Heat Transfer

Transient heat conduction using numerical methods

Lecturer: Jakob Hærvig

Notation for transient problems

  • On space axis $m,n,k$ denote node number in $x,y,z$-directions
    • Distance between nodes is $\Delta x$
  • On time axis $i$ denote node number in $t$-direction (time)
    • Time between nodes is $\Delta t$
  • General notation:

    • $$ T_{m,n,k}^i $$

The energy balance equation for transient problems

We sum up heat transfer to/from volume element:

    $$ \small \left( \begin{array}{ccc} \text{Sum of heat}\\ \text{transfer rate}\\ \text{across surfaces}\\ \text{of volume element} \end{array} \right) + \left( \begin{array}{ccc} \text{Rate of heat}\\ \text{generation}\\ \text{inside the}\\ \text{element} \end{array} \right) = \left( \begin{array}{ccc} \text{Rate of change}\\ \text{of the energy}\\ \text{content of}\\ \text{the element} \end{array} \right) $$

  • For transient heat transfer (temperature changes with time):

    $$ \small \sum_\text{all surfaces} \dot{Q} + \dot{G}_\text{element} = \frac{\Delta E_\text{element}}{\Delta t} $$

    $$ \small \sum_\text{all surfaces} \dot{Q} + \dot{G}_\text{element} = m c_p \frac{\Delta T_\text{element}}{\Delta t} $$

    $$ \small \sum_\text{all surfaces} \dot{Q} + \dot{G}_\text{element} = V_\text{element} \rho c_p \frac{\Delta T_\text{element}}{\Delta t} $$

    • Explicit and implicit solution methods

    Question is: How do we advance from time $t$ (timestep $i$) to timestep $t+\Delta t$ (timestep $i+1$)?

    Explicit method

    \(\displaystyle \sum_\text{all surfaces} \textcolor{#008000}{Q^i} + \textcolor{#008000}{\dot{G}_\text{element}^i} \) \(\displaystyle =\) \(\displaystyle \frac{\rho V_\text{element} c_p (T_m^{i+1}-T_m^i)}{\Delta t} \)

    Implicit method

    \(\displaystyle \sum_\text{all surfaces} Q^{i+1} + \dot{G}_\text{element}^{i+1} \) \(\displaystyle =\) \(\displaystyle \frac{\rho V_\text{element} c_p (T_m^{i+1}-T_m^i)}{\Delta t} \)

    Explicit: Use data from current time step $i$ (all known temperatures)

    Implicit: Use data from next time step $i+1$ (unknown temperatures)

    Interior nodes using explicit solution method

    Energy balance at interior node:

      $$\begin{aligned} kA\frac{\textcolor{#008000}{T_{m-1}^i}-\textcolor{#008000}{T_{m}^i}}{\Delta x} + kA\frac{\textcolor{#008000}{T_{m+1}^i}-\textcolor{#008000}{T_{m}^i}}{\Delta x}+ \dot{g}_\text{m}^iA\Delta x=\frac{\rho V c_p (T_m^{i+1}-T_m^i)}{\Delta t} \end{aligned}$$
    • Re-arranging and inserting thermal diffusivity $\alpha=k/(\rho c_p)$:
      • $$\begin{aligned} \textcolor{#008000}{T_{m-1}^i}-2\textcolor{#008000}{T_{m}^i}+\textcolor{#008000}{T_{m+1}^i} + \frac{\dot{g}_m^i \Delta x^2}{k} = \frac{\Delta x^2}{\alpha \Delta t}\left(T_m^{i+1}-T_m^i\right) \end{aligned}$$
    • Inserting mesh Fourier number $\tau=\alpha\Delta t/\Delta x^2$:
      • $$\begin{aligned} \textcolor{#008000}{T_{m-1}^i}-2\textcolor{#008000}{T_{m}^i}+\textcolor{#008000}{T_{m+1}^i} + \frac{\dot{g}_m^i \Delta x^2}{k} = \frac{1}{\tau}\left(T_m^{i+1}-T_m^i\right) \end{aligned}$$
    • Isolating the only unknown $\textcolor{#b52025}{T_m^{i+1}}$:
      • $$\begin{aligned} \textcolor{#b52025}{T_m^{i+1}} = \tau(T_{m-1}^i+T_{m+1}^i)+(1-2\tau)T_m^i+\tau\frac{\dot{g}_m^i \Delta x^2}{k} \end{aligned}$$
    • 1 equation for each node with only 1 unknown each 🤩

    Boundary nodes using explicit solution method

    • Energy balance for boundary node with convection:
      • $$\begin{aligned} hA(T_\infty-\textcolor{#008000}{T_0^i}) + kA\frac{\textcolor{#008000}{T_1^i}-\textcolor{#008000}{T_0^i}}{\Delta x}+ \dot{g}_\text{0}^iA\frac{\Delta x}{2}=\frac{\rho V c_p (T_0^{i+1}-T_0^i)}{\Delta t} \end{aligned}$$
    • Re-arranging and noting thermal diffusivity $\alpha=k/(\rho c_p)$:
      • $$\begin{aligned} \left(\frac{-2h\Delta x}{k}-2\right)\textcolor{#008000}{T_0^i}+2\textcolor{#008000}{T_{1}^i}+\left(\frac{-2h\Delta x}{k}\right)T_\infty+\dot{g}_0^i\frac{(\Delta x)^2}{k} = \frac{(\Delta x)^2}{\alpha \Delta t}\left(T_0^{i+1}-T_0^i\right) \end{aligned}$$
    • Defining mesh Fourier number $\tau=\alpha\Delta t/\Delta x^2$:
      • $$\begin{aligned} \left(\frac{-2h\Delta x}{k}-2\right)\textcolor{#008000}{T_0^i}+2\textcolor{#008000}{T_{1}^i}+\left(\frac{-2h\Delta x}{k}\right)T_\infty+\dot{g}_0^i\frac{(\Delta x)^2}{k} = \frac{1}{\tau}\left(T_0^{i+1}-T_0^i\right) \end{aligned}$$
    • Isolating the only unknown $\textcolor{#b52025}{T_m^{i+1}}$:
      • $$\begin{aligned} \textcolor{#b52025}{T_0^{i+1}} = \left(1-2\tau-2\tau \frac{h \Delta x}{k}\right)\textcolor{#008000}{T_0^i}+(2\tau)\textcolor{#008000}{T_1^i}+2\tau\frac{h\Delta x}{k}T_\infty+\frac{\tau\dot{g}_0^i(\Delta x)^2}{k} \end{aligned}$$
    • 1 equation for each boundary node with 1 unknown 😁