John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 37
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Plot Tcyl as a function of time if g = 0.76 m/s2and if h = [2.733 + 10.448(∆T ◦ C)1/6 ]2 W/m2 K. (Do it numerically if you cannot integrate the resulting equation analytically.)5.40The mechanical engineers at the University of Utah end springsemester by roasting a pig and having a picnic. The pig isroughly cylindrical and about 26 cm in diameter. It is roastedProblems261over a propane flame, whose products have properties similarto those of air, at 280◦ C. The hot gas flows across the pig atabout 2 m/s.
If the meat is cooked when it reaches 95◦ C, andif it is to be served at 2:00 pm, what time should cooking commence? Assume Bi to be large, but note Problem 7.40. The pigis initially at 25◦ C.5.41People from cold northern climates know not to grasp metalwith their bare hands in subzero weather. A very slightly frostedpeice of, say, cast iron will stick to your hand like glue in, say,−20◦ C weather and might tear off patches of skin.
Explain thisquantitatively.5.42A 4 cm diameter rod of type 304 stainless steel has a verysmall hole down its center. The hole is clogged with wax thathas a melting point of 60◦ C. The rod is at 20◦ C. In an attemptto free the hole, a workman swirls the end of the rod—andabout a meter of its length—in a tank of water at 80◦ C. If his 688 W/m2 K on both the end and the sides of the rod, plotthe depth of the melt front as a function of time up to say, 4cm.5.43A cylindrical insulator contains a single, very thin electrical resistor wire that runs along a line halfway between the centerand the outside.
The wire liberates 480 W/m. The thermal conductivity of the insulation is 3 W/m2 K, and the outside perimeter is held at 20◦ C. Develop a flux plot for the cross section,considering carefully how the field should look in the neighborhood of the point through which the wire passes. Evaluatethe temperature at the center of the insulation.5.44A long, 10 cm square copper bar is bounded by 260◦ C gas flowson two opposing sides. These flows impose heat transfer coefficients of 46 W/m2 K. The two intervening sides are cooled bynatural convection to water at 15◦ C, with a heat transfer coefficient of 30 W/m2 K.
What is the heat flow through the blockand the temperature at the center of the block? (This couldbe a pretty complicated problem, but take the trouble to thinkabout Biot numbers before you begin.)5.45Lord Kelvin made an interesting estimate of the age of the earthin 1864. He assumed that the earth originated as a mass ofChapter 5: Transient and multidimensional heat conduction262molten rock at 4144 K (7000◦ F) and that it had been cooledby outer space at 0 K ever since.
To do this, he assumedthat Bi for the earth is very large and that cooling had thusfar penetrated through only a relatively thin (one-dimensional)layer. Using αrock = 1.18 × 10−6 m2 /s and the measured sur1face temperature gradient of the earth, 27 ◦ C/m, Find Kelvin’svalue of Earth’s age. (Kelvin’s result turns out to be muchless than the accepted value of 4 billion years.
His calculation fails because internal heat generation by radioactive decay of the material in the surface layer causes the surfacetemperature gradient to be higher than it would otherwisebe.)5.46A pure aluminum cylinder, 4 cm diam. by 8 cm long, is initially at 300◦ C. It is plunged into a liquid bath at 40◦ C withh = 500 W/m2 K. Calculate the hottest and coldest temperatures in the cylinder after one minute. Compare these resultswith the lumped capacity calculation, and discuss the comparison.5.47When Ivan cleaned his freezer, he accidentally put a large canof frozen juice into the refrigerator. The juice can is 17.8 cmtall and has an 8.9 cm I.D.
The can was at −15◦ C in the freezer,but the refrigerator is at 4◦ C. The can now lies on a shelf ofwidely-spaced plastic rods, and air circulates freely over it.Thermal interactions with the rods can be ignored. The effective heat transfer coefficient to the can (for simultaneousconvection and thermal radiation) is 8 W/m2 K. The can hasa 1.0 mm thick cardboard skin with k = 0.2 W/m·K. Thefrozen juice has approximately the same physical propertiesas ice.a.
How important is the cardboard skin to the thermal response of the juice? Justify your answer quantitatively.b. If Ivan finds the can in the refrigerator 30 minutes afterputting it in, will the juice have begun to melt?5.48A cleaning crew accidentally switches off the heating systemin a warehouse one Friday night during the winter, just aheadof the holidays. When the staff return two weeks later, thewarehouse is quite cold. In some sections, moisture that con-Problems263densed has formed a layer of ice 1 to 2 mm thick on the concrete floor.
