Two examples illustrated here have demonstrated the usefulness of the proposed design criteria. T h e optimum designed volume for a reactor was obtained by the expected relative sensitivity criterion. T h e result so obtained gives a larger underdesign in the higher cost region and a smaller overdesign in the lower cost region in comparison with that obtained by expected cost criterion. By using this criterion, designers can make sure that the result obtained has a smallest average normalized deviation from the optimal behavior over the range of uncertainty. T h e optimal temperature for a reversible and exothermic reaction has been determined by the proposed design criterion. T h e optimal temperature to keep the deviation from the maximum reaction rate within a certain limit is found over the range of uncertainty in activation energy. I n addition, the necessary equations for finding a n appropriate temperature for a general homogeneous, reversible, and exothermic reaction by using the proposed design criterion have been obtained as indicated by Equations 23 and 25.
T h e authors express their gratitude to the Office of Coal Research, Department of the Interior, Washington, D. C., for financial support. Nomenclature
frequency factor for forward, backward reaction cost CA, C, concentration E, E’ activation energy for forward, backward reaction f(w) = density function for w FA0 = feed rate of component A J = objective function G
= = = =
= reaction rate constant for forward, backward reaction
= A/A’ = chemical equilibrium constant - reaction rate r R = gas constant s = relative sensitivity SE = expected value of S, Equation 7 T , T* = temperature T, = optimal temperature = reactor volume - parameter W - mole fraction x dimensionless extent of reaction = conversion of component A
v,v*
x x,
=
GREEKLETTERS pj
&
Ak e0 7
Acknowledgment
A,A’
k, k‘ K* K,
= = = = = = = =
stoichiometric coefficient of species j forward reaction order with respect to species j backward reaction order with respect to species j heat of reaction range o f k decision variable optimal decision dimensionless temperature
literature Cited
Kittrell, J. R., Watson, C. C., Chem. Eng. Progr. 62 (4), 79 (1966). Levenspiel, O., “Chemical Reaction Engineering,” p. 134, Wiley, New York. 1962. Luce, R. D.,’Raiffa, H., “Games and Decisions,” Chap. 13, Wiley, New York, 1957. Ray, W. H., Aris, R., IND.ENG. CHEM.FUNDAMENTALS 5 , 478 ( 1966). Rohrer,’ R. A., Sobral, M., Jr., ZEEE Trans. Automatic Control AC-10, No. I , 43 (1965). RECEIVED for review March 3, 1967 ACCEPTED July 31, 1967
OPTIMIZATION OF YIELD THROUGH
FEED COMPOSITION HCN Process B. Y . K . PAN AND R. G . ROTH Hydrocarbons and Polymers Division, Monsanto Co., Texas City,Tax.
Andrussow H C N process has been widely used to proThis is a n autothermal process involving the reactions of ammonia, methane, and air over a platinum and rhodium gauze catalyst. T h e over-all reaction can be expressed as: HE
T duce hydrogen cyanide. CHI
3 + NHI + 1
0 2 -+
HCN
+ 3Hz0
(1)
AH = -102,464 B.t.u./lb. mole of H C N a t 1150’ C. Several investigators, notably Chrttien and Thomas (1948), Maffezonni (1953), and Mihail (1957), have done a considerable amount of work to determine the effects of feed composition on H C N formation. They used various compositions by C H I , and air or adjusting the.relative concentrations of “3, oxygen in the feed. Lack of fundamental knowledge and
effective techniques, however, prevented optimization of yield. Usually a number of feed ratios were tested in the reactor, samples of both feed and off-gas were analyzed, and the reactor performance was evaluated from these data. A favorable feed composition was then chosen, based on these trial and error tests. Such techniques required much labor and time and gave no assurance of optimum conditions. This investigation was oriented to gaining insight into the reaction system and thereby establishing a method for the optimization of H C N yield. Reaction temperatures in conjunction with air/(CHI “3) mole ratio were the most reliable variables in controlling the reaction system, and a simple and effective method of optimization was developed. This was accomplished by theoretical considerations, followed by experimentation, mathematical correlation, statistical analysis, and chemical interpretation.
