type and distribution can thus be influenced by the rate of formation or removal of the products. This study shows that large yields of acetylene can be obtained from the pyrolysis of coals in the microwave discharge. Presence of low pressures of gases such as argon increases the rate of volatilization of coal, and fast quenching or removal of the primary products enhances the product yield. I n a more practical manner, the process can probably be best carried out in a fast flow system under low pressures of argon, hydrogen, or preferably synthesis gas. These prospects are being studied. Acknowledgment
The authors thank Irving Wender for his valuable discussions, Gus Pantages and George Kambic for their technical assistance, A. G. Sharkey, Jr., and Janet L. Shultz for their mass spectrometric analyses, and John Queiser for his infrared analysis.
Fu, Y. C., Blaustein, B. D., Chem. Ind. London. 1967, 1257. Fu, Y. C., Blaustein, B. D., Fuel, London 47, 463 (1968). Graves, R. D., Kawa, W., Hiteshue, .R. W., IND. ENG. CHEM.PROCESS DESIGNDEVELOP. 5, 59 (1966). Hawk, C. O., Schlesinger, M. D., Hiteshue, R . W., Bur. Mines, Rept. Invest. 6264 (1963). Karn, F. S., Friedel, R. A., Sharkey, A. G., Jr., Carbon 5, 25 (1967). Kawana, Y., Makino, M., Kimura, T., Kogyo Kagahu Zasshi 69, 1144 (1966); Intern. Chem. Eng. (Eng. Trans.) 7, 359 (1967). Krukonis, V. J., Schoenberg, T., “7th International Conference on Coal Science,” p. 5-132, Prague, 1968. Kewman, J. 0. H., Coldrick, A. J. T., Evans, P. L., “7th International Conference on Coal Science,” p. 4-26, Prague, 1968. Rau, E., Eddinger, R . T., Fuel, London 43, 246 (1964). Rau, E., Seglin, L., Fuel, London 43, 147 (1964). Shultz, J. L., Sharkey, A. G., Jr., Carbon 5, 57 (1967).
literature Cited
Blaustein, B. D., Fu, Y. C., Aduan. Chem. Ser., No. 80, 259 (1969). Bond, R . L., Ladner, W. R., McConnell, G. I. T., Fuel, London 45, 381 (1966).
RECEIVED for review June 17, 1968 ACCEPTED November 1, 1968
ELABORATION AND EXTENSION OF EXPERIMENTAL RESULTS THROUGH MATHEMATICAL CORRELATIONS AND FUNDAMENTAL KNOWLEDGE HCN Reactor Studies B I N G H A M Y .
K .
P A N
Hydrocarbons and Polymers Division, Monsanto Co., Texas City, Ten. 77590 After the effects of key variables are investigated, the performance of a system can be represented by a polynomial of an adequate order. In HCN reactor studies, a satisfactory correlation of HCN yields in terms of CHd/NHs mole ratio “3) mole ratio was obtained with a third-order polynomial. and air/(CHd Subsequently, the reactor performance and optimum feed ratios can be predicted for HCN production. However, extrapolation of the empirical correlation could be misleading, and extension by experimentation to extreme conditions was very difficult. Finally, the HCN reactor performance was expanded to CHI/”? = 0 and NH3/CHd = 0 using the existing mathematical correlation in conjunction with the fundamental knowledge of the reaction system.
+
PANand Roth
(1968) reported feed ratios in the H C S process to have a close relationship with the reaction temperatures which could be used t o maximize HCK yields. Twenty-five sets of data had been obtained in a pilot plant study, and a model of a second-order polynomial was developed from 24 sets of these data. However, no reason was given for the adoption of the polynomials, 262
I & E C PROCESS D E S I G N A N D DEVELOPMENT
and no effort was made to improve their correlations or to define the optimum yield directly from feed ratios. The purpose of this paper is to resolve these aspects. After a reliable mathematical equation of the reactor performance in terms of feed ratios was developed, interrelationship of the equation and extrapolation beyond the experimental range were also examined. Thereby, the
Table I. Reactor Performance
Calcd. Wt. %c
Yield Test. Wt. Fc ( N H d , (NHilo HCN
Yield Test. Mole % L C
Test No.
RI
R2
A1 A2 A3 A4 A5 A6
0.9 1.1 1.25 1.4 1.55 1.7
3.25 3.25 3.25 3.25 3.25 3.25
8.03 7.27 6.78 6.35 5.98 5.65
7.91 7.32 6.79 6.40 5.98 5.61
0.86 0.86 1.13 1.72 1.87 1.88
6.32 7.42 7.49 6.42 5.73 5.34
56.48 72.36 83.37 86.33 87.83 86.77
50.34 63.86 69.49 63.20 60.37 59.33
BO B1 B2 B3 B4 B5 B6 B7 B8
0.6 0.8 0.9 1.o 1.085 1.2 1.4 1.7 2.0
3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09
9.95 8.84 8.38 7.95 7.63 7.24 6.63 5.90 5.34
10.00 8.88 8.47 8.01 7.71 7.45 6.70 6.20 5.38
0.79 0.90 1.01 1.06 1.08 1.76 2.07 2.50 2.67
5.01 6.54 7.24 7.67 7.81 7.26 6.27 4.88 3.54
34.27 51.63 61.14 69.53 74.21 80.38 85.32 83.09 82.30
c1 c2 c3 c4 C5 C6 C7
0.7 0.9 1.o 1.1 1.25 1.4 1.7
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
D1 D2 D3 D4 D5
0.7 0.9 1.1 1.4 1.7
2.80 2.80 2.80 2.80 2.80
10.17 9.10 8.23 7.20 6.40
10.25 9.04 8.12 7.24 6.42
1.39 1.74 2.57 3.10 3.33
8.08 9.06 7.34 5.40 4.02
(“4
trends of the Andrussow HCN process were extended to a greater range, and the usefulness and limitations of the mathematical correlations were generalized. Development of Mathematical Equations
A basic approach to describe a reactor performance is to establish a mechanistic kinetic model. This requires a rather thorough study of the reaction mechanism and its corresponding rate equations. However, for practical use in process improvement, a less theoretical model made of empirical correlations from operating variables can serve the same purpose. I n the case of the HCN process, feed ratios are basic variables. After their effects on the system were investigated, there was no obvious mathematical model to fit the data. A reasonable approach (Ostle, 1964) was then to develop a reliable correlation with the polynomial of an adequate order. Table I shows the two newly added tests, BO and B8, together with those listed by Pan and Roth (1968). Two revisions were done to the second-order polynomial. The first was generated from 25 sets of data, which excluded tests BO and B8. The second was generated from all 27 sets. The basic model describing the performance of the H C N catalytic reactor is still a second-order polynomial, designated as the first model. =
a0
+ alxl f aZx2 + aSx: + a4x: + a5xlx2
where
Z = Y , C, L , o r D
x?
= air/ (CH,
+ NH,) = Rz
(1)
Temp., O C.
D
T/
T,,
10.87 11.75 16.64 26.80 31.27 33.16
38.79 24.39 13.87 10.00 8.37 7.51
1182 1140 1127 1122 1115 1125
978 958 947 950 953 960
31.56 46.40 53.85 60.33 63.82 61.39 59.90 49.59 41.45
7.90 10.14 11.92 13.23 14.01 23.62 30.90 40.65 49.73
60.54 43.46 34.23 26.44 22.17 14.99 9.20 9.76 8.32
1205 1162 1144 1130 1125 1104 1095 1105 1116
977 962 948 937 934 940 945 954 962
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
55.45 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
Y
The various coefficients and standard errors of estimate (standard deviations) of these revised equations, shown in Table 11, were obtained from multiple regression analysis using a digital computer. The accuracy of the correlated relationship is measured by a larger multiple correlation coefficient, R , and a smaller standard error of estimate, U. Table I1 shows that the over-all accuracy of the first revised equation is better than that of the response equation given by Pan and Roth (19681, because the revised equation used more data and should have resulted in a better correlation for the same range. Yet the second revised equation using two more sets of data was not as good as the first revised equation, because it covered a 40% wider range of R1 = CHI/KH3 mole ratio (from 0.6 to 2.0) than the first revised equation (from 0.7 to 1.7). But for the two extreme C H 4 / X H j ratios a t 0.6 and 2.0, the second revision should be better than the first. This was verified from the observed L‘S. predicted HCN yields-for example, the observed ultimate yields were 34.27 and 82.305 a t these two ratios. The predicted values from the first revised equation are 33.30 and 69.485, respectively, and from the second revised equation are 35.53 and 77.07‘C, respectively. These revised equations still result in rather large deviations when used in predicting reactor performance. In experience, the reactor performance had good reproducibility; the deviation of yields of duplicate runs usually was less than ~ 1 . 5 % absolute. Thus, the calculated deviations should be attributed primarily to the inadequacy of the assumed second-order polynomial. One logical approach (Ostle, 1964) to improving the fit of the model consisting of a polynomial is to increase its order (or degree). Thus, the above secnnd-order polynomial was VOL. 8 NO. 2 APRIL 1969
263
Table II. Coefficients and Standard Errors of Estimates
Y , Mole 5;
Item
C , Mole SC
L , Mole 7;
Table 111. Observed and Calculated HCN Ultimate Yield
D, Mole C;
1st Model
Test
Obsd Mole cc
Calcd. mole
a0 a! a? a3 a, a5
R d
294.51366 -35.95775 -120.71568 -67.19379 1.63966 76.53896
280.55652 -99.13774 -119.54477 -58.10257 6.38665 78.76918
-4.94227 77.45249 12.20150 0.81478 -5.37555 -14.04515
-172.84118 19.11541 106.17611 57.42092 -0.90444 -64.01003
((
1 2 3 4 5
56.48 72.36 83.37 86.33 87.83
0.97262 2.92040
0.90249 3.80621
0.98185 2.69422
0.98066 2.52438
6 7 8 9 10
First Model. Second Revision (Based on 27 Sets of Data) a0
a1
ai a3 ai as
R U
178.48500 -68.44356 -32.91055 -56.71263 -13.31402 79.47207
192.19963 -140.98831 -45.32485 -52.21124 -8.03299 89.09453
-2.87057 92.18345 4.55114 2.35609 -2.58449 -20.78894
-82.21566 47.50349 36.29909 49.60271 11.31302 -67.72648
0.97569 3.29439
0.90734 4.33168
0.98230 2.81156
0.98281 2.84397
Second Model (Based on 25 Sets of Data) -15673.02916 -2787.31256 -10845.54469 14273.24647 542.48229 499.57096 -157,91578 -396.10036 15532.54016 2684.21733 10897.51758 -14096.94857 -51.93045 -7.52308 -27.91141 34.12165 -5092.81356 -816.22943 -3656.29666 4635.42095 -342.52286 -400.60552 190.03063 247.58596 74.79909 124.05686 -69.44099 -56.80404 545.87186 68.83953 413.23335 -499.00467 n- -97.00320 -169.62774 95.24371 i 1.36508 111.29673 153.18009 -75.58608 -84.97564
a.
al
ar a; a?
ai a6
a:
a@ a9
R
0.99181 1.80604
U
0.97954 2.00135
0.99444 1.68377
Resid.
Resid.
56.59 72.28 80.51 85.71 87.90
-0.11 0.08 2.86 0.62 -0.07
55.11 74.24 82.07 85.99 87.53
1.37 -1.88 1.30 0.34 0.30
86.77 51.63 61.14 69.53 74.21
87.05 54.59 63.22 70.51 75.65
-0.03 -2.96 -2.08 -0.98 -1.44
88.20 51.66 62.79 70.92 75.81
-1.43 -0.03 -1.65 -1.39 -1.60
11 12 13 14 15
80.38 85.32 83.09 54.24 75.15
81.06 86.22 83.90 53.20 69.52
-0.68 -0.90 -0.81 1.04 5.63
79.99 82.47 81.68 52.89 73.38
0.39 2.85 1.41 1.Xi 1.77
16 17 18 19 20
80.59 82.41 84.49 83.59 81.80
75.66 80.45 85.13 86.78 81.01
4.93 1.96 -0.64 -3.19 0.79
79.34 82.05 85.37 85.14 83.10
1.25 -0.64 -0.88 -1.55 -1.30
21 22 23 24 25
57.45 78.19 83.19 82.17 81.96
61.29 75.45 84.25 87.36 78.37
-3.84 2.74 -1.06 -5.19 3.59
59.72 76.25 83.06 82.89 81.04
-2.27 1.94 0.13 -0.72 0.92
Table IV. Observed and Calculated HCN Once-Through Yield 1st
0.99286 1.73227
2=
a0
+ alxl + a2x2 + a3x: + + ajx1x2 + asx: + a-xi+ aax:x, + asxlx:
The same 25 sets of data which have the best correlation in the first model were again submitted to multiple regression analysis. The results of the third-order polynomial calculated by the computer are also given in Table 11. A great deal of improvement is gained over the first model. For instance, the standard error of estimate for the ultimate yield, Y , decreases from 2.920 to 1.806, and that for the once-through yield, C, from 3.806 to 2.001. Now the 95% confidence limits of the HCN yields become
Y . =tt95,c(15) X
u
= 3 2 . 1 3 1 x 1.806 = =t3.8486
C. + ~ & ~ ? (xl 5u )= 1 2 . 1 3 1 x 2.001 = +4.2641 These are significant improvements over the previous correlations (Pan and Roth, 1968), in which the 95.; confidence limits were ~ 6 . 3 0 1for Y and +8.223 for C. The improvement of the calculated individual values was also evaluated (Tables I11 and IV). From the statistics point of view, some of the terms contained in the model may contribute little t o the prediction and could be eliminated. This can be done with the test of significance for coefficients by calculating the t statistics. For example, based on the computer regression analysis of once-through yield: 264
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
2nd Model
Model
Calcd
Resid.
mole
D
-3.27 2.12 4.70 -2.02 -2.67
-49.91 64.18 66.72 65.04 61.66
0.43 -0.32 2.77 -1.84 -1.29
6 7 8 9 10
59.33 46.40 53.85 60.33 63.82
58.24 50.37 54.91 58.30 60.26
1.09 -3.97 -1.06 2.03 3.56
59.09 47.16 56.13 61.16 62.90
0.24 -0.76 -2.28 -0.83 0.92
11 12 13 14 15
61.39 59.90 49.59 47.07 61.39
61.59 60.23 49.47 48.54 56.43
-0.20 -0.33 0.12 -1.47 4.96
62.39 56.82 48.73 45.06 60.12
-1.00 3.08 0.86 2.01 1.27
16 17 18 19 20
63.60 60.51 54.82 50.11 42.07
58.64 59.68 59.07 55.84 41.54
4.96 0.82 -4.25 -5.73 0.53
62.13 61.47 57.08 50.84 42.87
1.47 -0.96 -2.26 -0.73 -0.80
21 22 23 24 25
49.66 63.14 56.91 46.99 39.45
52.42 58.11 59.16 52.00 34.39
-2.76 5.03 -2.25 -5.01 5.06
51.72 60.98 58.43 45.50 39.52
-2.06 2.16 -1.52 1.49 -0.07
t=-
ai
mole
Calcd
53.61 61.74 64.79 65.22 63.04
1 2 3 4
u4x:
(2)
Obsd Mole c( 50.34 63.86 69.49 63.20 60.37
Test
increased to a third-order polynomial, designated as the second model
2nd Model C'alcd mole
First Model. First Revision (Based on 25 Sets of Data)
-0
S
Ir
- -7.5231
-0
96.9010
Lc
Resid
= -0.0776
From the statistics t table, the absolute value of t a t 95'r confidence limits with 15 degrees of freedom (25 observations minus 10 coefficients estimated) must be great-
er than 2.131; thus a , is not significantly different from zero and can be eliminated. Subsequently, other coefficients of the proposed second model should be re-evaluated without the ai term. But the second model gives satisfactory results, and the major objective of a mathematical correlation is to provide a reliable equation even if one or two additional terms are included. Therefore, no further regression analysis was conducted to shorten the second model consisting of third-order polynomial terms. Instead, derivations were made on the existing model to give some insight into the correlations. Relationship Derived from Mathematical Correlations
Interesting relationships among NH 3 conversion, C, to HCK (once-through yield), N H I leakage, L , and N H I decomposition, D , can be derived from the following basic equation:
C + L + D = 100 or
C = 100 - ( L + D)
(3) Figure 1, the plots of L , D, C, and ( L + D ) for a constant R, = air (CH4 + NHI) = 3.09 mole ratio, shows that the maximum C occurs a t the minimum point of the sum of ( L + D ) , just as Equation 3 indicates. The observed values giving the maximum C are given in Table V. Furthermore, to obtain a maximum C for a certain R? = air' (CH, + N H I ) mole ratio, the following necessary condition should be met:
Table V. Values of R I Giving Maximum C Where dev. = obsd. R 1st
-
calcd.
