# 15.2: Untitled Page 236

- Page ID
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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

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*

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*

*c***v ****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

*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.

*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

*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

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

*M* *M*

(16a)

2

2

1

M

*M* *M*

(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*

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

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

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

*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

*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

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*.

*Material Balances *

487

Appendix F

Heterogeneous Reactions

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

*A* *N*

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*

*NJARA* *dV*

*A*1

*A*

*V*

*V*

1

(F‐3)

*A * *N*

*A * *N*

*N*

*JARA* *dV *

*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

*NJARA* *dV * *NJARA* *dV*

*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

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

*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.

*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

*A* *N*

*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

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 **j**v **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*

*A* *N*

, total mass density, kg/m3

*A*

*A*1

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

Amundson, N.R. 1966, *Mathematical Methods in Chemical Engineering: Matrices and* *Their Application*, Prentice‐Hall, Inc., Englewood Cliffs, New Jersey.

Aris, R. 1965, *Introduction to the Analysis of Chemical Reactors*, Prentice‐Hall, Inc., Englewood Cliffs, New Jersey.

Aris, R. 1965, Prolegomena to the rational analysis of systems of chemical reactions, Archive for Rational Mechanics and Analysis, **19**, 81‐99.

Bailey, J.E. and Ollis, D.F. 1986, *Biochemical Engineering Fundamentals*, Sec. 7.7, McGraw Hill Higher Education, 2nd Edition, New York.

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. 2002, *Transport Phenomena*, Second Edition, John Wiley & Sons, Inc., New York.

Birkhoff, G. 1960, *Hydrodynamics: A Study in Logic, Fact, and Similitude*, Princeton University Press, Princeton, New Jersey.

Bjornbom, P.H. 1977, The relation between the reaction mechanism and the stoichiometric behavior of chemical reactions, AIChE Journal **23**, 285 – 288.

Bodenstein, M. and Lind, S.C. 1907, Geschwindigkeit der Bildung der Bromwasserstoffes aus sienen Elementen, Z. physik. Chem. **57**, 168 – 192.

Bradie, B. 2006, *A Friendly Introduction to Numerical Analysis*, Pearson Prentice Hall, New Jersey.

Bureau International des Poids et Mesures, http://www.bipm.fr/enus/3_SI/si-

Corliss, G. 1977, Which root does the bisection method find?, SIAM Review **19**, 325 – 327.

Denn, M.M. 1980, Continuous drawing of liquids to form fibers, Ann. Rev. Fluid.

Mech. **12**, 365 – 387.

Dixon, D.C. 1970, The definition of reaction rate, Chem. Engr. Sci. **25**, 337‐338.

Feynman, R.P., Leighton, R.B. and Sands, M. 1963, *The Feynman Lectures on* *Physics*, Addison‐Wesley Pub. Co., New York.

Frank‐Kamenetsky, D.A. 1940, Conditions for the applicability of Bodenstein’s method in chemical kinetics, J. Phys. Chem. (USSR), 14, 695 – 702.

496

*Material Balances*

Gates, B.C. and Sherman, J.D. 1975, Experiments in heterogeneous catalysis: Kinetics of alcohol dehydration reactions, Chem. Eng. Ed. Summer, 124 – 127.

Gibbs, J.W. 1928, *The Collected Works of J. Willard Gibbs*, Longmans, Green and Company, London.

Gleick, J. 1988 *Chaos: Making a New Science*, Penguin Books, New York.

Herzfeld, K.F. 1919, The theory of the reaction speeds in gases, Ann. Physic **59**, 635 – 667.

Hinshelwood, C.N. and Askey, P.J. 1927, Homogeneous reactions involving complex molecules: The kinetics of the decomposition of gaseous dimethyl ether, Proc. Roy. Soc. **A115**, 215 – 226.

Horn, F. and Jackson, R. 1972, General mass action kinetics, Arch Rat Mech **47**, 81

– 116.

Hougen, O.A. and Watson, K.M. 1943, *Chemical Process Principles*, John Wiley & Sons, Inc., New York.

Hurley, J.P. and Garrod, C. 1978, *Principles of Physics*, Houghton Mifflin Co., Boston.

Kolman, B. 1997, *Introductory Linear Algebra*, Sixth Edition, Prentice‐Hall, Upper Saddle River, New Jersey,

Kvisle, S., Aguero, A. and Sneeded, R.P.A. 1988, Transformation of ethanol into 1,3‐butadiene over magnesium oxide/silica catalysts, Applied Catalysis, **43**, 117 –

121.

Lavoisier, A. L. 1777, Memoir on Combustion in General, *Mémoires de L’Academie* *Royal des Sciences* 592 – 600.

Levich, V.G. 1962, *Physicochemical Hydrodynamics*, Prentice‐Hall, Inc., Englewood Cliffs, N.J.

Lindemann, F.A. 1922, The radiation theory of chemical action, Trans. Faraday Soc. **17**, 598 – 606.

Michaelis, L. and Menten, M.L. 1913, Die Kinetik der Invertinwirkung, Biochem Z **49**, 333 – 369.

Mono Lake Committee, http://www.monolake.org

Monod, J. 1942, Recherche sur la Croissance des Cultures Bactériennes, Herman Editions, Paris.

498

Monod, J. 1949, The growth of bacterial cultures, Ann. Rev. Microbiol. **3**, 371 –

394.

National Institute of Standards and Technology,

http://physics.nist.gov/cuu/Units/prefixes.html.

Noble, B. 1969, *Applied Linear Algebra*, Prentice‐Hall, Inc., Englewood Cliffs, New Jersey.

