Advanced Heat and Mass Transfer - Chapter 1

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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, security , information i nformation 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 single topics developed over 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

Chapter 1 Introduction

Amir Faghri, Yuwen Zhang , and John Howell

Copyright © 2010 Globa l Digital Press

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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 t he 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, l iquid, 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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Advanced Heat and Mass Transfer



understand the operational principles of these physical systems and devices for further application in homeworks and examples in i n 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 T 1 to T 2 is proportional to the mass of the system and the temperature rise, i.e., (1.1) Q = mc (T2 − T1 )




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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 t he temperature of a unit mass of the substance by one degree. For example, for 1.00 kcal kcal/kg /kg--  C = 4.18 4.18 kJ/ kJ/kg kg-- C , which means that it takes 1 water at 15 °C, c = 1.00 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 c p − cv = Rg (1.2) where R g , the gas constant, and is related to the universal gas constant, Ru, by Rg = Ru / M with M being 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 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) (kg/kmol)

c p (kJ/ (kJ/kkg- o C)

(kJ/kg- o C) cv (kJ

28.97 26.9815 44.01 63.546

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

0.718

2.016 18.015 55.847 207.2 28.013 31.999 18.015 107.868 200.59 18.015

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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120 120 ) C o ( e r u t a r e p m e T

100 100 80 60 40 20 0 -20 0

500

1000

1500

2000

2500

3000

Heat Added Add ed (kJ) (kJ) Figure 1.2 Temperature profile for phase change from subcooled ice to superheated steam.

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

hv (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 hv . 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  v q = mh (1.3) where m is the mass of material changing phase per unit time. The latent heat of vaporization, hv , is the difference between the enthalpy of vapor and of liquid, i.e., (1.4) hv = hv − h 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)

The total volume of the liquid and vapor V = m v + mv vv

(1.6)

remains constant during the phase change process. Thus dmv dm v + vv = 0 dt dt

(1.7)

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 dE cv  v = q − mh dt

(1.10)

where the internal energy of the control volume is Ecv = m e + mv ev (1.11) Differentiating eq. (1.11) and considering eqs. (1.8) and (1.9), one obtains  v e − v ev dEcv   v  = −m dt  vv − v

  

(1.12)

Substituting eq. (1.12) into eq. (1.10) and considering eq. (1.4) yields 







 hv +  h − q=m

vv e − v ev 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 crit ical point. Therefore, eq. (1.3) is usually valid val id for constant-pressure liquid-vapor phase change processes. During melting and solidification processes, heat transfer can be expressed as  s q = mh (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 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 properties of the substance is a molecule of 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 10−10 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 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, 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 1.6605402 × 10−27 kg. This mass is 1/12 of the mass of the carbon-12 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. phase. Plasma is sometimes called the fourth phase of matter. In the description of matter, phase




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indicates how particles group group together to form a substance. 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. motion. This motion depends 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 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. ˚

10 Advanced Heat and Mass Transfer



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=

8kbT π m

(1.15)

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 unit area on one side is given by f =

1 N c 4

(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: τ =

λ c

(1.18)

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

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

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σ ×10 (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

c (m/s) 467 397 627 474 284 379 298 1769 1256 475 609 444 313

τ (ps) 148 182 84 78 569 119 119 100 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):  molecules ′′ = m

N cm

4

1/ 2

kT  =N m b   2π m 

1/ 2

 k Tm  = N  b   2π 

(1.19)

The pressure of gas in a container results from the large number of gas molecules colliding with the container wall. Although each molecule in the container collides with the container wall randomly and discontinuously, the collisions of a large number of molecules with the container wall impart a constant and continuous pressure on the wall. As expected, the pressure in a container is related to the number, and average velocity, of the molecules by 1 c2 p = Nm V 3

(1.20)

where N is the number of molecules in the container, m is the mass of each molecule, V is is the volume of the container, and c 2 is the average of the square of the molecular velocity. velocity. 2

c =

1

N

c N 

2

n

(1.21)

n =1

The average of the square of the molecules’ velocity is related to its three components by 1 3

u 2 = v2 = w 2 = c 2

(1.22)

The average kinetic energy of a molecule is defined as




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) pV = nRu T 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 E =

3 Ru T 2 N A

(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 k T 2 b

(1.27)

where the Boltzmann constant is Ru 8.3143 −23 = = × 1 . 3 8 1 0 J/K N A 6.022 × 1023 The specific heat at constant volume, cv, is given by kinetic theory as 3 kb cv = 2m kb =

(1.28)

(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 wate r 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 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 =

mkbT

2 3π 3/ 2

(1.31)

kb3T m

(1.32)

σ 2

1 π 3/ 2σ 2

2

kb3T 3

3π 3/ 2σ 2 P

m

(1.33)

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 t he 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.




