# Review of Convective Heat Transfer Modelling in CFD Simulations of Fire-Driven Flows
## 3 Main Modelling Approaches
### Experimental Correlations
#### 3.1.4 Correlations  
A brief overview of the most widely used experimental correlations for the Nusselt number, applicable to horizontal/vertical surfaces and spheres for both forced and natural convection scenarios, is given below. It is worth noting that these experimental correlations provide the average Nusselt number over the entire surface (i.e., use of the global characteristic length \(L_{c}\)), an approach that is most widely used in CFD modelling. No focus is given on experimental correlations based on the local Nusselt number (i.e., use of local length scales). Therefore, the non-dimensional numbers i.e., \(Re\) and \(Ra\) involved in Tables 2 and 3, respectively, have to be calculated based on the characteristic length \(L_{c}\). For completeness, a set of simplified correlations which calculate the convective heat transfer coefficients, \(h\), in the case of natural convection is included in Table 4. For a more complete set of empirical correlations, the reader is referred to [14, 32, 33, 34]. The characterisation of the type of flow for forced convection (i.e., in Table 2) and for natural convection (i.e., in Tables 3 and 4) implies the following:  
* Laminar flow: The flow is laminar over the entire surface. * Turbulent flow: The flow is turbulent over the entire surface or when the laminar region is too small relative to the turbulent region. * Combined flow: For cases when the surface is sufficiently long for the flow to become turbulent but not long enough to disregard the laminar region. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Geometry** & \(L_{c}\) & **Range of Validity** & \(Nu\) \\ \hline
**Parallel to a plate** & & & \\ Laminar flow & L & \(Re\leq 5\times 10^{5}\) & \(Nu=0.664 Re^{1/2}Pr^{1/3}\)[33] \\  & & \(0.6\leq Pr\leq 60\) & \\ Turbulent flow & L & \(5\times 10^{5}\leq Re\leq\times 10^{7}\) & \(Nu=0.037 Re^{4/5}Pr^{1/3}\)[33] \\  & & \(0.6\leq Pr\leq 60\) & \\ Combined flow & L & \(5\times 10^{5}\leq Re\leq\times 10^{7}\) & \(Nu=(0.037 Re^{4/5}-871)Pr^{1/3}\)[33] \\  & & \(0.6\leq Pr\leq 60\) & \\ \hline
**Around a sphere** & D & \(0\leq Re<200\) & \(Nu=2+0.6 Re^{1/2}Pr^{1/3}\)[35] \\  & & \(0\leq Pr\leq 250\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Empirical correlations for the average Nusselt number for forced convection over isothermal surfaces. The characteristic length, \(L_{c}\), is defined in Section 3.1.3. Figure 1: Characteristic length, \(L_{c}\), for (**a**) forced and (**b**) natural convection scenarios. \(A_{s}\) and \(p\) are the surface area and perimeter of the plate, respectively. Determination of whether the flow is laminar or turbulent is made by calculation of the Reynolds number for forced convection problems (e.g., [38]) and of the Rayleigh number for natural convection problems (e.g., [39]). Determination as to whether a combined flow exists over the surface, is made by comparing the critical distance, \(x_{cr}\), (i.e., based on either the critical Reynolds or the critical Rayleigh number depending on the type of flow) with the characteristic length of the problem, \(L_{c}\). If \(x_{cr}<L_{c}\) then the flow over the surface is both laminar and turbulent (i.e., combined flow) while if \(x_{cr}>L_{c}\) then the flow is laminar throughout the whole length of the surface. The convective heat transfer, \(h\), coefficient can be back-calculated from the Nusselt number by considering the fluid conductivity, \(k\), and the characteristic length of the problem, \(L_{c}\). Finally, the convective heat fluxes are then calculated based on [PERSON]'s law as:  
\[q_{c}^{\prime\prime}=h(T_{g}-T_{w}), \tag{16}\]  
where \(T_{g}\) is the first grid cell value in the gas phase and \(T_{w}\) is the surface temperature. Convective heat transfer, within the context of fire-driven flows, is often modelled considering experimental correlations based on the Nusselt number [14] for either natural or forced convection.