The tape and reel have a 4-in. OD and a 2-in. ID. Tape is about 0.315-in. wide. Neglect the aluminum hub.

The tape and reel have a 4-in. OD and a 2-in. ID. Tape is about 0.315-in. wide. Neglect the aluminum hub.


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 back-up 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 "warm-up" 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 warm-up problem presents a tape pack idealized as a vertical disk or plate. Dearth suggests neglecting the aluminum hub for the first-cut 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 heat-transfer problem has all the features of a real-life problem. Once confident with the method and procedures, tackle real-life 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/(hr-ft-°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 heattransfer-film 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/(hr-ft2-°F).

The partial differential equations for a time-dependent 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.

The FEA model shows only half the tape pack. Once confident of results, the model could be changed by including the aluminum hub or by using tape wound in larger spools.

The FEA model shows only half the tape pack. Once confident of results, the model could be changed by including the aluminum hub or by using tape wound in larger spools.



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 five-minute intervals.

For further reading

David Dearth suggests these texts for additional reading on heat transfer.

  1. E.R.G. Eckert & Robert M. Drake, Jr., Heat and Mass Transfer, 2nd Ed., McGraw-Hill Book Co., 1959.
  2. J.P. Holman, Heat Transfer, 5th Ed., Chapter 2-9, Conduction-Convection Systems, McGraw-Hill Book Co., 1981.
  3. W.M. Rohsenow & J.P. Hartnett, Handbook of Heat Transfer, McGraw-Hill Book Co., 1973.
  4. Donald R. Pitts and Leighton E. Sissom, Heat Transfer 2nd Ed. Schaum's Outline Series, McGraw-Hill Book Co., 1977.
  5. M. P. Heisler, Trans. The American Soc. of Mechanical Engineers, Trans. ASME, 69: 227-(1947).

 

Hand calculations, an FE model, and free software

Interested 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_TapePack-TransientHeat.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) 672-1549.

MAKE CONTACT

David Dearth, Applied Technology
AppliedAT@AOL.com
MSC.Software
(714) 540-8900
mscsoftware.com