# Kinetic Models of Wood Biomass Drying in Hot Airflow Systems
## 4 Theoretical Predictive Research  
Theoretical predictive research attempts to perform modelling based on the equations of mass and energy balances, together with mass and heat transfer models. The modelling must consider three balances:  
_Water mass balance_  
\[\dot{m}_{air}\cdot(\omega-\omega_{air1})=m_{s}\cdot(x-x_{e})\]  
_Energy balance in the air_  
\[\dot{m}_{air}\cdot[(Cp_{air}\cdot T+\omega\cdot(\lambda+Cp_{steam}\cdot T))-( Cp_{air}\cdot T_{1}+\omega_{air1}\cdot(\lambda+Cp_{steam}\cdot T_{1}))]=K\cdot A \cdot h_{w}\cdot(x-x_{e})+h\cdot A\cdot(T-T_{b})\]  
_Biomass energy balance_  
\[m_{s}\cdot(Cp_{b}+x\cdot Cp_{w})\frac{dT_{b}}{dt}+m_{s}\cdot\lambda\cdot\frac {dx}{dt}=\dot{m}_{air}\cdot[(\lambda+Cp_{v}\cdot T_{e})\cdot(\omega-\omega_{ air1})+(Cp_{air}+\omega Cp_{v})\cdot(T-T_{e})]\]  
\[\dot{m}_{w}=\frac{m_{s}}{A}\cdot\frac{dx}{dt}=K\cdot(x-x_{e})\]  
\(T\) is the air temperature and \(T_{b}\) is the biomass temperature. The unknown variables are \(\omega\), \(T\), \(x\), \(T_{b}\). [PERSON] et al. (2015) [58] and [PERSON] (2007) [59] already warn that to simulate and model wood drying, it is necessary to implement strong simplifications of the process and the factors that govern it, such as, for example, considering the wood to be dried as a continuous medium, eliminating the heterogeneity of the material, or making the air temperature of the air stream constant during the drying process. Some authors have designed kinetic models of drying based on mass transfer processes. [PERSON] et al. (2000) [23], using an experimental method, sought to determine the convective heat and mass transfer coefficient in _Pinus resinosa_ A. based on moisture content, establishing the difference in water potential between the wood surface and the air as the driving force for heat and mass exchange. Their conclusions were that the coefficient varies as a function of air velocity, the higher the velocity, the higher the coefficient, and it is constant up to wood surface moisture content values close to 80%. Other authors designed drying models based on [60; 61], developed by [62; 63; 64; 65; 66; 67; 68; 69]. Their models are based on the conservation of energy to model the drying of a pine particle of infinite cylindrical geometry as a porous medium. To do so, they use the following equation:  
\[\varepsilon_{g}+\varepsilon_{s}+\varepsilon_{fw}+\varepsilon_{bw}=1\]  
where the sum of the volume fractions of each phase (gas mixture: \(g\), solid: \(s\), free water: \(fw\), and bound water: \(bw\)) equals 1. Thus, they investigated nine physical properties (effective thermal conductivity, specific heat capacity, intrinsic permeabilities of gas and liquid, relative permeabilities of gas and liquid, capillary pressure, diffusivity of bound water, and effective diffusivity of water vapour) and the transport coefficients of the model. They concluded that these properties showed a noticeable impact on the predicted drying time, of which intrinsic gas permeability was the most influential in the model. [PERSON] et al. (2021) [3] developed a model capable of simultaneously simulating mass and heat transfer at low temperature in small wood masses of _Pinus radiata_ D. Don. species, predicting the mechanical stresses during the drying process. In particular, the values of the local heat transfer coefficient (\(h\)) were calculated by Colburn's Analogy, which allowed finding the values in terms of convective mass coefficients \(k_{r}\) (radial) and \(k_{t}\) (tangential), respectively ([PERSON] et al., 2020) [70]. \[h_{m}=k_{r}\left(\frac{K_{w}}{D_{air}}\right)\left(\frac{Sch}{Pr}\right)^{- \frac{1}{3}}\]  
where \(h_{m}\) is the local mass transfer coefficient (water), \(k_{r}\) is the radial conductivity, \(K_{w}\) is the permeability, \(D_{air}\) is the diffusivity, \(Sch\) is the [PERSON] number, and \(Pr\) is the [PERSON] number. [PERSON] et al. (2016) [22] modelled for two types of tropical wood (_Triplochiton scleroxylon_ K. [PERSON] and _Terminalia superha_ Engl. and Diels.) the heat and mass transfers during drying.