Question

A 10-cm-thick aluminum plate (α = 97.1 × 10−6 m2/s) is being heated in liquid with temperature of 475°C. The aluminum plate has a uniform initial temperature of 25°C. If the surface temperature of the aluminum plate is approximately the liquid temperature, determine the temperature at the center plane of the aluminum plate after 15 s of heating. Solve this problem using the analytical one-term approximation method.

Answer 1

Answer:

$T_0 = 338.916 Degree\ celcius$

Explanation:

Given data:

Thickness of aluminium sheet 10 cm

initial temperature = 25 degree celcius

Assumption

Thermal properties remain constant, transfer of heat by radiation is negligible.

from the information given in the question we have

T_S ≈T_∞ , it implies we have h → ∞

from table 4.2 Biot number → ∞ the value of

$\lambda_1 = 1.5708 and A_1= 1.2732$

The fourier number is

$t = \frac{\alpha t}{l^2} = \frac{97.1\times 10^{-6} \times 15}{0.05^2} = 0.5826$

Temperature at center after 15 second of heating

$\theta _{0 wall} = \frac{T_0 - T_{\infity}}{T_i -T_{\infity}} = A_i e^{\lambda_1^2 t}$

$T_0 = T_i -T_{\infity} \times A_i e^{\lambda_1^2 t}$

$T_0 = (25 - 475) 1.2732 e^{-1.5708^2 \times 0.5826} +  475  = 356 degree celcius$

$T_0 = 338.916 Degree\ celcius$