I n d . Eng. C h e m . Res. 1991,30, 414-420
414
formed. These findings are described by a mathematical model that is shown to represent the experimental data reasonably well. Acknowledgment This work was sponsored by NSF Grants CBT-8516874 and CTS-8905463. The assistance of Dr. R. H. Jean in the derivation of the mathematical model is greatly appreciated. Nomenclature A = cross-section area of test column, L2 d , = diameter of the particle, L D, = diameter of the test column, L Hpb= height of the packed bed, L Hpbi = initial height of the packed bed, L t = time, 0 Ugi= superficial gas velocity at the bottom of the packed bed, LIB Ugo= superficial gas velocity out of the top of the packed bed,
LIB
Uli = superficial liquid velocity at the bottom of the packed bed or initial start-up step change in liquid flow, L/8 U,,, = superficial liquid velocity out of the top of the packed bed, L/O Ut", = moving rate of the top of the packed bed, L/O Ubot = moving rate of the bottom of the packed bed, L/O U,; = linear gas velocity at the bottom of the packed bed, L/O U,,' = linear gas velocity out of the top of the packed bed, Ugp{ = linear gas velocity inside the packed bed, L/O Uli' = linear liquid velocity at the bottom of the packed bed,
LIB
Umf= minimum fluidization velocity, L/O V = the depletion rate of the moving packed bed, L/O x", = axial location of the bottom of the packed bed, L
Xf = axial location of the bottom of the foam layer, L Xt = axial location of the top of the packed bed, L Greek L e t t e r s tgbc = gas holdup in the bubble column region beneath the
packed bed liquid holdup in the bubble column region beneath the packed bed t,,b = solid holdup inside the packed bed tgpb = gas holdup inside the packed bed tlpb = liquid holdup inside the packed bed tgf = gas holdup in the top region above the packed bed tgl = liquid holdup in the top region above the packed bed 4 = sphericity p1 = liquid denstiy, M/L3 ps = solid density, MIL3 tlbc =
L i t e r a t u r e Cited Chen, Y. M.; Bavarian, F.; Fan, L.-S.; Buttke, R. D.; Beaton, W. I. Transients in Bed Expansion of a Three-phase Fluidized Bed. In Fluidization and Fluid-Particle Systems: Fundamental and Applications; Fan, L.-S., Ed.; AIChE Symposium Series 85 (No. 270); AIChE: New York, 1989; p 49. Didwania, A. K.; Homsey, G. M. Rayleigh-Taylor Instability in Fluidized Bed. Ind. Eng. Chem. Fundam. 1981,20, 318. Fan, L. T.; Schmitz, J. A.; Miller, E. N. Dynamic of Liquid-Solid Fluidized Bed Expansion. AIChE J . 1963, 9, 149. Fan, L.4.; Bavarian, F.; Gorowara, R. L.; Kreischer, B. E.; Buttke, R. D.; Peck, L. E. Hydrodynamics of Gas-Liquid-Solid Fluidization Under High Gas Holdup Conditions. Power Techonol. 1987, 53, 285. Nacef, S.;Wild, G.; Laurent, A.; Kim, S. D. Effect D'Echelle en Fluidisation Gaz-Liquide-Solide. Entropie 1988, No. 143-144, 83. Slis, P. L.; Willemse, T. W.; Kramers, H. The Response of the Level of a Liquid Fluidized Bed to a Sudden Change in the Fluidizing Velocity. Appl. Sci. Res. 1959, 8, 209.
Received f o r review January 29, 1990 Revised manuscript received August 6 , 1990 Accepted September 10, 1990
A Physical Theory Superimposed onto the Chemical Theory for Describing Vapor-Liquid Equilibria of Binary Systems of Formaldehyde in Active Solventst Stefan0 B r a n d a n i , Vincenzo B r a n d a n i , * and Gabriele Di Giacomo Dipartimento d i Chimica, Ingegneria Chimica e Materiali, Universitci d e L'Aquila, I-67040 Monteluco di Roio, L'Aquila, I t a l y
Vapor-liquid equilibria in the binary mixtures water-formaldehyde and methanol-formaldehyde are satisfactorily correlated by superimposing a physical model onto the chemical theory for describing t h e liquid phase. In previous works, t h e liquid phase for t h e binary systems water-formaldehyde and methanol-formaldehyde was described in terms of chemical theory only, while both physical a n d chemical forces were taken into account in describing the vapor phase. I n this work, we take into account the physical forces acting upon the molecules of the true species, which are present in the liquid phase, without modifying the description of the vapor phase. For the liquid phase, we use the UNIQUAC equation to express t h e effect of physical forces by using only one adjustable physical parameter independent of temperature for each binary system. Introduction
design of separation processes in the chemical industry.
