Diffusion mass transfer in fluid systems cambridge series in chemical engineering 3rd edition ebook

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Contents

List of Symbols

xiii

Preface to Third Edition

xix

Preface to Second Edition

xxi

1 Models for Diffusion

1

1.1 1.2 1.3 1.4

The Two Basic Models Choosing Between the Two Models Examples Conclusions Questions for Discussion

2 3 7 9 10

PART I Fundamentals of Diffusion 2 Diffusion in Dilute Solutions 2.1 2.2 2.3 2.4 2.5 2.6

Pioneers in Diffusion Steady Diffusion Across a Thin Film Unsteady Diffusion in a Semi-infinite Slab Three Other Examples Convection and Dilute Diffusion A Final Perspective Questions for Discussion Problems Further Reading

3 Diffusion in Concentrated Solutions 3.1 3.2 3.3 3.4 3.5 3.6

Diffusion With Convection Different Forms of the Diffusion Equation Parallel Diffusion and Convection Generalized Mass Balances A Guide to Previous Work Conclusions Questions for Discussion Problems Further Reading

4 Dispersion 4.1 Dispersion From a Stack 4.2 Dispersion Coefficients

vii

13 13 17 26 33 41 49 50 51 55 56 56 59 67 75 84 90 90 91 94 95 95 97

viii

Contents

4.3 Dispersion in Turbulent Flow 4.4 Dispersion in Laminar Flow: Taylor Dispersion 4.5 Conclusions Questions for Discussion Problems Further Reading

101 104 110 110 111 113

PART II Diffusion Coefficients 5 Values of Diffusion Coefficients 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Diffusion Coefficients in Gases Diffusion Coefficients in Liquids Diffusion in Solids Diffusion in Polymers Brownian Motion Measurement of Diffusion Coefficients A Final Perspective Questions for Discussion Problems Further Reading

6 Diffusion of Interacting Species 6.1 6.2 6.3 6.4 6.5

Strong Electrolytes Associating Solutes Solute–Solvent Interactions Solute–Boundary Interactions A Final Perspective Questions for Discussion Problems Further Reading

7 Multicomponent Diffusion 7.1 7.2 7.3 7.4 7.5 7.6

Flux Equations for Multicomponent Diffusion Irreversible Thermodynamics Solving the Multicomponent Flux Equations Ternary Diffusion Coefficients Tracer Diffusion Conclusions Questions for Discussion Problems Further Reading

117 117 126 134 135 139 142 156 157 157 159 161 161 172 183 190 205 206 206 209 211 211 214 218 224 225 231 232 232 234

PART III Mass Transfer 8 Fundamentals of Mass Transfer 8.1 A Definition of a Mass Transfer Coefficient 8.2 Other Definitions of Mass Transfer Coefficients

237 237 243

ix

Contents

8.3 8.4 8.5 8.6

Correlations of Mass Transfer Coefficients Dimensional Analysis: The Route to Correlations Mass Transfer Across Interfaces Conclusions Questions for Discussion Problems Further Reading

9 Theories of Mass Transfer 9.1 9.2 9.3 9.4 9.5 9.6

The Film Theory Penetration and Surface-Renewal Theories Why Theories Fail Theories for Solid–Fluid Interfaces Theories for Concentrated Solutions Conclusions Questions for Discussion Problems Further Reading

249 257 261 269 270 270 273 274 275 277 281 284 294 298 300 300 303

10 Absorption

304

10.1 10.2 10.3 10.4 10.5

305 307 314 321 326 326 327 331

The Basic Problem Absorption Equipment Absorption of a Dilute Vapor Absorption of a Concentrated Vapor Conclusions Questions for Discussion Problems Further Reading

11 Mass Transfer in Biology and Medicine 11.1 11.2 11.3 11.4

Mass Transfer Coefficients Artificial Lungs and Artificial Kidneys Pharmocokinetics Conclusions Questions for Discussion Problems Further Reading

12 Differential Distillation 12.1 12.2 12.3 12.4 12.5

Overview of Distillation Very Pure Products The Column’s Feed and its Location Concentrated Differential Distillation Conclusions Questions for Discussion Problems Further Reading

332 333 339 347 350 351 351 352 353 353 356 362 366 371 371 372 374

x

Contents

13 Staged Distillation 13.1 13.2 13.3 13.4 13.5

Staged Distillation Equipment Staged Distillation of Nearly Pure Products Concentrated Staged Distillation Stage Efficiencies Conclusions Questions for Discussion Problems Further Reading

375 376 379 385 393 400 400 401 403

14 Extraction

404

14.1 14.2 14.3 14.4 14.5 14.6

404 407 409 413 416 420 420 421 423

The Basic Problem Extraction Equipment Differential Extraction Staged Extraction Leaching Conclusions Questions for Discussion Problems Further Reading

15 Adsorption

424

15.1 15.2 15.3 15.4 15.5 15.6

425 427 431 439 443 450 450 450 452

Where Adsorption is Important Adsorbents and Adsorption Isotherms Breakthrough Curves Mass Transfer Effects Other Characteristics of Adsorption Conclusions Questions for Discussion Problems Further Reading

PART IV Diffusion Coupled With Other Processes 16 General Questions and Heterogeneous Chemical Reactions 16.1 16.2 16.3 16.4 16.5 16.6

Is the Reaction Heterogeneous or Homogeneous?

What is a Diffusion-Controlled Reaction?

Diffusion and First-Order Heterogeneous Reactions Finding the Mechanism of Irreversible Heterogeneous Reactions Heterogeneous Reactions of Unusual Stoichiometries Conclusions Questions for Discussion Problems Further Reading

455 456 457 459 465 469 473 473 474 477

xi

Contents

17 Homogeneous Chemical Reactions 17.1 17.2 17.3 17.4 17.5 17.6

Mass Transfer with First-Order Chemical Reactions Mass Transfer with Second-Order Chemical Reactions Industrial Gas Treating Diffusion-Controlled Fast Reactions Dispersion-Controlled Fast Reactions Conclusions Questions for Discussion Problems Further Reading

478 479 488 492 500 504 507 508 508 512

18 Membranes

513

18.1 18.2 18.3 18.4 18.5 18.5

Physical Factors in Membranes Gas Separations Reverse Osmosis and Ultrafiltration Pervaporation Facilitated Diffusion Conclusions Questions for Discussion Problems Further Reading

514 520 526 534 539 545 545 546 548

19 Controlled Release and Related Phenomena

549

19.1 19.2 19.3 19.4 19.5

Controlled Release by Solute Diffusion Controlled Release by Solvent Diffusion Barriers Diffusion and Phase Equilibrium Conclusions Questions for Discussion Problems Further Reading

20 Heat Transfer 20.1 20.2 20.3 20.4 20.5

Fundamentals of Heat Conduction General Energy Balances Heat Transfer Coefficients Rate Constants for Heat Transfer Conclusions Questions for Discussion Problems Further Reading

21 Simultaneous Heat and Mass Transfer 21.1 Mathematical Analogies Among Mass, Heat, and Momentum Transfer 21.2 Physical Equalities Among Mass, Heat, and Momentum Transfer

551 555 558 562 565 565 566 566 568 568 575 579 585 591 591 591 593 594 594 600

xii

Contents

21.3 21.4 21.5 21.6

Index

Drying Design of Cooling Towers Thermal Diffusion and Effusion Conclusions Questions for Discussion Problems Further Reading

