Development of an analytical model to predict the radiative heat flux to a vertical element due to a localised fire

Accepted Manuscript Development of an analytical model to predict the radiative heat flux to a vertical element due to a localised fire Nicola Tondini, Christophe Thauvoye, François Hanus, Olivier Vassart PII:

S0379-7112(18)30191-7

DOI:

https://doi.org/10.1016/j.firesaf.2019.03.001

Reference:

FISJ 2789

To appear in:

Fire Safety Journal

Received Date: 27 April 2018 Revised Date:

12 February 2019

Accepted Date: 3 March 2019

Please cite this article as: N. Tondini, C. Thauvoye, Franç. Hanus, O. Vassart, Development of an analytical model to predict the radiative heat flux to a vertical element due to a localised fire, Fire Safety Journal (2019), doi: https://doi.org/10.1016/j.firesaf.2019.03.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

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Development of an analytical model to predict the radiative heat flux to a vertical element due to a localised fire Nicola Tondini1*, Christophe Thauvoye2, François Hanus3, Olivier Vassart3 Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123, Trento, Italy. 2

CTICM, Espace technologique l’Orme des Merisiers, 91193, Saint-Aubin, France.

3

Long Product R&D, Arcelormittal, 66 Rue de Luxembourg, 4009, Esch-sur-Alzette, Luxembourg.

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Corresponding author: *[email protected]; ph: +39 0461 281976

Keyword: Localised fires; analytical method; radiative heat flux; radiation; vertical target; columns,

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computational fluid dynamics; structural fire engineering

Abstract

This paper describes an analytical model developed for design purposes with the aim to assess the radiative heat flux to a vertical element subjected to a localised fire. It focuses on the calculation of the radiative heat flux, which represents the main source of heating of

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vertical members that are not engulfed into the fire. The model exploits the concept of virtual solid flame whose height and temperature are determined according to the correlations provided in Annex C of EN1991-1-2 and relies on an analytical formulation

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easy to implement in any spreadsheet. A major enhancement of the proposed model is the

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capability of providing a way to predict the heat flux on all faces of a vertical member. Moreover, it is deemed valid for any range of heat fluxes that can be expected in a localised fire and it is inherently conservative to be used for design calculations. The proposed model was validated against several experimental tests performed within the European LOCAFI project, as well as against tests found in the literature. Moreover, a parametric numerical analysis was carried out by means of Computational Fluid Dynamics (CFD) numerical simulations that were calibrated on the basis of the experimental outcomes. In this way, the 1

ACCEPTED MANUSCRIPT validation of the analytical model was also performed numerically and useful information about the main parameters that significantly influenced the numerical simulation of localised fires fire was drawn. Conservative agreement in terms of heat flux predictions between the

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proposed model and experimental as well as numerical outcomes was found.

1 INTRODUCTION

For building typologies where a generalised fire cannot develop (e.g.: external structures,

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open car parks, atria, large industrial or transportation halls, etc.), as well as any fire in its early stage, localised fires represent an important issue. In fact, a localised fire can be

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defined as a fire involving only a limited part of a compartment. Zone models are not foreseen to incorporate the effects of localised fires on structures. The main challenge is the calculation of the thermal action to which the structural elements are exposed as a function of the fire properties that can evolve with time, location, size, rate of heat release

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(RHR), etc. Two analytical models, i.e. Hasemi and Heskestad, that are included in Annex C of EN1991-1-2 [1], exist but they have limited field of applicability, e.g. the Hasemi model is valid only for flame impacting the ceiling, and they are typically employed for horizontal

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structural members located at the ceiling level or for vertical elements fully engulfed into the localised fire. In this respect, a noteworthy example of use of the Hasemi and Heskestad

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models is the design of open car parks for which some European National Standards and Guidelines refer to [2-5]. However, due to the absence in the code of a suitable model able to provide the thermal action on vertical members external to the fire, safe assumptions, in the best cases, are taken. Therefore, the complexity of the phenomenon is such that the problem is more often tackled by means of numerical methods, e.g. Computational Fluid Dynamics [6,7]. Several experimental works have been carried out to determine the effect 2

ACCEPTED MANUSCRIPT of a localised fire on vertical structural members, particularly steel columns [8-13]. However, no simple model capable of estimating the thermal radiation was developed in these studies. Nonetheless, analytical methods conceived to estimate the thermal radiation to external targets from hydrocarbon pool fires have been proposed by various researchers:

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simple methods such as the point source model [14] and the Shokri and Beyler simple correlation [15], that was derived from the best fit of “effective” emissive power obtained from several experimental tests, as well as more detailed procedures like the ones

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proposed again by Shokri and Beyler [15] and Mudan [16], which are based on the computation of the configuration factor. However, their field of applicability is limited: the

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point source model is estimated accurate for heat fluxes lower than 5 kW/m2 [17], e.g. when human exposure analysis is needed; the simple Shokri and Beyler correlation is valid for vertical targets located at the ground level only, whereas the Shokri and Beyler detailed model is deemed suitable for heat fluxes larger than 5 kW/m2 [17]. The Mudan model can

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be applied for any range of heat fluxes but it exhibits high variation between the predictions and the experiments [17]. Moreover, all these models do not provide any information about the received flux of a target which does not have orientation with the normal pointing toward

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the centre of the fire. This represents a drawback in structural fire engineering problems

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because a column is generally composed of four faces with different orientations and the designer does not have guidance. Thus, there is a lack of a more general analytical model for determining the thermal radiation to a vertical target with different orientations that is applicable to any range of heat fluxes. On these premises, the European LOCAFI project [18] was funded with the aim to provide designers an analytical model that allow them computing the radiative heat flux to vertical structural elements subjected to localised fires. In this respect, a comprehensive series of experimental tests on mainly hydrocarbon pool 3

ACCEPTED MANUSCRIPT fires in well-ventilated conditions characterised by different size, fuel, layout, with/withoutengulfed/not engulfed steel column, with/without ceiling was envisaged [18,19]. They served as a means to perform several numerical CFD analyses to calibrate the numerical models that were used to validate the proposed analytical method through parametric

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analysis. In sum, the paper is organised as follows: Section 2 briefly describes the experimental tests; Section 3 presents the numerical calibration of the CFD models and provides insight into the parameters that more affect the simulations; Section 4 is devoted

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to the description of the new analytical method for the radiative heat flux prediction to vertical members; Section 5 shows the validation against both experimental tests and

are drawn.

2 EXPERIMENTAL TESTS

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numerical simulations through parametric analysis and finally in Section 6 the conclusions

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Herein a brief presentation of the experimental tests on mainly circular hydrocarbon pool fires carried out at the University of Liege (ULG) and at the University of Ulster (ULSTER) within LOCAFI is provided. For a comprehensive description of the 22 tests with liquid

at ULSTER to [18].

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combustibles carried out at ULG, the reader can refer to [19], whilst for the tests conducted

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2.1 Tests at ULG

The ULG tests were performed to provide evidence of the effect of steel structural members engulfed into localised fires. They were conducted in a large hall 30 m x 20 m x 15 m located at the Centre Européen Pour la Sécurité in Marchienne-au-Pont near Charleroi (Belgium). 24 tests were performed by varying: pool fire diameter (1 m - 2.2 m), fuel type (heptane, diesel and wood cribs), column presence (with / without engulfed steel column), 4

ACCEPTED MANUSCRIPT steel profile (HE and circular hollow section (CHS)). Since no calorimeter measuring the RHR was available, a hydraulic system was conceived in order to adjust the fuel flow into the pan and consequently the RHR. In this way, for each test the RHR per unit surface was set to about 500 kW/m2 by allowing for a combustion efficiency χ equal to 0.8 [19]. In Figure

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1, photos taken during the tests are shown.

(a)

(b)

(c)

2.2 Tests at ULSTER

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Figure 1. ULG tests: b) Test 14 – Heptane D = 1.4 m HE 300 A; b) Test 19 – Diesel D = 1.4 m; c) Test 21 – Diesel D = 2.2 m CHS 203 x 5 [19].