+Presented at the 11th IUPAC Conference on Chemical Thermodynamics, Como, Italy, Aug 26-31, 1990.
More recently, Brandani and Di Giacomo (1985) proposed a model, for describing the vapor and liquid phases for the systems water-formaldehyde and methanol-form-
0888-5885/91/2630-0414$02.50/0
0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 415 aldehyde, that takes only chemical forces into account in the liquid phase, while it considers the effect of both physical and chemical forces in the vapor phase. In this work, we present a further implementation of the model; in fact, we take into account the physical interactions between the true species in the liquid phase, using the UNIQUAC equation to express the nonideality of the mixtures. This description of the VLE behavior of the binary systems water-formaldehyde and methanol-formaldehyde is achieved with only one more adjustable parameter independent of temperature for each binary system. The improvement in the description of VLE for these systems is significant, and moreover, it will be easier to extend the model to the ternary system watermethanol-formaldehyde.
Theoretical Model Binary Systems Containing Formaldehyde. Let A be the active solvent (water or methanol) and F be formaldehyde. In the liquid phase, together with free molecules of solvent and formaldehyde, there are species AFi, which are formally constituted by i molecules of formaldehyde and one molecule of solvent (Walker, 1964; Brandani et al., 1980; Kogan and Ogorodnikov, 1980; Brandani and Di Giacomo, 1985; Maurer, 1986). Because of the low volatility of polymers AF;, we can assume, according to Hall and Piret (1949),that only AF is present together with free molecules of solvent and formaldehyde in the vapor phase. Although the mechanism of formation of polymers AFi is a subject under discussion (Molzahn and Wolf, 1982), from a thermodynamic point of view, we may consider them as formed by successive reactions of the type F + AFi-1 = AF; (1) As discussed elsewhere (Brandani et al., 1980), two equilibrium constants are sufficient to describe the species distribution in the liquid phase: K f and K$. K f is the thermodynamic equilibrium constant for eq 1when i = 1; K t is the thermodynamic equilibrium constant for eq 1 when i # 1. Of course the description of the equilibrium distribution in the vapor phase requires only the thermodynamic equilibrium constant KA, since in eq 1, according to our hypothesis, i may only assume the value of 1. Vapor Phase. In the vapor phase, equilibrium is of the type F+A=AF (2) To describe this chemical equilibrium quantitatively, we use a thermodynamic equilibrium constant
where f is the fugacity of the true molecular species, u is the true mole fraction, @+ is the fugacit coefficient of the true species, P is the total pressure, K i P! is the ratio of the partial pressures of the true species, and K$ is the ratio of the fugacity coefficients of the true species. To evaluate K A as a function of temperature for the systems water-formaldehyde and methanol-formaldehyde, we used the experimental data of Hall and Piret (1949) and the virial equation of state, neglecting all third virial coefficients. T o calculate the second virial coefficient of the mixture, we assumed that the second virial crosscoefficients are given by
B, = (B;i + B j j ) / 2
(4)
Table I. Values of the Parameters aAand BA Expressing the Temperature Dependence of the Equilibrium Solvation Constant KAof Formaldehyde with Active Solvent A in the Vapor Phase A CYA BA, K water -23.3 (0.6)' 7700 (200)" methanol -15.6 (0.5)' 6230 (170)'
'Standard deviations in parentheses. The true mole fractions of the three species are related to the apparent mole fractions y A and y F (Brandani et al., 1980; Brandani and Di Giacomo, 1985)
UA
- (YA - Y F ) / Y A + U F ( Y F / Y A )
(6)
= (YF - U F ) / Y A
(7)
UAF
