# Review of Convective Heat Transfer Modelling in CFD Simulations of Fire-Driven Flows
## 4 Specific Treatment of Burning Surfaces
### Modelling Approaches  
An overview of approaches related to convective heat transfer modelling which are applicable when burning surfaces are involved, and that have appeared in the literature in the past, are presented in this section. The modelling approaches, previously presented in Section 3, are also applicable here even though not explicitly developed for flows involving burning surfaces. * Based on the stagnant film theory [15; 16]: The model is based on a 1D steady stagnant film approach and describes the gas-phase combustion process by considering known heat transfer relationships. The convective heat fluxes can be calculated as [9]: \[q_{c}^{\prime\prime}=\frac{h}{c_{p}}\bigg{[}\bigg{(}\frac{\chi_{a}-\chi_{r}}{ \chi_{a}}\bigg{)}Y_{\text{O}_{2},\infty}\Delta H_{\text{O}_{2}}-c_{p}(T_{w}-T _{\infty})\bigg{]}\cdot f_{blowing},\] (40) where \(\chi_{a}\) is the combustion efficiency of the fuel, \(\chi_{r}\) is the (global) radiative fraction, \(Y_{\text{O}_{2},\infty}\) is the ambient oxygen mass fraction, \(\Delta H_{\text{O}_{2}}\) is the heat of combustion per kg of \(O_{2}\) consumed (i.e., approximately 13.1 MJ/kg for many hydrocarbon fuels), \(c_{p}\) is the heat capacity, \(T_{w}\) is the surface temperature which, in a fully developed fire, can be taken as the boiling point for a liquid or the ignition temperature for a solid, \(T_{\infty}\) is the ambient temperature and \(f_{blowing}\) is a correction for the 'blowing' effect (i.e., see Section 4.2). This expression stems from the classical B-number theory which describes liquid fuel evaporation due to convective heat transfer from nearby flames [9]. Estimation of the convection coefficients, \(h\), can be made through the use of empirical experimental correlations based on the Nusselt number (e.g., Tables 2 and 3). The calculation of the convective heat fluxes based on the stagnant film theory does not involve the flame temperature directly (i.e., Equation (40)) rather only indirectly through the temperature-dependent properties (e.g., convection coefficient \(h\), heat capacity \(c_{p}\), thermal conductivity \(k\), and Prandtl number \(Pr\)). For a detailed derivation of Equation (40) the reader is referred to [9]. It is interesting to note that Equation (40) can be re-written as: \[q_{c}^{\prime\prime}=h(T_{f}-T_{s}),\] (41) where \(T_{f}\) is a global flame temperature calculated as: \[T_{f}=T_{\infty}+\bigg{(}\frac{(\chi_{a}-\chi_{r})\Delta H_{\text{O}_{2}}}{ \chi_{a}c_{p}}\bigg{)}\bigg{(}\frac{Y_{\text{O}_{2},\infty}\Delta H_{\text{O }_{2}}}{c_{p}}\bigg{)}.\] (42) The concept of a global flame temperature is further discussed, in the context of liquid pool fires, in [42]. Employing the stagnant film theory for modelling convective heat transfer has been considered in literature in the past e.g., for predicting the mass loss rates of pool fires (e.g., [16; 42]), for determining the convective heat fluxes from pool fire experiments (e.g., [64]) and in numerical simulations involving liquid pool fires in mechanically ventilated compartments (e.g., [65; 66]). The method based on stagnant film theory, previously shown to be both accurate and grid insensitive for pool fires [67], has had limited applicability on CFD simulations of flame spread until recently, e.g., [12] but has been employed in a global analytical model [68] for determining the convective heat feedback from the gas-phase combustion to the surface of charring materials.