Consider the crude$-$oil preheater given in Figure below for different control configurations.

a$.$ Define configurations in I, II and III. Comment on all of them $($about difficulties in each and performances$).$

b$.$ If the following steady state energy balance is obtained for this system,

$Q_F=F{C}_p(T-T_i)$

Derive a control law to control exit temperature $T$ so that $T$ attains the desired value $T^{**}$ by measuring the disturbance variable, inlet temperature $T_i$ and by adjusting the heat input $Q_F$ as the manipulated variable above its initial steady state value, i$.$e$.$:

$Q_F=Q_F+K(T^{**}-T)$

Note that a control law is a relationship between the manipulated variable and the error $)=(T^{**}-T$ between the desired value and the controlled variable which can either be measured or can be found$/$calculated$/$estimated using other measured variables.

Define this control system and draw the block diagram.