15.3: Untitled Page 235
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- 18368
Chapter 9
k IV
Elementary chemical kinetic schema IV:
NO
NO
N O
(5)
2
3
2
5
Since neither NO nor NO appear in the global stoichiometric schema given by 3
Eq. 1, one can assume that these two compounds are reactive intermediates or Bodenstein products and their rates of production can be set equal to zero as an approximation
9‐15. In Problem 9‐14 we considered the decomposition of N O to produce 2
5
NO and O by means of the kinetic schemata illustrated by Eqs. 2 through 5 in 2
2
Problem 9‐14. The reactive intermediates were identified as NO and NO . The 3
rate equations for these two reactive intermediates and for N O are given by 2
5
R
k c
k c
c
k c
c
k c
c
(1)
NO3
I N2O5
II NO3 NO2
III NO NO3
IV NO2 NO3
R
k c
c
k c
c
(2)
NO
II NO3 NO2
III NO NO3
R
k c
k c
c
(3)
N2O5
I N2O5
IV NO2 NO3
If the condition of local reaction equilibrium is imposed according to Local reaction equilibrium:
R
0 ,
R
0
(4)
NO3
NO
one can obtain a simple representation for R
in terms of only c
. In this
N2O5
N2O5
problem you are asked to replace the assumptions given by Eqs. 4 with restrictions indicating that R
and R
are small enough so that Eqs. 4 become acceptable
NO3
NO
approximations.
Section 9.4
9‐16. If species C in Eq. 9‐88 is a reactive intermediate or Bodenstein product, identify the stoichiometric matrix and the Bodenstein matrix associated with Eq. 9‐88. Impose the condition of local reaction equilibrium on the Bodenstein product in order to derive an expression for R in terms of c , c and c .
E
A
B
D
Appendix A
A1 Atomic Mass of Common Elements Referred to Carbon‐12
(Details are available at http://www.nist.gov/physlab/data/comp.cfm)
Element
Symbol
Atomic mass, g/mol
Aluminum
Al
26.9815
Antimony
Sb
121.75
Argon
Ar
39.948
Arsenic
As
74.9216
Barium
Ba
137.34
Beryllium
Be
9.0122
Bismuth
Bi
208.980
Boron
B
10.811
Bromine
Br
79.904
Cadmium
Cd
112.40
Calcium
Ca
40.08
Carbon
C
12.01
Cerium
Ce
140.12
Cesium
Cs
132.905
Chlorine
Cl
35.453
Chromium
Cr
51.996
Cobalt
Co
58.9332
Copper
Cu
63.546
Fluorine
F
18.9984
Gallium
Ga
69.72
Germanium
Ge
72.59
Gold
Au
196.967
Hafnium
Hf
178.49
Helium
He
4.0026
Hydrogen
H
1.00797
Indium
In
114.82
Iodine
I
126.904
Iridium
Ir
192.2
Iron
Fe
55.847
Krypton
Kr
83.80
439
440
Appendices
Element
Symbol
Atomic mass, g/mol
Lead
Pb
207.19
Lithium
Li
6.939
Magnesium
Mg
24.312
Manganese
Mn
54.938
Mercury
Hg
200.59
Molybdenum
Mo
95.94
Neon
Ne
20.183
Nickel
Ni
58.71
Niobium
Nb
92.906
Nitrogen
N
14.0067
Oxygen
O
15.9994
Palladium
Pd
106.4
Phosphorus
P
30.9738
Platinum
Pt
195.09
Plutonium
Pu
242
Potassium
K
39.102
Radium
Ra
226
Radon
Rn
222
Rhodium
Rh
102.905
Rubidium
Rb
85.47
Selenium
Se
78.96
Silicon
Si
28.086
Silver
Ag
107.868
Sodium
Na
22.9898
Strontium
Sr
87.62
Sulfur
S
32.064
Tantalum
Ta
180.948
Tellurium
Te
127.60
Thallium
Tl
204.37
Thorium
Th
232.038
Tin
Sn
118.69
Titanium
Ti
47.90
Tungsten
W
183.85
Uranium
U
238.03
Vanadium
V
50.942
Xenon
Xe
131.30
441
Element
Symbol
Atomic mass, g/mol
Yttrium
Y
88.905
Zinc
Zn
65.37
Zirconium
Zr
91.22
442
Appendices
A2 Physical Properties of Various Chemical Compounds
Name
Formula
Molecular
Liquid @ T(K)
T melting T boiling
Mass, g/mol
density (g/L)
K
K
Argon
Ar
39.948
83.8
87.3
Acetaldehyde
C2H4O
44.054
778
293
150.2
293.6
Acetic acid
C2H4O2
60.052
1049
293
289.8
391.1
Acetone
C3H6O
58.08
790
293
178.2
329.4
Acetylene
C2H2
26.038
192.4
189.2
Acrylic acid
C3H4O2
72.064
1051
293
285
414
Ammonia
NH3
17.031
195.4
239.7
Aniline
C6H7N
93.129
1022
293
267
457.5
Benzaldehide
C7H6O
106.124
1045
293
216
452
Benzene
C6H6
78.114
885
289
278.7
353.3
Benzoic acid
C7H6O2
122.124
1075
403
395.6
523
Bromine
Br2
159.808
3119
293
266
331.9
1,2‐Butadiene
C4H6
54.092
137
284
1,3‐Butadiene
C4H6
54.092
164.3
268.7
n‐Butane
C4H10
58.124
134.8
272.7
i‐Butane
C4H10
58.124
113.6
261.3
n‐Butanol
C4H10O
74.123
810
293
183.9
390.9
1‐Butene
C4H8
56.108
87.8
266.9
i‐Butene
C4H8
56.108
132.8
266.3
Carbon
tetrachloride
CCl4
153.823
1584
298
250
349.7
Carbon dioxide
CO2
44.01
216.6
194.7
Carbon monoxide
CO
28.01
68.1
81.7
Chlorine
Cl2
70.906
172.2
238.7
Chlorobenzene
C6H5Cl
112.559
1106
293.
