A heat transfer textbook

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A HEAT TRANSFER THIRD TEXTBOOK EDITION John H Lienhard IV / John H Lienhard V A Heat Transfer Textbook  Lienhard & Lienhard Phlogiston Press ISBN 0-9713835-0-2 PSB 01-04-0249 A Heat Transfer Textbook A Heat Transfer Textbook Third Edition by John H Lienhard IV and John H Lienhard V Phlogiston Press Cambridge Massachusetts Professor John H Lienhard IV Department of Mechanical Engineering University of Houston 4800 Calhoun Road Houston TX 77204-4792 U.S.A Professor John H Lienhard V Department of Mechanical Engineering Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge MA 02139-4307 U.S.A Copyright ©2004 by John H Lienhard IV and John H Lienhard V All rights reserved Please note that this material is copyrighted under U.S Copyright Law The authors grant you the right to download and print it for your personal use or for non-profit instructional use Any other use, including copying, distributing or modifying the work for commercial purposes, is subject to the restrictions of U.S Copyright Law International copyright is subject to the Berne International Copyright Convention The authors have used their best efforts to ensure the accuracy of the methods, equations, and data described in this book, but they not guarantee them for any particular purpose The authors and publisher offer no warranties or representations, nor they accept any liabilities with respect to the use of this information Please report any errata to the authors Lienhard, John H., 1930– A heat transfer textbook / John H Lienhard IV and John H Lienhard V — 3rd ed — Cambridge, MA : Phlogiston Press, c2004 Includes bibliographic references and index Heat—Transmission Mass Transfer I Lienhard, John H., V, 1961– II Title TJ260.L445 2004 Published by Phlogiston Press Cambridge, Massachusetts, U.S.A This book was typeset in Lucida Bright and Lucida New Math fonts (designed by Bigelow & Holmes) using LATEX under the Y&Y TEX System For updates and information, visit: http://web.mit.edu/lienhard/www/ahtt.html This copy is: Version 1.22 dated January 5, 2004 Preface This book is meant for students in their introductory heat transfer course — students who have learned calculus (through ordinary differential equations) and basic thermodynamics We include the needed background in fluid mechanics, although students will be better off if they have had an introductory course in fluids An integrated introductory course in thermofluid engineering should also be a sufficient background for the material here Our major objectives in rewriting the 1987 edition have been to bring the material up to date and make it as clear as possible We have substantially revised the coverage of thermal radiation, unsteady conduction, and mass transfer We have replaced most of the old physical property data with the latest reference data New correlations have been introduced for forced and natural convection and for convective boiling The treatment of thermal resistance has been reorganized Dozens of new problems have been added And we have revised the treatment of turbulent heat transfer to include the use of the law of the wall In a number of places we have rearranged material to make it flow better, and we have made many hundreds of small changes and corrections so that the text will be more comfortable and reliable Lastly, we have eliminated Roger Eichhorn’s fine chapter on numerical analysis, since that topic is now most often covered in specialized courses on computation This book reflects certain viewpoints that instructors and students alike should understand The first is that ideas once learned should not be forgotten We have thus taken care to use material from the earlier parts of the book in the parts that follow them Two exceptions to this are Chapter 10 on thermal radiation, which may safely be taught at any point following Chapter 2, and Chapter 11 on mass transfer, which draws only on material through Chapter v vi We believe that students must develop confidence in their own ability to invent means for solving problems The examples in the text therefore not provide complete patterns for solving the end-of-chapter problems Students who study and absorb the text should have no unusual trouble in working the problems The