\[d_p^4 = (d_p^{t-1})^4-\sqrt{3}\beta^{2}{\Delta t}^2\]
In general, once a hot firebrand particle is generated and begins its traverse through the atmosphere, it cools due to convective and radiative heat losses. The temperature rate of change implementation for a spherical wooden particle is based on the influence of two heat transfer processes on the particle temperature using the energy equation: 
\[\rho_{ap} V_p c_p \frac{dT_{p}}{dt} = -S_{p}(\mathbf{q}_{conv} + \mathbf{q}_{rad}) \]
where \(V_p\text{, }c_p\)\(T_p\)  and \(S_p\)  are the volume, specific heat capacity, temperature, and the surface area of the firebrand particle, and \(\mathbf{q}_{conv}\)  and \(\mathbf{q}_{rad}\) are the convective and radiative heat fluxes, respectively, directed from the particle to the ambient air. 
The net flux of heat transferred from a firebrand particle to the ambient flow due to convective processes is represented by
\[\mathbf{q}_{conv} = \overline{h} \left(T_{p} - T_{a}\right)\]
where \(T_a\)  the ambient temperature near the particle and \(\overline{h}\)  is the average convection heat transfer coefficient,  determined using the Nuselt number, \(\overline{h}=\overline{Nu} k_{air}/d_{p}\), such that \(k_{air}\) is the air thermal conductivity (kair=27).  For a solid sphere, \(\overline{Nu}\) may be calculated as \[\overline{Nu} = 2 + 0.6Re^{1/2} Pr^{1/3}\]
where \(Pr\) is the Prandtl number (Pr = 0.7). 
The net flux of heat transferred from a firebrand particle to the ambient flow due to radiative processes is represented by
\[\mathbf{q}_{rad} = \sigma\epsilon\left(T^{4}_{p} - T^{4}_{a}\right) \]
where \(\sigma\) is the Stefan-Boltzmann constant and \(\epsilon\) is the emissivity of the firebrand.
We estimate cp as the weighted average between wood and charcoal 
\[c_p=\frac{d_{eff}}{d}c_{p_w} + \left(1-\frac{d_{eff}}{d}c_{p_c}\right)\]
were \(c_{p_w}\) is the specific heat capacity for wood (\(c_{p_w}\) = 1466 J kgK-1) and \(c_{p_c}\) for charcoal (\(c_{p_c}\) = 712 J kgK-1).