Answer:
t = 4.0 min
Explanation:
given data:
diameter of rod = 2 cm
T_1 = 100 degree celcius
Air stream temperature = 20 degree celcius
heat transfer coefficient = 200 W/m2. K
WE KNOW THAT
copper thermal conductivity = k = 401 W/m °C
copper specific heat Cp = 385 J/kg.°C
density of copper = 8933 kg/m3
charateristic length is given as Lc
[tex]Lc = \frac{V}{A_s}[/tex]
[tex]Lc = \frac{\frac{\pi D^2}{4} L}{\pi DL}[/tex]
[tex]Lc = \frac{D}{4}[/tex]
[tex]Lc = \frac{0.02}{6} = 0.005 m[/tex]
Biot number is given as [tex]Bi = \frac{hLc}{k}[/tex]
[tex]Bi = \frac{200*0.005}{401}[/tex]
Bi = 0.0025
As Bi is greater than 0.1 therefore lumped system analysis is applicable
so we have
[tex]\frac{T(t) - T_∞}{Ti - T_∞} = e^{-bt}[/tex] ............1
where b is given as
[tex]b = \frac{ hA}{\rho Cp V}[/tex]
[tex]b = \frac{ h}{\rho Cp Lc}[/tex]
[tex]b = \frac{200}{8933*385*0.005}[/tex]
b = 0.01163 s^{-1}
putting value in equation 1
[tex]\frac{25-20}{100-20} = e^{-0.01163t}[/tex]
solving for t we get
t = 4.0 min
