# Modelling of the Radiant Heat Flux and Rate of Spread of Wildfire within the Urban Environment

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## Abstract

**:**

## 1. Introduction

## 2. Wildfire Fuels

- Wildfire fuel is homogenous in structure with average heights and characteristics applied across a 100 m by 100 m assessment area;
- Wildfire intensity and production of radiant heat is dependent on average fuel load densities across the entire 100 m by 100 m assessment area.

- Canopy fuel is contained in the forest crown. The crown encompasses the leaves and fine twigs of the tallest layer of trees in a forest or woodland. Crown involvement may lead to erratic and extreme fire behaviour and contributes to spotting distances.
- Bark fuel is the flammable bark on tree trunks and upper branches that contributes to transference of surface fires into the canopy, embers and firebrands, and subsequent spot fires.
- Elevated fuel includes shrubs, scrub, and juvenile understory plants up to 2–3 m in height, however, canopy of heights less than 4 m can be included when there is no identifiable separation between the canopy and lower shrubs. The individual fuel components generally have an upright orientation and may be highly variable in ground coverage. Elevated fuels influence the flame height and rate of spread of a fire whilst also contributing to crown involvement by providing vertical fuel structure.
- Near-surface fuels include grasses, low shrubs, and heath, sometimes containing suspended components of leaves, bark, and twigs. This layer can vary from a few centimeters to up to 0.6 m in height. Near-surface fuel components include a mixture of orientations from horizontal to vertical. This layer may be continuous or have large gaps in ground coverage and influences both the rate of spread of a fire and flame height.
- Surface fuel includes leaves, twigs, and bark on the forest floor. Surface fuel (or litter) components are generally horizontally layered. Surface fuel usually contributes the greatest to fuel quantity and includes the partly decomposed fuel (duff) on the soil surface. This fuel layer influences the rate of spread of a fire and flame depth as well as contributing to the establishment of a fire post initial ignition.

## 3. Empirical Modelling of Wildfire

_{f}for forest, woodland, and rainforest are given by [2,3]

_{eff}is the effective slope (slope of land under the vegetation or fuel bed, and W is the overall fuel load (t/ha). The assumed geometry is commonly known as the “radiant heat panel”, with the horizontal position of the panel considered to be located below the midpoint between the base and tip of the flame front [10]. Both the flame temperature, nominally 1095 K [2], and emissivity, nominally 0.95 [2], are assumed to be consistent across the panel, whilst the receiving body is assumed to be aligned perpendicular to the approaching fire front.

## 4. Modelling of Fuel Beds that Restrict Fire Growth

- Regardless of the actual geometry and coverage of fuel within the assessment area, [2] assumes landscape scale wildfire behaviour with a 100% homogenous fuel loading within the assessment area and a head fire width of 100 m;
- When fuel bed geometry prevents a 100 m head fire or quasi-steady RoS being obtained, failure to adjust wildfire fuel inputs may result in significant overestimation of wildfire impact, particularly radiant heat flux; and
- In order to more accurately model wildfires in fuel beds that restrict fire growth, it is necessary to calculate available fuel loads that will contribute to fire behaviour over the area being assessed using the vegetation availability factor equation as described below.

_{F}), given by

_{A}(t/ha), and the available total fuel load W

_{A}(t/ha), are then defined as

## 5. Modelling Point Source Ignitions

^{−1}) is a constant related to how rapidly the head fire accelerates. Further, [18,28] suggest that a reasonable first estimate for β can be established using the assumption that the fire will accelerate to 90% of the equilibrium rate of spread in 30 min (i.e., 0.5 h) for treed vegetation structures, including forest and woodlands. The attainment of the 90% equilibrium rate of spread 30 min post ignition within treed fuel structures is supported by the findings of [17,23,28,28].

^{−1}). This would only be appropriate in the current setting if the $RoS$ were considered in km/min rather than km/h.

## 6. Modelling the Radiant Heat Flux of a Partially Shielded Fire Front

^{2}) of a wildfire is calculated as the product of the flame emissive power E (kW/m

^{2}), the atmospheric transmissivity $\tau $, and the view factor $\phi $ [2,3]. It is expressed as:

_{f}is the flame temperature.

