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INTRODUCTION
1.1 Introduction
Heat and mass transfer or transport phenomena can be encountered in many applications ranging from design and optimization of traditional engineering systems, such as heat exchangers, turbine, electronic cooling, heat pipes, and food processing equipment, to emerging technologies in sustainable energy, biological systems, security, information technology and nanotechnology. While some of these examples aim at transferring large quantities of heat over small temperature differences, others involve heat and mass transfer 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, heat and mass transfer 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 theory of advanced heat and mass transfer relies on the familiar and basic 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 development of reliable and efficient algorithms for advanced heat and mass transfer requires the use of many familiar analytical and numerical tools such as control volume and 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 mass transfer rates – all issues that are relevant regardless of the number of phases present in the system. Traditionally, heat and mass transfer at the graduate level is taught in four separate courses: heat conduction, convective heat transfer, mass transfer, and radiation. The materials covered in these courses are rather extensive and some of them are even irrelevant. Graduate students are not given appropriate exposure to topics related to modern emerging technologies. There are, of course, excellent generalized undergraduate textbooks, as well as advanced graduate level books on single topics developed over the last two decades. Due to curriculum limitations however, and small faculty size in thermal science areas, a number of universities are offering single courses that cover all of the intermediate and
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advanced heat and mass transfer principals. Thus, we have to provide advanced, relevant materials in heat and mass transfer in a single volume for undergraduate senior and graduate students instead of relying upon several books. The features of this textbook are as follows: 1. All relevant advanced heat and mass transfer topics in single-phase heat conduction, convection, radiation, and multi-phase transport phenomena, are covered in a single textbook, and are explained from a fundamental point of view. 2. The book presents the generalized integral, differential, and average formulations for the governing equations of transport phenomena. 3. The book employs a top-down approach. For example, it emphasizes the basic physics of the problem by beginning with a general governing equation and reducing it for the particular problem. 4. Rather than being contained in an individual chapter, mass transfer is integrated throughout the book. 5. Modern applications of heat and mass transfer, e.g. nanotechnology, biotechnology, energy, material processing etc., are emphasized via examples and homework problems. 6. The foundations of the numerical approach are discussed, so as to ensure that the student understands the basis and limitations of these methods. 7. Topics which are lacking in most other books are integrated into colloquial; e.g. porous media, micro-scale heat transfer, and multi-phase, multi-component systems. 8. The book presents all forms of phase changes, including boiling, condensation, melting, solidification, sublimation, and vapor deposition from one perspective in the context of the fundamental treatment. 9. The molecular approach to describe the transport phenomena is also discussed, along with the connection between the microscopic and molecular approaches. Traditionally, three approaches were used to present transport phenomena: microscopic (differential), macroscopic (integral) and molecular level. However, most heat transfer textbooks place the emphasis on microscopic and/or macroscopic. With the importance of microscale heat and mass transfer 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. This chapter begins with a review of the concept of phases of matter and a discussion of molecular level presentation. This is followed by a review of transport phenomena with detailed emphasis on 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 modern practical applications are described, which require students to
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understand the operational principles of these physical systems and devices for further application in homeworks and examples in future chapters.
1.2 Physical Concepts
1.2.1 Sensible Heat
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 two pieces of metal together for a 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 our current understanding 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 4.18 joules (J) of work always equals 1 calorie (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)
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Figure 1.1 Schematic of Joule’s experiment demonstrating the mechanical equivalent of heat.
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, for water at 15 °C, c = 1.00 kcal/kg- C = 4.18 kJ/kg- 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 specific heat for a gas differs from that of a liquid and solid because the value of specific heat for gases depend upon 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 specific heat at constant volume, cv , and at constant pressure, c p , for gases are quite different. The relationship between these two values of specific heat for an ideal gas is given by (1.2) c p − cv = Rg where Rg, the gas constant, and is related to the universal gas constant, Ru, by Rg = Ru / M with M being the molecular mass of the ideal gas. The molecular mass and specific heat for some selected substances are listed in Table 1.1. Clearly, the values of specific heat 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 heat of liquids and solids at constant pressure are assumed to apply to all real processes.
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Table 1.1 Specific heats of different substances at 20 °C Substance Air Aluminum Carbon dioxide Copper Glass Hydrogen Ice (–5 °C) Iron Lead Marble Nitrogen Oxygen Steam (100 °C) Silver Mercury Water M (kg/kmol) 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 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
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 precipitation of rain have been observed by 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 a change in 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 the 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 understanding 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%
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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
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 produces an increase in temperature of the now-liquid, up to 100 °C. Continued heating of the liquid water at 100 °C does not yield any increase in 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
hs (kJ/kg)
14 105 335 25 88 184
Boiling point (°C) –183 78 100 1750 2193 5900
hv (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 hs . The latent heat of fusion for water is about 335 kJ/kg. The heat required to vaporize a liquid substance of unit
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mass is defined as the latent heat of vaporization, and it is represented by hv . 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 heat of vaporization for all materials is much larger than their latent heat 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 a change in phase, the heat absorbed or released can be expressed as the product of the mass quantity and the latent heat of the material. For liquid-vapor phase change, such as vaporization or condensation, the heat transferred can be expressed as q = mhv (1.3) where m is the mass of material changing phase per unit time. The latent heat of vaporization, hv , is the difference between the enthalpy of vapor and of liquid, i.e., hv = hv − h (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 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 at 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, m and mv . The change in mass of the liquid and of the vapor satisfies
dm dmv + = −m dt dt
(1.5) (1.6) (1.7)
The total volume of the liquid and vapor
V = m v + mv vv
remains constant during the phase change process. Thus
dmv dm v + vv = 0 dt dt
Combining eqs. (1.5) and (1.7) yields
vv dm = −m dt vv − v
(1.8)
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q
Figure 1.3 Liquid-vapor phase change in rigid tank with a relief valve.
v dmv = m dt vv − v
(1.9)
The first law of thermodynamics for the control volume is
dEcv = q − mhv dt
(1.10)
where the internal energy of the control volume is (1.11) Differentiating eq. (1.11) and considering eqs. (1.8) and (1.9), one obtains
Ecv = m e + mv ev
v e − v ev dEcv = −m v dt vv − v
(1.12)
Substituting eq. (1.12) into eq. (1.10) and considering eq. (1.4) yields
v e − v ev q = m hv + h − v vv − v
(1.13)
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 = mhs (1.14)
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where hs 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 heat and mass transfer, and therefore they are briefly reviewed 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 which 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 expressed in atomic mass units. One atomic mass unit is equal to −27 1.6605402 × 10 kg. This mass is 1/12 of 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
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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. 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 ˚C 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 10−9 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 500 m/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 direction of the velocity changes 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.
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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 obey 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 = v2 = w 2 . Information concerning mean molecular velocity, frequency, mean free path, and density number with the above assumptions can be obtained using simple kinetic theory (Berry et al., 2000; Lide, 2009). The average magnitude of the molecular velocity is given by simple kinetic theory
c=
where kb is the 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 4
8kbT πm
(1.15)
(1.16)
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
λ=
1 2πσ 2N
(1.17)
where σ is the molecular diameter. The relaxation time, τ, which is the average time between two subsequent collisions, is: λ τ= (1.18)
c
The collision rate, τ −1 , is the average number of collisions an individual particle undergoes per unit time. After the last collision with other molecules, the molecule travels an average distance of 2λ / 3 before it collides with the plane. A
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table for the particle diameters, mean free path, mean velocity and relaxation time (τ ) between molecular collisions is given in Table 1.3 for some gases at 25°C and atmospheric pressure.
Table 1.3 Kinetic properties of gases at 25 °C and atmospheric pressure
Gas Air Ar CH4 C2H4 C6H6 CO2 Cl2 H2 He N2 NH3 O2 SO2
σ ×1010 (m) 3.66 3.58 3.82 4.52 5.27 4.53 4.40 2.71 2.15 3.70 4.32 3.55 4.29
λ × 108 (m)
6.91 7.22 5.25 3.74 16.2 4.51 2.99 12.6 20.0 6.76 4.97 7.36 2.99
c (m/s)
467 397 627 474 284 379 298 1769 1256 475 609 444 313
τ (ps)
148 182 84 78 569 119 100 71 159 142 82 166 96
The mass flux of molecules in one direction at a point in a gas is given as (Tien and Lienhard, 1979): 1/2 1/ 2 N cm kT k Tm ′′ (1.19) =N m b =N b mmolecules = 4 2π m 2π 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
p= 1 c2 Nm 3 V
(1.20)
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.
c2 = 1 N
c
n =1
N
2 n
(1.21)
The average of the square of the molecules’ velocity is related to its three components by
1 u 2 = v2 = w 2 = c 2 3
(1.22)
The average kinetic energy of a molecule is defined as
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E=
1 mc 2 2
(1.23)
Substituting eq. (1.23) into eq. (1.20) yields
pV = 2 NE 3
(1.24)
The monatomic ideal gas also satisfies the ideal gas law, i.e., (1.25) where Ru = 8.3143 kJ/kmol-K is the universal gas constant, which is the same for all gases. Combining eqs. (1.24) and (1.25) yields
pV = nRu T E= 3 Ru T 2 NA
(1.26)
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 (1.26) can also be rewritten as
E= 3 kbT 2
(1.27)
where the Boltzmann constant is
kb = Ru 8.3143 = = 1.38 × 10−23 J/K 23 N A 6.022 × 10 cv =
(1.28)
The specific heat at constant volume, cv, is given by kinetic theory as
3 kb 2m
(1.29)
From eq. (1.26) 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 (see Fig. 1.1), the paddle wheel 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 of all the energy of all the molecules in an object. The internal energy of an ideal gas equals the sum of all the kinetic energies of all its atoms. This sum can be expressed as the total number of molecules, N, times the average kinetic energy per atom, i.e.,
E = NE =
3 NkbT 2
(1.30)
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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. (1.26)], while internal, or thermal, energy is the total energy of all of the molecules in the object [see eq. (1.30)]. If two objects with equal mass 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. Similarly, viscosity, μ, thermal conductivity, k, and mass self-diffusion coefficient, D11, can be obtained using simple kinetic theory (see Problem 1.5 and 1.6). Following are the results:
μ=
k= D11 =
2 3π
3 /2
mkbT
σ2
3 kbT m 3 kb T 3 m
(1.31) (1.32) (1.33)
1
π 3/ 2σ 2
2
Equations (1.31) – (1.33) can be used for binary systems with components 1 and 2, if σ and m are replaced by (σ1 + σ 2 ) / 2 and m1m2 / (m1 + m2 ) , respectively. The significance of the above results should not be overlooked even though some simplified assumptions were used in their developments. Equations (1.31) and (1.32) for μ and k are independent of pressure for a gas. This is proven experimentally for pressure up to 10 atmospheric pressures. According to this prediction, viscosity and thermal conductivity are proportional to 1/2 power of absolute temperature while the diffusion coefficient is proportional to 3/2 power of absolute temperature. To better model the temperature effects, one needs to replace the rigid sphere model and the mean free path concepts and use the Boltzmann equation to describe the nonequilibrium phenomena accordingly. It is important to point out that equations described above using simple kinetic theory 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 energy, and their internal energy at a given temperature will be greater than that of a monatomic gas at the same temperature.
3π 3/2σ 2 P
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Figure 1.4 Modes of molecular kinetic energy: (a) rotational energy, (b) vibrational energy.
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. As noted above, the simple kinetic theory of ideal gas was based on the mean free path concept. While it provides the first order of magnitude approximation for several key transport phenomena properties, the simple kinetic theory is limited to local equilibrium and therefore it is for time durations much larger than relaxation time. The advanced kinetic theory is based on the Boltzmann transport equation, which is presented in the next section. Example 1.1: Calculate the time between subsequent collisions (relaxation time), τ , mean free path, λ , and the number of collisions each molecule experiences per second for air at 25 °C and 1 atm. Using the speed of sound show why we can smell odor very frequently even when we may be far from the source. Solution: The number density of air molecule is
N=
p 1.013 × 105 Pa = = 2.46 × 1025 molecules/m3 kbT (1.381× 10−23 J/K) × (298.15K)
The mean distance between molecules is L = N -1/3 = 3.4 × 10−9 m = 3.4nm The diameter of molecules, according to Table 1.3, is σ = 0.366nm = 3.7 × 10−10 m
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The mean free path can be obtained from eq. (1.17), i.e.
λ=
1 2πσ N
2
=
1 2π × (3.7 ×10
−10 2
) × 2.46 × 10
25
= 66 × 10−9 m = 66nm
The average magnitude of the molecular velocity can be obtained from eq. (1.15), i.e.,
c= 8kbT 8 × 1.381× 10−23 × 298.15 = = 465m/s πm π × 28.97 ×1.66 × 10-27
where the molecular mass of air is 28.97 atomic mass units (each atomic mass unit is 1.66 × 10-27 kg). The relaxation time is λ τ = = 0.14ns
c
The speed of sound in air is
Vsound = γ RgT = 345m/s
where γ = c p / cv = 1.4 for air. The speed of sound is less than
the
molecular velocity of 465m/s. The number of collisions for each molecule per second is τ −1 = 7 billion/s Since the average speed of molecules is higher than the speed of sound, and the number of collisions that each molecules experiences is high, it does not take very long for the nose to detect odor.
