Question

The convection coefficient for flow over a solid sphere may be determined by submerging the sphere, which is initially at 25 °C, into the flow, which is at 75 °C, and measuring its surface temperature at some time during the transient heating process. If the sphere has a diameter of 0.1 m, a thermal conductivity of 22 W/(m·K), and a thermal diffusivity of 9.0 ×10-5 m2/s, at what time will a surface temperature of 60 ºC be recorded if the convection coefficient is 300 W/(m2·K)?

Answer 1

Answer:

t= 17 .1 s

Explanation:

Given that

Ti= 25 C

T∞= 75°C

T= 60 ºC

K=22 W/(m·K),

$h=300 W/m^2.k$

$\alpha = 9\times 10^{-5}\ m^2/s$

d= 0.1 m

So for sphere  

Lc= d/6 = 0.0166 m

We know that

$Bi=\dfrac{hL_c}{K}$

$Bi=\dfrac{300\times 0.0166}{22}$

Bi = 0.22

$Fo=\dfrac{\alpha t}{L_c^2}$

$Fo=\dfrac{9\times 10^{-5}\times t}{0.0166^2}$

Fo = 0.32 t

Lets take

θo= Ti - T∞ =25 - 75 = 50 °C

θ = T-T∞ = 60 -75 = 15  °C

We know that

$\theta =\theta _oe^{{-Bi.Fo}}$

$15 =50e^{{-0.22\times 0.32t}}$

by solving this t= 17 .1 s