Fair, J. R., Lambright. A. J., Anderson, J. W., lnd Eng. Chem.. Process Des. Develop.. 1, 33 (1962). Hughrnark. G . A., lnd. Eng. Chem., Process Des. Develop.. 6, 218 (1967). Leibson. I.. Holcomb. E. G . . Cacoso, A G . , Jacmic. J. J.. AlChE J , , 2, 296 (1956). Levich. V G , “Physicochemical Hydrodynamics,” 2nd ed, Prentice-Hall, Englewood Cliffs, N J., 1962, pp 464-465
Towell, G. D., Strand, C. P., Ackerman, G. H., AlChE lnst Chem Eng Symp. Ser., No. 10, 10 (1965). Van Krevelen, D. W., Hoftijzer, P. J., Chem. Eng. Progr., 46, 29 (1950). Yoshida. F., Akita, K . . AlChE J., 11, 9 (1965).
Received f o r revieu, J u n e 25, 1973 Accepted October 10, 1973
Calculation of High-pressure Phase Equilibria and Molecular-Weight Distribution in Part ia1 Decompression of Polyet hy le ne- Et hyIe ne Mixt ures David C. Bonner, Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409
Dennis P. Maloney, and John M. Prausnitz” Department of Chem/cal Engmeerfng. Unfvers/ty of Calfforn/a, Berkeley, Cal/fornfa 94720
Following synthesis in a high-pressure reactor, low-density polyethylene is partially separated from ethylene by decompression. The statistical mechanical theory developed by Prigogine and Flory for solutions of chain molecules is used to calculate phase equilibria in the system polyethylene-ethylene at 260” and at 200, 500, and 900 atm. Attention is given to the effect of molecular-weight distribution and to the algorithm for calculating the results.
Introduction Low-density polyethylene is synthesized in reactors at relatively high temperatures and pressures (180-300” and 500-3000 atrn). The polyethylene-ethylene mixture in the reactor may consist of one or more phases, depending upon the molecular-weight distribution of the polymer and upon the reactor temperature and pressure. At 3000 atm and 300” the mixture exists as one phase. However, if the temperature and pressure are reduced, for example, to 900 atm a t 260”, two phases are formed; one phase contains primarily polyethylene and the other phase primarily ethylene. Decompression of the reacting mixture as it leaves the reactor is necessary to separate the product and t o recycle the unreacted ethylene. We show in this work how the equation-of-state theory (Prigogine. e t al., 1953, 1957; Flory, 1965, 1970; Patterson, 1969; Bonner and Prausnitz, 1973) of chain molecule solutions may be used to calculate phase equilibria between the polymer-rich phase (e)and the ethylene-rich phase ( p ) , taking into account the molecular-weight distribution of the polymer. To illustrate the calculations, we have obtained results at the arbitrarily chosen conditions 260” and 200,500, and 900 atm.
Solution Theory The first qualitatively correct theory of chain molecule solutions was proposed independently by Huggins (1941) and by Flory (1941). However, the Flory-Huggins theory often gives poor quantitative predictions of phase equilibria a t extreme conditions or near the critical solution temperature (Koningsveld and Staverman, 1968; Siow, et al., 1972). We have therefore chosen to use the more accurate equation-of-state theory of chain molecule solutions in this work. While that theory retains the original Flory ex-
pression for the combinatorial entropy of athermal mixing, it also contains important contributions to the Gibbs energy of mixing due to differences in free volume between the two components. These contributions are neglected in the older Flory-Huggins theory. The equationof-state theory is also called the free-volume theory. Using Flory’s notation, the configurational canonical partition function (62) for a binary mixture of solvent (1) and polymer (2) is
where
=
combinatorial factor, = ( A i i m i + Azzrlzz + A ~ ~ V ~ ZA )=/ geometrical Y, packing factor, Y = volume per segment = V/Nr, Y* = hard-core volume per segment, 3Nrc = 23N,r,c, = total external degrees of freedom, 3cL = number of degrees of freedom per segment of species i, r, = number of segments per molecule of species i, M, = total number of molecules of species i, A,,, = number of intermolecular potential interactions (“contacts”) between segments of types i and j , ~ r i= lenergy ~ per i-j “contact,” k = Boltzmann’s constant, T = absolute temperature, and V = total volume. Parameters c, r, 1112, and u are mixture parameters. The Appendix gives further explanation of the parameters used in the theory. Equation 1leads to a reduced equation of state Qfreevo,
h m b
=
Flory-Huggins 3 3.Lrc
~ ( ~ 1. 3
)
-
- 5
P
-
T
E
‘
u
, -Eo
l/.i
L
-
1/3
- 1 \
-1
-
UT
where p = p / p * , Y = Y / u * , and = TIT*. Equation 2 is formally the same for pure components and for mixtures. The parameters denoted by asterisks in eq 2 are characteristic segmental parameters. For mixtures, the characInd. Eng. C h e m . , Process D e s . Develop., Vol. 13, No. 1. 1974
91
51-
o
Experirnentol Theory
- Free-Volume
- 4
300 "C
% I
0
400
I
I
800 1200 1600 P R E S S U R E , atrn
I
2000
I 2400 1.1
F i g u r e 1, Specific volume us. pressure for ethylene.
