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Process Engineering And Design Using Visual Basic Arun K Datta

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Process Engineering And Design Using Visual Basic Arun K Datta Process Engineering And Design Using Visual Basic Arun K Datta Process Engineering And Design Using Visual Basic Arun K Datta

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Process Engineering And Design Using Visual Basic Arun K Datta
Process Engineering And Design Using Visual Basic Arun K Datta
Half title page PROCESS ENGINEERING and DESIGN USING VISUAL BASIC® 45423_C000.fm Page i Monday, August 27, 2007 11:16 AM
45423_C000.fm Page ii Monday, August 27, 2007 11:16 AM
Title Page CRC Press is an imprint of the Taylor & Francis Group, an informa business Boca Raton London New York ARUN DATTA PROCESS ENGINEERING and DESIGN USING VISUAL BASIC® 45423_C000.fm Page iii Monday, August 27, 2007 11:16 AM
Visual Basic and Excel are registered trademarks of Microsoft. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-4542-0 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse- quences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Datta, Arun. Process engineering and design using Visual BASIC / Arun Datta. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4200-4542-0 (alk. paper) ISBN-10: 1-4200-4542-3 (alk. paper) 1. Chemical processes--Mathematical models. 2. Chemical processes--Computer simulation. 3. Microsoft Visual BASIC. I. Title. TP155.7.D36 2006 660’.28--dc22 2006103307 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 45423_C000.fm Page iv Monday, August 27, 2007 11:16 AM
Dedication To my mother Smt. Narayani Datta 45423_C000.fm Page v Monday, August 27, 2007 11:16 AM
45423_C000.fm Page vi Monday, August 27, 2007 11:16 AM
vii Contents Chapter one Basic mathematics.......................................................................1 Introduction .............................................................................................................1 Physical constants...................................................................................................1 SI prefixes .......................................................................................................1 Mensuration.............................................................................................................2 Triangles..........................................................................................................2 Rectangles.......................................................................................................2 Parallelogram (opposite sides parallel).....................................................3 Rhombus (equilateral parallelogram)........................................................3 Trapezoid (four sides, two parallel)...........................................................3 Quadrilateral (four sided)............................................................................4 Regular polygon of n sides..........................................................................4 Circle................................................................................................................4 Ellipse..............................................................................................................6 Parabola ..........................................................................................................6 Prism................................................................................................................6 Pyramid...........................................................................................................7 Right circular cylinder..................................................................................7 Sphere..............................................................................................................7 Right circular cone ........................................................................................8 Dished end .....................................................................................................8 Irregular shape...............................................................................................8 Trapezoidal rule.............................................................................................8 Simpson’s rule ...............................................................................................8 Irregular volume ...........................................................................................9 Algebra .....................................................................................................................9 Factoring .........................................................................................................9 Arithmetic progression.................................................................................9 Geometric progression ...............................................................................10 Infinite series (in GP)..................................................................................10 Best fit straight line (least squares method) ...........................................10 Binomial equation ....................................................................................... 11 Polynomial equation................................................................................... 11 45423_C000.fm Page vii Monday, August 27, 2007 11:16 AM
viii Process engineering and design using Visual Basic Maxima/minima .........................................................................................12 Matrix............................................................................................................13 Addition and multiplication of matrices............................................13 Addition of matrices..............................................................................14 Multiplication of matrices.....................................................................14 Matrix properties involving addition .................................................14 Matrix properties involving multiplication .......................................15 Matrix properties involving addition and multiplication...............15 Transpose .................................................................................................15 Symmetric matrix...................................................................................16 Diagonal matrix......................................................................................16 Determinants................................................................................................16 Properties of determinants....................................................................17 Cofactor....................................................................................................18 Determinant and inverses.....................................................................19 Adjoint......................................................................................................19 Cramer’s rule...............................................................................................20 Trigonometry .........................................................................................................21 Functions of circular trigonometry ..........................................................21 Periodic functions .......................................................................................22 The magic identity ......................................................................................22 The addition formulas................................................................................23 Double angle and half angle.....................................................................23 Product and sum formulas........................................................................24 Relations between angles and sides of triangles ...................................25 Law of sines .................................................................................................26 Law of tangents...........................................................................................26 Law of cosines .............................................................................................26 Other relations .............................................................................................26 Inverse trigonometric functions................................................................27 Hyperbolic functions ..................................................................................28 Other hyperbolic functions........................................................................29 Inverse hyperbolic functions.....................................................................29 Analytical geometry .............................................................................................30 Straight line..................................................................................................30 Straight line through two points .........................................................30 Three points on one line .......................................................................31 The circle.......................................................................................................31 Tangent.....................................................................................................31 Normal .....................................................................................................32 Four points on a circle...........................................................................32 Circle through three points...................................................................33 Conic section................................................................................................33 Focus.........................................................................................................33 Eccentricity ..............................................................................................33 Directrix....................................................................................................33 45423_C000.fm Page viii Monday, August 27, 2007 11:16 AM
Contents ix Partial derivatives...................................................................................33 Parabola....................................................................................................34 Tangent line with a given slope, m .....................................................36 Ellipse .......................................................................................................36 Hyperbola................................................................................................38 Calculus..................................................................................................................40 Differential calculus ....................................................................................40 Understanding the derivatives .................................................................41 Standard derivatives...................................................................................42 Integral calculus ..........................................................................................44 Volume of horizontal dished end........................................................44 Volume of vertical dished end .............................................................46 Standard integrals .......................................................................................47 Differential equations...........................................................................................48 First-order differential equations..............................................................48 Separation of variables ..........................................................................49 Second-order differential equations.........................................................50 Bessel function ........................................................................................51 Partial differential equations...............................................................................52 Laplace transform .......................................................................................60 Standard Laplace transforms ....................................................................61 Fourier half-range expansions ..................................................................62 Fourier half-range cosine series................................................................63 Fourier half-range sine series....................................................................63 Numerical analysis ...............................................................................................65 Solving linear equations (Newton’s method).........................................65 Newton’s method in two variables..........................................................66 Numerical methods in linear algebra......................................................68 Gauss elimination...................................................................................68 Cholesky method....................................................................................70 Numerical integration ................................................................................72 Trapezoidal rule......................................................................................72 Simpson’s rule.........................................................................................72 Double integration using Simpson’s rule ...............................................75 Numerical solution of first-order differential equations ......................76 Euler’s method........................................................................................76 Improved Euler’s method.....................................................................76 Runge–Kutta method.............................................................................77 Second-order differential equations.........................................................79 Runge–Kutta–Nystrom method...........................................................79 Partial differential equations.....................................................................81 Heat conduction problem .....................................................................81 Numerical solution .....................................................................................83 Alternating direction implicit (ADI) method ....................................85 Equation of state ...................................................................................................90 Boyle’s law and Charles’ law....................................................................90 45423_C000.fm Page ix Monday, August 27, 2007 11:16 AM
x Process engineering and design using Visual Basic Equation of state for real gas ....................................................................92 Comparison between Peng–Robinson and SRK EOSs .........................92 Acentric factor..............................................................................................94 Vapor pressure of pure components........................................................94 SRK EOS .......................................................................................................94 PR EOS..........................................................................................................97 Alternate method to estimate vapor pressure of pure components................................................................................................97 Vapor pressure of pure water ...................................................................98 Equilibrium ratio (K) ..................................................................................98 Mixing rules .................................................................................................99 Estimate fugacity coefficient of vapor ...................................................101 Influence of interaction coefficient .........................................................102 Estimation of dew point ..........................................................................102 Unit conversions .................................................................................................103 Programming.......................................................................................................103 General notes for all programmers.............................................................. Vessel...........................................................................................................103 Program limitations ..................................................................................103 Horizontal...................................................................................................108 Data entry...................................................................................................108 Inclined................................................................................................... 110 Vertical.................................................................................................... 111 Conversion ................................................................................................. 112 Program limitations .................................................................................. 114 Procedure............................................................................................... 114 References/further reading............................................................................... 114 Chapter two Fluid mechanics.......................................................................117 Introduction ......................................................................................................... 117 Bernoulli’s theorem ............................................................................................ 118 Velocity heads............................................................................................ 119 Flow measurements............................................................................................120 Orifice/Venturi meter...............................................................................120 Thermal expansion factor (Fa).................................................................122 Coefficient of discharge CD ......................................................................122 Orifice meter..........................................................................................122 Restriction orifice.......................................................................................123 Venturi meter .............................................................................................124 Expansion factor (Y) .................................................................................124 Orifices ...................................................................................................124 Nozzles and Venturi ............................................................................124 Nonrecoverable pressure drop................................................................124 Orifices ...................................................................................................125 Venturi with 15° divergent angle.......................................................125 Venturi with 7° divergent angle.........................................................125 45423_C000.fm Page x Monday, August 27, 2007 11:16 AM
Contents xi Critical flow................................................................................................125 Area meter: rotameters.............................................................................125 Flow through an open channel...............................................................126 V notch........................................................................................................126 Rectangular notch .....................................................................................127 Frictional pressure drop.....................................................................................127 Darcy equation...........................................................................................127 Flow in open channel...............................................................................128 Estimation of friction factor.....................................................................129 Friction factor — laminar flow ..........................................................129 Friction factor — turbulent flow........................................................129 Two-K method.....................................................................................................130 K for reducer/expander...........................................................................130 Reducer ..................................................................................................130 Expander................................................................................................132 Pipe entrance..............................................................................................132 Pipe exit ......................................................................................................132 Split flow.....................................................................................................132 Split 1,3...................................................................................................133 Split 1,2...................................................................................................133 Split 3,1...................................................................................................134 Split 1,2,3................................................................................................134 Split 1,3,2................................................................................................134 Split 3,1,2................................................................................................134 Hydraulics — general guidelines ....................................................................134 Roughness of pipe wall............................................................................135 Control valve CV.......................................................................................135 Line sizing criteria for liquid lines.........................................................136 Line sizing for gravity flow lines ...........................................................136 Downpipe sizing .......................................................................................136 Line sizing criteria for vapor lines.........................................................137 Relief valve inlet line sizing ....................................................................137 Relief valve outlet line sizing..................................................................138 Line sizing criteria for two-phase flow .................................................138 Hydraulics — compressible fluids...................................................................139 Adiabatic flow in a pipe ..........................................................................139 Isothermal flow in a pipe.........................................................................142 Heat loss...............................................................................................................143 Types of cross-country buried pipelines ...............................................143 Yellow jacket ..............................................................................................143 Coating thickness .................................................................................143 Fusion-bonded epoxy coating.................................................................144 Rate of heat transfer .................................................................................144 Film resistance (Rfilm)............................................................................144 Resistance of pipe (Rpipe)......................................................................146 Resistance of coatings (Rcoating)............................................................146 45423_C000.fm Page xi Monday, August 27, 2007 11:16 AM
xii Process engineering and design using Visual Basic Resistance of environment (Renv)........................................................147 Viscosity of water......................................................................................149 Thermal conductivity of water ...............................................................149 Viscosity of air ...........................................................................................150 Thermal conductivity of air.....................................................................150 Choked flow...............................................................................................150 Limiting differential pressure..................................................................151 Limiting expansion factor Y....................................................................151 Hydraulics — two-phase flow..........................................................................153 Beggs and Brill correlations.....................................................................155 Step 1: Estimation of flow regime .....................................................155 Step 2: Estimation of horizontal holdup ..........................................155 Step 3: Estimation of uphill holdup..................................................156 Step 4: Estimation of downhill holdup ............................................156 Step 5: Estimation of friction factor...................................................157 Step 6: Estimation of pressure drop..................................................157 Mukherjee and Brill correlations............................................................158 Step 1: Estimation of flow regime .....................................................158 Step 2: Estimation of holdup..............................................................160 Step 3: Estimation of hydrostatic head.............................................160 Step 4: Estimation of acceleration head............................................160 Step 5: Estimation of friction factor...................................................161 Step 6: Estimation of frictional pressure drop.................................161 CO2 corrosion.............................................................................................164 CO2 corrosion mechanism .......................................................................164 NACE requirements..................................................................................165 Rate of corrosion .......................................................................................165 Constant Kt .................................................................................................165 Fugacity of CO2 .........................................................................................166 Calculation of pH factor f(pH)................................................................167 Calculation of shear stress.......................................................................168 Effect of temperature................................................................................169 Effect of glycol ...........................................................................................170 Effect of corrosion inhibitor.....................................................................170 Condensation factor..................................................................................170 Programming.......................................................................................................171 Program for flow elements......................................................................171 General overview .................................................................................171 Project details ........................................................................................172 Calculation form...................................................................................172 Program limitations and notes................................................................173 Program for hydraulic calculations........................................................176 General overview .................................................................................177 Project details ........................................................................................177 Program limitations and notes...........................................................178 Form incompressible fluid..................................................................178 45423_C000.fm Page xii Monday, August 27, 2007 11:16 AM
Contents xiii Form compressible fluid .....................................................................181 Pressure drop comparison ..................................................................184 Form for two-phase flow ....................................................................184 Horizontal pipe section .......................................................................187 Uphill pipe section...............................................................................188 Downhill pipe section..........................................................................190 General conclusion...............................................................................192 Program for corrosion calculations........................................................192 Program limitations and notes...........................................................193 Calculation of pH.................................................................................193 Nomenclature ......................................................................................................197 Greek characters..................................................................................................200 References.............................................................................................................200 Chapter three Separators...............................................................................203 Introduction .........................................................................................................203 General principles of separation ......................................................................203 Droplet in a vertical vessel......................................................................203 Droplet in a horizontal vessel.................................................................206 Gravity settling: limiting conditions......................................................206 Newton’s law.............................................................................................207 Stoke’s law..................................................................................................207 Intermediate law .......................................................................................207 Critical particle diameter .........................................................................207 Vertical vs. horizontal separators.....................................................................208 Advantages of the horizontal separator................................................209 Disadvantages of the horizontal separator...........................................209 Advantages of the vertical separator.....................................................209 Disadvantages of the vertical separator................................................209 Design of a gas–liquid separator .....................................................................209 Critical settling velocity ...........................................................................209 Design constant KD ...................................................................................210 API 521 method......................................................................................... 211 Design of liquid–liquid separators ..................................................................212 Mist eliminator....................................................................................................214 Wire mesh mist eliminator ......................................................................214 Efficiency of the mist eliminator........................................................214 Target collection efficiency..................................................................215 Inertial impaction......................................................................................215 Direct interception.....................................................................................215 Diffusion ................................................................................................215 Target collection efficiency..................................................................216 Pressure drop of mist eliminator............................................................216 Vane type mist eliminator........................................................................217 Efficiency of vane pack ............................................................................217 Terminal centrifugal velocity ..................................................................217 45423_C000.fm Page xiii Monday, August 27, 2007 11:16 AM