Obsd R1
Calcd HI
Dei
R1
DeL
3.25 3.09 2.94 2.80
1.24 1.14 1.05 0.94
1.35 1.24 1.14 1.04
-0.11 -0.10 -0.09 -0.10
1.25 1.12 1.02 0.94
-0.01 +0.02 +0.03 0
Calcd
I t means that the slope of curve L is equal and opposite to that of curve D a t the point of maximum C. Based on the shapes of these two curves in Figure 1, the former is decreasing and the latter is increasing as R , increases. Therefore, their respective slopes should be opposite to each other a t the maximum C. Besides, Equation 4 can be employed to estimate the maximum C as follows: From Equation 1 of the first model and Table 11:
dL ~
dR I
= a l + 2 a j R i+ a j R 2 = 77.45249
+ 2(0.81478)R1- 14.04515 R 2
-__ dD = - ( a , + 2 a r R 1+ aiR4 dR 1
Then
(4)
2nd Model
R
= -19.11541 - 2(57.42092)Ri
or
Rl
Model
R1can be
+ 64.01003R2
solved from Equation 4 and
R1 =
78.06518RZ - 96.56'790 116.47140
(5)
Substituting different values of R? from Table 11, the R , values which give the maximum C are calculated from Equation 5 and listed in Table V.
60
s
10
40
10
PO
IO
I
0
0
0 4
1
1
I
1
0 1
1.2
I 6
"' MOLE
2 0
Similarly, the estimated RIvalues giving the maximum C can be obtained from the second model, since the resultant equation of R: is quadratic and involves two roots for a given R? value. Only one root giving a more reasonable value is taken (Table V ) . As expected, the predicted R: values from the second model are closer to the observed values than those from the first model. The feed ratio giving the maximum C can also be obtained directly from its correlations in terms of R1and R?. As a matter of fact, the R , values obtained from d C dR; = 0 a t given R? values were the same as those calculated with Equation 4. Thus, the above derivations not only served as an additional means to ensure the exact feed composition for the maximum conversion, but also demonstrated the basic interrrelationship of mathematic correlations. In practice, these correlations have been translated to plant HCK production and have proved very useful in maximizing the feed composition.
