Heat Conduction Through a Wall

In this tutorial, we study steady-state heat flow through a wall. We can simplify this problem to one dimension considering that the wall's width and height are much larger than its thickness, and the temperature varies only across the thickness direction.

Mathematical Formulation

Steady heat conduction in one dimension is described by the Laplace equation: \(\frac{d^2T}{dx^2} = 0\), where \(T\) signifies the temperature values. The above is used to find the temperature distribution within common engineering components like house walls.

The above schematic illustrates the problem domain and outlines the associated boundary conditions. Our objective is to determine the temperature distribution within the wall itself. In this problem, a convection boundary condition, implemented as Robin type, is applied at the inner side of the wall (\(x = 0\)). The latter is expressed as \(\frac{dT}{dx}|_{x=0}=-{\frac{h}{k}}(T-T_{in})\), where \(h\) is the heat transfer coefficient, \(k\) the thermal conductivity, and \(T_{in}\) is the internal temperature of the house (room temperature). We assume here that \({\frac{h}{k}}\) = 1 m-1 and \(T_{in}\) = 25 °C. At the outer side of the wall (\(x = W\), where \(W\) = 0.15 m, is the width of the wall), we apply a constant temperature (\(T_0\) = 5 °C) boundary condition, representing the outdoor temperature, implemented as Dirichlet type in the finite element code.

Solving with FEAScript

Below is a demonstration of how to use the FEAScript library to solve this stationary heat transfer problem in your web browser. You only need a simple HTML page to run this example where the following code snippets should be included. First, we should load the required external libraries: 

<head>
    . . .
    <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjs/5.0.0/math.min.js"></script>
    <script src="https://cdnjs.cloudflare.com/ajax/libs/plotly.js/2.27.0/plotly.min.js"></script>
    . . .
</head>

We should then define the problem parameters, such as the solver type, the geometry configuration, and the boundary conditions. This is performed using JavaScript objects directly in the HTML file: 

      <body>
          . . .
          <script type="module">
              // Import the FEAScript library
              import { FEAScriptModel, plotSolution, printVersion } from "https://core.feascript.com/src/index.js";
              window.addEventListener("DOMContentLoaded", (event) => {
      
                // Print FEAScript version
                printVersion();
      
                // Create a new FEAScript model
                const model = new FEAScriptModel();
      
                // Set solver configuration
                model.setSolverConfig("solidHeatTransferScript");
      
                // Define mesh configuration
                model.setMeshConfig({
                  meshDimension: "1D",
                  elementOrder: "linear",
                  numElementsX: 10,
                  maxX: 0.15,
                });
      
                // Define boundary conditions
                model.addBoundaryCondition("0", ["convection", 1, 25]);
                model.addBoundaryCondition("1", ["constantTemp", 5]);

                // Set solver method
                model.setSolverMethod("lusolve");
 
                // Solve the problem and get the solution
                const { solutionVector, nodesCoordinates } = model.solve();
      
                // Plot the solution as a 1D line plot
                plotSolution(
                  solutionVector,
                  nodesCoordinates,
                  model.solverConfig,
                  model.meshConfig.meshDimension,
                  "line",
                  "solutionPlot"
                );
                
              });
          </script>
          . . .
      </body>

Results

Below is the 1D line plot of the computed temperature distribution along the wall thickness. This plot is generated in real time using FEAScript. You can find a minimal example of this tutorial in the example directory.

Cannot draw the results. Please turn your phone to horizontal position to see the results.

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