# Kinetic Models of Wood Biomass Drying in Hot Airflow Systems
## 3 Empirical Investigation of the Process  
Some experimental investigations, such as those of [PERSON] and [PERSON] (1995) [36], show different drying phases. First is the constant-rate period, when the moisture content is above the fibre saturation point (\(FSP\)). After this point, a period starts where the drying rate decreases rapidly, called the 'falling-rate period'. When the moisture content reaches the equilibrium point, the drying rate reaches zero. The definition of the FSP has been much debated so far. [PERSON] et al. (2016) [37] defined a new process point (based on thermodynamic solutions) different from the FSP, called critical moisture content \(x_{c}\), which is the most accurate in establishing the transition point from the period of constant drying rate to the period with decreasing drying rate. In the research of [PERSON] et al. (2010) [38], studying the effects of dry and wet bulb temperature on the drying times of materials in convective processes with forced air at ambient conditions, forced hot air and superheated steam, they indicated that the critical humidity of the material was not a fixed characteristic of the material, but depended on the air temperature and the drying rate in the constant period. The constant drying rate was calculated from the heat transfer, using Equation (5), where h is the combined heat transfer coefficient of convection and radiation (W/m\({}^{2}\)K), \(T_{bh}\) is the wet bulb temperature, \(T\) is the dry bulb temperature (K), and \(\lambda_{w}\) is the specific latent heat of evaporation at process temperature (J/kg). \[\dot{m}_{w}=\frac{h\cdot(T-T_{bh})}{\lambda_{w}} \tag{5}\]  
[PERSON] et al. (2010) [38] proposed a linear regression model for the calculation of critical humidity (\(x_{c}\)), obtaining a coefficient of determination \(R^{2}\) = 0.97. \[x_{c}=0.063829\cdot\dot{m}_{w}-0.00168\cdot T_{bh}+0.250371\]  
In doing so, they concluded that the critical moisture point is low at high wet bulb temperatures and constant at low temperatures. This study of the critical moisture content of the material studied is very important, as it can be very useful in helping to determine the end point of the drying process. Once the critical moisture content has been predicted for the period with constant drying rate, [PERSON] et al. (2020) [39] describe the kinetics as follows:  
\[\dot{m}_{w}=-\frac{m_{0}}{A}\frac{dx}{dt}\dot{\phi}\frac{dx}{dt}=-\dot{m}_{w} \frac{A}{M_{0}}=-k_{l}\]  
\[x=x_{0}-k_{l}t=x_{0}-\dot{m}_{w}\frac{A}{M_{0}}t\]  
where \(\dot{m}_{w}\) is the drying rate (g water/s m\({}^{2}\)), \(m_{0}\) and \(A\) are the dry mass (g) and its moisture transfer area (m\({}^{2}\)), and \(x\) is the moisture of the solid on a dry basis. For the period with decreasing drying rate, [PERSON] et al. (2020) [39] define the drying kinetics as follows:  
\[\dot{m}_{w}=-\frac{m_{0}}{A}\frac{dx}{dt}=K(x-x_{e})\;\;\text{for}\;\omega< \omega_{S}\]  
\[\frac{dx}{x-x_{e}}=-K\frac{A}{m_{0}}dt\]  
\[\ln\frac{x-x_{e}}{x_{c}-x_{e}}=-K\frac{A}{m_{0}}t\]  
\[\frac{x-x_{e}}{x_{c}-x_{e}}=e^{-K\frac{A}{m_{0}}t}\]  
where \(x_{c}\) is the critical moisture content that separates the period with constant drying rate from the period with decreasing drying rate, and \(x_{e}\) is the moisture at equilibrium. The first term of the equation is called the MR Moisture Ratio [40], such that:  
\[MR=\frac{x-x_{e}}{x_{c}-x_{e}}\]  
There are extensive studies relating the variation of moisture ratio (MR) with time in a drying process, such as, [17; 41; 42; 43; 44; 45; 46; 47]. All of them used empirical models, shown in Table 2,which are applicable in the drying period of decreasing speed, obtaining their coefficients for particular conditions. An example of empirical research is the work of [53], who tested all the empirical models in Table 1 for the calculation of rubber drying kinetics with a hot air dryer. From each, the parameters of the constants were obtained for particular cases.