Lecture notes on Convection Heat Transfer

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Part III Convective Heat Transfer 2676. 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 re- places it with cold air. In this chapter we learn to form analytical descrip- tions of these convective heating (or cooling) processes. Our aim is to predicth andh, 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 269270 Laminar and turbulent boundary layers §6.1 Figure 6.1 A boundary layer of thicknessδ. 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 1 immersed in the flow. The first step that has to be taken beforeh can be predicted is the mathematical description of the boundary layer. This description was 2 first made by Prandtl (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) ∞ wherex is the length along the surface andρ andµ are the fluid density 3 in kg/m and the dynamic viscosity in kg/m·s. We have five variables in 1 We qualify this remark when we treat the b.l. quantitatively. 2 Prandtl was educated at the Technical University in Munich and finished his doctor- ate 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.§6.1 Some introductory ideas 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 ρu x u x ∞ ∞ = fn(Re ) Re ≡ = (6.1) x x x µ ν where ν is the kinematic viscosityµ/ρ and Re is called the Reynolds x 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, whereu remains constant, is ∞ δ 4.92 3 = (6.2) x Re x 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 nearly272 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-to- turbulent 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,u , the liquid streamline av marked with ink would become wobbly and then break up into increas- ingly disorderly eddies, and it would finally be completely mixed into the§6.1 Some introductory ideas 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(u ) , the transitional av crit value of the average velocity, must depend on the pipe diameter,D,on µ, and onρ—four variables in kg, m, and s. There is therefore only one pi-group: ρD(u ) av crit Re ≡ (6.3) critical µ 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 onx: u x ∞ crit Re = (6.4) x critical ν At larger values ofx 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 5 For the boundary layer shown, Re = 3.5×10 , but in general the x critical 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 struc- tural vibrations 6.2, §5.5. On a flat plate, a boundary layer will remain 4 laminar even when such disturbances are very large if Re ≤ 6× 10 . x With relatively undisturbed conditions, transition occurs for Re in the x 5 5 range of 3× 10 to 5× 10 , and in very careful laboratory experiments, 6 turbulent transition can be delayed until Re ≈ 3× 10 or so. Turbulent x 6 transition is essentially always complete before Re = 4×10 and usually x 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 Re . x Thermal boundary layer If the wall is at a temperatureT , different from that of the free stream, w T , there is a thermal boundary layer thickness,δ —different from the ∞ t flow b.l. thickness,δ. A thermal b.l. is pictured in Fig. 6.5. Now, with ref- erence 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:   ∂T  −k  =h(T −T ) (6.5) f w ∞  ∂y y=0    conduction into the fluid wherek is the conductivity of the fluid. Notice two things about this f 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 ofq. The other point is that while eqn. (6.5) looks like a b.c. of the third kind, it is not. This condition definesh within the fluid instead of specifying it as known information on the boundary. Equation (6.5) can be arranged in the form     T −T w  ∂  hL T −T  w ∞  = = Nu , the Nusselt number (6.5a) L  ∂(y/L) k f   y/L=0§6.1 Some introductory ideas 275 Figure 6.5 The thermal boundary layer during the flow of cool fluid over a warm plate. whereL 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 Nu ≡hx/k . From Fig. 6.5 we x f see immediately that the physical significance of Nu is given by L Nu = (6.6) L  δ t In other words, the Nusselt number is inversely proportional to the thick- ness of the thermal b.l. 3 The Nusselt number is named after Wilhelm Nusselt, 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 ofh. And, as the preceding remarks make very clear, this evaluation will have to start with a devel- opment of the flow field in the boundary layer. 3 Nusselt finished his doctorate in mechanical engineering at the Technical Univer- sity 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 anal- ysis 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 conden- sation. 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 method- ology. 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 whereu,v, andw are thex,y, andz 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 stream- lines can be expressed in terms of a stream function, ψ(x,y) = con- stant, 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§6.2 Laminar incompressible boundary layer on a flat surface 277 Figure 6.7 A steady, incompressible, two-dimensional flow field represented by streamlines, or lines of constantψ. channel in a flux plot (Section 5.7); such channels are adiabatic—no heat flow can cross them. Therefore, we write the equation for the conserva- tion of mass by summing the inflow and outflow of mass on two faces of a triangular element of unit depth, as shown in Fig. 6.7: ρvdx−ρudy= 0 (6.7) If the fluid is incompressible, so thatρ= constant along each streamline, then −vdx+udy= 0 (6.8) But we can also differentiate the stream function along any streamline, ψ(x,y)= constant, in Fig. 6.7:     ∂ψ ∂ψ    dψ= dx+ dy= 0 (6.9)   ∂x ∂y y x If we compare eqns. (6.8) and (6.9), we immediately see that the coef- ficients ofdx anddy must be the same, so     ∂ψ ∂ψ    v=− and u= (6.10)   ∂x ∂y y x278 Laminar and turbulent boundary layers §6.2 Furthermore, 2 2 ∂ ψ ∂ ψ = ∂y∂x ∂x∂y so it follows that ∂u ∂v + = 0 (6.11) ∂x ∂y This is called the two-dimensional continuity equation for incompress- ible flow, because it expresses mathematically the fact that the flow is continuous; it has no breaks in it. In three dimensions, the continuity equation for an incompressible fluid is ∂u ∂v ∂w ∇·u= + + = 0 ∂x ∂y ∂z Example 6.1 Fluid moves with a uniform velocity,u , in thex-direction. Find the ∞ stream function and see if it gives plausible behavior (see Fig. 6.8). Solution. u=u andv= 0. Therefore, from eqns. (6.10) ∞     ∂ψ ∂ψ    u = and 0= ∞   ∂y ∂x y x Integrating these equations, we get ψ=u y+ fn(x) and ψ= 0+ fn(y) ∞ Comparing these equations, we get fn(x) = constant and fn(y) = u y+ constant, so ∞ ψ=u y+ constant ∞ This gives a series of equally spaced, horizontal streamlines, as we would expect (see Fig. 6.8). We set the arbitrary constant equal to zero in the figure.§6.2 Laminar incompressible boundary layer on a flat surface 279 Figure 6.8 Streamlines in a uniform horizontal flow field,ψ=u y. ∞ Conservation of momentum The momentum equation in a viscous flow is a complicated vectorial ex- pression called the Navier-Stokes equation. Its derivation is carried out in any advanced fluid mechanics text (see, e.g., 6.3, Chap. III). We shall offer a very restrictive derivation of the equation—one that applies only to a two-dimensional incompressible b.l. flow, as shown in Fig. 6.9. Here we see that shear stresses act upon any element such as to con- tinuously distort and rotate it. In the lower part of the figure, one such 4 element is enlarged, so we can see the horizontal shear stresses and the pressure forces that act upon it. They are shown as heavy arrows. We also display, as lighter arrows, the momentum fluxes entering and leaving the element. Notice that bothx- andy-directed momentum enters and leaves the element. To understand this, one can 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 momentum will push him back, but a component of momentum will also jar him toward the rear of the train, because of the relative motion. Particles of fluid entering elementA will likewise influence its motion, with theirx components of momentum carried into the element by both components of flow. The velocities must adjust themselves to satisfy the principle of con- servation of linear momentum. Thus, we require that the sum of the external forces in thex-direction, which act on the control volume, A, must be balanced by the rate at which the control volume,A, forcesx- 4 The stress,τ, is often given two subscripts. The first one identifies the direction normal to the plane on which it acts, and the second one identifies the line along which it acts. Thus, if both