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15.3: Untitled Page 235

  • Page ID
    18368
  • Chapter 9

    k IV

    Elementary chemical kinetic schema IV:

    NO

     NO

    

     N O

    (5)

    2

    3

    2

    5

    Since neither NO nor NO appear in the global stoichiometric schema given by 3

    Eq. 1, one can assume that these two compounds are reactive intermediates or Bodenstein products and their rates of production can be set equal to zero as an approximation

    9‐15. In Problem 9‐14 we considered the decomposition of N O to produce 2

    5

    NO and O by means of the kinetic schemata illustrated by Eqs. 2 through 5 in 2

    2

    Problem 9‐14. The reactive intermediates were identified as NO and NO . The 3

    rate equations for these two reactive intermediates and for N O are given by 2

    5

    R

    k c

    k c

    c

    k c

    c

    k c

    c

    (1)

    NO3

    I N2O5

    II NO3 NO2

    III NO NO3

    IV NO2 NO3

    R

    k c

    c

    k c

    c

    (2)

    NO

    II NO3 NO2

    III NO NO3

    R

      k c

    k c

    c

    (3)

    N2O5

    I N2O5

    IV NO2 NO3

    If the condition of local reaction equilibrium is imposed according to Local reaction equilibrium:

    R

     0 ,

    R

     0

    (4)

    NO3

    NO

    one can obtain a simple representation for R

    in terms of only c

    . In this

    N2O5

    N2O5

    problem you are asked to replace the assumptions given by Eqs. 4 with restrictions indicating that R

    and R

    are small enough so that Eqs. 4 become acceptable

    NO3

    NO

    approximations.

    Section 9.4

    9‐16. If species C in Eq. 9‐88 is a reactive intermediate or Bodenstein product, identify the stoichiometric matrix and the Bodenstein matrix associated with Eq. 9‐88. Impose the condition of local reaction equilibrium on the Bodenstein product in order to derive an expression for R in terms of c , c and c .

    E

    A

    B

    D

    Appendix A

    A1 Atomic Mass of Common Elements Referred to Carbon‐12

    (Details are available at http://www.nist.gov/physlab/data/comp.cfm)

    Element

    Symbol

    Atomic mass, g/mol

    Aluminum

    Al

    26.9815

    Antimony

    Sb

    121.75

    Argon

    Ar

    39.948

    Arsenic

    As

    74.9216

    Barium

    Ba

    137.34

    Beryllium

    Be

    9.0122

    Bismuth

    Bi

    208.980

    Boron

    B

    10.811

    Bromine

    Br

    79.904

    Cadmium

    Cd

    112.40

    Calcium

    Ca

    40.08

    Carbon

    C

    12.01

    Cerium

    Ce

    140.12

    Cesium

    Cs

    132.905

    Chlorine

    Cl

    35.453

    Chromium

    Cr

    51.996

    Cobalt

    Co

    58.9332

    Copper

    Cu

    63.546

    Fluorine

    F

    18.9984

    Gallium

    Ga

    69.72

    Germanium

    Ge

    72.59

    Gold

    Au

    196.967

    Hafnium

    Hf

    178.49

    Helium

    He

    4.0026

    Hydrogen

    H

    1.00797

    Indium

    In

    114.82

    Iodine

    I

    126.904

    Iridium

    Ir

    192.2

    Iron

    Fe

    55.847

    Krypton

    Kr

    83.80

    439

    440

    Appendices

    Element

    Symbol

    Atomic mass, g/mol

    Lead

    Pb

    207.19

    Lithium

    Li

    6.939

    Magnesium

    Mg

    24.312

    Manganese

    Mn

    54.938

    Mercury

    Hg

    200.59

    Molybdenum

    Mo

    95.94

    Neon

    Ne

    20.183

    Nickel

    Ni

    58.71

    Niobium

    Nb

    92.906

    Nitrogen

    N

    14.0067

    Oxygen

    O

    15.9994

    Palladium

    Pd

    106.4

    Phosphorus

    P

    30.9738

    Platinum

    Pt

    195.09

    Plutonium

    Pu

    242

    Potassium

    K

    39.102

    Radium

    Ra

    226

    Radon

    Rn

    222

    Rhodium

    Rh

    102.905

    Rubidium

    Rb

    85.47

    Selenium

    Se

    78.96

    Silicon

    Si

    28.086

    Silver

    Ag

    107.868

    Sodium

    Na

    22.9898

    Strontium

    Sr

    87.62

    Sulfur

    S

    32.064

    Tantalum

    Ta

    180.948

    Tellurium

    Te

    127.60

    Thallium

    Tl

    204.37

    Thorium

    Th

    232.038

    Tin

    Sn

    118.69

    Titanium

    Ti

    47.90

    Tungsten

    W

    183.85

    Uranium

    U

    238.03

    Vanadium

    V

    50.942

    Xenon

    Xe

    131.30

    Material Balances

    441

    Element

    Symbol

    Atomic mass, g/mol

    Yttrium

    Y

    88.905

    Zinc

    Zn

    65.37

    Zirconium

    Zr

    91.22

    442

    Appendices

    A2 Physical Properties of Various Chemical Compounds

    Name

    Formula

    Molecular

    Liquid @ T(K)

    T melting T boiling

    Mass, g/mol

    density (g/L)

