Momentum, Heat, and Mass Transfer Fundamentals David P. Kessler Robert A. Greenkorn MARCEL DEKKER, INC. Momentum, Heat, and Mass Transfer David I? Kessler Robert A. Greenkorn Purdue University West La fayette, Indiana MARCEL . . . . . . - - - MARCEL DEKKER, INC. DEKKER NEW YORK BASEL Library of Congress Cataloging-in-Publication Data Kessler, David P. Greenkorn. Momentum, heat, and mass transfer fundamentals / David P. Kessler, Robert A. p. cm. Includes bibliographical references and index. ISBN 0-8247-1972-7 (alk. paper) 1. Transport theory. 2. Heat-Transmission. 3. Chemical engineering. I. Greenkorn, Robert Albert. TP 156.T7K48 1999 11. Title. 66W.284242 I 99- 10432 CIP This book is printed on acid-free paper. 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Current printing (last digit): 10987654321 PRINTED IN THE UNITED STATES OF AMERICA PREFACE This text springs from our experience over the past 30+ years teaching the momentum, heat, and mass transferhansport sequence in the School of Chemical Engineering at Purdue University. As faculty members in a state land- grant institution, we encounter students with a wide variety of backgrounds planning for a wide variety of ultimate careers. We believe that with a fm grasp of engineering fundamentals, our graduates can readily progress to careers that involve either highly technical functions or broader responsibilities in management. Our objective with this volume is to provide a foundation in basic momentum, heat, and mass transfer/transport sufficient to permit the student to do elementary design and analysis, and adequate as a base from which to learn more advanced concepts. We present the fundamentals of both microscopic and macroscopic processes. The text is built around a large number of examples which are worked in detail. Many of the examples are, of course, idealized, because their objective is to illustrate elementary principles, but we have kept them as realistic as possible. Since the book is intended as a textbook, we have incorporated a high level of detail and fedundancy in an effort to make the text readable for those just being introduced to the area. At the expense of conciseness in many places, we have attempted to avoid those gaps in derivations that are obvious to one familiar to the area, but utterly opaque to the novice. We have included many references to more advanced material, or simply to other approaches to the material herein. Age and, we hope, wisdom has disabused us of any conceit that we possesses the only valid approach to the subject. To make access to other material easier for the student, we have included page numbers for most references to avoid the necessity of an index search in the citation. In the same vein, we have attempted to make our nomenclature conform to the most common usage in the area and have incorporated an extensive nomenclature table. An abbreviated thumb index permits rapid access to chapters and the more commonly accessed tables and figures. Problems in momentum, heat, and mass transfer in fluids are profoundly difficult because the most practical application of the area is to the turbulent flow regime. To date, very little in the way of rigorous solution to even the most elementary turbulent flow problems is possible, although the exponential increase in computing power with time holds great promise for the future. V vi Pre fuce We feel that dimensional analysis is, and for the foreseeable future will be, still crucial for design of experiments, scale-up of equipment, and simplifying differential equations and associated boundary conditions. Closed-form analytical solutions are still important for the insight that they bring; however, most applied problems seem clearly destined to be solved numerically. For this reason we have included a fair amount of numerical solution techniques. Emphasis is mostly on finite difference algorithms, which are often easily implemented on a spreadsheet. Finite element analysis is heavily reliant on software, which requires too heavy a time investment to permit other than the abbreviated treatment here, although