+
VOL. 7
NO. 1
JANUARY 1 9 6 8
53
The widely used process of making hydrogen cyanide from ammonia, methane, and air over platinum and rhodium gauze catalyst was investigated on a 4-inch i.d. reactor. For a certain feed throughput, the maximum HCN yield occurred at the minimum flame temperature, and the maximum NH3 conversion to HCN occurred a t the minimum off-gas temperature. Response equations of the reactor performance in terms of feed ratios and reaction temperatures were established. The best over-all correlation o f the equation reNHI) mole ratio, CH4/NH3 mole ratio, and flame temperature. Based on this dissulted from air/(CH4 covery, a simple and effective method o f optimization of HCN yield through feed composition was obtained. Theoretical considerations, experimentation method, mathematical correlation, and statistical analysis are included.
+
Theoretical Cansiderations
Initially, the theoretical basis was considered for a relationship between feed composition and yield determinable by some easily measured variable such as temperature. The basic equation for a catalytic flow reactor is:
Equation 3 shows that conversion is only a function of reaction temperature and concentrations of components. Consequently, in an autothermal system a relationship could be defined among feed composition, reaction temperature, and conversion or yield. In the three-component system, feed composition can be described by various combinations of two weight or mole feed ratios. A wise choice of feed ratios, however, can yield more information and better correlation. Based on the stoichiometric equation
dx
r(Cl---Cn,P,T,A)
(2)
where conversion, moles/mass of feed or a feed component w = mass of catalyst F = feed rate, or a feed component rate, mass/time r = reaction rate, moles/(mass of catalyst) (time) Cl,---Cn , . = concentrations of components P = reactor pressure T = reactor temDerature A = catalyst activity, moles of product/(mass of a t a given temperature and feed composition =
Y
+
the 02/(CH4 "3) mole ratio should be 0.75, or the air/ (CHI "3) mole ratio should be 3.75. But when oxygen or air is used in such an abundant quantity, combustion of C H I and N H I with 0 2 will take place preferentially and result in a poor conversion of NH3 with CH4 to form HCN. Therefore, a complex situation arises for maximizing HCN yield. Without reliable kinetic information it can be loosely stated that the a i r / ( C H ~f NH,) ratio should be large enough to supply heat for H C N formation, but not
+
I n the usual practice, the reactor pressure, P, and catalyst weight, w ,of a fixed-bed reactor are constant. After a period of catalyst on-stream time, the catalyst activity, A , of a new batch of catalyst becomes constant. T h e most needed information is how to optimize the conversion or yield for a certain rate of feed or a feed component, F. Consequently, Equation 2 can be simplified as shown below:
TO FLARE
wd'p
to h
CH4
AiR
t;
54
CEL-
EXCHANGER HEAT
I
I
I
d I
"3
Figure 1. F. Filter FRC. Flow recorder and controller TI. Thermocouple
A
Flow diagram of HCN unit
PI. Pressure indicator
l&EC PROCESS DESIGN A N D DEVELOPMENT
PRC. Pressure recorder and controller
R. Regulator
ST. Sample tap
so large as to cause unnecessary decomposition of CH4 and N H 3. Furthermore, if different CH4/NH3 mole ratios are used in "3) ratio and a conjunction with a given air/(CH4 constant air flow rate, the competitive reactions of C H I and NH3 with air can be revealed and the relations of reaction temperatures with H C N formation may be established. Thus, the following two ratios were chosen as basic parameters in this investigation:
+
RI=
CH4
__ mole ratio,
Rz
air
=
CH4
3"
+
mole ratio 3 "
Both NH3 and H C N in the feed and off-gas were analyzed by chemical titrations. Ammonia was titrated against HCl, and H C N was titrated against AgN03. Total feed throughput in different series of tests was within 100 f 5 pounds per hour. Feed composition was varied by NH,) mole ratio from 2.80 to 3.25 changing the air/(CH4 and the CH4/NH3 mole ratio from 0.7 to 1.7, which covered the feed range of greatest interest in the production of HCN.