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The aim of the tests at the FireSERT Lab of the University of Ulster (UK) was to provide

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evidence of the effect of localised fires on steel structural members in many different layouts, where the steel structural elements are not engulfed into the fire or, if they are, a ceiling was put in place so that the flame impacted it. More than 60 tests with localised fires were carried out by varying: pool fire diameter (0.7 m – 4 pans of diameter 0.7 m), fuel type (kerosene, diesel and wood cribs), steel column profile (HE, IPE and CHS) and position of the steel column with respect to the pool fire. Moreover, tests with and without ceiling were

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ACCEPTED MANUSCRIPT performed, as depicted in Figure 2. The calorimeter installed at FireSERT was exploited to

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estimate the RHR.

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(a) (b) Figure 2. ULSTER tests: a) Test I7 – Diesel D = 0.7 m; b) Test O29 – Diesel D = 0.7 m with ceiling [18].

3 NUMERICAL CALIBRATION OF THE CFD MODELS

An extensive numerical investigation was performed in order to calibrate CFD models able

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to reproduce the experimental outcomes. The CFD software that was employed for numerical simulations was FDS, developed by the National Institute of Standards and Technology (NIST) [20]. The analysis served as a means to identify the main parameters

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that allow a good representation of the experimental outcomes as well as to conduct a

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parametric numerical analysis to validate the proposed analytical model. Thus, the following steps were fulfilled:

1) Numerical analyses were performed to identify the main parameters that were significant in the calibration process of the experimental tests (Section 3.1); 2) Six meaningful experimental tests were calibrated in the most accurate way so as to understand the impact of each identified parameter (Section 3.2);

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ACCEPTED MANUSCRIPT 3) Forty-four experimental tests were then numerically simulated based on the findings of Step 1 and Step 2 to analyse how the CFD models could reproduce the experimental evidence in more general terms based on a large sample (Section 3.3). 3.1 Identification of the main parameters and assumptions in the numerical

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modelling

Version 6 of FDS was used throughout the calibration process. This version includes different models for the turbulent viscosity: Deardorff (default), Constant Smagorinsky,

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Dynamic Smagorinsky, and Vreman. All turbulent models were tried in the numerical simulations. Indeed, the Constant Smagorinsky model with constant Cs equal to 0.1

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provided the most accurate results in terms of flame characterisation with respect to the tests [18,19]. In the Smagorinsky model, the eddy viscosity is linked to the rate-of-strain tensor through the Cs constant. Cs typically varies from 0.1 to 0.24 [21]. The fuel properties, if not specifically provided by the supplier, were taken from literature,

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preferably with respect to overventilated conditions. The heat of combustion ∆Hc differs from the ideal value in case of incomplete combustion. Both ideal ∆Hc,ideal and typical ∆Hc values heat of combustion were used in numerical simulations. Moreover, due to

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incomplete combustion, in addition to CO2 and H2O, production of soot and CO occurs.

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Higher soot content entails lower flame temperature [14] and soot is the primary emitter and absorber of thermal radiation; thus, it provides the mechanism of radiative heat loss. This phenomenon was actually observed in the numerical simulations and the selection of an appropriate soot yield was based on literature [17,22]. Conversely, the CO production was found to be negligible and it was not considered in the remainder of the study. The Number of Radiation Angles (NRA) was an additional parameter that influenced the results. In fact, the Radiative Transfer Equation (RTE) for an absorbing, emitting and 7

ACCEPTED MANUSCRIPT scattering medium depends on both spatial coordinates and the direction. It is solved using the finite volume method. As the solution of the RTE depends on the direction, the space of 4π steradians is divided into elementary solid angles and the NRA is set to 100 by default. Using a finite volume method has some drawbacks: theoretically, in a field where a

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radiative point source is present, the iso-fluxes have a shape of a sphere centred on that source but by using a finite volume method leads to the occurrence of “waves”. Thus, if this number is too small and the measurement distance is too large with respect to the diameter

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of the fire source, no uniform distribution of heat flux at equal distances in an axi-symmetric problem, like a circular pool fire, is observed. This non-uniformity can be significant, as

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shown in Figure 3. In this respect, in order to even out this numerical discrepancy by keeping a reasonable computational effort, the number of radiation angles was increased up to 200. This numerical effect was more evident in the ULG tests (ULG 06 and ULG 14) characterised by small pool fires (diameter = 1-1.4 m) where the heat flux gauge was

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located at 3.75 m from the fire centre. In order to even out this discrepancy, several finelyspaced heat flux gauges oriented towards the fire centre were included in the computational domain to calculate the mean heat flux. Conversely, in the ULSTER tests being the heat

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flux gauge placed closer to the fire source (max at 1.5 m), the radiative heat flux was more

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uniform for small pool fire diameters (0.7 m).

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(b)

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(a)

Figure 3. Integrated intensity at 3.75 m from the pool fire axis: a) ULG 06 TEST – D = 1 m – NRA = 200; b) ULG 19 TEST – D = 2.2 m – NRA = 200.

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The radiative loss fraction χrad was another parameter that affected the results in terms of flame temperature and heat flux. Typical χrad values are comprised between 0.3 – 0.4 for typical sooty fuels with diameters of about 1 m. However, when the diameter increases smoke tends to shroud the flame and the radiative fraction progressively decreases until

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values as low as 0.05 for pool fires of diameter 50 m [17]. Values of radiative fractions were selected based on literature [17] by taking into account the type of fuel and the size of the fire. The dimension of the mesh grid is obviously paramount to provide accurate results and

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D* (m) defined as

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it was selected based on the maximum cell size δx (m) and the characteristic fire diameter

  Q& D* =   ρ c T g   ∞ p ∞ 

2/5

(1)

Where Q& is the total release rate (kW), ρ∞ is the air density in kg/m3, cp the air specific heat (kJ/kgK), T∞ is the ambient temperature (K) and g is the gravity acceleration (m/s2). In particular, the ratio between the two was selected D * / δ x > 10 ÷ 20

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(2)

ACCEPTED MANUSCRIPT in accordance with [20,23]. In both fire laboratory facilities some flame tilting was observed. In detail, in the ULG tests [19] tilting was less accentuated than in the ULSTER ones [18]. For this reason, an air draught was included as dynamic pressure in the numerical models to obtain a good agreement between experimental and numerical values. It is worth

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pointing out that the impact of the crosswind mainly depends on the power of the fire and in most cases its significance was low compared to other parameters. However, it served to better represent the flame shape in time and space. Moreover, in the numerical simulations

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the magnitude of the crosswind was sometimes higher, particularly when simulating ULSTER tests, than it was recorded in the laboratory to allow for the combination of light air

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draughts with possible uneven vaporization of fuel due to uneven radiant heat flux feedback to the fuel that can be a further cause of tilting during the tests. When a steel column was included in the CFD domain, its thermal properties, i.e. thermal conductivity and specific heat, were assigned according to EN1993-1-2 [24]. Moreover, each part of the cross

each obstruction.

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section was modelled with zero thick obstructions and the actual thickness was assigned to

In order to measure the heat flux, several devices were employed; however, the “Radiative

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Heat Flux Gas” device was mainly used because of its versatility [20]. A heat flux gauge

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records the radiative heat flux on a surface and in principle it needs a physical surface in the simulation. However, in this case, it is not possible to record the heat flux by locating several gauges along one line as the heat flux will be shielded by the first surface. Furthermore, as FDS 6 is based on structured mesh the orientation of any surface is limited to six directions, i.e. +x,-x,+y,-y,+z,-z. In order to overcome this shortcoming, “Radiative Heat Flux Gas” records the radiative heat flux away from a solid surface, i.e. no shielding, and with respect to any orientation. Air temperatures were measured with both 10

ACCEPTED MANUSCRIPT “Temperature” and “Thermocouple” quantities. The thermocouples were located in the same positions of the tests and some others were added in order to capture the temperature evolution. Model parameters were varied within their range of applicability and based on literature so that a meaningful model calibration could be achieved.