where apparent means that association and solvation have been neglected. According to Prausnitz et al. (1980), this calculation is achieved by utilizing the Lewis fugacity rule In
C # ]I
= BjP/(RT)
(8)
The second virial coefficient Bj is calculated by the generalized method of Hayden and O'Connell (1975), using the parameters reported by Brandani and Di Giacomo (1985). The functional form assumed for K A was In K A = aA
+ pA/T
(9)
where parameters aAand PAwere obtained by minimizing the objective function
eerp
where represents the experimental data of Hall and Piret (1949) for gaseous mixtures of formaldehyde and active solvent A, is the pressure calculated with the virial equation of state for the same system, and N is the number of the experimental data. In Table I, the values of aAand PA, together with their standard deviations, are reported for the water-formaldehyde and methanol-formaldehyde systems. The apparent fugacities in the vapor phase for the two components are
ccalc
f x = @AppYAp f 8 = @bppYFp
(11)
(12)
where, as shown by Nothnagel et al. (1973), the apparent fugacity coefficient of component j is given by
@p= ( U . /YI.)@! I
For formaldehyde, the ratio is given by
uF/YF
(13)
in the limit of y F = 0
where 6 is the vapor pressure of active solvent A. Liquid Phase. In the liquid phase, there are equilibria expressed by eq 1. To describe these chemical equilibria,
416
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991
Table 11. Values of the Parameters af,fif, a:, and fit Expressing the Temperature Dependence of the E uilibrium Solvation and Association Constants Kf and XI 1of Formaldehyde with Active Solvent A in the Liquid Phase A 4 P:, K 4 LJ$> K water -5.923 3997.1 (1.9)" -1.012 (0.009)" 2680.8 (3.1)" (0.006)"
methanol -3.837
3896.8 (1.9)" -7.290 (0.002)" 4525.8 (3.1)"
(0.005)"
Standard deviations in parentheses.
Table 111. Values of the Parameters a& fib and 8: Expressing the Temperature Dependence of Henry's Constant of Formaldehyde in Active Solvent A A aA, OA, SA, water -6.397 (0.003)O 20.622 (0.003)" -12.610 (0.003)' methanol 7.890 (0.004)' -7.477 (0.005)" Standard deviations in parentheses.
The apparent fugacities in the liquid phase for the two components are
f k = ~ P ' x ~ d d exp[uk(Pfi fi)/RTI
(25)
we use two thermodynamic equilibrium constants (Brandani et al., 1980; Brandani and Di Giacomo, 1985)
K$.=--=Y$,A
ZF,A
... =--- G,* zF,A
where solvent molar volumes in the liquid phase are calculated by the modified Rackett equation (Spencer and Danner, 1972) and the partial molar volume of formaldehyde a t infinite dilution in solvent A is calculated by the correlation proposed by Brelvi and O'Connell (1972). The required parameters are reported by Brandani and Di Giaoomo (1985). The apparent activity coefficients of the two components are given by
-
Y$Y$,.,A ZFZF,-~A
Y$Y$?A ++,A
Qe3Q0
for i L 3 (17)
where z is the true mole fraction and lim (In 7;)= 0
(18)
2,-0
lim (In Y ~ = )0 2 p l
(19)
where the two activity coefficients are normalized according to two different conventions
Equation 17 holds for i 1 3, since Qe3 = QG4 = ... = Q$
(20)
as will be shown in the section illustrating the model for the activity coefficients. The temperature dependence of these thermodynamic equilibrium constants is given by
Y~PP-
1 as
XA
yPpP=;,
1 as
XF
since
--
1
(29)
O
(30)
In Kf = af + P f / T
(21)
To express the effect of temperature on Henry's constant of formaldehyde in solvent A, H&, we have chosen the functional form
In K$. = af = @ / T
(22)
In (H&/Pc) = ah
cut, cut,
The values of parameters &', and 8; are reported in Table 11. For the water-formaldehyde system, the parameters were obtained by reducing the experimental VLE data of Brandani et al. (1980), while for the methanol-formaldehyde system the parameters were obtained by reducing the thermodynamically consistent (Brandani et al., 1987) experimental data of Kogan and Ogorodnikov (1980). The apparent mole fraction of the formaldehyde, xF, is related to the true mole fraction, zF, by the relationship
