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15.2: Untitled Page 236

  • Page ID
    18369
  • 14

    a

    a

    a

    a

     21

    22 

     23

    24 

    A

    A

    11

    12

    a

    a

    a

    a

    31

    32

    33

    34

    a

    a

    a

    a

     41

    42 

     43

    44 

    (C‐21)

    b

    b

    b

    b

    11

    12

    31

    32

    B

     

    B

    11

    21

    b

    b

    b

    b

     21

    22 

     41

    42 

    Use of these representations in Eq. C‐20 leads to

    B 

    11

    A

    A 

     

      C

    (C‐22)

    11

    12 B21

    and matrix multiplication in terms of the submatrices provides A B

     A B

     C

    (C‐23)

    11 11

    12 21

    In some cases, we will make use of a complete column partition of the matrix A which requires a complete row partition of the matrix B. This partition is illustrated by

    (C‐24)

    In terms of the submatrices it can be expressed as

    468

    Appendices

    B 

    11

    B 

    21

    A

    A

    A

    A 

    C

    (C‐25)

    11

    12

    13

    14  B 

    31

    B 

    41 

    and matrix multiplication leads to

    A B

     A B

     A B

     A B

     C

    (C‐26)

    11 11

    12 21

    13 31

    14 41

    Problems

    C‐1. Given a matrix equation of the form c  Ab having an explicit representation of the form,

    c

    a

    a

    a

    a

    1

    11

    12

    13

    14

     

      b

    1

    c

    a

    a

    a

    a

     2 

     21

    22

    23

    24  

    b

       

      2

    c

    a

    a

    a

    a

    (1)

    3

    31

    32

    33

    34

     

      b

    3

    c

    a

    a

    a

    a

     

      

    4

    41

    42

    43

    44

    b

     

      4 

    c

    a

    a

    a

    a

     5 

     51

    52

    53

    54 

    develop a partition that will lead to an equation for the column vector represented by

    c

    1

     

    c

    ?

    (2)

     2

    c

     3

    If the elements of c are related to the elements of b according to

    c

    b

    4

    3

        

    (3)

    c

    b

     5 

     4 

    what are the elements of the matrix normally identified as A

    ?

    21

    C‐2. Construct the complete column/row partition of the matrix equation given by

    Material Balances

    469

    a

    a

    a

    a   b

    c

    11

    12

    13

    14

    1

    1

      

     

    a

    a

    a

    a

    b

    c

     21

    22

    23

    24   2 

     2

    (1)

    a

    a

    a

    a   b

    c

    31

    32

    33

    34

    3

    3

      

     

    a

    a

    a

    a

    b

    c

     41

    42

    43

    44   4 

     4 

    and show how it can be represented in the form

     

     

     

     

     

     

     

     

     

     

    b

    b

    b

    b

    (2)

      1

      2

      3

      4

     

     

     

     

     

     

     

     

     

     

     

    470

    Appendices

    Appendix D

    Atomic Species Balances

    Throughout this text we have made use of macroscopic mass and mole balances to solve a variety of problems with and without chemical reactions.

    Our solutions have been based on the application of Axiom I and Axiom II, and our attention has been focused on mass flow rates, molar rates of production owing to chemical reaction, accumulation of mass or moles, etc. In this appendix we show how problems can be solved using macroscopic atomic species balances. This approach has some advantages when carrying out calculations by hand since the number of atomic species balance equations is almost always less than the number of molecular species balance equations.

    We begin our development of atomic species balance equations with Axioms I and II given by

    d

    Axiom I:

    c dV

    c v n dA

    R dV ,

    A  1 , 2 ,..., N

    (D‐1)

    A

    A A

    A

    dt V

    A

    V

    A N

    Axiom II

    N

    R

     0 ,

    J  1 , 2 ,..., T

    (D‐2)

    JA A

    A 1

    Here we should remember that the individual components of the atomic matrix,

    N  , are described by (see Sec. 6.2)

    JA

     number of moles of

    N

      J‐type atoms per mole ,

    J  1 , 2 ,...,T , and A  1 , 2 ,...N (D‐3) JA

     of molecular species A

    To develop an atomic species balance, we multiply Eq. D‐1 by N

    and sum over

    JA

    all molecular species to obtain

    A N

    A N

    A N

    d

    N

    c dV

    N

    c v n dA

    N

    R dV

     

     

     

    (D‐4)

    JA A

    JA A A

    JA

    A

    dt

    A  1

    A  1

    A  1

    V

    A

    V

    Material Balances

    471

    On the basis of Axiom II, we see that the last term in this result is zero and our atomic species balance takes the form

    Axioms I & II:

    A N

    A N

    d

    N

    c dV

    N

    c v n dA  0 ,

    J  1 , 2

     

     

    JA A

    JA A A

    , ... , T (D‐5)

    dt

    A  1

    A  1

    V

    A

    Here we have indicated explicitly that there are T atomic species balance equations instead of the N molecular species balance equations which are given by Eq. D‐1. When T N it may be convenient to solve material balance problems using atomic species balances.

    In many applications of Eq. D‐5, diffusive transport at the surface of the control volume is negligible and v n can be replaced by v n leading to A

    A N

    A N

    d

    N

    c dV

    N

    c v n dA  0 ,

    J  1 , 2

     

     

    (D‐6)

    JA A

    JA A

    , ... , T

    dt

    A  1

    A  1

    V

    A

    If the concentration is given in terms of mole fractions one can use c x c to A

    A

    express Eq. D‐6 as

    A N

    A N

    d

    N

    x

     

    c dV

    N

    x

     

    cv n dA  0 , J  1 , 2

    (D‐7)

    JA

    A

    JA

    A

    , ... , T

    dt

    A  1

    A  1

    V

    A

    For some reacting systems these results, rather that Eq. D‐1, provide the simplest algebraic route to a solution.

    When problems are stated in terms of species mass densities and species mass fractions, it is convenient to rearrange Eq. D‐5 in terms of these variables. To develop the forms of Eqs. D‐6 and D‐7 that are useful when mass flow rates are given, we begin by representing the species concentration in terms of the species density according to

    c

     

    MW

    (D‐8)

    A

    A

    A

    The species density is now expressed in terms of the mass fraction and the total mass density according to

      

    (D‐9)

    A

    A

    and when these two results are used in Eq. D‐6 we obtain

    472

    Appendices

    A N

    A N

    N  

    N

    d

     

    JA

    A

    JA

    A

     dV

    v n dA  0 , J  1 , 2 , ...,T

     

     

    (D‐10)

    dt

    MW

    MW

     1 

    A

     1 

    A

    A

    A

    V

    A

    This represents the “mass” analog of Eq. D‐7, and it will be convenient when working with problems in which mass flow rates and mass fractions are given.

