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1 INTRODUCTION TO TRANSPORT PHENOMENA 1.1 Introduction Multiphase transport phenomena must be considered in the design and optimization of many engineering systems, such as heat exchangers, heat pipes, electronics cooling devices, biotechnology, nanotechnology, food processing equipment, and fuel cells. While some of these examples make intentional use of phase change to transfer large quantities of heat over small temperature differences, others involve phase change as an inevitable consequence rather than an intended design feature of the process. In each of these cases, and in many others that will be cited in this text, the presence of multiple phases and of phase change has a profound impact on system performance, and must be accounted for in order to achieve the system design objectives in the most efficient manner. The presence of multiple phases within a single system may significantly alter the systems’ performance characteristics in terms of: (1) the pressure drop that determines a flowing system’s power requirements, (2) the heat transfer rates that control its capacity, and (3) the flow stability that in turn affects its operational characteristics. The presence of multiple phases inevitably makes systems more complicated and affects reliability. In making the progression to the more realistic, i.e., more complex, design space of multiphase systems, the student or practicing engineer confronts some characteristic features that, usually in some combination, distinguish multiphase systems from the single-phase, single-component world of typical undergraduate heat transfer courses and texts: 1. Thermodynamic equilibrium between phases. 2. Multiple sets of thermophysical properties and field variables. 3. Interfaces between phases including mass transfer where jump conditions prevail. 4. A prominent role for latent heat transfer. 5. Simultaneous coupled heat and mass transfer. 6. Complex dynamics such as bubble growth and collapse. Chapter 1 Introduction to Transport Phenomena 1 7. A wide range of physical scales within a single design problem. In spite of these complexities, the theory of multiphase systems relies on the familiar laws of thermodynamics, fluid mechanics, and heat transfer, such as the first and second laws of thermodynamics, Newton’s laws, Fourier’s law, etc. Moreover, the definition of reliable predictive algorithms for multiphase systems requires the use of many familiar analytical tools such as control volume or differential analysis; Lagrangian or Eulerian reference systems; and dimensional and scale analysis. Typically predicted events, as given or implied by the design problems posed, include heat transfer rates, temperature histories, and steady state temperature profiles, as well as rates of mass transfer – all issues that are relevant regardless of the number of phases present in the system. However, while most of the physical principles, analytical tools, and predicted outputs are common to single- and multiphase systems, the latter require more complex expressions of the basic laws, more elaborate application of the analytical tools, and some unique terms in order to accommodate distinguishing factors such as those listed above. Depending on the designer’s chosen modeling approach, this more complex expression may take the form of discrete sets of continuity, momentum, or energy equations, with one set for each phase in the system. Alternatively, the increased complexity may appear as added or modified terms – latent heat or homogeneous density, for example – within the same number of equations as a single-phase system would require. When applying analytical tools, special care must be taken to perform control volume analyses in the multiphase context within the limits of their applicability, i.e., within regions where suitable averaging schemes are feasible and in which the presence of interfaces between phases has been taken into account. The analyst must also develop interfacial equations that express jump conditions between phases. New concepts that will be encountered while deriving multiphase system models include terms such as void fraction, which describes the relationship between the volumes, and slip ratio, which describes velocities of the phases comprising the system. As is clear from these examples, readers may expect to encounter in multiphase heat transfer, much that is familiar from more basic subject matter, as well as new concepts that are challenging in nature. It is the aim of this book to develop, within a single volume, these more sophisticated expressions and analytical tools from the familiar physical laws, and then to demonstrate their use for all types of phase changes (solid ' liquid ' vapor ' solid) at a level suitable for senior undergraduate students, graduate students, scientists, and practicing engineers. In the process, the text aims to provide a thorough understanding of: (a) the physical principles governing the experimental and analytical bases of multiphase heat transfer; (b) the generalized governing equations in multiphase systems for physical and analytical modeling of multiphase heat and mass transfer; and (c) the analysis of phase change heat and mass transfer for various multiphase systems. Ttraditionally, three approaches were used to present transport phenomena: microscopic (differential), macroscopic (integral) and molecular level. However, 2 Transport Phenomena in Multiphase Systems most heat transfer textbooks place the emphasis on microscopic and/or macroscopic. With the importance of micro-scale heat transfer or transport phenomena in applications of nanotechnology and biotechnology, as well as molecular dynamic simulations, it is important to discuss the molecular approach and the connection between the molecular and microscopic approaches. In this textbook, an attempt is made to better describe this relationship. For example, the generalized conservation equations in Chapter 3 have been developed not only microscopically and macroscopically using the continuum approach, but also, using the Boltzmann equation. This chapter begins with a review of the concept of phases of matter and a discussion of the role of phases in systems that include, simultaneously, more than one phase. This is followed by a review of transport phenomena with detailed emphasis in multicomponent systems, microscale heat transfer, dimensional analysis, and scaling. The processes of phase change between solid, liquid, and vapor are also reviewed briefly, and the classification of multiphase systems is presented. Finally, in this chapter some typical practical applications are described, which require students to understand the operational principles of these multiphase devices for further understanding and application in homeworks and examples in future chapters. 1.2 Physical Concepts 1.2.1 Sensible Heat Multiphase heat and mass transfer is concerned primarily with the study of interactions between energy and matter. In light of this fact, it is instructive to briefly review the historical background of the concepts of sensible and latent heat. As will be explained in the subsequent development of this text, energy is a property possessed by particles of matter, and heat is the transfer of this energy between particles. It is common sense that heat flows from an object at a higher temperature to one at a lower temperature. Before the nineteenth century, it was believed that heat was a fluid substance named caloric. The temperature of an object was thought to increase when caloric flowed into an object and to decrease when caloric flowed out of the object. Combustion was believed to be a process during which a large amount of caloric was released. Because heat flow never produced a detectable change in mass, and because caloric could not be detected by any other means, it was logical to assume that caloric was massless, odorless, tasteless, and transparent. Although the caloric theory explained many observations, such as heat flow from an object with a high temperature to one at a lower temperature, it was unable to account for other phenomena, such as heat generated by friction. For example, one can rub together two pieces of metal for a Chapter 1 Introduction to Transport Phenomena 3 long time and generate heat indefinitely, a process that is inconsistent with a characterization of heat as a substance of finite quantity contained within an object. In the 1800s an English brewer, James Prescott Joule, established the correct concept of heat through a number of experiments. One of his experiments is demonstrated in Fig. 1.1. The paddle wheel turns when the weight lowers, and friction between the paddle wheel and the water causes the water temperature to rise. The same temperature rise can also be obtained by heating the water on a stove. From this and many other experiments, Joule found that one joule (J) of work always equals 4.18 calories (cal) of heat, which is well known today as the mechanical equivalent of heat. Therefore heat, like work, is a transfer of thermal energy rather than the flow of a substance. In a process where heat is transferred from a high-temperature object to a low-temperature object, thermal energy, not a substance, is transferred from the former to the latter. In fact, the unit for heat in the SI system is the joule, which is also the unit for work. The amount of sensible heat Q required to raise the temperature of a system from T1 to T2 is proportional to the mass of the system and the temperature rise, i.e., Q = mc(T2 − T1 ) (1.1) where the proportionality constant c is called the specific heat and is a property of the material. Specific heat is defined as the amount of heat required to raise the temperature of a unit mass of the substance by one degree. For example, g Figure 1.1 Schematic of Joule’s experiment demonstrating the mechanical equivalent of heat. 4 Transport Phenomena in Multiphase Systems c = 1.00 kcal/kg- D C = 4.18 kJ/kg-D C for water at 15 °C, which means that it takes 1 kcal of heat to raise the temperature of 1 kg of water from 15 °C to 16 °C. The definition of the specific heat for a gas differs from that of a liquid and solid because the value of the gas specific heat depends on how the process is carried out. The specific heat values for two particular processes are of special interest to scientists and engineers: constant volume and constant pressure. The values of the specific heats at constant volume, cv , and at constant pressure, c p , for gases are quite different. The relationship between these two specific heats for an ideal gas is given by c p − cv = Rg (1.2) where Rg, the gas constant, is related to the universal gas constant, Ru, by Rg = Ru / M with M being the molecular mass of the ideal gas. The molecular masses and specific heats for some selected substances are listed in Table 1.1. Clearly, the values of the specific heats for a given gas at constant pressure and constant volume are quite different. For liquids and solids, the specific heat can be assumed to be process-independent, because these phases are nearly incompressible. Therefore, the specific heats of liquids and solids at constant pressure are assumed to apply to all real processes. Table 1.1 Specific heats of different substances at 20 °C Substance M (kg/kmol) c p (kJ/kg- o C) 1.005 0.90 0.84 0.39 0.84 14.27 2.10 0.45 0.13 0.86 1.04 0.92 2.02 0.23 0.14 4.18 cv (kJ/kg- o C) 0.718 0.65 Air Aluminum Carbon dioxide Copper Glass Hydrogen Ice (–5 °C) Iron Lead Marble Nitrogen Oxygen Steam (100 °C) Silver Mercury Water 28.97 26.9815 44.01 63.546 2.016 18.015 55.847 207.2 28.013 31.999 18.015 107.868 200.59 18.015 10.15 0.74 0.66 1.47 1.2.2 Latent Heat Although phase change phenomena such as the solidification of lava, the melting of ice, the evaporation of water, and the fall of rain have been observed by Chapter 1 Introduction to Transport Phenomena 5 mankind for centuries, the scientific methods used to study phase change were not developed until the seventeenth century because a flawed understanding of temperature, energy, and heat prevailed. It was incorrectly believed that the addition or removal of heat could always be measured by the change of temperature. Based on the misconception that temperature change always accompanies heat addition, a solid heated to its melting point was thought to require only a very small amount of additional heat to completely melt. Likewise, it was thought that only a small amount of extra cooling was required to freeze a liquid at its melting point. In both of these examples, the heat transferred during phase change was believed to be very small, because the temperature of the substance undergoing phase change did not change by a significant amount. Between 1758 and 1762, an English professor of medicine, Dr. Joseph Black, conducted a series of experiments measuring the heat transferred during phase change processes. He found that the quantity of heat transferred during phase change was in fact very large, a phenomenon that could not be explained in terms of sensible heat. He demonstrated that the conventional wisdom about the amount of heat transferred during phase change was wrong, and he used the term “latent heat” to define heat transferred during phase change. Latent heat is a hidden heat, and it is not evident until a substance undergoes a phase change. Perhaps the most significant application of Dr. Black’s latent heat theory was James Watt’s 500% improvement of steam engine thermal efficiency. James Watt was an engineer and Dr. Black’s assistant for a time. The concept of latent heat can be demonstrated by tracing the phase change of water from subcooled ice below 0 °C, to superheated vapor above 100 °C. Let us consider a 1-kg mass of ice with an initial temperature of –20 °C. When heat is added to the ice, its temperature gradually increases to 0 °C, at which point the temperature stops increasing even when heat is continuously added. During the ensuing interval of constant temperature, the change of phase from ice to liquid water can be observed. After the entire mass of ice is molten, further heating 120 100 Temperature ( C) 80 60 40 20 0 -20 0 500 1000 1500 2000 2500 3000 Heat Added (kJ) Figure 1.2 Temperature profile for phase change from subcooled ice to superheated steam. o 6 Transport Phenomena in Multiphase Systems produces an increase of the now-liquid temperature up to 100 °C. Continued heating of the liquid water at 100 °C does not yield any increase of temperature; instead, the liquid water is vaporized. After the last drop of the water is vaporized, continued heating of the vapor will result in the increase of its temperature. The phase change process from subcooled ice to superheated vapor is shown in Fig. 1.2. It is seen that a substantial amount of heat is required during a change of phase, an observation consistent with Dr. Black’s latent heat theory. Table 1.2 Latent heat of fusion and vaporization for selected materials at 1 atm Substance Oxygen Ethyl Alcohol Water Lead Silver Tungsten Melting point (°C) –218.18 –114 0 327 961 3410 hsA (kJ/kg) 14 105 335 25 88 184 Boiling point (°C) –183 78 100 1750 2193 5900 hAv (kJ/kg) 220 870 2251 900 2300 4800 The heat required to melt a solid substance of unit mass is defined as the latent heat of fusion, and it is represented by hsA . The latent heat of fusion for water is about 335 kJ/kg. The heat required to vaporize a liquid substance of unit mass is defined as the latent heat of vaporization, and it is represented by hAv . The latent heat of vaporization for water is about 2251 kJ/kg. The latent heats of other materials, shown in Table 1.2, demonstrate that the latent heats of vaporization for all materials are much larger than their latent heats of fusion, because the molecular spacing for vapor is much larger than that for solid or liquid. The latent heat for deposition/sublimation, hsv , for water is about 2847 kJ/kg. 1.2.3 Phase Change When a process involves phase change, the heat absorbed or released can be expressed as the product of the mass quantity and the latent heat of the phase change material. For liquid-vapor phase change, such as evaporation, boiling, or condensation, the heat transferred can be expressed as q = mhAv (1.3) where m is the mass of material changing phase per unit time. The latent heat of vaporization, hAv , is the difference between the enthalpy of vapor and of liquid, i.e., hAv = hv − hA (1.4) For a pure substance, liquid-vapor phase change always occurs at the saturation temperature. Since the pressure for the saturated liquid-vapor mixture Chapter 1 Introduction to Transport Phenomena 7 q Figure 1.3 Liquid-vapor phase change in rigid tank with a relief valve. is a function of temperature only, liquid-vapor phase change usually occurs at a constant pressure. As will become evident, eq. (1.3) needs to be modified for a liquid-vapor phase change process occurring at a constant pressure (Bejan, 1993). Figure 1.3 shows a rigid tank filled with a mixture of liquid and vapor in equilibrium. To maintain a constant pressure during evaporation, a relief valve at the top of the tank is opened; the mass flow rate of vapor through the valve is m. At time t, the masses of the liquid and vapor are, respectively, mA and mv . The change of the masses of the liquid and vapor satisfies dmA dmv + = −m (1.5) dt dt The total volume of the liquid and vapor V = mA vA + mv vv (1.6) remains constant during the phase change process. Thus dm dmA (1.7) vA + v vv = 0 dt dt Combining eqs. (1.5) and (1.7) yields §v · dmA = −m ¨ v ¸ (1.8) dt © vv − vA ¹ §v · dmv = m¨ A ¸ dt © vv − vA ¹ (1.9) 8 Transport Phenomena in Multiphase Systems The first law of thermodynamics for the control volume (which will be presented in detail in Chapter 3) is dEcv = q − mhv (1.10) dt where the internal energy of the control volume is Ecv = mA eA + mv ev (1.11) Differentiating eq. (1.11) and considering eqs. (1.8) and (1.9), one obtains §v e −v e · dEcv (1.12) = −m ¨ v A A v ¸ dt © vv − vA ¹ Substituting eq. (1.12) into eq. (1.9) and considering eq. (1.4) yields ª § v e − v e ·º q = m « hAv + ¨ hA − v A A v ¸ » (1.13) vv − vA ¹ » « © ¬ ¼ where the terms in the parentheses are the correction on latent heat required for this specific process. This correction is necessary in order to adjust the internal energy and maintain constant pressure and temperature in the phase change process. It should be pointed out, however, that this correction is usually very insignificant except near the critical point. Therefore, eq. (1.3) is usually valid for constant-pressure liquid-vapor phase change processes. During melting and solidification processes, heat transfer can be expressed as q = mhsA (1.14) where hsA is the latent heat of fusion. Since the change of volume during solidliquid phase change is insignificant, a correction similar to that in eq. (1.13) usually is not necessary. 1.3 Molecular Level Presentation 1.3.1 Introduction The concepts associated with phases of matter at microscopic and macroscopic levels are important in the study of multiphase systems, and therefore they are reviewed briefly here. For this purpose, consider a lump of ordinary sugar. When the lump is broken into smaller pieces, each of these smaller pieces is still identifiable as a particle of sugar based on properties such as its color, density, and crystalline shape. If one continues to grind the sugar to a finer powder, the basic properties of the material remain the same except that the size of the particles is reduced. When the very fine powder sugar is dissolved in water, the particles are too small to be seen with a microscope, yet the taste of sugar persists. Evaporating the water from the sugar solution restores the original distinguishing properties of the solid sugar mentioned above. This simple experiment shows that matter is composed of particles and that the ultimate Chapter 1 Introduction to Transport Phenomena 9 particles of matter are extremely small. The smallest particle of sugar that retains any identifying properties of the substance is a molecule of sugar. A molecule is the smallest chemical unit of a substance that is capable of stable, independent existence; however, not all substances are composed of molecules. Some substances are composed of electrically-charged particles known as ions. To get an idea of the extremely small size of molecules, we can consider that a molecule of water is about 3×10-10 m (3 Angstroms, Å) in diameter. On the other hand, molecules of more complex substances may have sizes of more than 200 Å. If a molecule of sugar is analyzed further, it is found to consist of particles of three simpler kinds of matter: carbon, hydrogen and oxygen. These simpler forms of matter are called elements. An atom is the smallest unit of an element that can exist either alone or in combination with other atoms of the same or different elements. The smallest atom, an atom of hydrogen, has a diameter of 0.6 Å. The largest atoms are slightly larger than 6 Å in size. The atomic mass of an atom is One atomic mass unit is equal to expressed in atomic mass units. −27 1.6605402 × 10 kg. This mass is 1/12 the mass of the carbon-12 atom. The integer nearest to the atomic mass is called the mass number of an atom. The mass number for a hydrogen atom is 1; for a common uranium atom, one of the heaviest atoms, it is 238. Under normal conditions at the macroscopic level, there are three phases (or states) of matter: solid phase, liquid phase, and gaseous phase. Plasma is sometimes called the fourth phase of matter. In the description of matter, phase indicates how particles group together to form a substance. The structure of a substance can vary from compactly-arranged particles to highly-dispersed ones. In a solid, the particles are close together in a fixed pattern, while in a liquid the particles are almost as close together as in a solid but are not held in any fixed pattern. In a gas, the particles are also not held in any fixed pattern, but the average distance between particles is large. Both liquids and gases are called fluids. A gas that is capable of conducting electricity is called plasma. Gases do not normally conduct electricity, but when they are heated to high temperatures or collide intensely with each other, they form electrically-charged particles called ions. These ions give the plasma the ability to conduct an electrical current. Because of the high temperatures that prevail in the sun and other stars, their constituent matter exists almost entirely in the plasma phase. 1.3.2 Kinetic Theory According to the elementary kinetic theory of matter, the molecules of a substance are in constant motion. This motion depends on the average kinetic energy of molecules, which depends in turn on the temperature of the substance. Furthermore, the collisions between molecules are perfectly elastic except when chemical changes or molecular excitations occur. 10 Transport Phenomena in Multiphase Systems The concept of heat as the transfer of thermal energy can be explained by considering the molecular structures of a substance. At the standard reference state (25 ÛC at 1 atm), the density of a typical gas is about 1/1000 of that of the same matter as a liquid. If the molecules in the liquid are closely packed, the distance between gas molecules is about 10001/ 3 = 10 times the size of the molecules. Since the size of molecules is on the order of 10−10 m, the distance between the molecules of gas is on the order of 1 0 − 9 m . Therefore, a gas in the standard reference state can be viewed as a set of molecules with large distances between them. Since the distance between gas molecules is so large, the intermolecular forces are very weak, except when molecules collide with each other. The distance that a molecule travels between two collisions is on the order of 10−7 m, and the average velocity of molecules is about 500m/s, which means that the molecules collide with each other every 10−10 s, or at the rate of 10 billion collisions per second. The duration of each collision is approximately 10−13 s, a much shorter interval than the average time between two collisions. The movement of gas molecules can therefore be characterized as frequent collisions between molecules, with free movement between collisions. For any particular molecule, the magnitude and the direction of the velocity change arbitrarily due to frequent collision. The free path between collisions is also arbitrary and difficult to trace. Although the motion of the individual molecule is random and chaotic, the movement of the molecules in a system can be characterized using statistical rules. The following assumptions about the structure of the gases are made in order to investigate the statistical rules of the random motion of the molecules: 1. The size of the gas molecules is negligible compared with the distance between gas molecules. 