1D Steady-State Heat Conduction

It is applied to problems where a temperature gradient exists in only one direction, such as in flat plates, cylinders, and spherical shells. Typical examples include thermal insulation evaluation of walls, design of pipe insulation, and calculation of allowable current for electrical wires.

The governing equation for 1D steady-state with internal heat generation is as follows.

If $k$ is constant and there is no heat generation, it becomes $\frac{d^2T}{dx^2} = 0$, and the temperature distribution becomes linear.

For fixed surface temperatures $T(0)=T_1$, $T(L)=T_2$ and uniform $\dot{q}_v$,

A quadratic curve is superimposed. The maximum temperature is not necessarily at the center; it becomes biased when $T_1 \neq T_2$.

The electrical circuit analogy is very effective for 1D heat conduction. The thermal resistance for a flat plate is

The convective thermal resistance is $R_{conv} = \frac{1}{hA}$. These are connected in series or parallel to estimate the overall temperature drop.

Exactly. This concept of thermal resistance networks also forms the basis for FloTHERM's Compact Thermal Model and JEDEC's DELPHI model.

Analytical solutions cannot be obtained when there is temperature-dependent thermal conductivity $k(T)$ or non-uniform heat generation. Discretizing on a uniform grid using FDM gives:

Here, $k_{i+1/2}$ is evaluated using the harmonic mean at the cell interface. $k_{i+1/2} = \frac{2k_i k_{i+1}}{k_i + k_{i+1}}$

To maintain heat flux continuity at interfaces between different materials. Using the arithmetic mean could lead to discontinuous heat flux at the interface. This treatment is also used in CFD solvers.

In 1D FEM, two-node linear elements are used. The element thermal conductivity matrix is:

It has exactly the same form as a structural spring element. For convective boundaries, add the term $hA\begin{bmatrix}0&0\\0&1\end{bmatrix}$ to the right end.

Yes. For educational purposes, implementing about 10 elements in Excel is very effective for understanding the essence of FEM. In practice, of course, we use general-purpose solvers, but having a custom tool for verification can be very useful.

1D models are overwhelmingly efficient in the conceptual design stage. Determining insulation thickness for walls, selecting pipe insulation, and sizing electrical wires can be done with sufficient accuracy using 1D calculations.

For a steam pipe with an outer diameter of 50mm (150°C) wrapped with glass wool insulation ($k=0.04$ W/(m K)), with ambient air at 25°C and an external surface heat transfer coefficient $h=10$ W/(m²K):

With an insulation thickness of 50mm, $r_i=25$mm, $r_o=75$mm, the heat loss per unit length is approximately 20 W/m.

Exactly. However, pipe elbows, branches, and flange sections exhibit 2D/3D effects, so we use 1D for overall heat loss calculation and 3D for local temperature evaluation.

Using 1D theoretical solutions to verify 3D analysis results is very effective.

Just checking "if the order of magnitude is correct" can prevent 80% of design errors. That is the greatest value of 1D models.

Let's compare implementation methods in general-purpose FEM solvers.