2 4
Extrapolation beyond Experimental Range RATIO
"3
Figure 1 . Relations of conversion, leakage, and decomposition of I:"
Although a higher order polynomial can improve the fit of data within the experimental range, it does not necessarily result in improvement for extrapolation beyond that range. The performance of a system outside the VOL. 8 NO. 2 APRIL 1 9 6 9
265
experimental range could continuously increase, decrease, flatten out, or make a rapid change of direction. If there are no data available for correlation, it is impossible to know if one empirical equation is better than others. Usually, extrapolating an empirical correlation beyond the experimental range involves risk and uncertainty. Yet, sometimes extension by experimentation to extreme conditions is very difficult or is restricted by available time. Then the needed information should be obtained with the aid of fundamental knowledge of the system. One may ask what the HCN reactor performance would be if the C H , / S H s mole ratio and/or air/(CH, + NH1) mole ratio continues to increase and decrease. The investigated range for the a i r / ( C H 4 + NH,) mole ratio is from 2.80 to 3.25 mole ratios, which are of great importance to the HCK industry. If the air/ (CH, + NH?) mole ratio is much smaller than 2.80, HCK yields will be very low. Also, the catalyst used in the HCK reactor will be quickly deactivated by carbonaceous deposit because of the insufficient supply of air. If the air/(CH, + XH,) mole ratio is much larger than 3.25, the system becomes inflammable and explosive. But the effect of extending the C H , / K H I mole ratio beyond the current range is not so obvious. To simplify calculation, let the C H I / S H , mole ratio be extended to zero. Then, if the first model is used, a t air/(CH, + "I) = 3.09 and 3.25, the predicted values of C from both revised equations of this model are negative. These are impractical. If the second model is used, the predicted C will be larger than 1OO'C. This is impossible. Thus, these mathematical correlations should not be extrapolated to estimate the reactor performance far beyond the explored range. However, a reasonable approach can be achieved by combining the existing mathematical correlations with the aid of fundamental knowledge. The existing correlations indicate that the response curves C , L , and D are monotonic for a considerable extension-Le., they either gradually increase or decrease with C H i / N H ? mole ratio. Therefore, extrapolations can be accomplished, as demonstrated below.
W h e n CH,/NH,-+
0
In this condition. CH4 could be zero in feed gas. The amount of 0, for complete combustion of N H ? in the system should be 0.75 mole per mole of N H I , as the following reaction indicates:
2NH3 + !>O? + N,
+ 3H?O
Consequently, the trend of reactor performance for various air/ (CH, + S H I ) mole ratios should be as follows: At Air/(CH, + NHI) = 3.25 Mole Ratio. The feed then has 0 2 / N H 1 = 0.683 mole ratio (based on 2 1 mole Onin air). Oxygen is insufficient for a complete combustion of KH.3, and some unreacted NH3 (leakage) will be in the off-gas. Meanwhile, conversion to HCK should be zero, because there is no CH, in the feed to react with N H I . K H I decomposition should be very high, but should never reach loo(;,, since some N H , leaks through the catalyst without reacting. At .4ir/(CH, + " I ) = 3.09 lMole Ratio. The feed gas has On/KH.i= 0.649 mole ratio. Under these conditions, there is a greater insufficiency of oxygen for a complete combustion of KH.]. Consequently, more unreacted X H , (leakage) should be in the off-gas than in the previous case when air/(CH, + NH,) = 3.25. Conversion to HCN is still zero because there is no CH, in the feed. 3" 266
I & E C PROCESS D E S I G N A N D DEVELOPMENT
-20-
CH4 "3
MOLE R A T I O
Figure 2. Extrapolation of HCN reactor performance
decomposition should be lower than in the previous case because the leakage is larger. In the same manner, similar trends can be established for air/(CH, + NH,) = 2.94 and 2.80. I t is also reasonable to say that C H 4 / N H 3reaches zero and air/(CHI + KH3) approaches 3.571 mole ratio, NH3 leakage will approach zero and N H I decomposition will approach 100% as the limits. Needless to say, whenever CH4/XH3 reaches zero, conversion to HCK should be zero. The trends of the above extrapolations are shown in broken lines in Figure 2.
