Problem $2$: Electrically Heated Vessel Wall, Adapted from Deen $3-10(30$ POINTS$)$

To prevent heat loss from a fluid$-$filled vessel, it is decided to attach an electrical heater to the outside of the

wall, as shown in Figure $1.$ Passing a current through the heater yields a constant rate of heat generation,

$H_V.$ The bulk temperature and heat transfer coefficient inside the vessel are $T_i$ and $h_i,$ respectively. The

outside values are $T_0$ and $h_0,$ where $x{H}_{V}TT\frac{x}{L_H}\frac{T-T_i}{T_o-T_i}{H}_V{H}_V\frac{L_H^2}{k_H(T_o-T_i)}{T}_0.$ Assume that the steady$-$state temperature $is$ a function $ofx$

only.

Calculate the heating rate that $is$ just sufficient $to$ eliminate any heat loss from the inside $of$ the container.

What are the $limiting$ behaviors $of{H}_V$ and $Tin$ the heater when the Biot number $of$ the heater $is$ very small

and very large? $(Hint$: Describe the $limiting$ behavior $ofTin$ terms $of$ dimensionless variables$\frac{x}{L_H}$ and

$\frac{T-T_i}{T_o-T_i}$ and the $limiting$ behavior $of{H}_{V}in$ terms $of$ dimensionless variable $H_V\frac{L_H^2}{k_H(T_o-T_i)}.)$

Figure $1$: Electrical heater applied $to$ the outside $of$ a vessel.