2. The molecules collide infrequently because the collision time is much shorter than the free motion time. 3. The effects of gravity and any other field force are negligible, thus the molecules move along straight lines between collisions. The motion of gas molecules obeys Newton’s second law. 4. The collision of gas molecules is elastic, which means that the kinetic energy before and after a collision is the same. Therefore, the gas can be viewed as a set of elastic molecules that move freely and randomly. Any gas that satisfies the above assumptions is referred to as an ideal gas. At thermodynamic equilibrium, the density of the gas in a container is uniform. Therefore, it is reasonable to assume that the gas molecules do not prefer any particular direction over other directions. In other words, the average of the square of the velocity components of the gas in all three directions should be the same, i.e., u 2 = v 2 = w2 . The average magnitude of the molecular velocity is given by kinetic theory Chapter 1 Introduction to Transport Phenomena 11 8k b T (1.15) πm where kb is Boltzmann constant, and m is the mass of the molecule. For any stationary surface exposed to the gas, the frequency of the gas molecular bombardment per unit area on one side is given by 1 f = Nc (1.16) 4 where N is the number density of the molecules, defined as number of molecules per unit volume (N = N / V ). The mean free path, defined as average distance traveled by a molecule between collisions, is 1 (1.17) λ= 2πσ 2N where σ is the molecular diameter. After the last collision with other molecules, the molecule travels an average distance of 2λ / 3 before it collides with the plane. A table for the particle diameters, mean free path, mean velocity and mean time (τ ) between molecular collisions is given in Table 1.3 for some gases at 25° C and atmospheric pressure. c= Table 1.3 Kinetic properties of gases at 25 °C and atmospheric pressure (Lide, 2004)* λ × 108 (m) d ×1010 (m) Air 3.66 6.91 Ar 3.58 7.22 CO2 4.53 4.51 H2 2.71 12.6 He 2.15 20.0 Kr 4.08 5.58 N2 3.70 6.76 NH3 4.32 4.97 Ne 2.54 14.3 O2 3.55 7.36 Xe 4.78 4.05 * Reproduced by permission of Routledge/Taylor & Francis Group, LLC. Gas cm (m/s) τ (ps) 148 182 119 71 159 203 142 82 256 166 185 467 397 379 1769 1256 274 475 609 559 444 219 The pressure of gas in a container results from the large number of gas molecules colliding with the container wall. Although each molecule in the container collides with the container wall randomly and discontinuously, the collisions of a large number of molecules with the container wall impart a constant and continuous pressure on the wall. As expected, the pressure in a container is related to the number and average velocity of the molecules by 1 c2 (1.18) p = Nm 3 V 12 Transport Phenomena in Multiphase Systems where N is the number of molecules in the container, m is the mass of each molecule, V is the volume of the container, and c 2 is the average of the square of the molecular velocity 1N 2 c 2 = ¦ cn (1.19) N n =1 The average of the square of the molecules’ velocity is related to its three components by 1 u 2 = v 2 = w2 = c 2 (1.20) 3 The average kinetic energy of a molecule is defined as 1 e = mc 2 (1.21) 2 Substituting eq. (1.21) into eq. (1.18) yields 2 pV = Ne (1.22) 3 The monatomic ideal gas also satisfies the ideal gas law, i.e., pV = nRuT (1.23) where Ru = 8.3143 kJ/kmol-K is the universal gas constant, which is the same for all gases. Combining eqs. (1.22) and (1.23) yields 3 Ru e= T (1.24) 2 NA where N A = N / n is the number of molecules per mole, which is a constant that equals 6.022 × 1023 and is referred to as Avogadro’s number. Equation (1.24) can also be rewritten as 3 e = kbT (1.25) 2 where the Boltzmann constant is R 8.3143 = 1.38 × 10−23 J/K (1.26) kb = u = 23 N A 6.022 × 10 From eq. (1.24) it is evident that the average kinetic energy of molecules increases with increasing temperature. In other words, the molecules in a hightemperature gas have more kinetic energy than those in a low-temperature gas. When two objects at different temperatures come into contact, the higher-kineticenergy molecules of the high-temperature object collide with the lower-kineticenergy molecules of the low-temperature object. During these molecular collisions, some of the molecular kinetic energy of the high-temperature object is transferred to the molecules of the low-temperature object. Consequently, the molecules of the low-temperature object gain kinetic energy and its overall temperature increases. In the experiment conducted by Joule, the paddle wheel Chapter 1 Introduction to Transport Phenomena 13 collides with the water molecules and kinetic energy is transferred from the wheel to the water molecules, causing the water temperature to rise. Another important concept that can be illustrated using kinetic theory is internal energy, E, defined as the sum total of all the energy of all the molecules in an object. The internal energy of the ideal gas equals the sum of all the kinetic energies of all the atoms. This sum can be expressed as the total number of molecules, N, times the average kinetic energy per atom, i.e., 3 E = Ne = NkbT (1.27) 2 which shows that the internal energy of an ideal gas is only a function of mole number and temperature. Internal energy is also sometimes called thermal energy. It is very important to distinguish between temperature, internal energy, and heat. Temperature is related to the average kinetic energy of individual molecules [see eq. (1.24)], while internal, or thermal, energy is the total energy of all of the molecules in the object [see eq. (1.27)]. If two objects with equal masses of the same material and the same temperature are joined together, the temperature of the combined objects remains the same, but the internal energy of the system is doubled. Heat is a transfer of thermal energy from one object to another object at lower temperature. The direction of heat transfer between two objects depends solely on relative temperature, not on the amount of internal energy contained within each object. It is necessary to point out that eqs. (1.24) and (1.27) are valid only for an ideal monatomic gas. For ideal gas molecules containing more than one atom, the molecules can rotate and the different atoms in the molecule can vibrate around their equilibrium position (see Fig. 1.4). Therefore, the kinetic energy of molecules with more than one atom must include both rotational and vibrational Figure 1.4 Modes of molecular kinetic energy: (a) rotational energy, (b) vibrational energy. 14 Transport Phenomena in Multiphase Systems energy, and their internal energy at a given temperature will be greater than that of a monatomic gas at the same temperature. The internal energy of an ideal gas with a molecule containing more than one atom still depends solely on mole number and temperature. The internal energy of a real gas is a function of both temperature and pressure, which is a more complex condition than that of an ideal gas. The internal energies of liquids and solids are much more complicated, because the interactive forces between atoms and molecules also contribute to their internal energy. 1.3.3 Intermolecular Forces A keen understanding of intermolecular forces is imperative for discussing the different phases of matter. In general, the intermolecular forces of a solid are greater than those of a liquid. This trend can be observed when looking at the force it takes to separate a solid as compared to that required to separate a liquid. Also, the molecules in a solid are much more confined to their position in the solid’s structure as compared to the molecules of a liquid, thereby affecting their ability to move. Most solids and liquids are deemed incompressible. The underlying reason for their “incompressibility” is that the molecules repel each other when they are forced closer than their normal spacing; the closer they become, the greater the repelling force (Tien and Lienhard, 1979). A gas differs from both a solid and a liquid in that its kinetic energy is great enough to overcome the intermolecular forces, causing the molecules to separate without restraint. The intermolecular forces in a gas decrease as the distance between the molecules increases. Both gravitational and electrical forces contribute to intermolecular forces; for many solids and liquids, the electrical forces are on the order of 1029 times greater than the gravitational force. Therefore, the gravitational forces are typically ignored. To quantify the intermolecular forces, a potential function φ (r ) is defined as the energy required to bring two molecules, which are initially separated by an infinite distance, to a finite separation distance r. The form of the function always depends on the nature of the forces between molecules, which can be either repulsive or attractive depending on intermolecular spacing. When the molecules are close together, a repulsive electrical force is dominant. The repulsive force is due to interference of the electron orbits between two molecules, and it increases rapidly as the distance between two molecules decreases. When the molecules are not very close to each other, the forces acting between molecules are attractive in nature and generally fall into one of three categories. The first category is electrostatic forces, which occur between molecules that have a finite dipole moment, such as water or alcohol. The second category is induction forces, which occur when a permanently-charged particle or dipole induces a dipole in a nearby neutral molecule. The third category is Chapter 1 Introduction to Transport Phenomena 15 dispersion forces, which are caused by transient dipoles in nominally-neutral molecules or atoms. An accurate representation of the potential function φ (r ) should account for all of the forces discussed above. It should be able to reflect repulsive forces for small spacing and attractive forces in the intermediate distance. When the distance between molecules is very large, there should be no intermolecular forces. While the exact form of φ (r ) is not known, the following Lennard-Jones 6-12 potential provides a satisfactory empirical expression for nonpolar molecules: ª§ r ·12 § r ·6 º φ (r ) = 4ε «¨ 0 ¸ − ¨ 0 ¸ » (1.28) ©r¹ » «© r ¹ ¬ ¼ where ε and r0 are, respectively, energy of interaction and equilibrium distance, and both of them depend on the type of the molecules. Figure 1.5 shows the Lennard-Jones 6-12 potential as a function of distance between two molecules. When the distance between molecules is small, the Lennard-Jones potential decreases with increasing distance between molecules and the repulsive force dominates; it is necessary to add energy to the system in order to bring the molecules any closer. As the molecules separate, there is a distance, rmin , at which the Lennard-Jones potential becomes minimum. As the molecules move further apart, the Lennard-Jones potential increases with increasing distance between molecules, the attractive force dominates. The Lennard-Jones potential approaches zero when the molecular distance becomes very large. When the Lennard-Jones potential is minimal, the following condition is satisfied: dφ (rmin ) =0 (1.29) dr φ Molecules repulse one another for r < rmin 0 r0 r −2ε Molecules attract one another for r > rmin Figure 1.5 Lennard-Jones 6-12 potential vs. distance between two spherical, nonpolar molecules. 16 Transport Phenomena in Multiphase Systems Substituting eq. (1.28) into eq. (1.29), one obtains rmin = 21/ 6 r0 (1.30) When looking at the three phases of matter in the context of Fig. 1.5, some relationships can be described. For the solid state, the atoms are limited to vibrating about the equilibrium position, because they do not have enough energy to overcome the attractive force. The molecules in a liquid are free to move because they have a higher level of vibrational energy, but they have approximately the same molecular distance as the molecules of a solid. The energy required to overcome the attractive forces in a solid, thus allowing the molecules to move freely, corresponds to the latent heat of fusion. In the gaseous phase, on the other hand, the molecules are so far apart that they are virtually unaffected by intermolecular forces. The energy required to create vapor by separating closely-spaced molecules in a liquid corresponds to the latent heat of vaporization. The Lennard-Jones potential at this point is φ (rmin ) = −2ε (1.31) Simulation of phase change at the molecular level is not necessary for many applications in macro spatial and time scales. For heat transfer at micro spatial and time scales, the continuum transport model breaks down and simulation at the molecular level becomes necessary. One example that requires molecular dynamics simulation is heat transfer and phase change during ultrashort pulsed laser materials processing (Wang and Xu, 2002). This process is very complex because it involves extremely high rates of heating (on the order of 1016 K/s) and high temperature gradients (on the order of 1011 K/m). The motion of each molecule in the system is described by Newton’s second law, i.e., N d 2r Fij = mi 2i (1.32) ¦ dt j =1( j ≠ i ) where mi and ri are the mass and position of ith molecule in the system. In arriving at eq. (1.32), it is assumed that the molecules are monatomic and have only three degrees of freedom of motion. For molecules with more than one atom, it is also necessary to consider the effect of rotation. The force between the ith and jth molecules can be obtained from ∂φij Fij = − (1.33) ∂rij where rij is the distance between the ith and jth molecules. The Lennard-Jones potential between the ith and jth molecules is obtained by ª§ r ·12 § r ·6 º φij = 4ε «¨ 0 ¸ − ¨ 0 ¸ » ¨ rij ¸ » «¨ rij ¸ © ¹¼ ¬© ¹ (1.34) The transport properties can be obtained by the kinetic theory in the form of very complicated multiple integrals that involve intermolecular forces. For nonpolar substance that Lennard-Jones potential is valid, these integral can be Chapter 1 Introduction to Transport Phenomena 17 evaluated numerically. For a pure gas, the self-diffusivity, D, viscosity, μ , and thermal conductivity, k, are (Bird et al., 2002) 3 π mkbT 1 D= (1.35) 8 πσ 2 Ω D ρ μ= k= 5 π mkbT 16 πσ 2 Ω μ (1.36) 25 π mkbT cv (1.37) 32 πσ 2 Ω k where σ is collision diameter. The dimensionless collision integrals are related by Ω μ = Ω k ≈ 1.1Ω D and are slow varying functions of kbT / ε ( ε is a characteristic energy of molecular interaction). If all molecules can be assumed as rigid ball, all collision integrals will become unity. More information about collision diameter and integrals can be found in Section 7.3.3. cv in eq. (1.37) is the molar specific heat under constant volume. It follows from eqs. (1.35) – (1.37) that the Prandtl and Schmidt numbers are, respectively, 0.66 and 0.75, which is a very good approximation for monatomic gases (e.g., helium in Table B.3). The transport properties at system length scale of less than 10λ will be different from the macroscopic properties, because the gas molecules are not free to move as they naturally would. While the transport properties discussed here are limited to low-density monatomic gas, the discussion can also be extended to polyatomic gases, and monatomic and polyatomic liquids. For a nonequilibrium system, the mean free path theory is no longer valid, and the Boltzmann equation should be used to describe the molecular velocity distribution in the system. For low-density nonreacting monatomic gas mixtures, the random molecular movement can be described by the molecular velocity distribution function fi (c, x,t ) , where c is the particle velocity and x is the position vector in the mixture. At time t, the probable number of molecules of the ith species that are located in the volume element dx at position x and have velocity within the range d c about c is fi (c, x,t )d cd x . The evolution of the velocity distribution function with time can be described using the Boltzmann equation Df i ∂fi = + c ⋅ ∇ x f i + a ⋅ ∇ c f f = Ωi ( f ) (1.38) Dt ∂t where ∇ x and ∇c are ∇ operator with respect to x and c, respectively (see Appendix C), a is the particle acceleration (m/s2), and Ωi is a five-fold integral term that accounts for the effect of molecular collision on the change of velocity distribution function fi. The Boltzmann equation can also be considered as a continuity equation in six dimensional position-velocity space ( x and c ). The velocity distribution function is related to the number density by 18 Transport Phenomena in Multiphase Systems ³ f (c, x, t )dc = N (x, t ) i i (1.39) It can be demonstrated that the stress tensor, eq., (1.53), heat flux, eq. (1.64), and diffusive mass flux, eq. (1.115) can be obtained from the solution of velocity distribution function fi. More detailed information about Boltzmann equation and its applications related to transport phenomena in multiphase systems can be found in Section 4.7. In addition, the Boltzmann equation can also be used to describe transport of electrons and electron-lattice interaction and to yield the two-step heat conduction model for electron and lattice temperatures for nanoscale and microscale heat transfer (Qiu and Tien, 1993; Chen, 2004). 1.3.4 Cohesion and Adhesion Cohesion is the intermolecular attractive force between molecules of the same kind or phase. For a solid, cohesion is significant only when the molecules are extremely close together; for instance, once a crack forms in a metal structure, the two edges of the crack will not rejoin even if pushed together. The rejoining can not occur because gas molecules attach to the fractured surface, preventing the cohesive intermolecular attraction from occurring. The fundamental basis for viscosity observed in fluids is cohesion within the fluids. Viscosity is the resistance of a liquid or a gas to shear forces; it can be measured as a ratio of shear stress to shear strain. As a significant factor in the analysis of fluid flows, viscosity depends on temperature. Generally, as temperature increases the viscosity of a gas increases, while that of a liquid decreases. Adhesion is the intermolecular attractive force between molecules of different kinds or phase. An example of adhesion is the phenomenon of water wetting a glass surface. Intermolecular forces between the water and the glass cause the wetting. In this case, the adhesive force between the water and the glass is greater than the cohesive forces within the water. The opposite case can also occur, where the liquid is repelled from the surface, indicating that cohesion in the liquid is greater than adhesion between the liquid and the solid. For example, when a freshly waxed car sits in the rain, the raindrops bead on the surface and then easily flow off. 1.3.5 Enthalpy and Energy Phase change processes are always accompanied by a change of enthalpy, which we will now consider at the molecular level. Phase change phenomena can be viewed as the destruction or formation of intermolecular bonds as the result of changes in intermolecular forces. The intermolecular forces between the molecules in a solid are greater than those between molecules in a liquid, which Chapter 1 Introduction to Transport Phenomena 19 are in turn greater than those between molecules in a gas. This reflects the greater distance between molecules in a gas than in a liquid, and the greater distance between molecules in a liquid than in a solid. As a result, the intermolecular bonds in a solid are stronger than those in a liquid. In a gas, which has the weakest intermolecular forces of all three phases, intermolecular bonds do not exist between the widely separated molecules. When the intermolecular bonds between the molecules in a solid are completely broken, sublimation occurs. The approximate energy levels identified for the H2O molecule near 273 K are summarized in Table 1.4. Since it takes 0.29 eV (electron-volts) to break a hydrogen bond in the ice lattice, and there are two bonds per H2O molecule, the energy required to completely free a H2O molecule from its neighbor should be 0.58 eV. The energy required to break two hydrogen bonds, 0.58 eV, should be of the same order of magnitude as the enthalpy of sublimation, which is 0.49 eV, as shown in Table 1.4. The energy required to melt ice is 0.06 eV per molecule, while it takes 0.39 eV per molecule to vaporize liquid water. The difference between the enthalpy of melting and vaporization at the molecular level explains the difference in the latent heats of fusion and of vaporization, as summarized in Table 1.4. The internal energy of a substance with molecules containing more than one atom (such as H2O) is the sum of the kinetic, rotational, and vibrational energies. Since the molecules in a solid are held in a fixed pattern and are not free to move or rotate, the lattice vibrational energy is the primary contributor to the internal energy of the ice. As can be seen from Table 1.4, the enthalpy of melting and vaporization are much larger than the lattice vibrational energy, which explains why the latent heat is usually much greater than the sensible heat. Table 1.4 Energies of the H2O molecule in the vicinity of 273 K Types of energy Lattice vibration Intermolecular hydrogen bond breaking Enthalpy of melting Enthalpy of vaporization Enthalpy of sublimation Approximate magnitude per molecule (eV) 0.0054 0.58 0.06 0.39 0.49 Different phases are characterized by their bond energy and their molecular configurations. For example, the intermolecular bonds in a solid are very strong, and thus able to hold the molecules in a fixed pattern. The intermolecular bonds in a liquid are strong enough to hold the molecules together but not strong enough to hold them in a fixed pattern. The intermolecular bonds in a gas are completely broken and the molecules can move freely. Therefore, phase change can be viewed as conversion from one type of intermolecular ordering to another, i.e., a reordering process. From a microscopic point of view, the entropy of a system, S, is related to the total number of possible microscopic states of that system, known as the thermodynamic probability, P, by the Boltzmann relation: 20 Transport Phenomena in Multiphase Systems S = kb ln P −23 (1.40) where kb is the Boltzmann constant, 1.3806 × 10 J/K . Therefore, the entropy of a system increases when the randomness or thermodynamic probability of a system increases. Sublimation, melting, and vaporization are all processes that increase the randomness of the system and therefore produce increases of entropy. Since phase changes occur at constant temperature, one can express the increases of entropy in these processes as: Δh (1.41) Δs = = constant T where Δh is the change of enthalpy during phase change, i.e., the latent heat, and T is the phase change temperature. The constant in eq. (1.41) depends on the particular phase change process but is independent of the substance. For vaporization and condensation, Trouton’s rule is applicable h (1.42) sv − sA = Av  83.7J/(mol-K) Tsat while Richards’ rule is valid for melting and solidification h (1.43) sA − ss = sA  8.37J/(mol-K) Tm The thermodynamics of multiphase systems is presented in Chapter 2, which begins with a review of single-phase thermodynamics, including thermodynamic laws and relations, and proceeds to the concepts of equilibrium and stability. This is followed by discussion of thermodynamic surfaces and phase diagrams for single- and multicomponent systems. Also discussed are equilibrium criteria for single and multicomponent multiphase systems and the metastable equilibrium that exists in the multiphase system. Chapter 2 concludes with a discussion of thermodynamics at the interface and the effects of surface tension and disjoining pressure, including the superheat effect. 1.4 Review of Fundamentals of Transport Phenomena 1.4.1 Continuum Flow Limitations The transport phenomena are usually modeled in continuum states for most applications – the materials are assumed to be continuous and the fact that matter is made of atoms is ignored. Recent development in fabrication and utilization of nanotechnology, micro devices, and microelectromechanical systems (MEMS) requires noncontinuum modeling of transport phenomena in nano- and microchannels. When the dimension of the nano- or microchannel, D, is small compared to the molecular mean free path , which is defined as average distance between collisions for a molecule, the traditional Navier-Stokes equation and the Chapter 1 Introduction to Transport Phenomena 21 energy equation based on the continuum assumption have failed to provide accurate results. The continuum assumption may also not be valid in conventional systems – for example, the early stages of high-temperature heat pipe startup from a frozen state (Cao and Faghri, 1993) and microscale heat pipes (Cao and Faghri, 1994). During the early stage of startup of high-temperature heat pipes, the vapor density in the heat pipe core is very low and partly loses its continuum characteristics. The vapor flow in this condition is usually referred to as rarefied vapor flow. Because of the low density, the vapor in the rarefied state is somewhat different from the conventional continuum state. Also, the vapor density gradient is very large along the axial direction of the heat pipe. The vapor flow along the axial direction is caused mainly by the density gradient via vapor molecular diffusion. The validity of the continuum assumption can also be violated in micro heat pipes. As the size of the heat pipe decreases, the vapor in the heat pipe may lose its continuum characteristics. The heat transport capability of a heat pipe operating under noncontinuum vapor flow conditions is very limited, and a large temperature gradient exists along the heat pipe length. This is especially true for a miniature or micro heat pipe, whose dimensions may be extremely small. The continuum criterion is usually expressed in terms of the Knudsen number (1.44) D Based on the degree of rarefaction of gas, the flow regimes in microchannel can be classified into four regimes: 1. Continuum regime ( Kn < 0.01 ). The Navier-Stokes equation is valid. 2. Slip flow regime ( 0.01<Kn < 0.1 ). The Navier-Stokes equation can be used with the application of slip-condition, i.e., allowing non-zero axial fluid velocity along near the wall of the microchannel. 3. Transition regime ( 0.1<Kn < 3 ). The Navier-Stokes equation is not valid, and the flow must be solved using the Boltzmann equation or Direct Monte Carlo Simulation (DMCS). 4. Free molecular flow regime ( Kn>3 ). The collision between molecules can be neglected and a collisionless Boltzmann equation can be used. The mean free path, eq. (1.17), for dilute gases based on the kinetic theory can be rewritten in terms of temperature and pressure 1.051kbT λ= (1.45) 2πσ 2 p The transition density under which the continuum assumption is invalid can be obtained by combining eqs. (1.44) – (1.45) and Kn=0.01, i.e., Kn = λ 22 Transport Phenomena in Multiphase Systems ρtr = 1.051kb 2πσ 2 Rg DKn (1.46) where the ideal gas equation of state, p = ρ Rg T was used. Assuming that the vapor is in the saturation state, the transition vapor temperature Ttr corresponding to the transition density can be obtained by using the Clausius-Clapeyron equation (see Chapter 2) combined with the equation of state: ª h §1 p 1 ·º (1.47) Ttr = sat exp « − Av ¨ − ¸» ρ Rg « Rg © Ttr Tsat ¹ » ¬ ¼ where psat and Tsat are the saturation pressure and temperature, hAv is the latent heat of vaporization, and the vapor density ρ is given by eq. (1.46). Equation (1.47) can be rewritten as § Ttr ρ Rg · hAv § 1 1· (1.48) ln ¨ ¸+ ¨− ¸=0 © psat ¹ Rg © Ttr Tsat ¹ and solved iteratively for Ttr using the Newton-Raphson/secant method. The transition vapor temperature is the boundary between continuum regime and noncontinuum regime. 1.4.2 Transport Phenomena Transport phenomena include momentum transfer, heat transfer, and mass transfer, all of which are fundamental to an understanding of both single and multiphase systems. It is assumed that the reader has a good undergraduate-level understanding of transport phenomena as applied to single-phase systems, as well as the associated thermodynamics, fluid mechanics, and heat transfer. Therefore, this subsection provides only an overview of these transport phenomena. Fluid Mechanics A fluid at rest can resist a normal force but not a shear force, while fluid in motion can also resist a shear force. The fluid continuously deforms under the action of shear force. A fluid’s resistance to shear or angular deformation is measured by viscosity, which can be thought of as the internal “stickiness” of the fluid. The force and the rate of strain (i.e., rate of deformation) produced by the force are related by a constitutive equation. For a Newtonian fluid, the shear stress in the fluid is proportional to the time rate of deformation of a fluid element or particle. Figure 1.6 shows a Couette flow where the plate at the bottom is stationary and the fluid is driven by the upper moving plate. This flow Chapter 1 Introduction to Transport Phenomena 23 U u(y) y x τyx Figure 1.6 Couette flow. is one-dimensional since velocity does not vary in the x- and z-directions are zero. The constitutive relation for Couette flow can be expressed as du τ yx = − μ (1.49) dy where τ yx is the shear stress ( N/m 2 ), μ is the dynamic viscosity ( N-s/m 2 ), which is a fluid property, and du / dy is the velocity gradient in the y-direction, also known as the rate of deformation. If the shear stress τ y x and the rate of deformation du / dy have a linear relationship, as shown in eq. (1.49), the fluid is referred to as Newtonian and eq. (1.49) is called Newton’s law of viscosity. It is found that the resistance to flow for all gases and liquids with molecular mass less than 5000 is well presented by eq. (1.49). For non-Newtonian fluids, such as polymeric liquids, slurries, or other complex fluids, for example in biological applications, such as blood or operation of joints, drag-reducing slimes on marine animals, and digesting foodstuffs the shear stress and the rate of deformation no longer have a linear relationship. Bird et al. (2002) provided detailed information for treatment for non-Newtonian fluids. The viscous force comes into play whenever there is velocity gradient in a fluid. For a three-dimensional fluid flow problem, the stress ′, is a tensor of rank two with nine components. It can be expressed as summation of an isotropic, thermodynamic stress, − pI , and a viscous stress, : ′ = − pI + (1.50) where p is thermodynamic pressure, and I is the unit tensor defined as ª1 0 0 º I = «0 1 0 » (1.51) « » «0 0 1 » ¬ ¼ 24 Transport Phenomena in Multiphase Systems z τzz τzx τxz τxy τzy τyz τyy τyx y τxx x Figure 1.7 Components of the stress tensor in a fluid In a Cartesian coordinate system, the viscous stresses are ªτ xx τ xy τ xz º « » = «τ yx τ yy τ yz » (1.52) «τ zx τ zy τ zz » ¬ ¼ where the first subscript represents the axis normal to the face on which the stress acts, and the second subscript represents the direction of the stress (see Fig. 1.7). The components of the shear stresses are symmetric ( τ xy = τ yx , τ xz = τ zx , and τ yz = τ zy ). The viscous stress tensor can be expressed by Newton’s law of viscosity: 2 = 2 μ D − μ (∇ ⋅ V )I (1.53) 3 where ∇ ⋅ V is the divergence of the velocity (see Appendix C). The rate of deformation, or strain rate D, presented below for a Cartesian coordinate system in a three-dimensional flow is another tensor of rank two (see Appendix C). 1 § ∂u ∂v · 1 § ∂u ∂w · º ∂u ª ¨ + ¸» ¨+¸ « 2 © ∂y ∂x ¹ 2 © ∂z ∂x ¹ ∂x « » « 1 § ∂v ∂u · 1 1 § ∂v ∂w · » ∂v T D = ª ∇V + ( ∇V ) º = « ¨ + ¸ ¨ + ¸ » (1.54) ¼ 2¬ 2 © ∂z ∂y ¹ » ∂y « 2 © ∂x ∂y ¹ « 1 § ∂w ∂u · 1 § ∂w ∂v · » ∂w «¨ » + +¸ ¸ ¨ ∂z « » ¬ 2 © ∂x ∂z ¹ 2 © ∂y ∂z ¹ ¼ where ( ∇V ) is the transverse tensor of ∇V as defined in Appendix C. T Chapter 1 Introduction to Transport Phenomena 25 For example, in a Cartesian coordinate system, the normal and shear viscous stresses can be expressed as ∂u 2 § ∂u ∂v ∂w · τ xx = 2 μ − μ ¨ + + ¸ (1.55) ∂x 3 © ∂x ∂y ∂z ¹ § ∂u ∂v · (1.56) +¸ © ∂y ∂x ¹ For one-dimensional flow in Fig. 1.6, eq. (1.56) is reduced to eq. (1.49). The stress-strain rate relationships in eqs. (1.49) and (1.53) are valid for laminar flow only. For turbulent flow, eqs. (1.49) or (1.53) can still be used provided that the time-averaged velocity is used and the turbulent effects are included in the viscosity (White, 1991; Kays et al., 2004). For a multicomponent system, the viscosity of the mixture is related to the viscosity of the individual component by N xμ μ = ¦ Ni i (1.57) i =1 ¦ xφ j =1 i ij τ xy = μ ¨ where xi is molar fraction of component i and 1/ 4 §Mj · º (1.58) ¨ ¸» » © Mi ¹ ¼ where Mi is molecular mass of the ith species. Equation (1.57) can reproduce the viscosity for the mixtures with an averaged deviation of 2%. Additional correlations to estimate the viscosities of various gases and gas mixtures as well as liquids can be found from the standard reference by Poling et al. (2000). M· 1§ φij = ¨1 + i ¸ ¨ 8© Mj ¸ ¹ −1/ 2 ª §μ «1 + ¨ i « ¨μ ¬ ©j · ¸ ¸ ¹ 1/ 2 2 Heat Transfer Heat transfer is a process whereby thermal energy is transferred in response to a temperature difference. There are three modes of heat transfer: conduction, convection, and radiation. Conduction is heat transfer across a stationary medium, either solid or fluid. For an electrically nonconducting solid, conduction is attributed to atomic activity in the form of lattice vibration, while the mechanism of conduction in an electrically-conducting solid is a combination of lattice vibration and translational motion of electrons. Heat conduction in a liquid or gas is due to the random motion and interaction of the molecules. For most engineering problems, it is impractical and unnecessary to track the motion of individual molecules and electrons, which may instead be described using the macroscopic averaged temperature. The heat transfer rate is related to the temperature gradient by Fourier’s law. For the one-dimensional heat conduction 26 Transport Phenomena in Multiphase Systems y T2 L T(y) T1 T Figure 1. 8 One-dimensional conduction. problem shown in Fig. 1.8, in which temperature varies along the y-direction only, the heat transfer rate is obtained by Fourier’s law dT q′′ = − k (1.59) y dy where q′′ is the heat flux along the y-direction, i.e., the heat transfer rate in the yy direction per unit area (W/m2), and dT / dy (K/m) is the temperature gradient. The proportionality constant k is thermal conductivity (W/m-K), a property of the medium. For heat conduction in a multidimensional isotropic system, eq. (1.59) can be rewritten in the following generalized form: q′′ = −k ∇T (1.60) where both the heat flux and the temperature gradient are vectors, i.e., q′′ = iq′′ + jq′′ + kq′′ (1.61) x y z While the thermal conductivity for isotropic materials does not depend on the direction, it is dependent on direction for anisotropic materials. Unlike the isotropic materials whose thermal conductivity is a scalar, the thermal conductivity of the anisotropic material is a tensor of the second order: ª k xx k xy k xz º « » k = « k yx k yy k yz » (1.62) « k zx k zy k zz » ¬ ¼ and eq. (1.60) will become q′′ = −k ⋅ ∇T (1.63) Chapter 1 Introduction to Transport Phenomena 27 Equation (1.60) or (1.63) is valid in a system that is uniform in all aspects except for the temperature gradient, i.e., no gradients of mass concentration or pressure. In a multicomponent system, mass transfer can also contribute to the heat flux (Curtiss and Bird, 1999) N N N xi x j DiT § J i J j · (1.64) q′′ = −k ⋅ ∇T + hi J i + cRuT ¨− ¸ ¨ ρj ¸ i =1 i =1 j =1( j ≠ i ) ρ i Dij © ρ i ¹ where the second term on the right-hand side represents the interdiffusional ¦ ¦¦ convection term, which is not zero even ¦ J i = 0 . hi is partial enthalpy (J/kg) for i =1 N the ith species. The third term on the right-hand side is the contribution of concentration gradient to the heat flux, which is referred to as the diffusionthermo or Dufour effect. c is the molar concentration (kmol/m3) of the mixture, Ru is the universal gas constant, xi and xj are respectively molar fraction of the ith and jth components, DiT is the multicomponent thermal diffusivity (m2/s) (which will be considered in the discussion of mass transfer in the latter part of this section) and Dij is Maxwell-Stefan diffusivity, which is related to the multicomponent Fick diffusivity, Dij , defined in eq. (1.108). For a binary system D12 = x1 x2 ω1ω2 D12 (1.65) where ω1 and ω2 are mass fraction of component 1 and 2 respectively. For a ternary system D12D33 − D13D 23 xx D12 = 1 2 (1.66) ω1ω2 D12 + D33 − D13 − D 23 For a system of more than four components, the Maxwell-Stefan diffusivity can be obtained using methods described in Curtiss and Bird (1999; 2001). The interdiffusional convection term represented by the second term on the righthand side of eq. (1.64) is usually important for multicomponent diffusion system. While the Dufour energy flux represented by the third terms in the right-hand side of eq. (1.64) is negligible for many engineering problems, they may become important for the case with very large temperature gradient. J i is diffusive mass flux relative to mass-averaged velocity, which will be discussed in latter part of this section. The second mode of heat transfer is convection, which occurs between a wall at one temperature, Tw , and a moving fluid at another temperature, T∞ ; this is exemplified by forced convective heat transfer over a flat plate, as shown in Fig. 1.9. The mechanism of convection heat transfer is a combination of random molecular motion (conduction) and bulk motion (advection) of the fluid. Newton’s law of cooling is used to describe the heat transfer rate: q′′ = h(Tw − T∞ ) (1.67) 28 Transport Phenomena in Multiphase Systems U∞ U∞ , T∞ T∞ T(x,y) q" u(x,y) Tw Figure 1.9 Forced convective heat transfer. where h is the convective heat transfer coefficient (W/m2-K), which depends on many factors including fluid properties, flow velocity, geometric configuration, and any fluid phase change that may occur as a result of heat transfer. Unlike thermal conductivity, the convective heat transfer coefficient is not a property of the fluid. Typical values of mean convective heat transfer coefficients for various heat transfer modes are listed in Table 1.5. Convective heat transfer is often measured using the Nusselt number defined by hL Nu = (1.68) k where L and k are characteristic length and thermal conductivity of the fluid, respectively. Table 1.5 Typical values of mean convective heat transfer coefficients Mode Forced convection Free convection (ΔT = 20 o C) ) Evaporation Condensation of water at 1 atm Boiling of water at 1 atm Natural convectioncontrolled melting and solidification Geometry Air flows at 2 m/s over a 0.2 m square plate Air at 2 atm flowing in a 2.5 cm-diameter tube with a velocity of 10 m/s Water flowing in a 2.5 cm-diameter tube with a mass flow rate of 0.5 kg/s Airflow across 5 cm-diameter cylinder with velocity of 50 m/s Vertical plate 0.3 m high in air Horizontal cylinder with a diameter of 2 cm in water Falling film on a heated wall Vertical surface Outside horizontal tube Pool Forced convection Melting in a rectangular enclosure Solidification around a horizontal tube in a superheated liquid phase change material h (W/m 2 -K) 12 65 3500 180 4.5 890 6000-27000 4000-11300 9500-25000 2500-3500 5000-100000 500-1500 1000-1500 Chapter 1 Introduction to Transport Phenomena 29 The third mode of heat transfer is radiation. The transmission of thermal radiation does not require the presence of a material medium because radiation heat transfer can occur in a vacuum. Thermal radiation is a form of energy emitted by matter at a nonzero temperature and its wavelength is primarily in the range between 0.1 to 10 m. The emission can be from a solid surface as well as from a liquid or gas. Thermal radiation may be considered as the propagation of electromagnetic waves or alternately of a collection of particles, such as photons or quanta of photons. When matter is heated, some of its molecules or atoms are excited to a higher energy level. Thermal radiation occurs when these excited molecules or atoms return to lower energy states. Although thermal radiation can result from changes of the energy states of electrons, as well as vibrational and rotational energy of molecules or atoms, all of these radiant energies travel at the speed of light. The wavelength λ is related to the frequency, ν , by (1.69) λν = c 8 where c is the speed of light with a value of 2.998 × 10 m/s in a vacuum. A quantitative description of the mechanism of the thermal radiation requires quantum mechanics. An electromagnetic wave with frequency of ν can also be viewed as a particle –a photon– with energy of ε = hν (1.70) -34 2 where h = 6.626068 × 10 m -kg/s is the Planck’s constant. Both mass and charge of a photon are zero. For a blackbody, defined as an ideal surface that emits the maximum energy that can be emitted by any surface at the same temperature, the spectral emissive power, Eb ,λ (W/m3), can be obtained by Planck’s law Eb ,λ = λ (e 2 1 c2 /( λT ) c − 1) (1.71) radiation where c1 = 3.742 × 10−16 W-m 2 and c1 = 1.4388 × 10−2 m-K are constants. The unit of the surface temperature T in eq. (1.71) is K. The emissive power for a blackbody, Eb (W/m2), is Eb = ³ Eb ,λ d λ 0 ∞ (1.72) Substituting eq. (1.71) into eq. (1.72), Stefan-Boltzmann’s law is obtained Eb = σ SBT 4 (1.73) where, and σ SB is the Stefan-Boltzmann constant, 5.67 × 10−8 W/m 2 -K 4 . For a real surface, the emissive power is obtained by E = ε Eb (1.74) where is the emissivity, defined as the ratio of emissive power of the real surface to that of a blackbody at the same temperature. Since a blackbody is the best emitter, the emissivity of any surface must be less than or equal to 1. A simple but important case of radiation heat transfer is the radiation heat exchange between a small surface with area A, emissivity , and temperature Tw, and a much larger surface surrounding the small surface (see Fig. 1.10). If the 30 Transport Phenomena in Multiphase Systems Figure 1.10 Radiation heat transfer between a small surface and its surroundings. temperature of the surroundings is Tsur, the heat transfer rate per unit area from the small object is obtained by 4 q′′ = εσ SB (Tw4 − Tsur ) (1.75) For a detailed treatment of radiation heat transfer, including radiation of nongray surfaces and participating media, the readers should consult Siegel and Howell (2002). Mass Transfer When there is a species concentration difference in a multicomponent mixture, mass transfer occurs. There are two modes of mass transfer: diffusion and convection. Diffusion results from random molecular motion at the microscopic level, and it can occur in a solid, liquid or gas. Similar to convective heat transfer, convective mass transfer is due to a combination of random molecular motion at the microscopic level and bulk motion at the macroscopic level, and it can occur only in a liquid or gas. The species concentration in a mixture can be measured by concentration i, which is defined as the mass of species i per unit volume of the mixture (kg/m3). The density of the mixture equals the sum of the concentrations of all N species, i.e., ρ = ¦ ρi i =1 N (1.76) The concentration of the ith species can also be represented by the mass fraction of the ith species, defined as ωi = It follows from eq. (1.76) that ρi ρ (1.77) Chapter 1 Introduction to Transport Phenomena 31 ¦ω i =1 N i =1 (1.78) The concentration of the ith species can also be represented by molar concentration, defined as the number of moles of the ith species per unit volume, ci (kmol/m3), which is related to the mass concentration by (1.79) Mi The molar concentration of the mixture equals the sum of the molar concentrations of all N species, i.e., c = ¦ ci i =1 N ci = ρi (1.80) The molar fraction of the ith species is defined as c xi = i (1.81) c which is identical to the molecular number fraction – the fraction of the number of molecules of the ith species to the number of molecules of all species in a given volume. This concept is essential when kinetic theory is used to describe the mass transfer process. Equation (1.81) leads to ¦x i =1 N i =1 N (1.82) The mean molecular mass of the mixture can be expressed as c i =1 The mass fraction is related to the molar fraction by xM xM ωi = N i i = i i M ¦ xjM j j =1 M= ρ = ¦ xi M i (1.83) (1.84) The molar fraction is related to the mass fraction by ω / Mi xi = N i ¦ω j / M j j =1 (1.85) Mass diffusion of the i component in the mixture will result in a velocity, Vi , of the ith component relative to the stationary coordinate axes. The local mass-averaged velocity of all species, V , is defined as V= th ¦ ρ Vi ¦ ρ Vi i i =1 N N N ¦ρ i =1 = i i =1 ρ = ¦ ωi Vi i =1 N (1.86) i 32 Transport Phenomena in Multiphase Systems which demonstrates that the local mass flux due to diffusion, ρ V , is equal to the summation of mass flux for each species, ¦ ρi Vi . i =1 N The molar-averaged velocity can be defined in a similar manner: (1.87) c i =1 The velocity of the ith species relative to the mass or molar-averaged velocity, Vi − V or Vi − V * is defined as diffusion velocity. The mass flux and molar flux relative to stationary coordinate axes are defined as m′′ = ρi Vi (1.88) i ′′ = ci Vi ni (1.89) The fluxes defined in eqs. (1.88) and (1.89) are related by m′′ (1.90) n′′ = i i Mi Applying eqs. (1.88) and (1.89) into eqs. (1.86) and (1.87), the total mass flux and molar flux are obtained. m′′ = ¦ m′′ = ρ V i n′′ = ¦ n′′ = cV * i i =1 i =1 N N V* = ¦ c Vi i i =1 N = ¦ xi Vi N (1.91) (1.92) The mass flux relative to the mass-averaged velocity is J i = ρi (Vi − V ) and the molar flux relative to the molar averaged velocity is J * = ci ( Vi − V * ) i According to eqs. (1.86) and (1.87), we have (1.93) (1.94) (1.95) ¦J = ¦J i i =1 i =1 N N * i =0 Although any one of the four fluxes defined in eqs. (1.88) – (1.89) and (1.93) – (1.94) are adequate to describe mass diffusion under all circumstances, there is usually a preferred definition of flux that can lead to less algebraic complexity. When mass diffusion is coupled with advection, eq. (1.93) is preferred because the mass-averaged velocity, V , is the velocity used in the momentum and energy equations. On the other hand, eq. (1.94) is preferred for the multicomponent system with constant molar density c, resulting from constant pressure and temperature. According to eqs. (1.88) – (1.94), the following relationships between different fluxes are valid: m′′ = ρi V + J i = ωi ¦ m′′ + J i i j j =1 N (1.96) Chapter 1 Introduction to Transport Phenomena 33 y L ω1 ( y ) 0 ′′ m1 y Figure 1.11 One-dimensional mass diffusion. ω1 n′′ = ci V * + J * = xi ¦ n′′ + J * i i i i i =1 N (1.97) For a binary system that is uniform with all aspects except concentration, i.e., no temperature or pressure gradient, the diffusive mass flux can be obtained by Fick’s law: J1 = − ρ D12 ∇ω1 (1.98) or alternatively * J1 = −cD12∇x1 (1.99) 2 where D12 is binary diffusivity (m /s). The mass and molar-flux relative to a stationary coordinate axes are ′′ ′′ m1 = ω1 (m1 + m′′ ) − ρ D12∇ω1 (1.100) 2 ′′ ′′ 2 n1 = x1 (n1 + n′′ ) − cD12∇x1 (1.101) Equation (1.100) is widely applied in systems with constant density, while eq. (1.101) is more appropriate for systems with constant molar concentration. It should be noted that from eqs. (1.100) and (1.101) the absolute fluxes of species ′′ ′′ ( m1 or n1 ) for a binary system can always be presented as summation of two parts: one part due to convection [the first terms in eqs. (1.100) and (1.101)], and another part due to diffusion [the second terms in eqs. (1.100) and (1.101)]. For an isothermal and isobaric steady-state one-dimensional binary system shown in Fig. 1.11, in which surface (y =0) is impermeable to species 2, the mass flux of the species 1 at the surface (y = 0) is ρ D12 ∂ω1 ′′ m1 y = − (1.102) 1 − ω1 ∂y For a system with more than two components, Fick’s law is no longer appropriate and one must find other approaches to relate mass flux and concentration gradient. For a multicomponent low-density gaseous mixture, the 34 Transport Phenomena in Multiphase Systems following Maxwell-Stefan relation can be used to relate the molar fraction gradient of the ith component and the molar flux by: N xi x j ∇xi = − (Vi − V j ) j =1( j ≠ i ) Dij (1.103) N 1 =− ( x j n′′ − xin′′j ) , i = 1, 2,", N − 1 i j =1( j ≠ i ) cDij ¦ ¦ where Dij is binary diffusivity from species i to species j. Equation (1.103) was originally suggested by Maxwell for a binary mixture based on kinetic theory and extended to diffusion of gaseous mixture of N species by Stefan. For an Ncomponent system, N(N–1)/2 diffusivities are required. The diffusion in a multicomponent system is different from diffusion in a binary system, because the movement of the ith species is no longer proportional to the negative concentration gradient of the ith species. It is possible that (1) a species moves against its own concentration gradient, referred to as reverse diffusion; (2) a species can diffuse even when its concentration gradient is zero, referred to as osmotic diffusion; or (3) a species does not diffuse although its concentration gradient is favorable to such diffusion, referred to as diffusion barrier (Bird et al., 2002). The Maxwell-Stefan relation can also be rewritten in terms of mass fraction and mass flux M N ωi J j − ω j J i , i = 1, 2," , N − 1 (1.104) ∇ωi + ωi ∇(ln M ) = ρ j =1( j ≠i ) Dij M j ¦ where M and is the molar-averaged molecular mass of the mixture. Although eqs. (1.103) and (1.104) were originally developed for low-density gaseous mixtures, it has been shown that they are also valid for dense gases, liquids, and polymers, except Dij should be replaced by multicomponent Maxwell-Stefan diffusivity Dij . The diffusive mass fluxes for a system that contains N components can be obtained by solving a set of N − 1 equations (1.104) and eq. (1.95) (Kleijn et al., 1989) J i = − ρ Dim ∇ωi − ρωi Dim ∇(ln M ) + M ωi Dim j =1( j ≠ i ) ¦ N Jj M j Dij −1 , i = 1, 2," , N (1.105) where the effective mass diffusivity from species i to the mixture Dim is § N Mω j · Dim = ¨ (1.106) ¸ , i = 1, 2," , N ¨ j =1( j ≠i ) M j Dij ¸ © ¹ It is noted that eq. (1.103) is valid for a system that is uniform in all regards except for mass concentration. For a multicomponent system with concentration, temperature and pressure gradients, the mass flux can be expressed as a linear ¦ Chapter 1 Introduction to Transport Phenomena 35 combination of concentration gradients – referred to as generalized Fick’s law – or a linear combination of mass fluxes – referred to as Maxwell’s expression. The total diffusive mass flux vector of species i can be expressed as a linear form (Curtiss and Bird, 1999) J i = − DiT ∇ ln T + ρi ¦D d ij j =1 N j , i = 1, 2," , N (1.107) which is referred to as the generalized Fick equation, which is applicable to a system with concentration, temperature, and pressure gradients. The first term on the right-hand side represents thermal diffusion ( DiT is thermal diffusion coefficient). The second term is diffusion caused by all other driving forces, including concentration gradient, pressure, and body force; Dij is the multicomponent Fick diffusivities. Dij is obtained from ˆ ωiDij = Dij − n =1( n ≠ i ) ˆ ¦ ωn Din N (1.108) where ˆ Dij = ωiω j xi x j Dij (1.109) The multicomponent Fick diffusivities Dij are symmetric ( Dij = D ji ) and satisfy i =1 ¦ ωiDij = 1 . N Equation (1.107) can be rearranged to express the driving force, di, in terms of mass flux, i.e., N xi x j § DiT DT · j di = − − ¨ ¸ ∇(ln T ) ¨ ρi ρj ¸ j =1( j ≠ i ) Dij © ¹ (1.110) N xi x j § J i J j · − ¨− ¸ , i = 1, 2," , N ¨ ρj ¸ j =1( j ≠ i ) Dij © ρ i ¹ ¦ ¦ which are referred to as generalized Maxwell-Stefan equations. The diffusional driving force di is obtained by § 1 gi · hi cRuT 1 1N di = T ∇ ¨ ∇ ln T − ∇p − Xi + ¦ ρ j X j ¸+ ρi ρ ρ j =1 © T Mi ¹ Mi (1.111) where gi is partial molar Gibbs free energy (J/kmol), hi is partial molar enthalpy (J/kmol), and Xj is the body force per unit mass (m/s2) for the ith component. Introducing the activity of the ith component ai, dgi = RuTd ln ai (1.112) the diffusional driving force becomes cRuTdi = ci RuT ∇ ln ai + (ε i − ωi )∇p − ρi Xi + ωi ¦ ρ j X j j =1 N (1.113) 36 Transport Phenomena in Multiphase Systems where ε i is the volume fraction of the ith component. For an ideal gas mixture, the generalized driving force becomes cRuTdi = ∇pi − ωi ∇p − ρi Xi + ωi ¦ ρ j X j j =1 N (1.114) where the first two terms on the right-hand side are driving forces for ordinary diffusion and pressure diffusion. The last two terms are driving force for body force diffusion. If gravity is the only body force, the body force diffusion is zero because X j = g for any components. The thermal (Soret) diffusion was included in the first term in eq. (1.107). Substituting eq. (1.114) into eq. (1.107) and after some manipulations, the final form of mass flux in the Fick form is obtained as (Curtiss and Bird, 1999) N § · ρN J i = − DiT ∇ ln T + i Dij ¨ ∇pi − ω j ∇p − ρ j X j + ω j ρ n X n ¸ (1.115) cRuT j =1 © n =1 ¹ Similarly, substituting eq. (1.114) into eq. (1.110), the final form of MaxwellStefan equations becomes N xi x j § J j J i · − ¸ = ∇pi − ωi ∇p − ρi Xi cRuT ¨ ¨ ρi ¸ j =1( j ≠ i ) Dij © ρ j ¹ (1.116) N N xi x j § DT DiT · j +ωi ρ j X j − cRuT − ¨ ¸ ∇(ln T ), i = 1, 2," , N ¨ ρi ¸ j =1 j =1( j ≠ i ) Dij © ρ j ¹ ¦ ¦ ¦ ¦ ¦ For dilute monatomic gas mixtures, the Maxwell-Stefan diffusivities can be approximated by binary diffusivity, i.e., Dij ≈ Dij and eq. (1.116) is preferred over eq. (1.115) because Dij strongly depends on the mass concentrations. Equation (1.116) reduces to eq. (1.103) for the gaseous mixture with uniform temperatures and pressures. The multicomponent thermodiffusivity for species i is expressed as N ρMiM j T DiT = Dij kij (1.117) 2 M j =1( j ≠ i ) ¦ T