Ind. Eng. Chem. Process ~ e s Dev. . 1984, 23, 801-805
not well understood and needs to be investigated further. The temperature dependence of the reaction rate constant is shown in Figure 15 as a plot of In k2 vs. 1/T. From the slope of this plot, an activation energy of 13.27 kcal/mol was obtained. The magnitude of activation energy also indicates that the carbonylation of nitrobenzene to phenylisocyanate occurs in the kinetic regime.
Conclusions The palladium complexes consisting of N-containing heterocyclic ligands (pyridine, isoquinoline etc.) were found to be active catalysts for the carbonylation of nitrobenzene to phenylisocyanate. The effect of different ligands, oxide promoters, solvents, and process parameters on the conversion of nitrobenzene and selectivity to phenylisocyanate was studied. It was found that oxide promoters increased the selectivity to phenylisocyanate but had no significant effect on the activity. Selectivity was found to be a complex function of process variables. In most conditions, the selectivity vs. conversion plot was found to pass through a maximum. With increase in CO pressure and catalyst concentration, the selectivity was found to increase. The kinetics of carbonylation of nitrobenzene to phenylisocyanate was also studied using a Pd(Py),Cl, complex catalyst. It was found that the absorption of CO was kinetically controlled in the range of conditions studied. A rate equation has been proposed and an activation energy of 13.27 kcal/mol was obtained from the temperature dependence of the reaction rate constant. Nomenclature A l = concentration of dissolved CO in bulk liquid phase, mol/cm3 A* = concentration of CO at the gas-liquid interface, mol/cm3
801
C, = concentration of catalyst in bulk liquid phase, mol/cm3 D = diffusion coefficient, cm3/s k2 = second-order rate constant, cm3/(mol s) kLa = gas-liquid mass transfer coefficient, s-l N = agitation speed, rps RA = rate of absorption of CO, mol/(cm3 s) S = solubility of CO, g/(L atm) t = time, s T = temperature, K x = mole fraction of nitrobenzene CY = parameter defined as RA/kLa[A*] Registry No. PhN02,98-95-3;PhNCO, 103-71-9;Pd(Py),Clz, 14872-20-9;Pd(Isoq),Clz,23807-51-4. Literature Cited Chaudhari, R. V.; Doraiswamy, L. K. Chem. Eng Sci. 1974, 2 9 , 129. Choudhari, V. R.; Parande, M. G.; Brahme, P. H. Ind. Eng. Chem. Fundam. 1982, 21, 472. David, D. J.; Staley, H. B. "Analytical Chemistry of the Polyurethanes", Voi. XVI, Part 111; Wlley-Interscience: New York, 1969. Franco, N. 8.; Robinson, M. A. U.S. Patent 3600419, 1971. Hardy, W. 6.; Bennett, R. P. Tetrahedron Lett. 1967. 961. Hurley, T. J.; Robinson, M. A. U S . Patent 3585231, 1971. Itatani, H.; Bailer, J. C., Jr. J . Am. Oil Chem. SOC. 1967, 44, 147. Kajimoto. T.; Tsuji, J. Bull. Chem. SOC.J 1969, 42(3), 827. Kharasch. M. S.; Sevler. R. C.: Frank, R. M. J . Am. Shem. SOC.1938, 6 0 , 3 523 965, 1970. Nefedov, 6. K.; Manov-Yuevenskii, V. I.; Novikov, S. S. DOH. Chem. 1977, 234(4), 826. Ottmann, G. F.; Kober, E. H.; Gavin, D. F. U S . Patents 3481967, 1969; 3 523 . -- 962. ..-, 1970. . Prichard, W. W. U:S. Patent 3 576 836, 1971. Smith, E.; Schnabel, W. U S . Patent 3576835, 1971. Tietz, H.; Unverferth, K.; Sagasser, D.; Schwetlich, K. Z . Chem. 1978, 18, 142. Weigert, F. J. J . Org. Chem. 1973, 38, 1316. Yamahara, T.; Takamatsu, S.; Hirose, K. Japan Kokai 7364048, 1973. ~
Received for review January 18, 1983 Reuised manuscript received December 16, 1983
Generating Appropriate Density Values from a Cubic State Equation To Avoid False Unit K Values. Application to Distillation Problems Stevan Jovanovl6' and Ratomlr Paunovie Faculty of Technology, Instffute for Petrochemlstty, Gas, Oil and Chemical Engineering, Universlty of Novi Sad, 2 1000 Novi Sad, VeJka Vlahovi6a 2, Yugos&v&
A simple and reliable method is presented to avoid trivial solutions when a cubic equation of state is used for evaluating equilibrium constants in the course of distillation calculations. The failure is prevented by generating artificial density values leading to a correct solution. The validity of the method is demonstrated by solving some illustrative distillation problems.