1.3.3 Intermolecular Forces and Boltzmann Transport Equation
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.
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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 dispersion forces, which are caused by transient dipoles in nominally-neutral molecules or atoms. An accurate representation of the intermolecular 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 function provides a satisfactory empirical expression for nonpolar molecules:
φ (r ) = 4ε
r 12 r 6 0 − 0 r r
(1.34)
where ε is a constant and r0 is a characteristic length. 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 until a point is reached after which the repulsive force dominates, and 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 and the attractive force dominates. The Lennard-Jones potential approaches zero when the molecular distance becomes very large. When the Lennard-Jones potential is at minimum, the following condition is satisfied: dφ (rmin ) =0 (1.35)
dr
Substituting eq. (1.34) into eq. (1.35), one obtains,
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φ
Molecules repulse one another for r < rmin
F
r ←⎯ → ←⎯ • ⎯ • ⎯⎯ →
F = −∇φ (r )
F
0
r0
r
−ε
Molecules attract another for r > rmin
one
Figure 1.5 Lennard-Jones 6-12 potential vs. distance between two spherical, nonpolar molecules.
(1.36) For typical gas molecules, r0 ranges from 0.25 to 0.4 nm, which result a range from 0.28 to 0.45 nm for rmin. The Lennard-Jones potential at this point is φ (rmin ) = −ε (1.37) 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. 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 (i) in the system is described by Newton’s second law, i.e.,
j =1( j ≠ i )
rmin = 21/6 r0 ≈ 1.12r0
N
Fij = mi
d 2 ri dt 2
, i = 1, 2,3...n
(1.38)
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where mi and ri are the mass and position of the ith molecule in the system. In arriving at eq. (1.38), 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 Lennard-Jones potential between the ith and jth molecules is obtained by
r φij = 4ε 0 rij
24ε rο
r − 0 rij
12
6
(1.39)
The force between the ith and jth molecules can be obtained from
Fij = −∇φij = rο rij
13 7 r rij − ο rij rij
(1.40)
where rij is the distance between the ith and jth molecules. The transport properties can be obtained by the kinetic theory in the form of very complicated multiple integrals that involve intermolecular forces. For a nonpolar substance that Lennard-Jones potential is valid, these integrals can be 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= (1.41) 8 πσ 2 Ω D ρ
μ=
k=
5 π mkbT 16 πσ 2 Ω μ
(1.42) (1.43)
25 π mkbT cv 32 πσ 2 Ωk
where σ is collision diameter, and cv in is the molar specific heat under constant volume. The dimensionless collision integrals are related by Ωμ = Ωk ≈ 1.1ΩD and are slow varying functions of kbT/ε ( ε is a characteristic energy of molecular interaction). If all molecules can be assumed to be rigid balls, all collision integrals will become unity. It follows from eqs. (1.41) – (1.43) 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 C.3). The transport properties at system length scales 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 gases, 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
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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 dc about c is fi (c, x, t )dcd x . The evolution of the velocity distribution function with time can be described using the Boltzmann equation
Dfi ∂fi = + c ⋅ ∇ x f i + a ⋅ ∇c f f = Ωi ( f ) (1.44) Dt ∂t where ∇ x and ∇c are the ∇ operator with respect to x and c, respectively (see
Appendix G), 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 to be a continuity equation in a six-dimensional position-velocity space ( x and c ). The velocity distribution function is related to the number density (number of particles per unit volume) by (1.45) f i (c, x, t )dc = N i (x, t )
Note that the density is ρ (x, t ) = mN (x, t ) where m is the mass of the particle. The total number of particles, N, inside the volume, V, as a function of time is
N (t ) =
f (c, x, t )dcdx
V c
(1.46)
In thermodynamic equilibrium, f is independent of time and space, i.e., f (c, x, t ) = f (c) . It can be demonstrated that the stress tensor, eq. (1.60), heat flux, eq. (1.71), and diffusive mass flux, eq. (1.118) can be obtained from the solution of the velocity distribution function fi. More detailed information about the Boltzmann equation and its applications related to transport phenomena in multiphase systems can be found in Faghri and Zhang (2006). Most macroscopic transport equations, such as Fourier’s law of conduction, Navier-Stokes equation for viscous flow, or the equation of radiative transfer for photons and phonons can be developed from the Boltzmann transport equation using local equilibrium assumptions. More detailed information concerning formulation and solution techniques of the Boltzmann equation can be found in Tien and Lienhard (1979) and Ceracignani (1988). 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; Zhang and Chen, 2007; Zhang, 2007).
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,
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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 a different kind 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 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. The internal energy of a substance with molecules
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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: S = −kb ln ( P ) (1.47) where kb is the Boltzmann constant, 1.3806 × 10−23 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:
Δs = Δh = constant T
(1.48)
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. (1.48) depends on the particular phase change process but is independent of the substance. For vaporization and condensation, Trouton’s rule is applicable
sv − s = hv 83.7J/(mol-K) Tsat
(1.49)
while Richards’ rule is valid for melting and solidification
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s − ss =
hs 8.37J/(mol-K) Tm
(1.50)
1.4 Fundamentals of Momentum, Heat and Mass Transfer
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 characteristic dimension, L, is small compared to the molecular mean free path, λ, the traditional Navier-Stokes equation and the energy equation based on the continuum assumption have failed to provide accurate results. The continuum assumption also fails when the gas is at very low pressure (rarefied). The continuum assumption may also not be valid in conventional sized 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 miniature or micro heat pipes, whose dimensions may be extremely small. The continuum criterion is usually expressed in terms of the Knudsen number λ Kn = (1.51)
L
Based on the degree of rarefaction of gas or the dimension of the system, the flow regimes in various devices can be classified into four regimes:
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1. Continuum regime ( Kn < 0.001 ). The Navier-Stokes and energy equations are valid (with no-slip/no jump boundary conditions). 2. Slip flow regime ( 0.001<Kn < 0.1 ). The Navier-Stokes and energy equations can be used with the application of slip or jump boundary conditions, i.e., allowing non-zero axial fluid velocity near the wall of the object. 3. Transition regime ( 0.1<Kn < 10 ). The Navier-Stokes equation is not valid, and the flow must be solved using molecular based models such as the Boltzmann equation or Direct Simulation Monte Carlo (DSMC). 4. Free molecular flow regime ( Kn>10 ). The collision between molecules can be neglected and a collisionless Boltzmann equation can be used. In the slip flow regime, the slip boundary condition refers to circumstance when the tangential velocity of the fluid at the wall is not the same as the wall velocity. Temperature jump is similarly defined as when the temperature of the fluid next to the wall is not the same as the wall temperature. The mean free path, eq. (1.17), for dilute gases based on the kinetic theory can be rewritten in terms of temperature and pressure
λ=
1.051kbT 2πσ 2 p
(1.52)
The transition density under which the continuum assumption is invalid can be obtained by combining eqs. (1.51) and (1.52) and Kn = 0.001, i.e.,
ρtr =
1.051kb 2πσ 2 Rg DKn
(1.53)
where the ideal gas equation of state, p = ρ RgT 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 combined with the equation of state:
Ttr =
h 1 psat 1 exp − v − ρ Rg Rg Ttr Tsat
(1.54)
where psat and Tsat are the saturation pressure and temperature, hv is the latent heat of vaporization, and the vapor density, ρ, is given by eq. (1.53). Equation (1.54) can be rewritten as Ttr ρ Rg hv 1 1 ln − (1.55) + =0
Rg Ttr Tsat and solved iteratively for Ttr using the Newton-Raphson/secant method. The psat
transition vapor temperature is the boundary between the continuum and noncontinuum regimes.
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1.4.2 Momentum, Heat and Mass Transfer
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 basic undergraduate-level knowledge of transport phenomena as applied to single-phase systems, as well as the associated thermodynamics, fluid mechanics, and heat transfer. Momentum Transfer 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 is one-dimensional since velocity components in the x- and z-directions are zero. The constitutive relation for Couette flow can be expressed as
τ yx = − μ
du dy
(1.56)
where τ yx is the shear stress ( N/m 2 ) on fluid, μ is the dynamic viscosity ( N-s/m 2 ), which is a fluid property, and du / dy is the velocity gradient in the ydirection, also known as the rate of deformation. If the shear stress, τyx, and the rate of deformation, du / dy , have a linear relationship, as shown in eq. (1.56), the fluid is referred to as Newtonian and eq. (1.56) is called Newton’s law of viscosity. It is found that the resistance to flow for all gases and liquids with
U
u(y) y x
Figure 1.6 Couette flow.
τyx
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molecular mass less than 5000 is well presented by eq. (1.56). For nonNewtonian fluids, such as polymeric liquids, slurries, or other complex fluids, for example in biological applications, such as blood or operation of joints, dragreducing 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 a velocity gradient in a fluid. For a three-dimensional fluid flow problem, the stress τ ′, is a tensor of rank two with nine components. It can be expressed as the summation of an isotropic, thermodynamic stress, − pI , and a viscous stress, τ : τ ′ = − pI + τ (1.57) where p is thermodynamic pressure, and I is the unit tensor defined as
1 0 0 I = 0 1 0 0 0 1
(1.58)
In a Cartesian coordinate system, the viscous stresses are τ xx τ xy τ xz
τ = τ yx τ yy τ yz τ zx τ zy τ zz
(1.59)
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 ).
z
τzz τzx τxz τxy τzy τyz τyy τyx
y
τxx
x
Figure 1.7 Components of the stress tensor in a fluid
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The viscous stress tensor can be expressed by Newton’s law of viscosity:
2 τ = 2μ D − μ (∇ ⋅ V )I 3
(1.60)
where ∇ ⋅ V is the divergence of the velocity (see Appendix G). 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 G).
∂u ∂x 1 ∂v ∂u 1 T D = ∇V + ( ∇V ) = + 2 2 ∂x ∂y 1 ∂w ∂u + 2 ∂x ∂z
T
1 ∂u 2 ∂y
+
∂v ∂x
1 ∂u 2 ∂z 1 ∂v 2 ∂z
+ +
∂w ∂x ∂w
∂v ∂y 1 ∂w ∂v + 2 ∂y ∂z
∂y
(1.61)
∂w ∂z
where ( ∇V ) is the transverse tensor of ∇V as defined in Appendix G. For example, in a Cartesian coordinate system, the normal and shear viscous stresses can be expressed as
τ xx = 2 μ
∂u 2 ∂u ∂v ∂w − μ + + ∂x 3 ∂x ∂y ∂z
∂u ∂v + ∂y ∂x
(1.62) (1.63)
τ xy = μ
For one-dimensional flow in Fig. 1.6, eq. (1.63) is reduced to eq. (1.56). The stress-strain rate relationships in eqs. (1.56) and (1.60) are valid for laminar flow only. For turbulent flow, eqs. (1.56) or (1.60) 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 xi μi μ= (1.64) N xiφij i =1
j =1
where xi is the molar fraction of component i and
M 1 φij = 1 + i Mj 8
−1/2
1 + μi μj
1/2
Mj Mi
1/4
2
(1.65)
where Mi is the molecular mass of the ith species. Equation (1.64) 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).
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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 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
′′ qy = − k dT dy
(1.66)
′′ where qy is the heat flux along the y-direction, i.e., the heat transfer rate in the ydirection per unit area (W/m2), and dT / dy (K/m) is the temperature gradient. The proportionality constant k is thermal conductivity (W/m-K) and is a property of the medium. For heat conduction in a multidimensional isotropic system, eq. (1.66) can be rewritten in the following generalized form: q′′ = −k∇T (1.67)
y
T2 L T(y)
T1
Figure 1.8 One-dimensional conduction.
T
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where both the heat flux and the temperature gradient are vectors, i.e.,
′′ ′′ ′′ q′′ = iqx + jqy + kqz
(1.68)
While the thermal conductivity for isotropic materials does not depend on the direction, it is dependent on direction for anisotropic materials. Unlike isotropic material whose thermal conductivity is a scalar, the thermal conductivity of anisotropic material is a tensor of the second order:
k xx k = kyx kzx
kxy kyy kzy kxz kyz kzz
(1.69)
and eq. (1.67) will become: (1.70) Equation (1.67) or (1.70) 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).
q′′ = −k ⋅∇T + q′′ = −k ⋅∇T
h J
i =1
N
ii
+ cRuT
i =1
N
xi x j DiT ρi Dij j =1( j ≠ i )
N
Ji J j − ρi ρ j
(1.71)
where J i is diffusive mass flux relative to mass-averaged velocity, which will be discussed in the latter part of this section. The second term on the right-hand side represents the interdiffusional convection term, which is not zero even though
J
i =1
N
i
th = 0 . hi is partial enthalpy (J/kg) for the i species. The third term on the
right-hand side is the contribution of the concentration gradient to the heat flux, which is referred to as the diffusion-thermo or Dufour effect. c is the molar concentration (kmol/m3) of the mixture, Ru is the universal gas constant, xi and xj are molar fractions of the ith and jth components respectively, 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 the Maxwell-Stefan diffusivity, which is related to the multicomponent Fick diffusivity, ij , defined in eq. (1.113). A binary system gives
(1.72) ω1ω2 12 where ω1 and ω2 are mass fractions of components 1 and 2, respectively. For a ternary system:
D12 = D12 = x1 x 2
1233 − 13 23 ω1ω2 12 + 33 − 13 − 23
x1 x 2
(1.73)
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 right-
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hand side of eq. (1.71) is usually important for multicomponent diffusion systems. While the Dufour energy flux represented by the third term on the righthand side of eq. (1.71) is negligible for many engineering problems, it may become important for cases with a very large temperature gradient.
