1 Introduction

To adequately design and manage fire safety systems, it is necessary to properly understand fire dynamics and the conditions resulting from a compartment fire. In a pre-flashover compartment fire, hazardous gases and heat are accumulated in the upper portion of the room, which is then denominated as hot gas layer (HGL) or upper layer, and its composition influences on visibility and hazards for occupants.

The hot gas layer temperature (HGLT) in pre-flashover compartment fires is related to several safety issues, such as the occurrence of hazardous conditions for people, the fire spread to combustible items far from the fire source and the occurrence of flashover (when the entire room is involved in the fire) [1]. Knowing the HGLT in a certain fire scenario helps engineers to predict the onset of flashover and to design fire safety systems to prevent it. During the pre-flashover stage the fire can still be controlled, suppressed, and evacuation is still possible.

Although Computational Fluid Dynamics (CFD) based models can reproduce fire behavior to a considerable degree, the hand-calculation methods are still widely applied, since they are more time-efficient and cheaper when compared to numerical simulations and in general can provide a rough, but useful first estimate of the fire. Both, HGLT and interface height can be assessed through analytical equations (hand-calculation methods) or computational models (CFD), being HGLT the focus of the present work.

The McCaffrey, Quintiere and Harkleroad correlation [2] (known as MQH correlation) is considered the most well-established hand-calculation method to predict HGLT in pre-flashover compartment fires. However, this correlation was designed for fires positioned at the center of the room and on the ground level, which was proven to be the less hazardous condition. Nowadays, several other hand-calculation methods are available to predict fire safety parameters [2,3,4,5], but none of them takes into account the fire source location (walls proximity and vertical position) to determine the HGLT.

Several researchers have been analysing the influence of walls and corners on fire parameters, such as mass flow rate, HGLT and flame height [6,7,8,9,10,11,12,13,14]. Mowrer and Williamson [6] observed that fires positioned in corners and along walls have a restricted air entrainment and it results in higher HGLT than those predicted by the MQH correlation. Consequently, they developed modification factors to adjust that correlation and extend its applicability to wall and corner fires. Azhakesan et al. [10] undertook an experimental study of liquid pool fires in corner and center fire geometries. In an initial interrogation they found out that the modifications to MQH correlation suggested by Mowrer and Williamson [6] are reasonable. Some investigations about the influence of the elevation of the fire source on fire paremeters have been published. Backovski, Foote and Alvares [15] investigated among other things, the effect of elevated fires on the temperature profiles in forced-ventilation enclosure fires, observing that when the distance between the fire and the ceiling is shortened, the hotter become the gas temperature. Mounaud [16] conducted experiments to analyse the influence of the elevation of the fire source in compartment fires on the species generation and transport. Zhang et al. [17] conducted experiments with elevated fires in closed compartments (to represent ship fires); they studied the smoke filling processes in these cases, measuring parameters such as mass loss rate, light extinction coefficient, oxygen concentration and gas temperature profile. In a posterior work, Zhang et al. [18] presented a similar investigation on a ceiling vented compartment and observed that the fire location significantly impacted the light extinction coefficient, the oxygen concentration and the gas temperature, producing a less hazardous fire if the fire was elevated higher, which is the opposite behavior when comparing to elevated fires in closed compartments. Liu et al. [19] conducted experiments to investigate the fire source elevation effect on fire and smoke behavior, foccusing on the critical velocity, in a tunnel with longitudinal ventilation. As can be seen, the literature is still very scarce on this area.

In this work, new semi-empirical correlations are developed to predict the HGLT in pre-flashover compartment fires as a function of the fire source location, heat release rate, ventilation factor, surface area, effective heat transfer coefficient and ambient thermal properties. These correlations are based on numerical experiments, performed with a widely validated CFD fire model (FDS). The main justification for choosing this topic was the necessity of improvement on the hand-calculation methods to obtain pre-flashover compartment fire parameters, to allow the fire safety engineers to obtain rapidly, more reliable data to design fire safety systems. Knowing the HGLT helps engineers to predict the onset of flashover and to design fire suppression and smoke detection/extraction systems, which give occupants more time to leave the fire compartment, saving lives. Despite the existence of other hand-calculation methods to predict the HGLT, from the authors’ best knowledge, none of them take into account the position where the fire starts. Most of them were developed considering the fire starting in the middle of the room, at the ground level and away from any objects that may restrict the air entrainment into the flame. In this work, we showed that this is the scenario which results in the lowest HGLT, so the estimates obtained applying currently existing methods would probably underestimate the HGLT.