+
Operating conditions of tests are shown in Table I . The effects of feed composition on reactor performance were expressed in terms of yield, Y , conversion, C, which is oncethrough yield, N H I leakage, L, and NH3 decomposition, D. Their definitions are
Experimental Methods
Layout of the pilot plant unit is shown in Figure 1. A 4inch i.d. stainless steel reactor was equipped with six layers of gauze catalyst. The catalyst was made of 90% platinum and 10% rhodium and fabricated into 3 mil X 80 mesh. The reactor section was insulated with fiber flax and Kaylo to ensure adiabatic condition. An optical pyrometer was used to measure the flame temperature, which was sighted through a sight glass on top of the reactor. A shield Chromel-Alumel thermocouple was placed beneath the gauze catalyst to record the off-gas temperature. Feed gases enriched with air could be easily lighted over the catalyst surface by an electrically heated platinum wire. After ignition, the igniter was withdrawn and feed adjusted to the desired ratio. The reactor off-gas was quenched via a shell-and-tube heat exchanger to minimize thermal decomBy maintaining quench water a t position of H C N and "3. 70' to 90' C., the formation of solid condensation products was prevented. All tests were conducted at a constant reactor pressure (14 p.s.i.g.) and constant catalyst activity. After a new batch of catalyst was installed in the reactor, the activity was increased with the catalyst on-stream time until the peak activity was reached. Then it could remain constant for several hundred hours. The time required to activate a new gauze catalyst to constant catalyst activity was 60 to 65 hours. After each change of feed ratio, approximately 30 minutes were required to reach steady-state condition for analyzing.
Table 1. Test
Ria
.41 A2 A3 A4 A5 A6
c1
0.7
c2 c3 c4
0.9
D1 D2 D3 D4 D5 R1
1.1 1.25 1.4 1.55 1.7
0.8
C6 c7
a
0.9
B1 B2 B3 B4 B5 B6 B7
c5
=
R2b 3.25 3.25 3.25 3.25 3.25 3.25
0.9
1 .o 1.085 1.2 1.4 1.7
1 .o 1.1 1.25 1.4 1.7
0.7 0.9
1.1
1.4 1.7 C H I / N H ~mole ratio.
b
D where ("I), respectively.
= 100
- (C + L) mole %
and ("a),
= NH3 in feed and reactor off-gas,
Results and Discussion
Optimization Method. Table I1 shows the results of reactor performance obtained a t different operating conditions. Figures 2 through 7 are plots of reactor performance and reaction temperatures against feed ratios. For a constant "3) mole ratio, there was a C H ~ / N H Imole air/(CH4 ratio which gave the maximum yield and the minimum flame temperature, while another CH&H3 mole ratio gave the maximum conversion and the minimum off-gas temperature. Based on these findings, a desired feed ratio can be quickly located after several measurements of the reaction tempera-
+
Operating Conditions Lb. Mole/Hr. CHa Air 0.444 3.042 0.491 3.042 0.520 3,042 0.546 3.042 0.569 3.042 0.590 3.042
0.493 0.446 0.416 0.390 0.367 0.347
Throughput, Lb./Hr. 103.70 103.67 103.61 103.58 103.56 103.52
"a
3.09 3.09 3.09 3.09 3.09 3.09 3.09
0.419 0.446 0.472 0.490 0.515 0.550 0.593
2.914 2.914 2.914 2.914 2.914 2.914 2.914
0.524 0.496 0.472 0.452 0.429 0.393 0.349
100.12 100.08 100.08 100.04 100.04 100.00 99.93
2.94 2.94 2.94 2.94 2.94 2.94 2.94
0.390 0.448 0.474 0.496 0.526 0.553 0.597
2.784 2.784 2.784 2.784 2.784 2.784 2.784
0.557 0.498 0.474 0.451 0.421 0.395 0.351
96.45 96.38 96.37 96.34 96.31 96.30 96.26
2.80 2.80 2.80 2.80 2.80
0.402 0.463 0.512 0.570 0.614
2.732 2.732 2.732 2.732 2.732
0.574 0.514 0.465 0,407 0.361
95.43 95.37 95.33 95.27 95.19
NO. 1
JANUARY 1968
R2 = a i r / ( C H 1 f
"3)
mole ratio.