604 609 615 621 621 622 624 626

List of Symbols

surface area per volume major axis of ellipsoid (Section 5.2) constant area absorption factor (Chapters 13 and 14) constant minor axis of ellipsoid (Section 5.2) bottoms (Chapters 10, 12 and 13) boundary positions (Section 7.3) total molar concentration concentration of species 1, in either moles per volume or mass per volume critical micelle concentration (Section 6.2) total concentration (Chapter 6) concentration of species 1 averaged over time (Sections 4.3 and 17.4) concentration fluctuation of species 1 (Sections 4.3, 17.3, and 17.4) vector of concentrations (Section 7.3) concentration of species 1 at an interface i capacity factor (Section 13.1) molar and specific heat capacities respectively, at constant pressure molar and specific heat capacities respectively at constant volume diameter or other characteristic length binary diffusion coefficient distillate (Chapters 12 and 13) effective diffusion coefficient, for example, in a porous solid binary diffusion coefficient of species i binary diffusion coefficient corrected for activity effects multicomponent diffusion coefficient (Chapter 7) Knudsen diffusion coefficient of a gas in a small pore micelle diffusion coefficient (Section 6.2) intradiffusion coefficient (Section 7.5) dispersion coefficient extraction factor (Chapter 14) residence-time distribution (Section 9.2) friction coefficient for a diffusing solute (Section 5.2) friction factor for fluid flow (Chapter 21) packing factor (Section 10.2) feed (Chapters 12 and 13) Faraday’s constant (Section 6.1) solution to a binary diffusion problem (Section 7.3) acceleration due to gravity

a a a, ai A A b b B B, b c c1 cCMC cT c1 c10 c c1i C C~p , C^p C~t , C^t d D D Deff Di D0 Dij DKn Dm D* E E E(t) f f F F F F(D) g xiii

xiv

G G$ G# h h, hi H ~ H ^ H, i H HTU i jv jT ji jim ji* ð2Þ j1 a ji Js JT k kp kx , k y kB kT k0 k0 k# K KG, KL Kp Kx, Ky Kn l L L L$ L# Lij Lp m M M ~i M

List of Symbols

molar flux of gas mass flux of gas (Sections 10.2 and 13.1) molar flux of gas in stripping section (Chapters 12 and 13) reduced plate height (Section 15.5) heat transfer coefficients (Chapters 20 and 21) partition coefficient molar and specific enthalpies (Chapters 20–21 and Chapter 7, respectively) partial specific enthalpy (Chapter 7) height of transfer unit current density (Section 6.1) volume flux across a membrane (Section 18.3) total electrolyte flux (Section 6.1) diffusion flux of solute i relative to the volume average velocity diffusion flux of solute i relative to the mass average velocity diffusion flux relative to the molar average velocity diffusion flux of solute (1) relative to velocity of solvent (2) diffusion flux of solute i relative to reference velocity a entropy flux (Section 7.2) total solute flux in different chemical forms (Section 6.2) mass transfer coefficient based on a concentration driving force mass transfer coefficient based on a partial pressure driving force (Table 8.2-2) mass transfer coefficients based on mole fraction driving forces in liquid and gas, respectively (Table 8.2-2) Boltzmann’s constant thermal conductivity (Chapters 20–21) mass transfer coefficient at low transfer rate (Section 9.5) mass transfer coefficient without chemical reaction (Chapter 17) capacity factor (Sections 4.4 and 15.1) equilibrium constant for chemical reaction overall mass transfer coefficients based on concentration driving force in gas or liquid, respectively overall mass transfer coefficient based on partial pressure difference in gas overall mass transfer coefficient based on mole fraction driving force in liquid or gas, respectively Knudsen number (Section 6.4) length, e.g., of a membrane length, e.g., of a pipe molar flux of liquid mass flux of liquid (Sections 10.2 and 13.1) molar flux of liquid in stripping section (Sections 12.3 and 13.3) Onsager phenomenological coefficient (Section 7.2) solvent permeability (Section 18.3) partition coefficient relating mole fractions in gas and liquid mass total solute (Sections 4.2 and 5.5) molecular weight of species i

List of Symbols

n ni N Ñ Ni Ni NTU p P P Pij q q q q r r, ri R RD R0 s S^ S i t t ti t1/2 T ui U Û vr, vh v x, v y v va vo v# v* vi V V i Vij W W Ws x

micelle aggregation number or hydration number (Section 6.2) flux of species i relative to fixed coordinates number of ideal stages Avogadro’s number flux of species i at an interface number of moles of species i number of transfer units pressure power membrane permeability (Chapter 18) weighting factor (Section 7.3) scattering vector (Section 5.6) feed quality (Sections 12.3 and 13.3) solute concentration per volume adsorbent (Chapter 15) energy flux (Chapters 7, 20, and 21) radius rate of chemical reaction gas constant reflux ratio (Chapters 12 and 13) characteristic radius distance from pipe wall (Section 9.4) specific entropy (Chapter 7) partial specific entropy of species i time modal matrix (Section 7.3) transference number of ion i (Section 6.1) reaction half-life temperature ionic mobility (Section 6.1) overall heat transfer coefficient specific internal energy velocities in the r and h directions velocities in the x and y directions mass average velocity velocity relative to reference frame a volume average velocity velocity fluctuation (Sections 4.3 and 17.4) molar average velocity velocity of species i volume partial molar or specific volume of species i fraction of molecular volume (Section 5.1) width work (Section 20.2) shaft work (Section 20.2) mole fraction in liquid of more volatile species (Chapters 12 and 13)

xv

xvi

xB, xD, xF xi Xi y yi z |z| zi a a a aij b b c c ci d d(z) dij e e eij f g g h h h ji, j i k k ki K l li li m m n n Q q qi r

List of Symbols

mole fractions of more volatile species in bottoms, distillate and feed, respectively (Chapters 12 and 13) mole fraction of species i, especially in a liquid or solid phase generalized force causing diffusion (Section 7.2) mole fraction in vapor of more volatile species (Chapters 12 and 13) mole fraction of species i in a gas position magnitude of charge (Section 6.1) charge on species i thermal diffusivity (Chapters 20 and 21) thermal diffusion factor (Section 21.5) flake aspect ratio (Sections 6.4 and 9.5) conversion factor (Section 7.1) diaphragm cell calibration constant (Sections 2.2 and 5.5) pervaporation selectivity (Section 18.4) interfacial influence (Section 6.3) surface tension (Section 6.4) activity coefficient of species i thickness of thin layer, especially a boundary layer Dirac function of z Kronecker delta void fraction enhancement factor (Section 17.1) interaction energy between colliding molecules (Sections 5.1 and 20.4) combined variable Murphree efficiency (Section 13.4) effectiveness factor (Section 17.1) dimensionless concentration fraction of unused adsorption bed (Section 15.3) fraction of surface elements (Section 9.2) forward and reverse reaction rate constants respectively of reaction i length ratio (Section 6.4) heat of vaporization (Sections 12.3 and 13.3) equivalent ionic conductance of species i (Section 6.1) equivalent conductance viscosity chemical potential of species i partial specific Gibbs free energy of species i, i.e., the chemical potential divided by the molecular weight (Section 7.2) kinematic viscosity stoichiometric coefficient (Sections 16.5 and 17.2) dimensionless position correlation length (Section 6.3) osmotic pressure (Section 18.3) total density, i.e., total mass concentration mass concentration of species i rate of entropy production (Section 7.2)

List of Symbols

r r, r# r s ri rij s s s s s0 / /i w c x x x xi X

standard deviation (Sections 5.5 and 15.4) reflection coefficients (Section 18.3) Soret coefficient (Section 21) diagonal matrix of eigenvalues (Chapter 7) eigenvalue (Section 7.3) collision diameter characteristic time tortuosity (Section 6.4) residence time for surface element (Section 9.2) shear stress (Chapter 21) shear stress at wall (Section 9.4) Thiele modulus (Section 17.1) volume fraction of species i electrostatic potential combined concentration (Section 7.3) jump frequency (Section 5.3) regular solution parameter (Section 6.3) coefficient of solute permeability (Section 18.3) mass fraction of species i collision integral in Chapman–Enskog theory (Section 5.1)

xvii

Preface to the Third Edition

Like its earlier editions, this book has two purposes. First, it presents a clear description of diffusion, the mixing process caused by molecular motion. Second, it explains mass transfer, which controls the cost of processes like chemical purification and environmental control. The first of these purposes is scientific, explaining how nature works. The second purpose is more practical, basic to the engineering of chemical processes.

While diffusion was well explained in earlier editions, this edition extends and clarifies this material. For example, the Maxwell–Stefan alternative to Fick’s equation is now treated in more depth. Brownian motion and its relation to diffusion are explicitly described. Diffusion in composites, an active area of research, is reviewed. These topics are an evolution of and an improvement over the material in earlier editions.

