Problem $1.$ For a VAV system, the supply air temperature is controlled at a setpoint of $7C$ using

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

outdoor air is at $35C$ 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 $4k\frac{g}{s}.$ The supply chilled water temperature is a constant of $5C,$ and

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

constant of $4\mathrm{.}2k\frac{J}{k}gK.$

$(1)$ When the dry air mass flow rate is $1k\frac{g}{s},$ find the most suitable value of $u_0$ under this working

condition;

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

supply air temperature when the dry air mass flow rate is $1\mathrm{.}5k\frac{g}{s}.$ Please using the energy balance to calculate the Q instead of just using the specifice heat of air