+ PA(Tc/T) + 6A(Tc/T)'
(32)
where Pc and Tc are the critical pressure and the critical temperature of formaldehyde, respectively. The values of the parameters CUB,p i , and 6fj are reported in Table 111. In the case of the methanol-formaldehyde system, for which the experimental data used for the fitting range from 60 to 80 " C , we assumed S i = 0. To express the activity coefficients that take into account physical forces between molecules nf true species, we use the UNIQUAC model (Abrams and Prausnitz, 1975; Maurer and Prausnitz, 1978) with only one adjustable parameter, independent of temperature, for each binary system. The expression of the activity coefficient of solvent A is given by In yA (combinatorial) = In ( + A / z A ) + 1 - @ A / z A + 5qA[ln ( e A / @ A ) + * A / e A - 11 (33)
and In YA = In
(combinatorial)
+ In YA
(residual)
(35)
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 417 To simplify, we assume that Ukk is the same for all polymers AFi with i L 2. Denoting A with the suffix 1, F with 2, AF with 3, and AF2 with 4 results in
Table IV. Values of the Parameters Expressing the Physical Interactions between Formaldehyde and Active Solvent A ~~~
A water methanol
~
€FA
u12 = - ( ~ 1 1 ~ 2 2 ) ~ ’ ~-( 1[FA)
-0.030 (0.001)’ 0.0061 (0.0004)a
(48) (49)
’Standard deviations in parentheses. The expressions for the activity coefficients of solutes j are given by
(50)
In yj’ (combinatorial) = In
(51)
(@A/zA)+
(52) u34 = -(u33u44)1’2( 1 - ;[FA) 1
In 7: (residual) =
and In yj’ = In 77 (combinatorial) + In 77 (residual)
(53)
where [FA is the adjustable binary parameter. The values of the parameters [FA are reported in Table IV. As a consequence of our assumption, it can be shown that
(38)
where
and
0.78-(ln 1 qAFz
(55)
7lFP)residual
and therefore
Vapor-Liquid Equilibrium. By using the thermodynamic relationships for phase equilibria, for the total pressure we obtain
P =Y
A z A ~exp[(uk ~
- BA)(P - fi)/RT]% UA
The van der Waals properties of the solvents, rAand q A , were taken from the collection reported by Prausnitz et al. (1980), while those of formaldehyde are rF = 1.18 and qF = 1.20 and those of polymer AFi ri = rA 0.92i (42)
+ qi = q A + 0.78i
(43)
were calculated following Bondi (1968). Parameters of the UNIQUAC equation 7mk
= exP[-(u,k
- Ukk)/RTI
(44)
were calculated according to Abrams and Prausnitz (1975) (45)
where X is the latent heat of vaporization, XF = 5020 cal/mol, Xw = 10520 cal/mol, AM = 8410 cal/mol, and those of polymers are given with sufficient accuracy by
+ XF XAF2 = X A + 2XF XAF = A,
according to Bryant and Thompson (1971).
+
where the temperature dependence of f i and the vapor pressure of solvent A were expressed with the equation reported by Prausnitz et al. (1967). Results a n d Discussion The experimental data of Brandani et al. (1980) for the water-formaldehyde system and those of Kogan and Ogorodnikov (1980) for the methanol-formaldehyde system were used for the determination of parameters, minimizing the objective function
r (:
ssQ(aC,PA,aB,p8,aa,pa,6a,5~A) =
‘( k
Pk,calc
- Pk,exp
Pk,exp
YA,k,calc
+
where Pcalc is given by eq 56, while Y
- YA,k,exp
YA,k,exp
r
(58)
~ is given , ~ by ~
(46)
(47) (59)
~
418 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 Table V. Results of the Fit AAPD chem theory system water-formaldehyde methanol-formaldehyde methanol-formaldehyde
no. of data points
temp range, " C
P
63 27 20
40-90 60-80 60-80
0.72 3.28 2.56
this work
YA
P
YA
0.51 0.40
0.41 1.12 1.04
0.31 0.37
data source a b b, c
nBrandani et al. (1980). *Kogan and Ogorodnikov (1980). 'Brandani et al. (1987). 2.10
a
1.70
x
I PI
2
1.30
E
2 4 O
.90 nu
\
33
+
I
J
f .50
g; O l 2 *%
t
t i ; =0 , au -4s U
3
0
=
I -10 1.10
1.15
1.20
1.25
1.30
1,35
Reclprocol o f Rrduced Tenperatwe Figure 1. Reduced Henry's constant of formaldehyde vs reciprocal of reduced temperature (physical and chemical theory).
