#### 3.1. Flame morphology and Spreading Behavior

In the downward flame spreading process, the controlling mechanism is based on heat and mass transfer to the unburned area. The unique dynamic, parallel, and symmetric combustion scenario in this work resulted in morphological variations of the pyrolysis leading edge (or flame front) of the PUR foam as shown in

Figure 3. Images of typical sequential downward flame spreading are shown in

Figure 3a. These variations in the leading edge can be divided further into three stages (

Figure 3b), during which a distinctive inverted ‘W’ shape changes to an inverted ‘V’ shape. Initially, after the flame has travelled for about 10 cm, the flame spreading reaches to an approximate steady state, and the flame pyrolysis front show essentially one-dimensional linear flame spreading (Stage 1 in

Figure 3b). The flame front at the board edges spreads faster than at the center, so that the flame front becomes more irregular, with an inverted ‘V’ shape for each board (Stage 2 in

Figure 3b). Eventually, an inverted parallel symmetric ‘W’ shaped flame front composed of two inverted ‘V’ shapes is observed, indicating two-dimensional flame spreading. Finally, the lateral flame front spreading becomes more rapid, leading to a ‘slash’ shape (Stage 3 in

Figure 3b), and an inverted ‘V’ shape composed of two slash-shaped leading fronts emerges. The inverted ‘V’ shape appearing in our experiments is similar to the flame morphologies observed in the CCTV and Grenfell tower accidents (

Figure 4). Prior work has also demonstrated the formation of an inverted ‘V’ shape leading edge with solid fuel flame spreading, although not in all cases. In the case of wide boards, the leading edge has been found to exhibit an inverted ‘U’ morphology rather than a ‘V’ shape, indicating an edge effect caused by air entrainment from both sides, as reported by Gong using polymethyl methacrylate (PMMA,

Figure 5) [

19]. These variations in the flame leading edge are discussed in detail in

Section 3.3.

#### 3.2. Overall Comparison of Burning Rates

Figure 6 presents a comparison of mass loss data acquired during the relatively steady burning stage in conjunction with different adjacent façade configurations. The extent of complete combustion,

$\eta $, can be used to investigate the effects of a parallel symmetric flame and entrainment of the fire plume with changes in the adjacent angle. This term is defined as

$\eta =\frac{{m}_{r}}{{m}_{i}}$, where

${m}_{i}$ is the initial mass of the PUR foam board, and

${m}_{r}$ is the mass remained after extinguished or the mass of the char when the flame spreading process completed.

Table 2 summarizes the

$\eta $ values obtained for specific angles. The relationship between the PUR burning rate and the adjacent façade angle is seen to be nonlinear, i.e.,

$\eta $ first decreases with increases in the angle and then increases. This phenomenon is thought to result from a coupling effect based on heat feedback from the opposite flame and air entrainment due to the chimney effect. Both are modified by changing the adjacent angle, as discussed below.

The PUR burning rate for each angle is largely dependent on heat transfer from the flame. Comparing the data for $\vartheta ={60}^{\xb0}$ ($\eta =34.10\%$) and $\vartheta ={90}^{\xb0}$ ($\eta =31.78\%$), the two parallel flames evidently affect one another to different extents as the angle is changed by both radiative and convective heat transfer. When the angle is decreased, radiation heat feedback is strengthened. However, increased entrainment of cold air into the flame plume in the gap between the façades (due to the chimney effect) with decreases in the angle could cool the combustion zone, resulting in a decreased burning rate.

With the enlarged angle, such as $\vartheta ={120}^{\xb0}$ ($\eta =33.20\%$), it further decreased the heat transfer from the opposite flame even though a sufficient air supply was available, leading to weakened combustion compared with a more narrow façade construction. Above a specific angle, the mutual reinforcement effect was greatly decreased and become negligible, reducing the combustion intensity, e.g., occurred at 150° ($\eta =40.48\%$) and 180° ($\eta =51.52\%$).

