gain the flame envelopes. Fig. 6 presents the typical flame images in quasi-steady stage and their corresponding binary ones of 4 and 10 cm pool diameters under different ambient pressures for comparison. Visually, the flame dimensions for different pools are quite obvious, i.e. larger pool exhibits higher flame height. The flames of 4 cm pool seem to be dominated by the laminar flow regime viscous forces, characterizing as smooth flames with constant motion. In contrast, for 10 cm pool, the flames tend to be turbulent, shown as the very irregular flame geometries with chaotic eddies, vortices and other flow instabilities. In general, buoyancy is the dominating driving force in natural pool fire flames. The flames are more prone to be turbulent as the pool diameter increases. Based on the work by Blinov and Khudiakov [26], Drysdale [24] concluded that the flames are laminar for

< 0.03 m; a transitional behavior between laminar and turbulent exists in the range of

0.03 m M

M 1.0 m; the fully turbulent flames occur for

> 1.0 m. Therefore, the current pool

fires are assigned to be in the transitional region in spite of the smooth flame envelopes for 4 cm

pool fires. The shifting of the flame envelope is also affected by buoyancy, which is directly

related to Grashof number, i.e. NO = PQ∆ S T /U~

[27]. This implies that the decreasing

pressure may result in flames with less turbulence. Additionally, it is worthwhile to note the flame stretching at lower pressures in Fig. 6, which is usually explained by that the flame under reduced pressure has to extend to a higher position to engulf enough air to fulfill the stoichiometric reaction [28]. Previous tests at different altitudes have demonstrated that the diffusion flame height can be ∗

correlated with dimensionless heat release rate ∗

=

TW XY

in a unified way [14], which is defined as

W ZP

(3)

/

where TW is the air density, XY is the specific heat of air,

W

is the ambient temperature, P is

the gravitational acceleration. According to the ideal gas equation, TW

W

= /[\ , [\ /XY = 1 −

1/], where [\ is gas constant for air, and ] (= 1.4) is ratio of specific heats for air, and thus,

Eq. (3) can be transformed as ∗

Since

=

(1 − 1/])

=

ZP

/

(4)

∆`a ∙ b , where ∆`a is the heat of combustion of the fuel, b is the

_

combustion efficiency which is experimentally demonstrated to be slightly influenced by the ambient pressure [29], and it is further assumed to be constant in this work. Thus, it has ∗

c =4

b

∆`a (1 − 1/])

ZP

/

~

/

(5)

Zukoski [30] proposed a simplified equation to correlate the diffusion flame height for

∗

≥ 1, i.e. L / = 3.3

∗ /

. Subsequently, Heskestad [31] provided an explicit expression

based on the virtual origin analysis, i.e. L / = 3.7 of 0.12 <

∗

∗ /

< 1.2 × 10 . In current study, the values of

− 1.02, which is valid in a wider range ∗

range from 1.04~3.06, lying in the

application conditions of these two formulas. Thus, the non-dimensional flame height (L / ) can be correlated by the 2/5 power of

∗

L Fig. 7 plots L /

correlated by

, i.e. ~

∗

/

~f

g

/

(6)

, where the fitted results are quite well, further

demonstrating the applicability of the proposed equations by Zukoski and Heskestad in this work. Combining Eq. (2), the flame height can be further deduced as

L ~f

g

/

~f

g

/

=

(

)/

/

(7)

This implies that the flame height is a function of pressure and pool dimension. It should be noted that the flame dimension plays an important role in determining its radiated energy, and thus Eq. (7) will provide basic theory for the following correlation of radiative fraction.

Fig. 7. The dimensionless flame height correlated with (

/

)

/

3.2 Determination and correlation of radiative fraction The total radiative heat flux from a flame can be determined by integrating the measured spatial distribution of radiant flux over a surface surrounding the fire [32]. However, this method is at times impractical due to the requirement for multi-point measurements of radiant heat fluxes. Many estimates of radiative heat flux or heat loss fraction reported in the literature rely on a single-point measurement [3, 5, 33]. In this work, three radiometers with different spatial distributions are employed to estimate the radiative fraction separately. Considering the similar variation tendency shown in Fig. 2, the pressure effect on radiative heat flux is consistent with that on burning intensities, as shown in Fig. 8, where the results measured by R1, R2, and R3 radiometers are presented. Note that the results in Fig. 8 are also the mean values of repeated experiments with the relative uncertainty less than 5%. It can be observed that for any pool diameters, the radiative heat flux measured by R3 is always the smallest due to its relatively larger

distance from the center of the flame, while the relative magnitude of the values measured by R1 and R2 varies, depending on the flame height.