xiv Process engineering and design using Visual Basic Pressure drop through the vane pack ...................................................218 General dimensions and setting of levels.......................................................219 The horizontal separator..........................................................................219 Boot.........................................................................................................221 Vertical separator.......................................................................................222 Separator internals..............................................................................................223 The inlet nozzle .........................................................................................223 The vortex breaker....................................................................................224 Separator control.................................................................................................224 Pressure and flow control........................................................................226 Light liquid level control .........................................................................226 Heavy liquid level and slug control ......................................................226 High performance separator.............................................................................226 Salient features of GLCC .........................................................................228 Design parameters..............................................................................................228 Flow rates ...................................................................................................228 Slug length .................................................................................................228 Density ........................................................................................................230 Viscosity......................................................................................................231 Oil in gas droplet size ..............................................................................232 Oil in water droplet size ..........................................................................232 Water in oil droplet size...........................................................................233 Inlet nozzle velocity..................................................................................233 Gas outlet nozzle velocity........................................................................233 Liquid outlet velocity ...............................................................................233 Separator program..............................................................................................233 Program limitations/notes ......................................................................234 Horizontal separators...............................................................................234 Three-phase flooded weir ...................................................................234 Three-phase nonflooded-weir separator ..........................................234 Three-phase with boot separator.......................................................234 Two-phase vapor–liquid separator....................................................234 Two-phase liquid–liquid separator ...................................................235 Vertical separators.....................................................................................235 Two-phase vapor–liquid separator....................................................235 Two-phase liquid–liquid separator ...................................................235 General overview of the separator.exe program .................................235 Design .........................................................................................................240 Nomenclature ......................................................................................................244 Greek characters..................................................................................................245 References.............................................................................................................246 Chapter four Overpressure protection .......................................................247 Introduction .........................................................................................................247 Impact on plant design ............................................................................247 Impact on individual design...................................................................248 45423_C000.fm Page xiv Monday, August 27, 2007 11:16 AM
Contents xv Definition..............................................................................................................248 Accumulation.............................................................................................248 Atmospheric discharge.............................................................................249 Built-up back pressure..............................................................................249 General back pressure ..............................................................................249 Superimposed back pressure ..................................................................249 Balanced-bellows PRV..............................................................................250 Blowdown ..................................................................................................250 Closed discharge system..........................................................................250 Cold differential test pressure.................................................................250 Conventional PRV.....................................................................................250 Design capacity..........................................................................................250 Design pressure .........................................................................................250 Maximum allowable accumulated pressure.........................................251 Maximum allowable working pressure ................................................251 Operating pressure....................................................................................251 Overpressure..............................................................................................251 Pilot-operated PRV....................................................................................251 Pressure relief valve (PRV)......................................................................252 Pressure safety valve ................................................................................252 Rated relieving capacity...........................................................................252 Relief valve.................................................................................................252 Relieving conditions .................................................................................252 Rupture disk...............................................................................................252 Safety relief valve......................................................................................252 Safety valve ................................................................................................253 Set pressure ................................................................................................253 Vapor depressuring system.....................................................................253 Vent stack....................................................................................................253 Types of pressure relief valves .........................................................................254 Conventional pressure relief valve (vapor service).............................254 Conventional pressure relief valve (liquid service) ............................256 Balanced-bellows pressure relief valve .................................................256 Pilot-operated pressure relief valve .......................................................259 Rupture disk...............................................................................................262 Selection of pressure relief valves....................................................................262 Conventional pressure relief valve.........................................................262 Balanced-bellows pressure relief valve .................................................264 Pilot-operated pressure relief valve .......................................................264 Rupture disk...............................................................................................265 PRV installation and line sizing.......................................................................265 Compressors and pumps.........................................................................265 Fired heaters...............................................................................................266 Heat exchangers ........................................................................................267 Piping ..........................................................................................................267 Pressure vessels .........................................................................................267 45423_C000.fm Page xv Monday, August 27, 2007 11:16 AM
xvi Process engineering and design using Visual Basic PRV isolation valves .................................................................................268 Inlet piping to PRVs .................................................................................270 Discharge piping from PRVs...................................................................271 Contingency quantification...............................................................................272 General........................................................................................................272 Power failure..............................................................................................273 Local power failure ..............................................................................274 Failure of a distribution center ..........................................................274 Total power failure...............................................................................275 Cooling water failure................................................................................275 Instrument air failure ...............................................................................276 Steam failure ..............................................................................................277 Total steam failure................................................................................278 Loss of steam to specific equipment.................................................278 Partial steam failure.............................................................................278 Check valve failure ...................................................................................278 Blocked outlet ............................................................................................279 Pump or compressor discharge .........................................................279 Multiple outlet ......................................................................................280 Block valve downstream of control valve........................................280 Control valve failure.................................................................................280 Vapor breakthrough..................................................................................281 Maximum flow ..........................................................................................283 Thermal relief.............................................................................................283 Modulus of elasticity of pipe material (E) .......................................286 Coefficient of linear thermal expansion (α).....................................286 Valve leakage rate (q)...........................................................................287 Compressibility of liquid (Z)..............................................................288 Coefficient of cubic expansion of liquids (β)...................................288 Installation of thermal relief valve ....................................................288 Fire exposure..............................................................................................289 General guidelines ...............................................................................290 Estimation of wetted surface area .....................................................290 Fire circle................................................................................................292 Estimation of latent heat and physical properties..........................292 Liquid wet vessel..................................................................................293 Vessels with only gas...........................................................................295 Two liquid phases ................................................................................296 Heat exchanger tube rupture ..................................................................298 Contingency calculation......................................................................299 Reflux failure and overhead system ......................................................302 Loss of reboiler heat .................................................................................303 Venting of storage tank ............................................................................303 Venting due to liquid movements.....................................................304 Thermal venting ...................................................................................304 45423_C000.fm Page xvi Monday, August 27, 2007 11:16 AM
Contents xvii Fire exposure.........................................................................................304 Minimum flow area .............................................................................304 Sizing procedure .................................................................................................307 Sizing of liquid relief................................................................................307 Sizing of vapor relief ................................................................................309 Critical flow...........................................................................................309 Subcritical flow ..................................................................................... 311 Conventional and pilot-operated PRV.............................................. 311 Balanced-bellows PRV......................................................................... 311 Sizing for steam relief............................................................................... 311 Sizing for two-phase fluids .....................................................................313 Design of flare stack .................................................................................330 Minimum distance ....................................................................................330 Fraction of heat intensity transmitted, τ ...............................................331 Fraction of heat radiated, F .....................................................................331 Heat release, Q...........................................................................................332 Sizing of a flare stack: simple approach................................................332 Calculation of stack diameter.............................................................332 Calculation of flame length ................................................................333 Flame distortion caused by wind velocity.......................................333 Sizing of flare stack: Brzustowski and Sommer approach.................335 Calculation of flare stack diameter....................................................335 Location of flame center xc, yc ............................................................335 Lower explosive limit of mixtures ....................................................335 Vertical distance yc................................................................................338 Horizontal distance xc..........................................................................338 SIL analysis ..........................................................................................................343 Definitions ..................................................................................................344 Diagnostic coverage (DC) ...................................................................344 Final element.........................................................................................344 MooN......................................................................................................344 Programmable electronics (PE)..........................................................344 Programmable electronic system (PES)............................................344 Protection layer.....................................................................................344 Safety-instrumented function (SIF) ...................................................344 Safety-instrumented systems (SIS) ....................................................345 Safety integrity......................................................................................345 Safety integrity level (SIL) ..................................................................345 Safety life cycle .....................................................................................345 Matrix for SIL determination ..................................................................345 Probability of failure on demand ...........................................................347 ALARP model............................................................................................348 Determination of SIL ...........................................................................348 Financial.................................................................................................350 Health and safety .................................................................................351 Environment and asset........................................................................351 45423_C000.fm Page xvii Monday, August 27, 2007 11:16 AM
xviii Process engineering and design using Visual Basic Programming.......................................................................................................353 Program for pressure relief valve...........................................................353 Program limitations and notes...........................................................353 General overview .................................................................................354 Project details ........................................................................................355 File save..................................................................................................355 File open.................................................................................................355 File print.................................................................................................355 Exit ..........................................................................................................355 Specific message or warning: back pressure ...................................355 Back-pressure correction factor..........................................................356 Pilot-operated PRV...............................................................................356 Liquid .....................................................................................................357 Vapor ......................................................................................................357 Two-phase type 1 and type 2 calculations.......................................357 Program for flare stack estimation.........................................................362 Program limitations and notes...........................................................363 Specific message/warning..................................................................364 Nomenclature ......................................................................................................365 Greek characters..................................................................................................366 References.............................................................................................................366 Chapter five Glycol dehydration.................................................................369 Introduction .........................................................................................................369 Basic scheme........................................................................................................370 Advantages.................................................................................................370 Disadvantages............................................................................................370 Pre-TEG coalescer......................................................................................370 Contactor ....................................................................................................370 Flash separator...........................................................................................372 Filters...........................................................................................................372 Pumping .....................................................................................................372 Glycol/glycol exchanger..........................................................................373 Gas/glycol exchanger...............................................................................373 Regenerator ................................................................................................373 Physical properties .............................................................................................374 Selection of type of glycol .......................................................................374 Common properties of glycol .................................................................374 Densities of aqueous glycol solutions...............................................376 Solubility of various compounds.......................................................376 Fire hazard information ...........................................................................376 Viscosities of aqueous glycol solutions ............................................381 Specific heats of aqueous glycol solutions.......................................385 Thermal conductivities of aqueous glycol solutions......................387 Design aspects.....................................................................................................389 Water content in hydrocarbon gas .........................................................389 45423_C000.fm Page xviii Monday, August 27, 2007 11:16 AM
Contents xix Equilibrium dew point.............................................................................389 Minimum lean-TEG concentration.........................................................389 Number of theoretical stages of the contactor.....................................391 Design of contactor...................................................................................394 Type of internals...................................................................................394 Flash separator...........................................................................................397 Filters...........................................................................................................397 Particulate filter ....................................................................................398 Carbon filter ..........................................................................................398 Glycol/glycol exchanger..........................................................................398 Gas/glycol exchanger...............................................................................398 Regenerator ................................................................................................400 Still column............................................................................................402 Reboiler.......................................................................................................403 Fire tube heat density..........................................................................403 Fire tube heat flux ................................................................................403 Lean glycol storage...................................................................................404 Energy exchange pump ...........................................................................404 Burner management .................................................................................406 Specifications..............................................................................................412 Programming.......................................................................................................415 Program limitations ..................................................................................415 General overview......................................................................................416 File menu ...............................................................................................416 Unit menu..............................................................................................417 Project details ........................................................................................417 Data entry ..............................................................................................417 References.............................................................................................................418 Index.....................................................................................................................419 45423_C000.fm Page xix Monday, August 27, 2007 11:16 AM
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xxi Preface The availability of various design tools and software made process engineer- ing and design simple but also, in some cases, paradoxically led to inade- quate design. Unfortunately, not only young engineers but also reasonably experienced engineers are becoming more dependent on software tools with- out having a basic understanding of the design. I have seen people using HYSYS® to estimate water boiling temperature at atmospheric pressure. This reduced self-confidence is the key factor for inappropriate process design in many cases. As everything in process design can’t be covered by software tools like HYSYS®, HTRI®, etc., most design/consulting groups have developed their own design tools mostly using Excel® spreadsheets. Excel spreadsheets are extremely easy to develop and easy to corrupt through cut-and-paste mod- ifications as well as with attempts to modify the macros of the spreadsheet. Sometimes we also failed to recognize the limitations of Excel spreadsheets and develop something that does not produce the intended output. In spite of different procedures and design tools, we sometimes come across some typical design requirements that are not covered by any known procedure. Some understanding of basic mathematics and fundamental pro- cess engineering can solve a large number of problems without much diffi- culty. For example, a volume calculation of a horizontal vessel requires simple integration, volume calculation of an inclined vessel using Simpson’s rule, etc. The main purpose of this work is to identify small but important issues we normally come across during design and the best possible procedures to resolve them. This work gives a detailed analysis of the methodology used for a particular calculation and in some cases limitations of the proce- dure. This is done so that the user can check a calculation manually and develop an understanding of the basic design. This will improve the self- confidence of the user. The programs have been developed in Visual Basic® to avoid limitations of other programming tools, e.g., Excel. All programs have been developed to use both the International System of Units (SI) and English units (the 45423_C000.fm Page xxi Monday, August 27, 2007 11:16 AM
xxii Process engineering and design using Visual Basic default choice is SI); however, the programs do not allow changing of indi- vidual units. Though each program has been checked extensively for correctness, the possibility of program bugs can’t be totally eliminated. Any feedback on program malfunctioning will be highly appreciated, as well as any other comments to improve future editions. Arun K. Datta 45423_C000.fm Page xxii Monday, August 27, 2007 11:16 AM
xxiii Acknowledgements I wish to express my gratitude to WorleyParsons, Brisbane, for their strong support while this work was being produced. Special thanks go to my wife, Dr. Nivedita Datta, and my daughter, Raka Datta, whose support and encouragements were the key factors in the completion of this work. I am also very grateful to Malcolm Rough of WorleyParsons for his full review of the manuscript and many useful suggestions. I also appreciate the encouragement I have received from my mentors over many years, partic- ularly Tony Buckley and Ivan Broome of WorleyParsons. I have tried my best to thoroughly utilize my learning from them. I also acknowledge the help received from the American Petroleum Institute, the Gas Processors Suppliers Association, and John M. Campbell and Company for allowing me to publish their relevant figures and tables. Special thanks are also due to the editorial and production staff of CRC Press, a Taylor & Francis Company, particularly Allison Shatkin, Marsha Pronin, and Ari Silver for publishing this work with outstanding quality. I will thankfully acknowledge any suggestion to further improve future editions. Arun K. Datta Brisbane, Australia 45423_C000.fm Page xxiii Monday, August 27, 2007 11:16 AM
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xxv About the author Arun K. Datta, principal process engineer working with WorleyParsons, has twenty-seven years’ experience in the field of process engineering and design. He holds a master’s degree in chemical engineering from the Indian Institute of Technology, Delhi, and has worked with several process consul- tancy organizations in both India and Australia. Arun has been a consultant for a large number of process engineering organizations including refineries, oil and gas industries, fine chemicals, and pharmaceuticals. His fields of expertise include heat and mass transfer, pro- cess simulations, exchanger design, pressure vessel design, design of safety systems, and design of control systems. Some of his clients include BP Refinery, Caltex Refinery, Santos, ExxonMobil, ONGC, Indian Oil, Incitec, and Oil Search. Arun is a chartered professional engineer in Australia (Queensland chap- ter) and has membership with the Institute of Engineers, Australia. He is also a member of the interview committee for the Institute of Engineers. 45423_C000.fm Page xxv Monday, August 27, 2007 11:16 AM
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1 chapter one Basic mathematics Introduction Understanding mathematics is the most fundamental requirement in under- standing engineering. Some fields in chemical engineering require the solu- tion of complex mathematical equations. The purpose of this section is to get some idea in the field of mathematics commonly used for chemical engineering design and includes the following: • Physical constants • Mensuration • Algebra • Trigonometry • Analytical geometry • Calculus • Differential equations • Partial differential equations • Numerical analysis • Equation of states • Unit conversions • Programming Physical constants Commonly used physical constants are presented in Table 1.1. SI prefixes The International System of Units (SI) prefixes are presented in Table 1.2. 45423_book.fm Page 1 Monday, August 27, 2007 12:44 PM
2 Process engineering and design using Visual Basic Mensuration Triangles Area = 1/2 bh (1.1) where b = base and h = altitude. Rectangles Area = ab (1.2) where a and b are the length of the sides. Table 1.1 Commonly Used Physical Constants Name Symbol Unit Value Speed of light c cm/sec 2.99792458E10 Plank’s constant h erg.sec 6.6260755E-27 hbar erg.sec 1.05457266E-27 Gravitational constant G cm3/(g.sec2) 6.67259E-8 G Nm2/kg2 6.67259E-11 Electron charge e ESU 4.8032068E-10 Mass of electron me g 9.1093897E-28 Mass of proton mp g 1.6726231E-24 Mass of neutron mn g 1.6749286E-24 Mass of hydrogen mH g 1.6733E-24 Atomic mass unit amu g 1.6605402E-24 Avogadro’s number NA 6.0221367E23 Boltzmann constant k erg/k 1.380658E-16 Electron volt eV erg 1.6021772E-12 Radiation density constant a erg/(cm3.K4) 7.5646E-15 Stefan–Boltzmann constant sigma erg/(cm2.K4.sec) 5.67051E-5 Rydberg constant erg 2.1798741E-11 Gas constant R J/(mol.K) 8.31439 R cal/(mol.K) 1.98719 R l.atm/(mol.K) 0.0820567 Pi π 3.1415926536 Napierian (natural) logarithm base e 2.7182818285 Euler’s constant γ 0.5772156649 Logarithm conversion log x 0.4342944819 ln x Logarithm conversion ln x 2.302585093 log x Radian 57.2957795131° Degree 0.0174532925 rad Minute 0.0002908882 rad Second 0.0000048481 rad 45423_book.fm Page 2 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 3 Parallelogram (opposite sides parallel) Area = ah = ab sinα (1.3) where a and b are the lengths of the sides, h the height, and α the angle between the sides. See Figure 1.1. Rhombus (equilateral parallelogram) Area = 1/2 ab (1.4) where a and b are lengths of diagonals. Trapezoid (four sides, two parallel) Area = 1/2(a + b)h (1.5) where the lengths of parallel sides are a and b and h = height. Table 1.2 SI Prefixes Multiplication Factor Prefix Symbol 1 000 000 000 000 000 000 = 1018 exa E 1 000 000 000 000 000 = 1015 peta P 1 000 000 000 000 = 1012 tera T 1 000 000 000 = 109 giga G 1 000 000 = 106 mega M 1 000 = 103 kilo k 100 = 102 hecto h 10 = 101 deka da 0.1 = 10–1 deci d 0.01 = 10–2 centi c 0.001 = 10–3 milli m 0.000 001 = 10–6 micro μ 0.000 000 001 = 10–9 nano n 0.000 000 000 001 = 10–12 pico p 0.000 000 000 000 001 = 10–15 femto f 0.000 000 000 000 000 001 = 10–18 atto a Figure 1.1 Parallelogram. h a b α 45423_book.fm Page 3 Monday, August 27, 2007 12:44 PM
4 Process engineering and design using Visual Basic Quadrilateral (four sided) Area = 1/2 ab sinθ (1.6) where a and b are the lengths of the diagonals, and the acute angle between them is θ. Regular polygon of n sides (refer to Figure 1.2) Area = 1/4 nL2 cot(180°/n) (1.7a) R = L/2 csc(180°/n) (1.7b) r = L/2 cot(180°/n) (1.7c) β = 360°/n (1.7d) θ = (n – 2)180°/n (1.7e) L = 2r tan(β/2) = 2R sin(β/2) (1.7f) Circle (refer to Figure 1.3) Let C = circumference r = radius D = diameter A = area S = arc length subtended by θ L = chord length subtended by θ H = maximum rise of arc above chord, r – H = d θ = central angle (rad) subtended by arc S Figure 1.2 Regular polygon. L r R β θ 45423_book.fm Page 4 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 5 Then, C = 2πr = πD (1.8a) S = rθ = 1/2 Dθ (1.8b) (1.8c) (1.8d) θ = S/r = 2cos–1(d/r) = 2sin–1(L/D) (1.8e) A(circle) = πr2 = 1/4 πD2 (1.8f) A(sector) = 1/2 rS = 1/2 r2θ (1.8g) A(segment) = A(sector) – A(triangle) = 1/2 r2 (θ – sinθ) (1.8h) = r2 cos–1 (r – H)/r –(r – H) (2rH – H2)0.5 (1.8i) (1.8j) where h = H/D, h ≤ 0.5 φ = 2{90 – sin–1(1 – 2h)} Figure 1.3 Circle. D r H d S θ L r d r d = − = = 2 2 2 2 2 2 2 sin( / ) tan( / ) θ θ d r L L = − = 1 2 4 1 2 2 2 2 / / cot( / ) θ A A h h ( ) ( ) ( / )( / ) ( )( . segment total = ° − − φ π 360 4 0 5 2 − − h) / π 4 45423_book.fm Page 5 Monday, August 27, 2007 12:44 PM
6 Process engineering and design using Visual Basic Ellipse (refer to Figure 1.4) Area = πab (1.9a) Circumference = 2π{(a2 + b2)/2}0.5 (approximately) (1.9b) Parabola (refer to Figure 1.5) Length of arc EFG = ( 1.10a) Area of section EFG = 4/3 xy (1.10b) Prism Lateral surface area = (perimeter of right section) * (lateral edge) (1.11a) Volume = (area of base) * (altitude) (1.11b) Figure 1.4 Ellipse. Figure 1.5 Parabola. a b y x E G F 4 2 2 4 2 2 2 2 2 x y y x x x y y + ( )+ + + ( ) { } ln / 45423_book.fm Page 6 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 7 Pyramid Lateral area of a regular pyramid = 1/2 (perimeter of base) * (slant height) (1.12a) = 1/2 (number of sides)*(length of one side) * (slant height) (1.12b) Volume = 1/3 (area of base) * (altitude) (1.12c) Right circular cylinder Lateral surface area = 2π (radius) * (altitude) (1.13a) Volume = π (radius)2 * (altitude) (1.13b) Sphere (refer to Figure 1.6) Area (sphere) = 4πR2 = πD2 (1.14a) Area (zone) = 2πRh = πDh (1.14b) Volume (sphere) = 4/3πR3 = 1/6πD3 (1.14c) Volume (spherical sector) = 2/3πR2h = 1/6πh1(3r2 2 + h1 2) (1.14d) Volume (spherical segment of one base) = 1/6πh1(3r2 2 + h1 2) (1.14e) Volume (spherical segment of two bases) = 1/6πh(3r1 2 + 3r2 2 + h2 2) (1.14f) Figure 1.6 Sphere. r2 r1 h1 R h2 h D 45423_book.fm Page 7 Monday, August 27, 2007 12:44 PM
8 Process engineering and design using Visual Basic Right circular cone Curved surface area = πr(r2 + h2)0.5 (1.15a) Volume = 1/3 πr2h (1.15b) where r = radius h = height of the cone Dished end Surface area (general) = πa2 + π/2(b2/e)ln{(1 + e)/(1 – e)} (1.16a) Surface area (ellipsoidal, b = a/2) = πa2 + (πa2/8e)ln{(1 + e)/(1 – e)} (1.16b) = 4.336 a2 Surface area (hemispherical, b = a) = 2πa2 (1.16c) Volume (general) = 2/3 πa2b (1.16d) Volume (ellipsoidal) = 1/3 πa3 (1.16e) Volume (hemispherical) = 2/3 πa3 (1.16f) where a = semimajor axis b = semiminor axis e = eccentricity = (1 – b2/a2)0.5 Irregular shape Let yo, y1, y2, …, yn be the lengths of a series of equally spaced parallel chords and h be the distance between them. The approximate area of the figure is given by using the trapezoidal rule or by Simpson’s rule. Trapezoidal rule Area = h/2 {(yo + yn) + 2(y1 + y2 + y3 + … + yn–1)} (1.17) Simpson’s rule Area = h/3 {(yo + yn) + 4(y1 + y3 + y5 + … + yn–1) + (1.18) 2(y2 + y4 + … + yn–2)} 45423_book.fm Page 8 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 9 where n is even. The greater the value of n, the greater the accuracy of approximation. Irregular volume To find irregular volume, replace y’s by cross-sectional area Aj , and use the results in the preceding equation. Algebra Factoring a2 − b2 = (a + b)(a − b) (1.19a) a3 − b3 = (a − b)(a2 + ab + b2) (1.19b) a3 + b3 = (a + b)(a2 − ab + b2) (1.19c) an − bn = (a − b)(an–1 + an–2b + an–3b2 + … + bn–1) (1.19d) (a + b)2 = a2 + 2ab + b2 (1.19e) (a − b)2 = a2 − 2ab + b2 (1.19f) (a + b + c)2 =a2 + b2 + c2 + 2ab + 2ac + 2bc (1.19g) (a − b − c)2 =a2 + b2 + c2 − 2ab − 2ac + 2bc (1.19h) (a + b)3 = a3 + 3a2b + 3ab2 + b3 (1.19i) (a − b)3 = a3 − 3a2b + 3ab2 − b3 (1.19j) (a + b)n = an + nan–1b + {n(n − 1)/2!}an–2b2 + (1.19k) {n(n − 1)(n − 2)/3!}an–3b3 + … + bn Arithmetic progression Aseries is said to be in arithmetic progression (AP), if the algebraic difference between any two successive terms be the same throughout the series. The following series is in AP. a, a + b, a + 2b, a + 3b, …, nth term where a = first term b = common difference 45423_book.fm Page 9 Monday, August 27, 2007 12:44 PM
10 Process engineering and design using Visual Basic The nth term can be defined as tn = a + (n − 1)b (1.20a) Sum of the series S = (n/2){2a + (n − 1)b} (1.20b) Geometric progression A series is said to be in geometric progression (GP) if the ratio of any term to the preceding one is the same throughout the series. The following series is in GP. a, ar, ar2, ar3, …, nth term where a = first term r = common ratio The nth term can be defined as tn = arn–1 (1.21a) Sum of the series S = a(1 − rn)/(1 − r) (1.21b) Infinite series (in GP) When r < 1, the sum of the series = a/(1 − r) (1.21c) Example Series 1 + 1/3 + 1/9 + 1/27 + … + 1/∞. The sum = a/(1 – r) = 3/2. Best fit straight line (least squares method) The straight line y = a + bx should be fitted through given points (x1,y1), (x2,y2), …, (xn,yn) so that the sum of the squares of the distances of those points from the straight line is minimum, where the distances are measured in the vertical direction (y-direction). The values of a and b are calculated from the following equations: 45423_book.fm Page 10 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 11 an + bΣxj =Σyj (1.22a) and aΣxj + bΣxj 2 =Σxjyj (1.22b) where n is the number of points. Example 1.1 Let the four points be (−1,1), (−0.1,1.099), (0.2,0.808), and (1,1). n = 4 Σxj = 0.1 Σxj 2 = 2.05 Σyj = 3.907 Σxjyj = 0.0517 4a + 0.1b = 3.907 0.1a + 2.05b = 0.0517 The solution is a = 0.9773 b = –0.0224 Therefore, the equation of the best fit straight line is y = 0.9773 – 0.0224x. Binomial equation The general Equation is: ax2 + bx + c = 0 (1.23a) General solution of Equation 1.23a (1.23b) Polynomial equation General solution of the following type of polynomial equation: y = ax2 + bx + c (1.24a) If three values (x1,y1), (x2,y2), and (x3,y3) are known, then x b b ac a = − ± − 2 4 2 45423_book.fm Page 11 Monday, August 27, 2007 12:44 PM
12 Process engineering and design using Visual Basic (1.24b) (1.24c) (1.24d) Example 1.2 Solve the equation y = ax2 + bx + c, for three sets of values (1,10), (3,32), and (5,70). Solution c = 10 − 2 − 3 = 5 Therefore, the general solution will be y = 2x2 + 3x + 5. Maxima/minima Let a function be y = f(x) (1.25a) If a solution for x at y′ = 0 exists, then there will be either a maximum value or a minimum value for the function. At a value of x, where y′ = 0: If y″ is negative, then there will be a maximum value for the function. If y″ is positive, then there will be a minimum value for the function. Example 1.3a Let the equation be y = ax + b (1.25b) a y y x x y y x x x x x = − − − − − − ( )( ) ( )( ) ( )( 1 2 2 3 2 3 1 2 1 2 2 2 2 − − − − − x x x x x 3 2 2 3 2 1 2 ) ( )( ) b y y a x x x x = − − − { } − ( ) ( ) ( ) 1 2 1 2 2 2 1 2 c y ax bx = − − 1 1 2 1 a = − − − − − − − − − ( )( ) ( )( ) ( )( ) ( 10 32 3 5 32 70 1 3 1 9 3 5 9 25) )( ) 1 3 2 − = b = − − − { } − = ( ) ( ) 10 32 2 1 9 2 3 45423_book.fm Page 12 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 13 y′ = a = 0 No solution exists at y′ = 0; this indicates that there will be no maxima or minima for the function. This is the equation of a straight line. Example 1.3b Let the equation be y = x3 + x – 1 (1.25c) y′ = 3x2 + 1 = 0 (1.25d) or, x2 = −1/3 (1.25e) No general solution exists for Equation 1.25e, meaning Equation 1.25c will have no general maxima or minima. Example 1.3c Let the equation be y = 4x3 – 3x + 2 (1.25f) y′ = 12x2 − 3 = 0 x = ± 1/2 At x = 1/2, y″ = 12 and at x = −1/2, y″ = −12. Therefore at x = 1/2, the value of y will be minimum. Value of y = 1. At x = –1/2 the value of y will be maximum. Value of y = 3. The nature of this graph is such that for x values of less than –1/2, the value of y will decrease continuously, and for x values more than 1/2, the value of y will increase continuously. Matrix Addition and multiplication of matrices • If two m by n matrices A and B are given, the sum A + B can be defined as the m by n matrix adding corresponding elements, i.e., (A + B) [i,j] =A[i,j] + B[i,j] (1.26a) • If a matrix A and a number c are given, the multiplication of the matrix can be defined as cA = cA[i,j] (1.26b) 45423_book.fm Page 13 Monday, August 27, 2007 12:44 PM
14 Process engineering and design using Visual Basic Addition of matrices i. ii. Multiplication of matrices iii. iv. Multiplication of two matrices is well defined only if the number of columns of the first matrix is the same as the number of rows of the second matrix. If A is an m by n matrix (m rows, n columns) and B is an n by p matrix (n rows, p columns), then their product AB is the m by p matrix (m rows, p columns) given by (AB)[i,j] = A[i,1] * B[1,j] + A[i,2] * B[2,j] + … + A[i,n]*B[n,j] for each pair of i and j. Multiplication has the following properties: • (AB)C = A(BC) for all k by m matrices A, m by n matrices B, and n by p matrices C (associativity). • (A + B)C = AC + BC for all m by n matrices A and B and k by m matrices C (distributivity). • In general, commutativity does not hold, meaning generally AB ≠ BA. Matrix properties involving addition Let A, B, and C be m by n matrices. Then, a. A + B = B + A (1.27a) b. (A + B) + C = A + (B + C) (1.27b) c. A + 0 = A (1.27c) where 0 is the m by n zero matrix (all its entries are equal to 0). a b c d e f u v w x y z a u b v c w d x e y f ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + + + + + z z ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b c d e f a b c d e f a b c d e f ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ 2 2 2 2 2 2 ⎥ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 a b c d e f a b c d e f g h ae bg af bh ce dg cf dh ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + + + ⎡ * ⎣ ⎣ ⎢ ⎤ ⎦ ⎥ a b c d e f x y z ax by cz dx ey fz ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = + + + + * ⎡ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 45423_book.fm Page 14 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 15 d. A + B = 0 if and only if B = –A (1.27d) Matrix properties involving multiplication a. Let A, B, and C be three matrices. If the products AB, (AB)C, BC, and A(BC) exist then (AB)C = A(BC) (1.28a) b. If α and β are numbers and A is a matrix, then α(βA) = (αβ)A (1.28b) c. If α is a number, and A and B are two matrices such that the product AB is possible, then α(AB) = (αA)B = A(αB) (1.28c) d. If A is an n by m matrix and 0 is the m by k zero matrix, then A0 = 0 (1.28d) Matrix properties involving addition and multiplication a. Let A, B, and C be three matrices, then (A + B)C = AC + BC (1.29a) and A(B + C) = AB + AC (1.29b) b. If α and β are numbers and A and B are matrices, then α(A + B) = αA + αB (1.29c) and (α + β)A = αA + βA (1.29d) Transpose The transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. The transpose of an n by m matrix A is the n by m matrix AT defined by AT[i,j] = A[j,i] (1.30) For the matrix 45423_book.fm Page 15 Monday, August 27, 2007 12:44 PM
16 Process engineering and design using Visual Basic A = The transpose AT = Symmetric matrix A symmetric matrix is a matrix equal to its transpose. A symmetric matrix must be a square matrix, for example, and Diagonal matrix A diagonal matrix is a symmetric matrix with all of its entries equal to zero, except possibly the values of the diagonal, for example, and Determinants Determinants are important both in calculus and algebra. The determinant of a 1 × 1 matrix is the element itself: det[a] = a. The 2 × 2 matrix A = has determinant det (A) = ad – bc (1.31) a b c d e f g h i j k l ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ a d g j b e h k c f i l ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ a b b c ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b c b d e c e f ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ a b 0 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b 0 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 45423_book.fm Page 16 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 17 The 3 × 3 matrix A = has determinant det(A) = a(ei – fh) – b(di – fg) + c(dh – eg) (1.32) Properties of determinants 1. Any matrix A and its transpose have the same determinant, meaning det A = det AT. 2. If the elements of one row (or column) of a determinant are all zero, the value of the determinant is zero. 3. The determinant of a triangular matrix is the product of the entries on the diagonal, that is, = ad 4. If one determinant is obtained from another by interchanging any two rows (or columns), the value of either is the negative of the value of the other. 5. If the elements of one row (or column) of a determinant are multiplied by the same constant factor, the value of the determinant is multiplied by this factor. 6. If two rows (or columns) of a determinant are identical, the value of the determinant is zero. 7. If two determinants are identical except for one row (or column), the sum of their values is given by a single determinant obtained by adding corresponding elements of dissimilar rows (or columns) and leaving the remaining elements unchanged. a b c d e f g h i ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ a b d a b d 0 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b c d c d a b ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ na nb c d n a b c d a b nc nd ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 45423_book.fm Page 17 Monday, August 27, 2007 12:44 PM
18 Process engineering and design using Visual Basic 8. The value of a determinant is not changed, if to the elements of any row (or column) are added a constant multiple of the corresponding elements of any other row (or column). 9. If all elements but one in a row (or column) are zero, the value of the determinant is the product of that element and its cofactor. Cofactor Let A be an n × n matrix. The ij-th cofactor of A, denoted by Aij, will be Aij = (−1)i+j (1.33) Example 1.4 Find cofactors A32 and A24 of a 4 × 4 determinant A Solution 3 2 1 5 4 2 7 5 7 2 8 5 19 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = a nc b nd c d a b c d a b c na d nb + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎥ Mij A = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ 1 3 5 6 2 4 0 3 1 5 9 2 4 0 2 7 A M 32 3 2 32 1 1 5 6 2 0 3 4 2 7 8 = − = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − + ( ) A24 2 4 1 1 3 5 1 5 9 4 0 2 192 = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − + ( ) 45423_book.fm Page 18 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 19 Determinant and inverses Theorem 1 If A is invertible, then det A ≠ 0 and det A–1 = 1/det A (1.34) Adjoint Let a determinant be (1.35) Let A be an n × n matrix and let B, given in Equation 1.35, denote the matrix of its cofactors. Then the adjoint of A, written as adjA, is the transpose of the n × n matrix B. (1.36) Theorem 2 Let A be an n × n matrix. Then, (1.37) Theorem 3 Let A be an n × n matrix. Then A is invertible if and only if detA ≠ 0. If detA ≠ 0, then (1.38) B A A A A A A A A A n n n n nn = ⎡ 11 12 1 21 22 2 1 2 ... ... ... ⎣ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ adjA B A A A A A A A A T n n n n = = 11 21 1 12 22 2 1 2 ... ... . . .. Ann ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ( )( ) ... ... ... A adjA A A A = det det det 0 0 0 0 0 0 0 0 0 0 0 0 0 det det A A I ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = ( ) A A adjA − = 1 1 det 45423_book.fm Page 19 Monday, August 27, 2007 12:44 PM
20 Process engineering and design using Visual Basic Cramer’s rule Consider a system of n equations and n unknowns. a11x1 + a12x2 + … + a1nxn = b1 a21x1 + a22x2 + … + a2nxn = b2 (1.39a) an1x1 + an2x2 + … + annxn = bn which can be written in the form Ax = b (1.39b) If detA ≠ 0, then Equation 1.39b will have a unique solution given by x = b/A (1.39c) Let D be detA. The other matrix can be defined as Let D1 = detA1, D2 = detA2 …, Dn = detAn, then (1.40) Example 1.5 Solve using Cramer’s rule, the system 2x1 + 4x2 + 6x3 = 18 4x1 + 5x2 + 6x3 = 24 3x1 + x2 – 2x3 = 4 A b a a b a a b a a n n n n nn 1 1 12 1 2 22 2 2 = ⎡ ⎣ ⎢ ... ... ... ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = , ... ... A a b a a b a a n n n 2 11 1 1 21 2 2 1 b b a n nn ... , ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ etc. x D D x D D x D D n n 1 1 2 2 = = = , , , … 45423_book.fm Page 20 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 21 Solution The determinant is calculated first as D = = 2(−10 − 6) – 4(− 8 – 18) + 6(4 – 15)= −32 + 104 – 66 = 6 Because the value of D ≠ 0, the system will have a unique solution. D1 = = 18(−10 – 6) – 4 (−48 – 24) + 6(24 – 20) = 24 D2 = = 2(−48 – 24) − 18(−8 – 18) + 6(16 – 72) = −12 D3 = = 2(20 – 24) − 4(16 – 72) + 18(4 – 15) = 18 Therefore, x1 = D1/D = 24/6 = 4 x2 = D2/D = −12/6 = −2 x3 = D3/D = 18/6 =3 Trigonometry Functions of circular trigonometry The trigonometric functions are the ratio of various sides of the reference angles presented in Figure 1.7. The trigonometric relationships are: Sine of α = sin α = p/h; Secant of α = sec α = h/b Cosine of α = cos α = b/h; Cosecant of α = csc α = h/p 2 4 6 4 5 6 3 1 2 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 18 4 6 24 5 6 4 1 2 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 2 18 6 4 24 6 3 4 2 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 2 4 18 4 5 24 3 1 4 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 45423_book.fm Page 21 Monday, August 27, 2007 12:44 PM
22 Process engineering and design using Visual Basic Tangent of α = tan α = p/b; Cotangent of α = cot α = b/p The magnitude and sign of trigonometric functions are presented in Table 1.3. Periodic functions The following periodic functions are commonly used in trigonometry: cos (α + 2π) = cos α (1.41a) sin (α + 2π) = sin α (1.41b) cos (α + 2nπ) = cos α, n = 0, ±1, ±2, … (1.41c) sin (α + 2nπ) = sin α, n = 0, ±1, ±2, … (1.41d) The magic identity The following identities are very basic to the analysis of trigonometric expressions: sin2 α + cos2 α = 1 (1.42a) 1 + tan2 α = sec2 α (1.42b) 1 + cot2 α = csc2 α (1.42c) Figure 1.7 Triangles. Table 1.3 Magnitude of Trigonometric Functions Function 0–90° 90–180° 180–270° 270–360° sin α csc α cos α sec α tan α cot α +0 to +1 +∞ to +1 +1 to 0 +1 to +∞ +0 to +∞ +∞ to +0 +1 to +0 +1 to +∞ –0 to –1 –∞ to –1 –∞ to –0 –0 to –∞ 0 to –1 –∞ to –1 –1 to –0 –1 to –∞ +0 to +∞ +∞ to +0 –1 to –0 –1 to –∞ +0 to +1 +∞ to +1 –∞ to –0 –0 to –∞ b p h b b b p h h h p p α α α 45423_book.fm Page 22 Monday, August 27, 2007 12:44 PM
Chapter one: Basic mathematics 23 The addition formulas The following addition formulas are very important for the analysis of trig- onometric expressions: cos (a + b) = cos a cos b – sin a sin b (1.43a) cos (a − b) = cos a cos b + sin a sin b (1.43b) sin (a + b) = sin a cos b + cos a sin b (1.43c) sin (a − b) = sin a cos b – cos a sin b (1.43d) (1.43e) (1.43f) Example 1.6 Find the exact value of cos 165°, given that cos 120° = −1/2, sin 120° = , and cos 45° = sin 45° = . Solution cos 165° = cos (120° + 45°) = cos 120° cos 45° – sin 120° sin 45° = = Double angle and half angle Double-angle and half-angle formulas are very useful in trigonometry. The following equations are widely used: cos (2a) = cos2 a – sin2 a = 2 cos2 a – 1 = 1 – 2sin2 a (1.44a) tan( ) tan tan tan tan a b a b a b + = + − 1 tan( ) tan tan tan tan a b a b a b − = − + 1 3 2 / 2 2 / − − 1 2 2 2 3 2 2 2 * * − + 6 2 4 45423_book.fm Page 23 Monday, August 27, 2007 12:44 PM
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To AMCROSS, Paris: Answering Inquiry No. ——. Mother and baby son three months old well and happy. In this instance the worried fighter was an officer—a captain of infantry. During the time which elapsed between the two cablegrams he was wounded and the answer found him in a hospital, side by side with a French blessé. A Red Cross searcher acted as interpreter for their felicitations and in her official report of the incident included this notation: Captain W—— was much improved as a result of the good news. He is sitting up and eating roast chicken to-day. He says the American Red Cross has cured him. The Red Cross representatives here in America could not enter a home unless they were welcome; neither could they force their way into the hearts of men. They were compelled to wait until their help was sought. The growing mental depression of a certain major of a fighting division during those tense months of the midsummer of 1918 did not escape the attention of the American Red Cross man attached to that division. Suddenly the man, who had been marked because of his poise, became taciturn—isolated himself. A reference to the Red Cross Home Service which its division worker tactfully introduced into the table talk at the mess at which both sat, however, did elicit some trivial rejoinder from the man with the golden oakleaf upon his shoulder; while the following day that same major wrote a letter to the Red Cross man—and bared the reason for his most obvious melancholy. It seemed that back here in the United States he had a little son, from whom he had received no word whatsoever in more than six months. The child was with the major's divorced wife, and his father was more than anxious to know if he was regularly playing out of doors, if he was receiving his father's allotment, and if he was buying the promised Thrift Stamp each week. The army man already
had his second golden service stripe and greatly feared that his little son might be beginning to forget him. Under conditions such as these, visiting the boy was a diplomatic mission indeed. Finally it was intrusted to the wife of an army officer. And because army officers' wives are usually achieved diplomats if not born ones, the ultimate result came in weekly letters from the boy, which not only greatly relieved his father's mind but greatly increased the bonds of affection between the two. The Greatest Mother in the World is never above diplomacy—which is, perhaps, just another way of expressing tact and gentleness. There were many, many occasions, too, when the relatives at home depended upon that selfsame diplomacy of hers to tell the disagreeable stories of losses or perhaps to prepare the boys overseas to face an empty chair in the family circle. There was one particularly fearful moment when a brilliant young officer had to be told that the reason why his young wife had ceased to write was because she had gone insane and specialists believed that she could not recover. Boys were driven to Red Cross offices by hidden affairs that flayed them hideously and of which they wished to purge themselves. Some wanted to set old wrongs right. Others had fallen blindly into the hands of the unscrupulous and had only fully awakened to see their folly after they actually were upon the battlefields of France. Then there were the softer phases of life—the shy letters and the blushing visitors who wished to have a marriage arranged with Thérèse or Jeanne of the black eyes and the delicate oval face. I remember one of our boys who had fallen in love with a girl in Nancy. Theirs was a courtship of unspoken love, unless soft glances and gentle caresses do indeed speak more loudly than mere words; for they had no easy bond of a common tongue. His French was doughboy French, which was hardly French at all, and her English was limited. So that after he had gone on to the Rhine and the letter came from her to him in the delicate hand that the sisters at the convent had taught, he needs must seek out Red Cross Home Communication and intrust to it the task of uncommon delicacy,
which it fulfilled to the complete delight and satisfaction of both of them. For how could any mother, let alone the Greatest Mother in the World, blind her eyes entirely to love? She apparently had no intention of doing any such thing. For how about that good-looking doughboy from down in the Ozark country somewhere, who arrived in Paris on a day in the autumn of 1918 with the express intention of matrimony, if only he knew where he could get the license? French laws are rather fussy and explicit in such matters. Some one suggested the Home Service Bureau of the American Red Cross to the boy. He found his way quickly to it—with little Marie, or whatever her name really was, hanging on his arm. A Red Cross man prayerfully guided the pair through the legal mazes of the situation. First they went to a law office in the Avenue de l'Opéra where the necessary papers were made out; then the procession solemnly moved to the office of the United States Vice Consul at No. 1 Rue des Italiens, where the signature of the American official representative was duly affixed to each of the papers; after which to the foreign office, where the French went through all the elaborate processes of sealings and signatures which they seem to love so dearly, and then—the work of Mother Red Cross was finished. They were quite ready for the offices of the Church. With the signing of the armistice all this work was greatly increased —was, in fact, doubled and nearly trebled. When a man was fighting his physical needs seemingly were paramount; but once off the field, the worries that lurked in his subconscious mind seemed to rise quickly to the surface. He then recalled that long interval since last he heard from home. That troubled him, and he turned to the Red Cross—those pamphlets and posters did have a tremendous effect. And if he had no definite troubles over here, such as those we have just seen, he was apt to be just plain hungry for a sight of the home —and the loved ones that it held. It was in answer to a demand such as this last that a Red Cross representative right here in the United States took her motor car and
drove for a half day out to see a family of whose very existence she had never before even heard; and, as a result of her call, wrote back a letter from which the following excerpts are taken: I want to tell you about a never-to-be-forgotten trip that I took the other day out to see a one hundred per cent patriot; an American mother who has three sons in the service. The home is one of the coziest, homiest, friendliest places you can imagine; one story, with that cool spacious plan of construction that makes you want to get a book, capture a chair on the wide, comfortable porch, and forget the world and its dizzy rush; a great sweep of lawn and with some handsome Hereford calves browsing in one direction and a cluster of shade trees nearer the house. The hills surrounding the house make a lovely view and all were covered with grazing stock, also the fine Hereford cattle for which the place is known. But the best part of the home is the dear little woman who hung a service flag in the window with the name of a boy under each of the three stars. She is the type of mother that draws every one to her; tender, sensible, capable, broad-minded, and with a shrewd sense of humor that keeps things going and makes life worth living for the entire household. She took us to a roomy side porch where her sewing unit of the Red Cross meets each Tuesday. A marvelous amount of work has been turned out in that side porch, and I'll wager a dollar to a doughnut that I know the moving spirit of the workers. Off in a big, cool parlor bedroom there were stacked up several perfectly enchanting 'crazy quilts' made by these same busy women at odd moments. These are ready to be sent to Serbia or they may be sold at auction for the benefit of the Red Cross. We saw pictures of each boy in the service—one in the navy, one in the heavy artillery, and Milton, whom we all hope is not in the hospital by now. Each boy had in his eyes the same intrepid look that the mother has—one can tell that they made good soldiers. Knowing how busy farm folk are, we reluctantly took our leave after
seeing all these interesting things and, as we swung out into the country lane, we looked back and there stood the mother waving and smiling—the very best soldier of them all. Can you not see how very simple it all was—how very human, too? As you saw in one of the earlier chapters of this book, a fairly formal and elaborate plan of organization had been laid out for all this work; but, perhaps because war after all, is hardly more than a series of vast emergencies, the American Red Cross searchers, either in the field or in the hospitals, could hardly confine themselves to any mere routine of clerical organization or work in the great task that was thrust upon them. The unexpected was forever upon them. As a single instance of this take the time when, in the Verdun sector and in the hottest days of fighting that the American Army found there, so many demands were made upon our Red Cross by the officers and men of the A. E. F. for the purchase of necessities in Paris that a definite shopping service quite naturally evolved itself out of the situation. The man who initiated that service raced a motor car from Verdun to the Paris headquarters in order to secure the materials necessary for its inauguration. For when the American Red Cross made up its mind to do a thing, it did it—and pretty quickly too. So it went—a service complicatedly simple, if I may so express it. For, despite its own batteries of typewriters and card indexes, there was, at almost all times, that modicum of human sympathy that tempered the coldness of mere system and glorified what might otherwise have been a mere job of mechanical routine into a tremendously human and tender thing. The men and girls of the Home Communication Service had a task of real worth. Of a truth it was social service—of the most delicate nature. It included at all times not only the study of the physical needs of the soldier or sailor, but also at many times that of his mental needs as well. In reality, it
became a large part of the scheme of preserving and enlarging the morale of the A. E. F. Every time a soldier was freed of endless, nagging worry, he became a better soldier and so just that much more strength was added to the growing certainty of victory.
