The mass and energy balances for the outlet temperature $(T$ in $K)$ and concentration of reactant $A(C_A$ in moles $\frac{?}{f^3})$ from an adiabatic $($perfectly insulated$)$ continuously stirred tank reactor $($CSTR$)$ are as follows:

$f(C_A)=Q^{**}{C}_{A_0}-Q^{**}{C}_A-V**C_A**k_0^{**}$exp$(\frac{-E}{RT})=0$

$f^{\text{'}}(C_A)=-Q-V**k_0**exp(\frac{-E}{RT})=0$

$f(T)={}^{**}{Q}^{**}{C}_p^{**}(T_0-T)-V**C_A**k_0^{**}{H}_{R**N}**exp(\frac{-E}{RT})=0$

$f^{\text{'}}(T)={}^{**}{Q}^{**}{C}_p^{**}(-1)-V**C_A**k_0**H_{RKN}**\frac{E}{RT}$exp$(\frac{-E}{RT})=0$

where $C_{Al}=$ feed concentration of $A=200$mole$\frac{s}{f}{t}^3,Q=100f\frac{t^3}{h}r,V=30f{t}^3,R=1\mathrm{.}987ca\frac{l}{K}-$mole, $k0=6{10}^{11}h{r}^{-1},E=18500ca\frac{l}{m}ol,P=28330\frac{g}{f}{t}^3,H_{BxN}=-21111ca\frac{l}{m}$ole of $A,C_P=1\mathrm{.}0ca\frac{l}{g}-K,$ and $I_g=311K.$ Put all results from $4$A into a worksheet labeled problem $4A$ in Excel, save the pol file as yourlastnameproblem$4$a$.$pol, and put the results from problems $4$B and $4$C into a worksheet labeled problems$4$Band$4$C$.$ All of the Excel worksheets should go into the same file as the rest of the exam. Define as much as you can using the Insert Name Create function, as you will eventually find how the results for $C_A$ and I vary with flowrate $Q.$ If done properly, for the base case, you should get $C_A=195\mathrm{.}02$ moles $f{t}^3$ and $I=314\mathrm{.}71K.$ If the rate constant, $ko,$ is varied from $1{10}^{10}h{r}^{-1}$ to $1{10}^{12}h{r}^{-1},$ one will go from essentially no reaction $(C_A$ and $T$ are the same as $C_{A0}$ and $T_0)$ to the point where the temperature starts accelerating out of control.

A$)(8$ pts$.)$ Solve this problem in Polymath for the base case. A start is in the accompanying Polymath file. You will need both $f(CA)$ and $f(T).$

B$)(15$ pts$.)$ Solve this problem in Excel using Newton's method. When you solve this, copy and paste all the way down to row $65536,$ the maximum number of rows. Use the result from a$65536$ for your data table in part C$.$ Your answers are in a$65536$ and d$65536.$ Notice the VERY slow temperature convergence. In fact, the temperature has not totally converged by the final row. If you do this correctly, you will notice your answers for A and $B$ being very slightly different because Excel is limited in the number of iterations more than Polymath is$.$ This is a double iterative Newton's method, so the concentration of A will depend on $T,$ and vice versa. I have started your spreadsheet with initial guesses of temperature and concentration equal to the starting conditions.

C$).)$ Vary ko$,$ from $1{10}^{10}h{r}^{-1}$ to $1{10}^{12}h{r}^{-1}$ using the data table in the Excel file that accompanies $C_A$ and $T$ vary with $Q$ using a Data Table function. What happens to the data table values in the concentration of A column $($column $H)$ when $ko$ gets above $7\mathrm{.}2{10}^{11}hr?$ What does this mean?