Problem $5$: A $2-$cm diameter copper sphere with a thermocouple at its center is suddenly

immersed in liquid nitrogen contained in a Dewar flask. The temperature response is

determined using a digital data acquisition system that records the temperature every $0\mathrm{.}05$

s$.$ The maximum rate of temperature change dT$/$dt is found to occur when T $=92\mathrm{.}5$ K$,$

with a value of $19\mathrm{.}8$ K$/$s$.$ The saturation temperature of nitrogen at $1$ atm pressure is $77\mathrm{.}4$

K$.$ Take $=8930$ kg$/$m$3,$ c $=235$ J$/($kg K$),$ and k $=450$ W$/($m K$)$ for copper at $92\mathrm{.}5$ K$.$

$2/2$

$1)$ Using the lumped thermal capacity model, determine the corresponding heat

transfer coefficient.

$2)$ Check the Biot number to ensure that the model is valid