2 Numerical experiments methodology

A computational fluid dynamics (CFD) model, namely Fire Dynamics Simulator (FDS), was applied to simulate 252 pre-flashover compartment fire scenarios. These simulations reproduced the room geometry applied on Steckler et al. [20,21,22], considering several fire source positions (longitudinal, transversal and vertical).

FDS is a Large Eddy Simulation (LES) fire model, developed by the National Institute of Standards and Technology (NIST) and VTT Technical Research Center of Finland, which solves numerically the Navier–Stokes equations adapted for low-speed (Ma < 0.3). The core algorithm is an explicit predictor–corrector scheme, second order accurate in space and time [23]. The FDS version employed in the simulations was the FDS 6.6.0 and its default models were applied.

Firstly, 13 cases in 15 different fire heat source positions at the ground level were tested to determine their influence on the temperature results. The tested fire positions can be seen in Fig. 1a (indicated by the fuel pan position using letters A to Q) and the 13 studied cases are summarized in Table 1.

Fig. 1
figure 1

source ground positions b Fire source elevation

a Fire

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Table 1 List of openings and HRRs employed in the simulations
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Based on the findings for the HGLT of the 15 fire source positions on ground level, 19 different elevations on the positions A, B and C were simulated (see Fig. 1b). A total of 195 fire scenarios were simulated for the ground fire analysis, and 57 fire scenarios were simulated for the elevated fire source analysis, summing 252 numerical experiments.

A thermocouple tree in the front corner of the room measured the gas temperature profile (Fig. 2). The thermocouple tree was placed 0.305 m from the walls, with the lower thermocouple 0.057 m from the floor and the other thermocouple spaced equally 0.114 m from each other. The HGLT (\(T_{U}\)) was obtained through the average value of the temperature in the upper portion of the room, above the temperature profile curve inflection.

Fig. 2
figure 2

Room geometry and temperature measurement points

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The thermocouples were modelled to represent the characteristics of the physical ones in the experiments. The room geometry and the thermocouple tree can be seen in Fig. 2.

All cases were simulated for 900 s, and reached steady state before 800 s, this means that the hot gas temperature did not change significantly with time anymore. The temperature results were obtained through an average considering the results obtained between 800 and 900 s (steady state), to compensate the oscillatory results caused by the LES turbulence model.

2.1 Governing equations

The set of LES filtered equations which describe the fire phenomenon is composed by the Mass transport equation [Eq. (1)], the Species transport Eq. (2), the Momentum transport Eq. (3), the Energy (enthalpy) transport equation [Eq. (4)], and the equation of state, also known as ideal gas Law [Eq. (5)] [24].

$$ \frac{{\partial \bar{\rho }}}{{\partial t}} + \nabla \cdot \left( {\bar{\rho }{\tilde{\bf u}}} \right) = \left\langle {\dot{m}_{b}^{{\prime\prime\prime}} } \right\rangle $$
(1)
$$ \frac{{\partial \bar{\rho }\tilde{Y}_{i} }}{{\partial t}} + \nabla \cdot \left( {\bar{\rho }\tilde{Y}_{i} {\tilde{\bf u}}} \right) = - \nabla \cdot \left( {\overline{{J_{i} }} + J_{i}^{{sgs}} } \right) + \dot{m}_{i}^{{\prime\prime\prime}} + \langle\dot{m}_{{b,i}}^{{\prime\prime\prime}}\rangle _{{Vb}}$$
(2)
$$ \frac{{\partial \left( {\bar{\rho }{\tilde{\bf u}}} \right)}}{{\partial t}} + \nabla \cdot \left( {\bar{\rho }{\tilde{\bf u}\tilde{\bf u}}} \right) = - \nabla \bar{p} - \nabla \cdot \left( {\bar{\tau } + \tau ^{{sgs}} } \right) + \bar{\rho }{ \bf g} + \langle{\bf f}_{D}^{{\prime\prime\prime}}\rangle _{{Vb}}$$
(3)
$$ \frac{{\partial \left( {\bar{\rho }\tilde{h}} \right)}}{{\partial t}} + \nabla \cdot \left( {\bar{\rho }\tilde{h}{\tilde{\bf u}}} \right) = \frac{{\tilde{D}p_{0} }}{{\tilde{D}t}} - \nabla \cdot \left( {{\bar{\bf q}} + {\bf q}^{{sgs}} } \right) - \nabla \cdot \overline{{{\bf q}_{{\bf r}} }} + \langle\dot{q}_{{c,b}}^{{'''}}\rangle_{{Vb}} + \langle\dot{h}_{b}^{{'''}}\rangle_{{Vb}}$$
(4)
$$ \overline{\rho } = \frac{{p_{0} \overline{W}}}{{R\tilde{T}}} $$
(5)