teristic parameters p * , u*, and T* are functions of composition (Flory, 1965). The characteristic parameters p * , u* (or u s p * ) , and T* can be obtained from experimental pressure-volume-temperature (PVT)values (Orwoll and Flory, 1967). For pure components, the characteristic parameters are given by the relations (with subscripts omitted) p x = s7/2u*2 (3a)
T*
=
s7/2cku*
(3b)
u* = Mu,,*/NAr (3c) where s = number of surface sites of each segment exposed to intermolecular interactions with other, nonconnected segments, u S p = specific volume, usp* = hard-core volume of 1 g of pure substance, M = molecular weight, and N4 = Avogadro's number. We have evaluated the characteristic parameters for ethylene based on data given by Benzler and Koch (1955). The characteristic parameters for polyethylene are those of Orwoll and Flory (1967) evaluated a t low pressure. To represent accurately the experimental results, the parameters are temperature dependent. Smoothed results for the characteristic parameters of ethylene are given in the Appendix. Figure 1 shows the PVT data for ethylene of Benzler and Koch (1955). The solid lines in Figure 1 are optimal fits using eq 2 with the assumption that ethylene has only three external degrees of freedom, i . e . , c = 1. The fit of ethylene data is moderately good. At pressures greater than 400 atm, the maximum error in volume predicted by eq 2 is &3%. A t pressures of the order of 200 atm, the maximum error in volume is larger; however, data at pressures below 400 atm have little effect on the values of the ethylene parameters. The ethylene parameters pl*, VI,,,* and TI* are those obtained from the optimal fit of ethylene data from I to 2400 atm. The discrepancies among reported PVT data for liquid polyethylene a t high pressures are so large that characteristic parameters obtained from such data are highly uncertain. No trend of the data with chain branching was noted; therefore we assume that volumetric properties of liquid polyethylene are not strongly dependent on branching. The low-pressure parameters of Orwoll and Flory (1967), obtained from high-density polyethylene data, were used in this study and assumed to be valid at high pressures. Figure 2 shows low- and moderate-density polyethylene PVT data compared to calculations with eq 2 using the parameters of Orwoll and Flory (1967). Within the accuracy of the density data now available, no distinction can be made between high- and low-density molten polyethylene. The chemical potential ( p ) of ethylene or polyethylene in the mixture relative to that of the pure component a t 92
Ind. Eng. Chem., Process Des. Develop., Vol. 13,No. 1. 197d
I
I
0
400
1
I
I
800 1200 1600 PRESSURE, otm
I
I
2400
2000
Figure 2. Specific volume US. pressure for polyethylene at 160". Density of solid ( p ) has units g/cc.
