John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 28
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On the wall itself, hw is only 54 W/m2 K.Calculate heff for the wall with its fins. (heff = Qwall divided byAwall and [Twall − T∞ ].)4.34Evaluate d(tanh x)/dx.4.35An engineer seeks to study the effect of temperature on thecuring of concrete by controlling the temperature of curing inthe following way. A sample slab of thickness L is subjectedto a heat flux, qw , on one side, and it is cooled to temperatureT1 on the other.
Derive a dimensionless expression for thesteady temperature in the slab. Plot the expression and offera criterion for neglecting the internal heat generation in theslab.4.36Develop the dimensionless temperature distribution in a spherical shell with the inside wall kept at one temperature and theoutside wall at a second temperature. Reduce your solutionto the limiting cases in which routside rinside and in whichroutside is very close to rinside .
Discuss these limits.4.37Does the temperature distribution during steady heat transferin an object with b.c.’s of only the first kind depend on k?Explain.4.38A long, 0.005 m diameter duralumin rod is wrapped with anelectrical resistor over 3 cm of its length. The resistor impartsProblems189a surface flux of 40 kW/m2 .
Evaluate the temperature of therod in either side of the heated section if h = 150 W/m2 Karound the unheated rod, and Tambient = 27◦ C.4.39The heat transfer coefficient between a cool surface and a saturated vapor, when the vapor condenses in a film on the surface,depends on the liquid density and specific heat, the temperature difference, the buoyant force per unit volume (g[ρf −ρg ]),the latent heat, the liquid conductivity and the kinematic viscosity, and the position (x) on the cooler. Develop the dimensionless functional equation for h.4.40A duralumin pipe through a cold room has a 4 cm I.D. and a5 cm O.D.
It carries water that sometimes sits stationary. Itis proposed to put electric heating rings around the pipe toprotect it against freezing during cold periods of −7◦ C. Theheat transfer coefficient outside the pipe is 9 W/m2 K (includingboth convection and radiation).
Neglect the presence of thewater in the conduction calculation, and determine how farapart the heaters would have to be if they brought the pipetemperature to 40◦ C locally. How much heat do they require?4.41The specific entropy of an ideal gas depends on its specificheat at constant pressure, its temperature and pressure, theideal gas constant and reference values of the temperature andpressure. Obtain the dimensionless functional equation forthe specific entropy and compare it with the known equation.4.42A large freezer’s door has a 2.5 cm thick layer of insulation(kin = 0.04 W/m·K) covered on the inside, outside, and edgeswith a continuous aluminum skin 3.2 mm thick (kAl = 165W/m·K). The door closes against a nonconducting seal 1 cmwide.
Heat gain through the door can result from conductionstraight through the insulation and skins (normal to the planeof the door) and from conduction in the aluminum skin only,going from the skin outside, around the edge skin, and to theinside skin. The heat transfer coefficients to the inside, hi ,and outside, ho , are each 12 W/m2 K, accounting for both convection and radiation. The temperature outside the freezer is25◦ C, and the temperature inside is −15◦ C.a. If the door is 1 m wide, estimate the one-dimensional heatgain through the door, neglecting any conduction around190Chapter 4: Analysis of heat conduction and some steady one-dimensional problemsthe edges of the skin. Your answer will be in watts permeter of door height.b.
Now estimate the heat gain by conduction around theedges of the door, assuming that the insulation is perfectly adiabatic so that all heat flows through the skin.This answer will also be per meter of door height.4.43A thermocouple epoxied onto a high conductivity surface is intended to measure the surface temperature. The thermocouple consists of two each bare, 0.51 mm diameter wires. Onewire is made of Chromel (Ni-10% Cr with kcr = 17 W/m·K) andthe other of constantan (Ni-45% Cu with kcn = 23 W/m·K). Theends of the wires are welded together to create a measuringjunction having has dimensions of Dw by 2Dw . The wires extend perpendicularly away from the surface and do not touchone another.
A layer of epoxy (kep = 0.5 W/m·K separatesthe thermocouple junction from the surface by 0.2 mm. Airat 20◦ C surrounds the wires. The heat transfer coefficient between each wire and the surroundings is h = 28 W/m2 K, including both convection and radiation. If the thermocouplereads Ttc = 40◦ C, estimate the actual temperature Ts of thesurface and suggest a better arrangement of the wires.4.44The resistor leads in Example 4.10 were assumed to be “infinitely long” fins. What is the minimum length they each musthave if they are to be modelled this way? What are the effectiveness, εf , and efficiency, ηf , of the wires?References[4.1] V. L. Streeter and E. B.