Table 1.5 Typical values of mean convective heat transfer coefficients
Mode
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 of a 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
Forced convection
Free convection
(ΔT = 20 C) )
Evaporation Condensation of water at 1 atm Boiling of water at 1 atm Natural convectioncontrolled melting and solidification
o
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 rate of heat transfer: q′′ = h (Tw − T∞ ) (1.74) 2 where h is the convective heat transfer coefficient (W/m -K), which depends on many factors including fluid properties, flow velocity, geometric configuration, U∞ , T∞ U∞ T∞
T(x,y) q"
u(x,y)
Figure 1.9 Forced convective heat transfer.
Tw
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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:
Nu = hL k
(1.75)
where L and k are characteristic length and thermal conductivity of the fluid, respectively. The third mode of heat transfer is radiation. The transmission of thermal radiation does not require the presence of a propagating medium and, therefore 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 μm. Emission can be from a solid surface as well as from a liquid or gas. Thermal radiation may be considered to be the propagation of electromagnetic waves or alternately as the propagation 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 in the energy states of electrons, as well as changes in vibrational or rotational energy of molecules or atoms, all of these radiant energies travel at the speed of light. The wavelength λ of radiative emissions is related to their frequency, ν , by λν = c (1.76) 8 where c is the speed of light and has a value of 2.998 × 10 m/s in a vacuum. A quantitative description of the mechanism of 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 (1.77) where h = 6.626068×10-34 m2-kg/s is 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
5
c1 c2 /( λT )
− 1)
(1.78)
where c1 = 3.742 × 10−16 W-m 2 and c2 = 1.4388 × 10−2 m-K are radiation constants. The unit of the surface temperature T in eq. (1.78) is K. The emissive power for a blackbody, Eb (W/m2), is
Eb =
∞
0
Eb ,λ d λ
(1.79)
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Figure 1.10 Radiation heat transfer between a small surface and its surroundings.
Substituting eq. (1.78) into eq. (1.79), Stefan-Boltzmann’s law is obtained Eb = σ SB T 4 (1.80) where σ SB = 5.67 × 10−8 W/m 2 -K 4 is the Stefan-Boltzmann constant. For a real surface, the emissive power is obtained by E = ε Eb (1.81) where ε 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 ε, and temperature Tw, and a much larger surface surrounding the small surface (see Fig. 1.10). If the temperature of the surroundings is Tsur, the heat transfer rate per unit area from the small object is obtained by 4 4 q′′ = εσ SB (Tw − Tsur ) (1.82) For a detailed treatment of radiation heat transfer, including radiation of nongray surfaces and participating media, 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. It can occur only in a liquid or gas.
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The species concentration in a mixture, ρi, 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=
ρ
i =1
N
i
(1.83)
The concentration of the ith species can also be represented by the mass fraction of the ith species, defined as ρ ωi = i (1.84) ρ It follows from eq. (1.83) that
ω
i =1
N
i
=1
(1.85)
The concentration of the ith species can also be represented as 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 ρ ci = i (1.86)
Mi
The molar concentration of the mixture equals the sum of the molar concentrations of all N species, i.e.,
c=
c
i =1
N
i
(1.87)
The molar fraction of the ith species is defined as
xi = ci c
(1.88)
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 (1.88) leads to
x
i =1
N
i
=1
(1.89)
The mean molecular mass of the mixture can be expressed as ρN
M= c
=
x Mi
i i =1
(1.90)
The mass fraction is related to the molar fraction by
ωi =
xi M i
j =1
N
=
xjMj
xi M i M
(1.91)
The molar fraction is related to the mass fraction by
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Amir Faghri, Yuwen Zhang, and John Howell
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xi =
ωi / M i
ω
j =1
N
(1.92)
j
/ Mj
Mass diffusion of the ith 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=
ρ Vi
i i =1
N
ρ
=
ω Vi
i i =1
N
(1.93)
which demonstrates that the local mass flux due to diffusion, ρ V , is equal to the
summation of mass flux for each species,
N
ρ Vi .
i i =1
N
The molar-averaged velocity can be defined in a similar manner:
V* =
c Vi
i i =1
c
=
x Vi
i i =1
N
(1.94)
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 i (1.95) m′′ = ρi Vi i (1.96) n′′ = ci Vi The fluxes defined in eqs. (1.95) and (1.96) are related by
i n′′ = i m′′ Mi
(1.97)
Applying eqs. (1.95) and (1.96) into eqs. (1.93) and (1.94), the total mass flux and molar flux are obtained.
m′′ = m′′ = ρ V
i i =1 N N
(1.98) (1.99)
n′′ =
n′′ = cV
i i =1
*
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
(1.100) (1.101) (1.102)
According to eqs. (1.93) and (1.94), we have
i =1
N
Ji =
J
i =1
N
* i
=0
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Although any one of the four fluxes defined in eqs. (1.95) – (1.96) and (1.100) – (1.101) 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. (1.100) is preferred because the mass-averaged velocity, V , is the velocity used in the momentum and energy equations. On the other hand, eq. (1.101) is preferred for the multicomponent system with constant molar density, c, resulting from constant pressure and temperature. According to eqs. (1.95)–(1.101), the following relationships between different fluxes are valid:
i m′′ = ρi V + J i = ωi i n′′ = ci V* + J* = xi i m′′ + J
j j =1 N N i
(1.103) (1.104)
n′′ + J
i i =1
* i
For a binary system that is uniform in 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 (1.105) or alternatively * J1 = −cD12 ∇x1 (1.106) 2 where D12 is binary diffusivity (m /s). The mass and molar flux relative to a stationary coordinate axes are ′′ ′′ 2 m1 = ω1 (m1 + m ′′ ) − ρ D12 ∇ω1 (1.107) ′′ ′′ 2 n1 = x1 (n1 + n′′ ) − cD12 ∇x1 (1.108) Equation (1.107) is widely applied in binary systems with constant density, while eq. (1.108) is more appropriate for systems with constant molar concentration. It should be noted that from eqs. (1.107) and (1.108) the absolute ′′ ′′ fluxes of species ( m1 or n1 ) for a binary system can always be presented as a summation of two parts: one part due to convection [the first term in eqs. (1.107) and (1.108)], and another part due to diffusion [the second term in eqs. (1.107) and (1.108)]. 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 species 1 at the surface (y = 0) is ρ D12 ∂ω1 ′′ (1.109) m1y = − 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 following Maxwell-Stefan relation can be used to relate the molar fraction gradient of the ith component and the molar flux:
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y
L
ω1 ( y )
0
′′ m1 y
Figure 1.11 One-dimensional mass diffusion.
ω1
∇x i = −
j =1( j ≠ i )
N
xi x j Dij
(Vi − V j )
1 i j x j n′′ − xi n′′ , i = 1, 2, , N − 1 =− cDij j =1( j ≠ i )
N
(1.110)
(
)
where Dij is the binary diffusivity from species i to species j. Equation (1.110) was originally suggested by Maxwell for a binary mixture based on kinetic theory and was extended to diffusion of gaseous mixtures of N species by Stefan. For an N-component 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 N ωi J j − ω j J i M ∇ωi + ωi ∇(ln M ) = , i = 1, 2, , N − 1 (1.111) ρ j =1( j ≠ i ) Dij M j
where M is the molar-averaged molecular mass of the mixture. Although eqs. (1.110) and (1.111) were originally developed for low-density gaseous mixtures, it has been shown that they are also valid for dense gases, liquids, and polymers, except that Dij should be replaced by the multicomponent Maxwell-Stefan diffusivity Dij .
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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
ij j =1
N
j
, i = 1, 2, , N
(1.112)
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, where DiT is thermal diffusion coefficient. The second term is diffusion caused by all other driving forces, including concentration gradient, pressure, and body force, where ij is the multicomponent Fick diffusivity. ij is obtained from
ˆ ωiij = Dij −
n =1( n ≠ i )
N
ˆ ωn Din
(1.113)
where
ˆ Dij =
ωi ω j
xi x j
Dij
(1.114)
The multicomponent Fick diffusivities, ij , are symmetric ( ij = ji ) and satisfy
ω
i i =1
N
ij
=1.
Equation (1.112) can be rearranged to express the driving force, di, in terms of mass flux, i.e.,
di = − xi x j DiT DT j ∇(ln T ) − ρi ρj Dij j =1( j ≠ i ) N xi x j J i J j − − , i = 1, 2, , N Dij ρi ρ j j =1( j ≠ i )
N
(1.115)
which are referred to as generalized Maxwell-Stefan equations. The diffusional driving force di is obtained by
cRuT
ρi
1 gi hi 1 1 di = T ∇ ∇ ln T − ∇p − Xi + + ρ ρ T Mi Mi
ρ X
j j =1
N
j
(1.116)
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. For an ideal gas mixture, the generalized driving force becomes
cRu T d i = ∇pi − ωi ∇p − ρi Xi + ωi
ρ X
j j =1
N
j
(1.117)
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
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because X j = g for any component. The thermal (Soret) diffusion was included in the first term in eq. (1.112). Substituting eq. (1.117) into eq. (1.112) and after some manipulations, the final form of mass flux in the Fick form is obtained as (Curtiss and Bird, 1999) N ρi N J i = − DiT ∇ ln T + ij ∇pi − ω j ∇p − ρ j X j + ω j ρn Xn (1.118)
cRuT
j =1
n =1
Similarly, substituting eq. (1.117) into eq. (1.115), the final form of the Maxwell-Stefan equations becomes
cRuT xi x j J j J i − = ∇pi − ωi ∇p − ρi Xi Dij ρ j ρi j =1( j ≠ i )
N
xi x j DT DiT j ∇(ln T ) , i = 1, 2, , N ρ j X j − cRuT +ωi − ρj ρi Dij j =1 j =1( j ≠ i )
N
N
(1.119)
For dilute monatomic gas mixtures, the Maxwell-Stefan diffusivities can be approximated by binary diffusivity, i.e., Dij ≈ Dij and eq. (1.119) is preferred over eq. (1.118) because ij strongly depends on the mass concentrations. Equation (1.119) reduces to eq. (1.110) for the gaseous mixture with uniform temperatures and pressures. The multicomponent thermodiffusivity for species i is expressed as N ρ Mi M j T DiT = Dij kij (1.120) 2
j =1( j ≠ i )
M
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 M1 M 2 T DiT = D12 k12 (1.121) 2
M
The thermal diffusion ratio in a binary system depends on both temperature and concentration, and selected values for liquids and gases are given in Table 1.6. 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. (1.118) can be simplified to get the diffusive mass flux in this case, the multicomponent Fick diffusivities, ij , are strongly dependent on the concentration as evidenced by eqs. (1.113) and (1.114). An alternative approach that utilizes eq. (1.119) was developed in Faghri and Zhang (2006). With gravity as the only body force and the total pressure approximately constant, then eq. (1.119) can be represented in terms of mass fraction by a matrix.
38 Advanced Heat and Mass Transfer
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Table 1.6 Thermal diffusion ratios for selected liquids and low-density gases (Bird et al., 2002)
Components C2H2Cl4 – n-C6H14 C2H4Br2 – C2H4Cl2 C2H2Cl4 – CCl4 CBr4 – CCl4 CCl4 – CH3OH CH3OH – H2O cyclo-C6H12 – C6H6 Ne-He Gases N2 – H2 D2 –H2
T(K)
298 298 298 298 313 313 313 330 264 327
x1
0.5 0.5 0.5 0.09 0.5 0.5 0.5 0.2 0.6 0.294 0.775 0.10 0.50 0.90
T k12
Liquids
1.08 0.225 0.060 0.129 1.23 -0.137 0.100 0.0531 0.1004 0.0548 0.0663 0.0145 0.0432 0.0166
[J] =
ρ
M
A −1B −1 [∇ω ] − DT ∇ ln T
(1.122)
The components of matrices A and B can be found below. N ωj ωi
Aii = − + M i M N DiN j =1( j ≠ i ) M i M j Dij 1 1 Aij = ωi − M i M j Dij M i M N DiN Bii = (1 − ωi ) M i + ωi M N
Bij = ωi M N − M j
(1.123) (1.124) (1.125) (1.126)
(
)
Equation (1.122) is in a form that can be very easily programmed and it can be simplified for a variety of cases. For a binary mixture – a simplest mixture, A and B are both a single value.