Fig. 8. The radiative heat fluxes measured by the radiometers The radiative heat fluxes received by the sensors can be illustrated as Fig. 9. Modak [34] indicated that if the radiation receiver is mounted at a distance more than five times pool radii away from the axis of the burner, the flame radiation can be assumed as the radiative heat flux originating from a point source. In current study, the horizontal distance (i) between sensors and the axis of the burner is fixed as 0.6 m, which is much larger than five times pool radii even for 10 cm pool. Thus, the flame radiation is regarded as a point source in the mid-point height of the flame height above the burner [5]. In Fig. 9, ` is the vertical distance of the sensors away from

the upper rim of the burner, and [ the straight-line distance between the point source and the receiver, which can be calculated by ,i + HL /2 − `I . In this case, previous researchers [3, 5,

6, 34, 35] recommend the following equation to estimate the radiative fraction based on the measured radiation flux (j ), =

4c[ j

(8)

Fig. 9. Calculation of flame radiation fraction based on single-point method Hamins [3] found that a single location radiance measurement with a correction for anisotropic effects yielded estimates of

O

within 13% of that determined by multi-location

measurements. For point source model, Modak [34] indicated that estimation of

single-point measurement are correct within 10% at a measurement distance of 2.5

2% at a distance of 5 . Thus, the estimates of

O

O

based on and within

in this work should be correct with less than 2%

uncertainty since the measurement distance is larger than 5

even for 10 cm pool. All the

radiative heat fluxes measured in current work are employed to quantify the radiative fraction, and the average value for the measurements of three radiometers is regarded as the ultimate outcome for each configuration. The radiative fractions in current work are plotted in Fig. 10 against pressure, where the uncertainty of measurements is reflected as the error bars. It is shown that

O

exhibits a positive relationship with the increasing pressure. This is consistent with the findings at high [36, 37] and low pressure environments [5, 14], and is usually attributed to the decreasing soot formation at reduced pressure [14, 25]. Although the fitting result is unsatisfactory, a power law relationship between them, i.e.

∝

.

, is obtained.

Fig. 10 Pressure effect on

in this work

Under atmospheric pressure, Hamins et al. [3] found that the radiative fractions non-alcohol fuels (n-heptane included) range from 0.27 to 0.33. Raj and Prabhu [16] also presented the similar finding for n-heptane fires with pool diameters ranging from 0.3~1.0 m. Moreover, the radiative fraction will markedly decrease as the pool diameter is larger than 5 m, because eddies of black soot can obscure the flame radiation for larger-diameter fires, and the eddy size or soot path length increases as the fire diameter increases, causing the transmittance of the external eddies to decrease and block radiation from leaving the flame [38, 39]. The radiative fractions of n-heptane pool fires at 101 kPa in this work are found to increase from 0.27 to 0.37 for pool diameters considered, as depicted in Fig. 11 together with the radiative fractions reported in the literature. The current results are very close to that by Hamins et al. for pool diameters of 4.6 and 7.1 cm [3], while for pool diameters of 0.3~5 m, the radiative fraction seems irregular, varying in the range of 0.25~0.4. As discussed in Section 3.1, the dominated heat feedback mechanism for large-scale

pool fires ( ≥ 0.2 m) is the radiation originating from soot in the flame, which essentially differs

from that of small-scale pool fires ( M 0.1 m) in current study.

