Question

A thick plate made of stainless steel is initially at a uniform temperature of 300 C. The surface is suddenly exposed to a coolant at 20 C with a convective surface coefficient of 110 W/m2⋅K. Evaluate the temperature after 3 min of elapsed time
(a) at the surface
(b) a depth of 50 mm
Work this problem both analytically and numerically.

Answer 1

We convert from degrees Celsius to degrees Kelvin,

• Initial Temperature $T_i = 300\° C =573.15K$
• Coolent Temperature $T_{\infty}=20\°c  = 293.15K$

Convective temperature coefficient, $h=110W/m2-K$

For steel we have to,

$\rho = 8055kg/m$

$C_p = 480J/Kg-K$

$k=15.1$

$\alpha = 3.91*10^{-6}m^2/s$

Given the error equation, then

$\frac{T-T_i}{T_{\infty}-T_i}= e(\frac{x}{2\sqrt{\alpha t}})$

A)

At x=0

$\frac{T-T_i}{T_{\infty}-T_i}=e(0)$

From the tables,

$e(0)=1$

$\frac{T-T_i}{T_{\infty}-T_i}=1$

$T=T_{\infty}=20\° C$

B)

At $x=50mm=0.5m$

$\frac{T-T_i}{T_{\infty}-T_i}= e(\frac{x}{2\sqrt{\alpha t}})$

$\eta=\frac{x}{2\sqrt{\alpha t}}=\frac{0.05}{2\sqrt{3.91*10^{-6}*60}} = 1.63$

At this value

$e(1.63)=0.02196$

$\frac{T-300}{20-300}=0.02196$

$T=293.85 \°c$