1. Transient Heat Conduction

• Consider a planar wall at an initial uniform temperature $T_i$
• At t=0, wall is exposed to fluid at Temperature $T_\infty$
• Time-dependent changes in wall temp from $T_i$ to $T_\infty$

<aside> 💡 How can we describe these time-dependent changes??

• Solve heat equation (PDE)
• Lumped Capacitance Method </aside>

2. Lumped Capacitance Method

If $Biot(H.T.)\ll1$

• Conduction within solid is very fast
• Convection from surface is very slow

<aside> 💡 Temperature is nearly uniform within solid at each point in time.

</aside>

This can simplify transient problems:

$$ T(x,t) \approx T(t) $$

Recall the Integral Form of Heat Transport:

$$ \frac{\partial}{\partial t}\int\rho c_p T dV

\cancel{\int_\text{CV}\dot S_V dV}

\int_\text{CS}(\vec q \cdot \hat n)dA

\cancel{\int_\text{CS}\rho c_p T(\vec v \cdot \hat n)dA} $$

• The second term is cancelled as NO heat generation
• The final term is cancelled as NO fluid motion within CV

$$ \frac{\partial}{\partial t}\int\rho c_p T dV

\int_\text{CS}(\vec q \cdot \hat n)dA =-q_hA $$

As $\rho$, $c_p$ and $T$ are uniform within CV, only $T$ changes with time:

$$ \rho c_p V \frac{\partial T}{\partial t} = -q_hA $$

Therefore,

$$ \rho c_p V\frac{\partial T}{\partial t} = -hA(T-T_\infty) $$

In words:

$$ \text{Rate of change of Thermal Energy} = \text{Convective Heat Transfer from surface} $$

Rearrange:

$$ \frac{\partial T}{\partial t} = -\frac{hA}{\rho c_pV}(T-T_\infty) $$

Here, the constant term in words:

$$ \frac{hA}{\rho c_pV} = \frac{\text{Conduction Conduvtivity} (hA)}{\text{Heat Capacitance} (\rho c_p V)} $$