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3.2 Detailed analysis of a limited number of tests

The selection of the tests to be accurately simulated was based on the following criteria: •

in order to compute a meaningful average value of the measured air temperatures

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and heat fluxes, tests that exhibited long steady state conditions were preferred; tests with pool fires ranging from small to large diameters;

•

tests with different combustibles;

•

tests without column;

•

tests with column engulfed/not engulfed in the pool fires and without/with ceiling.

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•

Therefore, the tests that were selected and simulated with FDS are reported in Table 1.

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0.7

diesel

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ULG 06 ULG 14 ULG 19 ULG 20 ULSTER I7 ULSTER O29

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Test

Table 1. Tests numerically and accurately simulated in FDS Column Estimated Diameter position experimental Fuel Column Ceiling* (m) relative to RHR (kW) the fire axis [18,19] 1.0 heptane 393 1.4 heptane HE 300 A CENTRED 771 2.2 diesel 1901 2.2 heptane 1899

0.7

diesel

IPE 300

0.5 m

-

466

CHS 219x10

1.0 m

Yes

490

* Ceiling located at 3.5 m from the floor

3.2.2 Results of the numerical simulations Herein a description of the main outcomes of the numerical simulations with focus on the heat flux is summarised. In particular, only the simulations of two tests, one of ULG and one

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ACCEPTED MANUSCRIPT of ULSTER, are here described. For a deeper description of more numerical results, the reader may also refer to [25]. 3.2.2.1 ULG 19 In the ULG 19 test, the ideal heat of combustion for a complete combustion was taken

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equal to 44000 kJ/kg: a common value for hydrocarbon fuels. The soot yield was 0.1, which is typical for diesel pool fires. Some variation has been encountered in literature [26]. As highlighted in [19] the RHR of the pool fire was controlled by a special system conceived

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to inject the fuel into the pool by adjusting its flow. The fuel flow was estimated to obtain an RHR of about 500 kW/m2 with a combustion efficiency of 0.8. The uncertainty was

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estimated around ±15%. In the numerical simulation, a combustion efficiency of 0.9, resulting in Q& '' = 581 kW/m2, i.e. Q& = 2208 kW, was considered. The CFD domain was 8.5 m x 8.0 m x 6.0 m with a grid mesh of 10 cm x 10 cm x 10 cm that entailed a value of D*/δx ≈ 13. The radiative loss fraction was decreased down to 0.30 from the default value of 0.35.

diameter.

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A decrease in radiative loss fraction is somehow expected for pool fires with increasing

An air draught was included in the direction N-S with approximated speed of 0.45 m/s. The

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flame shape is shown in Figure 4a and the results in terms of temperatures are illustrated in

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Figure 4b. By comparing the average numerical and experimental temperatures in Figure 4b, it is possible to observe that a reasonable good agreement was achieved. Furthermore, comparison between the average numerical and experimental heat fluxes is shown in Table 2, being former conservative by about +14%.

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(a) (b) Figure 4: ULG T19: a) flame shape; b) comparison between numerical and experimental values of the average air temperatures along the pool fire axis and 15 cm apart. Table 2. ULG 19: Comparison between numerical and experimental heat flux Height Distance EXP NUM NUM/EXP 2 2 (m) (m) (kW/m ) (kW/m ) (-) 1.86 3.75 2.98 3.39 1.14

3.2.2.2 ULSTER I7

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The ULSTER I7 test consisted of a diesel pool fire of diameter equal to 0.7 m located at 0.5 m from the weak axis of an IPE 300, as shown in Figure 5. The 20-cm high pan containing the fuel was located at 12 cm from the floor. The volume of fuel was 15 litres; thus, about 4

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cm thick when poured into the pan. Consequently, it was decided to include the pan in the CFD model so as to take into account the actual height of the fire from the floor and the

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effect of the pan edges on combustion. Hence, the pan was modelled with obstructions made of steel to which a thickness of 2.5 mm was assigned. From the photos supplied by the University of Ulster, it was noted that, due to an air draught, the flame was tilted parallel to the column web, as depicted in Figure 2a. Therefore, a “wind” speed of 0.8 m/s was included in the direction consistent with experimental observation. It is worth pointing out that the wind speed introduced in the numerical model was higher than that recorded in the 13

ACCEPTED MANUSCRIPT lab (each horizontal component less than 0.5 m/s [18]) as was the resultant of the two measured wind speed components in the lab plus the possible effect of uneven vaporization of fuel due to uneven radiant heat flux feedback to the fuel, which can also contribute to tilting. The RHR was measured with a calorimeter and it was then corrected as

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described in [18]. The steady state average value then yielded 464 kW considering a combustion efficiency equal to 1.0. Indeed, due to the magnitude of the air draught relative to the pool fire size, more efficient combustion may be envisaged because of improved

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mixing [27]. The CFD domain was 4 m x 4 m x 4 m discretised with a mesh grid 5 cm x 5 cm x 5 cm that corresponded to D*/δx ≈ 14. Radiative loss fraction was increased up to

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0.45, which is a value consistent with pool fires of similar fuel and size [17]. The values of heat of combustion, fuel density and soot yield were the same as the ones used in the ULG T19 test. Figure 5a indicates that the pool fire was well represented in FDS (compare with Figure 2a). The comparison in terms of temperature (Figure 6) shows good agreement with

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a maximum error of about -19% when considering temperatures higher than 100°C. The meaning of top and bottom flange as well as back and front web is shown in Figure 5b. The radiative heat flux at height from the floor z = 1 m and z = 2 m and at distance 1.5 m from

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the centre of the column is shown in Table 3. The orientation of the G1 and G2 heat flux

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gauges was towards the centre of the column. The numerical predictions are, in general, conservative with respect to the experimental values.

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α

TF FW BW BF

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G2

BF = Bottom Flange BW = Back Web FW = Front Web TF = Top Flange

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(a) (b) Figure 5: ULSTER I7: a) flame shape; b) location of heat flux gauges G1 and G2.

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(a) (b) Figure 6: ULSTER I7: a) comparison of the vertical evolution of average air temperatures along the top flange (TF); b) comparison of the vertical evolution of average air temperatures along the front web (FW).

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Table 3. ULSTER I7: Comparison between numerical and experimental heat flux Height (m)

G1

EXP

G2

NUM 2

NUM/EXP 2

(kW/m ) (kW/m )

(-)

EXP

NUM 2

NUM/EXP 2

(kW/m ) (kW/m )

(-)

1

6.15

7.47

1.21

10.64

10.32

0.97

2

2.28

3.62

1.59

2.66

3.45

1.30

3.2.3 Summary

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ACCEPTED MANUSCRIPT The numerical simulations showed that the main parameters to be considered when calibrating the localised fires under study were: •

The turbulence model. The Constant Smagorinsky with Cs = 0.1 allowed a better representation of the flame shape in time and in space.

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The radiative loss fraction. An appropriate value of radiative loss fraction that depends on the type of fuel and pool fire dimensions was central for obtaining a good agreement between experimental and numerical radiative heat flux values. As

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mentioned before, representative values found in literature were chosen. However, it is worth to present here a sensitivity analysis performed on the simulation of two

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tests, as shown in Table 4. From Table 4 it is possible to observe that by increasing the radiation loss fraction by 43% (from 0.35 to 0.50 in the ULSTER I7 test) the increase in recorded heat flux is about 47%.

0.30 0.35 0.40

0.00 0.00 0.00

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χrad

Table 4. Sensitivity analysis of the radiation loss fraction impact ULG 14 WIND (m/s) NRA RHR° (kW) HEAT FLUX (kW/m2) VARIATION 100 100 100

943 943 943

0.48 0.58 0.67

20.8% 39.6%

° The sensitivity analysis was performed with combustion efficiency equal to 1

0.35 0.40 0.45 0.50

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χrad

ULSTER I7 WIND (m/s) NRA RHR* (kW) HEAT FLUX (kW/m2) VARIATION 0.00 0.00 0.00 0.00

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•

200 200 200 200

550 550 550 550

6.41 7.08 8.40 9.41

10.5% 31.1% 46.8%

* The sensitivity analysis was performed before the RHR correction

However, since in the design practice the fire can be a combination of different burning fuel materials, e.g. burning a car, or no detailed information on the characteristics of the fire is available, the default value equal to 0.35 is recommended. This assumption, that seems in contradiction with the previous 16

ACCEPTED MANUSCRIPT findings, is supported by the fact that radiation loss fraction values larger than 0.35 are typically associated with small fires, i.e. D ≤ 1 m, whereas fires relevant for structural fire engineering applications are typically larger than 1.5 m, e.g. a car is about 4 m, for which the value of 0.35 is generally representative or conservative, to

•

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become very conservative for large and sooty fires (D > 10 m).

