Fundamentals of Convection 6

https://ebook2book.ir/ CHAPTER 6 F U N D A M E N TA L S O F CONVECTION S o far, we have considered conduction, which is the mechanism of heat transfer through a solid or a quiescent fluid. We now consider convection, which is the mechanism of heat transfer through a fluid in the presence of bulk fluid motion. Convection is classified as natural (or free) and forced convection, depending on how the fluid motion is initiated. In forced convection, the fluid is forced to flow over a surface or in a pipe by external means such as a pump or a fan. In natural convection, any fluid motion is caused by natural means such as the buoyancy effect, which manifests itself as the rise of warmer fluid and the fall of the cooler fluid. Convection is also classified as external and internal, depending on whether the fluid is forced to flow over a surface or in a pipe. We start this chapter with a general physical description of the convection mechanism. We then discuss the velocity and thermal boundary layers and laminar and turbulent flows. We continue with the discussion of the dimensionless Reynolds, Prandtl, and Nusselt numbers and their physical significance. Next we derive the convection equations on the basis of mass, momentum, and energy conservation, and we obtain solutions for flow over a flat plate. We then nondimensionalize the convection equations and obtain functional forms of friction and convection coefficients. Finally, we present analogies between momentum and heat transfer. OBJECTIVES When you finish studying this chapter, you should be able to: ■ ■ ■ ■ ■ ■ ■ https://ebook2book.ir/ Understand the physical mechanism of convection and its classification, Visualize the development of velocity and thermal boundary layers during flow over surfaces, Gain a working knowledge of the dimensionless Reynolds, Prandtl, and Nusselt numbers, Distinguish between laminar and turbulent flows, and gain an understanding of the mechanisms of momentum and heat transfer in turbulent flow, Derive the differential equations that govern convection on the 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. https://ebook2book.ir/ FUNDAMENTALS OF CONVECTION 20°C 5 m/s 6–1 Air . Q1 50°C (a) Forced convection . Q2 Air Warmer air rising (b) Natural convection . No convection Q3 currents Air (c) Conduction FIGURE 6–1 Heat transfer from a hot surface to the surrounding fluid by convection and conduction. Hot plate Fluid Q Heat transfer through the fluid Cold plate FIGURE 6–2 Heat transfer through a fluid sandwiched between two parallel plates. ■ PHYSICAL MECHANISM OF CONVECTION We mentioned in Chap. 1 that there are three basic mechanisms of heat transfer: 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. 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 or gas, however, can be by conduction or convection, depending on the presence 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 convection, corresponding to the case of quiescent fluid (Fig. 6–1). 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 brings warmer and cooler chunks of fluid into contact, initiating higher rates 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. To clarify this point further, consider steady heat transfer through a fluid contained between two parallel plates maintained at different temperatures, as shown in Fig. 6–2. The temperatures of the fluid and the plate are the same at the points of contact because of the continuity of temperature. Assuming no fluid motion, the energy of the hotter fluid molecules near the hot plate is transferred to the adjacent cooler fluid molecules. This energy is then transferred 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 surface. We know that heat is transferred from the hot block to the surrounding 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. Experience shows that convection heat transfer strongly depends on the fluid properties dynamic viscosity ​μ,​ thermal conductivity k, density ​ρ,​ and specific heat ​c​ρ​,​as well as the fluid velocity V. It also depends on the geometry and the roughness of the solid surface, in addition to the type of fluid flow (such as being streamlined or turbulent). Thus, we expect the convection heat transfer relations to be rather complex because of the dependence of convection on so many variables. This is not surprising, since convection is the most complex mechanism of heat transfer. Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference and is conveniently expressed by Newton’s law of cooling as ​​​q​​ • ​​conv​​  = h(​T​s​​  − ​T​∞​​) https://ebook2book.ir/ (W/​m2​​ ​)​ (6–1) https://ebook2book.ir/ CHAPTER 6 or • ​​​​Q ​​​​ conv​​  = h​A​s​​(T ​ ​s​​  − ​T​∞​​)​  (W)​​ (6–2) where h = convection heat transfer coefficient, W/​m2​​ ​⋅K ​As​​​ = heat transfer surface area, ​m2​​ ​     ​​           ​  ​  ​​ ​ ​​ ​T​s​​ = 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 layer. The fluid property responsible for the no-slip condition and the development of the boundary layer is viscosity and is discussed briefly in Sec. 6–2. A fluid layer adjacent to a moving surface has the same velocity as the surface. 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 surface to the fluid layer adjacent to the surface is by pure conduction, since the fluid layer is motionless, and it can be expressed as | ∂T ​ ​​q​​ • ​​conv​​  = ​​q• ​​​​ cond​​  = −​k​fluid​​ ​​​ ___ ​    ​​ ​​   ∂y y = 0 (W/​m2​​ ​) https://ebook2book.ir/ Courtesy of IIHR-Hydroscience & Engineering Uniform approach velocity, V (6–3) ​where T represents the temperature distribution in the fluid and ​(∂ T/∂ y)​y=0​​ is the temperature gradient at the surface. Heat is then convected away from the surface as a result of fluid motion. Note that convection heat transfer from a 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 Eqs. 6–1 and 6–3 for the heat flux to obtain −kfluid(∂ T/∂ y)y = 0 ​ h = ​ ______________     ​  (W/m2⋅K)​ Ts − T∞ FIGURE 6–3 The development of a velocity profile due to the no-slip condition as a fluid flows over a blunt nose. (6–4) y x Relative velocities of fluid layers Zero velocity at the surface Plate FIGURE 6–4 A fluid flowing over a stationary surface comes to a complete stop at the surface because of the no-slip condition. https://ebook2book.ir/ 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 convection heat transfer coefficients over the entire surface. Nusselt Number In convection studies, it is common practice to nondimensionalize the governing equations and combine the variables, which group together into dimensionless 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 h​L​ ​​ ​Nu = ___ ​   ​​c k FIGURE 6–5 Wilhelm Nusselt (1882–1957), was a German engineer, born in Nuremberg, Germany. He studied machinery at the Technical Universities of BerlinCharlottenburg and Munchen and conducted advanced studies in mathematics and physics. His doctoral thesis was on the “Conductivity of Insulating Materials,” which he completed in 1907. In 1915, Nusselt published his pioneering paper: “The Basic Laws of Heat Transfer,” in which he first proposed the dimensionless groups now known as the principal parameters in the similarity theory of heat transfer. His other famous works were concerned with the film condensation of steam on vertical surfaces, the combustion of pulverized coal, and the analogy between heat and mass transfer in evaporation. Among his well-known mathematical works are the solutions for laminar heat transfer in the entrance region of tubes and for heat exchange in crossflow, and the basic theory of regenerators. Source: Wikipedia (6–5) where k is the thermal conductivity of the fluid and ​L​c​ 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 the 20th century, and it is viewed as the dimensionless convection heat transfer coefficient. To understand the physical significance of the Nusselt number, consider a fluid layer of thickness L and temperature difference ​ΔT = ​T​2​− ​T​1​,​as shown in Fig. 6–6. Heat transfer through the fluid layer is by convection when the fluid involves some motion and by conduction when the fluid layer is motionless. Heat flux (the rate of heat transfer per unit surface area) in either case is ​​​q​​ • ​​conv​​ = hΔT​ (6–6) and ΔT ​​​q​​ • ​​cond​​ = k __________ ​   ​​ L (6–7) Taking their ratio gives ​​q• ​​​​ conv​​ _ _ _hΔT hL ​​ _____  ​  = ​  _ _ _ _ _ _ _ _ _ _ _ _ _ ​  = ​ ___ ​ = Nu​ ​​q• ​​​​ cond​​ kΔT/L k (6–8) which is the Nusselt number. Therefore, the Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer. The larger the Nusselt number, the more effective the convection. A Nusselt number of N ​ u = 1​for a fluid layer represents heat transfer across the layer by pure conduction. We use forced convection in daily life more often than you might think (Fig. 6–7). We resort to forced convection whenever we want to increase the rate of heat transfer from a hot object. For example, we turn on the fan on hot 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 packaging is to use a large enough fan. https://ebook2book.ir/