mixture temperature and pressure can be calculated from the canonical partition function using the relation
J+l
J+l
(4)
The ethylene activity, a1 (the standard state is taken to be pure ethylene a t system temperature and pressure), is given by the free-volume theory as
-P1-- P1° kT
+
h a , = In*, -t (I - r l / r 2 ) q 2
where X12 = p i * + ( s i / s z ) p z * - 2 [ p i * p z * ( ~ i / ~ z ) ] ~/~(l A ) , 9, = [N,rL/(N1rl+ N z r z ) ] = segment fraction = [m,u,,,*/(mluls,* + m2uzsp*)l1 0, = [N,r,s,/(Nlrlsl + N2r2s2)]= site fraction, A = fractional deviation of binary, 1-2, interaction from the geometric mean of the purecomponent interactions, and m , = mass of component i. The polyethylene activity can also be obtained from eq 4. The activity given by eq 5 differs from that of Flory (1965) by the term @/T)(Z - tl), because Flory (1965) assumed that this term is negligible at low pressure. Solutions Containing Polydisperse Polymers We now turn our attention to the case of a mixture of ethylene ( I ) and polyethylene ( 2 ) in which the polyethylene is polydisperse. When both phases are at the same temperature and pressure, phase equilibrium for phases a and p, containing ethylene and n species of polyethylene, can be determined by solution of the n + 1equations P I g = Pi0
(6) PZIU = P2,p P2nU
=
P2n
B
The second subscript in eq 6 refers to a polymer with a particular molecular weight. Equations 6 can be rewritten by subtracting the pure-component chemical potential ( P O ) from each side of the equation. Since both phases are a t high densities, we assume that for each polymer species, the contributions to the chemical potential from
internal degrees of freedom are the same in both phases (Bonner, et al., 1973). We further assume that for the densities considered here, ethylene has only three external degrees of freedom, i.e., ethylene's two rotational and one vibrational modes are not affected by density and are therefore internal degrees of freedom. This results in (PI
-
PI0)O
(A1
- P210)"
(Pa
-
= (PI
- P,V
(7a)
(PZl
-
P2l")d
(7b)
P2io)a = (P2,
-
PAOY
(7C)
=
(7d) ( P h - P2no)a = (PL" - P?,O)d The term ( - E o / h T ) in eq 1 is unchanged by polymer polydispersity since it is formulated for a random system of energetic contacts (Flory, 1965). However, the combinatorial factor in eq 1, Qcomb, is changed slightly for polydisperse systems (Flory, 1944). We can use eq 5 for the ethylene chemical potential by modifying the second term in 5 to read [l - r 1 / ( ? ~ ) ~ ] * 2 , where ( 7 ~is)the ~ number-average chain length of polymer in the phase being considered. The chemical potential difference for polymer Ppecies f of molecular weight M , is given by
where
the quantitative results are sensitive to the parameters, especially A (or X12). Calculation of P h a s e Equilibria
+
The n 1 equations (eq 7) prove to be intractable for a polymer having a broad molecular-weight distribution, because n is a very large number for such a case. We have, therefore, adopted the technique of Flory (1944) who solves the mass balance and chemical potential equations for all species simultaneously. We assume that the molecularweight distribution can be represented by a continuous, normalized mathematical function w ( M )where 0 < M < m . In principle, the molecular-weight distribution for a polymer consists of discrete values rather than a continuous function. Therefore the integrals presented in eq 11. 12. and 13 which follow should properly be summations over all n. polymer species represented by chemical potential ey 7b, 7c, and 'id. However, since it is common practice to report molecular-weight distributions as continuous functions, we have adopted integral notation. A mass balance for each polymer species of molecular weight M , (corresponding to polymer species i) results in the equations
where rnza(M,j = mass of polymer of molecular weight M , in phase a , m2(M,) = total mass of polymer of molecular weight M , = [ m p ( M , ) m&(M,j],and \ k p ( M , ) = @ 2 r a and * $ ( M , ) = g2,J. Then the total mass of polymer in phase 01 is found by integrating eq 10 from M , = 0 t o M , = m and similarly for phase d
