Heat and Mass Transfer Fundamentals and Applications PDF

https://ebook2book.ir/ CHAPTER F U N D A M E N TA L S O F CONVECTION 6 S o far, we have considered conduction, which is...

https://ebook2book.ir/ CHAPTER F U N D A M E N TA L S O F CONVECTION 6 S o far, we have considered conduction, which is the mechanism of heat transfer through a solid or a quiescent fluid. We now consider convec- OBJECTIVES tion, which is the mechanism of heat transfer through a fluid in the When you finish studying this chapter, presence of bulk fluid motion. you should be able to: Convection is classified as natural (or free) and forced convection, depend- Understand the physical ing on how the fluid motion is initiated. In forced convection, the fluid is mechanism of convection and forced to flow over a surface or in a pipe by external means such as a pump its classification, or a fan. In natural convection, any fluid motion is caused by natural means Visualize the development of such as the buoyancy effect, which manifests itself as the rise of warmer fluid velocity and thermal boundary and the fall of the cooler fluid. Convection is also classified as external and layers during flow over surfaces, internal, depending on whether the fluid is forced to flow over a surface or in a pipe. Gain a working knowledge of the dimensionless Reynolds, We start this chapter with a general physical description of the convec- Prandtl, and Nusselt numbers, tion mechanism. We then discuss the velocity and thermal boundary layers Distinguish between laminar and laminar and turbulent flows. We continue with the discussion of the and turbulent flows, and dimensionless Reynolds, Prandtl, and Nusselt numbers and their physical gain an understanding of the significance. Next we derive the convection equations on the basis of mass, mechanisms of momentum and momentum, and energy conservation, and we obtain solutions for flow over heat transfer in turbulent flow, a flat plate. We then nondimensionalize the convection equations and obtain Derive the differential equations functional forms of friction and convection coefficients. Finally, we present that govern convection on the analogies between momentum and heat transfer. basis of mass, momentum, and energy balances, and solve these equations for some simple cases such as laminar flow over a flat plate, Nondimensionalize the convection equations and obtain the functional forms of friction and heat transfer coefficients, and Use analogies between momentum and heat transfer, and determine heat transfer coefficient from your knowledge of friction coefficient. 391 https://ebook2book.ir/ https://ebook2book.ir/ 392 FUNDAMENTALS OF CONVECTION 20°C 5 m/s 6–1 PHYSICAL MECHANISM OF CONVECTION Air. We mentioned in Chap. 1 that there are three basic mechanisms of heat trans- Q1 50°C fer: conduction, convection, and radiation. Conduction and convection are similar in that both mechanisms require the presence of a material medium. But they are different in that convection requires the presence of fluid motion. (a) Forced convection Heat transfer through a solid is always by conduction, since the molecules of a solid remain at relatively fixed positions. Heat transfer through a liquid Warmer air or gas, however, can be by conduction or convection, depending on the pres- rising Air. Q2 ence of any bulk fluid motion. Heat transfer through a fluid is by convection in the presence of bulk fluid motion and by conduction in the absence of it. Therefore, conduction in a fluid can be viewed as the limiting case of convec- tion, corresponding to the case of quiescent fluid (Fig. 6–1). (b) Natural convection Convection heat transfer is complicated by the fact that it involves fluid motion as well as heat conduction. The fluid motion enhances heat transfer, since it. No convection brings warmer and cooler chunks of fluid into contact, initiating higher rates Q3 currents Air of conduction at a greater number of sites in a fluid. Therefore, the rate of heat transfer through a fluid is much higher by convection than it is by conduction. In fact, the higher the fluid velocity, the higher the rate of heat transfer. (c) Conduction To clarify this point further, consider steady heat transfer through a fluid contained between two parallel plates maintained at different temperatures, as FIGURE 6–1 shown in Fig. 6–2. The temperatures of the fluid and the plate are the same Heat transfer from a hot surface to the at the points of contact because of the continuity of temperature. Assuming surrounding fluid by convection and no fluid motion, the energy of the hotter fluid molecules near the hot plate is conduction. transferred to the adjacent cooler fluid molecules. This energy is then trans- ferred to the next layer of the cooler fluid molecules. This energy is then transferred to the next layer of the cooler fluid, and so on, until it is finally transferred to the other plate. This is what happens during conduction through a fluid. Now let us use a syringe to draw some fluid near the hot plate and inject it next to the cold plate repeatedly. You can imagine that this will speed up the heat transfer process considerably, since some energy is carried to the other side as a result of fluid motion. Consider the cooling of a hot block with a fan blowing air over its top sur- face. We know that heat is transferred from the hot block to the surround- ing cooler air, and the block eventually cools. We also know that the block cools faster if the fan is switched to a higher speed. Replacing air with water enhances the convection heat transfer even more. Hot plate Experience shows that convection heat transfer strongly depends on the fluid properties dynamic viscosity μ, thermal conductivity k, density ρ, and specific heat cρ,as well as the fluid velocity V. It also depends on the geometry Heat transfer and the roughness of the solid surface, in addition to the type of fluid flow Fluid through the (such as being streamlined or turbulent). Thus, we expect the convection heat fluid Q transfer relations to be rather complex because of the dependence of convec- tion on so many variables. This is not surprising, since convection is the most complex mechanism of heat transfer. Cold plate Despite the complexity of convection, the rate of convection heat transfer is FIGURE 6–2 observed to be proportional to the temperature difference and is conveniently Heat transfer through a fluid expressed by Newton’s law of cooling as sandwiched between two parallel plates. q   conv  = h(Ts  − T∞) (W/m2 ) (6–1) https://ebook2book.ir/ https://ebook2book.ir/ 393 CHAPTER 6 or Q  conv  = hAs(T s  − T∞)  (W) (6–2) where h = convection heat transfer coefficient, W/m2 ⋅K As = heat transfer