    We should note that Eq. D‐10 actually represents a molar balance on the th J

    atomic species since an examination of the units will indicate that

    A N

    N

     

    moles per

    JA

    A

     

     v n dA  

    (D‐11)

    MW

    unit time

    1

    A

    A

    A

    To obtain the actual atomic species mass balance one would multiply Eq. D‐10 by the atomic mass of the th

    J atomic species, AW .

    J

    At steady state Eq. D‐4 represents a system of homogeneous equations in terms the net atomic species molar flow rates leaving the control volume. For there to be a nontrivial solution, the rank  r of  N

    JA  must be less than N.

    Consequently there are at most r linearly independent atomic species balances (see Sec. 6.2.3 for details).

    EXAMPLE D.1: Production of Sulfuric Acid

    In this example we consider the production of sulfuric acid illustrated in Figure D.1. The mass flow rate of the 90% sulfuric acid stream is specified as m

      100 lb / h , and we are asked to determine the mass flow

    1

    m

    rate of the pure sulfur trioxide stream, m

     . As is often the custom with

    2

    liquid systems the percentages given in Figure D.1 refer to mass fractions, thus we want to create a final product in which the mass fraction of sulfuric acid is 0.98.

    In order to analyze this process in terms of the atomic species mass balance, we use the steady state form of Eq. D‐10 given here as A N N  

    JA

    A

    v n dA  0 ,

    J  S , H , O

      

    (1)

    MW

     1 

    A

    A

    A

    For the system shown in Fig. D.1 we see that Eq. 1 takes the form

    index-482_1.png

    index-482_2.png

    Material Balances

    473

    A N

    A N

    N  

     

    JA

    A

    NJA A

    m

      

    1

    m 2

     

    A

    MW

     

    A

    MW

    A 1

    1

    A 1

    2

    (2)

    A N N  

    JA

    A

    m

    3

    ,

    J

    S , H , O

    MW

     1 

    A

    A

    3

    In the process under investigation there are three atomic species indicated by S, H and O , and there are three molecular species indicated by H SO , H O and SO . Since only sulfur and oxygen appear in the 2

    4

    2

    3

    stream for which we want to determine the mass flow rate, we must make use of either a sulfur balance or an oxygen balance. However, sulfur Figure D.1. Sulfuric acid production

    appears in only two of the molecular species while oxygen appears in all three. Because of this a sulfur balance is preferred. At this point we recall the definition of N

    given by Eq. D‐3

    JA

    474

    Appendices

     number of moles of 

    N

      J‐type atoms per mole , J  1 , 2 , ... ,T, and A  1 , 2 , ... ,N (3) JA

     of molecular species A

    which leads to

    N

     1 ,

    J  S , A  H SO

    JA

    2

    4

    (4)

    N

     1 ,

    J  S , A  SO

    JA

    3

    Letting J in Eq. 2 represent sulfur we have

    m

     (

    )

    m

    m

     (

    )

    1

    H SO

    1

    2

    3

    H SO

    3

    2

    4

    2

    4

    (5)

    MW

    MW

    MW

    H2SO4

    SO3

    H2SO4

    and an overall mass balance

    m

    m

    m

    (6)

    1

    2

    3

    can be used to eliminate m

     in terms of m and m . This allows us to 3

    1

    2

    solve for the mass flow rate of sulfur trioxide that is given by

    (

    )  (

    )  

    H

    m

    2SO4 3

    H2SO4 1 1

    m

    (7)

    2

    MW

    H2SO4

     (

    ) 

    H2SO4 3

    MW

    SO

    3

    In this case we have been able to obtain a solution using only a single atomic element balance along with the overall mass balance. Given this solution, we can easily compute the mass flow rate of pure sulfur trioxide to be

    m

     32 65

    .

    lb / h

    (8)

    2

    m

    In the previous example we solved a problem in which there were three molecular species and three atomic species and we found that a solution could be obtained easily using an atomic species balance. When the number of atomic species is less that the number of molecular species, there is a definite computational advantage in using the atomic species balance. However, the physical description of most processes is best given in terms of molecular species and this often controls the choice of the method used to solve a particular problem.

    index-484_1.png

    index-484_2.png

    Material Balances

    475

    EXAMPLE D.2: Combustion of carbon and air

    Carbon is burned with air, as illustrated in Figure D.2, with all the carbon oxidized to CO and CO . The ratio of carbon dioxide produced 2

    to carbon monoxide produced is 2:1. In this case we wish to determine Figure D.2. Combustion of carbon and air

    the flue gas composition when 50% excess air is used. The percent excess air is defined as

    percentage of

     

     excess air

    (1)

     molar flow 

     molar rate of

    rate of oxygen  consumption of oxygen

     entering

    

     owing to reaction

     100

     molar rate of

    consumption of oxygen

     owing to reaction

    

    476

    Appendices

    and this definition requires a single application of Eq. D‐1 in order to incorporate the global molar rate of production of oxygen, R

    , into the

    O2

    analysis.

    Atomic Species Balances

    The steady‐state form of Eq. D‐7 is given by

    A N

    N

    x

     

    c v n dA  0 ,

    J  C , O , N

    (2)

    JA

    A

    A  1

    A

    and for the system illustrated in Figure D.2 we obtain

    A N

    A N

    A N

    N

    x

    M  

    N

    x

    M

     

    N

    x

    M (3)

    JA

    A

    1

    JA

    A

    2

    JA

    A

    3

    A  1

    A  1

    A  1

    1

    2

    3

    We begin by choosing the Jth atomic species to be carbon so that Eq. 3

    takes the form

    Atomic Carbon Balance:

    M

     ( x )  ( x

    )  

    M

    (4)

    1

    CO 3

    CO

    2 3

    3

    and we continue with this approach to obtain the atomic oxygen and atomic nitrogen balance equations given by

    2( x

    ) M

    O

    2

    2

    Atomic Oxygen Balance:

    2

    (5)

     2( x )  2( x

    )  ( x

    )  

    M

    O

    2 3

    CO2 3

    CO 3

    3

    Atomic Nitrogen Balance:

    2( x

    ) M

     2( x ) M

    (6)

    N2 2

    2

    N

    3

    3

    2

    Directing our attention to the percentage of excess air defined by Eq. 1, we define the numerical excess air as

    (

    M

      

    O )

    R

    2

    O

    2

    2

     

     0 5

    .

