# Review of Convective Heat Transfer Modelling in CFD Simulations of Fire-Driven Flows
## 4 Specific Treatment of Burning Surfaces
### Special Topic: Convective Heat Transfer within the Liquid  
When computing the convective heat flux at the liquid surface, besides the convective heat transfer coefficient, \(h\), and the gas phase temperature, \(T_{g}\), the liquid surface temperature is required. In general, when steady-state in fully-developed conditions is considered, the surface temperature is generally taken as the boiling temperature of the liquid. However, when transient calculations are sought, the surface temperature is calculated using a heat balance, which, in the most advanced approach, is coupled to the convective motion of the liquid by solving the Navier-Stokes equations as in e.g., [69; 70]. The convective motion of the liquid may be caused by several mechanisms which, most likely, occur simultaneously. The first mechanism is due to the non-uniform heat flux distribution at the surface of the liquid which yields a non-uniform surface temperature distribution and consequently surface tension gradients therein. The liquid moves then from low surface tension regions to high surface tension regions. This is known as the Marangoni or capillary effect. The convective motion of the liquid may also be caused by a temperature difference between the liquid and the burner walls. A third potential cause for liquid motion is in-depth radiation, which causes the liquid temperature slightly beneath the surface to be higher than the surface temperature. This results in a hydrodynamic instability (called the Rayleigh-Benard instability), making the hot liquid rise above the slightly less hot liquid. Detailed experimental data on the convective motion of the liquid in a pool fire remain very scarce. To the best of our knowledge, the most comprehensive experimental work on this topic has been carried out by [PERSON] and his co-workers (e.g., [71]) where the flow field within the liquid has been measured using PIV and the thermal structure (of the liquid) has been characterised using thermocouples. However, the work remained limited to a 9 cm-diameter methanol pool fire. From a modelling perspective, solving the Navier-Stokes equations, as done by [PERSON] et al. [69], remains a very computationally intensive approach for 'practical' fire dynamics simulations where a lot of resources are mainly devoted to the gas phase, given its importance with respect, for example, to tenability and structural integrity. The most common approach is to treat the liquid as a 'thermally-thick solid' by solving a 1D Fourier's equation. In [43], convective motion of the liquid and the subsequent increased heat-up are indirectly accounted for by using the 'effective thermal conductivity', \(k_{eff}\), which is calculated as:  
\[k_{eff}=Nu_{i}\cdot k, \tag{45}\]  
where \(k\) is the actual thermal conductivity of the liquid and \(Nu_{i}\) is the 'internal' Nusselt number (within the liquid), not to be confused with the Nusselt number used for convective heat transfer at the interface between the liquid and the surrounding gas. In [43], \(Nu_{i}\) is calculated from a correlation for internally heated horizontal plane layer with isothermal top boundary and thermally insulated bottom boundary:  
\[Nu_{i}=0.338 Ra_{i}^{0.227} \tag{46}\]  
The internal Rayleigh number, \(Ra_{i}\), is calculated as:  
\[Ra_{i}=\frac{g\beta_{tr}^{\alpha^{m}}L^{5}}{k\alpha}f(\eta), \tag{47}\]  
where \(g\) is the gravitational acceleration while \(\beta\), \(\
u\) and \(\alpha\) are the coefficient of thermal expansion, the kinematic viscosity and the thermal diffusivity of the liquid, respectively. The variable \(\dot{q}_{r}^{\alpha^{m}}\) is a volumetric source term for in-depth radiation, \(L\) is the liquid depth, \(\eta\) is the non-dimensional length-scale and \(f\) is a pre-defined function describing the distribution of radiation over the liquid depth (see [43] for more details).