surface area, m2                    Ts = temperature of the surface, ° C T∞ = temperature of the fluid sufficiently far from the surface, ° C Judging from its units, the convection heat transfer coefficient h can be defined as the rate of heat transfer between a solid surface and a fluid per unit surface area per unit temperature difference. You should not be deceived by the simple appearance of this relation, because the convection heat transfer coefficient h depends on several of the mentioned variables, and thus is difficult to determine. Fluid flow is often confined by solid surfaces, and it is important to understand how the presence of solid surfaces affects fluid flow. Consider the flow of a fluid in a stationary pipe or over a solid surface that is nonporous (i.e., impermeable to the fluid). All experimental observations indicate that a fluid in motion comes to a complete stop at the surface and assumes a zero velocity relative to the surface. That is, a fluid in direct contact with a solid “sticks” to the surface due to viscous effects, and there is no slip. This is known as the no-slip condition. Figure 6–3, obtained from a video clip, clearly shows the evolution of a velocity gradient as a result of the fluid sticking to the surface of a blunt nose. The layer that sticks to the surface slows the adjacent fluid layer because of viscous forces between the fluid layers, which slows the next layer, and so on. Therefore, the no-slip condition is responsible for the development of the velocity profile. The flow region adjacent to the wall in which the viscous effects (and thus the velocity gradients) are significant is called the boundary FIGURE 6–3 layer. The fluid property responsible for the no-slip condition and the devel- The development of a velocity profile due to the no-slip condition as a fluid opment of the boundary layer is viscosity and is discussed briefly in Sec. 6–2. flows over a blunt nose. A fluid layer adjacent to a moving surface has the same velocity as the surface. Courtesy of IIHR-Hydroscience & Engineering A consequence of the no-slip condition is that all velocity profiles must have zero values with respect to the surface at the points of contact between a fluid and a solid surface (Fig. 6–4). Another consequence of the no-slip condition is the surface drag, which is the force a fluid exerts on a surface in the flow direction. An implication of the no-slip condition is that heat transfer from the solid Relative surface to the fluid layer adjacent to the surface is by pure conduction, since Uniform velocities approach the fluid layer is motionless, and it can be expressed as velocity, V of fluid layers ∂T | q   conv  = q   cond  = −kfluid  ___      ∂y y = 0 (W/m2 ) where T represents the temperature distribution in the fluid and (∂ T/∂ y)y=0 is (6–3) Zero velocity at the the temperature gradient at the surface. Heat is then convected away from the y x surface surface as a result of fluid motion. Note that convection heat transfer from a Plate solid surface to a fluid is merely the conduction heat transfer from the solid surface to the fluid layer adjacent to the surface. Therefore, we can equate FIGURE 6–4 Eqs. 6–1 and 6–3 for the heat flux to obtain A fluid flowing over a stationary surface comes to a complete stop at −kfluid(∂ T/∂ y)y = 0 the surface because of the no-slip h =  ______________      (W/m2⋅K) (6–4) Ts − T∞ condition. https://ebook2book.ir/ https://ebook2book.ir/ 394 FUNDAMENTALS OF CONVECTION for the determination of the convection heat transfer coefficient when the temperature distribution within the fluid is known. The convection heat transfer coefficient, in general, varies along the flow (or x-) direction. The average or mean convection heat transfer coefficient for a surface in such cases is determined by properly averaging the local convec- tion heat transfer coefficients over the entire surface. Nusselt Number In convection studies, it is common practice to nondimensionalize the govern- ing equations and combine the variables, which group together into dimen- sionless numbers in order to reduce the number of total variables. It is also common practice to nondimensionalize the heat transfer coefficient h with the Nusselt number, defined as hL Nu = ___    c (6–5) k where k is the thermal conductivity of the fluid and Lc is the characteristic length. The Nusselt number is named after Wilhelm Nusselt (Fig. 6–5), who made significant contributions to convective heat transfer in the first half of FIGURE 6–5 the 20th century, and it is viewed as the dimensionless convection heat trans- Wilhelm Nusselt (1882–1957), was a fer coefficient. German engineer, born in Nuremberg, To understand the physical significance of the Nusselt number, consider a Germany. He studied machinery at fluid layer of thickness L and temperature difference ΔT = T2− T1,as shown the Technical Universities of Berlin- in Fig. 6–6. Heat transfer through the fluid layer is by convection when the Charlottenburg and Munchen and fluid involves some motion and by conduction when the fluid layer is motion- conducted advanced studies in mathe- less. Heat flux (the rate of heat transfer per unit surface area) in either case is matics and physics. His doctoral thesis was on the “Conductivity of Insulating q   conv = hΔT (6–6) Materials,” which he completed in and 1907. In 1915, Nusselt published his pioneering paper: “The Basic Laws ΔT q   cond = k __________     (6–7) of Heat Transfer,” in which he first L proposed the dimensionless groups now known as the principal param- Taking their ratio gives eters in the similarity theory of heat q   conv _ _ _hΔT hL transfer. His other famous works were  _____    =   _ _ _ _ _ _ _ _ _ _ _ _ _   =  ___  = Nu (6–8) concerned with the film condensa- q   cond kΔT/L k tion of steam on vertical surfaces, the which is the Nusselt number. Therefore, the Nusselt number represents the combustion of pulverized coal, and enhancement of heat transfer through a fluid layer as a result of convection the analogy between heat and mass relative to conduction across the same fluid layer. The larger the Nusselt num- transfer in evaporation. Among his ber, the more effective the convection. A Nusselt number of N u = 1for a fluid well-known mathematical works are layer represents heat transfer across the layer by pure conduction. the solutions for laminar heat transfer We use forced convection in daily life more often than you might think in the entrance region of tubes and for heat exchange in crossflow, and the (Fig. 6–7). We resort to forced convection whenever we want to increase the basic theory of regenerators. rate of heat transfer from a hot object. For example, we turn on the fan on hot Source: Wikipedia summer days to help our bodies to cool more effectively. The higher the fan speed, the better we feel. We stir our soup and blow on a hot slice of pizza to make them cool faster. The air on windy winter days feels much colder than it actually is. The simplest solution to heating problems in electronics packag- ing is to use a large enough fan. https://ebook2book.ir/ https://ebook2book.ir/ 395 CHAPTER 6 T2 EXAMPLE 6–1 H eat Transfer Calculation from Temperature Profile Fluid · During the flow of air at T∞= 20° Cover a plate surface maintained at a constant Q L layer temperature of Ts= 160° C,the dimensionless temperature profile within the air layer over the plate is determined to be T1 ΔT = T2 – T1 T(y) − T∞ −ay _________ = e Ts− T∞ FIGURE 6–6 Heat transfer through a fluid layer of where a = 3200 m−1and y is the vertical distance measured from the plate surface in thickness L and temperature m (Fig. 6–8). Determine the heat flux on the plate surface and the convection heat difference Δ T. transfer coefficient. SOLUTION Airflow over a flat plate has a given temperature profile. The heat flux on the plate surface and the convection heat transfer coefficient are to be determined. Assumptions 1 The given nondimensional temperature profile is representative of the variation of temperature over the entire plate. 2 Heat transfer by radiation is negligible. Blowing Properties The thermal conductivity of air at the film temperature of Tf  = on food (Ts+ T∞ )/2 = (160°C + 20°C)/2 = 90°Cis k = 0.03024 W/m⋅K(Table A–15). Analysis Noting that heat transfer from the plate to air at the surface is by conduction, heat flux from the solid surface to the fluid layer adjacent to the surface is determined from ∂T q = q cond= −kfluid___ |     ∂y y = 0 FIGURE 6–7 where the temperature gradient at the plate surface is We resort to forced convection whenever we need to increase the ∂T ___ |     = (Ts− T∞)a[e−ay ∂ y y = 0          ]y = 0= (Ts− T∞)(−a) = (160° C − 20° C)(−3200 m ) = −4.48 × 10 ° C/m −1 5 rate of heat transfer. T(y) T ay Air, T∞ e Substituting, the heat flux is determined to be Ts T q = −(0.03024 W/m⋅K)(−4.48 × 105 ° C/m) = 1.35 × 104 W/m2 y Ts x Then the convection heat transfer coefficient becomes FIGURE 6–8 −k (∂ T/∂ y)y = 0 ___________________________ Schematic for Example 6–1. fluid h =    Ts− T∞    −(0.03024 W/m⋅K)(−4.48 × 105° C/m) _______________________________________________________ =        = 96.8 W/m ·K 2 (160° C − 20° C) Discussion The convection heat transfer coefficient could also be determined from Newton’s law of cooling, q = h(Ts− T∞). https://ebook2book.ir/ https://ebook2book.ir/ 396 FUNDAMENTALS OF CONVECTION 6–2 CLASSIFICATION OF FLUID FLOWS Convection heat transfer is closely tied with fluid mechanics, which is the sci- ence that deals with the behavior of fluids at rest or in motion, and the interac- tion of fluids with solids or other fluids at the boundaries. A wide variety of fluid flow problems is encountered in practice, and it is usually convenient to classify them on the basis of some common characteristics to make it feasible to study them in groups. There are many ways to classify fluid flow problems, and here we present some general categories. Viscous Versus Inviscid Regions of Flow When two fluid layers move relative to each other, a friction force develops between them, and the slower layer tries to slow down the faster layer. This internal resistance to flow is quantified by the fluid property viscosity, which is a measure of internal stickiness of the fluid. Viscosity is caused by cohesive forces between the molecules in liquids and by molecular collisions in gases. There is no fluid with zero viscosity, and thus all fluid flows involve viscous Inviscid flow region effects to some degree. Flows in which the frictional effects are significant are called viscous flows. However, in many flows of practical interest, there Viscous flow are regions (typically regions not close to solid surfaces) where viscous forces region are negligibly small compared to inertial or pressure forces. Neglecting the viscous terms in such inviscid flow regions greatly simplifies the analysis Inviscid flow without much loss in accuracy. region The development of viscous and inviscid regions of flow as a result of inserting a flat plate parallel into a fluid stream of uniform velocity is shown in Fig. 6–9. The fluid sticks to the plate on both sides because of the no-slip FIGURE 6–9 condition, and the thin boundary layer in which the viscous effects are signifi- Flow of an originally uniform fluid cant near the plate surface is the viscous flow region. The region of flow on stream over a flat plate, and the both sides away from the plate and unaffected by the presence of the plate is regions of viscous flow (next to the the inviscid flow region. plate on both sides) and inviscid flow (away from the plate). Internal Versus External Flow A fluid flow is classified as being internal or external, depending on whether the fluid is forced to flow in a confined channel or over a surface. The flow of an unbounded fluid over a surface such as a plate, a wire, or a pipe is external flow. The flow in a pipe or duct is internal flow if the fluid is completely bounded by solid surfaces. Water flow in a pipe, for example, is internal flow, and airflow over a ball or over an exposed pipe during a windy day is external flow (Fig. 6–10). The flow of liquids in a duct is called open-channel flow if the duct is only partially filled with the liquid and there is a free surface. The flows of water in rivers and irrigation ditches are examples of such flows. Internal flows are dominated by the influence of viscosity throughout the flow field. In external flows, the viscous effects are limited to boundary lay- ers near solid surfaces and to wake regions downstream of bodies. FIGURE 6–10 External flow over a tennis ball, and Compressible Versus Incompressible Flow the turbulent wake region behind. A flow is classified as being compressible or incompressible, depending on Courtesy NASA Ames Research Center and the level of variation of density during flow. Incompressibility is an approxi- Cislunar Aerospace, Inc. mation, and a flow is said to be incompressible if the density remains nearly https://ebook2book.ir/ https://ebook2book.ir/ 397 CHAPTER 6 constant throughout. Therefore, the volume of every portion of fluid remains unchanged over the course of its motion when the flow (or the fluid) is incompressible. The densities of liquids are essentially constant, and thus the flow of liq- uids is typically incompressible. Therefore, liquids are usually referred to as incompressible substances. A pressure of 210 atm, for example, causes the density of liquid water at 1 atm to change by just 1 percent. Gases, on the other hand, are highly compressible. A pressure change of just 0.01 atm, for example, causes a change of 1 percent in the density of atmospheric air. Liquid flows are incompressible to a high level of accuracy, but the level of variation in density in gas flows and the consequent level of approximation made when modeling gas flows as incompressible depends on the Mach number, defined as Ma = V/c, where c is the speed of sound Laminar whose value is 346 m/s in air at room temperature at sea level. Gas flows can often be approximated as incompressible if the density changes are under about 5 percent, which is usually the case when M a < 0.3.Therefore, the compressibility effects of air can be neglected at speeds under about 100 m/s. Note that the flow of a gas is not necessarily a compressible flow. Transitional Small density changes of liquids corresponding to large pressure changes can still have important consequences. The irritating “water hammer” in a water pipe, for example, is caused by the vibrations of the pipe generated by the reflection of pressure waves following the sudden closing of the valves. Laminar Versus Turbulent Flow Turbulent Some flows are smooth and orderly while others are rather chaotic. The highly ordered fluid motion characterized by smooth layers of fluid is called FIGURE 6–11 laminar. The word laminar comes from the movement of adjacent fluid Laminar, transitional, and turbulent flows. particles together in “laminates.” The flow of high-viscosity fluids such as ©Henri Werlé–ONERA, the French Aerospace Lab oils at low velocities is typically laminar. The highly disordered fluid motion that typically occurs at high velocities and is characterized by velocity fluctuations is called turbulent (Fig. 6–11). The flow of low-viscosity fluids such as air at high velocities is typically turbulent. The flow regime greatly L = 0.2 m influences the required power for pumping. A flow that alternates between being laminar and turbulent is called transitional. Natural (or Unforced) Versus Forced Flow V0 = 0.2 m/s A fluid flow is said to be natural or forced, depending on how the fluid motion is initiated. In forced flow, a fluid is forced to flow over a surface or in a pipe by external means such as a pump or a fan. In natural flows, any fluid motion is due to natural means such as the buoyancy effect, which manifests itself as Re = 3 × 103 the rise of the warmer (and thus lighter) fluid and the fall of cooler (and thus denser) fluid (Fig. 6–12). In solar hot-water systems, for example, the ther- FIGURE 6–12 mosiphoning effect is commonly used to replace pumps by placing the water In this schlieren image, the tank sufficiently above the solar collectors. rise of lighter, warmer air adjacent to the person’s body indicates Steady Versus Unsteady Flow that humans and warm-blooded animals are surrounded by thermal The terms steady and uniform are used frequently in engineering, and thus it plumes of rising warm air. is important to have a clear understanding of their meanings. The term steady G.S. Settles, Gas Dynamics Lab, Penn State Uni- implies no change at a point with time. The opposite of steady is unsteady. versity. Used by permission. https://ebook2book.ir/ https://ebook2book.ir/ 398 FUNDAMENTALS OF CONVECTION The term uniform implies no change with location over a specified region. These meanings are consistent with their everyday use (steady girlfriend, uni- form distribution, etc.). The terms unsteady and transient are often used interchangeably, but these terms are not synonyms. In fluid mechanics, unsteady is the most general term that applies to any flow that is not steady, but transient is typically used for developing flows. When a rocket engine is fired up, for example, there are transient effects (the pressure builds up inside the rocket engine, the flow accelerates, etc.) until the engine settles down and operates steadily. The term periodic refers to the kind of unsteady flow in which the flow oscillates about a steady mean. Many devices such as turbines, compressors, boilers, condensers, and heat exchangers operate for long periods of time under the same conditions, and they are classified as steady-flow devices. (Note that the flow field near the rotating blades of a turbomachine is of course unsteady, but we consider the overall flow field rather than the details at some localities when we classify devices.) During steady flow, the fluid properties can change from point to point within a device, but at any fixed point they remain constant. Therefore, the volume, the mass, and the total energy content of a steady-flow device or flow section remain constant in steady operation. Steady-flow conditions can be closely approximated by devices that are intended for continuous operation such as turbines, pumps, boilers, con- densers, and heat exchangers of power plants or refrigeration systems. Some cyclic devices, such as reciprocating engines or compressors, do not satisfy the steady-flow conditions since the flow at the inlets and the exits is pulsat- ing and not steady. However, the fluid properties vary with time in a periodic manner, and the flow through these devices can still be analyzed as a steady- flow process by using time-averaged values for the properties. One-, Two-, and Three-Dimensional Flows A flow field is best characterized by the velocity distribution, and thus a flow is said to be one-, two-, or three-dimensional if the flow velocity varies in one, two, or three primary dimensions, respectively. A typical fluid flow involves a three-dimensional geometry, and the velocity→ may vary in all three dimensions, → rendering the flow three-dimensional [ V ( x, y, z)in rectangular or V(r, θ, z)in cylindrical coordinates]. However, the variation of velocity in cer- tain directions can be small relative to the variation in other directions and can be ignored with negligible error. In such cases, the flow can be modeled conveniently as being one- or two-dimensional, which is easier to analyze. Consider steady flow of a fluid through a circular pipe attached to a large tank. The fluid velocity everywhere on the pipe surface is zero because of the no-slip condition, and the flow is two-dimensional in the entrance region of the pipe since the velocity changes in both the r- and z-directions. The veloc- ity profile develops fully and remains unchanged after some distance from the inlet (about 10 pipe diameters in turbulent flow, and less in laminar pipe flow, as in Fig. 6–13), and the flow in this region is said to be fully developed. The fully developed flow in a circular pipe is one-dimensional since the velocity varies in the radial r-direction but not in the angular θ - or axial z-directions, as shown in Fig. 6–13. That is, the velocity profile is the same at any axial z-location, and it is symmetric about the axis of the pipe. https://ebook2book.ir/ https://ebook2book.ir/ 399 CHAPTER 6 Developing velocity Fully developed FIGURE 6–13 profile, V(r, z) velocity profile, V(r) The development of the velocity profile in a circular pipe. V = V(r, z) r and thus the flow is two-dimensional in the entrance region, and it becomes one-dimensional downstream when the velocity profile fully develops z and remains unchanged in the flow direction, V = V(r). Note that the dimensionality of the flow also depends on the choice of coordinate system and its orientation. The pipe flow discussed, for example, is one-dimensional in cylindrical coordinates, but two-dimensional in Cartesian coordinates—illustrating the importance of choosing the most appropriate coordinate system. Also note that even in this simple flow, the velocity cannot be uniform across the cross section of the pipe because of the no-slip condition. However, at a well-rounded entrance to the pipe, the velocity profile may be approximated as being nearly uniform across the pipe, since the velocity is nearly constant at all radii except very close to the pipe wall. 6–3 VELOCITY BOUNDARY LAYER Consider the parallel flow of a fluid over a flat plate, as shown in Fig. 6–14. Surfaces that are slightly contoured such as turbine blades can also be approx- imated as flat plates with reasonable accuracy. The x-coordinate is measured along the plate surface from the leading edge of the plate in the direction of the flow, and y is measured from the surface in the normal direction. The fluid approaches the plate in the x-direction with a uniform velocity V, which is practically identical to the free-stream velocity over the plate away from the surface (this would not be the case for crossflow over blunt bodies such as a cylinder). For the sake of discussion, we can consider the fluid to consist of adjacent layers piled on top of each other. The velocity of the particles in the first fluid V Laminar boundary Transition Turbulent boundary layer region layer V V Turbulent y u region u Buffer layer 0 Viscous sublayer x xcr Boundary layer thickness, δ FIGURE 6–14 The development of the boundary layer for flow over a flat plate, and the different flow regimes. Courtesy of University of Delaware. https://ebook2book.ir/ Confirming Pages https://ebook2book.ir/ 400 FUNDAMENTALS OF CONVECTION Relative layer adjacent to the plate becomes zero because of the no-slip condition. This velocities of fluid layers motionless layer slows down the particles of the neighboring fluid layer as V V a result of friction between the particles of these two adjoining fluid layers at different velocities. This fluid layer then slows down the molecules of the u Zero 0.99 V velocity next layer, and so on. Thus, the presence of the plate is felt up to some nor- at the mal distance δfrom the plate beyond which the free-stream velocity remains δ surface essentially unchanged. As a result, the x-component of the fluid velocity, u, varies from 0 at y = 0to nearly V at y = δ(Fig. 6–15). FIGURE 6–15 The region of the flow above the plate bounded by δin which the effects The development of a boundary layer of the viscous shearing forces caused by fluid viscosity are felt is called on a surface is due to the no-slip the velocity boundary layer. The boundary layer thickness, δ ,is typically condition and friction. defined as the distance y from the surface at which u = 0.99V. The hypothetical line of u = 0.99Vdivides the flow over a plate into two regions: the boundary layer region, in which the viscous effects and the velocity changes are significant, and the irrotational flow region, in which the frictional effects are negligible and the velocity remains essentially constant. Wall Shear Stress Consider the flow of a fluid over the surface of a plate. The fluid layer in contact with the surface tries to drag the plate along via friction, exerting a friction force on it. Likewise, a faster fluid layer tries to drag the adjacent slower layer and exert a friction force because of the friction between the two layers. Friction force per unit area is called shear stress and is denoted by τ. Experimental studies indicate that the shear stress for most fluids is pro- portional to the velocity gradient, and the shear stress at the wall surface is expressed as | Liquids ∂u Viscosity τw  = μ  ___     (N/m2 ) (6–9) ∂y y = 0 where the constant of proportionality μ is the dynamic viscosity of the fluid, whose unit is k g/m⋅s(or equivalently, N ⋅s/m2 ,or P a⋅s,or p oise = 0.1 Pa⋅s). Gases The fluids that obey the linear relationship above are called Newtonian fluids, after Sir Isaac Newton, who expressed it first in 1687 (Fig. 1–36). Most common fluids such as water, air, gasoline, and oils are Newtonian fluids. Blood and liquid plastics are examples of non-Newtonian fluids. In Temperature this text we consider Newtonian fluids only. FIGURE 6–16 In fluid flow and heat transfer studies, the ratio of dynamic viscosity to The viscosity of liquids decreases and density appears frequently. For convenience, this ratio is given the name the viscosity of gases increases with kinematic viscosity ν and is expressed as ν = μ/ρ.Two common units of temperature. kinematic viscosity are m2 /s and stoke (1 stoke = 1 cm2/s = 0.0001 m2 /s). The viscosity of a fluid is a measure of its resistance to deformation, and it is a strong function of temperature. The viscosities of liquids decrease with temperature, whereas the viscosities of gases increase with tem- perature (Fig. 6–16). The viscosities of some fluids at 2 0° Care listed in Table 6–1. Note that the viscosities of different fluids differ by several orders of magnitude. https://ebook2book.ir/ cen98195_ch06_391-438.indd 400 06/04/19 12:36 PM https://ebook2book.ir/ 401 CHAPTER 6 The determination of the wall shear stress τw from Eq. 6–9 is not practical TABLE 6–1 since it requires a knowledge of the flow velocity profile. A more practical approach in external flow is to relate τwto the upstream velocity V as Dynamic viscosities of some fluids at 1 atm and 2 0° C(unless otherwise stated) ρV 2 τw  = Cf  ____     (N/m2 ) (6–10) Dynamic Viscosity 2 Fluid μ, kg/m⋅s Glycerin: where Cfis the dimensionless friction coefficient or skin friction coefficient, −20° C 134.0 whose value in most cases is determined experimentally, and ρ is the density  0° C 10.5 of the fluid. Note that the friction coefficient, in general, varies with location 20° C 1.52 along the surface. Once the average friction coefficient over a given surface is 40° C 0.31 available, the friction force over the entire surface is determined from Engine oil: SAE 10W 0.10 SAE 10W30 0.17 ρV 2 Ff  = Cf As   ____   (N) (6–11) SAE 30 0.29 2 SAE 50 0.86 Mercury 0.0015 where Asis the surface area. Ethyl alcohol 0.0012 The friction coefficient is an important parameter in heat transfer stud- Water: ies since it is directly related to the heat transfer coefficient and the power 0° C 0.0018 20° C 0.0010 requirements of the pump or fan. 100° C (liquid) 0.00028 100° C (vapor) 0.000012 Blood, 3 7° C 0.00040 6–4 THERMAL BOUNDARY LAYER Gasoline 0.00029 Ammonia 0.00015 We have seen that a velocity boundary layer develops when a fluid flows over Air 0.000018 a surface as a result of the fluid layer adjacent to the surface assuming the sur- Hydrogen, 0 ° C 0.0000088 face velocity (i.e., zero velocity relative to the surface). Also, we defined the velocity boundary layer as the region in which the fluid velocity varies from zero to 0.99V. Likewise, a thermal boundary layer develops when a fluid at a specified temperature flows over a surface that is at a different temperature, as shown in Fig. 6–17. Consider the flow of a fluid at a uniform temperature of T∞over an isother- mal flat plate at temperature Ts.The fluid particles in the layer adjacent to T∞ Free-stream T∞ the surface reach thermal equilibrium with the plate and assume the surface temperature Ts.These fluid particles then exchange energy with the particles in the adjoining fluid layer, and so on. As a result, a temperature profile devel- ops in the flow field that ranges from Ts at the surface to T∞ sufficiently far T∞ Thermal from the surface. The flow region over the surface in which the temperature boundary variation in the direction normal to the surface is significant is the thermal δt Ts layer x boundary layer. The thickness of the thermal boundary layer δt at any loca- tion along the surface is defined as the distance from the surface at which the Ts + 0.99(T∞ – Ts ) temperature difference T − Ts equals 0.99(T∞− Ts).Note that for the special case of Ts= 0,we have T = 0.99T∞at the outer edge of the thermal boundary FIGURE 6–17 layer, which is analogous to u = 0.99Vfor the velocity boundary layer. Thermal boundary layer on a flat The thickness of the thermal boundary layer increases in the flow direction, plate (the fluid is hotter than the plate since the effects of heat transfer are felt at greater distances from the surface surface). further downstream. The convection heat transfer rate anywhere along the surface is directly related to the temperature gradient at that location. Therefore, the shape of the temperature profile in the thermal boundary layer dictates the convection heat transfer between a solid surface and the fluid flowing over it. In flow over a heated (or cooled) surface, both velocity and thermal boundary layers develop https://ebook2book.ir/ https://ebook2book.ir/ 402 FUNDAMENTALS OF CONVECTION simultaneously. Noting that the fluid velocity has a strong influence on the tem- perature profile, the development of the velocity boundary layer relative to the thermal boundary layer will have a strong effect on the convection heat transfer. Prandtl Number The relative thickness of the velocity and the thermal boundary layers is best described by the dimensionless parameter Prandtl number, defined as Molecular diffusivity of momentum __ _____________________________ ν μc Pr =            =      =  ___ p (6–12) Molecular diffusivity of heat α k It is named after Ludwig Prandtl (Fig. 6–18), who introduced the concept of boundary layer in 1904 and made significant contributions to boundary layer theory. The Prandtl numbers of fluids range from less than 0.01 for liquid FIGURE 6–18 metals to more than 100,000 for heavy oils (Table 6–2). Note that the Prandtl Ludwig Prandtl (1875–1953), was number is in the order of 10 for water. a German physicist famous for his The Prandtl numbers of gases are about 1, which indicates that both momen- work in aeronautics, born in Freising, tum and heat dissipate through the fluid at about the same rate. Heat diffuses Bavaria. His discovery in 1904 of the very quickly in liquid metals ( Pr ⪡ 1)and very slowly in oils ( Pr ⪢ 1) relative boundary layer which adjoins the sur- to momentum. Consequently the thermal boundary layer is much thicker for face of a body moving in a fluid led to liquid metals and much thinner for oils relative to the velocity boundary layer. an understanding of skin friction drag Liquid metals are a special class of fluids with very low Prandtl numbers and of the way in which streamlining (Table 6–2). The very low Prandtl number is due to the high thermal conduc- reduces the drag of airplane wings and tivity of these fluids, since the specific heat and viscosity of liquid metals other moving bodies. Prandtl’s work are very comparable to other common fluids. Considerable interest has been and decisive advances in boundary placed on liquid metals as coolants in applications where large amounts of layer and wing theories became the heat must be removed from a relatively small space, as in a nuclear reactor. basic material of aeronautics. He also Liquid metals, aside from having high thermal conductivity values, have high made important contributions to the thermal capacity, low vapor pressure, and low melting point. They remain in theories of supersonic flow and of tur- the liquid state at higher temperatures than conventional fluids. This makes bulence, and he contributed much to them more attractive for use in compact heat exchangers. However, liquid the development of wind tunnels and metals are corrosive in nature, and their contact with air or water may result other aerodynamic equipment. The in violent action. Suitable measures for handling them have been developed. dimensionless Prandtl number was named after him. ©ullstein bild Dtl./Getty Images 6–5 LAMINAR AND TURBULENT FLOWS If you have been around smokers, you probably noticed that the cigarette smoke rises in a smooth plume for the first few centimeters and then starts fluctuating randomly in all directions as it continues its rise. Other plumes behave similarly (Fig. 6–19). Likewise, a careful inspection of flow in a pipe TABL E 6–2 reveals that the fluid flow is streamlined at low velocities but turns chaotic as Typical ranges of Prandtl numbers for the velocity is increased above a critical value, as shown in Fig. 6–20. The flow common fluids regime in the first case is said to be laminar, characterized by smooth stream- Fluid Pr lines and highly ordered motion, and turbulent in the second case, where it Liquid metals 0.004–0.030 is characterized by velocity fluctuations and highly disordered motion. The Gases 