problems vary in the demand that they lay on the student, and we hope that each instructor will select those that best challenge their own students The first three chapters form a minicourse in heat transfer, which is applied in all subsequent chapters Students who have had a previous integrated course thermofluids may be familiar with this material, but to most students it will be new This minicourse includes the study of heat exchangers, which can be understood with only the concept of the overall heat transfer coefficient and the first law of thermodynamics We have consistently found that students new to the subject are greatly encouraged when they encounter a solid application of the material, such as heat exchangers, early in the course The details of heat exchanger design obviously require an understanding of more advanced concepts — fins, entry lengths, and so forth Such issues are best introduced after the fundamental purposes of heat exchangers are understood, and we develop their application to heat exchangers in later chapters This book contains more material than most teachers can cover in three semester-hours or four quarter-hours of instruction Typical onesemester coverage might include Chapters through (perhaps skipping some of the more specialized material in Chapters 5, 7, and 8), a bit of Chapter 9, and the first four sections of Chapter 10 We are grateful to the Dell Computer Corporation’s STAR Program, the Keck Foundation, and the M.D Anderson Foundation for their partial support of this project JHL IV, Houston, Texas JHL V, Cambridge, Massachusetts August 2003 Contents I The General Problem of Heat Exchange Introduction 1.1 Heat transfer 1.2 Relation of heat transfer to thermodynamics 1.3 Modes of heat transfer 1.4 A look ahead 1.5 Problems Problems References Heat conduction concepts, thermal resistance, and the overall heat transfer coefficient 2.1 The heat diffusion equation 2.2 Solutions of the heat diffusion equation 2.3 Thermal resistance and the electrical analogy 2.4 Overall heat transfer coefficient, U 2.5 Summary Problems References Heat exchanger design 3.1 Function and configuration of heat exchangers 3.2 Evaluation of the mean temperature difference in a heat exchanger 3.3 Heat exchanger effectiveness 3.4 Heat exchanger design Problems References 3 10 35 36 37 46 49 49 58 62 78 86 86 96 99 99 103 120 126 129 136 vii Contents viii II Analysis of Heat Conduction Analysis of heat conduction and some steady one-dimensional problems 4.1 The well-posed problem 4.2 The general solution 4.3 Dimensional analysis 4.4 An illustration of dimensional analysis in a complex steady conduction problem 4.5 Fin design Problems References III Transient and multidimensional heat conduction 5.1 Introduction 5.2 Lumped-capacity solutions 5.3 Transient conduction in a one-dimensional slab 5.4 Temperature-response charts 5.5 One-term solutions 5.6 Transient heat conduction to a semi-infinite region 5.7 Steady multidimensional heat conduction 5.8 Transient multidimensional heat conduction Problems References Convective Heat Transfer 139 141 141 143 150 159 163 183 190 193 193 194 203 208 218 220 235 247 252 265 267 Laminar and turbulent boundary layers 269 6.1 Some introductory ideas 269 6.2 Laminar incompressible boundary layer on a flat surface 276 6.3 The energy equation 292 6.4 The Prandtl number and the boundary layer thicknesses 296 6.5 Heat transfer coefficient for laminar, incompressible flow over a flat surface 300 6.6 The Reynolds analogy 311 6.7 Turbulent boundary layers 313 6.8 Heat transfer in turbulent boundary layers 322 Problems 330 References 338 610 An introduction to mass transfer §11.3 A kinetic model of diffusion Diffusion coefficients depend upon composition, temperature, and pressure Equations that predict D12 and Dim are given in Section 11.4 For now, let us see how Fick’s law arises from the same sort of elementary molecular kinetics that gave Fourier’s and Newton’s laws in Section 6.4 We consider a two-component dilute gas (one with a low density) in which the molecules A of one species are very similar to the molecules A of a second species (as though some of the molecules of a pure gas had merely been labeled without changing their properties.) The resulting process is called self-diffusion If we have a