where \(\rho\) is the density, \(t\) is the time, \({\varvec{u}}\) is the velocity vector, \(\dot{m}_{b}^{{\prime\prime\prime}}\) is the source term of mass, \(Y_{i}\) is the mass fraction of species i, \(J_{i}\) is the diffusive mass flux of species i, \(\dot{m}_{i}^{{\prime\prime\prime}}\) is the mass production rate per unit volume of species i by chemical reactions, \(\dot{m}_{b,i}^{{\prime\prime\prime}}\) is the mass production rate per unit volume of species i by evaporating droplets/particles, \({\text{p}}\) is the pressure, \({\uptau }\) is the viscous stress tensor, \(g\) is the gravity vector, \({\text{f}}_{b}^{{\prime\prime\prime}}\) represents the drag force per unit volume, \(h\) is the sensible enthalpy, \(p_{0}\) is the thermodynamic pressure, \({\varvec{q}}\) is the heat flux vector, \({\varvec{q}}_{{\varvec{r}}}\) is the radiative heat flux vector, \(\dot{q}_{c,b}^{{\prime\prime\prime}}\) is the convective heat transfer source term of the fuel, \(\dot{h}_{b}^{{\prime\prime\prime}}\) is the fuel enthalpy source term, \(T\) is the temperature, \(R\) is the universal gas constant, and \({\overline{\text{W}}}\) is the molecular weight of the gas mixture.

In FDS, the governing equations are approximated using finite differences on the uniformly spaced three-dimensional numerical grid, since for LES models, uniform meshing is always preferred [25].

The solution of the momentum equation requires the solution of an elliptic partial differential equation (PDE) for the pressure, so, before the components of velocity can be advanced in time, this elliptic PDE (known as a Poisson equation) must be solved for the pressure term. It is solved using a direct Fast Fourier Transforms (FFT) based solver that is part of a library of routines for solving elliptic PDEs called CRAYFISHPAK [25].

The turbulence model is based on Large Eddy Simulation (LES) and the subgrid-scale turbulent viscosity, \(\mu_{t}\), is obtained with a modified Deardorff Model (FDS turbulence default model), given by:

$$ \mu_{t} = \rho C_{V} \Delta \sqrt {k_{sgs} } $$
(6)

where \(\rho\) is the density, \(C_{V}\) is the model constant, set to the value \(C_{V} = 0.1\), \(\Delta\) is the width filter and \(k_{sgs}\) is the subgrid kinetic energy.

In FDS, due to difficulties defining a consistent test filter for use with the Deardorff’s turbulence model near the wall, at corners, and inside cavities, the turbulent viscosity of the first off-wall cell is obtained from the Smagorinsky model with Van Driest damping [25].

More information about the mathematical model solved by FDS can be found in McGrattan et al. [25].

2.2 Initial and boundary conditions

For the validation step, the initial ambient temperature (\(T_{\infty }\)) was considered the same of the experiments, varying for each case according to Steckler et al. [20]. However, after the validation step, for the numerical experiments, the initial ambient temperature was specified as 24 °C (297 K) for all the cases, so the HGLT could be analysed properly, once it depends on this initial condition. The initial and ambient pressure (\(p_{\infty }\)) was specified as 101, 325 kPa for all the cases.

The lateral and top domain boundaries were assumed “OPEN”, which means that the fluid is allowed to enter or exit the computational domain based on local pressure gradients. The gradients of the tangential components of velocity are set to zero at an open boundary.

The bottom domain boundary was assumed as a smooth solid insulated boundary (floor). The compartment boundaries (walls and ceiling) were modelled as 0.1 m thick smooth solids with the properties described in Table 2, which are based on the properties ranges described in [26]. These surfaces have insulation characteristics of ceramic fiber boards and the back side of these wall obstructions was set as “EXPOSED”, allowing the model to compute the heat flux to, and temperature of, the walls.

Table 2 Walls and ceiling properties
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For the modeling of boundary layer flows, FDS uses LES with near-wall modeling (wall functions) [25]. The wall model used is the logarithmic law of the wall, so, the viscous stress at the wall, \(\tau_{w}\), is modeled with a logarithmic velocity profile. In FDS, the law of the wall is approximated by:

$$ u^{ + } = y^{+} \,\,\,\, {\text{for}}\,\,\,\, y^{+} < 11.81 $$
(7)
$$ u^{ + } = \frac{1}{\kappa }\ln y^{ + } + B \,\,{\text{for}}\,\, y^{ + } \ge 11.81 $$
(8)

where \(\kappa = 0.41\) is the von Kármán constant and \(B = 5.2\). The friction velocity is defined as \(u_{\tau } \equiv \sqrt {\tau_{w} /\rho }\), and from the friction velocity the nondimensional streamwise velocity is defined as \(u^{ + } \equiv u/u_{\tau }\) and the nondimensional wall-normal distance is defined as \(y^{ + } \equiv y/\delta_{\nu }\), where \(\delta_{\nu } = \nu /u_{\tau } = \mu /\left( {\rho u_{\tau } } \right)\) and represents the viscous length scale.