0.7–1.0 transition from laminar to turbulent flow does not occur suddenly; rather, it Water 1.7–13.7 occurs over some region in which the flow fluctuates between laminar and Light organic fluids 5–50 turbulent flows before it becomes fully turbulent. Most flows encountered Oils 50–100,000 in practice are turbulent. Laminar flow is encountered when highly viscous Glycerin 2000–100,000 fluids such as oils flow in small pipes or narrow passages. https://ebook2book.ir/ https://ebook2book.ir/ 403 CHAPTER 6 We can verify the existence of these laminar, transitional, and turbulent flow regimes by injecting some dye streak into the flow in a glass tube, as Turbulent flow the British scientist Osborn Reynolds (Fig. 6–21) did over a century ago. We observe that the dye streak forms a straight and smooth line at low velocities when the flow is laminar (we may see some blurring because of molecular diffusion), has bursts of fluctuations in the transitional regime, and zigzags Laminar flow rapidly and randomly when the flow becomes fully turbulent. These zigzags and the dispersion of the dye are indicative of the fluctuations in the main flow and the rapid mixing of fluid particles from adjacent layers. Typical average velocity profiles in laminar and turbulent flow are also given in Fig. 6–14. Note that the velocity profile in turbulent flow is much fuller than that in laminar flow, with a sharp drop near the surface. The turbu- lent boundary layer can be considered to consist of four regions, characterized by the distance from the wall. The very thin layer next to the wall where vis- cous effects are dominant is the viscous sublayer. The velocity profile in this layer is very nearly linear, and the flow is streamlined. Next to the viscous sublayer is the buffer layer, in which turbulent effects are becoming signifi- cant, but the flow is still dominated by viscous effects. Above the buffer layer is the overlap layer, in which the turbulent effects are much more significant, but still not dominant. Above that is the turbulent layer, in which turbulent effects dominate over viscous effects. FIGURE 6–19 The intense mixing of the fluid in turbulent flow as a result of rapid fluctua- Laminar and turbulent flow tions enhances heat and momentum transfer between fluid particles, which regimes of candle smoke. increases the friction force on the surface and the convection heat transfer rate. It also causes the boundary layer to enlarge. Both the friction and heat transfer coefficients reach maximum values when the flow becomes fully tur- bulent. So it will come as no surprise that a special effort is made in the design Dye trace of heat transfer coefficients associated with turbulent flow. The enhancement in heat transfer in turbulent flow does not come for free, however. It may be necessary to use a larger pump to overcome the larger friction forces accom- Vavg panying the higher heat transfer rate. Reynolds Number Dye injection The transition from laminar to turbulent flow depends on the surface geom- (a) Laminar flow etry, surface roughness, flow velocity, surface temperature, and type of fluid, among other things. After exhaustive experiments in the 1880s, Osborne Reynolds discovered that the flow regime depends mainly on the ratio Dye trace of the inertia forces to viscous forces in the fluid. This ratio is called the Reynolds number (Fig. 6–21), which is a dimensionless quantity, and which Vavg is expressed for external flow as (Fig. 6–22) Inertia forces VLc _____ ____________ ρVL      =  ____ Re =        =    c (6–13) Viscous forces ν μ Dye injection where V is the upstream velocity (equivalent to the free-stream velocity for a (b) Turbulent flow flat plate), Lc is the characteristic length of the geometry, and ν = μ/ρis the kinematic viscosity of the fluid. For a flat plate, the characteristic length is FIGURE 6–20 the distance x from the leading edge. Note that kinematic viscosity has the The behavior of colored fluid injected unit m2 /s, which is identical to the unit of thermal diffusivity and which can into the flow in laminar and be viewed as viscous diffusivity or diffusivity for momentum. turbulent flows in a pipe. https://ebook2book.ir/ https://ebook2book.ir/ 404 FUNDAMENTALS OF CONVECTION At large Reynolds numbers, the inertia forces, which are proportional to the density and the velocity of the fluid, are large relative to the viscous forces, and thus the viscous forces cannot prevent the random and rapid fluctuations of the fluid. At small or moderate Reynolds numbers, however, the viscous forces are large enough to suppress these fluctuations and to keep the fluid “in line.” Thus the flow is turbulent in the first case and laminar in the second. The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number. The value of the critical Reynolds number is dif- ferent for different geometries and flow conditions. For flow over a flat plate, the generally accepted value of the critical Reynolds number is Recr= V xcr/v = 5 × 105 ,where xcr is the distance from the leading edge of the plate at which transition from laminar to turbulent flow occurs. The value of Recrmay change substantially, however, depending on the level of turbulence in the free stream. 6–6 HEAT AND MOMENTUM TRANSFER IN TURBULENT FLOW Most flows encountered in engineering practice are turbulent, and thus FIGURE 6–21 it is important to understand how turbulence affects wall shear stress and Osborne Reynolds (1842–1912), an heat transfer. However, turbulent flow is a complex mechanism dominated English engineer and physicist best by fluctuations, and despite tremendous amounts of work done in this area known for his work in the fields of by researchers, the theory of turbulent flow is still not fully understood. hydraulics and hydrodynamics, was Therefore, we must rely on experiments and the empirical or semi-empirical born in Belfast, Ireland. Reynolds’ studies of condensation and the correlations developed for various situations. transfer of heat between solids and Turbulent flow is characterized by disorderly and rapid fluctuations of swirling fluids brought about radical revisions regions of fluid, called eddies, throughout the flow. These fluctuations provide an in boiler and condenser design, and additional mechanism for momentum and energy transfer. In laminar flow, fluid his work on turbine pumps laid the particles flow in an orderly manner along pathlines, and momentum and energy foundation for their rapid develop- are transferred across streamlines by molecular diffusion. In turbulent flow, the ment. His classical paper on “The Law swirling eddies transport mass, momentum, and energy to other regions of flow of Resistance in Parallel Channels” much more rapidly