When NH,/CH,-
0
I n this condition, NH, gradually diminishes in the feed, and the CH4/NHq ratio could approach infinity as the limit. The amount of O2 for ultimate combustion of CH, in the feed should be 2 moles per mole of CH,, as shown in the following reaction:
CH,
+ 202-
CO2 + 2HjO
O? content could be much less than this value, but there is very little chance for the small amount of S H Y to react with CH, because of the greater reaction rate of CH, with On (Pan and Roth, 1968) and a greater number of CHI molecules to compete with NH,. In the practical situation, CH4 can be converted into a mixture of CO, C o n ,and H,O a t a smaller amount of 02. Therefore, the small amount of KH3, if any, in the feed will pass through the reactor without a chance of reaction, resulting in nearly 100% leakage. Meanwhile, conversion to HCN will be drastically decreased with
D = NH, decomposition, mole % L = N H ? leakage (unreacted), mole % R i = C H 4 / N H 3mole ratio Rr = air/ (CH, + N H ? ) mole ratio S = variance Y = ultimate yield = HCN/[(,UH3), - (NH7)"I, mole L& 2 = reactor performance, C , D , L , or Y u = standard error of estimate i, 0 = feed inlet and reactor outlet
the decreasing of N H s in the feed. NH3 decomposition will also decrease to zero as the limit. These extremities become more pronounced when XH3/CH4 becomes smaller, as shown in broken lines a t the right of Figure 2 . Since the above extrapolations are based on R1 = C H 4 / NH3 from 0.7 to 1.7, the two additional sets of data of R I = 0.6 and 2.0 a t Rz = air/(CH, + N H 3 ) = 3.09 can be regarded as confirmations of the correct trends of the above extrapolations. Acknowledgment
The author thanks Fred Applegath for many suggestions and computer work. Nomenclature an, a l
---- as C
Literature Cited
Ostle, B., "Statistics in Research," 2nd ed., pp. 190-1, Iowa State University Press, Ames, Iowa, 1964. Pan, B. Y. K., Roth, R . G., IND. ENG.CHEM.PROCESS DESIGNDEVELOP.7, 1-53 (1968).
= coefficientsof polynomials = once-through yield = HCi%/(i%H3),, mole %
RECEIVED for review June 20, 1968 ACCEPTED December 16, 1968
HEAT TRANSFER BETWEEN COILS AND NON-NEWTONIAN FLUIDS
WITH PROPELLER AGITATION A .
H .
P .
S K E L L A N D
A N D
G .
R .
D I M M I C K '
University of Notre Dame, Notre Dame, Ind. 46556
THEdirect involvement in non-Xewtonian products and processing in the chemical process industries of the United States probably exceeds 12 to 15 billion dollars annually (Brasie, 1964). I t is therefore somewhat remarkable that so few papers have appeared in established journals on heat transfer to or from non-Newtonian materials in agitated vessels. Carreau et al. (1966) studied heating and cooling of power law pseudoplastic non-Newtonian fluids in a jacketed vessel agitated by a four-bladed, 45" pitched turbine. They correlated about 109 data points with a mean deviation of 19.3% by the equation
D2"L h,Di= 1.474 K k
[
"'(=)I
n
"
for D J D T = 315; 0.34 5 n i 0.63; 100 2 "NRegen" - 5000. The differential viscosity pd = d.r,,g,/d ( d u l d y ) , 5 and subscript 03 denotes high shear rates. These authors expressed doubts about the validity of their Prandtl num. ber evaluated using p d - . Hagedorn and Salamone (1967) correlated extensive heating and cooling data, again specifically for power law non-Sewtonian liquids in a jacketed vessel using, in turn, anchor, paddle, propeller, and turbine agitation. They Present address, Atomic Energy of Canada, Ltd., Chalk River, Ontario, Canada.
achieved correlation with a mean deviation of 20% using the equation
for 1.56 5 D f D r 5 3.5; 0.36 5 n 5 1.0; 35 5 D ' N ? - " p / K 5 680,000. The constants C' and a to g were tabulated for the various impeller types used. The quantities K and n appearing in Equations 1 and 2 are those used to characterize the well-known empirical Ostwald-deWaele or power law model:
No studies on heat transfer between coils and nonKewtonian fluids have yet appeared. I n addition to being confined to jacketed vessels, Equations 1 and 2 are further restricted to non-Newtonian materials of the power law type. The present study was therefore undertaken to provide data on coil heat transfer and to obtain a generalized correlation applicable to all time-independent nonNewtonian fluids, thereby eliminating restrictions associated with particular empirical relationships such as the Bingham plastic, Ellis, or power law models. VOL. 8 NO. 2 A P R I L 1 9 6 9
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