Dynamic Model of the Reactor

Consider the following Ordinary Differential Equations model of the ammonia reactor:

$\frac{d{T}_f}{dx}=-\frac{u{S}_1}{W{C}_{pf}}(T_g-T_f)$

$\frac{d{T}_g}{dx}=-\frac{u{S}_1}{W{C}_{pg}}(T_g-T_f)+\frac{(-H)S_2}{W{C}_{pg}}$

$\frac{d{N}_{N_2}}{dx}=-$

where $=f(k_1\frac{p_{N_2}{p}_{H_2}^{1\mathrm{.}5}}{p_{N{H}_3}}-k_2\frac{p_{N{H}_3}}{p_{H_2}^{1\mathrm{.}5}})$

$k_1=1\mathrm{.}78954{10}^4$exp$(\frac{-20800}{R{T}_g})$

$k_2=2\mathrm{.}5714{10}^{16}$exp$(\frac{-47400}{R{T}_g})$

$p_{N_2}=\frac{286N_{N_2}}{2\mathrm{.}598N_{N_{2,0}}+2N_{N_2}}$

$p_{H_2}=3p_{N_2}$

Page $3$ of $4$

$($C$)$ Department of Chemical Engineering, University College London, $2024$

$p_{N{H}_3}=\frac{286(2\mathrm{.}23N_{N_{2,0}}-2N_{N_2})}{2\mathrm{.}598N_{N_{2,0}}+2N_{N_2}}$

$N_{N_2}$ is the molar flowrate of $N_2$ per area catalyst in $kg$ moles $m^{-2}{h}^{-1},T_f$ is temperature of feed gas in $K,T_g$ is

temperature of reacting gas in $K,x$ is reactor length in $m.$ The constant parameters in the model are $C_{pf}($heat

capacity of the feed gas, $0\mathrm{.}707kcalk{g}^{-1}{K}^{-1}),C_{pg}($heat capacity of reacting gas, $0\mathrm{.}719kcalk{g}^{-1}{K}^{-1}),$ catalyst

activity, $1),H($heat of reaction, $-26000$kca$\frac{l}{k}gmol{N}_2,p($partial pressure of a component, psi$),R($ideal gas

constant, $1\mathrm{.}987kcalkgmo{l}^{-1}{K}^{-1}),S_l($surface area of catalyst tubes per unit length of reactor, $10m),S_2($cross

sectional area of catalyst zone, $0\mathrm{.}78m^2),u($overall heat transfer coefficient, $500kcal{h}^{-1}{m}^{-2}{K}^{-1}),W($total

mass flowrate, $264000kg{h}^{-1}).$

Q$3$: Steady$-$state operation: Assume that the reactor is operating at steady$-$state, i$.$e$.,$ ODE equations $(1)-(3)$

and can be replaced by the following algebraic equations:

$0=-\frac{u{S}_1}{W{C}_{pf}}(T_g-T_f)$

$0=-\frac{u{S}_1}{W{C}_{pg}}(T_g-T_f)+\frac{(-H)S_2}{W{C}_{pg}}$

$0=-$

Solve these nonlinear algebraic equations for $N_{N_{2,0}}=701\mathrm{.}2kmol{m}^{-2}{h}^{-1},$ using the CNS solver in GAMS.

Submit the GAMS $(.$gms$)$ file and report the solution obtained. Do not submit the $.$lst file.