John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 49
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Basic Engr., 91:371–378, 1969.[6.11] J. Boussinesq. Théorie de l’écoulement tourbillant. Mem. Pres.Acad. Sci., (Paris), 23:46, 1877.[6.12] F. M. White. Viscous Fluid Flow. McGraw-Hill Book Company, NewYork, 1974.References[6.13] B. S. Petukhov. Heat transfer and friction in turbulent pipe flowwith variable physical properties. In T.F. Irvine, Jr. and J. P.
Hartnett, editors, Advances in Heat Transfer, volume 6, pages 504–564.Academic Press, Inc., New York, 1970.[6.14] A. A. Žukauskas and A. B. Ambrazyavichyus. Heat transfer froma plate in a liquid flow. Int. J. Heat Mass Transfer, 3(4):305–309,1961.[6.15] A. Žukauskas and A. Šlanciauskas. Heat Transfer in TurbulentFluid Flows.
Hemisphere Publishing Corp., Washington, 1987.[6.16] S. Whitaker. Forced convection heat transfer correlation for flowin pipes past flat plates, single cylinders, single spheres, and forflow in packed beds and tube bundles. AIChE J., 18:361, 1972.[6.17] S. W. Churchill. A comprehensive correlating equation for forcedconvection from flat plates. AIChE J., 22:264–268, 1976.[6.18] S.
B. Pope. Turbulent Flows. Cambridge University Press, Cambridge, 2000.[6.19] P. A. Libby. Introduction to Turbulence. Taylor & Francis, Washington D.C., 1996.3397.Forced convection in a variety ofconfigurationsThe bed was soft enough to suit me. . .But I soon found that there camesuch a draught of cold air over me from the sill of the window that thisplan would never do at all, especially as another current from the ricketydoor met the one from the window and both together formed a series ofsmall whirlwinds in the immediate vicinity of the spot where I had thoughtto spend the night.Moby Dick, H.
Melville, 18517.1IntroductionConsider for a moment the fluid flow pattern within a shell-and-tube heatexchanger, such as that shown in Fig. 3.5. The shell-pass flow moves upand down across the tube bundle from one baffle to the next. The flowaround each pipe is determined by the complexities of the one before it,and the direction of the mean flow relative to each pipe can vary.
Yetthe problem of determining the heat transfer in this situation, howeverdifficult it appears to be, is a task that must be undertaken.The flow within the tubes of the exchanger is somewhat more tractable,but it, too, brings with it several problems that do not arise in the flow offluids over a flat surface. Heat exchangers thus present a kind of microcosm of internal and external forced convection problems. Other suchproblems arise everywhere that energy is delivered, controlled, utilized,or produced. They arise in the complex flow of water through nuclearheating elements or in the liquid heating tubes of a solar collector—inthe flow of a cryogenic liquid coolant in certain digital computers or inthe circulation of refrigerant in the spacesuit of a lunar astronaut.We dealt with the simple configuration of flow over a flat surface in341342Forced convection in a variety of configurations§7.2Chapter 6.
This situation has considerable importance in its own right,and it also reveals a number of analytical methods that apply to otherconfigurations. Now we wish to undertake a sequence of progressivelyharder problems of forced convection heat transfer in more complicatedflow configurations.Incompressible forced convection heat transfer problems normallyadmit an extremely important simplification: the fluid flow problem canbe solved without reference to the temperature distribution in the fluid.Thus, we can first find the velocity distribution and then put it in theenergy equation as known information and solve for the temperaturedistribution.
Two things can impede this procedure, however:• If the fluid properties (especially µ and ρ) vary significantly withtemperature, we cannot predict the velocity without knowing thetemperature, and vice versa. The problems of predicting velocityand temperature become intertwined and harder to solve. We encounter such a situation later in the study of natural convection,where the fluid is driven by thermally induced density changes.• Either the fluid flow solution or the temperature solution can, itself,become prohibitively hard to find. When that happens, we resort tothe correlation of experimental data with the help of dimensionalanalysis.Our aim in this chapter is to present the analysis of a few simpleproblems and to show the progression toward increasingly empirical solutions as the problems become progressively more unwieldy.
We beginthis undertaking with one of the simplest problems: that of predictinglaminar convection in a pipe.7.2Heat transfer to and from laminar flows in pipesNot many industrial pipe flows are laminar, but laminar heating and cooling does occur in an increasing variety of modern instruments and equipment: micro-electro-mechanical systems (MEMS), laser coolant lines, andmany compact heat exchangers, for example. As in any forced convectionproblem, we first describe the flow field. This description will include anumber of ideas that apply to turbulent as well as laminar flow.§7.2Heat transfer to and from laminar flows in pipesFigure 7.1 The development of a laminar velocity profile in a pipe.Development of a laminar flowFigure 7.1 shows the evolution of a laminar velocity profile from the entrance of a pipe.