    K

    K

    Argon

    Ar

    39.948

    83.8

    87.3

    Acetaldehyde

    C2H4O

    44.054

    778

    293

    150.2

    293.6

    Acetic acid

    C2H4O2

    60.052

    1049

    293

    289.8

    391.1

    Acetone

    C3H6O

    58.08

    790

    293

    178.2

    329.4

    Acetylene

    C2H2

    26.038

    192.4

    189.2

    Acrylic acid

    C3H4O2

    72.064

    1051

    293

    285

    414

    Ammonia

    NH3

    17.031

    195.4

    239.7

    Aniline

    C6H7N

    93.129

    1022

    293

    267

    457.5

    Benzaldehide

    C7H6O

    106.124

    1045

    293

    216

    452

    Benzene

    C6H6

    78.114

    885

    289

    278.7

    353.3

    Benzoic acid

    C7H6O2

    122.124

    1075

    403

    395.6

    523

    Bromine

    Br2

    159.808

    3119

    293

    266

    331.9

    1,2‐Butadiene

    C4H6

    54.092

    137

    284

    1,3‐Butadiene

    C4H6

    54.092

    164.3

    268.7

    n‐Butane

    C4H10

    58.124

    134.8

    272.7

    i‐Butane

    C4H10

    58.124

    113.6

    261.3

    n‐Butanol

    C4H10O

    74.123

    810

    293

    183.9

    390.9

    1‐Butene

    C4H8

    56.108

    87.8

    266.9

    i‐Butene

    C4H8

    56.108

    132.8

    266.3

    Carbon

    tetrachloride

    CCl4

    153.823

    1584

    298

    250

    349.7

    Carbon dioxide

    CO2

    44.01

    216.6

    194.7

    Carbon monoxide

    CO

    28.01

    68.1

    81.7

    Chlorine

    Cl2

    70.906

    172.2

    238.7

    Chlorobenzene

    C6H5Cl

    112.559

    1106

    293.

    227.6

    404.9

    Chloroform

    CHCl3

    119.378

    1498

    293

    209.6

    334.3

    Cyclobutane

    C4H8

    56.108

    182.4

    285.7

    Cyclohexane

    C6H12

    84.162

    779

    293

    279.7

    353.9

    Cyclohexanol

    C6H12O

    100.161

    942

    303

    298

    434.3

    Cyclopentane

    C5H10

    70.135

    745

    293

    179.3

    322.4

    Material Balances

    443

    Name

    Formula

    Molecular

    Liquid @ T(K)

    T melting T boiling

    Mass, g/mol

    density (g/L)

    K

    K

    Cyclopentene

    C5H8

    68.119

    772

    293

    138.1

    317.4

    Ethane

    C2H6

    30.07

    89.9

    184.5

    Ethanol

    C2H6O

    46.069

    789

    293

    159.1

    351.5

    Ethyl amine

    C2H7N

    45.085

    683

    293

    192

    289.7

    Ethyl acetate

    C4H8O2

    88.107

    901

    293

    89.6

    350.3

    Ethylbenzene

    C8H10

    106.168

    867

    293

    178.2

    409.3

    Ethylendiamine

    C2H8N2

    60.099

    896

    293

    284

    390.4

    Ethyl ether

    C4H10O

    74.123

    713

    293

    156.9

    307.7

    Ethyl propionate

    C5H10O2 102.134

    895

    293

    199.3

    372

    Ethylene

    C2H4

    28.054

    104

    169.4

    Ethylene Glycol

    C2H6O2

    62.069

    1,114

    293

    260.2

    470.4

    Ethylene oxide

    C2H4O

    44.054

    161

    283.5

    Fluorine

    F2

    37.997

    53.5

    85

    Formaldehide

    CH2O

    30.026

    156

    254

    Formic acid

    CH2O2

    46.025

    1,226

    288

    281.5

    373.8

    Glycerol

    C3H8O3

    92.095

    1,261

    293

    291

    563

    n‐Heptane

    C7H16

    100.205

    684

    293

    182.6

    371.6

    1‐Heptanol

    C7H16O

    116.204

    822

    293

    239.2

    449.5

    1‐Heptene

    C7H14

    98.189

    697

    293

    154.3

    366.8

    n‐Hexane

    C6H14

    86.178

    659

    293

    177.8

    341.9

    1‐Hexanol

    C6H14O

    102.177

    819

    293

    229.2

    430.2

    Hydrogen

    H2

    2.016

    14.0

    20.4

    Hydrogen bromide HBr

    80.912

    187.1

    206.1

    Hydrogen chloride

    HCl

    36.461

    159.0

    188.1

    Hydrogen cyanide

    CHN

    27.026

    688

    293

    259.9

    298.9

    Hydrogen sulfide

    H2S

    34.08

    187.6

    212.8

    Iodine

    I2

    253.808

    3,740

    453

    386.8

    457.5

    Isopropyl alcohol

    C3H8O

    60.096

    786

    293

    184.7

    355.4

    Maleic anhydride

    C4H2O3

    98.058

    1,310

    333

    326

    472.8

    Methane

    CH4

    16.043

    90.7

    111.7

    Mercury

    Hg

    200.59

    13,546

    293

    234.3

    630.1

    Methanol

    CH4O

    32.042

    791

    293

    175.5

    337.8

    Methyl acetate

    C3H6O2

    74.08

    934

    293

    175

    330.1

    444

    Appendices

    Name

    Formula

    Molecular

    Liquid @ T(K)

    T melting T boiling

    Mass, g/mol

    density (g/L)

    K

    K

    Methyl acrylate

    C4H7O2

    86.091

    956

    293

    196.7

    353.5

    Methyl amine

    CH5N

    31.058

    179.7

    266.8

    Methyl benzoate

    C8H8O2

    136.151

    1,086

    293

    260.8

    472.2

    Methyl ethyl ketone C4H8O

    72.107

    805

    293

    186.5

    352.8

    Naphtalene

    C10H8

    128.174

    971

    363

    353.5

    491.1

    Nitric oxide

    NO

    30.006

    109.5

    121.4

    Nitrogen

    N2

    28.013

    63.3

    77.4

    Nitrogen dioxide

    NO

    30.01

    112.2

    122.2

    Nitrogen tetroxide

    N2O4

    46.006

    261.9

    294.3

    Nitrous oxide

    N2O

    44.013

    182.3

    184.7

    Oxygen

    O2

    31.999

    54.4

    90.2

    n‐Pentane

    C5H12

    72.151

    626

    293

    143.4

    309.2

    1‐Pentanol

    C5H12O

    88.15

    815

    293

    195

    411

    1‐Pentyne

    C5H8

    68.119

    690

    293

    167.5

    313.3

    1‐Pentene

    C5H10

    70.135

    640

    293

    107.9

    303.1

    Phenol

    C6H6O

    94.113

    1059

    314

    313

    455

    Propane

    C3H8

    44.097

    85.5

    231.1

    1‐Propanol

    C3H8O

    60.096

    804

    293

    146.9

    370.4

    Propionic acid

    C3H6O2

    74.08

    993

    293

    252.5

    414

    Propylene

    C3H6

    42.081

    87.9

    225.4

    Propylene oxide

    C3H6O

    58.08

    829

    293

    161

    307.5

    Styrene

    C8H8

    104.152

    906

    293

    242.5

    418.3

    Succinic acid

    C4H6O4

    118.09

    456

    508

    Sulfur dioxide

    SO2

    64.063

    197.7

    263

    Sulfur trioxide

    SO3

    80.058

    1780

    318

    290

    318

    Toluene

    C7H8

    92.141

    867

    293

    178

    383.8

    Trimethyl amine

    C3H9N

    59.112

    156

    276.1

    Vinyl chloride

    C2H3Cl

    62.499

    119.4

    259.8

    Water

    H2O

    18.015

    998

    293

    273.2

    373.2

    o‐Xylene

    C8H10

    106.168

    880

    293

    248

    417.6

    m‐Xylene

    C8H10

    106.168

    864

    293

    225.3

    412.3

    p‐Xylene

    C8H10

    106.168

    861

    293

    286.4

    411.5

    Material Balances

    445

    A3 Constants for Antoine’s Equation

    o

    log p

    A B / (  T) , p is in mm Hg and T is in C

    vap

    vap

    Compound

    Formula

    A

    B

    

    Acetaldehyde

    CH3CHO

    7.0565

    1070.6

    236.0

    Acetic acid

    CH3COOH

    7.29963

    1479.02

    216.81

    Acetone

    CH3COCH3

    7.23157

    1277.03

    237.23

    Acetylene

    C2H2

    7.0949

    709.1

    253.2

    Acrylic acid

    C2H3COOH

    7.1927

    1441.5

    192.66

    Ammonia

    NH3

    7.36050

    926.132

    240.17

    Aniline

    C6H7N

    7.2418

    1675.3

    200.01

    Benzaldehide

    C7H6O

    7.1007

    1628.00

    207.04

    Benzene

    C6H6

    6.90565

    1211.03

    220.790

    Benzoic acid

    C7H6O2

    7.45397

    1820

    147.96

    1,2‐Butadiene

    C4H6

    7.1619

    1121.0

    251.00

    1,3‐Butadiene

    C4H6

    6.85941

    935.531

    239.554

    n‐Butane

    C4H10

    6.83029

    945.90

    240.00

    n‐Butanol

    C3H7CH2OH

    7.4768

    1632.39

    178.83

    i‐Butane

    C4H10

    6.74808

    882.80

    240.00

    n‐Butene

    C4H8

    6.84290

    926.10

    240.00

    i‐Butene

    C4H8

    6.84134

    923.200

    240.00

    Carbon

    tetrachloride

    CCl4

    6.8941

    1219.58

    227.17

    Chlorobenzene

    C6H5Cl

    6.9781

    1431.05

    217.56

    Chloroform

    CHCl3

    6.93707

    1171.2

    236.01

    Cyclobutane

    C4H8

    6.92804

    1024.54

    241.38

    Cyclohexane

    C6H12

    6.84498

    1203.526

    222.863

    Cyclopentane

    C5H10

    6.88678

    1124.16

    231.37

    Cyclopentene

    C5H8

    6.92704

    1121.81

    233.46

    Ethane

    C2H6

    6.80266

    656.40

    256.00

    Ethanol

    CH3CH2OH

    8.16290

    1623.22

    228.98

    Ethyl amine

    C2H7N

    7.38618

    1137.3

    235.86

    Ethyl acetate

    C4H8O2

    7.01455

    1211.9

    216.01

    Ethene (Ethylene)

    C2H4

    6.74756

    585.00

    255.00

    446

    Appendices

    Compound

    Formula

    A

    B

    

    Ethylbenzene

    C6H5C2H5

    6.95719

    1424.55

    213.206

    Ethylenediamine

    C2H8N2

    7.12599

    1350.0

    201.03

    Ethyl ether

    C4H10O

    6.98467

    1090.64

    231.21

    Ethyl propionate

    C5H10O2

    7.01907

    1274.7

    209.0

    Ethylene glycol

    C2H6O2

    8.7945

    2615.4

    244.91

    Formaldehyde

    HCHO

    7.1561

    957.24

    243.0

    Formic acid

    HCOOH

    7.37790

    1563.28

    247.06

    Glycerol

    C3H8O3

    7.48689

    1948.7

    132.96

    n‐ Heptane

    C7H16

    6.90240

    1268.115

    216.900

    1‐Heptanol

    C7H16O

    6.64766

    1140.64

    126.56

    1‐Heptene

    C7H14

    6.90068

    1257.5

    219.19

    n‐ Hexane

    C6H14

    6.87776

    1171.530

    224.366

    Hydrogen bromide HBr

    6.28370

    539.62

    225.30

    Hydrogen chloride

    HCl

    7.167160

    744.49

    258.704

    Hydrogen cyanide

    HCN

    7.17185

    1123.0

    236.01

    Hydrogen sulfide

    H2S

    6.99392

    768.13

    247.093

    Iodine (c)

    I2

    9.8109

    2901.0

    256.00

    Isopropyl alcohol

    C3H8O

    8.11822

    1580.92

    219.62

    Maleic anhydride

    C4H2O3

    7.06801

    1635.4

    191.01

    Methane

    CH4

    6.61184

    389.93

    266.00

    Methanol

    CH3OH

    8.07246

    1574.99

    238.86

    Methyl acetate

    C3H6O2

    7.00495

    1130.0

    217.01

    Methyl acrylate

    C4H7O2

    6.99596

    1211.0

    214.01

    Methyl amine

    CH5N

    7.49688

    1079.15

    240.24

    Methyl benzoate

    C8H8O2

    7.04738

    1629.4

    192.01

    Methyl ethyl ketone C4H8O

    7.20868

    1368.21

    236.51

    Napthalene

    C10H8

    6.84577

    1606.5

    187.227

    Nitric oxide

    NO

    8.74300

    682.94

    268.27

    Nitrogen tetroxide

    N2O4

    7.38499

    1185.72

    234.18

    Nitrous oxide

    N2O

    7.00394

    654.26

    247.16

    n‐ Pentane

    C5H12

    6.85221

    1064.63

    232.000

    1‐Pentanol

    C5H12O

    7.17758

    1314.56

    168.16

    1‐Pentyne

    C5H8

    6.96734

    1092.52

    227.19

    1‐Pentene

    C5H10

    6.84648

    1044.9

    233.53

    Material Balances

    447

    Compound

    Formula

    A

    B

    

    Phenol

    C6H5OH

    7.13457

    1516.07

    174.569

    Phosphorus

    trichloride

    PCl3

    6.8267

    1196

    227.0

    Phosphine

    PH3

    6.71559

    645.512

    256.066

    Propane

    C3H8

    6.82973

    813.20

    248.00

    1‐Propanol

    CH3CH2CH2OH

    6.79498

    969.27

    150.42

    Propionic acid

    C3H6O2

    7.57456

    1617.06

    205.68

    Propene

    (Propylene)

    C3H6

    6.81960

    785.00

    247.00

    Propylene oxide

    C3H6O

    6.65456

    915.31

    208.29

    n‐Propionic

    Acid

    CH3CH2COOH

    7.54760

    1617.06

    205.67

    Styrene

    C8H8

    6.95709

    1445.58

    209.44

    Sulfur dioxide

    SO2

    7.28228

    999.900

    237.190

    Sulfur trioxide

    SO3

    9.05085

    1735.31

    236.50

    Toluene

    C6H5CH3

    6.95464

    1344.800

    219.482

    Trimethyl amine

    C3H9N

    6.97038

    968.7

    234.01

    Vynil chloride

    C2H3Cl

    6.48709

    783.4

    230.01

    Water

    H2O

    7.94915

    1657.46

    227.03

    o‐Xylene

    C6H5(CH3)2

    6.99891

    1474.679

    213.686

    p‐Xylene

    C6H5(CH3)2

    6.99052

    1453.430

    215.307

    448

    Appendices

    Appendix B

    Iteration Methods

    B1. Bisection method

    Given some function of x such as H( x) , we are interested in the solution of the equation

    H( x)  0 ,

    x x

    (B‐1)

    Here we have used x to represent the solution. For simple functions such as

    H( x)  x b we obtain a single solution given by x b , while for a more complex function such as

    2

    H( x)  x b we obtain more that one solution as

    indicated by x   b . In many cases there is no explicit solution for Eq. B‐1. For example, if H( x) is given by

    H( x)  a sin ( x 2 )  b cos(2 x

     )

    (B‐2)

    we need to use iterative methods to determine the solution x x .

    The simplest iterative method is the bisection method (Corliss, 1977) that is illustrated in Figure B‐1. This method begins by locating x and x such that o

    1

    H( x ) and H( x ) have different signs. In Figure B‐1 we see that x and x have o

    1

    o

    1

    been chosen so that there is a change of sign for H( x) , i.e., H( x )  0 ,

    H( x )  0

    (B‐3)

    o

    1

    Thus if H( x) is a continuous function we know that a solution H( x )  0 exists somewhere between x and x . We attempt to locate that solution by means of o

    1

    a guess (i.e., the bisection) indicated by

    x x

    o

    1

    x

    2

    (B‐4)

    2

    As illustrated in Figure B‐1, this guess is closer to the solution, x x , than either x or x , and if we repeat this procedure we will eventually find a value of x o

    1

    index-458_1.png

    index-458_2.png

    index-458_3.png

    index-458_4.png

    Material Balances

    449

    Figure B‐1. Illustration of the bisection method

    that produces a value of H( x) that is arbitrarily close to zero. In terms of the particular graph illustrated in Figure B‐1, it is clear that x will be located 3

    between x and x ; however, this need not be the case. For example, in 1

    2

    Figure B‐2 we have represented a slightly different function for which x will be 3

    located between x and x . The location of the next guess is based on the idea o

    2

    that the function H( x) must change sign. In order to determine the location of the next guess we examine the products H( x )  H( x

    ) and H( x )  H( x

    ) in

    n

    n1

    n

    n2

    order to make the following decisions:

    n

    x

    n

    x 1

    if H( x ) H(

    n

    n

    x 1)

    0 ,

    then

    n

    x 1

    2

    (B‐5)

    n

    x

    n

    x 2

    if H( x ) H(

    n

    n

    x 2)

    0 ,

    then n

    x 1

    2

    Since these two choices are mutually exclusive there is no confusion about the next choice of the dependent variable. The use of Eqs. B‐5 is crucial when the details of H( x) are not clear, and a program is written to solve the implicit equation.

    index-459_1.png

    index-459_2.png

    index-459_3.png

    index-459_4.png

    450

    Appendices

    Figure B‐2. Alternate choice for the second bi‐section B2. False position method

    The false position method is also known as the method of interpolation (Wylie, 1951) and it represents a minor variation of the bisection method. Instead of bisecting the distance between x and x in Figure B‐1 in order to locate the o

    1

    point x , we use the straight line indicated in Figure B‐3. Sometimes this line is 2

    called the secant line. The definition of the tangent of the angle  provides 0  H( x )

    H( x )  H( x )

    1

    o

    1

    tan  

    (B‐6)

    x

    2

    1

    x

    o

    x

    1

    x

    and we can solve for x to obtain

    2

    x x H( x )

    o

    1 

    1

    x

    2

    1

    x

    (B‐7)

    H(

    o

    x )

    H( 1

    x )

    This replaces Eq. B‐4 in the bisection method and it can be generalized to obtain

    x x

    H( x

    )

    n

    n1 

    n 1

    n

    x 2

    n

    x 1

    (B‐8)

    H( x )  H(

    n

    n

    x 1)

    index-460_1.png

    index-460_2.png

    index-460_3.png

    index-460_4.png

    Material Balances

    451

    Application of successive iterations will lead to a value of x that approaches x shown in Figure B‐3.