Wylie. Fluid Mechanics. McGraw-Hill BookCompany, New York, 7th edition, 1979. Chapter 4.[4.2] E. Buckingham. Phy. Rev., 4:345, 1914.[4.3] E. Buckingham. Model experiments and the forms of empirical equations. Trans. ASME, 37:263–296, 1915.[4.4] Lord Rayleigh, John Wm. Strutt. The principle of similitude. Nature,95:66–68, 1915.References[4.5] J. O. Farlow, C. V. Thompson, and D. E. Rosner.
Plates of the dinosaurstegosaurus: Forced convection heat loss fins? Science, 192(4244):1123–1125 and cover, 1976.[4.6] D. K. Hennecke and E. M. Sparrow. Local heat sink on a convectivelycooled surface—application to temperature measurement error. Int.J. Heat Mass Transfer, 13:287–304, 1970.[4.7] P. J. Schneider. Conduction Heat Transfer. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1955.[4.8] A. D.
Kraus, A. Aziz, and J.R. Welty. Extended Surface Heat Transfer.John Wiley & Sons, Inc., New York, 2001.1915.Transient and multidimensionalheat conductionWhen I was a lad, winter was really cold. It would get so cold that if youwent outside with a cup of hot coffee it would freeze. I mean it would freezefast. That cup of hot coffee would freeze so fast that it would still be hotafter it froze. Now that’s cold!Old North-woods tall-tale5.1IntroductionJames Watt, of course, did not invent the steam engine. What he did dowas to eliminate a destructive transient heating and cooling process thatwasted a great amount of energy. By 1763, the great puffing engines ofSavery and Newcomen had been used for over half a century to pump thewater out of Cornish mines and to do other tasks. In that year the younginstrument maker, Watt, was called upon to renovate the Newcomen engine model at the University of Glasgow.
The Glasgow engine was thenbeing used as a demonstration in the course on natural philosophy. Wattdid much more than just renovate the machine—he first recognized, andeventually eliminated, its major shortcoming.The cylinder of Newcomen’s engine was cold when steam entered itand nudged the piston outward. A great deal of steam was wastefullycondensed on the cylinder walls until they were warm enough to accommodate it. When the cylinder was filled, the steam valve was closed andjets of water were activated inside the cylinder to cool it again and condense the steam. This created a powerful vacuum, which sucked thepiston back in on its working stroke.
First, Watt tried to eliminate thewasteful initial condensation of steam by insulating the cylinder. Butthat simply reduced the vacuum and cut the power of the working stroke.193194Transient and multidimensional heat conduction§5.2Then he realized that, if he led the steam outside to a separate condenser,the cylinder could stay hot while the vacuum was created.The separate condenser was the main issue in Watt’s first patent(1769), and it immediately doubled the thermal efficiency of steam engines from a maximum of 1.1% to 2.2%. By the time Watt died in 1819, hisinvention had led to efficiencies of 5.7%, and his engine had altered theface of the world by powering the Industrial Revolution.
And from 1769until today, the steam power cycles that engineers study in their thermodynamics courses are accurately represented as steady flow—ratherthan transient—processes.The repeated transient heating and cooling that occurred in Newcomen’s engine was the kind of process that today’s design engineermight still carelessly ignore, but the lesson that we learn from historyis that transient heat transfer can be of overwhelming importance. Today, for example, designers of food storage enclosures know that suchsystems need relatively little energy to keep food cold at steady conditions.
The real cost of operating them results from the consumptionof energy needed to bring the food down to a low temperature and thelosses resulting from people entering and leaving the system with food.The transient heat transfer processes are a dominant concern in the design of food storage units.We therefore turn our attention, first, to an analysis of unsteady heattransfer, beginning with a more detailed consideration of the lumpedcapacity system that we looked at in Section 1.3.5.2Lumped-capacity solutionsWe begin by looking briefly at the dimensional analysis of transient conduction in general and of lumped-capacity systems in particular.Dimensional analysis of transient heat conductionWe first consider a fairly representative problem of one-dimensional transient heat conduction:⎧⎪i.c.: T (t = 0) = Ti⎪⎪⎪⎨21 ∂T∂ Tb.c.: T (t > 0, x = 0) = T1=with⎪∂x 2α ∂t⎪∂T ⎪⎪⎩ b.c.: − k= h (T − T1 )x=L∂x x=LLumped-capacity solutions§5.2195The solution of this problem must take the form of the following dimensional functional equation:T − T1 = fn (Ti − T1 ), x, L, t, α, h, kThere are eight variables in four dimensions (K, s, m, W), so we look for8−4 = 4 pi-groups.
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