

Take recording tape, for instance. Consulting engineer David Dearth tells of a job that dealt with early tape drives for computers. "My job was to set operational standards for tapebased backup peripherals shortly after they began appearing on the market. A problem with tape, however, is that data reliability is sensitive to temperature extremes. So I was asked to estimate the time it took for tape packs to return to room temperature after extended exposure to high temperatures, as might happen if the tapes were left in a delivery vehicle all day," says Dearth.
Conventional hand solutions and FEA techniques can both be used to estimate midpack temperature response (a function of time) as the polyester tape cools to room temperature. The goal is to estimate the time needed for the tape pack to return to about 72°F measured at its center, after "soaking" at 130°F.
The tape winds around an aluminum hub, which makes hand solutions for transient heat transfer difficult. Therefore, FEA is the best alternative. "But prior to developing an FEA idealization, I recommend sample "warmup" problems with known textbook solutions. This test problem has been extensively investigated. It's also a "sanity" check, a hand calculation to ensure I was using the right heattransfer parameters before testing the cooling on an FEA version of the problem," he says.
The warmup problem presents a tape pack idealized as a vertical disk or plate. Dearth suggests neglecting the aluminum hub for the firstcut investigation. A search through engineering literature on heat transfer will identify several similar problems. "Of course, no engineering reference contains a complete approach from beginning to end. That would be too easy. But this sample transient heattransfer problem has all the features of a reallife problem. Once confident with the method and procedures, tackle reallife problems in FEA," he adds.
A sketch of the simplified tape pack has its initial temperature Ti = 130°F. Then it's exposed to still air (Tair) at 72°F. The tape pack is essentially polyester with thermal conductivity K = 0.02168 Btu/(hrft°F). For now, ignore the aluminum hub and the tape's oxide coating. The goal is to estimate the time required it takes the tape pack to cool to almost room temperature.
First determine the average heat transferfilm coefficient for free ( natural) convection to the surrounding air. Assuming the vertical tape pack is similar to a vertical plane, an estimate for the average film coefficient is 1.6985 Btu/(hrft2°F).
The partial differential equations for a timedependent conductionconvection system for the model appear in Tape pack with aluminum hub and are found in the first reference in For further reading. "By solving an approximation to the exact series solution, you'll find that the time required for the center of the tape pack to return to 72.72°F is about 78.29 min. Try using a spreadsheet to minimize round offs in the math. An accompanying table compares temperature estimates at the tapepack center versus time. One column is for hand solutions and the other for FEA results from MSC/Nastran. The percent difference is no larger than ±0.001% during the initial transient response and almost no difference between solutions after cooling for about 15 min.


RETURNING TO ROOM TEMPERATURE 

TIME (min)

FEA

HAND


START 
0

130

130

5

120.876

120.760


10

108.692

108.574


15

99.484

99.434


20

92.586

92.578


25

87.419

87.435


30

83.549

83.578


35

80.650

80.684


40

78.479

78.514


45

76.853

76.886


50

75.635

75.665


55

74.723

74.749


60

74.039

74.062


65

73.528

73.547


70

73.144

73.160


Range 99% 
75

72.857

72.870

room temp. 
80

72.642

72.653

85

72.481

72.490


END 
90

72.360

72.367

The table shows temperatures from hand calculations and FEA at fiveminute intervals.

For further readingDavid Dearth suggests these texts for additional reading on heat transfer.

Hand calculations, an FE model, and free softwareInterested readers can obtain a zip file, containing hand calculations and run notes for the FEA model from www.machinedesign.com/md/misc/feaupdate.zip. HandCalcs_TapePackTransientHeat.pdf contains detailed hand calculations with summary spreadsheet arithmetic. RunNotes_TapePackTransientHeat.pdfare run notes and keystroke summary for the FEA model in MSC/Nastran format. And the FEA model TapePackTransientHeat_v2004.mod is small enough to process using the limited node version of MSC/Nastran v2004. For a free copy of this demo software, log on to mscsoftware.com/offers/master/contact.cfm or call MSC Software at (866) 6721549. 
MAKE CONTACT
David Dearth, Applied Technology
[email protected]
MSC.Software
(714) 5408900
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