The fuel applied to all cases was methane with heat release rate (HRR) prescribed as constant, according to Table 1. The data obtained through the present simulations were employed to analyze the influence of the fire heat source location on the HGLT in pre-flashover compartment fires and to develop improved correlations to predict this parameter.

2.3 Mesh resolution

Three different methods were applied to ensure that the proper mesh resolution was employed in the numerical simulations. The first was the analysis of the non-dimensional parameter \(D^{*} /\delta_{x}\), largely employed for FDS simulations; the second was a sensitivity analysis, and the third was the Measurement of the Turbulence Resolution (MTR), which is the recommended method to verify the adequacy of the mesh resolution in LES simulations [23, 27,28,29].

For simulations involving buoyant plumes, the non-dimensional parameter \(D^{*} /\delta_{x}\) is a good way to determine how well the flow field is resolved [30, 31]. \(D^{*}\) is the characteristic fire diameter given by Eq. (9) and \({\updelta }_{{\text{x}}}\) is the nominal size of a mesh cell.

$$ D^{*} = \left( {\frac{{\dot{Q}}}{{\rho_{{\infty C_{p} T_{\infty } \surd g}} }}} \right)^{\frac{2}{5}} $$
(9)

where \(\dot{Q}\) is the HRR, \(\rho_{\infty }\) is the ambient air density, \(c_{p}\) in the ambient air specific heat, \(T_{\infty }\) is the ambient temperature and g is the acceleration of gravity.

As a Rule of Thumb, McDermott et al. [29] suggest that values of \(D^{*} /\delta_{x}\) of the order of 10 provide an adequate grid resolution for the plume, while the validation study sponsored by the U.S. Nuclear Regulatory Commission [32], suggested \(D^{*} /\delta_{x}\) values ranging from 4 to 16.

Table 3 presents the values of \(D^{*}\) and \(D^{*} /\delta_{x}\) for four mesh sizes (\(\delta_{x} = \) 6 cm, 5 cm, 4 cm and 3 cm). These meshes are equally spaced in all directions x–y-z, and they are uniform all over the computational domain. As can be observed in Table 3, all meshes present \(D^{*} /\delta_{x}\) contained inside the recommended range.

Table 3 Values of \(D^{*}\) and \(D^{*} /\delta_{x}\) for meshes 6 cm, 5 cm, 4 cm and 3 cm
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The second criteria used to evaluate the mesh discretization was a comparison between the results (temperature profiles, average hot gas layer temperature \(T_{U}\), average lower layer temperature \(T_{L}\) and computational time) obtained for the four different meshes applied to test 14 from Steckler et al. [20] (according to Table 6). The results are shown in Fig. 3 and Table 4.

Fig. 3
figure 3

Mesh sensitivity analysis (Steckler’s test 14)

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Table 4 Mesh sensitivity analysis and computational time
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As can be observed in Fig. 3 and Table 4, meshes \({\updelta }_{{\text{x}}} = \) 5 cm, 4 cm and 3 cm present similar results, all of them agreeing well with the experimental data. So, the mesh \({\updelta }_{{\text{x}}} = \) 5 cm has been chosen to be applied for the numerical experiments, as a matter of computational time and result stability.

To ensure the right selection of the mesh, the MTR for the 5 cm mesh has been calculated for cases 5, 11, 12 and 13 (described in Table 1). The MTR is a posteriori analysis that gives a measure of how well the turbulence is being resolved in the domain in LES simulations. It is a scalar quantity defined locally by Eq. (10).

$$ M\left( x \right) = \frac{{\langle k_{sgs}\rangle }}{{\langle TKE \rangle + \langle k_{sgs} \rangle}} $$
(10)

where the angled brackets denote a time-average, the resolved turbulent kinetic energy per unit mass (TKE) is given by Eq. (11) and the subgrid kinetic energy (\({\text{k}}_{{{\text{sgs}}}}\)) is obtained directly from FDS for each point of interest.

$$ TKE = \frac{1}{2}(\left( {\tilde{u} - \langle \tilde{u}\rangle} \right)^{2} + \left( {\tilde{v} - \langle \tilde{v}\rangle} \right)^{2} + \left( {\tilde{w} - \langle \tilde{w}\rangle} \right)^{2} $$
(11)

where ũ, ṽ, w̃ are the resolved LES velocity components and are also obtained from FDS. The MTR value was calculated based on 15 points measured in regions of interest inside the compartment. The results presented in Table 5 are the mean MTR values.