    Figure B‐3. False position construction

    B3. Newton’s method

    Newton’s method (Ypma, 1995), which is also known as the Newton‐Raphson method, is named for Sir Isaac Newton and is perhaps the best known method for finding roots of real valued functions. The method is similar to the false position method in that a straight line is used to locate the next estimate of the root of an equation; however, in this case it is a tangent line and not a secant line.

    This is illustrated in Figure B‐4a where we have chosen x as our first estimate of o

    the solution to Eq. B‐1 and we have constructed a tangent line to H( x) at x . The o

    slope of this tangent line is given by

    dH

    H( x )  0

    o

    (B‐9)

    dx

    x x

    o

    x

    1

    x

     o

    and we can solve this equation to produce our next estimate of the root. This new estimate is given by

    index-461_1.png

    index-461_2.png

    index-461_3.png

    index-461_4.png

    452

    Appendices

    H( x )

    o

    x

    x

    (B‐10)

    1

    o

    ( dH dx) x o x

    and we use this result to determine H( x ) as indicated in Figure B‐4a.

    1

    Figure B‐4a. First estimate using Newton’s method

    Given H( x ) and x we can construct a second estimate as indicated in 1

    1

    Figure B‐4b, and this process can be continued to find the solution given by x .

    The general iterative procedure is indicated by

    H( x )

    n

    x

    x

    ,

    n  0 , 1 , 2 ,...,

    (B‐11)

    n1

    n

    ( dH dx) xnx

    Newton’s method is certainly an attractive technique for finding solutions to implicit equations; however, it does require that one know both the function and its derivative. For complex functions, calculating the derivative at each step in the iteration may require more effort than that associated with the bisection method or the false position method. In addition, if the derivative of the function is zero in the region of interest, Newton’s method will fail.

    index-462_1.png

    index-462_2.png

    index-462_3.png

    index-462_4.png

    Material Balances

    453

    Figure B‐4b. Second estimate using Newton’s method

    B4. Picard’s method

    Picard’s method for solving Eq. B‐1 begins by defining a new function according to Definition:

    f ( x)  x H( x)

    (B‐12)

    Given any value of the dependent variable, x , we define a new value, x

    , by

    n

    n1

    Definition:

    x

    f ( x ) ,

    n

    0 , 1 , 2 , 3 ,...

    (B‐13)

    n

    1

    n

    This represents Picard’s method or the method of direct substitution or the method of successive substitution. If this procedure converges, we have

    f ( x )  x H( x )  x (B‐14)

    In Eq. B‐13 we note that the function f ( x ) , maps the point x to the new point n

    n

    x

    . If the function f x maps the point x to itself, i.e., f ( x)  x , then x is n

    ( )

    1

    called the fixed point of f ( x) . In Figure B‐5 we again consider the function represented in Figures B‐1, B‐3 and B‐4, and we illustrate the functions f ( x) , y( x)

    and H( x) . The graphical representation of the fixed point, x , is the intersection

    index-463_1.png

    index-463_2.png

    index-463_3.png

    index-463_4.png

    index-463_5.png

    454

    Appendices

    Figure B‐5. Picard’s method

    of the function of f ( x) with the line y x . Note that not all functions have fixed points. For example if f ( x) is parallel to the line y x there can be no intersection and no fixed point. Given our first estimate, x , we use Eq. B‐13 to o

    compute x according to

    1

    x

    f ( x )

    (B‐15)

    1

    o

    Clearly x is further from the solution, x , than x and we can see from the 1

    o

    graphical representation in Figure B‐5 that Picard’s method diverges for this case.

    If x were chosen to be less than the solution, x , we would also find that the o

    iterative procedure diverges. If the slope of f ( x) were less than the slope f o

    y( x) , we would find that Picard’s method converges. This suggests that the method is useful for “weak” fu c

    n tions of x , i.e., df dx  1 and this is confirmed in Sec. B6.

    index-464_1.png

    index-464_2.png

    index-464_3.png

    index-464_4.png

    Material Balances

    455

    B

    5. Wegstein’s method

    In Figure B‐6 we have

    strated

    illu

    the same function, f ( x) , that appears in Figure B‐5. For some point x in the neighborhood of x we can approximate

    1

    o

    Figure B‐6. Wegstein’s method

    the derivative of f ( x) according to

    df

    f ( x )  f ( x )

    1

    o

     slope  S

    dx

    (B‐16)

    1

    x

    o

    x

    and we can use this result to obtain an approximation

    for the function f ( x ) .

    1

    f ( x ) 

    f ( x )  S x x

    (B‐17)

    1

    o

     1 o

    At this point we recall Eq. B‐14 in the form

    456

    Appendices

    f ( x )  x

    (B‐18)

    nd

    a

    note that if x is

    in the neighborhood of x we obtain the approximation 1

    f ( x )  x

    (B‐19)

    1

    1

    e

    W use this result in Eq. B‐17 to produce an equation

    x

    f ( x )  S x x

    (B‐20)

    1

    o

     1 o

    in which S is an adjustable parameter that is used to determine the next step in the iterative procedure. It is traditional, but not necessary, to define a new adjustable arameter

    p

    according to

    S

    q

    S

    (B‐21)

    1

    se

    U of this representation in Eq. B‐20 leads to

    x

     (1  q) f ( x )  q x

    (B‐22)

    1

    o

    o

    nd

    a

    we can generalize this result to Wegstein’s method given by

    x

    (1 q) f ( x )

    q x ,

    n  0 , 1 , 2 , 3 , e

    ... tc

    n

    1

    n

    n

    (B‐23)

    When the adjustable parameter is equal to zero, q  0 , we obtain Picard’s method described in Sec. B4. When the adjustable parameter greater than zero and less than one, 0  q  1 , we obtain a damped successive substitution process that improves stability for nonlinear systems. When the adjustable parameter is egative,

    n

    q  0 , we obtain an accelerated successive substitution that may lead to an unstable procedure.