Mass transfer is much better explained here than it was earlier. I believe that mass transfer is often poorly presented because it is described only as an analogue of heat transfer. While this analogue is true mathematically, its overemphasis can obscure the simpler physical meaning of mass transfer. In particular, this edition continues to emphasize dilute mass transfer. It gives a more complete description of differential distillation than is available in other introductory sources. This description is important because differential distillation is now more common than staged distillation, normally the only form covered. This edition gives a much better description of adsorption than has been available. It provides an introduction to mass transfer applied in biology and medicine.

The result is an engineering book which is much more readable and understandable than other books covering these subjects. It provides much more physical insight than conventional books on unit operations. It explores the interactions between mass transfer and chemical reaction, which are omitted by many books on transport phenomena.

The earlier editions are good, but this one is better.

The book works well as a text either for undergraduates or graduate students. For a one-semester undergraduate chemical engineering course of perhaps 45 lectures plus recitations, I cover Chapter 2, Sections 3.1 to 3.2 and 5.1 to 5.2, Chapters 8 to 10, 12 to 15, and 21. If there is time, I add Sections 16.1 to 16.3 and Sections 17.1 to 17.3. If this course aims at describing separation processes, I cover crystallization before discussing membrane separations. We have successfully taught such a course here at Minnesota for the last 10 years.

For a one semester graduate course for students from chemistry, chemical engineering, pharmacy, and food science, I plan for 45 lectures without recitations. This course covers Chapters 2 to 9 and Chapters 16 to 19. It has been a mainstay at many universities for almost 30 years.

This description of academic courses should not restrict the book’s overall goal.

Diffusion and mass transfer are often interesting because they are slow. Their rate controls many processes, from the separation of air to the spread of pollutants to the size of a human sperm. The study of diffusion is thus important, but it is also fun. I hope that this book catalyzes that fun for you. xix

Preface to Second Edition

The purpose of this second edition is again a clear description of diffusion useful to engineers, chemists, and life scientists. Diffusion is a fascinating subject, as central to our daily lives as it is to the chemical industry. Diffusion equations describe the transport in living cells, the efficiency of distillation, and the dispersal of pollutants. Diffusion is responsible for gas absorption, for the fog formed by rain on snow, and for the dyeing of wool. Problems like these are easy to identify and fun to study.

Diffusion has the reputation of being a difficult subject, much harder than, say, fluid mechanics or solution thermodynamics. In fact, it is relatively simple. To prove this to yourself, try to explain a diffusion flux, a shear stress, and chemical potential to some friends who have little scientific training. I can easily explain a diffusion flux: It is how much diffuses per area per time. I have more trouble with a shear stress. Whether I say it is a momentum flux or the force in one direction caused by motion in a second direction, my friends look blank. I have never clearly explained chemical potentials to anyone.

However, past books on diffusion have enhanced its reputation as a difficult subject.

These books fall into two distinct groups that are hard to read for different reasons. The first group is the traditional engineering text. Such texts are characterized by elaborate algebra, very complex examples, and turgid writing. Students cheerfully hate these books; moreover, they remember what they have learned as scattered topics, not an organized subject.

The second group of books consists of texts on transport processes. These books present diffusion by analogy with fluid flow and heat transfer. They are much more readable than the traditional texts, especially for the mathematically adroit. They do have two significant disadvantages. First, topics important to diffusion but not to fluid flow tend to be omitted or deemphasized. Such cases include simultaneous diffusion and chemical reaction. Second, these books usually present diffusion last, so that fluid mechanics and heat transfer must be at least superficially understood before diffusion can be learned. This approach effectively excludes students outside of engineering who have little interest in these other phenomena. Students in engineering find difficult problems emphasized because the simple ones have already been covered for heat transfer.

Whether they are engineers or not, all conclude that diffusion must be difficult.

In the first edition, I tried to describe diffusion clearly and simply. I emphasized physical insight, sometimes at the loss of mathematical rigor. I discussed basic concepts in detail, without assuming prior knowledge of other phenomena. I aimed at the scope of the traditional texts and at the clarity of books on transport processes. This second edition is evidence that I was partly successful. Had I been completely successful, no second edition would be needed. Had I been unsuccessful, no second edition would be wanted.

In this second edition, I’ve kept the emphasis on physical insight and basic concepts, but I’ve expanded the book’s scope. Chapters 1–7 on diffusion are largely unchanged, though some description of diffusion coefficients is abridged. Chapter 8 on mass transfer xxi

xxii

Preface to Second Edition

is expanded to even more detail, for I found many readers need more help. Chapters 9–12, a description of traditional chemical processes are new. The remaining seven chapters, a spectrum of topics, are either new or significantly revised. The result is still useful broadly, but deeper on engineering topics.

I have successfully used the book as a text for both undergraduate and graduate courses, of which most are in chemical engineering. For an undergraduate course on unit operations, I first review the mass transfer coefficients in Chapter 8, for I find that students’ memory of these ideas is motley. I then cover the material in Chapters 9–12 in detail, for this is the core of the subject. I conclude with simultaneous heat and mass transfer, as discussed in Chapters 19–20. The resulting course of 50 classes is typical of many offered on this subject. On their own, undergraduates have used Chapters 2–3 and 8–9 for courses on heat and mass transfer, but this book’s scope seems too narrow to be a good text for that class.

For graduate students, I give two courses in alternate years. Neither requires the other as a prerequisite. In the first graduate course, on diffusion, I cover Chapters 1–7, plus Chapter 17 (on membranes). In the second graduate course, on mass transfer, I cover Chapters 8–9, Chapters 13–16, and Chapter 20. These courses, which typically have about 35 lectures, are an enormous success, year after year. For nonengineering graduate students and for various short courses, I’ve usually used Chapters 2, 8, 15–16, and any other chapters specific to a given discipline. For example, for those in the drug industry, I might cover Chapters 11 and 18.

I am indebted to many who have encouraged me in this effort. My overwhelming debt is to my colleagues at the University of Minnesota. When I become disheartened, I need simply to visit another institution to be reminded of the advantages of frank discussion without infighting. My students have helped, especially Sameer Desai and Diane Clifton, who each read large parts of the final manuscript. Mistakes that remain are my fault.

Teresa Bredahl typed most of the book, and Clover Galt provided valuable editorial help. Finally, my wife Betsy gives me a wonderful rich life.

CH A PT E R 1

Models for Diffusion

If a few crystals of a colored material like copper sulfate are placed at the bottom of a tall bottle filled with water, the color will slowly spread through the bottle.

At first the color will be concentrated in the bottom of the bottle. After a day it will penetrate upward a few centimeters. After several years the solution will appear homogeneous.

The process responsible for the movement of the colored material is diffusion, the subject of this book. Diffusion is caused by random molecular motion that leads to complete mixing. It can be a slow process. In gases, diffusion progresses at a rate of about 5 cm/ min; in liquids, its rate is about 0.05 cm/min; in solids, its rate may be only about 0.00001 cm/min. In general, it varies less with temperature than do many other phenomena.

This slow rate of diffusion is responsible for its importance. In many cases, diffusion occurs sequentially with other phenomena. When it is the slowest step in the sequence, it limits the overall rate of the process. For example, diffusion often limits the efficiency of commercial distillations and the rate of industrial reactions using porous catalysts. It limits the speed with which acid and base react and the speed with which the human intestine absorbs nutrients. It controls the growth of microorganisms producing penicillin, the rate of the corrosion of steel, and the release of flavor from food.

In gases and liquids, the rates of these diffusion processes can often be accelerated by agitation. For example, the copper sulfate in the tall bottle can be completely mixed in a few minutes if the solution is stirred. This accelerated mixing is not due to diffusion alone, but to the combination of diffusion and stirring. Diffusion still depends on random molecular motions that take place over smaller distances. The agitation or stirring is not a molecular process, but a macroscopic process that moves portions of the fluid over much larger distances. After this macroscopic motion, diffusion mixes newly adjacent portions of the fluid. In other cases, such as the dispersal of pollutants, the agitation of wind or water produces effects qualitatively similar to diffusion; these effects, called dispersion, will be treated separately.

The description of diffusion involves a mathematical model based on a fundamental hypothesis or ‘‘law.’’ Interestingly, there are two common choices for such a law. The more fundamental, Fick’s law of diffusion, uses a diffusion coefficient. This is the law that is commonly cited in descriptions of diffusion. The second, which has no formal name, involves a mass transfer coefficient, a type of reversible rate constant.