The number of radiation angles can be also important owing to numerical effects introduced by FDS in solving RTE, as reported in Table 5, and it should be increased

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if the measurement point is located fairly away from the fire source. Roughly speaking, from simulations with NRA = 200 it was observed that if the ratio between

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distance (L) of the heat flux gauge from the centre of the fire and the pool fire diameter (D) was less than 1.7, this numerical effect was not particularly significant (ULG 19, ULG 20, ULSTER I7), whereas if L/D > 2.5 it was not negligible (ULG 06

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and ULG 14).

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Table 5. Example of impact of NRA in numerical analyses ULG 14

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NRA WIND (m/s) RHR* (kW) χrad HEAT FLUX (kW/m2) VARIATION 100 0.00 943 0.45 0.69 200 0.00 943 0.45 1.08 56.5% 400 0.00 943 0.45 1.08 56.5% *The sensitivity analysis was performed with combustion efficiency equal to 1

Regarding the impact of the crosswind, it was found that in the numerical modelling the air draught was important - variation of heat flux of more than +40% with respect to the case without crosswind - to accurately capture the experimental heat flux when tests with RHR of 17

ACCEPTED MANUSCRIPT about 500 kW and wind speed above 0.5 m/s were being calibrated, i.e. ULSTER tests. It should be noted that an RHR = 500 kW entails small flames that are more sensitive to a wind speed of 0.5 m/s. Thus, a sensitivity analysis on the crosswind impact is shown in Table 6. To give a quantitative measure, the dimensionless wind V can be computed

V=

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according to Eq. (3) [14]

umeasured umeasured = u0.max 1.9Q& 1/5

(3)

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Where u is the wind speed (m/s), u0.max is the maximum characteristic plume velocity (m/s) and Q& is the average experimental RHR (kW). From Table 6, when V ≥ 0.1 the crosswind in

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the numerical simulations becomes relevant, whereas in literature [14] a V value greater or equal of 0.2 is deemed high enough for the air movement to be significant. Only in the CFD

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models of ULSTER I7 and O29 tests V was greater than 0.1.

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Table 6. Examples of crosswind impact in numerical analyses

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WIND (m/s) RHR° (kW) 0.15 2208 0.28 2208 0.40 2208 0.45 2208 0.50 2208

V 0.02 0.03 0.05 0.05 0.06

ULG 19 χrad NRA HEAT FLUX (kW/m2) VARIATION 0.30 100 2.92 0.30 100 2.93 0.3% 0.30 100 2.98 2.1% 0.30 100 3.04 4.1% 0.30 100 3.01 3.1%

° The sensitivity analysis was performed with combustion efficiency equal to 0.9

ULSTER I7 WIND (m/s) RHR* (kW) V χrad NRA HEAT FLUX (kW/m2) VARIATION 0.00 550 0.00 0.35 200 6.41 0.15 550 0.02 0.35 200 7.12 11.1% 18

ACCEPTED MANUSCRIPT 0.35 0.60 1.00 2.00

550 550 550 550

0.05 0.09 0.15 0.30

0.35 0.35 0.35 0.35

200 200 200 200

7.33 9.19 11.54 14.13

14.3% 43.3% 80.0% 120.5%

* The sensitivity analysis was performed before the RHR correction

The detailed model calibration of the all 6 tests was reasonable accurate, above all for heat

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fluxes, as reported in Table 7. In fact, the numerical heat flux predictions, excluding one value, were within ± 30% of the experimental values, which is not unusual owing to the uncertainty levels present in fire characterisation [22]. Moreover, the predictions tend to be

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on the safe side. In sum, this analysis provides valuable information on the significant parameters to be used in FDS modelling aimed at determining the heat flux radiated away

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from a localised fire. In addition, these findings will be exploited in the numerical analysis of a large sample of tests described in Section 3.3 and in the parametric analysis that will serve as a means to estimate the heat flux in the validation process of the proposed

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analytical model, as shown in Section 5.

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Table 7. Comparison between experimental and numerical outcomes for the detailed analysis

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TEST HEAT FLUX (kW/m2) ULG EXP NUM NUM/EXP 06 0.9 1.06 1.18 14 1.29 1.27 0.98 19 2.98 3.39 1.14 20 4.12 4.04 0.98 ULSTER I7 6.15 7.47 1.21 2.28 3.62 1.59 10.64 10.32 0.97 2.66 3.45 1.30 O29 15.45 17.35 1.12 5.32 4.71 0.89 19

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3.3 Simulation of a large sample of experimental tests Experimental tests are always subjected to small perturbations that influence them in a specific way. For example, the combustion efficiency can be affected by fuel impurities and

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atmospheric pressure variations may influence the flow through crosswind effect. Moreover, turbulent flows are highly sensitive to these small variations and they have a tendency to amplify them. Thus, it is more relevant to perform an analysis based on a large sample. In

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this respect, many configurations were studied by ULSTER and some were specific to a column shape [18]. As the column is outside the fire, it has no impact on the flame and on

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radiative heat fluxes. Therefore, it was decided to group the simulations according to two criteria: i) the type of column and ii) the position of the pan(s), as reported in Table 8. Table 8. A large sample of tests numerically simulated in FDS

1 1 1 1 1 1 1 1 1 1 1 2 2 3 1 1 1 1 1 1 1

Column shape

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I-1 I-1 I-1 I-1 I-1 I-2 I-2 I-2 I-2 I-2 I-2 I-3 I-3 I-4 I-5 I-5 H-1 H-1 H-2 H-2 H-3

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I1 I3 I5 I7 I13 I2 I4 I6 I8 I14 I15 I12 I16 I11 I9 I10 H1 H3 H2 H4 H8

Number of fires

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 1.6 1.6 0.7 0.7 0.7 0.7 0.7

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Test Group

Fire diameter (m)

IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 IPE 300 HE 200 B HE 200 B HE 200 B HE 200 B HE 200 B 20

Column position relative to the fire axis (m) 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 CENTRED CENTRED 0.5 0.5 1.0 1.0 0.7

Ceiling

Corrected experimental RHR (kW) [18]

-

529 559 578 466 570 484 637 513 484 525 520 1192 1114 1872 3750 3201 438 458 514 484 512

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O1

O-1

1

0.7

O3

O-1

1

0.7

O7

O-1

1

0.7

O2

O-2

1

0.7

O4

O-2

1

0.7

O9

O-3

4

0.7

O21

OC-1

1

0.7

O23

OC-1

1

0.7

O27

OC-1

1

0.7

O22

OC-2

1

0.7

O24

OC-2

1

0.7

O28

OC-2

1

0.7

O29

OC-2

1

O25

OC-3

1

O30

OC-3

1

O31

OC-4

2

O32

OC-4

2

O37

OC-5

0.7

0.7

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0.7

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HE 200 B HE 200 B HE 200 B HE 200 B HE 200 B CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10 CHS 219x10

0.7

0.7

0.7

0.7 1.4 0.5 0.5 0.0

-

465 411 1106 1771 2955

0.5

-

503

0.5

-

468

0.5

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0.7 0.7 0.7 0.7 1.6

-

493

-

515

-

442

-

2665

0.5

Yes

563

0.5

Yes

511

0.5

Yes

496

1.0

Yes

575

1.0

Yes

607

1.0

Yes

468

1.0

Yes

490

1.4

Yes

512

1.4

Yes

472

1.0

Yes

1074

1.0

Yes

952

0.5

Yes

2506

1.0 1.0 0.5

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1 1 2 3 1

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H-3 H-4 H-5 H-6 H-7

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H10 H11 H5 H6 H7

Therefore, 44 ULSTER tests (out of more than 60) were reproduced numerically with FDS using the numerical parameters based on the considerations made in Section 3.1 and Section 3.2. In this respect, a radiation fraction loss of 0.35 was assumed for all simulations and no effects of possible air draughts were included. Figure 7 shows the flame shape 21