- Equations (9) and (12) impose the assumption that the site is horizontally central with respect to the fire front. This assumption will be relaxed to allow the calculation of view factors for obstructions and fire fronts which are not centrally aligned to the site.
- Equations (8)–(12) are formulated in terms of parameters specifically referencing the fire front (not an obstruction). Furthermore, although convenient from a computational perspective, they are not presented in a means that offers significant geometrical insight. The equations will be reformulated in terms of view angles from the site to the fire front or obstruction(s).

#### 6.1. Generalisation and Reformulation of the View Factor Formulae

#### 6.2. Calculating the View Factor Subject to Shielding Obstructions

- Calculate the view factor ${\phi}_{F}$ of the unobstructed vertical approximation to the fire front by setting $i=1$, $j=n$, ${\nu}_{i}^{U}={\nu}_{1}^{FU}$ and ${\nu}_{i}^{L}={\nu}_{1}^{FL}$ in Equation (14), and then substituting the resulting angles into Equation (13).
- In order to accommodate non-rectangular obstructions, the obstructed view factor ${\phi}_{O}$ is calculated by approximating the obstructions using thin rectangles defined within the angular increments from ${\beta}_{i}$ to ${\beta}_{i+1}$ for $i=1,2,\dots ,n-1$. For each angular increment, the obstructed view factor ${\phi}_{O}^{i}$ is calculated by determining the maximum value of ${\nu}_{i}^{jU}$ and minimum value of ${\nu}_{i}^{jL}$ for the obstructions that lie between the flame front and the site. If ${\nu}_{i}^{jU}>{\nu}_{i}^{FU}$, then ${\nu}_{i}^{FU}$ is used to denote the top of the obstructing rectangle, as any part of the obstruction extending above the flame front does not actual block the view of the flame front. This is illustrated in Figure 13.

- Calculate the total obstructed view factor$${\phi}_{O}={\displaystyle \sum _{i=1}^{n-1}{\phi}_{O}^{i}}.$$
- Calculate the view factor of the partially obstructed flame front$$\phi ={\phi}_{F}-{\phi}_{O}$$

#### 6.3. Modifications to the Optimisation Algorithm

- In the original algorithm the initial value (lowest value) of the flame angle considered in the optimisation algorithm is the site slope $\theta $. This is not a valid angle in the case that an obstruction exists between the flame front and the site, as it effectively allows the fire front to penetrate the obstruction. To avoid this situation it is necessary to set the initial flame angle such that the fire front would clear the obstruction. This amounts to setting$${\alpha}_{0}={\mathrm{tan}}^{-1}\left(\mathrm{tan}\left(\theta \right)+\mathrm{max}\left\{\frac{{h}_{O}^{j}\left({\beta}_{i}\right)}{x-{x}_{O}^{j}\left({\beta}_{i}\right)}:{n}_{1}^{j}\le i\le {n}_{2}^{j},j=1,2,\dots ,M\right\}\right),$$A further complication could arise if the center of the fire front lies in front of the obstruction when the base of the fire front lies behind the obstruction. The issue in this instance is that the obstruction would not have an impact on the view factor. To avoid this situation the minimum flame angle is required to satisfy$${\alpha}_{0}\ge \mathrm{max}\left\{{\mathrm{cos}}^{-1}\left(\frac{2(x-{x}_{O}^{j}\left({\beta}_{i}\right)-\epsilon )}{{L}_{f}}\right):{n}_{1}^{j}\le i\le {n}_{2}^{j},j=1,2,\dots ,M\right\},$$
^{−6}). - If the fire front is positioned on top of an obstruction, the flame angle ${\alpha}_{0}$ is set to 90 degrees to effectively consider the fire front as being behind the obstruction. In this case, the algorithm is not required to proceed further to determine an optimal value of ${\alpha}_{0}$.
- Since the algorithm does not start with the flame angle ${\alpha}_{0}$ equal to the site slope $\theta $, it is possible that the initial value of ${\alpha}_{0}$ could turn out to be the flame angle that optimises the view factor. The standard optimisation algorithm of [2,3] terminates or refines its search increment when the view factors ${\phi}_{0}$, ${\phi}_{1}$, and ${\phi}_{2}$, which correspond to the flame angles ${\alpha}_{0}<{\alpha}_{1}<{\alpha}_{2}$ satisfy ${\phi}_{1}\ge {\phi}_{0}$ and ${\phi}_{1}>{\phi}_{2}$, however, if ${\phi}_{0}>{\phi}_{1}$ at the first step the algorithm will not terminate. Hence the additional termination or refinement criteria, ${\phi}_{0}>{\phi}_{1}$ must be added to the algorithm in addition to the existing criteria (i.e., (${\phi}_{1}\ge {\phi}_{0}$ and ${\phi}_{1}>{\phi}_{2}$) or ${\phi}_{0}>{\phi}_{1}$).
- In the case that the obstruction completely obscures the line of sight from the building site to the top of the flame front, the optimisation algorithm will never terminate as it will not be able to identify a non-zero view factor no matter how much the flame angle ($\alpha $) is increased. In order to avoid this situation, an additional condition is added to both loops of the algorithm. Specifically, if ${\alpha}_{1}>90$° during the iteration then the algorithm will terminate immediately, and the flame angle will be set to ${\alpha}_{1}=90$°. This measure is only required to avoid an infinite loop, and will not affect the outcome of the calculation.

