# The Influence of Water on the Heat Loss of Hot Mix Asphalt
## 2 Heat Flow in the HMA Layer-Theoretical Model  
The heat flow in the HMA layer is caused by transient (unsteady state) heat conduction. The rate of heat propagation depends on the thermal conductivity \(\lambda\) of HMA and the heat transfer coefficient \(\alpha\) which is a measure of the ability of the surrounding medium to absorb heat (Figure 2).  
The process is governed by the Fourier law in the differential form expressed as a product of two functions: Time and space (1) [12]:  
\[T_{(x)}=T_{med}+\left[\left(\sum_{n=1}^{n=\infty}e^{-\phi_{n}^{2}Fo}\frac{2 sin \delta_{n}}{\delta_{n}+sin\delta_{n}cos\delta_{n}}cos\delta_{n}\frac{x}{s_{m}} \right)\times\left(T_{in}-T_{med}\right)\right] \tag{1}\]  
where: \(T_{(x)}\)--temperature inside the layer at time \(\tau\) and at a distance \(x\) from the layer middle plane of the layer [K], \(T_{in}\)--initial temperature of the layer at time \(\tau=0\) [K], \(T_{med}\)--temperature of the medium surrounding the body [K], \(s_{m}\)--perpendicular distance between the layer surface and middle plane (m), \(x\)--distance of the point in consideration from the middle plane of the layer [m], \(Fo\)--Fourier number.  
The Fourier number is determined with the (2) formula [12]:  
\[Fo=\frac{\lambda\times\tau}{\rho_{b}\times c\times s_{m}^{2}} \tag{2}\]  
where \(\lambda\)--thermal conductivity coefficient of the HMA layer [W/(m\(\cdot\)K)], \(\tau\)--time from the beginning of the heat transfer process [s], \(\rho_{b}\)--bulk density of the HMA layer [kg/m\({}^{3}\)], \(c\)--specific heat of HMA [J/(kg\(\cdot\)K)].  
The values of \(\delta_{n}\) correspond to the points of intersection of the following two functions: \(y_{1}=ctg\delta\) and \(y_{2}=\delta/Bi\). The Biot number is determined from the formula (3) [12; 13; 14; 15; 16]:  
\[Bi=\frac{\alpha\times s_{m}}{\lambda} \tag{3}\]  
where: \(Bi\)--Biot number, \(\alpha\)--heat transfer coefficient of the surrounding medium [W/(m\({}^{2}\)-K)].  
Figure 2: Heat flow in the Hot Mix Asphalt (HMA) layer and heat transfer during the cooling process.  
It is quite easy to define most parameters of the theoretical model described by equation (1). Two of them, however, require more attention, namely: Thermal conductivity \(\lambda\) of HMA and the heat transfer coefficient \(\alpha\) of the surrounding medium (3).  
HMA does not classify as a good heat conductor. Based on the author's own studies their value of \(\lambda\) ranges from ca. 0.50 to 1.30 W/(m\(\cdot\)K) [17]. The exact value depends on a number of factors, including proportions of ingredients (binder, aggregate), rock characteristics (density and porosity), content and type of voids: Connected (open) or isolated (closed). Relatively small values of \(\lambda\) indicate that in the case of the high amount of heat transfer to the surrounding media temperature stratification will occur in the layer with the high cooling rate on the surfaces and small temperature drops inside the layer. This is caused by a much slower propagation of heat from the inside to the surfaces of the layer in relation to its release rate to the surrounding media (air, underlying courses of pavement, etc.).