Spatiotemporal measurement of light extinction coefficients in compartment fires

In case of fire, the visibility plays a major role as it limits the occupants’ orientation capabilities and the perception of signs. These effects are determined by the light extinction due to smoke or other aerosols produced in fires. The presented method is based on the optical observation of an array of light sources during a fire in a laboratory experiment. The smoke induced into the compartment leads to a drop in intensity of each individual light source. This information is used to deduce the extinction along the line‐of‐sight to the camera. Once the data are captured, an automated processing is used to locate the diodes on the images and determine their intensity. Here, the optical image of the small diodes is assumed to have a known shape, so that the optimisation algorithm is capable to identify the location of the diode's centre and quantify the luminosity in a sub‐pixel range. The result is a time series for each diode, indicating the change of the relative luminosity, w.r.t. the initial values. Finally, a model for the extinction along each line‐of‐sight is formulated. It assumes that the light extinction coefficient is distributed in homogeneous layers. The number of layers is a free model parameter. Given this spatial distribution of the extinction coefficient and the experimental geometry, each line‐of‐sight is impacted by a number of layers, of yet unknown coefficient values. An inverse modelling approach is used here to find coefficient values that match the modelled line‐of‐sight extinction with the observed luminosity drops. The final result is a time‐ and height‐dependent distribution of the light extinction coefficient during the full experiment.


Summary
In case of fire, the visibility plays a major role as it limits the occupants' orientation capabilities and the perception of signs. These effects are determined by the light extinction due to smoke or other aerosols produced in fires. The presented method is based on the optical observation of an array of light sources during a fire in a laboratory experiment. The smoke induced into the compartment leads to a drop in intensity of each individual light source. This information is used to deduce the extinction along the line-of-sight to the camera. Once the data are captured, an automated processing is used to locate the diodes on the images and determine their intensity.
Here, the optical image of the small diodes is assumed to have a known shape, so that the optimisation algorithm is capable to identify the location of the diode's centre and quantify the luminosity in a sub-pixel range. The result is a time series for each diode, indicating the change of the relative luminosity, w.r.t. the initial values. Finally, a model for the extinction along each line-of-sight is formulated. It assumes that the light extinction coefficient is distributed in homogeneous layers. The number of layers is a free model parameter. Given this spatial distribution of the extinction coefficient and the experimental geometry, each line-of-sight is impacted by a number of layers, of yet unknown coefficient values. An inverse modelling approach is used here to find coefficient values that match the modelled line-of-sight extinction with the observed luminosity drops. The final result is a time-and height-dependent distribution of the light extinction coefficient during the full experiment.

