$8\mathrm{.}15.$ For the series reactors in Figure Q$8\mathrm{.}15,$ the outlet concentration is controlled

at $0\mathrm{.}414$mol$\frac{e}{m^3}$ by adjusting the inlet concentration with a proportional$-$

only feedback controller. At the initial base case operation, the valve is

$50$ percent open, giving $C_{A0}=0\mathrm{.}925$mol$\frac{e}{m^3}.$ One first$-$order reaction

$A$ B occurs; the data are $V=1\mathrm{.}05m^3,F=0\mathrm{.}085\frac{m^3}{m}in,$ and $k=$

$0\mathrm{.}040mi{n}^{-1}.$ The process transfer function is derived in Example $4\mathrm{.}2$ as

$C_{A2}\frac{s}{C_{A0}}(s)=\frac{0\mathrm{.}447}{(8\mathrm{.}25s+1{)}^2}$; the additional model relates the

valve to inlet concentration, which for a linear valve and small flow of $A$

$(F>F_A)$ gives $C_{A0}\frac{s}{v}(s)=\frac{0\mathrm{.}925}{50}=0\mathrm{.}0185\frac{\frac{\text{m}\text{o}\text{l}\text{e}}{m^3}}{\%}$ open;

you may assume for this question that the sensor dynamics are negligible.

$($a$)$ Determine whether the reactors are stable without feedback control.

$($b$)$ Determine the closed$-$loop transfer function for a set point response.

$($c$)$ By analyzing the denominator of the transfer function $($the character$-$

istic polynomial$),$ determine the stability of the feedback system for

controller gain, $K_c,$ values of $($i$)0\mathrm{.}0,($ii$)121,($iii$)605,$ and $($iv$)2420$

$($in $\%$valve opening $\frac{?}{\text{m}\text{o}\text{l}}\frac{e}{m^3}).$

$($d$)$ By analyzing the total closed$-$loop transfer function, determine the

steady$-$state offset for a set point change with controller gain, $K_c,$

values of $($i$)0\mathrm{.}0,($ii$)121,($iii$)605,$ and $($iv$)2420($in $\%$valve

opening $\frac{?}{\text{m}\text{o}\text{l}}\frac{e}{m^3}).$

$($e$)$ Without simulating, sketch the general shape of the dynamic response

for a set point step change for each of the cases in $(c)$ and $(d)$ above.

FIGURE Q$8\mathrm{.}158\mathrm{.}15.$ For the series reactors in Figure Q$8\mathrm{.}15,$ the outlet concentration is controlled

at $0\mathrm{.}414$ mole$/$m$3$ by adjusting the inlet concentration with a proportionalonly feedback controller. At the initial base case operation, the valve is

$50$ percent open, giving Cao $=0\mathrm{.}925$ mole$/$m$3.$ One first$-$order reaction

A $->$ B occurs; the data are V $=1\mathrm{.}05$ m$3,$ F $=0\mathrm{.}085$ m$3/$min$,$ and k $=$

$0\mathrm{.}040$ min$-1.$ The process transfer function is derived in Example $4\mathrm{.}2$ as

CA$2($s$)/$CA$0($s$)=0\mathrm{.}447/(8\mathrm{.}25^+$ l$)2$; the additional model relates the

valve to inlet concentration, which for a linear valve and small flow of A

$($F $$ FA$)$ gives CA$0($s$)/$v$($s$)=0\mathrm{.}925/50=0\mathrm{.}0185($mole$/$m$3)/\%$open;

you may assume for this question that the sensor dynamics are negligible.

$($a$)$ Determine whether the reactors are stable without feedback control.

$($b$)$ Determine the closed$-$loop transfer function for a set point response.

$($c$)$ By analyzing the denominator of the transfer function $($the character

istic polynomial$),$ determine the stability of the feedback system for

controller gain, Kc$,$ values of $($i$)0\mathrm{.}0,($ii$)121,($iii$)605,$ and $($iv$)2420$

$($in $\%$ valve opening$/$mole$/$m$3).$

$($d$)$ By analyzing the total closed$-$loop transfer function, determine the

steady$-$state offset for a set point change with controller gain, Kc$,$

values of $($i$)0\mathrm{.}0,($ii$)121,($iii$)605,$ and $($iv$)2420($in $\%$valve

opening$/$mole$/$m$3).$

$($e$)$ Without simulating, sketch the general shape of the dynamic response

for a set point step change for each of the cases in $($c$)$ and $($d$)$ above.