Throughout the length of the pipe, the mass flow rate,ṁ (kg/s), is constant, of course, and the average, or bulk, velocity uav isalso constant:ρu dAc = ρuav Ac(7.1)ṁ =Acwhere Ac is the cross-sectional area of the pipe. The velocity profile, onthe other hand, changes greatly near the inlet to the pipe. A b.l. buildsup from the front, generally accelerating the otherwise undisturbed core.The b.l. eventually occupies the entire flow area and defines a velocity profile that changes very little thereafter. We call such a flow fully developed.A flow is fully developed from the hydrodynamic standpoint when∂u=0∂xorv=0(7.2)at each radial location in the cross section. An attribute of a dynamicallyfully developed flow is that the streamlines are all parallel to one another.The concept of a fully developed flow, from the thermal standpoint,is a little more complicated.
We must first understand the notion of themixing-cup, or bulk, enthalpy and temperature, ĥb and Tb . The enthalpyis of interest because we use it in writing the First Law of Thermodynamics when calculating the inflow of thermal energy and flow work to opencontrol volumes. The bulk enthalpy is an average enthalpy for the fluid343344Forced convection in a variety of configurations§7.2flowing through a cross section of the pipe:ρuĥ dAcṁ ĥb ≡(7.3)AcIf we assume that fluid pressure variations in the pipe are too small toaffect the thermodynamic state much (see Sect.
6.3) and if we assume aconstant value of cp , then ĥ = cp (T − Tref ) andṁ cp (Tb − Tref ) =ρcp u (T − Tref ) dAc(7.4)Acor simplyTb =Acρcp uT dAcṁcp(7.5)In words, then,Tb ≡rate of flow of enthalpy through a cross sectionrate of flow of heat capacity through a cross sectionThus, if the pipe were broken at any x-station and allowed to dischargeinto a mixing cup, the enthalpy of the mixed fluid in the cup would equalthe average enthalpy of the fluid flowing through the cross section, andthe temperature of the fluid in the cup would be Tb .
This definition of Tbis perfectly general and applies to either laminar or turbulent flow. Fora circular pipe, with dAc = 2π r dr , eqn. (7.5) becomesRρcp uT 2π r dr(7.6)Tb = 0 Rρcp u 2π r dr0A fully developed flow, from the thermal standpoint, is one for whichthe relative shape of the temperature profile does not change with x. Westate this mathematically as∂Tw − T=0(7.7)∂x Tw − Tbwhere T generally depends on x and r . This means that the profile canbe scaled up or down with Tw − Tb . Of course, a flow must be hydrodynamically developed if it is to be thermally developed.§7.2Heat transfer to and from laminar flows in pipesFigure 7.2 The thermal development of flows in tubes witha uniform wall heat flux and with a uniform wall temperature(the entrance region).Figures 7.2 and 7.3 show the development of two flows and their subsequent behavior. The two flows are subjected to either a uniform wallheat flux or a uniform wall temperature. In Fig.
7.2 we see each flow develop until its temperature profile achieves a shape which, except for alinear stretching, it will retain thereafter. If we consider a small length ofpipe, dx long with perimeter P , then its surface area is P dx (e.g., 2π R dxfor a circular pipe) and an energy balance on it is1dQ = qw P dx = ṁdĥb= ṁcp dTb(7.8)(7.9)so thatqw PdTb=ṁcpdx1(7.10)Here we make the same approximations as were made in deriving the energy equation in Sect. 6.3.345346Forced convection in a variety of configurations§7.2Figure 7.3 The thermal behavior of flows in tubes with a uniform wall heat flux and with a uniform temperature (the thermally developed region).This result is also valid for the bulk temperature in a turbulent flow.In Fig. 7.3 we see the fully developed variation of the temperatureprofile.
If the flow is fully developed, the boundary layers are no longergrowing thicker, and we expect that h will become constant. When qw isconstant, then Tw − Tb will be constant in fully developed flow, so thatthe temperature profile will retain the same shape while the temperaturerises at a constant rate at all values of r . Thus, at any radial position,dTbqw P∂T=== constant∂xdxṁcp(7.11)In the uniform wall temperature case, the temperature profile keepsthe same shape, but its amplitude decreases with x, as does qw . Thelower right-hand corner of Fig. 7.3 has been drawn to conform with thisrequirement, as expressed in eqn. (7.7).Heat transfer to and from laminar flows in pipes§7.2The velocity profile in laminar tube flowsThe Buckingham pi-theorem tells us that if the hydrodynamic entry length,xe , required to establish a fully developed velocity profile depends onuav , µ, ρ, and D in three dimensions (kg, m, and s), then we expect tofind two pi-groups:xe= fn (ReD )Dwhere ReD ≡ uav D/ν.
The matter of entry length is discussed by White[7.1, Chap. 4], who quotesxe 0.03 ReDD(7.12)The constant, 0.03, guarantees that the laminar shear stress on the pipewall will be within 5% of the value for fully developed flow when x >xe . The number 0.05 can be used, instead, if a deviation of just 1.4% isdesired.
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