CHAPTER XI WHEN JOHNNY CAME MARCHING HOME On November 11, 1918, the armistice was signed and the fighting of the Great War ceased—almost as abruptly as it had begun. And the ebb tide of American troops from Europe back to the United States began; almost at once. For a time it was an almost imperceptible tide; in the following month but 75,000 soldiers all told—officers and enlisted men—were received through the port of New York, at all times the nation's chief war gateway; yet this was but the beginning. Each month of the early half of 1919 registered an increase of this human tide inflowing as against the preceding months, until May, with 311,830 troops received home, finally beat, by some 5,000 men, the record outgoing month of July, 1918, when under the terrific pressure induced by the continued German drive, 306,731 officers and men had been dispatched from these shores. Yet June, 1919, overtopped May. In that month 342,686 troops passed not only under the shadow of the beloved statue of Liberty, but also into the friendly and welcoming ports of Boston, Newport News, and Charleston, while the Secretary of War promised that the midsummer months that were immediately to follow would break the June record. A promise which was fulfilled. Long before the signing of the armistice, Pershing had ruled that the work of the American Red Cross with the well men of the A. E. F. was specifically to be limited to them while they were en route from one point to another—along the lines of communication, as you already have seen in an earlier chapter. To the Young Men's Christian Association was intrusted the chief burden of caring for them in their more or less permanent camps. This meant for our Red Cross in the final months of the war—before peace was actually signed and
declared—a task almost exactly like that which had confronted it in its very first months of war experience in France. The stations along the railroad lines of eastern France, Luxembourg, and the Moselle Valley—the lines of communication between our French base ports and the occupied districts of the German states—offered to the American Red Cross the very same canteen problems as had once faced it at Châlons-sur-Marne and Épernay. Treves and Coblenz were hardly different from either of these—save perhaps in their increased size. Because Coblenz is rather more closely connected in the mind of the average American with our Army of Occupation, let us begin with it, here and now. It was, in fact, the easternmost outpost of the work of our Red Cross with our army over there. There the lines of communication officially began, and ran up the railway which ascends the beautiful but extremely tortuous valley of the Moselle. And where the lines of communication began—in the great railroad station of Coblenz—the American Red Cross also began. It had two canteens in that station; one just off the main waiting room, and the other, for the convenience of troops who were merely halted in the train shed of the station while going to and from the other American mobilization centers in that Rhine bridgehead, right on the biggest and the longest of the train platforms. Both were busy canteens; never more so, however, than just before 10:30 o'clock in the morning, which was the stated hour for the departure of the daily leave-train toward the border lines of France. Then it was the Red Cross coffee and sandwiches, tobacco and chewing gum were in greatest demand; for the long leave-train boasted no such luxury as dining cars, and there was scarce enough time at the noonday stop at Treves for one to avail oneself of the lunch-room facilities in the station there. Yet Treves for the American Red Cross was a far, far more important point than Coblenz. It was the headquarters of all its work in Germany, and boasted in addition to the large American Red Cross canteens in each of the two railroad stations, on either bank of the
Moselle, and the recreation huts at the base hospitals—for that matter, there were also recreation huts at the base hospitals in and about Coblenz—well-equipped clubs for both enlisted men and officers. Of these the club for the enlisted men—for the rank and file of doughboy—quite properly was the best equipped. In the beginning it had been one of those large combination beer gardens and music halls that always have been so very dear to the heart of the German. It was the very sort of plant that could be, and was, quickly adapted to the uses of a really big group of men. Its main bierhalle made a corking dining room for the doughboys. The meals kept pace with the apartment. Three times a day they appeared—feeding daily from 600 to 1,600 boys—and they were American meals—in fact, for the most part composed of American food products—meats from Chicago, butter and cheese from New York State, flour from Minnesota, and the like. For each of these a flat charge of two marks—at the rate of exchange then prevailing, about eighteen cents—was made. But if a doughboy could not or would not pay, no questions were asked. The Treves Enlisted Men's Club which the American Red Cross gave the A. E. F. was not a commercial enterprise. It was run by an organization whose funds were the gift of the American people—given and given freely in order that their boys in khaki might have every comfort that money might provide. The great high-ceilinged halle held more than a restaurant. It was a reading room as well, stocked with many hundreds of books and magazines. In fact a branch of the American Library Association operated—and operated very successfully—a small traveling loan library in one of the smaller rooms of the club. Upon the walls of the vast room were pictures and many maps—maps of the valley of the Moselle, of that of the Rhine, of the Saar basin, of the operations in France. These last held much fascination for the doughboys. The most of them were of divisions which had led in the active and hard fighting, and the tiny flags and the blue-chalk marks on the operation maps were in reality placed there by their own efforts—but
a few weeks and months before. It was real fun to fight the old actions over and over again—this time with talk and a pointing stick. There were, of course, such fundamental conveniences for roaming doughboys as baths, a bootblack and a barber shop—this last equipped with chairs which the boys themselves invented and constructed; a plain stout wooden armchair, into the back of which a board—not unlike an old-fashioned ironing board—was thrust at an angle. When turned one way this board formed just the proper headrest for a shave; in the other direction it was at exactly the right angle for haircutting. For the Officers' Club of our Red Cross at Treves, the Casino in the Kornmarkt, the heart of the city, was taken over. The fact that this was in the beginning a well-equipped club made the problem of its adaption a very slight one indeed. And the added fact that officers require, as a rule, far less entertainment than the enlisted men also simplified its operation. As it was, however, the officers were usually given a dance or a show each week—in the comfortable, large hall of the Casino. In the Enlisted Men's Club there was hardly a night, however, without some sort of an entertainment in its halle; and the vast place packed to the very doors. The next stop after Treves in the eastbound journey from the Rhine of the man in khaki was usually Nancy. And here there were not only canteen facilities at the railroad station, but a regular Red Cross hotel—situated in the Place Stanislas, in the very heart of the town. In other days this had been the Grand Hotel, and the open square that it faced has long been known as one of the handsomest in all France. In fact, Nancy itself is one of the loveliest of all French towns; and despite the almost constant aërial bombardments that were visited upon it, escaped with comparatively minor damage.
NEVER SAY DIE Sorely wounded, our boys at the great A. R. C. field hospital in the Auteuil race track outside of Paris, kept an active interest in games and sports The Red Cross hotel there was opened on September 30, 1918, and closed on the tenth of April of the following spring—had eighty-eight rooms, capable of accommodating one hundred guests, and two dormitories capable of providing for some forty more. The room charges were invariably five francs for a room—with the exception of one, usually reserved for generals or other big wigs—which rented at eight francs a night. For the dormitory beds an even charge of two francs (forty cents) nightly was made, while in the frequent event of all these regular accommodations of the hotel being engaged and the necessity arising of placing cots in its broad hallways, no charge whatsoever was made for these emergency accommodations. For the excellent meals—served with the fullness of a good old- fashioned Yankee tavern—a progressive charge of four francs for breakfast, five francs for lunch, and six francs for dinner was made. Surely no one could fairly object to the restaurant prices, which, even in France in war-time stress, ranged from eighty cents to a dollar and twenty! In fact it was a bonanza for the American officers who formed the chief patrons of the place—although a bit of
thoughtfulness on the part of some one had provided this particular hostelry with a dormitory of twelve beds and a single room with three which was held reserved for American women war workers; an attention which was tremendously appreciated by them. Eleven miles distant from Nancy was Toul; but Toul we have already visited in the pages of this book. We know already the comfortable accommodations that the traveler in khaki found in the group of hotels and canteens which our Red Cross operated there. There were many of these, even outside of Paris; one of the largest the tavern at the badly overcrowded city of Bordeaux. That tavern had been little to boast of, in the beginning. It was an ancient inn indeed; but good taste—the purchase of some few dozen yards of cretonne, and cleanliness—the unrelenting use of mop and broom and soap—had accomplished wonders with it. There were others of these American Red Cross hotels in France during the fighting period —the ones at Dijon, Is-sur-Tille, and Marseilles were particularly popular. But it was in Paris itself that the Red Cross accommodations for the itinerant doughboy in the final months of the war, as in the long and difficult half year that intervened between the signing of the armistice and the signing of peace, reached their highest development. In the beginning these had taken form in canteens which were operated night and day at each of the important railroad stations. These were all right—so far as they went. Their one-franc or seventy-five centime meals were wonderful indeed. I have eaten in these canteens many times myself—and always eaten well. I have been seated between a doughboy from North Carolina and one from North Dakota and been served by a society woman in steel-gray uniform—a woman whose very name was a thing to be emblazoned in the biggest headline type of the New York newspapers, but who was working week in and week out harder than the girls in busy restaurants back home are usually wont to work.
If you would see these canteens as they really worked, gaze upon them through the eyes of a brilliant newspaper woman from San Francisco, who took the time and the trouble to make a thorough study of them. She wrote: A brown puddle of coffee was spreading over the white oilcloth. The girl from home sopped it up with her dish towel. She brushed away messy fragments of food and bread crumbs. Again there were few vacant places for American soldiers on the benches at the long table in the canteen at the Gare St. Lazare. The canteen, one of a circuit of thirteen maintained by the Red Cross in Paris, had formerly been the corner of a baggage room in one of the most important Paris terminals. The concrete floor bruised her feet. She was as conscious of them as Alice in Wonderland who discovered her own directly beneath her chin after she nibbled the magic toadstool. The girl was tired, but she smiled. It was really a smile within a smile. There was one on her lips which seemed to sparkle and glance, waking responsive smiles on the faces of the men. At once the gob who was born down in Virginia and had trained at Norfolk, decided that she was from his own South. The six-foot doughboy from California knew that she came from some small town in the Sierras. To each of the men she suddenly represented home. That smile stays in place each day until she reaches her room in a pension across the Seine on the Rue Beaux Arts. There, closing the door upon the world with its constant pageant of uniformed men who seem forever hungry and thirsty, she lets her smile fade away for the first time that day. The smile within is tucked away in her heart with the memory of agonizing moments aboard an ocean liner when she felt her exalted desire for service ebbing away because she feared she would not be needed. Needed! Now she wonders who else could have managed so tactfully the boy who had been at sea for one year and
discovered that he had forgotten how to talk to an American woman. His diffidence was undermined with another dish of rice pudding and an extra doughnut. He became a regular boarder at the canteen where breakfast costs nine cents and any other man's size meal may be had for thirteen cents. His leave ended in a half day of excited shopping for which his younger sister will always be grateful. The girl from home had been one of those solemn creatures who was called to the Overseas Club in New York for service abroad. She was one of hundreds who had clinched their own faith in their ideals by pledging such service. It had been a wrench, saying good-bye at the station in the Middle West. There were no boys in the family, and her father had made a funny little joke which betrayed his pride about 'hanging out a service flag now.' Armed with interminable lists which called for supplies for twelve months, she bought her equipment. All the time she was saying to herself: 'I am ready to give all of my youth and my strength to the cause and to hasten victory.' Then the armistice was signed. The wireless instrument sang with the message. There was a celebration. The ship remained dark, still sliding through the nights warily, but her next trip would be made with decks ablaze and portholes open. The war was ended. It seemed to the girl that in the silence of the aftermath she could hear once more the wings of freedom throbbing above the world. She was glad and she was sorry. Her fear was that after all the Red Cross would not need her because she came too late. Canteen service—she pictured the work minus the tonic of danger as a social job. Dressed in a blue smock and white coif she would bid a graceful farewell to the A. E. F. as it filtered out of Europe. Now she smiles. Needed? Her fingers are scarred and she wonders if she ever will be able to pour one thousand bowls of coffee from the gigantic white porcelain pitcher without blistering her hands.
Each day she looks at the line of men jostling one another at the door. She listens to their interminable questions and comes to the full realization that she is one of the most important people in Paris, one of two hundred girls feeding thirty-five thousand soldiers daily. As some workers leaving for home after more than a year of service tell of making sandwiches under shell fire, of sleeping by the roadside in the woods to fool the boche flyers who bombed the Red Cross buildings, she still feels the sly nip of envy. But soldiers do not cease to be soldiers and heroes when the war is done. Other puddles formed on the table and she mopped them up. She had used three towels during her eight-hour shift. A soldier, one of the thousands passing daily through the six Paris stations on their way home, journeying to leave areas, going to join the Army of Occupation or assigned to duty in the city, called to her. 'Sister, I want to show you something,' he said, and unwrapped a highly decorative circlet of aluminum. It was a napkin ring which he had bought from a poilu who made it of scraps from the battlefield. There was an elaborate monogram engraved on a small copper shield. 'For my mother,' he explained. 'If you don't think it is good enough I will get something else.' At once fifty rival souvenirs were produced. Men came from other tables to exhibit their own. There was the real collector who bemoaned the theft of a 'belt made by a Russian prisoner in Germany and decorated with the buttons of every army in the world including the fire department of Holland.' One of the new arrivals had hands stiffened from recently healed wounds. She brought his plate of baked beans, roast meat, potatoes, a bowl of coffee, and pudding. A young Canadian with flaming, rosy cheeks divided the last doughnut with his friend, the Anzac. Crullers are the greatest influence in canteen for the general friendliness among soldiers of different armies. A League of Nations
could be founded upon them if negotiations were left to the privates about the oilcloth-covered tables. The boy with the crippled hands protested that he did not want to accept a dinner for which there was so little charge. 'Say, Miss,' he said, 'I can pay more. I don't have to be sponging.' 'You have folks in the states?' she asked. He had. 'Then,' she explained, 'they are the ones who support the American Red Cross. When you come here it is because the folks asked you in to dinner.' 'But I haven't any folks,' announced a sailor. 'I'm from the States, so I am your folks,' she retorted, 'and the Red Cross is your folks. We invite you to three meals a day as long as you stay in Paris.' 'You are my folks,' said the boy who was only a youngster, 'and you sure look like home to me.' The soldier with the crippled hands wanted to describe his wounds. Like hundreds of others he began with the sensations in the field, 'when he got his.' Deftly as she had learned to do during hundreds of such recitals, she cleaned up the table and stacked the plates without seeming to interrupt. It was three o'clock, the end of her day. She had reported at seven in the morning. The following week she would report with the other members of the staff at eleven at night because the doors of a canteen must never be closed. The boy talked on. He was explaining homesickness, the sort which drives men from cafés where the food is unfamiliar and the names on the menus cannot be translated into 'doughboy French' to such places as the little room in the Gare St. Lazare. She discovered that her habitual posture was with arms akimbo and hands spread out over her hips. This position seemed to rest the
ache in her shoulders. Through her memory flashed pictures of waitresses in station eating houses who stood that way while tourists fought for twenty minutes' worth of ham and eggs between trains. Red Cross after-war canteens were a social center for pretty idlers in smart blue smocks? The smile on her lips never faltered and the hidden smile in her heart became a little song of laughter. She was 'helping'—helping in an 'eating joint,' some of the boys called it. But it was an eating joint with a soul. What more could one ask of an eating-house? From the canteen at the railroad terminals—which were all right so far as they went—it was an easy step of transition to the establishment of hotels for the enlisted men in the accessible parts of Paris—until there was a total of six of these last, in addition to the five railway station canteens—at Gare St. Lazare, Gare du Nord, Gare d'Orsay, Gare d'Orléans, and Gare Montparnasse. The winter- time hotels were in the Avenue Victor Emanuel, Rue Traversière, Rue la Victoire, Rue St. Hyacinthe, and the Rue du Bac. These were all, in the beginning, small Parisian taverns of the pension type, which were rather quickly and easily adapted to their war-time uses. The great difficulty with the first five of these American Red Cross doughboy hotels was their extreme popularity. They could hardly keep pace with the demands made upon them—in the last weeks that preceded and immediately following the signing of the armistice; while, with the coming of springtime and the granting of wholesale leaves of absence by the army, an immediate and most pressing problem confronted the American Red Cross in Paris. The boys were coming into the town—almost literally in whole regiments, and the provisions for their housing and entertainment there were woefully inadequate—to say the least. Not only were these
accommodations, as furnished by the French, inadequate and poor, but the charges for them often were outrageous. Yet to furnish hotel accommodations in the big town, even of the crudest sort, for a thousand—perhaps two thousand—doughboys a night was no small problem. There were no more hotels, large or small, available for commandeering in Paris; the various allied peace commissions had completely exhausted the supply. Yet our Red Cross, accustomed by this time to tackling big problems—and the solution of this was, after all, but part of the day's work, and because there were no more hotels or apartment houses or dormitories or barracks of any sort whatsoever available in the city of more than two million folks—our Red Cross decided to build a hotel. And so did—almost overnight. It was a summer hotel, that super-tavern for our doughboys, and it stood squarely in the center of that famous Parisian playground, the Champs de Mars—and almost within stone throw of the Eiffel Tower and the Ecole Militaire. To create it several dozen long barracks—like American Red Cross standard khaki tents—were erected in a carefully planned pattern. Underneath these were builded wooden floors and they were furnished with electric lights and running water. A summer hotel could not have been more comfortable; at least few of them are. The Tent City, as it quickly became known, was opened about March 4, 1919, with bed accommodations for 1,400 men, while preparations were quickly made to increase this capacity by another five hundred, for the latest and the biggest of American Red Cross hotels in Paris had leaped into instant popularity. Between six and nine-thirty in the morning and ten-thirty and midnight in the evening, the boys would come streaming in to the registry desk, like commercial travelers into a popular hostelry in New York or Philadelphia or Chicago. They would sleep—perhaps for the first time in many, many months—in muslin sheets. And these were as immaculate as those of any first-class hotel in the States.
There was no charge whatsoever for these dormitory accommodations. For the meals—simple but good and plentiful—the normal price of fifty centimes (nine or ten cents) was asked, but never demanded; while merely for the asking any of our boys in khaki could have at any hour the famous Red Cross sandwiches of ham or salmon or beef mixture or jam—chocolate or coffee or lemonade a-plenty to wash it down. Definite provision was made for their amusement; there were rubberneck wagons to take them afield to the wonderful and enduring tourist sights of Paris and her environs—and at the Tent City itself a plenitude of shows and dances as well as the more quiet comfort of books or magazines, or the privilege and opportunity of writing a letter home. Of what use these last in Paris? you ask. Your point is well taken. I would have taken it myself—before I first went to the Tent City. When I did it was a glorious April day, the sun shone with an unaccustomed springtime brilliancy over Paris, and yet the air was bracing and fit for endeavor of every sort. Yet the big reading room tent of the Red Cross hotel in the Champs de Mars was completely filled—with sailor boys or boys in khaki reading the books or paper most liked by them. The sight astonished me. Could these boys—each on a leave of but three short days—be blind to the wonders of Paris? Or was their favorite author particularly alluring that week? I decided to ask one of them about it. I saw Paris yesterday—Notre Dame, the Pantheon, Napoleon's Tomb, the Opera House, the Louvre, the Follies—the whole blame business. It's some hike. But I did it. An' to-day I'm perfectly satisfied to sit here and read these guys a-telling of how they would have fought the war. Of such was the nature of the American doughboy.
