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Ellipsoidal Solid Flame Model for Structures Under Localized Fire

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Abstract

This paper presents a model to evaluate the thermal energy transfer between a localized fire and the surfaces exposed to it, without the flame impinging the ceiling of the semi-open compartment. Although this type of fire may not have significant consequences for the structure as a whole, it is capable of triggering other disasters such as explosions and larger fires, which is why its study becomes increasingly important. Currently, this accident is analyzed using either sophisticated or semi-empirical numerical models available in the literature. The former uses computational fluid dynamics (CFD), which acceptably reproduces the fire, although with high computational cost. In turn, the semi-empirical models generally present conservative results. The proposed model presents variants in classic simple models available in the literature with the aim of being a tool that allows designers to estimate the thermal fields resulting from this type of fires at the preliminary structure design stage. In this model, the thermal analysis is performed using a finite element program, considering relevant parameters that characterize the fire such as: heat release rate, location and equivalent diameter of the fire source, among others. Through subroutines, the finite element model considers (a) a modification of hot gases temperature field based in a classic simple model and (b) proposition of a new geometry of the flame. The estimated radiative heat flux employs a solid ellipsoidal flame whose height changes according to the heat release rate. The convective heat flux is evaluated using a model for localized fire. Efficiency and accuracy of the methodology are checked by comparing the simulation results with those obtained by sophisticated models developed in fire dynamic simulator (FDS). The cases studied consider: (a) the replication of the experimental test conducted at Luleå University and (b) an offshore platform deck under localized fire action. The results of the first case confirm that the FDS replicates the experimental measurements with high accuracy. Finally, the results show that the proposed model allows to realistically represent the temperature fields generated by the fire, with relatively low computational cost compared to the CFD models for cases (a) and (b), therefore it is possible to use it to develop preliminary analyses in other fire scenarios.

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Abbreviations

\( b \) (m):

Characteristic plume radius

\( c_{p} \) (J/g K):

Specific heat

\( e_{r,abs}^{''} \) (\( {\text{kW/m}}^{2} \)):

Radiative energy absorbed

\( e_{r,emi}^{''} \) (\( {\text{kW/m}}^{2} \)):

Radiative energy emitted

\( e_{r,inc}^{''} \) (\( {\text{kW/m}}^{2} \)):

Radiative incident energy

\( e_{r,ref}^{''} \) (\( {\text{kW/m}}^{2} \)):

Radiative energy reflected

\( g \) (m/s2):

Gravitational acceleration

\( h_{net}^{AST} \) (\( \frac{\text{kW}}{{{\text{m}}^{2} \;{\text{K}}}} \)):

Net adiabatic heat transfer coefficient

\( h_{conv} \) (\( \frac{\text{kW}}{{{\text{m}}^{2} \;{\text{K}}}} \)):

Convective heat transfer coefficient

\( h_{rad} \) (\( \frac{\text{kW}}{{{\text{m}}^{2} \;{\text{K}}}} \)):

Adiabatic radiation heat transfer coefficient

\( k \) (\( {\text{W/mK}} \)):

Thermal conductivity

\( m^{{\prime \prime }} \) (kg/m2 s):

Burning rate per unit area

\( q_{conv }^{''} \) (kW/m2):

Convective heat flux

\( q_{f} \) (kW/m2):

Heat release rate per unit area

\( q_{net}^{''} \) (\( {\text{kW/m}}^{2} \)):

Net heat flux

\( q_{rad}^{''} \) (kW/m2):

Radiative heat flux

\( r \) (m):

Radial distance to flame axis

\( \bar{r} \) (–):

Normalized flame axis distance

\( t_{d} \) (s):

Fire decay time for NFSC curve

\( t_{g} \) :

Fire growth time for NFSC curve

\( t_{s} \) :

Fire constant behavior time for NFSC curve

\( z \) (m):

Height of the selected point

\( z_{o} \) (m):