    (7)

    

    RO

    2 

    At this point we must make use of the mole balance represented by Eq. D‐1 in order to represent the global net rate of production of oxygen as Molecular Oxygen Balance:

    R

     ( M )  ( M )

    (8)

    O2

    O2 3

    O2 2

    Material Balances

    477

    This can be used to obtain a relation between the numerical excess air and the molar flow rates of oxygen given by

     ( M )

    1   ( M

    )

    (9)

    O

    2 2

    O2 3

    For use with the other constraining equations, this can be expressed in terms of mole fractions to obtain

    ( x ) M

    1   ( x

    ) M

    (10)

    O

    2

    2

    2

    O2 3

    3

    This result, along with the atomic species balances and the input data, can be used to determine the flue gas composition

    Analysis

    We are given that the molar rate of production of carbon dioxide is two times the molar rate of production of carbon monoxide. We can express this information as

    M

      M

    ,

      2

    (11)

    CO2

    CO

    and in terms of mole fractions this leads to

    ( x

    )

      ( x )

    (12)

    CO2 3

    CO 3

    The final equation required for the solution of this problem is the constraint on the sum of the mole fractions in Stream #3 that is given by ( x

    )  ( x

    )  ( x

    )  ( x

    )

     1

    (13)

    O2 3

    N2 3

    CO 3

    CO2 3

    In addition to this constraint on the mole fractions in Stream #3, we assume that the air in Stream #2 can be described by

    ( x

    )

     0 . 21 ,

    ( x

    )

     0 . 79

    (14)

    O2 2

    N2 2

    Use of Eq. 12 in Eqs. 4 through 6 gives

    Atomic Carbon Balance:

    M

     ( x ) 1   M

    (15a)

    1

    CO 3 

     3

    Atomic Oxygen Balance:

     

    1

    ( x

    ) M

    ( x

    )   

    ( x

    )  M

    O

    2

    2

    O

    3

    2 

    2

    2

    CO 3

    3

    (15b)

    Atomic Nitrogen Balance:

    ( x

    ) M

     ( x ) M

    (15c)

    N2 2

    2

    N2 3

    3

    478

    Appendices

    At this point we note that we have eight unknowns and seven equations; however, one of the molar flow rates can be eliminated to develop a solution.

    Algebra

    In order to determine the flue gas composition, we do not need to determine the absolute values of M

     , M , and M but only two

    1

    2

    3

    dimensionless flow rates. These are defined by

    M

    MM

    (16a)

    2

    2

    1

    M

    MM

    (16b)

    3

    3

    1

    so that Eqs. 15 and Eq. 10 take the form

    1  ( x

    )

    1  

    CO 3 

     M

    (17a)

    3

    M

     

    1

    ( x

    )

    ( x

    )   

    ( x

    ) 

    O

    2

    2

    O

    3

    M

    (17b)

    2 

    2

    2

    CO 3

    3

    ( x

    ) M

     ( x ) M

    (17c)

    N2 2

    2

    N2 3

    3

     ( x ) M

    1   ( x

    )

    O

    2

    2

    M

    (17d)

    2

    O2 3

    3

    From these four equations we need to eliminate M and M in order to 2

    3

    determine the mole fractions and thus the composition of the flue gas.

    Equations 17a and 17c can be used to obtain the following representations for M and M

    2

    3

    1

    M

    (18a)

    3

    ( x

    ) 1  

    CO 3 

    ( x

    )

    N

    3

    M

    (18b)

    2

    ( x

    ) 1   ( x

    )

    CO 3 

    2  N2 2

    Use of these two results in Eq. 17b leads to

    1

    ( x

    )

    ( x

    )  

     N 2 

     N 2 

    2  

    2

    2

    ( x

    )

     

    ( x )  

    ( x )

    (19a)

    N2 3

    O2 3

    CO 3

    ( x

    )

    ( x

    )

     O

    

    2 2

    O

    2 2

    

    

    known

    known

    Material Balances

    479

    and when we eliminate M and M from Eq. 17d we find

    2

    3

    ( x

    ) 

    1  

    N

    2 

    2

    ( x

    )

     

    ( x )

    (19b)

    N2 3

    O

    2 3

    ( x

    )

    O

    2 2 

    

    known

    These two equations, along with the constraint on the mole fractions given by

    ( x

    )  ( x

    )  1   ( x

    )

     1

    (20)

    O

    3

    N

    3

    2

    2

    CO 3

    are sufficient to determine ( x

    ) , ( x

    ) and ( x

    ) . These mole

    O2 3

    N2 3

    CO 3

    fractions can be expressed as

    1

    ( x

    )

    (21a)

    O2 3

    1  K  1   K K

    K

    3

     3 1 2

    K

    3

    ( x

    )

    (21b)

    N2 3

    1  K  1   K K

    K

    3

     3 1 2

    K K K

    3

    1 

    2

    ( x

    )

    (21c)

    CO 3

    1  K  1   K K

    K

    3

     3 1 2

    ( x

    )

      ( x )

    (21d)

    CO2 3

    CO 3

    where the numerical values of K , K and K are given by 1

    2

    3

    ( x

    )

    N

    2

    2

    K

     3 . 7619

    (22a)

    1

    ( x

    )

    O2 2

    ( x

    )

    N

    2

    2

    1

    K

     

     9 . 4048

    (22b)

    2

     2

    ( x

    )

    O2 2

    ( x

    )

    N

    2  1   

    2

    K

    11 . 2857

    (22c)

    3

    ( x

    )

     

    O

    2 2

    Use of these values in Eqs. 21 gives the following values for the mole fractions

    index-489_1.png

    480

    Appendices

    ( x

    )  0 0681

    .

    , ( x

    ) 

    O

    0 . 7685 ,

    2 3

    N2 3

    (23)

    ( x

    )  0 0545

    .

    , ( x

    ) 

    CO 3

    CO

    0 1080

    .

    2 3

    The sum of the mole fractions is 0.999 indicating the presence of a small numerical error that could be diminished by carrying more significant figures. On the other hand, the input data for problems of this type is never accurate to  0 . 1 % and it would be misleading to develop a more accurate solution.

