$1$ Task: Evaporation and Condensation $(20$ Points$)$

A new process for computer cooling should be investigated. A chip is connected to a closed, cup$-$shaped vessel in which a cooling fluid $($grey fluid in the sketch below$)$ at saturation temperature was filled. The dissipated heat flux $q^{}$ of the chip is removed by evaporation of the cooling fluid at the bottom of the vessel and subsequent condensation at the thin vessel wall to the cooling water. There are no temperature differences in the vessel wall where condensation takes place. The curvature of the vessel has no influence on the heat transfer or the condensate thickness

Calculate generally and numerically for a steady state condition:

a$.$ Calculate the chip temperature at the bottom of the vessel $T_B$ for a given heat flux which is $90\%$ of the maximum heat load $q_{\text{m}\text{a}\text{x}}^{}.$ How much heat is transferred $(Q^{}).$

b$.$ Determine the height of the condensator which is necessary to remove this heat flow $Q^{}.$

c$.$ Calculate the mean velocity $u$ of the condensate film at position $x=0,5H.$

Given parameters and properties: