Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 6
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From this point of view, Eq. 1.18 can be taken as the generalizeddefinition of the heat transfer coefficient.Boundary Layer Concept.The transfer of heat between a solid body and a liquid or gas flowis a problem whose consideration involves the science of fluid motion. On the physical motionof the fluid there is superimposed a flow of heat, and the two fields interact. In order to determine the temperature distribution and then the heat transfer coefficient (Eq. 1.14) it is necessary to combine the equations of motion with the energy conservation equation.
However, acomplete solution for the flow of a viscous fluid about a body poses considerable mathematical difficulty for all but the most simple flow geometries. A great practical breakthrough wasmade when Prandtl discovered that for most applications the influence of viscosity is confinedto an extremely thin region very close to the body and that the remainder of the flow fieldcould to a good approximation be treated as inviscid, i.e., could be calculated by the methodof potential flow theory.The thin region near the body surface, which is known asthe boundary layer, lends itself to relatively simple analysisPotentialby the very fact of its thinness relative to the dimensions offlow, .-" "region 1 1 1the body. A fundamental assumption of the boundary layeru~l~....~-~approximation is that the fluid immediately adjacent to the,,~bo~r~ ~ y ~ ,body surface is at rest relative to the body, an assumption"~, , , " v-"-j~lu req~n~n,r[fffflllll//'"that appears to be valid except for very low-pressure gases,when the mean free path of the gas molecules is large rela,y////I///"tive to the body [6].
Thus the hydrodynamic or velocityboundary layer 5 may be defined as the region in which thefluid velocity changes from its free-stream, or potential flow,value to zero at the body surface (Fig. 1.3). In reality there isF I G U R E 1.3 Boundary layer flow past an extemalno precise "thickness" to a boundary layer defined in thissurface.manner, since the velocity asymptotically approaches thefree-stream value.
In practice we simply imply that theboundary layer thickness is the distance in which most of the velocity change takes place.The viscous forces within the boundary layer region are described in terms of the shearstress x between the fluid layers. If this stress is assumed to be proportional to the normalvelocity gradient, we have the defining equation for viscosityduT.=~t dy(1.20)The constant of proportionality la is called the dynamic viscosity (Pa-s), and Eq.
1.20 is sometimes referred to as Newton's law of shear [7] for a simple flow in which only the velocity component u exists. The ratio of the viscosity l.t to the density p is known as the kinematic viscosity(m2/s) and is defined asv-~t(1.21)PFlow inside a tube is a form of boundary layer problem in which, near the tube entrance,the boundary layer grows in much the same manner as over an external surface until itsgrowth is stopped by symmetry at the centerline of the tube (Fig. 1.4).
Thus the tube radiusbecomes the ultimate boundary layer thickness.When there is heat transfer or mass transfer between the fluid and the surface, it is alsofound that in most practical applications the major temperature and concentration changesoccur in a region very close to the surface. This gives rise to the concept of the thermal boundary layer ~)rand the concentration boundary layer ?h~.The influence of thermal conductivity kand mass diffusivity D is confined within these regions.
Outside the boundary layer region theflow is essentially nonconducting and nondiffusing. The thermal (or concentration) boundarylayer may be smaller than, larger than, or the same size as the velocity boundary layer. Thedevelopment of the thermal boundary layer in the entrance region of a tube is shown in Fig. 1.5.BASIC CONCEPTS OF HEAT TRANSFER1.7layerinletflow-~>--- ---- -...._>~ -8---- - - . . . . . ~ . ~ , .~---~-~,"L.F"F I G U R E 1.4flow~r~d-"r.3IEntrance lengthVelocity profile for laminar flow in a tube.Inletflowat uniformtemperature/ - - - Thermal boundary layer~ T.-~_•/ I -'•/---ri.,/-Tm~T(r,x)I_F"Entrance length.3r1F I G U R E 1.5 The development of temperature profile in the entranceregion of a tube.It is important to notice the similarity between Eqs. 1.1, 1.6, and 1.20.
The heat conductionequation, Eq. 1.1, describes the transport of energy; the diffusion law, Eq. 1.6, describes thetransport of mass; and the viscous shear equation, Eq. 1.20, describes the transport of momentum across fluid layers. We note also that the kinematic viscosity v, the thermal diffusivity o~,and the diffusion coefficient D all have the same dimensions L2/t. As shown in Table 1.10, adimensionless number can be formed from the ratio of any two of these quantities, which willgive relative speeds at which momentum, energy, and mass diffuse through the medium.Laminar and Turbulent Flows.
There are basically two different types of fluid motion,identified as laminar and turbulent flow. In previous sections we referred basically to laminarflow.In the case of flow over a flat plate (Fig. 1.6), the flow near the leading edge is smooth andstreamlined. Locally within the boundary layer the velocity is constant and invariant withtime.
