Acknowledgments
Callcott, T. G . . Lynch, A. J., Proc. A u s t . lnst. Min. Met., 209, 109-131
This work was supported by a n Environmental Protec. ~801218, tion A~~~~~ ~~~~~~~hGrant, ~~~~tN ~ EPA "Size Reduction in Solid Waste Processing." The financia1 aid of the African-American Institute in supporting Dennis Obeng's graduate studies is appreciated. P u ~ f e s sor Thomas Mika of the Material Science Department gave advice concerning some of the analytical aspects of this work.
Gaudin. A. M . , Trans. A I M € , 73,253 (1926). Gaudin, A. M . . Meloy. T. p., Trans. A I M € . 223,40-43 (19621. Harris, C. C., Trans. A I M € , 241,343-358 (1968). Harris, C. C . , Trans. A I M € , 244,187-190(1969) Harris. C. C . , Trans. Inst. Mlniflg Met.. 79,C157-158 (1970) Patrick, P. K . . "Waste Volume Reduction by Pulverization, Crushing. and Shearina," The institute of Public Ciearino 69th Annual Conference. 5th to 9tk,June 1967. Rosin. P., Rammler. E., lnst. Fuel, 7,29-36 (1933) Savage, G., Trezek, G. J . , "Size Reduction in Solid Waste ProcessingSize Reduction Facility," Report prepared for Environmental Protection Agency, Grant No. EPA R801218,May 1972 Schuhmann, R , Tech. Publ. A I M € , 7189, 1 1 (Juiy 1940) Trezek, G. J . , 'Refuse Comminution." Compost Sci.. 13 (4).13-15
Literature Cited Austin, L G., Power Techno/., 5, 1-17 (1971/1972). Bergstrom. B. H . , Trans. A I M € , 235,45 (1966) Broadbent, S.R.. Callcott. T. G., Phil. Trans. Roy. SOC.,249,99 (1956a). Broadbent, S.R . , Callcott, T. G., J . lnst. Fuel, 20. 524-539 (1956b):30,
(19.641
(1972).
Received f o r recielc ,January 29, 1974 Accepted November 27, 1974
21-25 (1957) _. Callcott,
?. G:, j. lnst
Fuel. 33,529-539 (1960)
Kinetics of the Hydrodealkylation of Methylnaphthalenes in a
Nonisother mal Flow Reactor Paolo Beltrarne,* Bruno Marongiu, Vincenzo Solinas, Sergio Torrazza Istituto Chimico. Universita. 09700 Cagiiari. lfaiy
Lucio Forni and Sandro Mori lstituto d! Chimica Fisica. Universita, 20133 Milano. ltaly
The thermal hydrodealkylation of 1-methylnaphthalene ( A M N ) and of 2-methylnaphthalene ( B M N ) has been studied in a tubular flow reactor markedly deviating from isothermal conditions. Complete temperature profiles in the reactor were recorded: maximum values of temperature were kept at ca. 630, 660. and 700°C for A M N runs and at ca. 700, 730, and 760°C for BMN. Total pressure ranged from 4.9 to 39.7 atm, and H2:hydrocarbon molar ratio in the feed was varied from 3 to ca. 9. Assuming the rate equation r = ~ . C M N ~ - C HArrhenius ,~, parameters and partial reaction orders were obtained by a nonlinear optimization procedure. Results are: ( A M N ) rn = 0.25; n = 0.8; A = 4.8 X l o 9 1,0~05/rno10~05 sec, E = 50800 cal/mol; ( B M N ) rn = 0.35; n = 0.75; A = 4.2 X l o 8 l.O.l/rnolo.' sec, E = 46900 cal/ mol. Experimental molar conversions are reproduced with relative mean square deviations of 12% ( A M N ) and 6% ( B M N ) .
