Applying the Theory Of Corresponding States to Multicomponent Systems
THOMAS w. LELAND, jr., and WILLIAM H. MUELLER
I
Department of Chemical Engineering, The Rice Institute, Houston, Tex.
Pseudo-critical values from theoretical concepts are possible
ALTHOUGH the theory of coriesponding states as originally stated by van der Waals in 1881 applies only to pure substances, more recent work (5. 20, 27) has provided a theoretical basis for extending it to multicomponent systems. A method suggested by this theoretical approach has been investigated for calculating hypothetical or pseudo-critical conditions for both liquid and vapor phases. These values are then used to calculate reduced conditions a t tvhich therniody-namic properties of mixtures may be found from generalized correlations for pure substances. or from either a n equation of state or tabulated properties of a single closely related pure substance chosen as a reference. This approach does not require a knowledge of equation of state constants and their combination rules for all components. I t permits the use of different equations of state for liquid and vapor phases or any number of different equations of state each fitted to various rrgions of the observed P-V-T properties of the reference substance chosen. K a y (8) defined the pseudo criticals for a mixture of n components as:
Suppose that for each pure component the corresponding states theorem is applicable. Pitzer (75) has shown that pure substances which obey the corresponding states principle are those for which: molecular energies can be treated classically with negligible quantum effects; the rotational and internal energy states of each molecule are independent of molecular positions so that the internal energy changes for the assembly may be derived from the cumulative effects of molecular translation and intermolecular potential; and the intermolecular potential between two molecules depends only on the distance between them and can be expressed as: LYr) =
E
lf(./.)l
(5)
where E and cr are t\io parameters \vith dimensions of energy and length. The Lennard-Jones porential ( 7 7 ) meets this requirement, although any two-parameter potential like Equation 5 ma)- be used The Lennard-Jones equation : U(r)
-46
[ ( u / Y ) ~- (u/Y)”]
(6)
form of Equation 5. If a and t\io potential parameters,
f:
=
v=o
Collecting terms in Equation 10 gives:
- Z Z X ~ X ~ E : ~ U+: fi(~ ] T ) x - Z Z x i x , E y c r , ] + . . . . = 0 (11)
fo( T )[ti3g1/4 1531314
For a given mixture, the terms in the brackets in Equation 11 are functions only of its composition. Functionsf, X ( T ) are such that Equation 11 is not a homogeneous function of the temperature since each f Y ( T ) function generated by the Lennard-Jones potential has the forin Cv[ T
-(q2)]. If the corresponding
states principle is to hold over a range (of temperatures for a fixed composition, each term in brackets in Equation 11 must equal 0, which gives these equations from the second virial coefficient: n
n
n
=
X,(Pc)i
are its
2,+1
fy(T)&
is used as an illustration. The second virial coefficient for a purc substance is
n
PC’
e
n
(2)
i=l
Other empirical definitions of pseudo criticals have been proposed by Joffe (7). These methods are effective for predicting the behavior of nonpolar gaseous mixtures but become more in error for saturated vapors and liquids.