ACCEPTED MANUSCRIPT obtained from the numerical simulations (bottom) compared with the experimental evidence (top) for ULSTER tests I1, I3 and I5. A simple comparison of the flame length can be done by taking into account that the simulation devices are clearly identifiable at heights 1 m, 2 m and 3 m from the floor that correspond to the two horizontal sticks visible in the photos

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taken during the tests. Since the flame length varies in time, the mean luminous flame length L was determined according to the definition given by Zukoski et al. [28]. The simulations provided a good agreement in terms of flame length. In fact, the flame length is

numerical simulations. Test I3

Test I5

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Test I1

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in the range of 1.8 m – 2.0 m in the experiments and in the range of 1.9 m – 2.1 m in

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(a)

(b) Figure 7: Comparison of flame shape for tests I1, I3 and I5 of Group I-1: a) experimental; b) numerical.

22

ACCEPTED MANUSCRIPT Then, the analysis focussed on the comparison between the heat flux predicted by CFD simulations and the one measured in the tests. In this respect, Figure 8 shows the results in terms of predicted vs. measured heat flux for each of the 4 large groups, i.e. I-shape column, H-shape column, O-shape column without ceiling and OC-shape column with a

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ceiling. It is possible to observe that the difference between numerical and experimental results mainly lies within ±30%, which, as noted before, is a common level of uncertainty in fire characterisation. Thus, this analysis demonstrated the ability of the model to reproduce

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reasonably well the experimental tests. However, it is worth pointing out some limitations of the numerical modelling to reproduce the variability of the experimental data. In fact, even

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in the case of two equivalent tests, e.g. same pan position, the outcomes were barely the same due to several phenomena like: slight flame tilting in different directions due to crosswind and variable efficiency of combustion. Furthermore, for the same combustible it can be observed that variations occurred and gave different mass fuel flows. For example,

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tests I1 and I2 should be identical, i.e. each time a 0.7 m pan was filled with 15 l of kerosene, but the resulting RHR values were different, as reported in Table 8. For these reasons, tests such as I1 (kerosene) and I5 (diesel) were grouped together. Furthermore,

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as the highest heat fluxes were measured for gauges close to the pan, this effect was

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amplified if flame tilting toward the gauge occurred, leading to numerically underpredicted heat fluxes. As a final note, in the tests, if flame tilting was observed, it occurred most of the times in the same direction and, as already mentioned, no crosswind was included in the simulations.

23

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(b)

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(a)

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(c) (d) Figure 8: Comparison between the predicted heat flux obtained from numerical simulations and experimental outcomes: a) I-shape column; b) H-shape column; c) O-shape column without ceiling; d) OC-shape column with ceiling

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4 DEVELOPMENT OF THE ANALYTICAL MODEL One of the LOCAFI goals was to provide a design tool for practitioners to conservatively

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estimate the radiative heat flux on all four surfaces of a vertical member subjected to and not engulfed into one or multiple localised fires with limits in terms of fire diameter and RHR consistent with Annex C of EN1991-1-2 [1], i.e. D ≤ 10 m and RHR ≤ 50 MW. Thus, through validation against experimental outcomes and numerical simulations an analytical formulation, called LOCAFI model, was developed by relying on: •

The virtual solid flame shape concept; 24

ACCEPTED MANUSCRIPT •

The existing formulae of Annex C of EN1991-1-2 [1] used to estimate: i) the flame length and ii) the temperature along the localised fire axis.

4.1 Virtual solid flame concept The concept of virtual solid flame assumes that the flame be a “solid” surface that radiates

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toward the element. Moreover, it is considered that the fire area has a circular shape. If the main combustible involved in the localised fire is not circular, it is modelled as a circle with equivalent area. This is a good approximation when the ratio of the two sides is no greater

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than 2. As will be shown in Section 5, the LOCAFI model provides safe predictions when

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TE D

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the virtual solid flame is modelled as a cone, as illustrated in Figure 9.

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Figure 9: Example of virtual solid flame modelled according to a conical shape and temperature along the height in °C

4.2 Existing localised fire correlations in EN1991-1-2 As stated before, the analytical model relies on the existing correlations provided by Annex C of EN1991-1-2 [1] that deals with localised fires. In particular, the temperature at which the localised fire radiates away as a function of the height will be computed by exploiting Eq. (4) 25

ACCEPTED MANUSCRIPT T ( z ) = 20 + 0.25Qc2 / 3 ( z − z0 )

−5 / 3

≤ 900

(4)

z0 = −1.02 D + 0.00524Q 2 / 5

where T (°C) is the temperature along the centreline of the pool fire, Qc is the convective part of the RHR Q in (W) taken as 0.8 times the RHR, z is the height along the fire

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centreline (m) and z0 is the virtual origin (m) and D is the diameter (or its equivalent) of the localised fire (m) [1]. In Tondini and Franssen [19] it was shown that this correlation was intended to be also extended into the combusting zone, where temperatures as high as

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900°C can prevail. It is worth noting that the virtual solid flame is intended as the visible part of the flame. Therefore, it has a lower temperature than the temperature along the

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centreline expressed by the correlation of Eq. (4). In this way, the proposed analytical model was conceived to include a degree conservativeness that was investigated in Section 5. Moreover, in order to determine the whole size of the cone, the mean flame length calculated according to Eq. (5) will be employed

(5)

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L f = − 1.02 D + 0.0148Q 2 / 5

where D is the diameter (or its equivalent) of the localised fire (m) and Q is the RHR (W). 4.3 Member modelling

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Since the incident radiative heat flux depends on the relative location of the target with

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respect to the position of the fire, to have an accurate estimate of the radiative heat flux it is necessary to subdivide the vertical member into different parts along its height, as shown in Figure 10. Moreover, it is assumed that the sections of the vertical member are modelled with a rectangular shape, as illustrated in Figure 10, independently of their original shape, e.g. I, H or O. This approach is consistent with the assumptions given in Annex G of EN1991-1-2 [1] and it is conservative because avoids the calculation of configuration

26

ACCEPTED MANUSCRIPT factors by considering possible shadow effects. For the rectangular section, the radiative

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heat flux can be then determined for the four faces.

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Figure 10: Example of modelling of an H-column

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4.4 Solid flame discretisation

The incident radiative heat flux emitted by a black body surface A1 received by an surface A2 is given by:

'' q&inc . A1 → A2 = FA1 → A2 σ ( T + 273.15 )

4

(6)

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Where FA2→A1 is the configuration factor, σ is the Stefan-Boltzmann constant (W/m2K4) and T the temperature (°C) of surface A1. In the remainder of the paper, the emissivity of the flame will be conservatively assumed equal to 1, like for a black body, and this assumption is also

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fairly accurate for fire larger than 1 m.

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In order to determine the radiative heat flux received by the vertical element, it is necessary to know the configuration factor and for a conical shape no analytical formula is available. Thus, in principle, numerical integration is required. However, for simple geometrical shapes, analytical formulae to obtain the configuration factor exist, e.g. for cylinders and rings. Therefore, the conical shape can be approximated as a succession of cylinders, as illustrated in Figure 11, where the discretisation is done in such a way to be safe sided, see Figure 11b. At each time instant, the whole size of the cone can be determined by knowing 27

ACCEPTED MANUSCRIPT the diameter of the fire D (or its equivalent) and the flame length Lf computed according to Eq. (5). Then, a uniform temperature Ti (°C) is associated with each cylinder and ring, as schematically reported in Figure 11b. In this respect, Eq. (4) is used to estimate such temperatures along the height of the localised fire. If the flame touches the ceiling, the cone

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SC

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will be truncated at the ceiling height.

