Page 413 - Bird R.B. Transport phenomena

Problems 395
PROBLEMS 12A.1. Unsteady-state heat conduction in an iron sphere. An iron sphere of 1-in. diameter has the
following physical properties: к = 30 Btu/hr • ft • F, C p = 0.12 Btu/lb wl • F. and p = 436 lb /ft .
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w
Initially the sphere is at a temperature of 70°F.
(a) What is the thermal diffusivity of the sphere?
(b) If the sphere is suddenly plunged into a large body of fluid of temperature 270°F, how
much time is needed for the center of the sphere to attain a temperature of 128°F?
(c) A sphere of the same size and same initial temperature, but made of another material, re-
quires twice as long for its center to reach 128°F. What is its thermal diffusivity?
(d) The chart used in the solution of (b) and (c) was prepared from the solution to a partial
differential equation. What is that differential equation?
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Answers: (a) 0.574 ftVhr; (b) 1.1 sec; (c) 0.287 ft /hr
12A.2 Comparison of the two slab solutions for short times. What error is made by using Eq. 12.1-8
(based on the semi-infinite slab) instead of Eq. 12.1-31 (based on the slab of finite thickness),
when at/b 2 = 0.01 and for a position 0.9 of the way from the midplane to the slab surface?
Use the graphically presented solutions for making the comparison.
Answer: 4%
12A.3 Bonding with a thermosetting adhesive 1 (Fig. 12A.3). It is desired to bond together two
sheets of a solid material, each of thickness 0.77 cm. This is done by using a thin layer of thermo-
setting material, which fuses and forms a good bond at 160°C The two plates are inserted in a
press, with both platens of the press maintained at a constant temperature of 220°C. How
long will the sheets have to be held in the press, if they are initially at 20°C? The solid sheets
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have a thermal diffusivity of 4.2 X 10~ cm /s.
Answer: 85 s
12A.4. Quenching of a steel billet. A cylindrical steel billet 1 ft in diameter and 3 ft long, initially at
1000°F, is quenched in oil. Assume that the surface of the billet is at 200°F during the quench-
ing process. The steel has the following properties, which may be assumed to be independent
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of the temperature: к = 25 Btu/hr • ft • F, p = 77 g/cm , and C p = 0.12 cal/g • C.
Estimate the temperature of the hottest point in the billet after five minutes of quenching.
Neglect end effects; that is, make the calculation for a cylinder of the given diameter but of in-
finite length. See Problem 12C.1 for the method for taking end effects into account.
Answer: 750°F

12A.5. Measurement of thermal diffusivity from amplitude of temperature oscillations.
(a) It is desired to use the results of Example 12.1-3 to measure the thermal diffusivity a =
k/pC p of a solid material. This may be done by measuring the amplitudes Д and A 2 at two

Thermosetting adhesive
/ Upper platen (heated)
Г i

2b
L
Lower platen (heated) Fig. 12A.3. Two sheets of solid material with a thin
layer of adhesive in between.

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This problem is based on Example 10 of J. M. McKelvey, Chapter 2 of Processing of Thermoplastic
Materials (E. C. Bernhardt, ed.), Reinhold, New York (1959), p. 93.