the area is certainly of growing importance. We have attempted to give an overview sufficient to enable the student to read further on hidher own. At Purdue we cover the material in this text in two three-credit required undergraduate courses. In general, depending on the background of the students, the material in Chapter 5 on systems of units will receive more or less emphasis. Similarly, the transfer of heat by radiation stands alone and can be tailored to the desires of the particular professor. Perry’s ChernicuZ Engineer’s Hundbookl remains a useful supplemental reference for physical properties arid empirical correlations. One of the commercial design packages can give help with physical property estimation, pumps and pipe networks, and more detailed heat exchanger design. David P. Kessler Robert A. Greenkorn 1 Perry, R. H. and D. W. Green, Eds. (1984). Clzernicul Engineer’s Hwuibook. New York, NY, McGraw-Hill. THUMB INDEX CONTENTS 1 ESSENTIALS 1.6.2 Types of derivatives 2 THE MASS BALANCES 2.1.1 The macroscopic total mass balance 2.1.2 The macroscopic species mass balance 2.2.1 The microscopic total mass balance (continuity equation) 2.2.2 The microscopic species mass balance 3 THE ENERGY BALANCES 3.1.2 The macroscopic total energy balance 3.1.3 The macroscopic mechanical energy balance 3.1.4 The macroscopic thermal energy balance 3.2.1 The microscopic total energy balance 3.2.2 The microscopic mechanical energy balance 3.2.3 The microscopic thermal energy balance 4 THE MOMENTUM BALANCES 4.1 The Macroscopic Momentum Balance 4.2 The Microscopic Momentum Balance 4.3 Summary of Balance Equations and Constitutive Relationships 5 APPLICATION OF DIMENSIONAL ANALYSIS 6 MOMENTUM TRANSFER IN FLUIDS 6.7 Drag Coefficients Table 6.3-1 Elementary plane flows Table 6.7.2-1 Properties of pipe Figure 6.7.2-3 Moody friction factor chart xi 1 63 73 74 86 96 103 113 114 141 149 150 157 158 169 169 196 199 211 281 302 371 386 399 vii Thumb Index ix 7 HEAT TRANSFER MODELS Table 7.2.1 - 1 Components of Fourier Equation 7.2.4 One-dimensional steady-state conduction in rectangular coordinates 7.2.5 One-dimensional steady-state conduction in cylindrical coordinates 7.2.6 One-dimensional steady-state conduction in spherical coordinates 7.2.8 One-dimensional unsteady-state conduction Semi-infinite slab Finite slab Infinite cylinder and sphere 7.2.9 Multi-dimensional unsteady-state conduction 7.3.2 Heat transfer coefficients 7.4 Conduction and Convection in Series 7.5 Radiation Heat Transfer Models Heisler charts Reciprocity relation Summation rule 7.7.3 NTU methd for design of heat exchangers 7.7.4 F-factor method for design of heat exchangers 8 MASS TRANSFER MODELS Table 8.2.3-1 Equivalent forms of Fick's Law 8.3 Convective Mass Transfer Models Height of transfer unit models 8.7 Design of Mass Transfer Columns 8.8 Mass Transfer with Chemical Reaction APPENDIX A: VECTOR AND TENSOR OPERATIONS APPENDIX C: NOMENCLATURE INDEX 517 = 530 = 536 = 574 = 581 = 629 630 638 = 651 667 - 675 = 719 = 734 = 741 = 759 - 760 = 806 = 822 = 843 = 860 882 = 903 = 923 - 963 = 989 = 997 - 1009 = TABLE OF CONTENTS Preface Thumb Index 1 ESSENTIALS 1.1 Models Figure 1.1- 1 Modeling the weather Figure 1.1-2 A poor model of the weather 1.1.1 Mathematical models and the real world 1.1.2 Scale of the model 1.2 The Entity Balance Example 1.2-1 An entity balance 1.2.1 Conserved quantities 1.2.2 S teady-state processes 1.3 The Continuum Assumption 1.4 Fluid Behavior Figure 1.3-1 Breakdown of continuum assumption 1.4.1 Laminar and turbulent flow 1.4.2 Newtonian fluids Figure 1.4.1-1 Injection of dye in pipe flow Figure 1.4.2- 1 Shear between layers of fluid Figure 1.4.2-2 Momentum transfer between layers of fluid Figure 1.4.2-3 Sign convention for momentum flux between layers of fluid Figure 1.4.2-4 Sign convention for shear stress on surface layers of fluid Table 1.4.2- 1 Summary of sign convention for stresslmomentum flux tensor Figure 1.4.2-5 Migration of momentum by molecular motion Figure 1.4.2-6 Viscosity of common fluids Example I .4.2-1 Flow offluids between frxed parallel plates 1.4.3 