227.6
404.9
Chloroform
CHCl3
119.378
1498
293
209.6
334.3
Cyclobutane
C4H8
56.108
182.4
285.7
Cyclohexane
C6H12
84.162
779
293
279.7
353.9
Cyclohexanol
C6H12O
100.161
942
303
298
434.3
Cyclopentane
C5H10
70.135
745
293
179.3
322.4
443
Name
Formula
Molecular
Liquid @ T(K)
T melting T boiling
Mass, g/mol
density (g/L)
K
K
Cyclopentene
C5H8
68.119
772
293
138.1
317.4
Ethane
C2H6
30.07
89.9
184.5
Ethanol
C2H6O
46.069
789
293
159.1
351.5
Ethyl amine
C2H7N
45.085
683
293
192
289.7
Ethyl acetate
C4H8O2
88.107
901
293
89.6
350.3
Ethylbenzene
C8H10
106.168
867
293
178.2
409.3
Ethylendiamine
C2H8N2
60.099
896
293
284
390.4
Ethyl ether
C4H10O
74.123
713
293
156.9
307.7
Ethyl propionate
C5H10O2 102.134
895
293
199.3
372
Ethylene
C2H4
28.054
104
169.4
Ethylene Glycol
C2H6O2
62.069
1,114
293
260.2
470.4
Ethylene oxide
C2H4O
44.054
161
283.5
Fluorine
F2
37.997
53.5
85
Formaldehide
CH2O
30.026
156
254
Formic acid
CH2O2
46.025
1,226
288
281.5
373.8
Glycerol
C3H8O3
92.095
1,261
293
291
563
n‐Heptane
C7H16
100.205
684
293
182.6
371.6
1‐Heptanol
C7H16O
116.204
822
293
239.2
449.5
1‐Heptene
C7H14
98.189
697
293
154.3
366.8
n‐Hexane
C6H14
86.178
659
293
177.8
341.9
1‐Hexanol
C6H14O
102.177
819
293
229.2
430.2
Hydrogen
H2
2.016
14.0
20.4
Hydrogen bromide HBr
80.912
187.1
206.1
Hydrogen chloride
HCl
36.461
159.0
188.1
Hydrogen cyanide
CHN
27.026
688
293
259.9
298.9
Hydrogen sulfide
H2S
34.08
187.6
212.8
Iodine
I2
253.808
3,740
453
386.8
457.5
Isopropyl alcohol
C3H8O
60.096
786
293
184.7
355.4
Maleic anhydride
C4H2O3
98.058
1,310
333
326
472.8
Methane
CH4
16.043
90.7
111.7
Mercury
Hg
200.59
13,546
293
234.3
630.1
Methanol
CH4O
32.042
791
293
175.5
337.8
Methyl acetate
C3H6O2
74.08
934
293
175
330.1
444
Appendices
Name
Formula
Molecular
Liquid @ T(K)
T melting T boiling
Mass, g/mol
density (g/L)
K
K
Methyl acrylate
C4H7O2
86.091
956
293
196.7
353.5
Methyl amine
CH5N
31.058
179.7
266.8
Methyl benzoate
C8H8O2
136.151
1,086
293
260.8
472.2
Methyl ethyl ketone C4H8O
72.107
805
293
186.5
352.8
Naphtalene
C10H8
128.174
971
363
353.5
491.1
Nitric oxide
NO
30.006
109.5
121.4
Nitrogen
N2
28.013
63.3
77.4
Nitrogen dioxide
NO
30.01
112.2
122.2
Nitrogen tetroxide
N2O4
46.006
261.9
294.3
Nitrous oxide
N2O
44.013
182.3
184.7
Oxygen
O2
31.999
54.4
90.2
n‐Pentane
C5H12
72.151
626
293
143.4
309.2
1‐Pentanol
C5H12O
88.15
815
293
195
411
1‐Pentyne
C5H8
68.119
690
293
167.5
313.3
1‐Pentene
C5H10
70.135
640
293
107.9
303.1
Phenol
C6H6O
94.113
1059
314
313
455
Propane
C3H8
44.097
85.5
231.1
1‐Propanol
C3H8O
60.096
804
293
146.9
370.4
Propionic acid
C3H6O2
74.08
993
293
252.5
414
Propylene
C3H6
42.081
87.9
225.4
Propylene oxide
C3H6O
58.08
829
293
161
307.5
Styrene
C8H8
104.152
906
293
242.5
418.3
Succinic acid
C4H6O4
118.09
456
508
Sulfur dioxide
SO2
64.063
197.7
263
Sulfur trioxide
SO3
80.058
1780
318
290
318
Toluene
C7H8
92.141
867
293
178
383.8
Trimethyl amine
C3H9N
59.112
156
276.1
Vinyl chloride
C2H3Cl
62.499
119.4
259.8
Water
H2O
18.015
998
293
273.2
373.2
o‐Xylene
C8H10
106.168
880
293
248
417.6
m‐Xylene
C8H10
106.168
864
293
225.3
412.3
p‐Xylene
C8H10
106.168
861
293
286.4
411.5
445
A3 Constants for Antoine’s Equation
o
log p
A B / ( T) , p is in mm Hg and T is in C
vap
vap
Compound
Formula
A
B
Acetaldehyde
CH3CHO
7.0565
1070.6
236.0
Acetic acid
CH3COOH
7.29963
1479.02
