John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 16
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If the interior temperature is 20◦ C and the outdoor temperature is −5◦ C, find the heat loss through the wall inwatts and the heat flux in W/m2 .d. Which of the five thermal resistances is dominant?2.43We found that the thermal resistance of a cylinder was Rtcyl =(1/2π kl) ln(ro /ri ). If ro = ri + δ, show that the thermal resistance of a thin-walled cylinder (δ ri ) can be approximatedby that for a slab of thickness δ. Thus, Rtthin = δ/(kAi ), whereAi = 2π ri l is the inside surface area of the cylinder.
Howmuch error is introduced by this approximation if δ/ri = 0.2?[Hint: Use a Taylor series.]2.44A Gardon gage measures a radiation heat flux by detecting atemperature difference [2.10]. The gage consists of a circularconstantan membrane of radius R, thickness t, and thermalconductivity kct which is joined to a heavy copper heat sinkat its edges. When a radiant heat flux qrad is absorbed by themembrane, heat flows from the interior of the membrane tothe copper heat sink at the edge, creating a radial temperature gradient.
Copper leads are welded to the center of themembrane and to the copper heat sink, making two copperconstantan thermocouple junctions. These junctions measurethe temperature difference ∆T between the center of the membrane, T (r = 0), and the edge of the membrane, T (r = R).The following approximations can be made:•The membrane surface has been blackened so that it absorbs all radiation that falls on it•The radiant heat flux is much larger than the heat lostfrom the membrane by convection or re-radiation.
Thus,all absorbed radiant heat is removed from the membraneby conduction to the copper heat sink, and other losescan be ignored•The gage operates in steady state•The membrane is thin enough (t R) that the temperature in it varies only with r , i.e., T = T (r ) only.Answer the following questions.96Chapter 2: Heat conduction, thermal resistance, and the overall heat transfer coefficienta. For a fixed copper heat sink temperature, T (r = R), sketchthe shape of the temperature distribution in the membrane, T (r ), for two arbitrary heat radiant fluxes qrad 1and qrad 2 , where qrad 1 > qrad 2 .b.
Find the relationship between the radiant heat flux, qrad ,and the temperature difference obtained from the thermocouples, ∆T . Hint: Treat the absorbed radiant heatflux as if it were a volumetric heat source of magnitudeqrad /t (W/m3 ).2.45You have a 12 oz. (375 mL) can of soda at room temperature(70◦ F) that you would like to cool to 45◦ F before drinking. Yourest the can on its side on the plastic rods of the refrigeratorshelf. The can is 2.5 inches in diameter and 5 inches long.The can’s emissivity is ε = 0.4 and the natural convection heattransfer coefficient around it is a function of the temperaturedifference between the can and the air: h = 2 ∆T 1/4 for ∆T inkelvin.Assume that thermal interactions with the refrigerator shelfare negligible and that buoyancy currents inside the can willkeep the soda well mixed.a. Estimate how long it will take to cool the can in the refrigerator compartment, which is at 40◦ F.b.
Estimate how long it will take to cool the can in the freezercompartment, which is at 5◦ F.c. Are your answers for parts 1 and 2 the same? If not, whatis the main reason that they are different?References[2.1] W. M. Rohsenow and J. P. Hartnett, editors. Handbook of HeatTransfer. McGraw-Hill Book Company, New York, 1973.[2.2] R.
F. Wheeler. Thermal conductance of fuel element materials.USAEC Rep. HW-60343, April 1959.[2.3] M. M. Yovanovich. Recent developments in thermal contact, gapand joint conductance theories and experiment. In Proc. Eight Intl.Heat Transfer Conf., volume 1, pages 35–45.
San Francisco, 1986.References[2.4] C. V. Madhusudana. Thermal Contact Conductance. SpringerVerlag, New York, 1996.[2.5] American Society of Heating, Refrigerating, and Air-ConditioningEngineers, Inc. 2001 ASHRAE Handbook—Fundamentals. Altanta,2001.[2.6] R. K. Shah and D. P. Sekulic. Heat exchangers. In W. M. Rohsenow,J. P. Hartnett, and Y. I. Cho, editors, Handbook of Heat Transfer,chapter 17. McGraw-Hill, New York, 3rd edition, 1998.[2.7] Tubular Exchanger Manufacturer’s Association. Standards ofTubular Exchanger Manufacturer’s Association. New York, 4th and6th edition, 1959 and 1978.[2.8] H.
Müller-Steinhagen. Cooling-water fouling in heat exchangers. InT. F. Irvine, Jr., J. P. Hartnett, Y. I. Cho, and G. A. Greene, editors,Advances in Heat Transfer, volume 33, pages 415–496. AcademicPress, Inc., San Diego, 1999.[2.9] W. J. Marner and J. W. Suitor. Fouling with convective heat transfer.In S.
Kakaç, R. K. Shah, and W. Aung, editors, Handbook of SinglePhase Convective Heat Transfer, chapter 21. Wiley-Interscience,New York, 1987.[2.10] R. Gardon. An instrument for the direct measurement of intensethermal radiation. Rev. Sci. Instr., 24(5):366–371, 1953.Most of the ideas in Chapter 2 are also dealt with at various levels inthe general references following Chapter 1.973.Heat exchanger designThe great object to be effected in the boilers of these engines is, to keepa small quantity of water at an excessive temperature, by means of asmall amount of fuel kept in the most active state of combustion. . .Nocontrivance can be less adapted for the attainment of this end than one ortwo large tubes traversing the boiler, as in the earliest locomotive engines.The Steam Engine Familiarly Explained and Illustrated,Dionysus Lardner, 18363.1Function and configuration of heat exchangersThe archetypical problem that any heat exchanger solves is that of getting energy from one fluid mass to another, as we see in Fig.