The boundary layer thickness grows with increasing distance from the leading edge, andat some critical distance the inertial effects become sufficiently large compared to the viscousdamping action that small disturbances in the flow begin to grow. As these disturbancesLaminar "---~Transition ~--~TurbulentIF I G U R E 1.6 Laminar, transition, and turbulent boundary layer flowregimes in flow over a flat plate.1.8CHAPTER ONEbecome amplified, the regularity of the viscous flow is disturbed and a transition from laminar to turbulent flow takesplace. (However, there still must be a very thin laminar sub,,~mV-,vr-wna-W Tlayer next to the wall, at least for a smooth plate.) These disturbances may originate from the free stream or may beinduced by surface roughness.~In the turbulent flow region a very efficient mixing takestplace, i.e., macroscopic chunks of fluid move across streamFIGURE 1.7 Property variation with time at somelines and transport energy and mass as well as momentumpoint in a turbulent boundary layer.vigorously.
The most essential feature of a turbulent flow isthe fact that at a given point in it, the flow property X (e.g.,velocity component, pressure, temperature, or a species concentration) is not constant withtime but exhibits very irregular, high-frequency fl__uctuations (Fig. 1.7). At any instant, X maybe represented as the sum of a time-mean value X and a fluctuating component X'. The average is taken over a time that is large compared with the period of typical fluctuation, and if Xis independent of time, the time-mean flow is said to be steady.The existence of turbulent flow can be advantageous in the sense of providing increasedheat and mass transfer rates.
However, the motion is extremely complicated and difficult todescribe theoretically [3, 8]. In dealing with turbulent flow it is customary to speak of a totalshear stress and total fluxes normal to the main flow direction (the main flow is in the x direction, and the y axis is normal to the flow direction), which are defined asa~"r.,= la --~y - pU"v"(1.22)aT - pCpv'T')q;'= -(k -~y(1.23)jl.,=-(D ~OC'- -~C--~)(1.24)where the first term on the right side of Eqs. 1.22-1.24 is the contribution due to moleculardiffusion and the second term is the contribution due to turbulent mixing.
For example, u'v' isthe time average of the product of u' and v'.A simple conceptual model for turbulent flow deals with eddies, small portions of fluid inthe boundary layer that move about for a short time before losing their identity [8]. The transport coefficient, which is defined as eddy diffusivity for momentum transfer ~M, has the form8~M -b-Y-y= -u'v'(1.25)Similarly, eddy diffusivities for heat and mass transfer, ~n and ~m, respectively, may be definedby the relationsbTeI4 ~ =-v'T"bC1Em - ~ y-" --I) t f t l(1.26)(1.27)Hence the total shear stress and total fluxes may be expressed, with the help of the relationsof Eqs.
1.5 and 1.21, asb~x, : p(v + ,M) by(1.28)BASIC CONCEPTS OF HEAT TRANSFER1.9n3Tqt = -pcp(ct + if,) by(1.29)3C1jl,t = -(]D + ifm) -~y(1.30)In the region of a turbulent boundary layer far from the surface (the core region), the eddydiffusivities are much larger than the molecular diffusivities. The enhanced mixing associatedwith this condition has the effect of making velocity, temperature, and concentration profilesmore uniform in the core. This behavior is shown in Fig.
1.8, which gives the measured velocity distributions for laminar and turbulent flow where themass flow is the same in both cases [7]. It is evident from Fig.1.8 that the velocity gradient at the surface, and therefore thesurface shear stress, is much larger for turbulent flow thanfor laminar flow. Following the sameargument, the tempera(a)(b)ture or concentration gradient at the surface, and thereforeF I G U R E 1.8 Velocity distribution in a tube: (a) lamthe heat and mass transfer rates, are much larger for turbuinar; (b) turbulent.lent than for laminar flow.
When the flow in the tube is turbulent, the mean velocity is about 83 percent of the centervelocity. For laminar flow, the profile has a parabolic shape and the mean velocity is one-halfthe value at the center.A fundamental problem in performing a turbulent flow analysis involves determining theeddy diffusivities as a function of the mean properties of the flow. Unlike the molecular diffusivities, which are strictly fluid properties, the eddy diffusivities depend strongly on thenature of the flow; they can vary from point to point in a boundary layer, and the specific variation can be determined only from experimental data.For flow in circular tubes, the numerical value of the Reynolds number (defined in Table1.10), based on mean velocity at which transition from laminar to turbulent flow occurs, wasestablished as being approximately 2300, i.e.,Reef=( VmDcr=2300(1.31)There exists, however, as demonstrated by numerous experiments [7], a lower value for Recrthat is approximately at 2000.
Below this value the flow remains laminar even in the presenceof very strong disturbances. If the Reynolds number is greater than 10,000, the flow is considered to be fully turbulent. In the 2300 to 10,000 region, the flow is often described as transition flow. It is possible to shift these values by minimizing the disturbances in the inlet flow,but for general engineering application the numbers cited are representative.For a flow over a flat plate, the transition to turbulent flow takes place at distance x, measured from the leading edge, as determined byRecr :aP:SX 10Sto 106(1.32)aPOx < OIj ~ - > 0Iu~(x)IIFlow reversal J"~"VorticesF I G U R E 1.9 Velocity profile associated with separation on a circular cylinder in cross flow.but the values are dependent on the level of turbulence inthe main stream.
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