heat transfer

Thermal Deflection of Bimetallic Beam

  • Overview: This example illustrates the solution to a coupled thermo-elasticity problem, focusing on thermal expansion and contraction in mechanical components due to temperature changes.

  • Key Concepts:

    • Thermal stress arises when structures experience constraints that prevent free thermal expansion or contraction.

    • A bimetallic beam is utilized in experiments, consisting of two materials with significantly different coefficients of thermal expansion (CTE).

Beam Geometry

  • Materials:

    • Invar, with thermal properties noted as AT = 100.

    • Copper, noted for its different thermal expansion characteristics.

  • Geometry Parameters:

    • Length (L) = 0.1 m

    • Width (W) = 5e-3 m

    • Height (H) = 1e-3 m

    • Beam Creation Command:

      • gm = multicuboid(L, W, [H, H], Zoffset=[0, H])

  • Visualization:

    • Use command figure pdegplot(gm) to plot the geometry.

Material Properties and Analysis Setup

  • Cell Identification: Identify cell labels for specifying material properties using the command:

    • figure pdegplot(gm, CellLabel='on')

  • FEM Model Setup:

    • Create an femodel object for static structural analysis:

      • model = femodel(AnalysisType='structuralStatic', Geometry=gm)

    • Material Properties for Copper:

      • Young's Modulus (Ec) = 137e9 N/m^2

      • Poisson's Ratio (nuc) = 0.28

      • CTE for Copper (CTEc) = 20.00e-6 m/m-°C

      • Assignment:

        • model.MaterialProperties(1) = materialProperties(YoungsModulus=Ec, PoissonsRatio=nuc, CTE=CTEc)

Assigning Material Properties

  • Material Properties for Invar:

    • Young's Modulus (Ei) = 130e9 N/m^2

    • Poisson's Ratio (nui) = 0.354

    • CTE for Invar (CTEi) = 1.2e-6 m/m-°C

    • Assignment:

      • model.MaterialProperties(2) = materialProperties(YoungsModulus=Ei, PoissonsRatio=nui, CTE=CTEi)

  • Boundary Conditions: Imposing a fixed boundary condition on the left end of the beam:

    • Visualize with figure pdegplot(gm, FaceLabels='on')

Applying Loads and Mesh Generation

  • Thermal Load: Assume temperature change (ΔT) = 100°C, reference temperature = 25°C, operating temperature = 125°C.

    • Command:

      • model.CellLoad = cellLoad(Temperature=125)

      • model.ReferenceTemperature = 25;

  • Mesh Generation: Generate a mesh and solve the model. Use:

    • model = generateMesh(model, Hmax=H/2)

    • R = solve(model)

Visualization of Results

  • Deflected Shape Plot: Visualize using displacement as colormap data:

    • Command:

      • figure pdeplot3D(R.Mesh, ColorMapData=R.Displacement.Magnitude, Deformation=R.Displacement, DeformationScaleFactor=2)

  • Live Script Visualization: For enhanced visualization, create a new live script in the Live Editor:

    • Follow steps to visualize PDE results by selecting R from the results section and setting data parameters for displacement.

Analytical Comparison

  • Calculate Analytical Deflection: Use the formula for deflection due to temperature difference:

    • deflectionAnalytical = 3*(CTEc - CTEi)*100*2*H*L^2/(H^2*K1)

    • Define K1 as:

      • K1 = 14 + (Ec/Ei) + (Ei/Ec)

  • Results Comparison:

    • Calculate the percentage error between PDE Toolbox results and analytical results:

      • percentError = 100 * (PDEToobox_Deflection - deflectionAnalytical) / PDEToobox_Deflection

    • Present results in a table with columns for PDE Toolbox results, analytical results, and percentage error.

Summary of Results

  • Findings: The results from the thermal deflection analysis are comparable, confirming the validity of the approach due to the large aspect ratio of the beam.