John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 36
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The figure shows the thermocouple reading as a function of time during the quenching process. If the Biot number is small, the center temperature can be interpreted as the uniform temperature of thesphere during the quench. First draw tangents to the curve,and graphically differentiate it. Then use the resulting valuesof dT /dt to construct a graph of the heat transfer coefficientas a function of (Tsphere − Tsat ).
The result will give actualvalues of h during boiling over the range of temperature differences. Check to see whether or not the largest value of theBiot number is too great to permit the use of lumped-capacitymethods.5.7A butt-welded 36-gage thermocouple is placed in a gas flowwhose temperature rises at the rate 20◦ C/s. The thermocouple steadily records a temperature 2.4◦ C below the known gasflow temperature. If ρc is 3800 kJ/m3 K for the thermocouplematerial, what is h on the thermocouple? [h = 1006 W/m2 K.]5.8Check the point on Fig. 5.7 at Fo = 0.2, Bi = 10, and x/L = 0analytically.5.9Prove that when Bi is large, eqn.
(5.34) reduces to eqn. (5.33).5.10Check the point at Bi = 0.1 and Fo = 2.5 on the slab curve inFig. 5.10 analytically.Chapter 5: Transient and multidimensional heat conduction254Figure 5.28Problem 5.65.11Configuration and temperature response forSketch one of the curves in Fig. 5.7, 5.8, or 5.9 and identify:• The region in which b.c.’s of the third kind can be replacedwith b.c.’s of the first kind.• The region in which a lumped-capacity response can beassumed.• The region in which the solid can be viewed as a semiinfinite region.5.12Water flows over a flat slab of Nichrome, 0.05 mm thick, whichserves as a resistance heater using AC power.
The apparentvalue of h is 2000 W/m2 K. How much surface temperaturefluctuation will there be?Problems2555.13Put Jakob’s bubble growth formula in dimensionless form, identifying a “Jakob number”, Ja ≡ cp (Tsup − Tsat )/hfg as one ofthe groups. (Ja is the ratio of sensible heat to latent heat.) Becertain that your nondimensionalization is consistent with theBuckingham pi-theorem.5.14A 7 cm long vertical glass tube is filled with water that is uniformly at a temperature of T = 102◦ C. The top is suddenlyopened to the air at 1 atm pressure. Plot the decrease of theheight of water in the tube by evaporation as a function of timeuntil the bottom of the tube has cooled by 0.05◦ C.5.15A slab is cooled convectively on both sides from a known initial temperature.
Compare the variation of surface temperature with time as given in Fig. 5.7 with that given by eqn. (5.53)if Bi = 2. Discuss the meaning of your comparisons.5.16To obtain eqn. (5.62), assume a√ complex solution of the typeΘ = fn(ξ)exp(iΩ), where i ≡ −1. This will assure that thereal part of your solution has the required periodicity and,when you substitute it in eqn. (5.60), you will get an easy-tosolve ordinary d.e. in fn(ξ).5.17A certain steel cylinder wall is subjected to a temperature oscillation that we approximate at T = 650◦ C + (300◦ C) cos ωt,where the piston fires eight times per second.
For stress design purposes, plot the amplitude of the temperature variationin the steel as a function of depth. If the cylinder is 1 cm thick,can we view it as having infinite depth?5.18A 40 cm diameter pipe at 75◦ C is buried in a large block ofPortland cement. It runs parallel with a 15◦ C isothermal surface at a depth of 1 m. Plot the temperature distribution alongthe line normal to the 15◦ C surface that passes through thecenter of the pipe.
Compute the heat loss from the pipe bothgraphically and analytically.5.19Derive shape factor 4 in Table 5.4.5.20Verify shape factor 9 in Table 5.4 with a flux plot. Use R1 /R2 =2 and R1 /L = ½. (Be sure to start out with enough blank papersurrounding the cylinders.)Chapter 5: Transient and multidimensional heat conduction256Eggs cook as theirproteins denature andcoagulate. The time tocook depends onwhether a soft or hardcooked egg desired.Eggs may be cooked byplacing them (cold orwarm) into cold waterbefore heating starts orby placing warm eggsdirectly into simmeringwater [5.20].5.21A copper block 1 in. thick and 3 in.
square is held at 100◦ Fon one 1 in. by 3 in. surface. The opposing 1 in. by 3 in.surface is adiabatic for 2 in. and 90◦ F for 1 inch. The remaining surfaces are adiabatic. Find the rate of heat transfer.[Q = 36.8 W.]5.22Obtain the shape factor for any or all of the situations pictured in Fig. 5.29a through j on pages 258–259. In each case,present a well-drawn flux plot. [Sb 1.03, Sc Sd , Sg =1.]5.23Two copper slabs, 3 cm thick and insulated on the outside, aresuddenly slapped tightly together.