than molecular diffusion, greatly enhancing mass, momen- (1883) investigated the transition tum, and heat transfer. As a result, turbulent flow is associated with much higher from smooth, or laminar, to turbulent values of friction, heat transfer, and mass transfer coefficients (Fig. 6–23). flow. In 1886 he also formulated “The Even when the average flow is steady, the eddy motion in turbulent flow Theory of Lubrication,” and later in causes significant fluctuations in the values of velocity, temperature, pressure, 1889, he developed a mathematical and even density (in compressible flow). Figure 6–24 shows the variation of framework which became the standard the instantaneous velocity component u with time at a specified location, as in turbulence work. His other work can be measured with a hot-wire anemometer probe or other sensitive device. included the explanation of the radi- We observe that the instantaneous values of the velocity fluctuate about an ometer and an early absolute determi- average value, which suggests that the velocity can be expressed as the sum of nation of the mechanical equivalent an average value u  ¯and a fluctuating component u′, of heat. The dimensionless Reynolds number, which provides a criterion ¯ + u′ u =  u  (6–14) for dynamic similarity and for correct This is also the case for other properties such as the velocity component v modeling in many fluid flow experi- in the y-direction, and thus v =  ¯v+ v′, P =  ¯ =  ¯ P+ P′, and T T+ T ′. The aver- ments, is named after him. age value of a property at some location is determined by averaging it over ©Paul Fearn/Alamy a time interval that is sufficiently large so that the time average levels off to a constant. Therefore, the time average of fluctuating components is zero, e.g., ¯  u′   = 0.The magnitude of u′is usually just a few percent of  u  ¯,but the high frequencies of eddies (in the order of a thousand per second) makes https://ebook2book.ir/ https://ebook2book.ir/ 405 CHAPTER 6 them very effective for the transport of momentum, thermal energy, and mass. Inertia forces In time-averaged stationary turbulent flow, the average values of properties Re = Viscous forces (indicated by an overbar) are independent of time. The chaotic fluctuations ρV 2avg L2 of fluid particles play a dominant role in pressure drop, and these random Vavg = μVavg L motions must be considered in analyses together with the average velocity. L ρVavg L Perhaps the first thought that comes to mind is to determine the shear stress = μ in an analogous manner to laminar flow from τ = −μ d¯   /dr, where ¯ u  u(r)is the Vavg L average velocity profile for turbulent flow. But the experimental studies show = ν that this is not the case, and the shear stress is much larger due to the turbulent fluctuations. Therefore, it is convenient to think of the turbulent shear stress FIGURE 6–22 as consisting of two parts: the laminar component, which accounts for the The Reynolds number can be viewed friction between layers in the flow direction (expressed as τlam= μ(d  ¯ u/dr), as the ratio of inertial forces to viscous and the turbulent component, which accounts for the friction between the forces acting on a fluid element. fluctuating fluid particles and the fluid body (denoted as τturband is related to the fluctuation components of velocity). Consider turbulent flow in a horizontal pipe and the upward eddy motion of fluid particles in a layer of lower velocity to an adjacent layer of higher veloc- ity through a differential area dA as a result of the velocity fluctuation v′, as shown in Fig. 6–25. The mass flow rate of the fluid particles rising through dA is ρv′   dA,and its net effect on the layer above dA is a reduction in its aver- 2 2 2 2 2 12 2 5 7 5 5 5 5 5 5 2 5 7 2 12 age flow velocity because of momentum transfer to the fluid particles with 7 7 7 7 7 7 12 7 5 12 lower average flow velocity. This momentum transfer causes the horizontal 12 12 12 12 12 2 7 5 12 2 velocity of the fluid particles to increase by u′,and thus causes its momentum in the horizontal direction to increase at a rate of (ρv′ dA)u′ ,which must be (a) Before turbulence (b) After turbulence equal to the decrease in the momentum of the upper fluid layer. FIGURE 6–23 Noting that force in a given direction is equal to the rate of change of The intense mixing in turbulent flow momentum in that direction, the horizontal force acting on a fluid element brings fluid particles at different tem- above dA due to the passing of fluid particles through dA is δ F = (ρv′ dA)(−u′ ) peratures into close contact and thus = −ρu′ v′ dA.Therefore, the shear force per unit area due to the eddy motion enhances heat transfer. of fluid particles δF/dA = −ρu′ v′ can be viewed as the instantaneous tur- bulent shear stress. Then the turbulent shear stress can be expressed as τturb= −ρ ¯ u′ v′ where ¯ u  ′ v′ is the time average of the product of the fluctuat- ing velocity components u ′ and v′ .Similarly, considering that h = cpT rep- resents the energy of the fluid and T′is the eddy temperature relative to the mean value, the rate of thermal energy transport by turbulent eddies is q turb= ρ cp¯  v′ T′ . Note that ¯   v′  ≠ 0, even though ¯ u  ′   = 0 and ¯ v  ′   = 0 (and thus ¯ u u′ u  ′ v′   = 0), and experi- mental results show that ¯ u  ′ v′ is usually a negative quantity. Terms such as −¯  u′ v′ or −ρ ¯ ρ u′ 2 are called Reynolds stresses or turbulent stresses. u' The random eddy motion of groups of particles resembles the random motion u– of molecules in a gas—colliding with each other after traveling a certain dis- tance and exchanging momentum and heat in the process. Therefore, momen- tum and heat transport by eddies in turbulent boundary layers is analogous to the molecular momentum and heat diffusion. Then turbulent wall shear stress and turbulent heat transfer can be expressed in an analogous manner as ∂ ¯ u ∂T Time, t τturb  = −ρ ¯ u′v′  = μt  ___   and  q  turb     = ρcp¯ v  ′ T′  = −kt  ___  (6–15) ∂y ∂y FIGURE 6–24 where μ tis called the turbulent (or eddy) viscosity, which accounts for momen- Fluctuations of the velocity tum transport by turbulent eddies, and k tis called the turbulent (or eddy) component u with time at a specified thermal conductivity, which accounts for thermal energy transport by location in turbulent flow. https://ebook2book.ir/ https://ebook2book.ir/ 406 FUNDAMENTALS OF CONVECTION y turbulent eddies. Then the total shear stress and total heat flux can be expressed conveniently as ∂ ¯ u ∂ ¯ u τtotal  = (μ + μt)  ___  = ρ(ν + νt)  ___  (6–16) ρv'dA u(y) ∂y ∂y dA and v' ∂ ¯ T ∂ ¯ T q   total  = −(k + kt)  ___  = −ρcp(α + αt)  ___  u' (6–17) ∂y

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