Ind. Eng. Chem. Res. 1988,27, 1487-1493 p
= density, lb/ft3 = viscosity, lb/(ft-h)
SubscriDts
A = component A B = component B c = critical g = gas phase i = component i 1 = liquid phase o = overall t = total Registry No. Methanol, 67-56-1; water, 7732-18-5. Supplementary Material Available: The Fortran program for the algorithm described in the text (3 pages). Ordering information is given on any current masthead page. Literature Cited Baker, D. R.; Shyrock, H. A. J. Heat Transfer 1961,83,339. Bromsley, L. A.; Wilke, C. R. Ind. Eng. Chem. 1951,43,1641. Chen, N.H.;Othmer, D. H. J. Chem. Eng. Data 1962,7,37. Eldridge, R. B. M.S. Thesis, University of Arkansas, Fayetteville, 1981. King, C. J.; Hsuek, L.; Mao, K. W. J.Chem. Eng. Data 1965,10,348.
1487
Moncada, D. M. M.S. Thesis, University of Arkansas, Fayetteville, 1978. Perry, R. H.; Chilton, C. H. Chemical Engineer’s Handbook, 5th ed.; McGraw-Hill: New York, 1973. Pittaway, K. R.; Thibodeaux, L. J. Znd. Eng. Chem. Process Des. Deu. 1980,19,40. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Sherwood, T. K.; Holloway, F. A. L. Trans. AZChE 1940,36, 39. Schiebel, E. G. Znd. Eng. Chem. 1959,46,1959. Thibodeaux, L.J. Chem. Eng. 1965. June 2. 165. Thibodeaux; L. J. Znd. Eng. Chem. Process Des. Dev. 1980,19,33. Thibodeaux, L.J.; Moncada, D. Paper presented at the symposium on “Recent Advances in Separation Technology“, National Meeting of the American Institute of Chemical Engineers, Chicago, IL, Nov 16-20, 1980. Thibodeaux, L.J.; Daner, D. R.; Kimura, A,; Millican, J. D.; Parikh, R. J. Ind. Eng. Chem. Process Des. Deu. 1977,16, 325. Treybal, R. E. Mass Transfer Operations, 3rd ed.; McGraw-Hill: New York, 1980. Velaga, A. Ph.D. Thesis, University of Arkansas, Fayetteville, 1986. Yen, L. C.; Woods, S. S. AZChE J. 1966,12,95. Yoshida, A. Chem. Eng. B o g . Symp. Ser. 1955,51,59.
Received for review April 23, 1987 Revised manuscript received November 20, 1987 Accepted April 15, 1988
GENERAL RESEARCH Cyclopentadiene-Dicyclopentadiene Phase Equilibria and Reaction Rate Behavior Using the Transient Total Pressure Method Colin S. Howat* and George W. S w i f t Kurata Thermodynamics Laboratory, Department of Chemical and Petroleum Engineering, University of Kansas, Lawrence, Kansas 66045
This article presents new phase equilibria and dimerization rate data for the cyclopentadiene-dicyclopentadiene binary system using the transient total pressure experimental method. This experimental method can be applied to reacting systems to determine simultaneously phase equilibria and reaction rate behavior without phase sampling when it is coupled with a suitable experimental design method. Phase equilibria and reaction rate data are reported a t 313 K. Reaction rate parameters based on an autocatalytic, second-order model are also reported. The initial rate of dimerization is equivalent to those reported in the literature. Cyclopentadiene (CPD) is a contaminant and catalyst poison in the process for the polymerization of isoprene. The acceptable limit for CPD in polymerization-grade isoprene is approximately 1ppm. Raw isoprene feedstocks contain about l0-20% CPD. Measurement of the phase equilibria of chemical mixtures containing CPD is complicated by the spontaneous dimerization of CPD to dicyclopentadiene (DCPD). Relative volatilities of CPDisoprene mixtures are near unity, which mitigates against using the traditional method of sampling coexisting phases for reacting systems. Also, the uncertainty in the phase equilibria is increased further by the chemical-reactioninduced nonequilibrium. Consequently, a new experimental/numerical method was required to measure the phase equilibria of CPD-containing mixtures. This article presents a numerical method founded upon maximum likelihood estimation (MLE) which can be applied to transient total pressure data to simultaneously 0888-5885/88/2627-1487$01.50/0
determine the phase and kinetic behaviors. This analysis procedure is used in conjunction with the experimental design procedure of Howat and Swift (1983) to ensure that systematic differences between the resultant model and the data are minimized. In this new experimental procedure, the pressure transients of samples with different starting compositions are measured over a suitable time interval. The transient data are then analyzed by using maximum likelihood estimation to determine the rate constants, solution model parameters, and vapor pressures. The experimental design is then reanalyzed to minimize systematic differences between the model and the data. Discussion of t h e Method The governing equation for vapor-liquid equilibria is y.@P $ 2 = 3tiyiPioq+” exp(ui(P- P,O)/RT) (1) provided that (1) the component partial molar volume is 0 1988 American Chemical Society
