In a materials$-$processing experiment on a space station, a $1$ cm $-$diameter sphere of

alloy is to be cooled from $600$ K to $400$ K $.$ The sphere is suspended in a test chamber

by three jets of nitrogen at $300$ K $.$ The equivalent nitrogen jet velocity is approximately

at $5\mathrm{.}8($m$)/($s$).$ Take the alloy density to be $\backslash$rho $=14,000$k$($g$)/($m$^(3)),$ specific heat c$=140($J$)/($k$)$gK$,$

and thermal conductivity k$=240($W$)/($m$)$K$.$ Since the emittance of the alloy is very small,

the radiation contribution to heat loss can be ignored. Calculate the time required for

the cooling process and the minimum quenching rate, $($d$($T$)/($d$)$t$)\_($min$).(1^(\text{'})).$

Nitrogen properties at $400$ K :

k$=0\mathrm{.}0326($W$)/($mK$),\backslash$rho $=0\mathrm{.}854$k$($g$)/($m$^(3)),$c$=1047($J$)/($kgK$),\backslash$mu $=21\mathrm{.}5*10^(-6)$k$($g$)/($ms$),$Pr$=0\mathrm{.}69()/($b$)$

ar $(\backslash$Nu $\_($D$))=2+(0\mathrm{.}4$Re$\_($D$)^((1)/(2))+0\mathrm{.}06$Re$\_($D$)^((2)/(3)))$Pr$^(0\mathrm{.}4),3\mathrm{.}5$