    B6. Stability of iteration methods

    In this section we

    r

    conside the linear stability characteristics of Newton’s method, Picard’s method, and Wegstein’s method that have been used to solve e

    th implicit equation given by

    H( x)  0 ,

    x x

    (B‐24)

    The constraint associated with the linear analysis will be listed below and it must be kept in mind when interpreting results such as those presented in Chapter 7.

    We begin by

    g

    recallin the three iterative methods as

    Material Balances

    457

    H( x )

    Newton’s method:

    n

    x

    x

    ,

    n  0 , 1 , 2 ,..., (B‐25)

    n1

    n

    ( dH dx) xnx

    Picard’s method:

    x

    f ( x ) ,

    n

    0 , 1 , 2 ,....

    (B‐26)

    n

    1

    n

    Wegstein’s method:

    x

    (1 q) f ( x )

    q x ,

    n

    0 , 1 , 2 , ...

    (B‐27)

    n

    1

    n

    n

    in which the auxiliary function, f ( x) , is defined by Definition:

    f ( x)  x H( x)

    (B‐28)

    The general form of these three iterative methods is given by x

    (

    G x ) ,

    n

    0 , 1 , 2 ,.....

    n

    (B‐29)

    1

    n

    and for each of the three methods on seeks

    to find the fixed point x of G(x)

    such that

    x

    (

    G x )

    (B‐30)

    Our stability analysis of Eqs. B‐25 through B‐27 is based on linearizing (

    G x)

    bout

    a

    the fixed point x

    . We let  x and  x

    be small perturbations from the

    n

    n1

    fixed point as indicated by

    x

    x   x ,

    x

    x  

    n

    n

    n

    x

    1

    n

    1

    (B‐31)

    This allows us to express Eq. B‐29 as

    x

    x

    (

    G x   x ) ,

    n  0 , 1 , 2 ,.....

    (B‐32)

    n1

    n

    and a Taylor series expansion (See Problems 5‐30 and 5‐31 in Chapter 5) leads to

     2 

    dG

    d G

    1

    2

    (

    G x x

     ) 

    (

    G x ) 

    x

    x

     

    .....

    (B‐33)

    n

    n

    2

    n

    2

    dx

    x

    dx x

    On the basis of Eq. B‐30 this infinite series simplifies to

     2 

    dG

    d G

    1

    2

    (

    G x   x )  x

    x

    x

     

    .....

    (B‐34)

    n

    n

    2

    n

    2

    dx

    x

    dx x

    nd

    a

    we can use Eq. B‐32 to represent the left hand side in a simpler form to obtain

    458

    Appendices

     2 

    dG

    d G

    1

    2

    x   x

    x   x

    x

     

    .....

    (B‐35)

    n1

    n

    2

    n

    2

    dx

    x

    dx x

    At this point we impose a constraint on the higher order terms expressed as

     2

    dG

    d G

    Constraint:

    1

    2

    x

    

    x

    .....

    (B‐36)

    n

    2

    n

    2

    dx

    x

    dx x

    so that Eq. B‐35 takes the form

    dG

    x

    x

    ,

    n

    0 , 1 , 2 , 3 , 4 ,.....

    (B‐37)

    n1

    n dx x

    If we write a few of these equations explicitly as

    dG

    x

    x

    (B‐38a)

    1

    o 

    dx x

    dG

    x

    x

    (B‐38b)

    2

    1 

    dx x

    dG

    x

      x

    (B‐38c)

    3

    2 

    dx x

    ……………………

    dG

    x

    x

    (B‐38d)

    n

    n1 

    dx x

    it becomes clear that they can be used to provide a general representation given by

    n

    x

    x

    dG dx  ,

    n  0 , 1 , 2 ,.....

    n

    o 

    x

    (B‐39)

    At this point we see that x

      0 when

    n   provided

    that

    n

    dG dx   1

    (B‐39)

    x

    When x

      0 as n   the system converges and one says that the fixed point n

    x is attracting. The three special cases represented by Eq. B‐39

    can be expressed

    as

    Material Balances

    459

    dG dx 

     1 ,

    the fixed point x is attracting

    x

    I.

    (B‐41)

    and the iteration converges

    dG dx 

     1 ,

    the fixed point x is repelling

    x

    II.

    (B‐42)

    and the iteration diverges

    dG dx 

     1 ,

    the fixed point x is neither

    III.

    x

    (B‐43)

    attracting nor repelling

    It is extremely important to note that the stability analysis leading to these three results is based on the linear approximation associated with Eq. B‐36. In this development, the word attracting is used for a system that converges since x n

    moves toward x as n increases, while the word repelling is used for a system that

    diverges since x moves away from x as n increases. The case in which the fixed n

    point is neither attracting nor repelling can lead to chaos (Gleick, 1988; Peitgen et al., 1992).

    At this point we are ready to return to Eqs. B‐25, B‐26 and B‐27 in order to determine the linear stability characteristics of Newton’s method, Picard’s method, and

    Wegstein’s method.

    Newton’s method

    In this case we have

    H( x)

    (

    G x)  x  

    (B‐44)

    dH dx

    and the derivative that is required to determine the stability is given by 2

    H( x)

    d H

    dG dx  

    (B‐45)

    dH dx2

    2

    dx

    Evaluation of this derivative at the fixed point where H( x )  0 leads to

    dG dx   0

    (B‐46)

    x

    460

    Appendices

    This indicates that Newton’s method will converge provided that  dH dx  0

    and provided that the initial guess, x , is close

    gh

    enou

    to x so that Eq. B‐36 is

    o

    atisfied.

    s

    If Eq. B‐36 is not satisfied, the linear

    stability analysis leading to

    Eqs. B‐41 through B‐43 is not valid.