Just as it was necessary at Treves and Bordeaux and elsewhere— because of the very volume of the problem—to separate his entertainment from that of his officers, so it became necessary to effect a similar solution in Paris; for the officer is quite as much a ward of our Red Cross as the doughboy, himself. And so early in the solution of this entire great problem a superb home in the very heart of Paris—the town residence of the Prince of Monaco at No. 4 Avenue Gabriel and just a step from the Place de la Concorde—was secured and set aside as an American Red Cross Officers' Club. Lovely as this was, and seemingly more than generous in its accommodations, these were soon overwhelmed by the demands placed upon them, and steps were taken toward finding a real officers' hotel for the men of the A. E. F. when they should come to Paris. These led to the leasing of the Hotel Louvre, at the head of the Avenue de l'Opéra and almost adjoining the Comédie Française, the American University Union, and the Louvre. After being rapidly redecorated and otherwise transformed to meet the necessities of the A. E. F. it was reopened on the sixth of January, 1919, as the American Officers' Hotel in charge of Mr. L. M. Boomer, the directing genius of several large New York hotels. Mr. Boomer brought to the Red Cross a great practical hotel experience, and the house under his management quickly attained an overwhelming success. It had, in the first instance, been charmingly adapted to its new uses. Its rather stiff and old-fashioned interior had been completely transformed; there was all through the building an indefinable but entirely unmistakable home atmosphere. Our American officers fairly reveled in it. Into this setting was placed good operation—a high-grade American- operated hotel, if you please, in the very heart of Paris and all her stout traditions. Petit déjeuners begone! They are indeed starvation diet for a hungry Yank. The breakfast in the American Officers' Hotel, which our Red Cross set up and operated, cost a uniform five francs
(one dollar) and had the substantial quality of a regular up-and- doing tavern on this side of the Atlantic. Before we rest, here are three typical bills of fare of a single ordinary day in this A. R. C.-A. E. F. establishment. The day was the nineteenth of April, 1919, and the three meals were as follows: Breakfast Five Francs—($1.00). Bananas Quaker Oats Eggs and Bacon Griddle Cakes with Sirup Confiture Coffee, Cocoa, or Chocolate Luncheon Eight Francs—($1.60). Oyster Soup, with Okra Scollops of Veal, Dewey Nouilles, Milanaise Cold Meats, with Jelly Russian Salad Assorted Eclairs Raspberry Ice Cream Coffee Dinner Ten Francs—($2.00). Crème St. Cloud Rouget Portugaise
Roasted Filet of Beef, Cresson Pommes Château Endive Flamandes Salade de Saison Candied Fruits Coffee Ice Cream Coffee Yet the charm of the American Officers' Hotel in Paris rested not alone in the real excellence of its cuisine, nor in the comfort of its cleanly sleeping rooms. It carried its ideals of genuine service far beyond these mere fundamentals. It recognized the almost universal Yankee desire to have one's shoes shined in a shop and so set up a regular American boot-blacking stand in one of its side corridors, a thing which every other Parisian hotel would have told you was quite impossible of accomplishment. It recognized the inconvenience of tedious waiting and long queues at the box office of the Paris theaters by setting up a theater ticket office in its lobby, which made no extra charge for the distinct service rendered. Nor was there a charge for the services of Miss Curtis, the charming little Red Cross girl, who went shopping with a fellow or for him, and who had a knack of getting right into those perplexing Paris shops and getting just what a fellow wanted at an astonishingly low price—for Paris in war times, anyway. Her range of experience was large; from the man with a silver star on each shoulder who wanted to buy a modish evening gown for his wife at a price not to exceed forty dollars, to the chunky Nevada lieutenant who had won three thousand francs at redeye on the preceding evening and was anxious to blow it all in the next morning in buying souvenirs for mother. With both she did her best. Her motto was that of the successful shop keeper: We aim to please. When Mr. Boomer had this hotel set up and running and turned his attention to some other housing problems of our Red Cross, the management fell to Major H. C. Eberhart, who had been his assistant in Paris and before that had been affiliated in a managerial
capacity with several large American houses. He carried forward the job so well begun. With the slow but very sure movement of our doughboys back from eastern France and Germany toward the base ports along the westerly rim of France, where they were embarking in increasing numbers for the blessed homeland, it became necessary for General Pershing to establish concentration areas, or reservoir camps, well back from the Atlantic Coast but convenient to it. By far the largest and most important of these was in the neighborhood of the city of Le Mans, some one hundred and fifty miles southwest of Paris, which meant in turn that what was finally destined to be the largest of the canteens of our American Red Cross in France outside of Paris was the final one established. It was known as the American Red Cross Casual Canteen and, situated within three blocks to the east of the railroad station at Le Mans, was a genuine headquarters for all the American soldiers for ten or fifteen or twenty miles roundabout. And in the bare chance that there might not be a doughboy who had chanced to hear of it, it was well indicated—by day, by a huge sign of the crimson cross, and by night that emblem blazing forth in all the radiance of electricity. When the doors were finally opened—about the middle of March, 1919—there were sleeping quarters under its hospitable roof for 250 enlisted men and forty officers. In the canteen portion of the establishment, 200 men could be served at a single sitting; in all 500 at each of the three meals a day. The comforts of this place almost approximated those of a hotel. When the men rose from their beds in the morning—clean sheets and towels and pillowcases, of course, even though it did mean that the Red Cross had to establish its own laundry in the establishment—they could step, quickly and easily, into a commodious washroom and indulge, if they so chose, in a
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Process Engineering And Design Using Visual Basic Arun K Datta

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  • 6. Half title page PROCESS ENGINEERING and DESIGN USING VISUAL BASIC® 45423_C000.fm Page i Monday, August 27, 2007 11:16 AM
  • 7. 45423_C000.fm Page ii Monday, August 27, 2007 11:16 AM
  • 8. Title Page CRC Press is an imprint of the Taylor & Francis Group, an informa business Boca Raton London New York ARUN DATTA PROCESS ENGINEERING and DESIGN USING VISUAL BASIC® 45423_C000.fm Page iii Monday, August 27, 2007 11:16 AM
  • 9. Visual Basic and Excel are registered trademarks of Microsoft. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-4542-0 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse- quences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Datta, Arun. Process engineering and design using Visual BASIC / Arun Datta. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4200-4542-0 (alk. paper) ISBN-10: 1-4200-4542-3 (alk. paper) 1. Chemical processes--Mathematical models. 2. Chemical processes--Computer simulation. 3. Microsoft Visual BASIC. I. Title. TP155.7.D36 2006 660’.28--dc22 2006103307 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 45423_C000.fm Page iv Monday, August 27, 2007 11:16 AM
  • 10. Dedication To my mother Smt. Narayani Datta 45423_C000.fm Page v Monday, August 27, 2007 11:16 AM
  • 11. 45423_C000.fm Page vi Monday, August 27, 2007 11:16 AM
  • 12. vii Contents Chapter one Basic mathematics.......................................................................1 Introduction .............................................................................................................1 Physical constants...................................................................................................1 SI prefixes .......................................................................................................1 Mensuration.............................................................................................................2 Triangles..........................................................................................................2 Rectangles.......................................................................................................2 Parallelogram (opposite sides parallel).....................................................3 Rhombus (equilateral parallelogram)........................................................3 Trapezoid (four sides, two parallel)...........................................................3 Quadrilateral (four sided)............................................................................4 Regular polygon of n sides..........................................................................4 Circle................................................................................................................4 Ellipse..............................................................................................................6 Parabola ..........................................................................................................6 Prism................................................................................................................6 Pyramid...........................................................................................................7 Right circular cylinder..................................................................................7 Sphere..............................................................................................................7 Right circular cone ........................................................................................8 Dished end .....................................................................................................8 Irregular shape...............................................................................................8 Trapezoidal rule.............................................................................................8 Simpson’s rule ...............................................................................................8 Irregular volume ...........................................................................................9 Algebra .....................................................................................................................9 Factoring .........................................................................................................9 Arithmetic progression.................................................................................9 Geometric progression ...............................................................................10 Infinite series (in GP)..................................................................................10 Best fit straight line (least squares method) ...........................................10 Binomial equation ....................................................................................... 11 Polynomial equation................................................................................... 11 45423_C000.fm Page vii Monday, August 27, 2007 11:16 AM
  • 13. viii Process engineering and design using Visual Basic Maxima/minima .........................................................................................12 Matrix............................................................................................................13 Addition and multiplication of matrices............................................13 Addition of matrices..............................................................................14 Multiplication of matrices.....................................................................14 Matrix properties involving addition .................................................14 Matrix properties involving multiplication .......................................15 Matrix properties involving addition and multiplication...............15 Transpose .................................................................................................15 Symmetric matrix...................................................................................16 Diagonal matrix......................................................................................16 Determinants................................................................................................16 Properties of determinants....................................................................17 Cofactor....................................................................................................18 Determinant and inverses.....................................................................19 Adjoint......................................................................................................19 Cramer’s rule...............................................................................................20 Trigonometry .........................................................................................................21 Functions of circular trigonometry ..........................................................21 Periodic functions .......................................................................................22 The magic identity ......................................................................................22 The addition formulas................................................................................23 Double angle and half angle.....................................................................23 Product and sum formulas........................................................................24 Relations between angles and sides of triangles ...................................25 Law of sines .................................................................................................26 Law of tangents...........................................................................................26 Law of cosines .............................................................................................26 Other relations .............................................................................................26 Inverse trigonometric functions................................................................27 Hyperbolic functions ..................................................................................28 Other hyperbolic functions........................................................................29 Inverse hyperbolic functions.....................................................................29 Analytical geometry .............................................................................................30 Straight line..................................................................................................30 Straight line through two points .........................................................30 Three points on one line .......................................................................31 The circle.......................................................................................................31 Tangent.....................................................................................................31 Normal .....................................................................................................32 Four points on a circle...........................................................................32 Circle through three points...................................................................33 Conic section................................................................................................33 Focus.........................................................................................................33 Eccentricity ..............................................................................................33 Directrix....................................................................................................33 45423_C000.fm Page viii Monday, August 27, 2007 11:16 AM
  • 14. Contents ix Partial derivatives...................................................................................33 Parabola....................................................................................................34 Tangent line with a given slope, m .....................................................36 Ellipse .......................................................................................................36 Hyperbola................................................................................................38 Calculus..................................................................................................................40 Differential calculus ....................................................................................40 Understanding the derivatives .................................................................41 Standard derivatives...................................................................................42 Integral calculus ..........................................................................................44 Volume of horizontal dished end........................................................44 Volume of vertical dished end .............................................................46 Standard integrals .......................................................................................47 Differential equations...........................................................................................48 First-order differential equations..............................................................48 Separation of variables ..........................................................................49 Second-order differential equations.........................................................50 Bessel function ........................................................................................51 Partial differential equations...............................................................................52 Laplace transform .......................................................................................60 Standard Laplace transforms ....................................................................61 Fourier half-range expansions ..................................................................62 Fourier half-range cosine series................................................................63 Fourier half-range sine series....................................................................63 Numerical analysis ...............................................................................................65 Solving linear equations (Newton’s method).........................................65 Newton’s method in two variables..........................................................66 Numerical methods in linear algebra......................................................68 Gauss elimination...................................................................................68 Cholesky method....................................................................................70 Numerical integration ................................................................................72 Trapezoidal rule......................................................................................72 Simpson’s rule.........................................................................................72 Double integration using Simpson’s rule ...............................................75 Numerical solution of first-order differential equations ......................76 Euler’s method........................................................................................76 Improved Euler’s method.....................................................................76 Runge–Kutta method.............................................................................77 Second-order differential equations.........................................................79 Runge–Kutta–Nystrom method...........................................................79 Partial differential equations.....................................................................81 Heat conduction problem .....................................................................81 Numerical solution .....................................................................................83 Alternating direction implicit (ADI) method ....................................85 Equation of state ...................................................................................................90 Boyle’s law and Charles’ law....................................................................90 45423_C000.fm Page ix Monday, August 27, 2007 11:16 AM
  • 15. x Process engineering and design using Visual Basic Equation of state for real gas ....................................................................92 Comparison between Peng–Robinson and SRK EOSs .........................92 Acentric factor..............................................................................................94 Vapor pressure of pure components........................................................94 SRK EOS .......................................................................................................94 PR EOS..........................................................................................................97 Alternate method to estimate vapor pressure of pure components................................................................................................97 Vapor pressure of pure water ...................................................................98 Equilibrium ratio (K) ..................................................................................98 Mixing rules .................................................................................................99 Estimate fugacity coefficient of vapor ...................................................101 Influence of interaction coefficient .........................................................102 Estimation of dew point ..........................................................................102 Unit conversions .................................................................................................103 Programming.......................................................................................................103 General notes for all programmers.............................................................. Vessel...........................................................................................................103 Program limitations ..................................................................................103 Horizontal...................................................................................................108 Data entry...................................................................................................108 Inclined................................................................................................... 110 Vertical.................................................................................................... 111 Conversion ................................................................................................. 112 Program limitations .................................................................................. 114 Procedure............................................................................................... 114 References/further reading............................................................................... 114 Chapter two Fluid mechanics.......................................................................117 Introduction ......................................................................................................... 117 Bernoulli’s theorem ............................................................................................ 118 Velocity heads............................................................................................ 119 Flow measurements............................................................................................120 Orifice/Venturi meter...............................................................................120 Thermal expansion factor (Fa).................................................................122 Coefficient of discharge CD ......................................................................122 Orifice meter..........................................................................................122 Restriction orifice.......................................................................................123 Venturi meter .............................................................................................124 Expansion factor (Y) .................................................................................124 Orifices ...................................................................................................124 Nozzles and Venturi ............................................................................124 Nonrecoverable pressure drop................................................................124 Orifices ...................................................................................................125 Venturi with 15° divergent angle.......................................................125 Venturi with 7° divergent angle.........................................................125 45423_C000.fm Page x Monday, August 27, 2007 11:16 AM
  • 16. Contents xi Critical flow................................................................................................125 Area meter: rotameters.............................................................................125 Flow through an open channel...............................................................126 V notch........................................................................................................126 Rectangular notch .....................................................................................127 Frictional pressure drop.....................................................................................127 Darcy equation...........................................................................................127 Flow in open channel...............................................................................128 Estimation of friction factor.....................................................................129 Friction factor — laminar flow ..........................................................129 Friction factor — turbulent flow........................................................129 Two-K method.....................................................................................................130 K for reducer/expander...........................................................................130 Reducer ..................................................................................................130 Expander................................................................................................132 Pipe entrance..............................................................................................132 Pipe exit ......................................................................................................132 Split flow.....................................................................................................132 Split 1,3...................................................................................................133 Split 1,2...................................................................................................133 Split 3,1...................................................................................................134 Split 1,2,3................................................................................................134 Split 1,3,2................................................................................................134 Split 3,1,2................................................................................................134 Hydraulics — general guidelines ....................................................................134 Roughness of pipe wall............................................................................135 Control valve CV.......................................................................................135 Line sizing criteria for liquid lines.........................................................136 Line sizing for gravity flow lines ...........................................................136 Downpipe sizing .......................................................................................136 Line sizing criteria for vapor lines.........................................................137 Relief valve inlet line sizing ....................................................................137 Relief valve outlet line sizing..................................................................138 Line sizing criteria for two-phase flow .................................................138 Hydraulics — compressible fluids...................................................................139 Adiabatic flow in a pipe ..........................................................................139 Isothermal flow in a pipe.........................................................................142 Heat loss...............................................................................................................143 Types of cross-country buried pipelines ...............................................143 Yellow jacket ..............................................................................................143 Coating thickness .................................................................................143 Fusion-bonded epoxy coating.................................................................144 Rate of heat transfer .................................................................................144 Film resistance (Rfilm)............................................................................144 Resistance of pipe (Rpipe)......................................................................146 Resistance of coatings (Rcoating)............................................................146 45423_C000.fm Page xi Monday, August 27, 2007 11:16 AM
  • 17. xii Process engineering and design using Visual Basic Resistance of environment (Renv)........................................................147 Viscosity of water......................................................................................149 Thermal conductivity of water ...............................................................149 Viscosity of air ...........................................................................................150 Thermal conductivity of air.....................................................................150 Choked flow...............................................................................................150 Limiting differential pressure..................................................................151 Limiting expansion factor Y....................................................................151 Hydraulics — two-phase flow..........................................................................153 Beggs and Brill correlations.....................................................................155 Step 1: Estimation of flow regime .....................................................155 Step 2: Estimation of horizontal holdup ..........................................155 Step 3: Estimation of uphill holdup..................................................156 Step 4: Estimation of downhill holdup ............................................156 Step 5: Estimation of friction factor...................................................157 Step 6: Estimation of pressure drop..................................................157 Mukherjee and Brill correlations............................................................158 Step 1: Estimation of flow regime .....................................................158 Step 2: Estimation of holdup..............................................................160 Step 3: Estimation of hydrostatic head.............................................160 Step 4: Estimation of acceleration head............................................160 Step 5: Estimation of friction factor...................................................161 Step 6: Estimation of frictional pressure drop.................................161 CO2 corrosion.............................................................................................164 CO2 corrosion mechanism .......................................................................164 NACE requirements..................................................................................165 Rate of corrosion .......................................................................................165 Constant Kt .................................................................................................165 Fugacity of CO2 .........................................................................................166 Calculation of pH factor f(pH)................................................................167 Calculation of shear stress.......................................................................168 Effect of temperature................................................................................169 Effect of glycol ...........................................................................................170 Effect of corrosion inhibitor.....................................................................170 Condensation factor..................................................................................170 Programming.......................................................................................................171 Program for flow elements......................................................................171 General overview .................................................................................171 Project details ........................................................................................172 Calculation form...................................................................................172 Program limitations and notes................................................................173 Program for hydraulic calculations........................................................176 General overview .................................................................................177 Project details ........................................................................................177 Program limitations and notes...........................................................178 Form incompressible fluid..................................................................178 45423_C000.fm Page xii Monday, August 27, 2007 11:16 AM