Fig. 11 Radiative fraction of n-heptane pool fires at atmospheric pressure In the altitude tests by Tu and Zhou et al. [12, 14], they claimed the radiative fraction of n-heptane pool fires should decrease slightly at lower pressure. Due to the limitation of available altitude pressures, the pressure effect on

O

seems not obvious, and thus,

O

is usually treated as

constant. However, Fig. 10 clearly shows a positive effect of pressure on radiative fraction. Apart from that, it also indicates that for a fixed pressure, the pool diameter may have a positive effect on the radiative fraction in the range of

=4~10 cm. In effect, for a given fuel,

O

is a very weak

function of pool size (7.6~122 cm) for both luminous and nonluminous flames [40]. However, this influence seems to be more apparent at low pressures (shown in Fig. 10), and needs further examination. Theoretically, the thermal radiation from a fire emanates from both gaseous species such as water vapor, carbon dioxide and carbon monoxide as well as from luminous soot particles [4]. Markstein [41] indicated that the irradiance of the burnt gas (non-visible) plume above a fire accounts for less than 10% of the mean irradiance of the visible fire. N-heptane is usually regarded as a moderately sooting fuels, which tend to produce relatively sooty flames even at reduced pressures, as exemplified in Figs. 6 and 9. Thus, it is assumed that the flame radiation mainly originates from the soot in the flame. For optically thin flames, the incident flame radiative heat flux (j ) can be estimated by [42]

j =l

m

n1 − exp(−ri )s (9)

where l

is the view factor from fire to the surface of the radiometer, m is the

Stefan-Boltzmann constant, r is the soot absorption coefficient, i

is the mean beam length. In

this work, all the sensors rightly face the flame. With an assumption of cylindrical flame envelope, the view factor to its parallel receiver can be estimated as [43] l

=

1 tan cu

(u − 1) 1 u−1 l l − 2u x z+ { tan } − tan } ~ (10) |(u + 1) u u+1 c u√ | √u − 1

= (1 + u) + l , | = (1 − u) + l . In current study,

where u = 2i/ , l = 2L / ,

u = 12~30 and l = 8.24~15.72. Based on Eq. (10), the view factor can be obtained, i.e. in the

range of 0.0123~0.0271, depending on the pool diameter and flame height. It also indicates that the view factors in this work vary slightly and can be assumed as constant for all the tests. Moreover, ri

is usually assumed as a very small value, especially at reduced pressure where

soot yield is even lower [14, 42]. Thus, Eq. (9) can be approximated as j ≈l

m

ri ~m

ri (11)

Specifically, r is positively related to the soot volume fraction (ۥ ), which depends on the

characteristic flow time (‚€ ) and the characteristic soot formation time (‚ƒ ) [6, 35, 44], i.e. r~€„ ~T

‚ ‚ ~ (12) ‚E ‚E

where ‚E is inversely proportional to pressure, ‚E ~

; for buoyant turbulent diffusion flame,

‚ is related to the Kolmogorov time, and it has been deduced as ‚ ~

/

S

/

[35], where the

characteristic flame length scale (S) has a positive relationship with flame height [5], i.e. S~L .

Thus, it has

Moreover, i

r~

/

S

/

=

/

L

/

(13)

is usually given with an optimal factor of 3.6, and conventionally expressed as [42,

44]

where …€ is the flame volume, †

i = 3.6

… (14) †

is the flame surface area. Further assuming that the

time-average flame shape is a cylinder (as evidenced by the flame contour in Fig. 9) with the diameter of

and a height of L€ [27], it has …~

L (15)

According to De Ris [42], if one has a convex flame volume … surrounded by a flame

surface † , the total radiation leaving the flame volume can be estimated by = † j (16)

Substituting Eqs. (11)~(15) into Eq. (16), we can obtain ~m

/

L

/

… ~m

L

/

/

(17)

Thus, the flame radiative fraction can be estimated as =

m ~c 4

/

L

/

∆`a ∙ b

~

m

L

/

/

(18)

It also should be noted that the temperature decreases as the radiation fraction increases according to previous work [5, 45], i.e.

m

~1/

(19)

Substituting Eqs. (7) and (19) into Eq. (18), the radiative fraction can be theoretically predicted by Eq. (20) in terms of pressure and pool diameter, ~‡

/

H

(

)/

/

I

/

ˆ

/

=

.

.

.