+
= segment fraction of polymer species i and \Ir2 = Z , = l n 9 ~ , The . superscript R in eq 8 stands for "residual;" it refers to all contributions other than those from the combinatorial entropy of athermal mixing. We have assumed here that pz*, Tz*, and U Z * (or U Z , , , * ) are independent of polymer chain length and are therefore the same for all polymer species. Orwoll and Flory (1967) show that this is a good assumption provided that the polymer chain length is large. We have also assumed that the deviation parameter A is a function of temperature only and, to reduce the number of adjustable parameters, we have set r1 = 1 and sl/sz = 1. Estimates of sI/sz based on lattice models and van der Waals radii vary from 0.7 to 1.4; therefore, setting s l / s g = 1 provides a reasonable estimate. The characteristic parameters p * , T*, and uSl)* for ethylene and polyethylene a t 260" are listed in the Appendix along with the value of A at 260". The characteristic parameters and A are assumed to be independent of pressure. The Appendix also gives the parameter X z l which follows immediately from A as indicated by eq 8b. Since s1 = s2 in this study. XIZ-- X Z I . The parameter Xlz, assumed to be a linear function of temperature, was obtained by fitting the cloud-point data of Ehrlich (1965) a t 130 and 200" assuming that the polyethylene is monodisperse and of infinite molecular weight. The data of Ehrlich do not agree well with those of Steiner and Hod6 (1972). One must therefore regard the yuantitative results of this study with caution. Unfortunately, 92,
where m2 = mZa + m2P = total mass of polymer. Note that the quantity * & / 9 z a can be calculated from eq 7a. The quantity * Z ~ ( M , ) / * ~ J (can M ~be ) calculated from eq 7c. Algorithm for Solution of Simultaneous Equilibrium a n d Mass Balance A molecular-weight distribution function u(M,)is chosen and normalized so that
m2 = L m u l ( M . ) d M j
(13)
where m2 is the total weight of polymer in the system. In order to fix ideas. let us consider a typical phase equilibrium diagram for monodisperse polyethylene-ethylene as shown in Figure 3. A phase equilibrium diagram such as Figure 3 exists for each molecular weight of polyethylene present in the mixture. The pressure and temperature of interest are fixed. If the pressure is low enough, there are two phases. Simultaneous solution of the mass balance Ind. Eng. Chern., Process Des. Develop., Vol. 13, No. 1 , 1974
93
I
I
I
/Critical
I
I
point
2
1
I
1
ORIGINAL DISTRIBUTION
I
0 0
0.2 0.4 0.6 0.8 W E IG HT FR A C T ION POLY ETHYL E N E
1.0
Figure 3. Ethylene-polyethylene binary coexistence curve (constant Tand molecular weight). I
I
-
POLYETHYLENE M O L E C U L A R WEIGHT, M
Figure 5 . Molecular weight distributions in equilibrium phases (at 260" and 900 atrn).
Table I. Pressure Effect on Equilibrium Phases"
I
M,, = 13.170
M W / i J N = 10 MZ /Mw= IO MASS=/w(MI dM
01
0
I
25,000
I 50,000
I
100,ooO
75,000
POLYETHYLENE MOLECULAR WEIGHT,
-
M
Figure 4 . Log normal molecular weight distribution. and the equations of phase equilibrium for the polymer of continuous molecular-weight distribution yield the composition of each phase and the molecular-weight distribution of polyethylene in each phase. The procedure we have used is outlined as follows. (1)Choose a value of *zo(*iO = 1 - 9 2 " ) . (2) Assume ( l / i i ~=) ~ 0 in eq 5 and 7a. (3) Use eq 5 and 7a to determine 3 z a ( * l a = 1 - * z a ) . (4) Calculate mla, mza, mid, and mz@using * l a , 3 f , 910, 3213 and the characteristic parameters from the definition of (5) Calculate * z a ( M L ) / 3 2 @ ( Musing L ) eq 7c. (6) Calculate mza from eq 11 using m2a from step 4 in eq 11. ( 7 ) If r n p from step 6 is not equal to mza from step 4, return to step 1 but now use the calculated value of ( f p ) n in each phase. (?)n is equal to
*.
r l h*sp i - 4 M o d M I M 1 u l * s p ~ m M L - l ~d(M M,L ) The iteration is continued until mza from step 6 is equal to mza from step 4, within a desired tolerance. The molecular-weight distribution of polymer in each phase can then be calculated using eq 9 and 10.