    Picard’s method

    In this case Eqs. B‐26 and B‐29 provide (

    G x)  f ( x) and

    dG dx    df dx

    (B‐47)

    x

    x

    and from Eq. B‐41 we conclude that Picard’s method is stable when

    df dx   1

    (B‐48)

    x

    In Example 7.7 of Chapter 7 we used the fixed point iteration (see Eq. 17) that can be expressed as

    x

    f ( x )

    1

    C 1

    x

    ,

    C

    0 3

    . 0 ,

    n

    0 , 1 , 2 , 3 ,..... (B‐49)

    n

    1

    n

    

    n

    This leads to the condition

    df dx  1  C  1

    (B‐50)

    that produces the stable iteration illustrated in Table 7.7a. In Example 7.8 of Chapter 7 we find another example of Picard’s method (see Eq. 40) that we repeat here as

    15 1678

    .

    x

    x

    f ( x )

    x

    37 . 6190

    n ,

    n

    0 , 1 , 2 , 3 ,... (B‐51)

    n

    1

    n

    n

    0 . 8571  x

    n

    The solution is

    given by x  0 . 6108 and this leads to

    df dx

    .

    (B‐52)

    x

      213 1733

    dicating

    in

    that Picard’s method is unstable for this particular fixed point iteration. This result is consistent with the entries in Table 7.8a.

    egstein’s

    W

    method

    In this case Eqs. B

    ‐27 and B‐29 provide

    (

    G x)  (1  q) f ( x)  q x (B‐53)

    Material Balances

    461

    which leads to

    (

    G x)  (1  q) f ( x)  q x (B‐54)

    From this we have

    dG

    df

     (1  q)

    q

    (B‐55)

    dx

    dx

    and the stability condition given by Eq. B‐41 indicates that Wegstein’s method will converge provided that

    df

    (1  q)

    q  1

    (B‐56)

    dx

    Here one can see that the adjustable parameter q can often be chosen so that this inequality is satisfied and Wegstein’s method will converge as illustrated in Examples 7.7 and 7.8 of Chapter 7.

    462

    Appendices

    Appendix C

    Matrices

    Matrix Methods and Partitioning

    In order to support the results obtained for the atomic matrix studied in Chapter 6 and for the mechanistic matrix studied in Chapter 9, we need to consider that matter of partitioning matrices. All the information necessary for our studies of stoichiometry is contained in Eq. 6‐22; however, that information can be presented in different forms depending on how the atomic matrix and the column matrix of net rates of production are partitioned. In our studies of reaction kinetics, all the information that we need is contained in the mechanistic matrix; however, that information can also be presented in different forms depending on presence or absence of Bodenstein products. In this appendix we review the methods required to develop the desired different forms.

    Matrix addition

    We begin our study of partitioning with the process of addition (or subtraction) as illustrated by the following matrix equation

    a

    a

    a

    a

    b

    b

    b

    b

    c

    c

    c

    c

    11

    12

    13

    14

    11

    12

    13

    14

    11

    12

    13

    14

    a

    a

    a

    a

    b

    b

    b

    b

    c

    c

    c

    c

     21

    22

    23

    24 

     21

    22

    23

    24 

     21

    22

    23

    24

     (C‐1)

    a

    a

    a

    a

    b

    b

    b

    b

    c

    c

    c

    c

    31

    32

    33

    34

    31

    32

    33

    34

    31

    32

    33

    34

    a

    a

    a

    a

    b

    b

    b

    b

    c

    c

    c

    c

     41

    42

    43

    44 

     41

    42

    43

    44 

     41

    42

    43

    44 

    This can be expressed in more compact nomenclature according to A  B

     C

    (C‐2)

    The fundamental meaning of Eqs. C‐1 and C‐2 is given by the following sixteen (16) equations:

    index-472_1.png

    index-472_2.png

    index-472_3.png

    Material Balances

    463

    a

    b

    c

    a

    b

    c

    11

    11

    11

    21

    21

    21

    a

    b

    c

    a

    b

    c

    12

    12

    12

    22

    22

    22

    a

    b

    c

    a

    b

    c

    13

    13

    13

    23

    23

    23

    a

    b

    c

    a

    b

    c

    14

    14

    14

    24

    24

    24

    (C‐3)

    a

    b

    c

    a

    b

    c

    31

    31

    31

    41

    41

    41

    a

    b

    c

    a

    b

    c

    32

    32

    32

    42

    42

    42

    a

    b

    c

    a

    b

    c

    33

    33

    33

    43

    43

    43

    a

    b

    c

    a

    b

    c

    34

    34

    34

    44

    44

    44

    These equations represent a complete partitioning of the matrix equation given by Eq. C‐1, and we can also represent this complete partitioning in the form (C‐4)

    Here we have shaded the particular partition that represents the first of Eqs. C‐3.

    The complete partitioning illustrated by Eq. C‐4 is not particularly useful; however, there are other possibilities that we will find to be very useful and one example is the row/column partition given by

    (C‐5)

    Each partitioned matrix can be expressed in the form

    (C‐6)

    and the partitioned matrix equation is given by

    A

    A

    B

    B

    C

    C

    11

    12

    11

    12

    11

    12

      

      

    (C‐7)

    A

    A

    B

    B

    C

    C

    21

    22 

     21 22 

     21

    22 

    464

    Appendices

    We usually think of the elements of a matrix as numbers such as a , a , etc.; 11

    12

    however, the elements of a matrix can also be matrices as indicated in Eq. C‐7.

    The usual rules for matrix addition lead to

    A

     B

     C

    (C‐8a)

    11

    11

    11

    A

     B

     C

    (C‐8b)

    12

    12

    12

    A

     B

     C

    (C‐8c)

    21

    21

    21

    A

     B

     C

    (C‐8d)

    22

    22

    22

    and the details associated with Eq. C‐8a are given by

    a

    a

    b

    b

    c

    c

    11

    12

    11

    12

    11

    12

      

      

    (C‐9)

    a

    a

    b

    b

    c

    c

     21

    22 

     21

    22 

     21

    22 

    A little thought will indicate that this matrix equation represents the first four equations given in Eqs. C‐3. Other partitions of Eq. C‐1 are obviously available and will be encountered in the following paragraphs.

