FEATool Multiphysics  v1.10
Finite Element Analysis Toolbox
Shrink Fitting of an Assembly

FEATool supports modeling heat transfer through both conduction, that is heat transported by a diffusion process, and also convection, which is heat transported through a fluid through convection by a velocity field. The heat transfer physics mode supports both these processes, and defines the following equation

\[ \rho C_p\frac{\partial T}{\partial t} + \nabla\cdot(-k\nabla c) = Q - \rho C_p\mathbf{u}\cdot\nabla T \]

where \(\rho\) is the density, \(C_p\) the heat capacity, \(k\) is the thermal conductivity, \(Q\) heat source term, and \(\mathbf{u}\) a vector valued convective velocity field.

ex_ht5_geom_50.svg

This example models heat conduction in the form of transient cooling for shrink fitting of a two part assembly. A tungsten rod heated to 84 °C is inserted into a -10 °C chilled steel frame part. The time when the maximum temperature has cooled to 70 °C should be determined. The assembly is cooled due to convection through a surrounding medium kept at Tinf = 17 °C and a heat transfer coefficient of h = 750 W/m2K. The surrounding cooling medium is not modeled directly, thus the convective term is omitted, but the effects are incorporated into the model by the use of natural convection boundary conditions [1].

Tutorial

This section describes how to set up and solve the thermal shrink fitting example with the FEATool graphical user interface (GUI).

This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Heat Transfer > Shrink Fitting of an Assembly from the File menu. Or alternatively, follow the step-by-step instructions below.

  1. To start a new model click the New Model toolbar button, or select New Model... from the File menu.
  2. Select the Heat Transfer physics mode from the Select Physics drop-down menu.

    heat_transfer3_02_50.png
  3. Press OK to finish the physics mode selection.
  4. To create a rectangle, first click on the Create square/rectangle Toolbar button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the shape outline, and release the button to finalize the shape.
  5. Select R1 in the geometry object Selection list box.
  6. To modify and edit the selected rectangle, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
  7. Enter 0 into the xmin edit field.
  8. Enter 0.11 into the xmax edit field.
  9. Enter 0 into the ymin edit field.
  10. Enter 0.12 into the ymax edit field.

    heat_transfer3_10_50.png
  11. Press OK to finish and close the dialog box.
  12. To create a circle or ellipse, first click on the Create circle/ellipse Toolbar button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the shape outline, and release the button to finalize the shape.
  13. Select E1 in the geometry object Selection list box.
  14. To modify and edit the selected ellipse, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
  15. Enter 0.065 0 into the center edit field.
  16. Enter 0.015 into the xradius edit field.
  17. Enter 0.015 into the yradius edit field.

    heat_transfer3_17_50.png
  18. Press OK to finish and close the dialog box.
  19. To create a circle or ellipse, first click on the Create circle/ellipse Toolbar button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the shape outline, and release the button to finalize the shape.
  20. Select E2 in the geometry object Selection list box.
  21. To modify and edit the selected ellipse, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
  22. Enter 0.11 0.12 into the center edit field.
  23. Enter 0.035 into the xradius edit field.
  24. Enter 0.035 into the yradius edit field.

    heat_transfer3_24_50.png
  25. Press OK to finish and close the dialog box.
  26. To create a circle or ellipse, first click on the Create circle/ellipse Toolbar button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the shape outline, and release the button to finalize the shape.
  27. Select E3 in the geometry object Selection list box.
  28. To modify and edit the selected ellipse, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
  29. Enter 0 0.06 into the center edit field.
  30. Enter 0.025 into the xradius edit field.
  31. Enter 0.025 into the yradius edit field.

    heat_transfer3_31_50.png
  32. Press OK to finish and close the dialog box.

    heat_transfer3_32_50.png
  33. Select Combine Objects... from the Geometry menu.
  34. Enter R1 - E1 - E2 - E3 into the Geometry Formula edit field.

    heat_transfer3_34_50.png
  35. Press OK to finish and close the dialog box.

    heat_transfer3_35_50.png
  36. To create a rectangle, first click on the Create square/rectangle Toolbar button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the shape outline, and release the button to finalize the shape.
  37. Select R2 in the geometry object Selection list box.
  38. To modify and edit the selected rectangle, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
  39. Enter 0.065 into the xmin edit field.
  40. Enter 0.16 into the xmax edit field.
  41. Enter 0.05 into the ymin edit field.
  42. Enter 0.07 into the ymax edit field.

