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.

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 *T _{inf} = 17 °C* and a heat transfer coefficient of

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.

- To start a new model click the
**New Model**toolbar button, or select*New Model...*from the*File*menu. Select the

**Heat Transfer**physics mode from the*Select Physics*drop-down menu.- Press
**OK**to finish the physics mode selection. - 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. - Select
**R1**in the geometry object*Selection*list box. - 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. - Enter
`0`

into the*x*edit field._{min} - Enter
`0.11`

into the*x*edit field._{max} - Enter
`0`

into the*y*edit field._{min} Enter

`0.12`

into the*y*edit field._{max}- Press
**OK**to finish and close the dialog box. - 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. - Select
**E1**in the geometry object*Selection*list box. - 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. - Enter
`0.065 0`

into the*center*edit field. - Enter
`0.015`

into the*x*edit field._{radius} Enter

`0.015`

into the*y*edit field._{radius}- Press
**OK**to finish and close the dialog box. - 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. - Select
**E2**in the geometry object*Selection*list box. - 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. - Enter
`0.11 0.12`

into the*center*edit field. - Enter
`0.035`

into the*x*edit field._{radius} Enter

`0.035`

into the*y*edit field._{radius}- Press
**OK**to finish and close the dialog box. - 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. - Select
**E3**in the geometry object*Selection*list box. - 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. - Enter
`0 0.06`

into the*center*edit field. - Enter
`0.025`

into the*x*edit field._{radius} Enter

`0.025`

into the*y*edit field._{radius}Press

**OK**to finish and close the dialog box.- Select
**Combine Objects...**from the*Geometry*menu. Enter

`R1 - E1 - E2 - E3`

into the*Geometry Formula*edit field.Press

**OK**to finish and close the dialog box.- 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. - Select
**R2**in the geometry object*Selection*list box. - 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. - Enter
`0.065`

into the*x*edit field._{min} - Enter
`0.16`

into the*x*edit field._{max} - Enter
`0.05`

into the*y*edit field._{min} Enter

`0.07`

into the*y*edit field._{max}- Press
**OK**to finish and close the dialog box. - 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. - Select
**E4**in the geometry object*Selection*list box. - 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. - Enter
`0.065 0.06`

into the*center*edit field. - Enter
`0.01`

into the*x*edit field._{radius} Enter

`0.01`

into the*y*edit field._{radius}- Press
**OK**to finish and close the dialog box. Select

**R2**and**E4**in the geometry object*Selection*list box.- Press the
**+ / Add geometry objects***Toolbar*button. - Select
**CJ1**in the geometry object*Selection*list box. - Press the
**Copy and/or transform selected geometry object***Toolbar*button. Select the

**Make copy of geometry object**check box.- Press
**OK**to finish and close the dialog box. Select

**CS3**and**CJ1**in the geometry object*Selection*list box.Press the

**- / Subtract geometry objects***Toolbar*button.- 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.

Enter

`0.0025`

into the*Grid Size*edit field and press the**Generate**button to call the grid generation algorithm.- 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.

- Select
**1**in the*Subdomains*list box. - Enter
`rho_tungsten`

into the*Density*edit field. - Enter
`cp_tungsten`

into the*Heat capacity*edit field. - Enter
`k_tungsten`

into the*Thermal conductivity*edit field. Enter

`84`

into the*Initial condition for T*edit field.- Select
**2**in the*Subdomains*list box. - Enter
`rho_steel`

into the*Density*edit field. - Enter
`cp_steel`

into the*Heat capacity*edit field. - Enter
`k_steel`

into the*Thermal conductivity*edit field. Enter

`-10`

into the*Initial condition for T*edit field.- 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.

- 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.

Name | Expression |
---|---|

rho_tungsten | 19000 |

cp_tungsten | 134 |

k_tungsten | 163 |

rho_steel | 7500 |

cp_steel | 470 |

k_steel | 44 |

- Switch to
**Boundary**mode by clicking on the corresponding*Mode Toolbar*button. 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*T*._{inf}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.- Press the
**Plot Options***Toolbar*button. - Select the
**Contour Plot**check box. Enter

`20`

into the*Number or specified vector of contour levels to plot*edit field.Press

**Apply**to plot and visualize the selected postprocessing options.

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.

- 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.

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