1488 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988
equivalent to the pure-component liquid volume and (2) the pure liquids are incompressible. Summing over all components results in the equation for the total pressure: m
P = Xx1-y,P,O(4,O/4Jexp(u,(P- P,O)/RT)
(2)
and governing rate equation, assuming a second-order model, are
2C,H,
-
ClOH12
- ( l / V ) dn,/dt = k(nl/V)’
(11) (12)
1
Equation 2 is the basis for the Barker method. It shows that P = fb,,T 1 (3) Given composition, temperatures, and pressures, eq 3 can be used to determine solution model or equation of state interaction parameters and pure-component vapor pressures by using nonlinear regression analysis or MLE. Equation 4 gives the objective function for MLE applied to total pressure data. n
s = X((P,e - P;)2/Sp,Z + (T,”- T1C)2/STZ2+
- x,c)/s,82) (4)
Model equations are required for the activity coefficients, vapor pressures, fugacity coefficients, and liquid densities. Activity coefficients can be described by using the modified Wilson solution model (Tsuboka and Katayama, 1975): In Y,= -[In ( C ( x J A J r-) )C ( X k h c k ) / X ( X J A J k ) l + k
PLJ =
=
P ~ J
(6)
ul/u]
exp(-(&j - Xjj)/RT)
(7)
The Miller (1964) equation can be used to describe vapor pressure as a function of temperature: In P = A , / T + B, + CIT + D,T2 (8) The pure-component and mixture fugacity coefficients required for eq 2 can be estimated by using the Graboski and Daubert (1978) equation of state. Liquid densities needed for eq 2 and 6 can be estimated by using the Hankinson and Thomson (1979) method. Application of eq 4 to mixtures undergoing chemical reaction is not possible since the liquid composition changes as the reaction proceeds and is unknown. Making two primary assumptions allows the development of a similar numerical analysis technique for transient total pressure data. First, the reaction is slow compared to the time required to reach equilibrium (reaction rate limiting): under this assumption, eq 2 describes the total pressure. Second, the experimental procedure is such that no vapor phase is present, thus eliminating a simultaneous vapor reaction. The functional dependence of pressure for reacting systems is
P = f(t,T,n,(O)) (9) where n,(” is the molar charge of component i at time zero. An equivalent form to eq 4 for a reacting system results in the following MLE objective function: n
S = C((P,“- PLc)2/~pc2 + (T,”- T,C)2/~T,2 + 1
( t , e-
t1C)/stg2+
- ( l / V ) dn,/dt = k’/(l - k ” ( ~ ~ z / V ) ) ( n l /(13) V)~
k = k’/(l
1
liij
where component 1 is CPD and component 2 is DCPD. The reverse reaction is negligible at temperatures near 300 K, and consequently, the rate equation is based on the forward reaction only. Assuming that hydrocarbon volumes and isothermal conditions allows eq 12 to be integrated easily and n, determined. Schmid et al. (1948) conclude that the CPD dimerization reaction is autocatalytic. The rate expression for this type of reaction, again ignoring the reverse reaction, is
m
( C ( n k ( 0 ) ek
nk(o)c)2/snk2(o))1) (10)
Equation 10 is used to determine the parameters for the vapor pressure, solution model, and rate models. The CPD-DCPD system is used as an example to present the details of the method. The chemical reaction
- k”(n,/V))
(14)
The transient total pressure method must be able to discriminate among kinetic models if it is to be viable. The uncertainty in the CPD dimerization rate expression provides the opportunity to test model discrimination capability. The data required for this procedure consist of initial molar charges, temperatures, times, and total pressures with experimental uncertainties in these quantities. The regression procedure begins by assuming values for k,the solution model interaction parameters, and if desired, the component vapor pressure(s). Equation 12 is solved for nl, which by material balance gives n2. Mole fractions can then be determined with resultant determination of activity coefficients and total pressures. The residuals are determined by using eq 10, and all parameters are then reestimated. The procedure is then repeated until estimates for the parameters stop changing.