Table 5 MTR results for mesh size \(\delta_{x}\) = 5 cm
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According to Pope [27], the MTR value must be less than or equal to 0.2, which corresponds to the resolution of 80% of the turbulent kinetic energy. Additionally, as showed by McDermott [29], a MTR mean value near 0.2 provides satisfactory results for mean velocities and species concentrations in non-reacting, buoyant plumes. So, the 5 cm size mesh is capable to resolve more than 80% of the kinetic energy of the flow field.

2.4 Comparison with experimental data

To validate the numerical model, the results of HGLT (\(T_{U}\)) were compared to experimental data presented by Steckler et al. [20] for 40 fire scenarios. The description of the investigated scenarios and its results can be seen in Table 6. On experiments 160, 163 and 164 the fire sources were raised 0.3 m from the floor, while it was on the ground for the other experiments.

Table 6 Comparison between numerical (FDS, present study) and experimental (Steckler et al. [20]) results
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As can be observed in Table 6, the numerical model can be considered validated, since for the HGLT the maximum relative deviation found was 13.98% and the mean relative error was 3.17%,

The comparison between experimental temperature profiles and the ones obtained by FDS for tests 14, 160, 410, 163, 610 and 164 can be seen in Fig. 4. Similar results were obtained for all 40 tests described in Table 6. A good agreement between experimental data and numerical results was found, so it can be concluded that the mathematical model employed on FDS was validated and can be applied to generate reliable data for pre-flashover compartment fire scenarios.

Fig. 4
figure 4

Temperature profiles comparison between numerical results (FDS) and experimental data (Steckler et al. [20])

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3 Results and discussions

3.1 Fire source position influence: ground level

The influence of the fire source location at the ground level was analysed varying the fire position along the axis x (longitudinal direction) and y (transversal direction), as shown in Fig. 1a. The 13 studied cases (Table 1) have been simulated in 15 different positions (Fig. 1a), resulting 195 different scenarios.

The numbers inside the circles in Fig. 5a–f show the HGLT (\(T_{U}\)) obtained through FDS simulations for each one of the studied ground level positions (Letters A to Q in Fig. 1a) for cases 1, 5, 9, 11, 12 and 13 (according to Table 1). Each plot (a to f) represents the upper view of the room for the indicated cases and the HGLT (\(T_{U}\)) results were placed on the position inside the room where the fire source was placed (these temperatures must be assumed as homogeneous over all the HGL), so each square (plots a to f) presents the condensed results of the 15 tested fire positions (Fig. 1a).

Fig. 5
figure 5

source position on the ground level. Cases numeration according to Table 1 and fire positions according to Fig. 1a

Average HGLT [°C] in each fire

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As can be observed, the highest HGLT were found for the corner fire locations (positions B and N), represented by the red circles, followed by the near wall fire location (positions C, D, E, H and K) represented by the yellow circles. The positions away from walls (center positions—A, F, G, J, L, M, P and Q), represented by the green circles, are the ones which produced the lowest HGLT, especially those near the opening doorway. This behaviour was expected since corner and wall fires are subjected to reduced air entrainment.

According to Zukoski et al. [33], in the early stages of a fire (pre-flashover) in a building, the rate of production of hot gases by a fire and the temperature of these gases will depend very strongly on the rate of entrainment in the fire plume and in the flame itself. According to Mowrer and Williamson [6], the fire plume temperature will significantly increase for corner and near wall fire plumes; they stated that the convective energy flux through the openings would not depend on the fire source location in the room. This statement was confirmed by the results shown in Fig. 5a–f, once fire source position in the same group of interest (away from walls (center), along walls or near corners) showed small variations.

An interesting observation can be made through Fig. 5c, which represents case 9 (window opening). For the window cases (cases 8, 9 and 10), when the fire source is in position Q (Fig. 1a), the upper layer temperature is higher than the other center positions (green circles), this is different from the cases with a doorway, which presented the lowest HGLT when the fire source was in position Q. This is explained by the reduction of air entrainment caused by the presence of the wall in the lower part of the fire source in the window cases for position Q. Another pattern can be observed, once the ventilation factor (opening size) is augmented, lower become the HGLT for each fire position.

So, from this analysis, we can conclude that what causes the increase in the HGLT is the presence of an obstruction, that can be the walls or even a piece of furniture, which restricts the air entrainment into the flame. This confirmed a weak influence of the fire source location for the cases when the fire source is at the floor level and at the same position group (away from walls (center), along walls or near corners), as suggested by Mowrer and Williamson [6].