Several kinetic measurements have been performed on toluene hydrodealkylation and their results have been critically examined in reviews (Asselin, 1964; Benson and Shaw, 1967). Few studies on the analogous reaction of methylnaphthalenes have been reported (Asselin, 1964; Gonikberg, e t al., 1964, 1965) and some discrepancies exist among published results, mainly concerning the form of the rate equation. Therefore further kinetic measurements about l-methylnaphthalene (AMN) and 2 methylnaphthalene (BMN) have been undertaken by using a tubular flow reactor. Isothermal conditions would have been difficult to a t tain all along the reactor a t the temperatures required for thermal hydrodealkylation (Amano and Uchiyama, 1963; Zimmerman and York. 1964; Shull and Hixson, 1966). On the other hand, with the present computing facilities, a problem of nonisothermal kinetics can be easily handled by nonlinear regression analysis.
Experimental Section Materials. Merck-Schuchardt 97% l-methylnaphthalene and 98% 2-methylnaphthalene ( m p 35°C) and cylinder 99.99% hydrogen were employed. In the conditions of the glc analysis described below, both methglnaphthalenes gave a single peak. A p p a r a t u s a n d Procedure. A scheme of the apparatus is presented in Figure 1. The tubular flow reactor, made of stainless steel (Incoloy), was fitted with two contacting axial concentric thermowells: the annular reaction tube was 49 cm long, 1.2 cm o.d., and 0.6 cm i.d. A vaporizerpreheater was also provided. Reactor and preheater were heated by means of four electric furnaces. The hydrocarbon reservoir, the pump, and the connecting pipes were equipped with a heating jacket in order to avoid crystallization of BMN. Axial temperature profiles were measured by placing thin (1.6 mm 0.d.) iron-constantan thermocouples at variInd. Eng. C h e m . , Process Des. Dev., Vol. 14, No. 2, 1975
117
I I I
I I
! !
I I I
I
1
LF;;;
! ! !
I
I
HFLYLT -1
!
!
I
! !
!-
AP
&-- -E-
Lia Figure 1. Apparatus: SIN, methylnaphthalene reservoir; MP, micrometric metering pump; 31, manometer; HF, heating furnace; TC. thermocouples; TR. temperature recorder.
ous (15-16) positions along the wells, and were recorded by a potentiometric strip-chart recorder. Hydrocarbon flow rates were measured on the graduated glass reservoir. The reactor effluent, after passing through a suitable series of condensing traps, was sent to a wet test meter in order to measure gas flow rates. All measurements were performed a t least 30 min after steady-state conditions had been attained. Sampling time ranged from 15 to 60 min according to the flow rates, so that 10-15 g of liquid products were collected during each run. Flow Conditions. In the conditions of the kinetic runs (20 < Re < 100) the flow in the reactor was considered of laminar type. This implies that, neglecting radial temperature gradients, the radial velocity profile should be that for laminar flow in annuli (Knudsen and Katz, 1958)
where R1 and Rz are the inner and outer radii, respectively, urn is the mean velocity, and u is the velocity a t a distance R from the annulus axis. In order to check the influence of the radial temperature gradients on the flow, a longitudinal well was drilled in the outer wall of the reactor, and the temperature profile of the tube was recorded by means of a n additional thermocouple. By comparing the profiles of the axial well and of the outer tube it was observed that the difference (Tout,, - T,,,,,eI)ranged from +10 to +1"C before the maximum temperature zone, and from +1 to +3"C after it. For the heating portion of the reactor, the observed heat transmission is justified by a mechanism of radiant heat transfer from the outer wall to the axial thermowell through the 3 m m thick annular space, and of pseudoconvection from both metal surfaces to the flowing gas. Calculations based on this model showed t h a t in a typical 118
