
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 heattransferfilm 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. 
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David Dearth, Applied Technology
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