Consider a double$-$glazed window. The frame has a thermal resistance of $0\mathrm{.}35($m$2$ K$)/$W$,$ while the glazed section is made of two panels of glass with a thickness of $6$ mm and a thermal conductivity of $1$ W$/($m K$).$ The cavity between the two panels is $16$ mm thick and is filled with Argon $(\backslash$lambda $=0\mathrm{.}018),$ a gas which can be considered still. The thermal bridge between glass and frame is $0\mathrm{.}02$ W$/($m K$),$ the frontal dimension of the glass are $0\mathrm{.}6$ x $1\mathrm{.}2$ m and the frame has a width around the glass of $9$ cm$.$ Assume surface heat transfer coefficient equal to $25$ W$/($m$2$ K$)$ for the outdoor surface and $8$ W$/($m$2$ K$)$ for the indoor surface.

Determine:

$1.$ The transmittance of the glass assuming that the radiative heat transfer coefficient in the cavity is $0\mathrm{.}25$ W$/($m$2$ K$)$

$3.$ The surface temperature of the frame, when the indoor temperature is $20\backslash$deg C and the outdoor temperature is $0\backslash$deg C

$4.$ The maximum relative humidity which can be accepted in the indoor environment to avoid surface condensation on the window.