Characterizing Heat Release Rates Using an Inverse Fire Modeling Technique
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Abstract
A ubiquitous source of uncertainty in fire modeling is specifying the proper heat release rate (HRR) for the fuel packages of interest. An inverse HRR calculation method is presented to determine an inverse HRR solution that satisfies measured temperature data. The methodology uses a predictorcorrected method and the Consolidated Model of Fire and Smoke Transport (CFAST) zone model to calculate hot gas layer (HGL) temperatures in single compartment configurations. The inverse method runs at superrealtime speeds while calculating an inverse HRR solution that reasonably matches the original HRR curve. Examples of the inverse method are demonstrated by using a multiple step HRR case, complex HRR curves, experimental temperature data with a constant HRR, and a case with an experimentally measured HRR. In principle, the methodology can be applied using any reasonably accurate fire model to invert for the HRR.
Keywords
Compartment fires Fire growth Fire modeling Heat release rate Inverse fire modeling problems1 Introduction
Currently, the use of fire models in scenarios involving firefighter injuries, lineofduty deaths, or forensic applications requires a tedious and manual iterative process of modifying the input parameters to create the desired or expected results from a zone or field fire model and comparing the results to a timeline of observations. This process can result in significant errors or nonphysical results from fire models and might not include a sufficiently wide range of conditions that adequately describe the fire effects or fire behavior for a given scenario.
Previous work by Jahn et al. [1] has demonstrated a method to forecast the fire size in an enclosure using sensordriven inputs. That study used realtime sensor data (e.g., heat detectors, smoke detectors) to steer a fire model and account for changes in the environment of a fire scenario. The goal of that study was to use information from the evolving fire scenario to accelerate model predictions. A study by Cowlard et al. [2] describes the process of using realtime sensor data to assist firefighting operations through the use of high performance computers running numerous fire simulations in parallel and fetching precomputed scenarios. That study also demonstrated the sensitivity of the model results to the input parameters and how sensor data could be used to steer and correct the simulations.
Additional studies have been performed on sensordriven fire simulations to determine the location and size of the fire. A paper by Davis and Forney [3] outlines a process for using correlations and zone models as a sensordriven zone model. A study by Koo et al. [4] used a sensordriven steering method that performed at superrealtime speeds using high performance computing resources with the ability to run 1,000 scenarios per minute. A study by Richards et al. [5] used transient temperature data from ceiling sensors to determine the heat release rate (HRR) and location of fires in largescale compartments, but the inverse HRR solution had an error of 300% to 500% of the measured HRR. Studies by Neviackas [6], Neviackas and Trouvé [7], and Leblanc and Trouvé [8] used hot gas layer (HGL) temperatures in an enclosure (single and multiple compartments) and a genetic algorithm to search for an average inverse HRR. The genetic algorithm required multiple hours of runtime, and the solution was limited to a constant, timeaveraged HRR. In general, the approaches used in these studies are infeasible for general applications because of the amount of computational expenditure required or the inaccuracy of their inversions. However, the need for such inversion capability is evident. The focus of this study is to develop a quick, inexpensive method to compute transient HRR data using known temperature data in an enclosure.
A study by Lee and Lee [9] demonstrated the use of a sequential inverse method to determine the size and location of a compartment fire. In that study, the HRR in a compartment was calculated sequentially by using a discretized form of Alpert’s correlation [10] for gas temperatures in ceiling jet flows. The results exhibit a large amount of noise (±100% error in the resulting HRR solution), and the correlations apply only to a limited scope of physical scenarios. However, these types of correlations can still be useful in the predictor step of an inverse HRR methodology, which was implemented in the inverse method in this paper and is described in the following section.
An inverse recovery methodology is presented that uses a fire model to search for a HRR that satisfies sampled temperature data in an enclosure. The Consolidated Model of Fire and Smoke Transport (CFAST) zone model [11], which is maintained by the National Institute of Standards and Technology, was used to reconstruct a timevarying inverse HRR solution.
First, a simple case with step increments in the HRR and cases with various complex HRR curves are used to demonstrate the inverse solution method. Then, temperatures from experimental enclosure fire tests are used to determine an inverse HRR solution. Finally, a case with an experimentally measured HRR are used to demonstrate the robustness and accuracy of the method for the calculation of transient HRR solutions.
2 Inverse Heat Release Rate Solution Methodology

Step 1: For a temperature difference (Y − T) between the measured and predicted temperatures, the predictor step computes \(\Updelta \dot{\bf Q}\) for all times t _{ i } by using the sensitivity, J, (i.e., dT/dQ) found from the MQH correlation in Equation 4. An intermediate value of \({\dot {\bf{Q}}}^{k+1}\) based on the MQH correlation is then computed using Equation 3.