Application to Numerical Examples This section develops the importance of coupling the analysis method with an experimental design procedure. It must be recognized that the pressure residuals are relied upon to identify systematic errors in the model and data. Detailed results and analysis are found in Howat (1983). The objective of the ultimate experimental program is to determine accurate estimates of the CPD vapor pressure, the solution model interaction parameters, and the CPD dimerization kinetic parameters. The experimental design procedure of Howat and Swift (1983) is used to develop and test the experimental design. This procedure requires a parameter base which is initially assumed to be canonical. It also requires estimated experimental uncertainties. Table I summarizes the parameter values and the experimental uncertainties used in the experimental design analysis. The experimental design procedure can be summarized as follows: 1. Develop a proposed experimental design including temperatures, starting compositions, duration, and number of measurements. 2. Preform a Monte Carlo simulation of the proposed design, examine pressure residuals for unwanted systematic behavior, and compare calculated parameter values to the canonical values. 3. If the residual behavior and parameter values are acceptable, propose expected systematic errors in the data and/or models. 4. Perform Monte Carlo simulations with the proposed systematic differences between the model used to set up the data and the one used in the analysis.
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1489 Table I. Parameter Values Used in the Design of the Cvclouentadiene-DicvcloDentadiene ExDeriments _ _ _ _ _ _ _ _ ~ ~ ~ Miller coeff CPD DCPD A -3.66508 X loa 1.47671 X lo' B 1.74693 X 10' -2.02863 X lo2 C -4.31966 X 3.99403 X lo-' D 2.02846 X lo* -3.43201 X lo4 ref Hull et al. (1965) Howat and Swift (1985) modified Wilson coeff 423.02 XZ1-Xll, cal/mol -269.66 X12-X22, cal/mol Howat and Swift (1985); ref isoprene-DCPD kinetic constants 4.7520 X lo7 ko,L/mol/min E , cal/mol 1.6343 X lo3 Howat (1983) ref measurement uncertainties st, min 1 0.02 ST,K sp, kPa 0.08 1 x 104 s,(O), mol
5. Analyze the pressure residuals and, possibly, the parameter values for characteristic trends which would identify model or data error. 6. If no trends are present and the simulated systematic error is likely, then the experiment must be redesigned so that the behavior of the pressure residuals will identify that error. The examples discussed in this section are restricted to the analysis of a single isotherm (313.15 K) in order to simplify the discussion. As an example, the first experimental design tested had a 12-h duration with measurement intervals of 0.5 h. The initial CPD mole fraction was 1.0. Analysis conducted with no systematic errors resulted in parameter values consistent with the canonical values (Howat, 1983). Pressure residuals randomly scattered around zero. Even though the mole fraction after 12 h was 0.54, this experimental design appeared to be acceptable. However, simulations with two systematic errors indicated its inadequacy. First, one problem with the transient total pressure method is pressure lag: this is potentially caused by the CPD charge temperature being lower than the bath temperature. The initial rate of reaction is thus lower than the reaction rate that would be indicated by the bath temperature. Simulations with pressure lag did not result in systematic trends in the pressure residuals. Further, the vapor pressure estimate for CPD becomes biased. The other parameters were unaffected. Another systematic error is that caused by the possible autocatalytic rate of reaction, e.g., eq 12 vs eq 13. For example, developing the canonical data with eq 13 but analyzing it with eq 12 represents a model systematic error. The parameter k" was set such that the rate in pure DCPD was 40% higher than that in pure CPD. Examination of the RSME and bias from those simulations indicated that the estimated solution nonideality has compensated for the model error. There are no nonrandom trends in the total pressure residuals. Consequently, this experimental design is unacceptable because it cannot identify possible systematic errors in the model or data. The more significant problem is the systematic error in the model since the phase equilibria and kinetic parameters are biased. A second design consists of two data sets each with 12-h duration with measurement intervals of 0.5 h. The starting compositions for the two sets are 1.0 and 0.5 mole fraction CPD, respectively. Even though fewer measurements are made compared to the first design, the precision in the parameter estimates im-
0.600
I I
0.400
3
I
0.200
0 V
u
a 0.000 I
n
a
i
-0.400
-0.600 0.00
0.20
0.40
0.60
nola F r a c t l n n
0.80
1.00
CPD
Figure 1. Comparison between calculated and simulated total pressures using five mixture designs with no systematic error (symbols correspond to data sets). 1.00
0.60
n'Bo
5
0.40
0
1
*
-
0
a 0.00 -0 - 0 . 2 0 a -0.40 -
2
o
0.20
I
I
V n
-0,60
-0.BO
0.00
V I
0.20
0.40
0.60
0.60
1.00
nolo Fractlon CPD
Figure 2. Comparison between calculated and simulated total pressures using five mixture designs with model systematic error (symbols correspond to data sets).
