John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook, страница 8

Quite often the student is inclined toview as “theoretical” a problem that does not involve numbers or thatrequires the development of algebraic results.The problems assigned in this book are all intended to be useful inthat they do one or more of five things:1. They involve a calculation of a type that actually arises in practice(e.g., Problems 1.1, 1.3, 1.8 to 1.18, and 1.21 through 1.25).2. They illustrate a physical principle (e.g., Problems 1.2, 1.4 to 1.7,1.9, 1.20, 1.32, and 1.39). These are probably closest to having a“theoretical” objective.3.

They ask you to use methods developed in the text to develop otherresults that would be needed in certain applied problems (e.g., Problems 1.10, 1.16, 1.17, and 1.21). Such problems are usually the mostdifficult and the most valuable to you.4. They anticipate development that will appear in subsequent chapters (e.g., Problems 1.16, 1.20, 1.40, and 1.41).5.

They require that you develop your ability to handle numerical andalgebraic computation effectively. (This is the case with most of theproblems in Chapter 1, but it is especially true of Problems 1.6 to1.9, 1.15, and 1.17).Problems37Partial numerical answers to some of the problems follow them inbrackets. Tables of physical property data useful in solving the problemsare given in Appendix A.Actually, we wish to look at the theory, analysis, and practice of heattransfer—all three—according to Webster’s definitions:Theory: “a systematic statement of principles; a formulation of apparentrelationships or underlying principles of certain observed phenomena.”Analysis: “the solving of problems by the means of equations; the breaking up of any whole into its parts so as to find out their nature,function, relationship, etc.”Practice: “the doing of something as an application of knowledge.”Problems1.1A composite wall consists of alternate layers of fir (5 cm thick),aluminum (1 cm thick), lead (1 cm thick), and corkboard (6cm thick).

The temperature is 60◦ C on the outside of the forand 10◦ C on the outside of the corkboard. Plot the temperature gradient through the wall. Does the temperature profilesuggest any simplifying assumptions that might be made insubsequent analysis of the wall?1.2Verify eqn. (1.15).1.3q = 5000 W/m2 in a 1 cm slab and T = 140◦ C on the cold side.Tabulate the temperature drop through the slab if it is madeof• Silver• Aluminum• Mild steel (0.5 % carbon)• Ice• Spruce• Insulation (85 % magnesia)• Silica aerogelIndicate which situations would be unreasonable and why.Chapter 1: Introduction381.4Explain in words why the heat diffusion equation, eqn.

(1.13),shows that in transient conduction the temperature dependson the thermal diffusivity, α, but we can solve steady conduction problems using just k (as in Example 1.1).1.5A 1 m rod of pure copper 1 cm2 in cross section connectsa 200◦ C thermal reservoir with a 0◦ C thermal reservoir.

Thesystem has already reached steady state. What are the ratesof change of entropy of (a) the first reservoir, (b) the secondreservoir, (c) the rod, and (d) the whole universe, as a result ofthe process? Explain whether or not your answer satisfies theSecond Law of Thermodynamics. [(d): +0.0120 W/K.]1.6Two thermal energy reservoirs at temperatures of 27◦ C and−43◦ C, respectively, are separated by a slab of material 10cm thick and 930 cm2 in cross-sectional area.

The slab hasa thermal conductivity of 0.14 W/m·K. The system is operating at steady-state conditions. What are the rates of change ofentropy of (a) the higher temperature reservoir, (b) the lowertemperature reservoir, (c) the slab, and (d) the whole universeas a result of this process? (e) Does your answer satisfy theSecond Law of Thermodynamics?1.7(a) If the thermal energy reservoirs in Problem 1.6 are suddenlyreplaced with adiabatic walls, determine the final equilibriumtemperature of the slab.

(b) What is the entropy change for theslab for this process? (c) Does your answer satisfy the SecondLaw of Thermodynamics in this instance? Explain. The densityof the slab is 26 lb/ft3 and the specific heat is 0.65 Btu/lb·◦ F.[(b): 30.81 J/K].1.8A copper sphere 2.5 cm in diameter has a uniform temperatureof 40◦ C. The sphere is suspended in a slow-moving air streamat 0◦ C.

The air stream produces a convection heat transfer coefficient of 15 W/m2 K. Radiation can be neglected. Since copper is highly conductive, temperature gradients in the spherewill smooth out rapidly, and its temperature can be taken asuniform throughout the cooling process (i.e., Bi 1). Writethe instantaneous energy balance between the sphere and thesurrounding air. Solve this equation and plot the resultingtemperatures as a function of time between 40◦ C and 0◦ C.Problems391.9Determine the total heat transfer in Problem 1.8 as the spherecools from 40◦ C to 0◦ C.

Plot the net entropy increase resulting from the cooling process above, ∆S vs. T (K). [Total heattransfer = 1123 J.]1.10A truncated cone 30 cm high is constructed of Portland cement. The diameter at the top is 15 cm and at the bottom is7.5 cm. The lower surface is maintained at 6◦ C and the top at40◦ C. The other surface is insulated. Assume one-dimensionalheat transfer and calculate the rate of heat transfer in wattsfrom top to bottom.

