$\mu \left(\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {v}_{z}}{\partial {r}_{s}}\right)\right)+\frac{{P}_{in}-{P}_{out}}{L}=\rho \frac{\partial {v}_{z}}{\partial t}$ $-2{v}_{m}\left[1-{\left(\frac{r}{R}\right)}^{2}\right]\frac{\partial {C}_{s}}{\partial z}+\mathfrak{D}\frac{{\partial }^{2}{C}_{s}}{\partial {z}^{2}}+\mathfrak{D}\left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {C}_{s}}{\partial r}\right)\right]-\left(-{r}_{s}\right)=\frac{\partial {C}_{s}}{\partial t}$ $-2\rho cp{v}_{m}\left[1-{\left(\frac{r}{R}\right)}^{2}\right]\frac{\partial T}{\partial z}+k\frac{{\partial }^{2}T}{\partial {z}^{2}}+k\left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)\right]-\left(-{r}_{s}\right)\left(-\Delta {H}_{r}\right)+16\mu {v}_{m}^{2}\frac{{r}^{2}}{{R}^{4}}=\rho cp\frac{\partial T}{\partial t}$ where $\left(-{r}_{s}\right)=\frac{{K}_{3}{C}_{E}{C}_{S}}{{k}_{m}\left(1+\frac{{C}_{S0}-{C}_{S}}{{k}_{I}}\right)+{C}_{S}+\frac{{C}_{S}^{2}}{{k}_{III}}}$ $t=0$ ${v}_{z}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{S}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}T={T}_{0}$ $\forall z,\forall r$ $r=0$ $\frac{\partial {v}_{z}}{\partial r}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {C}_{s}}{\partial r}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial T}{\partial r}=0$ $\forall z,\forall t>0$ $r=R$ ${v}_{z}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {C}_{s}}{\partial r}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial T}{\partial r}=0$ $\forall z,\forall t>0$ $z=0$ ${C}_{s}={C}_{s0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}T={T}_{0}$ $\forall r,\forall t>0$ $z=L$ $\frac{\partial {C}_{s}}{\partial z}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial T}{\partial z}=0$ $\forall r,\forall t>0$