(20)

Fig. 12. The radiative fractions in this work correlated with Eq. (20) Fig. 12 depicts the radiative fractions of n-heptane pool fires in this work together with that available in the literature [3, 12, 16, 33, 38, 39] based on Eq. (20). It can be observed that the current results can well collapse, exhibiting a linear relation, better than that in Fig. 10. Meanwhile,

the radiative fractions of small-scale n-heptane pool fires with equivalent diameter (

G)

of 6.77 cm

at different altitudes (Lhasa and Hefei) by Hu et al. [33] and that with diameters of 4.6 and 7.1 cm at normal pressure by Hamins et al. [3] can also be unified by Eq. (20) . In these cases, the A

values are roughly estimated as the linear insertion values between the two adjacent pools in

current study. The linear fitting for these small-scale pool fires ( M 10 cm), shown as the hollow data points in Fig. 12, is relatively good with [ = 0.883, expressed as = 0.08

.

.

.

+ 0.11 (21)

However, it is also notable that the results for pool diameters larger than 0.3 m, the data points with dot center in Fig. 12, are still scattered, even for that at reduced pressure [12]. The value of .

.

.

largely depends upon the determination of A, while in most situations, A

values are not available, especially for larger pool dimensions at normal pressure. Fang et al. [6]

provided relatively abundant A values covering the pool diameters of 4.5~37.2 cm based on the

experimental results at two altitudes, as presented in Fig. 5. Their inference that A roughly

approaches 1 for pool diameter larger than 10 cm is also directly employed to estimate .

.

.

simplified as

for large-scale pool fires ( ≥ 0.3 ). In this case, Eq. (20) can be further

~

.

.

, which shows a weak dependence on both pressure and diameter,

but cannot interpret the existing data at normal pressure well. Thus, this model may not be applicable to the radiation-dominated pool fires, or A values for larger pool diameters need to be

determined more accurately to validate the feasibility of Eq. (20). As for the pool diameters within

10~30 cm, although relevant pool fire tests at different pressures are available in the Refs. [6, 11, 14], the radiative fraction is not involved in their work. As a whole, the model in Eq. (21) can well interpret the current results together with the small-scale n-heptane pool fires in the literature [3, 33], which covers the pressure range of 40~101 kPa and pool diameters of 4~10 cm. 3.3 Discussion The pool fires in this work are assigned to be small-scale ones, the dominated heat feedback modes of which are mainly conduction or convection. For conduction-dominated pool fires, the heat feedback (ja‰Š‹ ) is directly proportional to the excess flame temperature, i.e. ja‰Š‹ ~H Œ I,

where

Œ

−

is the boiling point of the fuel [24]. As the radiation heat loss decreases at lower

pressure due to the decrease in soot formation, flame temperature

increases with the

decreasing pressure, which has been confirmed by numerous experiments at both sub-atmospheric

and elevated pressures [28, 36, 37]. Additionally, to the increase in H

−

ΠI.

Œ

decreases at lower pressures, further leading

Therefore, the resultant relationship between the conductive item and

pressure can be estimated as ja‰Š‹ ~

with a negative A value. In the work by Zhou et al. [14],

the n-heptane pool fire tests in Lhasa and Hefei present the mean flame temperatures of 692 and 678 oC, respectively. Considering the difference in the boiling points at two altitudes (89.0 oC in Lhasa and 98.5 oC in Hefei), the variation in H

−

ŒI

is only 23.5 oC, which is quite small,

reasonably resulting in a relatively smaller A. Hu et al. [33] directly neglected this insignificant influence and claimed that ja‰Š‹ ~

. The experimental result of 4×4 cm n-heptane pool fires at

two altitudes gives a negative A [6], as shown in Fig. 5. Based on the very small ethanol pool

fires at elevated pressures, Chen et al. [10] also found that ja‰Š‹ ~ 1~2.5 atm and ja‰Š‹ ~

.

.

for 1.5 cm pool fires at

for 2.0 cm ones at 1~1.5 atm. Thus, 4 cm pool fires in current work

should be conduction-dominated ones with a relatively reliable A value (-0.12). With respect to

the convection-dominated pool fires, the heat feedback ja‰Š„ can be estimated as ja‰Š„ = ℎH

−

ΠI,

where ℎ is the convective heat transfer coefficient related to the Grashof number.