| INTRODUCTION
In the assessment of life safety in complex building structures in case of fire visibility is one of the main tenability criteria in performancebased design. Especially if the orientation of the occupants depends on the perception of signs and doors, 1 a sufficient visibility is crucial.
Both aspects, that is, visibility as a tenability criterion and as a limiting factor for the orientation, manifest in the ASET-RSET 2 concept. Here, visibility will reduce the available safe egress time 3,4 and due to reduction of walking speed or prolonged orientation and route evaluation, 5 it will increase the required safe egress time. As modern fire safety analysis is based on computer simulations, for example, using the fire dynamics simulator (FDS 6 ), a valid prediction of the smoke spread and especially its light attenuation must be achieved. Past computations with FDS with the aim to validate the prediction of smoke density and light extinction, like References [7,8], indicate significant deviations and the need for further research.
The copyrightline for this article was changed on 19 December 2020 after original online publication.
The determination of visibility is basically based on a threshold for the intensity of the perceived light. 9 As the light travels from the object, for example, an illuminated sign, to the observer, its intensity may be reduced due to an interaction with the medium between both. The kind of interaction, for example, scattering or absorption, as well as its amplitude and dependence on wavelength are subject to the type and number of particles in the medium. While the particle density n has mainly an impact on the amplitude, the predominant is given by the properties of the particles. The latter aspect is effectively described by an interaction, here mainly the extinction crosssection c ext . Its value and the value of quantities defined further down depend on the wavelength of the light. However, this will not be further distinguished in this article and implicitly assumed that they all are wavelength dependent, without additional indexing or labelling in the text.
As a light ray travels along the line-of-sight, its initial intensity I 0 is in general reduced. The transmission T describes the ratio of the reduced intensity I and I 0 . Using the Beer-Lambert law (equation 1, cf. 10 ), the transmission of monochromatic light can be expressed as a function of the optical depth τ.
In a homogeneous medium with a particle density n and an extinction cross-section c ext the optical depth τ is proportional to the distance along the line-of-sight Δs (Equation 2).
A further simplification is the definition of the extinction coefficient σ, which represents the product of n and c ext (see Equation (2)).
This leads to only a single physical quantity to be measured in order to describe the light attenuation, as the other one, the distance Δs, is purely geometrical. A summary of the relevant processes and optical properties of smoke can be found in the SFPE handbook of fire protection engineering. 11,12 It should be noted that the reported extinction coefficients in the SFPE handbook are close to constant and depend only on the type of combustion, that is, flaming vs smouldering.
A common approach to measure the light extinction is an apparatus that combines a light transmitter and receiver. Eventually, both are close to each other, while a light beam is sent out and reflected back with a mirror. This is the basic setup of the MIREX 13 measurement system. It uses infrared light, which is send across a distance of 2 m. The specification for the light spectrum is that 50% of the power is emitted between 800 and 950 nm. This apparatus can be used to precisely measure the light extinction at a localised, yet 1 m wide, position. The usage of multiple devices would allow a spatially resolved measurement; however, it would increase the cost and impact the fluid dynamics due to the physical extension of the apparatus. However, the measured data can be used for localised comparison with the results of the presented approach. Fundamental challenges and uncertainties in the cause of experimentally investigating the light extinction and scattering during compartment fires are outlined in References [14,15].
In this article, a simple technique to measure the extinction coefficients σ in the range of visible light is proposed. The implicit measurement leads to spatially, here in vertical direction, and temporally resolved data. One of the fundamental assumptions is a layered structure of the smoke. The final goal is to provide spatiotemporal data as a validation basis for the computation of light extinction processes in compartment fires. A validation case, similar to Reference [16], will be conducted in future work.
The outline of this article is as follows. First, one of the conducted experiments is presented. Its data will be used as example data for the following methodology section. The presented methodology is split into the raw data capturing, the data post-processing and the determination of the extinction coefficients. Finally, two application examples are shown and conclusions and future work are stated.

| EXPERIMENTAL SET-UP
This section outlines the main characteristics of the experimental setup. The experimental data are used in the following sections to demonstrate the methodology and to provide an application example.
The experiment took place in the Heinz-Luck fire detection laboratory  Figure 1A depicts the location of the objects of interest: a pool fire, a camera, a MIREX apparatus and a vertical LED strip. In the following, only the placement of the camera is important, which is at a height of 2.3 m and at a distance of 4.4 m from the LED strip. A photo of one of the experiments is shown in Figure 1B.
The LED strip used in the experiment is a common consumer product. It is build out of 5 cm long parts that contain three LED units (see Figure 2A). The units separation distance is 5/3 cm, that is, about 1.67 cm. Each of the units has three individual LEDs that emit red, green and blue light and are therefore capable to produce light in combination of those (see Figure 2B). In addition, it is possible to control the intensity of a LED. In the presented experiments, the LED units are set to white, that is, all LEDs are on, with the maximal intensity. The length of the vertically aligned strip was 2.35 m, starting about 5 cm below the ceiling, and it contained 141 LED units.
The fire was one of the fires defined in EN 54 as TF5. It is a pool fire with n-heptane as fuel. In contrast to the norm, the amount of fuel was reduced to 500 g, which lead to a burning duration of about 3 minutes. The choice of this fire type is due to the fact that it is the most simple one to be modelled with numerical simulation methods, as it does not involve pyrolysis and the combustion reaction is consuming a single pure fuel. Figure 3 shows the time line of the experimental procedure. The timing in this run was as follows: • t = 0 seconds start of the experiment • t~30 seconds ignition of the pool fire • t~200 seconds all fuel is consumed, the fire is off • t~420 seconds hall's ventilation system is turned on • t~1200 seconds end of the measurements, end of the experiment The examples in the following methodology section are based on experimental data gathered in the above-stated experimental run.

| METHODOLOGY
The presented methodology is split into three aspects. The first subsection outlines a few recommendations of the raw data acquisition, while the second one describes the procedure to model the image of each LED unit and thus find its position and determine its intensity in an image frame. In the last subsection, a method to discretise the compartment volume into homogeneous smoke layers and the determination of the light attenuation coefficients is introduced.

