ProbProblem $-2($Water$-$gas shift reactor unit of a NG reforming process $-$ mole balance,

extent of the reaction at equilibrium and equilibrium conversion calculation$)$

Consider that the equilibrium state of the water$-$gas shift reaction $(CO+H_{2}OharrC{O}_2+H_2)$ approximately

described by the following relation

$\frac{y_{C{O}_2}{y}_{H_2}}{y_{CO}{y}_{H_{2}O}}=K_e(T)=0\mathrm{.}0247$exp$[\frac{4020}{T}(K)]$

where $K_e$ is the reaction equilibrium constant, and $yi$ is the mole fraction of species $i$ in the reactor

contents at equilibrium.

The feed to a shift reactor contains $20\mathrm{.}0$ mole $\%CO,10\mathrm{.}0\%C{O}_2,40\mathrm{.}0\%$ water, and the balance an inert

gas. The reactor is maintained at $T=1123K.$

i$.$ For a basis of $1$mol feed and draw and label a flowchart. Carry out a degree$-$of$-$freedom analysis

of the reactor based on extents of reaction and use it to prove that you have enough information

to calculate the composition of the reaction mixture at equilibrium. You do not need to do

calculation for this part $($i$)$ of this question.

ii$.$ Estimate the total moles of gas in the reactor at equilibrium and then the equilibrium mole fraction

of hydrogen in the product.

iii. A gas sample is drawn from the reactor and analyzed shortly after startup and the mole fraction

of hydrogen is significantly different from the calculated value. Assuming that no calculation

mistakes or measurement errors have been made, what is a likely explanation for the discrepancy

between the calculated and measured hydrogen yields?

iv$.$ Construct a spreadsheet to take as input the reactor temperature and the feed component mole

fractions $y_{CO}$;$y_{H2O},$ and $y_{CO2}($assume no hydrogen is fed$)$ and to calculate the mole fraction $y_{H2}$

in the product gas when equilibrium is reached. The spreadsheet column headings should be

The columns between $Ke$ and $y(H2)$ may contain intermediate quantities in the calculation of

$yH2.$ First test your program for the conditions of Part $($a$)$ and verify that it is correct. Then try a

variety of values of the input variables and draw conclusions about the conditions $($reactor

temperature and feed composition$)$ that maximize the equilibrium yield of hydrogen.

$($ii$)$ Suppose the feed to the reactor consists of $n_{A0}$;$n_{B0}$;$n_{C0}$ and $n_{D0},$ moles of $A,B,C,$ and $D,$

respectively. Write expressions for the moles of each species in the final product, $n_A,n_B,n_C,$ and

$n_D$ in terms of incoming moles and the extent $()$ be of the reaction.

$($iii$)$ Then use these expressions and the given equilibrium constant to derive an equation for ${}_e,$ the

equilibrium extent of reaction, in terms of $n_{A0}$;$n_{B0}$;$n_{C0}$ and $n_{D0}.$

$($iv$)$ If the feed to the reactor contains equimolar quantities of mercaptan $(C{H}_3-S-H)$ and hydrogen,

calculate the equilibrium fractional conversion.

$($v$)($iv$)$ It is desired to produce $70$mol of methane starting with $75$mol of mercaptan $(C{H}_3-S-H).$ If

the reaction proceeds to equilibrium, how much hydrogen must be fed? What is the composition

of the final product?