216.81
Acetone
CH3COCH3
7.23157
1277.03
237.23
Acetylene
C2H2
7.0949
709.1
253.2
Acrylic acid
C2H3COOH
7.1927
1441.5
192.66
Ammonia
NH3
7.36050
926.132
240.17
Aniline
C6H7N
7.2418
1675.3
200.01
Benzaldehide
C7H6O
7.1007
1628.00
207.04
Benzene
C6H6
6.90565
1211.03
220.790
Benzoic acid
C7H6O2
7.45397
1820
147.96
1,2‐Butadiene
C4H6
7.1619
1121.0
251.00
1,3‐Butadiene
C4H6
6.85941
935.531
239.554
n‐Butane
C4H10
6.83029
945.90
240.00
n‐Butanol
C3H7CH2OH
7.4768
1632.39
178.83
i‐Butane
C4H10
6.74808
882.80
240.00
n‐Butene
C4H8
6.84290
926.10
240.00
i‐Butene
C4H8
6.84134
923.200
240.00
Carbon
tetrachloride
CCl4
6.8941
1219.58
227.17
Chlorobenzene
C6H5Cl
6.9781
1431.05
217.56
Chloroform
CHCl3
6.93707
1171.2
236.01
Cyclobutane
C4H8
6.92804
1024.54
241.38
Cyclohexane
C6H12
6.84498
1203.526
222.863
Cyclopentane
C5H10
6.88678
1124.16
231.37
Cyclopentene
C5H8
6.92704
1121.81
233.46
Ethane
C2H6
6.80266
656.40
256.00
Ethanol
CH3CH2OH
8.16290
1623.22
228.98
Ethyl amine
C2H7N
7.38618
1137.3
235.86
Ethyl acetate
C4H8O2
7.01455
1211.9
216.01
Ethene (Ethylene)
C2H4
6.74756
585.00
255.00
446
Appendices
Compound
Formula
A
B
Ethylbenzene
C6H5C2H5
6.95719
1424.55
213.206
Ethylenediamine
C2H8N2
7.12599
1350.0
201.03
Ethyl ether
C4H10O
6.98467
1090.64
231.21
Ethyl propionate
C5H10O2
7.01907
1274.7
209.0
Ethylene glycol
C2H6O2
8.7945
2615.4
244.91
Formaldehyde
HCHO
7.1561
957.24
243.0
Formic acid
HCOOH
7.37790
1563.28
247.06
Glycerol
C3H8O3
7.48689
1948.7
132.96
n‐ Heptane
C7H16
6.90240
1268.115
216.900
1‐Heptanol
C7H16O
6.64766
1140.64
126.56
1‐Heptene
C7H14
6.90068
1257.5
219.19
n‐ Hexane
C6H14
6.87776
1171.530
224.366
Hydrogen bromide HBr
6.28370
539.62
225.30
Hydrogen chloride
HCl
7.167160
744.49
258.704
Hydrogen cyanide
HCN
7.17185
1123.0
236.01
Hydrogen sulfide
H2S
6.99392
768.13
247.093
Iodine (c)
I2
9.8109
2901.0
256.00
Isopropyl alcohol
C3H8O
8.11822
1580.92
219.62
Maleic anhydride
C4H2O3
7.06801
1635.4
191.01
Methane
CH4
6.61184
389.93
266.00
Methanol
CH3OH
8.07246
1574.99
238.86
Methyl acetate
C3H6O2
7.00495
1130.0
217.01
Methyl acrylate
C4H7O2
6.99596
1211.0
214.01
Methyl amine
CH5N
7.49688
1079.15
240.24
Methyl benzoate
C8H8O2
7.04738
1629.4
192.01
Methyl ethyl ketone C4H8O
7.20868
1368.21
236.51
Napthalene
C10H8
6.84577
1606.5
187.227
Nitric oxide
NO
8.74300
682.94
268.27
Nitrogen tetroxide
N2O4
7.38499
1185.72
234.18
Nitrous oxide
N2O
7.00394
654.26
247.16
n‐ Pentane
C5H12
6.85221
1064.63
232.000
1‐Pentanol
C5H12O
7.17758
1314.56
168.16
1‐Pentyne
C5H8
6.96734
1092.52
227.19
1‐Pentene
C5H10
6.84648
1044.9
233.53
447
Compound
Formula
A
B
Phenol
C6H5OH
7.13457
1516.07
174.569
Phosphorus
trichloride
PCl3
6.8267
1196
227.0
Phosphine
PH3
6.71559
645.512
256.066
Propane
C3H8
6.82973
813.20
248.00
1‐Propanol
CH3CH2CH2OH
6.79498
969.27
150.42
Propionic acid
C3H6O2
7.57456
1617.06
205.68
Propene
(Propylene)
C3H6
6.81960
785.00
247.00
Propylene oxide
C3H6O
6.65456
915.31
208.29
n‐Propionic
Acid
CH3CH2COOH
7.54760
1617.06
205.67
Styrene
C8H8
6.95709
1445.58
209.44
Sulfur dioxide
SO2
7.28228
999.900
237.190
Sulfur trioxide
SO3
9.05085
1735.31
236.50
Toluene
C6H5CH3
6.95464
1344.800
219.482
Trimethyl amine
C3H9N
6.97038
968.7
234.01
Vynil chloride
C2H3Cl
6.48709
783.4
230.01
Water
H2O
7.94915
1657.46
227.03
o‐Xylene
C6H5(CH3)2
6.99891
1474.679
213.686
p‐Xylene