## 7. Case Studies

#### 7.1. Case Study 1

_{f}= 1. Suppose that the radiant heat flux of a fire in the bush land behind the houses is to be estimated at a site or house on the opposite side of the street. The geometry of the specific case considered here is provided in Figure 14.

- The method outlined in [2], ignoring the obstructions presented by the houses located between the site and vegetation fuel bed.
- The method outlined in [2] with the receiver height $h$ set to 3 m (instead of the mid-level of the flame front).
- The method outlined in this paper, where each of the four houses is considered to reduce the view factor of the flame front.
- A simplified method in which the four obstructions are considered as a single rectangular obstruction with height 5 m (i.e., the height of the tallest house), and width equal to the combined width of the four houses. The combined width is the distance from the westernmost edge of the westernmost structure to the easternmost edge of the easternmost structure.

#### 7.2. Case Study 2

_{f}= 0.2 scales back the surface and overall fuel loads as defined in Equations (2) and (3). The fire is assumed to ignite from a point source at the edge of the Freeway, 30 m from the site/receiver. The fire is assumed to spread perpendicular to the Freeway at an accelerating rate $Ro{S}_{a}$, which is related to the distance from the ignition point D by Equation (7). The rate parameter $\beta =2\mathrm{ln}(10)$ h

^{−1}, as suggested by [18,28], is utilised. Figure 17 provides a plot of the accelerating rate of spread $Ro{S}_{a}$ and the equilibrium rate of spread $RoS$ against the distance from the ignition point D.

- The fire front is modelled with a constant (equilibrium) rate of spread from the ignition point, a width of 100 m, a vegetation factor of V
_{f}= 1, and the obstruction (wall) is ignored (the model of [2]). - The fire front is modelled with a constant (equilibrium) rate of spread from the ignition point, a width of 100 m, a vegetation factor of V
_{f}= 0.2, and the obstruction (wall) is ignored. - The fire front is modelled with an accelerating rate of spread from the ignition point, a vegetation factor of V
_{f}= 1, and the obstruction (wall) is ignored. - The fire front is modelled with an accelerating rate of spread from the ignition point, a vegetation factor of V
_{f}= 0.2, and the obstruction (wall) is ignored. - The fire front is modelled with a constant (equilibrium) rate of spread from the ignition point, a width of 100 m, a vegetation factor of V
_{f}= 1, and the obstruction (wall) is included. - The fire front is modelled with a constant (equilibrium) rate of spread from the ignition point, a width of 100 m, a vegetation factor of V
_{f}= 0.2, and the obstruction (wall) is included. - The fire front is modelled with an accelerating rate of spread from the ignition point, a vegetation factor of V
_{f}= 1, and the obstruction (wall) is included. - The fire front is modelled with an accelerating rate of spread from the ignition point, a vegetation factor of V
_{f}= 0.2, and the obstruction (wall) is included (i.e., the Case Study 2 scenario).

_{f}= 1 exceeded those with V

_{f}= 0.2. All of the models that include the modelling of acceleration start from a flux of zero, which increases as the rate of spread, length, and width increase (in addition to the increase from the larger view factor as the front closes on the site). Significantly, in Figure 19 the yellow line corresponding to the Case Study 2 scenario is not visible as the heat flux at the site remains zero. This is because the fuel load and rate of spread are not sufficient to create a front with sufficient height to be visible above the 3 m obstruction after 20 m of spreading, with the flame height reaching only 2.4 m.