proved (Howat, 1983). Simulating the experiment with model systematic error resulted in pressure residuals showing moderate systematic behavior in the high CPD composition region, indicating model systematic error. This characteristic residual behavior was enhanced by using five starting mixtures ( x = 1.0, 0.65,0.50, 0.35, and 0.20) with experimental durations of 8 h with 0.5-h measurement intervals. These starting compositions were chosen because they represent the approximate ending composition of the data set with the next higher starting composition. Figures 1and 2 present scatter diagrams of this design with and without the kinetic model systematic error. Different symbols are used to represent each of the five data sets. Systematic trends in the residuals are apparent for each data set. This indicates the suitability of the design for identifying probable model systematic error. Systematic error in the data due to pressure lag did not appear as systematic trends in the residuals. Consequently, this experimental design cannot be used to recognize systematic error in the data. However, the only parameter affected is the estimated vapor pressure. The major conclusions of this section are as follows: 1. Solution model coefficients, vapor pressures, and rate constants can be determined simultaneously by regression analysis given that there are no systematic errors in the model or in the data. 2. The likely type of kinetic model systematic errors to be encountered with the CPD/DCPD binary mixture will result in systematic trends in the residuals of total pressure. 3. The likely type of systematic data error does not result in systematic trends in the residuals. The only parameter affected is the estimated vapor pressure.
1490 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 Table 11. Experimental Cyclopentadiene-Dicyclopentadiene Data = 1.0; symbol 0
xl(0)
T 313.19 313.19 313.19 313.17 313.18 313.18 313.19 313.19 313.19 313.18 313.18 313.18 313.18 313.18 313.19 313.19
t 31 62 89 118 151 182 210 241 271 300 331 363 389 419 450 478
P 98.34 94.58 87.73 88.07 84.47 81.23 78.31 75.51 72.77 70.26 67.81 65.39 63.41 61.37 59.29 57.56
xl(0)
t
28 57 86 117 148 176 210 238 269 301 330 361 389 419 448 476
= 1.0; symbol 0
T 313.20 313.19 313.18 313.19 313.19 313.19 313.19 313.18 313.18 313.19 313.18 313.18 313.18 313.18 313.18 313.18
xl(0)
P
t 30 54 85 116 149 180 209 238 266 297 328 354 387 416 445 476
98.75 95.02 91.58 88.06 84.70 81.81 78.46 75.75 72.99 70.35 67.86 65.53 63.37 61.43 59.45 57.74
= 0.451; symbol 0
T 313.18 313.18 313.17 313.18 313.18 313.18 313.19 313.19 313.18 313.18 313.18 313.19 313.18 313.19 313.18 313.18
P 45.18 44.21 42.94 41.73 40.48 39.56 38.54 37.82 36.82 35.98 35.13 34.42 33.60 32.93 32.24 31.54
t
31 56 97 120 146 175 209 236 268 299 328 356 387 418 449 484
= 0.450; symbol 0 T P 313.17 44.36 313.17 43.34 41.72 313.16 313.17 40.88 313.17 40.02 313.17 39.21 313.17 38.00 313.17 37.22 313.17 36.33 35.42 313.18 34.69 313.17 313.17 33.94 313.17 33.21 32.49 313.17 313.17 31.80 31.24 313.18
supplement xl(0) = 0.651; symbol A t T P 33 313.18 62.91 313.17 60 60.98 92 313.18 58.92 122 313.18 57.08 313.19 150 55.44 183 313.19 53.58 212 313.18 52.10 239 313.19 50.66 270 313.19 49.18 301 313.19 47.77 313.18 46.41 331 361 313.18 45.10 394 313.18 43.88 423 313.18 42.81 451 313.18 41.74 481 313.18 40.62
1.20
1.10
1
Estimate
-
- - - - Estimate
( T a b l e I) [AutOCatl
I
1.00
2
0.90
-L \
000
0.70
0.60 I
0.50 0.0
2.0
4.0
Time
6.0 hours
8.0
/
10.0
Figure 3. Comparison between experimental and estimated pressure transient with initial composition of 100% CPD/O% DCPD.