    In this appendix we have illustrated how problems can be solved using atomic species balances instead of molecular species balances. In general, the number of atomic species balances is smaller than the number of molecular species balances, thus the use of atomic species balances leads to fewer equations. However, information is often given in terms of molecular species, such as the percent excess oxygen in Example D.2, and this motivates the use of molecular species balances. Nevertheless, the algebraic effort in Example D.2 is less than one would encounter if the problem were solved using molecular species balances.

    Problems

    D‐1. A stream of pure methane ( CH ) is partially burned with air in a furnace at 4

    a rate of 100 moles of methane per minute. The air is dry, the methane is in excess, and the nitrogen is inert in this particular process. The reactants and products of the reaction are illustrated in Figure D‐1. The exit gas contains a 1:1

    ratio of H O : H and a 10:1 ratio of CO : CO . Assuming that all of the oxygen 2

    2

    2

    and 94% of the methane are consumed by the reactions, determine the flow rate and composition of the exit gas stream.

    Figure D‐1. Combustion of methane

    Material Balances

    481

    D‐2. A fuel composed entirely of methane and nitrogen is burned with excess air.

    The dry flue gas composition in volume percent is: CO , 7.5%, O , 7%, and the 2

    2

    remainder nitrogen. Determine the composition of the fuel gas and the percentage of excess air as defined by

     molar flow 

     molar rate of

    rate of oxygen

     consumption of oxygen

     entering

    owing to reaction

    percent of

     

     100

    excess air 

     molar rate of

    consumption of oxygen

     owing to reaction

    482

    Appendices

    Appendix E

    Conservation of Charge

    In Chapter 6 we represented conservation of atomic species by A N

    Axiom II:

    N

    R

     0 ,

    J  1 , 2 , ... ,T

    (E‐1)

    JA A

    A  1

    and in matrix form we expressed this result as

    R

    N

    N

    N

    ...... N

    N

    1

    11

    12

    13

    1 ,N1 ,

    1 N

    R

     

    2

    0

    N

    N

    ...... N

    N

     21

    22

    2 ,N1

    2 N

     

    R

    3

    0

     

    N

    N

    ......

    Axiom II:

     31

    31

    .

     0

    (E‐2)

    .

    ......

     

    .

    .

     

    .

    ......

    R

     

    N1

    0

    N

    N

    ...... N

    N

    T 1

    T 2

    T ,N1

    TN   R

    N

    If some of the species undergoing reaction are charged species (ions), we need to impose conservation of charge (Feynman, et al., 1963) in addition to conservation of atomic species. This is done in terms of the additional axiomatic statement given by

    A N

    Axiom III:

    N

    R

     0

    (E‐3)

     e A A

    A  1

    in which N

    represents the electronic charge associated with molecular species e A

    A. In terms of matrix representation, Axiom III can be added to Eq. E‐2 to obtain a combined representation for conservation of atomic species and conservation of charge. This combined representation is given by

    index-492_1.png

    Material Balances

    483

    N

    N

    N

    ...... N

    N

    11

    12

    13

    1 ,N1 ,

    1 N R

    1

     

    0

    N

    N

    .

    ...... N

    N

    21

    22

    2 ,N1

    2 N

    R

    2 

     

    0

     

    N

    N

    ......

    .

    .

    R

    3

    31

    32

    0

    .

    .

    ......

    .

    .

    .

     

    (E‐4)

    .

     

    .

    .

    ......

    .

    .

    .

     

    .

    N

    .

    ......

    .

    .   R

     

    N1

    T 1

    0

    N

    N

    ......

    N

    N

    R

     e1

    e

     

    2

    e 1

    e

    N

    N

    N

    Here the elements in the last row of the ( T  1)  N matrix take on the values associated with the charge on species 1, 2, … N such as N

     0 ,

    non‐ionic species

    e1

    =

    N

      2 , ionic species such as SO

    (E‐5)

    e 2

    4

    +

    N

      1 , ionic species such as Na

    e 3

    As an example of competing reactions in a redox system (Porter, 1985) we

    consider a mixture consisting of ClO ,

    +

    H O , Cl , H O , ClO , and ClO .

    2

    3

    2

    2

    3

    2

    The visual representation for the atomic / electronic matrix is given by (E‐6)

    and use of this result with Eq. E‐4 leads to

    R

     

    ClO

    2

    R

      H

    1

    0

    2

    0

    1

    1

     

    3O

    0

     

     

    2

    1

    0

    1

    3

    2

    R

    0

    Axiom II & III:

    Cl

     

    2

       

    (E‐7)

     0

    3

    0

    2

    0

    0   R

    0

    H

    2O

     

    1

    1

    0

    0 

    1

    0 

    0

    R

     

    ClO

    3

    R

    ClO

    2 

    484

    Appendices

    At this point we follow the developments given in Chapters 6 through 9 and search for the optimal form of the atomic / electronic matrix. We begin with

     1

    0

    2

    0

    1

    1 

    2

    1

    0

    1

    3

    2

    A

    (E‐8)

    e

     0

    3

    0

    2

    0

    0 

    1 1

    0

    0 

    1

    0 

    and apply a series of elementary row operations to find the row reduced echelon form given by

     1

    0

    0

    0

    5 3

    4 3 

    0

    1

    0

    0

    2 3

    4 3

    A

    (E‐9)

    e

     0

    0

    1

    0

     1 3 1 6 

     0

    0

    0

    1

     1

    2 

    Use of this result in Eq. E‐7 leads to

    R

     

    ClO

    2 

    R

    H

    1

    0

    0

    0

    5 3

    4 3

     

    3O

    0

     

     

    0

    1

    0

    0

    2 3

    4 3

    R

    0

    Cl

     

    2

       

    (E‐10)

    0

    0

    1

    0

     1 3 1 6   R

    0

    H

    2O

     

    0

    0

    0

    1

     1

    2

    0

    R

     

    ClO

    3 

    R

    ClO

    2 

    We now follow the type of analysis given in Sec. 6.4 and apply the obvious column / row partition to obtain

    R

     

    ClO

    1 0 0 0 

    2 

     5 3

    4 3

    0

     

     

     

    0 1 0 0

    R

    2 3

    4 3

    R

    0

      H O

    ClO

     

    3

    3 

    (E‐11)

    0 0 1 0 

    1 3 1 6

     