To do this, note that the heat transfer, Q,must be the same at every cross section. Write Fourier’s lawlocally, and integrate it from top to bottom to get a relationbetween this unknown Q and the known end temperatures.[Q = −0.70 W.]1.11A hot water heater contains 100 kg of water at 75◦ C in a 20◦ Croom. Its surface area is 1.3 m2 . Select an insulating material,and specify its thickness, to keep the water from cooling morethan 3◦ C/h. (Notice that this problem will be greatly simplifiedif the temperature drop in the steel casing and the temperaturedrop in the convective boundary layers are negligible. Can youmake such assumptions? Explain.)Figure 1.17 Configuration forProblem 1.121.12What is the temperature at the left-hand wall shown in Fig. 1.17.Both walls are thin, very large in extent, highly conducting, andthermally black.

[Tright = 42.5◦ C.]1.13Develop S.I. to English conversion factors for:• The thermal diffusivity, α• The heat flux, q• The density, ρChapter 1: Introduction40• The Stefan-Boltzmann constant, σ• The view factor, F1–2• The molar entropy• The specific heat per unit mass, cIn each case, begin with basic dimension J, m, kg, s, ◦ C, andcheck your answers against Appendix B if possible.Figure 1.18 Configuration forProblem 1.141.14Three infinite, parallel, black, opaque plates transfer heat byradiation, as shown in Fig.

1.18. Find T2 .1.15Four infinite, parallel, black, opaque plates transfer heat byradiation, as shown in Fig. 1.19. Find T2 and T3 . [T2 = 75.53◦ C.]1.16Two large, black, horizontal plates are spaced a distance Lfrom one another. The top one is warm at a controllable temperature, Th , and the bottom one is cool at a specified temperature, Tc .

A gas separates them. The gas is stationary becauseit is warm on the top and cold on the bottom. Write the equation qrad /qcond = fn(N, Θ ≡ Th /Tc ), where N is a dimensionless group containing σ , k, L, and Tc . Plot N as a function ofΘ for qrad /qcond = 1, 0.8, and 1.2 (and for other values if youwish).Now suppose that you have a system in which L = 10 cm,Tc = 100 K, and the gas is hydrogen with an average k of0.1 W/m·K . Further suppose that you wish to operate in such away that the conduction and radiation heat fluxes are identical.Identify the operating point on your curve and report the valueof Th that you must maintain.Problems41Figure 1.19 Configuration forProblem 1.151.17A blackened copper sphere 2 cm in diameter and uniformly at200◦ C is introduced into an evacuated black chamber that ismaintained at 20◦ C.• Write a differential equation that expresses T (t) for thesphere, assuming lumped thermal capacity.• Identify a dimensionless group, analogous to the Biot number, than can be used to tell whether or not the lumpedcapacity solution is valid.• Show that the lumped-capacity solution is valid.• Integrate your differential equation and plot the temperature response for the sphere.1.18As part of a space experiment, a small instrumentation package is released from a space vehicle.

It can be approximatedas a solid aluminum sphere, 4 cm in diameter. The sphere isinitially at 30◦ C and it contains a pressurized hydrogen component that will condense and malfunction at 30 K. If we takethe surrounding space to be at 0 K, how long may we expect theimplementation package to function properly? Is it legitimateto use the lumped-capacity method in solving the problem?(Hint: See the directions for Problem 1.17.) [Time = 5.8 weeks.]1.19Consider heat conduction through the wall as shown in Fig.

1.20.Calculate q and the temperature of the right-hand side of thewall.1.20Throughout Chapter 1 we have assumed that the steady temperature distribution in a plane uniform wall in linear. ToChapter 1: Introduction42Figure 1.20 Configuration forProblem 1.19prove this, simplify the heat diffusion equation to the formappropriate for steady flow. Then integrate it twice and eliminate the two constants using the known outside temperaturesTleft and Tright at x = 0 and x = wall thickness, L.1.21The thermal conductivity in a particular plane wall depends asfollows on the wall temperature: k = A + BT , where A and Bare constants.

The temperatures are T1 and T2 on either sideif the wall, and its thickness is L. Develop an expression for q.Figure 1.21 Configuration forProblem 1.221.22Find k for the wall shown in Fig. 1.21. Of what might it bemade?1.23What are Ti , Tj , and Tr in the wall shown in Fig. 1.22? [Tj =16.44◦ C.]1.24An aluminum can of beer or soda pop is removed from therefrigerator and set on the table. If h is 13.5 W/m2 K, estimateProblems43Figure 1.22 Configuration for Problem 1.23when the beverage will be at 15◦ C.

Ignore thermal radiation.State all of your other assumptions.1.25One large, black wall at 27◦ C faces another whose surface is127◦ C. The gap between the two walls is evacuated. If the second wall is 0.1 m thick and has a thermal conductivity of 17.5W/m·K, what is its temperature on the back side? (Assumesteady state.)1.26A 1 cm diameter, 1% carbon steel sphere, initially at 200◦ C, iscooled by natural convection, with air at 20◦ C. In this case, h isnot independent of temperature. Instead, h = 3.51(∆T ◦ C)1/4W/m2 K.