Question $2(50$ points$)$: Let$$s suppose that, for the first experiment of your thesis, you are obligated to analyze the solubility of colemanite $($Ca$2$B$6$O$115($H$2$O$))$ in different temperatures. In order to stabilize the temperature of your solvent, water, you are using a heating wire. Neglect the curve of the wire and the change of its thermal conductivity with temperature. Also, assume that enough time has passed for the system to reach steady$-$state, and the outer surface of the wire has the same temperature with water. Note that, the condition for outer surface of the wire is uniform through its length and therefore the temperature gradience along $-$z direction is negligible. a$)$ Suggest your assumptions for this system. Derive an expression for the differential equation that represents the temperature profile along $-$r direction, T$($r$).(15$ pt$)$ b$)$ State your boundary conditions. Find the constants of the temperature profile by substituting these boundary conditions into the equation. $(15$ pt$)$ c$)$ Suppose that the heating wire is stainless steel $($AISI $302),$ has a $4$ cm radius, $50$ cm length and provides $12$ kW of energy. The temperature of water is $30$C degrees. Find the temperature value in $$C at r$=$R$/2.($Hint: You need to find thermal conductivity constant, kss$,$ for AISI$302$ material by yourself$)(15$ pt$)$ d$)$ In a heat transfer perspective, would it make more sense to use a carbon steel wire? Only comment, do not calculate. $(5$ pt$)$