  • 18. Contents xiii Form compressible fluid .....................................................................181 Pressure drop comparison ..................................................................184 Form for two-phase flow ....................................................................184 Horizontal pipe section .......................................................................187 Uphill pipe section...............................................................................188 Downhill pipe section..........................................................................190 General conclusion...............................................................................192 Program for corrosion calculations........................................................192 Program limitations and notes...........................................................193 Calculation of pH.................................................................................193 Nomenclature ......................................................................................................197 Greek characters..................................................................................................200 References.............................................................................................................200 Chapter three Separators...............................................................................203 Introduction .........................................................................................................203 General principles of separation ......................................................................203 Droplet in a vertical vessel......................................................................203 Droplet in a horizontal vessel.................................................................206 Gravity settling: limiting conditions......................................................206 Newton’s law.............................................................................................207 Stoke’s law..................................................................................................207 Intermediate law .......................................................................................207 Critical particle diameter .........................................................................207 Vertical vs. horizontal separators.....................................................................208 Advantages of the horizontal separator................................................209 Disadvantages of the horizontal separator...........................................209 Advantages of the vertical separator.....................................................209 Disadvantages of the vertical separator................................................209 Design of a gas–liquid separator .....................................................................209 Critical settling velocity ...........................................................................209 Design constant KD ...................................................................................210 API 521 method......................................................................................... 211 Design of liquid–liquid separators ..................................................................212 Mist eliminator....................................................................................................214 Wire mesh mist eliminator ......................................................................214 Efficiency of the mist eliminator........................................................214 Target collection efficiency..................................................................215 Inertial impaction......................................................................................215 Direct interception.....................................................................................215 Diffusion ................................................................................................215 Target collection efficiency..................................................................216 Pressure drop of mist eliminator............................................................216 Vane type mist eliminator........................................................................217 Efficiency of vane pack ............................................................................217 Terminal centrifugal velocity ..................................................................217 45423_C000.fm Page xiii Monday, August 27, 2007 11:16 AM
  • 19. xiv Process engineering and design using Visual Basic Pressure drop through the vane pack ...................................................218 General dimensions and setting of levels.......................................................219 The horizontal separator..........................................................................219 Boot.........................................................................................................221 Vertical separator.......................................................................................222 Separator internals..............................................................................................223 The inlet nozzle .........................................................................................223 The vortex breaker....................................................................................224 Separator control.................................................................................................224 Pressure and flow control........................................................................226 Light liquid level control .........................................................................226 Heavy liquid level and slug control ......................................................226 High performance separator.............................................................................226 Salient features of GLCC .........................................................................228 Design parameters..............................................................................................228 Flow rates ...................................................................................................228 Slug length .................................................................................................228 Density ........................................................................................................230 Viscosity......................................................................................................231 Oil in gas droplet size ..............................................................................232 Oil in water droplet size ..........................................................................232 Water in oil droplet size...........................................................................233 Inlet nozzle velocity..................................................................................233 Gas outlet nozzle velocity........................................................................233 Liquid outlet velocity ...............................................................................233 Separator program..............................................................................................233 Program limitations/notes ......................................................................234 Horizontal separators...............................................................................234 Three-phase flooded weir ...................................................................234 Three-phase nonflooded-weir separator ..........................................234 Three-phase with boot separator.......................................................234 Two-phase vapor–liquid separator....................................................234 Two-phase liquid–liquid separator ...................................................235 Vertical separators.....................................................................................235 Two-phase vapor–liquid separator....................................................235 Two-phase liquid–liquid separator ...................................................235 General overview of the separator.exe program .................................235 Design .........................................................................................................240 Nomenclature ......................................................................................................244 Greek characters..................................................................................................245 References.............................................................................................................246 Chapter four Overpressure protection .......................................................247 Introduction .........................................................................................................247 Impact on plant design ............................................................................247 Impact on individual design...................................................................248 45423_C000.fm Page xiv Monday, August 27, 2007 11:16 AM
  • 20. Contents xv Definition..............................................................................................................248 Accumulation.............................................................................................248 Atmospheric discharge.............................................................................249 Built-up back pressure..............................................................................249 General back pressure ..............................................................................249 Superimposed back pressure ..................................................................249 Balanced-bellows PRV..............................................................................250 Blowdown ..................................................................................................250 Closed discharge system..........................................................................250 Cold differential test pressure.................................................................250 Conventional PRV.....................................................................................250 Design capacity..........................................................................................250 Design pressure .........................................................................................250 Maximum allowable accumulated pressure.........................................251 Maximum allowable working pressure ................................................251 Operating pressure....................................................................................251 Overpressure..............................................................................................251 Pilot-operated PRV....................................................................................251 Pressure relief valve (PRV)......................................................................252 Pressure safety valve ................................................................................252 Rated relieving capacity...........................................................................252 Relief valve.................................................................................................252 Relieving conditions .................................................................................252 Rupture disk...............................................................................................252 Safety relief valve......................................................................................252 Safety valve ................................................................................................253 Set pressure ................................................................................................253 Vapor depressuring system.....................................................................253 Vent stack....................................................................................................253 Types of pressure relief valves .........................................................................254 Conventional pressure relief valve (vapor service).............................254 Conventional pressure relief valve (liquid service) ............................256 Balanced-bellows pressure relief valve .................................................256 Pilot-operated pressure relief valve .......................................................259 Rupture disk...............................................................................................262 Selection of pressure relief valves....................................................................262 Conventional pressure relief valve.........................................................262 Balanced-bellows pressure relief valve .................................................264 Pilot-operated pressure relief valve .......................................................264 Rupture disk...............................................................................................265 PRV installation and line sizing.......................................................................265 Compressors and pumps.........................................................................265 Fired heaters...............................................................................................266 Heat exchangers ........................................................................................267 Piping ..........................................................................................................267 Pressure vessels .........................................................................................267 45423_C000.fm Page xv Monday, August 27, 2007 11:16 AM
  • 21. xvi Process engineering and design using Visual Basic PRV isolation valves .................................................................................268 Inlet piping to PRVs .................................................................................270 Discharge piping from PRVs...................................................................271 Contingency quantification...............................................................................272 General........................................................................................................272 Power failure..............................................................................................273 Local power failure ..............................................................................274 Failure of a distribution center ..........................................................274 Total power failure...............................................................................275 Cooling water failure................................................................................275 Instrument air failure ...............................................................................276 Steam failure ..............................................................................................277 Total steam failure................................................................................278 Loss of steam to specific equipment.................................................278 Partial steam failure.............................................................................278 Check valve failure ...................................................................................278 Blocked outlet ............................................................................................279 Pump or compressor discharge .........................................................279 Multiple outlet ......................................................................................280 Block valve downstream of control valve........................................280 Control valve failure.................................................................................280 Vapor breakthrough..................................................................................281 Maximum flow ..........................................................................................283 Thermal relief.............................................................................................283 Modulus of elasticity of pipe material (E) .......................................286 Coefficient of linear thermal expansion (α).....................................286 Valve leakage rate (q)...........................................................................287 Compressibility of liquid (Z)..............................................................288 Coefficient of cubic expansion of liquids (β)...................................288 Installation of thermal relief valve ....................................................288 Fire exposure..............................................................................................289 General guidelines ...............................................................................290 Estimation of wetted surface area .....................................................290 Fire circle................................................................................................292 Estimation of latent heat and physical properties..........................292 Liquid wet vessel..................................................................................293 Vessels with only gas...........................................................................295 Two liquid phases ................................................................................296 Heat exchanger tube rupture ..................................................................298 Contingency calculation......................................................................299 Reflux failure and overhead system ......................................................302 Loss of reboiler heat .................................................................................303 Venting of storage tank ............................................................................303 Venting due to liquid movements.....................................................304 Thermal venting ...................................................................................304 45423_C000.fm Page xvi Monday, August 27, 2007 11:16 AM
  • 22. Contents xvii Fire exposure.........................................................................................304 Minimum flow area .............................................................................304 Sizing procedure .................................................................................................307 Sizing of liquid relief................................................................................307 Sizing of vapor relief ................................................................................309 Critical flow...........................................................................................309 Subcritical flow ..................................................................................... 311 Conventional and pilot-operated PRV.............................................. 311 Balanced-bellows PRV......................................................................... 311 Sizing for steam relief............................................................................... 311 Sizing for two-phase fluids .....................................................................313 Design of flare stack .................................................................................330 Minimum distance ....................................................................................330 Fraction of heat intensity transmitted, τ ...............................................331 Fraction of heat radiated, F .....................................................................331 Heat release, Q...........................................................................................332 Sizing of a flare stack: simple approach................................................332 Calculation of stack diameter.............................................................332 Calculation of flame length ................................................................333 Flame distortion caused by wind velocity.......................................333 Sizing of flare stack: Brzustowski and Sommer approach.................335 Calculation of flare stack diameter....................................................335 Location of flame center xc, yc ............................................................335 Lower explosive limit of mixtures ....................................................335 Vertical distance yc................................................................................338 Horizontal distance xc..........................................................................338 SIL analysis ..........................................................................................................343 Definitions ..................................................................................................344 Diagnostic coverage (DC) ...................................................................344 Final element.........................................................................................344 MooN......................................................................................................344 Programmable electronics (PE)..........................................................344 Programmable electronic system (PES)............................................344 Protection layer.....................................................................................344 Safety-instrumented function (SIF) ...................................................344 Safety-instrumented systems (SIS) ....................................................345 Safety integrity......................................................................................345 Safety integrity level (SIL) ..................................................................345 Safety life cycle .....................................................................................345 Matrix for SIL determination ..................................................................345 Probability of failure on demand ...........................................................347 ALARP model............................................................................................348 Determination of SIL ...........................................................................348 Financial.................................................................................................350 Health and safety .................................................................................351 Environment and asset........................................................................351 45423_C000.fm Page xvii Monday, August 27, 2007 11:16 AM
  • 23. xviii Process engineering and design using Visual Basic Programming.......................................................................................................353 Program for pressure relief valve...........................................................353 Program limitations and notes...........................................................353 General overview .................................................................................354 Project details ........................................................................................355 File save..................................................................................................355 File open.................................................................................................355 File print.................................................................................................355 Exit ..........................................................................................................355 Specific message or warning: back pressure ...................................355 Back-pressure correction factor..........................................................356 Pilot-operated PRV...............................................................................356 Liquid .....................................................................................................357 Vapor ......................................................................................................357 Two-phase type 1 and type 2 calculations.......................................357 Program for flare stack estimation.........................................................362 Program limitations and notes...........................................................363 Specific message/warning..................................................................364 Nomenclature ......................................................................................................365 Greek characters..................................................................................................366 References.............................................................................................................366 Chapter five Glycol dehydration.................................................................369 Introduction .........................................................................................................369 Basic scheme........................................................................................................370 Advantages.................................................................................................370 Disadvantages............................................................................................370 Pre-TEG coalescer......................................................................................370 Contactor ....................................................................................................370 Flash separator...........................................................................................372 Filters...........................................................................................................372 Pumping .....................................................................................................372 Glycol/glycol exchanger..........................................................................373 Gas/glycol exchanger...............................................................................373 Regenerator ................................................................................................373 Physical properties .............................................................................................374 Selection of type of glycol .......................................................................374 Common properties of glycol .................................................................374 Densities of aqueous glycol solutions...............................................376 Solubility of various compounds.......................................................376 Fire hazard information ...........................................................................376 Viscosities of aqueous glycol solutions ............................................381 Specific heats of aqueous glycol solutions.......................................385 Thermal conductivities of aqueous glycol solutions......................387 Design aspects.....................................................................................................389 Water content in hydrocarbon gas .........................................................389 45423_C000.fm Page xviii Monday, August 27, 2007 11:16 AM
  • 24. Contents xix Equilibrium dew point.............................................................................389 Minimum lean-TEG concentration.........................................................389 Number of theoretical stages of the contactor.....................................391 Design of contactor...................................................................................394 Type of internals...................................................................................394 Flash separator...........................................................................................397 Filters...........................................................................................................397 Particulate filter ....................................................................................398 Carbon filter ..........................................................................................398 Glycol/glycol exchanger..........................................................................398 Gas/glycol exchanger...............................................................................398 Regenerator ................................................................................................400 Still column............................................................................................402 Reboiler.......................................................................................................403 Fire tube heat density..........................................................................403 Fire tube heat flux ................................................................................403 Lean glycol storage...................................................................................404 Energy exchange pump ...........................................................................404 Burner management .................................................................................406 Specifications..............................................................................................412 Programming.......................................................................................................415 Program limitations ..................................................................................415 General overview......................................................................................416 File menu ...............................................................................................416 Unit menu..............................................................................................417 Project details ........................................................................................417 Data entry ..............................................................................................417 References.............................................................................................................418 Index.....................................................................................................................419 45423_C000.fm Page xix Monday, August 27, 2007 11:16 AM
  • 25. 45423_C000.fm Page xx Monday, August 27, 2007 11:16 AM
  • 26. xxi Preface The availability of various design tools and software made process engineer- ing and design simple but also, in some cases, paradoxically led to inade- quate design. Unfortunately, not only young engineers but also reasonably experienced engineers are becoming more dependent on software tools with- out having a basic understanding of the design. I have seen people using HYSYS® to estimate water boiling temperature at atmospheric pressure. This reduced self-confidence is the key factor for inappropriate process design in many cases. As everything in process design can’t be covered by software tools like HYSYS®, HTRI®, etc., most design/consulting groups have developed their own design tools mostly using Excel® spreadsheets. Excel spreadsheets are extremely easy to develop and easy to corrupt through cut-and-paste mod- ifications as well as with attempts to modify the macros of the spreadsheet. Sometimes we also failed to recognize the limitations of Excel spreadsheets and develop something that does not produce the intended output. In spite of different procedures and design tools, we sometimes come across some typical design requirements that are not covered by any known procedure. Some understanding of basic mathematics and fundamental pro- cess engineering can solve a large number of problems without much diffi- culty. For example, a volume calculation of a horizontal vessel requires simple integration, volume calculation of an inclined vessel using Simpson’s rule, etc. The main purpose of this work is to identify small but important issues we normally come across during design and the best possible procedures to resolve them. This work gives a detailed analysis of the methodology used for a particular calculation and in some cases limitations of the proce- dure. This is done so that the user can check a calculation manually and develop an understanding of the basic design. This will improve the self- confidence of the user. The programs have been developed in Visual Basic® to avoid limitations of other programming tools, e.g., Excel. All programs have been developed to use both the International System of Units (SI) and English units (the 45423_C000.fm Page xxi Monday, August 27, 2007 11:16 AM
  • 27. xxii Process engineering and design using Visual Basic default choice is SI); however, the programs do not allow changing of indi- vidual units. Though each program has been checked extensively for correctness, the possibility of program bugs can’t be totally eliminated. Any feedback on program malfunctioning will be highly appreciated, as well as any other comments to improve future editions. Arun K. Datta 45423_C000.fm Page xxii Monday, August 27, 2007 11:16 AM
  • 28. xxiii Acknowledgements I wish to express my gratitude to WorleyParsons, Brisbane, for their strong support while this work was being produced. Special thanks go to my wife, Dr. Nivedita Datta, and my daughter, Raka Datta, whose support and encouragements were the key factors in the completion of this work. I am also very grateful to Malcolm Rough of WorleyParsons for his full review of the manuscript and many useful suggestions. I also appreciate the encouragement I have received from my mentors over many years, partic- ularly Tony Buckley and Ivan Broome of WorleyParsons. I have tried my best to thoroughly utilize my learning from them. I also acknowledge the help received from the American Petroleum Institute, the Gas Processors Suppliers Association, and John M. Campbell and Company for allowing me to publish their relevant figures and tables. Special thanks are also due to the editorial and production staff of CRC Press, a Taylor & Francis Company, particularly Allison Shatkin, Marsha Pronin, and Ari Silver for publishing this work with outstanding quality. I will thankfully acknowledge any suggestion to further improve future editions. Arun K. Datta Brisbane, Australia 45423_C000.fm Page xxiii Monday, August 27, 2007 11:16 AM
