Problem $3(15$ pts$)$

Given:

A CAV system is used to control the room air at a temperature of $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and relative humidity of $\backslash (50\backslash \%\backslash ).$ The outdoor air temperature and relatively humidity is $\backslash (5^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and $\backslash (20\backslash \%\backslash ),$ respectively. The return air mixes with the fresh air at a mass flow rate ratio of $\backslash (4$: $1\backslash )($return air: fresh air$)$ before entering the air handling unit. A heating coil followed by a humidifier is used. The total air mass flow rate is $1\backslash (\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$s$\}\backslash ).$ The supply air temperature is $\backslash (30^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ and relative humidity is $\backslash (40\backslash \%\backslash ).$ The pressure is a constant at $101\mathrm{.}3$ kPa $.$

Find:

$(1)$ The temperature and relative humidity of the air entering the heating coil. $(4$ pts$)$

$(2)$ The heat transfer rate at the heating coil, in kW $.(7$ pts$)$

$(3)$ The mass flow rate of water vapor added to the air by the humidifier. $(4$ pts$)$