one-dimensional concentration distribution, as shown in Fig 11.5, molecules of A diffuse down their concentration gradient in the x-direction This process is entirely analogous to the transport of energy and momentum shown in Fig 6.13 We take the temperature and pressure of the mixture (and thus its number density) to be uniform and the mass-average velocity to be zero Individual molecules move at a speed C, which varies randomly from molecule to molecule and is called the thermal or peculiar speed The average speed of the molecules is C The average rate at which molecules cross the plane x = x0 in either direction is proportional to N C, where N is the number density (molecules/m3 ) Prior to crossing the x0 -plane, the molecules travel a distance close to one mean free path, —call it a, where a is a number on the order of one The molecular flux travelling rightward across x0 , from its plane of origin at x0 − a, then has a fraction of molecules of A equal to the value of NA /N at x0 − a The leftward flux, from x0 + a, has a fraction equal to the value of NA /N at x0 + a Since the mass of a molecule of A is MA /NA (where NA is Avogadro’s number), the net mass flux in the x-direction is then   jA  x0   NA   − N x0 +a x0 −a  M  N  A A  = η NC N N  A (11.30) where η is a constant of proportionality Since NA /N changes little in a distance of two mean free paths (in most real situations), we can expand the right side of eqn (11.30) in a two-term Taylor series expansion about Diffusion fluxes and Fick’s law §11.3 611 Figure 11.5 One-dimensional diffusion x0 and obtain Fick’s law:   jA  x0   d(NA /N )   = η NC −2a  NA dx x0  dmA   = −2ηa(C)ρ dx  M   A (11.31) x0 (for details, see Problem 11.6) Thus, we identify DAA = (2ηa)C (11.32) and Fick’s law takes the form jA = −ρDAA dmA dx (11.33) The constant, ηa, in eqn (11.32) can be fixed only with the help of a more detailed kinetic theory calculation [11.2], the result of which is given in Section 11.4 The choice of ji and mi for the description of diffusion is really somewhat arbitrary The molar diffusion flux, Ji∗ , and the mole fraction, xi , are often used instead, in which case Fick’s law reads ∗ J
i = −cDim ∇xi (11.34) Obtaining eqn (11.34) from eqn (11.27) for a binary mixture is left as an exercise (Problem 11.4) 612 An introduction to mass transfer §11.3 Typical values of the diffusion coefficient Fick’s law works well in low density gases and in dilute liquid and solid solutions, but for concetrated liquid and solid solutions the diffusion coefficient is found to vary with the concentration of the diffusing species In part, the concentration dependence of those diffusion coefficients reflects the inadequacy of the concentration gradient in representing the driving force for diffusion in nondilute solutions Gradients in the chemical potential actually drive diffusion In concentrated liquid or solid solutions, chemical potential gradients are not always equivalent to concentration gradients [11.3, 11.4, 11.5] Table 11.1 lists some experimental values of the diffusion coefficient in binary gas mixtures and dilute liquid solutions For gases, the diffusion coefficient is typically on the order of 10−5 m2 /s near room temperature For liquids, the diffusion coefficient is much smaller, on the order of 10−9 m2 /s near room temperature For both liquids and gases, the diffusion coefficient rises with increasing temperature Typical diffusion coefficients in solids (not listed) may range from about 10−20 to about 10−9 m2 /s, depending upon what substances are involved and the temperature [11.6] The diffusion of water vapor through air is of particular technical importance, and it is therefore useful to have an empirical correlation specifically for that mixture:   2.072 −10 T for 282 K ≤ T ≤ 450 K (11.35) DH2 O,air = 1.87 × 10 p where DH2 O,air is in m2 /s, T is in kelvin, and p is in atm [11.7] The scatter of the available data around this equation is about 10% Coupled diffusion phenomena Mass diffusion can be driven by factors other than concentration gradients, although the latter are of primary importance For example, temperature gradients can induce mass diffusion in a process known as thermal diffusion or the Soret effect The diffusional mass flux resulting from both temperature and concentration gradients in a binary gas mixture is then [11.2] M1 M2 k ∇ ln(T ) (11.36) j