  • 29. 45423_C000.fm Page xxiv Monday, August 27, 2007 11:16 AM
  • 30. xxv About the author Arun K. Datta, principal process engineer working with WorleyParsons, has twenty-seven years’ experience in the field of process engineering and design. He holds a master’s degree in chemical engineering from the Indian Institute of Technology, Delhi, and has worked with several process consul- tancy organizations in both India and Australia. Arun has been a consultant for a large number of process engineering organizations including refineries, oil and gas industries, fine chemicals, and pharmaceuticals. His fields of expertise include heat and mass transfer, pro- cess simulations, exchanger design, pressure vessel design, design of safety systems, and design of control systems. Some of his clients include BP Refinery, Caltex Refinery, Santos, ExxonMobil, ONGC, Indian Oil, Incitec, and Oil Search. Arun is a chartered professional engineer in Australia (Queensland chap- ter) and has membership with the Institute of Engineers, Australia. He is also a member of the interview committee for the Institute of Engineers. 45423_C000.fm Page xxv Monday, August 27, 2007 11:16 AM
  • 31. 45423_C000.fm Page xxvi Monday, August 27, 2007 11:16 AM
  • 32. 1 chapter one Basic mathematics Introduction Understanding mathematics is the most fundamental requirement in under- standing engineering. Some fields in chemical engineering require the solu- tion of complex mathematical equations. The purpose of this section is to get some idea in the field of mathematics commonly used for chemical engineering design and includes the following: • Physical constants • Mensuration • Algebra • Trigonometry • Analytical geometry • Calculus • Differential equations • Partial differential equations • Numerical analysis • Equation of states • Unit conversions • Programming Physical constants Commonly used physical constants are presented in Table 1.1. SI prefixes The International System of Units (SI) prefixes are presented in Table 1.2. 45423_book.fm Page 1 Monday, August 27, 2007 12:44 PM
  • 33. 2 Process engineering and design using Visual Basic Mensuration Triangles Area = 1/2 bh (1.1) where b = base and h = altitude. Rectangles Area = ab (1.2) where a and b are the length of the sides. Table 1.1 Commonly Used Physical Constants Name Symbol Unit Value Speed of light c cm/sec 2.99792458E10 Plank’s constant h erg.sec 6.6260755E-27 hbar erg.sec 1.05457266E-27 Gravitational constant G cm3/(g.sec2) 6.67259E-8 G Nm2/kg2 6.67259E-11 Electron charge e ESU 4.8032068E-10 Mass of electron me g 9.1093897E-28 Mass of proton mp g 1.6726231E-24 Mass of neutron mn g 1.6749286E-24 Mass of hydrogen mH g 1.6733E-24 Atomic mass unit amu g 1.6605402E-24 Avogadro’s number NA 6.0221367E23 Boltzmann constant k erg/k 1.380658E-16 Electron volt eV erg 1.6021772E-12 Radiation density constant a erg/(cm3.K4) 7.5646E-15 Stefan–Boltzmann constant sigma erg/(cm2.K4.sec) 5.67051E-5 Rydberg constant erg 2.1798741E-11 Gas constant R J/(mol.K) 8.31439 R cal/(mol.K) 1.98719 R l.atm/(mol.K) 0.0820567 Pi π 3.1415926536 Napierian (natural) logarithm base e 2.7182818285 Euler’s constant γ 0.5772156649 Logarithm conversion log x 0.4342944819 ln x Logarithm conversion ln x 2.302585093 log x Radian 57.2957795131° Degree 0.0174532925 rad Minute 0.0002908882 rad Second 0.0000048481 rad 45423_book.fm Page 2 Monday, August 27, 2007 12:44 PM
  • 34. Chapter one: Basic mathematics 3 Parallelogram (opposite sides parallel) Area = ah = ab sinα (1.3) where a and b are the lengths of the sides, h the height, and α the angle between the sides. See Figure 1.1. Rhombus (equilateral parallelogram) Area = 1/2 ab (1.4) where a and b are lengths of diagonals. Trapezoid (four sides, two parallel) Area = 1/2(a + b)h (1.5) where the lengths of parallel sides are a and b and h = height. Table 1.2 SI Prefixes Multiplication Factor Prefix Symbol 1 000 000 000 000 000 000 = 1018 exa E 1 000 000 000 000 000 = 1015 peta P 1 000 000 000 000 = 1012 tera T 1 000 000 000 = 109 giga G 1 000 000 = 106 mega M 1 000 = 103 kilo k 100 = 102 hecto h 10 = 101 deka da 0.1 = 10–1 deci d 0.01 = 10–2 centi c 0.001 = 10–3 milli m 0.000 001 = 10–6 micro μ 0.000 000 001 = 10–9 nano n 0.000 000 000 001 = 10–12 pico p 0.000 000 000 000 001 = 10–15 femto f 0.000 000 000 000 000 001 = 10–18 atto a Figure 1.1 Parallelogram. h a b α 45423_book.fm Page 3 Monday, August 27, 2007 12:44 PM
  • 35. 4 Process engineering and design using Visual Basic Quadrilateral (four sided) Area = 1/2 ab sinθ (1.6) where a and b are the lengths of the diagonals, and the acute angle between them is θ. Regular polygon of n sides (refer to Figure 1.2) Area = 1/4 nL2 cot(180°/n) (1.7a) R = L/2 csc(180°/n) (1.7b) r = L/2 cot(180°/n) (1.7c) β = 360°/n (1.7d) θ = (n – 2)180°/n (1.7e) L = 2r tan(β/2) = 2R sin(β/2) (1.7f) Circle (refer to Figure 1.3) Let C = circumference r = radius D = diameter A = area S = arc length subtended by θ L = chord length subtended by θ H = maximum rise of arc above chord, r – H = d θ = central angle (rad) subtended by arc S Figure 1.2 Regular polygon. L r R β θ 45423_book.fm Page 4 Monday, August 27, 2007 12:44 PM
  • 36. Chapter one: Basic mathematics 5 Then, C = 2πr = πD (1.8a) S = rθ = 1/2 Dθ (1.8b) (1.8c) (1.8d) θ = S/r = 2cos–1(d/r) = 2sin–1(L/D) (1.8e) A(circle) = πr2 = 1/4 πD2 (1.8f) A(sector) = 1/2 rS = 1/2 r2θ (1.8g) A(segment) = A(sector) – A(triangle) = 1/2 r2 (θ – sinθ) (1.8h) = r2 cos–1 (r – H)/r –(r – H) (2rH – H2)0.5 (1.8i) (1.8j) where h = H/D, h ≤ 0.5 φ = 2{90 – sin–1(1 – 2h)} Figure 1.3 Circle. D r H d S θ L r d r d = − = = 2 2 2 2 2 2 2 sin( / ) tan( / ) θ θ d r L L = − = 1 2 4 1 2 2 2 2 / / cot( / ) θ A A h h ( ) ( ) ( / )( / ) ( )( . segment total = ° − − φ π 360 4 0 5 2 − − h) / π 4 45423_book.fm Page 5 Monday, August 27, 2007 12:44 PM
  • 37. 6 Process engineering and design using Visual Basic Ellipse (refer to Figure 1.4) Area = πab (1.9a) Circumference = 2π{(a2 + b2)/2}0.5 (approximately) (1.9b) Parabola (refer to Figure 1.5) Length of arc EFG = ( 1.10a) Area of section EFG = 4/3 xy (1.10b) Prism Lateral surface area = (perimeter of right section) * (lateral edge) (1.11a) Volume = (area of base) * (altitude) (1.11b) Figure 1.4 Ellipse. Figure 1.5 Parabola. a b y x E G F 4 2 2 4 2 2 2 2 2 x y y x x x y y + ( )+ + + ( ) { } ln / 45423_book.fm Page 6 Monday, August 27, 2007 12:44 PM
  • 38. Chapter one: Basic mathematics 7 Pyramid Lateral area of a regular pyramid = 1/2 (perimeter of base) * (slant height) (1.12a) = 1/2 (number of sides)*(length of one side) * (slant height) (1.12b) Volume = 1/3 (area of base) * (altitude) (1.12c) Right circular cylinder Lateral surface area = 2π (radius) * (altitude) (1.13a) Volume = π (radius)2 * (altitude) (1.13b) Sphere (refer to Figure 1.6) Area (sphere) = 4πR2 = πD2 (1.14a) Area (zone) = 2πRh = πDh (1.14b) Volume (sphere) = 4/3πR3 = 1/6πD3 (1.14c) Volume (spherical sector) = 2/3πR2h = 1/6πh1(3r2 2 + h1 2) (1.14d) Volume (spherical segment of one base) = 1/6πh1(3r2 2 + h1 2) (1.14e) Volume (spherical segment of two bases) = 1/6πh(3r1 2 + 3r2 2 + h2 2) (1.14f) Figure 1.6 Sphere. r2 r1 h1 R h2 h D 45423_book.fm Page 7 Monday, August 27, 2007 12:44 PM
  • 39. 8 Process engineering and design using Visual Basic Right circular cone Curved surface area = πr(r2 + h2)0.5 (1.15a) Volume = 1/3 πr2h (1.15b) where r = radius h = height of the cone Dished end Surface area (general) = πa2 + π/2(b2/e)ln{(1 + e)/(1 – e)} (1.16a) Surface area (ellipsoidal, b = a/2) = πa2 + (πa2/8e)ln{(1 + e)/(1 – e)} (1.16b) = 4.336 a2 Surface area (hemispherical, b = a) = 2πa2 (1.16c) Volume (general) = 2/3 πa2b (1.16d) Volume (ellipsoidal) = 1/3 πa3 (1.16e) Volume (hemispherical) = 2/3 πa3 (1.16f) where a = semimajor axis b = semiminor axis e = eccentricity = (1 – b2/a2)0.5 Irregular shape Let yo, y1, y2, …, yn be the lengths of a series of equally spaced parallel chords and h be the distance between them. The approximate area of the figure is given by using the trapezoidal rule or by Simpson’s rule. Trapezoidal rule Area = h/2 {(yo + yn) + 2(y1 + y2 + y3 + … + yn–1)} (1.17) Simpson’s rule Area = h/3 {(yo + yn) + 4(y1 + y3 + y5 + … + yn–1) + (1.18) 2(y2 + y4 + … + yn–2)} 45423_book.fm Page 8 Monday, August 27, 2007 12:44 PM
  • 40. Chapter one: Basic mathematics 9 where n is even. The greater the value of n, the greater the accuracy of approximation. Irregular volume To find irregular volume, replace y’s by cross-sectional area Aj , and use the results in the preceding equation. Algebra Factoring a2 − b2 = (a + b)(a − b) (1.19a) a3 − b3 = (a − b)(a2 + ab + b2) (1.19b) a3 + b3 = (a + b)(a2 − ab + b2) (1.19c) an − bn = (a − b)(an–1 + an–2b + an–3b2 + … + bn–1) (1.19d) (a + b)2 = a2 + 2ab + b2 (1.19e) (a − b)2 = a2 − 2ab + b2 (1.19f) (a + b + c)2 =a2 + b2 + c2 + 2ab + 2ac + 2bc (1.19g) (a − b − c)2 =a2 + b2 + c2 − 2ab − 2ac + 2bc (1.19h) (a + b)3 = a3 + 3a2b + 3ab2 + b3 (1.19i) (a − b)3 = a3 − 3a2b + 3ab2 − b3 (1.19j) (a + b)n = an + nan–1b + {n(n − 1)/2!}an–2b2 + (1.19k) {n(n − 1)(n − 2)/3!}an–3b3 + … + bn Arithmetic progression Aseries is said to be in arithmetic progression (AP), if the algebraic difference between any two successive terms be the same throughout the series. The following series is in AP. a, a + b, a + 2b, a + 3b, …, nth term where a = first term b = common difference 45423_book.fm Page 9 Monday, August 27, 2007 12:44 PM
  • 41. 10 Process engineering and design using Visual Basic The nth term can be defined as tn = a + (n − 1)b (1.20a) Sum of the series S = (n/2){2a + (n − 1)b} (1.20b) Geometric progression A series is said to be in geometric progression (GP) if the ratio of any term to the preceding one is the same throughout the series. The following series is in GP. a, ar, ar2, ar3, …, nth term where a = first term r = common ratio The nth term can be defined as tn = arn–1 (1.21a) Sum of the series S = a(1 − rn)/(1 − r) (1.21b) Infinite series (in GP) When r < 1, the sum of the series = a/(1 − r) (1.21c) Example Series 1 + 1/3 + 1/9 + 1/27 + … + 1/∞. The sum = a/(1 – r) = 3/2. Best fit straight line (least squares method) The straight line y = a + bx should be fitted through given points (x1,y1), (x2,y2), …, (xn,yn) so that the sum of the squares of the distances of those points from the straight line is minimum, where the distances are measured in the vertical direction (y-direction). The values of a and b are calculated from the following equations: 45423_book.fm Page 10 Monday, August 27, 2007 12:44 PM
  • 42. Chapter one: Basic mathematics 11 an + bΣxj =Σyj (1.22a) and aΣxj + bΣxj 2 =Σxjyj (1.22b) where n is the number of points. Example 1.1 Let the four points be (−1,1), (−0.1,1.099), (0.2,0.808), and (1,1). n = 4 Σxj = 0.1 Σxj 2 = 2.05 Σyj = 3.907 Σxjyj = 0.0517 4a + 0.1b = 3.907 0.1a + 2.05b = 0.0517 The solution is a = 0.9773 b = –0.0224 Therefore, the equation of the best fit straight line is y = 0.9773 – 0.0224x. Binomial equation The general Equation is: ax2 + bx + c = 0 (1.23a) General solution of Equation 1.23a (1.23b) Polynomial equation General solution of the following type of polynomial equation: y = ax2 + bx + c (1.24a) If three values (x1,y1), (x2,y2), and (x3,y3) are known, then x b b ac a = − ± − 2 4 2 45423_book.fm Page 11 Monday, August 27, 2007 12:44 PM
  • 43. 12 Process engineering and design using Visual Basic (1.24b) (1.24c) (1.24d) Example 1.2 Solve the equation y = ax2 + bx + c, for three sets of values (1,10), (3,32), and (5,70). Solution c = 10 − 2 − 3 = 5 Therefore, the general solution will be y = 2x2 + 3x + 5. Maxima/minima Let a function be y = f(x) (1.25a) If a solution for x at y′ = 0 exists, then there will be either a maximum value or a minimum value for the function. At a value of x, where y′ = 0: If y″ is negative, then there will be a maximum value for the function. If y″ is positive, then there will be a minimum value for the function. Example 1.3a Let the equation be y = ax + b (1.25b) a y y x x y y x x x x x = − − − − − − ( )( ) ( )( ) ( )( 1 2 2 3 2 3 1 2 1 2 2 2 2 − − − − − x x x x x 3 2 2 3 2 1 2 ) ( )( ) b y y a x x x x = − − − { } − ( ) ( ) ( ) 1 2 1 2 2 2 1 2 c y ax bx = − − 1 1 2 1 a = − − − − − − − − − ( )( ) ( )( ) ( )( ) ( 10 32 3 5 32 70 1 3 1 9 3 5 9 25) )( ) 1 3 2 − = b = − − − { } − = ( ) ( ) 10 32 2 1 9 2 3 45423_book.fm Page 12 Monday, August 27, 2007 12:44 PM
  • 44. Chapter one: Basic mathematics 13 y′ = a = 0 No solution exists at y′ = 0; this indicates that there will be no maxima or minima for the function. This is the equation of a straight line. Example 1.3b Let the equation be y = x3 + x – 1 (1.25c) y′ = 3x2 + 1 = 0 (1.25d) or, x2 = −1/3 (1.25e) No general solution exists for Equation 1.25e, meaning Equation 1.25c will have no general maxima or minima. Example 1.3c Let the equation be y = 4x3 – 3x + 2 (1.25f) y′ = 12x2 − 3 = 0 x = ± 1/2 At x = 1/2, y″ = 12 and at x = −1/2, y″ = −12. Therefore at x = 1/2, the value of y will be minimum. Value of y = 1. At x = –1/2 the value of y will be maximum. Value of y = 3. The nature of this graph is such that for x values of less than –1/2, the value of y will decrease continuously, and for x values more than 1/2, the value of y will increase continuously. Matrix Addition and multiplication of matrices • If two m by n matrices A and B are given, the sum A + B can be defined as the m by n matrix adding corresponding elements, i.e., (A + B) [i,j] =A[i,j] + B[i,j] (1.26a) • If a matrix A and a number c are given, the multiplication of the matrix can be defined as cA = cA[i,j] (1.26b) 45423_book.fm Page 13 Monday, August 27, 2007 12:44 PM
  • 45. 14 Process engineering and design using Visual Basic Addition of matrices i. ii. Multiplication of matrices iii. iv. Multiplication of two matrices is well defined only if the number of columns of the first matrix is the same as the number of rows of the second matrix. If A is an m by n matrix (m rows, n columns) and B is an n by p matrix (n rows, p columns), then their product AB is the m by p matrix (m rows, p columns) given by (AB)[i,j] = A[i,1] * B[1,j] + A[i,2] * B[2,j] + … + A[i,n]*B[n,j] for each pair of i and j. Multiplication has the following properties: • (AB)C = A(BC) for all k by m matrices A, m by n matrices B, and n by p matrices C (associativity). • (A + B)C = AC + BC for all m by n matrices A and B and k by m matrices C (distributivity). • In general, commutativity does not hold, meaning generally AB ≠ BA. Matrix properties involving addition Let A, B, and C be m by n matrices. Then, a. A + B = B + A (1.27a) b. (A + B) + C = A + (B + C) (1.27b) c. A + 0 = A (1.27c) where 0 is the m by n zero matrix (all its entries are equal to 0). a b c d e f u v w x y z a u b v c w d x e y f ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + + + + + z z ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b c d e f a b c d e f a b c d e f ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ 2 2 2 2 2 2 ⎥ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 a b c d e f a b c d e f g h ae bg af bh ce dg cf dh ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + + + ⎡ * ⎣ ⎣ ⎢ ⎤ ⎦ ⎥ a b c d e f x y z ax by cz dx ey fz ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = + + + + * ⎡ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 45423_book.fm Page 14 Monday, August 27, 2007 12:44 PM
  • 46. Chapter one: Basic mathematics 15 d. A + B = 0 if and only if B = –A (1.27d) Matrix properties involving multiplication a. Let A, B, and C be three matrices. If the products AB, (AB)C, BC, and A(BC) exist then (AB)C = A(BC) (1.28a) b. If α and β are numbers and A is a matrix, then α(βA) = (αβ)A (1.28b) c. If α is a number, and A and B are two matrices such that the product AB is possible, then α(AB) = (αA)B = A(αB) (1.28c) d. If A is an n by m matrix and 0 is the m by k zero matrix, then A0 = 0 (1.28d) Matrix properties involving addition and multiplication a. Let A, B, and C be three matrices, then (A + B)C = AC + BC (1.29a) and A(B + C) = AB + AC (1.29b) b. If α and β are numbers and A and B are matrices, then α(A + B) = αA + αB (1.29c) and (α + β)A = αA + βA (1.29d) Transpose The transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. The transpose of an n by m matrix A is the n by m matrix AT defined by AT[i,j] = A[j,i] (1.30) For the matrix 45423_book.fm Page 15 Monday, August 27, 2007 12:44 PM
  • 47. 16 Process engineering and design using Visual Basic A = The transpose AT = Symmetric matrix A symmetric matrix is a matrix equal to its transpose. A symmetric matrix must be a square matrix, for example, and Diagonal matrix A diagonal matrix is a symmetric matrix with all of its entries equal to zero, except possibly the values of the diagonal, for example, and Determinants Determinants are important both in calculus and algebra. The determinant of a 1 × 1 matrix is the element itself: det[a] = a. The 2 × 2 matrix A = has determinant det (A) = ad – bc (1.31) a b c d e f g h i j k l ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ a d g j b e h k c f i l ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ a b b c ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b c b d e c e f ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ a b 0 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b 0 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ a b c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 45423_book.fm Page 16 Monday, August 27, 2007 12:44 PM
  • 48. Chapter one: Basic mathematics 17 The 3 × 3 matrix A = has determinant det(A) = a(ei – fh) – b(di – fg) + c(dh – eg) (1.32) Properties of determinants 1. Any matrix A and its transpose have the same determinant, meaning det A = det AT. 2. If the elements of one row (or column) of a determinant are all zero, the value of the determinant is zero. 3. The determinant of a triangular matrix is the product of the entries on the diagonal, that is, = ad 4. If one determinant is obtained from another by interchanging any two rows (or columns), the value of either is the negative of the value of the other. 5. If the elements of one row (or column) of a determinant are multiplied by the same constant factor, the value of the determinant is multiplied by this factor. 6. If two rows (or columns) of a determinant are identical, the value of the determinant is zero. 7. If two determinants are identical except for one row (or column), the sum of their values is given by a single determinant obtained by adding corresponding elements of dissimilar rows (or columns) and leaving the remaining elements unchanged. a b c d e f g h i ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ a b d a b d 0 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b c d c d a b ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ na nb c d n a b c d a b nc nd ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 45423_book.fm Page 17 Monday, August 27, 2007 12:44 PM
  • 49. 18 Process engineering and design using Visual Basic 8. The value of a determinant is not changed, if to the elements of any row (or column) are added a constant multiple of the corresponding elements of any other row (or column). 9. If all elements but one in a row (or column) are zero, the value of the determinant is the product of that element and its cofactor. Cofactor Let A be an n × n matrix. The ij-th cofactor of A, denoted by Aij, will be Aij = (−1)i+j (1.33) Example 1.4 Find cofactors A32 and A24 of a 4 × 4 determinant A Solution 3 2 1 5 4 2 7 5 7 2 8 5 19 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = a nc b nd c d a b c d a b c na d nb + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎥ Mij A = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ 1 3 5 6 2 4 0 3 1 5 9 2 4 0 2 7 A M 32 3 2 32 1 1 5 6 2 0 3 4 2 7 8 = − = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − + ( ) A24 2 4 1 1 3 5 1 5 9 4 0 2 192 = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − + ( ) 45423_book.fm Page 18 Monday, August 27, 2007 12:44 PM
  • 50. Chapter one: Basic mathematics 19 Determinant and inverses Theorem 1 If A is invertible, then det A ≠ 0 and det A–1 = 1/det A (1.34) Adjoint Let a determinant be (1.35) Let A be an n × n matrix and let B, given in Equation 1.35, denote the matrix of its cofactors. Then the adjoint of A, written as adjA, is the transpose of the n × n matrix B. (1.36) Theorem 2 Let A be an n × n matrix. Then, (1.37) Theorem 3 Let A be an n × n matrix. Then A is invertible if and only if detA ≠ 0. If detA ≠ 0, then (1.38) B A A A A A A A A A n n n n nn = ⎡ 11 12 1 21 22 2 1 2 ... ... ... ⎣ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ adjA B A A A A A A A A T n n n n = = 11 21 1 12 22 2 1 2 ... ... . . .. Ann ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ( )( ) ... ... ... A adjA A A A = det det det 0 0 0 0 0 0 0 0 0 0 0 0 0 det det A A I ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = ( ) A A adjA − = 1 1 det 45423_book.fm Page 19 Monday, August 27, 2007 12:44 PM
  • 51. 20 Process engineering and design using Visual Basic Cramer’s rule Consider a system of n equations and n unknowns. a11x1 + a12x2 + … + a1nxn = b1 a21x1 + a22x2 + … + a2nxn = b2 (1.39a) an1x1 + an2x2 + … + annxn = bn which can be written in the form Ax = b (1.39b) If detA ≠ 0, then Equation 1.39b will have a unique solution given by x = b/A (1.39c) Let D be detA. The other matrix can be defined as Let D1 = detA1, D2 = detA2 …, Dn = detAn, then (1.40) Example 1.5 Solve using Cramer’s rule, the system 2x1 + 4x2 + 6x3 = 18 4x1 + 5x2 + 6x3 = 24 3x1 + x2 – 2x3 = 4 A b a a b a a b a a n n n n nn 1 1 12 1 2 22 2 2 = ⎡ ⎣ ⎢ ... ... ... ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = , ... ... A a b a a b a a n n n 2 11 1 1 21 2 2 1 b b a n nn ... , ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ etc. x D D x D D x D D n n 1 1 2 2 = = = , , , … 45423_book.fm Page 20 Monday, August 27, 2007 12:44 PM
  • 52. Chapter one: Basic mathematics 21 Solution The determinant is calculated first as D = = 2(−10 − 6) – 4(− 8 – 18) + 6(4 – 15)= −32 + 104 – 66 = 6 Because the value of D ≠ 0, the system will have a unique solution. D1 = = 18(−10 – 6) – 4 (−48 – 24) + 6(24 – 20) = 24 D2 = = 2(−48 – 24) − 18(−8 – 18) + 6(16 – 72) = −12 D3 = = 2(20 – 24) − 4(16 – 72) + 18(4 – 15) = 18 Therefore, x1 = D1/D = 24/6 = 4 x2 = D2/D = −12/6 = −2 x3 = D3/D = 18/6 =3 Trigonometry Functions of circular trigonometry The trigonometric functions are the ratio of various sides of the reference angles presented in Figure 1.7. The trigonometric relationships are: Sine of α = sin α = p/h; Secant of α = sec α = h/b Cosine of α = cos α = b/h; Cosecant of α = csc α = h/p 2 4 6 4 5 6 3 1 2 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 18 4 6 24 5 6 4 1 2 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 2 18 6 4 24 6 3 4 2 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 2 4 18 4 5 24 3 1 4 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 45423_book.fm Page 21 Monday, August 27, 2007 12:44 PM
  • 53. 22 Process engineering and design using Visual Basic Tangent of α = tan α = p/b; Cotangent of α = cot α = b/p The magnitude and sign of trigonometric functions are presented in Table 1.3. Periodic functions The following periodic functions are commonly used in trigonometry: cos (α + 2π) = cos α (1.41a) sin (α + 2π) = sin α (1.41b) cos (α + 2nπ) = cos α, n = 0, ±1, ±2, … (1.41c) sin (α + 2nπ) = sin α, n = 0, ±1, ±2, … (1.41d) The magic identity The following identities are very basic to the analysis of trigonometric expressions: sin2 α + cos2 α = 1 (1.42a) 1 + tan2 α = sec2 α (1.42b) 1 + cot2 α = csc2 α (1.42c) Figure 1.7 Triangles. Table 1.3 Magnitude of Trigonometric Functions Function 0–90° 90–180° 180–270° 270–360° sin α csc α cos α sec α tan α cot α +0 to +1 +∞ to +1 +1 to 0 +1 to +∞ +0 to +∞ +∞ to +0 +1 to +0 +1 to +∞ –0 to –1 –∞ to –1 –∞ to –0 –0 to –∞ 0 to –1 –∞ to –1 –1 to –0 –1 to –∞ +0 to +∞ +∞ to +0 –1 to –0 –1 to –∞ +0 to +1 +∞ to +1 –∞ to –0 –0 to –∞ b p h b b b p h h h p p α α α 45423_book.fm Page 22 Monday, August 27, 2007 12:44 PM
  • 54. Chapter one: Basic mathematics 23 The addition formulas The following addition formulas are very important for the analysis of trig- onometric expressions: cos (a + b) = cos a cos b – sin a sin b (1.43a) cos (a − b) = cos a cos b + sin a sin b (1.43b) sin (a + b) = sin a cos b + cos a sin b (1.43c) sin (a − b) = sin a cos b – cos a sin b (1.43d) (1.43e) (1.43f) Example 1.6 Find the exact value of cos 165°, given that cos 120° = −1/2, sin 120° = , and cos 45° = sin 45° = . Solution cos 165° = cos (120° + 45°) = cos 120° cos 45° – sin 120° sin 45° = = Double angle and half angle Double-angle and half-angle formulas are very useful in trigonometry. The following equations are widely used: cos (2a) = cos2 a – sin2 a = 2 cos2 a – 1 = 1 – 2sin2 a (1.44a) tan( ) tan tan tan tan a b a b a b + = + − 1 tan( ) tan tan tan tan a b a b a b − = − + 1 3 2 / 2 2 / − − 1 2 2 2 3 2 2 2 * * − + 6 2 4 45423_book.fm Page 23 Monday, August 27, 2007 12:44 PM
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  • 56. To AMCROSS, Paris: Answering Inquiry No. ——. Mother and baby son three months old well and happy. In this instance the worried fighter was an officer—a captain of infantry. During the time which elapsed between the two cablegrams he was wounded and the answer found him in a hospital, side by side with a French blessé. A Red Cross searcher acted as interpreter for their felicitations and in her official report of the incident included this notation: Captain W—— was much improved as a result of the good news. He is sitting up and eating roast chicken to-day. He says the American Red Cross has cured him. The Red Cross representatives here in America could not enter a home unless they were welcome; neither could they force their way into the hearts of men. They were compelled to wait until their help was sought. The growing mental depression of a certain major of a fighting division during those tense months of the midsummer of 1918 did not escape the attention of the American Red Cross man attached to that division. Suddenly the man, who had been marked because of his poise, became taciturn—isolated himself. A reference to the Red Cross Home Service which its division worker tactfully introduced into the table talk at the mess at which both sat, however, did elicit some trivial rejoinder from the man with the golden oakleaf upon his shoulder; while the following day that same major wrote a letter to the Red Cross man—and bared the reason for his most obvious melancholy. It seemed that back here in the United States he had a little son, from whom he had received no word whatsoever in more than six months. The child was with the major's divorced wife, and his father was more than anxious to know if he was regularly playing out of doors, if he was receiving his father's allotment, and if he was buying the promised Thrift Stamp each week. The army man already
  • 57. had his second golden service stripe and greatly feared that his little son might be beginning to forget him. Under conditions such as these, visiting the boy was a diplomatic mission indeed. Finally it was intrusted to the wife of an army officer. And because army officers' wives are usually achieved diplomats if not born ones, the ultimate result came in weekly letters from the boy, which not only greatly relieved his father's mind but greatly increased the bonds of affection between the two. The Greatest Mother in the World is never above diplomacy—which is, perhaps, just another way of expressing tact and gentleness. There were many, many occasions, too, when the relatives at home depended upon that selfsame diplomacy of hers to tell the disagreeable stories of losses or perhaps to prepare the boys overseas to face an empty chair in the family circle. There was one particularly fearful moment when a brilliant young officer had to be told that the reason why his young wife had ceased to write was because she had gone insane and specialists believed that she could not recover. Boys were driven to Red Cross offices by hidden affairs that flayed them hideously and of which they wished to purge themselves. Some wanted to set old wrongs right. Others had fallen blindly into the hands of the unscrupulous and had only fully awakened to see their folly after they actually were upon the battlefields of France. Then there were the softer phases of life—the shy letters and the blushing visitors who wished to have a marriage arranged with Thérèse or Jeanne of the black eyes and the delicate oval face. I remember one of our boys who had fallen in love with a girl in Nancy. Theirs was a courtship of unspoken love, unless soft glances and gentle caresses do indeed speak more loudly than mere words; for they had no easy bond of a common tongue. His French was doughboy French, which was hardly French at all, and her English was limited. So that after he had gone on to the Rhine and the letter came from her to him in the delicate hand that the sisters at the convent had taught, he needs must seek out Red Cross Home Communication and intrust to it the task of uncommon delicacy,
  • 58. which it fulfilled to the complete delight and satisfaction of both of them. For how could any mother, let alone the Greatest Mother in the World, blind her eyes entirely to love? She apparently had no intention of doing any such thing. For how about that good-looking doughboy from down in the Ozark country somewhere, who arrived in Paris on a day in the autumn of 1918 with the express intention of matrimony, if only he knew where he could get the license? French laws are rather fussy and explicit in such matters. Some one suggested the Home Service Bureau of the American Red Cross to the boy. He found his way quickly to it—with little Marie, or whatever her name really was, hanging on his arm. A Red Cross man prayerfully guided the pair through the legal mazes of the situation. First they went to a law office in the Avenue de l'Opéra where the necessary papers were made out; then the procession solemnly moved to the office of the United States Vice Consul at No. 1 Rue des Italiens, where the signature of the American official representative was duly affixed to each of the papers; after which to the foreign office, where the French went through all the elaborate processes of sealings and signatures which they seem to love so dearly, and then—the work of Mother Red Cross was finished. They were quite ready for the offices of the Church. With the signing of the armistice all this work was greatly increased —was, in fact, doubled and nearly trebled. When a man was fighting his physical needs seemingly were paramount; but once off the field, the worries that lurked in his subconscious mind seemed to rise quickly to the surface. He then recalled that long interval since last he heard from home. That troubled him, and he turned to the Red Cross—those pamphlets and posters did have a tremendous effect. And if he had no definite troubles over here, such as those we have just seen, he was apt to be just plain hungry for a sight of the home —and the loved ones that it held. It was in answer to a demand such as this last that a Red Cross representative right here in the United States took her motor car and
  • 59. drove for a half day out to see a family of whose very existence she had never before even heard; and, as a result of her call, wrote back a letter from which the following excerpts are taken: I want to tell you about a never-to-be-forgotten trip that I took the other day out to see a one hundred per cent patriot; an American mother who has three sons in the service. The home is one of the coziest, homiest, friendliest places you can imagine; one story, with that cool spacious plan of construction that makes you want to get a book, capture a chair on the wide, comfortable porch, and forget the world and its dizzy rush; a great sweep of lawn and with some handsome Hereford calves browsing in one direction and a cluster of shade trees nearer the house. The hills surrounding the house make a lovely view and all were covered with grazing stock, also the fine Hereford cattle for which the place is known. But the best part of the home is the dear little woman who hung a service flag in the window with the name of a boy under each of the three stars. She is the type of mother that draws every one to her; tender, sensible, capable, broad-minded, and with a shrewd sense of humor that keeps things going and makes life worth living for the entire household. She took us to a roomy side porch where her sewing unit of the Red Cross meets each Tuesday. A marvelous amount of work has been turned out in that side porch, and I'll wager a dollar to a doughnut that I know the moving spirit of the workers. Off in a big, cool parlor bedroom there were stacked up several perfectly enchanting 'crazy quilts' made by these same busy women at odd moments. These are ready to be sent to Serbia or they may be sold at auction for the benefit of the Red Cross. We saw pictures of each boy in the service—one in the navy, one in the heavy artillery, and Milton, whom we all hope is not in the hospital by now. Each boy had in his eyes the same intrepid look that the mother has—one can tell that they made good soldiers. Knowing how busy farm folk are, we reluctantly took our leave after
  • 60. seeing all these interesting things and, as we swung out into the country lane, we looked back and there stood the mother waving and smiling—the very best soldier of them all. Can you not see how very simple it all was—how very human, too? As you saw in one of the earlier chapters of this book, a fairly formal and elaborate plan of organization had been laid out for all this work; but, perhaps because war after all, is hardly more than a series of vast emergencies, the American Red Cross searchers, either in the field or in the hospitals, could hardly confine themselves to any mere routine of clerical organization or work in the great task that was thrust upon them. The unexpected was forever upon them. As a single instance of this take the time when, in the Verdun sector and in the hottest days of fighting that the American Army found there, so many demands were made upon our Red Cross by the officers and men of the A. E. F. for the purchase of necessities in Paris that a definite shopping service quite naturally evolved itself out of the situation. The man who initiated that service raced a motor car from Verdun to the Paris headquarters in order to secure the materials necessary for its inauguration. For when the American Red Cross made up its mind to do a thing, it did it—and pretty quickly too. So it went—a service complicatedly simple, if I may so express it. For, despite its own batteries of typewriters and card indexes, there was, at almost all times, that modicum of human sympathy that tempered the coldness of mere system and glorified what might otherwise have been a mere job of mechanical routine into a tremendously human and tender thing. The men and girls of the Home Communication Service had a task of real worth. Of a truth it was social service—of the most delicate nature. It included at all times not only the study of the physical needs of the soldier or sailor, but also at many times that of his mental needs as well. In reality, it
  • 61. became a large part of the scheme of preserving and enlarging the morale of the A. E. F. Every time a soldier was freed of endless, nagging worry, he became a better soldier and so just that much more strength was added to the growing certainty of victory.