(a) (b) Figure 11: Flame modelling a) conical approximation; b) shape and temperature discretisation

4.4.1 Cylinder

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The configuration factor between an infinitesimal plane and a finite cylinder, as represented in Figure 12a, can be computed according to Eq. (7) [29]:

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   Y 2 − B +1  −1  C − B + 1  cos −1     + cos    C + B −1   A −1        Y 2 − B + 1    A +1 −1   cos    −Y  2  B ( A − 1)    2 A − 1 + 4 Y S S  ( )      FdA1 → A2 = −  B 2 Bπ   C − B + 1  C + B +1 cos −1  − C  2    ( C + B − 1) + 4C  B ( C + B − 1)       −1  1  + H cos      B   2 where S = s / r X = x / r Y = y / r H = h / r A = X + Y ² + S ² B = S ² + X ² C = ( H − Y ) ²

28

(7)

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ACCEPTED MANUSCRIPT

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(a) (b) Figure 12: a) Configuration between an infinitesimal surface and a finite cylinder; b) schematic view of the contribution to the heat flux received by facej from cylinder czi

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Thus, the incident radiative heat flux (W/m2) received by the surface facej from the cylinder at height zi (czi), as shown in Figure 12b, is given in Eq. (8)

'' q&inc .czi → facej = Fczi → facejσ ( T ( czi ) + 273.15 )

4

(8)

where Fzi→facej is the configuration factor between the radiating cylindrical surface czi and the

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receiving surface facej and T(czi) is the temperature (°C) of the cylinder solid flame czi. The property of additivity of the configuration factor applies.

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There is a restriction for the application of Eq. (7): the plane, defined by dA1, cannot cut the cylinder. Therefore, additional rules are needed in order to handle all possible

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configurations. Indeed, in the case shown in Figure 13, Face 1 sees the cylinder but Face 2 and Face 4 see it only partially, while no radiative heat flux from the virtual solid flame reaches Face 3. Thus, Face 1 corresponds to the situation described by Eq. (7) and can be handled. Face 3 does not pose any problem because the incident radiative heat flux is zero. The case of Face 2 and Face 4 is more complex and the formula cannot be directly applied because the plane, where the face is located, cuts the cylinder. 29

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Figure 13: Example of interaction between a cylinder and the vertical member (top view)

As the radiative heat flux is mainly controlled by the solid angle between the radiative

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source and the target, the proposed solution is to adopt a surface shape which can lead to a similar configuration factor. In fact, two different surfaces with same temperature and emissivity, which are viewed under the same solid angle, give the same configuration factor. Therefore, a cylinder with a modified geometry is used, as indicated in Figure 14. In

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this respect, the modified cylinder is seen by the target through a solid angle that is close to the solid angle under which the initial cylinder was seen by the target itself. This is particularly true when the target is further away from the source. The diameter of the

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cylinder is then reduced so that the modified cylinder is fully visible by the target face in

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such a way that the analytical formula given in Eq. (7) can be applied.

30

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ACCEPTED MANUSCRIPT

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Figure 14: Adaptation of the cylinder modelling for Face 4

4.4.2 Ring

The conical shape modelled with superposition of cylinders with decreasing radius implies the presence of rings. Thus, it is necessary to compute the radiative heat flux coming from the ring surfaces that are located at heights below the target. In fact, all ring surfaces that

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are located above the target cannot be seen by the latter. In this respect, the configuration factor between an infinitesimal plane element and a ring in a perpendicular plane (see

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Figure 15) is given by Eq. (9) [29]

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 H H 2 + R22 + 1 FdA1 → A2 =  − 2  H 2 + R2 + 1 2 − 4R 2 ) 2 2  ( where R = r / l H = h/l

The formula is valid if l > r2.

31

   2 ( H 2 + R12 + 1) − 4 R12  H 2 + R12 + 1

(9)

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Figure 15: Configuration between an infinitesimal surface and a finite ring in a perpendicular plane

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Moreover, it is valid only for a ring centred in a plane perpendicular to the target with the normal to the target pointing the centre of the ring, which is not always the case, as shown

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in Figure 16. However, the analytical formula of Eq. (9) provides the highest configuration factor because the cosine value of the angle is equal to 1. Consequently, it is on the safe

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side and Eq. (9) is used with l the distance between the face and the ring centre.

Figure 16: Influence of the target orientation

Furthermore, the ring, as for the cylinder, can be only partially visible, as shown in Figure 17. Then, the exterior and even the inner radius of the ring are reduced in a similar way as 32

ACCEPTED MANUSCRIPT the cylinder. In the example of Figure 17, the circles defined by the inner radius rzi+1 and the outer radius rzi delimit a ring partially visible by the target. Therefore, they are both adjusted to have a ring defined by the two circles with radius rzi_adjusted and rzi+1_adjusted fully visible by

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the target, as shown in Figure 17.

Figure 17: Adjustment of a ring partial visible to the target surface

4.4.3 Total incident heat flux

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The radiative heat flux received by a vertical face is the sum of the radiative heat flux emitted by all cylinders and rings visible by facej. Hence, the total incident heat flux (W/m2) yields n

m

i =1

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'' q&inc . flame → facej = ∑ Fi → facejσ ( T ( czi ) + 273.15 ) + ∑ Fk → facejσ ( T ( rzk ) + 273.15 ) 4

4

(10)

k =1

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Where n is the number of cylinders, Fi→facej is the configuration factor between the radiating cylindrical surface czi and the receiving surface facej, σ is the Stefan-Boltzmann constant (W/m2K4), T(czi) is the temperature in °C of the cylinder solid flame czi, m is the number of rings that are located at lower heights than facej, Fk→facej is the configuration factor between the radiating ring surface rzk and the receiving surface facej and T(rzk) is the temperature in °C of the ring solid flame rzk. 33

ACCEPTED MANUSCRIPT 4.4.4 Practical observations •

In general, the orientation of the cross section with respect to the fire can be any, as illustrated in Figure 18. Solutions to sort this situation out have already been given in

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the previous sections.

nFace.2

nFace.3

A

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FIRE

nFace.4

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nFace.1

Figure 18: General orientation between the element cross-section and the fire.

Moreover, the flux computation can be further simplified assuming that the normal of Face 1 points to the centre of the fire. With the normal to the element surface

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pointing to the centre of the circular fire (see Figure 19a), only the radiative heat fluxes received by Face 1 can be actually computed. In fact, this orientation entails

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zero radiative heat flux received by Face 3 and the heat flux received by Face 2 and Face 4 can be conservatively assumed as 50% of the heat flux computed on Face 1. In this respect, in Figure 19b reports the ratio between the actual configuration factor

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•

of Face 2 (or Face 4) computed according to numerical integration and the configuration factor of Face 1 divided by two for several L/D values with varying diameters and distances by keeping constant the height of the target with respect to the fire. This assumption becomes rather safe-sided if the ratio L/D is larger than 2.5; where L is the distance between the target and centre of the fire characterised by 34

ACCEPTED MANUSCRIPT diameter D. Conversely, it is a better and conservative estimate for L/D values close to 0.5, i.e. when the heat flux magnitude is expected to be more significant for structural or ignition problems. Thus, in order to avoid an overconservative prediction, it is recommended to use this simplification for low L/D values. In structural fire engineering problems typically characterised by localised fires with

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•

flame heights in the range of 2 m - 20 m, it is recommended to subdivide both the virtual solid flame and the vertical member into the same number of parts, with a

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maximum height of each part equal to 50 cm in order to well reproduce the shape of flame. However, the proposed analytical model can be used, in principle, for any

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types of analysis involving the computation of the radiative heat flux from a localised fire. Therefore, the degree of refinement of the subdivision may depend on the problem nature.

nFace.3

nFace.2

nFace.1

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TE D

nFace.4 A

Face 1

Face 2 Face 3 Face 4

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FIRE

(a) (b) Figure 19: a) Target orientation towards the centre of the fire; b) ratio between the actual configuration factor of Face 2 (or Face 3) computed according to numerical integration and the configuration factor of Face 1 divided by two as a function of L/D.

