Problem #$3$

$6$ points

A very long, $1-$cm$-$diameter copper rod is exposed to ambient air at $22C.$ One of the rod's ends is

attached to a hot$-$plate maintained at $150C.$ The heat transfer coefficient between the surface of

the rod and the air is $11\frac{W}{m^2}C,$ and the thermal conductivity of the rod is $377\frac{W}{m}C.$

Determine:

a$)$ The heat transfer rate from the rod to the surrounding air $(1$ point$)$

b$)$ Assuming a rod length of $30$ cm $,$ and a convective tip with heat transfer coefficient of $11$

$\frac{W}{m^2}C,$ determine the heat transfer rate from the rod to the surrounding air $(2$ points$)$

c$)$ Repeat case b$)$ using a rod length of $15$ cm $(1$ point$)$

d$)$ What rod length would you choose, if your goal is to optimize the heat transfer vs$.$ rod

length? Justify your answer $(2$ points$)$