Problem $1$: 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$.$

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

coefficient.

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