Choosing between these two models is the subject of this chapter. Choosing Fick’s law leads to descriptions common to physics, physical chemistry, and biology. These descriptions are explored and extended in Chapters 2–7. Choosing mass transfer coefficients produces correlations developed explicitly in chemical engineering and used implicitly in chemical kinetics and in medicine. These correlations are described in Chapters 8–15.

Both approaches are used in Chapters 16–21.

We discuss the differences between the two models in Section 1.1 of this chapter.

In Section 1.2 we show how the choice of the most appropriate model is determined.

1

1 / Models for Diffusion

2

In Section 1.3 we conclude with additional examples to illustrate how the choice between the models is made.

1.1 The Two Basic Models In this section we want to illustrate the two basic ways in which diffusion can be described. To do this, we first imagine two large bulbs connected by a long thin capillary (Fig. 1.1-1). The bulbs are at constant temperature and pressure and are of equal volumes. However, one bulb contains carbon dioxide, and the other is filled with nitrogen.

To find how fast these two gases will mix, we measure the concentration of carbon dioxide in the bulb that initially contains nitrogen. We make these measurements when only a trace of carbon dioxide has been transferred, and we find that the concentration of carbon dioxide varies linearly with time. From this, we know the amount transferred per unit time.

We want to analyze this amount transferred to determine physical properties that will be applicable not only to this experiment but also in other experiments. To do this, we first define the flux:

amount of gas removed ðcarbon dioxide fluxÞ ¼ ð1:1-1Þ time ðarea capillaryÞ In other words, if we double the cross-sectional area, we expect the amount transported to double. Defining the flux in this way is a first step in removing the influences of our particular apparatus and making our results more general. We next assume that the flux is proportional to the gas concentration: 0 1 carbon dioxide ðcarbon dioxide fluxÞ ¼ k@ concentration A ð1:1-2Þ difference

N2

CO2 Concentration here

The proportionality constant k is called a mass transfer coefficient. Its introduction signals one of the two basic models of diffusion. Alternatively, we can recognize

Time CO2

Fig. 1.1-1. A simple diffusion experiment. Two bulbs initially containing different gases are connected with a long thin capillary. The change of concentration in each bulb is a measure of diffusion and can be analyzed in two different ways.

1.2 / Choosing Between the Two Models

3

that increasing the capillary’s length will decrease the flux, and we can then assume that

carbon dioxide concentration difference ð1:1-3Þ ðcarbon dioxide fluxÞ ¼ D capillary length The new proportionality constant D is the diffusion coefficient. Its introduction implies the other model for diffusion, the model often called Fick’s law.

These assumptions may seem arbitrary, but they are similar to those made in many other branches of science. For example, they are similar to those used in developing Ohm’s law, which states that 0 1 0 1

voltage; or current; or 1 @ area times flux A ¼ @ potential A ð1:1-4Þ resistance of electrons difference Thus, the mass transfer coefficient k is analogous to the reciprocal of the resistance. An alternative form of Ohm’s law is 0 1 1

potential current density 1 B difference C @ or flux of A ¼ @ A resistivity length electrons 0

ð1:1-5Þ

The diffusion coefficient D is analogous to the reciprocal of the resistivity.

Neither the equation using the mass transfer coefficient k nor that using the diffusion coefficient D is always successful. This is because of the assumptions made in their development. For example, the flux may not be proportional to the concentration difference if the capillary is very thin or if the two gases react. In the same way, Ohm’s law is not always valid at very high voltages. But these cases are exceptions; both diffusion equations work well in most practical situations, just as Ohm’s law does.

The parallels with Ohm’s law also provide a clue about how the choice between diffusion models is made. The mass transfer coefficient in Eq. 1.1-2 and the resistance in Eq. 1.1-4 are simpler, best used for practical situations and rough measurements. The diffusion coefficient in Eq. 1.1-3 and the resistivity in Eq. 1.1-5 are more fundamental, involving physical properties like those found in handbooks. How these differences guide the choice between the two models is the subject of the next section.

1.2 Choosing Between the Two Models The choice between the two models outlined in Section 1.1 represents a compromise between ambition and experimental resources. Obviously, we would like to express our results in the most general and fundamental ways possible. This suggests working with diffusion coefficients. However, in many cases, our experimental measurements will dictate a more approximate and phenomenological approach. Such approximations often imply mass transfer coefficients, but they usually still permit us to reach our research goals.

1 / Models for Diffusion

4

Analyze as mass transfer Flux = k Δ(concentration) Hydrogen gas

k is not constant; variation with time correlated; variation with position ignored

Metal

Hydrogen concentration vs. time

Analyze as diffusion ∂ (concentration) Flux = –D ∂z

z

D is constant; variation with time and position predicted

Fig. 1.2-1. Hydrogen diffusion into a metal. This process can be described with either a mass transfer coefficient k or a diffusion coefficient D. The description with a diffusion coefficient correctly predicts the variation of concentration with position and time, and so is superior.

This choice and the resulting approximations are best illustrated by two examples. In the first, we consider hydrogen diffusion in metals. This diffusion substantially reduces a metal’s ductility, so much so that parts made from the embrittled metal frequently fracture. To study this embrittlement, we might expose the metal to hydrogen under a variety of conditions and measure the degree of embrittlement versus these conditions.

Such empiricism would be a reasonable first approximation, but it would quickly flood us with uncorrelated information that would be difficult to use effectively.

As an improvement, we can undertake two sets of experiments. First, we can saturate metal samples with hydrogen and determine their degrees of embrittlement. Thus we know metal properties versus hydrogen concentration. Second, we can measure hydrogen uptake versus time, as suggested in Fig. 1.2-1, and correlate our measurements as mass transfer coefficients. Thus we know average hydrogen concentration versus time.

To our dismay, the mass transfer coefficients in this case will be difficult to interpret.

They are anything but constant. At zero time, they approach infinity; at large time, they approach zero. At all times, they vary with the hydrogen concentration in the gas surrounding the metal. They are an inconvenient way to summarize our results. Moreover, the mass transfer coefficients give only the average hydrogen concentration in the metal. They ignore the fact that the hydrogen concentration very near the metal’s surface will reach saturation but the concentration deep within the metal will remain zero. As a result, the metal near the surface may be very brittle but that within may be essentially unchanged.

We can include these details in the diffusion model described in the previous section.

This model assumed that

hydrogen flux

¼D

hydrogen hydrogen

concentration at z ¼ 0 concentration at z ¼ l ðthickness at z ¼ lÞ ðthickness at z ¼ 0Þ ð1:2-1Þ

1.2 / Choosing Between the Two Models

5

or, symbolically, j1 ¼ D

c1 jz¼0 c1 jz¼l l 0

ð1:2-2Þ

where the subscript 1 symbolizes the diffusing species. In these equations, the distance l is that over which diffusion occurs. In the previous section, the length of the capillary was appropriately this distance; but in this case, it seems uncertain what the distance should be. If we assume that it is very small, c1 jz¼z c1 jz¼zþl dc1 ¼ D l!0 zjzþl zjz dz

j1 ¼ D lim

ð1:2-3Þ

We can use this relation and the techniques developed later in this book to correlate our experiments with only one parameter, the diffusion coefficient D. We then can correctly predict the hydrogen uptake versus time and the hydrogen concentration in the gas. As a dividend, we get the hydrogen concentration at all positions and times within the metal.

Thus the model based on the diffusion coefficient gives results of more fundamental value than the model based on mass transfer coefficients. In mathematical terms, the diffusion model is said to have distributed parameters, for the dependent variable (the concentration) is allowed to vary with all independent variables (like position and time).

In contrast, the mass transfer model is said to have lumped parameters (like the average hydrogen concentration in the metal).

These results would appear to imply that the diffusion model is superior to the mass transfer model and so should always be used. However, in many interesting cases the models are equivalent. To illustrate this, imagine that we are studying the dissolution of a solid drug suspended in water, as schematically suggested by Fig. 1.2-2. The dissolution of this drug is known to be controlled by the diffusion of the dissolved drug away from the solid surface of the undissolved material. We measure the drug concentration versus time as shown, and we want to correlate these results in terms of as few parameters as possible.