i = −ρD12 ∇m1 + T M2 Diffusion fluxes and Fick’s law §11.3 Table 11.1 Typical diffusion coefficients for binary gas mixtures at atm and dilute liquid solutions [11.4] Gas mixture air-carbon dioxide air-ethanol air-helium air-napthalene air-water T (K) 276 313 276 303 313 D12 (m2/s) 1.42×10−5 1.45 6.24 0.86 2.88 argon-helium 295 628 1068 8.3 32.1 81.0 (dilute solute, 1)-(liquid solvent, 2) T (K) D12 (m2/s) ethanol-benzene benzene-ethanol water-ethanol carbon dioxide-water ethanol-water 288 298 298 298 288 2.25×10−9 1.81 1.24 2.00 1.00 methane-water 275 333 0.85 3.55 pyridene-water 288 0.58 where kT is called the thermal diffusion ratio and is generally quite small Thermal diffusion is occasionally used in chemical separation processes Pressure gradients and body forces acting unequally on the different species can also cause diffusion Again, these effects are normally small A related phenomenon is the generation of a heat flux by a concentration gradient (as distinct from heat convected by diffusing mass), called the diffusion-thermo or Dufour effect In this chapter, we deal only with mass transfer produced by concentration gradients 613 An introduction to mass transfer 614 11.4 §11.4 Transport properties of mixtures6 Direct measurements of mixture transport properties are not always available for the temperature, pressure, or composition of interest Thus, we must often rely upon theoretical predictions or experimental correlations for estimating mixture properties In this section, we discuss methods for computing Dim , k, and µ in gas mixtures using equations from kinetic theory—particularly the Chapman-Enskog theory [11.2, 11.8, 11.9] We also consider some methods for computing D12 in dilute liquid solutions The diffusion coefficient for binary gas mixtures As a starting point, we return to our simple model for the self-diffusion coefficient of a dilute gas, eqn (11.32) We can approximate the average molecular speed, C, by Maxwell’s equilibrium formula (see, e.g., [11.9]):  C= 8kB NA T πM 1/2 (11.37) where kB = R ◦ /NA is Boltzmann’s constant If we assume the molecules to be rigid and spherical, then the mean free path turns out to be = kB T = √ 2 π 2d p π 2N d √ (11.38) where d is the effective molecular diameter Substituting these values of C and  in eqn (11.32) and applying a kinetic theory calculation that shows 2ηa = 1/2, we find DAA = (2ηa)C = (kB /π )3/2 d2  NA M 1/2 T 3/2 p (11.39) The diffusion coefficient varies as p −1 and T 3/2 , based on the simple model for self-diffusion To get a more accurate result, we must take account of the fact that molecules are not really hard spheres We also have to allow for differences in the molecular sizes of different species and for nonuniformities This section may be omitted without loss of continuity The property predictions of this section are used only in Examples 11.11, 11.14, and 11.16, and in some of the end-of-chapter problems Transport properties of mixtures §11.4 Figure 11.6 potential 615 The Lennard-Jones in the bulk properties of the gas The Chapman-Enskog kinetic theory takes all these factors into account [11.8], resulting in the following formula for DAB : DAB = (1.8583 × 10−7 )T 3/2 AB pΩD (T ) 1 + MA MB where the units of p, T , and DAB are atm, K, and m2/s, respectively The AB (T ) describes the collisions between molecules of A and B function ΩD It depends, in general, on the specific type of molecules involved and the temperature The type of molecule matters because of the intermolecular forces of attraction and repulsion that arise when molecules collide A good approximation to those forces is given by the Lennard-Jones intermolecular potential (see Fig 11.6.) This potential is based on two parameters, a molecular diameter, σ , and a potential well depth, ε The potential well depth is the energy required to separate two molecules from one another Both constants can be inferred from physical property data Some values are given in Table 11.2 together with the associated molecular weights (from [11.10], with values for calculating the diffusion coefficients of water from [11.11]) An introduction to mass transfer 616 §11.4 Table 11.2 Lennard-Jones constants and molecular weights of selected species  