VOL. 7
55
Table 11.
8.03 7.27 6.78 6.35 5.98 5.65
Y 56.48 72.36 83.37 86.33 87.83 86.77
Yield Test, Mole 70 C L 50.34 10.87 63.86 11.75 16.64 69.49 63.20 26.80 60.37 31.27 59.33 33.16
D 38.79 24.39 13.87 10.00 8.37 7.51
Temp., Tf 1182 1140 1127 1122 1117 1125
C. To 978 958 947 950 953 960
3.09 3.09 3.09 3.09 3.09 3.09 3.09
8.84 8.38 7.95 7.63 7.24 6.63 5.90
8.88 8.47 8.01 7.71 7.45 6.70 6.20
0.90 1.01 1.06 1.08 1.76 2.07 2.50
6.54 7.24 7.67 7.81 7.26 6.27 4.88
51.63 61.14 69.53 74.21 80.38 85.32 83.09
46.40 53.85 60.33 63.82 61.39 59.90 49.59
10.14 11.92 13.23 14.01 23.62 30.90 40.65
43.46 34.23 26.44 22.17 14.99 9.20 9.76
1162 1144 1130 1125 1104 1095 1105
962 948 937 934 940 945 954
2.94 2.94 2.94 2.94 2.94 2.94 2.94
9.87 8.83 8.38 7.99 7.45 6.98 6.21
9.77 8.83 8.34 8.09 7.36 6.89 6.32
1.29 1.62 1.75 2.15 2.57 2.76 3.07
7.30 8.60 8.43 7.77 6.37 5.48 4.22
54.24 75.15 80.59 82.41 84.49 83.59 81.80
47.07 61.39 63.60 60.51 54.82 50.11 42.07
13.20 18.35 19.14 25.58 35.11 40.06 48.58
39.73 20.29 17.26 12.91 10.07 9.83 9.35
1135 1098 1088 1085 1084 1092 1097
945 922 920 926 936 940 950
10.17 10.25 1.39 8.08 1.74 9.10 9.04 9.06 8.23 2.57 8.12 7.34 3.10 7.24 7.20 5.40 6.42 6.40 3.33 4.02 b R P = air/(CH4 NHs) mole ratio.
57.45 78.19 83.19 82.17 81.96
49.66 63.14 56.91 46.99 39.45
13.56 19.25 31.58 42.82 51.89
36.78 17.61 11.50 10.19 8.17
1100 1080 1071 1084 1090
924 910 924 936 945
R in
R zb
0.9 1.1 1.25 1.4 1.55 1.7
3.25 3.25 3.25 3.25 3.25 3.25
B1 B2 B3 B4 B5 B6 B7
0.8 0.9 1.o 1.085 1.2 1.4 1.7
c1 c2 c3 c4 c5 C6 c7
0.7 0.9 1.o 1.1 1.25 1.4 1.7
2.80 0.7 2.80 0.9 2.80 1.1 2.80 1.4 2.80 1.7 'Ri = CHd/NHa mole ratio.
0
Reactor Performance
Yield Test, W t . 7 0 (NH3)o HCN 7.91 0.86 6.32 0.86 7.32 7.42 6.79 1.13 7.49 6.40 1.72 6.42 5.98 1.87 5.73 5.61 1.88 5.34
Test No. A1 A2 A3 A4 A5 A6
D1 D2 D3 D4 D5
Calcd. wt. % ("a)$
(NH3)i
+
1240
1000-
1220
990-
1200
980 -
1180
970-
I I60
9600
0
n 1140
3
w
F
I +
1120
AIR
W
I
MOL.
