The steam methane reforming $($SMR$)$ reaction is commonly used to produce hydrogen:

$C{H}_4+2H_{2}OharrC{O}_2+4H_2$

The reaction occurs at high temperatures, so you can assume that all species are in the gas phase.

The real reaction typically takes place over a $N\frac{i}{A}{l}_2{O}_3$ catalyst and is extremely complex to

model ${?}^1,$ but here we will greatly simplify it while using numbers from the real reaction wherever

possible. In this problem, you will assume the reaction is in a batch reactor and use a

combination of materials balances, rate laws, and numerical integration of differential equations

to evaluate the conversion water to hydrogen via the water$-$gas shift reaction. You will work with

partial pressures and can assume that the total pressure has been included in the rate constants.

Assume that the reaction occurs reversibly, with a forward rate constant of $k_f=0\mathrm{.}32at{m}^{-2}$

$**min-1,$ and a reverse rate constant of $kr=0\mathrm{.}001at{m}^{-4}**mi{n}^{-1}$ and the reaction order of

each species corresponds to its stoichiometric coefficient. Use initial partial pressures of

$p_{CH4}=p_{H2O}=0\mathrm{.}5,$ and $p_{CO2}=p_{H2}=0.$ Derive all relevant equations and numerically solve

the resulting initial value problem to plot the pressures of $H_{2}O$ and $H_2$ from $t=0$ to $60$

minutes. You should use the most accurate built$-$in solver available $($e$.$g$.$ ODE $45$ in

Matlab, solve$\_$ivp in Python$)$ for this part of problem.

a$.$ Setup: Derive equations, place them in standard form, and indicate initial

conditions.

b$.$ Results: A single plot with curves for each partial pressure.

c$.$ Analysis: Briefly discuss the observed behavior.

Repeat problem $1$ with Euler's method. You should use a write a code that executes

Euler's method and create a "convergence plot" that shows the error at $t=60$min as a

function of step size. The convergence plot should have at least $5$ different step sizes and

reach a percent error $(100**\frac{|p_{\text{t}\text{r}\text{u}\text{e}}-p_{\text{E}\text{u}\text{l}\text{e}\text{r}}|}{p_{\text{t}\text{r}\text{u}\text{e}}})$ of $0\mathrm{.}01\%$ or lower.

a$.$ Setup: Provide the equations for Euler's method and use them to manually solve

for one time step.

b$.$ Results: A single plot with at least $5$ points connected by a line.

c$.$ Analysis: Briefly discuss the time requirements for Euler's method as a function

of step size and as compared to ODE$45.$