One way to correlate the dissolution results is to use a mass transfer coefficient. To do this, we write a mass balance on the solution: 0 1

accumulation @ of drug in A ¼ total rate of dissolution solution V

dc1 ¼ Aj1 dt ¼ Ak½c1 ðsatÞ c1

ð1:2-4Þ

where V is the volume of solution, A is the total area of the drug particles, c1(sat) is the drug concentration at saturation and at the solid’s surface, and c1 is the concentration in the bulk solution. Integrating this equation allows quantitatively fitting our results with one parameter, the mass transfer coefficient k. This quantity is independent of drug solubility, drug area, and solution volume, but it does vary with physical properties like stirring rate and solution viscosity. Correlating the effects of these properties turns out to be straightforward.

1 / Models for Diffusion

6

Analyze as chemical reaction dc 1 = κ [c1(sat) – c1] dt

κ is reaction rate constant for a fictitious reaction

Solid drug

Drug concentration

Analyze as mass transfer dc

Saturation

V dt1 =kA [c1(sat) – c1] k varies with stirring.

Note that kA/V = κ Analyze as diffusion dc

Time

V dt1 = D A [c1(sat) –c1] l l varies with stirring and with D. Note that D/ l = k

Fig. 1.2-2. Rates of drug dissolution. In this case, describing the system with a mass transfer coefficient k is best because it easily correlates the solution’s concentration versus time.

Describing the system with a diffusion coefficient D gives a similar correlation but introduces an unnecessary parameter, the film thickness l. Describing the system with a reaction rate constant k also works, but this rate constant is a function not of chemistry but of physics.

The alternative to mass transfer is diffusion theory, for which the mass balance is

dc1 D ½c1 ðsatÞ c1 ¼A V ð1:2-5Þ l dt in which l is an unknown parameter, equal to the average distance across which diffusion occurs. This unknown, called a film or unstirred layer thickness, is a function not only of flow and viscosity but also of the diffusion coefficient itself.

Equations 1.2-4 and 1.2-5 are equivalent, and they share the same successes and shortcomings. In the former, we must determine the mass transfer coefficient experimentally; in the latter, we determine instead the thickness l. Those who like a scientific veneer prefer to measure l, for it genuflects toward Fick’s law of diffusion. Those who are more pragmatic prefer explicitly recognizing the empirical nature of the mass transfer coefficient.

The choice between the mass transfer and diffusion models is thus often a question of taste rather than precision. The diffusion model is more fundamental and is appropriate when concentrations are measured or needed versus both position and time. The mass transfer model is simpler and more approximate and is especially useful when only average concentrations are involved. The additional examples in section 1.3 should help us decide which model is appropriate for our purposes.

Before going on to the next section, we should mention a third way to correlate the results other than the two diffusion models. This third way is to assume that the dissolution shown in Fig. 1.2-2 is a first-order, reversible chemical reaction. Such a reaction might be described by dc1 ¼ jc1 ðsatÞ jc1 dt

ð1:2-6Þ

1.3 / Examples

7

In this equation, the quantity jc1(sat) represents the rate of dissolution, jc1 stands for the rate of precipitation, and j is a rate constant for this process. This equation is mathematically identical with Eqs. 1.2-4 and 1.2-5 and so is equally successful. However, the idea of treating dissolution as a chemical reaction is flawed. Because the reaction is hypothetical, the rate constant is a composite of physical factors rather than chemical factors. We do better to consider the physical process in terms of a diffusion or mass transfer model.

1.3 Examples In this section, we give examples that illustrate the choice between diffusion coefficients and mass transfer coefficients. This choice is often difficult, a juncture where many have trouble. I often do. I think my trouble comes from evolving research goals, from the fact that as I understand the problem better, the questions that I am trying to answer tend to change. I notice the same evolution in my peers, who routinely start work with one model and switch to the other model before the end of their research.

We shall not solve the following examples. Instead, we want only to discuss which diffusion model we would initially use for their solution. The examples given certainly do not cover all types of diffusion problems, but they are among those about which I have been asked in the last year.

Example 1.3-1: Ammonia scrubbing Ammonia, the major material for fertilizer, is made by reacting nitrogen and hydrogen under pressure. The product gas can be washed with water to dissolve the ammonia and separate it from other unreacted gases. How can you correlate the dissolution rate of ammonia during washing?

Solution The easiest way is to use mass transfer coefficients. If you use diffusion coefficients, you must somehow specify the distance across which diffusion occurs. This distance is unknown unless the detailed flows of gases and the water are known; they rarely are (see Chapters 8 and 9).

Example 1.3-2: Reactions in porous catalysts Many industrial reactions use catalysts containing small amounts of noble metals dispersed in a porous inert material like silica.

The reactions on such a catalyst are sometimes slower in large pellets than in small ones.

This is because the reagents take longer to diffuse into the pellet than they do to react.

How should you model this effect?

Solution You should use diffusion coefficients to describe the simultaneous diffusion and reaction in the pores in the catalyst. You should not use mass transfer coefficients because you cannot easily include the effect of reaction (see Sections 16.1 and 17.1).

Example 1.3-3: Corrosion of marble Industrial pollutants in urban areas like Venice cause significant corrosion of marble statues. You want to study how these pollutants penetrate marble. Which diffusion model should you use?

Solution The model using diffusion coefficients is the only one that will allow you to predict pollutant concentration versus position in the marble. The model using

8

1 / Models for Diffusion

mass transfer coefficients will only correlate how much pollutant enters the statue, not what happens to the pollutant (see Sections 2.3 and 8.1).

Example 1.3-4: Protein size in solution You are studying a variety of proteins that you hope to purify and use as food supplements. You want to characterize the size of the proteins in solution. How can you use diffusion to do this?

Solution Your aim is determining the molecular size of the protein molecules.

You are not interested in the protein mass transfer except as a route to these molecular properties. As a result, you should measure the protein’s diffusion coefficient, not its mass transfer coefficient. The protein’s diffusion coefficient will turn out to be proportional to its radius in solution (see Section 5.2).

Example 1.3-5: Antibiotic production Many drugs are made by fermentations in which microorganisms are grown in a huge stirred vat of a dilute nutrient solution or ‘‘beer.’’

Many of these fermentations are aerobic, so the nutrient solution requires aeration. How should you model oxygen uptake in this type of solution?

Solution Practical models use mass transfer coefficients. The complexities of the problem, including changes in air bubble size, flow effects of the non-Newtonian solution, and foam caused by biological surfactants, all inhibit more careful study (see Chapter 8).

Example 1.3-6: Facilitated transport across membranes Some membranes contain a mobile carrier, a reactive species that reacts with diffusing solutes, facilitating their transport across the membrane. Such membranes can be used to concentrate copper ions from industrial waste and to remove carbon dioxide from coal gas. Diffusion across these membranes does not vary linearly with the concentration difference across them. The diffusion can be highly selective, but it is often easily poisoned.

Should this diffusion be described with mass transfer coefficients or with diffusion coefficients?

Solution This system includes not only diffusion but also chemical reaction.

Diffusion and reaction couple in a nonlinear way to give the unusual behavior observed.

Understanding such behavior will certainly require the more fundamental model of diffusion coefficients (see Section 18.5).

Example 1.3-7: Flavor retention When food products are spray-dried, they lose a lot of flavor. However, they lose less than would be expected on the basis of the relative vapor pressures of water and the flavor compounds. The reason apparently is that the drying food often forms a tight gellike skin across which diffusion of the flavor compounds is inhibited. What diffusion model should you use to study this effect?

Solution Because spray-drying is a complex, industrial-scale process, it is usually modeled using mass transfer coefficients. However, in this case you are interested in the inhibition of diffusion. Such inhibition will involve the sizes of pores in the food and of molecules of the flavor compounds. Thus you should use the more basic diffusion model, which includes these molecular factors (see Section 6.4).

1.4 / Conclusions

9

Example 1.3-8: The smell of marijuana Recently, a large shipment of marijuana was seized in the Minneapolis–St. Paul airport. The police said their dog smelled it. The owners claimed that it was too well wrapped in plastic to smell and that the police had conducted an illegal search without a search warrant. How could you tell who was right?