C H 4 t NH, RATIO
1100
AIR / C H 4 tNH, MOL. RATIO
950-
2 940Y
k 930-
LL
920 -
1080
I060
910-
1040
0 Figure 2.
0.4 0 . 8 1.2 1.6 2.0 2 . 4 C H 4 / N H 3 MOL. RATIO Flame temperatures at various feed ratios
tures a t different feed ratios. Only one or two yield tests by chemical titrations are needed to determine the precise HCN yield and conversion at optimum conditions. Specific advantages are offered by this technique, since the method of optimization is more accurate, more economical, faster, and simpler. I n the temperature range of 1000' to 1300' C., an experi56
I&EC PROCESS DESIGN A N D DEVELOPMENT
9001 0 Figure 3.
,
0.4 0.8 1.2 1.6 2.0 2.4 C H 4 / N H 3 MOL. RATIO Off-gas temperatures at various feed ratios
enced operator using an optical pyrometer can make measurements with a deviation of only & 3' C . Meanwhile, the optimum feed ratio is determined by varying only C H 4 and "8 at a constant air setting; thus, a frequent variation of air flow in a series of runs is eliminated. Manpower can be saved because fewer yield tests are required. Operation becomes faster and simpler because tem-
1240
1001 I220
90-
MOL. RATIO
80 -
70 60
H C N Y I E L D 70
HCN 1200 1180
0
3'25-\
0
a
-
60%
1160
I W
#
1140
I-
i 50-
W
0
I
CONVERSION
40-
z
5
\L 2.94 C2.80
1120
LL
I100
3020
-
1080
I
o
0.4
0
Figure 4. ratios
0.8
b
1040
'
HCN yields and conversions at various feed
I
90-
990
80
-
980
70
-
y
60 -
5
I
2.4
CONVERSION '10
970
t i
NH
DECOMPOSITION
NH,- LEAKAGE
960
i-
950
50-
(3
-
I
-3.25
-
AIR
MOL.
I
2
940
0
930 6 3 . 5 ?o'
920 6 3 o/'
9 IO 900
0
I
I
0.8 1.2 1.6 2.0 CH4 / N H 3 M O L . RATIO
Figure 6. Relation of yields and flame temperatures at various feed ratios
IO00
30
I
1
0.4
I
I
1.2 1.6 2.0 2.4 CH4 / N H 3 MOL. RATIO
loot
'2
8 3 . 2 '10
I060
10-
0.4
0.8 1.2 1.6 2.0 2.4 CH4 / NHS MOL, RATIO
Figure 5. Ammonia leakages and decompositions at various feed ratios
perature measurements replace tedious chemical titrations, and no sampling problem is encountered. I n the study of feed ratios, it is desirable to avoid any possible effect associated with variation of total feed rate (throughput). However, this investigation showed that a variation of + 5% of feed throughput had a negligible effect on reaction temperature. T h e throughput has already been large enough to eliminate
I
I
0.4
I
I
I
0.8 1.2 1.6 2 . 0 CH4 / N H 3 MOL. RATIO
I
2.4
Figure 7. Relation of conversions and temperatures at various feed ratios
off-gas
the external diffusion resistance between the bulk gas phase and catalytic interface. For example, a constant feed composition consisting of a CH4,"HS = 1.085 mole ratio and an "8) = 3.09 mole ratio was tested at various air/(CHI feed throughputs. The corresponding reaction temperatures are shown in Figure 8. When feed throughput was varied within 100 i 5 pounds per hour, the variations of flame and
+
VOL 7
NO. 1
JANUARY 1968
57
OC 1130
CH4
-=
1.085 MOL R A T I O
"3
AIR =3.09MOL RATIO CH4+NH3
OC
I120
