A steel ball of 10 mm diameter at 1000 K is required to

A steel ball of 10 mm diameter at 1000 K is required to be cooled to 350 K by immersing it in a water environment at 300 K. The convective heat transfer coefficient is 1000 W/m²K. Thermal conductivity of steel is 40 W/m K. The time constant for the cooling process is 16 s. The time required (in s) to reach the final temperature is

Biot Number =	hL_c	= 0.458
	K

For sphere L_c =	Volume	=	d
	surface area		6

∴ Bi =	hd	=	1000 × 0.01
	6K		6 × 40

= 0.0416 < 0.1
Hence lumped heat analysis is used

	T - T_∞	= exp		-hA_st		= e	-t
	T_i - T_∞			_{ρ V C_p}			^ṫ

Thermal time constant,

t^* =	ρ V C_p	= 16 sec
	hA_s

∴	350 - 300	= e	_-t
∴	1000 - 300	= e	₁₆

⇒ t = 42.2249 sec