A=− B = ω2 M1 + ω1 M 2 =
1 M1 M 2 D12
(1.127) (1.128)
x2 M 2 xM MM M1 + 1 1 M 2 = 1 2 M M M
Therefore, the diffusion mass flux of a binary mixture is: J i = − ρ D12∇ωi − DiT ∇ ( ln T ) (1.129) 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
M1 0 B≈ 0 0 M2 0 0 M N −1 0
(1.130)
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and the A matrix is approximately
( M M D )−1 1 N 1N 0 A≈ 0 0
( M 2 M N D2 N )−1
…
0 −1 0 ( M N −1 M N DN -1, N ) … 0
(1.131)
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: J i = − ρ D1N ∇ωi − DiT ∇ ( ln T ) (1.132) 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 MMD ω J1 = − ρ D12∇ω1 + 1 2 12 1 ∇p p (1.133) M1 M 2 D12ω1ω2 T
+ Ru T ( X1 − X 2 ) − D1 ∇(ln T )
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, α= k/(ρcp). The binary diffusivity at 1 atm for selected gases is listed in Tables E.1-E.3, Appendix E. It should be pointed out that the pressure diffusion, body force diffusion, and Soret diffusion represented by the second to fourth terms in eq. (1.133) is negligible for most applications. Similarly one can reduce the generalized Maxwell-Stefan equation (1.119) to the following form by neglecting pressure, body force, and thermal diffusion effects and assuming the mean molecular mass is constant [see. eq. (1.111)]: N ωi J j − ω j J i M ∇ωi = , i = 1, 2, , N − 1 (1.134) ρ j =1( j ≠ i ) Dij M j
In order to solve for the mass flux, J, eq. (1.134) can be rearranged to get
Ji = −
ρD
j =1
N −1
eff ,ij ∇ω j
, i = 1, 2, , N − 1
(1.135)
where
[ Deff ,ij ] = F −1
Fii =
(1.136)
ωk M
Dik M k
ωi M
DiN M N
+
k =1( k ≠ i )
N
(1.137)
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1 1 Fij = ωi M − Dij M j DiN M N
, i ≠ j
(1.138)
The discussions thus far have been limited to diffusion only, when bulk velocity of the multicomponent fluid is not zero, the second mode of mass transfer, convective mass transfer, also play a significant role. The mass flux due to convection may be expressed in a manner analogous to eq. (1.74): ′′ m1 = hm ( ρ1, w − ρ1,∞ ) = ρ hm (ω1, w − ω1,∞ ) (1.139) ′′ where the species mass flux, m1 , is again exemplified by transport from a flat surface to a vapor stream flowing over that surface. The term hm (m/s) in eq. (1.139) 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:
Sh = hm L D12
(1.140)
For convenience and physical analogy between various diffusion transport processes, the three rates of diffusion transport equations for mass, momentum, and heat for one-dimensional, constant properties are summarized below: dω ′′ (1.141) m1, y = − ρ D12 1 , Fick's law (binary diffusion)
dy
τ yx = −ν
′′ qy = −α
d ( ρ u) , dy dy
Newton's law viscosity (viscous fluid shear)
(1.142) (1.143)
d ( ρ c pT )
, Fourier's law (heat conduction)
It is apparent that all the rate equations are of the same form, where flux equals a constant times potential gradient. The proportionality constant is a function of materials involved in the transport process. This analogous relationship can be used to predict a 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 correlation of friction or similarly mass transfer coefficients. In general, when an analogy exists and it does not always, information obtained from a simple experimental setup can be applied to more complex physical experimental setups. In subsequent chapters, we see circumstances in which two or more processes are governed by the same dimensionless governing equations and boundary conditions. In these circumstances, one can develop these analogous relations from a more accurate
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Table 1.7 Summary of fundamental laws in momentum, heat and mass transfer
Equations Newton’s law of viscosity, eqs. (1.60) and (1.57) 2 T τ′ = − pI + μ ∇V + ( ∇V ) − μ (∇ ⋅ V )I 3 Momentum Simplified Newton’s law of viscosity τ′ = − pI + μ ∇V + ( ∇V )T Fourier’s law, eq. (1.67) q′′ = −k∇T Fourier’s law, eq. (1.70) q′′ = −k ⋅ ∇T Energy Equation (1.71) q′′ = −k ⋅ ∇T +
N
Flux
Requirements Newtonian fluid Laminar Newtonian fluid Laminar Single component Single component
Comments/Assumptions Single or 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
h J
i =1
N
ii
+cRuT
i =1
xi x j DiT J i J j − ρi Dij ρi ρ j j =1( j ≠ i )
N
Multicomponent
Fick’s law, eqs. (1.105) and (1.106) J1 = − ρ D12∇ω1
* J1 = −cD12∇x1
Binary only
Maxwell-Stefan equation (1.110)
∇x i = −
j =1( j ≠ i )
N
xi x j Dij
( Vi − V j )
=−
j =1( j ≠ i )
N
1 i x j n′′ − xi n′′j cDij
(
)
Multicomponent gaseous mixture
No temperature and pressure gradients. Same body forces for all N components
Maxwell-Stefan equation (1.111) ∇ωi + ωi ∇(ln M ) Mass = M
ρ
j =1( j ≠ i )
N
N
ωi J j − ω j J i
Dij M j
Multicomponent gaseous mixture
No temperature and pressure gradients. Same body forces for all N components
Maxwell-Stefan equation (1.119)
cRuT xi x j J j J i − = ∇pi − ωi ∇p − ρi Xi Dij ρ j ρi j =1( j ≠ i )
+ωi
j =1
N
ρ j X j − cRuT
xi x j DT DiT j− ∇(ln T ) ρi Dij ρ j j =1( j ≠ i )
N
Multicomponent system
Including ordinary, pressure, body force, and thermal diffusion
Mass flux in binary system, eq. (1.133) MMD ω J1 = − ρ D12∇ω1 + 1 2 12 1 ∇p p M M D ωω T + 1 2 12 1 2 ( X1 − X2 ) − D1 ∇(ln T ) RT
Binary only
Including ordinary, pressure, body force, and thermal diffusion
42 Advanced Heat and Mass Transfer
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basis. As a first approximation, the convective heat and mass transfer coefficients for laminar and turbulent heat transfer can be related by the following equation:
h h k = = ρ c p Le2 /3 = hm hm D12 Le1/3
(1.144)
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 to E. Various empirical equations for transport properties of gases and liquids can also be found in Poling et al. (2000). Chapter 2 presents the generalized integral and differential governing equations for transport phenomena in local-instance formulations using various fluxes presented in Table 1.7. The generalized governing equations for multiphase systems in averaged formulations are also presented in Chapter 2. The averaged formulations are obtained by averaging the governing equations within a small time interval (time average) or a small control volume (spatial average). Chapter 3 also covers single- and multiphase transport phenomena in porous media, including multi-fluid and mixture models.
Example 1.2. Show that the general Maxwell-Stefan equation (1.110) for the mass flux in a multicomponent gas system can be simplified for a binary system to Fick’s law, eq. (1.106). Solution: For a binary system, the number of components, N, is equal to 2. The Maxwell-Stefan equation, (1.110), for the first component (i=1) becomes
∇x1 = −
1 ′′ 2 ( x2n1 − x1n′′ ) cD12
(1.145)
The molar fluxes of component 1 relative to stationary coordinate axes can be obtained by eq. (1.104), i.e., * ′′ ′′ 2 n1 = x1 (n1 + n′′ ) + J1 (1.146) which can be rearranged to * ′′ 2 (1 − x1 )n1 − x1n′′ = J1 (1.147) Since x 2 = 1 − x1 , eq. (1.147) can be rearranged to * ′′ 2 x 2 n1 − x1n′′ = J1 (1.148) Substituting eq. (1.148) into eq. (1.145), one obtains * J1 = −cD12 ∇x1 (1.149) which is identical to Fick’s law, eq. (1.106).
1.4.3 Microscale and Nanoscale Transport Phenomena
Transport phenomena at dimensions between 1 and 100 μm 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 systems is surface tension. In
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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 lasers) the fundamental theory used in larger scale systems breaks down because of fundamental differences in the physics. The purpose of research in micro/nanoscale heat transfer is to exploit their 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. Therefore, the temperature difference between the turbine and the compressor is much less, which decreases the system’s 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, α2 (1.150) Z=
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
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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 (1.151) From kinetic theory, the thermal conduction of each component is given by (Flik et al. 1992),
1 ke− = c p,e − ce− λe − 3 1 k ph = c p, ph c ph λ ph 3
(1.152) (1.153)
The subscripts e- and ph refer to an electron and a phonon, 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.
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Defects, dislocations, impurities and boundaries within a solid structure all have an effect 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 (1.154) Z = ρ c ph The speed of sound (mean velocity of a phonon) is related to the elastic stiffness of a chemical bond (E) by (1.155) 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
c ph = E
λ ph ≈
hc ph
3kB T
(1.156)
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.
(a)
(b)
Figure 1.12 Mean free path of electrons or phonons (a) far away from boundaries and (b) in the presence of boundaries.
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PA → B =
4Z A Z B
( Z A + Z B )2
(1.157)
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. (1.152) and (1.153) 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− (1.158) k 3δ keff ,t 2λ = 1− (1.159) 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 due to a decrease in temperature. This equation is valid because, in a highly electrically conductive material, essentially all of the thermal transport is carried through electrons. (1.160) k = σ e −1 L0T 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 size on the thermal conductivity by considering only the phonon transport. Such analyses were 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. 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
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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
Heat and mass transfer are, 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 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. Dimensional analysis is very important to interpolate the experimental laboratory results (prototype models) to full scale system. Two criteria must be fulfilled to perform such an objective. First, dimensional similarity, in which all dimensions of the prototype to full scale system must be in the same ratio, should be fulfilled. Secondly, dynamic similarity should be met in which relevant dimensionless groups are the same between the prototype model and full scale system. 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
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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 (1.161) h = h ( k , μ , c p , ρ , U , ΔT , D , L ) 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 (1.161) can also be rewritten as (1.162) F (h, k , μ , c p , ρ , U , ΔT , D, L ) = 0 It can be seen from eq. (1.162) that nine dimensional parameters are required to describe the convection problem. To determine the functions in eq. (1.161) or (1.162), 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 Π theorem (Buckingham, 1914), the number of dimensionless variables required to describe the problem equals the number of primary dimensions required to describe the problem minus the number of dimensional variables. 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 Π theorem and are formed from products of powers of certain original dimensional variables. Any of such dimensionless groups can be written as (1.163) Π = h a k b μ c c d ρ eU f (ΔT ) g D h Li p Substituting dimensions (units) of all variables into eq. (1.163) yields
Q Q M Qt M L ghi Π= 2 T LL L T LT Lt MT L3 t
c − d + e −2 a − b − c −3e + f + h + i − c + d − f a b c d e f
(1.164)
be summed to zero, i.e.:
L t T =M Q For Π to be dimensionless, the components of each primary dimension must
− a −b − d + g
a +b+ d
This gives a set of five equations with nine unknowns. According to linear algebra, the number of distinctive solutions of eq. (1.165) is four ( 9 − 5 = 4 ), which coincides with the Π 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. (1.165) become
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
(1.165)
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b = 0, c = −1, e = 1, g = 0, and h = 1 , which give us the first nondimensional
variable
Π1 =
ρUD μ
(1.166)
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. (1.165) as b = −1, d = 0, e = 0, g = 0, and h = 1 . The second nondimensional variable becomes
hD k which is the Nusselt number ( Π 2 = Nu ). Π2 =
(1.167)
k L Π4 = (1.169) D which are the Prandtl number ( Π 3 = Pr = ν / α ) and the aspect ratio of the tube.
Following a similar procedure, we can get two other dimensionless variables: cpμ Π3 = (1.168)
Equation (1.162) can be rewritten as
F (Π1 , Π 2 , Π 3 , Π 4 ) = 0
(1.170)
or
Nu = f (Re, Pr, L / D) (1.171) Furthermore, if the flow and heat transfer in the tube are fully developed, meaning no change of flow and heat transfer in the axial direction, eq. (1.171) can be simplified further: Nu = f (Re, Pr) (1.172) It can be seen that the number of nondimensional variables is three, as opposed to the nine dimensional variables in eq. (1.161).
Example 1.3 For boiling and condensation processes, obtain the appropriate dimensionless parameters using a similar procedure to the one above. Solution: 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, ( ρ − ρ v ) g , latent heat of vaporization, hv , and surface tension, σ : (1.173) h = h[k , μ , c p , ρ , L, ΔT , ( ρ − ρ v ) g, hv , σ ]
There are 10 dimensional variables in eq. (1.173) 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 liquidvapor phase change process, i.e.: Nu = f ( Gr, Ja, Pr, Bo ) (1.174)
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where the Grashof number is defined as: ρ g( ρ − ρ v ) L3 (1.175) Gr = μ2 The new dimensionless parameters introduced in eq. (1.174) are the Jakob number, Ja, and the Bond number, Bo, which are defined as c p ΔT Ja = (1.176) hv
Bo = ( ρ − ρ v ) gL2
σ
(1.177)
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. 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. (1.172) and (1.174). 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, thermal properties of the fluid, and flow properties.