3.2 Fire source position influence: elevated fire

For analysing the influence of the vertical position of the fire source on the HGLT, 19 elevations (ranging from z = 0 m to z = 1.8 m with 0, 1 m increments) have been simulated for positions A (center fire), B (corner fire) and C (back-wall fire), as can be seen in Fig. 1a. These simulations were conducted only for case 5 (Table 1), generating data for 57 numerical simulations.

Figure 6 presents the results obtained from these numerical experiments for the HGLT as a function of the fire source elevation (z) for positions A, B and C.

Fig. 6
figure 6

source elevation and Hot gas layer temperature

Relation between fire

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As can be observed, as the height of the fire source location was augmented, so was the HGLT. This behaviour was observed for all positions (center, wall and corner fires). However, the influence of the elevation was greater for fires in the center of the room, followed by fires along walls, and finally fires near corners. This may be explained by the reduction of the air entrainment rate. When the fire source is elevated, it becomes closer to the hot gas layer interface and its plume and flame have a smaller region to entrain fresh air (Figs. 7a–c). With this reduction of air entrainment, the HGLT rises. Once, fires in corners and along walls already have an reduced air entrainment of approximately one quarter and one half (as a consequence of the walls restriction in the perimeter of the flame/plume), respectively, when elevated they suffer less influence then the ones away from walls.

Fig. 7
figure 7

source elevations: a z = 0 m, b z = 0.6 m, c z = 1.2 m

Illustration of the distance between the fire plume and upper gas layer for different fire

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Significant differences in the HGLT between subsequent levels (elevations) were observed, being most of them of the order of 10–20 °C (considering increasing steps of Δz = 0.1 m between elevations), while when comparing HGLT for the fire source at the floor level and at the highest studied level (z = 1.8 m), the difference in the HGLT were of 110 °C when the fire is placed in a corner, 180 °C when the fire is place near a wall and 242 °C when the fire is placed in the center of the room.

Although the influence of the elevation on the HGLT is not perfectly linear, a linear dependence can be considered to correlate the HGLT and the elevation of the fire source.

3.3 Semi-empirical Correlations to Predict HGLT considering the fire position

As it can be observed in the previous analysis, the fire source location has an important influence on the HGLT. It is noticed that walls or other objects adjacent to the fire will cause an augmentation on the plume temperature, and consequently in the HGLT. It is also noticed the great influence that the vertical fire source position has on the HGLT, which follows a quite linear trend.

As the available correlations in the literature do not take into account properly the fire source location, the present work developed improved semi-empirical correlations to predict the HGLT in pre-flashover compartment fires, depending on the fire position. As well as the MQH correlation [2], the present correlations were derived from an energy balance on the well-stirred gas layer in the upper portion of the room, resulting in a function of two dimensionless terms presented in Eq. (12).

$$ \frac{\Delta T}{{T_{\infty } }} = f\left( {\frac{{\dot{Q}}}{{\sqrt g C_{p} \rho_{\infty } T_{\infty } A_{0} \sqrt {H_{0} } }},\,\,\frac{{h_{k} A_{T} }}{{\sqrt g C_{p} \rho_{\infty } A_{0} \sqrt {H_{0} } }}} \right) $$
(12)

where \(\Delta T \) corresponds to the upper layer temperature rise \(\left( {\Delta T = TU - T\infty } \right)\), the first term in parenthesis represents the ratio of the energy released by the fire to the energy convected through the openings, and the second term stands for the energy lost through the walls divided by the energy convected. Where \(\dot{Q}\) is the HRR [kW], g is the acceleration of gravity [m/s2], \(C_{p}\) is the specific heat of the ambient air [\({\text{kJ}}/\left( {{\text{kg}} \cdot {\text{K}}} \right)\)], \(\rho_{\infty }\) is the density of the ambient air [kg/m3], \(T_{\infty }\) is the ambient air temperature [K], \(A_{0} \sqrt {H_{0} }\) is the ventilation factor [m5/2], \(h_{k}\) is the effective heat transfer coefficient [\({\text{kW}}/\left( {{\text{m}}^{2} \cdot {\text{K}}} \right)\)] and \(A_{T}\) is the total area of the compartment surface [m2]. These are the same dimensionless variables employed in the MQH correlation [2], however, the MQH correlation was developed only considering fires at the center of the room and does not account for the fire source elevation effect on the HGLT.

