Thermal Contact Between Plain Plates

Consider the problem of the steady heat flux in the complex plate with width ∑hi and thermal conductivity coefficients ki, whose first and last surface maintain temperatures t1 and tn+1 and between the plates with numbers m-1 and m+1, there is a thermal contact with specific resistance Rm(see figure).

The temperature change and heat flux along the width of the complex plate, consisting of n sheets with widths h1, h2,... hn and thermal conductivity coefficients k1, k2,... kn , respectively, for each sheet, fi, i=1,2,..., n are defined using the formula:

Let all sheets, save two ones, are in the ideal thermal contact along boundary surfaces. Put thermal resistance Rm between sheets numbered m-1 and m+1, then the heat flux will be uninterrupted at the transition from one area to the other, and in this case, will be the same in each point (i.e. f1=f2=...=fn=f). The change of temperature between the opposite surfaces of the entire complex plate will equal the sum of temperature changes in separate sheets:

Then:

,

Let accept the following initial data: number of sheets is n=2 , length and width of each sheet are 100 mmand 50 mm respectively, and widths of the sheets h1, h2 , are 8 mm, 10 mm . The applied temperatures t1and t2 equal 373K and 273K respectively.

Thermal conductivity coefficients are:

,

Thermal resistance of the contact

(it is approximately equivalent to the thermal resistance of the plain air layer with the width 0.05 mm with

)

Thus,

.

By calculation with the help of AutoFEM Analysis, we have obtained the following results (for the lower and upper sides of the contact, respectively):

Table 1*

FEA Mesh Parameters: Type of finite element linear tetrahedron (4-nodes) |
|||

Results |
Numerical result, w* |
Analytical result, w |
Error , % |

Temperature, t2, K |
327.2 |
340.7 |
4 |

Temperature, t3, K |
285.3 |
281.68 |
1.27 |

Table 2*

FEA Mesh Parameters: Type of finite element quadratic tetrahedron (10-nodes) |
|||

Results |
Numerical result, w* |
Analytical result, w |
Error , % |

Temperature, t2, K |
327.2 |
340.7 |
4 |

Temperature, t3, K |
285.3 |
281.68 |
1.27 |

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.

Read more about AutoFEM Thermal Analysis