  • 62. CHAPTER XI WHEN JOHNNY CAME MARCHING HOME On November 11, 1918, the armistice was signed and the fighting of the Great War ceased—almost as abruptly as it had begun. And the ebb tide of American troops from Europe back to the United States began; almost at once. For a time it was an almost imperceptible tide; in the following month but 75,000 soldiers all told—officers and enlisted men—were received through the port of New York, at all times the nation's chief war gateway; yet this was but the beginning. Each month of the early half of 1919 registered an increase of this human tide inflowing as against the preceding months, until May, with 311,830 troops received home, finally beat, by some 5,000 men, the record outgoing month of July, 1918, when under the terrific pressure induced by the continued German drive, 306,731 officers and men had been dispatched from these shores. Yet June, 1919, overtopped May. In that month 342,686 troops passed not only under the shadow of the beloved statue of Liberty, but also into the friendly and welcoming ports of Boston, Newport News, and Charleston, while the Secretary of War promised that the midsummer months that were immediately to follow would break the June record. A promise which was fulfilled. Long before the signing of the armistice, Pershing had ruled that the work of the American Red Cross with the well men of the A. E. F. was specifically to be limited to them while they were en route from one point to another—along the lines of communication, as you already have seen in an earlier chapter. To the Young Men's Christian Association was intrusted the chief burden of caring for them in their more or less permanent camps. This meant for our Red Cross in the final months of the war—before peace was actually signed and
  • 63. declared—a task almost exactly like that which had confronted it in its very first months of war experience in France. The stations along the railroad lines of eastern France, Luxembourg, and the Moselle Valley—the lines of communication between our French base ports and the occupied districts of the German states—offered to the American Red Cross the very same canteen problems as had once faced it at Châlons-sur-Marne and Épernay. Treves and Coblenz were hardly different from either of these—save perhaps in their increased size. Because Coblenz is rather more closely connected in the mind of the average American with our Army of Occupation, let us begin with it, here and now. It was, in fact, the easternmost outpost of the work of our Red Cross with our army over there. There the lines of communication officially began, and ran up the railway which ascends the beautiful but extremely tortuous valley of the Moselle. And where the lines of communication began—in the great railroad station of Coblenz—the American Red Cross also began. It had two canteens in that station; one just off the main waiting room, and the other, for the convenience of troops who were merely halted in the train shed of the station while going to and from the other American mobilization centers in that Rhine bridgehead, right on the biggest and the longest of the train platforms. Both were busy canteens; never more so, however, than just before 10:30 o'clock in the morning, which was the stated hour for the departure of the daily leave-train toward the border lines of France. Then it was the Red Cross coffee and sandwiches, tobacco and chewing gum were in greatest demand; for the long leave-train boasted no such luxury as dining cars, and there was scarce enough time at the noonday stop at Treves for one to avail oneself of the lunch-room facilities in the station there. Yet Treves for the American Red Cross was a far, far more important point than Coblenz. It was the headquarters of all its work in Germany, and boasted in addition to the large American Red Cross canteens in each of the two railroad stations, on either bank of the
  • 64. Moselle, and the recreation huts at the base hospitals—for that matter, there were also recreation huts at the base hospitals in and about Coblenz—well-equipped clubs for both enlisted men and officers. Of these the club for the enlisted men—for the rank and file of doughboy—quite properly was the best equipped. In the beginning it had been one of those large combination beer gardens and music halls that always have been so very dear to the heart of the German. It was the very sort of plant that could be, and was, quickly adapted to the uses of a really big group of men. Its main bierhalle made a corking dining room for the doughboys. The meals kept pace with the apartment. Three times a day they appeared—feeding daily from 600 to 1,600 boys—and they were American meals—in fact, for the most part composed of American food products—meats from Chicago, butter and cheese from New York State, flour from Minnesota, and the like. For each of these a flat charge of two marks—at the rate of exchange then prevailing, about eighteen cents—was made. But if a doughboy could not or would not pay, no questions were asked. The Treves Enlisted Men's Club which the American Red Cross gave the A. E. F. was not a commercial enterprise. It was run by an organization whose funds were the gift of the American people—given and given freely in order that their boys in khaki might have every comfort that money might provide. The great high-ceilinged halle held more than a restaurant. It was a reading room as well, stocked with many hundreds of books and magazines. In fact a branch of the American Library Association operated—and operated very successfully—a small traveling loan library in one of the smaller rooms of the club. Upon the walls of the vast room were pictures and many maps—maps of the valley of the Moselle, of that of the Rhine, of the Saar basin, of the operations in France. These last held much fascination for the doughboys. The most of them were of divisions which had led in the active and hard fighting, and the tiny flags and the blue-chalk marks on the operation maps were in reality placed there by their own efforts—but
  • 65. a few weeks and months before. It was real fun to fight the old actions over and over again—this time with talk and a pointing stick. There were, of course, such fundamental conveniences for roaming doughboys as baths, a bootblack and a barber shop—this last equipped with chairs which the boys themselves invented and constructed; a plain stout wooden armchair, into the back of which a board—not unlike an old-fashioned ironing board—was thrust at an angle. When turned one way this board formed just the proper headrest for a shave; in the other direction it was at exactly the right angle for haircutting. For the Officers' Club of our Red Cross at Treves, the Casino in the Kornmarkt, the heart of the city, was taken over. The fact that this was in the beginning a well-equipped club made the problem of its adaption a very slight one indeed. And the added fact that officers require, as a rule, far less entertainment than the enlisted men also simplified its operation. As it was, however, the officers were usually given a dance or a show each week—in the comfortable, large hall of the Casino. In the Enlisted Men's Club there was hardly a night, however, without some sort of an entertainment in its halle; and the vast place packed to the very doors. The next stop after Treves in the eastbound journey from the Rhine of the man in khaki was usually Nancy. And here there were not only canteen facilities at the railroad station, but a regular Red Cross hotel—situated in the Place Stanislas, in the very heart of the town. In other days this had been the Grand Hotel, and the open square that it faced has long been known as one of the handsomest in all France. In fact, Nancy itself is one of the loveliest of all French towns; and despite the almost constant aërial bombardments that were visited upon it, escaped with comparatively minor damage.
  • 66. NEVER SAY DIE Sorely wounded, our boys at the great A. R. C. field hospital in the Auteuil race track outside of Paris, kept an active interest in games and sports The Red Cross hotel there was opened on September 30, 1918, and closed on the tenth of April of the following spring—had eighty-eight rooms, capable of accommodating one hundred guests, and two dormitories capable of providing for some forty more. The room charges were invariably five francs for a room—with the exception of one, usually reserved for generals or other big wigs—which rented at eight francs a night. For the dormitory beds an even charge of two francs (forty cents) nightly was made, while in the frequent event of all these regular accommodations of the hotel being engaged and the necessity arising of placing cots in its broad hallways, no charge whatsoever was made for these emergency accommodations. For the excellent meals—served with the fullness of a good old- fashioned Yankee tavern—a progressive charge of four francs for breakfast, five francs for lunch, and six francs for dinner was made. Surely no one could fairly object to the restaurant prices, which, even in France in war-time stress, ranged from eighty cents to a dollar and twenty! In fact it was a bonanza for the American officers who formed the chief patrons of the place—although a bit of
  • 67. thoughtfulness on the part of some one had provided this particular hostelry with a dormitory of twelve beds and a single room with three which was held reserved for American women war workers; an attention which was tremendously appreciated by them. Eleven miles distant from Nancy was Toul; but Toul we have already visited in the pages of this book. We know already the comfortable accommodations that the traveler in khaki found in the group of hotels and canteens which our Red Cross operated there. There were many of these, even outside of Paris; one of the largest the tavern at the badly overcrowded city of Bordeaux. That tavern had been little to boast of, in the beginning. It was an ancient inn indeed; but good taste—the purchase of some few dozen yards of cretonne, and cleanliness—the unrelenting use of mop and broom and soap—had accomplished wonders with it. There were others of these American Red Cross hotels in France during the fighting period —the ones at Dijon, Is-sur-Tille, and Marseilles were particularly popular. But it was in Paris itself that the Red Cross accommodations for the itinerant doughboy in the final months of the war, as in the long and difficult half year that intervened between the signing of the armistice and the signing of peace, reached their highest development. In the beginning these had taken form in canteens which were operated night and day at each of the important railroad stations. These were all right—so far as they went. Their one-franc or seventy-five centime meals were wonderful indeed. I have eaten in these canteens many times myself—and always eaten well. I have been seated between a doughboy from North Carolina and one from North Dakota and been served by a society woman in steel-gray uniform—a woman whose very name was a thing to be emblazoned in the biggest headline type of the New York newspapers, but who was working week in and week out harder than the girls in busy restaurants back home are usually wont to work.
  • 68. If you would see these canteens as they really worked, gaze upon them through the eyes of a brilliant newspaper woman from San Francisco, who took the time and the trouble to make a thorough study of them. She wrote: A brown puddle of coffee was spreading over the white oilcloth. The girl from home sopped it up with her dish towel. She brushed away messy fragments of food and bread crumbs. Again there were few vacant places for American soldiers on the benches at the long table in the canteen at the Gare St. Lazare. The canteen, one of a circuit of thirteen maintained by the Red Cross in Paris, had formerly been the corner of a baggage room in one of the most important Paris terminals. The concrete floor bruised her feet. She was as conscious of them as Alice in Wonderland who discovered her own directly beneath her chin after she nibbled the magic toadstool. The girl was tired, but she smiled. It was really a smile within a smile. There was one on her lips which seemed to sparkle and glance, waking responsive smiles on the faces of the men. At once the gob who was born down in Virginia and had trained at Norfolk, decided that she was from his own South. The six-foot doughboy from California knew that she came from some small town in the Sierras. To each of the men she suddenly represented home. That smile stays in place each day until she reaches her room in a pension across the Seine on the Rue Beaux Arts. There, closing the door upon the world with its constant pageant of uniformed men who seem forever hungry and thirsty, she lets her smile fade away for the first time that day. The smile within is tucked away in her heart with the memory of agonizing moments aboard an ocean liner when she felt her exalted desire for service ebbing away because she feared she would not be needed. Needed! Now she wonders who else could have managed so tactfully the boy who had been at sea for one year and
  • 69. discovered that he had forgotten how to talk to an American woman. His diffidence was undermined with another dish of rice pudding and an extra doughnut. He became a regular boarder at the canteen where breakfast costs nine cents and any other man's size meal may be had for thirteen cents. His leave ended in a half day of excited shopping for which his younger sister will always be grateful. The girl from home had been one of those solemn creatures who was called to the Overseas Club in New York for service abroad. She was one of hundreds who had clinched their own faith in their ideals by pledging such service. It had been a wrench, saying good-bye at the station in the Middle West. There were no boys in the family, and her father had made a funny little joke which betrayed his pride about 'hanging out a service flag now.' Armed with interminable lists which called for supplies for twelve months, she bought her equipment. All the time she was saying to herself: 'I am ready to give all of my youth and my strength to the cause and to hasten victory.' Then the armistice was signed. The wireless instrument sang with the message. There was a celebration. The ship remained dark, still sliding through the nights warily, but her next trip would be made with decks ablaze and portholes open. The war was ended. It seemed to the girl that in the silence of the aftermath she could hear once more the wings of freedom throbbing above the world. She was glad and she was sorry. Her fear was that after all the Red Cross would not need her because she came too late. Canteen service—she pictured the work minus the tonic of danger as a social job. Dressed in a blue smock and white coif she would bid a graceful farewell to the A. E. F. as it filtered out of Europe. Now she smiles. Needed? Her fingers are scarred and she wonders if she ever will be able to pour one thousand bowls of coffee from the gigantic white porcelain pitcher without blistering her hands.
  • 70. Each day she looks at the line of men jostling one another at the door. She listens to their interminable questions and comes to the full realization that she is one of the most important people in Paris, one of two hundred girls feeding thirty-five thousand soldiers daily. As some workers leaving for home after more than a year of service tell of making sandwiches under shell fire, of sleeping by the roadside in the woods to fool the boche flyers who bombed the Red Cross buildings, she still feels the sly nip of envy. But soldiers do not cease to be soldiers and heroes when the war is done. Other puddles formed on the table and she mopped them up. She had used three towels during her eight-hour shift. A soldier, one of the thousands passing daily through the six Paris stations on their way home, journeying to leave areas, going to join the Army of Occupation or assigned to duty in the city, called to her. 'Sister, I want to show you something,' he said, and unwrapped a highly decorative circlet of aluminum. It was a napkin ring which he had bought from a poilu who made it of scraps from the battlefield. There was an elaborate monogram engraved on a small copper shield. 'For my mother,' he explained. 'If you don't think it is good enough I will get something else.' At once fifty rival souvenirs were produced. Men came from other tables to exhibit their own. There was the real collector who bemoaned the theft of a 'belt made by a Russian prisoner in Germany and decorated with the buttons of every army in the world including the fire department of Holland.' One of the new arrivals had hands stiffened from recently healed wounds. She brought his plate of baked beans, roast meat, potatoes, a bowl of coffee, and pudding. A young Canadian with flaming, rosy cheeks divided the last doughnut with his friend, the Anzac. Crullers are the greatest influence in canteen for the general friendliness among soldiers of different armies. A League of Nations
  • 71. could be founded upon them if negotiations were left to the privates about the oilcloth-covered tables. The boy with the crippled hands protested that he did not want to accept a dinner for which there was so little charge. 'Say, Miss,' he said, 'I can pay more. I don't have to be sponging.' 'You have folks in the states?' she asked. He had. 'Then,' she explained, 'they are the ones who support the American Red Cross. When you come here it is because the folks asked you in to dinner.' 'But I haven't any folks,' announced a sailor. 'I'm from the States, so I am your folks,' she retorted, 'and the Red Cross is your folks. We invite you to three meals a day as long as you stay in Paris.' 'You are my folks,' said the boy who was only a youngster, 'and you sure look like home to me.' The soldier with the crippled hands wanted to describe his wounds. Like hundreds of others he began with the sensations in the field, 'when he got his.' Deftly as she had learned to do during hundreds of such recitals, she cleaned up the table and stacked the plates without seeming to interrupt. It was three o'clock, the end of her day. She had reported at seven in the morning. The following week she would report with the other members of the staff at eleven at night because the doors of a canteen must never be closed. The boy talked on. He was explaining homesickness, the sort which drives men from cafés where the food is unfamiliar and the names on the menus cannot be translated into 'doughboy French' to such places as the little room in the Gare St. Lazare. She discovered that her habitual posture was with arms akimbo and hands spread out over her hips. This position seemed to rest the
  • 72. ache in her shoulders. Through her memory flashed pictures of waitresses in station eating houses who stood that way while tourists fought for twenty minutes' worth of ham and eggs between trains. Red Cross after-war canteens were a social center for pretty idlers in smart blue smocks? The smile on her lips never faltered and the hidden smile in her heart became a little song of laughter. She was 'helping'—helping in an 'eating joint,' some of the boys called it. But it was an eating joint with a soul. What more could one ask of an eating-house? From the canteen at the railroad terminals—which were all right so far as they went—it was an easy step of transition to the establishment of hotels for the enlisted men in the accessible parts of Paris—until there was a total of six of these last, in addition to the five railway station canteens—at Gare St. Lazare, Gare du Nord, Gare d'Orsay, Gare d'Orléans, and Gare Montparnasse. The winter- time hotels were in the Avenue Victor Emanuel, Rue Traversière, Rue la Victoire, Rue St. Hyacinthe, and the Rue du Bac. These were all, in the beginning, small Parisian taverns of the pension type, which were rather quickly and easily adapted to their war-time uses. The great difficulty with the first five of these American Red Cross doughboy hotels was their extreme popularity. They could hardly keep pace with the demands made upon them—in the last weeks that preceded and immediately following the signing of the armistice; while, with the coming of springtime and the granting of wholesale leaves of absence by the army, an immediate and most pressing problem confronted the American Red Cross in Paris. The boys were coming into the town—almost literally in whole regiments, and the provisions for their housing and entertainment there were woefully inadequate—to say the least. Not only were these
  • 73. accommodations, as furnished by the French, inadequate and poor, but the charges for them often were outrageous. Yet to furnish hotel accommodations in the big town, even of the crudest sort, for a thousand—perhaps two thousand—doughboys a night was no small problem. There were no more hotels, large or small, available for commandeering in Paris; the various allied peace commissions had completely exhausted the supply. Yet our Red Cross, accustomed by this time to tackling big problems—and the solution of this was, after all, but part of the day's work, and because there were no more hotels or apartment houses or dormitories or barracks of any sort whatsoever available in the city of more than two million folks—our Red Cross decided to build a hotel. And so did—almost overnight. It was a summer hotel, that super-tavern for our doughboys, and it stood squarely in the center of that famous Parisian playground, the Champs de Mars—and almost within stone throw of the Eiffel Tower and the Ecole Militaire. To create it several dozen long barracks—like American Red Cross standard khaki tents—were erected in a carefully planned pattern. Underneath these were builded wooden floors and they were furnished with electric lights and running water. A summer hotel could not have been more comfortable; at least few of them are. The Tent City, as it quickly became known, was opened about March 4, 1919, with bed accommodations for 1,400 men, while preparations were quickly made to increase this capacity by another five hundred, for the latest and the biggest of American Red Cross hotels in Paris had leaped into instant popularity. Between six and nine-thirty in the morning and ten-thirty and midnight in the evening, the boys would come streaming in to the registry desk, like commercial travelers into a popular hostelry in New York or Philadelphia or Chicago. They would sleep—perhaps for the first time in many, many months—in muslin sheets. And these were as immaculate as those of any first-class hotel in the States.
  • 74. There was no charge whatsoever for these dormitory accommodations. For the meals—simple but good and plentiful—the normal price of fifty centimes (nine or ten cents) was asked, but never demanded; while merely for the asking any of our boys in khaki could have at any hour the famous Red Cross sandwiches of ham or salmon or beef mixture or jam—chocolate or coffee or lemonade a-plenty to wash it down. Definite provision was made for their amusement; there were rubberneck wagons to take them afield to the wonderful and enduring tourist sights of Paris and her environs—and at the Tent City itself a plenitude of shows and dances as well as the more quiet comfort of books or magazines, or the privilege and opportunity of writing a letter home. Of what use these last in Paris? you ask. Your point is well taken. I would have taken it myself—before I first went to the Tent City. When I did it was a glorious April day, the sun shone with an unaccustomed springtime brilliancy over Paris, and yet the air was bracing and fit for endeavor of every sort. Yet the big reading room tent of the Red Cross hotel in the Champs de Mars was completely filled—with sailor boys or boys in khaki reading the books or paper most liked by them. The sight astonished me. Could these boys—each on a leave of but three short days—be blind to the wonders of Paris? Or was their favorite author particularly alluring that week? I decided to ask one of them about it. I saw Paris yesterday—Notre Dame, the Pantheon, Napoleon's Tomb, the Opera House, the Louvre, the Follies—the whole blame business. It's some hike. But I did it. An' to-day I'm perfectly satisfied to sit here and read these guys a-telling of how they would have fought the war. Of such was the nature of the American doughboy.
  • 75. Just as it was necessary at Treves and Bordeaux and elsewhere— because of the very volume of the problem—to separate his entertainment from that of his officers, so it became necessary to effect a similar solution in Paris; for the officer is quite as much a ward of our Red Cross as the doughboy, himself. And so early in the solution of this entire great problem a superb home in the very heart of Paris—the town residence of the Prince of Monaco at No. 4 Avenue Gabriel and just a step from the Place de la Concorde—was secured and set aside as an American Red Cross Officers' Club. Lovely as this was, and seemingly more than generous in its accommodations, these were soon overwhelmed by the demands placed upon them, and steps were taken toward finding a real officers' hotel for the men of the A. E. F. when they should come to Paris. These led to the leasing of the Hotel Louvre, at the head of the Avenue de l'Opéra and almost adjoining the Comédie Française, the American University Union, and the Louvre. After being rapidly redecorated and otherwise transformed to meet the necessities of the A. E. F. it was reopened on the sixth of January, 1919, as the American Officers' Hotel in charge of Mr. L. M. Boomer, the directing genius of several large New York hotels. Mr. Boomer brought to the Red Cross a great practical hotel experience, and the house under his management quickly attained an overwhelming success. It had, in the first instance, been charmingly adapted to its new uses. Its rather stiff and old-fashioned interior had been completely transformed; there was all through the building an indefinable but entirely unmistakable home atmosphere. Our American officers fairly reveled in it. Into this setting was placed good operation—a high-grade American- operated hotel, if you please, in the very heart of Paris and all her stout traditions. Petit déjeuners begone! They are indeed starvation diet for a hungry Yank. The breakfast in the American Officers' Hotel, which our Red Cross set up and operated, cost a uniform five francs
  • 76. (one dollar) and had the substantial quality of a regular up-and- doing tavern on this side of the Atlantic. Before we rest, here are three typical bills of fare of a single ordinary day in this A. R. C.-A. E. F. establishment. The day was the nineteenth of April, 1919, and the three meals were as follows: Breakfast Five Francs—($1.00). Bananas Quaker Oats Eggs and Bacon Griddle Cakes with Sirup Confiture Coffee, Cocoa, or Chocolate Luncheon Eight Francs—($1.60). Oyster Soup, with Okra Scollops of Veal, Dewey Nouilles, Milanaise Cold Meats, with Jelly Russian Salad Assorted Eclairs Raspberry Ice Cream Coffee Dinner Ten Francs—($2.00). Crème St. Cloud Rouget Portugaise
  • 77. Roasted Filet of Beef, Cresson Pommes Château Endive Flamandes Salade de Saison Candied Fruits Coffee Ice Cream Coffee Yet the charm of the American Officers' Hotel in Paris rested not alone in the real excellence of its cuisine, nor in the comfort of its cleanly sleeping rooms. It carried its ideals of genuine service far beyond these mere fundamentals. It recognized the almost universal Yankee desire to have one's shoes shined in a shop and so set up a regular American boot-blacking stand in one of its side corridors, a thing which every other Parisian hotel would have told you was quite impossible of accomplishment. It recognized the inconvenience of tedious waiting and long queues at the box office of the Paris theaters by setting up a theater ticket office in its lobby, which made no extra charge for the distinct service rendered. Nor was there a charge for the services of Miss Curtis, the charming little Red Cross girl, who went shopping with a fellow or for him, and who had a knack of getting right into those perplexing Paris shops and getting just what a fellow wanted at an astonishingly low price—for Paris in war times, anyway. Her range of experience was large; from the man with a silver star on each shoulder who wanted to buy a modish evening gown for his wife at a price not to exceed forty dollars, to the chunky Nevada lieutenant who had won three thousand francs at redeye on the preceding evening and was anxious to blow it all in the next morning in buying souvenirs for mother. With both she did her best. Her motto was that of the successful shop keeper: We aim to please. When Mr. Boomer had this hotel set up and running and turned his attention to some other housing problems of our Red Cross, the management fell to Major H. C. Eberhart, who had been his assistant in Paris and before that had been affiliated in a managerial
  • 78. capacity with several large American houses. He carried forward the job so well begun. With the slow but very sure movement of our doughboys back from eastern France and Germany toward the base ports along the westerly rim of France, where they were embarking in increasing numbers for the blessed homeland, it became necessary for General Pershing to establish concentration areas, or reservoir camps, well back from the Atlantic Coast but convenient to it. By far the largest and most important of these was in the neighborhood of the city of Le Mans, some one hundred and fifty miles southwest of Paris, which meant in turn that what was finally destined to be the largest of the canteens of our American Red Cross in France outside of Paris was the final one established. It was known as the American Red Cross Casual Canteen and, situated within three blocks to the east of the railroad station at Le Mans, was a genuine headquarters for all the American soldiers for ten or fifteen or twenty miles roundabout. And in the bare chance that there might not be a doughboy who had chanced to hear of it, it was well indicated—by day, by a huge sign of the crimson cross, and by night that emblem blazing forth in all the radiance of electricity. When the doors were finally opened—about the middle of March, 1919—there were sleeping quarters under its hospitable roof for 250 enlisted men and forty officers. In the canteen portion of the establishment, 200 men could be served at a single sitting; in all 500 at each of the three meals a day. The comforts of this place almost approximated those of a hotel. When the men rose from their beds in the morning—clean sheets and towels and pillowcases, of course, even though it did mean that the Red Cross had to establish its own laundry in the establishment—they could step, quickly and easily, into a commodious washroom and indulge, if they so chose, in a
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