    R

    R

    0

    Cl

    ClO

     

     

    2

    2

     

    0 0 0 1

    1

    2

    

     

    0

    R

    

    H

    2O 

    non‐pivot

    pivot

    submatrix

    submatrix

    Making use of the property of the identity matrix leads to

    Material Balances

    485

    R

     

    R

     

    ClO

    1 0 0 0 

    2

    ClO

    2 

     

    0 1 0 0

    R

    R

    H

    3O

    H

      

    3O 

    (E‐12)

    0 0 1 0  R

    R

    Cl

     

    2

    Cl

    2 

    0 0 0 1  R

    R

    H

    2O

    H

    2O 

    and substituting this result in Eq. E‐11 provides the desired result

    R

     

    ClO

    2 

    5 3

    4

     3

    R

     

    2 3 4 3

    R

    H O

    ClO

    3

    3 

    (E‐13)

     1 3

    1 6

    R

    R

    Cl

    ClO

     

    2

    2

    

     1

    2

    R

    

    pivot

    H

    2O 

    

    species

    pivot matrix

    non‐pivot

    species

    As discussed in Chapter 6, the choice of pivot and non‐pivot species is not completely arbitrary. Thus one must arrange the atomic / electronic matrix in row reduced echelon form as illustrated in Eq. E‐9 in order to make use of the pivot theorem indicated by Eq. E‐13.

    Use of Eq. E‐1 with Eq. E‐3 is a straightforward matter leading to Eq. E‐4.

    Within the framework of Chapter 6, one can apply Eq. E‐4 in a routine manner in order to solve problems in which charged species are present. In Chapter 6 and in Chapter 7 we dealt with problems in which net rates of production had to be measured experimentally, and Eq. E‐13 is an example of this type of situation.

    There we see that the net rates of production of the pivot species ( CLO and 3

    CLO ) must be determined experimentally so that the pivot theorem can be 2

    used to determine the net rates of production of the non‐pivot species ( ClO , 2

    +

    H O , Cl and H O ).

    3

    2

    2

    Mechanistic Matrix

    In our studies of reaction kinetics in Chapter 9 we made use of chemical reaction rate expressions so that all the net rates of production could be calculated in terms of a series of reference reaction rates. These reaction rates were developed on the basis of mass action kinetics and thus contained rate coefficients and the concentrations of the chemical species involved in the

    486

    Appendices

    reaction. That development made use of elementary stoichiometry which we express as

    A N

    Elementary stoichiometry:

    k

    N R

     0 , J  1 , 2 ,..,T , k  I, II, .., K (E‐14)

    JA A

    A  1

    This result insures that atomic species are conserved in each elementary kinetic step, and Eq. E‐1 is satisfied by imposition of the condition (see Eqs. 9‐46 and 9‐47)

    k  K

    k

    R

    R ,

    k  I, II, .., K

    (E‐15)

    A

    A

    k  I

    When confronted with charged species (ions) in a study of reaction kinetics, one makes use of elementary conservation of charge as indicated by A N

    Elementary conservation of charge:

    k

    N

    R

     0 , k  I, II, .., K

    (E‐16)

     e A A

    A  1

    Thus charge is conserved in each elementary step of a chemical kinetic schema, and total conservation of charge indicated by Eq. E‐3 is automatically achieved in the construction of a mechanistic matrix.

    index-496_1.png

    index-496_2.png

    Material Balances

    487

    Appendix F

    Heterogeneous Reactions

    Our analysis of the stoichiometry of heterogeneous reactions is based on conservation of atomic species expressed as

    AN

    Axiom II:

    N R

     0

    (F‐1)

    JA A

    A1

    We follow the classic continuum point of view (Truesdell and Toupin, 1960) and assume that this result is valid everywhere. That is to say that Axiom II is valid in homogeneous regions where quantities such as R change slowly and it is A

    valid in interfacial regions where R changes rapidly. We follow the work of A

    Wood et al (2000) and consider the    interface illustrated in Figure F‐1. The volume V encloses the    interface and

    Figure F‐1. Catalytic surface

    extends into the homogeneous regions of both the  ‐phase and the  ‐phase.

    The total net rate of production of species A in the volume V is represented by

    488

    Appendices

    R dV

    R

    dV

    R

    dV

    R

    dA

    (F‐2)

    A

    A

    A

    s

    A

    V

    V

    V

    A

    

    Here the dividing surface that separates the  ‐phase from the  ‐phase is represented by A and the heterogeneous rate of production of species A is identified by R

    . This quantity is also referred to as the surface excess reaction As

    rate (Whitaker, 1992). Multiplying Eq. F‐2 by the atomic species indicator and summing over all molecular species leads to

    A N

    A N

    N R dV

      JA A

      NJARAdV

    A1

    A

    V

    V

    1

    (F‐3)

    A N

    A N

       N

    JARAdV

      N R dA , J 1 , 2

    JA As

    ,...,T

    A  1

    A

    V

    A 

    1

    From Eq. F‐1 we see that the left hand side of this result is zero and we have A N

    A N

    0 

      NJARAdV    NJARAdV

    A  1

    A

    V

    V

    1

    (F‐4)

    A N

    N R

    dA ,

    J

     

    1 , 2

    JA As

    ,...,T

    A

    A 

    1

    At this point we require that the homogeneous net rates of production satisfy the two constraints given by

    A N

    A N

    N

    R

     0 ,

    N

    R

     0 , J  1 , 2 ,...,T

    (F‐5)

    JA A

    JA A

    A  1

    A  1

    and this leads to the following form of Eq. F‐4

    A N

    N R

    dA  0 ,

    J  1 , 2 ,...,T

    (F‐6)

      JA As

    A

    A 

    1

    Material Balances

    489

    Catalytic surfaces consist of catalytic sites where reaction occurs and non-catalytic regions where no reaction occurs. Because the heterogeneous rate of production is highly non‐uniform, it is appropriate to work in terms of the area average

    1

    R

    R

    dA

    (F‐7)

    As 

    As

    A  A

    so that Eq. F‐6 takes the form

    A N

    N

    R

     0 .

