Sampling Results and Conclusions Askins, et al. (1951), describe gas sampling from Shell Oil Company’s FCC regenerator. They withdrew gas samples from various points in the fluid catalyst bed of a 40-ft diameter catalytic cracking regenerator. A filter in their sampling tube prevented catalyst contamination of their gas samples. Catalyst samples were not taken. Gas analyses indicated that a high degree of gas downmixing exists, since essentially no variation in gas composition was found a t any point. They also concluded that most of the gas introduced passed through the bed in the form of bubbles, rather than flowing around and fluidizing the catalyst particles. Preferential CO2 adsorption on catalyst was also found. Our sampling system successfully sampled partially regenerated catalyst and flue gas and measured temperatures a t six locations in the dense bed. Results are shown in Table I1 and Figures 2-4. Operating conditions for the Richmond FCC reactor and regenerator are shown in Tables I11 and IV. Carbon analyses show that mixing is better than expected. Most of the catalyst is fairly well regenerated 3 ft above the grid. Some catalyst samples collected in the no. 5 sampler contain less coke than the “fully” regenerated catalyst. This suggests some “bypass” of unregenerated catalyst (perhaps under the baffle between the spent catalyst riser and the overflow well). Catalyst samples collect-
ed from right under diplegs have higher carbon levels than those collected away from diplegs (Figure 2 ) . Gas analyses sh0.w that the 0 2 concentration 3 ft above the grid near the‘spent catalyst riser varies from 1% to 370,whereas closer to the overflow well, 0 2 concentrations vary between 3% and 8% (Figure 3). CO2/CO ratios are very close to that of the final flue gas (1.28) in every sample. This is very close to the thermodynamic equilibrium value. This suggests that the CO2/CO ratio in an FCC regenerator does not depend on 0 2 concentration. The similarity of CO2/CO ratio 3 ft above the grid and in the final flue gas (1.28) also shows that there was little CO afterburning. Temperatures vary less than 40°F between the wall and the center of the regenerator 3 ft above grid level (Figure 4). Acknowledgment H.F. Mason, E.H. Doberenz, R.L. Flanders, B.G. Spars, aad R.R. Tarica contributed with helpful suggestions. The samplers were manufactured in the Chevron Research Machine Shop. Literature Cited Askins. J.W., Hinds, G.P., Jr., Kunreuther, (8),401 (1951).
F.. Chem. Eng.
Progr.. 47
Receicedfor recieu. July 12, 1974 Accepted October 7, 1974
COMMUNICATIONS A Correlating Equation for Dewpoint Temperature-Pressure-Composition Data of the Methane-Ethane System between 100 and 600 Psia Application of a previously reported technique of using Chebyshev polynomials has resulted in the correlation of P-T-y data for the methane-ethane system over a range of 100 to 600 psia. The average of absolute errors of the calculated pressures relative to 165 data points of 15 compositions is 0.807%. The average of absolute differences between ( J P / J T ) ) , calculated from this correlation equation and those corresponding values of 95 points in the literature is 4.37%.
Dewpoint temperature and pressure data of vapor mixtures are needed in the design of partial and total condensers as well as other types of process equipment. While these can be estimated by using an equation of state with appropriate mixing rules, each calculation to determine the dewpoint temperature, pressure, or composition involves a trial-and-error or repetitive search. Therefore, it is frequently desirable to correlate a set of such estimated results or measured data points in the form of a single relationship to cover a wide range of each of the variables. This alternative avoids direct interpolation or repetitive use of an equation of state. Such a relationship thus would provide coherence and can facilitate the retrieval of variables in an optimization type of design calculations. It also could aid the computation of partial derivatives (e g , Houser and Weber, 1961) in estimating thermodynamic properties. Due to their complexities, correlations of dewpoint temperatures in terms of a single equation have not been found in the literature. This work is a continuation of using a previously reported technique (Corn, e t a l , 1974) of correlating the bubble point temperature-vapor pressure data of the methane-ethane system. The 187 data points representing data of 13 vapor mixtures and 2 pure components in the methane-ethane 96
Ind. Eng. Chem., Process Des. Develop., Vol. 14,
No. 1, 1975
system from 100 to 600 psia were correlated by using the orthogonal Chebyshev polynomials. Dewpoint temperatures at 100 and 600 psia were a t first correlated as T,,, and T,,,. These were then used for computing the normalized temperature parameter X to calculate coefficients of eq 1 for each vapor composition by minimizing the sum of squares of pressure differences between the calculated values and the data. Finally, the coefficients were correlated with the vapor composition by using the leastsquares method. Correlation Equation The correlating expression, eq 1, is the same as that of the bubble temperature data except that different equations were found to represent various coefficients. Higher order Chebyshev coefficients were dropped since they did not improve the correlation.