Example 1.4 Dry air at 35 °C and velocity of 1.5 m/s flows over a flat plate of 2 m × 2 m covered with a thin layer of water (see Fig. 1.13) . The plate surface temperature is kept at 30 °C. Calculate the heat transfer due to convection and evaporation from the plate to air. The radiation heat transfer from the liquid surface and the heat conduction in the liquid can be neglected. Solution: The properties of the air are taken as the average of the free surface and free stream. These properties at T=32.5°C are, ρ = 1.1431 kg/m3 ,
ν = 16.44 ×10−6 m 2 /s, k = 26.7 × 10−3 W/m-K, water vapor in the air is (see Appendix E)
DH 2O, air = 1.87 × 10-10 Pr=0.706.
The diffusivity of
T 2.072 = 2.64 × 10−5 m 2 /s p
Reynolds number is
Re L =
u∞ L
ν
=
1.5 × 2 16.44 × 10
−6
= 1.825 × 105
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Table 1.8 Summary of dimensionless numbers for transport phenomena
Name Bond number Brinkman number Capillary number Eckert number Fourier number Froude number
Grashof number Jakob number Kapitza number Knudsen number Lewis number Mach number Nusselt number Peclet number Prandtl number Rayleigh number Reynolds number Schmidt number Sherwood number Stanton number Stanton number (mass transfer) Stefan number Strouhal number Weber number Womersley number
Symbol Definition Bo Br Ca Ec Fo Fr
Gr
Ja Ja v
Physical interpretation
2
( ρ − ρ v ) gL / σ
Buoyancy force/surface tension Viscous dissipation /enthalpy change We/Re Kinetic energy/enthalpy change Dimensionless time Interfacial force/gravitational force
Buoyancy force/viscous force Sensible heat/latent heat Surface tension/viscous force Mean free path /characteristic length Ratio between thermal and mass diffusivities 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 Time characteristics of fluid flow Inertial force/surface tension force Radial force /viscous force
Area of significance Boiling and condensation High-speed flow Two-phase flow High speed flow Transient problems Flow with a free surface
Natural convection Film condensation and boiling Wave on liquid film Noncontinuum flow Mass transfer 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 Oscillating flow /bioengineering Liquid-vapor phase change Bioengineering
μU 2 / (kΔT ) μU / σ
U
2
(c p ΔT )
α t / L2
U 2 / ( gL ) gβ ΔTL3 / ν 2 c p ΔT / hv c pv ΔT / hv
4 μ g / [( ρ − ρ v )σ 3 ]
Ka Kn Le Ma Nu Pe Pr Ra Re Sc Sh St Stm Ste Sr We Wr
λ/L α/D
U /c hL / k UL / α
ν /α
gβ ΔTL3 / (να ) UL / ν
ν /D
hm L / D12
h / ( ρ c pU )
hm / U
c p ΔT / hs Lf / U
ρU 2 L / σ
ω R2 / ν
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Table 1.9 Correlations for convective heat transfer for various modes and geometries
Heat transfer mode
Geometry
Nusselt number Nu x = 0.332Re1/2 Pr1/3 x (Pr > 0.6)
Comments and restrictions Isothermal surface
Re x < 5 × 105 (laminar)
Dimensionless numbers
Flow parallel to a flat plate
Nu = 0.664Re1/2 Pr1/3 L (Pr > 0.6) Nu x = 0.565Re1/2 Pr1/2 x (Pr ≤ 0.05)
Nu = 0.037(Re0.8 − 871) Pr 0.33 L (0.6 ≤ Pr ≤ 60)
Nu x = Re x = Nu =
hx k u∞ x
ν
5 × 105 < Re L < 108 (Turbulent)
hL k uL Re 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 Re > 10,000 (Fully developed turbulent) μ w is viscosity evaluated at Tw
Nu = Re =
u
Nu = 0.027 Re0.8 × Pr 0.33 ( μ / μ w ) (0.7 ≤ Pr ≤ 16700) Flow in a pipe (miniature)
0.14
hD k uD
Forced convection
u is mean velocity
ν
Nu = (1 + F ) ( f / 8)(Re− 1000) Pr × 1 + 12.7( f / 8)0.5 (Pr 2/3 − 1) f = [1.82log(Re) − 1.64]−2 F = 7.6 × 10−5 Re × [1 − ( D / D0 ) ]
2
u
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
Flow between parallel plates
Nu = 7.54 + 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 = 0.023Re0.8 Pr 0.33 ( Pr > 0.5)
Re > 10,000 (Turbulent)
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Table 1.9 Correlations for convective heat transfer for various modes and geometries (cont’d)
Heat transfer mode
Geometry Flow across a circular cylinder
Nusselt number
Nu = 0.3 + 0.62Re1/2 Pr1/3 [1 + (0.4 / Pr /) 2/3 ]1/4
4/5
Comments and restrictions
Dimensionless numbers
Re 5/8 × 1 + 282000 Flow across a sphere
Nu = 2 + (0.4 Re0.5
Re Pr > 0.2 (Both laminar and turbulent) Nu =
Re =
3.5 < Re < 76000
1
hD k u∞ D
ν
Forced convection
0.71 ≤ Pr ≤ 380
+0.06 Re 2/3 ) Pr 0.4 ( μ / μ w ) 4
μw is viscosity evaluated at Tw
Flow through a packed bed of spheres Nu = 1.625Re1/2 Pr1/3
15 ≤ Re ≤ 120 D – diameter of sphere A – bed crosssectional area
Nu = Re =
hD k mD Aμ
On a vertical surface ΔT = Tw − T∞ Applicable to both laminar and turbulent
hL k gβΔTL3
Nu
1/2
= 0.825 + 0.387Ra1/6
Nu = Ra =
[1 + (0.492 / Pr)9/16 ]8/27
να
Free convection
On a horizontal heated square facing up Nu = 0.54(Gr Pr)1/4
Isothermal
surface
Nu = Gr =
105 ≤ Gr ≤ 7 × 107 For rectangle, use shorter side of L
hL k gβΔTL3
ν2
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Table 1.9 Correlations for convective heat transfer for various 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 = hL k gβΔTL3
Nu = 0.27(Gr Pr)1/4
≤ 3 × 1010 For rectangle, use shorter side of L
ν2
On a horizontal cylinder Free convection
Nu
1/2
= 0.60 +
0.387Ra1/6 Ra < 1012
Nu = Ra =
hD k gβΔTD3
+
[1 + (0.559 / Pr)9/16 ]8/27
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 gβΔTD3
να
Falling film evaporation
Laminar
− Nu = 1.10Reδ 1/3
(Reδ ≤ 30) Wavy laminar
Evaporation
− Nu = 0.828Reδ 0.22
Nu − local Nusselt number
1
(30 ≤ Reδ ≤ 1800) Turbulent
0 Nu = 0.0038Reδ .4 Pr 0.65
2 h (ν / g) 3 Nu = k Γ − mass flow rate 4Γ per unit width of the Reδ = vertical surface μ
(Reδ > 1800) Laminar (Nusselt) On a vertical surface
− Nu = 1.10Reδ 1/3
(Reδ ≤ 30) Wavy laminar Re Nu = 1.22 δ Reδ − 5.22 (30 ≤ Reδ ≤ 1800) Turbulent
Nu = 0.023 Reδ
0.25 −0.5
Nu − local Nusselt number
1
Condensation
2 h (ν / g) 3 Nu = k Γ − mass flow rate 4Γ per unit width of the Reδ = vertical surface μ
Pr
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Table 1.9 Correlations for convective heat transfer for various modes and geometries (cont’d)
Heat transfer mode
Geometry On tubes
Nusselt number
Comments and restrictions
Dimensionless numbers
Nu = 0.729
D3hv g ( ρ − ρ v ) 4 × nkν ΔT
1
ΔT = Tsat − Tw
n − number of tubes
Nu =
hD k
Condensation In microscale channel ( Dh < 1.5mm )
Y = 1.3 for Re ≤ 65 Y = (0.5 Dh − 1)
We =
Ja =
ρV 2 L σ
hv
c p (Tsat − Tw )
Nu = We − Ja Re PrY
/ (2 Dh ) for Re > 65
Re =
m′′Dh
μ
m′′ –mass flux (kg/s-m2)
hLc k
σ
g( ρ − ρ v )
Nucleate, saturated pool boiling Nu =
m C 3 Pr
2 Ja
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 =
Ja =
c p , ΔT hv
ΔT = Tw − Tsat
Boiling
Nu = hLc kv
σ
g( ρ − ρ v )
Film boiling on a horizontal plate
Nu = 0.425
1 + 0.4 Jav × Gr Prv Jav
1 4
Lc =
Term in parentheses accounts for sensible heating effect in vapor film
Gr = g[( ρ − ρ v ) / ρ v ]Lc
2 νv 3
Ja v =
c p, v ΔT hv
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Table 1.9 Correlations for convective heat transfer for various 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
1 + 0.4 Jav 4 × Gr Prv Jav
1
D film thickness
Nu =
Gr =
hD kv
3
g[( ρ − ρ v ) / ρ v ] D
Film boiling on a sphere Boiling
νv
2
Nu = 0.4
1 + 0.4Ja v 3 × Gr Prv Ja v
Boiling in microchannel (D=1.39 – 1.69 mm)
1
D film thickness
Ja v =
c p , v ΔT hv
Nu = 30 Re ×Bo
0.714
0.857
−0.143
(1 − x )
Correlation obtained by using Freon ® 141 x is quality
Nu =
hD k
Bo =
q′′ hvm′′
m′′ – mass flux (kg/s-m2)
Nu = hH k gβ ΔTH 3
Melting in a rectangular cavity Melting
Nu = (2τ ) −1/2 +[c1Ra1/4 − (2τ )−1/2 ] ×[1 + (c2 Ra
n = −2
3/4 3/2 n 1/ n
τ
)]
Nusselt number is function of time
c1 = 0.35, c2 = 0.175
να τ = SteFo α t
Fo = H2
Ra =
Solidification around a horizontal tube Solidification Nu = 0.52Ra1/4
D is transient equivalent outer diameter of the
Nu = Ra =
hD k gβΔTD3
solid Ra ≤ 109
να
u ∞ , T∞ , ω ∞
Sublimation
Nu x = 0.458Re1/2 Pr1/3 x
Uniform heat flux surface
Re x < 5 × 105
′′ qw
Sh x = 0.459 Re1/2 Sc1/3 x
hx k hm x Sh x = D Nu x =
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Air
u∞ = 1.5m/s T∞ = 35 o C qeva qconv
o
2m
Water Tw = 30 C
2m
Figure 1.13 Dry air flows over a flat plate covered with water.
According to Table 1.9, for flow parallel to a flat plate with Pr > 0.6 , the Nusselt number is
Nu = 0.664 Re0.5 Pr1/3 = 252.57 L
The heat transfer coefficient is The heat transfer from the plate to the air due to convection is The Schmidt number is
Sc = h = Nuk / L = 252.57 × 26.7 × 10−3 / 2 = 3.37W/m 2 -K qconv = hA(Tw − T∞ ) = −3.37 × 2 × 2 × 5 = −67.44W
ν
D
=
16.44 × 10−6 2.64 × 10−5
= 0.623
0.33
The Sherwood number is
Sc Sh = Nu Pr
0.33
0.623 = 252.57 × 0.706
= 242.26
The mass transfer coefficient is
hm = ShD12 / L = 242.26 × 2.64 × 10−5 / 2 = 0.0032m/s
The concentration of the water vapor at the surface of the flat plate is ρ v,w = ρ v,sat @30 C = 0.03420kg/m3 The concentration of the water vapor in the incoming air is ρv,∞ = 0 since the air is dry. The latent heat for evaporation is taken as that at 30°C, and is hv = 2430.2 kJ/kg . The heat transfer due to evaporation is therefore
qeva = hm A( ρ v,w − ρ v,∞ )hv
= 0.0032 × 2 × 2 × (0.03420 − 0) × 2430.2 × 1000 = 1060W
The heat transfer due to convection is significantly less than the heat transfer due to evaporation.
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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 a few percentage points of the results produced by the exact 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. 1.14. 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. The temperature satisfies the following two conditions T (δ , t ) = Ti (1.178)
∂T ∂x =0
x =δ
(1.179)
The energy equation for this problem and the corresponding initial and boundary conditions are:
1 ∂T = ∂x 2 α ∂t T ( x , t ) = T0
∂ 2T
x > 0, t > 0 x = 0, t > 0
(1.180)
(1.181) T ( x, t ) = Ti x > 0, t = 0 (1.182) Since temperature difference occurs only within 0 ≤ x < δ , the order of magnitude of x is the same as δ , i.e., x δ (1.183)
Figure 1.14 Thermal penetration depth for conduction in a semi-infinite solid.
