Problem $1.$ Adiabatic CSTR$.($Example $8-8,$ Fogler $4$th Ed$,$ p$\mathrm{.}526-531$; Example $12-3$ Fogler $5$th Ed$,$ p$\mathrm{.}568-572,6$th Ed$,$ in $12\mathrm{.}4$ CSTR with Heat Effects.

a$).$ Reproduce the sequence of calculations presented in Example $8-8(12-3)($Eqs numbers are given for $4$th Ed$).$ Plot the mass balance and energy balance dependencies of the conversion versus temperature, as shown in Fig.E$8-8\mathrm{.}2($E$12-3\mathrm{.}2).$ You will see that at the conditions given in this example there is only one steady state at T$=613\backslash$deg R$.$ these conditions imply the space time $\backslash (\backslash$tau $=0\mathrm{.}123\backslash )$ h$,$ heat of reaction $\backslash (\backslash$Delta H$^$o $=-36,400\backslash )$ Btu$/$lb mol, and the initial temperature $\backslash ($ T$\_0=535\backslash )\backslash$deg R$.($Note: $\backslash$deg R means the Rankine scale of temperature, use proper conversion$).$ Note for this and following tasks, you will have to write a code to solve equations E$\mathrm{.}8-8\mathrm{.}5$ for mole balance and $8-29$ for energy balance, allowing for variation the space time $\backslash (\backslash$tau $\backslash ),$ heat of reaction $\backslash (\backslash$Delta H$^$o $\backslash )$ and initial temperature $\backslash ($ T$\_0\backslash ).$ You may ignore $\backslash (\backslash$Delta C$\_$p$($T$-$T$\_$R$)\backslash )$ term as it is small as seen from E$8-8\mathrm{.}12.$

b$)$ Fix all parameters, including the space time $\backslash (\backslash$tau $=0\mathrm{.}123\backslash )$ h and heat of reaction $\backslash (\backslash$Delta H$^$o $=-36,400\backslash )$ Btu$/$lb mol, except for the initial temperature $\backslash ($ T$\_0\backslash ).$ By varying the initial temperature $\backslash ($ T$\_0\backslash ),$ find the region of $\backslash ($ T$\_0\backslash )$ that corresponds to multiple steady states. Present plots at different $\backslash ($ T$\_0\backslash )$ that correspond to the regions of single and multiple steady states. Find the boundaries of the region of multiple steady states from the conditions of tangent. Graphical solution is acceptable. Solution using a solver is a plus.

c$)$ Fix all parameters, including the space time $\backslash (\backslash$tau $=0\mathrm{.}123\backslash )$ h and the initial temperature $\backslash ($ T$\_0=535\backslash )\backslash$deg R$,$ except for the heat of reaction $\backslash (\backslash$Delta H$^$o $\backslash ).$ By varying $\backslash (\backslash$Delta H$^$o $\backslash ),$ find the region of $\backslash (\backslash$Delta H$^$o $\backslash )$ that corresponds to the region of multiple states. Present plots at different $\backslash (\backslash$Delta H$^$o $\backslash )$ that correspond to the regions of single and multiple steady states. Find the boundaries of the region of multiple states from the conditions of tangent. Graphical solution is acceptable. Solution using a solver is a plus.

d$)$ Explore how the variation of the space time $\backslash (\backslash$tau $\backslash )($at all other parameters fixed$)$ affects the steady state conditions.

e$)$ Construct the Levenspiel plot for the conditions of Example $8-8.$ Determine conditions of the minimum $($if the minimum exists$).$ Compare this plot to an isothermal CSTR kept at $\backslash ($ T$\_0=535\backslash )\backslash$deg R$.$