Species σ (Å) ε/kB (K) Al Air Ar Br2 C CCl2 F2 CCl4 CH3 OH CH4 CN CO CO2 C H6 C2 H5 OH CH3 COCH3 C H6 Cl2 F2 2.655 3.711 3.542 4.296 3.385 5.25 5.947 3.626 3.758 3.856 3.690 3.941 4.443 4.530 4.600 5.349 4.217 3.357 2750 78.6 93.3 507.9 30.6 253 322.7 481.8 148.6 75.0 91.7 195.2 215.7 362.6 560.2 412.3 316.0 112.6 a b M kg kmol  26.98 28.96 39.95 159.8 12.01 120.9 153.8 32.04 16.04 26.02 28.01 44.01 30.07 46.07 58.08 78.11 70.91 38.00  Species σ (Å) ε/kB (K) H2 H2 O H2 O H O2 H2 S He Hg I2 Kr Mg NH3 N2 N2 O Ne O2 SO2 Xe 2.827 2.655a 2.641b 4.196 3.623 2.551 2.969 5.160 3.655 2.926 2.900 3.798 3.828 2.820 3.467 4.112 4.047 59.7 363a 809.1b 289.3 301.1 10.22 750 474.2 178.9 1614 558.3 71.4 232.4 32.8 106.7 335.4 231.0 M kg kmol  2.016 18.02 34.01 34.08 4.003 200.6 253.8 83.80 24.31 17.03 28.01 44.01 20.18 32.00 64.06 131.3 Based on mass diffusion data Based on viscosity and thermal conductivity data AB An accurate approximation to ΩD (T ) can be obtained using the Lennard-Jones potential function The result is  AB (T ) = σAB ΩD kB T εAB ΩD where, the collision diameter, σAB , may be viewed as an effective molecular diameter for collisions of A and B If σA and σB are the cross-sectional diameters of A and B, in Å,7 then (11.40) σAB = (σA + σB ) The collision integral, ΩD is a result of kinetic theory calculations calculations based on the Lennard-Jones potential Table 11.3 gives values of One Ångström (1 Å) is equal to 0.1 nm Transport properties of mixtures §11.4 617 ΩD from [11.12] The effective potential well depth for collisions of A and B is √ (11.41) εAB = εA εB Hence, we may calculate the binary diffusion coefficient from DAB (1.8583 × 10−7 )T 3/2 = pσAB ΩD 1 + MA MB (11.42) where, again, the units of p, T , and DAB are atm, K, and m2/s, respectively, and σAB is in Å Equation (11.42) indicates that the diffusivity varies as p −1 and is independent of mixture concentrations, just as the simple model indicated that it should The temperature dependence of ΩD , however, increases the overall temperature dependence of DAB from T 3/2 , as suggested by eqn (11.39), to approximately T 7/4 Air, by the way, can be treated as a single substance in Table 11.2 owing to the similarity of its two main constituents, N2 and O2 Example 11.3 Compute DAB for the diffusion of hydrogen in air at 276 K and atm Solution Let air be species A and H2 be species B Then we read from Table 11.2 εA εB = 78.6 K, = 59.7 K σA = 3.711 Å, σB = 2.827 Å, kB kB and calculate these values σAB = (3.711 + 2.827)/2 = 3.269 Å εAB kB = (78.6)(59.7) = 68.5 K Hence, kB T /εAB = 4.029, and ΩD = 0.8822 from Table 11.3 Then (1.8583 × 10−7 )(276)3/2 + m2 /s DAB = (1)(3.269) (0.8822) 2.016 28.97 = 6.58 × 10−5 m2 /s An experimental value from Table 11.1 is 6.34 × 10−5 m2 /s, so the prediction is high by 5% Table 11.3 Collision integrals for diffusivity, viscosity, and thermal conductivity based on the Lennard-Jones potential kB T /ε 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.10 2.20 2.30 2.40 2.50 2.60 618 ΩD 2.662 2.476 2.318 2.184 2.066 1.966 1.877 1.798 1.729 1.667 1.612 1.562 1.517 1.476 1.439 1.406 1.375 1.346 1.320 1.296 1.273 1.253 1.233 1.215 1.198 1.182 1.167 1.153 1.140 1.128 1.116 1.105 1.094 1.084 1.075 1.057 1.041 1.026 1.012 0.9996 0.9878 Ωµ = Ωk 2.785 2.628 2.492 2.368 2.257 2.156 2.065 1.982 1.908 1.841 1.780 1.725 1.675 1.629 1.587 1.549 1.514 1.482 1.452 1.424 1.399 1.375 1.353 1.333 1.314 1.296 1.279 1.264 1.248 1.234 1.221 1.209 1.197 1.186 1.175 1.156 1.138 1.122 1.107 1.093 1.081 kB T /ε 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 6.00 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.0 400.0 ΩD Ωµ = Ωk 0.9770 0.9672 0.9576 0.9490 0.9406 0.9328 0.9256 0.9186 0.9120 0.9058 0.8998 0.8942 0.8888 0.8836 0.8788 0.8740 0.8694 0.8652 0.8610 0.8568 0.8530 0.8492 0.8456 0.8422 0.8124 0.7896 0.7712 0.7556 0.7424 0.6640 0.6232 0.5960 0.5756 0.5596 0.5464 0.5352 0.5256 0.5170 0.4644 0.4360 0.4172 1.069 1.058 1.048 1.039 1.030 1.022 1.014 1.007 0.9999 0.9932 0.9870 0.9811 0.9755 0.9700 0.9649 0.9600 0.9553 0.9507 0.9464 0.9422 0.9382 0.9343 0.9305 0.9269 0.8963 0.8727 0.8538 0.8379 0.8242 0.7432 0.7005 0.6718 0.6504 0.6335 0.6194 0.6076 0.5973 0.5882 0.5320 0.5016 0.4811 §11.4 Transport properties of mixtures Limitations of the diffusion coefficient