off-gas temperatures were only +1.2' and * 2 O C., respectively. The off-gas had a little higher temperature drop than the flame when the feed decreased from 110 to 75 pounds per hour. The slight difference could be attributed to thermal reactions occurring in the void space between the bottom of the catalyst bed and the thermocouple. These variations, however, had a negligible effect in defining the minimum flame and off-gas temperatures. If a drastic change of feed throughput needs to be made, such as a n increase from 100 to several hundred pounds per hour, the optimum condition can be achieved in the same mannervarying the CH4,"Ha mole ratio for a constant a i r / ( C H ~ "3) mole ratio while keeping the feed throughput around the given value; then the maximum yield or conversion will be obtained a t the feed ratios which result in the minimum flame temperature or the minimum off-gas temperature. Mathematical Correlation. RESPONSE EQUATIONS. The variation of feed throughput of 100 f 5 pounds per hour had a negligible effect on reaction temperature; thus, the reactor performance expressed as second-order equations in terms of R1 and R P ,R1 and T,, or R I and TOis suggested as:
+
1950
1110
X 1100 t
I
I
I
I
70 80 90 100 THROUGHPUT LBS / HR
1
110
Figure 8. Effect of feed throughput on flame and offgas temperatures
Y, C. 1120,220 289.877 -263.980 131.572 112.279 -217.680 TI,
ao a1
a2 a3
a4 a5
O
Response equations X = a0 alRl f azRz
+
T o , C. 833.157 250.286 -111.744 72.811 59,522 -140.658
Mole 7 0 274.981 -34.032 -108.693 -67.837 -0,276 76.402
+ a&i* + a4Ra2 f a5RlRz
(4)
where X can be H C N yield, NH3 conversion to HCN, NH3 leakage, NHz decomposition, and H C N and NHI concentrations in the off-gas. The constants aa, a l , a2, a3, (14, and a5 were evaluated by multiple regression analysis using the GE 235 computer (Tables I11 and IV).
Coefficients of Response Equations in Terms of
Table 111.
Coeficient
= a0 f aiRi f aaRp
C, Mole % 362.906 -101.166 -173.502 -57.406 15.282 78.910
L, Mole -116.616 81.449 84,704 -0,542 -17.271 -14.325
Rl and Rz
D, Mole yo -142.731 17.093 87.135 58.099 2.176 -63.867
HCN, wt. 70 36,134 -18.956 -9.883 -5.371 -0.376 9.596
"3
Wt. 7% -7.213 5.161 6.510 - 0,784 -1.523 -0.443
+ a3R? + ~ 4 R 2+~ a5RiR~
where X = T,, To,Y , C, L,D,HCN, or NHI R , = 4CH mole ratio "3
R* =
air CHd
+
mole ratio "3
Table IV.
Coeficient 00 a1
a2
aa a4 a5
a0 a1
a2 ai a4 05
58
Y , Mole 1241.535 -427.517 -1.581 -39.170 412 X 10-6 0.493 2315.376 - 447.907 -4.070 -58.024 1667 X 10-6 0.656
C, Mole
Coefficients of Response Equations
yo
L , Mole
yo
In terms of R1 and T t 170.220 975.877 -875.145 604.905 0.562 -2.089 -46.587 17.000 -647 X 10-6 1100 x 100.888 -0.556
1655.898 - 1708.345 -1,646 - 94.466 185 X 10-6 2.041
D , Mole
- 1065,429
264.227 1.566 29.624 -472 X 10-6 -0.327
In terms of R, and To 274.481 - 1849.455 1557.830 126.558 -1,881 3.589 54.510 39,430 1737 X 10-6 -1596 X 10-6 -1,742 -0.272
I h E C PROCESS DESIGN A N D DEVELOPMENT
HCN, Wt. %
"3,
52.466 -95.488 0.019 -2.707 -54 x 10-6 0,089
99,680 43.191 -0,198 0.965 988 X 10-6 -0.040
310.567
54.035 121.976 -0.211 3.839 167 X 10-6 -0,136
- 157.372
-0,455 -7.064 135 X 10-6 0.181
Wt. %
These response equations generated by multiple regression analysis have the following uses: (1) A quantitative relationship between reactor performance and a wide range of feed ratios is established. These equations can be used to predict the reactor performance with assured confidence within the experimental range. (2) A definite order of a mass of data is found. For instance, the superimposed Figures 6 and 7 can be expressed by Equations 5 and 6, respectively:
70)
I’IHCN yield, mole
1.581 TI
1241.535 - 427.517 R1 -
=
- 39.170 Ri2 + 412 10-6 TI2
X
+ 0.493 RiT,
(5)
1708.345 R1 - 1.646 Tu - 94.466 RiZ +
C(HCN conv., mole
%)
=
1655.898
185 X 10-6 T,2
+ 2.041 RlT,
(6)