Throughout this work, the burning rates for comparison were selected from the relatively steady flame spreading period, as obtained using the linear fitting method depicted in

Figure 7a (based on data acquired at a 90° adjacent façade angle). During this stage of combustion, a plot of the pyrolysis front position against time is nearly linear at first. However, a sudden, unexpected increase in the mass loss rate was observed with phenomenological two-pass processing, dividing the data into two stages with different slopes. This increase indicates an acceleration of the downward flame spreading during the later period of combustion over a narrow range of angles (

$\vartheta ={60}^{\xb0}$ and

$\vartheta ={90}^{\xb0}$), as shown in

Figure 7b. In such cases, the data plot is closer to parabolic than linear over a sufficiently long time scale. Taking

$\vartheta ={90}^{\xb0}$ as an example, if flame spreading is monitored over approximately 300 s, the mass loss rate is constant at 0.146 g/s during the initial stage but later abruptly increases to 0.181 g/s. This effect is attributed to preheating of the unburned region of the PUR via heat transfer from the flame, leading to a widened preheating zone and less heat is required for pyrolysis of the PUR by preheated. Thus, the flame front reaches the pyrolysis temperature more quickly during the later period, which in turn accelerates the flame spreading.

#### 3.3. Flame Height and Flame Front Variations

The average flame heights at various façade angles are plotted in

Figure 8, and this plot shows that the flame height first increased and then decreased with enlarged angle. A critical angle

${\vartheta}_{c}$ is noticed at approximately

$\vartheta ={90}^{\xb0}$ in this plot, similar to the trend exhibited by the burning rate vs. angle, which is not surprising because the burning rate is the key parameter determining the flame height, spreading velocity and other factors according to classical fire dynamic theory. The variation in the average flame height is significant at

$\vartheta <{\vartheta}_{c}$ but little change is seen above this value. These data can be explained based on radiation, chimney, and restriction effects. Considering that the radiation effect is the most important thermal feedback parameter in this situation, it changes not linearly with the angle (see

Figure 9). The radiation heat feedback

${q}_{r}$ to the preheating zone for each board consists of that from the sample itself

${q}_{{}_{rs}}^{\u2033}$, and that from the adjacent façade

${q}_{ra}$. However,

${q}_{{}_{rs}}^{\u2033}$ is negligible because the view factor

${F}_{s}$ is small enough. Therefore, we can write

The approximately value of ${q}_{{}_{ra}}^{\u2033}$ can be obtained from Equation (3), where ${Q}_{ra}$ is the total heat flux transfer to the adjacent façade and $R$ is the length from fire center to the burning zone of opposite adjacent façade. $\alpha $ is the approximate radiation angle between the opposite flame and the board pyrolysis zone. Based on fire dynamics, the flame radiation can be assumed to be emitted in a spherical pattern from a geometrical center. Due to the relatively small PUR board width, the radiative heat flux at this point can be considered to be equal to the average flux received from the opposite flame. According to Equation (1), with decreases in $\vartheta $, the flame radiation output increases. If the adjacent angle is $\vartheta {=180}^{\xb0}$, the façades are parallel to one another and ${q}_{{}_{ra}}^{\u2033}$ will be low. In contrast, at $\vartheta {=90}^{\xb0}$, the heat flux between adjacent façades will be the highest and the mutual fire sources will impact each other to the maximum extent.

As the adjacent façade angle increases, the chimney effect decreases and the flame height drops. Thus, when $\vartheta ={\vartheta}_{c}$ at a value of approximately 90°, the maximum burning rate and flame height are obtained. In the case that $\vartheta >{\vartheta}_{c}$, the radiative and chimney effects are both greatly decreased, while the restriction effect is also weakened such that the flame height slightly decreases. Therefore, the flame height first increases and then drops with increases in the adjacent façade angle.

The flame front or pyrolysis edge are determined by several factors, including flame height, air entrainment, and the time over which the fire has developed. As aforementioned, the leading edge changed from an inverted ‘W’ shape to an inverted ‘V’ shape. This phenomenon can be interpreted based on Gollner’s [

20] boundary layer theory which has its basis in the work of Chilton and Colburn [

21] and Silver [

22], whose extension to the Reynolds’ analogy established a relationship between mass, momentum, and heat transfer in a boundary layer over the fuel surface. The associated equation is

where the air shear stress,

${\tau}_{s}$, is the viscosity coefficient multiplied by the derivative of velocity. This term can be written as

In addition,

${\mu}_{\infty}$ is the free-stream velocity,

$v$ is the kinematic viscosity or momentum diffusivity,

${\dot{m}}_{t}{}^{\u2033}$ is the mass transfer caused by shear flow,

$D$ is the species diffusivity, and

$B$ is the Spalding number is the mass transfer coefficient. This value is determined using the equation

