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

`(T-T_i)/(T_(\infty)-T_i)= e((x)/(2√(\alpha t)))`

A)

At x=0

`(T-T_i)/(T_(\infty)-T_i)=e(0)`

From the tables,

`e(0)=1`

`(T-T_i)/(T_(\infty)-T_i)=1`

`T=T_(\infty)=20\° C`

B)

At
`x=50mm=0.5m`

`(T-T_i)/(T_(\infty)-T_i)= e((x)/(2√(\alpha t)))`

`\eta=(x)/(2√(\alpha t))=\frac{0.05}{2\sqrt{3.91*10^(-6)*60}} = 1.63`

At this value

`e(1.63)=0.02196`

`(T-300)/(20-300)=0.02196`

`T=293.85 \°c`