A$1.$ A large cylindrical vessel is used as a batch reactor and is perfectly mixed. The liquid inside is heated to keep the reactants at an elevated temperature. Heat is lost through the portion of the vessel wall that is in contact with its liquid contents, corresponding to an area of $25m^2.$ The overall heat$-$transfer coefficient, $U,$ is $200W{m}^{-2}{K}^{-1}.$

The temperature of the ambient surroundings is $20C,$ the mass of liquid in the tank is ${10}^{4}kg,$ and its specific heat capacity is $3000Jk{g}^{-1}{K}^{-1}.$ There is no flow into or out of the tank. The contribution of any chemical reactions to the heat balance is very small, and should be neglected.

$($a$)$ Determine the rate of heating required to keep the contents of the tank at a steady$-$state temperature of $100C.$

$[4$ marks$]$

$($b$)$ At a given moment $($at $t=0),$ the rate of heat input to the tank undergoes a step$-$increase by $10\%($due to an increased flow of steam$).$ The ambient temperature has not changed, nor has the overall heat$-$transfer coefficient.

i$.$ Formulate a dynamic model of the system, take deviation variables, and take Laplace transforms to obtain an expression with the following general form:

$T^{**}(s)=\frac{K_d}{s+1}{Q}^{**}(s)$

Evaluate the constants $$ and $K_d.$

$[8$ marks$]$

ii$.$ Find a time$-$domain expression for the dynamic response of the output variable $T^{**}(t)$ after the step$-$change in heat input and thus, determine how much time has elapsed when the temperature of the tank contents reaches $102C.$

$[8$ marks$]$

Note, you may use the identity, $L^{-1}(\frac{1}{s+1}\frac{1}{s})=1-$exp$(-\frac{t}{})$