Introduction In the course of solving energy and material balances of a separation unit, the VLE ratios Kiare obtained from the phase equilibrium conditions xiv: = yiviv (i = 1, 2, ..., N) (1) When an equation of state is used to evaluate both the liquid and vapor phase fugacity coefficients, the trivial solution (TS) of the nonlinear system of eq 1may occur. Namely, if the initial estimates are such that the equation of state indicates single-phase condition, the compressi0196-4305/84/1123-0801$01.50/0
bilities of the two phases are calculated to be the same (ZL = Zv). As a result, unit Kivalues for all the components are obtained. In the past few years, several authors have suggested some ideas for avoiding TS, the problem first being stated by Harmens (1973, 1975). There are two aproaches to ensure the proper solution. The first is to make good initial estimates for VLE iterations, used by following authors. Asselineau et al. (1979) and Michelsen (1980) presented algorithms for VLE calculations based on the Newton-Raphson method for so@ 1984 American Chemical Society
802
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
lution of material balance and equilibrium equations. The good initial estimates are provided by linear (Asselineau et al., 1979) or cubic (Michelsen, 1980) extrapolation from the previously calculated solution. In addition, a method for the automatic selection of the specification variable is proposed by Michelsen (1980) that ensures a stable proceeding along the phase envelope. Thanks to the described initialization procedure, methods by Asselineau et al. and Michelsen provide the rapid and reliable construction of the vapor-liquid phase envelopes up to the critical point (CP), being, however, somewhat cumbersome for incorporation into process simulators. Michelsen (1982a,b) presented an efficient procedure for the isothermal flash calculations, which is reliable up to the CP. The method could be easily adjusted for tray by tray distillation computations. The second aproach to the problem is to force a sa&factory compressibility factor to be generated by state equation. Huron et al. (1977/78) have proposed a step by step procedure for searching a pressure interval where the state equation has three real roots. The procedure seems to be time consuming and, as reported, not always reliable. Recently, Poling et al. (1981) have presented a simple and efficient method for producing adequate Z values which is suitable for incorporation into programs for separation column computations. The authors have suggested pressure reduction to generate vapor phase properties and adjustment of composition for the liquid phase. No recommendations for a magnitude of corrections have been proposed. In addition, the effect of pressure reduction on possible simultaneous disappearance of liquid phase properties has not been studied. The scope of this paper is to present a simple and reliable method for avoiding trivial roots by generating appropriate Z values. The procedure is intended to be used in modules for process simulations. The utility of the proposed method is demonstrated for the Soave-Redlich-Kwong (SRK) equation of state (Soave, 1972),which may be rewritten as f(z)= 23 - 2 2 B ( A - 1 - B)Z - AB^ = o (2)
+
where Z is the compressibility factor and A and B are given by A = a(T)/bRT; B = b P / R T (3)
Analysis The appearance of TS is affected by producing unrealistic values of vapor and/or liquid densities from a state equation. The possible forms of the curve f ( Z ) (eq 2) are presented in Figure 1. When eq 2 has three real roots (form 111),the largest represents the vapor phase and the smallest represents the liquid phase. Evidently, the problem arises when the vapor compressibility factor Zv is to be predicted from curves IV or V. The similar situation appears for the liquid phase with cases I or 11. It should be noted that a liquid mixture compressibility factor value (ZL)greater then 1/3, predicted as a single real root of the curves I and 11, might be considered satisfactory as well. This calls for an appropriate criterion. A loose empirical criterion has been proposed by Poling et al. (1981). The following four actions for controlling and avoiding trivial roots are suggested (JovanoviE et al., 1982): (a) adjustment of pressure value; (b) generation of artificial density values by the local extrema in f(Z); (c) pressure correction to produce artificial vapor density value; and (d) temporary conservation of liquid phase density value. (a) Adjustment of Pressure Value. As a result of iterative K value calculations. the close values for ZLand
Figure 1. Possible forms of f(Z).
