ρ=8954 Kg ⁄ $m^3$ ,$C_p$=383 J ⁄ KgK,k=386 W ⁄ mK initially at uniform temperature ti=250℃, is suddenly immered in a fluid which is maintained at a uniform temperature ta= 50℃. The heat transfer coefficient between the sphere and the fluid is h=200W⁄($m^2$ K.) Determine the temperature of the copper block at τ=5 min after the immersion.

D=10cm=0.1m,$T_i$=250℃,$T_∞$=50℃,h=200w/$(m^2$ k),t=5min=300sec

First we have to check whether lumped system analysis satisfies or not

Bi= $\frac{hl}{k}=\frac{(200×0.1)}{(386×6)}$=0.0086

Which is less than 0.1 therefore lumped system analysis satisfies

$\frac{(T-T_∞)}{(T_i-T_∞ )}=exp⁡-\frac{ht}{ρlc}$

$\frac{(T-50)}{(250-50)}=exp-\frac{(200×300×6)}{(8954×0.1×383)}$

T=120℃