Solution In this case, you are concerned with the diffusion of odor across the thin plastic film. The diffusion rate is well described by either mass transfer or diffusion coefficients. However, the diffusion model explicitly isolates the effect of the solubility of the smell in the film, which dominates the transport. This solubility is the dominant variable (see Section 2.2). In this case, the search was illegal.

Example 1.3-9: Scale-up of wet scrubbers You want to use a wet scrubber to remove sulfur oxides from the flue gas of a large power plant. A wet scrubber is essentially a large piece of pipe set on its end and filled with inert ceramic material. You pump the flue gas up from the bottom of the pipe and pour a lime slurry down from the top. In the scrubber, there are various reactions, such as CaO þ SO2 ! CaSO3

ð1:2-6Þ

The lime reacts with the sulfur oxides to make an insoluble precipitate, which is discarded. You have been studying a small unit and want to use these results to predict the behavior of a larger unit. Such an increase in size is called a scale-up. Should you make these predictions using a model based on diffusion or mass transfer coefficients?

Solution This situation is complex because of the chemical reactions and the irregular flows within the scrubber. Your first try at correlating your data should be a model based on mass transfer coefficients. Should these correlations prove unreliable, you may be forced to use the more difficult diffusion model (see Chapters 9, 16, and 17).

1.4 Conclusions This chapter discusses the two common models used to describe diffusion and suggests how you can choose between these models. For fundamental studies where you want to know concentration versus position and time, use diffusion coefficients. For practical problems where you want to use one experiment to tell how a similar one will behave, use mass transfer coefficients. The former approach is the distributed-parameter model used in chemistry, and the latter is the lumped-parameter model used in engineering. Both approaches are used in medicine and biology, but not always explicitly.

The rest of this book is organized in terms of these two models. Chapters 2–4 present the basic model of diffusion coefficients, and Chapters 5–7 review the values of the diffusion coefficients themselves. Chapters 8–15 discuss the model of mass transfer coefficients, including their relation to diffusion coefficients. Chapters 16–19 explore the coupling of diffusion with heterogeneous and homogeneous chemical reactions, using both models. Chapters 20–21 explore the simpler coupling between diffusion and heat transfer.

1 / Models for Diffusion

10

In the following chapters, keep both models in mind. People involved in basic research tend to be overcommitted to diffusion coefficients, whereas those with broader objectives tend to emphasize mass transfer coefficients. Each group should recognize that the other has a complementary approach that may be more helpful for the case in hand.

Questions for Discussion 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

What are the dimensions in mass M, length L, and time t of a diffusion coefficient?

What are the dimensions of a mass transfer coefficient?

What volume is implied by Ficks’s law?

What volume is implied when defining a mass transfer coefficient?

Can the diffusion coefficient ever be negative?

Give an example for a diffusion coefficient which is the same in all directions.

Give an example when it isn’t.

When a silicon chip is doped with boron, does the doping involve diffusion?

Does the wafting of smells of a pie baking in the oven involve diffusion?

How does breathing involve diffusion?

How is a mass transfer coefficient related to a reaction rate constant?

Will a heat transfer coefficient and a mass transfer coefficient be related?

Will stirring a suspension of sugar in water change the diffusion coefficient?

Will it change the density? Will it change the mass transfer coefficient?

PART I

Fundamentals of Diffusion

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DANCE ON STILTS AT THE GIRLS’ UNYAGO, NIUCHI

Newala, too, suffers from the distance of its water-supply—at least the Newala of to-day does; there was once another Newala in a lovely valley at the foot of the plateau. I visited it and found scarcely a trace of houses, only a Christian cemetery, with the graves of several missionaries and their converts, remaining as a monument of its former glories. But the surroundings are wonderfully beautiful. A thick grove of splendid mango-trees closes in the weather-worn crosses and headstones; behind them, combining the useful and the agreeable, is a whole plantation of lemon-trees covered with ripe fruit; not the small African kind, but a much larger and also juicier imported variety, which drops into the hands of the passing traveller, without calling for any exertion on his part. Old Newala is now under the jurisdiction of the native pastor, Daudi, at Chingulungulu, who, as I am on very friendly terms with him, allows me, as a matter of course, the use of this lemon-grove during my stay at Newala.

FEET MUTILATED BY THE RAVAGES OF THE “JIGGER” (Sarcopsylla penetrans)

The water-supply of New Newala is in the bottom of the valley, some 1,600 feet lower down. The way is not only long and fatiguing, but the water, when we get it, is thoroughly bad. We are suffering not only from this, but from the fact that the arrangements at Newala are nothing short of luxurious. We have a separate kitchen—a hut built against the boma palisade on the right of the baraza, the interior of which is not visible from our usual position. Our two cooks were not long in finding this out, and they consequently do—or rather neglect to do—what they please. In any case they do not seem to be very particular about the boiling of our drinking-water—at least I can attribute to no other cause certain attacks of a dysenteric nature, from which both Knudsen and I have suffered for some time. If a man like Omari has to be left unwatched for a moment, he is capable of anything. Besides this complaint, we are inconvenienced by the state of our nails, which have become as hard as glass, and crack on the slightest provocation, and I have the additional infliction of pimples all over me. As if all this were not enough, we have also, for the last week been waging war against the jigger, who has found his Eldorado in the hot sand of the Makonde plateau. Our men are seen all day long—whenever their chronic colds and the dysentery likewise raging among them permit—occupied in removing this scourge of Africa from their feet and trying to prevent the disastrous consequences of its presence. It is quite common to see natives of this place with one or two toes missing; many have lost all their toes,

or even the whole front part of the foot, so that a well-formed leg ends in a shapeless stump. These ravages are caused by the female of Sarcopsylla penetrans, which bores its way under the skin and there develops an egg-sac the size of a pea. In all books on the subject, it is stated that one’s attention is called to the presence of this parasite by an intolerable itching. This agrees very well with my experience, so far as the softer parts of the sole, the spaces between and under the toes, and the side of the foot are concerned, but if the creature penetrates through the harder parts of the heel or ball of the foot, it may escape even the most careful search till it has reached maturity.

Then there is no time to be lost, if the horrible ulceration, of which we see cases by the dozen every day, is to be prevented. It is much easier, by the way, to discover the insect on the white skin of a European than on that of a native, on which the dark speck scarcely shows. The four or five jiggers which, in spite of the fact that I constantly wore high laced boots, chose my feet to settle in, were taken out for me by the all-accomplished Knudsen, after which I thought it advisable to wash out the cavities with corrosive sublimate. The natives have a different sort of disinfectant—they fill the hole with scraped roots. In a tiny Makua village on the slope of the plateau south of Newala, we saw an old woman who had filled all the spaces under her toe-nails with powdered roots by way of prophylactic treatment. What will be the result, if any, who can say?

The rest of the many trifling ills which trouble our existence are really more comic than serious. In the absence of anything else to smoke, Knudsen and I at last opened a box of cigars procured from the Indian store-keeper at Lindi, and tried them, with the most distressing results. Whether they contain opium or some other narcotic, neither of us can say, but after the tenth puff we were both “off,” three-quarters stupefied and unspeakably wretched. Slowly we recovered—and what happened next? Half-an-hour later we were once more smoking these poisonous concoctions—so insatiable is the craving for tobacco in the tropics.

Even my present attacks of fever scarcely deserve to be taken seriously. I have had no less than three here at Newala, all of which have run their course in an incredibly short time. In the early afternoon, I am busy with my old natives, asking questions and making notes. The strong midday coffee has stimulated my spirits to

an extraordinary degree, the brain is active and vigorous, and work progresses rapidly, while a pleasant warmth pervades the whole body. Suddenly this gives place to a violent chill, forcing me to put on my overcoat, though it is only half-past three and the afternoon sun is at its hottest. Now the brain no longer works with such acuteness and logical precision; more especially does it fail me in trying to establish the syntax of the difficult Makua language on which I have ventured, as if I had not enough to do without it. Under the circumstances it seems advisable to take my temperature, and I do so, to save trouble, without leaving my seat, and while going on with my work. On examination, I find it to be 101·48°. My tutors are abruptly dismissed and my bed set up in the baraza; a few minutes later I am in it and treating myself internally with hot water and lemon-juice.