Ind. Eng. Chern., Process Des. Dev., Vol. 14, No. 2, 1975
run the radiant heat transfer from the outer wall to the thermowell, a t the steady state, amounted to ca 1 cal/ sec. The heat losses by conduction along the thermowell were evaluated of the order of one-tenth of such a value. The heat required for heating the gas, also taking into account the heat of reaction (AH = ca -12 kcal/mol), did not exceed 2 cal/sec, of which the largest portion should be provided by the reactor wall. A heat transfer coefficient between metal surfaces and gas h = 6 x 10 3 cal/sec cmz K was computed by standard formulas for pseudo-convection associated with laminar flow. Using this value, one calculates that the transmission to the gas of 2 cal/sec requires temperature differences between metal surfaces and gas of 2-3°C. Therefore it was evaluated that the radial temperature gradient in the gas was ca. 10°C a t the inlet of the reaction zone, rapidly decreasing thereafter. Since the flowing gas is mainly hydrogen, whose viscosity in the thermal conditions of interest is scarcely dependent on temperature (Landolt, 1969), such a small radial gradient cannot appreciably affect (McAdams, 1954) the velocity profile (1).Moreover, the mentioned calculations make it possible to accept the temperatures measured by the axial thermocouples as a good approximation of the actual gas temperatures. Analysis. A gas chromatograph equipped with a thermal conductivity detector was used throughout. Condensed products were analyzed on a 1.2 m X 3 m m i.d. SS colurnn, packed with 100-200 mesh DURAPAK (Carbowax 400/Porasil F), carrier gas hydrogen (20 cm3/min), column temperature 140°C. Gaseous products were analyzed on a 2.5 m X 3 m m i.d. SS column, packed with 70-325 mesh silica gel, carrier gas hydrogen (20 cm3/min), column temperature 70°C. By-products were not considered in the quantitative analysis of liquids. Gases were analyzed by comparison with standard mixtures of composition very close to that of the sample.
Results Analyses of the condensed reactor effluent revealed unreacted methylnaphthalene, naphthalene, and traces of heavier by-products. The gaseous reaction product was methane containing usually 1-770ethane. Therefore the observed stoichiometry was close to the theoretical equation C,,H;-CH,
+
H,
-
C,,H,
+ CH,
(2)
which has been employed for the kinetic computations Kinetic runs were carried out a t various total pressures from 4.9 to 39.7 atm. The hydr0gen:hydrocarbon molar ratio in the feed ranged from 3 to ca. 9. Runs can be grouped together in sets characterized by similar axial temperature profiles. In order to smooth out experimental errors, mean profiles were calculated by fitting temperature readings for a whole set of runs to a 5th-order polynomial T = T ( r ) . Temperatures were usually reproduced &1O"C in the central portion of the reactor, responsible for most of the observed conversion. An example is given in Figure 2. Mean temperature profiles, labeled A, B. . . . , P, are shown in Figure 3. Conditions of the kinetic runs and measured conversions are summarized in Table I for AMY (36 runs) and in Table I1 for B M S (45 runs). The mass balance equation for laminar flow in a n annular ring of thickness dR may be written as follows, applying the ideal gas approximation
(3)
Table I. Kinetic Results for 1-Methylnaphthalenea Molar conversion
Hydrocarbon feed r a t e , mol/hr
700.
0.146 0.217 0.283 0.362 0.442 I
I
0.141 0.217 0.283 0.362 0.435
Figure 2. Recorded temperatures for the central portion of the reactor tube (AMN runs at 4.9 a t m ) . Curve: mean temperature profile A.