    J  1 , 2 , ... ,T

    (F‐8)

    JA As 

    A  1

    We summarize our results associated with Axiom II as

    A N

    Axiom II (general)

    N

    R

     0 , J  1 , 2 ,...,T

    (F‐9)

    JA A

    A  1

    A N

    Axiom II (  ‐phase)

    N

    R

     0 , J  1 , 2 ,...,T

    (F‐10)

    JA A

    A  1

    A N

    Axiom II (  ‐phase)

    N

    R

     0 , J  1 , 2 ,...,T

    (F‐11)

    JA A

    A  1

    A N

    Axiom II (    interface)

    N

    R

     0 , J  1 , 2 ,...,T

    (F‐12)

    JA As 

    A  1

    For a reactor in which only homogeneous reactions occur, we make use of Eq. F‐9

    in the form

    R

    A

    R

    B

    Axiom II:

    A R

     0 ,

    R

      .

    (F‐13)

    .

    R

    N

    in which A is the atomic matrix. For a catalytic reactor in which only heterogeneous reactions occur at the    interface, we make use of Eq. F‐12 in the form

    490

    Appendices

     R  

    As 

     R

    Bs  

    Axiom II:

    A R 

     0

    R 

    s

    ,

    

    s 

    .

    (F‐14)

    .

    R

     Ns  

    The pivot theorem associated homogeneous reactions is obtained from Eq. F‐13

    and the analysis leads to Eq. 6‐80 which is repeated here as

    Pivot Theorem (homogeneous reactions):

    R

     P R

    (F‐15)

    NP

    P

    The pivot theorem associated with heterogeneous reactions is obtained from Eq. F‐14 and is given here as

    Pivot Theorem (heterogeneous reactions):

    R 

     P R 

    s  

    s  (F‐16)

    NP

    P

    The fact that the axiom and the application take exactly the same form for both homogeneous and heterogeneous reactions has led many to ignore the difference between these two distinct forms of chemical reaction.

    In general, measurement of the net rates of production are carried out at the macroscopic level, thus we generally obtain experimental information for the global net rate of production. For a homogeneous reaction, this takes the form R

    R dA ,

    A  1 , 2 , ... , N

    (F‐17)

    A

    A

    V

    while the global net rate of production for a heterogeneous reaction is given by R

    R dA ,

    A  1 , 2 , ... , N

    (F‐18)

    A

    As 

    A 

    Here we note that the global net rates of production for both homogeneous reactions and heterogeneous reactions have exactly the same physical significance, thus it is not unreasonable to use the same symbol for both quantities. Given this simplification, the global version of the pivot theorem can be expressed as

    Pivot Theorem (global form):

    R

     P R

    (F‐19)

    NP

    P

    for both homogeneous and heterogeneous reactions.

    Nomenclature

    A

    area, m2; surface area of the control volume V, m2; absorption factor A ( t)

    a

    surface area of an arbitrary, moving control volume

    ( ) , m2

    a

    V t

    (

    A t)

    surface area of a specific moving control volume V( t) , m2

    A

    area of the entrances and exits for the control volume V , m2

    e

    A ( t)

    area of the entrances and exits for the control volume

    , m2

    e

    a

    V (t)

    A ( t)

    surface area of a body, m2

    m

    AW

    atomic mass of the Jth atomic species, g/mol

    J

    A

    atomic matrix, also identified as  N

    JA

    A

    row reduced echelon form of the atomic matrix, also identified as

    N

    JA

    B

    Bodenstein matrix that maps r onto R

    B

    c

    / MW , molar concentration of species A, mol/m3

    A

    A

    A

    A N

    c

    c

     , total molar concentration, mol/m3

    A

    A 1

    C

    conversion

    D

    diameter, m

    f

    force vector, N

    f

    magnitude of the force vector, N

    g

    gravity vector, m/s2

    g

    magnitude of the gravity vector, m/s2

    h

    height, m

    i, ,

    j k

    unit base vectors

    I

    unit matrix

    K

    equilibrium coefficient for species A

    eq,A

    491

    492

    Material Balance

    L

    length, m

    m

    mass, kg

    m

    mass flow rate, kg/s

    m

    mass of species A, kg

    A

    m

    mass flow rate of species A, kg/s

    A

    M

    molar flow rate of species A, mol/s

    A

    A N

    M

    M

      , total molar flow rate, mol/s

    A

    A  1

    M

    dimensionless molar flow rate

    A

    M

    dimensionless molar flow rate of species A

    MW

    molecular mass of species A, g/mol

    A

    M

    mechanistic matrix that maps r onto R

    M

    n

    outwardly directed unit normal vector

    n

    number of moles of species A

    A

    A N

    n

    n

     , total number of moles

    A

    A  1

    N

    number of molecular species in a multicomponent system

    NJA

    number of J‐type atoms associated with molecular species A

    N

    JA

    atomic matrix, also identified as A

    N

    JA

    row reduced echelon form of the atomic matrix, also

    identified as A

    AN

    p

    p

     , total pressure, N/m2

    A

    A1

    p

    partial pressure of species A, N/m2

    A

    p

    vapor pressure of species A, N/m2

    A,vap

    p

    p p , gauge pressure, N/m2

    g

    o

    Material Balances

    493

    p

    reference pressure (usually atmospheric), N/m2

    o

    P

    pivot matrix that maps R on to R

    P

    NP

    Q

    volumetric flow rate, m3/s

    r

    radial position, m

    r

    rank of a matrix

    r

    column matrix of elementary reaction rates, mol/m3s

    R

    universal gas constant, see Table 5‐1 for units

    r

    net mass rate of production of species A per unit volume, kg/m3s A

    R

    r / MW , net molar rate of production of species A per unit A

    A

    A

    volume, mol/m3s

    A

    R

    global net molar rate of production of species A, mol/s R

    column matrix of net molar rates of production, mol/m3s

    R

    column matrix of non‐pivot species net molar rates of production, NP

    mol/m3s

    NP

    R

    column matrix of non‐pivot species global net molar rates of production, mol/s

    R

    column matrix of pivot species net molar rates of production, P

    mol/m3s

    P

    R

    column matrix of pivot species global net molar rates of production, mol/s

    R

    column matrix of all net rates of production, mol/m3s

    M

    R

    column matrix of net rates of production of Bodenstein products, B

    mol/m3s

    S

    stoichiometric matrix

    S

    selectivity

    T

    temperature, K

    t

    time, s

    u

    v v , mass diffusion velocity of species A, m/s A

    A

    u

    v v , molar diffusion velocity of species A, m/s A

    A

    v

    species A velocity, m/s

    A

    494

    Material Balance

    A N

    v

    v , mass average velocity, m/s

    A A

    A  1

    A N

    v

    x

    v , molar average velocity, m/s

    A A

    A  1

    v

    magnitude of velocity vector, m/s

    v r

    relative velocity, m/s

    V

    volume, m3; volume of a fixed control volume, m3

    ( )

