For a VAV system, the supply air temperature is controlled at a setpoint of $7$ oC using

the proportional control with a Kp of $0\mathrm{.}5$ per K$.$ The room air is controlled at $24$ oC and $60\%.$ The

outdoor air is at $35$ oC and $70\%.$ The mass flow rate ratio of fresh air to return air is a constant of

$1$:$4($fresh air : return air$).$ The mass flow rate of chilled water is linear to the water valve opening

with maximum capacity of $4$ kg$/$s$.$ The supply chilled water temperature is a constant of $5$ oC$,$ and

the return chilled water temperature is a constant of $12$ oC$.$ The specific heat of liquid water is

constant of $4\mathrm{.}2$ kJ$/$kgK$.$

$(1)$ When the dry air mass flow rate is $1$ kg$/$s$,$ find the most suitable value of uo under this working

condition;

$(2)$ Use the uo determined in question $(1)$ in the proportional control, calculate the offset of the

supply air temperature when the dry air mass flow rate is $1\mathrm{.}5$ kg$/$s$.$