Step 2: For the corrector step, the CFAST model is run with the MQHderived HRR values \({\dot {\bf{Q}}}^{k+1}\) to generate temperatures T ^{ k+1} at the next iteration.

Step 3: If the error is less than a specified tolerance (\(S({\dot {\bf{Q}}})\le 1 \times10^{3}\)), then the resulting \({\dot {\bf{Q}}}\) is returned. Otherwise, Steps 1 and 2 are repeated as the predictorcorrector procedure iterates. The result of the inverse HRR method is a piecewise linear function of HRR versus time, as shown in Equation 1.
The Python programming language, which is a highlevel scripting language, was used to generate CFAST input files, run CFAST multiple times while searching for a HRR solution, parse the output from CFAST, and repeat this process to create an inverse HRR solution. This method is demonstrated with various examples in the following sections.
3 Zone Model Setup
The zone model, CFAST version 6.2.0, was used in this study. The source code for version 6.2.0 of CFAST was used to compile the CFAST program for the Mac OS X operating system, and the command line binary was controlled by an automatic script rather than using the graphical interface. This approach allowed for the inverse search method to perform efficiently and autonomously.
Thermal Properties Used for Various Material Boundary Conditions
Material  k (W/mK)  c _{ p } (J/kgK)  ρ (kg/m^{3})  δ (cm)  \(\varepsilon\) (−) 

Gypsum  0.16  900  790  1.6  0.9 
Type X gypsum  0.14  900  770  1.3  0.9 
Aluminum  231  1,033  2,702  0.3  0.9 
Glass fiberboard  0.04  720  105  8.8  0.9 
4 Multiple Step Function Increments in the Heat Release Rate
For comparison, the MQH correlation was also used with the same inputs and boundary conditions as the inverse HRR method. Equation 4 was used to compute the HRR at each time step, and the results are shown in Figure 2b as a dashdot line, which has a relative error of 0.25. Although the inverse method is a better approximation to the actual HRR, the MQH correlation is still useful in the predictor step of the inverse HRR method, as described in the previous section. By using the predictorcorrector method, this inverse HRR method can be extended to problems that exceed the limitations of the existing correlations by using physicsbased models such as CFAST or Fire Dynamics Simulator (FDS) [16].
Maximum Change in Temperature and Inverse HRR for Various Amounts of Noise
Amount of noise (%)  Max. \(\Updelta T\) (°C)  Max. \(\Updelta Q\) (kW) 

5  6  42 
10  13  129 
15  18  228 
5 Complex Heat Release Rate Curves
To evaluate the ability of the inverse method to determine a solution for complex HRR curves, three example HRR curves from CFAST were used. The original HRR was input into an initial CFAST run to generate synthetic temperature data, and the resulting HGL temperatures were used as inputs for the inverse method to recover the original CFAST HRR curve. The sample resolution of the input HGL temperature data was 10 s for all of the cases.
For simplicity, the gas phase combustion parameters were the same as in the previous section (i.e., methane with a heat of combustion of 50 MJ/kg); therefore, the HRR curve was the only independent search parameter. The enclosure dimensions were the same for all of the cases (6.1 m × 4.9 m × 2.4 m enclosure).
6 Experimentally Measured Compartment Temperature Data
The inverse method was applied to various scenarios involving actual fire conditions by using experimentally measured temperatures from enclosure fire experiments as inputs to the inverse method. The experimental setup, input values, and resulting inverse HRR solutions are described in the following sections.
6.1 Steckler Compartment Data
The experimental steadystate compartment temperatures from 11 tests with various ventilation areas from the Steckler compartment fire data [17] were used as inputs to the inverse method.
Error in Inverse HRR Solution Versus Vent Width From the Steckler Experiments
Vent width (m)  Reported HRR (kW)  Inverse HRR (kW)  HRR error (%) 