where kij is the thermal diffusion ratio and it is related to the thermal diffusion T T T factor, α ij by kij = α ij xi x j (Bird et al., 2002) where xi and xj are molar fractions of component i and j, respectively. For applications that involve mass diffusion in a binary mixture containing species 1 and 2, the multicomponent thermodiffusivity becomes ρ M 1M 2 T DiT = D12 k12 (1.118) 2 M The thermal diffusion ratio in a binary system depends on both temperature and concentration, and some selected values for liquids and gases are given in Table 1.6. Chapter 1 Introduction to Transport Phenomena 37 Table 1.6 Thermal diffusion ratios for selected liquids and low-density gases (Bird et al., 2002)a Components C2H2Cl4 – n-C6H14 C2H4Br2 – C2H4Cl2 C2H2Cl4 – CCl4 CBr4 – CCl4 CCl4 – CH3OH CH3OH – H2O cyclo-C6H12 – C6H6 Ne-He T(K) 298 298 298 298 313 313 313 330 x1 0.5 0.5 0.5 0.09 0.5 0.5 0.5 0.2 T k12 Liquids 1.08 0.225 0.060 0.129 1.23 -0.137 0.100 0.0531 N2 – H2 Gases 264 327 D2 –H2 a 0.6 0.294 0.775 0.10 0.50 0.90 0.1004 0.0548 0.0663 0.0145 0.0432 0.0166 Reproduced by permissions of John Wiley and Sons, Inc. For transport phenomena in applications such as biotechnology, fuel cells and many others, it is usually assumed that the gas is ideal, the only body force is gravity, and the mixture pressure gradient is negligible. Although eq. (1.115) can be simplified to get the diffusive mass flux in this case, the multicomponent Fick diffusivities Dij are strongly dependent on the concentration as evidenced by eqs. (1.108) and (1.109). An alternative approach that utilizes eq. (1.116) will be developed here (Rice, 2005). With gravity as the only body force and the total pressure approximately constant, the first four terms in the right-hand side of eq. (1.116) become ∇pi − ωi ∇p − ρi Xi − ωi ¦ ρ j X j ≈ ∇pi = cRT ∇xi j =1 N (1.119) Substituting eq. (1.119) into eq. (1.116) yields N N xi x j § J j J i · xi x j § DT DiT · j − ¸ = ∇xi − ¦ − (1.120) ¨ ¸ ∇ ln T ¨ ¦ D ¨ρ ρ ¸ ρi ¸ Dij ¨ ρ j j =1( j ≠ i ) j =1( j ≠ i ) ij © j i¹ © ¹ where the Maxwell-Stefan diffusivity Dij has been replaced by binary diffusivity Dij for ideal gas mixture. Equation (1.120) can be represented by a matrix. A [ J ] = [∇x ] − A ª DT º ∇ ln T (1.121) ¬¼ i.e., [ J ] = A −1 [∇x] − ª DT º ∇ ln T ¬¼ (1.122) where the second term on the right-hand side represents the effect of thermal diffusion. It is also known that the summation of the mass diffusion fluxes over all the species is zero, and that the summation of the mole fractions is unity, 38 Transport Phenomena in Multiphase Systems ¦ J i = 0, and ¦ ωi = ¦ xi =1 . Therefore, we can reduce the A matrix to a N N N ( N − 1) × ( N − 1) ∇xi = i =1 i =1 i =1 matrix, with the balance solved by continuity. The mole fraction gradient, ∇xi , must be related to the mass fraction gradient, ∇ωi . ωN 1 1 M ∇ (ωi M ) = ( M ∇ωi + ωi∇M ) = M ∇ωi + Mi ¦1 M j ∇x j (1.123) Mi Mi i i j= Since the sum of the mole fraction is unity, the last term can be rewritten as: ¦ M j ∇x j = ¦ ( M j − M N ) ∇x j N j =1 j =1 N −1 (1.124) Substituting eq. (1.124) into eq. (1.123), one obtains ª º ¬ M i + ωi ( M N − M i ) ¼ ∇xi + ωi N −1 j =1( j ≠ i ) ¦ ª M − M j ∇x j º = M ∇ωi ¬N ¼ ( ) (1.125) which can be written in a matrix form B [ ∇x ] = M [ ∇ ω ] i.e., [∇x] = MB−1 [∇ω ] where, Bii = (1 − ωi ) M i + ωi M N (1.126) (1.127) (1.128) (1.129) Bij = ωi M N − M j ( ) and [∇ω ] are ( N − 1) . The left-hand side of eq. (1.120) can be rewritten in terms of the mass fraction. N N xi x j § J j J i · ωiω j M 2 § J j J i · − ¸= ¦ −¸ ¨ ¨ ¦ ¨ ρi ¸ j =1( j ≠i ) M i M j Dij ¨ ρ j ρi ¸ j =1( j ≠ i ) Dij © ρ j ¹ © ¹ (1.130) 2 N 1 M = ω J − ω j Ji ¦ ρ j =1( j ≠ i ) M i M j Dij i j Note that the size of the matrix B is ( N − 1) × ( N − 1) , and the arrays [∇x] ( ) Defining Cij = M2 ρ M i M j Dij (1.131) eq. (1.130) becomes N xi x j § J j J · ¦ D ¨ ρ − ρi ¸ = ¨j ¸ j =1( j ≠ i ) ij © i¹ = ωi j =1( j ≠ i ) j =1( j ≠ i ) ¦ N Cij ωi J j − ω j J i ( ) (1.132) ¦ N Cij J j − J i j =1( j ≠ i ) ¦ N Cijω j Considering that the summation of the diffusive mass fluxes is zero and the sum of the mass fraction is unity, we have Chapter 1 Introduction to Transport Phenomena 39 ωi j =1( j ≠ i ) ¦ N Cij J j − J i N −1 j =1( j ≠ i ) ¦ N Cijω j = ωi N −1 j =1( j ≠ i ) ¦ (C ij − CiN J j (1.133) ) −ωi CiN J i − J i j =1( j ≠ i ) ¦ (C ij − CiN ω j + J i CiN (ωi − 1) ) Therefore the components of matrix A are N ª º ωj ωi Aii = − « +¦ » « M i M N DiN j =1( j ≠i ) M i M j Dij » ¬ ¼ § 1 1 Aij = ωi ¨ − ¨ M i M j Dij M i M N DiN © Substituting eq. (1.127) into eq. (1.122), multicomponent becomes (1.134) · (1.135) ¸ ¸ ¹ the diffusive mass flux for the A −1B −1 [∇ω ] − ª DT º ∇ ln T (1.136) ¬¼ M where the components of matrices A and B can be found in eqs. (1.134), (1.135), (1.128) and (1.129). Equation (1.136) is in the form that can be very easily programmed. Equation (1.136) can be simplified for a variety of cases. The simplest case is a binary mixture. For a binary mixture, A and B are both a single value. 1 A=− (1.137) M 1M 2 D12 xM xM MM B = ω2 M 1 + ω1M 2 = 2 2 M 1 + 1 1 M 2 = 1 2 (1.138) M M M Therefore, the diffusion mass flux of a binary mixture is: J i = − ρ D12∇ωi − DiT ∇ ( ln T ) (1.139) For a mixture of several species that are all very dilute in species N ( ωi  1, for i = 1, 2,..., N − 1 and ω N ≈ 1 ), the B matrix is approximately [J] = ρ ª M1 0 … «0 M % 2 B≈« «# %% « ¬0 … 0 and the A matrix is approximately ª ( M M D ) −1 0 « 1 N iN −1 « 0 ( M 2 M N D2 N ) A≈« # % « « … 0 « ¬ º » » » » M N −1 ¼ 0 # 0 º » » % # » % 0 » −1 » 0 ( M N −1M N DN -1, N ) » ¼ … 0 (1.140) (1.141) 40 Transport Phenomena in Multiphase Systems The mixture molecular mass is also approximately equal to the molecular mass of the Nth component, M ≈ M N . Therefore the diffusion flux for each component is: (1.142) J i = − ρ D1N ∇ωi − DiT ∇ ( ln T ) While the above discussions provided the generalized descriptions for mass transfer in a multicomponent system, the mass flux in a nonisothermal and nonisobaric binary system has a much simpler expression. The mass flux of species 1 due to ordinary, pressure, body force, and thermal diffusion for a binary system becomes MM D ω J1 = − ρ D12 ∇ω1 + 1 2 12 1 ∇p p (1.143) M 1 M 2 D12ω1ω2 T + ( X1 − X 2 ) − D1 ∇(ln T ) RuT where D12 is the binary diffusion coefficient (mass diffusivity, m 2 /s ) of species 1 in a mixture of species 1 and 2. The unit for the mass diffusivity, D12 , is the same as the kinematic viscosity, ν = μ / ρ , and the thermal diffusivity, = k/( cp). The binary diffusivity at 1 atm for selected gases is listed in Tables B.71B73, Appendix B. It should be pointed out that the pressure diffusion, body force diffusion, and Soret diffusion represented by the second to fourth terms in eq. (1.143) is negligible for most applications. Similarly one can reduce the generalized Maxwell-Stefan equation (1.116) to the following form by neglecting pressure, body force, and thermal diffusion effects and assuming the mean molecular mass is constant [see. eq. (1.104)]: M N ωi J j − ω j J i , i = 1, 2," , N − 1 (1.144) ∇ωi = ρ j =1( j ≠i ) Dij M j ¦ In order to solve for the mass flux J, eq. (1.144) can be rearranged to get Ji = − where [ Deff ,ij ] = F −1 ¦ρD j =1 N −1 eff ,ij ∇ ω j , i = 1, 2,", N − 1 (1.145) (1.146) Fii = ωi M DiN M N + k =1( k ≠ i ) ¦ N ωk M Dik M k (1.147) §1 1· (1.148) Fij = ωi M ¨ − , i≠ j ¨ Dij M j DiN M N ¸ ¸ © ¹ The second mode of mass transfer, convective mass transfer, may be expressed in a manner analogous to eq. (1.67): m′′ = hm ( ρ1, w − ρ1,∞ ) = ρ hm (ω1, w − ω1,∞ ) (1.149) Chapter 1 Introduction to Transport Phenomena 41 Table 1.7 Summary of fundamental laws in transport phenomena Flux Equations Momen- Newton’s law of viscosity, eqs. (1.53) and (1.50) tum 2 T ′ = − pI + μ ª∇V + ( ∇V ) º − μ (∇ ⋅ V )I ¬ ¼3 Requirements Comments/Assumptions Newtonian Single or multicomponent fluid Laminar Newtonian fluid Laminar Single component Single component Multicomponent Single or multicomponent Incompressible Isotropic Anisotropic Anisotropic The second and third terms account for, respectively, the interdiffusion convection and thermodiffusion effects No temperature and pressure gradients. Same body force for both components No temperature and pressure gradients. Same body forces for all N components Simplified Newton’s law of viscosity ′ = − pI + μ ª∇V + ( ∇V ) º ¬ ¼ T Energy Fourier’s law, eq. (1.60) q′′ = −k ∇T Fourier’s law, eq. (1.63) q′′ = −k ⋅ ∇T Equation (1.64) q′′ = −k ⋅ ∇T + + cRuT Mass ¦h J i =1 N ii ¦¦ i =1 N xi x j DiT j =1( j ≠ i ) ρi Dij N § Ji J j · ¨−¸ ¨ ρi ρ j ¸ © ¹ Binary only Fick’s law, eqs. (1.98) and (1.99) J1 = − ρ D12∇ω1 * J1 = −cD12∇x1 Maxwell-Stefan equation (1.103) ∇xi = − =− j =1( j ≠ i ) ¦ N xi x j Dij (Vi − V j ) Multicomponent gaseous mixture 1 x j n′′ − xi n′′ i j j =1( j ≠ i ) cDij ¦ N ( ) Multicomponent gaseous mixture Multicomponent system No temperature and pressure gradients. Same body forces for all N components Including ordinary, pressure, body force, and thermal diffusion Maxwell-Stefan equation (1.104) ∇ωi + ωi ∇ (ln M ) = M ρ j =1( j ≠ i ) ¦ N ωi J j − ω j J i Dij M j Maxwell-Stefan equation (1.116) N xi x j § J j J i · − ¸ = ∇pi − ωi ∇p − ρi Xi cRuT ¨ ¨ ρi ¸ j =1( j ≠ i ) Dij © ρ j ¹ ¦ j xi x j § DT DiT · j − ¨ ¸ ∇ (ln T ) ρi ¸ Dij ¨ ρ j j =1 j =1( j ≠ i ) © ¹ Mass flux in binary system, eq. (1.143) +ωi ¦ρ X N j − cRuT ¦ N MM D ω J1 = − ρ D12 ∇ω1 + 1 2 12 1 ∇p p M 1 M 2 D12ω1ω2 + ( X1 − X 2 ) − D1T ∇(ln T ) RT Binary only Including ordinary, pressure, body force, and thermal diffusion 42 Transport Phenomena in Multiphase Systems where the species mass flux, m′′, is again exemplified by transport from a flat surface to a vapor stream flowing over that surface. The term hm (m/s) in eq. (1.149) is the convection mass transfer coefficient, ρ1, w is the species mass concentration at the surface, ρ1,∞ is the species mass concentration in the free stream, and ω1, w and ω1,∞ are the species mass fractions at the surface and in the free stream, respectively. As is the case for the convective heat transfer coefficient, the convective mass transfer coefficient is a function of fluid properties, the flow field characteristics, and the geometric configuration. Results are frequently expressed in a dimensionless form that also reflects the analogy between heat and mass transfer, the Sherwood number: hL Sh = m (1.150) D12 Equations (1.49) and (1.59) can be rewritten in the forms analogous to eq. (1.102), i.e., ∂( ρu) (1.151) τ yx = −ν ∂y ∂ ( ρ c pT ) q′′ = −α (1.152) y ∂y While eq. (1.102) states that mass transfer occurs due to a concentration ( ρ1 ) gradient, momentum and heat transfer are due to momentum ( ρ u ) and energy ( ρ c pT ) gradients, as indicated in eqs. (1.151) and (1.152). Therefore, momentum, heat, and mass transfer are analogous to each other. This analogous relationship can be used to predict one transport phenomenon on the basis of knowledge of another transport phenomenon. For example, the empirical correlation of turbulent heat transfer can be obtained by applying correlations of friction through the Reynolds analogy (Incropera and DeWitt, 2001). Various fluxes in terms of the transport properties in multicomponent systems are summarized in Table 1.7. The transport properties are presented in Appendix B. Various empirical equations for transport properties of gases and liquids can also be found in Poling et al. (2000). Chapter 3 presents the generalized integral and differential governing equations for transport phenomena in multiphase systems in local-instance formulations using various fluxes presented in Table 1.7. The instantaneous formulation requires a differential balance for each phase, combined with appropriate jump and boundary conditions to match the solution of these differential equations at the interfaces. Also discussed in Chapter 3 are a rarefied vapor self-diffusion model and the application of the differential formulations to combustion. The generalized governing equations for multiphase systems in averaged formulations are presented in Chapter 4. The averaged formulations are obtained by averaging the governing equations within a small time interval (time Chapter 1 Introduction to Transport Phenomena 43 average) or a small control volume (spatial average). The governing equations for the multidimensional multi-fluid and homogeneous models, as well as areaaveraged governing equations for one-dimensional flows, are also discussed. Chapter 4 also covers single- and multiphase transport phenomena in porous media, including multi-fluid and mixture models. Finally, Boltzmann statistical averaging, including a detailed discussion of the Boltzmann equation and the Lattice Boltzmann method (LBM) for modeling both single and multiphase systems is presented in Chapter 4. 1.4.3 Microscale and Nanoscale Transport Phenomena Transport phenomena at dimensions between 1 and 100 μ m are different from those at larger scales. At these scales, phenomena that are negligible at larger scales become dominant, but the macroscopic transport theory is still valid. One example of these phenomena in multiphase system is surface tension. In larger scale systems, the hydrostatic pressure or dynamic pressure effects may dominate over pressure drops caused by surface tension, but at smaller scales, the pressure drops caused by surface tension can dominate over hydrostatic and dynamic pressure effects. Transport phenomena in these scales are still regarded as macroscale, because classical theory of transport theory is still valid. As systems scale down even further to the nanoscale, 1-100 nm, or for ultrafast process (e.g., materials processing using picosecond or femtosecond laser) the fundamental theory used in larger scale systems breaks down because of fundamental differences in the physics. The purpose of research in microscale/nanoscale heat transfer is to exploit the differences from the macroscopic systems to create and improve materials, devices and systems. It is also the objective of this research to understand when system performance will begin to degrade because of the adverse effects of scaling. One large field of interest in which microscale heat transfer is of great interest is energy and thermal systems. Microscale energy and thermal systems include thin film fuel cells, thin film electrochemical cells, photon-to-electric devices, micro heat pipes, bio-cell derived power, and microscale radioisotopes (Peterson, 2004). Other devices that are of a slightly larger scale but require components on the microscale include miniaturized heat engines as well as certain combustion-driven thermal systems. These energy systems can power anything from MEMS sensors and actuators to cell phones. Thermal management of thermally based energy systems is extremely important. To emphasize its importance, consider a micro heat engine. The most general components of a micro heat engine are a compressor, a combustor, and a turbine. Generally, the combustor runs much cooler than the turbine, because the gases must expand in the combustor to drive the turbine’s shaft. However, the turbine and the compressor are linked by this shaft. The shorter the distance between the turbine and the compressor, the less thermal resistance through the shaft. 44 Transport Phenomena in Multiphase Systems Therefore, the temperature difference between the turbine and the compressor is much less, which decreases the systems efficiency. Another form of energy conversion is thermoelectric energy conversion, in which thermal energy is directly converted to electricity. There are three thermoelectric effects: the Seebeck effect, the Peltier effect, and the Thomson effect. The Seebeck effect occurs when electrons flow from the hot side to the cold side of a material under a temperature gradient. The result is an electric potential field (voltage) balancing the electron diffusion. The Peltier effect occurs when heat is carried by electrons in an electrical current in a material held at a constant temperature. The Thomson effect is seen when current flows through a conductor under a temperature gradient. The suitability of a thermoelectric material for energy conversion is based on the figure of merit Z (1.153) Re k where the Seebeck coefficient, electrical resistivity and thermal conductivity are α , Re and k, respectively. Materials with a high figure of merit are difficult to find in bulk form, therefore, nanostructures provide additional parameter space (Chen et al., 2004). To manipulate the nanostructures of certain materials, the electron and phonon thermoelectric transport must first be understood. Conduction heat transfer in a solid at a small scale is analogous to the kinetic theory of gases (see Section 1.3.2). However, instead of molecules transporting momentum, there are electrons and phonons transferring heat. The classical heat conduction theory is a macroscopic model based on the equilibrium assumption. From the microscopic perspective, the energy carriers in a substance include phonons, photons, and electrons. Depending on the nature of heating and the structure of the materials, the energy can be deposited into materials in different ways: it can be simultaneously deposited to all carriers by direct contact, or only to a selected carrier by radiation (Qiu and Tien, 1993). For short-pulsed laser heating of metal, the energy deposition involves three steps: (1) deposition of laser energy on electrons, (2) exchange of energy between electrons, and (3) propagation of energy through media. If the pulse width is shorter than the thermalization time, which is the time it takes for the electrons and lattice to reach equilibrium, the electron and lattice are not in thermal equilibrium, and a two-temperature model is often used. If the laser pulse is shorter than the relaxation time, which is the mean time required for electrons to change their states, the hyperbolic conduction model must be used. More insights about the microscale heat transfer can be found in Tzou (1996) and Majumdar (1998). In addition to the two-temperature and hyperbolic models, which are still continuum models, another approach is to understand heat and mass transfer at the molecular level using the molecular dynamics method (Maruyama, 2001). The conduction of heat is caused by electrons and by phonons. Therefore, the thermal conductivity, k, in solids can be broken into two components, the thermal conduction by electrons (ke-) and by phonons (kph). k = ke− + k ph (1.154) Z= α2 Chapter 1 Introduction to Transport Phenomena 45 From kinetic theory, the thermal conduction of each component is given by (Flik et al. 1992), 1 ke− = c p ,e− ce− λe− (1.155) 3 1 ki , ph = c ph c ph λ ph 3 (1.156) The subscripts e- and ph refer to an electron and a proton, respectively. The specific heats of the electron and the phonon are c p ,e− and c p , ph , respectively. The average velocities of an electron and a phonon are ce− and c ph . A phonon is a sound particle, and the speed at which it travels is the speed of sound in that material. Finally, the mean free paths of an electron and a phonon are λe− and λ ph , respectively. The mean free path is the distance one of the conduction energy carriers (electrons or phonons) travels before it collides with an imperfection in a material. Defects, dislocations, impurities and boundaries within a solid structure all have an affect on the phonon transport in a solid. The effect of impurities in a solid material can be described in terms of the acoustic impedance (Z) of the lattice waves (Huxtable et al., 2004). The acoustic impedance is Z = ρ c ph (1.157) The speed of sound (mean velocity of a phonon) is related to the elastic stiffness of a chemical bond (E) by E c ph = (1.158) ρ The acoustic impedance is a function of both the stiffness of a bond and the density. When a phonon encounters a change in the acoustic impedance, it may scatter. Scattering of this nature can change the direction in which a phonon is traveling and also its mean free path or wavelength. The wavelength or mean free path of a material also has temperature dependence that can be approximated by hc ph λ ph ≈ (1.159) 3k BT where h is Planck’s constant. From this equation, it can be seen that the mean free path has an inverse dependence on temperature. Therefore, with a decrease in temperature, the wavelength will increase, which increases the probability that a particle will be affected by either an imperfection or a boundary. Physical boundaries in a material have the effect of scattering the energy carriers. The effect of a boundary on the mean free path is presented in Fig. 1.12. The scattering characteristics of boundaries either reflect or transmit energy carriers. The probability that a phonon will transmit from material A to material B PA→ B at normal incidence is a function of the impedance of both materials. 46 Transport Phenomena in Multiphase Systems PA→ B = 4Z A Z B ( Z A + Z B )2 (1.160) The effect of boundaries can reduce the effective mean free path in the vertical and horizontal directions for phonons traveling at all incident angles. Examining eqs. (1.155) and (1.156) indicates that reducing the mean free path will reduce the thermal conductivity. Physical boundaries in a system can be grain boundaries or the surfaces of an extremely thin solid film. The effective conductivity in a thin film was related to the isotropic bulk conductivity of a material by Flik and Tien (1990). The effective conductivities normal to the thin film keff,n and along the thin film layer keff,t are: keff , n λ =1− (1.161) k 3δ keff ,t 2λ =1− (1.162) 3πδ k where δ is the film thickness. The electrical component of thermal conductivity in a solid can also be approximated by the Wiedemann-Franz law, which is valid up to the metalinsulator transition (Castellani et al., 1987). This transition occurs when the electrical conductivity of a metal suddenly changes from high to low conductivity with the decrease in temperature. This equation is valid because, in a highly electrically conductive material, essentially all of the thermal transport is carried through electrons. k = σ e−1 L0T (1.163) where L0 is the Lorenz number and σ e− is the electrical conductivity. For semiconductor materials, it is expected that the contribution to heat conduction by the electrons will be small, because the electrical conductivity is small. Therefore, molecular dynamics simulations have been used to predict the effect of system (a) (b) Figure 1.12 Mean free path of electrons or phonons (a) far away from boundaries and (b) in the presence of boundaries Chapter 1 Introduction to Transport Phenomena 47 size on the thermal conductivity by considering only the phonon transport. Such analyses are done by Petzsch and Böttger (1994), Schelling et al. (2002) and McGaughey and Kaviany (2004). The methods used in these studies are nonequilibrium molecular dynamics (NEMD) or equilibrium molecular dynamics (EMD). The NEMD approach to computing thermal conductivity is called the “direct method,” which imposes a temperature gradient across a simulation cell and is analogous to the experimental set-up. However, due to computational capacity, the cell is very small, resulting in extremely high temperature gradients in which Fourier’s law of heat conduction may break down. A commonly used approach to EMD is the Green-Kubo method, which uses current fluctuations to compute thermal conductivity by the fluctuation-dissipation theorem. This theorem captures the linear response of a system subjected to an external perturbation and is expressed in terms of fluctuation properties of the thermal equilibrium. For more detail on molecular dynamics simulations, refer to section 5.7.2. Heat conduction in a liquid is a combination of the random movement of molecules, similar to the kinetic theory of gases, as well as the movement of phonons and/or electrons. Therefore, thermal transport through conduction is much more complex in the liquid state than the gaseous or solid states; the theory is in its infancy, and is therefore not included. The discussion of microscale and nanoscale systems thus far has been primarily theoretical. Therefore, a discussion of experimental measurements of these systems is needed. The most common tool for sensing and actuating at the nanometer scale is atomic force microscopy (AFM) (King and Goodson, 2004). The basic structural design of an AFM consists of a micromachined tip at the end of a cantilever beam. A motion control stage is attached to the cantilever beam. The motion control stage brings the tip into contact with a surface, and moves the tip laterally over the surface. As the tip follows the surface, small changes in the vertical position of the cantilever beam are detected. The result is a topographic map of a surface with a resolution as good as 1 nm. The AFM can be used to intentionally modify a surface over which it scans in order to study the effects of certain modifications. This research includes local chemical delivery, thermallyassisted indentation of soft materials, direct indentation of soft materials, and guiding electromagnetic radiation into photoreactive polymers. 