Zv, both greater then 1/3, may arise. To separate 2 values it is necessary to reduce pressure by a step by step procedure pnew = aPoM ( a < 1) (4) until the condition Z"/ZL > p > 1
(5)
is met. The recommended values for the constants a and
p are 0.9 and 1.2, respectively. (b) Artificial Density Values. In early column iterations, cases IV and I1 for vapor and liquid phase, respectively, may occur (see Figure 1). It is suggested to use artificial compressibility factor values instead of that produced by the state equation. The local minimum Z value (2,") is adopted as the vapor compressibility factor 1 ZV , = -(1 q q (6) 3
+
where q = 1 - 3B(A - 1 - B )
In the absence of the true ZLvalue less then the artificial one is calculated from
Z,L = 1(1- q'P) 3
(7) 1/3
(case 11), (8)
The idea of adopting local extremum 2 values (JovanoviE et al., 1982) was independently proposed by Gundersen (1982) for the isothermal flash calculations. When applying a derivative exploiting method for column calculations, the use of artificial Z values gives rise to false zero af/aZ values. To prevent failure, it is suggested to calculate af/aZ with ZLsomewhat lower then ZmL and/or Zv somewhat higher then ZmV.The recommended values are ZL = O.85ZmLand 2" = 1.152,'. (c) Producing Artificial Vapor Density Values. If the local extrema are absent (curve V), it is necessary to find out conditions for generating them, i.e., for the transition of curve V to curve IV. In this work, the pressure correction is proposed via the necessary condition for the existence of three real roots of a cubic equation. For eq 2 the necessary condition is stated as follows q>o
(9)
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
,
9 CASE T
f
ACTION
T b
IV j
F
T R U E 2’
,
Table I. Selected Examples (mol % in Feed) examp1e component A B
NZ COZ CH4
7 x 104 1.76 1.25
C2H4 C2H6
C3H6 C3H8
i-C4HIo L
n-C4H10
i-CSHIZ n-CsH12
T I
C7H16
0
i
Figure 2. K values generator (refer to Figure 1 for cases I-V).
Further, if the term 3B2 is neglected, relation 9 may be replaced by the conservative one
Thus, the corrected pressure can be explicitly found as
RT “Or
C 17.7
H2
C U L A
803
79.37 13.42 0.71 1.15 0.29 0.31 0.53 31.7 100.0
pressure, bar feed rate, mol/h distillate 82.4 flow, mol/h reflux flow, 131.9 mol/h 18 number of stages 7 feed stage feed bubble point condition liquid cond en ser totala type
36.2 37.4 9.0 15.7 1.7
29.8 30.8 7.4 12.9 1.4
30.4 100.0
30.4 100.0
37.5
47.5
33.0
41.8
16
16
12 30.7% vapor partial
12 61.9% vapor partial
Total condenser is approximated to a partial condenser.
= 3b(A - 1)
The proposed algorithm for calculating appropriate K values is presented in Figure 2. For a detailed discussion of the described actions the reader is referred to the paper by JovanoviE et al. (1982), where the method is applied to bubble point pressure calculations. It is to be noted that the presented procedure for generating Z values is applied until a pre-specified loose convergence criterion of material and energy balances is satisfied. Namely, at some point in the calculation, when TQ,x and/or y become sufficiently close to their true values, there will be no need to apply actions (a)-(d).
Example A describes a distillation column used as deethanizer (Shah and Bishnoi, 1978). In the example, during the course of establishing initial composition profile, the true vapor densities at the stages 13, 14, and 15 had to be replaced by the artificial ones (action b). The subsequent iterations required no adjustments and the column calculation successfully converged in four iterations. In this way, the improvement of the initial temperature profile by the bubble-point temperature method, as originaly suggested by Shah and Bishnoi (1978), may be avoided. Examples B and C are the distillation column presented by Poling et al. (1981). For example B, temperature profiles initially, after first and second iteration and at convergence, are given in Figure 3. Artificial ZmLvalues were used (action b) at the very start of the calculation for the first six stages. In the first iteration the pressure reduction was required to produce the artificial Zv values (action c ) at stages 3 and 4. In the second iteration the artificial Zv values were adopted (action b) at the same stages. Finally, the calculation terminated at the sixth iteration. Example C required some more interventions due to the presence of H2as presented in Table I1 and Figure 3. The total number of iterations was 13.
Results The proposed method was tested on several distillation problems for the systems shown in Table I. The calculated method along with the convergence criterion exploited was that proposed by Ishii and Otto (19731, with an additional constraint regarding maximum allowable stage temperature correction ATj I100 “C. The initial temperature profile is obtained by linear interpolation between the estimated bubble-point temperatures on the top stage, feed stage, and bottom stage. To establish the initial vapor flow profile, constant molar overflow is assumed.