Generally, it has ja‰Š„ ~ℎ~NO

/

~

/

for laminar flow and ja‰Š„ ~ℎ~NO

/

~

/

for

turbulent flow [24]. The increasing pressure may change the flow of fuel vapor layer above the

liquid surface from laminar to turbulent, and the exponent A should be within the range of 1/2~2/3. In current study, A values for 6, 8, 10 cm pool fires are smaller than 1/2, approaching to

the laminar flame regime. This may be attributed to two reasons: 1) the weak dependence of H

−

ŒI

on pressure; 2) heat feedback partially influenced by the conduction regimes. The

effect of the former is insignificant, while the latter is identified as the coupling effects of conduction and convection on the heat feedback regimes [17]. The current fitting results in Fig. 4 are obtained through well-designed and reproducible tests at four static pressures, which is similar

to the work by Wieser et al. [13], producing more credible A values than that based on the tests at

two altitudes [5, 6, 14].

As claimed before, the feasibility of Eq. (20) relies on the accuracy of A value, which needs

to be determined by the tests at multi-pressure environments. However, such data are still lacking

because of the difficulties of conducting larger pool fires at various altitudes. It should be noted that the recently flourishing fire tests at high altitude in Lhasa are due to the completion of plateau fire laboratory in 2009 [46]. Since then, a large number of pool fire tests were conducted in the

EN54 standard combustion room in this laboratory, e.g. [5, 6, 11, 12, 14, 25], while the works are mainly limited to the tests at two altitudes. Under such circumstances, the A values obtained in

Fig. 5 seem to be questionable, which are also somewhat distinct from that by multi-pressure tests. In fact, compared with conducting fire tests at actual altitudes, researchers prefer the altitude chamber with precisely adjustable pressure range, which can provide consecutive static low pressure environments for fire tests. However, the confined and sealed environment in the altitude chamber also imposes restriction on its application range for the tested fire size. For the altitude chamber in this work, we have performed comparative fire tests with pool diameters of 6~14 cm at high altitude (Lhasa) and in the chamber with corresponding ambient pressure (64.3 kPa) in order to explore whether the altitude chamber can faithfully replicate the experimental conditions at high altitude [18]. The results indicate that only for pool diameter smaller than 12 cm, the burning intensity in quasi-steady stage for chamber test can well simulate that for corresponding altitude test, and the enclosure effect related to the influence of the entrapped combustion product on the combustion process and fire plume characteristics cannot be neglected for larger pool dimension [18]. This is also the reason why only 4~10 cm pool fires are investigated in this work.

To acquire more specific A values of convection-dominated pool fires (10~20 cm), an altitude

chamber with larger volume is necessary to conduct the experiments, which deserves further examination in future work.

For pool fires with diameters larger than 20 cm, the heat feedback (j

D‹ )

should be

dominated by radiation, which is directly related to the soot formation or soot volume fraction. Both the experiments at high altitude [11, 25] and elevated pressures [47] have confirmed the power law relationship between €„ and

, i.e. j

D‹ ~€„ ~

with 0.9 < A < 2.0. In Fig. 5, the

fully turbulent combustion region involving boiling phenomenon is intercepted to determine A

values by Fang et al. [6], as shown in Fig. 3. This may results in the relatively smaller A values

since the burning with the presence of boiling stage is more prone to be conductively dominated [20]. Thus, the approximation of A = 1 for pool fires with

≥ 30 cm may be not adequate for

the validation of Eq. (20) in Fig. 12. Pool fire tests at different pressures in such scales are still necessary to obtain the accurate A values.

As for the scale effect on radiative fraction, Hamins [3] found that for a particular fuel, only

small changes in

occur for pool sizes of 4.6~30 cm, which is consistent with the results of

Burgess and Hertzberg [40] for both luminous (benzene) and nonluminous (methanol) flames. Yang et al. [48] derived a piecewise correlation for the radiative fraction, i.e.

0.1 m <

< 1.0 m and

~

.

for

for

≥ 1.0 m. Based on previous tests on n-heptane pool

fires [3, 38, 39], the former is more precisely fitted as on pool diameter. In this work,

~

~

.