Using Lennard-Jones potential in Equation 7 gives for a pure substance (6) : B
=
fa( T ) U 3 € ” 4
+ fl(
2v+l
fy(T)u3e
Derivation of Pseudo-Critical Properties
The problem of defining pseudocritical values for a multicomponent phase is analogous to the problem of Lvriting the virial coefficients for a solution in the virial coefficients for the pure components. The second virial coefficient for a n n-component mixture (70) is related to the pure component virials by :
Third virials are expressed :
+ .+ + . ... (8)
T)U3€34
If Equation 8 is used to \\rite the second virial for each component in mixture in accordance with Equation 3, n
n
m
2”+ 1
If the pseudo-critical concept is valid, then for each constant composition of the mixture there must exist a hypothetical pure substance with criticals Pc‘> Vc’, and T,’ which has thermodynamic properties identical with those of the mixture a t the same P, V , and T conditions. If the corresponding states principle is to be upheld, this pure substance must also have a potential function in the
Other sets may be xvritten by applying this same technique to third and higher virials. Howehrer, as any t\vo of these equations \vi11 determine a and a, no single pair of E and z values can satisfy all of them. Because 3 and e are functions of critical properties, there are no rigorous pseudo-critical conditions for a multicomponent system even though the corresponding states principle applies to each individual pure component. Suppose two of the equations in the second virial set for rhe best form of the two-parameter potential function in Equation 5 are used to define a and 1: n
n
n
n
VOL. 51, NO. 4
APRIL 1959
597
All the other equations in the second virial set will have the form:
From the equation of state a and b may be evaluated in terms of the criticals : b
cc
(21 1
V, a u a
a a V,T, a u 3 c
(22)
Substituting Equations 21 and 22 in Equations 19 and 20 and solving simultaneously gives :
in the third virial set, the form:
+e
defining the pseudo-critical temperature and pressure. The terms in the summations in Equations 15 and 16 when i # j are based on approximations to the interaction constants between unlike molecules : Considering only dispersion forces betu een nonpolar molecules :
# 0
= 1/(€"u3)L(e"u"),
(€%3)tI
(33)
(14)
and so forth for all virial sets. E and i are determined by Equations 12 and 13 in which a and 0 are ideally the exponents in two equations of the second virial set for the best two-parameter potential function causing minimum values in residuals &,, +€, etc. Simultaneous solution of Equations 12 and 13 gives:
With all of these substitutions, Equations 15 and 16 become, for /3 = 0:
(24) i=l j=1
which shows that the values of in Equations 15 and 16 are:
CY
and
/3
(35)
T,'
a = l
p = o
L=l j = 1
This same procedure applied to a recent two-constant equation of state P,' = (2) gives: ra = 5/3
J
n
p = o
a and /3 must be determined experimentally. A good idea of their range can be obtained from two-constant equations of state. Both the Dieterici and van der Waals equations have second virial coefficients: B
=
b - a/RT
(17)
The pseudo constants for the mixture evaluated from Equation 3 are :
22
XLX,azj
(18)
In this work the value zcro and the value of empirically by using Equations 15 and 16 to predict experimentally measured compressibility factors in saturated vapors and liquids. The resulting values of are correlated empirically in Table I as a func-
LT r
tion of
n x , ( ~ , ) i
i=l
n
n
n
n
Table 1. (Y Values in Equations for Pseudo-Critical Properties Function
a
Function
a
50.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100 1.200
2.20 2.06 1.98 1.91 1.84 1.76 1.69 1.61 1.54
1.300 1.400 1.500 1.600 1.700 1.800 1.900 j2.000
1.47 1.39 1.32 1.24 1.17 1.09 1.02 1.00
598
n
7
xi(
T.iil.
i=l
The empirically evaluated a values are seen to fall in the range predicted by Equations 25 and 27. The pseudo criticals for an n-component phase can now be written from Equations 15 and 16 setting 0 = 0 and substituting : E a T, (29) us oc
temperatures a t a given composition,
Ipc I
,=I
If Equation 18 is to hold over a range of
t =
v,
(30)
This procedure defines a pseudo-critical volume and temperature. However, because of the awkwardness in dealing with systems described by reduced temperatures and volumes and the uncertainty in the critical volume data, it is convenient to change the pseudo-critical volume variable to a pseudo-critical pressure by defining an empirical pseudocritical compressibility factor for the mixture as &'=
2
XI(ZC)E
(31 1
i=l
The term V, in Equation 30 is replaced by z,T,!P, so that:
Substituting Equations 32 and 29 in Equations 15 and 16 produces relations
INDUSTRIAL AND ENGINEERING CHEMISTRY
1
The Reference Substance
Using the corresponding states principle, thermodynamic properties of the mixture can be equated to the known properties of a pure reference substance a t reduced conditions equal to the pseudo reduced conditions of the mixture. According to the law of corresponding states as originally stated, there exists a single function: VB
=
f( T R ?P R )
(37)
bvhich holds for all fluids. As all actuaI fluids fail to conform to the assumutions inherent in Equation 37, the choice of the reference substance is very important. .4 number of authors have investigated the addition of a third parameter to the right side of Equation 37: VR = ~ ( T R P R, . w )
(38)
where w has been defined in various ways (3, 73, 74, 76). The reference substance chosen should have the same value of w as the mixture. In applying corresponding states to mixtures, the ideal w is one which is related to molecular properties and appears in a known intermolecular potential function. A pseudo W can be evaluated for the mixture like the evaluation of Z and 2. Possibilities for this type of w function are Kihara's molecular eccentricity factors (6, 9),or some function of Corner's molecular lengthwidth ratios (4, 6). I n the usual definition of w as z.,
M U L T I C O M P O N E N T SYSTEMS
Table
II.