0.141 0.210 0.283 0.362 0.435 0.146 0.210 0.290 0.362 0.435 0.145 0.217 0.290 0.377 0.435 0.145 0.217 0.283 0.362 0.435
Rm (Profile A; P 6.4 6 .O 6.5 6 .O 5.7 (Profile B; P 6 .O 6.3 6.5 6.2 5.9 (Profile C ; P 3.2 2.9 2.9 2.9 3.1 (Profile D; P 9.1 9.9 9 .o 9.1 8.9 (Profile E: P 5.8 6.2 6.2 5.6 6.1 (Profile F: P 5.6 5.7 6 .O 5.6 5.6 (Profile G ; P 7.9 8.9 6.8 6.2 7.2 6.9
Exptl
=
=
=
=
=T
=
4.9 atm) 0.105 0.075 0.063 0.055 0.050 12.6 atm) 0.239 0.180 0.149 0.112 0.101 12.6 atm) 0.222 0.173 0.140 0.116 0.09 5 12.6 atm) 0.248 0.191 0.172 0.149 0.111 39.7 atm) 0.382 0.334 0.268 0.232 0.202 39.7 atm) 0.530 0.487 0.428 0.388 0.351 39.7 atm) 0.822 0.713 0.675 0.632 0.589 0.532
(v)
Calcd 0.106 0.073 0.056 0.044 0.036 0.278 0.192 0.1 50 0.119 0.101 0.275 0.196 0.149 0.118 0.100 0.297 0.221 0.167 0.136 0.114 0.408 0.316 0.259 0.208 0.182 0.583 0.444 0.375 0.324 0.288
0020 20
zccrn]
0.145 0.217 0.304 0.413 0.442 0.576
40
Figure 3. Mean temperature profiles. Left, AMN runs. Right, BMN runs. Profiles B, C, D, and F are similar to A; K, L, and P are similar to H; N is similar to M .
The rate equation was taken as y
= ko ' exp
E
[-m(f-
+-)I
a
Cm"
(5) Remembering t h a t (u,,,/u) = f(R) (see eq 1) and t h a t T = T ( z ) ,substituting eq 4 and 5 into 3, integrating eq 3, and applying the ideal gas approximation to express CH, and CMNO,one obtains
(1 where
1-m
3)
= 2 ' f(R)
For temperature profiles, see Figure 3 .
CH2" (4)
where the Arrhenius equation has been written in the form suitable for optimization procedures (Dye and Nicely, 1971). The reference temperature To was chosen as 903 K for AMN and 973 K for BMN. Since hydrogen is an excess reagent, one can take
f
0.938 0.786 0.662 0.548 0.520 0.437
z=-. (1
- 172)
1000
PS (-P , , ) ~ ( P ~ ~ ) ~ - ' 0 .,,O,m+n k F,
Averaging C,,l~fover the entire reactor section, the overall molar conversion
y = 1 - (cmf)a"/cMpJ0
(8)
may be obtained
(9) Integrals appearing in eq 7 and 9 have been numerically calculated by the 16-points Gauss method. The parameInd. Eng. Chern., Process Des. Dev., Vol. 14, No. 2, 1975
119
Table 111. Optimized Kinetic Parameters.a ( r = A . e - E / 1 . 9 8 7 T . CMN"'. C H ~ "
Table 11. Kinetic Results for 2-Methylnaphthalene= ~~
~
Hydrocarbon feed rate, mol/hr
Molar conversion (v)
Rm
Exptl
1-Methylnaphthalene
Calcd
(Profile H ; P = 4.9 atm) 0.166 0.217 0.289 0.326
5.6 5.8 '6.5 6.1
0.121 0.090 0.069 0.062
A , (mol/l.)'-"-"/sec E , cal/mol
0.121 0.093 0.069 0.063
n2
0.145 0.217 0.297 0.398 0.564 0.217 0.318 0.369 0.441 0.145 0.210 0.268 0.384 0.478 0.630 0.141 0.210 0.290 0.348 0.464 0.558 0.150 0.296 0.359 0.435 0.145 0.225 0.293 0.348 0.435 0.166 0.224 0.289 0.384 0.507 0.586 a
6.4 6 .O 5.8 5.3 6.1