1.4.4 Dimensional Analysis Multiphase heat transfer is, like all engineering disciplines, quantitative in nature. We seek to define, for example, the amount of energy that can be transferred by a given heat pipe design in order to determine its suitability for a particular application. Or we seek to restrict the dimensions of a certain flat-plate heat exchanger in order to maintain a stable flow in a system. The student or practicing engineer is well advised to pursue such “bottom line” answers deliberately and systematically, so that all of the physics relevant to a given 48 Transport Phenomena in Multiphase Systems problem can be identified and prioritized early in the problem-solving process. If attention has not been first paid to the “big picture,” plunging hastily into “crunching the numbers” is counterproductive. Two highly-recommended analytical tools, reviewed in this subsection, can accomplish the critical task of organizing and filtering information so the design process can be efficiently and effectively executed. These tools are dimensional analysis and scale analysis, and they are applicable to both single- and multiphase systems. The convective heat transfer coefficient is a function of the thermal properties of the fluid, the geometric configuration, flow velocities, and driving forces. As a simple example, let us consider forced convection in a circular tube with a length L and a diameter D. The flow is assumed to be incompressible and natural convection is negligible compared with forced convection. The heat transfer coefficient can be expressed as h = h(k , μ , c p , ρ ,U , ΔT , D, L) (1.164) where k is the thermal conductivity of the fluid, μ is viscosity, c p is specific heat, ρ is density, U is velocity, and ΔT is the temperature difference between the fluid and tube wall. Equation (1.164) can also be rewritten as F (h, k , μ , c p , ρ ,U , ΔT , D, L) = 0 (1.165) It can be seen from eq. (1.165) that eight dimensional parameters are required to describe the convection problem. To determine the functions in eq. (1.164) or (1.165), it is necessary to perform a large number of experiments, which is not practical. The theory of dimensional analysis shows – as will become evident later – that it is possible to use fewer dimensionless variables to describe the convection problem. According to Buckingham’s theorem (Buckingham, 1914), the number of dimensionless variables required to describe the problem equals the number of dimensional variables [eight as indicated by eq. (1.165)], less the number of primary dimensions required to describe the problem. Since there are five primary dimensions or units required to describe the problem – mass M (kg), length L (m), time t (sec), temperature T (k), and heat transfer rate Q (W) – four dimensionless variables are needed to describe the problem. These dimensionless variables can be identified using Buckingham’s theorem and are formed from products of powers of certain original dimensional variables. Any of such dimensionless groups can be written as Π = h a k b μ c c d ρ eU f (ΔT ) g D h Li (1.166) p Substituting dimensions (units) of all variables into eq. (1.166) yields a b c d e f § Q · § Q · § M · § Qt · § M · § L · g h i Π =¨ 2 ¸ ¨ ¸¨ ¸¨ ¸ ¨ 3¸ ¨ ¸ T LL (1.167) © L T ¹ © LT ¹ © Lt ¹ © MT ¹ © L ¹ © t ¹ = M c − d + e L−2 a −b − c −3e + f + h + i t − c + d − f T − a −b − d + g Q a + b + d For Π to be dimensionless, the components of each primary dimension must be summed to zero, i.e., Chapter 1 Introduction to Transport Phenomena 49 c−d +e=0 −2a − b − c − 3e + f + h + i = 0 −c + d − f = 0 −a − b − d + g = 0 a+b+d =0 which is a set of five equations with nine unknowns. According to linear algebra, the number of distinctive solutions of eq. (1.168) is four ( 9 − 5 = 4 ), which coincides with the theorem. In order to obtain the four distinctive solutions, we have free choices on four of the nine components. If we select a = d = i = 0 and f = 1 , the solutions of eq. (1.168) become b = 0, c = −1, e = 1, g = 0, and h = 1 , which give us the first nondimensional variable ρUD Π1 = (1.169) (1.168) μ which is the Reynolds number ( Π1 = Re ). Similarly, we can set a = 1, c = f = 0 and i = 0 and get the solutions of eq. (1.168) as b = −1, d = 0, e = 0, g = 0, and h = 1 . The second nondimensional variable becomes hD Π2 = (1.170) k which is the Nusselt number ( Π 2 = Nu ). Following a similar procedure, we can get two other dimensionless variables: cp μ Π3 = (1.171) k L Π4 = (1.172) D which are the Prandtl number ( Π 3 = Pr = ν / α ) and the aspect ratio of the tube. Equation (1.165) can be rewritten as F ( Π1 , Π 2 , Π 3 , Π 4 ) = 0 (1.173) or Nu = f (Re, Pr, L / D) (1.174) Furthermore, if the flow and heat transfer in the tube are fully developed, no change of flow and heat transfer in the axial direction, eq. (1.174) can be simplified further: Nu = f (Re, Pr) (1.175) It can be seen that the number of nondimensional variables is three, as opposed to the nine dimensional variables in eq. (1.164). If the flow is assumed to be compressible and natural convection is not negligible compared with forced convection, the heat transfer coefficient can be expressed as 50 Transport Phenomena in Multiphase Systems h = h(k , μ , c p , ρ ,U , g , β , ΔT , D, L, c) (1.176) where β is thermal expansion coefficient, and c is local speed of sound. Equation (1.176) can also be rewritten as F (h, k , μ , c p , ρ ,U , g , β , ΔT , D, L, c) = 0 (1.177) It follows from Buckingham’s Π theorem that seven dimensionless variables are needed to describe the compressible mixed convection problem. These dimensionless variables can be identified using a similar procedure and eq. (1.176) can be rewritten in terms of dimensionless variables: Nu = f (Re,Gr, Pr, L / D, Ma, Fr) (1.178) Except the dimensionless variables defined previously, Gr, Ma, and Fr, are the Grashof number, Mach number, and Froude number, respectively, i.e., ρ 2 g βΔTD 3 (1.179) Gr = 2 μ U (1.180) c U2 Fr = (1.181) gL The Froude number can also find its application in an external free surface flow, such as liquid impinging on a disk and falling films. For boiling and condensation processes, the appropriate dimensionless parameters can be obtained using a similar procedure. The heat transfer coefficient for boiling or condensation depends on the properties of the fluid, a characteristic length L, the temperature difference ΔT , buoyancy force ( ρA − ρv ) g , latent heat of vaporization hAv , and surface tension, σ : Ma = h = h[k , μ , c p , ρ , L, ΔT ,( ρA − ρ v ) g , hAv ,σ ] (1.182) There are 10 dimensional variables in eq. (1.182) and there are five primary dimensions in the boiling and condensation problem. Therefore, it will be necessary to use (10 − 5) = 5 dimensionless variables to describe the liquid-vapor phase change process, i.e., Nu = f ( Gr, Ja, Pr, Bo ) (1.183) where the Grashof number is defined as ρ g ( ρA − ρv ) L3 (1.184) Gr = 2 μ The new dimensionless parameters introduced in eq. (1.183) are the Jakob number, Ja, and the Bond number, Bo, which are defined as c p ΔT hAv ( ρ − ρ v ) gL2 Bo = A σ Ja = (1.185) (1.186) Chapter 1 Introduction to Transport Phenomena 51 Table 1.8 Summary of dimensionless numbers for transport phenomena in multiphase systems Name Bond number Brinkman number Capillary number Eckert number Fourier number Symbol Bo Br Ca Ec Fo Definition ( ρ A − ρ v ) gL2 / σ Physical interpretation μU 2 /(k ΔT ) μU / σ U 2 (c p ΔT ) α t / L2 U 2 /( gL) g βΔTL3 /ν 2 Froude number Grashof number Jakob number Fr Gr Ja A Ja v Buoyancy force/surface tension Viscous dissipation /enthalpy High-speed flow change We/Re Two-phase flow Kinetic energy/enthalpy change High speed flow Transient Dimensionless time problems Interfacial force/gravitational Flow with a free force surface Buoyancy force/viscous force Natural convection Area of significance Boiling and condensation c pA ΔT / hAv c pv ΔT / hAv Sensible heat/latent heat Film condensation and boiling Wave on liquid film Noncontinuum flow Compressible flow Single and multiphase convective heat transfer Forced convection Single and multiphase convection Natural convection Forced convection Convective mass transfer Convective mass transfer Forced convection Mass transfer Melting and solidification Liquid-vapor phase change Kapitza number Knudsen number Mach number Ka Kn Ma μA4 g /[( ρA − ρv )σ 3 ] λ/L U /c Surface tension/viscous force Mean free path /characteristic length Velocity/speed of sound Thermal resistance of conduction/thermal resistance of convection RePr Rate of diffusion of viscous effect/rate of diffusion of heat GrPr Inertial force/viscous force Rate of diffusion of viscous effect/rate of diffusion of mass Resistance of diffusion /resistance of convection Nu/(Re Pr) Sh/(Re Sc) Sensible heat/latent heat Inertial force/surface tension force Nusselt number Peclet number Prandtl number Rayleigh number Reynolds number Schmidt number Sherwood number Stanton number Stanton number (mass transfer) Stefan number Weber number Nu Pe Pr Ra Re Sc Sh St Stm Ste We hL / k UL / α ν /α g βΔTL3 /(να ) UL /ν ν /D hm L / D12 h /( ρ c pU ) hm / U c p ΔT / hsA ρU 2 L / σ 52 Transport Phenomena in Multiphase Systems Table 1.9 Convective heat transfer coefficients for various heat and mass transfer modes and geometries Heat transfer mode Geometry Nusselt number Nu x = 0.332 Re1 / 2 Pr1/ 3 x Comments and restrictions Isothermal surface Re x < 5 × 105 (laminar) Dimensionless numbers Nu x = Re x = hx k u∞ x Flow parallel to a flat plate (Pr > 0.6) Nu x = 0.565Re1/ 2 Pr1/ 2 x (Pr ≤ 0.05) ν Nu = 0.037(Re0.8 − 871) Pr 0.33 L (0.6 ≤ Pr ≤ 60) 5 × 105 < Re L < 108 (Turbulent) Nu = Re L = hL k u∞ L ν Flow in a pipe (conventional size) Nu = 3.66 0.0668( D / L) Re Pr + 1 + 0.04[( D / L) Re Pr]2 / 3 Isothermal surface Re ≤ 2300 Thermal entry region L / D ≥ 10 Nu = Re = u Nu = 0.027 Re 0.8 0.14 hD k uD × Pr 0.33 ( μ / μ w ) Forced convection Flow in a pipe (miniature) Nu = (1 + F ) × (0.7 ≤ Pr ≤ 16700) Re > 10,000 (Fully developed turbulent) μ w is viscosity evaluated at Tw ν u is mean velocity ( f / 8)(Re− 1000) Pr 1 + 12.7( f / 8)0.5 (Pr 2 / 3 − 1) u f = [1.82log(Re) − 1.64]−2 F = 7.6 × 10−5 Re × [1 − ( D / D0 ) ] 2 D0=1.164 mm is reference diameter. Correlation was obtained for water at D=0.102, 0.76 and 1.09 mm. Nu = Re = hD k uD ν Nu = 7.54 Flow between parallel plates + 0.03( Dh / L ) Re Pr 1 + 0.016[( Dh / L ) Re Pr] 2/3 Isothermal Re ≤ 2800 (Laminar) surface Nu = Re = hDh k uDh ν Nu = 0.023Re Pr ( Pr > 0.5) 0.8 0.33 Re > 10,000 (Turbulent) Chapter 1 Introduction to Transport Phenomena 53 Table 1.9 Convective heat transfer coefficients for various heat and mass transfer modes and geometries (cont’d) Heat transfer mode Geometry Flow across a circular cylinder Nusselt number Nu = 0.3 + 0.62 Re1 / 2 Pr1/ 3 [1 + (0.4 / Pr/) 2 / 3 ]1/ 4 4/5 Comments and restrictions Dimensionless numbers ª § Re ·5 / 8 º × «1 + ¨ ¸» « © 282000 ¹ » ¬ ¼ Re Pr > 0.2 (Both laminar and turbulent) Nu = 3.5 < Re Flow across a sphere Re = hD k u∞ D ν Forced convection Nu = 2 + (0.4 Re 0.5 1 4 +0.06Re 2 / 3 ) Pr 0.4 ( μ / μ w ) 0.71 ≤ Pr ≤ 380 μ w < 76000 is viscosity evaluated at Tw 15 ≤ Re ≤ 120 D – diameter of sphere A – bed crosssectional area hD k mD Re = Aμ Flow through a packed bed of spheres Nu = 1.625Re1 / 2 Pr1/ 3 Nu = On a vertical surface ΔT = Tw − T∞ Applicable to both laminar and turbulent Nu 1/ 2 = 0.825 + Nu = Ra = 0.387Ra1/ 6 [1 + (0.492 / Pr)9 /16 ]8 / 27 hL k g βΔTL3 να Free convection On a horizontal heated square facing up Nu = 0.54(Gr Pr)1 / 4 Isothermal surface 105 ≤ Gr ≤ 7 × 107 For rectangle, use shorter side of L Nu = Gr = hL k g β ΔTL3 ν2 54 Transport Phenomena in Multiphase Systems Table 1.9 Convective heat transfer coefficients for various heat and mass transfer modes and geometries (cont’d) Heat transfer mode Geometry On a horizontal heated square facing down Nusselt number Comments and restrictions Isothermal surface 3 × 105 ≤ Gr Dimensionless numbers Nu = Gr = Nu = 0.27(Gr Pr)1 / 4 ≤ 3 × 1010 For rectangle, use shorter side of L hL k g β ΔTL3 ν2 On a horizontal cylinder Nu Free convection + 1/ 2 = 0.60 + Ra < 1012 Nu = Ra = 0.387Ra1/ 6 [1 + (0.559 / Pr)9 /16 ]8 / 27 hD k g βΔTD 3 να On a sphere Nu = 2 + + 0.589Ra1 / 4 [1 + (0.469 / Pr)9 /16 ]4 / 9 ΔT = Tw − T∞ Ra < 1011 Pr ≥ 0.7 Nu = Ra = hD k g βΔTD 3 να Falling film evaporation Laminar − Nu = 1.10 Reδ 1 / 3 (Reδ ≤ 30) Nu − local Nusselt number Evaporation Wavy laminar − Nu = 0.828Reδ 0.22 (30 ≤ Reδ ≤ 1800) Turbulent 0 Nu = 0.0038Reδ .4 Pr 0.65 (Reδ > 1800) h(ν A2 / g ) 3 Nu = k Γ − mass flow rate 4Γ per unit width of the Re = δ μ vertical surface 1 On a vertical surface Laminar (Nusselt) − Nu = 1.10 Reδ 1 / 3 (Reδ ≤ 30) Condensation Wavy laminar Re Nu = 1.22 δ Reδ − 5.22 (30 ≤ Reδ ≤ 1800) Nu − local Nusselt number h(ν A2 / g ) 3 Nu = k Γ − mass flow rate per unit width of the Re = 4Γ δ μ vertical surface 1 Turbulent Nu = 0.023Reδ Pr 0.25 −0.5 Chapter 1 Introduction to Transport Phenomena 55 Table 1.9 Convective heat transfer coefficients for various heat and mass transfer modes and geometries (cont’d) Heat transfer mode Geometry On tubes Nusselt number Comments and restrictions Dimensionless numbers Nu = 0.729 ª D 3hAv g ( ρ A − ρ v ) º 4 ׫ » nkAν A ΔT ¬ ¼ 1 ΔT = Tsat − Tw n − number of tubes Nu = hD kA Condensation In microscale channel ( Dh < 1.5mm ) Nu = We − Ja Re PrY Y = 1.3 for Re ≤ 65 Y = (0.5Dh − 1) /(2 Dh ) for Re > 65 We = ρAV 2 L σ hAv m′′Dh Ja = Re = c pA (Tsat − Tw ) μA m′′ –mass flux (kg/s-m2) Nucleate, saturated pool boiling Nu = 2 Ja A C 3 PrAm m=2 for water m=4.1 for other fluids C=0.013 watercopper or stainless steel C=0.006 for waternickel or brass Nu = Lc = hLc kA σA g ( ρA − ρv ) Ja A = c p , A ΔT hAv ΔT = Tw − Tsat Boiling Nu = hLc kv σA g ( ρA − ρv ) g [( ρ A − ρ v ) / ρ v ] Lc 3 Film boiling on a horizontal plate Nu = 0.425 ª § 1 + 0.4 Jav · º 4 × «Gr Prv ¨ ¸» Jav « © ¹» ¬ ¼ 1 Term in parentheses accounts for sensible heating effect in vapor film Lc = Gr = νv 2 Ja v = c p , v ΔT hAv 56 Transport Phenomena in Multiphase Systems Table 1.9 Convective heat transfer coefficients for various heat and mass transfer modes and geometries (cont’d) Heat transfer mode Geometry Film boiling on a horizontal cylinder Nusselt number Comments and restrictions Dimensionless numbers Nu = 0.62 ª § 1 + 0.4 Jav · º 4 × «Gr Prv ¨ ¸» Jav « © ¹» ¬ ¼ 1 D  film thickness Nu = Gr = hD kv 3 g [( ρ A − ρ v ) / ρ v ] D Film boiling on a sphere Boiling νv 2 Nu = 0.4 ª § 1 + 0.4Ja v · º × «Gr Prv ¨ ¸» « © Ja v ¹» ¬ ¼ Boiling in microchannel (D=1.39 – 1.69 mm) 1 3 D  film thickness Ja v = c p , v ΔT hAv Nu = 30 Re0.857 ×Bo 0.714 (1 − x ) −0.143 Correlation obtained by using Freon ® 141 x is quality hD kA q′′ Bo = hAv m′′ Nu = m′′ – mass flux (kg/s-m2) hH Nu = k g βΔTH 3 Ra = Melting in a rectangular cavity Melting Nu = (2τ ) −1 / 2 +[c1Ra1/ 4 − (2τ ) −1/ 2 ] ×[1 + (c2 Ra 3 / 4τ 3 / 2 ) n ]1/ n c1 = 0.35, c2 = 0.175 n = −2 Nusselt number is function of time να τ = SteFo α At Fo = H2 Solidification around a horizontal tube Solidification Nu = 0.52Ra1 / 4 D is transient equivalent outer diameter of the solid Ra ≤ 109 Nu = Ra = hD k g βΔTD 3 να u ∞ , T∞ , ω ∞ Sublimation Nu x = 0.458Re1 / 2 Pr1/ 3 x ′′ qw Sh x = 0.459 Re1/ 2 Sc1/ 3 x Uniform heat flux surface Re x < 5 × 105 hx k hx Sh x = m D Nu x = Chapter 1 Introduction to Transport Phenomena 57 In a convection-controlled melting or solidification process, the heat transfer coefficient is a function of the thermal properties of the Phase Change Materials (PCMs), a characteristic length L, the temperature difference ΔT , the buoyancy force g βΔT , and the latent heat of fusion hsA h = h(k , μ , c p , ρ , L, ΔT , g βΔT , hsA , t ) (1.187) where hsA is latent heat of melting, and t is time. Equation (1.187) has 10 dimensional variables, and the number of basic dimensions in the melting or solidification problem is 5. Therefore, it will be necessary to use (10 − 5) = 5 dimensionless variables to describe the solid-liquid phase change process, i.e., Nu = f ( Gr, Pr,Ste, Fo ) (1.188) where Ste and Fo are the Stefan number and Fourier number, respectively, defined as c ΔT (1.189) Ste = p hsA kt Fo = (1.190) ρ c p L2 Table 1.8 provides a summary of the definitions, physical interpretations, and areas of significance of the important dimensionless numbers for transport phenomena in multiphase systems. At reduced-length scale, the effects of gravitational and inertial forces become less important while the surface tension plays a dominant role (Eijkel and van den Berg, 2005). The convective heat transfer coefficient can be obtained analytically, numerically, or experimentally, and the results are often expressed in terms of the Nusselt number, in a fashion similar to eqs. (1.175), (1.183), and (1.188). Table 1.9 summarizes the existing correlations in literature for various heat transfer modes for both single-phase and two-phase systems in different geometric configurations. It can be seen that the heat transfer coefficient depends on surface geometry, the driving force of the fluid motion, and thermal properties of the fluid, as well as flow properties. 1.4.5 Scaling Scaling, or scale analysis, is a process that uses the basic principles of heat transfer (or other engineering disciplines) to provide order-of-magnitude estimates for quantities of interest. For example, scale analysis of a boundarylayer type flow can provide the order of magnitude of the boundary layer thickness. In addition, scale analysis can provide the order of magnitude of the heat transfer coefficient or Nusselt number, as well as the form of the functions that describe these quantities. Scale analysis confers remarkable capability because its result is within percentage points of the results produced by the exact 58 Transport Phenomena in Multiphase Systems solution (Bejan, 2004). Scaling will be demonstrated by analyzing a heat conduction problem and a contact melting problem. The first example is a scale analysis of a thermal penetration depth for conduction in a semi-infinite solid as shown in Fig. 1.13. The initial temperature of the semi-infinite body is Ti . At t=0 the surface temperature is suddenly increased to T0 . At a given time t, the thermal penetration depth is δ , beyond which the temperature of the solid is not affected by the surface temperature, i.e., the temperature satisfies the following two conditions at the thermal penetration depth T (δ , t ) = Ti (1.191) ∂T =0 (1.192) ∂x x =δ The energy equation for this problem and the corresponding initial and boundary conditions are: ∂ 2T 1 ∂T = x > 0, t > 0 (1.193) ∂x 2 α ∂t T ( x, t ) = T0 x = 0, t > 0 (1.194) T ( x, t ) = Ti x > 0, t = 0 (1.195) Since temperature difference occurs only within 0 ≤ x < ∞ , the order of magnitude of x is the same as δ , i.e., (1.196) x δ The order of magnitude of the term on the left-hand side of eq. (1.193) is ∂ 2T ∂ § ∂T · 1 ΔT ΔT (1.197) =¨ =2 ¸ ∂x 2 ∂x © ∂x ¹ δ δ δ where ΔT = T0 − Ti . The order of magnitude of the right-hand side of eq. (1.193) is 1 ∂T 1 ΔT  (1.198) α ∂t α t Figure 1.13 Thermal penetration depth for conduction in a semi-infinite solid. Chapter 1 Introduction to Transport Phenomena 59 Equation (1.193) requires that the two orders of magnitude represented by eqs. (1.197) and (1.198) equal each other, i.e., ΔT 1 ΔT  (1.199) δ2 α t The order of magnitude of the thermal penetration depth is then δ  αt (1.200) Equation (1.200) indicates that the thermal penetration depth is proportional to the square root of time, which agrees with the results obtained by the integral approximate solution (see Chapter 6). This simple example demonstrates the additional capability of scale analysis over dimensional analysis, which could not provide the form of the functions. Scale analysis can also be used to analyze heat transfer problems with phase change. Figure 1.14 shows a schematic of a contact melting problem in which a solid PCM at its melting point, Tm, sits on top of a heating surface at temperature Tw ( Tw > Tm ). The width of the PCM is L. Melting occurs at the contact area between the PCM and heating surface. The liquid PCM produced by melting is in the form of a thin layer, since gravitational force acts on the solid PCM. The solid PCM melts as it contacts the heating surface and thus the entire solid moves downward at a velocity of Vs. p u μ 2 (1.201) L δ The continuity and momentum equations of the liquid phase are ∂u ∂v + =0 (1.202) ∂x ∂y ∂p ∂ 2u +μ 2 (1.203) ∂x ∂y where the inertia terms in eq. (1.203) have been neglected because the liquid velocity is very low. 0=− Figure 1.14 Contact melting. 60 Transport Phenomena in Multiphase Systems The orders of magnitude of the two terms in eq. (1.202) must be the same, i.e., u Vs  (1.204) Lδ where Vs and δ are solid downward velocity and the thickness of the liquid film underneath the solid, respectively. Similarly, the following relationship can be obtained by equating orders of magnitude of two terms in eq. (1.203): Supposing that the heat transfer in the thin liquid film is due only to conduction, then the energy balance at the solid-liquid interface gives us ∂T (1.205) − kA = ρ s hsAVs ∂y The scale analysis of eq. (1.205) yields ΔT  ρ s hsAVs (1.206) kA where ΔT = Tw − Tm . Combining eqs. (1.204) and (1.201) yields § μV L2 · (1.207) δ ¨ s ¸ © Δp ¹ Substituting eq. (1.207) into eq. (1.206), the solid PCM velocity is obtained: § k ΔT · § p · Vs  ¨ A ¸ ¨ 2¸ © ρ s hsA ¹ © μ L ¹ Equation (1.208) can be nondimensionalized as Vs L 3/ 4 1/ 4 3/ 4 1/ 4 δ 1/ 3 (1.208) §ρ · § pL2 ·  ¨ Ste ¸ ¨ (1.209) ¸ α © ρs ¹ © μα ¹ where the left-hand side is the Peclet number based on L. The order of magnitude of the pressure in the liquid layer is defined as the net weight of the solid PCM over the horizontal projected area of the solid PCM. The horizontal projected length of the PCM is L in this example. The order of magnitude of the downward velocity obtained from scaling, eq. (1.208), has the same form as the results obtained by analytical solution as will be shown in Chapter 6. In a similar analysis for contact melting of a PCM encapsulated in a circular tube, the horizontal projected length of the PCM has the same order of magnitude as the diameter of the circular tube. Chapter 1 Introduction to Transport Phenomena 61 1.5 Multiphase Systems and Phase Changes 1.5.1 Overview and Classifications A multiphase system is one characterized by the simultaneous presence of several phases, the two-phase system being the simplest case. The term ‘two-component’ is sometimes used to describe flows in which the phases consist of different chemical substances. For example, steam-water flows are two-phase, while airwater flows are two-component. Some two-component flows (mostly liquidliquid) technically consist of a single phase but are identified as two-phase flows in which the term “phase” is applied to each of the components. Since the same mathematics describes two-phase and two-component flows, the two expressions will be treated as synonymous. This book deals with a variety of multiphase systems, in which the phases passing through the system may be solid, liquid or gas, or a combination of these three. The analysis of multiphase systems can include consideration of multiphase flow and multiphase heat and mass transfer. When all of the phases in a multiphase system exist at the same temperature, multiphase flow is the only concern. However, when the temperatures of the individual phases are different, interphase heat transfer also occurs. If different phases of the same pure substance are present in a multiphase system, interphase heat transfer will result in a change of phase, which is always accompanied by interphase mass transfer. The combination of heat transfer with mass transfer during phase change makes multiphase systems distinctly more challenging than simpler systems. Based on the phases that are involved in the system, phase change problems can be classified as: (1) solid-liquid phase change (melting and solidification), (2) solid-vapor phase change (sublimation and deposition), and (3) liquid-vapor phase change (boiling/evaporation and condensation). Melting and sublimation are also referred to as fluidification because both liquid and vapor are regarded as fluids. Phase change problems can also be classified on the basis of the system’s geometric configurations and the structures of the interfaces separating different phases. From the geometric configuration of the system, one can classify multiphase problems as (1) external phase change problems in which one phase extends to infinity, and (2) internal phase change problems in which the different phases are confined to a limited space. Examples belonging to the former class include melting and solidification in semi-infinite regions, pool boiling, and film condensation. Some examples belonging to the latter class are melting and solidification in finite slabs, forced convective boiling, and condensation in channels. Another method for classifying multiphase systems considers the structure of the interfaces. Multiphase systems can be classified as (1) separated phase, (2) mixed phase, and (3) dispersed phase, as summarized in Table 1.10. The separated phase case has two immiscible phases separated by a clearly-defined 62 Transport Phenomena in Multiphase Systems Table 1.10 Classification of multiphase systems Type Case Typical regimes Phase change on plane surface Geometry Configuration (a) Liquid layer in vapor (b) Vapor layer in liquid (c) Solid layer in liquid (d) Liquid layer in solid (e) Solid layer in vapor Examples (a) Film condensation (b) Film boiling (c) Solidification (d) Melting (e) Sublimation and deposition A B Liquid-gas jet flow (a) Liquid jet in gas (b) Gas jet in liquid (a) Atomization (b) Jet condenser Separated phases C Liquid-vapor annular flow (a) Liquid core and vapor film (b) Vapor core and liquid film (a) Film boiling (b) Film condensation or evaporation D Melting at a single melting point Solid core and liquid annular layer Melting of ice in a duct E Solidification at a single melting point Liquid core and solid annular layer Freezing water in a duct F Mixed phases Slug or plug flow Large vapor bubbles in a continuous liquid Pulsating heat pipes G Bubbly annular flow Vapor bubbles in liquid film with vapor core Film evaporation with wall nucleation Chapter 1 Introduction to Transport Phenomena 63 Table 1.10 Classification of multiphase systems (cont’d) H Droplet annular flow Vapor core with liquid droplets and annular liquid film Steam generator in boiler Mixed Phases (Continued) I Bubbly droplet annular flow Vapor bubbles in liquid film with vapor core Boiling nuclear reactor channel J Melting over a temperature range Solid and mushy zone in liquid Melting of binary solid K Solidification over a temperature range Liquid core with layer of solid and mushy zone Freezing of binary solution L Liquid-vapor (gas) bubbly flow Discrete vapor bubbles in a liquid Chemical reactors Absorbers Evaporators Separating devices Dispersed Phases M Liquid-vapor (gas) droplet flow Discrete liquid droplets in a vapor Spray cooling Atomizers Combustors N Particulate flow (a) Solid particles in liquid (Slurry Flow) (b) Discrete solid particles in gas (c) Fluidized beds (a) Melting, solidification of PCM suspension in liquid (b) Combustion of solid fuels (c) Fluidized bed reactors 64 Transport Phenomena in Multiphase Systems geometrically-simple interface (Cases A through E). Such systems can be further classified according to whether phase change occurs on a plane surface or inside a channel. Phase change occurring on a plane surface can