Chapter 10, Problem 10.4P

(a)

Interpretation Introduction

Interpretation:

The time taken for the significant change in the temperature at the sheet centerline is to be calculated.

Concept Introduction:

For a solid that is semi-infinite with the penetration distance $x_p$, the time required for the arbitrary change of 1 percent in the surface temperature is calculated as:

  $x_p=3.64\sqrt{\alpha t}$   ....... (1)

Here, $\alpha$ is the thermal diffusivity of the solid.

(b)

Interpretation Introduction

Interpretation:

The bulk average temperature of the plastic at the time calculated in part (a) is to be calculated.

Concept Introduction:

For a semi-infinite plate, the bulk temperature of the slab when it is heated from both the sides, is calculated by the formula:

  $\frac{T_s-{\overline{T}}_b}{T_s-T_a}=\frac{8}{{\pi }^2}\left[e^{-a_1\text{Fo}}+\frac{1}{9}{e}^{-9a_1\text{Fo}}+\frac{1}{25}{e}^{-25a_1\text{Fo}}+\cdots \right]$   ....... (2)

Here,

  $\begin{array}{c}{T}_s=\text{ Constant slab surface temperature}\\ T_a=\text{ Initial slab temperature}\\ {\overline{T}}_b=\text{ Bulk temperature}\\ \text{Fo}=\text{Fourier number}=\alpha t_T/s^2\\ \alpha =\text{ Thermal diffusivity of the slab}\\ t_T=\text{ Time of heating or cooling}\\ s=\text{ Half of the slab thickness}\\ a_1={\left(\pi /2\right)}^2=2.467\end{array}$