Conclusion The simple and reliable method for avoiding trivial roots in VLE calculations with a cubic equation of state, developed previously (JovanoviE et al., 1982), has proved to be applicable in the simulation of multistage separation processes. The computational results indicate the reliability of the algorithm, even at awkward operating conditions. The general character of the algorithm allows for its application with any cubic equation of state. Its simplicity and compactness makes it suitable for incorporations into the process simulators.
Similar expressions for other frequently used cubic equations of state are given in the Appendix and Table 111. (d) Conservation of Liquid Phase Density Value. It should be noted that the pressure reduction may cause the loss of the proper liquid phase density values, Le., transition of form V to form I (Figure l),leading to a “vapor like” (2 > l/J TS. In order to avoid failure, the previously calculated pL value at the specified column pressure P is conserved, which gives the following simple correction
804
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
Table 11. Results of Column Calculations-Example C“ iteration no. 1 2 3 4 0 1 2
0
0
0
3 4
0 0 0
0 0
0
5 6
Am Am A08 0
I 8 9-13‘ a
5
( 0 )Action
b for ZL; (H)action b for Zv; (A)action c.
7
8
9
15
16
0 0
0 0
0 0
0
0
0
0
0 0
0 0
0
0 0
0
Am
’No interventions. Acknowledgment
Table I11 analytic expression for pressure
state equation
The authors wish to thank Mrs. A. Mihajlov and Mrs. G. CiriE-MatijeviE for providing assistance on the example problems, and the Petroleum Research Fund (administered by the American Chemical Society) for financial support.
W
Redlich and Kwong -RT __ _a 0 (1949) V - b V(V + b ) Peng and Robinson ~RT a (1976) v - b v(v + b ) t b(v- b )
Appendix The general expression for determining the corrected pressure value, which produces artificial vapor density, can be written as
2c
(b1113 RT 2v + b a ‘I< u 2 ~ b- V ( U + b ) RT U v 2 + c b v - ( c - - l ) b a (‘u- b
-___
Ishikawa e t al. (1980) Harmens and Knapp (1980); Schmidt and Wenzel ( 1 9 8 0 ) 280
~
F
The values of coefficient W for frequently used cubic equations of state are given in Table 111. P
Example
B
x
-
220
+ Y / 200
I80
2
L
---
INITIAL
-
A F TT EE RR
1
6
8 Stage
FINAL
ESIIMATES FSI R ES C TO N Dl T El R l EARl lAOTNl O N RESULTS
10 Number
12
16
1L
/
2 80
7
Example Example
C C
260
Nomenclature A, B = parameters (eq 3) a, b, c = equation of state parameters K = equilibrium ratio vector N = number of components in the system P = pressure q = parameter (eq 7) R = universal gas constant T = temperature W = parameter (eq A-1) x = mole fraction of liquid phase y = mole fraction of vapor phase 2 = compressibility factor a,p = constants (eq 4 and 5) cp = fugacity coefficient p = density Subscripts i = component number j = stage number m = value at local extremum Superscripts L = liquid phase V = vapor phase
240
-
stage no. 6
Y
220
Literature Cited
b-
200
-
---
-
I80
INITIAL
ESlIMATES
AFTER
sixrn
FINAL
UESULIS
ITERATION
Asseiineau, L.; Bogdani6, 0.; Vldai, J. J . FluU Phase Equ/l/b. 1979, 3 , 273. Gundersen, T. Comput. Chem. €ng. 1982, 6 . 245. Harmens, A. “Proceedlngs of the Brltlsh Cryogenic Council Conference”; BrigMon, 1973, p 91. Harmens, A. Cryogenics 1075, 75. 217. Harmens, A.; Knapp, H. Ind. Eng. Chem. fundam. 1080, f s , 291. Hwon, M. J.; Dufow, 0. N.; V!dal, J. J . Fluid phase Equlllb. 1977178. 1 247. Ishii, Y.; Otto, D. Can. J . Chem. Eng. 1978, 51, 601. Ishikawa, T.; Chung, W. K.; Lu, B. C.-Y. AIChE J . 1980, 26, 372. JovanoviE. S.; Paunovl6, R.;Mihajlov, A. “Proceedings of the Thhd AustrianItalian-Yugoslav Chemical Engineering Conference”, Graz, Austria, Sept 1982; Vol. I , p 80. Martin, J. J. Ind. Eng. Chem. Fundam. 1979, 78, 81. Michelsen, M. L. J . Fluid Phase Equillb. 1980, 4 , 1. I
160
I
2
4
6
8 Stage
,
10 Number
1
12
l
14
.
I
16
Figure 3. Temperature profiles for examples B and C: ( 0 )action b for ZL;(m) action b for Zv; (A)action c.