The one on the left side isinitially at 100◦ C and the one on the right side at 0◦ C. Determine the left-hand adiabatic boundary’s temperature after 2.3s have elapsed. [Twall 80.5◦ C]5.24Estimate the time required to hard-cook an egg if:• The minor diameter is 45 mm.• k for the entire egg is about the same as for egg white.No significant heat release or change of properties occursduring cooking.• h between the egg and the water is 1000 W/m2 K.• The egg has a uniform temperature of 20◦ C when it is putinto simmering water at 85◦ C.• The egg is done when the center reaches 75◦ C.5.25Prove that T1 in Fig.
5.5 cannot oscillate.5.26Show that when isothermal and adiabatic lines are interchangedin a two-dimenisonal body, the new shape factor is the inverseof the original one.5.27A 0.5 cm diameter cylinder at 300◦ C is suddenly immersedin saturated water at 1 atm. If h = 10, 000 W/m2 K, find thecenterline and surface temperatures after 0.2 s:a. If the cylinder is copper.b.
If the cylinder is Nichrome V. [Tsfc 200◦ C.]c. If the cylinder is Nichrome V, obtain the most accuratevalue of the temperatures after 0.04 s that you can.Problems2575.28A large, flat electrical resistance strip heater is fastened to afirebrick wall, unformly at 15◦ C. When it is suddenly turned on,it releases heat at the uniform rate of 4000 W/m2 . Plot the temperature of the brick immediately under the heater as a function of time if the other side of the heater is insulated. Whatis the heat flux at a depth of 1 cm when the surface reaches200◦ C.5.29Do Experiment 5.2 and submit a report on the results.5.30An approximately spherical container, 2 cm in diameter, containing electronic equipment is placed in wet mineral soil withits center 2 m below the surface.
The soil surface is kept at 0◦ C.What is the maximum rate at which energy can be released bythe equipment if the surface of the sphere is not to exceed30◦ C?5.31A semi-infinite slab of ice at −10◦ C is exposed to air at 15◦ Cthrough a heat transfer coefficient of 10 W/m2 K. What is theinitial rate of melting of ice in kg/m2 s? What is the asymptotic rate of melting? Describe the melting process in physical terms.
(The latent heat of fusion of ice, hsf = 333, 300J/kg.)5.32One side of an insulating firebrick wall, 10 cm thick, initiallyat 20◦ C is exposed to 1000◦ C flame through a heat transfercoefficient of 230 W/m2 K. How long will it be before the otherside is too hot to touch, say at 65◦ C? (Estimate properties at500◦ C, and assume that h is quite low on the cool side.)5.33A particular lead bullet travels for 0.5 sec within a shock wavethat heats the air near the bullet to 300◦ C. Approximate thebullet as a cylinder 0.8 cm in diameter.
What is its surfacetemperature at impact if h = 600 W/m2 K and if the bullet wasinitially at 20◦ C? What is its center temperature?5.34A loaf of bread is removed from an oven at 125◦ C and set onthe (insulating) counter to cool in a kitchen at 25◦ C. The loafis 30 cm long, 15 cm high, and 12 cm wide. If k = 0.05 W/m·Kand α = 5 × 10−7 m2 /s for bread, and h = 10 W/m2 K, whenwill the hottest part of the loaf have cooled to 60◦ C? [About 1h 5 min.]Figure 5.29 Configurations for Problem 5.22258Figure 5.29 Configurations for Problem 5.22 (con’t)259Chapter 5: Transient and multidimensional heat conduction2605.35A lead cube, 50 cm on each side, is initially at 20◦ C. The surroundings are suddenly raised to 200◦ C and h around the cubeis 272 W/m2 K.
Plot the cube temperature along a line fromthe center to the middle of one face after 20 minutes haveelapsed.5.36A jet of clean water superheated to 150◦ C issues from a 1/16inch diameter sharp-edged orifice into air at 1 atm, moving at27 m/s. The coefficient of contraction of the jet is 0.611. Evaporation at T = Tsat begins immediately on the outside of the jet.Plot the centerline temperature of the jet and T (r /ro = 0.6) asfunctions of distance from the orifice up to about 5 m. Neglectany axial conduction and any dynamic interactions betweenthe jet and the air.5.37A 3 cm thick slab of aluminum (initially at 50◦ C) is slappedtightly against a 5 cm slab of copper (initially at 20◦ C). The outsides are both insulated and the contact resistance is neglible.What is the initial interfacial temperature? Estimate how longthe interface will keep its initial temperature.5.38A cylindrical underground gasoline tank, 2 m in diameter and4 m long, is embedded in 10◦ C soil with k = 0.8 W/m2 K andα = 1.3 × 10−6 m2 /s.
water at 27◦ C is injected into the tankto test it for leaks. It is well-stirred with a submerged ½ kWpump. We observe the water level in a 10 cm I.D. transparentstandpipe and measure its rate of rise and fall. What rate ofchange of height will occur after one hour if there is no leakage? Will the level rise or fall? Neglect thermal expansion anddeformation of the tank, which should be complete by the timethe tank is filled.5.39A 47◦ C copper cylinder, 3 cm in diameter, is suddenly immersed horizontally in water at 27◦ C in a reduced gravity environment.
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