# Polyisocyanurate Foam Pyrolysis and Flame Spread Modeling
## 2 Materials and Methods
### Modeling
#### 2.4.3 Modeling of Full-Scale Experiments  
Modeling of the full-scale experiments performed on the PIR foam panels without facers was carried out using a recently developed, semi-empirical approach [32]. In this approach, each panel forming the corner wall was divided into 28 rectangular elements, as shown in Figure 2. For each of these elements, an empirical relation between the incident heat flux measured by a water-cooled heat flux gauge (\(q^{\prime\prime}\)) and the total HRR was established using the results of experiments performed in the same setup, at the same burner setting, on PMMA. An example of such relation obtained for one particular element is shown in Figure 2. It was assumed that these relations are not significantly affected by material composition and used here to simulate flame spread on the PIR foam. The simulation was initiated by guessing the HRR evolution with time. The initial guess was taken as twice the HRR of the burner, HRR\({}_{\text{b}}\), measured in the absence of any flammable wall material. Using this guessed HRR profile (labeled as Input HRR in Figure 2), \(\dot{q}^{\prime\prime}\) dependencies on time were generated for each of the 28 elements of the corner wall from the empirical relations between \(\dot{q}^{\prime\prime}\) and HRR. These dependencies were subsequently converted to front surface boundary conditions for 28 instances of one-dimensional ThermaKin2 Ds pyrolysis simulations, each of which represented an element of the PIR foam panel. In the current simulations, an assumption was made that \(\dot{q}^{\prime\prime}\) was purely radiative in nature and, thus, the front surface thermal boundary conditions of each element consisted of the time-dependent incident radiant heat flux from the flame (\(\dot{q}^{\prime\prime}\)) and radiative losses from the heated PIR foam. Heat transfer through the Marinite I calcium silicate board that was located behind the foam panels (see Figure 1) was modeled explicitly using the material properties provided by the manufacturer [40]. The results of the pyrolysis simulations were used to compute Simulated HRR by adding up contributions from individual elements and the burner:  
\[\text{Simulated HRR}(t)\ =\ \text{HRR}_{\text{b}}(t)+2\sum_{j\ =\ 1}^{28}A_{j}\left[\sum_{i\ =\ 1}^{Ng}\dot{m}^{\prime\prime}_{ij}(t)\Delta H^{i}_{c}\right]. \tag{1}\]  
In this equation, \(i\) index is used to add contributions from all \(Ng\) gaseous decomposition products, while \(j\) is used to sum the contributions from all panel elements, each of which had front surface area \(A_{j}\). \(t\) is time from the start of the burner; \(\dot{m}^{\prime\prime}_{ij}\) is time-dependent mass flux of a gaseous decomposition product \(i\) from element \(j\) (computed by  
Figure 2: Workflow diagram for the corner wall flame spread modeling. Note that the heat flux gauge locations shown here represent locations from a previous study performed on PMMA [32]. ThermaKin2 Ds); and \(\Delta H_{c}^{i}\) is the heat combustion of product \(i\). The Simulated HRR was subsequently compared with the Input HRR using a criterion CR,  
\[\text{CR }=\ \frac{1}{t_{e}}\int_{0}^{t_{e}}\abs{\text{Input HRR}\big{(}t^{\prime}\big{)}-\text{Simulated HRR}\big{(}t^{\prime}\big{)}}dt^{\prime}. \tag{2}\]  
If, for any \(t_{e}\) between 1 s after the burner ignition and the end of the simulation time, the CR was found to be less than 5 kW, the Simulated HRR was declared to be converged and the simulation was stopped. On the other hand, if CR was found to be equal or larger than 5 kW for any \(t_{e}\), the Simulated HRR was taken as an Input HRR for the next iteration. The overall workflow of this iterative process is further illustrated in Figure 2. Typically, two to three iterations were required to achieve convergence. ThermaKin2 Ds calculations performed as a part of each iteration were carried out using the same integration parameters as were used for the modeling of the gram-scale experiments.