## 8. Discussion

- Firefighters not being deployed to suppress wildfires and defend homes as a result of over-estimation of wildfire behaviour that indicates suppression efforts are not suitable, resulting in avoidable house loss and impacts on communities. This may occur as firefighting suppression thresholds are related to wildfire behaviour parameters throughout jurisdictions internationally [31]. Where inappropriate predictions fail to consider vegetation geometry that does not support the assumptions of landscape wildfire modelling, otherwise defendable areas may be left unguarded due inappropriate evaluation of suppression strategies;
- Inappropriate modelling of wildfire through landscaped gardens, public open space, road reserves, and residential areas within urban areas. In turn, land that is actually suitable for development may be identified as being subject to overestimated wildfire impact which restricts or prohibits development altogether. Typically, this may occur in urban settings where a small unmanaged vacant residential lot is modelled as supporting a landscape scale wildfire, in turn restricting or prohibiting development on adjacent and near-by lots; and
- Unnecessary requirements for over engineering and wildfire resistant construction standards of affected dwellings and structures that hinders development through either misidentification of land as being subject to unacceptable levels of wildfire impact, or through making development cost-prohibitive as a result of the level of wildfire resistant engineering and construction required.

## Author Contributions

## Funding

## Acknowledgements

## Conflicts of Interest

## Nomenclature

${V}_{f}$ | = | Vegetation availability factor |

$w$ | = | Surface fuel load (t/ha) |

$W$ | = | Total fuel load (t/ha) |

${w}_{A}$ | = | Available surface fuel load (t/ha) |

${W}_{A}$ | = | Available total fuel load (t/ha) |

$RoS$ | = | Equilibrium rate of spread (km/h) |

$Ro{S}_{a}$ | = | Accelerating rate of spread (km/h) |

$t$ | = | Time since ignition (h) |

$\beta $ | = | Fire acceleration parameter (h^{−1}) |

$D$ | = | Distance travelled by accelerating fire (km) |

$q$ | = | Radiant heat flux (kW/m^{2}) |

$E$ | = | Flame emissive power (kW/m^{2}) |

$\tau $ | = | Atmospheric transmissivity |

$\phi $ | = | View factor |

$\sigma $ | = | Stefan-Boltzman constant (kW/m^{2}/K^{4}) |

$\epsilon $ | = | Flame emissivity |

${T}_{f}$ | = | Flame temperature (K) |

$d$ | = | Horizontal distance from receiver to base of flame front (m) |

${L}_{f}$ | = | Flame length (m) |

$\alpha $ | = | Flame angle (°) |

$h$ | = | Elevation of receiver (m) |

${W}_{f}$ | = | Flame width (m) |

$\theta $ | = | Site slope (°) |

$FDI$ | = | Fire danger index |

$H$ | = | Heat of combustion (kJ/kg) |

${T}_{a}$ | = | Ambient temperature (K) |

$RH$ | = | Relative humidity |

## References

- Australian Building Codes Board (ABCB). National Construction Code; Australian Building Codes Board: Canberra, Australia, 2016; Volume 2.
- SAI Global. AS3959:2009 Construction of Buildings in Bushfire-Prone Areas; Standards Australia: Sydney, Australia, 2009. [Google Scholar]
- SAI Global. AS3959:2018 Construction of Buildings in Bushfire-Prone Area; Standards Australia: Sydney, Australia, 2018. [Google Scholar]
- Penney, G. Bushfire Fuels—Representation in Empirical and Physics Based Models. Master’s Thesis, Victoria University, Melbourne, Australia, 2017. [Google Scholar]
- Richardson, S. BALc Report for Rural Fire Risk Consultancy Pty Ltd., 5th ed.; RUIC Fire: Perth, Australia, unpublished.
