In this problem we consider the heat transfer that takes place between a travel mug featuring vacuum insulation and its surrounding. We can model the mug as shown in the below figure. Given that point A is located in the volumetric center of the travel mug, point B located at the at the bottom of the countertop vertically from A$,$ and point C a distance d from the outer layer of the mug horizontally from A$.$ Your travel mug, made with a polymer of thermal conductivity k$,$ has a radius r$,$ height H$,$ and a wall thickness t that is the same for the inner, vacuum, and outer layers. Additionally, assume that the lid of your mug is perfectly insulating. The air surrounding the bucket has a temperature T$\backslash$infty and a heat transfer coefficient of ha$.$ The countertop is $3$ cm thick and has a uniform temperature Tcounter as well as a thermal conductivity of kcounter. If there is steady state heat transfer, $($assume$,$ for the purpose of the problem, that there is no heat transfer or activity in the corner patterned areas$).$

a$)$ Draw two resistance diagrams for A $->$ B and A $->$ C

b$)$ List the BCs and IC$($s$)$ and set up the mathematical equations to find the rate of heat transfer into the bucket.

c$)$ Now given that the mixture is $70\%$ water and $30\%$ ice, as well as the numerical values of the variables stated in the problem, what is the total rate of energy absorption into the bucket?

r $=3$ cm

H $=23$ cm

k $=1$ W$/$mK

k counter $=\mathrm{.}2$ W$/$mK

t $=\mathrm{.}25$ cm

T counter $=30$ C

tcounter $=3$ cm

T infinity $=30$ C

ha $=2\mathrm{.}5$ W$/$m$^2$K

d $=2$ cm