include combinations of different phases, as indicated in Table 1.10. Liquid-gas jet flow may involve a liquid jet in a gas phase or a gas jet in a liquid phase, while phase change in a channel includes liquid-vapor annular flow as well as melting and solidification occurring at a single temperature. At the other extreme of interfacial complexity are the dispersed phases (cases L through N), including bubbly flow – discrete gaseous bubbles in a continuous fluid; droplet flow – discrete fluid droplet in a continuous liquid-vapor (gas) system; and solid-particle flow – discrete particles in a liquid or gas carrier. Change in an interfacial structure from separated phase to dispersed phase can occur gradually; as a result, there are mixed phases (Cases F through K) in which both separated and dispersed phases coexist. For a liquidvapor annular flow with a vapor core surrounded by a liquid film, thin film evaporation occurs when heat is applied to the external surface of the tube – Case C of the separated-phase type. If the wall temperature is increased to a sufficient level, vapor bubbles can be generated in the liquid layer, so the system transforms to case G of the mixed-phase type: bubbly annular flow. If the wall temperature is further increased, the flow changes to a liquid-vapor droplet form – Case M of the dispersed-flow type. Characteristic features may be associated with the behavior of each of the three possible phases comprising the multiphase systems of Table 1.10. The solid phase can be regarded as incompressible because the density of the solid phase can be treated as constant for most cases. In cases where no fluidification or solidification occurs, the solid phase has a non-deformable interface with the fluid phase, or phases, flowing over it. The flow characteristics depend strongly on the size of the individual solid elements and on the motion of the associated fluids. When melting and solidification are involved, the volume and shape of the solid can change with time. For melting and solidification occurring at a single melting point, the liquid phase is continuous, while the solid phase is discontinuous in a mushy zone formed by melting or solidification of a binary substance. The solid phase is also discontinuous in cases of particulate flow, because the solid particles are dispersed in either liquid or gas phases. In multiphase systems containing a liquid phase, the liquid can be the continuous phase, containing dispersed elements of solids (particles), gases (bubbles), or other liquids (drops). The liquid can also be discontinuous, for example, in the form of drops suspended in a gas or in another liquid such as in liquid-vapor droplet flow. A liquid also differs from a solid because its interface with other fluids (gases or other liquids) is readily deformable. The vapor (gas) phase in a multiphase system can be continuous, as in film evaporation or condensation, or as in liquid-vapor annular flow. It can also be discontinuous, as in liquid-vapor bubbly flow. Compared with the liquid phase, a vapor (gas) is highly compressible because its density is a strong function of the temperature and pressure. Notwithstanding this behavior, many multiphase flows containing vapor (gases) can be treated as essentially incompressible, especially if the Chapter 1 Introduction to Transport Phenomena 65 pressure is reasonably high and the Mach number for the gas phase is low, say less than 0.3. A multiphase system with separated phases can be considered as a field that is divided into single-phase regions with interfaces between the phases. The governing equations for a multiphase system with separated phases can be written using the standard local instantaneous differential balance for each singlephase region, with appropriate jump conditions to match the solution of these differential equations at the interfaces. This method, which is referred to as the interface tracking method, involves solving the single-phase equations in each separate phase (Chapter 3). By contrast, explicit tracking of the interfaces in mixed-phase and dispersed-phases is more complex and sometimes even impossible. In this case, spatial averaging of the governing equations is performed over each phase or simultaneously over the phases within a multiphase control volume (Chapter 4). Fundamentals of various phase change heat transfer problems will be reviewed in the remainder of this section. 1.5.2 Solid-Liquid Phase Change Including Melting and Solidification The molecules in a solid phase are compactly arranged in a closely-spaced, fixed pattern, and are bonded together by intermolecular attractive forces. When heat is applied to a solid, the temperature of the solid rises as the vibrational energy of the molecules also increases. When the temperature of the solid reaches its melting point, continued heating increases the vibrational energy of the molecules to a level that overcomes the intermolecular attractive force, so that the molecules no longer remain in a fixed pattern. As a result, relative motion among molecules becomes possible and the solid melts. On the other hand, after a liquid is cooled to the melting point, continued cooling reduces the vibrational energy of the molecules to a level that can no longer overcome the intermolecular attractive force, and the molecules are bonded together and held in a fixed pattern, so that the liquid phase becomes solidified. Melting and solidification of single-component substances occur at a single temperature – the melting point – which is a property of the substance. In a solidliquid phase change problem, different phases, which possess different thermophysical properties, are separated by an interface. The location of the interface moves as a result of the phase changes occurring in the substance. As will become evident in Chapter 6, the interfacial velocity is determined by the transient energy balance in the substance, since phase change is always accompanied by the absorption or release of latent heat. Successful solution of melting and solidification problems yields the temperature distribution in both the liquid and solid phases as well as the locations of the solid-liquid interfaces. For a solid-liquid phase change occurring at a single temperature, the interfacial structure can be classified as separated phase, because its location can always be identified. 66 Transport Phenomena in Multiphase Systems Since the melting point differs between substances, melting and solidification of multicomponent substances occur over a range of temperatures instead of at a single melting point, as is the case for a single-component substance. When the temperature of a multicomponent substance is within the phase-change temperature range, the substance is a mixture of solid and liquid which is termed the mushy zone. Melting and solidification of multicomponent substances are complicated by the fact that a mushy zone exists between the solid and liquid phases and so there are two moving interfaces: one between the solid and mushy zones, and another between the mushy and liquid zones. Successful solution of melting and solidification in multicomponent systems yields the temperature and concentration distribution in all three zones, as well as the locations of two moving interfaces. Solid-liquid phase change, including melting and solidification, is treated in detail Chapter 6, starting with the classification of solid-liquid phase changes and generalized boundary conditions at the interface. Different approaches to the solution of melting and solidification problems, including exact, integral approximate, and numerical solutions, are introduced. Solidification in binary solution systems, contact melting, melting and solidification in porous media, applications of solid-liquid phase change, and microscale solid-liquid phase change are also presented in Chapter 6. 1.5.3 Solid-Vapor Phase Change Including Sublimation and Vapor Deposition Phase change between solid and liquid occurs when the system pressure is above the triple point pressure. When the system pressure is below the triple point pressure, direct phase change between solid and vapor can occur. The process whereby solid is vaporized without going through the liquid phase is referred to as sublimation. When heat is applied to the solid with the pressure below the triple point pressure, the vibrational energy of the molecules will be increased and the temperature of the solid will be raised. When the temperature of the solid reaches the sublimation temperature, continued heating increases the vibrational energy of the molecules to a level that overcomes the intermolecular attractive force and breaks bonds between molecules. At this point the solid is vaporized. When a vapor with pressure below the triple point pressure is cooled, decreasing the vapor temperature results in decreasing molecular kinetic energy and decreasing the distance between molecules. When the vapor reaches a certain temperature, further cooling significantly reduces the intermolecular distance to a level at which the molecules are bonded together and held in a fixed pattern. The process whereby a vapor phase turns to solid is referred to as deposition, which is the opposite process of sublimation. The deposition process discussed above involves physical processes only. There is also a deposition process involving chemical reaction, which is referred to as chemical vapor deposition (CVD). When the gaseous precursors are Chapter 1 Introduction to Transport Phenomena 67 absorbed by a heated substrate – where a chemical reaction takes place – the nucleation and lattice incorporation leads to formation of a solid film. CVD can find applications in coating as well as laser-assisted manufacturing processes. Solid-vapor phase change, including sublimation and vapor deposition, is introduced in Chapter 7. The discussion begins with a brief overview of solidvapor phase change and proceeds to detailed analyses on sublimation without and with chemical reaction, as well as physical and chemical vapor deposition. 1.5.4 Interfacial Phenomena Interfacial phenomena are extremely important for various multiphase systems, especially two-phase liquid-vapor processes. Some examples of such processes in nature are the formation of clouds and vertical pumping of water in trees. Industrial drying of wood and other porous materials is a two-phase liquidvapor process of the kind also exploited in industrial large-scale devices (such as boilers and condensers) for boiling, evaporation, and condensation. Heat pipes, heat pumps, and heat sinks for cooling of electronic components operate on similar principles as well. Interfacial phenomena are vitally important for compact two-phase heat exchangers, especially for those containing capillaryporous and enhanced heat transfer structures. In fact, the developments of miniature high-heat-flux heat transfer devices – for example, those used for cooling computer chips – are impossible without careful consideration of the physical phenomena at liquid-vapor-solid interfaces. Certain phenomena can be observed at the interfacial region between two distinct material regions. These can be demonstrated by considering the idealized problems shown in Fig. 1.15. The first two problems, (a) and (b), use classical methods of fluid mechanics and thermodynamics that result in closed-form solutions with appropriate assumptions. Study of the combustion problem presented in (c) has produced meaningful solutions, but usually in the form of more complex versions of the idealized problem stated above. The feature common to these problems is that all of them require input information at the boundary between the two regions. Problems (b), (c), and (d) all involve mass transfer across the boundary. The distinguishing feature of these problems, which is also the primary source of complication in the final formulation, is the introduction of additional terms through the boundary conditions. These additional terms account for the flux of mass, momentum, and energy from one region to another; hence, the influence of interfacial phenomena on these fluxes at the boundary becomes part of the problem. Many devices utilizing two-phase heat transfer are designed so that these terms are the most significant ones; therefore, they create the driving potential as well as the limiting conditions for performance. A typical approach to treating interfacial phenomena effects in two-phase heat transfer is to apply the kind of knowledge used in formulating and solving 68 Transport Phenomena in Multiphase Systems Figure 1.15 Examples of interfacial phenomena: (a) Flow of two immiscible fluids between two parallel plates; (b) Pure substance in two phases at equilibrium; (c) Combustion of a liquid fuel droplet in gas; (d) Solid surface reacting with gas in a surrounding atmosphere. problems (a) and (b). Chapter 5 introduces the concepts of surface tension, wetting phenomena, and contact angle, which are followed by a discussion on motion induced by capillarity. Additional detailed descriptions are presented for interfacial balances and boundary conditions for mass, momentum, energy, and species for multicomponent and multiphase interface. Also considered in Chapter 5 are heat and mass transfer through the thin film region during evaporation and condensation, including the effect of interfacial resistance and disjoining pressure. The dynamics of interfaces, including stability and wave effects, are presented. Numerical simulations of interfaces and free surfaces using both continuum and non-continuum approaches are provided in Section 5.7. 1.5.5 Condensation The distance between molecules in a vapor (gas) is the greatest among all three phases; therefore, the intermolecular forces in vapor (gas) have virtually no effect on the motions of the molecules. The molecules can move randomly because their kinetic energy is sufficiently high. Decreasing the vapor temperature results in decreasing molecular kinetic energy and decreasing distance between molecules. When the vapor temperature reaches the saturation temperature, further cooling significantly reduces the intermolecular distance to a level where the intermolecular attractive force is sufficient to restrict the motion of the molecules. The vapor phase turns to liquid and condensation occurs . From a macroscopic viewpoint, condensation typically occurs when a saturated vapor – pure or multi-component – contacts an object such as a wall or other contaminant that has a temperature below the saturation temperature. In a Chapter 1 Introduction to Transport Phenomena 69 Figure 1.16 Condensation on a surface: (a) filmwise condensation, (b) dropwise condensation. multicomponent vapor, the saturation temperature is referred to as the dew point. In most applications, the cooler object is a solid wall such as those found in the condensers of most industrial applications. The vapor condenses on the cooled surface as a thin film, or as droplets, depending on whether the surface is wettable or nonwettable with the condensate [Fig. 1.16 (a) and (b)]. As the condensate forms on the cooler wall it becomes subcooled and additional condensate begins to form on the surface of the existing condensate. Condensation is a mass transfer process in which vapor transfers from the bulk fluid to the interface between the vapor and the condensate film. Energy is transferred from the vapor to the cooler wall surface through condensation. The condensation problem is complicated by the fact that although both the liquid and the vapor phases are fluids, their densities are significantly different. To solve the condensation problem, it is necessary to obtain the temperature distribution in the condensate as well as the vapor phase. The condensation rate is obtained from an energy balance at the liquid-vapor interface. For multicomponent condensation problems, it is also necessary to obtain the concentration distribution in the vapor phase adjacent to the condensate. Film-wise or drop-wise condensation, where the vapor and the heat sink (cold wall) are separated by the condensate, are referred to as indirect contact condensation. By contrast, when a vapor makes contact with a low-temperature liquid that is not attached to the cold wall, condensation occurs on the surface of the low-temperature liquid. Since the vapor is in direct contact with the heat sink – the low-temperature liquid in this case – this mode is referred to as direct contact condensation. Direct condensation is limited by the heat capacity of the low temperature liquid. Direct contact condensation includes two noteworthy cases: (1) vapor condensing on the surface of liquid droplets that are suspended in a gas phase, thus forming a fog, and (2) vapor bubbles dispersed throughout the bulk liquid. In the first case, condensation occurs on the surface of the liquid droplet and the size of the droplet grows. In the second case, the vapor bubbles surrounded by the cold liquid shrink and eventually collapse due to condensation. Solutions are possible for condensation at the outside surface of an individual liquid droplet surrounded by vapor, or for condensation at the inner surface of an 70 Transport Phenomena in Multiphase Systems individual bubble surrounded by bulk liquid, but an analytical solution at the system level is still impossible, and one must rely on numerical solutions or experiments. Chapter 8 begins with a discussion of two main modes of liquid droplet embryo formation in condensation: homogeneous and heterogeneous, followed by a detailed examination of dropwise and filmwise condensation at both macroand microscale levels. Applications of condensation in microgravity and condensation in porous media are also discussed in Chapter 8. 1.5.6 Evaporation and Boiling While the intermolecular attractive force in a liquid can hold together the molecules within, it cannot hold the molecules in a fixed pattern because the vibrational energy of the molecules is still high enough to allow them to move freely. After the liquid temperature reaches the saturation temperature, further heating of the liquid significantly increases the molecular vibrational energy, breaking up the bonds between molecules, and the liquid is vaporized. Depending on the geometric configuration and the temperature of a given system, the vaporization process – opposite of condensation – can be evaporation or boiling. Evaporation is a liquid-to-vapor transformation process that occurs across a liquid-vapor interface; it is different from boiling that occurs at a solid-liquid (heating) interface (see Fig. 1.17). Another distinction between evaporation and boiling is that there are no vapor bubbles formed during the evaporation process. As will be shown in Chapter 9, evaporation can occur on liquid films, drops, and jets. Films flow on a heated or adiabatic surface as a result of gravity or vapor shear. Drops may evaporate from a heated substrate, or they may be suspended in (a) (b) Figure 1.17 Boiling and evaporation: (a) evaporation occurring at a liquid-vapor interface, and (b) boiling occurring at a solid-liquid interface Chapter 1 Introduction to Transport Phenomena 71 a gas mixture or immiscible fluid. Jets may be cylindrical in shape or elongated (ribbon-like). To avoid nucleate boiling, liquid superheating should not exceed 3 to 40 K, depending on the liquid. Boiling is the liquid-vapor phase change that occurs at a solid-liquid interface when the surface temperature of the solid exceeds the saturation temperature of the liquid. This process is characterized by the formation of vapor bubbles, which are initiated at the solid surface and then grow and detach. The bubbles’ growth and dynamics are influenced by many factors, including temperature, surface conditions, and the properties of the fluid. Depending on whether the bulk liquid is quiescent or moving, boiling can be classified as pool boiling or flow boiling, respectively. Chapter 9 presents criteria and classification of evaporation, evaporation from an adiabatic wall, evaporation from a heated wall, evaporation in porous media, evaporation in micro/miniature channels, as well as direct-contact evaporation. Chapter 10 introduces the pool boiling curve and characterizes the various boiling regimes (free convection, nucleate, transition, and film boiling), followed by detailed discussions of each of the four pool boiling regimes, critical heat flux, minimum heat flux, and direct numerical simulation. Also discussed in Chapter 10 are the Leidenfrost phenomena as well as physical phenomena of boiling in porous media. 1.5.7 Two-Phase Flow Two-phase flow refers to the interactive flow of two distinct phases with common interfaces in a channel, with each phase representing a mass or volume of matter. The two phases can exist as combinations of solid, gas and/or liquid phases. Although multiphase flow involving three phases can also exist, most multiphase engineering applications are two-phase flow. Flow in any channel requires design, development, and optimization. It is important to predict the flow phases as well as the flow regimes, i.e., characteristic flow patterns based on the interfaces formed between the phases. This knowledge enables prediction of the pressure drop and heat transfer characteristics based on the flow rate, channel size and operating conditions. From the pressure drop data, the proper flow characteristics can be determined to minimize the occurrence of corrosion, erosion, or scale formation, all of which can lead to excessive friction. For example, in the case of oil and gas flow in pipelines or chemical process industries, prediction of flow regimes permits the elimination of excessive gas pressure buildup, meltdowns, and potential explosions. The information derived from heat transfer analysis can be used to design the flow channels and operating conditions as well as to predict and avoid flow instabilities. Two-phase flow, which involves fluid flow of a mixture of two phases, can be (1) liquid-vapor flow, (2) liquid-liquid, (3) liquid-solid particles, and (4) gassolid particles. Two-phase flow involving phase change between the liquid and vapor phases of a single substance is of particular interest to the heat transfer 72 Transport Phenomena in Multiphase Systems community and to practicing engineers. Forced convective condensation and boiling fit into this category. The fact that each phase in the two-phase flow problem has its own properties, velocity, and temperature makes the solution of two-phase flow and heat transfer problems very challenging. Two-phase flow models can be classified into two types: (1) separated flow models, and (2) homogeneous flow models. The separated-flow model allows each phase to be assigned a particular region in the flow field where it has its own velocity and temperature. To solve problems involving two-phase flow and phase-change heat transfer, it is necessary to obtain the flow field and temperature distribution of both phases; correct matching conditions at the interfaces between phases must also be applied. The governing equations for the two phases must be given separately, and boundary conditions for each phase at the interface must be matched. In the homogeneous flow model, however, it is assumed that the velocities and temperatures of the two phases are identical, and therefore the liquid and vapor phases share one set of governing equations. Chapter 11 starts with definitions of various parameters for two-phase flow and flow patterns in vertical and horizontal tubes. This is followed by two-phase flow models as well as prediction of pressure drops and void fractions. Finally, the two-phase flow regimes and heat transfer characteristics for forced convective condensation and boiling at both macro- and microscale levels are presented in Chapter 11. 1.6 Applications of Transport Phenomena in Multiphase Systems Applications of melting and solidification include ice-making, food processing, solidification of pure metals or alloys in casting; melting and solidification of metals in welding; thermal sprays; purification of metals; crystal growth in semiconductor and optical components; nuclear reactor safety; atmospheric reentry by spacecraft; and thermal control of spacecraft. While liquid-vapor phase change finds application in traditional technologies such as power generation, refrigeration, and gas turbine combustion, it is also widely used in emerging technologies such as electronics cooling, laser machining, heat pipes, and fuel cells. Selected applications of multiphase systems in these technologies are reviewed in this section. 