# A Mathematical Model of Industrial Waste-Derived Fuel Droplet Combustion in High-Temperature Air
## 2 Materials and Methods
### Mathematical Model  
The structure of CLF is a rather complex system. When CLFs are heated, a set of interdependent processes, phase transformations and chemical reactions occurs [41]: passive warming, near-surface water vaporization, thermolysis of coal and municipal waste (solid combustible components), blending of ignitable gases with the oxidizing medium, gas blend firing and burnout, heating of the solid remainder followed by its heterogeneous ignition and combustion. These processes were described in the development of the mathematical model in Ansys Fluent. The discrete particle model (DPM) for composite fuels has been developed (Eulerian-Lagrangian model). The gas is defined as a stationary medium. The gas phase is described by a stationary Reynolds-averaged Navier-Stokes (RANS) model (1)-(3) [42]. \[\
abla\cdot\left(\rho\overset{\rightarrow}{u}\right)=0 \tag{1}\]  
\[\
abla\cdot\left(\rho\overset{\rightarrow}{u}\overset{\rightarrow}{u} \right)=-\
abla p+\
abla\cdot\left(\tau\right)+\rho\overset{\rightarrow}{g}+ \overset{\rightarrow}{F} \tag{2}\]  
\[\
abla\cdot\left[\overset{\rightarrow}{u}(\rho H+p)\right]=-\
abla\cdot \left(\sum_{i}h_{i}l_{i}\right) \tag{3}\]  
where \(\
abla\)--nabla operator; \(\overset{\rightarrow}{F}\)--vector of volumetric fluid-particle interaction force, N; \(H\)--enthalpy, W; \(\overset{\rightarrow}{\tau}\)--gas viscous stress, Pa; \(J\)--mass flux, kg/(m\({}^{2}\cdot\)s); \(h\)--sensible heat, W; and subscripts \(i\)--species. One of the most important parameters characterizing turbulent motion is turbulent kinetic energy \(k\). The transfer equation for turbulent kinetic energy is derived from the Navier-Stokes and Reynolds equations and is defined as (4):  
\[\text{div}\left(\rho k\overset{\rightarrow}{u}\right)=\text{div}\left[\left( \mu+\frac{\mu_{t}}{\sigma_{k}}\right)\text{grad}\,k\right]+G_{k}-\rho\varepsilon \tag{4}\]  
For finding viscous dissipation \(\varepsilon\), the transfer equation (\(k\)-\(\varepsilon\) model) is used (5):  
\[\text{div}\left(\rho\varepsilon\overset{\rightarrow}{u}\right)=\text{div} \left[\left(\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\right)\text{grad}\, \varepsilon\right]+C_{\varepsilon 1}\frac{\varepsilon}{k}G_{k}-C_{\varepsilon 2} \rho\frac{\varepsilon^{2}}{k} \tag{5}\]  
where \(G_{k}\)--the generation of turbulence kinetic energy related to the mean velocity gradient and the turbulent model constants \(C_{\varepsilon 1}=1.44\); \(C_{\varepsilon 2}=1.92\); \(\sigma_{k}=1\); \(\sigma_{\varepsilon}=1.3\). The gas species equation is defined as (6):  
\[\
abla\cdot\left(\rho\overset{\rightarrow}{u}Y\right)=-\
abla\cdot\overset{ \rightarrow}{J}+R_{p} \tag{6}\]  
where \(\
abla\)--nabla operator; \(R_{p}\)--rate of production, units vary. The combusting particle is used as a physical model in the discrete particle model (DPM) to define a single CLF droplet in a gaseous flow. The combusting particle is described by a system of the following laws: heating/cooling, release of volatiles, heterogeneous surface combustion [42]. Meanwhile, at each moment of time there is no temperature gradient inside the combustible particle. It is considered a body with an almost uniform temperature distribution. Since the model is developed for slurry fuel, it is necessary to use a wet combustion model, so the combusting particle has been extended to include the fraction of the liquid component capable of boiling and vaporizing. The vaporization rate is defined as:\[N_{i}=k_{c}(C_{i,s}-C_{i,\infty}) \tag{7}\]  
where \(N_{i}\)--polar flux of vapor, kg\(\cdot\)mol/(m\({}^{2}\cdot\)s); \(k_{c}\)--mass transfer coefficient, m/s; \(C_{i,s}\)--vapor concentration at the droplet surface, kg\(\cdot\)mol/m\({}^{3}\); and \(C_{i,\infty}\)--vapor concentration in the bulk gas, kg\(\cdot\)mol/m\({}^{3}\). [PERSON]'s law determines the vapor concentration near the surface of a fuel droplet:  
\[C_{i,s}=\frac{p_{\rm sat}(T_{p})}{RT_{p}} \tag{8}\]  
where \(R\)--the universal gas constant; \(p_{\rm sat}\)--pressure of saturated vapor; and \(T_{p}\)--droplet temperature.