subscripts are the same, the stress must act normal to a surface—it must be a pressure or tension instead of a shear stress.280 Laminar and turbulent boundary layers §6.2 Figure 6.9 Forces acting in a two-dimensional incompressible boundary layer. directed momentum out. The external forces, shown in Fig. 6.9, are     ∂τ ∂p yx τ + dy dx−τ dx+pdy− p+ dx dy yx yx ∂y ∂x   ∂τ ∂p yx = − dxdy ∂y ∂x The rate at whichA losesx-directed momentum to its surroundings is     2 ∂ρu ∂ρuv 2 2 ρu + dx dy−ρu dy+ u(ρv)+ dy dx ∂x ∂y   2 ∂ρu ∂ρuv −ρuvdx= + dxdy ∂x ∂y§6.2 Laminar incompressible boundary layer on a flat surface 281 We equate these results and obtain the basic statement of conserva- tion ofx-directed momentum for the b.l.:   2 ∂τ dp ∂ρu ∂ρuv yx dydx− dxdy= + dxdy ∂y dx ∂x ∂y The shear stress in this result can be eliminated with the help of Newton’s law of viscous shear: ∂u τ =µ yx ∂y so the momentum equation becomes     2 ∂ ∂u dp ∂ρu ∂ρuv µ − = + ∂y ∂y dx ∂x ∂y Finally, we remember that the analysis is limited toρ constant, and we limit use of the equation to temperature ranges in whichµ constant. Then 2 2 ∂u ∂uv 1dp ∂ u + =− +ν (6.12) 2 ∂x ∂y ρdx ∂y This is one form of the steady, two-dimensional, incompressible bound- ary layer momentum equation. Although we have takenρ constant, a more complete derivation reveals that the result is valid for compress- ible flow as well. If we multiply eqn. (6.11)byu and subtract the result from the left-hand side of eqn. (6.12), we obtain a second form of the momentum equation: 2 ∂u ∂u 1dp ∂ u u +v =− +ν (6.13) 2 ∂x ∂y ρdx ∂y Equation (6.13) has a number of so-called boundary layer approxima- tions built into it:         • ∂u/∂x is generally ∂u/∂y . • v is generallyu. • p ≠ fn(y)282 Laminar and turbulent boundary layers §6.2 The Bernoulli equation for the free stream flow just above the bound- ary layer where there is no viscous shear, 2 p u ∞ + = constant ρ 2 can be differentiated and used to eliminate the pressure gradient, 1dp du ∞ =−u ∞ ρdx dx so from eqn. (6.12): 2 2 ∂u ∂(uv) du ∂ u ∞ + =u +ν (6.14) ∞ 2 ∂x ∂y dx ∂y And if there is no pressure gradient in the flow—ifp andu are constant ∞ as they would be for flow past a flat plate—then eqns. (6.12), (6.13), and (6.14) become 2 2 ∂u ∂(uv) ∂u ∂u ∂ u + =u +v =ν (6.15) 2 ∂x ∂y ∂x ∂y ∂y Predicting the velocity profile in the laminar boundary layer without a pressure gradient Exact solution. Two strategies for solving eqn. (6.15) for the velocity profile have long been widely used. The first was developed by Prandtl’s 5 student, H. Blasius, before World War I. It is exact, and we shall sketch it only briefly. First we introduce the stream function,ψ, into eqn. (6.15). This reduces the number of dependent variables from two (u andv)to just one—namely,ψ. We do this by substituting eqns. (6.10) in eqn. (6.15): 2 2 3 ∂ψ ∂ ψ ∂ψ∂ ψ ∂ ψ − =ν (6.16) 2 3 ∂y∂y∂x ∂x∂y ∂y It turns out that eqn. (6.16) can be converted into an ordinary d.e. with the following change of variables: 9 √ u ∞ ψ(x,y)≡ u νxf(η) where η≡ y (6.17) ∞ νx 5 Blasius achieved great fame for many accomplishments in fluid mechanics and then gave it up. He is quoted as saying: “I decided that I had no gift for it; all of my ideas came from Prandtl.”§6.2 Laminar incompressible boundary layer on a flat surface 283 wheref(η) is an as-yet-undertermined function. This transformation is rather similar to the one that we used to make an ordinary d.e. of the heat conduction equation, between eqns. (5.44) and (5.45). After some manipulation of partial derivatives, this substitution gives (Problem 6.2) 2 3 d f d f f + 2 = 0 (6.18) 2 3 dη dη and   u df v 1 df 3 = = η −f (6.19) u dη u ν/x 2 dη ∞ ∞ The boundary conditions for this flow are  ⎫  ⎪ df  ⎪ ⎪  u(y= 0)=0or = 0 ⎪ ⎪  ⎪ dη ⎪ η=0 ⎪  ⎬  df  (6.20) u(y=∞)=u or  = 1 ⎪ ∞ ⎪  ⎪ dη ⎪ η=∞ ⎪ ⎪ ⎪ ⎪ ⎭ v(y= 0)=0or f(η= 0)= 0 The solution of eqn. (6.18) subject to these b.c.’s must be done numeri- cally. (See Problem 6.3.) The solution of the Blasius problem is listed in Table 6.1, and the dimensionless velocity components are plotted in Fig. 6.10. Theu com- ponent increases from zero at the wall(η= 0) to 99% ofu atη= 4.92. ∞ Thus, the b.l. thickness is given by δ 3 4.92= νx/u ∞ or, as we anticipated earlier eqn. (6.2), δ 4.92 4.92 3 3 = = x u x/ν Re ∞ x Concept of similarity. The exact solution