1. Transient Conduction through Semi-Infinite Solid

Recall Differential Form of Heat Transfer Equation:

$$ \rho c_p \left[ \frac{\partial T}{\partial t} + (\cancel{\vec{v}} \cdot \vec \nabla) T \right]

k\vec \nabla^2T + \cancel{\dot S_v} $$

• The second term is cancelled as NO Advective Flow within the solid
• The final term is cancelled as NO Heat Generation

Therefore:

$$ \frac{\partial T}{\partial t}

\alpha \frac{\partial^2 T}{\partial x^2} $$

where:

$$ \alpha = \frac{k}{\rho c_p} $$

Initial Condition:

At $t = 0$, $T = T_i \quad \forall x\ge 0$

Boundary Conditions:

At $x = 0$, $T=T_s \quad \forall t\ge 0^+$

At $x\to \infty$, $T\to T_i \quad \forall t\ge 0^+$

Variable Similarity / Order of Magnitude Analysis

Self-Similarity

<aside> 💡 Over time, the temperature profile is simply the same shape but ‘stretched’ in the x-direction - SELF-SIMILARITY

</aside>

For:

$$ \frac{\partial T}{\partial t}

\alpha \frac{\partial^2 T}{\partial x^2} $$

Normalise the variables:

$$ T^* = \frac{T}{T_c}\\ t^* = \frac{t}{t_c}\\ x^* = \frac{x}{x_c} $$

• subscript $c$ means characteristic quantity, a normalisation factor
• superscript $*$ means non-dimensionalised (normalised) quantity

Therefore:

$$ \frac{T_c}{t_c}\frac{\partial T^}{\partial t^}

\alpha \frac{T_c}{x_c^2}\frac{\partial^2T^*}{\partial x^{*2}}\\

\frac{1}{t_c}\frac{\partial T^}{\partial t^}

\alpha \frac{1}{x_c^2}\frac{\partial^2T^*}{\partial x^{*2}} $$

Choose characteristic quantities to be same order of magnitude as real quantities. Then:

$$ T^* \thicksim \, t^* \thicksim \, x^* \thicksim \,0(1) \qquad \text{Order of magnitude similar as 1}\\ \therefore \frac{\partial T^}{\partial t^} \thicksim \, \frac{\partial^2T^*}{\partial x^{*2}} \thicksim \, 0(1) $$

$$ \frac{1}{t_c}\frac{\partial T^}{\partial t^}

\alpha \frac{1}{x_c^2}\frac{\partial^2T^*}{\partial x^{*2}} \quad \to \quad \frac{1}{t_c}\thicksim \, \frac{\alpha}{x_c^2} \quad \to \quad x_c \thicksim \, \sqrt{\alpha t_c} $$

Units: $[m]$