Three hours later, the thermometer marks nearly 104°, and I make them carry me back into the tent, bed and all, as I am now perspiring heavily, and exposure to the cold wind just beginning to blow might mean a fatal chill. I lie still for a little while, and then find, to my great relief, that the temperature is not rising, but rather falling. This is about 7.30 p.m. At 8 p.m. I find, to my unbounded astonishment, that it has fallen below 98·6°, and I feel perfectly well. I read for an hour or two, and could very well enjoy a smoke, if I had the wherewithal—Indian cigars being out of the question.

Having no medical training, I am at a loss to account for this state of things. It is impossible that these transitory attacks of high fever should be malarial; it seems more probable that they are due to a kind of sunstroke. On consulting my note-book, I become more and more inclined to think this is the case, for these attacks regularly follow extreme fatigue and long exposure to strong sunshine. They at least have the advantage of being only short interruptions to my work, as on the following morning I am always quite fresh and fit.

My treasure of a cook is suffering from an enormous hydrocele which makes it difficult for him to get up, and Moritz is obliged to keep in the dark on account of his inflamed eyes. Knudsen’s cook, a raw boy from somewhere in the bush, knows still less of cooking than Omari; consequently Nils Knudsen himself has been promoted to the vacant post. Finding that we had come to the end of our supplies, he began by sending to Chingulungulu for the four sucking-pigs which we had

bought from Matola and temporarily left in his charge; and when they came up, neatly packed in a large crate, he callously slaughtered the biggest of them. The first joint we were thoughtless enough to entrust for roasting to Knudsen’s mshenzi cook, and it was consequently uneatable; but we made the rest of the animal into a jelly which we ate with great relish after weeks of underfeeding, consuming incredible helpings of it at both midday and evening meals. The only drawback is a certain want of variety in the tinned vegetables. Dr. Jäger, to whom the Geographical Commission entrusted the provisioning of the expeditions—mine as well as his own—because he had more time on his hands than the rest of us, seems to have laid in a huge stock of Teltow turnips,[46] an article of food which is all very well for occasional use, but which quickly palls when set before one every day; and we seem to have no other tins left. There is no help for it—we must put up with the turnips; but I am certain that, once I am home again, I shall not touch them for ten years to come.

Amid all these minor evils, which, after all, go to make up the genuine flavour of Africa, there is at least one cheering touch: Knudsen has, with the dexterity of a skilled mechanic, repaired my 9 × 12 cm. camera, at least so far that I can use it with a little care.

How, in the absence of finger-nails, he was able to accomplish such a ticklish piece of work, having no tool but a clumsy screw-driver for taking to pieces and putting together again the complicated mechanism of the instantaneous shutter, is still a mystery to me; but he did it successfully. The loss of his finger-nails shows him in a light contrasting curiously enough with the intelligence evinced by the above operation; though, after all, it is scarcely surprising after his ten years’ residence in the bush. One day, at Lindi, he had occasion to wash a dog, which must have been in need of very thorough cleansing, for the bottle handed to our friend for the purpose had an extremely strong smell. Having performed his task in the most conscientious manner, he perceived with some surprise that the dog did not appear much the better for it, and was further surprised by finding his own nails ulcerating away in the course of the next few days. “How was I to know that carbolic acid has to be diluted?” he mutters indignantly, from time to time, with a troubled gaze at his mutilated finger-tips.

Since we came to Newala we have been making excursions in all directions through the surrounding country, in accordance with old habit, and also because the akida Sefu did not get together the tribal elders from whom I wanted information so speedily as he had promised. There is, however, no harm done, as, even if seen only from the outside, the country and people are interesting enough.

The Makonde plateau is like a large rectangular table rounded off at the corners. Measured from the Indian Ocean to Newala, it is about seventy-five miles long, and between the Rovuma and the Lukuledi it averages fifty miles in breadth, so that its superficial area is about two-thirds of that of the kingdom of Saxony. The surface, however, is not level, but uniformly inclined from its south-western edge to the ocean. From the upper edge, on which Newala lies, the eye ranges for many miles east and north-east, without encountering any obstacle, over the Makonde bush. It is a green sea, from which here and there thick clouds of smoke rise, to show that it, too, is inhabited by men who carry on their tillage like so many other primitive peoples, by cutting down and burning the bush, and manuring with the ashes. Even in the radiant light of a tropical day such a fire is a grand sight.

Much less effective is the impression produced just now by the great western plain as seen from the edge of the plateau. As often as time permits, I stroll along this edge, sometimes in one direction, sometimes in another, in the hope of finding the air clear enough to let me enjoy the view; but I have always been disappointed.

Wherever one looks, clouds of smoke rise from the burning bush, and the air is full of smoke and vapour. It is a pity, for under more favourable circumstances the panorama of the whole country up to the distant Majeje hills must be truly magnificent. It is of little use taking photographs now, and an outline sketch gives a very poor idea of the scenery. In one of these excursions I went out of my way to make a personal attempt on the Makonde bush. The present edge of the plateau is the result of a far-reaching process of destruction through erosion and denudation. The Makonde strata are everywhere cut into by ravines, which, though short, are hundreds of yards in depth. In consequence of the loose stratification of these beds, not only are the walls of these ravines nearly vertical, but their upper end is closed by an equally steep escarpment, so that the

western edge of the Makonde plateau is hemmed in by a series of deep, basin-like valleys. In order to get from one side of such a ravine to the other, I cut my way through the bush with a dozen of my men.

It was a very open part, with more grass than scrub, but even so the short stretch of less than two hundred yards was very hard work; at the end of it the men’s calicoes were in rags and they themselves bleeding from hundreds of scratches, while even our strong khaki suits had not escaped scatheless.

NATIVE PATH THROUGH THE MAKONDE BUSH, NEAR MAHUTA

I see increasing reason to believe that the view formed some time back as to the origin of the Makonde bush is the correct one. I have no doubt that it is not a natural product, but the result of human occupation. Those parts of the high country where man—as a very slight amount of practice enables the eye to perceive at once—has not yet penetrated with axe and hoe, are still occupied by a splendid timber forest quite able to sustain a comparison with our mixed forests in Germany. But wherever man has once built his hut or tilled his field, this horrible bush springs up. Every phase of this process

may be seen in the course of a couple of hours’ walk along the main road. From the bush to right or left, one hears the sound of the axe— not from one spot only, but from several directions at once. A few steps further on, we can see what is taking place. The brush has been cut down and piled up in heaps to the height of a yard or more, between which the trunks of the large trees stand up like the last pillars of a magnificent ruined building. These, too, present a melancholy spectacle: the destructive Makonde have ringed them— cut a broad strip of bark all round to ensure their dying off—and also piled up pyramids of brush round them. Father and son, mother and son-in-law, are chopping away perseveringly in the background—too busy, almost, to look round at the white stranger, who usually excites so much interest. If you pass by the same place a week later, the piles of brushwood have disappeared and a thick layer of ashes has taken the place of the green forest. The large trees stretch their smouldering trunks and branches in dumb accusation to heaven—if they have not already fallen and been more or less reduced to ashes, perhaps only showing as a white stripe on the dark ground.

This work of destruction is carried out by the Makonde alike on the virgin forest and on the bush which has sprung up on sites already cultivated and deserted. In the second case they are saved the trouble of burning the large trees, these being entirely absent in the secondary bush.

After burning this piece of forest ground and loosening it with the hoe, the native sows his corn and plants his vegetables. All over the country, he goes in for bed-culture, which requires, and, in fact, receives, the most careful attention. Weeds are nowhere tolerated in the south of German East Africa. The crops may fail on the plains, where droughts are frequent, but never on the plateau with its abundant rains and heavy dews. Its fortunate inhabitants even have the satisfaction of seeing the proud Wayao and Wamakua working for them as labourers, driven by hunger to serve where they were accustomed to rule.

But the light, sandy soil is soon exhausted, and would yield no harvest the second year if cultivated twice running. This fact has been familiar to the native for ages; consequently he provides in time, and, while his crop is growing, prepares the next plot with axe and firebrand. Next year he plants this with his various crops and

lets the first piece lie fallow. For a short time it remains waste and desolate; then nature steps in to repair the destruction wrought by man; a thousand new growths spring out of the exhausted soil, and even the old stumps put forth fresh shoots. Next year the new growth is up to one’s knees, and in a few years more it is that terrible, impenetrable bush, which maintains its position till the black occupier of the land has made the round of all the available sites and come back to his starting point.