- DFES. Incident Reporting System Database 1998–2008 Vegetation fires within the Perth metropolitan area less than 1 hectare. Perth, Australia, 2016; (unpublished). [Google Scholar]
- Cruz, M.; Sullivan, A.; Leonard, R.; Malkin, S.; Matthews, S.; Gould, J.; McCaw, W.; Alexander, M. Fire Behaviour Knowledge in Australia; Bushfire Cooperative Research Centre: Melbourne, Australia, 2014; ISBN 987-0-9925684-2-9. [Google Scholar]
- Cruz, M.; Gould, J.; Alexander, M.; Sullivan, A.; McCaw, L.; Matthews, S. Empirical-based models for predicting head-fire rate of spread in Australian fuel types. Aust. For.
**2015**, 78, 118–158. [Google Scholar] [CrossRef] - Cruz, M.; Alexander, M.; Sullivan, A. Mantras of wildland fire behaviour modelling: Facts of fallacies? Int. J. Wildl. Fire
**2017**, 26, 973–981. [Google Scholar] [CrossRef] - Sullivan, A.; Ellis, P.; Knight, I. A review of radiant heat flux models used in bushfire applications. Int. J. Wildl. Fire
**2003**, 12, 101–110. [Google Scholar] [CrossRef] - Tan, Z.; Midgley, Y.; Douglas, G. A Computerised Model for Bushfire Attack Assessment and Its Applications in Bushfire Protection Planning. In Proceedings of the Congress of the Modelling and Simulation Society of New Zealand, Auckland, New Zealand, 12–15 December 2005; ISBN 0-9758400-2-9. [Google Scholar]
- Mendham, F. An engineered approach to bushfire management. In Proceedings of the 13th Coal Operators Conference, Wollongong, Australia, 14–15 February 2003; ISBN 0-8-6418-7785. [Google Scholar]
- WAPC. State Planning Policy 3.7 Planning in Bushfire Prone Areas; Department of Planning: Perth, Australia, 2015.
- Noble, I.; Bary, G.; Gill, A. McArthur’s fire-danger meters expressed as equations. Aust. J. Ecol.
**1980**, 5, 201–203. [Google Scholar] [CrossRef] - Catchpole, W.; Bradstock, R.; Choate, J.; Fogarty, L.; Gellie, N.; McCarthy, G.; McCaw, W.; Marsden-Smedley, J.; Pearce, G. Co-operative Development of Equations for Heathland Fire Behaviour. In Proceedings of the 3rd International Conference of Forest Fire Research, Luso, Portugal, 16–20 November 1998. [Google Scholar]
- Hines, F.; Tolhurst, K.; Wilson, A.; McCarthy, G. Overall Fuel Hazard Assessment Guide, 4th ed.; Victorian Government Department of Sustainability and Environment: Melbourne, Australia, 2010; ISBN 978-1-74242-677-8.
- Gould, J.; McCaw, W.; Cheney, N.; Ellis, P.; Knight, I.; Sullivan, A. Project Vesta—Fire in Dry Eucalypt Forest: Fuel Structure, Dynamics and Fire Behaviour; CSIRO: Perth, Australia, 2007; ISBN 978-0643065345. [Google Scholar]
- McAlpine, R. Acceleration of Point Source Fire to Equilibrium Spread. Master’s Thesis, University of Montana, Missoula, MT, USA, 1998. [Google Scholar]
- Sullivan, A. Wildland surface fire spread modelling, 1990–2007. 1: Physical and quasi-physical models. Int. J. Wildl. Fire
**2009**, 18, 349–369. [Google Scholar] [CrossRef] - Sullivan, A. Wildland surface fire spread modelling, 1990–2007. 2: Empirical and quasi-empirical models. Int. J. Wildl. Fire
**2009**, 18, 369–386. [Google Scholar] [CrossRef] - Frankman, D.; Webb, B.; Butler, B.; Jimenez, D.; Forthofer, J.; Sopko, P.; Shannon, K.; Hiers, K.; Ottmar, R. Measurements of convective and radiative heating in wildland fires. Int. J. Wildl. Fire
**2013**, 22, 157–167. [Google Scholar] [CrossRef] - Newnham, G.; Blanchi, R.; Leonard, J.; Opie, K.; Siggins, A. Bushfire Decision Support Toolbox Radiant Heat Flux Modelling: Case Study Three, 2013 Springwood Fire, New South Wales, CSIRO Report to the Bushfire CRC; CSIRO: Canberra, Australia, 2014. [Google Scholar]
- Kucuk, O.; Bilgili, E.; Baysal, I. Fire Development from a Point Source in Surface Fuels of a Mature Anatolian Black Pine Stand. Turk. J. Agric. For.