The results of the fit are reported in Table V. The results of this work are compared with those (Brandani et al., 1980, Brandani and Di Giacomo, 1985) obtained with the liquid phase described only by the chemical theory. For the water-formaldehyde system, the introduction of one more adjustable parameters to take into account physical forces in the liquid phase reduces the average absolute percent deviation (AAPD) in pressure by about 60%. For the methanol-formaldehyde system (27 data points), the introduction of the adjustable parameter that takes into account physical forces in the liquid phase, eliminating one parameter (Sf = 0) in the correlation of Henry's constant, reduces the AAPD in pressure by about 200%, while the AAPD in Y A is reduced by about 65%. For the methanol-formaldehyde system (20 data points), taking into account only the thermodynamically consistent data (Brandani et al., 19871, the reduction of the AAPD is about 146% in pressure and about 8% in yA. However, the parameters, obtained from the thermodynamically consistent data, present a standard deviation that is about 3 times less than that of the parameters obtained by the fit of all the data. The analysis of residuals, for the water-formaldehyde system, shows that they are randomly distributed with a mean near zero (pp = 0.22 Torr, up = 0.53 Torr), while for the methanol-formaldehyde system F~ = 1.90 Torr (op = 5.98 Torr) and pLy= -0.0008 Torr (ay = 0.0034 Torr). Maurer (19861, with his model, for the data of Kogan and Ogorodnikov (1980), finds an AAPD in pressure of 5.7% and an AAPD in Y A of 3.370, values that are much higher than our results. Figure 1 shows the behavior of the reduced Henry's constant of formaldehyde in the two solvents with the reciprocal of the reduced temperature. It can be seen that
1.10
1.15
1.20
1.25
Reclprocol of Reduced Tenperatwe
Figure 2. Reduced Henry's constant of formaldehyde vs reciprocal of reduced temperature (chemical theory only).
I
I
I
I
I
I
1
t
; i L
.a5.OO
.OS
.lo
.IS
.20
.25
.30
Vapor-Phase M e fractton Formldrhyd.
Figure 3. Apparent fugacity coefficients a t saturation vs vaporphase mole fraction of formaldehyde. Water-formaldehyde system a t 70 "C.
the Henry's constant of formaldehyde in water is much higher than the same property in methanol. The parameters used to draw the two curves are those of Table 111. Figure 2 shows the same property as in Figure 1, but in this case, the chemical theory was only used to describe the behavior of the liquid phase. The Henry's constant of formaldehyde in water shows the same behavior as in Figure 1, even though its numerical values are about 40% lower. The Henry's constant of formaldehyde in methanol,
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 419 1.00
7.10
6.10
0
P
5.10 4.10 3.10
.90
3 Y d .EO v)
Y
z-
,70
6
-U
-u ce 4 g
.60
2.10
1.10
u
x
f P
U
a F
c d
a
a a .LO ,DO
.EO
.lo
30
.40
,SO
.60
Vopor-Phare W e Frochon Fornoldmhyd.
Figure 4. Apparent fugacity coefficients a t saturation vs vaporphase mole fraction of formaldehyde. Methanol-formaldehyde system a t 70 "C. 1.10
1.00
i ,go i j i;
cc b
2 .EO
u .70
a
i$
-60
8 .so -00 .OS
10
.15
20
.25
30
35
,40
Llqud-Phore Mole Froctlon Fornrldehyde Figure 5. Apparent activity coefficients vs liquid-phase mole fraction of formaldehyde. Water-formaldehyde system a t 70 "C.
however, shows a minimum that has no physical significance. Figures 3 and 4 show, at 70 O C , the apparent fugacity coefficients at saturation as a function of vapor-phase mole fraction of formaldehyde for the water-formaldehyde and methanol-formaldehyde systems, respectively. Because of stronger interactions between methanol and formaldehyde, the apparent fugacity coefficients of these two components, although far below atmospheric pressure, are well removed from unity. At low mole fractions of formaldehyde, the effect of solvation is the dramatic reduction in the true mole fraction of this component and its apparent fugacity coefficient is very small. For the waterformaldehyde system, the effect of solvation is less pronounced and the apparent fugacity coefficients are much higher. Figures 5 and 6 show, at 70 "C, the apparent activity coefficients as a function of the liquid-phase mole fraction of formaldehyde for the water-formaldehyde and metha-
.OO
.lo
.20
-30 .40
SO
.60
-70
-80
Llquld-Phase Hde Froctlon Fornotdehyde
Figure 6. Apparent activity coefficients vs liquid-phase mole fraction of formaldehyde. Methanol-formaldehyde system a t 70 "C.
