Mathematical model of process of coal self-heating accompanied by chemical reactions and change of humidity (Q2768804)

The author proposes the following mathematical model of a coal self-heating process. Let \(\Omega=\{0<x<L, -\infty<y<\infty, |z|<l\}\) be a coal layer with boundary \(\partial\Omega\), let \(T(x,z,t)\) be a temperature and let \(C(x,z,t)\) be an oxygen concentration in the coal layer, respectively. Then the considered process is described by the following initial-boundary value problem NEWLINE\[NEWLINEc\rho{\partial T\over\partial t}-\lambda\left({\partial^2 T\over\partial x^2}+{\partial^2 T\over\partial z^2}\right)+c_{g}\rho_{g}\vec v_{f}{\partial T\over\partial x}=g(T,C,t),\quad P\in\Omega,\quad t>0;NEWLINE\]NEWLINE NEWLINE\[NEWLINEm{{\partial C}\over{\partial t}}-mD\left({\partial^2 C\over\partial x^2}+{\partial^2 C\over\partial z^2}\right)+\vec v_{f}{\partial C\over\partial x}=f(T,C,t),\quad 0<x<L,\quad |z|<l,\quad t>0;NEWLINE\]NEWLINE NEWLINE\[NEWLINET(x,z,0)=T_0,\quad C(x,z,0)=0,\quad 0<x<L,\;|z|<l;NEWLINE\]NEWLINE NEWLINE\[NEWLINET(0,z,t)=T_0,\;C(0,z,t)=0,\;T(L,z,t)=T_{L},\;{\partial C(L,z,t)\over\partial x}=0,\;|z|<l,\;t>0;NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lambda{\partial T\over\partial z}-\alpha_1(T-T_{L})=0,\quad mD{\partial C\over\partial z}-\beta_1C=0,\;0<x<L,\;z=-l,\;t>0;NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lambda{\partial T\over\partial z}+\alpha_2(T-T_{L})=0,\quad mD{\partial C\over\partial z}+\beta_2C=0,\;0<x<L,\;z=l,\;t>0,NEWLINE\]NEWLINE where \(\alpha_{i}\), \(i=1,2\), are the heat transfer coefficients, \(\beta_{i}\), \(i=1,2\), are the absorption coefficients on \(z=-l\) and \(z=l\), respectively; \(\lambda\) is the heat conduction coefficient; \(m\) is a porosity coefficient; \(D\) is a diffusion coefficient; \(\vec v_{f}\) is a velocity of heat transfer; \(c_{g}\) and \(\rho_{g}\) are gas heat capacity and density, respectively.