0.231 0.160 0.148 0.121 0.108 (Profile J; P = 4.9 atm) 6.4 0.376 8.3 0.287 5.1 0.250 6.4 0.209 5.7 0.144 (Profile K; P = 8.7 atm) 6.9 0.168 5.3 0.120 5.2 0.107 5.4 0.089 (Profile L ; P = 20.4 atm) 8.7 0.415 7.1 0.344 7.1 0.310 5.6 0.244 5.3 0.208 5.7 0.173 (Profile M; P = 20.4 atm) 8.5 0.398 8.8 0.327 8.3 0.277 8.1 0.241 0.169 8 .O 8.2 0.159 (Profile N ; P = 20.4 atm) 3.2 0.431 0.272 3 .O 3 .O 0.265 3.3 0.227 (Profile 0; P = 20.4 atm) 5.3 0.871 6 .O 0.739 5.9 0.713 5.8 0.684 5.7 0.574 (Profile P; P = 39.7 atm) 6.4 0.664 6.1 0.578 5.9 0.524 6.7 0.407 5.3 0.370 5.7 0.362
0.247 0.164 0.1 53 0.112 0.101
0.170 0.125 0.109 0.092
,
0.422 0.329 0.264 0.229 0.178 0.150 0.459 0.303 0.262 0.223 0.973 0.848 0.749 0.686 0.604 0.680 0.570 0.486 0.402 0.347 0.314
For temperature profiles, see Figure 3.
ters to be optimized for each hydrocarbon were four ( m , n. ko, E). In order to reduce this number, calculations were performed for several couples of m, n values, in each case optimizing ko and E by the method of Powell (1965) on a Univac 1106 computer. Input variables were the six coefficients of the polynomial T ( z ) ,pressure, hydrocarbon feed 120
Ind. Eng. Chern., Process Des. Dev., Vol. 14, No. 2, 1975
(4.2 46900 0.35 0.75 1.10
i 1.3)x10R i 1300 i 0.15 i 0.15 i 0.05
rate, Hz/hydrocarbon feed molar ratio, and observed conversion. The sum of squares of relative residuals
0.386 0.289 0.243 0.182 0.135
0.436 0.356 0.305 0.240 0.200 0.155
lo9
n m+n a Uncertainties represent 95% confidence limits.
(Profile I; P = 4.9 atm) 0.152 0.246 0.268 0.380 0.413
(4.8 i 2.0) X 50800 f 1500 0.25 f 0.20 0.80 f 0.20 1.05 i 0.15
2-Methylnaphthalene
calculated for the N experimental runs, was minimized with the convergence criterion of a relative variation 510-3 for both parameters between two consecutive iterations. For each hydrocarbon, a n absolute minimum of S with respect to all parameters was found by drawing a map of the relative minima, obtained a t fixed values of m and n, on the ( m , n ) plane. This procedure gave the best values of ko, E, m, and n. The 95% confidence regions of such parameter estimates were calculated according to Mezaki and Kittrell (1967). Values of m, n, and E , together with the Arrhenius frequency factors, A calculated from ko's are reported in Table 111. Overall molar conversions, calculated for every kinetic run using the optimized parameters, are given in Tables I and 11. Equation 9 was employed to obtain conversion profiles along the reactor, substituting Z values calculated by carrying the integration of eq 7 from z = 0 to different z values. It has been so verified t h a t 199% of the reaction takes place in the reactor portion where temperature exceeds 560°C (AMN) or 600°C (BMN).