**2007**, 31, 263–273. [Google Scholar] - Poon, S. Predicting Radiation Exposure from an Advancing Bushfire Flame Front; Warrington Fire: Victoria, Australia, 2003. [Google Scholar]
- Rossi, J.; Simeoni, A.; Moretti, B.; Leroy-Cancellieri, V. An analytical model based on radiative heating for the determination of safety distances in wildland fires. Fire Saf. J.
**2011**, 46, 520–527. [Google Scholar] [CrossRef] - Alexander, M. Calculating and interpreting forest fire intensities. Can. J. Bot.
**1982**, 60, 349–357. [Google Scholar] [CrossRef] - Alexander, M.; Cruz, N. Crown Fire Dynamics in Conifer Forests. In Synthesis of Knowledge of Extreme Fire Behavior: Volume I for Fire Managers; United States Department of Agriculture: Portland, OR, USA, 2016; pp. 107–142. [Google Scholar]
- Van Wagner, C.; Canadian Forestry Service, Canada; Kourtz, P.; National Forestry Institute, Canada. Personal Communication, 14 January 1985.
- Cheney, N. Fire behaviour. In Fire and the Australian Biota; Australian Academy of Science: Canberra, Australia, 1981; pp. 151–175. [Google Scholar]
- Cohen, D.; Butler, B. Modeling Potential Structure Ignitions from Flame Radiation Exposure with Implications for Wildland/Urban Interface Fire Management. In Proceedings of the 13th Fire and Forest Meteorology Conference; International Association of Wildland Fire: Lorne, Australia, 1996; pp. 81–86. [Google Scholar]
- Penney, G.; Habibi, D.; Cattani, M. Firefighter tenability and its influence on siege wildfire suppression. Fire Saf. J. under review.
- Hilton, J.; Leonard, J.; Blanchi, R.; Newnham, G.; Opie, K.; Rucinski, C.; Swedosh, W. Dynamic modelling of radiant heat from wildfires. In Proceedings of the 22nd International Congress on Modelling and Simulation (MODSIM2017), Tasmania, Australia, 3–8 December 2017; pp. 1104–1110. [Google Scholar]
- Dietenberger, M.; Boardman, C. EcoSmart fire as structure ignition model in wildland urban interface: Predictions and validations. Fire Technol.
**2017**, 53, 577–607. [Google Scholar] [CrossRef] - Caton, S.; Hakes, R.; Gorham, D.; Zhou, A.; Gollner, M. Review of pathways for building fire spread in the wildland urban interface part I: Exposure conditions. Fire Technol.
**2017**, 53, 429–473. [Google Scholar] [CrossRef] - Cohen, J. Relating flame radiation to home ignition using modeling and experimental crown fires. Can. J. For. Res.
**2004**, 34, 1616–1626. [Google Scholar] [CrossRef] - Cohen, J. Preventing disaster: Home ignitability in the wildland-urban interface. J. For.
**2000**, 98, 15–21. [Google Scholar] [CrossRef] - Finney, M.; Cohen, J.; Forthofer, J.; McAllister, S.; Gollner, M.; Gorham, D.; Saito, K.; Akafuah, N.; Brittany, A.; English, J. Role of buoyant flame dynamics in wildfire spread. Proc. Natl. Acad. Sci. USA
**2015**, 112, 9833–9838. [Google Scholar] [CrossRef] [PubMed][Green Version] - Leonard, J.; Blanchi, R.; White, N.; Bicknell, A.; Sargeant, A.; Reisen, F. Research and Investigation into the Performance of Residential Boundary Fencing Systems in Bushfires; Bushfire CRC Report CMIT(C)-2006-186; Bushfire CRC: Canberra, Australia, 2006. [Google Scholar]
- Leonard, J. Report to the 2009 Victorian Bushfires Royal Commission Building Performance in Bushfires; CSIRO: Highett, VIC, Australia, 2009; TEN-066-001-0001. [Google Scholar]
- Penney, G.; Habibi, D.; Cattani, M.; Carter, M. Calculation of Critical Water Flow Rates for Wildfire Suppression. Fire
**2019**, 2, 3. [Google Scholar] [CrossRef]