Ind. Eng. Chem. Process Des. Dev. 1984, 23, 805-808 Mlchelsen, M. L. J . Fluid Phese €9u/llb. 1982a, 9 ,1. Mlchelsen, M. L. J . FluM Phase Equilib. 1982b, 9 ,21. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15. 59. Prausnitz, J. M. Ind. Eng. Chem. Process Des Poling, B. E.; Edward, A. 0.; D e v . 1981. 20, 127. Redllch, 0.; Kwong, J. N. S. Chem. Rev. 1949, 4 4 , 233. Schmidt, G.; Wenzel, H. Chem. Eng. Sci. 1980, 35, 1503. Shah, M. K.; Bishnoi, P. R. Can. J . Chem. Eng. 1978. 56, 470.
805
Soave, G. Chem. Eng. Sei. 1972, 2 7 , 1197.
Received for review December 29, 1982 Accepted January 16,1984
Part of this work was presented at the Third Austrian-ItalianYugoslav Chemical Engineering Conference, Graz, Austria, Sept 1982.
Prediction of Gas-Particle Heat Transfer Coefficients by Pulse Technique Abdelhafeez N. Mousa Chemical Engineering Department, Kuwait Universlty, Kuwait
The heat transfer coefficient between a gas and a solid was studied by Introducing a pulse of heat in the stream of a gas flowing through a bed packed with solids. The variation of temperature with respect to time was measured at the inlet and outlet of the packed bed. The first moments were calculated from the temperature-time curves. The DC model (dispersion concentric model) was used to predict the heat transfer coefflclent. The particles used were spherical glass beads of 2.1 mm diameter, and air was used as the gas. The particle Reynolds number was varied from 14 to 50.
Introduction The determination of heat transfer coefficients between a flowing fluid and the particles of a packed bed by direct steady-state techniques is usually very difficult to realize. As an alternative, heat transfer coefficients may be obtained either by transient heat-transfer techniques or by performing mass transfer experiments from which heat transfer coefficients may be predicted by employing the analogy between the transfer of heat and mass. Gas-particle heat transfer coefficients in packed beds are available at high Reynolds numbers ranging from about 100 to 100000. Dayton et al. (1952), Meek (1961), and Shearer (1962) have successfully used sinusoidal input technique a t high Reynolds numbers to predict heat transfer coefficients. However, at low Reynolds numbers data are not only scarce, but their reliability is questionable. The reason for the scarcity of data at low Reynolds numbers may be due to the fact that the method of Gamson et al. (1943) is inapplicable in this region. Data for Reynolds numbers of less than 100 have been reported by Kunii and Smith (1960, 1961), Eichorn and While (19521, Pulsifer (1965), Wilke and Hougen (1945), Glaser and Thodos (1958) and De Acetis and Thodos (1960). Several models have been suggested for analyzing sinusoidal inputs. Gunn and Pryce (1969) used a one-parameter model, while models with four parameters have been used by Asbjornsen and Wang (1971) and Asbjornsen and Amundsen (1970). The dispersion concentric model (DC model), based on a fluid existing in dispersed plug flow with intraparticle temperatures leaving in radial symmetry, was used by Turner and Otten (1973) and Bradshaw et al. (1970), who reported heat transfer coefficients by step response measurements. The continuous solid-phase model (CS model), based on the assumptions of a fluid being in dispersed plug flow and a solid through which axial heat conduction is taking place in continuous phase, was used by Littman et al. (1968). Discussions of the DC
model and CS model are given by Kaguei et al. (1977). Wakao et al. (1979) have shown that, for the determination of heat transfer coefficients, the DC model is perhaps better. In the present study, the pulse technique was applied to determine the heat transfer coefficients between a flowing gas and a packed bed. The gas used was air and the bed was packed with spherical glass beads. The DC model was used to analyze the results. Theory According to the DC model proposed by Kaguei et al. (1977) and Wakao (1981) the instantaneous heat balance between a flowing fluid and a packed bed gives the following equations
with the following prevailing boundary conditions
aTP dr
k - = h,(TF - Tp)(atr = R) TF =
Tp = 0
TF = TF1(t) TF
= T&t) TF
=0
(3)
(at t = 0)
(44
(at x = XI)
(4b)
(at x = XI+ L )
(44
(atx =
(44
m)
aTP (at r = 0) (44 ar The solution of eq 1and 2 with the boundary conditions represented by eq 3 and 4 in the Laplace domain between -= 0
0 1984 American Chemical Society