, exhibiting a weak dependence

is theoretically derived as 0.15 power of pool diameter due to

its direct relevance with flame height and flame volume. This is also a weak function of pool size, but slightly higher than that obtained from the tests at normal pressure. Moreover, the fitting result in Fig. 12 is quite well for pool diameters not larger than 10 cm, reflecting the feasibility of Eq. (20) for small-scale cases. It is true that the current pool fires are limited to the diameters of 4~10 cm, which are too small to reach the fully turbulent flame regime. This is mainly restricted by the difficulty of conducting fire tests at various actual altitudes and the internal size of the available altitude chamber, which are also future challenges for large-scale pool fire research under low pressure

environments. Based on the accurate estimation of A values of small-scale pool fires, the

proposed theoretical model for radiative fraction in Eq. (20) has been validated for current tests as well as that at low [33] and normal [3] pressures. In pool fire research, conduction- and convection-dominated pool fires are critical parts. Many efforts to understand their heat feedback mechanisms under low pressures and further establish corresponding models have been made in previous work [5, 6, 12, 14, 17]. This work establishes a dimensional correlation for radiative fraction, providing a tentative method for interpreting this parameter, which may be helpful in understanding the radiative fraction of larger pool fires. Practically, the proposed model is applicable to predict the radiative fraction of small-scale fires, which can be applied to estimate the radiative heat flux more accurately, further conducing to the design of fire detector for the incipient fire at high altitude or under other low pressure environments.

4. Conclusions In this work, small-scale n-heptane pool fires were performed at low pressures by means of an altitude chamber, seeking to identify the key parameters affecting the flame radiative fraction. Based on the experiments, the exponential relationship between burning intensity and pressure is confirmed as

∝

, with A value positively increasing with pool diameter. Flame heights

increase with the decreasing pressure, and the dimensionless flame height (L / ) can be

uniformly correlated by (

/

)

/

. It is also observed that the radiative fractions calculated

based on the point source assumption exhibit a decreasing tendency with the reduced pressures, and are related to the pool diameter at the same time. In order to interpret this phenomenon, a dimensional scaling model is proposed based on the fundamentals of flame radiation, revealing the dependence of radiative fraction on both pressure and pool diameter, i.e.

~

.

.

.

.

The current results together with that of small-scale n-heptane pool fires in the literature can be well correlated by this model proposed.

Acknowledgements This research was financially supported by Shanghai Sailing Program (Grant No. 18YF1409600) and National Natural Science Foundation of China (No. 51909152).

Nomenclature Af cp D u l

G

fv l

g NO ℎ H l L i m 6 P j j ja‰Š‹ ja‰Š„ j D‹

Flame surface area (m2) Specific heat at constant pressure (J/kg· K) Fuel pan diameter (m, cm) Equivalent diameter (m) 2i/ in Eq. (10) 2L / in Eq. (10)

Soot volume fraction View factor from fire to the receiver

Gravitational acceleration (m/s2) Grashof number Convective heat transfer coefficient (W/m2· K) Vertical height of the radiometer away from the burner (m) Characteristic flame length scale (m) Horizontal distance between radiometer and the axis of the burner (m) Mean beam length of the flame (m) Total mass of the fuel consumed over time (g) Burning intensity or mass loss rate per unit area (g/s· m2 ) Repeated times of the experiments Ambient pressure (kPa) Measured radiation flux (kW/m2) Flame radiative heat flux (kW/m2) Conductive heat feedback from flame to fuel surface (kW/m2) Convective heat feedback from flame to fuel surface (kW/m2) Radiative heat feedback from flame to fuel surface (kW/m2) Actual heat release rate (kW)

∗

Rf [\ Œ

W

Vf *̅

Xr | zf

Total radiation from the flame (kW) Dimensionless heat release rate Straight-line distance between the point source and the receiver (m) Gas constant for air (J/kg· K) Boiling point of the fuel (K) Flame temperature (K) Ambient temperature (K) Flame volume (m3) Average value of the repeated tests (1 + u) + l in Eq. (10) Residual oxygen mole fraction in the altitude chamber Initial oxygen mole fraction Flame radiant fraction (1 − u) + l in Eq. (10) Flame height (m)

Greek symbols A Exponential factor in Eq. (2) Q Expansion coefficient of the gas ] The ratio of specific heats of air ) Standard deviation of the tests b Combustion efficiency r Soot absorption coefficient (m-1) TW Ambient air density (kg/m3) m Stefan-Boltzmann constant (W/m2·K4) ‚€ Characteristic flow time of the soot (s) ‚ƒ

Characteristic soot formation time (s)

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