prediction Equation (11.42) is not valid for all gas mixtures We have already noted that concentration gradients cannot be too steep; thus, it cannot be applied in, say, the interior of a shock wave when the Mach number is significantly greater than unity Furthermore, the gas must be dilute, and its molecules should be, in theory, nonpolar and approximately spherically symmetric Reid et al [11.4] compared values of D12 calculated using eqn (11.42) with data for binary mixtures of monatomic, polyatomic, nonpolar, and polar gases of the sort appearing in Table 11.2 They reported an average absolute error of 7.3 percent Better results can be obtained by using values of σAB and εAB that have been fit specifically to the pair of gases involved, rather than using eqns (11.40) and (11.41), or by constructing AB (T ) [11.13, Chap 11] a mixture-specific equation for ΩD The density of the gas also affects the accuracy of kinetic theory predictions, which require the gas to be dilute in the sense that its molecules interact with one another only during brief two-molecule collisions Childs and Hanley [11.14] have suggested that the transport properties of gases are within 1% of the dilute values if the gas densities not exceed the following limiting value (11.43) ρmax = 22.93M (σ Ωµ ) Here, σ (the collision diameter of the gas) and ρ are expressed in Å and kg/m3 , and Ωµ —a second collision integral for viscosity—is included in Table 11.3 Equation (11.43) normally gives ρmax values that correspond to pressures substantially above atm At higher gas densities, transport properties can be estimated by a variety of techniques, such as corresponding states theories, absolute reaction-rate theories, or modified Enskog theories [11.13, Chap 6] (also see [11.4, 11.8]) Conversely, if the gas density is so very low that the mean free path is on the order of the dimensions of the system, we have what is called free molecule flow, and the present kinetic models are again invalid (see, e.g., [11.15]) Diffusion coefficients for multicomponent gases We have already noted that an effective binary diffusivity, Dim , can be used to represent the diffusion of species i into a mixture m The preceding equations for the diffusion coefficient, however, are strictly applicable only when one pure substance diffuses through another Different equations are needed when there are three or more species present 619 620 An introduction to mass transfer §11.4 If a low concentration of species i diffuses into a homogeneous mix∗ ture of n species, then J
j for j ≠ i, and one may show (Problem 11.14) that D−1 im = n $ xj j=1 j≠i Dij (11.44) where Dij is the binary diffusion coefficient for species i and j alone This rule is sometimes called Blanc’s law [11.4] If a mixture is dominantly composed of one species, A, and includes only small traces of several other species, then the diffusion coefficient of each trace gas is approximately the same as it would be if the other trace gases were not present In other words, for any particular trace species i, Dim DiA (11.45) Finally, if the binary diffusion coefficient has the same value for each pair of species in a mixture, then one may show (Problem 11.14) that Dim = Dij , as one might expect Diffusion coefficients for binary liquid mixtures Each molecule in a liquid is always in contact with several neighboring molecules, and a kinetic theory like that used in gases, which relies on detailed descriptions of two-molecule collisions, is no longer feasible Instead, a less precise theory can be developed and used to correlate experimental measurements For a dilute solution of substance A in liquid B, the so-called hydrodynamic model has met some success Suppose that, when a force per molecule of FA is applied to molecules of A, they reach an average steady speed of vA relative to the liquid B The ratio vA /FA is called the mobility of A If there is no applied force, then the molecules of A diffuse as a result of random molecular motions (which we call Brownian motion) Kinetic and thermodynamic arguments, such as those given by Einstein [11.16] and Sutherland [11.17], lead to an expression for the diffusion coefficient of A in B as a result of Brownian motion: DAB = kB T (vA /FA ) Equation (11.46) is usually called the Nernst-Einstein equation (11.46)