1.6.1 Energy Systems, Including Fuel Cells and Combustors Thermal Energy Storage Interest in alternative energy technologies, such as solar energy, is growing due to increasing recognition of environmental considerations. The major barrier Chapter 1 Introduction to Transport Phenomena 73 Figure 1.18 Schematic of latent heat thermal energy storage system. to more widespread use of solar energy is its periodic nature, as it is available only during daytime and a heat storage device is needed to store energy for use at night. The latent heat thermal energy storage system, which utilizes phase-change materials (PCMs) to absorb and release heat, is widely used for this purpose (Viskanta, 1983). The PCM in the thermal energy storage system is molten when the system absorbs heat, and it solidifies when the system releases heat. The advantages of the latent heat thermal energy storage system are that a large amount of heat can be absorbed and released at a constant temperature, and the latent heat thermal energy system is considerably smaller than its counterpart using sensible heat thermal energy storage. A conceptual design of the latent heat thermal energy storage system is illustrated in Fig. 1.18 (Zhang and Faghri, 1996). The structure of the thermal storage system is similar to that of a conventional shell-and-tube heat exchanger, with the PCM on the shell side and the transfer fluid flowing inside the tube. The outside of the shell is insulated to prevent heat loss to the environment. The PCM fills the space between the tube and a shell in the shape of a hollow cylinder. A transfer fluid flowing inside the tube exchanges heat with the PCM. During the heat storage process, the fluid is at a temperature higher than the PCM’s melting point, and flows through the tube to melt the initially-solid PCM. In the heatrelease process, the fluid is at a temperature below the PCM’s melting point, and flows through the tube to solidify the liquid PCM. In a latent heat thermal energy storage system, these melting and solidification processes occur alternately to store and release heat. Power and Refrigeration Cycles Condensers are major components in power plants as well as in air conditioning units and refrigerators. The condenser converts exhaust 74 Transport Phenomena in Multiphase Systems Figure 1.19 Water-cooled condenser (Lock, 1994; Reprinted with permission from Oxford University Press). Figure 1.20 Feedwater heater concept. steam/refrigerant vapor into liquid by rejecting heat to the ambient environment. Fig. 1.19 shows a schematic diagram of a typical water-cooled condenser for a modern power plant. The cooling water flows inside the tubes and the exhaust steam condenses on the outside surfaces of the tubes. To improve the efficiency of the steam power plant, it is desirable to increase the average heat-addition temperature. One practical way to increase the Chapter 1 Introduction to Transport Phenomena 75 Figure 1.21 Boiler (Lock, 1994; Reprinted with permission from Oxford University Press). temperature of feedwater entering the boiler is to incorporate a feedwater heater, which uses steam extracted from various points of the steam turbine to heat the feedwater. The conceptual design of the feedwater heater is shown in Fig. 1.20. The steam introduced into the cold feedwater makes direct contact with it. Direct contact condensation occurs in the feedwater heater, and the mass flow rate of the feedwater at the outlet equals the sum of the inlet mass flow rates of feedwater and steam. In the ideal case, the feedwater leaves the heater as a saturated liquid at the heater pressure. Typical applications of evaporation and boiling in energy systems include the steam generator in a boiler, the evaporator in an air conditioner or a refrigerator, as well as components used in nuclear reactor heat transfer. The heat exchangers in the boiler of a modern power plant include an economizer, a steam generator (evaporator), and a superheater. A steam generator, as shown Fig. 1.21, includes three parts: (1) a water drum to store the saturated water, (2) risers where evaporation/boiling take place, and (3) a steam drum to store saturated vapor. A steam generator is also used in nuclear reactors to extract heat from the nuclear reaction. In an air conditioner or refrigerator, the evaporator absorbs heat to maintain a lower temperature in the room or the refrigerator. The heat transfer mechanism inside steam generators or evaporators is essentially forced convective boiling inside the tubes. 76 Transport Phenomena in Multiphase Systems Phase Change during Combustion Combustion is a chemical reaction process between a fuel (which can be solid, liquid or gas) and an oxidant that produces high-temperature gases; these can be used to generate steam in a boiler, drive a gas turbine, or melt metals in a metallurgical process. Except in cases where gaseous fuels are used, the combustion process always involves phase changes. In a typical combustor using liquid fuel, the fuel first breaks up into a finedroplet spray and then vaporizes. Combustion of liquid fuel is a very complex process because it involves interaction between a spray of multiple drops and a turbulent flow field, as well as the transient effect of a rapidly-growing flame front that provides the latent heat for the fuel vaporization. Successful modeling of combustion requires careful consideration of simultaneous heat and mass transfer between the fuel drops and the mainstream flow, conductive and convective heat transfer, the multi-component nature of real fuels, and the effect of realistic pressure and temperature levels on thermophysical properties such as latent heat and diffusion coefficients. In combustion applications the fluid mechanics and heat transfer processes, rather than the chemical reactions, control the rate of combustion and therefore the rate of energy addition to the cycle. As a result, accurate prediction of phenomena such as droplet breakup and evaporation is critical to successful combustor design. When solid fuel, such as coal, is used in the combustor, combustion occurs on the surface of the solid fuel. To increase the contact area between the coal and the oxidant, the coal is ground into fluidized particles (very small particles that can flow with the oxidant gas) that are consumed during the combustion. Combustion of solid fuel involves gas-solid two-phase flow, interaction between solid particles, diffusion of oxidant near the particle surfaces, conduction heat transfer in the solid particles, and convective heat transfer in the gas, as well as chemical reactions on the particle surface that consume the solid particles. Since the densities of solid fuel and oxidant are significantly different, resulting flow patterns are usually not homogeneous because the solid particles and the oxidant possess different velocities. These complexities greatly complicate the modeling combustion of solid fuel. Turbofan Engine Combustion The turbofan engine has evolved from its introduction in the 1950s to its current role as the primary power source for today’s commercial aviation fleet. A typical modern, high-bypass turbofan, shown in schematic cross-section in Fig. 1.22, produces thrust to power aircraft by ingesting ambient air, compressing the air, undergoing combustion, and expanding the hot gas through thrust-producing exhaust nozzles (Oates, 1984). Conventional practice divides the incoming flow into two streams: a primary or core stream, and a fan stream. The flow of the primary stream experiences compression and expansion, as the expansion process Chapter 1 Introduction to Transport Phenomena 77 Fan Fan Stream Compressors Combustor Primary Stream Turbines Figure 1.22 Components and main flows of the turbofan engine (courtesy of GE-P&W Engine Alliance, LLC). Diffuser Fuel Nozzle Case Liner Figure 1.23 Schematic of typical turbofan combustor. occurs both in the thrust-producing nozzle and, immediately upstream of the nozzle, in power-producing turbines that drive the compression systems of both streams. The flow of the fan stream also undergoes compression and expansion, but in this example stream expansion occurs only through the thrust-producing nozzle. Energy addition to drive the turbines occurs in the combustor, a typical example of which is shown in Fig. 1.23, wherein liquid fuel is added to the flow through a set of circumferentially-arrayed fuel nozzles. Naturally, the maximum temperature reached in the turbofan cycle occurs as the fuel burns in the combustor, so this is a site of significant heat transfer that must be accounted for 78 Transport Phenomena in Multiphase Systems in the design of the flow field and the hardware. The combustion process is accompanied by phase change because the liquid fuel is vaporized before it combines with oxygen and releases the chemical energy that drives the turbofan cycle. Accordingly, the combustor is the primary application of multiphase heat transfer in this field of technology. Fuel Cells A fuel cell is an electrochemical energy device that converts the chemical energy in the fuel directly into electrical energy. It is becoming an increasingly attractive alternative to other conversion technologies, from small-scale passive devices like batteries to large-scale thermodynamic cycle engines. Unlike conventional power devices, i.e., steam turbines, gas turbines, and internal combustion engines, which are based on certain thermal cycles, the maximum efficiency of fuel cells is not limited by the Carnot cycle principle. Figure 1.24 is a schematic of a general fuel cell (Faghri and Guo, 2005). A fuel cell generally functions as follows: electrons are released from the oxidation of fuel at the anode, protons (or ions) pass through a layer of electrolyte, and the electrons are used in reduction of an oxidant at the cathode. The desired output is the largest possible flow of electrons over the highest electric potential. Although other oxidants such as halogens have been used where high efficiency is critical, oxygen is the standard because it is readily available in the atmosphere. Fuel cells typically use hydrogen, carbon monoxide, or hydrocarbon fuels (i.e., methane, methanol). The hydrogen and carbon monoxide fuels may be the products of catalytically-processed hydrocarbons. Hydrogen from processed ammonia is also used as fuel. Oxidant (e.g., Oxygen, Air….) Reactant Products (e.g., H2O, CO2 … .) & Heat Cathode Load Electrolyte e- Anode Fuel (e.g., H2, CH4, CO, CH 3OH….) Figure 1.24 Fuel cell schematic. Chapter 1 Introduction to Transport Phenomena 79 Two cases are possible for the electrolyte: it may be a conductor for anions or for cations. In the first case, the oxidant at the cathode combines with electrons, which tend to circumvent the electrolyte, become anions, which travel through the electrolyte to the anode (Fuel Cell Handbook, 2000). At the anode, the anions give up their electrons and combine with hydrogen to formwater. The water, depleted fuel, and products are exhausted from the anode surface, and the depleted oxidant and products are exhausted from the cathode surface. In the second case, where the electrolyte conducts cations, the hydrogen-containing fuel is decomposed electrochemically, releasing electrons and leaving hydrogen cations to travel through the electrolyte. Upon reaching the cathode, the cations combine electrochemically with the oxidant and electrons, which tend to circumvent the electrolyte to form water. The water, depleted oxidant, and other gases present are exhausted from the cathode, while the depleted fuel and product gases are exhausted from the anode. There are several types of fuel cells, and they each belong to one of the two cases just described. Anion-conducting electrolyte fuel cells are (Larminie and Dicks, 2000): alkaline fuel cells – for example, those using potassium hydroxide molten carbonate that operates at about 650 °C – and solid oxide fuel cells that operate to 1000 °C. Cation-conducting electrolyte fuel cells include phosphoric acid fuel cells and polymer electrolyte membrane fuel cells. The latter, with power capacities from small batteries to automotive use, are receiving the most commercial and research attention. Among different types of fuel cells, the proton exchange membrane fuel cell (PEMFC) is one of the best candidates as an alternative energy source in the future because it offers advantages in light weight, durability, high power density and rapid adjustment to power demand. The fuel for PEMFCs can be either hydrogen or methanol. Figure 1.25 shows the basic structure of a PEMFC, which can be subdivided into three parts: the membrane electrode assemblies (MEAs), the gas diffusion layers (GDLs), and bipolar plates (Faghri, 2006; Faghri and Guo, 2005). The key component of the PEMFCs is the MEA, which is composed of a proton exchange membrane sandwiched between two fuel cell electrodes: the anode, where hydrogen is oxidized, and the cathode, where oxygen from air is reduced. A gas diffusion layer is formed from a porous material that must have high electric conductivity, high gas permeability, high surface area and good water management characteristics. One side of the bipolar plate is next to the cathode of a cell, while the other side is next to the anode of the neighboring cell. The fuel cell stack consists of a repeated, interleaved structure of MEAs, GDLs and bipolar plates. It is evident that flow channels are an essential component for flow distribution in many PEMFC designs. The flow channels in a PEMFC are typically on the order of a 1 mm hydraulic diameter, which falls into the range of minichannels (i.e., hydraulic diameters from 200 m to 3 mm). As shown in Fig. 1.25, one channel wall is porous (gas diffusion layer); mass transfer occurs on this wall along its length. Hydrogen is consumed on the anode side along the main flow dimension in minichannels. Oxygen from 80 Transport Phenomena in Multiphase Systems Gas Diffusion Layer Catalyst Layer Proton Exchange Membrane Bipolar Plate H2O Bipolar Plate - e H2O H + e- H+ H2 O2 H2 O2 Figure 1.25 Basic construction of a typical PEM fuel cell stack (Faghri and Guo, 2005; Reprinted with permission of Elsevier). Hydrogen (H2) CO2 + H2 Gas Purification Fresh Air Fuel Cells Reformer and Catalytic Burner Cooling Pump Compressor/ Expander Vaporizer Cooler Cooling Air Out Cooling Air In Water produced by fuel cells Water Tank Methanol Tank Figure 1.26 Methanol powered fuel cell system (Thomas and Zalbowitz, 1999). Chapter 1 Introduction to Transport Phenomena 81 air is introduced on the cathode side to form water at catalyst sites at the cathode; this water is transported into the minichannels through the gas diffusion layer, and eventually it is removed from the cell by the gas flow – and gravity – if so oriented. Several factors affect the efficiency of fuel cells. The operating temperature determines the maximum theoretical voltage at which a fuel cell can operate. Higher temperatures correspond to lower theoretical maximum voltages and lower efficiencies. However, a higher temperature at the electrodes increases electrochemical activity, which in turn increases efficiency. A higher operational temperature also improves the quality (exergy) of the waste heat. In addition, increasing pressure increases both maximum theoretical voltage and electrochemical activity. However, electrical resistance in the electrodes and corresponding connections, ionic resistance, and electrical conductivity in the electrolyte all lower cell efficiency. Efficiency is also affected by mass transport of products to the electrodes, as well as product permeation through the electrolyte. A practical fuel cell requires a support system, which has a related efficiency of its own. Figure 1.26 shows a support system for a methanol-powered fuel cell (Thomas and Zalbowitz, 1999). Every component in the system has an efficiency that affects the total system efficiency. The compressor, for example, has an isentropic efficiency, and the motor that turns the compressor also has an associated efficiency. As shown in Fig. 1.26, fuel cells require cooling because the chemical energy not converted to electrical energy in a fuel cell is converted to heat. For example, a fuel cell operating at 100 W and 50% efficiency generates 100 W of heat. This heat may be dissipated by convection, conduction, radiation and phase change. The heat generated in a fuel cell stack may be dumped into the atmosphere, but often it is utilized for other system components requiring heat. In some cases the heat is used to run a thermodynamic cycle for additional power generation. Figure 1.26 also provides an example of how many processes requiring heat transfer, particularly via phase changes, are present in a fuel cell system. The process begins in the methanol tank, where the methanol – along with water – is vaporized. The resulting mixture is heated, then reacts to form hydrogen and carbon dioxide. The hydrogen likely requires cooling before it arrives at the stack. In the fuel cell stack, water or steam is generated along with heat. A coolant pumped through the fuel cell stack is used to remove some of the heat generated. Heat is also removed from the stack in the product streams. Any water vapor exiting the stack must be condensed before arriving at the water tank. The aforementioned are just some of the possibilities of heat transfer and phase change processes in fuel cell stacks and systems. 82 Transport Phenomena in Multiphase Systems 1.6.2 Food and Biological Material Processing Food Freezing and Thawing A significant amount of food, including meat, fish, and vegetable products, is routinely frozen for preservation and transportation. According to the American Frozen Food Institute, the total retail sales of frozen foods in the U.S. reached more than $26.6 billion in 2001, so clearly this is a significant area of research and commercial interest. Food material that is subjected to this process begins as a combination of solid and liquid, and the liquid portion of the food is frozen by the end of the process. The liquid in the food cannot be considered a pure substance because there are different components in it, so from the heat transfer point of view, this food-freezing process can be classified as solidification of a multicomponent liquid in porous media. In order to preserve the structure of the food cells throughout the process, the temperature of the freezer should be low enough to achieve rapid freezing. However, lower freezer temperature means higher cost, especially for large industrial processes, so it is crucial to control the freezing rate in order to assure both fresh food and a cost-effective operation. The opposite of freezing is, of course, the process of thawing. The time required to thaw frozen materials constitutes a significant constraint on industrial applications. A traditional approach to thawing frozen materials is convection thawing, which uses hot fluid to convectively heat the frozen food materials. This process involves melting a multicomponent material in porous media so that a portion of the frozen food material becomes molten, and the food reverts to a combination of liquid and solid. The disadvantages of convection thawing are its requirements for long processing times and large space, and the potential for chemical and biological deterioration. An alternative method of thawing food material employs microwave heating; this is an attractive approach due to its rapid and uniform heating effects, and its high energy efficiency. From the heat transfer point of view, microwave heating can be considered volumetric heating instead of surface heating, which occurs in convective heating. Therefore, microwave thawing of food material can be considered to be melting in porous media with an internal heat source. The challenge of microwave thawing is that it requires a thorough understanding of the interaction between microwave radiation and the food materials undergoing phase change (Zeng and Faghri, 1994). The physical, thermal, and electrical properties of the food materials have a significant effect on the transient temperature distributions. Since there are so many types of food materials, these properties are not always readily available. Thermal modeling of the freezing and thawing of food materials remains a challenging area for the heat transfer community and food industry. Chapter 1 Introduction to Transport Phenomena 83 Cryopreservation of Biological Materials Cryopreservation uses liquid nitrogen to deep-freeze, and thus preserve, biological materials. Among the biological materials that can be preserved through this process are oocytes, embryos, tissues, and even entire organs. Preservation of embryos is necessary, for example, if couples have medical reason to delay their reproductive choice. Such situations may occur when a woman has to undergo a cancer treatment that may risk her subsequent ability to give birth to a healthy child. The doctor can preserve her oocytes or embryos and let her conceive after she is recovered. Cryopreservation of tissues (bones, tendons, corneas, heart valves, etc.) permits storage of deep-frozen biomaterials in a storage bank until they are needed for transplant. Recently, Wang et al. (2002) reported that transplant of a whole organ after cryopreservation and thawing is feasible. Cryopreservation of biological materials presents special challenges because both freezing and thawing can cause severe damage to the cells. In terms of heat transfer, cryopreservation involves solidification of multicomponent substances similar to those in food freezing, but the temperature is much lower (-196 °C). The cells in the biological materials must function properly after freezing and thawing. For this reason, it is very important to control the cooling rate and prevent severe cell damage. The mechanisms of damage for cells in suspension and in tissues are different (Asymptote Ltd., 2002). For cells in suspension, cell death can occur if the cooling rate is too low; in such situations, cells are exposed to a hypertonic condition for a long period and they become pickled. However, if the cooling rate is too high, intracellular ice formation can occur. The mechanism of cell damage in tissue is different from that in suspension and is still not fully understood. The water in tissues can exist in two different forms: (1) a continuous liquid phase in a small extracellular compartment, and (2) noncontinuous phases within the individual cells. The manner by which ice forms in the extracellular compartment of tissues, and the process of cellular dehydration, play significant roles in cell damage during freezing (Asymptote Ltd., 2002). Thawing is the opposite process of freezing biological materials. As with the freezing process, it is necessary to find the optimum heating rate to minimize the cell damage during thawing. 1.6.3 Laser-Assisted Manufacturing Selective Laser Sintering (SLS) of Metal Powder SLS is an emerging technology of solid freeform fabrication (SFF) that allows three-dimensional parts to be built from CAD data (Beaman et al., 1997). A schematic of the SLS process is illustrated in Fig. 1.27 (Marcus et al., 1993). SLS involves fabrication of near-full-density objects from powdered material via 84 Transport Phenomena in Multiphase Systems layer-by-layer sintering or via melting induced by a directed laser beam (generally CO2 or YAG). Thin (100–250 μm) powder layers are laser-scanned to fuse a densified two-dimensional slice to an underlying solid piece that consists of a series of stacked and fused two-dimensional slices. After laser scanning, the part bin is lowered by one layer-thickness, a fresh powder layer is spread, and the scanning process is repeated. Loose powder is removed after the part is extracted from its bin. Extremely complex part shapes can be formed from a variety of materials, which may be amorphous (e.g., polycarbonate), semi-crystalline (e.g., nylon), or crystalline (e.g., metal). The advantage of the SLS process is that complex parts can be made in a single step without any part-specific tooling or human intervention. However, this technology is still in its infancy, and many parts produced by the SLS process do not meet the requirements for functional strength. Further efforts at modeling, and experimental investigation of this technique, are still urgently needed in order to expand its application. For sintering of a metal powder, the latent heat of fusion usually is very large. Therefore, melting and resolidification phenomena have a significant effect on the temperature distribution in the parts and powder, the residual stress in the part, local sintering rates, and the final quality of the parts. A significant change of density accompanies the melting process, because the volume fraction of gas(es) in the powder decreases from a value as large as 0.6 to nearly zero after melting. In addition, the liquid metal infiltrates the unsintered region due to capillary and gravitational forces. The modeling of SLS of metal powder is a very challenging task because the melting and resolidification processes are highly nonlinear, and the process is further complicated by the shrinkage Figure 1.27 Selective laser sintering (Marcus et al., 1993). Chapter 1 Introduction to Transport Phenomena 85 phenomena. Successful modeling of SLS of metal powder can provide the transient temperature distribution in sintered and unsintered regions, the local sintering rate, and the distribution of concentration for cases involving more than one component powder. A thorough survey of the existing literature indicates that scant attention has been paid to thermal modeling of the sintering of metal powders, and it remains an ongoing effort (Zhang and Faghri, 1998; 1999a; Zhang et al., 2000; Chen and Zhang, 2006). Laser Machining Laser machining includes laser drilling and laser cutting, which are processes important to the automotive, aerospace, electronics, and materials-processing industries. Laser machining involves removing material by vaporizing the portion of the workpiece that interacts with the laser beam. The mechanism of vaporization is different for metal and ceramic workpiece materials. Figure 1.28 shows the physical model of laser drilling on a metal substrate (Ganesh et al., 1997). A laser beam is directed toward a solid target material at an initial temperature of Ti, which is below the melting point of the metal. The lasermaterial interaction can be divided into three stages (Zhang and Faghri, 1999b). During the first stage, the temperature of the solid remains below the melting point so that no melting or vaporization occurs. The solid absorbs thermal energy and its temperature increases with time. When the highest temperature of the solid – located at the center of the laser beam – reaches the target material’s melting point, continued laser beam irradiation results in melting of the target material; at this point, the process enters its second stage. In the second stage, the surface temperature of the liquid is below the saturation temperature, and the vaporization required by thermodynamic equilibrium is negligible. When the liquid surface’s highest temperature reaches the vaporization temperature of the Figure 1.28 Laser drilling physical model (Ganesh et al., 1997; Reprinted with permission of Elsevier). 