A 1D plane wall is defined as being infinitely long (i.e. heat transfer only occurs in one direction)

Position is denoted by $x$

• $x=0$ at left side of wall
• $x=L$ at right side of wall

Node number is denoted by $m$

• $m=1$ at left side of wall
• $m=M$ at right side of wall

Notation:

• $T_5$: Temperature at $m=5$
• $ \rho_5 $: Density at node $m=5$
• ...

Consider a function $f$ that depends on $x$. The first derivative is then defined as the slope of the tangent:

\( \frac{\text{d}f(x)}{\text{d}x} \) \(= \text{lim}_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\)

Leaving out the limit, we get an approximate solution:

\( \frac{\text{d}f(x)}{\text{d}x} \) \(\approx \frac{f(x+\Delta x)-f(x)}{\Delta x}\)

Remember, we are looking for:

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

We may approximate first derivative at midpoints as:

\( \frac{\partial T}{\partial x} \big|_{x_{m-1/2}} \) \( \approx \frac{T_{m}-T_{m-1}}{\Delta x} \) \( ~~~\text{and} ~~~ \) \( \frac{\partial T}{\partial x} \big|_{x_{m+1/2}} \) \( \approx \frac{T_{m+1}-T_{m}}{\Delta x} \)

... now, we can approximate the second derivative using two first-order derivatives:

\( \frac{\partial^2 T}{\partial x^2} \big|_{x_m} \) \( \approx \frac{\frac{\partial T}{\partial x} \big|_{x_{m+1/2}} - \frac{\partial T}{\partial x} \big|_{x_{m-1/2}}}{\Delta x} \) \(= \frac{\frac{T_{m+1}-T_{m}}{\Delta x} - \frac{T_{m}-T_{m-1}}{\Delta x}}{\Delta x} \)

... which reduces to:

\( \frac{\partial^2 T}{\partial x^2} \big|_{x_m} \) \( \approx \frac{T_{m-1}-2T_{m}+T_{m+1}}{\Delta x^2} \)

The equation for steady one-dimensional heat conduction now becomes:

\( \displaystyle \frac{T_{m-1}-2T_{m}+T_{m+1}}{\Delta x^2} + \frac{\dot{g}}{k}= 0 \)

A 1D plane wall is defined as being infinitely long (i.e. heat transfer only occurs in one direction)

Position is denoted by $x$

• $x=0$ at left side of wall
• $x=L$ at right side of wall

Node number is denoted by $m$

• $m=1$ at left side of wall
• $m=M$ at right side of wall

Volume elements with width $\Delta x$

• Properties are assumed constant within each volume element
• The finer the mesh (smaller $\Delta x$), the more accurate the solution

Notation:

• $T_5$: Temperature at $m=5$
• $ \rho_5 $: Density at node $m=5$
• ...

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 steady heat transfer (no change in time):

$$ \small \sum_\text{all surfaces} \dot{Q} + \dot{G}_\text{element} = \frac{\Delta E_\text{element}}{\Delta t} = 0 $$

• $\dot{G}_\text{element}$ is the heat generation rate in the element (W):

$$ \dot{G}_\text{element} = \dot{g}V_\text{element} $$

• $\dot{g}$ is the volumetric heat generation rate in the element ($\text{W}/\text{m}^3$)

Energy balance on control volume: $$\small kA \frac{T_{m-1}-T_m}{\Delta x}-kA\frac{T_m-T_{m+1}}{\Delta x} + \dot{g}_m A \Delta x=0$$

Conclusion: In both cases, it reduces to the same expression: $$\small T_{m-1}-2T_m+T_{m+1}+\frac{\dot{g}_m\Delta x^2}{kA}=0$$

Energy balance on control volume: $$\small kA \frac{T_{m-1}-T_m}{\Delta x}+kA\frac{T_{m+1}-T_{m}}{\Delta x} + \dot{g}_m A \Delta x=0$$

We have to specify boundary conditions on each non-internal node.

Four common boundary condition types:

1. Constant temperature

$T_0 = c$

2. Constant heat flux

$\dot{Q}_\text{left,surface}=\dot{q}A=c$

3. Convection

$\dot{Q}_\text{left,surface}=hA\left(T_\infty-T_0 \right)=c$

4. Radiation

$\dot{Q}_\text{left,surface}=\epsilon \sigma A \left(T_\text{surr}^4-T_0^4\right)=c$

.. or any combination of type 2, 3 and 4

Type 1 (constant temperature):

• We simply set the temperature of the node, e.g. $T_0 = c$

Type 2-4:

