# Review of Convective Heat Transfer Modelling in CFD Simulations of Fire-Driven Flows
## 3 Main Modelling Approaches
### Experimental Correlations
#### 3.1.4 Correlations
Nevertheless, there are some limitations associated with the use of such approaches, including that they have been developed for non-reacting flows, considering either iso-thermal or constant flux surfaces, and that they ignore mass transpiration effects. In addition, when employed within the context of CFD the first grid cell values are used as input in these correlations and not the free stream values since the latter are not well-defined. For this reason, the predicted rates of heat transfer can be highly dependant on the grid size [40]. It is also worth noting that uncertainties up to 25% have been reported in the resulting values of the convective heat transfer coefficient based on the free-stream properties and the roughness of the surface [14]. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Geometry** & \(\mathbf{L_{c}}\) & **Range of Validity** & \(\mathbf{h}\) \\ \hline
**Vertical plate** & & & \\ Laminar flow & L & \(10^{4}\leq Ra\leq 10^{9}\) & \(h=1.42(\Delta T/L_{c})^{1/4}\) \\ Turbulent flow & - & \(10^{9}\leq Ra\leq 10^{13}\) & \(h=1.31(\Delta T)^{1/3}\) \\ \hline
**Horizontal plate** & & & \\ Laminar flow & \(A_{s}/p\) & \(10^{4}\leq Ra\leq 10^{9}\) & \(h=1.32(\Delta T/L_{c})^{1/4}\) \\ Turbulent flow & - & \(10^{9}\leq Ra\leq 10^{13}\) & \(h=1.52(\Delta T)^{1/3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simplified correlations for the average Nusselt number for natural convection over isothermal surfaces [34]. The characteristic length, \(L_{c}\), is defined in Section 3.1.3. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Geometry** & \(\mathbf{L_{c}}\) & **Range of Validity** & \(\mathbf{Nu}\) \\ \hline
**Vertical plate** & & & \\ Laminar flow & L & \(10^{4}\leq Ra\leq 10^{9}\) & \(Nu=0.59 Ra^{1/4}\)[32] \\ Turbulent flow & L & \(10^{9}\leq Ra\leq 10^{13}\) & \(Nu=0.1 Ra^{1/3}\)[32] \\ Any type of flow & L & \(Ra\leq 10^{12}\) & \(Nu=\left(0.825+\frac{0.387 Ra^{1/6}}{(1+(0.492/Pr)^{9/16})^{7/2}}\right)^{2}\)[36] \\ \hline
**Horizontal plate** & & & \\ Laminar flow & \(A_{s}/p\) & \(10^{4}\leq Ra\leq 10^{7}\) & \(Nu=0.54 Ra^{1/4}\)[32] \\ Turbulent flow & \(A_{s}/p\) & \(10^{7}\leq Ra\leq 10^{11}\) & \(Nu=0.15 Ra^{1/3}\)[32] \\ \hline
**Sphere** & D & Ra \(\leq 10^{11}\) & \(Nu=2+\frac{0.589 Ra^{1/4}}{(1+(0.469/Pr)^{9/16})^{4/9}}\)[37] \\  & & \(Pr\geq 0.7\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Empirical correlations for the average Nusselt number for natural convection over isothermal surfaces. The characteristic length, \(L_{c}\), is defined in Section 3.1.3. Both natural convection and forced convection correlations have been applied in literature in the past in order to calculate the Nusselt number and then estimate \(h\). For example, in [41; 42], it is assumed that natural convective heat transfer occurs at the surface of a burning liquid similarly to a hot surface facing upwards (see horizontal plate correlations in Table 2). On the other hand, a forced convection regime is assumed in [43]. It is of interest here to note that for natural convection, the Grashof number is proportional to the cubic power of the length scale \(L\), i.e., \(Gr\propto L^{3}\). Thus, \(Nu\propto L^{3/4}\) in the laminar regime and \(Nu\propto L\) in the turbulent regime. As \(h\) is inversely proportional to \(L\), one obtains a \(-1/4\) power dependence on \(L\) in the laminar regime and no dependence of \(h\) on \(L\) in the turbulent regime. A similar reasoning for the forced convection regime (where \(Re\) is used) yields a \(-1/2\) power law dependence in the laminar regime and a \(-1/5\) power law dependence in the turbulent regime. Based on this dimensional analysis, the natural convection correlations are particularly interesting from a modelling point of view because they do not involve a length scale in the turbulent regime. This aspect would mainly be important in scenarios involving complex geometries where a single characteristic length scale would not be easily determined.