C6H5(CH3)2
6.99052
1453.430
215.307
448
Appendices
Appendix B
Iteration Methods
B1. Bisection method
Given some function of x such as H( x) , we are interested in the solution of the equation
H( x) 0 ,
x x
(B‐1)
Here we have used x to represent the solution. For simple functions such as
H( x) x b we obtain a single solution given by x b , while for a more complex function such as
2
H( x) x b we obtain more that one solution as
indicated by x b . In many cases there is no explicit solution for Eq. B‐1. For example, if H( x) is given by
H( x) a sin ( x 2 ) b cos(2 x
)
(B‐2)
we need to use iterative methods to determine the solution x x .
The simplest iterative method is the bisection method (Corliss, 1977) that is illustrated in Figure B‐1. This method begins by locating x and x such that o
1
H( x ) and H( x ) have different signs. In Figure B‐1 we see that x and x have o
1
o
1
been chosen so that there is a change of sign for H( x) , i.e., H( x ) 0 ,
H( x ) 0
(B‐3)
o
1
Thus if H( x) is a continuous function we know that a solution H( x ) 0 exists somewhere between x and x . We attempt to locate that solution by means of o
1
a guess (i.e., the bisection) indicated by
x x
o
1
x
2
(B‐4)
2
As illustrated in Figure B‐1, this guess is closer to the solution, x x , than either x or x , and if we repeat this procedure we will eventually find a value of x o
1
Material Balances
449
Figure B‐1. Illustration of the bisection method
that produces a value of H( x) that is arbitrarily close to zero. In terms of the particular graph illustrated in Figure B‐1, it is clear that x will be located 3
between x and x ; however, this need not be the case. For example, in 1
2
Figure B‐2 we have represented a slightly different function for which x will be 3
located between x and x . The location of the next guess is based on the idea o
2
that the function H( x) must change sign. In order to determine the location of the next guess we examine the products H( x ) H( x
) and H( x ) H( x
) in
n
n1
n
n2
order to make the following decisions:
n
x
n
x 1
if H( x ) H(
n
n
x 1)
0 ,
then
n
x 1
2
(B‐5)
n
x
n
x 2
if H( x ) H(
n
n
x 2)
0 ,
then n
x 1
2
Since these two choices are mutually exclusive there is no confusion about the next choice of the dependent variable. The use of Eqs. B‐5 is crucial when the details of H( x) are not clear, and a program is written to solve the implicit equation.
450
Appendices
Figure B‐2. Alternate choice for the second bi‐section B2. False position method
The false position method is also known as the method of interpolation (Wylie, 1951) and it represents a minor variation of the bisection method. Instead of bisecting the distance between x and x in Figure B‐1 in order to locate the o
1
point x , we use the straight line indicated in Figure B‐3. Sometimes this line is 2
called the secant line. The definition of the tangent of the angle provides 0 H( x )
H( x ) H( x )
1
o
1
tan
(B‐6)
x
2
1
x
o
x
1
x
and we can solve for x to obtain
2
x x H( x )
o
1
1
x
2
1
x
(B‐7)
H(
o
x )
H( 1
x )
This replaces Eq. B‐4 in the bisection method and it can be generalized to obtain
x x
H( x
)
n
n1
n 1
n
x 2
n
x 1
(B‐8)
H( x ) H(
n
n
x 1)
Material Balances
451
Application of successive iterations will lead to a value of x that approaches x shown in Figure B‐3.
