$3\mathrm{.}4\mathrm{.}10$ Figure P$3\mathrm{.}4\mathrm{.}10$ illustrates a system and fixed control volume at time $t$ and the system at a short time $t$ later. The system temperature is $T=310\mathrm{.}9K$ at time $t$ and $T=312\mathrm{.}6K$ at time $t+t,$ where $t=0\mathrm{.}1s.$ The system mass, $m,$ is $29\mathrm{.}2$ kg $,$ and $10$ percent of it moves out of the control volume in $t=0\mathrm{.}1s.$ The energy per unit mass $u^{}$ is $c_{v}T,$ where $c_v=1\mathrm{.}34{10}^5\frac{J}{k}g*K.$ The energy $U$ of the system at any time $t$ is mtilde$(u).$ Use the system or Lagrangian approach to evaluate $D\frac{U}{D}t$ of the system. Compare this result with the $D\frac{U}{D}t$ evaluated with that of the material time derivative and flux terms.

System

I has $10\%$ of system mass at time $t.$

Il has $90\%$ of system mass at times $t$ and $t+t.$

III has $10\%$ of system mass at time $t+t.$

at time $t+t$