| Raw data acquisition
The methodology is based on the analysis of photo images captured by a camera. During the experiment, the camera continuously  Figure 10 captures the LED strip. As the scientific quality of the images is crucial, the settings of the camera are stated and discussed here.
The used camera was a Canon 80D with a Canon 18 to 35 mm lens.
It was run in full manual mode, that is, without any brightness corrections or other adjustments. The choice of a manual mode is important as otherwise the camera would adjust its parameters in order to take wellbalanced images. This would introduce luminosity changes of the LEDs, which are not due to light attenuation but due to changes in the camera settings. The images are stored in the common jpeg format as well as in a raw format. The latter one can be used to further increase the accuracy of the analysis as it offers a higher luminosity resolution.
In the manual mode of the camera, all relevant parameters must  The central region has a radius r 0 and transits smoothly towards zero within the width w. However, the LED unit images are asymmetric.
Therefore, the radius r 0 and the width w can be defined as two components in the x-and y-direction, that is, r 0, x , r 0, y , w x and w y . In addition, a rotational angle α can be defined. In total there are eight parameters: A 0 , x 0 , y 0 , r 0, x , r 0, y , w x , w y and α. The explicit formula for the value A m of a pixel at position x, y in the model image is given in Equation (3).
With the radius r defined as the central region r 0 (x, y) and the width w(x, y) can be computed as a function of the above-defined polar coordinates In addition, penalty terms are defined to ensure that the central points are within the search area and the amplitude, radii and widths are positive.

| Computation of extinction coefficients
The final goal of the presented method is to gain spatially resolved information of the extinction coefficients. As described in the following, a simple model based on spatial discretisation, here in horizontal layers, is outlined. Using the captured intensities, based on the amplitude parameter A 0 in Equation (3), a best matching set of extinction coefficients is determined. This approach applies the line-of-sight integral of the Beer-Lambert law in an inhomogeneous medium.
The proposed model has the following assumptions: 1. the light absorption properties, that is, σ, in a layer is homogeneous, 2. the plume region is neglected, 3. light paths are linear, for example, refraction due to varying gas temperatures is neglected, and 4. only the product σ = c ext Á n in Equation (2) is determined.
The severity of these assumptions is different for each one. While the assumption of homogeneous layers may be critical for complex compartments, neglecting of the plume region will be a minor issue for sufficiently large spaces. Refraction will have a minor role for low energy fires, like smouldering, as the involved temperatures are just above the ambient. Finally, the capability to distinguish between c ext and n is not critical for the computation of visibility, as the optical depth depends only of their product σ.
The model's data structure is a set of N layers extinction coefficients σ i . They represent the values at homogeneous horizontal layers, which fill the domain of interest, for example, a compartment (see The usage of discrete layers with constant values simplifies this integral into a finite sum over all layers, using the computed travel paths Δs i, j and the yet to be determined extinction coefficients σ i (see Equation (10)).
F I G U R E 6 Fitting example for a single LED unit F I G U R E 7 Temporal evolution of model parameters for three selected LED units In the following all intensities are scaled to their initial values I 0 and thus range from zero to one. The experimental intensities I e are scaled by the mean of 10 reference images taken just before the experiment started. The modelled intensities I m, j of L j are given by Equation (11).
Equation (12) poses an equation system, containing N LEDs nonlinear equations, to be solved for σ i . This system has to be solved for each frame; however, the travel paths Δs i, j are computed only once, as they do not change in time.
I m,j σ 0 , …, σ Nlayers À Á = I e,j for allj∈ 1, N LEDs ½ : In order to solve the target equation system, a minimisation procedure is used. This has the advantage that additional criteria, but solely the distance between the experimental and modelled data, can be defined.
Therefore, a cost function Ω σ is defined as shown in Equation (13).
It is a sum of three contributions. The first one computes the For example, if the LED signal was damped to nearly zero, there is a minimal value for the extinction coefficient to achieve these results.
However, the actual value may be significantly higher. In order to quantify these effects, the sign of ϕ a can be accordingly chosen.
An example of the outcome of the above optimisation procedure is given in Figure 9. This figure contains a set of results, which are distinguished by the number of extinction coefficient layers N layers in Equation (10). For each of the results, the vertical distribution of extinction coefficients is shown (left-hand side) as well as the corresponding modelled intensity I m (right-hand side). First, the model is capable to represent the experimental intensities (grey dots) within the variation of the data. All N layers lead to very similar curves. Second, also the extinction coefficients follow a similar shape. The curves with higher numbers of layers show more features, which are not necessary physical but may be artefacts of a more accurate fitting of the noisy data. Third, the extinction coefficients above about 3.35 m drop towards zero. This is due to the fact that the cost function, Equation (13), favours small extinction coefficients. As there is no experimental data for this heights-the top LED unit is at about 3.35 m-this leads to small or even zero coefficients. Each layer of the low N layers numbers covers a larger portion of the space and therefore does not include layers, which are not crossed by any light paths.