• We apply an energy balance on the volume element:

$$ \small \begin{aligned} \small \sum_\text{all surfaces} \dot{Q} + \dot{G}_\text{element} = 0, \end{aligned} $$

$$\small \dot{Q}_\text{left,surface} + kA \frac{T_1-T_0}{\Delta x}+\dot{g}_0\left(\frac{\Delta x}{2}A\right)=0.$$

• Expression for $\dot{Q}_\text{left,surface}$ depends on boundary condition type type

All BC types (heat flux, convection and radiation) can easily be added to form mixed types:

\( \dot{Q}_\text{left,surface}\) \(= \) \( \dot{q}_0 A \) \( ~+ ~\) \(h A \left(T_\infty-T_0\right) \) \( ~+ ~\) \( \epsilon \sigma A \left(T_\text{surr}^4-T_0^4\right) \)

Insulated boundary (zero heat flux) can be modelled using mirror nodes:

For interior nodes:

\( \displaystyle \frac{T_{m+1}-2T_m+T_{m-1}}{\Delta x^2} + \frac{\dot{g}}{k} = 0 \)

For insulated boundaries:

\( \displaystyle \frac{T_{1}-2T_0+T_{1}}{\Delta x^2} + \frac{\dot{g}}{k} = 0 \)

Starting with the general energy balance:

$$ \displaystyle \begin{aligned} \sum_\text{all surfaces} \dot{Q} + \dot{G}_\text{element} = 0, \end{aligned} $$

Assuming perfect contact (no air gaps) at the interface, the energy balance becomes:

\( \displaystyle k_1 A \frac{T_{m-1}-T_m}{\Delta x_1} + k_2 A \frac{T_{m+1}-T_m}{\Delta x_2}\) \(\displaystyle+ \dot{g}_A A \Delta \frac{x}{2} + \dot{g}_B A \Delta \frac{x}{2}\) \( \displaystyle= 0\)

Double subscripts ($m$,$n$) to denote two dimensions (triple for three dimensions)

• $m$ used for $x$-direction
• $n$ used for $y$-direction

• Temperature at node ($m=1$, $n=0$) is written $T_{1,0}$
• Distances between nodes are $\Delta x$ and $\Delta y$
• Form volume elements around each node
• Convert from node number to actual coordinates by:
• $x=m\Delta x$
• $y=n\Delta y$

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 steady heat transfer (no changes in time):

$$ \small \sum_\text{all surfaces} \dot{Q} + \dot{G}_\text{element} = \frac{\Delta E_\text{element}}{\Delta t} = 0 $$

Sum up heat transfer to/from volume element:

\( \dot{Q}_\text{left,cond} \) \( + \) \( \dot{Q}_\text{right,cond} \) \( + \) \( \dot{Q}_\text{bottom,cond} \) \( + \) \( \dot{Q}_\text{top,cond} \) \( + \) \( \dot{G}_\text{element} \) \( = 0 \)

• .. or in terms of node numbers:

\( \displaystyle k A \frac{T_{m-1,n}-T_{m,n}}{\Delta x}\) \( + \) \( \displaystyle k A \frac{T_{m+1,n}-T_{m,n}}{\Delta x}\)
\( + \) \( \displaystyle k A \frac{T_{m,n-1}-T_{m,n}}{\Delta y}\) \( + \) \( \displaystyle k A \frac{T_{m,n+1}-T_{m,n}}{\Delta y}\) \( + \) \( \dot{g}_{m,n}(\Delta x \Delta y) \) \( = 0 \)

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 a volume element on the left boundary:

\( \dot{Q}_\text{left,surface} \) \( + \) \( \dot{Q}_\text{right,cond} \) \( + \) \( \dot{Q}_\text{bottom,cond} \) \( + \) \( \dot{Q}_\text{top,cond} \) \( + \) \( \dot{G}_\text{element} \) \( = 0 \)

• .. or in terms of node numbers:

$\displaystyle \dot{Q}_\text{left surface}$ $~+~$ $\displaystyle k\Delta y\frac{T_{1,n}-T_{0,n}}{\Delta x}$ $~+~$ $\displaystyle k\frac{\Delta x}{2}\frac{T_{0,n-1}-T_{0,n}}{\Delta y}$
$~+~$ $\displaystyle k\frac{\Delta x}{2}\frac{T_{0,n+1}-T_{0,n}}{\Delta y}$ $~+~$ $ \displaystyle \dot{g}_{0,n}(\frac{\Delta x}{2} \Delta y)$ $~=~$ $0$

• The expression for $\dot{Q}_\text{left surface}$ depends on the type of boundary condition

• Baseline: Approximate the shape using regular grid (stair-step approach)
• Use smaller volume elements near the boundary to better approximate the shape
• Use irregular boundary volume elements that conform to the shape