Figure B‐3. False position construction
B3. Newton’s method
Newton’s method (Ypma, 1995), which is also known as the Newton‐Raphson method, is named for Sir Isaac Newton and is perhaps the best known method for finding roots of real valued functions. The method is similar to the false position method in that a straight line is used to locate the next estimate of the root of an equation; however, in this case it is a tangent line and not a secant line.
This is illustrated in Figure B‐4a where we have chosen x as our first estimate of o
the solution to Eq. B‐1 and we have constructed a tangent line to H( x) at x . The o
slope of this tangent line is given by
dH
H( x ) 0
o
(B‐9)
dx
x x
o
x
1
x
o
and we can solve this equation to produce our next estimate of the root. This new estimate is given by
452
Appendices
H( x )
o
x
x
(B‐10)
1
o
( dH dx) x o x
and we use this result to determine H( x ) as indicated in Figure B‐4a.
1
Figure B‐4a. First estimate using Newton’s method
Given H( x ) and x we can construct a second estimate as indicated in 1
1
Figure B‐4b, and this process can be continued to find the solution given by x .
The general iterative procedure is indicated by
H( x )
n
x
x
,
n 0 , 1 , 2 ,...,
(B‐11)
n1
n
( dH dx) x nx
Newton’s method is certainly an attractive technique for finding solutions to implicit equations; however, it does require that one know both the function and its derivative. For complex functions, calculating the derivative at each step in the iteration may require more effort than that associated with the bisection method or the false position method. In addition, if the derivative of the function is zero in the region of interest, Newton’s method will fail.
Material Balances
453
Figure B‐4b. Second estimate using Newton’s method
B4. Picard’s method
Picard’s method for solving Eq. B‐1 begins by defining a new function according to Definition:
f ( x) x H( x)
(B‐12)
Given any value of the dependent variable, x , we define a new value, x
, by
n
n1
Definition:
x
f ( x ) ,
n
0 , 1 , 2 , 3 ,...
(B‐13)
n
1
n
This represents Picard’s method or the method of direct substitution or the method of successive substitution. If this procedure converges, we have
f ( x ) x H( x ) x (B‐14)
In Eq. B‐13 we note that the function f ( x ) , maps the point x to the new point n
n
x
. If the function f x maps the point x to itself, i.e., f ( x) x , then x is n
( )
1
called the fixed point of f ( x) . In Figure B‐5 we again consider the function represented in Figures B‐1, B‐3 and B‐4, and we illustrate the functions f ( x) , y( x)
and H( x) . The graphical representation of the fixed point, x , is the intersection
454
Appendices
Figure B‐5. Picard’s method
of the function of f ( x) with the line y x . Note that not all functions have fixed points. For example if f ( x) is parallel to the line y x there can be no intersection and no fixed point. Given our first estimate, x , we use Eq. B‐13 to o
compute x according to
1
x
f ( x )
(B‐15)
1
o
Clearly x is further from the solution, x , than x and we can see from the 1
o
graphical representation in Figure B‐5 that Picard’s method diverges for this case.
If x were chosen to be less than the solution, x , we would also find that the o
iterative procedure diverges. If the slope of f ( x) were less than the slope f o
y( x) , we would find that Picard’s method converges. This suggests that the method is useful for “weak” fu c
n tions of x , i.e., df dx 1 and this is confirmed in Sec. B6.
Material Balances
455
B
5. Wegstein’s method
In Figure B‐6 we have
strated
illu
the same function, f ( x) , that appears in Figure B‐5. For some point x in the neighborhood of x we can approximate
1
o
Figure B‐6. Wegstein’s method
the derivative of f ( x) according to
df
f ( x ) f ( x )
1
o
slope S
dx
(B‐16)
1
x
o
x
and we can use this result to obtain an approximation
for the function f ( x ) .
1
f ( x )
f ( x ) S x x
(B‐17)
1
o
1 o
At this point we recall Eq. B‐14 in the form
456
Appendices
f ( x ) x
(B‐18)
nd
a
note that if x is
in the neighborhood of x we obtain the approximation 1
f ( x ) x
(B‐19)
1
1
e
W use this result in Eq. B‐17 to produce an equation
x
f ( x ) S x x
(B‐20)
1
o
1 o
in which S is an adjustable parameter that is used to determine the next step in the iterative procedure. It is traditional, but not necessary, to define a new adjustable arameter
p
according to
S
q
S
(B‐21)
1
se
U of this representation in Eq. B‐20 leads to
x
(1 q) f ( x ) q x
(B‐22)
1
o
o
nd
a
we can generalize this result to Wegstein’s method given by
x
(1 q) f ( x )
q x ,
n 0 , 1 , 2 , 3 , e
... tc
n
1
n
n
(B‐23)
When the adjustable parameter is equal to zero, q 0 , we obtain Picard’s method described in Sec. B4. When the adjustable parameter greater than zero and less than one, 0 q 1 , we obtain a damped successive substitution process that improves stability for nonlinear systems. When the adjustable parameter is egative,
n
q 0 , we obtain an accelerated successive substitution that may lead to an unstable procedure.