Results The molecular-weight distribution shown in Figure 4, distribution w 1 of Koningsveld and Staverman (1968), was chosen to illustrate our calculations. We used an overall polyethylene composition of 12.5 wt YC at a temperature of 260" and pressures of 200, 500, and 900 atm. The results are shown in Table I. At 200 atm (a pressure similar to that used industrially in the decompression chamber upon leaving the reactor) the light phase is nearly pure ethylene plus some light polyethylene fractions (called "oils" in the industry); the heavy phase is nearly pure polyethylene with almost the original molecular-weight distribution. However, as the pressure is increased to 900 atm, the light phase contains a significant amount of polyethylene, and the heavy phase retains a significant amount of ethylene. The average molecular weight of polyethylene in the 94
Wt % of total polyethylene retained in light phase Wt % of total ethylene retained in - heavy phase -E, light phase Mn, heavy phase
Ind. Eng. Chern., Process D e s . Develop., Vol. 13, No. 1, 1974
a
200 a t m
500 atm
900 atrn
0.01
0.30
7.50
1.10 110 13,350
4.80 440 14,350
21.80 2600
19,500
At 260' and 12.5 wt 3' % polymer initially.
heavy phase also increases markedly over that of the original distribution as does the average molecular weight of polyethylene in the light phase. Figure 5 shows the molecular-weight distributions of the polymer in the original mixture and the polymer in both phases after phase splitting a t 260" and 900 atm.
Conclusion The free-volume theory of polymer solutions, coupled with a mass balance, can be used to calculate phase equilibria in ethylene-polyethylene mixtures at pressures and temperatures encountered in the manufacture of low-density polyethylene by the high-pressure process. The calculations take into account the effect of polymer dispersity but do not explicitly consider chain-branching effects. The limited experimental evidence now available indicates that chain-branching has little influence on volumetric properties (Chung, 1971) and on phase equilibria (Ehrlich, 1965) in systems containing molten polyethylene. Calculated results are sensitive to the parameters in the theory of polymer solutions. The parameters used in the calculations shown here are only estimates based on limited experimental data. To obtain better parameters, further computational and experimental work is required. Efforts toward that end are now in progress.
Appendix Mixture Parameters for the Free-Volume Theory. The mixture parameters required are p*, T*, and u * . In Flory's development (1965), one sets v* = eter c is assumed to be given by
c
+
= *IC1
VI*.
The param-
*2c2
(A-1)
The average chain length in the mixture, r, is given by
+
r = (rlN1 r2N2)/N (A-2) I V I ~ on ~ *eq ) . A-1, p* and where r2 = M Z U ~ ~ ~ , * / ( M Based T* are given by P* and
=
*12~1*
+
*,*P,*
+ 2\k1\k,pI2*
(A-3)
T*
= p*/('Pk,pl*/T,
+ ?Fl,p,*/T,*)
(A-4)
where pI2* = ( P ~ * P , * ) ~ ' ~-( ~A)
(A-5)
The reduced volume of the mixture, U , is obtained from eq 2 once the pressure, temperature, composition, and is (or X z l ) of the mixture are known. A more detailed discussion of mixture parameters is given by Bonner and Prausnitz (1973). The pure-component parameters for ethylene and polyethylene used in this study are functions of temperature and are given by the following equations
+
pz*
pl* = 5764.1 7.848T - 6.9 x 10-'T2 atm = 5124.1 - 5.257T 7.418 X 10-3T2 a t m
+
+ 3.3835T + 1.635 X K T: = 6151.3 + 9.7962T - 1.1742 X 10-*T2 K = 1.2419 + 9.74 X 10-5T + 3.76 X 10-7T2 cm3/g vzsp* = 0.9970 + 1.406 X 10-'T + 2.956 X 10-'T2 cm3/g T,*
= 2454.0
plSp*
where T is in "C. The value of 1 a t 260" is -0.05045. The value of X z l a t 260" is -96.5 atm. Acknowledgment