0 50 L 0 0
--
I
2 0
6 0 Time h o u r s
4 0
E O
I 0
Figure 4. Comparison between experimental and estimated pressure transient with initial composition of 45% CPD/55% DCPD.
4. More starting compositions per isotherm in a design result in clearer trends in the residuals. Experimental Results This section provides the analysis of experimental CPD/DCPD data measured nominally at 313.1 K. The data are analyzed to determined estimates for the parameters. The experimental design is then analyzed to assist in the search for the model that minimizes systematic trends in the residuals. The experimental equipment used in this work is similar to that described in detail in Shanker et al. (1981). The experimental procedure begins by charging DCPD to the equipment, and a precise amount is then metered into the test cell. CPD is removed from storage over dry ice, charged, and metered. The precise metering gives the initial moles of each component charged to the test cell. The total pressure, temperature, and time are measured at various intervals for suitable total time duration. These data then consist of one data set. Other mixtures are made up and their transient behavior measured. DCPD used in this work was purified as discussed in Howat and Swift (1985). The CPD was produced by cracking DCPD with simultaneous degassing and distillation. The CPD was collected in a dry ice/acetone bath and stored over dry ice to prevent dimerization. Table I1 lists the experimental CPD/DCPD data. The graphic symbology given in the table for each of the five data sets will be used throughout this section for that data set and for simulations of it. Support for subsequent maximum likelihood conclusions can be obtained by analyzing plots of normalized total pressure against time. This is done in Figures 3-5. Also
0 60
1
O5OL-0 0
- L -
2 0
_ _L A 4 0 6 0
Time
I-
E O
i 10 0
hours
Figure 5. Comparison between experimental and estimated pressure transient with the initial composition of 65% CPD/35% DCPD.
plotted in the figures are two estimated curves: the first (solid line) is based on the Table I model and the second (dashed line) is based on the assumption of ideal liquid solution with eq 13 rate equation assuming a 40% increase in k . Extrapolating the first five points of Figure 2 to zero indicates that the estimated vapor pressure will be high compared to the literature value (by about 1% ). Extrapolating the data of Figures 3 and 4 to zero indicates that the system is less nonideal than the Table I estimate: the extrapolated points fall below the solid line in the figures. The slope of the solid line is generally less than the slope of the data. This indicates that the rate of reaction is greater than the initial rate of reaction given in Table I. The coincidence of the dashed line and data indicates that eq 13 is applicable. Consequently,the conclusion from this
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1491 0.80 0.60 0.40
I
1
,
:::1
2.0
Q
c
I
00
0.8
-1,2
- -1.6 -2.0
-1 40
I
0.00
0.20
0.40
0.60
0.80
1
I
Y
B
X”
I 1.00
Molm Fraction C P D
Figure 6. Comparison between calculated and experimental pressures assuming a constant k .
0’40
t
0
0 40
0 O
-
I 0 2 o c
2-0.20
-0.40
1
--‘pa
0-
I
@
0 -0 40
0
O I
-0.60 0.00
0.20
0.40
0.60
0.80
1.00
M o l e Fractlon C P O
Figure 7. Comparison between calculated and simulated pressures where k increases by 40% and the assumed k is constant.
