This book is meant for students in their introductory heat transfer course — students who have learned calculus (through ordinary differential equations) and basic thermodynamics. We include the needed background in fluid mechanics, although students will be better off if they have had an introductory course in fluids. An integrated introductory course in thermofluid engineering should also be a sufficient background for the material here. Our major objectives in rewriting the 1987 edition have been to bring the material up to date and make it as clear as possible. We have substantially revised the coverage of thermal radiation, unsteady conduction, and mass transfer. We have replaced most of the old physical property data with the latest reference data. New correlations have been introduced for forced and natural convection and for convective boiling. The treatment of thermal resistance has been reorganized. Dozens of new problems have been added. And we have revised the treatment of turbulent heat transfer to include the use of the law of the wall. In a number of places we have rearranged material to make it flow better, and we have made many hundreds of small changes and corrections so that the text will be more comfortable and reliable. Lastly, we have eliminated Roger Eichhorn’s fine chapter on numerical analysis, since that topic is now most often covered in specialized courses on computation.
Part III Convective Heat Transfer 267 Laminar and turbulent boundary layers In cold weather, if the air is calm, we are not so much chilled as when there is wind along with the cold; for in calm weather, our clothes and the air entangled in them receive heat from our bodies; this heat .brings them nearer than the surrounding air to the temperature of our skin But in windy weather, this heat is prevented .from accumulating; the cold air, by its impulse .both cools our clothes faster and carries away the warm air that was entangled in them notes on “The General Effects of Heat”, Joseph Black, c 1790s 6.1 Some introductory ideas Joseph Black’s perception about forced convection (above) represents a very correct understanding of the way forced convective cooling works When cold air moves past a warm body, it constantly sweeps away warm air that has become, as Black put it, “entangled” with the body and replaces it with cold air In this chapter we learn to form analytical descriptions of these convective heating (or cooling) processes Our aim is to predict h and h, and it is clear that such predictions must begin in the motion of fluid around the bodies that they heat or cool It is by predicting such motion that we will be able to find out how much heat is removed during the replacement of hot fluid with cold, and vice versa Flow boundary layer Fluids flowing past solid bodies adhere to them, so a region of variable velocity must be built up between the body and the free fluid stream, as 269 Laminar and turbulent boundary layers 270 Figure 6.1 §6.1 A boundary layer of thickness δ indicated in Fig 6.1 This region is called a boundary layer, which we will often abbreviate as b.l The b.l has a thickness, δ The boundary layer thickness is arbitrarily defined as the distance from the wall at which the flow velocity approaches to within 1% of u∞ The boundary layer is normally very thin in comparison with the dimensions of the body immersed in the flow.1 The first step that has to be taken before h can be predicted is the mathematical description of the boundary layer This description was first made by Prandtl2 (see Fig 6.2) and his students, starting in 1904, and it depended upon simplifications that followed after he recognized how thin the layer must be The dimensional functional equation for the boundary layer thickness on a flat surface is δ = fn(u∞ , ρ, µ, x) where x is the length along the surface and ρ and µ are the fluid density in kg/m3 and the dynamic viscosity in kg/m·s We have five variables in We qualify this remark when we treat the b.l quantitatively Prandtl was educated at the Technical University in Munich and finished his doctorate there in 1900 He was given a chair in a new fluid mechanics institute at Göttingen University in 1904—the same year that he presented his historic paper explaining the boundary layer His work at Göttingen, during the period up to Hitler’s regime, set the course of modern fluid mechanics and aerodynamics and laid the foundations for the analysis of heat convection Some introductory ideas §6.1 271 Figure 6.2 Ludwig Prandtl (1875–1953) (Courtesy of Appl Mech Rev [6.1]) kg, m, and s, so we anticipate two pi-groups: δ = fn(Rex ) x Rex ≡ u∞ x ρu∞ x = µ ν (6.1) where ν is the kinematic viscosity µ/ρ and Rex is called the Reynolds number It characterizes the relative influences of inertial and viscous forces in a fluid problem The subscript on Re—x in this case—tells what length it is based upon We discover shortly that the actual form of eqn (6.1) for a flat surface, where u∞ remains constant, is 4.92 δ = x Rex (6.2) which means that if the velocity is great or the viscosity is low, δ/x will be relatively small Heat transfer will be relatively high in such cases If the velocity is low, the b.l will be relatively thick A good deal of nearly 272 Laminar and turbulent boundary layers §6.1 Osborne Reynolds (1842 to 1912) Reynolds was born in Ireland but he taught at the University of Manchester He was a significant contributor to the subject of fluid mechanics in the late 19th C His original laminar-toturbulent flow transition experiment, pictured below, was still being used as a student experiment at the University of Manchester in the 1970s Figure 6.3 Osborne Reynolds and his laminar–turbulent flow transition experiment (Detail from a portrait at the University of Manchester.) stagnant fluid will accumulate near the surface and be “entangled” with the body, although in a different way than Black envisioned it to be The Reynolds number is named after Osborne Reynolds (see Fig 6.3), who discovered the laminar–turbulent transition during fluid flow in a tube He injected ink into a steady and undisturbed flow of water and found that, beyond a certain average velocity, uav , the liquid streamline marked with ink would become wobbly and then break up into increasingly disorderly eddies, and it would finally be completely mixed into the Some introductory ideas §6.1 273 Figure 6.4 Boundary layer on a long, flat surface with a sharp leading edge water, as is suggested in the sketch To define the transition, we first note that (uav )crit , the transitional value of the average velocity, must depend on the pipe diameter, D, on µ, and on ρ—four variables in kg, m, and s There is therefore only one pi-group: Recritical ≡ ρD(uav )crit µ (6.3) The maximum Reynolds number for which fully developed laminar flow in a pipe will always be stable, regardless of the level of background noise, is 2100 In a reasonably careful experiment, laminar flow can be made to persist up to Re = 10, 000 With enormous care it can be increased still another order of magnitude But the value below which the flow will always be laminar—the critical value of Re—is 2100 Much the same sort of thing happens in a boundary layer Figure 6.4 shows fluid flowing over a plate with a sharp leading edge The flow is laminar up to a transitional Reynolds number based on x: Rexcritical = u∞ xcrit ν (6.4) At larger values of x the b.l exhibits sporadic vortexlike instabilities over a fairly long range, and it finally settles into a fully turbulent b.l 274 Laminar and turbulent boundary layers §6.1 For the boundary layer shown, Rexcritical = 3.5 × 105 , but in general the critical Reynolds number depends strongly on the amount of turbulence in the freestream flow over the plate, the precise shape of the leading edge, the roughness of the wall, and the presence of acoustic or structural vibrations [6.2, §5.5] On a flat plate, a boundary layer will remain laminar even when such disturbances are very large if Rex ≤ × 104 With relatively undisturbed conditions, transition occurs for Rex in the range of × 105 to × 105 , and in very careful laboratory experiments, turbulent transition can be delayed until Rex ≈ × 106 or so Turbulent transition is essentially always complete before Rex = 4×106 and usually much earlier These specifications of the critical Re are restricted to flat surfaces If the surface is curved away from the flow, as shown in Fig 6.1, turbulence might be triggered at much lower values of Rex Thermal boundary layer If the wall is at a temperature Tw , different from that of the free stream, T∞ , there is a thermal boundary layer thickness, δt —different from the flow b.l thickness, δ A thermal b.l is pictured in Fig 6.5 Now, with reference to this picture, we equate the heat conducted away from the wall by the fluid to the same heat transfer expressed in terms of a convective heat transfer coefficient: −kf ∂T ∂y = h(Tw − T∞ ) (6.5) y=0 conduction into the fluid where kf is the conductivity of the fluid Notice two things about this result In the first place, it is correct to express heat removal at the wall using Fourier’s law of conduction, because there is no fluid motion in the direction of q The other point is that while eqn (6.5) looks like a b.c of the third kind, it is not This condition defines h within the fluid instead of specifying it as known information on the boundary Equation (6.5) can be arranged in the form ∂ Tw − T Tw − T ∞ ∂(y/L) = y/L=0 hL = NuL , the Nusselt number kf (6.5a) §6.1 Some introductory ideas 275 Figure 6.5 The thermal boundary layer during the flow of cool fluid over a warm plate where L is a characteristic dimension of the body under consideration— the length of a plate, the diameter of a cylinder, or [if we write eqn (6.5) at a point of interest along a flat surface] Nux ≡ hx/kf From Fig 6.5 we see immediately that the physical significance of Nu is given by NuL = L δt (6.6) In other words, the Nusselt number is inversely proportional to the thickness of the thermal b.l The Nusselt number is named after Wilhelm Nusselt,3 whose work on convective heat transfer was as fundamental as Prandtl’s was in analyzing the related fluid dynamics (see Fig 6.6) We now turn to the detailed evaluation of h And, as the preceding remarks make very clear, this evaluation will have to start with a development of the flow field in the boundary layer Nusselt finished his doctorate in mechanical engineering at the Technical University in Munich in 1907 During an indefinite teaching appointment at Dresden (1913 to 1917) he made two of his most important contributions: He did the dimensional analysis of heat convection before he had access to Buckingham and Rayleigh’s work In so doing, he showed how to generalize limited data, and he set the pattern of subsequent analysis He also showed how to predict convective heat transfer during film condensation After moving about Germany and Switzerland from 1907 until 1925, he was named to the important Chair of Theoretical Mechanics at Munich During his early years in this post, he made seminal contributions to heat exchanger design methodology He held this position until 1952, during which time his, and Germany’s, great influence in heat transfer and fluid mechanics waned He was succeeded in the chair by another of Germany’s heat transfer luminaries, Ernst Schmidt 276 Laminar and turbulent boundary layers §6.2 Figure 6.6 Ernst Kraft Wilhelm Nusselt (1882–1957) This photograph, provided by his student, G Lück, shows Nusselt at the Kesselberg waterfall in 1912 He was an avid mountain climber 6.2 Laminar incompressible boundary layer on a flat surface We predict the boundary layer flow field by solving the equations that express conservation of mass and momentum in the b.l Thus, the first order of business is to develop these equations Conservation of mass—The continuity equation A two- or three-dimensional velocity field can be expressed in vectorial form: u = iu + jv + kw where u, v, and w are the x, y, and z components of velocity Figure 6.7 shows a two-dimensional velocity flow field If the flow is steady, the paths of individual particles appear as steady streamlines The streamlines can be expressed in terms of a stream function, ψ(x, y) = constant, where each value of the constant identifies a separate streamline, as shown in the figure The velocity, u, is directed along the streamlines so that no flow can cross them Any pair of adjacent streamlines thus resembles a heat flow ... time his, and Germany’s, great influence in heat transfer and fluid mechanics waned He was succeeded in the chair by another of Germany’s heat transfer luminaries, Ernst Schmidt 276 Laminar and turbulent... envision a boxcar moving down the railroad track with a man standing, facing its open door A child standing at a crossing throws him a baseball as the car passes When he catches the ball, its... closely at the implications of the similarity between the velocity and thermal boundary layers We first ask what dimensional analysis reveals about heat transfer in the laminar b.l We know by now that