The effective heat transfer coefficient must be obtained in the same manner that in the MQH correlation [2], depending on the exposure time (\({\text{t}})\) and thermal penetration time (\({\text{tp}})\) of the wall, which is given by \(tp = \delta_{w}^{2} /\left( {4\alpha_{w} } \right)\), where \({\upalpha }_{{\text{w}}}\) is the wall thermal diffusivity [m2/s] and \(\delta_{w}\) is the wall thickness [m]. For exposure time smaller than the penetration time \((t < tp)\), the effective heat transfer coefficient is given by \(h_{k} = \sqrt {k\rho c/t}\) while for an exposure time larger than the penetration time \((t > tp)\), it is \(h_{k} = k/\delta_{w}\), where \({\text{k}}\) is the wall thermal conductivity [\({\text{kW}}/\left( {{\text{m}} \cdot {\text{K}}} \right)\)], \({\uprho }\) is the wall density [kg/m3], \({\text{c}}\) is the specific heat of the wall [\({\text{kJ}}/\left( {{\text{kg}} \cdot {\text{K}}} \right)\)].

Figure 8 shows the dimensionless temperature (\(\Delta T/T_{\infty }\)) obtained by FDS for all the ground level simulations as a function of the first (Fig. 8a) and second (Fig. 8b) dimensionless terms in parenthesis in Eq. 12, showing a power law relationship, so Eq. 12 can assume the functional form presented in Eq. (13).

$$ \frac{\Delta T}{{T_{\infty } }} = \gamma \left( {\frac{{\dot{Q}}}{{\sqrt g C_{p} \rho_{\infty } T_{\infty } A_{0} \sqrt {H_{0} } }}} \right)^{\alpha } \cdot \left( {\frac{{h_{k} A_{T} }}{{\sqrt g C_{p} \rho_{\infty } A_{0} \sqrt {H_{0} } }}} \right)^{\beta } $$
(13)

where the constant \(\gamma\) and the exponents \(\alpha\) and \(\beta\) are determined from a best fit of the data from each group of fire positions (center, wall and corner) at the ground level, since it was observed a great influence of these fire positions on the HGLT. This means that 3 variants of the correlation are obtained for the HGLT prediction, one for each position of the fire source.

Fig. 8
figure 8

Dimensionless temperatures as a function of the dimensionless terms of Eq. 12a First term b second term

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The influence of the fire elevation on the HGLT did not show a power law relation, but a linear tendency instead. These effects then considered as a correction term (\(\delta\)) summed to the HGLT correlation when the elevation is different from zero, according to Eq. (14).

$$ \frac{\Delta T}{{T_{\infty } }} = \gamma \left( {\frac{{\dot{Q}}}{{\sqrt g C_{p} \rho_{\infty } T_{\infty } A_{0} \sqrt {H_{0} } }}} \right)^{\alpha } \cdot \left( {\frac{{h_{k} A_{T} }}{{\sqrt g C_{p} \rho_{\infty } A_{0} \sqrt {H_{0} } }}} \right)^{\beta } + \delta $$
(14)

This correction term (\(\delta\)) is defined using the dimensionless parameter \(z^{*}\):

$$ z^{*} = \frac{z}{H} $$
(15)

where z is the vertical fire position in m and H is the height of the room in m.

The correlation of the correction term (\(\delta\)) for Eq. 14 was obtained through a simple linear regression of the difference between the dimensionless temperature at each level \(\left( {\Delta T/T_{\infty } } \right)_{{{\text{level}}}}\) and at the ground level \(\left( {\Delta T/T_{\infty } } \right)_{{{\text{ground}}}}\).

The fitting of the data for terms \(\alpha\), \(\beta\) and \(\gamma\) was obtained through the software SPSS applying a multiple linear regression.

Three different sets of coefficients for the correlation (Eq. 14) were obtained according to the fire position inside the room (away from walls, near a corners and near a wall), see Table 7. Observe that the term \(\delta\) must be set as zero if the fire occurs at the ground level (z = 0), while if the fire is elevated, it must be calculated according to its dimensionless height (z*). All the temperatures in the correlation (Eq. 14) must be entered as absolute temperatures [K], so the HGLT (TU) obtained through the use of this correlation will also be given in K, but can easily be converted to °C.

Table 7 Coefficients for the new correlation (Eq. 14)
Full size table

Figure 9 presents a comparison between the predicted values of the HGLT applying Eq. (14) with the coefficients presented in Table 7 against the numerical data obtained with FDS. As FDS provides all temperatures in °C, the temperatures obtained through the correlations were also converted from K to °C. The dashed lines represent a 10% tolerance. As can be observed, an excellent fitting was obtained applying the newly developed correlation for all the fire positions.