    heat_transfer3_42_50.png
  43. Press OK to finish and close the dialog box.
  44. To create a circle or ellipse, first click on the Create circle/ellipse Toolbar button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the shape outline, and release the button to finalize the shape.
  45. Select E4 in the geometry object Selection list box.
  46. To modify and edit the selected ellipse, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
  47. Enter 0.065 0.06 into the center edit field.
  48. Enter 0.01 into the xradius edit field.
  49. Enter 0.01 into the yradius edit field.

    heat_transfer3_49_50.png
  50. Press OK to finish and close the dialog box.
  51. Select R2 and E4 in the geometry object Selection list box.

    heat_transfer3_51_50.png
  52. Press the + / Add geometry objects Toolbar button.
  53. Select CJ1 in the geometry object Selection list box.
  54. Press the Copy and/or transform selected geometry object Toolbar button.
  55. Select the Make copy of geometry object check box.

    heat_transfer3_55_50.png
  56. Press OK to finish and close the dialog box.
  57. Select CS3 and CJ1 in the geometry object Selection list box.

    heat_transfer3_57_50.png
  58. Press the - / Subtract geometry objects Toolbar button.

    heat_transfer3_58_50.png
  59. Switch to Grid mode by clicking on the corresponding Mode Toolbar button.

The default grid may be too coarse ensure an accurate solution. Decreasing the grid size and generating a finer grid can resolve curved boundaries better.

  1. Enter 0.0025 into the Grid Size edit field and press the Generate button to call the grid generation algorithm.

    heat_transfer3_61_50.png
  2. Switch to Equation mode by clicking on the corresponding Mode Toolbar button.

Equation and material coefficients are specified in Equation/Subdomain mode. In the Equation Settings dialog box enter the coefficients for density, heat capacity, thermal conductivity, and initial temperature for each material.

  1. Select 1 in the Subdomains list box.
  2. Enter rho_tungsten into the Density edit field.
  3. Enter cp_tungsten into the Heat capacity edit field.
  4. Enter k_tungsten into the Thermal conductivity edit field.
  5. Enter 84 into the Initial condition for T edit field.

    heat_transfer3_67_50.png
  6. Select 2 in the Subdomains list box.
  7. Enter rho_steel into the Density edit field.
  8. Enter cp_steel into the Heat capacity edit field.
  9. Enter k_steel into the Thermal conductivity edit field.
  10. Enter -10 into the Initial condition for T edit field.

    heat_transfer3_72_50.png
  11. Press OK to finish the equation and subdomain settings specification.

The Model Constants and Expressions functionality can be used to define and store convenient expressions which then are available in the point, equation, boundary coefficients, and as postprocessing expressions. Here it is used to define the material parameters.

  1. Press the Constants Toolbar button, or select the corresponding entry from the Equation menu, and enter the following variables in the Model Constants and Expressions dialog box. Press Enter after the last expression or use the Add Row button to expand the expression list.
NameExpression
rho_tungsten19000
cp_tungsten134
k_tungsten163
rho_steel7500
cp_steel470
k_steel44


heat_transfer3_74_50.png
  1. Switch to Boundary mode by clicking on the corresponding Mode Toolbar button.
  2. In the Boundary Settings dialog box, select the Heat flux boundary condition for all the boundaries. Enter k_tungsten and k_steel in the edit field for the convective transfer coefficient h for the boundaries corresponding to each material, and also enter 17 for the surrounding reference temperature Tinf.

    heat_transfer3_76_50.png
  3. Now that the problem is specified, press the Solve mode button to switch to solve mode. Since this is a time dependent study, open the solver settings and select the Time-Dependent solver. Set the Time step to 0.25, Simulation time to 16, Time stopping criteria to 0, then press Solve to start the solution process.

    heat_transfer3_77_50.png
  4. Press the Plot Options Toolbar button.
  5. Select the Contour Plot check box.
  6. Enter 20 into the Number or specified vector of contour levels to plot edit field.

    heat_transfer3_80_50.png
  7. Press Apply to plot and visualize the selected postprocessing options.

    heat_transfer3_81_50.png

Look back through the solutions and verify that the assembly has cooled to a temperature of 70 degrees around t = 13 s. Note that both the colorbar and Limits field will show the minimum and maximum surface plot value.

  1. Select 12.75 from the Available solutions/times drop-down menu, and press OK to plot and visualize the temperature at t = 12.75 s.

The shrink fitting of an assembly heat transfer model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB m-script text file, or GUI script (.fes) file.

Reference

[1] Krysl P. A Pragmatic Introduction to the Finite Element Method for Thermal and Stress Analysis. Pressure Cooker Press, USA, 2005.