The Makonde are, body and soul, so to speak, one with this bush.

According to my Yao informants, indeed, their name means nothing else but “bush people.” Their own tradition says that they have been settled up here for a very long time, but to my surprise they laid great stress on an original immigration. Their old homes were in the south-east, near Mikindani and the mouth of the Rovuma, whence their peaceful forefathers were driven by the continual raids of the Sakalavas from Madagascar and the warlike Shirazis[47] of the coast, to take refuge on the almost inaccessible plateau. I have studied African ethnology for twenty years, but the fact that changes of population in this apparently quiet and peaceable corner of the earth could have been occasioned by outside enterprises taking place on the high seas, was completely new to me. It is, no doubt, however, correct.

The charming tribal legend of the Makonde—besides informing us of other interesting matters—explains why they have to live in the thickest of the bush and a long way from the edge of the plateau, instead of making their permanent homes beside the purling brooks and springs of the low country.

“The place where the tribe originated is Mahuta, on the southern side of the plateau towards the Rovuma, where of old time there was nothing but thick bush. Out of this bush came a man who never washed himself or shaved his head, and who ate and drank but little.

He went out and made a human figure from the wood of a tree growing in the open country, which he took home to his abode in the bush and there set it upright. In the night this image came to life and was a woman. The man and woman went down together to the Rovuma to wash themselves. Here the woman gave birth to a stillborn child. They left that place and passed over the high land into the valley of the Mbemkuru, where the woman had another child, which

was also born dead. Then they returned to the high bush country of Mahuta, where the third child was born, which lived and grew up. In course of time, the couple had many more children, and called themselves Wamatanda. These were the ancestral stock of the Makonde, also called Wamakonde,[48] i.e., aborigines. Their forefather, the man from the bush, gave his children the command to bury their dead upright, in memory of the mother of their race who was cut out of wood and awoke to life when standing upright. He also warned them against settling in the valleys and near large streams, for sickness and death dwelt there. They were to make it a rule to have their huts at least an hour’s walk from the nearest wateringplace; then their children would thrive and escape illness.”

The explanation of the name Makonde given by my informants is somewhat different from that contained in the above legend, which I extract from a little book (small, but packed with information), by Pater Adams, entitled Lindi und sein Hinterland. Otherwise, my results agree exactly with the statements of the legend. Washing?

Hapana—there is no such thing. Why should they do so? As it is, the supply of water scarcely suffices for cooking and drinking; other people do not wash, so why should the Makonde distinguish himself by such needless eccentricity? As for shaving the head, the short, woolly crop scarcely needs it,[49] so the second ancestral precept is likewise easy enough to follow. Beyond this, however, there is nothing ridiculous in the ancestor’s advice. I have obtained from various local artists a fairly large number of figures carved in wood, ranging from fifteen to twenty-three inches in height, and representing women belonging to the great group of the Mavia, Makonde, and Matambwe tribes. The carving is remarkably well done and renders the female type with great accuracy, especially the keloid ornamentation, to be described later on. As to the object and meaning of their works the sculptors either could or (more probably) would tell me nothing, and I was forced to content myself with the scanty information vouchsafed by one man, who said that the figures were merely intended to represent the nembo—the artificial deformations of pelele, ear-discs, and keloids. The legend recorded

by Pater Adams places these figures in a new light. They must surely be more than mere dolls; and we may even venture to assume that they are—though the majority of present-day Makonde are probably unaware of the fact—representations of the tribal ancestress.

The references in the legend to the descent from Mahuta to the Rovuma, and to a journey across the highlands into the Mbekuru valley, undoubtedly indicate the previous history of the tribe, the travels of the ancestral pair typifying the migrations of their descendants. The descent to the neighbouring Rovuma valley, with its extraordinary fertility and great abundance of game, is intelligible at a glance—but the crossing of the Lukuledi depression, the ascent to the Rondo Plateau and the descent to the Mbemkuru, also lie within the bounds of probability, for all these districts have exactly the same character as the extreme south. Now, however, comes a point of especial interest for our bacteriological age. The primitive Makonde did not enjoy their lives in the marshy river-valleys.

Disease raged among them, and many died. It was only after they had returned to their original home near Mahuta, that the health conditions of these people improved. We are very apt to think of the African as a stupid person whose ignorance of nature is only equalled by his fear of it, and who looks on all mishaps as caused by evil spirits and malignant natural powers. It is much more correct to assume in this case that the people very early learnt to distinguish districts infested with malaria from those where it is absent.

This knowledge is crystallized in the ancestral warning against settling in the valleys and near the great waters, the dwelling-places of disease and death. At the same time, for security against the hostile Mavia south of the Rovuma, it was enacted that every settlement must be not less than a certain distance from the southern edge of the plateau. Such in fact is their mode of life at the present day. It is not such a bad one, and certainly they are both safer and more comfortable than the Makua, the recent intruders from the south, who have made USUAL METHOD OF good their footing on the western edge of the CLOSING HUT-DOOR plateau, extending over a fairly wide belt of country. Neither Makua nor Makonde show in their dwellings anything of the size and comeliness of the Yao houses in the plain, especially at Masasi, Chingulungulu and Zuza’s. Jumbe Chauro, a Makonde hamlet not far from Newala, on the road to Mahuta, is the

most important settlement of the tribe I have yet seen, and has fairly spacious huts. But how slovenly is their construction compared with the palatial residences of the elephant-hunters living in the plain.

The roofs are still more untidy than in the general run of huts during the dry season, the walls show here and there the scanty beginnings or the lamentable remains of the mud plastering, and the interior is a veritable dog-kennel; dirt, dust and disorder everywhere. A few huts only show any attempt at division into rooms, and this consists merely of very roughly-made bamboo partitions. In one point alone have I noticed any indication of progress—in the method of fastening the door. Houses all over the south are secured in a simple but ingenious manner. The door consists of a set of stout pieces of wood or bamboo, tied with bark-string to two cross-pieces, and moving in two grooves round one of the door-posts, so as to open inwards. If the owner wishes to leave home, he takes two logs as thick as a man’s upper arm and about a yard long. One of these is placed obliquely against the middle of the door from the inside, so as to form an angle of from 60° to 75° with the ground. He then places the second piece horizontally across the first, pressing it downward with all his might.

It is kept in place by two strong posts planted in the ground a few inches inside the door. This fastening is absolutely safe, but of course cannot be applied to both doors at once, otherwise how could the owner leave or enter his house? I have not yet succeeded in finding out how the back door is fastened.

MAKONDE LOCK AND KEY AT JUMBE CHAURO

This is the general way of closing a house. The Makonde at Jumbe Chauro, however, have a much more complicated, solid and original one. Here, too, the door is as already described, except that there is only one post on the inside, standing by itself about six inches from one side of the doorway. Opposite this post is a hole in the wall just large enough to admit a man’s arm. The door is closed inside by a large wooden bolt passing through a hole in this post and pressing with its free end against the door. The other end has three holes into which fit three pegs running in vertical grooves inside the post. The door is opened with a wooden key about a foot long, somewhat curved and sloped off at the butt; the other end has three pegs corresponding to the holes, in the bolt, so that, when it is thrust through the hole in the wall and inserted into the rectangular opening in the post, the pegs can be lifted and the bolt drawn out.[50]

MODE OF INSERTING THE KEY

With no small pride first one householder and then a second showed me on the spot the action of this greatest invention of the Makonde Highlands. To both with an admiring exclamation of “Vizuri sana!” (“Very fine!”). I expressed the wish to take back these marvels with me to Ulaya, to show the Wazungu what clever fellows the Makonde are. Scarcely five minutes after my return to camp at Newala, the two men came up sweating under the weight of two heavy logs which they laid down at my feet, handing over at the same time the keys of the fallen fortress. Arguing, logically enough, that if the key was wanted, the lock would be wanted with it, they had taken their axes and chopped down the posts—as it never occurred to them to dig them out of the ground and so bring them intact. Thus I have