86 Transport Phenomena in Multiphase Systems material, vaporization occurs at the liquid surface and the third stage starts. During the third stage, the locations of both the solid-liquid and liquid-vapor interfaces are unknown and must be determined. Vaporization creates a backpressure on the free surface of the liquid, which pushes the melt away in the radial direction. Thus the material is removed through a combination of vaporization and liquid expulsion. Due to strong evaporation occurring in the laser drilling process, the gas near the liquid-vapor interface is not in translational equilibrium; the translational equilibrium is achieved within a few mean free paths by collisions between particles in a thin region referred to as the Knudsen layer. Above this layer, lying stacked in the vertical direction, are the layers of vapor, disturbed air, and undisturbed quiescent air. The phase changes occurring in laser cutting are similar to those in the laser drilling process, but the problem can no longer be modeled as axisymmetric because the laser beam is moving. Although the workpiece material is removed via melting and vaporization, many researchers have assumed that phase change occurs in a single step, directly from solid to vapor (Kim and Majumdar, 1995; Modest 1996). This assumption is acceptable for a number of ceramics and other nonmetals, such as graphite and silicon nitride. However, this assumption may be inappropriate for metal, because in this case melting always occurs before vaporization takes place. Selective Area Laser Deposition (SALD) and SALD Vapor Infiltration (SALDVI) SALD and SALDVI are methods of building functional structures by using a laser beam to deposit solid materials from gas precursors in an environmentallycontrolled chamber (Fig. 1.29; Jakubenas et al., 1997). Both techniques utilize laser chemical vapor deposition (LCVD; Mazumder and Kar, 1995), which can be based on reactions initiated pyrolytically, photolytically, or a combination of both (Marcus et al., 1993), to deposit film to a desired location or to join powder particles together. While the SALD technique uses precursors to directly create free-standing parts, or to join together simple shapes to create parts with higher complexity, the SALDVI uses gas precursors and powder particles to build threedimensional parts. This is similar to other SFF techniques, such as SLS. The advantages of SALDVI over SALD include (a) uninfiltrated powder provides necessary support for producing overhangs, (b) confining the deposition to thin powder layers provides dimensional control in the direction of growth, and (c) it allows for tailoring of local chemistry and micro structures. The manufacturing process using SALDVI is very similar to that using SLS because both of them fabricate objects from powdered material via a layer-by-layer process induced by a directed laser beam. The only difference is that the powder particles in SALDVI are bonded together by LCVD, a process that occurs on the surface of the powder particles, while binding of powder particles in SLS is accomplished through sintering or melting. Chapter 1 Introduction to Transport Phenomena 87 (a) SALD (b) SALDVI Figure 1.29 Schematic of SALD and SALDVI (Jakubenas et al., 1997). Since the temperature gradient in the precursor near the laser spot is very high, and the consumption of the reactants near the laser spot creates a concentration gradient, the natural convection driven by both temperature and concentration gradients occurs. In addition, the precursors are usually transported to the reaction zone by convection. Therefore, convection plays a significant role in the SALD/SALDVI processes. The phase change process involved in SALD/SALDVI is completed by a chemical reaction that takes place on the substrate/powder particle surface. 1.6.4 Heat Pipes Heat pipes are devices used to transfer heat via the processes of evaporation and condensation (Faghri, 1995). In comparison to highly-conductive materials like copper, heat pipes can be designed to move larger quantities of heat over longer distances, through narrower spaces and at lower temperature differentials. With some exceptions, they are typically designed as passive devices that require no external power or control to perform their function. Heat pipes have resulted from a progression of refinements of other devices employing evaporation and condensation processes. The sealed Perkins tube, which was patented in the United Kingdom by Jacob Perkins (1836), has the structure that is closest to a modern gravity-assisted heat pipe – a thermosyphon, as shown in Fig. 1.30. Sealing the process from air allows the thermosyphon to function more efficiently over a larger range of temperature. One limitation to the thermosyphon, however, is that gravity can distribute the liquid only when the warm section lies below the cool section. The wick heat pipe, on the other hand, can operate in any orientation because it uses a wick to distribute the liquid. The principle of wick heat pipe operation is 88 Transport Phenomena in Multiphase Systems Figure 1.30 Thermosyphon. illustrated with the aid of Fig. 1.31. Heat is applied to the evaporator section and is conducted through the wick and liquid. Liquid evaporates at its interface with vapor as it absorbs the applied heat. In the condenser section, the vapor releases heat to its cooler interface with liquid as it condenses. In the wick, the menisci are increasingly pronounced approaching the evaporator end, due to the growing pressure drop required to draw the liquid through the increasing length of wick. There are additional contributions to pressure drop, such as friction of the vapor flow, adverse orientation against gravity, or other acceleration sources. Subsequently, the vapor pressure drops as it flows from the evaporator to the condenser. As stated above, the friction of the liquid flow through the wick causes the liquid pressure to drop from the condenser to the evaporator. If the heat pipe is to function, all pressure drop sources must be balanced by the capillary pressure differential provided at the menisci in the capillary wick. Provided the above requirements are met, there is considerable flexibility for heat pipe geometry, operating temperature, and heat transfer rate. Even though the term “heat pipe” is used, the container may take other shapes such as a flat plate heat pipe or a heat pipe with appendages (Faghri and Buchko, 1991). Bundles of carbon fibers may be used as a wick and held to the heat pipe container wall by a spring, thus allowing for flexible heat pipes (Faghri, 1995). With an appropriate choice of working fluid, heat pipes may operate in different Chapter 1 Introduction to Transport Phenomena 89 Figure 1.31 Principle of heat pipe with wick (Faghri, 1995; Reprinted by permission from Routledge/Taylor & Francis Group, LLC). temperature ranges. Liquid helium and its vapor are used in the lowesttemperature cryogenic heat pipes, while silver and its vapor are used in the highest-temperature heat pipes, with many choices of fluids available between these extremes (Faghri, 1995). Binary liquid heat pipes or gas-loaded heat pipes are used to maintain a certain temperature range despite variations in the heat rate. Heat pipe design employs the theory of both hydrodynamic and heat transfer processes. The hydrodynamic component of the theory addresses pressure drop 90 Transport Phenomena in Multiphase Systems in the vapor space and wick structure. Heat transfer processes include conduction and convection through the heat pipe container, wick, and liquid, as well as evaporation, boiling, or condensation along the heat pipe (Faghri, 1995). In the case of high-temperature heat pipes, melting and solidification are encountered during heat pipe start-up and shut-off (Faghri et al., 1991). Heat pipes currently are used extensively in some real applications. Along the Trans-Alaska pipeline, for example, more than 150,000 heat pipes provide for permafrost stabilization. These heat pipes are used to divert geothermal heat to the atmosphere in order to maintain a frozen and stable footing for the pipeline support structure. In most laptop computers, heat pipes are used extensively to eliminate bulky finned “heat sinks” and electrically-parasitic fans. Heat pipes also find increasing applications in HVAC heat recovery, where they offer a more robust alternative to counter-flow heat exchangers. In addition to the conventional heat pipes described above, it is worthwhile to mention a new type of heat pipe, the pulsating heat pipe (PHP), which is made from a long capillary tube bent into many turns, with the evaporator and condenser sections located at these turns. There are two types of PHPs: looped and unlooped (Fig. 1.32; Shafii et al., 2001), which are classified according to whether the two ends of a PHP are connected to each other. To operate, the PHP is partially charged with working fluid and the charge ratio is 40 ~ 60%. Since the diameter of the PHP is very small (less than 5 mm), vapor plugs and liquid slugs are formed as a result of capillary action. Heat input causes evaporation or boiling, which increases the pressure of the vapor plug in the heating section. Simultaneously, the pressure in the cooling section decreases due to condensation. This pressure difference pushes the liquid slug and vapor plug into the cooling section. The liquid slug and vapor plug in the cooling section are then pushed into the next heating section, which will push the liquid slug and vapor plug back to the cooling section. This process enables self-excited oscillatory motion of liquid slugs and vapor plugs. Heat is transported from the heating section to the cooling section via the pulsation of the working fluid in the axial direction of the tube. The unique feature of PHPs, as compared with conventional heat pipes, is that there is no wick structure to return the condensate to the heating section. Therefore, there is no countercurrent flow between the liquid and vapor. The entrainment limit in the conventional heat pipe does not have any effect on the capacity of heat transport by a PHP. With this simple structure, the PHP weighs less than a conventional heat pipe, and is an ideal candidate for space applications. Since the diameter of the PHP is very small, surface tension plays a greater role in the dynamics of PHP than gravitational force does, and it enables successful operation in a microgravity environment. This feature makes PHPs even more attractive for space applications. Other applications of PHPs include thermal control of electrical and electronic devices and components, as well as cooling of thyristors, diodes and ceramic resistors. Chapter 1 Introduction to Transport Phenomena 91 Figure 1.32 Pulsating heat pipes: (a) unlooped, (b) looped (Shafii et al., 2001). 1.6.5 Electronics Cooling Condensation in Miniature Tubes It is very important to develop new cooling strategies for electronic devices in response to the trend toward more compact equipment that requires higher levels of rejection heat flux. The trend creates a demand for more compact heat transfer devices capable of removing large amounts of heat over small temperature drops. Miniature and micro heat pipes have been and are being used Figure 1.33 Condensation in a miniature tube. 92 Transport Phenomena in Multiphase Systems in electronic cooling. In these applications, the heat generated by a chip is transported away and rejected from the system through condensation. To optimize the electronic cooling system performance, it is necessary to have a fundamental understanding of condensation in miniature tubes. The term “miniature tube” refers to a tube with a diameter of 3.5 mm or less (Begg et al., 1999). Unlike a conventionally-sized passage, in which surface tension effects are of limited importance, surface tension in a miniature tube can have a significant impact on the overall hydrodynamics and in particular on the thin films. Capillary blocking can occur in forced convective condensation in miniature tubes as a result of surface tension, in which case the liquid blocks the tube cross-section at some distance from the condenser entrance (Fig. 1.33). As a result, the part of the tube that is blocked by the liquid can make no contribution to heat removal. Accurate prediction of the condition of capillary blocking and the length of the effective condensation is crucial for the design of an electronic cooling system that involves condensation in miniature channels (Begg et al., 1999; Zhang et al., 2001). Heat Sinks A heat sink is a device that absorbs heat generated by electronic components or chips. Among the different types of heat sinks, two-phase forced-convection cooling of high-heat-flux/high-power electronic devices is one of the most effective means of thermal management. This method becomes especially important alongside the ongoing trend toward miniaturization, and the increase in Figure 1.34 Flat miniature heat sinks (Hopkins et al., 1999). Chapter 1 Introduction to Transport Phenomena 93 power dissipation per unit surface area of modern electronic devices, which already has reached 300 W/cm² (Cao et al., 1996). Various two-phase miniature heat sinks have been investigated and presented in the literature to meet this challenging demand. Hopkins et al. (1999), for example, presented flat miniature heat sinks with enhanced inner surfaces ranging from smooth, to shallow and wide trapezoidal micro grooves, to deep and narrow rectangular grooves (Fig. 1.34). The heat transfer mechanism in the heat sink can be evaporation or/and boiling in the micro grooves. It is imperative that the heat flux applied to the heat sink does not exceed the critical heat flux (CHF); above the CHF, the temperature of the electronic device increases significantly and may result in failure of the heat sink. Micro Heat Pipes and Heat Spreaders In addition to the limitations on maximum chip temperature, further constraints may be imposed on the level of temperature uniformity in electronic components. The micro heat pipe is a very promising technology for achieving Figure 1.35 Schematic of the micro heat pipe (Khrustalev and Faghri, 1994). 94 Transport Phenomena in Multiphase Systems high local heat removal rates and uniform temperatures in computer chips. Micro heat pipe structures can be fabricated on the substrate surfaces of electronic chips by using the same technology that forms the circuitry. These thermal structures can be an integral part of the electronic chip and remove heat directly from the area where the maximum dissipation occurs. The micro heat pipe is defined as a heat pipe in which the mean curvature of the liquid vapor interface is comparable in magnitude to the reciprocal of the hydraulic radius of the total flow channel. Typically, micro heat pipes have convex but cusped cross-sections (for example, a polygon), with hydraulic diameters ranging from 10 to 500 m. Figure 1.35 presents cross-sections of a triangular micro heat pipe in which the corners act as a wick (Khrustalev and Faghri, 1994). The heat load is uniformly distributed among all corners. Near the evaporator end cap, if the heat load is sufficiently high, the liquid meniscus is depressed in the corner, and its cross-sectional area, as well as the radius of curvature of the free surface, is extremely small. Most of the wall is dry or is covered by a nonevaporating liquid film. In the adiabatic section, the liquid cross-sectional area is comparatively larger. The inner wall surface may be covered with a thin liquid film due to the disjoining pressure. At the beginning of the condenser, a film of condensate is present on the wick, and surface tension drives the liquid flows through this film toward the meniscus region. The possibility of liquid blocking the end of the condenser is also shown in Fig. 1.35. Attempts were also made to etch micro heat pipes directly into silicon and use them as thermal spreaders (Peterson, 1996; Benson et al., 1998). These heat pipes have hydraulic diameters on the order of 10 m, which are classified as “true” micro heat pipes. The performance characteristics of the micro heat pipes are different from those of conventional heat pipes. Figure 1.36 Heat spreader (Adkins et al., 1994). Chapter 1 Introduction to Transport Phenomena 95 One limitation of the micro heat pipe is that the individual heat pipe can transfer heat only in the axial direction. To overcome this drawback, microscale flat plate heat spreaders fabricated as an integral part of large-scale silicon wafers have been proposed (Benson et al., 1998; Adkins et al., 1994). The heat spreaders are capable of transferring heat in any direction across the wafer surface. Figure 1.36 shows a typical structure of a 5×5 mm2 flat heat spreader in a silicon multichip module substrate. A wick structure is incorporated into the heat spreader, and the heat transfer is achieved via evaporation of the working fluid charged into the spreader. 1.6.6 Microscale Phase Change Heat Transfer Rapid Melting/Resolidification of Metal Irradiated by an Ultrafast Laser When the laser pulse is reduced to a nanosecond (10-9 sec) or less, the heat flux of the laser beam can be as high as 1012 W/m2 when the metal surface is molten. When the femtosecond pulse laser is used, the laser intensity can be up to 1021 W/m2. During ultrafast laser-metal interaction, the laser deposits energy to the free electrons and the energy is further transferred to the lattice through electron-phonon interaction. If the laser pulse width is shorter than the time required for the electron and lattice to achieve thermal equilibrium (thermalization time), the electrons and lattices can no longer be treated as being Figure 1.37 Ultrafast laser surface melting of a metallic material (Wang and Prasad, 2000; Reprinted with permission from Elsevier). 96 Transport Phenomena in Multiphase Systems in thermal equilibrium (Grigoropoulos and Ye, 2000). The energy equations for the electrons and lattice must be specified separately and coupled through a coupling factor (see Section 6.10). In this case, the motion of the solid-liquid interface is no longer governed by the energy balance at the interface; instead, it is governed by the nonequilibrium kinetics of phase change. The solid phase during melting can be significantly overheated, while the liquid temperature during the resolidification stage can be significantly undercooled. A schematic diagram of short-pulsed laser surface melting of a metallic material is shown in Fig. 1.37 (Wang and Prasad, 2000). Explosive Boiling during Ultrafast Laser-Materials Interaction The material removal can be achieved by liquid-vapor phase change by using an ultrafast laser at an intensity higher than that discussed above. The material removal can be achieved by (Chen and Beraun, 2003): (a) normal evaporation, (b) normal boiling, or (c) explosive boiling (or phase explosion). Normal evaporation occurs on the liquid surface without nucleation and is significant only for long-pulsed lasers (pulse width greater than 1 ns). Normal boiling requires higher laser fluence and a sufficiently long laser pulse (> 100 ns) to allow heterogeneous bubble nucleation of the vapor bubble. For an ultrashort pulsed laser, a melted material on and underneath the laser-irradiated surface cannot boil because the time scale does not allow the necessary heterogamous nuclei to form. Instead, the liquid is superheated to a degree past the normal saturation temperature and approaching the thermodynamic critical temperature, Tcr. At the temperature close to the critical temperature, homogeneous vapor bubble nucleation takes place at an extremely high rate, which in turn results in the near-surface region of the irradiated materials being ejected explosively. This Figure 1.38 Frozen waves on the silicon surface (Craciun et al., 2002; Reprinted with permission from Elsevier) Chapter 1 Introduction to Transport Phenomena 97 process is referred to as explosive boiling or phase explosion. The explosion creates waves on the liquid surface, and the waves can be “frozen” upon rapid cooling of the surface. Figure 1.38 shows a high-magnification SEM image of a crater formed on a single crystal silicon surface by a single 5 ns laser pulse at 2 J/cm2 at a wavelength of 1064 nm (Craciun et al., 2002). The frozen waves can be clearly seen in the figure. 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What is the final temperature of the final mixture? 1.3. An ice skater moving at 8m/s glides to a stop. The ice in immediate contact with the skates absorbs the heat generated by friction and melts. If the temperature of the ice is 0 °C and the weight of the ice skater is 60 kg, how much ice melts? 1.4. A lead bullet with 30-g mass traveling at 600 m/s hits a thin iron wall and emerges at a speed of 300 m/s. Suppose 50% of the heat generated is absorbed by the bullet, and the initial temperature of the bullet is 20°C. Find the final phase and temperature of the lead bullet after the impact. The specific heat of the liquid lead can be assumed to be the same as that of the solid lead. 1.5. The Lennard-Jones potential depends mainly on the distance between molecules. Under what conditions is the Lennard-Jones potential for molecules not appropriate? Chapter 1 Introduction to Transport Phenomena 103 1.6. Assume an ideal two-component gas mixture A and B, and develop a relation for mole and mass fraction for A and B in terms of partial pressure and total pressure for the mixture. 1.7. Verify that the following equation is valid for a binary system: J1 J* =1 ρω1ω2 cx1 x2 1.8. Prove that the sum of mass fluxes by diffusion for a multicomponent mixture is zero. i =1 ¦ Ji = 0 N Show that the sum of mass flux relative to stationary coordinate axes for multicomponent mixtures is different from zero. 1.9. Show that the Maxwell-Stefan equation (1.103) for multicomponent system can be reduced to Fick’s law, eq. (1.99), for a binary system. 1.10. The diffusive heat and mass flux vectors in a multicomponent system were expressed by eqs. (1.64) and (1.115), respectively. What are the heat and mass fluxes across a surface whose normal direction is n? 1.11. Develop the mass flux relationship for component A, leaving from a solid or liquid wall made of component A only to a binary (Fig. P1.1) mixture of gases A and B in terms of a) mass fractions and mass density of component A, and b) molar fractions and molar concentration of component A. Describe the assumptions in arriving at your final result for each case. y L Mixture of gases A+B m′′ or n′′ A A Figure P1.1 1.12. Perform a scale analysis of forced convective heat transfer over a flat plate, as shown in Fig. 1.9, to obtain the order of magnitude of the boundary layer thickness and the Nusselt number. 1.13. A solid PCM with a uniform initial temperature at the melting point, Tm , is in a half space, x > 0. At time t = 0, the temperature at the boundary x = 0 is suddenly increased to a temperature, T0 , which is above the melting point of the PCM. Perform a scale analysis to obtain the order of magnitude of the location of the melting front. 104 Transport Phenomena in Multiphase Systems 1.14. Air with a velocity of 10 m/s, temperature 22 °C at 50% relative humidity, flows over a swimming pool (40 m by 20 m) along its length direction. The water temperature in the pool is 32 °C. Calculate the average heat loss due to (a) sensible heat and (b) latent heat of vaporization. 1.15. A full water container with a 1 m2 surface area is maintained at 60°C with surrounding air at 30 °C and 20% relative humidity. Calculate the convection and evaporation heat losses. 1.16. Determine the thickness of an ice layer on a large rectangular swimming pool of 7×10 m2 in winter when air blows over the pool at 16 km/h and -23 °C. Assume the pool temperature is 5°C and the inner ice surface temperature is at 0°C. The convection heat transfer coefficient at the bottom of the ice is 50 W/m2-K. Use Table 1.9 to calculate the convective heat transfer coefficient on the top of the ice layer. What is the surface temperature on the outer layer of ice? 1.17. A water droplet is placed in dry stagnant air of uniform temperature, T∞ . Show that (a) kair hAv ρ air Nu (T∞ − Tsat ) = D12Sh = Le (b) c p ,air (T∞ − Tsat ) ω1δ − ω1∞ 1 − ω1δ hAv What are the assumptions made to produce the results found in (a) and (b)? ω1δ − ω1∞ 1 − ω1δ 1.18. Compare the orders of magnitude of the Prandtl, Schmidt, and Lewis numbers for various gases and liquids. Can any conclusions be made? 1.19. Perform an Internet search with the phrase “multiphase heat transfer” and write a brief summary of what you find. 1.20. Plasma is regarded as a state of matter in addition to gas, liquid, and solid. Go to your library and use the Internet to find information about plasmas. How do you compare the properties of plasma with those of a gas, liquid and solid? 1.21. The thermal conductivities of most PCMs are very low, which is disadvantageous for thermal energy storage systems using PCMs. What improvements can be made for the latent heat thermal energy storage system shown in Fig. 1.17? Explain how your design will be better than the one shown. 1.22. The initial state of the metal powder bed for SLS is usually room temperature. It is proposed to preheat the metal powder bed to a temperature close to the melting point of the metal powder. Discuss the impact of preheating on the SLS process. Chapter 1 Introduction to Transport Phenomena 105 1.23. The absorptivity of the powder bed is defined as the ratio of the laser energy absorbed by the powder bed over the total laser power. For both CO2 (wavelength 10.6 μm) and YAG (wavelength 1.06 μm) lasers, find absorptivities of the following powder materials: Cu, Al, Fe, and Ni. Discuss which laser is preferable. 1.24. The temperature and dew point of a city in summer are 80 °F and 40 °F, respectively. What is the relative humidity? 1.25. A two-phase heat sink is a very effective device for electronic cooling. What are the advantages and disadvantages of a two-phase heat sink in comparison with a single-phase heat sink? 1.26. Give an example of two-phase flow from everyday life or industrial applications, and briefly discuss the physical phenomena involved in your example. 1.27. Laser welding is a process that uses a laser beam to join two metal workpieces together. Analyze the phase changes involved in the laser welding process. You can use your library and the Internet to find related information. 1.28. The working fluid for a high-temperature heat pipe is liquid metal, which is solid at room temperature. Qualitively analyze the phase change phenomena involved during the start-up and shut-down of the high temperature heat pipe. 1.29. Compare the operational principles of pulsating heat pipes with conventional heat pipes. 1.30. What is the advantage of a fuel cell over the conventional battery and thermodynamic cycle? Why is heat transfer important in a fuel cell? 106 Transport Phenomena in Multiphase Systems