Peitgen, H‐O., Jürgens, H., and Saupe, D., 1992, *Chaos and Fractals. New Frontiers* *of Science*, Springer‐Verlag, New York.

Perry, R. H., Green, D. W., and Maloney, J. O., 1984, *Perryʹs Chemical Engineerʹ *

*Handbook, 6th Edition, * New York, McGraw‐Hill Books.

Perry, R. H., Green, D. W., and Maloney, J. O., 1997, *Perryʹs Chemical Engineerʹ *

*Handbook, 7th Edition, * New York, McGraw‐Hill Books.

Polanyi, M. 1920, Reaction isochore and reaction velocity from the standpoint of statistics, Z. Elektrochem. **26**, 49 – 54.

Porter, S.K. 1985, How should equation balancing be taught?, J. Chem. Education **62**, 507 – 508.

Prigogine, I. and Defay, R. 1954, *Chemical Thermodynamics*, Longmans Green and Company, London.

Reklaitis, G.V. 1983, *Introduction to Material and Energy Balances*, John Wiley & Sons, Inc. New York

Ramkrishna, D. and Song, H‐S, 2008, A Rationale for Monod’s Biochemical Growth Kinetics, Ind. Eng. Chem. Res. 47, 9090 – 9098.

Ramsperger, H.C. 1927, Thermal decomposition of azomethane over a large range of pressures, J. Am. Chem. Soc. **49**, 1495 – 1512.

Reid, R. C., Prausnitz, J. M., and Sherwood, T. K., 1977, *The Properties of Gases and* *Liquids*, Sixth Edition, New York, McGraw‐Hill Books.

Reppe, W.J. 1892 – 1969. A German scientist notable for his contributions to the chemistry of acetylene.

Rodgers, A. and Gibon, Y. 2009, Enzyme Kinetics: Theory and Practice, Chapter 4

in *Plant Metabolic Networks*, edited by J. Schwender, Springer, New York.

Rouse, H. and Ince, S. 1957, *History of Hydraulics*, Dover Publications, Inc., New York.

*Material Balances*

Rucker, T.G., Logan, M.A., Gentle, T.M., Muetterties, E.L. and Somorjai, G.A.

1986, Conversion of acetylene to Benzene over palladium single‐crystal surfaces.

1. The low‐pressure stoichiometric and high‐pressure catalytic reactions, J. Phys.

Chem. **90**, 2703 – 2708.

Sandler, S.I. 2006, *Chemical, Biochemical, and Engineering Thermodynamics*, 4th edition, John Wiley and Sons, New York.

Sankaranarayanan, T.M., Ingle, R.H., Gaikwad, T.B., Lokhande, S.K., Raja. T., Devi, R.N., Ramaswany, V., and Manikandan, P. 2008, Selective oxidation of ethane over Mo‐V‐Al‐O oxide catalysts: Insight to the factors affecting the selectivity of ethylene and acetic acid and structure‐activity correlation studies, Catal. Lett., **121**: 39 – 51.

Segel, I. 1993, *Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and* *Steady‐State Enzyme Systems*, Wiley‐Interscience, New York.

*Shreve’s Chemical Process Industries*, 1998, 5th edition, edited by J. Saeleczky and R.

Margolies, McGraw‐Hill Professional, New York.

Steding, D.J., Dunlap, C.E. and Flegal, A.R. 2000, New isotopic evidence for chronic lead contamination in the San Francisco Bay estuary system: Implications for the persistence of past industrial lead emissions in the biosphere, *Proceedings* *of the National Academy of Science* **97** (19).

Stein, S.K. and Barcellos, A. 1992, *Calculus and Analytic Geometry*, McGraw‐Hill, Inc., New York.

Tanaka, M., Yamamote, M. and Oku, M. 1955, Preparation of styrene and benzene from acetylene and vinyl acetylene, USPO, 272 – 299.

Toulmin, S.E. 1957, Crucial Experiments: Priestley and Lavoisier, Journal of the History of Ideas, **18**, 205 – 220.

Truesdell, C. 1968, *Essays in the History of Mechanics*, Springer‐Verlag, New York.

Truesdell, C. and Toupin, R. 1960, The Classical Field Theories, in *Handbuch der* *Physik*, Vol. III, Part 1, edited by S. Flugge, Springer‐Verlag, New York.

Wegstein, J.H. 1958, Accelerating convergences of iterative processes, Comm.

ACM **1**, 9 – 13.

Whitaker, S. and Cassano, A.E. 1986, *Concepts and Design of Chemical Reactors*, Gordon and Breach Science Publishers, New York.

500

Whitaker, S. 1988, Levels of simplification: The use of assumptions, restrictions, and constraints in engineering analysis, Chem. Eng. Ed. **22**, 104 – 108.

Whitaker, S. 1992, The species mass jump condition at a singular surface, Chem.

Engng. Sci. ** 47, ** 1677 – 1685.

Wisniak, J. 2001, Historical development of the vapor pressure equation from Dalton to Antoine, J. Phase Equilibrium **22**, 622 – 630.

Wood, B.D., Quintard, M. and Whitaker, S. 2000, Jump conditions at non-uniform boundaries: The catalytic surface, Chem. Engng. Sci. **55**, 5231 – 5245.

Wylie, C.R., Jr. 1951, *Advanced Engineering Mathematics*, McGraw‐Hill Book Co., Inc., New York.

Ypma, T.J. 1995, Historical development of the Newton‐Raphson method, SIAM

Review **37**, 531 – 551.

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

*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

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

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