analysis is that (1)the estimated vapor pressure from MLE will be above the literature value, (2) the system is more ideal than the model, and (3) the reaction is autocatalytic. Figure 6 is a residual plot of the data. The supplemental data set given in Table I1 is not included due to difficulties experienced during the experimental measurements. Parameter estimates resulting from the regression are given in the figure. Clearly, there is some systematic difference between the model and the data. The analysis of the data without the CPD vapor pressure as a regression variable resulted in equivalent systematic trends and parameter estimates for the kinetics and solution model parameters. Therefore, these trends cannot be attributed to errors in the CPD vapor pressure. The search for similar residual behavior through experimental design simulation is necessary to determine the cause. The probable cause is that the kinetic model is in error. The experimental design was analyzed by using eq 13 to generate the transient behavior, but the regression analysis of the simulated data was done assuming that eq 12 applied. Two simulations were run. In the first, k”is set such that k increases 40%, and in the second, it increases 300% between the composition limits of pure CPD and DCPD. These cases bracket the range of the composition dependence discussed in the literature. Figures 7 and 8 provide residual plots for these two cases. Figure 7 shows the same, if less pronounced, trend in the residuals as is shown in Figure 6. Figure 8 residuals have a more pronounced systematic deviation. These observations coupled with the normalized pressure plots lead to the conclusion that eq 13 is applicable with an apparent increase in k between 40% and 300%. The data were analyzed with the regression parameters of the CPD vapor pressure, k’, k”, and the solution model parameters. The results of that analysis are given in Table
-
v p ‘bc
-0.60 i . 2 0.00 0.20 0.40
0.60
0.00
1 10
Mole Fraction C P O
Figure 9. Comparison between calculated and experimental pressures for CPD/DCPD binary system. Table 111. Results of the CPD-DCPD Mixture Data Analvsis no. of points 64 313.18 av temp, K expt 1 uncertainties st, min 1 ST?
regression parameters
regression statistic expected uncertainty computed results
sp, kPa s,,(O), mol sn2(0),mol Po, kPa k’, L/mol/min k“, mol/L hzl-XI1, cal/mol X12-X22, cal/mol sp, kPa sp, kPa 71x2-
0.02 0.08 1.0 x 10-4 1.0 x 10-4 102.4 2.168 X 5.831 X 1053.18 -805.39 0.19 0.15 1.053 1.023
I11 and Figure 9. Expected error bands have been included in the figure for reference. The results in the table are in good agreement with the expected results with the exception of the vapor pressure. The resultant rate equation for the dimerization of CPD is
- ( l / V ) dn,/dt = 2.168 X 10-4/(1 - 5.831 X 10-2(n2/V))(nl/V)z (15) The literature value for the initial rate (k’) is 1.9 X mol/L/min with an uncertainty of 22%. The regressed value is 14% different, indicating that the two values agree. The effect of k”is such that the apparent rate of dimerization changes by 74% as the solution changes from pure CPD to pure DCPD. This is less than the 150% increase predicted by using the results of Schmid et al. (1948). However, their results are at 293 K. The estimated solution model coefficients result in infinite dilution activity
1492 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 Table IV. Provisional Estimate of the Cyclopentadiene-Dicyclopentadiene Phase Equilibria at 313.15 K Xt
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
P 1.1 10.9 20.5 30.2 40.2 50.2 60.3 70.4 80.4 90.5 100.6
Y1
@le
0.000 0.907 0.956 0.974 0.983 0.989 0.992 0.995 0.997 0.999 1.000
92.1 87.4 85.9 85.6 85.6 85.6 85.5 85.1 84.3 83.1 81.5
coefficients which are near ideal. This is in agreement with the expectation resulting from analysis of Figures 3-5. The uncertainty in total pressure based on the regression results (1.4%) is approximately equal to the expected value (1.170). The difference is statistically significant, which is probably due to the systematic difference between the two data sets having a nominal initial concentration of 45% CPD. The 0.7 kPa difference in the pressure at the initial reading corresponds to a difference of less than 0.007 mole fraction. The probable cause is nonequilibrium conditions a t the time of metering CPD. The CPD vapor pressure of 102.4 kPa is not in good agreement with the literature-based value of 100.8 kPa. The Table I correlation is based on the data of Hull et al. (1965). A possible experimental procedure problem could explain the discrepancy. The regression analysis uses the assumption that the bath and CPD are at the set temperature at the time of charge. The CPD was charged at 190 K, and although this had little effect on the bath, the likely