When a wide range of feed throughput is used, the response equations can be modified to incorporate throughput as another independent variable or established for different levels of throughput. ESTIMATION OF CH4/”H3 MOLERATIO TO GIVEMAXIMUM KHa) YIELD AND COSVERSION.For a given air/(CHi mole ratio: the value of C H J / N H ~giving the maximum yield, conversion, and minimum reaction temperature can be estimated through a n appropriate response equation. air ; find R1 giving ihe For example, R2 = 3.09 = CH4 “3 maximum yield. From Equation 4, the maximum condition is
+
+
or
(7) Substituting numerical values in Table I11 into Equation 7 :
+
-34.032 76.402(3.09) = 1.49 2( -67.837)
R1= --
T h e observed value is 1.45. Values are in good agreement. MAXIMUM O R MINIhlUM VALUE OF THE WHOLE SYSTEM. T h e question arises as to whether there are maximum H C N yield and conversion over the whole range of feed ratios; consequently, the following equations should be applied :
(E’)(E)- (mz)
Simi-
- (78.910)’ < 0
A = 4(-57.406)(15.282)
again there can be neither a maximum nor a minimum conversion. At first it appears anomalous that there are neither a maximum nor a minimum yield and conversion over the range of correlation. T h e reason for such an anomaly can be better understood through a geometrical view (Figure 9). The response surface consisting of two independent variables is shown in this figure. A maximum or minimum value is a relative one in a continuous function. A maximum must be larger than any values immediately preceding and following. A minimum must be smaller than any values immediately preceding and following. Figure 9 shows that M is the maximum of the continuous curve with an air/(CHd ”3) = 3.25 mole ratio, but M is not a maximum on the continuous curve M S which is also a curve of the response surface. Similarly, .I’is the maximum “3) = 2.80 of the continuous curve with an airt’(CH4 mole ratio, yet is neither a maximum nor a minimum along curve M.V. Therefore, there are no maximum and minimum in the whole response surface. This indicates that the H C N yield and conversion will increase toward M and decrease toward ‘V. Statistical Analysis. T o determine the effectiveness of the mathematical correlations and the predictabilities within the experimental range, a statistical analysis was made. A brief summary of these statistics is shown in Table V . Their meaning and significance are given below. MULTIPLECORRELATION COEFFICIENT, R. The multiple correlation coefficient, R, varies from 0 to 1.0. If there is no relationship between the variables, it has a value of zero. If there is a perfect relationship, it has a value of 1.0. O b viously, the correlation of the dependent and independent variables of this reaction system is very good because values of R shown in Table V are larger than 0.87. The quantity RZ can be considered as the fraction of the variability that can be attributed to the relationship between the dependent variable and independent variables. T h e quantity (1 - R2) is the fraction attributed to experimental error, random fluctuations, and deviation from the assumed mathematical model. STANDARD DEVIATION, U , AND CONFIDENCE LIMIT. T h e standard deviation is evaluated from error, E, and degree of freedom (d.f.).
+
+
32Y
d2Y
A =
there can be neither a maximum or a minimum yield. larly, substituting values for conversion:
>O,
bRlZ < 0 max. M 32Y
< 0 min.
3.09 0 - _ _ - /
Substituting numerical values from Table I11 into Equation 8 yields: A = 4(-67.837)(-0.276)
-
(76.402)z