nol-formaldehyde systems, respectively. The waterformaldehyde system shows positive deviations from ideal behavior, while the methanol-formaldehyde system shows negative deviations. In fact, in the methanol-formaldehyde system, in the liquid phase, the effect of solvation prevails over that of association; the opposite occurs in the water-formaldehyde system. This behavior explains the use of methanol as a stabilizer; in fact, methanol increases the solution stability owing to the formation of hemiformal. The effect of physical forces on the apparent activity coefficients can be deduced by the comparison of Figure 5 with Figure 2 reported by Brandani et al. (1980) and of Figure 6 with Figure 3 reported by Brandani and Di Giacomo (1985). The physical interaction parameters, tiA, are small. As pointed out by Ikonomou and Donohue (1988), in phase equilibria calculations, often the magnitude of this parameter is examined in an attempt to draw some conclusion about how well the theory models the physics involved. A large value of tiAis an indication that the theory has some fundamental inadequacy and needs to be improved. On the other hand, a small tiAdoes not necessarily indicate that the theory is correct. It is possible, for example, that the approximations used in the theory produce errors of roughly equal magnitude but of opposite direction that cancel each other out and thus result in a small tiA.Therefore, small values of tiAcan be misleading, and no conclusion can be drawn about the validity of the theory by looking only at tiAvalues. tiAvalues obtained in this work are not large, and this fact when coupled with the values obtained for the Henry's constants of formaldehyde can be taken as proof of the absence of major errors.
Conclusions The solution model developed in this work takes into account both physical and chemical forces in the vapor as well as in the liquid phase. It provides a representation of water-formaldehyde and methanol-formaldehyde solution behavior for a wide range of temperatures and compositions. It has been shown that consideration of physical forces provides significant improvement over the model based only on the chemical theory.
Ind. Eng. Chem. Res 1991, 30, 420-427
420
Acknowledgment We are indebted to the Italian Minister0 dell'universiti e della Ricerca Scientifica e Tecnologica for financial support. Registry No. HCHO, 50-00-0; MeOH, 67-56-1. Literature Cited Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J . 1975,21,62. Bondi, A. Physical Properties of Molecular Crystals, Liquids & Gases; John Wiley & Sons: New York, 1968;pp 450-471. Brandani, V.; Di Giacomo, G. Effect of Small Amounts of Methanol on the Vapour-Liquid Equilibrium for the Water-Formaldehyde System. Fluid Phase Equilib. 1985,24,307. Brandani, V.; Di Giacomo, G.; Foscolo, P. U. Isothermal VaporLiquid Equilibria for the Water-Formaldehyde System. A Predictive Thermodynamic Model. Ind. Eng. Chem. Process Des. Dev. 1980,19,179. Brandani, V.; Di Giacomo, G.; Mucciante, V. A Test for the Thermodynamic Consistency of VLE Data for the Systems WaterFormaldehyde and Methanol-Formaldehyde. Ind. Eng. Chem. Res. 1987,26, 1162. Brelvi, S. W.; O'Connell, J. P. Corresponding States Correlations for Liquid Compressibility and Partial Molar Volumes of Gases at Infinite Dilution in Liquids. AIChE J . 1972,18, 1239. Bryant, W. M.; Thompson, J. B. Chemical Thermodynamics of Polymerization of Formaldehyde in an Aqueous Environment. J . Polym. Sci.: Part A-1 1971,9, 2523. Hall, M. W.; Piret, E. L. Distillation Principles of Formaldehyde Solutions. Ind. Eng. Chem. 1949,41,1277.
