Thermal Contact Between Spherical Surfaces

 

Consider the problem of the steady heat flux in the complex spherical plate with the width ∑0.5Δd  and thermal conductivity  coefficients ki , whose first and last surfaces maintain temperatures t1 and tn+1 , and between the sheets with numbers m-1 and m+1, there is the thermal contact with specific resistance Rm (see figure).

The temperature change and thermal flux along the width of the complex plate, consisting of sheets with diameters d1, d2, ..., dn and thermal conductivity coefficients k1, k2,... kn, respectively, for each sheet fi, i=1,2,..., n are determined using the formula below:

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

Hence

,

Assume the following initial data: number of sheets is n=2 , diameters d1, d2, d3 of each sheet equal 350; 380; and 420 mm, respectively. Applied temperatures t1 and t4 are 473K and 273K, respectively. Thermal conductivity coefficients are:

(steel and aluminum alloy).

Thermal resistance (it is approximately equivalent to the thermal resistance of the spherical air layer with the width 0.05 mm and )

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)
      Number of corner nodes         7759
      Number of arguments:            7759
      Number of finite elements      35203

Results

Numerical result, w*

Analytical result, w

Error , %

Temperature, t2, K

411.6

424.7

8.3

Temperature, t3, K

287.8

290.8

3.05

Table 2*

FEA Mesh Parameters:
      Type of finite element            quadratic tetrahedron (10-nodes)
      Number of nodes                    54253
      Number of arguments:            54253
      Number of finite elements      37413

Results

Numerical result, w*

Analytical result, w

Error , %

Temperature, t2, K

411.3

424.7

-3.2

Temperature, t3, K

287.7

290.8

-1

 

 

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

 

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