The concrete floor is 25 cm thick and sits on compacted earth. Both the slab and the ground below it are nowat 20◦ F. The building operator turns on the heating system,quickly warming the air to 60◦ F. If the heat transfer coefficientbetween the air and the floor is 15 W/m2 K, how long will it takefor the ice to start melting? Take αconcr = 7.0 × 10−7 m2 /s andkconcr = 1.4 W/m·K, and make justifiable approximations asappropriate.5.49A thick wooden wall, initially at 25◦ C, is made of fir. It is suddenly exposed to flames at 800◦ C. If the effective heat transfercoefficient for convection and radiation between the wall andthe flames is 80 W/m2 K, how long will it take the wooden wallto reach its ignition temperature of 430◦ C?5.50Cold butter does not spread as well as warm butter.
A smalltub of whipped butter bears a label suggesting that, beforeuse, it be allowed to warm up in room air for 30 minutes afterbeing removed from the refrigerator. The tub has a diameter of 9.1 cm with a height of 5.6 cm, and the properties ofwhipped butter are: k = 0.125 W/m·K, cp = 2520 J/kg·K, andρ = 620 kg/m3 . Assume that the tub’s cardboard walls offer negligible thermal resistance, that h = 10 W/m2 K outsidethe tub.
Negligible heat is gained through the low conductivitylip around the bottom of the tub. If the refrigerator temperature was 5◦ C and the tub has warmed for 30 minutes in aroom at 20◦ C, find: the temperature in the center of the butter tub, the temperature around the edge of the top surface ofthe butter, and the total energy (in J) absorbed by the buttertub.5.51A two-dimensional, 90◦ annular sector has an adiabatic innerarc, r = ri , and an adiabatic outer arc, r = ro . The flat surface along θ = 0 is isothermal at T1 , and the flat surface alongθ = π /2 is isothermal at T2 .
Show that the shape factor isS = (2/π ) ln(ro /ri ).5.52Suppose that T∞ (t) is the time-dependent environmental temperature surrounding a convectively-cooled, lumped object.Chapter 5: Transient and multidimensional heat conduction264a. Show that eqn. (1.20) leads toddT∞(T − T∞ )=−(T − T∞ ) +dtTdtwhere the time constant T is defined as usual.b. If the initial temperature of the object is Ti , use eitheran integrating factor or a Laplace transform to show thatT (t) istd−t/τ−t/τT∞ (s) ds.−ees/τT (t) = T∞ (t)+[Ti − T∞ (0)] eds05.53Use the result of Problem 5.52 to verify eqn. (5.13).5.54Suppose that a thermocouple with an initial temperature Ti isplaced into an airflow for which its Bi 1 and its time constant is T . Suppose also that the temperature of the airflowvaries harmonically as T∞ (t) = Ti + ∆T cos (ωt).a.
Use the result of Problem 5.52 to find the temperature ofthe thermocouple, Ttc (t), for t> 0.; (If you wish, notethat the real part of eiωt is Re eiωt = cos ωt and usecomplex variables to do the integration.)b. Approximate your result for t T . Then determine thevalue of Ttc (t) for ωT 1 and for ωT 1. Explainin physical terms the relevance of these limits to the frequency response of the thermocouple.c.
If the thermocouple has a time constant of T = 0.1 sec,estimate the highest frequency temperature variation thatit will measure accurately.5.55A particular tungsten lamp filament has a diameter of 100 µmand sits inside a glass bulb filled with inert gas. The effective heat transfer coefficient for conduction and radiation is750 W/m·K and the electrical current is at 60 Hz. How muchdoes the filament’s surface temperature fluctuate if the gastemperature is 200◦ C and the average wire temperature is 2900◦ C?5.56The consider the parameter ψ in eqn. (5.41).a.
If the timescale for heat to diffuse a distance δ is δ2 /α, explain the physical significance of ψ and the consequenceof large or small values of ψ.References265b. Show that the timescale for the thermal response of a wirewith Bi 1 is ρcp δ/(2h). Then explain the meaning ofthe new parameter φ = ρcp ωδ/(4π h).c. When Bi 1, is φ or ψ a more relevant parameter?References[5.1] H. D.
Baehr and K. Stephan. Heat and Mass Transfer. SpringerVerlag, Berlin, 1998.[5.2] A. F. Mills. Basic Heat and Mass Transfer. Prentice-Hall, Inc., UpperSaddle River, NJ, 2nd edition, 1999.[5.3] L. M. K. Boelter, V. H. Cherry, H. A. Johnson, and R. C. Martinelli.Heat Transfer Notes. McGraw-Hill Book Company, New York, 1965.[5.4] M. P. Heisler. Temperature charts for induction and constant temperature heating.
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