c1 + where A = 418.6937
A
T ) log P = a0
+
alX
3551.9761q1 - 31586.77Oql2
(1)
+
+
1 5 2 0 8 5 . 5 8 ~~ ~3 4 8 8 8 0 . 2 ~ ~ ~3 8 5 9 1 0 . 9 8 ~~~ 16 10 57.1311 16
(2)
a, = 1259.2095
+ 8167.3544~1 - 74154.5253,'
t
-IZ'
3 5 5 7 9 2 . 4 6 ~ 1~ 815891.47~1+ ~ 904665.53~1' 3 7 8 9 5 7 . 6 4 ~ ~ ~(3)
al = 342.95426 66O89.4O5yl3
+
1429.7751~1,- 1 4 1 7 4 . 8 1 9 ~ t~ ~
- 145679.7411~4+ 154814.681'1~6 2 5 8 1 . 5 2 6 ~ ~ ~(4)
x T,,
= (2T - (Tmu
= 535.5 - 198.53,
(- 0.00319
Tmin =z 413.7 - 159.53,
+ TmiJ)/(Tma + yt(1 - VI)/
- Tm,A
+ 0.01778 exp(+ ~ 1 ( -1 ~t)/(0.01027 -
(5) (6)
0 . 0 0 9 4 1 8 ~ ~ ) (7)
Discussion Dewpoint temperatures of the methane-ethane system were found to be more difficult to correlate than the bubblepoint temperatures. This is reflected here by the more complex forms of eq 6 and 7 relative to corresponding ones reported for bubblepoint temperatures (Corn, et al., 1974). Also, the average of absolute errors of correlated pressure is 0.807% relative to 0.55% for the bubble temperature correlation. Large errors are confined in the range of y1 = 0.95 to 1.0. However, at y1 = 1.0 (pure methane) this correlation agrees well with the vapor pressure data (Matthews and Hurd, 1946) and has an average error of 0.64%. A more sensitive test is to compare slopes determined from correlation with those in the literature (Houser and Weber, 1961). The slopes are calculated by using eq 8 and pressures from eq 1.
These slopes have a range of 2 to 20 psia/"R which is twice as large as that of the bubblepoint temperature data. This is likely a factor causing difficulty in obtaining a good correlation. The average of absolute difference between the calculated values and the values of Houser and Weber (1961) for nine pressure levels of 90 points is 4.3770, which is higher than 1.6% reported for those from the bubblepoint temperature correlation (Corn, et a1 , 1974). Again, large errors appear in the range ofyl = 0.95 to 1.0. To demonstrate graphically the data fitting of this equation €or dewpoint temperatures and that for bubblepoint temperatures of the methane-ethane system (Corn, et al., 1974), correlation curves are plotted with measured points of Bloomer, e t a1 (1953), in Figure 1. Correlations of dewpoint temperatures and bubblepoint temperatures (Corn, et al., 1974) were not forced to converge into an identical equation for y1 = 0 and 1.0. For methane, these two correlations result in an average of absolute differences of pressures as 0.36% and the maximum temperature difference as 0.5"R, approximately. Corresponding values for ethane are 0.98% and 1"R. respectively. These correlating equations can be stored in a digital computer for solving T , P, x, or y in design or operations computations. Relative to the conventional methods of interpolating the data, these equations provide a coherent, smooth, and simple representation to reduce the calculation time in a computer as well as to eliminate the storage space required for data input.
30 0
20 0
400
500
60 0
TEMPERATURE, 'R
Figure 1. A comparison of data and correlation equatiors, P-T-x and P-T-y.
Conclusions A useful equation is proposed to represent the P-T-y relationship for the saturated vapors of the methane-ethane system over a pressure range of 103 to 600 psia. Nomenclature ao, a1 = Chebyshev polynomial coefficient A = coefficient in eq 1 P = pressureinpsia T = absolute temperature in "R
x1 = mole fraction of methane in liquid
X = normalized temperature defined by eq 5 y1 = mole fraction of methane in vapor
Literature Cited Barkelew, C. H., Valentine, J. L., Hurd, C. O . , Trans. Inst. Chem. Eng.. 43, 25 (1947). Bloomer, 0 . T., Gami, D. C.. Parent, J. D. , Inst. Gas Techno/. Res. Bull., 22 (1953). Corn, B. R.. Young, M. D.,Weber, J. H., Tao. L. C , lnd. Eng. Chem., Process Des. Develop., 13, 95 (1974). Houser. C. G., Weber, J. H., J. Chem. Eng. Data. 6, 510 (1961). Matthews. C. J., Hurd. C. 0.. Trans. Inst. Chem. Eng.. 42, 55 (1946).
Department of Chemical Engineering University of Nebraska Lincoln, Nebraska 68508
Bruce R. Corn J a m e s H . Weber Luh C. Tao*
Received for reuipu, J u n e 10, 1974 Accc.pted October 28, 1974 The authors acknowledge the financial aid from Phillips Petroleum Company to Bruce R. Corn as well as the support of the Engineering Research Center. I n d . Eng. Chem.,
Process Des. Develop., Vol. 14, No. 1, 1975
97
