# Review of Convective Heat Transfer Modelling in CFD Simulations of Fire-Driven Flows
## 2 Newton's Law and CFD Equations
### [PERSON]'s Law of Cooling  
Convection is the process of heat transfer between a surface and a fluid in motion and is governed by [PERSON]'s law of cooling which is expressed as:  
\[q_{c}^{{}^{\prime\prime}}=\frac{\dot{q}_{c}}{A}=h(T_{w}-T_{\infty}), \tag{1}\]  
where \(q_{c}^{{}^{\prime\prime}}\) is the convective heat flux, \(\dot{q}_{c}\) is the rate of convective heat transfer, \(A\) is the area where heat transfer occurs, \(h\) is the convective heat transfer coefficient, \(T_{w}\) is the surface temperature and \(T_{\infty}\) is the temperature of the surrounding fluid. The convective heat transfer coefficient, \(h\), can be defined as the rate of heat transfer between a surface and a fluid per unit surface area per unit temperature difference.  
The classification of convective heat transfer typically includes:
* Natural or free convection: fluid motion results from heat transfer and its resulting density differences (i.e., buoyancy),
* Forced convection: flow motion produced by an external agent (e.g., fan, pump),
* Mixed convection: combination of both forced and natural convection. In general, the complexities associated with modelling convection are due to the dependency of the convective heat transfer coefficient, \(h\), not only on the fluid properties (e.g., thermal conductivity, dynamic viscosity, density) but also on the surface geometry (e.g., horizontal, vertical) as well as on the type of flow (e.g., natural or forced convection). Typical values of the convective heat transfer coefficient encountered in different types of applications can be found Table 1. It is worth noting that the presence of a medium is a prerequisite for convection to take place (i.e., as well as for conduction but not necessarily for radiation). Additionally, any convection scenario involves conductive heat transfer as well, occurring in a thin layer in the vicinity of the surface, since the fluid velocities there are zero (i.e., non-slip boundary condition between the fluid and the surface).  
A brief overview of convective heat transfer, \(h\), values reported in literature in the past for pool fires and flame spread scenarios, is presented below.  
* Pool fires: Values in the order of \(h/c_{p}=0.008\) kg/(m\({}^{2}\cdot\) s) have been reported for pools without lips while \(h/c_{p}=0.0065\) kg/(m\({}^{2}\cdot\) s) for more turbulent fires with lip [15] (i.e., where \(c_{p}=1\) kJ/(kg \(\cdot\) K). The higher reported values for the cases without lip stem from their higher laminar heat transfer near the base of the pool [16].
* Flame spread: Under steady-state conditions, values of \(h=15.9\) W/(m\({}^{2}\cdot\) K) have been reported by [PERSON] et al. [17], in the order of \(h=17\)-18.2 W/(m\({}^{2}\cdot\) K) by [PERSON] et al. [18] while an asymptotic value of \(h=18\) W/(m\({}^{2}\cdot\) K) was obtained from measurements conducted by [PERSON] et al. [15; 16]. Finally, a value of \(h=14\) W/(m\({}^{2}\cdot\) K) (i.e., a typical value corresponding to natural convection) has been used by [PERSON] [19] in his modelling of the burning behaviour of PMMA. During steady-state conditions, it has been reported that radiative heat transfer accounted for approximately 75-80% [20] (or 75-87% in [21]) of the total heat fluxes in scenarios involving upward flame spread over PMM slabs. This implies that convection is expected to contribute to approximately 15-20% in the total heat transfer.