Figure 1.4 Modes of molecular kinetic energy: (a) rotational energy, (b) vibrational 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 equili brium and therefore it is for time ti me 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 p 1.013 × 105 Pa = = 2.46 × 1025 molecules/m3 N = −23 kbT (1.381× 10 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 2N

=

1

−9 = × 6 6 1 0 m = 66nm 2π × (3.7 × 10 10−10 )2 × 2.46 × 10 1025

The average magnitude of the molecular velocity can be obtained from eq. (1.15), i.e., c=

8 × 1.381× 10−23 × 298.15 = = 465m/s π m 1.66 × 10-27 π × 28.97 × 1.66

8kbT

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 τ =

λ = 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 forces of a solid are greater than those of a liquid. This trend can be observed when looking looking at the force it takes to separate a solid as compared to that required to separate a liquid. Also, the molecules in a solid are much more confined to their position in the solid’s structure as compared to the molecules of a liquid, thereby affecting their ability to move. Most solids and liquids are deemed incompressible. The underlying reason for their “incompressibility” is that the molecules repel each other when they are forced closer than their normal spacing; the closer they become, the greater the repelling force (Tien and Lienhard, 1979). A gas differs from both a solid and a liquid in that its kinetic energy is great enough to overcome the intermolecular forces, causing the molecules to separate without restraint. The intermolecular forces in a gas decrease as the distance between the molecules increases. Both gravitational and electrical forces contribute to intermolecular forces; for many solids and liquids, the electrical forces are on the order of 1029 times greater than the gravitational force. Therefore, the gravitational forces are typically ignored.




To quantify the intermolecular forces, a potential function φ (r ) is defined as the energy required to bring 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:  r 12  r 6  φ (r ) = 4ε  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 dr

(1.35)

Substituting eq. (1.34) into eq. (1.35), one obtains,




17

φ

Molecules repulse one another for r < < r min min

r ← → ← • •  →

F

0

F = −∇ φ (r )

r 0

−ε

F

r

Molecules attract another for r > > r min min

one

Figure 1.5 Lennard-Jones 6-12 potential vs. distance between two spherical, nonpolar molecules.

rmin = 21/6 r0 ≈ 1.12r0

(1.36) For typical gas molecules, r 0 ranges from 0.25 to 0.4 nm, which result a range from 0.28 to 0.45 nm for r min min. 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 example that requires 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 i nvolves extremely high rates of heating (on the t he order of 1016 K/s) and high temperature gradients (on the order of 10 11 K/m). The motion of each molecule (i) in the system is described by Newton’s second law, i.e., N



Fij = mi

j =1( j ≠i )

d 2ri dt 2

, i = 1,2,3...n

(1.38)




where mi and ri are the mass and position of the 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 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 

12 6   r0    −     rij       The force between the ith and jth molecules can be obtained from  r 13  r 7  r 24ε  ο ij Fij = −∇φ ij =   −  ο   rο  rij   rij   rij  where rij is the distance between the ith and jth molecules.

(1.39)

(1.40)

The transport properties can be obtained by the kinetic theory in the form of very complicated multiple integrals that involve intermolecular forces. For a non polar 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) D =

3 π mkbT 1 8 πσ 2 Ω D ρ

(1.41)

µ =

5 π mkbT 16 πσ 2 Ω µ

(1.42)

25 π mkbT cv 32 πσ 2 Ωk

(1.43)

k=

where σ is collision diameter, and cv in is the molar specific heat under constant volume. The dimensionless collision integrals are related by Ω μ = Ωk ≈ 1.1Ω D and are slow varying functions of kbT /ε ( ε is a characteristic energy of molecular interaction). If all molecules can be assumed 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 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 nonreacting monatomic gas gas mixtures, the random molecular movement can be described by the molecular velocity distribution function f i (c , x, t ) , where c is the particle velocity and x is the