35

ACCEPTED MANUSCRIPT 5 VALIDATION OF THE PROPOSED MODEL In order to investigate the effectiveness of the LOCAFI model a comprehensive validation

•

Experimental data obtained in the LOCAFI project

•

Experimental data taken from literature

•

Numerical parametric analysis

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5.1 Experimental data obtained in the LOCAFI project

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study was carried out through comparison with:

The radiative heat flux data obtained from tests performed at both ULG and ULSTER were

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exploited. The predicted heat flux relative to the ULG tests was determined by relying on the estimate of the average experimental RHR whose values can be found in [19]. Only the tests with reliable heat flux measurements were used, i.e. readings that exhibited sufficiently long steady-state conditions [19]. The comparison between the LOCAFI model

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predictions and the average experimental values are reported in Table 9 and are largely safe-sided. This is also due to the fact that in terms of temperatures along the centreline of the fire plume, the temperature correlation of Eq. (4) overpredicted, on average, the

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experimental values by 27% for diesel pool fires and by 20% for heptane pool fires, with

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and without column [19].

The ULSTER test results were grouped depending on the location of the pans with respect to the steel column. Although many experimental configurations were investigated only the case with one pan of 0.7 m diameter is presented here and other analyses are available in [18]. Two gauge positions and two pan locations were considered. The average experimental values were compared to those calculated by the LOCAFI model and reported 36

ACCEPTED MANUSCRIPT in Table 10. It is possible to observe that the LOCAFI model provides predictions that are, on average, safe sided for both series of tests. The different degree of conservativeness between the ULG and ULSTER tests can be mainly explained by different test conditions.

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Table 9. ULG tests: Comparison between the experimental and the analytical heat flux prediction determined at distance 3.75 m from the fire centreline and 1.86 m from the floor Experimental LOCAFI Model Estimated experimental MOD/EXP TEST heat flux heat flux RHR (kW) [19] (-) (kW/m²) (kW/m²) 1 87 0.25 0.74 3.01 2 141 0.41 0.92 2.24 3 87 0.21 0.74 3.45 4 121 0.36 0.84 2.33 7 373 0.48 2.20 4.55 8 372 0.78 2.19 2.82 10 776 1.76 3.91 2.22 11 750 1.23 3.86 3.13 12 752 1.54 3.86 2.51 13 769 1.20 3.89 3.25 14 776 1.29 3.91 3.03 15 1272 2.11 5.94 2.82 16 1281 2.59 5.96 2.30 17 1229 1.63 5.83 3.57 18 1236 2.30 5.85 2.54 19 1901 2.97 8.23 2.77 20 1911 4.12 8.24 2.00 21 1858 2.69 8.15 3.03 22 1871 3.83 8.17 2.13 Table 10. ULSTER tests: Comparison between the experimental and the analytical heat flux prediction

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Height Distance EXP LOCAFI Model MOD/EXP (m) (m) (kW/m²) (kW/m²) (-) 1 0.5 30.61 39.00 1.27 1 1.0 13.78 17.95 1.30 1 1.6 5.86 8.51 1.45 1 1.8 4.17 5.97 1.43 2 0.5 6.18 5.88 0.95 2 1.0 4.51 5.51 1.22 2 1.6 3.01 4.11 1.37 2 1.8 2.31 3.27 1.42 37

ACCEPTED MANUSCRIPT 5.2 Experimental tests taken from literature The effectiveness of the analytical model was also verified against 80 experimental tests taken from the literature [30-37] that involve different fuel types. Figure 20a clearly shows

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that the proposed LOCAFI model predicted conservative heat fluxes. Indeed, only less than 5% of the sample was underpredicted by more than 10%, with all six underpredicted values lying within the -30% margin. Moreover, the Shokri and Beyler detailed model [15] was used for comparison purposes to estimate the radiative heat flux to a vertical target. In fact,

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the Shokri and Beyler model is recommended by the SFPE Handbook [17] for the

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prediction of heat fluxes greater than 5 kW/m2, e.g. when the ignition of combustibles has to be estimated, because of low variation in the estimate. Similarly to the proposed LOCAFI model, it is also based on the determination of a configuration factor but relative to a cylindrical shape of the flame, whose height is computed according to Eq. (5). The radiative

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heat flux reads

q& '' = EF1→2 = 58 (10−0.00823 D ) F1→2

(11)

Where E is the emissive power (kW/m2), D is the diameter of the fire (m) and F1→2 is the

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configuration factor from vertical cylindrical radiation sources. Figure 20b shows the results

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based on the same sample of experimental data as of Figure 20a. From Figure 20b it is possible to observe that the Shokri and Beyler model provides good predictions. However, they are less conservative than the proposed LOCAFI model, particularly for heat fluxes less than 5 kW/m2, which perhaps may be the reason why the SFPE Handbook recommends its use for larger heat fluxes. In addition, in order to guarantee an adequate level of safety, Shokri and Beyler recommended to apply to the predicted heat flux a safety factor of two for design purposes [15]. In this respect, the LOCAFI model provides a heat 38

ACCEPTED MANUSCRIPT flux prediction that is directly suitable for design applications. In fact, it is, on average, more than 1.4 times higher than the measured heat flux values with no statistically significant underprediction for heat fluxes larger than 5 kW/m2, i.e. significant for structural fire engineering applications. Indeed, only one single experimental value was underpredicted

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out of 31 values characterised by heat fluxes greater than 5 kW/m2. This means about 3% of the considered subsample. In addition, the underprediction was about -3%, i.e. 10.2 kW/m2 (LOCAFI model) compared to 10.5 kW/m2 (experimental). In total, 6 experimental

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data out 80 were underpredicted by the LOCAFI model. Among these 6 values, the average underprediction was -12.8% and two values were characterised by underestimate lower

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than -15%, i.e. -18% and -28%. At the same time these two heat fluxes were in the order of 1.0 kW/m2 – 1.5 kW/m2; thus, low heat fluxes relatively to structural engineering applications. Similarly to the Shokri and Beyler model, the LOCAFI model exhibits low variation in the data. Therefore, the LOCAFI model can be applied without applying safety

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factors to predict the radiative heat flux in a large range. Finally yet importantly, differently from the Shokri and Beyler model, the major enhancement of the LOCAFI model consists in

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the capability of providing a way to predict the heat flux on all faces of a vertical member.

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(a)

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(b) Figure 20: a) Comparison between the LOCAFI model and experimental data; b) comparison between the LOCAFI model and the Shokri and Beyler model against experimental data. Average (AVG) ratio between model (MOD) and experimental (EXP) predictions; COV: coefficient of variation

5.3 Numerical parametric analysis The aim of the numerical parametric study was to confirm the effectiveness of the proposed analytical model with in mind that the present version of Annex C of EN 1991-1-2 is limited 40

ACCEPTED MANUSCRIPT to a maximum fire diameter of 10 m and a maximum RHR of 50 MW. These analyses were also carried out to provide an idea on how the model would perform if an FDS user wanted to simulate larger fires according to the indications given in Section 3. In particular no wind effect and a radiative loss fraction equal to 0.35 were taken. As a result, the radiative heat

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flux was measured at several distances from the fire centreline and at different heights. The simulations were grouped in four sets:

Group 1: heat release rate of 500 kW/m² and 4 m diameter fire

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Group 2: heat release rate of 500 kW/m² and 8 m diameter fire

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Group 3: heat release rate of 1000 kW/m² and 4 m diameter fire

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Group 4: heat release rate of 1000 kW/m² and 8 m diameter fire

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From Table 11 and Table 12, it is possible to observe that the LOCAFI model provided conservative predictions with respect to the outcomes obtained through FDS simulations.

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As shown in Section 5.2, this is a further confirmation that the LOCAFI model can be considered a suitable tool to predict the heat flux received by a vertical element for design purposes.