Society, and to Gulf Oil Chemicals Company for financial support and to the computer centers a t the University of California, Berkeley, and a t Texas Tech University for the use of their facilities. One of us (D. C. B.) also gratefully acknowledges the support of the National Science Foundation through Grant No. GK-37059. Literature Cited Benzler, H., Koch, A. V., Chem.-ing.-Tech., 27, 71 (1955). Bonner, D. C., BazQa, E. R., Prausnitz, J. M., ind. €ng. Chem., Fundam., 12,254 (1973). Bonner, D. C.. Prausnitz, J. M.. Amer. lnst. Chem. Eng. J.. 19, 943 (1973). Chung, C. I., J. Appl. Polym. Sci., 15, 1277 (1971). Ehrlich, P., J. Polym. Sci., PartA-3, 131 (1965). Flory, P. J., J. Chem. Phys., 9,660 (1941). Flory, P. J., J. Chem. Phys., 12,425 (1944). Flory, P. J., J . Amer. Chem. SOC.,87, 1833 (1965) Flory. P. J . , Discuss. faraday Soc., 49, 7 (1970). Foster, G. N. I l l . Waldman, N., Griskey. R. G., J. Appi. Poiym. Sci.. 10, 201 (1966). Hellwege, von K.-H., Knappe, H.. Lehmann, P., Kolloid-Z. 2. Poiym., 183,110 (1962). Huggins, M. L.,J. Chem. Phys.. 9,440 (1941). Koningsveld, R., Staverman, A . J.. J. Polym. Sci., Part A-2, 6 , 305 (1968). Matusoka, S., J. Polym. Sci.. 57, 569 (1962). Orwoll, R . A., Flory, P. J., J. Amer. Chem. SOC.,89, 6814 (1967) Parks, W . , Richards, R. B., Trans. faraday SOC.,45,203 (1949). Patterson, D.,Macromoiecules, 2, 672 (1969). Prigogine, I . , Trappeniers, N., Mathot, V . , Discuss. faraday SOC..15, 93 (1953). Prigogine, I., "Molecular Theory of Solutions," North-Holland Press, Amsterdam, 1957, Chapter X V I . Siow, K. S., Delmas, G.. Patterson, D . , Macromoiecules, 5, 79 (1972). Steiner, R.. Horle, K., Chem.-/ng.-Tech.,44, 1010 (1972).
The authors are grateful to the donors of the Petroleum Research Fund, administered by the American Chemical
Receiced for reuieu, J u l y 26, 1973 Accepted October 23, 1973
COMMUNlCATlONS
A Correlation Equation for Vapor Pressure-Bubblepoint Temperature Data for the Methane-Ethane System
An equation is developed to correlate the vapor ethane system between approximately 100 to 700 byshev polynomial. The average absolute error of including pure components is 0.55%. Slopes of pare well with literature values.
Vapor pressure data of many pure substances have been compiled (Jordan, 1954; Nesmeyanov, 1963) and correlation equations for narrow pressure ranges may be represented by simple equations such as the Antoine equation. Correlation equations for pure substances over a wide pressure range up to the vicinity of critical conditions are often very complex in form ( e . g , Martin, 1959; Strobridge, 1962). It is, therefore, understandable that correlation equations for mixtures t o cover a wide vapor pressure range are almost nonexistent. This work presents the results of a study to obtain such an equation for the methane-ethane system. Such an equation has the potential applications in design and operation of distillation equipment and storage tanks as well as in thermodynamic calculations (Houser and Weber, 1961).
pressure-bubblepoint temperatures for the methanepsia for the entire composition range by using a Chepredicted pressures of 200 points for 1 6 compositions ( a P / d T ) ( X I ) calculated from this equation also com-
For pure substances, Gibson (1967) used Chebyshev polynomials to correlate the vapor pressure of water and Ambrose, Counsell, and Davenport (1970) extended its usage to oxygen, nitrogen, and six organic compounds. They demonstrated the advantages of such orthogonal polynomials over other vapor-pressure equations. Equation l is their best choice with X as the normalized tem-
T log P = a ,
+
a,E,(X)
+
a,E,(X)
+
. . . + a , E , ( X ) (1)
perature between -1 and 1. Equation 1 was applied to vapor pressure-bubblepoint temperature data for mixtures of given compositions of the methane-ethane system (Bloomer, Gami. and Parent, 1953). However, the coeffiInd. Eng. Chem., P r o c e s s D e s . D e v e l o p . , Vol. 13, No. 1, 1974
95