B6. Stability of iteration methods
In this section we
r
conside the linear stability characteristics of Newton’s method, Picard’s method, and Wegstein’s method that have been used to solve e
th implicit equation given by
H( x) 0 ,
x x
(B‐24)
The constraint associated with the linear analysis will be listed below and it must be kept in mind when interpreting results such as those presented in Chapter 7.
We begin by
g
recallin the three iterative methods as
457
H( x )
Newton’s method:
n
x
x
,
n 0 , 1 , 2 ,..., (B‐25)
n1
n
( dH dx) x nx
Picard’s method:
x
f ( x ) ,
n
0 , 1 , 2 ,....
(B‐26)
n
1
n
Wegstein’s method:
x
(1 q) f ( x )
q x ,
n
0 , 1 , 2 , ...
(B‐27)
n
1
n
n
in which the auxiliary function, f ( x) , is defined by Definition:
f ( x) x H( x)
(B‐28)
The general form of these three iterative methods is given by x
(
G x ) ,
n
0 , 1 , 2 ,.....
n
(B‐29)
1
n
and for each of the three methods on seeks
to find the fixed point x of G(x)
such that
x
(
G x )
(B‐30)
Our stability analysis of Eqs. B‐25 through B‐27 is based on linearizing (
G x)
bout
a
the fixed point x
. We let x and x
be small perturbations from the
n
n1
fixed point as indicated by
x
x x ,
x
x
n
n
n
x
1
n
1
(B‐31)
This allows us to express Eq. B‐29 as
x
x
(
G x x ) ,
n 0 , 1 , 2 ,.....
(B‐32)
n1
n
and a Taylor series expansion (See Problems 5‐30 and 5‐31 in Chapter 5) leads to
2
dG
d G
1
2
(
G x x
)
(
G x )
x
x
.....
(B‐33)
n
n
2
n
2
dx
x
dx x
On the basis of Eq. B‐30 this infinite series simplifies to
2
dG
d G
1
2
(
G x x ) x
x
x
.....
(B‐34)
n
n
2
n
2
dx
x
dx x
nd
a
we can use Eq. B‐32 to represent the left hand side in a simpler form to obtain
458
Appendices
2
dG
d G
1
2
x x
x x
x
.....
(B‐35)
n1
n
2
n
2
dx
x
dx x
At this point we impose a constraint on the higher order terms expressed as
2
dG
d G
Constraint:
1
2
x
x
.....
(B‐36)
n
2
n
2
dx
x
dx x
so that Eq. B‐35 takes the form
dG
x
x
,
n
0 , 1 , 2 , 3 , 4 ,.....
(B‐37)
n1
n dx x
If we write a few of these equations explicitly as
dG
x
x
(B‐38a)
1
o
dx x
dG
x
x
(B‐38b)
2
1
dx x
dG
x
x
(B‐38c)
3
2
dx x
……………………
dG
x
x
(B‐38d)
n
n1
dx x
it becomes clear that they can be used to provide a general representation given by
n
x
x
dG dx ,
n 0 , 1 , 2 ,.....
n
o
x
(B‐39)
At this point we see that x
0 when
n provided
that
n
dG dx 1
(B‐39)
x
When x
0 as n the system converges and one says that the fixed point n
x is attracting. The three special cases represented by Eq. B‐39
can be expressed
as
459
dG dx
1 ,
the fixed point x is attracting
x
I.
(B‐41)
and the iteration converges
dG dx
1 ,
the fixed point x is repelling
x
II.
(B‐42)
and the iteration diverges
dG dx
1 ,
the fixed point x is neither
III.
x
(B‐43)
attracting nor repelling
It is extremely important to note that the stability analysis leading to these three results is based on the linear approximation associated with Eq. B‐36. In this development, the word attracting is used for a system that converges since x n
moves toward x as n increases, while the word repelling is used for a system that
diverges since x moves away from x as n increases. The case in which the fixed n
point is neither attracting nor repelling can lead to chaos (Gleick, 1988; Peitgen et al., 1992).
At this point we are ready to return to Eqs. B‐25, B‐26 and B‐27 in order to determine the linear stability characteristics of Newton’s method, Picard’s method, and
Wegstein’s method.
Newton’s method
In this case we have
H( x)
(
G x) x
(B‐44)
dH dx
and the derivative that is required to determine the stability is given by 2
H( x)
d H
dG dx
(B‐45)
dH dx2
2
dx
Evaluation of this derivative at the fixed point where H( x ) 0 leads to
dG dx 0
(B‐46)
x
460
Appendices
This indicates that Newton’s method will converge provided that dH dx 0
and provided that the initial guess, x , is close
gh
enou
to x so that Eq. B‐36 is
o
atisfied.
s
If Eq. B‐36 is not satisfied, the linear
stability analysis leading to
Eqs. B‐41 through B‐43 is not valid.