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The order of magnitude of the term on the left-hand side of eq. (1.180) is
∂ 2T ∂x
2
=
∂ ∂T ∂x ∂x
where ΔT = T0 − Ti . The order of magnitude of the right-hand side of eq. (1.180) is
(1.185) α ∂t α t Equation (1.180) requires that the two orders of magnitude represented by eqs. (1.184) and (1.185) equal each other, i.e.,
1 ∂T 1 ΔT
1 ΔT ΔT =2 δ δδ
(1.184)
(1.186) δ The order of magnitude of the thermal penetration depth is then δ αt (1.187) Equation (1.187) 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 3). This simple example demonstrates the additional capability of scale analysis over dimensional analysis, which could not provide the form of the functions.
2
ΔT
1 ΔT αt
Example 1.5 Consider transient heat conduction in a rectangular domain with dimensions of L by H (see Figure 1.15). Discuss the condition under which the problem can be simplified to one-dimension. Solution: The governing equation for transient heat conduction in a twodimensional domain is
∂ 2T ∂x
2
+
∂ 2T ∂y
2
=
1 ∂T α ∂t
(1.188)
The orders of magnitude of x and y are, respectively, the same as L and H, i.e., x L, y ~ H (1.189) The orders of magnitude of the terms on the left-hand side of eq. (1.188) are
∂ 2T ∂x 2 ∂ 2T ∂y
2
= =
∂ ∂T ∂x ∂x
1 ΔT ΔT =2 L LL
(1.190) (1.191)
∂ ∂T 1 ΔT ΔT =2 ∂y ∂y H H H
where ΔT = T1 − T2 . The condition under which the problem can be simplified as 1-D is
∂ 2T ∂x 2
∂ 2T ∂y 2
(1.192) (1.193)
i.e.,
HL
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y H
T = T1
T = T2
0
L
x
Figure 1.15 Heat conduction in a rectangular domain
Example 1.6 In order to study blood steady flow in the coronary arteries, a lab model prototype with all dimensions three times larger than the original dimensions is tested. The working fluid is a glycol-water mixture that has the same density and viscosity of the blood. What is the velocity of the scale model relative to that of the original model? Solution: Since the flow is steady state, the Strouhal and Womersley numbers are irrelevant. The dynamic similarity only requires that the Reynolds numbers for the original model and prototype model be equal, i.e., (1.194) Reo = Re p
Substituting the definition of the Reynolds number to eq. (1.194), one obtains
uL uL = ν o ν p
(1.195)
Since the viscosities of the original and prototype models are the same, ν o = ν p , and the dimension of the prototype model is three times that of the original model, 3Lo = L p , eq. (1.195) requires that u p = uo / 3.
1.5 Modern Applications of Heat and Mass Transfer
Traditionally, most of the applications of heat and mass transfer over the past four decades pertained to traditional power and thermal engineering, refrigeration, conventional energy systems including turbines as well as space
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activities. In recent years, modern and emerging technologies (i.e., nanotechnology, biotechnology, energy, and information technology) have started to play major roles. The technological advances over the last decade also provide new multidisciplinary opportunities in heat and mass transfer. Although transport phenomena in biological systems are of obvious relevance, thermal sciences research and development have traditionally been directed to biomedical applications. There are new biotechnological areas that would benefit from contributions of the thermal science community. Recently, increased attention has been paid to issues involving security, with large investments being made both by governments and the private sector. Although it may be argued that information technology and nanotechnology are becoming mature areas of research, integration of these topics into design platform and the classroom has yet to be achieved on a widespread basis. Selected applications of heat and mass transfer in these technologies are reviewed in this section. The examples presented in this section are illustrative, rather than inclusive.
1.5.1 Energy Systems
It is generally acknowledged that energy is one of humanity’s top problems for the next 50 years, in addition to water, food, environment, poverty, terrorism and war, disease, education, democracy and population. The single biggest challenge for mankind during the next few decades may be to provide energy for an estimated 10 billion people by the year 2050 on a sustainable basis. Sustainable and renewable sources that loom on the horizon include wind, geothermal, biomass, photosynthesis, hydro, photovoltaic and others. The ability to increase the generation of hydropower, for example, has been virtually exhausted as dams have already been built where they can exist. Many renewable and/or sustainable energy sources or conversion systems present numerous challenges and applications for the heat and mass transfer community. One of the sustainable and clean energy sources is solar energy. The major barrier 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 with solar systems. 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 photovoltaic cell (solar cell) is a device that directly converts light energy into electrical energy without going through thermal energy. However, cooling of photovoltaic cells is one of the main concerns when designing concentrating photovoltaic systems (Royne et al., 2005). If the cells are not properly cooled,
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they may experience both short-term (efficiency loss) and long-term (irreversible damage) degradation. It is imperative to keep the cell temperature low and uniform, and minimize the power consumption of the cooling system. While passive cooling is sufficient for single cells, even for very high solar concentrations, an active cooling system is necessary for densely packed cells under high concentrations. The existing high-capacity cooling technologies, such as liquid impingement, microchannels, and two-phase cooling devices can be applied to cool the photovoltaic cells. Combustion applications are found in steam generation in power plant boilers, gas turbine systems, and to melt metals in metallurgical processes. It is a chemical reaction process between a fuel (which can be solid, liquid or gas) and an oxidant that produces high-temperature gases. In a typical combustor using liquid fuel, the fuel first breaks up into a fine-droplet 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 of solid fuel combustion. Switching from oil or coal to hydrogen or alcohol for various applications using fuel cell technologies is becoming promising. The heat and transfer community has been actively involved over the last decade in research and development of fuel cell technology and therefore a presentation of this technology will be discussed in more detail. 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
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Oxidant (e.g., Oxygen, Air….) Reactant Products (e.g., H2O, CO 2 … .) & H eat
Cathode Load Electrolyte
e-
Anode
Fuel (e.g., H 2,
CH4, CO, CH3OH….)
Figure 1.16 Fuel cell schematic.
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 1.16 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. 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, and 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 form water. 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 hydrogencontaining 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
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Figure 1.17 Basic construction of a typical PEM fuel cell stack (Faghri and Guo, 2005; Reprinted with permission of Elsevier).
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 1.17 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 PEMFC 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
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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 0.2 to 3 mm). As shown in Fig. 1.17, 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 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. Development and application of fuel cell technology has increased significantly through analysis and improvement of the heat and mass transfer in the fuel cell stack and auxiliary components and by implementation of innovative heat transfer systems that address these issues. A detailed review of these problems related to fuel cell technology, including challenges and opportunities is presented by Faghri and Guo (2005). Current technical limitations can be overcome through a combination of novel technologies, increased efficiencies of components, decreased component size and mass and detailed modeling. There has been extensive development of fuel cell modeling that has been reviewed by Wang (2004), and Faghri and Guo (2005). A successful fuel cell model will capture all of the physics of the problem including reaction kinetics, the electric potential field, the fluid flow as well as thermal field (Rice and Faghri 2006; 2008).
1.5.2 Biological and Biomedical Systems
Transport phenomena research in biological systems may be at the dawn of a new age, with, for example, new opportunities to understand and control transport phenomena at the cellular level. By doing so, novel targeted therapies,
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including therapies based upon advances in nanotechnology, might be developed. Advances in pharmacology including drug discovery and screening, pharmaceutical manufacturing and drug delivery are widespread, and have already had a profound effect on the quality of life globally. Other areas such as tissue engineering are replete with transport phenomena of several types. In these and other emerging areas as well as in traditional areas of bioheat and mass transfer, the thermal sciences will significantly contribute to improved health and environment. A cell is a membrane-bound soup of water, proteins, lipids and nucleic acids that grows, reproduces and interacts with its environment. Cells are regarded as the basic organizing unit of life and cell biology plays essential roles in human disease, such as bacterial infections and cancer. Depending on the complexity of a cell, it can be either a prokaryote or eukaryote. While the former can be found in bacteria and only has a cell membrane and cytoplasm and no other organelles, the latter exists in plants, animals, fungi, and protista and has a number of different cell organelles. The cells are usually in thermal equilibrium and the transport phenomena encountered in the cells are mainly momentum and mass transfer. While most existing models treat a cell as a well-mixed homogeneous system, cell heterogeneity is essential to capture many important phenomena at the cellular level. Due to barriers between biology and thermal sciences, the role and importance of mass transfer in information processing is not realized by biologists. To fully understand the role of transport phenomena at the cellular level, it is essential to develop more sophisticated models incorporating nonNewtonian mechanics, heterogeneities, time-dependence and stochastic fluctuations. Heat and mass transfer in a single cell, tissue, and organ has been investigated. At the organ level, such as lung, liver, and kidney, their structure must allow rapid exchange of molecules between blood and tissues (Truskey et al., 2004). Although transport process occurs in all organs, the effect is very significant in some organs such as the cardiovascular and respiratory systems, the gastrointestinal tract, the liver and the kidneys. For example, the cardiovascular system consists of the heart, blood vessels and blood. The primary function of the cardiovascular system is to transport oxygen within red cells and remove carbon dioxide. In the respiratory system, oxygen is delivered to tissues when it is metabolized and carbon dioxide is removed. Digestion and absorption of nutrition are done in the gastrointestinal track. The liver performs metabolic functions such as carbohydrate storage and release, cholesterol metabolism, synthesis of plasma and transport proteins including removing toxic molecules from the blood. Kidneys are responsible for removal of urine and water products. In a number of diseases, such as atherosclerosis, cancer, and kidney diseases, alternation of transport processes are very important factors. In biomedical engineering, understanding of transport phenomena plays an essential role on design of replacement tissues and delivery of drugs. For example, while titanium is the most widely used metallic biomaterial for orthopedic implant, one major problem is the mismatch of the Young’s modulus between the bone (10-30 GPa)
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and the titanium implant (110 GPa); this mismatch causes the bone to be insufficiently loaded (referred to as stress-shielding) and retards bone remodeling and healing (Oh et al., 2003; Thieme et al., 2001). The mismatch can be overcome by introducing a graded porosity near the implant surface so that the Young’s modulus of the titanium implant can change continuously. The Drug Delivery Devices (DDDs) are polymer porous devices with pores for drug loading and release. These DDDs can be implanted or taken orally by the patient. The drug can be either released continuously or distributed in discrete alternating sections to achieve a pulse release. The pattern for drug release can be controlled by varying local porosity within the DDDs. Understanding of transport phenomena during the manufacturing processes is the key to achieve desired porosity and porous structure in the above examples. In addition, growth of new bone into the porous implant and release of drug from DDD are also dominated by the transport phenomena. Lasers have been widely used in medical applications for more than three decades. The majority of those applications involve thermal effects. For example, in laser hyperthermia, the temperature of a pathological tissue is often elevated to 42 ~ 45°C so that the growth of malignant tumor can be retarded. In laser coagulation, laser beams can cause immediate irreversible damage to pathological cells by heating them to above 60oC. In laser surgery, the laser beam can vaporize and cut tissues like a scalpel when tissue is heated to a temperature of as high as 100°C. No matter which treatment a doctor performs, a thorough understanding of the damage distribution within both the pathological tissue and the surrounding healthy tissue is imperative. During laser processing of pathobiological materials, such as ophthalmic microsurgery, laser lithotripsy, and angioplasty, plasma can be formed and it will interact with the laser beam. In such applications, a phenomenon called optical breakdown will occur as an extremely high intensity laser beam is focused onto the tissues. During the optical breakdown, a very high free electron density, i.e., plasma, on the order of 1018-1021 electrons/cm3 is produced. Two mechanisms are responsible for the optical breakdown: multiphoton absorption and avalanche ionization (also called cascade or impact ionization). Both mechanisms require a minimum threshold intensity before optical breakdown can be initiated. Once breakdown occurs, it leads to a rapid heating of the material in the focal volume, followed by its explosive expansion and the emission of a shock wave. The expansion of the heated volume further results in the formation of a cavity if it occurs in solids or of a cavitation bubble if it takes place in liquids. Heat and mass transfer during laser-pathobiological material processing play a dominant role and must be investigated. Cryopreservation that uses liquid nitrogen to deep-freeze, and thus preserve, biological materials is another example that heat and mass transfer is dominant. 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
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cancer treatment that may risk her subsequent ability to give birth to a healthy child. The doctor can preserve her oocytes or embryos to help her conceive after she has 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. 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 to 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.