Complex fluids Figure 1.4.3-1 Complex fluids Figure 1.4.3-2 Mechanical analog of viscoelasticity V vii 1 1 1 2 5 8 12 14 15 16 17 18 19 19 20 21 21 24 24 25 26 26 28 28 29 30 31 xi xii Tuble of Contents 1.4.4 Compressible vs. incompressible flows 1.5.1 General cone t of average Figure 1.5.1-1 Time-average speed for travel between two points Figure 1.5.1-2 Distance-average speed for travel between two points 1.5 Averages Example I. P .1 -I Time-average vs distance-average speed 1.5.2 Velocity averages Area-averaged velocity Example 1.5.2-1 Area-averuged velocity for luminar pipe POW Figure 1.5.2-1 Velocity profile Time-averaged velocity Exumple 1.5.2-2 Time-uveraged velocity for turbulent POW Example 1.5.2-3 Area-averuge of time-averaged velocity for turbulent pipe flow 1.5.3 Temperature averages Example 1.5.3-1 Area-uverage temperuture vs. bulk temperature Example 1.5.3-2 Bulk temperuture for quadratic temperuture profile, laminar pipe flow Example 1.5.4-1 Bulk concentration Example I S.5-1 Case examples of logarithmic mean Example 1.5.5-2 Approximation of logarithmic mean by urithmetic mean 1.5.4 Concentration averages 1.5.5 Arithmetic, logarithmic, and geometric means 1.6 Scalars, Vectors, Tensors and Coordinate Systems 1.6.1 The viscous stress tensor Components of the viscous stress tensor Figure 1.6.1-1 (a) Vectors associated by a particukv viscous stress tensor with the direction of the rectangular Cartesian axes Figure 1.6.1-1 (b) Vector associated with the 3-direction decomposed into its components 1.6.2 Types of derivatives Partial derivative Total derivative Substantial derivative, material derivative, derivative following the motion Example 1.6.2-1 Rute of change of pollen density 1.6.3 Transport theorem 32 33 33 34 35 36 37 37 39 39 41 42 42 44 46 50 52 54 57 59 59 60 60 61 62 63 63 63 64 64 65 66 Table of Contents xiii Figure 1.6.3-1 Motion of continuum Chapter 1 Problems 2 THE MASS BALANCES 2.1 The Macroscopic Mass Balances Figure 2.1 - 1 System for mass balances 2.1.1 The macroscopic total mass balance Accumulation of mass Input and output of mass Simplified forms of the macroscopic total mass balance Example 2.1.1-1 Mass balance on a surge tank Figure 2.1.1 - 1 Surge tank Example 2.1.1 -2 Volumetricjlow rate offluid in laminar flow in circular pipe Example 2. I. 1-3 Air storage tank Example 2. I. 1-4 Water manifold 2.1.2 The macroscopic species mass balance Generation of mass of a species Accumulation of mass of a species Input and output of mass of a species Example 2.1.2-1 Macroscopic species mass balance with zero -0 rde r irreversible reaction Example 2. I .2-2 Macroscopic species mass balance with .first-order irreversible reaction Figure 2.1.2-1 Perfectly mixed tank with reaction 2.2.1 The microscopic total mass balance (continuity equation) Special cases of the continuity equation Continuity equation in different coordinate systems 2.2 The Microscopic Mass Balances Table 2.2.1-1 Continuity equation (microscopic total mass balance) in rectangular, cylindrical, and spherical coordinate tiames Example 2.2. I -I Velocity components in two-dimensional steady incompressible jlow, rectangular coordinates Example 2.2.1-2 Velocity components in two-dimensional steady incompressible jlow, cylindrical coordinates Example 2.2.1-3 Compression of air Figure 2.2.1-1 Air compression by pisiston 2.2.2 The microscopic species mass balance Diffusion Chapter 2 Problems 3 THE ENERGY BALANCES 3.1 The Macroscopic Energy Balances 67 69 73 73 73 74 74 75 77 78 78 79 81 82 86 87 87 88 90 94 94 96 96 98 99 99 99 101 102 102 103 105 105 113 113 [...]