for u(x,y) reveals a most useful fact—namely, thatu can be expressed as a function of a single variable,η:  9  u u ∞   =f (η)=f y u νx ∞284 Laminar and turbulent boundary layers §6.2 Table 6.1 Exact velocity profile in the boundary layer on a flat surface with no pressure gradient 3 3 y u /νx u u v x/νu ∞ ∞ ∞    η f(η) f (η) (ηf −f) 2 f (η) 0.00 0.00000 0.00000 0.00000 0.33206 0.20 0.00664 0.06641 0.00332 0.33199 0.40 0.02656 0.13277 0.01322 0.33147 0.60 0.05974 0.19894 0.02981 0.33008 0.80 0.10611 0.26471 0.05283 0.32739 1.00 0.16557 0.32979 0.08211 0.32301 2.00 0.65003 0.62977 0.30476 0.26675 3.00 1.39682 0.84605 0.57067 0.16136 4.00 2.30576 0.95552 0.75816 0.06424 4.918 3.20169 0.99000 0.83344 0.01837 6.00 4.27964 0.99898 0.85712 0.00240 − 8.00 6.27923 1.00000 0.86039 0.00001 This is called a similarity solution. To see why, we solve eqn. (6.2) for 9 u 4.92 ∞ = νx δ(x) 3  and substitute this inf (y u /νx). The result is ∞ u y  f = = fn (6.21) u δ(x) ∞ The velocity profile thus has the same shape with respect to the b.l. thickness at eachx-station. We say, in other words, that the profile is similar at each station. This is what we found to be true for conduction √ into a semi-infinite region. In that case recall eqn. (5.51),x/ t always had the same value at the outer limit of the thermally disturbed region. Boundary layer similarity makes it especially easy to use a simple approximate method for solving other b.l. problems. This method, called the momentum integral method, is the subject of the next subsection. Example 6.2 ◦ Air at 27 C blows over a flat surface with a sharp leading edge at 1 1.5m/s. Find the b.l. thickness m from the leading edge. Check the 2 b.l. assumption thatu v at the trailing edge.§6.2 Laminar incompressible boundary layer on a flat surface 285 Figure 6.10 The dimensionless velocity components in a lam- inar boundary layer. Solution. The dynamic and kinematic viscosities areµ = 1.853× −5 −5 2 10 kg/m·s andν= 1.566× 10 m /s. Then u x 1.5(0.5) ∞ Re = = = 47, 893 x −5 ν 1.566× 10 The Reynolds number is low enough to permit the use of a laminar flow analysis. Then 4.92x 4.92(0.5) 3 3 δ= = = 0.01124= 1.124 cm Re 47, 893 x (Remember that the b.l. analysis is only valid ifδ/x 1. In this case, δ/x = 1.124/50= 0.0225.) From Fig. 6.10 or Table 6.1, we observe thatv/u is greatest beyond the outside edge of the b.l, at large η. Using data from Table 6.1 atη= 8,v atx= 0.5mis 2 −5 0.8604 (1.566)(10 )(1.5) 3 v= = 0.8604 x/νu (0.5) ∞ = 0.00590 m/s286 Laminar and turbulent boundary layers §6.2 or, sinceu/u → 1 at largeη ∞ v v 0.00590 = = = 0.00393 u u 1.5 ∞ Sincev grows larger asx grows smaller, the conditionvu is not sat- isfied very near the leading edge. There, the b.l. approximations them- selves break down. We say more about this breakdown after eqn. (6.34). 6 Momentum integral method. A second method for solving the b.l. mo- mentum equation is approximate and much easier to apply to a wide range of problems than is any exact method of solution. The idea is this: We are not really interested in the details of the velocity or temperature profiles in the b.l., beyond learning their slopes at the wall. These slopes give us the shear stress at the wall, τ = µ(∂u/∂y) , and the heat w y=0 flux at the wall,q =−k(∂T/∂y) . Therefore, we integrate the b.l. w y=0 equations from the wall,y= 0, to the b.l. thickness,y=δ, to make ordi- nary d.e.’s of them. It turns out that while these much simpler equations do not reveal anything new about the temperature and velocity profiles, they do give quite accurate explicit equations forτ andq . w w Let us see how this procedure works with the b.l. momentum equa- tion. We integrate eqn. (6.15), as follows, for the case in which there is no pressure gradient(dp/dx= 0):    δ 2 δ δ 2 ∂u ∂(uv) ∂ u dy+ dy=ν dy 2 ∂x ∂y ∂y 0 0 0 Aty=δ,u can be approximated as the free stream value,u , and other ∞ quantities can also be evaluated aty=δ just as thoughy were infinite: ⎡ ⎤      δ 2 ∂u ∂u ∂u ⎢ ⎥ dy+ (uv) −(uv) =ν ⎣ − ⎦ y=δ y=0 ∂x       ∂y ∂y 0 y=δ y=0 =u v ∞ ∞ =0    0 (6.22) The continuity equation (6.11) can be integrated thus:  δ ∂u v −v =− dy (6.23) ∞ y=0    ∂x 0 =0 6 This method was developed by Pohlhausen, von Kármán, and others. See the dis- cussion in 6.3, Chap. XII.

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