1.5.3 Security
Security involves homeland/defense security, food/water security, and economic security. Efforts to enhance security provide unique opportunities in research and development involving significant transport phenomena components. Topics related to transport phenomena include but are not limited to biological and chemical threat detection, biosensors, aerosol generation and dispersion, distributed power generation, portable power infrastructure, fire protection and particle transport. A few fundamental transport phenomena problems related to security are presented below. It is imperative to create a real time capability of emergency response against chemical, biological, or radioactive attacks or accidents so that the resultant fatalities can be reduced to minimum (Pepper, 2007). The long term effects of these events on the environment are also crucial as different substances behave differently. The effects of chemical agents are strongest at its initial release and decay with time. The effects of biological agents initially increase with time until reaching a peak, after which the effects gradually decay. On the contrary, it can
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(a) Office configuration
u = 1 m/s
(b) Path of containment particles
Figure 1.18 Containment particle dispersion traces (courtesy of Dr. D.W. Pepper).
take a relative long time to see the effects of radioactive agents by which time it is often too late to undo the damages. Diffusive and convective mass transfer plays a significant role on prediction of propagations of these agents under different weather conditions. Wang et al. (2005) developed a 3-dimensional finite element model for using Lagrangian particle transport technique to simulate contaminant transport for emergency response. The real-time simulation will allow instant prediction of trajectory and risk of contamination of the chemical, biological and radioactive agents. Figure 1.18 shows how containment particles introduced at a corner of a table disperse in an office suite (Pepper, 2007). The door is open and the air inlet velocity at the door is 1 m/s. The containment particles are transported through diffusion and convection from the source in the secretary’s room to the manager’s room. Another area of security that transport phenomena play a significant role in is fire safety (Atreya, 2007). While both combustion and fire are exothermic chemical reaction processes between fuel and oxidant, combustion is usually a useful process where chemical energy in the fuel is extracted and converted to heat, whereas fires are often results of natural or human disasters. Due to the
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small scale involved in combustion, detailed numerical modeling of the physical and chemical processes at highly resolved temporal and spatial scales is possible. On the contrary, the large temporal and spatial scales involved in fire simulation do not permit simulation of the onset of fire because the transport phenomena and chemical reactions during onset of fires occurs at temporal and spatial scales below the resolution limits of the most practical calculations (McGrattan et al., 2002). Another difference between combustion and fire is that combustion takes place in a relatively small scale but fire usually occurs at a very large scale. Fire is not considered as a design load in the prediction and evaluation of structural performance in current design practices. For example, the World Trade Center (WTC) towers could have sustained the impact of the planes during the attack on September 11, 2001, but the resulting fires caused structure failures which lead to total collapse (Usmani et al., 2003). In order to consider fire as a design load, it is imperative to develop a science-based set of verified tools to evaluate the performance of the entire structure under realistic fire conditions. Since the building materials are not intended for use as fuel, the data to characterize the fuel and the fire environment are not available (Baum, 2000); this makes development of reliable simulation tools more challenging. The physical and chemical phenomena that occur during fire that cause structure damage include fire dynamics, thermal response, and structural response; these phenomena cross many temporal and spatial orders of magnitude (Prasad and Baum, 2005). Fire dynamics includes modeling and simulations of combustion, fluid mechanics, convective heat and mass transfer, and most importantly, radiation heat transfer in the gases (air and/or smoke). Thermal analysis involves simulation of heating and cooling of the building materials. Finally, the structural analysis involves modeling displacements, stresses, and loss of load carrying capacity of the structure. Heat and mass transfer play dominant roles in fire dynamics and thermal responses analysis, which subsequently provide the needed data for structure analysis.
1.5.4 Information Technology
Due to the information technology revolution of the last two decades, strong industrial productivity growth has brought an improved quality of life worldwide. Inevitably, heat fluxes and volumetric energy generation rates keep increasing in many applications; but the operating temperatures of the devices must be held to reasonably low values to ensure their reliability. Although much advancement has been made in traditional areas such as electronics cooling, it is not enough. New challenges will surface in the manufacturing of information technology related equipment and materials that require further research and development by the transport phenomena community. As modern computer chips and power electronics become smaller and more densely packed, the need for more efficient cooling systems increases. The new
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design of a computer chip at Intel, for instance, will produce localized heat flux over 100 W/cm2, with the total power exceeding 300 W. In addition to the limitations on maximum chip temperature, further constraints may be imposed on the level of temperature uniformity in electronic components. Heat pipes are a very promising technology for achieving high local heat removal rates and uniform temperatures in computer chips. Miniature and micro heat pipes have been and are being used in electronic cooling. For example, a majority of laptop computers use heat pipes to get rid of heat produced in chip processors. The wick heat pipe can operate in any orientation because it uses a wick to distribute the liquid. The principle of wick heat pipe operation is illustrated with the aid of Fig. 1.19. Heat is applied to the evaporator section and is conducted
Figure 1.19 Principle of heat pipe with wick.
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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. 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 and miniature heat pipes are very promising technologies for achieving high local heat removal rates and uniform temperatures in computer chips (Faghri, 1995). 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 μm. Figure 1.20 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. 1.20. 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 μm, which are classified as “true” micro heat pipes. The performance characteristics of the micro heat pipes are different from those of conventional heat pipes.
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Figure 1.20 Schematic of the micro heat pipe
1.5.5 Nanotechnology
Evolving opportunities in nanotechnology include nanoscale devices and systems, identification and understanding of fundamental nanoscale phenomena and processes, nanomaterials, development of new major research facilities, societal dimensions, instrumentation research, and nanomanufacturing. Meanwhile, evolving activities in nanomedicine are characterized by challenging issues related to this multidisciplinary effort. The following examples will illustrate some applications and roles of heat and mass transfer at the nanoscale in nanotechnologies.
Ultrashort Pulse Laser Melting and Resolidification of a Thin Metal Film
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
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molten. When a femtosecond pulse laser is used, the laser intensity can be up to 1021 W/m2. Compared to long pulsed laser processing, short-pulsed laser processing enables users to precisely control the size of the heat-affected zone, the heat rate, and the interfacial velocity. During laser-metal interaction, the laser energy is first deposited into electrons on the metal surface. The excited electrons move into deeper parts of the metal, by ballistic motion, with velocity close to the Fermi velocity (~106 m/s). Meanwhile, those hot electrons are diffused into a deeper part of the electron gas at a speed (<104 m/s) much lower than that of the ballistic motion. The hot electrons also collide with lattices – referred to as electron-lattice coupling – and transfer energy to the lattices (Hohlfeld et al., 2000). 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 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. 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. 1.21 (Wang and Prasad, 2000).
Figure 1.21 Ultrafast laser surface melting of a metallic material (Wang and Prasad, 2000; Reprinted with permission from Elsevier).
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Explosive Boiling during Ultrafast Laser-Materials Interaction
Material removal can be achieved by liquid-vapor phase change by using an ultrafast laser at 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 longpulsed 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 a 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 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 1.22 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.
Figure 1.22 Frozen waves on a silicon surface (Craciun et al., 2002; Reprinted with permission from Elsevier)
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Heat and Mass Transfer in Nanofluid
While most research on heat transfer focuses on enhancement of heat transfer using various techniques, very few people pay attention to the inherently low thermal conductivity of the working fluid. It was demonstrated that dispersion of a tiny amount of nanoparticles in traditional fluids, which results in nanofluids, dramatically increases their thermal conductivities. For example, a small amount (less than 1% volume fraction) of copper nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil can increase their inherently poor thermal conductivity by 40% and 150%, respectively (Eastman et al., 2001; Choi et al., 2001). Keblinski et al. (2002) explored the possible factors influencing the heat transport capability of nanofluids that include (1) Brownian motion of nanoparticles, (2) molecular-level layering of the liquid at the nanoparticle surface, (3) nature of heat transport in nanoparticles, and (4) the effects of nanoparticles clustering. An order of magnitude analysis showed that the Brownian motion of nanoparticles is too slow to transport a significant amount of heat through a nanofluid; this conclusion was also supported by their results of Molecular Dynamics simulation. On the contrary, several empirical correlations to predict nanofluid thermal conductivity have been developed based on Brownian motion of nanoparticles (e.g., Jiang and Choi, 2004; Prasher et al., 2005). The viewpoints about the role of Brownian motion on the thermal conductivity of the nanofluids are still controversial at this time. Xue and Xu (2005) proposed a model of thermal conductivity of nanofluids with interfacial shells by considering the temperature distribution and liquid layering. They predicted the thermal conductivities of Al2O3 /water, CuO/water, and CuO/ethylene glycol nanofluids using a 3-nm interfacial shell thickness. Good agreement with experiments was obtained. The shortcoming of the above model is that the liquid layering thickness cannot be determined by these models and must be obtained by matching the experimental data. On the other hand, Xue et al. (2004) suggested that the liquid layering was not responsible for the large enhancement of the nanofluid thermal conductivity. The role of liquid layering at the nanoparticle surface on the heat transfer enhancement is still debatable. The carriers of heat in the crystalline solid are phonons, i.e., propagation of lattice vibrations. The commonly used diffusive heat transport theory is valid only if the mean free path of the phonons is much less than the size of the crystalline solid. If the mean free path of the crystalline solid is comparable to or greater than the size of the crystalline solid, which is the case for heat conduction in nanoparticles, the diffusive heat transport mechanism is no longer valid and ballistic transport is more realistic. The role of ballistic phonon motion on the enhancement of thermal conductivity has received scant attention. Another possible mechanism for enhancement of thermal conductivity is clustering of nanoparticles in the nanofluids, which could occur if the nanoparticles are not finely dispersed in the base fluid. Xuan et al. (2003) pointed out that during stochastic motion of the suspended nanoparticles, aggregation and
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dispersion may occur among nanoparticle clusters and individual nanoparticles. The role of nanoparticle clustering on the enhancement of the thermal conductivity needs further investigation. Ma et al. (2006) charged nanofluids (HPLC grade water containing 1.0 vol.% 5 – 50 nm of diamond nanoparticles) in a closed-loop copper pulsating heat pipe (PHP) and found that nanofluids significantly enhance the heat transport capability in the PHP. When the power input added on the evaporator is 100 W, the temperature difference between the evaporator and condenser can be reduced from 42 °C to 25 °C by addition of nanoparticles. The synthesis, heat transfer mechanism, and applications of nanofluids have been thoroughly reviewed by Das et al. (2006).
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Cao, Y., and Faghri, A., 1994, “Micro/Miniature Heat Pipes and Operating Limitations,” Journal of Enhanced Heat Transfer, Vol. 1, pp. 265-274. Castellani, C., DiCastro, C., Kotliar, G., and Lee, P.A., 1987, “Thermal Conductivity in Disordered Interacting-Electron Systems,” Physical Review Letters, Vol. 59, pp. 477-480. Cercignan C., 1988, The Boltzmann Equation and its Applications, Springer, New York. Chen, G., 2004, Nano-To-Macro Thermal Transport, Oxford University Press. Chen, G., Yang, B. and Liu, W., 2004, “Nanostructures for Thermoelectric Energy Conversion,” Heat and Fluid Flow in Microscale and Nanoscale Structures, eds. Faghri, M. and Sundén, B., Chapter 2, Southampton, UK. Chen, J.K. and Beraun, J.E., 2003, “Modeling of Ultrashort Laser Ablation of Gold Films in Vacuum,” Journal of Optics. A: Pure Applied Optics, Vol. 5, pp. 168-173. Choi, S. U. S., Zhang, Z. G., Yu, W., Lockwood, F. E. and Grulke, E. A., 2001, “Anomalous Thermal Conductivity Enhancement in Nano-tube Suspensions,” Applied Physics Letters, Vol. 79, pp. 2252-2254. Craciun, V., Bassim, N., Singh, R. K., Craciun, D., Hermann, J., BoulmerLeborgne, C., 2002, “Laser-Induced Explosive Boiling During Nanosecond Laser Ablation of Silicon,” Applied Surface Science, Vol. 186, pp. 288-292. Curtiss, C. F., and Bird, R. B., 1999, “Multicomponent Diffusion,” Industrial and Engineering Chemistry Research, Vol. 38, pp. 2115-2522. Curtiss, C. F., and Bird, R. B., 2001, “Errata,” Industrial and Engineering Chemistry Research, Vol. 40, p. 1791. Das, S.K., Choi, S.U.S., and Patel, H.E., 2006, “Heat Transfer in Nanofluids – a Review,” Heat Transfer Engineering, Vol. 27, pp. 3-9. Eastman, J. A., Choi, S. U. S., Li, S., Yu, W. and Thompson, L. J., 2001, “Anomalously Increased Effective Thermal Conductivities of Ethylene GlycolBased Nano-Fluids Containing Copper Nano-Particles,” Applied Physics Letters, Vol. 78, pp. 718-720. Eijkel, J. C. T., and van den Berg, A., 2005, “Nanofluidics: What Is It and What Can We Expect from It?” Microfluidics and Nanofluidics, Vol. 1, pp. 249-267. Faghri, A., 1995, Heat Pipe Science and Technology, Taylor & Francis, New York. Faghri, A., 2006, “Unresolved Issues in Fuel Cell Modeling,” Heat Transfer Engineering, Vol. 27, pp. 1-3.