Fig. 9
figure 9

Comparison between upper layer temperatures predicted with Eq. (14) and numerical data from FDS for fires sources a away from walls, b near a corner and c near a wall

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3.4 External validation of the new semi-empirical correlations: comparison with experimental data

To ensure the quality and applicability of the newly obtained correlations, the predictions obtained from Eq. 14 with the coefficients from Table 7 are then compared to different sets of experimental data available in the literature, to represent several fire scenarios, different from those employed to fit the new correlation.

Table 8 describes the characteristics of each experimental data set employed in the external validation (for more information about the fire scenarios, the references must be consulted).

Table 8 Experimental data sets employed in the external validation of the new correlation
Full size table

Figure 10a–c show the comparison of the HGLT predicted with the new correlation and the experimental data for fires away from walls (center), near corners and near walls, respectively. The continuous line represents that the predicted HGLT values are equal to the experimental ones, while the dashed lines represent a 10% tolerance.

Fig. 10
figure 10

Comparison between HGLT predicted by the newly designed correlation and experimental data from several authors: a center fires, b near-corner fires, and c near-wall fires

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As can be observed in Fig. 10a, even with a great variety of fire scenarios, an excellent agreement was obtained for most of the experimental data. A slight less precise agreement was found to Johansson et al. [32], where the predicted results were found to be a bit lower than the experiments. This may be explained by the fact that these data were obtained from a multi-room experiment, where the fire room opening was connected to another room instead of being connected to the exterior environment. This usually reduces the flow of air into and out the room, which in turn, increases the upper layer temperature.

In Fig. 10b, a very good agreement was obtained for experimental data from McCaffrey and Rockett [35], Li and Hertzberg [36], and Steckler’s experiments [20]. The data from Mowrer and Williamson [6], showed a slight higher variation, but still of the order of 20%, which is still a very reasonable agreement, considering that experimental data always present some measurement uncertainty and that there was a lack of information about the ambient temperature during experiments and precise wall and linen material properties that had to be estimated to apply in the correlation.

As can be seen in Fig. 10c, a very good agreement was obtained for most of the experimental data, most of them showing differences smaller than 10%. The highest differences were found for very wide line fire source (from Quintiere et al. [37]), which may be expected, once the correlation was designed with data from circular burners, and again for the data from Mowrer and Williamson [6], which can be related as discussed before to the experimental uncertainty or even to the lack of information.

Although the numerical results applied to develop the correlation presented small heat release rates, which would represent the maximum heat release rate of a small pool fire (31.6 and 62.9 kW) or small wood furniture (105.3 and 158 kW) (e.g. a wood framed chair with Polyurethane foam and cover), the comparison to other sets of experiments showed that these newly developed correlations stand for higher heat release hates as well as to different fire scenarios, including room sizes, ventilation factors, construction materials, etc.

4 Conclusions

In this work, numerical experiments were performed using the software FDS, in order to study the influence of the fire source position (transversal, longitudinal and vertical directions) on the upper layer temperature for pre-flashover compartment fires. It was confirmed that the HGLT does not depend on the fire source position when the fire occurs away from walls or obstructions (i.e. pieces of furniture), and that there is an augmentation on those temperatures when the fire occurs at corners or walls (at the ground level). Fires near corners presented the highest temperatures, followed by near wall fires, and the lowest temperatures were observed for fire sources away from walls.

The present paper also showed a great influence of the fire source elevation on the HGLT. An increase on the HGLT, mostly on the range of 10–20 °C, was observed between subsequent vertical levels (Δz = 0.1 m), while when comparing the HGLT for the fire source at the floor level (z = 0 m) and at the highest level (z = 1.8 m) the difference in the HGLT was of more than 100 ºC in all cases, reaching 242 °C for the fire at the center of the room. It was also observed that fires away from obstructions or walls suffered more influence from the elevation of the fire source, followed by fires near wall and the lowest influence was presented by fires near corners. It was noted that for fires occurring near the ceiling (above 50–55% the room high) the behaviour was the opposite than for fires on the floor, presenting the highest temperatures for fires at the center and the lowest for fires near corners.

Based on these findings, improved correlations have been designed to predict the HGLT, considering the fire source position. The correlations were developed based on numerical data generated using the software FDS. A good agreement between the HGLT predicted by the newly developed correlations and the numerical data from FDS have been found. As an external validation, predictions from the newly developed correlations were also compared to experimental data from different fire scenarios, showing a good agreement. So, the new correlations can be considered as validated for pre-flashover compartment fires and are capable to predict the upper layer temperature considering the fire source location (including its vertical position). This is an important achievement once it was observed that fires near corner or at higher elevations produce higher upper layer temperatures than those at the ground and away from walls, which are the ones predicted by the conventional correlations previously available in the literature.