Hayden, L. V.; O'Connell, J. P. A Generalized Method for Predicting Second Virial Coefficients. Ind. Eng. Chem. Process Des. Deu. 1975,14,209. Ikonomou, G. D.; Donohue, M. D. Extension of the Associated Perturbed Anisotropic Chain Theory to Mixtures with More Than One Associating Component. Fluid Phase Equilib. 1988,39,129. Kogan, L. V.; Ogorodnikov, S. K. Liquid-Vapor Equilibrium in the Formaldehyde-Methanol System. Zh. Prikl. Khim. 1980,53,115. Maurer, G. Vapor-Liquid Equilibrium of Formaldehyde- and Water-Containing Multicomponent Mixtures. AIChE J. 1986,32, 932. Maurer, G.; Prausnitz, J. M. On the Derivation and Extension of the UNIQUAC Equation. Fluid Phase Equilib. 1978,2,91. Molzahn, M.; Wolf, D. Distillation, Absorption and Extraction-Is There Any Scope for Research Left? Ger. Chem. Eng. 1982,5, 221. Nothnagel, K. H.; Abrams, D. S.; Prausnitz, J. M. Generalized Correlation for Fugacity Coefficients in Mixtures a t Moderate Pressures. Application of Chemical Theory of Vapor Imperfections. Ind. Eng. Chem. Process Des. Dev 1973,12,25. Prausnitz, J. M.; Eckert, C. A,; Orye, R. V.; O'Connell, J. P. Computer Calculations for Multicomponent Vapor-Liquid Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1967;p 218. Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A.; Hsieh, R. O'Connell, J. P. Computer Caculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1980;pp 221-270. Spencer, C. F.; Danner, R. P. Improved Equation for Prediction of Saturated Liquid Density. J . Chem. Eng. Data. 1972,17, 236. Walker, J. F. Formaldehyde; ACS Monograph Series; American Chemical Society: Washington, DC, 1964;p 103.
Received for review April 9, 1990 Revised manuscript received July 16,1990 Accepted August 1, 1990
A Generalized Viscosity Equation for Pure Heavy Hydrocarbons Ani1 K.Mehrotra Department of Chemical and Petroleum Engineering, The University of Calgary, Calgary, Alberta, Canada T2N 1N4
A new method is presented for the correlation and prediction of the viscosity of pure heavy hydrocarbons listed in API Research Project 42. The 273 heavy hydrocarbons in the database include branched/unbranched paraffins and olefins together with a variety of complex nonfused/fused aromatic and naphthenic compounds. A generalized one-parameter viscosity-temperature equation, log ( k + 0.8) = 100(O.OIT)b,is proposed (overall AAD < 7-1070) for all heavy hydrocarbons in the database. For each hydrocarbon, an optimum value of parameter b is provided. It is shown that parameter b varies linearly with the logarithm of molar mass as well as the inverse of boiling temperature (at 10 mmHg). This important observation leads to the development of a predictive method for the liquid-phase viscosity of pure heavy hydrocarbons. Numerous correlative and predictive methods for liquid viscosity and its variation with temperature have been proposed in the literature. A review of viscosity prediction methods is given by Reid et al. (1986). The TRAPP method (Ely and Hanley, 1981) for viscosity calculation uses a simple n-alkane, such as methane or propane, as the reference fluid in the corresponding states framework. Twu (1985) presented a method for the viscosity of petroleum fractions, which uses n-alkanes as the reference substances and requires boiling point and specific gravity as input parameters. However, the use of an n-alkane as the reference fluid in methods based on the corresponding states principle gives inadequate viscosity predictions for most non-paraffinic hydrocarbons (Johnson et al., 1987; Pedersen et al., 1984; Teja and Rice, 1981). Instead, as shown by Johnson et al. (1987) and Mehrotra and Svrcek 0888-5885/91/2630-0420$02.50/0
(1987), the use of a non-paraffinic hydrocarbon as the reference fluid give satisfactory predictions for the viscosity of bitumens, which are mixtures of mostly aromatic and naphthenic compounds. Bitumens from oil sands of Alberta (Canada) are extremely complex and viscous crude oils comprising high molecular weight polymeric asphaltenes and resins (Strausz, 1989). The SARA chemical analysis of bitumens and heavy oils is performed routinely to divide the constituent hydrocarbons into four groups: saturates, aromatics, resins, and asphaltenes. Most of the saturate fraction (14-28 mass %) of Alberta bitumens is composed of alkylcycloalkanes, involving one- to six-ring structures. The n-alkanes and other paraffinic hydrocarbons that exist in abundance in conventional crude oils are almost totally nonexistent in bitumens. The aromatic fraction (13-30 0 1991 American Chemical Society