Picard’s method
In this case Eqs. B‐26 and B‐29 provide (
G x) f ( x) and
dG dx df dx
(B‐47)
x
x
and from Eq. B‐41 we conclude that Picard’s method is stable when
df dx 1
(B‐48)
x
In Example 7.7 of Chapter 7 we used the fixed point iteration (see Eq. 17) that can be expressed as
x
f ( x )
1
C 1
x
,
C
0 3
. 0 ,
n
0 , 1 , 2 , 3 ,..... (B‐49)
n
1
n
n
This leads to the condition
df dx 1 C 1
(B‐50)
that produces the stable iteration illustrated in Table 7.7a. In Example 7.8 of Chapter 7 we find another example of Picard’s method (see Eq. 40) that we repeat here as
15 1678
.
x
x
f ( x )
x
37 . 6190
n ,
n
0 , 1 , 2 , 3 ,... (B‐51)
n
1
n
n
0 . 8571 x
n
The solution is
given by x 0 . 6108 and this leads to
df dx
.
(B‐52)
x
213 1733
dicating
in
that Picard’s method is unstable for this particular fixed point iteration. This result is consistent with the entries in Table 7.8a.
egstein’s
W
method
In this case Eqs. B
‐27 and B‐29 provide
(
G x) (1 q) f ( x) q x (B‐53)
461
which leads to
(
G x) (1 q) f ( x) q x (B‐54)
From this we have
dG
df
(1 q)
q
(B‐55)
dx
dx
and the stability condition given by Eq. B‐41 indicates that Wegstein’s method will converge provided that
df
(1 q)
q 1
(B‐56)
dx
Here one can see that the adjustable parameter q can often be chosen so that this inequality is satisfied and Wegstein’s method will converge as illustrated in Examples 7.7 and 7.8 of Chapter 7.
462
Appendices
Appendix C
Matrices
Matrix Methods and Partitioning
In order to support the results obtained for the atomic matrix studied in Chapter 6 and for the mechanistic matrix studied in Chapter 9, we need to consider that matter of partitioning matrices. All the information necessary for our studies of stoichiometry is contained in Eq. 6‐22; however, that information can be presented in different forms depending on how the atomic matrix and the column matrix of net rates of production are partitioned. In our studies of reaction kinetics, all the information that we need is contained in the mechanistic matrix; however, that information can also be presented in different forms depending on presence or absence of Bodenstein products. In this appendix we review the methods required to develop the desired different forms.
Matrix addition
We begin our study of partitioning with the process of addition (or subtraction) as illustrated by the following matrix equation
a
a
a
a
b
b
b
b
c
c
c
c
11
12
13
14
11
12
13
14
11
12
13
14
a
a
a
a
b
b
b
b
c
c
c
c
21
22
23
24
21
22
23
24
21
22
23
24
(C‐1)
a
a
a
a
b
b
b
b
c
c
c
c
31
32
33
34
31
32
33
34
31
32
33
34
a
a
a
a
b
b
b
b
c
c
c
c
41
42
43
44
41
42
43
44
41
42
43
44
This can be expressed in more compact nomenclature according to A B
C
(C‐2)
The fundamental meaning of Eqs. C‐1 and C‐2 is given by the following sixteen (16) equations:
Material Balances
463
a
b
c
a
b
c
11
11
11
21
21
21
a
b
c
a
b
c
12
12
12
22
22
22
a
b
c
a
b
c
13
13
13
23
23
23
a
b
c
a
b
c
14
14
14
24
24
24
(C‐3)
a
b
c
a
b
c
31
31
31
41
41
41
a
b
c
a
b
c
32
32
32
42
42
42
a
b
c
a
b
c
33
33
33
43
43
43
a
b
c
a
b
c
34
34
34
44
44
44
These equations represent a complete partitioning of the matrix equation given by Eq. C‐1, and we can also represent this complete partitioning in the form (C‐4)
Here we have shaded the particular partition that represents the first of Eqs. C‐3.
The complete partitioning illustrated by Eq. C‐4 is not particularly useful; however, there are other possibilities that we will find to be very useful and one example is the row/column partition given by
(C‐5)
Each partitioned matrix can be expressed in the form
(C‐6)
and the partitioned matrix equation is given by
A
A
B
B
C
C
11
12
11
12
11
12
(C‐7)
A
A
B
B
C
C
21
22
21 22
21
22
464
Appendices
We usually think of the elements of a matrix as numbers such as a , a , etc.; 11
12
however, the elements of a matrix can also be matrices as indicated in Eq. C‐7.