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Faghri, A., and Guo, Z., 2005, “Challenges and Opportunities of Thermal Management Issues Related to Fuel Cell Technology and Modeling,” International Journal of Heat Mass Transfer, Vol. 48, pp. 3891-3920. Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Burlington, MA. Flik, M.I., Choi, B.I. and Goodson, K.E., 1992, “Heat Transfer Regimes in Microstructures,” ASME Journal of Heat Transfer, Vol. 114, pp. 666-674. Flik, M.I. and Tien, C.L., 1990, “Size Effect on the Thermal Conductivity of High-Tc Thin-Film Superconductors,” ASME Journal of Heat Transfer, Vol. 112, pp. 872-881. Fuel Cell Handbook, CD, 5th edition, U.S. Department of Energy, Office of Fossil Energy, Federal Energy Technology Center, October 2000. Hohlfeld, J., Wellershoff, S. S., Gudde, J., Conrad, U., Jahnke, V., and Matthias, E., 2000, “Electron and Lattice Dynamics Following Optical Excitation of Metals,” Chemical Physics, Vol. 251, pp. 237-258. Grigoropoulos, C.P., and Ye, M., 2000, “Numerical Methods in Microscale Heat Transfer: Modeling of Phase-Change and Laser Interactions with Materials,” Advances of Numerical Heat Transfer, Vol. 2, eds. W.J. Minkowycz and E.M. Sparrow, pp. 227-257, Taylor & Francis, London, Great Britain. Huxtable, S.T., Abramson, A.R. and Majumdar, A., 2004, “Heat Transport in Superlattices and Nanowires,” Heat and Fluid Flow in Microscale and Nanoscale Structures, eds. Faghri, M. and Sundén, B., Chapter 3, Southampton, UK. Jiang, S.P., and Choi, S.U.S., 2004, “Role of Brownian Motion in the Enhanced Thermal Conductivity of Nanofluids,” Applied Physics Letters, Vol. 84, pp. 4316-4318. Kays, W.M., Crawford, M.E., and Weigand, B., 2004, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY. Keblinksi, P., Phillpot, S.R., Choi, S.U.S., and Eastman, J.A., 2002, “Mechanisms of Heat Flow in Suspensions of Nano-sized Particles (Nanofluids), Int. J. Heat Mass Transfer, Vol. 45, pp. 855-863. Kleijn, C.R., van Der Meer, H., and Hoogendoorn, C.J., 1989, “A Mathematical Model for LPCVD in a Single Wafer Reactor,” Journal of Electrochemical Society, Vol. 136, pp. 3423-3432. Khrustalev, D., and Faghri, A., 1994, “Thermal Analysis of a Micro Heat Pipe,” ASME Journal of Heat Transfer, Vol. 116, pp. 189-198. Kim, M.J., and Majumdar, P., 1995, “Computational Model for High-Energy Laser-Cutting Process,” Numerical Heat Transfer, Part A., Vol. 27, pp. 717-733.
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King, W.P. and Goodson, K.E., 2004, “Thermomechanical Formation and Thermal Detection of Polymer Nanostructures,” Heat and Fluid Flow in Microscale and Nanoscale Structures, eds. Faghri, M. and Sundén, B., Chapter 4, Southampton, UK. Larminie, J. and Dicks, A., 2000, Fuel Cell Systems Explained, Wiley, New York. Lide, D.R. ed., 2009, CRC Handbook of Chemistry and Physics, 89th Ed., CRC Press, Boca Raton, FL. Ma, H. B., Bogmeyer, B., Wison, C, Park, H, Yu, Q, Tirumala, M., Choi, S., 2006, “Nanofluid Effect on the Heat Transport Capability in an Oscillating Heat Pipe,” Applied Physical Letters, Vol. 88, pp. 143116(1-3) Majumdar, A., 1998, “Microscale Energy Transport in Solids,” Microscale Energy Transport, edited by Majumdar, A., Gerner, F., and Tien, C. L., Taylor & Francis, New York. Maruyama, S., 2001, “Molecular Dynamics Method for Microscale Heat Transfer,” Advances in Numerical Heat Transfer, edited by Minkowycz, W.J., and Sparrow, E.M., Taylor & Francis, New York, pp. 189-226. McGrattan, K. B., Baum, H. R., Rehm, R., Forney, G. and Prasad, K., 2002, “The Future of Fire Simulation,” Fire Protection Engineering, Vol. 13, pp. 24-36. McGaughey, A.J.H. and Kaviany, M., 2004, “Quantitative Validation of the Boltzmann Transport Equation Phonon Thermal Conductivity Model under the Single-Mode Relaxation Time Approximation,” Physical Review B, Vol., 69 pp. (094303-1)-(094303-12). Oh, I.H., Nomura, N., Masahashi, N., Hanada, S., 2003, “Mechanical Properties of Porous Titanium Compacts Prepared by Powder Sintering,” Scripta Materialia, Vol. 49, pp. 1197-1202. Pepper, D., 2007, “Creating a Real Time Emergency Response Capability,” Presentation on National Science Foundation Workshop: Frontiers in Transport Phenomena Research and Education, Storrs, CT. Peterson, G.P., 1996, “Modeling, Fabrication, and Testing of Micro Heat Pipes: An Update,” Applied Mechanics Review, Vol. 49, Part 2, pp. 175-183. Peterson, R.B., 2004, “Miniature and Microscale Energy Systems,” Heat and Fluid Flow in Microscale and Nanoscale Structures, eds. Faghri, M. and Sundén, B., Chapter 1, Southampton, UK. Poetzsch, R.H.H. and Böttger, H., 1994, “Interplay of Disorder and Anharmonicity in Heat Conduction: Molecular-Dynamics Study,” Physical Review B, Vol. 50, pp.15757-15763. Poling, B.E., Prausnitz, J. M., and O’Connell, J.P., 2000, The Properties of Gases and Liquids, 5th edition, McGraw-Hill, New York, NY.
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Problems
1.1. Why can severe skin burns be caused by hot steam? 1.2. When 0.01 kg of ice at 0 °C is mixed with 0.1 kg of steam at 100 °C and 1 atm, what is the phase of the final mixture? 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 of 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.
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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. Use simple kinetic theory of gases discussed in Section 1.3.2 as well as Newton’s law of viscosity to prove eq. (1.31). 1.6. Use simple kinetic theory of gases discussed in Section 1.3.2 as well as Fourier’s law of conduction to prove eq. (1.32). 1.7. In the final moments of Julius Caesar’s life, he gasped to his assailant, “Et tu, Brute!” 2000 years later, the inert N2 molecules that he released in his final breath have diffused uniformly throughout the world. What are the odds that people today are breathing those same molecules? Assume the tidal capacity of the human lung to be 500cm3 for average breathing and the world air volume to equal the volume at standard conditions of a spherical shell 13,000 km in diameter and 10 km thick (= 5 ×1024 cm3) . 1.8. Compare the mass fluxes of an ideal gas from a tank in the case of a very large hole and that of a very small hole. If the hole is much larger than the mean free path, a gross motion of the gas will be induced and kinetic theory will not be valid, so you will need to use the Bernoulli equation, which gives:
Δp m′′large=ρ 2 ρ
1.9. A common device to measure fluid viscosity consists of an annulus cylinder where the inner cylinder is rotating with an angular velocity Ω and the outer cylinder is stationary. Determine an equation relating the power required to turn the cylinder as a function of the fluid viscosity (μ), the inner and outer radius (Ri and Ro) and the annulus length (L). 1.10. A room wall consists of a layer of dry wall 3/8’’ thick, a layer of stationary air 3.5’’ thick and a 3/8’’ thick layer of plywood. The inside room temperature is 300K and the outside room temperature is 270K. Determine the heat flux through the wall assuming the dry wall and plywood thermal conductivity are 0.17 W/m-K and 0.12 W/m-K, respectively. 1.11. The Lennard-Jones potential depends primarily on the distance between molecules. Under what conditions is the Lennard-Jones potential for molecules not appropriate? 1.12. 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. 1.13. Verify that the following equation is valid for a binary system:
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J1
ρω1ω2
=
* J1 cx1 x 2
1.14. Prove that the sum of mass fluxes by diffusion for a multicomponent mixture is zero.
J
i =1
N
i
=0
Show that the sum of mass fluxes relative to a stationary coordinate axes for multicomponent mixtures is nonzero. 1.15. Show that the Maxwell-Stefan equation (1.111) for a multicomponent system can be reduced to Fick’s law, eq. (1.105), for a binary system. 1.16. The diffusive heat and mass flux vectors in a multicomponent system were expressed by eqs. (1.71) and (1.118), respectively. What are the heat and mass fluxes across a surface whose normal direction is n? y L Mixture of gases A+B
A A m′′ or n′′
Figure P1.1
1.17. 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. 1.18. Hydrogen gas is contained in a silicon spherical tank with a 0.05’’ thickness. The inside tank temperature is 1500K and a pressure of 10 atm. Determine the mass flux through the wall of the tank assuming the mass diffusion coefficient D =2 x 10-8 m2/sec and the outside pressure is 2 atm. 1.19. The heat transfer coefficient of a mixed forced and natural convection of a compressible fluid in a tube can be expressed as h = h(k, μ , c p , ρ , U ,
g, β , ΔT , D, L, c ) where β is thermal expansion coefficient, and c is local
speed of sound. Using the Buckingham’s Π theorem to show that that seven dimensionless variables are needed to describe the compressible mixed convection problem, i.e., Nu = f (Re, Gr, Pr, L / D, Ma, Fr) . 1.20. 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
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Amir Faghri, Yuwen Zhang, and John Howell
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difference ΔT , the buoyancy force gβΔT , and the latent heat of fusion hs , i.e., h = h(k, μ , c p , ρ , L, ΔT , gβ ΔT , hs , t ) . Using the Buckingham’s Π theorem show that the Nusselt number can be expressed as Nu = f(Gr, Pr, Ste, Fo). 1.21. 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. 1.22. Figure P1.2 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. Perform a scale analysis and show that the order of the magnitude of this velocity is
k ΔT Vs ρ s hs
3/ 4
p 2 μL
1/4
Figure P1.2
1.23. A solid PCM with a uniform initial temperature at the melting point, Tm , is in a space defined by, 0 ≤ x < ∞. 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. 1.24. 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 40m length. 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.
86 Advanced Heat and Mass Transfer
Amir Faghri, Yuwen Zhang, and John Howell
Copyright © 2010 Global Digital Press
1.25. 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. 1.26. 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? 1.27. Calculate the sublimation rate (in kg / s -m ) for a moth repellant to air. The moth repellant is a long solid naphthalene cylinder, with circular cross section of a 15 mm diameter. Naphthalene has a vapor molar concentration on the outer surface of 5 ⋅10−6 kmole/m3 and has a molecular weight of 128 kg/kmole. Assume an average mass transfer coefficient of 0.6m/s. 1.28. It is important to control the temperature of a fuel cell on the cathode side so that it contains saturated water vapor and therefore no flooding occurs and air passes through the cathode gas diffusion layer. Calculate the heat transfer coefficient, as well as the fuel cell efficiency, for a PEM fuel cell having 60mm × 60mm MEA and operates with a cell voltage of 0.65 volts at 14A current. The fuel cell’s operating temperature on the cathode side is 60°C and the fuel cell losses 10W of heat to the ambient at 25°C. 1.29. Calculate the heat flux due to convection, evaporation, and radiation from a plate wetted with a thin liquid water layer left outside with ambient air temperature of 17o C and humidity of 70% blowing over the plate. Do your results suggest why the sidewalks/driveways are sometimes wet instead of dry during the spring in the morning when it has not rained during the night? Assume average heat transfer coefficient due to wind motion is 50W/m2-k, ε = 0.95 , the liquid layer surface temperature is 2 C and effective sky temperature is −30o C . 1.30. Calculate the highest air temperature that will cause a thin layer of water to freeze outside on a clear night, when the sky temperature is −30o C and the mean heat transfer coefficient due to wind motion is h = 35W/m 2 -K . Assume air is dry, ε = 1 , and the heat transfer between the water and ground is negligible. Do your results suggest why, on a clear night, the air temperature need not drop below 0o C before a thin layer of water outside will freeze? 1.31. A water droplet is placed in dry stagnant air of uniform temperature, T∞ . Show that
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Amir Faghri, Yuwen Zhang, and John Howell
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(a)
kair ω − ω1∞ Nu ( T∞ − Tsat ) = D12Sh 1δ hv ρair 1 − ω1δ
(b) c p,air
ω1δ − ω1∞ 1 − ω1δ What are the assumptions made to produce the results found in (a) and (b)?
hv
(T∞ − Tsat )
= Le
1.32. Compare the orders of magnitude of the Prandtl numbers for various gases and liquids. Can any conclusions be made? 1.33. Example 1.6 considered the similarity between scale and original models for steady flow. For oscillatory blood flow, it is also necessary for the Wormersley numbers (see Table 1.7) of original and scaled models to be the same. If the frequencies of the oscillatory flow for the original and the scaled models are the same, discuss if the fluid (glycol-water mixture) in Example 1.6 can be used. 1.34. 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? 1.35. The temperature and dew point of a city in summer are 80 °F and 40 °F, respectively. What is the relative humidity? 1.36. 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? 1.37. The working fluid for a high-temperature heat pipe is liquid metal, which is solid at room temperature. Qualitatively analyze the phase change phenomena involved during the start-up and shut-down of the high temperature heat pipe. 1.38. What is the advantage of a fuel cell over the conventional battery and thermodynamic cycle? Why is heat transfer important in a fuel cell?
88 Advanced Heat and Mass Transfer
Amir Faghri, Yuwen Zhang, and John Howell
Copyright © 2010 Global Digital Press