Can you please solve these tasks step by step $???$ a$)$ A solar heating system produces hot water for a family house $.$ Derive a mathematical model which estimates the heat flux $($per collector area$)$ to water that flows in the flat absorber plates in a vacuum collector with a single glass, for a given incident radiation flux, qs$.$ b$)$ Implement the model as a computer program and apply it to the data in Table $1$ to estimate the recovered heat during the two days reported. The collector surface area is $5$ m$2$ and $0\mathrm{.}25$ kg$/$s water is fed to the system at $50$C$.$ c$)$ How is the heat flow to the water affected if the feed water temperature is raised to $60$C$?$ In the modeling the following simplifying assumptions can be made: $$ The incident short$-$wavelength radiation is mainly transmitted through the glass $(=0\mathrm{.}85,=0\mathrm{.}1)$ while the background radiation from the sky as well as the radiation emitted and reflected by the absorber plate is not transmitted through the glass $(=0,=0\mathrm{.}95).$ For the absorber plate $=0,=0\mathrm{.}15.$ The convective heat transfer coefficient is $$conv $=10$ W$/($m$2$K$).$ The collector is large enough to make it justified to neglect end effects and there are negligible heat losses from the collector bottom $$ The thermal resistance between the absorber plate and the water may be neglected. Table $1$ Incident radiation and ambient temperature during two days. Incident radiation flux, qs W$/$m$2$ Hour Day $1$ Day $27-8308-9809-1013010-114020011-128028012-1311038013-1414050014-1517061015-1610055016-177050017-185045018-1933019-2022020-2111021-226022-2330$ T$0($K$)270285$ Tsky $($K$)247268$