The elementary irreversible second$-$order gas$-$phase reaction $2$AlongrightarrowB$+C$ is carried out adiabatically in Plug flow reactor. Pure methanol $(A)$ enters the reactor with volumetric flow rate of $20d\frac{m^3}{s}$ at pressure of $2$atm and a temperature of $225C.$ Assume, $A$:$C{H}_{3}OH,B$:$C{H}_{3}OC{H}_3$ and $C$:$H_{2}O.$

Find the parameters and the mathematical relations between conversion, temperature and volume of the PFR$.$

Additional information:

$C_{PA}(\frac{J}{m}ol-K)=72\mathrm{.}62+0\mathrm{.}209T+5{10}^{-4}{T}^2$

$H_A^0=-238\mathrm{.}42k\frac{J}{m}ol$

$C_{pB}(\frac{J}{m}ol-K)=31\mathrm{.}70+0\mathrm{.}103T+6{10}^{-5}{T}^2$

$H_B^0=-184\mathrm{.}1k\frac{J}{m}ol$

$C_{PC}(\frac{J}{m}ol-K)=76\mathrm{.}72+0\mathrm{.}082T+1{10}^{-4}{T}^2$

$H_C^0=-285\mathrm{.}83k\frac{J}{m}ol$

$k=0\mathrm{.}1795\frac{L}{\text{m}\text{o}\text{l}-h}$ at $225C$

$E=1\mathrm{.}044{10}^{3}ca\frac{l}{m}ol$

$R=8\mathrm{.}314$kPa$-\frac{m^3}{\text{k}\text{m}\text{o}\text{l}-K}$

$1$atm$=1\mathrm{.}01325{10}^{5}Pa$

$70\%$