initial temperature of the CPD at time zero was not the bath temperature. Consequently, it is possible that the pressure at the time of the first reading (30-min mark) is higher than it would be had the bath and CPD come to temperature equilibrium instantaneously. This is the “pressure lag” discussed in the experimental design discussion. Multiple simulations of the experimental design were run to determine the impact of the pressure lag with various delays. In all cases, the residuals did not display any systematic trends. Only the vapor pressure estimate is affected by the temperature transient. All other parameters were unaffected. Consequently, an initial temperature transient could explain the high estimate for the vapor pressure. The kinetic parameters and solution model parameters coupled with the literature vapor pressure can be used to estimate the phase equilibria and reaction behavior of the CPDiDCPD system. Table IV provides the estimate for the phase behavior of the CPD/DCPD binary. This table should be viewed with some skepticism. The study of pressure lag suggests that this might be the cause of the high vapor pressure estimate, but that study is not definitive. Conclusions This article presents the foundation of an experimental and numerical procedure which can be used to determine simultaneously component vapor pressures, solution model interaction parameters, and reaction rate constants without phase sampling. The experimental procedure evolved from the total pressure method. Data consist of total pressure transients as a function of temperature and initial composition. The numerical procedure is based on maximum likelihood estimation and on suitable experimental design studies. It is shown that systematic errors in the kinetic model can be identified by resultant systematic trends in
the pressure residuals. Systematic errors in the data due to early temperature transients do not appear as systematic trends in the residuals. New data and the analysis thereof are presented for the cyclopentadiene-dicyclopentadienebinary system at 313.1 K. The resultant initial rate constant is equivalent to that based on literature data. The analysis indicates that the dimerization is autocatalytic, which is also consistent with the literature. The liquid phase is near ideal. The estimated vapor pressure for cyclopentadiene is above that based on the literature probably due to early temperature transients not accounted for in the analysis procedure. Acknowledgment The authors thank the National Science Foundation for partial support of this work under Grant INT-76-22712 A01. C.S.H. received partial support from the Amoco Foundation. Nomenclature A , B , C, D = Miller coefficients E = activation energy, cal/mol k = second-order rate constant, L/mol/min k’ = initial rate of dimerization, L/mol/min k” = autocatalytic rate constant, L/mol k o = specific rate constant (Arrhenius equation), L/mol/min m = number of components n = moles or number of measurements P = pressure (in general), kPa P = vapor pressure, kPa R = gas constant, cal/mol/K RSME = root mean square error s = standard deviation, units vary T = temperature, K t = time, min V = total volume, cm3 u = component molar volume, cm3/mol x = liquid-phase mole fraction y = vapor-phase mole fraction Greek Symbols a = relative volatility, dimensionless y = liquid-phase activity coefficient, dimensionless
A, = solution model interaction parameter, dimensionless X,,-X, = solution model interaction parameter, cal/mol 4 = fugacity coefficient in mixture, dimensionless qbo = pure-component fugacity coefficient, dimensionless pV = ratio of molar volumes, dimensionless
Superscripts c = calculated value
e = experimental value = infinite dilution (0) = zero time Subscripts P = pressure T = temperature t = time n = composition 1 = component 1: cyclopentadiene 2 = component 2: dicyclopentadiene Registry No. Cyclopentadiene,542-92-7; dicyclopentadiene, 77-73-6.
Literature Cited Barker, J. A. Aust. J . Chem. 1953,6, 207-210. Graboski, M. S.; Daubert, T. E. Znd. Eng. Chem. Process Des. Deu. 1978, 17(4),443-448. Hankinson, R. W.;Thomson, G . H. AZChE J . 1979,25(4), 653-663. Howat, C. S. Ph.D. Dissertation, University of Kansas, 1983.
Ind. Eng. Chem. Res. 1988,27, 1493-1496 Howat, C. S.; Swift, G . W. Fluid Phase Equilib. 1983,14,289-301. Howat, C. S.; Swift, G. W. J. Chem. Eng. Data 1985,30, 281-285. Hull, H. S.; Reid, A. F.; Turnbull, A. G. Aust. J. Chem. 1965, 18,
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Schmid, H.; Kubassa, F.; Herdy, R. Monat. Chem. (Wien) 1948, 79, 430-438.
Tsuboka, T.; Katayama, T. J. Chem. Eng. Jpn. 1975,8(3), 181-187.
549-552. Miller, D. G. Ind. Eng. Chem. 1964, 56(3), 46-57. Shanker, G.; Howat, C. S., 111; Torres, R. R.; Swift, G . W. Fluid Phase Equilib. 1981, 5, 305-321.