| APPLICATION EXAMPLE
As the focus of this contribution is the methodology, this section presents just two selected aspects of the data analysis. These are the temporal evolution of the extinction coefficient layers and the comparison to a MIREX measurement.  capture the dynamics as well as the amplitude of the extinction coefficient the same way as the MIREX measurement. However, the MIREX system seems to be more sensitive, that is, indicates higher extinction coefficients, in the initial phase and in the decaying phase.

| CONCLUSIONS
This article presents an approach to capture the light attenuation of individual LED units in a compartment fire. Using this approach, multiple quantities, like mainly the amplitude but also the position and widening of the signal, can be captured on a scale, which are finer than the colour level or the pixel width. This accuracy allows to include further effects, like refraction, into the analysis. The determination of the extinction coefficients is based on a simple method and is therefore expected to be robust with respect to the failure of individual LEDs or small movements of the strip. However, the validation needs further investigation to find a more conclusive comparison method with other techniques. Although the comparison with the MIREX data is promising, more suitable measurement locations must be revised as well as the impact of the wavelength, here between infrared and red, must be evaluated. Eventually, the analysis with the LED light, which targets wavelengths in the visible spectrum, may lead to more applicable extinction coefficients.
The instantaneous capturing of spatially resolved extinction coefficients, especially in the visible spectrum, will provide a validation basis for numerical tools. Especially as it generates data for inhomogeneous distributions, so that transport processes can be evaluated as well. The general approach will allow an application on small scale as well as on large scale, as long as the assumptions are expected to be valid. This shows the need of providing a method to check the validity with experimental methods.
In terms of applicability, the proposed method is easily applicable at low cost. This is due to the usage of consumer products and no need for fine tuning or calibration with external sources. The most part of the process is covered by the data analysis. As the analysis is written in Python and can be freely shared with others-please contact the authors to gain access-it can be directly applied by others researchers.

| OUTLOOK
As pointed out in the conclusions, there exists a range of topics to improve the validity and applicability of the proposed method and a few of them are subject of current research.
One of the main tasks is to investigate other fire types, but solely TF5. In this contribution, it was chosen as one of the simplest. However, especially in case of weak thermal drivers, like in case of smouldering, the dynamics will be more challenging to be captured and the gained spatial resolution will be of high importance for model validation.
To address the question of homogeneous layers, multiple LED strips and two (or more) cameras can be used. This will allow investigating the variations along the smoke layers. In combination with additional MIREX measurements at multiple heights, a wider understanding of the smoke spread, for example, light extinction coefficients, will be achieved. The precision of the measurements so far is limited by the production quality and potentially strong temperature dependence of the used commodity LEDs. Future experiments will be conducted with more stable devices accompanied with a study of the impact of temperature on luminosity and emission spectrum.
F I G U R E 1 1 Comparison of a MIREX measurement with the results of the presented approach. The data based on the LED measurement shows the results of an optimisation towards low as well as high extinction coefficients Further measurements of other quantities, like gas temperatures, will help to investigate other effects, like refraction. More information on the particles, using measurements of particle density and size distribution at few locations in the compartment during the fire, will allow to estimate the extinction cross-sections. In this way, different particle types, due to different combustion, pyrolysis or transport processes, will be distinguished for the various fire types. Thus, this information can be used to estimate the particle numbers, with the assumption of a known c ext , as well as the difference of infrared and red light measurements. In addition, there exist already approaches to characterise the particle properties based on their interaction with light at three selected wavelengths. 18,19 Similar analysis will be conducted with the presented method.
Finally, numerical simulations with FDS will be carried out. The results of these simulations will be used to investigate the homogeneity assumptions in combination with a comparison the measured (LED and MIREX) extinction coefficients. In case of fire-type TF5, all relevant quantities, like the mass loss rate, have been captured.
Thus, a design fire will be sufficient to represent the evaporation process.