    a

    V t

    volume of an arbitrary moving control volume, m3

    V( t)

    volume of a specific moving control volume, m3

    ( )

    m

    V t

    volume of a body also referred to as a material volume, m3

    v x, v y, v z

    components of the velocity vector, v i v  jv  k v , m/s x

    y

    z

    w n

    speed of displacement of the surface of the arbitrary moving

    control volume

    ( ) , m/s

    a

    V t

    x

    c / c , mole fraction of species A in the liquid phase A

    A

    y

    c / c , mole fraction of species A in the gas phase A

    A

    x, y, z

    rectangular Cartesian coordinates, m

    Y

    yield

    Greek Letters

    unit tangent vector.

    /

    , specific gravity

    H2O

    angle, radians

      , kinematic viscosity, m2/s

    mass density of species A, kg/m3

    A

    AN

     , total mass density, kg/m3

    A

    A1

    Material Balances

    495

    /  , mass fraction of species A

    A

    A

    void fraction, volume of void space per unit volume

    viscosity, P

    specific growth rate,

    3

    1

    ( m )

    s

    surface tension, N/m

    residence time, s

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    499

    Material Balances

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    Review 37, 531 – 551.

    Author Index

    Aguero, A., 497

    Herzfeld, K.F., 407, 497

    Amundson, N., iii, 232, 496

    Hinshelwood, C.N., 362, 497

    Aris, R., iii, 227, 431, 496

    Horn, F., 395, 497

    Askey, P.J., 362, 497

    Hougen, O.A., iii, 497

    Bailey, J.E., 373, 496

    Hurley, J.P., 12, 39, 497

    Barcellos, A., 49, 52, 223, 499

    Ince, S. 16, 64, 87, 499

    Bird, R.B., 379, 496

    Ingle, R.H., 499

    Birkhoff, G., 412, 496

    Jackson, R., 395, 497

    Bjornbom, P.H., 417, 420, 496

    Jürgens, H., 498

    Bodenstein, M., 400, 427, 496

    Kolman, B., 238, 245, 497

    Bradie, B., 324, 496

    Kvisle, S., 254, 497

    Bureau International

    Lavoisier, A.L., 99, 497

    des Poids et Mesures, 19, 496

    Leighton, R.B., 496

    Cassano, A.E., 395, 500

    Levich, V.G., 91, 497

    Corliss, G., 448, 496

    Lightfoot, E.N., 496

    Defay, R., 168, 498

    Lind, S.C., 400, 427, 496

    Denn, M.M., 45, 496

    Lindemann, F.A., 408, 497

    Devi, R.N., 499

    Logan, M.A., 499

    Dixon, D.C., 367, 496

    Lokhande, S.K., 499

    Dunlap, C.E., 10, 499

    Maloney, J. O., 498

    Feynman, R.P., 13, 482, 496

    Manikandan, P., 499

    Flegal, A.R., 10, 499

    Margolies, R., 499

    Frank‐Kamenetsky, D.A., 407, 497

    Menten, M.L., 414, 497

    Gaikwad, T.B., 499

    Michaelis, L., 414, 497

    Garrod, C., 12, 39, 497

    Mono Lake Committee, 81, 93, 498

    Gates, B.C., 393, 497

    Monod, J., 375, 498

    Gentle, T.M., 499

    Muerrerties, E. L., 499

    Gibbs, J.W., 168, 497

    National Institute

    Gibon, Y., 369, 434, 498

    of Science & Technology, 15, 498

    Gleick, J., 459, 497

    Noble, B., 245, 498

    Green, D. W., 498

    Oku, M., 499

    501

    502

    Material Balances

    Ollis, D.F., 373, 496

    Sherman, J.D., 393, 497

    Peitgen, H‐O, 459, 498

    Sherwood, T. K., 164, 498

    Perry, R. H., 19, 84, 498

    Segel, I., 413, 499

    Polanyi, M., 407, 498

    Shreve, 5, 499

    Porter, S.K., 483, 498

    Somorjai, G.A., 499

    Prausnitz, J. M., 498

    Song, H‐S, 416, 498

    Prigogine, I., 168, 498

    Sneeded, R.P.A., 497

    Quintard, M., 500

    Steding, D.J., 10, 499

    Raja. T., 499

    Stein, S.K., 49, 52, 223, 499

    Ramaswany, V., 499

    Stewart, W.E., 496

    Ramkrishna, D., 416, 498

    Tanaka, M., 267, 499

    Ramsperger, H.C., 403, 407, 411, 498

    Toulmin, S.E., 99, 499

    Reid, R. C., 164, 498

    Toupin, R., 233, 499

    Reklaitis, G.V., iii, 498

    Truesdell, C., 18, 233, 499

    Reppe, W., 267, 498

    Watson, K.M., iii, 497

    Rodgers, A., 369, 434, 499

    Wegstein, J.H., 325, 500

    Rouse, H., 17, 64, 87, 499

    Whitaker, S., 189, 395, 488, 500

    Rucker, T.G., 267, 499

    Wisniak, J., 168, 500

    Saeleczky, J., 499

    Wood, B.D., 487, 500

    Sandler, S.I., 36, 61, 162, 499

    Wylie, C.R., 450, 500

    Sands, M., 496

    Yamamote, M., 499

    Sankaranarayanan, T.M., 268, 499

    Ypma, T.J., 451, 500

    Saupe, D, 498

    Subject Index

    A

    C

    absorption, 3, 109

    capillary rise, 91

    absorption factor, 178, 195

    cell growth, 4, 368, 413

    air, molecular mass, 162

    charged species, 482

    air, theoretical, 280

    chemical kinetics, 239, 391, 395

    air conditioner, 184

    chemical reaction rate, 228, 360

    air dryer, 189

    chemical reaction rate equation, 229,

    Amagat’s law, 158

    400

    chemical potential, 169, 220

    Andrussov process, 341

    chemostat, 4, 413

    Antoine’s equation, 166, 445

    Clausius‐Clapeyron equation, 165

    area average concentration, 116

    coating flow, 59, 79

    array operations, 24

    coefficient of compressibility, 163

    atmospheric pressure, 22, 167, 171

    coefficient of thermal expansion, 163

    atomic mass, 439

    combustion, 2, 99, 277

    atomic matrix, 232, 258

    conservation of charge, 482

    atomic species, 231

    conservation of mass, 39

    contamination, lead, 10

    atomic species balance, 470

    control volume, 9, 43, 96

    atomic species indicator, 232

    control volume, construction, 55

    axioms, 1, 96, 231

    control volume, fixed, 44, 55, 67, 96

    B

    control volume, moving, 64, 67, 101

    conversion, 271

    batch distillation, 375