Height z Distance x (m) 2 3 5 8 12 2 3 5 8 12 2

NUM

LOCAFI Model MOD/NUM

2

(kW/m ) 43.2 22.9 8.3 2.3 0.8 27.1 17.4 8.8 4.5 1.5 16.9

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(m) 1 1 1 1 1 2 2 2 2 2 3

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Table 11. Comparison between the numerical and the analytical heat flux – Diameter = 4 m 2

(kW/m ) 88.0 45.0 14.0 4.9 2.0 59.0 36.0 14.0 5.1 2.0 35.0

(-) 2.04 1.96 1.69 2.10 2.64 2.18 2.06 1.59 1.13 1.35 2.07 41

NUM

LOCAFI Model MOD/NUM 2

(kW/m ) 85.5 45.8 17.0 5.0 1.4 60.2 38.1 18.3 8.7 2.6 40.8

(kW/m2) 98.0 54.0 19.0 7.4 3.2 82.0 51.0 20.0 7.8 3.3 66.0

(-) 1.15 1.18 1.12 1.48 2.30 1.36 1.34 1.09 0.90 1.26 1.62

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2.08 1.96 1.56 1.12 1.65 1.88 1.91 1.96 2.36

28.7 14.8 7.3 3.8 17.1 14.1 9.9 5.2 1.9

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41.0 19.0 7.8 3.4 29.0 22.0 13.0 7.0 3.4

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12.5 6.6 3.5 2.1 6.4 5.8 4.5 2.4 1.0

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3 5 8 12 2 3 5 8 12

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3 3 3 3 5 5 5 5 5

1.43 1.29 1.07 0.89 1.70 1.56 1.31 1.36 1.81

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(kW/m2) 51.4 30.0 8.6 2.5 37.4 27.4 13.4 5.3 27.2 21.5 10.9 7.3 15.2 13.7 8.6 4.0

(kW/m2) 96.0 58.0 16.4 6.3 72.0 51.0 18.0 6.8 47.0 42.0 18.0 7.2 29.0 26.0 16.0 7.5

6 CONCLUSIONS

(-) 1.87 1.93 1.91 2.55 1.93 1.86 1.34 1.28 1.73 1.95 1.65 0.99 1.91 1.90 1.87 1.88

(kW/m2) 99.8 59.1 18.5 6.0 76.7 55.8 26.2 10.4 60.1 46.8 22.8 13.5 38.1 32.4 18.9 8.8

(kW/m2) 100.0 70.0 24.0 10.0 97.0 69.0 26.0 11.0 84.0 63.0 27.0 11.0 61.0 48.0 24.0 11.0

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(m) 4 5 8 12 4 5 8 12 4 5 8 12 4 5 8 12

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(m) 1 1 1 1 2 2 2 2 3 3 3 3 5 5 5 5

RHR = 1000 kW/m2 - D = 8 m NUM LOCAFI Model MOD/NUM

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Height z Distance x

RHR = 500 kW/m2 - D = 8 m NUM LOCAFI Model MOD/NUM

(-) 1.00 1.18 1.30 1.67 1.27 1.24 0.99 1.06 1.40 1.35 1.18 0.81 1.60 1.48 1.27 1.25

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The article presents the development, for design purposes, of an analytical model, called LOCAFI model, that is able to predict the radiative heat flux received by a vertical element

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owing to a localised fire. The model, that exploits the correlations given in Annex C of EN1991-1-2 and limits the fire diameter equal to 10 m and RHR to 50 MW, is based on the

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concept of the virtual solid flame approximated with a conical shape by means of a succession of cylinders. Moreover, it is not only able to predict the radiative heat flux to a vertical surface that typically points toward the centre of the fire, but also to the other surfaces that compose, for instance, the section of a column. This is a major enhancement because in this way the proposed model represents a valuable tool to be employed in the design practice applied to structural fire engineering problems. In addition, as shown in the 43

ACCEPTED MANUSCRIPT validation process and differently from other available models, it was derived to be applied for all range of heat fluxes by keeping conservativeness and low variation between predicted and measured data. In greater detail, a comprehensive numerical investigation was performed with the aim to calibrate CFD models, developed with FDS, against the

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experimental data obtained in the LOCAFI project. Despite intrinsic numerical modelling limitations to reproduce the variability of experimental tests, reasonable good agreement in terms of flame shape in time and in space as well as in terms of the radiative heat flux,

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within the typical uncertainty in fire characterisation, i.e. ±30%, was obtained considering the turbulence model as the Constant Smagorinsky with constant Cs = 0.1 and a number of

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radiation angles (NRA) equal to 200. An appropriate value of radiative loss fraction that depends on the type of fuel and pool fire dimensions was either important for obtaining a good agreement between experimental and numerical values of radiative heat flux. However, if the characteristics of the fire are not fully known a priori, the default value equal

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to 0.35 is recommended, which a representative or conservative assumption for localised fires characterised by D ≥ 1 m. The LOCAFI model conservatively predicted the radiative heat flux of a large sample of experimental tests of various fuels. In particular, for heat

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fluxes larger than 5 kW/m2 the overprediction was on average more than 1.4, associated

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with low variation in the data (COV = 0.3) that suggests its application without any safety factor. This was also confirmed by the parametric numerical analysis carried out on representative large pool fires.

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ACCEPTED MANUSCRIPT ACKNOWLEDGEMENTS This work was carried out with a financial grant from the Research Fund for Coal and Steel of the European Community, within the LOCAFI project: "Temperature assessment of a vertical steel member subjected to localised fire", Grant N0 RFSR-CT-2012-00023.

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Nicola Tondini also acknowledges funding from the Italian Ministry of Education, University and Research (MIUR) in the frame of the "Departments of Excellence" grant L. 232/2016.

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[2] Ministero dell’Interno, D.M. 3 agosto 2015 “Approvazione di norme tecniche di prevenzione incendi”, ai sensi dell'articolo 15 del decreto legislativo 8 marzo 2006, n. 139, Roma, 2015.

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[21] Tannehill J.C., Anderson D.A., Pletcher R.H., Computional Fluid Mechanics and Heat Transfer, 2nd Edition’, Taylor & Francis, Washington, DC, USA, 1997. [22] Quintiere J.G. Principles of Fire Behavior, Delmar, Clifton Park, NY, USA, 1998

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[24] EN1993-1-2, Eurocode 3: Design of steel structures – Part 1-2: General rules – Structural fire design, CEN, Brussel, 2005. [25] Tondini N.; Thauvoye C., Hanus F. and Zhao B., Numerical Calibration of Hydrocarbon Localised Fire Tests by Means of Fire Dynamics Simulator, Proceedings of the 9th International Conference on Structures in Fire (SiF) Location: Princeton Univ, Princeton, NJ, USA, June 8-10, 2016.

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ACCEPTED MANUSCRIPT [32] Fu T.T., Heat Radiation from Fires of Aviation Fuels, Paper No. 10, Fall Meeting Eastern Section, The Combustion Institute, Princeton University, Princeton, NJ, 7-8 December, 1972. [33] Yumoto T., Fire Spread between Two Oil Tanks, Journal of Fire and Flammability, 8: 494-505, 1977.

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[37] Sautot C., Hanus F., Thauvoye C., Erez G. and Thiry A., Radiative flux affecting vertical steel member away from the fire – Simplified method Locafi, Proceedings of the 10th Conference on Structures in Fire, Belfast, UK, 6-8 June, 2018.

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Nicola Tondini, PhD Assistant professor Department of Civil, Env. & Mechanical Engineering, Via Mesiano 77- 38123 Trento, Italy. Tel. +39-0461-281976 e-mail: [email protected]

Trento, 12 February 2019

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Analytical model to predict radiative heat flux. Heat flux prediction for faces composing a vertical element. Flame shape approximated according to a cone. Analytical model provided conservative results in terms of predicted heat flux. Analytical model can be applied for design purposes.

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Dear Editor, The research highlights of the paper ‘Development of an analytical model to predict the radiative heat flux to a vertical element due to a localised fire’ are listed below:

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Thank you in advance for your co-operation in the matter.

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Yours sincerely,

The Authors

Via Mesiano, 77 – 38123 Trento, Italy - Tel. +39 0461/282669, Fax +39 0461/282599 e-mail [email protected]