Imagine that you are an engineer tasked with designing a solar thermal energy system in Madrid, Spain. Madrid is located at a latitude ranging from $40{25}^{\text{'}}$ to $40{34}^{\text{'}}N,$ and a longitude ranging from $3{41}^{\text{'}}$ to $3{47}^{\text{'}}W.$

$($a$)$ Create a sun$-$earth position diagram to illustrate the solar altitude angle and solar azimuth angle. Determine the maximum solar altitude angle that occurs at any time during the year in Madrid. Take the latitude of Madrid at $40N.$

$(7$ marks$)$

$($b$)$ Calculate the incident angle on a vertical surface facing south$-$southwest at $10$:$00$ am $($LST$)$ on January $21$ in Madrid.

$(11$ marks$)$

$($c$)$ If the flat$-$plate collectors are to be considered for a solar thermal heating system in Madrid, sketch the main solar thermal energy losses of a basic single glazing flat$-$plate collector. Discuss briefly about how the solar heat is eventually transferred to the fluid in the collector.

$(7$ marks$)$