Height of the virtual source

\( A_{fs} \) (m2):

Free surface area of the pool fire

\( D^{*} \) (m):

Plume characteristic diameter

D (m):

Equivalent diameter

\( E_{av} \) (kW/m2):

Flame average emissive power

\( E_{b} \) (kW/m2):

Black body emissive power

\( E_{fl} \) (kW/m2):

Flame emissive power

\( F_{ij} \) (–):

Geometric view factor

\( H_{c} \) (m):

Height of the compartment

\( H_{fl} \) (m):

Flame height

\( \bar{H}_{fl} \) (–):

Normalized height

\( HRR \) (kW):

Heat Release Rate

\( R^{*} \) :

Numerical grid spatial resolution

\( \alpha_{s} \) (–):

Absorptivity

\( \beta \) (–):

Mean beam length corrector

\( \delta_{x} \) (m):

Characteristic length

\( \varepsilon_{s} \) (–):

Surface emissivity

\( \varepsilon_{fl} \) (–):

Flame emissivity

\( \theta_{AST} \) (°C):

Adiabatic Surface Temperature

\( \bar{\theta }_{{\left( {\omega ,t} \right)}} \) (°C):

Normalized temperature

\( \theta_{fl} \) (K):

Real flame temperature

\( \theta_{fl,eq} \) (K):

Equivalent flame temperature

\( \theta_{g} \) (°C):

Gas temperature

\( \theta_{o} \) (°C):

Plume centerline temperature

\( \theta_{s} \) (°C):

Exposed surface’s temperature

\( \theta_{\infty } \) (K):

Ambient temperature

\( \kappa \) (1/m):

Absorption–extinction coefficient of the flame

\( \rho \) (kg/m3):

Specific mass

\( \rho_{\infty } \) (kg/m3):

Air density

\( \sigma \) (\( \frac{\text{kW}}{{{\text{m}}^{2} \;{\text{K}}^{4} }} \)):

Stefan–Boltzmann constant

\( \tau \) (–):

Atmospheric transmissivity

\( \varphi \) (–):

Fire intensity coefficient

\( \chi_{r} \) (–):

HRR’s radiative fraction

\( \chi_{lum} \) (–):

Percentage of visible flame

\( \omega \) :

Dependent variable (Eq. 21)

\( \phi \) (–):

Temporal integration factor

\( \Delta H_{c,eff} \) (kJ/kg):

Combustion specific heat

\( \Delta t \) :

Time interval

\( \left[ C \right] \) :

Capacity matrix

\( \left[ H \right] \) :

Boundary convection matrix

\( \left[ {K_{T} } \right] \) :

Total conductivity matrix

\( \left[ K \right] \) :

Conductivity matrix

\( \left\{ {R_{B} } \right\} \) :

Heat flux vector

\( \left\{ {R_{H} } \right\} \) :

Boundary convection vector

\( \left\{ {R_{T} } \right\} \) :

Vector of total nodal heat flux

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Acknowledgment

The authors wish to express their gratitude to the Brazilian Petroleum Corporation—PETROBRAS, the National Agency of Petroleum, Natural Gas and Biofuels of Brazil (ANP) and National Council of Scientific and Technological Development of Brazil (CNPq) for their support to the development of this work.

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Authors and Affiliations

  1. Ocean Engineering Program, COPPE/UFRJ, Rio de Janeiro, Brazil

    Miguel R. Manco, Murilo A. Vaz & Julio C. R. Cyrino

  2. Civil Engineering Program, COPPE/UFRJ, Rio de Janeiro, Brazil

    Alexandre Landesmann

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  1. Miguel R. Manco
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Correspondence to Murilo A. Vaz.

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Manco, M.R., Vaz, M.A., Cyrino, J.C.R. et al. Ellipsoidal Solid Flame Model for Structures Under Localized Fire. Fire Technol 54, 1505–1532 (2018). https://doi.org/10.1007/s10694-018-0750-y

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