The usual rules for matrix addition lead to
A
B
C
(C‐8a)
11
11
11
A
B
C
(C‐8b)
12
12
12
A
B
C
(C‐8c)
21
21
21
A
B
C
(C‐8d)
22
22
22
and the details associated with Eq. C‐8a are given by
a
a
b
b
c
c
11
12
11
12
11
12
(C‐9)
a
a
b
b
c
c
21
22
21
22
21
22
A little thought will indicate that this matrix equation represents the first four equations given in Eqs. C‐3. Other partitions of Eq. C‐1 are obviously available and will be encountered in the following paragraphs.
Matrix multiplication
Multiplication of matrices can also be represented in terms of submatrices, provided that one is careful to follow the rules of matrix multiplication. As an example, we consider the following matrix equation
a
a
a
a b
b
c
c
11
12
13
14
11
12
11
12
a
a
a
a
b
b
c
c
21
22
23
24 21
22
21
22
(C‐10)
a
a
a
a b
b
c
c
31
32
33
34
31
32
31
32
a
a
a
a
b
b
c
c
41
42
43
44 41
42
41
42
which conforms to the rule that the number of columns in the first matrix is equal to the number of rows in the second matrix. Equation C‐10 represents the eight (8) individual equations given by
a b
a b
a b
a b
c
(C‐11a)
11 11
12 21
13 31
14 41
11
a b
a b
a b
a b
c
(C‐11b)
11 12
12 22
13 32
14 42
12
a b
a b
a b
a b
c
(C‐11c)
21 11
22 21
23 31
24 4
21
a b
a b
a b
a b
c
(C‐11d)
21 12
22 22
23 32
24 42
22
a b
a b
a b
a b
c
(C‐11e)
31 11
32 21
33 31
34 41
31
Material Balances
465
a b
a b
a b
a b
c
(C‐11f)
31 12
32 22
33 32
34 42
32
a b
a b
a b
a b
c
(C‐11g)
41 11
42 21
43 31
44 41
41
a b
a b
a b
a b
c
(C‐11h)
41 12
42 22
43 32
44 42
42
which can also be expressed in compact form according to
AB
C
(C‐12)
Here the matrices A, B, and C are defined explicitly by
a
a
a
a
b
b
c
c
11
12
13
14
11
12
11
12
a
a
a
a
b
b
c
c
21
22
23
24
21
22
21
22
A
B
C
(C‐13)
a
a
a
a
b
b
c
c
31
32
33
34
31
32
31
32
a
a
a
a
b
b
c
c
41
42
43
44
41
42
41
42
In Eqs. C‐1 through C‐9 we have illustrated that the process of addition and subtraction can be carried out in terms of partitioned matrices. Matrix multiplication can also be carried out in terms of partitioned matrices; however, in order to conform to the rules of matrix multiplication, we must partition the matrices properly. For example, a proper row partition of Eq. C‐10 can be expressed as
(C‐14)
In terms of the submatrices defined by
a
a
a
a
a
a
a
a
11
12
13
14
31
32
33
34
A
,
A
11
21
a
a
a
a
a
a
a
a
21
22
23
24
41
42
43
44
(C‐15)
c
c
c
c
11
12
31
32
C
C
11
21
c
c
c
c
21
22
41
42
we can represent Eq. C‐14 in the form
A
A B
C
11
11
11
B
(C‐16)
A
A
B
C
21
21
21
466
Appendices
Often it is useful to work with the separate matrix equations that we have created by the partition, and these are given by
A
B
C
(C‐17)
11
11
A
B
C
(C‐18)
21
21
The details of the first of these can be expressed as
b
b
11
12
a
a
a
a b
b
c
c
11
12
13
14 21
22
11
12
A
B
,
C
(C‐19a)
11
11
a
a
a
a
b
b
c
c
21
22
23
24
31
32
21
22
b
b
41
42
Multiplication can be carried out to obtain
a b a b a b a b
a b a b a b a b
11 11
12 21
13 31
14 41
11 12
12 22
13 32
14 42
a b a b a b a b
a b a b a b a b
21 11
22 21
23 31
24 41
21 12
22 22
23 32
24 42
(C‐19b)
c
c
11
12
c
c
21
22
and equating the four elements of each matrix leads to
a b
a b
a b
a b
c
11 11
12 21
13 31
14 41
11
a b
a b
a b
a b
c
11 12
12 22
13 32
14 42
12
(C‐19c)
a b
a b
a b
a b
c
21 11
22 21
23 31
24 41
21
a b
a b
a b
a b
c
21 12
22 22
23 32
24 42
22
Here we see that these four individual equations (associated with the partitioned matrix equation) are those given originally by Eqs. C‐11a through C‐11d. A little thought will indicate that the matrix equation represented by Eq. C‐18 contains the four individual equations represented by Eqs. C‐11e through C‐11h. All of the information available in Eq. C‐10 is given explicitly in Eqs. C‐11 and partitioning of the original matrix equation does nothing more than arrange the information in a different form.
Material Balances
467
If we wish to obtain a column partition of the matrix A in Eq. C‐10, we must also create a row partition of matrix B in order to conform to the rules of matrix multiplication. This column/row partition takes the form (C‐20)
and the submatrices are identified explicitly according to
a
a
a
a
11
12
13