19

position vector in the mixture. At time t , the probable number of molecules of the ith species that are located in the volume element d x at position x and have velocity within the range d c 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 ∂f i = + c ⋅ ∇ x fi + a ⋅ ∇ c f f = Ω i ( f ) ∂t Dt

(1.44)

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 f i. 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 )dcd x (1.46)

 V

c

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 f i. 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 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 cohesion is significant only when when the molecules are are extremely close together; for instance, once a crack forms in a metal structure,




the two edges of the crack will not rejoin even if pushed together. The rejoining can not occur because gas molecules attach to the fractured surface, preventing the cohesive intermolecular attraction from occurring. The fundamental basis for viscosity observed in fluids is cohesion within the fluids. Viscosity is the resistance of a liquid or a gas to shear forces; it can be 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 phenomenon of water wetting a glass surface. Intermolecular forces between the water 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 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 adhesion between the liquid and 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 H 2O 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




21

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

Approximate magnitude per molecule (eV) 0.0054 0.58 0.06 0.39 0.49

Types of energy Lattice vibration Intermolecular hydrogen bond breaking Enthalpy of melting Enthalpy of vaporization Enthalpy of sublimation

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 k b 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: ∆s =

∆h

T

= constant

(1.48)

where ∆h 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 =

hv T sat

 83.7J/(mol-K)

(1.49)

while Richards’ rule is valid for melting and solidification




s − ss =

hs  8.37J/(mol-K) T m

(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 Kn =

λ L

(1.51)

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:




23

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 ( 0.00110 ). 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, 0.001, i.e., ρ tr =

1.051kb 2πσ 2 Rg D Kn

(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, T tr tr , corresponding to the transition density can be obtained by using the Clausius-Clapeyron equation combined with the equation of state: T tr =

 h  1 psat 1 − exp  − v  ρ Rg  Rg  Ttr T sa sat

   

(1.54)

where p sat and T sat are the saturation pressure and temperature, hv 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  hv  1 1 − ln  +  T sat tr  psat  Rg  Ttr

  = 0 

(1.55)

and solved iteratively for T tr using the Newton-Raphson/secant method. The transition vapor temperature is the boundary between the continuum and noncontinuum regimes.




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. plat e. 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/m2 ), 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, τ yx yx , 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)

y x

τ yx Figure 1.6 Couette flow.




25

molecular mass less than 5000 is well presented by eq. (1.56). For non Newtonian fluids, such as polymeric liquids, slurries, or other complex fluids, for example in biological applications, such as blood or operation of joints, 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, − I , and a viscous stress, τ : (1.57) τ ′ = − pI + τ 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 τ τ  yx yy yz   τ zx τ zy τ zz 

(1.59)

where the first subscript represents the axis axi s normal to the face on which the t he 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 y = τ yx , τ xz = τ zx , and τ yz = τ zy ). τ zz τ zy

τ zx

τ yz τ xz

τ yy τ xy

τ xx

τ yx y

x

Figure 1.7 Components of the stress tensor in a fluid




The viscous stress tensor can be expressed by Newton’s law of viscosity: τ

= 2 µ D −

2 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 1  ∂u ∂v   +  2  ∂y ∂x  ∂ x   1  ∂v ∂u  ∂v 1 T + D = ∇V + ( ∇V )  =     ∂y 2  2  ∂ x ∂y   1  ∂w ∂u  1  ∂w ∂v    + +   2  ∂ x ∂z  2  ∂y ∂z 

1  ∂u ∂ w    +  2  ∂z ∂x  

 1  ∂v ∂w   +  2  ∂z ∂y    ∂w  ∂z 

(1.61)

where ( ∇V )T 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

−

∂ x

2  ∂u ∂v ∂w  + + µ   3  ∂x ∂y ∂z 

(1.62)

 ∂u ∂v  + τ xy = µ   ∂ ∂ y x  

(1.63)

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

µ =

 i =1

x i µ i



N

(1.64)

x φ j =1 i ij

where xi is the molar fraction of component i and 2

  1/ 2  M 1/4  1 +  µ i   j   (1.65)   µ j   M i     th where M i is the molecular mass of the i species. Equation (1.64) can reproduce

1  M i φ ij = 1 + 8  M j

   

−1/ 2

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).




27

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 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

T 2 L T ( y y)

T 1

T

Figure 1.8 One-dimensional conduction.




Advanced Heat and Mass Transfer - Chapter 1

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