Received for review December 16, 1985 Revised manuscript received March 18, 1988 Accepted April 19, 1988
Minimum Fluidization Velocities of Wet Coal Particles Jaroslav Pata,* Milan h s k j l , Miloslav Hartman, and VBclav Veseljl Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, 165 02 Prague, Czechoslovakia
Minimum fluidization velocities of wet lignite coal beds were measured. The experiments were carried out in an 8.5 X m i.d. column with particles of average size ranging from 0.15 X to 4.25 X m, using air as a fluidizing gas. The experimental results were compared to the values predicted by commonly used correlations. On the basis of data obtained, a simple correction factor was derived. T h e factor makes it possible to calculate the minimum fluidization velocities of wet coal from equations, originally developed for dry beds if relative moisture content of the bed and average particle diameter are known. Good agreement with experimental data has been found in all cases. An experimental fluidized bed boiler of 20-MW heat output was usually supplied with wet lignite coal which contained sometimes as much as 35% (by weight) water. Owing to the fact that coal transportation to the boiler was based also on the fluidized bed principle, a necessity of knowledge of minimum fluidization arose. A number of correlations has been published to predict minimum fluidization velocities of dry materials (e.g., Ergun (1952), Leva (1959),Goroshko et al. (1958), Borodulya et al. (1982), Broadhurst and Becker (1975)). Unfortunately all these equations have shown discrepancy between the predicted and measured velocities if applied to beds of wet material. That is why the relation between the minimum fluidization velocity and relative moisture content of lignite coal bed has been investigated. At first the experimental minimum fluidization velocities of various wet monodisperse beds of “Chabafovice” lignite coal were determined. These experimental data were starting points for derivation of a simple formula which makes it possible to convert results of the published equations that can be used for prediction of “wet” minimum fluidization velocities of coal if average particle diameter and relative moisture content of the bed are known. The present paper is a part of our study on materials which are used at fluidized bed combustion and simultaneous desulfurization (Pata and Hartman, 1978, 1980; Svoboda and Hartman, 1981a,b; Svoboda et al., 1983, 1984).
Experimental Section The experimental setup was described in detail previously (Pata and Hartman, 1978). The apparatus consisted of a glass column of inside diameter of 8.5 X m and height of 0.9 m. The column was equipped with a plate distributor of 1% free area. The pressure drop across the bed was measured by a U manometer, and the velocity of air was checked by a rotameter. Experiments were carried out using air as a fluidizing gas at a temperature of 295 f 3 K. The lignite coal of ps = 1.61 X lo3 kg m-3 at 16.6-37.5% w t moisture content from the Chabafovice open pit mine was crushed and sieved. Eight differently sized fractions of coal, 0.05-0.25, 0.25-0.50, 0.50-0.63, 0.63-0.80,0.80-1.0, 1.0-1.6, 1.6-2.8, 3.5-5.0 mm, were used 0888-5885/88/262~-1493$01.50/0
Table I. Physical Properties of Lignite Coal Particles fraction 1 2 3 4 5 6 7 8 103d,, m 0.15 0.38 0.57 0.72 0.90 1.30 2.20 4.25 0.65 0.64 0.63 0.62 0.61 0.60 0.58 0.57 0.23 0.33 0.37 0.37 0.37 0.35 0.32 0.21
7
in these experiments. The properties of coal particles are summarized in Table I. The same quantity of each coal fraction was put into a glass-stoppered bottle, and a measured quantity of water was gradually added to each bottle after the previous measurement of Umfhad been concluded. Then the bottles were closed tightly, and their contents were regularly mixed for 2-day intervals. After 14-20 days, an equilibrium state at room temperature was achieved, and the next minimum fluidization velocity was measured. The moisture content of the fractions was computed from the known quantity of coal and water addition. A t moisture contents of 16.6%, 27.5%, 37.5% by weight, the calculated concentrations were checked experimentally by distillation of wet coal samples with toluene. The computed and experimental results agreed within f5%. The minimum fluidization velocities were determined from the plots of superficial velocity vs pressure drop on log-log paper. The velocity corresponding to the point of intersection of the two differently sloping portions of the plot was taken as the minimum fluidization velocity. Measurement started at a well fluidized state; afterwards air velocity was gradually reduced to zero. The ratio h l D for the fixed bed was within 1-2. For each minimum fluidization velocity, the height of the bed hmfwas simultaneously recorded. The part of the pressure drop vs velocity plot under minimum fluidization velocity (providing R e < 10) was used for calculation of particle sphericity.
Results and Discussion Density of Coal Particles. Particle densities were measured by the methanol displacement method. The true density of coal was calculated by Ps = w,
W,PMe
+ w,- wb
0 1988 American Chemical Society
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