Discussion The employed reactor presents a metal surface (wall + thermowells) of 0.028 m2 and a volume of 42 cm3. The resulting surface/volume ratio is large relative to a commercial reactor, but still very small compared with the specific surface of catalyst supports like alumina and silica, typically employed for hydrodealkylation (Asselin, 1964). Therefore the empirical rate equation (4) can be considered as referring to a purely thermal reaction. One of the kinetic equations proposed for the thermal hydrodealkylation of methylnaphthalenes includes a term corresponding to a reverse reaction (Bixel, e t al., 1964). This has not been considered in the formulation of rate equation (4)since A G O values of about -10 kcal/mol can be calculated for the hydrodealkylation of both AMN and BMN from the available values of the standard free energies of formation (Stull, et al., 1969). Experimental results were interpreted by assuming a perfect laminar flow in the reactor (eq 1). It has been ascertained t h a t this is a much better approximation than the plug flow model. In fact, the plug flow hypothesis was introduced for BMN, by putting f(R) = 1 in eq 6 and 9, and applying the described optimization procedure. P a rameters so "optimized" gave rise to a mean relative square deviations of calculated from experimental conversions of 11.570, while a deviation of 5.9% was obtained with the perfect laminar flow assumption. Examination of Tables I and I1 shows that calculated
conversions are closer to the experimental ones for BMN than for AMN: mean relative square deviations were 5.9 and 12.2'70, respectively. In both cases absolute deviations are particularly large for the high conversion runs Cv > 0.7), where they seem to be systematic. This can be attributed in part to secondary reactions, not taken into account by our kinetic model, in part to deviations from a perfect laminar flow. Since the gas in the reactor was flowing downward in the presence of large axial temperature gradients, a flow instability could have occurred, due to natural convection phenomena in the upper part of the reactor, giving rise to backmixing, particularly when flow rates were kept low in order to attain high conversions. However, the high conversion runs were very few and the form chosen for the objective function (eq lo), based on relative rather than absolute residuals, was apt to give only moderate weight to such runs, so t h a t in any case the parameters evaluation should have been only slightly affected. On the whole, the general agreement between calculated and experimental conversions can be regarded as satisfactory, if one considers t h a t the dependence of rate on temperature, total pressure, and reactants' ratio has been simultaneously accounted for. In particular, the overall reaction order ( m + n ) has been determined with limited margins of uncertainty: 1.05 f 0.15 for AMN, and 1.10 f 0.05 for BMN, that is the same, within confidence limits, for the two reactants. A similar value of the total reaction order, i.e., 1.2, has been previously reported by Morii, et al. (1962), for both isomers. The present results are also interesting for the magnitude of partial reaction orders (rn = 0.3; n = 0.8). These findings indicate that the reaction mechanism for methylnaphthalenes is probably different from those that imply a 1.5 reaction order, like that generally accepted for toluene (partial orders 1.0 in hydrocarbon and 0.5 in hydrogen) (Asselin, 1964) and that proposed for ethylbenzene (partial orders 1.0 in hydrogen and 0.5 in hydrocarbon) (Fujii and Kato, 1962). While the results for toluene can be justified by a simple radical chain mechanism initiated by hydrogen atoms (Benson and Shaw, 1967) and those for ethylbenzene by an analogous chain initiated by alkyl radicals, only a more complex mechanism, including several contributions to the chain initiation, would account for the kinetics of the methylnaphthalenes hydrodealkylation. Published values of the activation energy for