3.1. A simpleor composite wall of some kind divides the two flows and provides anelement of thermal resistance between them. There is an exception tothis configuration in the direct-contact form of heat exchanger. Figure3.2 shows one such arrangement in which steam is bubbled into water.The steam condenses and the water is heated at the same time. In otherarrangements, immiscible fluids might contact each other or noncondensible gases might be bubbled through liquids.This discussion will be restricted to heat exchangers with a dividingwall between the two fluids. There is an enormous variety of such configurations, but most commercial exchangers reduce to one of three basictypes.
Figure 3.3 shows these types in schematic form. They are:• The simple parallel or counterflow configuration. These arrangements are versatile. Figure 3.4 shows how the counterflow arrangement is bent around in a so-called Heliflow compact heat exchangerconfiguration.• The shell-and-tube configuration. Figure 3.5 shows the U-tubes ofa two-tube-pass, one-shell-pass exchanger being installed in the99100Heat exchanger design§3.1Figure 3.1 Heat exchange.supporting baffles. The shell is yet to be added. Most of the really large heat exchangers are of the shell-and-tube form.• The cross-flow configuration. Figure 3.6 shows typical cross-flowunits.
In Fig. 3.6a and c, both flows are unmixed. Each flow muststay in a prescribed path through the exchanger and is not allowedto “mix” to the right or left. Figure 3.6b shows a typical plate-fincross-flow element. Here the flows are also unmixed.Figure 3.7, taken from the standards of the Tubular Exchanger Manufacturer’s Association (TEMA) [3.1], shows four typical single-shell-passheat exchangers and establishes nomenclature for such units.These pictures also show some of the complications that arise intranslating simple concepts into hardware.
Figure 3.7 shows an exchanger with a single tube pass. Although the shell flow is baffled so that itcrisscrosses the tubes, it still proceeds from the hot to cold (or cold tohot) end of the shell. Therefore, it is like a simple parallel (or counterflow) unit. The kettle reboiler in Fig. 3.7d involves a divided shell-passflow configuration over two tube passes (from left to right and back to the“channel header”). In this case, the isothermal shell flow could be flowingin any direction—it makes no difference to the tube flow.
Therefore, thisexchanger is also equivalent to either the simple parallel or counterflowconfiguration.Function and configuration of heat exchangers§3.1Figure 3.2 A direct-contact heat exchanger.Notice that a salient feature of shell-and-tube exchangers is the presence of baffles. Baffles serve to direct the flow normal to the tubes. Wefind in Part III that heat transfer from a tube to a flowing fluid is usuallybetter when the flow moves across the tube than when the flow movesalong the tube. This augmentation of heat transfer gives the complicatedshell-and-tube exchanger an advantage over the simpler single-pass parallel and counterflow exchangers.However, baffles bring with them a variety of problems.
The flow patterns are very complicated and almost defy analysis. A good deal of theshell-side fluid might unpredictably leak through the baffle holes in theaxial direction, or it might bypass the baffles near the wall. In certainshell-flow configurations, unanticipated vibrational modes of the tubesmight be excited. Many of the cross-flow configurations also baffle thefluid so as to move it across a tube bundle. The plate-and-fin configuration (Fig. 3.6b) is such a cross-flow heat exchanger.In all of these heat exchanger arrangements, it becomes clear that adramatic investment of human ingenuity is directed towards the task ofaugmenting the heat transfer from one flow to another.
The variationsare endless, as you will quickly see if you try Experiment 3.1.Experiment 3.1Carry a notebook with you for a day and mark down every heat exchanger you encounter in home, university, or automobile. Classify eachaccording to type and note any special augmentation features.The analysis of heat exchangers first becomes complicated when weaccount for the fact that two flow streams change one another’s temper-101Figure 3.3 The three basic types of heat exchangers.102§3.2Evaluation of the mean temperature difference in a heat exchangerFigure 3.4 Heliflow compact counterflow heat exchanger.(Photograph coutesy of Graham Manufacturing Co., Inc.,Batavia, New York.)ature.
It is to the problem of predicting an appropriate mean temperature difference that we address ourselves in Section 3.2. Section 3.3 thenpresents a strategy to use when this mean cannot be determined initially.3.2Evaluation of the mean temperature differencein a heat exchangerLogarithmic mean temperature difference (LMTD)To begin with, we take U to be a constant value. This is fairly reasonablein compact single-phase heat exchangers. In larger exchangers, particularly in shell-and-tube configurations and large condensers, U is apt tovary with position in the exchanger and/or with local temperature. Butin situations in which U is fairly constant, we can deal with the varyingtemperatures of the fluid streams by writing the overall heat transfer interms of a mean temperature difference between the two fluid streams:Q = U A ∆Tmean(3.1)103Figure 3.5 Typical commercial one-shell-pass, two-tube-passheat exchangers.104a.
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