metal pellets $($length $10$ mm$,$ diameter $10$ mm$),$ initially at $300\backslash$deg C$,$ are quenched by immersing them in a liquid coolant bath at $30\backslash$deg C$.$ The pellets are made of copper $($thermal conductivity k $=400$ W$/$mk$,$

density p $=8900$ kg$/$m$\text{'},$ specific heat c $=385$ J$/$kg K$).$ The coefficient of convective heat transfer is $45$ W$/$m$\text{'}$K$.$

The pellets are kept immersed in the bath for $10$ minutes.

$($a$)$ Check if the 'lumped mass' assumption is valid for this cooling process.

$($b$)$ Calculate the time constant t $($sec$)$ for this cooling process.

$($c$)$ Determine the temperature $(\backslash$deg C$)$ of the pellets after $10$ minutes.

State clearly any assumptions you may make.