0.24  62.9  63.5  0.9 
0.36  62.9  62.5  0.7 
0.49  62.9  63.1  0.3 
0.49  62.9  66.4  5.6 
0.62  62.9  61.4  2.4 
0.74  62.9  61.4  2.4 
0.74  62.9  60.4  3.9 
0.74  62.9  61.1  2.9 
0.74  62.9  65.8  4.7 
0.86  62.9  61.2  2.7 
0.99  62.9  59.5  5.4 
6.2 UT Austin Experimental Data
7 Experimentally Measured Heat Release Rate Data
A representative test from the full set of furniture experiments was selected for this paper in which a mockup furniture specimen was burned in the enclosure. In that test, the furniture item was a threeseat sofa with cotton fabric and low density polyurethane foam padding placed on a steel frame. The specimen was ignited on the front using a CAL TB 133 gas burner (19 kW). The combustion products from the enclosure were collected in a furniture calorimeter hood, and the HRR was measured using oxygen consumption calorimetry. For the inverse HRR method, the sample resolution for the temperature inputs was 10 s. Additionally, the ambient temperature in the CFAST model was matched to that of the experiment.
8 Future Extensions of Inverse Fire Modeling Techniques
As faster computing resources become more readily available, these methods will become more important in the application of inverse fire modeling problems (IFMP). Additionally, this method can be used to quickly determine a unique HRR curve that corresponds to an observed fire timeline (e.g., timetemperature history, heat flux measurements, fire service events, ventilation events), which describes a complex IFMP scenario. Fire Dynamics Simulator and CFAST models can be used with various time dependent observations such as the time of window breakages, time of ventilation events, amount of smoke from ventilation openings, and time to flashover to better determine an inverse solution by using physical changes in the environment as bounding conditions. Additional measurements from experiments or fire incidents, such as heat fluxes and smoke layer heights, can be used to improve the inverse solution by imposing physical bounds on the inverse solution.
While the CFAST zone model is relatively inexpensive for this inverse HRR method, the results are based upon assumptions and simplifications of the underlying physics. In principle, this inverse method could be used with more complex fire models such as FDS to determine the resulting enclosure conditions (e.g., temperatures, heat fluxes) and further improve the inverse solution. Automated CFAST runs could be used to vary the fire size and location in the enclosure, and the resulting scenario and HRR could then be simulated in FDS to verify the physics with more fidelity. Previous related work has been performed by Hostikka et al. [19] regarding the probabilistic simulation of CFAST using the Monte Carlo method. That study utilized rank order correlations to identify model parameters that significantly affect the results. Because CFAST is computationally inexpensive compared to FDS, the predictor step of the inverse solution could quickly be computed using CFAST, and the results from CFAST could be used to steer subsequent FDS simulations in the corrector step.
9 Conclusion
A method for recovering transient HRR based upon measured transient compartment fire temperatures was presented. The inverse method required about 5 to 10 s of total run time on an Apple Macbook Pro computer with a 2.2 GHz processor to calculate a transient inverse HRR solution for each case; each case required between 10 and 30 CFAST runs for each case. For all of the cases described in this paper, the inverse HRR solution had a relative error between 0.04 and 0.24 compared to the actual HRR. The implementation of the loworder MQH correlation for the predictor step allows for a quick calculation of the update (predictor) step because it has the advantage of being directly invertible for the HRR. Use of the predictor step reduced the total number of computational (CFAST) iterations required to generate an inverse HRR solution that meets the specified convergence criterion.
For the multiple step function increment cases, the inverse solution adequately detected changes in the HRR steps. For the experimental enclosure temperature case, the inverse method effectively captured the activation of the gas burners. However, because the HRR was not measured directly, it is difficult to quantify the amount of error in the inverse solution. Qualitatively, this method captured a change in the HRR and exhibits potential for obtaining an inverse solution from these types of scenarios in which the measured HRR is unknown and only temperature data are available. For the complex HRR cases and the experimentally measured HRR case, the inverse method performed well and the inverse HRR solution was in good agreement with the actual HRR, which demonstrates the versatility and accuracy of the inverse HRR method.
One limitation of this methodology is that the material properties of the boundary conditions must be prespecified, and the inverse HRR solution is sensitive to the selection of boundary conditions, as shown in Figure 3. However, in the United States, most of the compartment configurations in which this method can potentially be applied (e.g., residential and commercial occupancies, fire experiments, fire investigations) are limited to certain types of boundary conditions such as gypsum wallboard or similar types of insulating building materials. Thus, it is believed that a computationally inexpensive methodology for the transient HRR solution for such cases is a valid contribution of this study.
Footnotes
 1.
This section summarizes partial results from SwRI Project No. 15998. This project was supported by Award No. 2010DNUXK221, awarded by the National Institute of Justice, Office of Justice Programs, U.S. Department of Justice. The opinions, findings, and conclusions or recommendations expressed in this paper are those of the author and do not necessarily reflect those of the Department of Justice.
Notes
Acknowledgments
This work was funded by the National Institute of Standards and Technology Dept. of Commerce Grant No. 60NANB7D6122.
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