hydrodealkylation of methylnaphthalenes (Asselin, 1964; Gonikberg, et al., 1964) are in the range 46-52 kcal/mol for AMN and 45-54 kcal/mol for BMN. Our values (Table 111) fall in the same ranges. An analogous comparison with the literature data may be done for reaction rates in suitably chosen "standard conditions" such as those given by Asselin (1964), that is pi12 = 15 and P \ I N = 3 atm, temperatures 600 and 800°C. In these conditions, rates calculated from the parameters in Table I11 are (in mmol/l. sec): AMN, 0.12 and 22; BMN, 0.08 and 9.8. a t 600 and 800"C, respectively. The corresponding literature values lie in the following ranges: AMN, 0.17-0.45 and 26-86; BMN, 0.08-0.29 and 19-98, respectively. On the basis of the present results, the reactivity of both methylnaphthalenes seems to be slightly lower than previously reported. On the other hand, it is
known t h a t the hydrodealkylation rate is affected by the presence in the feed of impurities that are potential sources of radicals (Zimmerman and York, 1964). Acknowledgment We are grateful to the Michigan State University Computer Laboratory for kindly providing the optimization program. Prof. S. Carrh is thanked for stimulating discussion. Nomenclature A = Arrhenius frequency factor [(mol/l.)l--m-n/sec] C, = molar concentration of i (mol/l.) E = Arrhenius activation energy (cal/mol) F , = total feed rate (mol/sec) G o = standard Gibbs free energy (kcal/mol) k = rate coefficient [(mol/l.)l-m-n /=I k o = rate coefficient a t the reference temperature m = reaction order with respect to methylnaphthalene n = reaction order with respect to Hz P = total pressure ( a t m ) Pi = partial pressure of i ( a t m ) r = reaction rate of methylnaphthalene (mol/l. sec) R = radius Rn, = (Hz/hydrocarbon) feed molar ratio S = objective function in the optimization procedure SIC= area of the reactor section (cm2) T = temperature ( K ) To = reference temperature u = gasvelocity Uni = mean gas velocity VLC= reactor volume (1.) x L = molar fraction of i in the reaction mixture y = molarconversion z = position along the reactor axis (cm) tr = reactor length (cm) Superscripts = inlet value of variable f = outlet value of variable Literature Cited Amano. A , Uchiyama, M., J. Phys. Chem.. 67, 1242 (1963). Asselin, G F., Advan. Petrol. Chem. Refining. 9, 47 (1964). Benson, S W.. Shaw. R . i.Chem Phys.. 47, 4052 (1967). Bixel, J. C., Merrill, L. S Jr., Allred. V . D , Benham, A. L., Ind. Eng Chem., Process Des. Develop., 3, 78 (1964). Dye. J. L , Nicely, V . A,, J. Chem Educ.. 48, 443 (1971). Fujii. S., Kato, S..Kogyo Kagaku Zassh!. 65. 1396 (1962); Chem. Abstr., 58, 6667 (1963). Gonikberg. M. G.. Gavrilova. A. E., Alekseev. E. F . , Komanenkova. R. A.. Neftekhimiya, 4, 252 (1964); Chem. Abstr.. 61, 3039 (1964). Gonikberg. M. G., Gavrilova. A E., Komanenkova, R. A , , Neftekhimiya. 5, 489 (1965); Chem. Abstr.. 63, 17815 (1965) Knudsen. J. G., Katz. D. L.. "Fluid Dynamics and Heat Transfer," p 92, McGraw-Hill, New York. N.Y., 1958. "Landolt-Bornstein Zahlenwerte und Funktionen.' Band I I , 5a. p 20. Springer. Berlin, 1969. McAdams. W. H . . "Heat Transmission,'' 3rd ed, p 229, McGraw-Hill. New York, N.Y., 1954. Mezaki, R . Kittrell, J . R . . Ind. Eng. Chem , 59 ( 5 ) ,63 (1967) Morii, H., Hashimoto, A.. Tominaga, H.. Ueno. K., Bull. Jap Petrol. l n s t , 4, 28 (1962) (see also Asselin (1964)). Powell, M J. D., Comput. J . 7 , 155. 303 (1965). Shull. S . E , Hixson, A N.. lnd. Eng. Chem.. Process Des Develop.. 5, 146 (1966). Stull, D. R.. Westrum. E. F. Jr , Sinke. G. C , "The Chemical Thermodynamics of Organic Compounds," Wiley, New York, N.Y.. 1969. Zimmerrnan, C C.. York. R . , Ind Eng. C h e m , Process Des. Develop.. 3, 254 (1964)
Received for reuieic F e b r u a r y 11, 1974 A c c e p t e d N o v e m b e r 6, 1974 T h i s w o r k was p a r t i a l l y s u p p o r t e d u n d e r a research c o n t r a c t w i t h Societh I t a l i a n a Resine S.I.R., M i l a n o , I t a l y .
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