... Table of Contents Mass transfer in flow in pipes Mass transfer from spheres, drops, and bubbles Example 8.4.1-2 Comparison of mars transfer coeffxient models Example 8.4.1-3 Mass transfer coefficient for dissolution of a sphere Packedbeds Height of transfer unit models 8.4.2 Dimensional analysis of mass transfer by convection 8.5 Overall Mass Transfer Coefficients Figure 8.5-1 Mass transfer concentrations... composition 8.5.1 Incorporation of overall mass transfer coefficient into height of transfer unit model Example 8.5.1-1 Overall transfer units 8.6 Relationship of Overall and Single-PhaseMass Transfer Coefficients Figure 8.6-1Assumption necessary to utilize overall m s as transfer coefficient Example 8.6-1 Controlling resistancefor mass trunsfer 8.7 Design of Mass Transfer Columns Figure 8.7-1 Typical countercumnt... Dispersion and diffusion as a function of Peclet number 8.3 ConvectiveMass Transfer Models 8.3.1 The concentration boundary layer Figure 8.3.1-1Concentration boundary layer Figure 8.3.1-2 Boundary layer Figure 8.3.1-3 Boundary layer solution for a flat plate 8.3.2 Film theory and penetration-renewaltheory 8.4 The Mass Transfer Coefficient for a Single Phase Example 8.4-1 Calculation offluxfrom a mass transfer. .. Phase Example 8.4-1 Calculation offluxfrom a mass transfer coefficient Exutizple 8.4-2 Mass transfer using partial pressure as a driving force Exunzple 8.4-3 Mass transfer using species m s s density as driving force 8.4.1 Design equations for single-phasemass transfer coefficients Flat plates Example 8.4.1-1 Average mass transfer coeficient j?om loca1 coefJicient xxvii 858 858 858 860 861 862 862 863 864... element method 7.3 Convection Heat Transfer Models 7.3.1 The thermal boundary layer Figure 7.3.1-1 Solution of Equation (7.3.1-1) 7.3.2 Heat transfer coefficients Single-phase heat transfer coefficients Figure 7.3.2- 1 Single-phaseheat transfer coefficients Correlationsfor prediction of heat transfer Average heat transfer coefficients Example 7.3.2-I Average heat transfer coefficientsfor pipe flow... temperatures using NTU and E for a heat exchanger of known area 7.7.4 F-factor method for design of heat exchangers Figure 7.7.4-1 Correction factor to log mean temperature difference - one shell pass, 2" tube passes Emmple 7.7.4-1 Use of F Fuctor compured to effect iveness/NTU met hod Chapter 7 Problems 8 MASS TRANSFER MODELS 8.1 The Nature of Mass Transfer 8.2 Diffusive Mass Transfer Models 8.2.1... average temperature, depending on whether external or internal flows are being modeled; and Tsis the temperature at the surface The third, k, is for mass transfer: , r 1 (1.1.2-3) where NA is the mass flux of species A, k is the single-phase , mass transfer coefficient, Z A ~ the concentration of A at the is surface, and ZA is either the freestream concentration or an average concentration of A depending... crude oil with orifice Chapter 6 Problems 7 HEAT TRANSFER MODELS 7.1 The Nature of Heat 7.1.1 Forced convection heat transfer 7.1.2 Free convection heat transfer Table 7.1.2-1 Dimensionless Fonns: Mass, Energy, and Momentum Equations for Natural and Forced Convection 7.2 Conduction Heat Transfer Models 7.2.1 Three-dimensionalconduction in isotropic media Table 7.2.1-1Components of Fourier Equation in... instantaneous irreversible reaction i a membrane n Example 8.8-1 Acidization of an oil well Figure 8.8-3 M s transfer with slow or reversible chemical as reaction Example 8.8-2 Mass transfer with heterogeneous reaction Example 8.8-3 Mass transfer with homgeneous reaction Chapter 8 Problems APPENDIX A: VECTOR AND TENSOR OPERATIONS A.l Symbolic Notation Table A.l Operational properties of the del operator in... convective heat transfer Forced convection in laminar f o lw Table 7.3.2-1 Nusselt number limit for laminar flow in ducts with various cross-sections Forced convection in turbulent flow Example 7.3.2-2 Comparison of the Dittus-Boelter, Colburn, and Sieder-Tate equations Heat transfer in non-circular conduits and annular flow External flows, natural and f o r d convection Table 7.3.2-2 Values of b and n for . Momentum, Heat, and Mass Transfer Fundamentals David P. Kessler Robert A. Greenkorn MARCEL DEKKER, INC. Momentum, Heat, and Mass Transfer David I? Kessler. MASS TRANSFER MODELS Table 8.2.3-1 Equivalent forms of Fick's Law 8.3 Convective Mass Transfer Models Height of transfer unit models 8.7 Design of Mass Transfer Columns 8.8 Mass. is to provide a foundation in basic momentum, heat, and mass transfer/ transport sufficient to permit the student to do elementary design and analysis, and adequate as a base from which to