    conversion factors, 20

    batch reactor, 360

    conversion of units, 27

    bisection method, 73, 448

    D

    biomass production, 368

    Dalton’s laws, 157

    Bodenstein matrix, 417, 427, 432

    degrees of freedom, 121

    Bodenstein products, 230, 407

    density, 42, 102

    body, 42

    density, bulk, 57

    Boudouard reaction, 341

    density, cup mixed, 57

    bubble point, 170

    density, species, 97, 102

    density, total, 104

    buoyancy force, 93

    dew point, 170

    diffusive flux, 112

    diffusion velocity, 110

    503

    504

    Material Balances

    dimensional homogeneity, law of, 19

    I

    discharge coefficient, 64

    ideal gas, 61, 156

    distribution coefficient, 177

    ideal gas mixtures, 156

    distillation, 119, 218

    ideal liquid mixtures, 164

    drying, 189

    identity matrix, 138

    E

    integrating factor, 359

    electric charge, 13

    ionic species, 483

    elementary row operations 140, 241

    iteration methods, 448

    energy, 17

    iteration methods, stability, 456

    equilibrium, 168

    K

    equilibrium coefficient, 177, 207, 366

    kinetics, mass action, 404

    equilibrium line, 203

    kinetics, reaction, 395

    equilibrium relation, 168, 177, 179

    L

    equilibrium stage, 176, 195

    Lavoisier, 99

    Euler cut principle, 40

    lead contamination, 11

    extraction, 177

    length, standard of, 12

    extraction, multistage, 193

    liquid‐liquid extraction, 177, 193

    F

    local reaction equilibrium, 410, 426

    local thermodynamic equilibrium,

    false position method, 450

    168

    flowsheet, 6, 319

    force, 17

    M

    mass, 1, 12, 39

    G

    mass action kinetics, 395, 404

    Gaussian elimination, 140

    mass average velocity, 106, 110

    Gauss‐Jordan method, 144

    mass diffusion velocity, 110

    graphical analysis, 205

    mass balance, species, 97

    mass flow rate, 56, 118

    H

    mass flux, 53, 106

    Henry’s law, 169, 212

    mass fraction, 104, 156

    heterogeneous reaction, 2, 487

    matrices, 29, 135, 428, 462,

    homogeneous reaction, 2, 489

    matrix algebra, 135, 462

    humidity, 172

    matrix operations, 29, 135, 462

    hydraulic ram, 90

    matrix partitioning, 252, 462

    hydrogen bromide, 422

    mechanistic matrix, 417, 425

    Michaelis‐Menten kinetics, 413

    mixer‐settler, 180

    Material Balances

    505

    mixer, 290, 356

    pivot theorem, 261, 431

    molar flux, 112, 377

    pivot theorem, global form, 431

    molar average velocity, 112

    pressure 17, 20, 22

    mole, 14

    pressure, gauge, 23

    process equilibrium relation, 179,

    mole balance, 113

    195, 379

    mole fraction, 104, 156

    projected area theorem, 49

    mole fraction, modified, 175

    purge stream, 296

    molecular mass, 15, 228

    R

    Mono Lake, 4, 81

    rank, 233, 258

    Monod equation, 375, 413

    Raoult’s law, 169

    moving control volumes, 64, 101

    reaction, chemical 228

    multi‐component systems, 96

    reaction, heterogeneous, 2, 487

    reaction, homogeneous, 2, 489

    N

    reaction kinetics, 395

    net rate of production, 99, 233, 246

    reaction rate, 367

    newton, 18

    reactive intermediates, 407

    Newton’s method, 451

    recycle streams, 287, 296

    non‐pivot species, 250, 254

    relative humidity, 173, 352

    non‐pivot submatrix, 257

    relative volatility, 169

    residence time, 357

    O

    reversible reaction, 364

    operating line, 203

    row echelon form, 243, 255

    optical isomers, 261, 268

    row reduced echelon form, 243

    P

    row reduced matrix, 265

    partial pressure, 157, 168

    S

    partial molar Gibbs free energy, 168

    saturation, 170

    perfect mixer, 355

    scrubber, 3

    performance, 7

    selectivity, 271

    phase equilibrium, 177

    separator, 269, 297, 320

    physical properties, 442

    sequential analysis, 203, 318

    slide coating, 78

    Picard’s method, 324, 334, 453

    slot die coating, 80

    pinch point, 209

    solubility, 220

    pivot matrix, 250, 261

    species mass density, 102

    pivot species, 241, 250

    species mole/mass balance, 119

    pivot submatrix, 261

    species molar concentration, 99

    506

    Material Balances

    species velocity, 107, 112

    Taylor series, 212, 221

    specific gravity, 33

    temperature, 14

    specific growth rate, 374

    theoretical air, 278

    splitter, 290

    Torricelliʹs law, 87

    stability of iteration methods, 456

    transient processes, 354

    standard cubic foot, 37

    U

    steady state hypothesis, 410

    units, 12

    stoichiometric equations, 239

    units, convenience, 22

    stoichiometric coefficients, 229, 238,

    units, derived, 16

    257

    units, SI, 13

    stoichiometric matrix, 417, 427, 432

    V

    stoichiometric schema, 229

    vapor pressure, 164

    stoichiometric schemata, 408

    velocity vector, 51

    stoichiometric skepticism, 230, 395

    velocity, average, 47

    velocity, mass average, 106, 110

    stoichiometry, 403

    velocity, molar average, 112

    stoichiometry, elementary, 403

    velocity, relative, 66, 106

    stoichiometry, global, 397

    velocity, species, 107, 112

    stoichiometry, local, 397, 403

    structure, 6

    W

    synthesis gas, 283

    Wegstein’s method, 325, 335, 455

    Système International, 13

    Y

    T

    yield, 271,

    tank, perfectly mixed, 355

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