    Matrix multiplication

    Multiplication of matrices can also be represented in terms of submatrices, provided that one is careful to follow the rules of matrix multiplication. As an example, we consider the following matrix equation

    a

    a

    a

    a   b

    b

    c

    c

    11

    12

    13

    14

    11

    12

    11

    12

     

    a

    a

    a

    a

    b

    b

    c

    c

     21

    22

    23

    24   21

    22 

     21

    22

    (C‐10)

    a

    a

    a

    a   b

    b

    c

    c

    31

    32

    33

    34

    31

    32

    31

    32

     

    a

    a

    a

    a

    b

    b

    c

    c

     41

    42

    43

    44   41

    42 

     41

    42 

    which conforms to the rule that the number of columns in the first matrix is equal to the number of rows in the second matrix. Equation C‐10 represents the eight (8) individual equations given by

    a b

    a b

    a b

    a b

    c

    (C‐11a)

    11 11

    12 21

    13 31

    14 41

    11

    a b

    a b

    a b

    a b

    c

    (C‐11b)

    11 12

    12 22

    13 32

    14 42

    12

    a b

    a b

    a b

    a b

    c

    (C‐11c)

    21 11

    22 21

    23 31

    24 4

    21

    a b

    a b

    a b

    a b

    c

    (C‐11d)

    21 12

    22 22

    23 32

    24 42

    22

    a b

    a b

    a b

    a b

    c

    (C‐11e)

    31 11

    32 21

    33 31

    34 41

    31

    index-474_1.png

    Material Balances

    465

    a b

    a b

    a b

    a b

    c

    (C‐11f)

    31 12

    32 22

    33 32

    34 42

    32

    a b

    a b

    a b

    a b

    c

    (C‐11g)

    41 11

    42 21

    43 31

    44 41

    41

    a b

    a b

    a b

    a b

    c

    (C‐11h)

    41 12

    42 22

    43 32

    44 42

    42

    which can also be expressed in compact form according to

    AB

     C

    (C‐12)

    Here the matrices A, B, and C are defined explicitly by

    a

    a

    a

    a

    b

    b

    c

    c

    11

    12

    13

    14

    11

    12

    11

    12

    a

    a

    a

    a

    b

    b

    c

    c

     21

    22

    23

    24 

     21

    22 

     21

    22

    A

    B

    C

    (C‐13)

    a

    a

    a

    a

    b

    b

    c

    c

    31

    32

    33

    34

    31

    32

    31

    32

    a

    a

    a

    a

    b

    b

    c

    c

     41

    42

    43

    44 

     41

    42 

     41

    42 

    In Eqs. C‐1 through C‐9 we have illustrated that the process of addition and subtraction can be carried out in terms of partitioned matrices. Matrix multiplication can also be carried out in terms of partitioned matrices; however, in order to conform to the rules of matrix multiplication, we must partition the matrices properly. For example, a proper row partition of Eq. C‐10 can be expressed as

    (C‐14)

    In terms of the submatrices defined by

    a

    a

    a

    a

    a

    a

    a

    a

    11

    12

    13

    14

    31

    32

    33

    34

    A

     

    ,

    A

    11

    21

    a

    a

    a

    a

    a

    a

    a

    a

     21

    22

    23

    24 

     41

    42

    43

    44 

    (C‐15)

    c

    c

    c

    c

    11

    12

    31

    32

    C

     

    C

    11

    21

    c

    c

    c

    c

     21

    22 

     41

    42 

    we can represent Eq. C‐14 in the form

    A 

    A B

    C 

    11

    11

    11

    B  

      

    (C‐16)

    A

    A

    B

    C

     21

     21 

     21

    466

    Appendices

    Often it is useful to work with the separate matrix equations that we have created by the partition, and these are given by

    A

    B

     C

    (C‐17)

    11

    11

    A

    B

     C

    (C‐18)

    21

    21

    The details of the first of these can be expressed as

    b

    b

    11

    12

    a

    a

    a

    a b

    b

    c

    c

    11

    12

    13

    14  21

    22 

    11

    12

    A

    B

     

    ,

    C

    (C‐19a)

    11

    11

    a

    a

    a

    a

    b

    b

    c

    c

     21

    22

    23

    24 

    31

    32

     21

    22 

    b

    b

     41

    42 

    Multiplication can be carried out to obtain

    a b a b a b a b

    a b a b a b a b

    11 11

    12 21

    13 31

    14 41

    11 12

    12 22

    13 32

    14 42

    a b a b a b a b

    a b a b a b a b

     21 11

    22 21

    23 31

    24 41

    21 12

    22 22

    23 32

    24 42 

    (C‐19b)

    c

    c

    11

    12

     

    c

    c

     21

    22 

    and equating the four elements of each matrix leads to

    a b

    a b

    a b

    a b

    c

    11 11

    12 21

    13 31

    14 41

    11

    a b

    a b

    a b

    a b

    c

    11 12

    12 22

    13 32

    14 42

    12

    (C‐19c)

    a b

    a b

    a b

    a b

    c

    21 11

    22 21

    23 31

    24 41

    21

    a b

    a b

    a b

    a b

    c

    21 12

    22 22

    23 32

    24 42

    22

    Here we see that these four individual equations (associated with the partitioned matrix equation) are those given originally by Eqs. C‐11a through C‐11d. A little thought will indicate that the matrix equation represented by Eq. C‐18 contains the four individual equations represented by Eqs. C‐11e through C‐11h. All of the information available in Eq. C‐10 is given explicitly in Eqs. C‐11 and partitioning of the original matrix equation does nothing more than arrange the information in a different form.

    index-476_1.png

    index-476_2.png

    Material Balances

    467

    If we wish to obtain a column partition of the matrix A in Eq. C‐10, we must also create a row partition of matrix B in order to conform to the rules of matrix multiplication. This column/row partition takes the form (C‐20)

    and the submatrices are identified explicitly according to

    a

    a

    a

    a

    11

    12

    13