**Figure 5.**Geometrical representation of the side view of the site and vertical approximation of a fire front.

**Figure 6.**Geometrical representation of the birds-eye view of the site and vertical approximation of a fire front.

**Figure 7.**Geometrical representation of the upper-left quadrant of the fire front relative to the site.

**Figure 9.**The discretisation of the fire front with respect to $\beta $ using 6 uniformly distributed values ${\left\{{\beta}_{i}\right\}}_{i=1}^{6}$ looking from above. Note that ${\beta}_{1},{\beta}_{2},{\beta}_{3}>0$ while ${\beta}_{4},{\beta}_{5},{\beta}_{6}<0$.

**Figure 10.**Any rectangle specified by a set of angles ${\beta}_{i}$ , ${\beta}_{j}$, ${\nu}^{U}$, and ${\nu}^{L}$ will have the same view factor relative to the site. Note that ${\nu}_{i}^{U}>0$ and ${\nu}_{i}^{L}<0$, while ${\beta}_{i}>0$ and ${\beta}_{j}>0$.

**Figure 11.**A flame front with top and bottom edge coordinates ${\left\{\left({r}_{i}^{F},{\beta}_{i},{\nu}_{i}^{FU}\right)\right\}}_{i=1}^{4}$ and ${\left\{\left({r}_{i}^{F},{\beta}_{i},{\nu}_{i}^{FL}\right)\right\}}_{i=1}^{4}$ respectively.

**Figure 12.**An obstruction may only partially obstruct the fire front and will only obstruct the fire front if it lies within the angular region.

**Figure 13.**(

**Left**) Obstruction 2 completely blocks the fire front from the site, so ${\nu}_{i}^{U}={\nu}_{i}^{FU}$ as the part of Obstruction 2 that extends above the view line of the top of the fire front does not contribute to blocking the fire front. (

**Right**) Obstruction 1 partially blocks the fire front from the site, so ${\nu}_{i}^{U}={\nu}_{i}^{1U}$.

**Figure 14.**A bird’s-eye view of the case study 1 scenario. The measurements within the house boxes denote the height of each house.

**Figure 15.**The radiant heat flux at the site as a function of the distance to the vegetation fuel bed.

**Figure 17.**The accelerating rate of spread $RoSa$ and the equilibrium rate of spread $RoS$ against the distance from the ignition point D.

**Figure 18.**The radiant heat flux for models ignoring the 3 m obstructing wall. The yellow line represents the Case Study 2 scenario.

**Figure 19.**The radiant heat flux for models including the 3 m obstructing wall. The yellow line represents the Case Study 2 scenario.

**Figure 20.**The progression of the fire front for modelling scenarios 5 through to 8. The yellow line represents the Case Study 2 scenario.

i | j | Quadrant |
---|---|---|

1 | 1 | Upper-left |

1 | 2 | Upper-right |

2 | 2 | Lower-right |

2 | 1 | Lower-left |

Parameter | Value | Parameter | Value |
---|---|---|---|

Effective slope | 0° | Flame temperature (T_{f}) | 1090 K |

Site slope (θ) | 0° | Ambient temperature (T_{a}) | 308 K |

Vegetation class | Forest | Relative humidity (RH) | 25% |

Fire Danger Index (FDI) | 80 | Flame width (W_{f}) | 100 m |

Surface fuel load (w) | 25 t/ha | Flame emissivity (ε) | 0.95 |

Overall fuel load (W) | 35 t/ha | Stefan Boltzman constant (σ) | 5.67 × 10^{−11} kW/m^{2}/K^{4} |

Heat of Combustion (H) | 18,600 kJ/kg |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Penney, G.; Richardson, S. Modelling of the Radiant Heat Flux and Rate of Spread of Wildfire within the Urban Environment. *Fire* **2019**, *2*, 4.
https://doi.org/10.3390/fire2010004

**AMA Style**

Penney G, Richardson S. Modelling of the Radiant Heat Flux and Rate of Spread of Wildfire within the Urban Environment. *Fire*. 2019; 2(1):4.
https://doi.org/10.3390/fire2010004

**Chicago/Turabian Style**

Penney, Greg, and Steven Richardson. 2019. "Modelling of the Radiant Heat Flux and Rate of Spread of Wildfire within the Urban Environment" *Fire* 2, no. 1: 4.
https://doi.org/10.3390/fire2010004