The hydrolysis of $1-$kestose, key step in the production of fructo$-$oligosaccarides which found in many healthy food due to the lack of calori and non$-$carcinogenic structure. In given continuous stirred and heat jacket tank reactor $($CSTR$),$ the hydrolysis of $1-$kestose is carried out by $-$ fruktosfuranosidase enzyme $($as catalyst$).1-$kestose solution $($substrate$)$ is fed to the reactor at $20C(T_0)$ with the flow rate of $2\frac{m^3}{\text{m}\text{i}\text{n}}(F_0).$ Enzyme solution is fed to the reactor at $25C(T_1)$ with the flow rate of $3\frac{m^3}{\text{m}\text{i}\text{n}}(F_1).$ The density of substrate and enzyme solution $({}_0$ and ${}_1)$ is $880\frac{kg}{m^3}$ and the heat capacity of substrate and enzyme solution $(C{p}_0$ and $C{p}_1)$

is $1\mathrm{.}2\frac{\text{k}\text{c}\text{a}\text{l}}{kgC}$; and they do not change with the temperature.

Although the enzymatic reaction is carried out at $35C,$ the reaction is not endothermic or exothermic. The required energy for the reaction is provided by saturated steam with the flow rate of $M(\frac{kg}{h})$ passing through a serpentine pipe in the tank. The volume of the tank is $20m^3(V),$ the enthalphy

is $640\frac{\text{k}\text{c}\text{a}\text{l}}{kg}(Hb).$

a$)(10P)$ Find the flow rate of saturated steam at steady state.

b$)(20P)$ Derive the transfer function gives the temperature $T(K)$ change of process stream at the exit of the reactor.

c$)(20P)$ If temperature of the feed temperature $T_1$ rises to $50C,$ find the fluid temperatures at the exit of the

tank at $1,2,4$ and $15$ minutes.