XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX''"> E.3 Problem 2 - Parallel Semi-Infinite Plates
This example is of two parallel plates of finite width and modeled as infinitely long. The boundary conditions are independent of the location along the infinite length of the plates. This allows us to model the problem in 2-D XY space. In this example, the plates have equal width and are directly opposed to each other. You may wish to consider other arrangements such as directly opposed plates of unequal width or equal width plates offset from direct opposition. You may also vary the separation between the plates and the width of the plates.
The problem is shown schematically in
Figure E‑1, along with a typical finite element model of the problem.
Figure E‑1 Example Problem, Parallel Semi-infinite Plates
with Opening to Space
By appropriate specification of boundary conditions, we will create symmetry in the heat flux from the plates to the ambient environment from the openings to the left and right. Only the facing surfaces of the plates will be able to radiate thermal energy. All other surfaces will be perfectly insulated. Also, the plates will be thin and good insulators themselves. Thus, other modes of heat transfer, such as conduction and convection, may be neglected. A heat flux varying along the width of each plate will be imposed on the radiating face of each plate. The variation in heat flux from left to right on the bottom plate’s face will be the same as the variation from right to left on the top plate’s face. In this example, we will use a linear ramp variation.
The plates will radiate to each other and to the ambient environment at the left and right. By symmetry arguments, the total energy radiated to the ambient environment on the left and on the right should be equal. By conservation of energy their sum must equal the heat flux applied to the plates’ surfaces.
As in the previous example, we will model one of the openings to the ambient environment as an ambient surface. Note that this ambient surface is not connected to the plates. The other opening in the enclosure to the ambient environment will be modeled with an ambient node.
Note that each surface, the top plate, the bottom plate, and the ambient surface, is convex. The material with MPID 693 used for this problem is a mica brick. The model files for this example were delivered with Patran and should be available on your computer system by typing ‘get_view’ and selecting the directory ‘pplate’. For assistance in locating these files, please contact your system administrator. You may wish to experiment with other geometric dimensions, heat fluxes, and ambient temperatures.
A Patran Thermal TEMPLATEDAT file is needed. You might use the one from the previous example, editing the comments. A TEMPLATEDAT file is shown below.
Likewise a Patran Thermal MATDAT file is needed. Use the MATDAT file from the previous example. Copy that file to the present directory for this problem. Otherwise you will need to create a new MATDAT file. The resulting Patran model is translated into Patran Thermal data files and Viewfactor input data files by clicking Apply on the Analysis menu. Be sure to choose the X-Y pick for the dimensionality of the problem under Translation Parameters.
Remember to check the VFMSG file for error messages when the Viewfactor analysis is completed. The VFDIAG file from this analysis is shown below. Since this is not a closed enclosure, the viewfactors to each surface do not sum to one. The ambitious user may wish to calculate the views from various surfaces to the opening between the plates and compare these calculated values to the one minus sum values in the VFDIAG file.
VFDIAG File |
$TITLE: PDA Viewfactor VER. 2.5 4-APR-91 17:18:55
$TITLE: PARALLEL SEMI-INFINITE PLATES OPEN TO THE LEFT AND RIGHT.
$TITLE: PARALLEL PLATES, SIMPLE, FINE, LINEAR HEAT AT NODES.
$TITLE: 22-MAR-91 16:51:24 2.5 $ENCL: 1 18 1
1 0.8375167251E+00 0.1624832749E+00 0.0000000000E+00
0.8469785452E+00 0.8280549049E+00
2 0.7995741367E+00 0.2004258633E+00 0.0000000000E+00
0.8067734241E+00 0.7923747897E+00
3 0.7514496446E+00 0.2485503554E+00 0.0000000000E+00
0.7604730725E+00 0.7424262762E+00
4 0.6918255091E+00 0.3081744909E+00 0.0000000000E+00
0.7027461529E+00 0.6809048057E+00
5 0.6208978891E+00 0.3791021109E+00 0.0000000000E+00
0.6335697174E+00 0.6082262397E+00
6 0.5413827300E+00 0.4586172700E+00 0.0000000000E+00
0.5551127791E+00 0.5276526213E+00
7 0.8375174403E+00 0.1624825597E+00 0.0000000000E+00
0.8280556202E+00 0.8469793200E+00
8 0.7995739579E+00 0.2004260421E+00 0.0000000000E+00
0.7923744321E+00 0.8067733645E+00
9 0.7514500618E+00 0.2485499382E+00 0.0000000000E+00
0.7424265146E+00 0.7604734898E+00
10 0.6918258667E+00 0.3081741333E+00 0.0000000000E+00
0.6809051633E+00 0.7027463317E+00
11 0.6208983064E+00 0.3791016936E+00 0.0000000000E+00
0.6082265377E+00 0.6335700154E+00
12 0.5413829088E+00 0.4586170912E+00 0.0000000000E+00
0.5276528001E+00 0.5551129580E+00
13 0.6211039424E+00 0.3788960576E+00 0.0000000000E+00
0.6090477705E+00 0.6331601143E+00
14 0.5795322657E+00 0.4204677343E+00 0.0000000000E+00
0.5738915801E+00 0.5851728916E+00 |
15 0.5567277074E+00 0.4432722926E+00 0.0000000000E+00
0.5546888113E+00 0.5587666035E+00
16 0.5567269325E+00 0.4432730675E+00 0.0000000000E+00
0.5587658286E+00 0.5546880364E+00
17 0.5795311928E+00 0.4204688072E+00 0.0000000000E+00
0.5851718783E+00 0.5738904476E+00
18 0.6211042404E+00 0.3788957596E+00 0.0000000000E+00
0.6331602335E+00 0.6090482473E+00
0.4586173E+00 0.3333322E+00 0.1052847E+00 0.3333322E+00 0.1052847E+00
0.4723472E+00 0.3333322E+00 0.1057708E+00 0.3333322E+00 0.1057708E+00
0.4723474E+00 0.3333322E+00 0.1057709E+00 0.3333322E+00 0.1057709E+00
$ENDENCL:
$EOF: |
$QTRAN
Finally, you will want to look at the Patran Thermal output data in the QOUTDAT file. Use the system editor to find the first occurrence of the string '1TIME'. Note the system heat balance. This is approximately the imposed heat flux to the plates’ surfaces. Page down through the data to node 1000, the ambient node. Note that the heat flux at this node is approximately half of the imposed heat flux as expected. Sum up the heat fluxes to the ambient surface and note that this sum is approximately half of the imposed total heat flux. The nodes numbered above 1000 are the radiosity nodes created by Viewfactor.
This is a simple problem and you may wish to try numerous variations on it, such as refining the mesh, changing the convergence criteria, changing the double area parameter of the $GAUSS_ORDER, changing the imposed heat flux, changing the ambient environment temperature, specifying the plate surface temperatures, and entering more complicated descriptions of the radiative surface properties.