Hence, the mass transfer rate is positively correlated with ${\tau}_{s}$**,** the effect of which is obvious during solid combustion due to an additional induced effect. The inverted ‘W’ flame front is attributed to three effects. First, the flame size will be larger at the specimen sides than in the center of the board. Second, shear entrainment from the lateral sides will accelerate the flame spreading velocity in these locations. At last, as the flame height and temperature reach their maximum values at the center of the adjacent board, the burning rate will be increased. In the later stage of combustion, due to the enhanced burning at the board sides by the effect of ${\tau}_{s}$, the spreading velocity at the sides will become significantly faster so as to form the inverted ‘V’ shape.

The angle

$\mathsf{\Theta}$ of the inverted ‘V’ shape was found to decrease as flame spreading progressed. This effect can be expressed by the equation simplified from Gong’s research [

19]

where

${q}_{\delta}^{\u2033}$ is the radiative heat feedback in the preheating zone,

${q}_{p}^{\u2033}$ is the thermal feedback in the combustion zone, and

${V}_{f}$ is the flame spreading rate, defined as the propagation speed of the flame front along the sample surface. The smallest value of

$\mathsf{\Theta}$ was observed at

$\vartheta {=90}^{\xb0}$ also associated with tangential entrainment effect, plus parallel fire thermal feedback is the largest resulting in peak combustion rate. Meanwhile, the flame height and maximum entrainment strength were enhanced further, and positively feedback the combustion efficiency of board edges, finally resulting in a sharp pyrolysis front.

#### 3.4. Flame Spreading Rates

The flame spreading rate is impacted by the disturbance associated with the ignition source in the early stage of combustion and the accelerated flame spreading during the later stages. Hence, the stable flame spreading stage was employed when determining the

${V}_{f}$ values, as shown in

Figure 10.

Quintiere [

11] proposed a simplified theory to predict the downward flame spreading rate over a thermally thick charring solid, based on the equation

where

$\rho c$ is the density multiplied by the specific heat,

$wd$ is the fuel width multiplied by the thickness,

$p$ is the pyrolysis length, and

$\delta $ is the preheating length. In the case of downward flame spreading with adjacent materials at various angles, the heat feedback is largely determined by the radiative heat flux

${q}_{r}$ (which could be negligible for single board flame spreading), convective heat flux

${q}_{cv}$ and conductive heat flux,

${q}_{cd}$. A diagram depicting downward parallel, symmetric flame spreading is presented in

Figure 11. Also, de Ris [

5] and Bhattacharjee et al. [

6,

7] proposed a formula to predict the downward flame spread rate of thermal thick solid, as shown in Equation (9).

which is an empirical relationship only for single board condition, without considering more radiation interaction.

Anyhow, the convective heat flux can be expressed as

As the adjacent angle or space between the two boards decreased, the stack effect will become prominent, such that upward air entrainment is strengthened. As a result of the cold air cooling effect, the convective heat transfer is reduced, although the effect of this heat transfer mechanism is minimal in the case of a narrow board. It is only when the sample width is extended that the weakening effect of the convective heat feedback will have a significant effect on flame spreading.

The value of ${q}_{cd}$ is approximately 0.03$w/(m\cdot k)$ for PUR foam, which affects the flame spreading behavior to a greater extent as the preheating length $\delta $. Due to the increase in the radiation heat feedback when the surface temperature of the entire board is sufficiently high, the flame spreading characteristics of a single board and of two adjacent boards become quite different especially in the later period. The initial temperature of the unburned region rises significantly, and so the heat input required to achieve vaporization and ignition is reduced. This effect could increase the depth to which the conductive heat from the flame front penetrates, leading to a larger preheating zone.

Because radiation heat feedback is stronger during downward flame spreading in the case of two parallel adjacent façades, this mechanism will be more important than convective and conductive heat transfer. Thus, the flame spreading rate will be largely determined by radiative feedback, which varies in a similar trend to the burning rate. In spite of internal radiant heat feedback, according to the measurement results by radiation flux meter, the radiation heat flux to external environment were compared as shown in

Figure 12. It can be seen that the maximum value represents largest thermal hazard also appears at 90° condition, which is also consistent with the mass loss rate and flame height trends.