Calculated Compressibility Factors
Error, yo ___ Molal ~~
Mole Press., Fraction Temp., P.S. CompoF. I.A. nent 1 System
ci-cs
CI-ncI
C:-?lcj
Cl-nClo
c1-c3
.CI-nC?
Cl-nCs
ZCslod
.
av.
This This critiZ E ~ , , ~ ~Methodmethod . cal
Binary Paraffin Hydrocarbon Systems' 0.192 800 L 0.2418 0.195 100 0.711 V 0.6321 0.707 1000 L 0.3271 0.251 0.245 100 V 0.6635 0.677 0.670 0.306 1200 L 0.4226 0.320 100 V 0.6779 0.625 0.633 500 L 0.0433 0.134 0.132 160 V 0.1550 0.660 0.613 1000 L 0.2800 0.336 0.321 160 V 0.3558 0.453 0.453 100 L 0.0175 0.0274 0.0278 100 V 0.4596 0.896 0.924 130 1000 L 0.2933 0.247 0.250 V 0.8270 0.782 0.804 1800 L 0.5683 0.492 160 0.480 V 0.6440 0.559 0.553 500 L 0.1004 0.132 0.135 190 V 0.5580 0.751 0.807 190 1000 L 0.2521 0.259 0.258 V 0.6718 0.714 0.743 1500 L 0.4263 0.400 190 0.397 V 0.6517 0.636 0.638 500 L 0.0783 0.136 220 0.135 V 0.4200 0.693 0.760 220 1000 L 0.2356 0.272 0.269 V 0.5630 0.658 0.679 220 1500 L 0.4376 0.456 0.505 V 0.4987 0.520 0.543 250 500 L 0.0530 0.145 0.141 V 0.2650 0.660 0.686 1000 L 0.2192 0.303 250 0.289 V 0.4327 0.565 0.596 160 2000 L 0.5460 0.494 0.478 V 0.8558 0.751 0.765 1500 L 0.4076 0.455 0.438 280 V 0.6010 0.662 0.651 5000 L 0.8064 1.105 100 1.074 0.960 V 0.9514 0.971 160 2000 L 0.4234 0.682 0.672 V 0.9963 0.896 0.904 220 2000 L 0.4091 0.661 0.655 V 0.9921 0.932 0.937 220 4000 L 0.6782 0.985 0.968 V 0.9657 0.951 0.955 280 2000 L 0.4028 0.644 0.639 V 0.9845 0.952 0.956 280 4000 L 0.6912 0.960 0.939 V 0.9430 0.973 0.968 340 4000 L 0.7358 0.944 0.922 V 0.8823 0.969 0.977 400 2000 L 0.4119 0.630 0.617 V 0.9351 0.952 0.967 460 1000 L 0.2378 0.362 0.369 V 0.8803 0.947 0.942 460 2000 L 0.4280 0.636 0.620 V 0.8815 0.932 0.941 460 2500 L 0.5379 0.746 0.714 V 0.8534 0.920 0.947 500 L 0.1235 0.124 0.122 100 V 0.5209 0.762 0.764 100 500 L 0.1556 0.130 0.133 V 0.8473 0.860 0.844 100 800 L 0.2450 0.200 0.204 V 0.8774 0.826 0.843 500 L 0.1201 0.130 0.134 160 V 0.6796 0.796 0.847 100 200 L 0.0625 0.0613 0.0612 V 0.8940 0.955 0.957 100 800 L 0.2508 0.222 0.222 V 0.9460 0.883 0.880 100 2000 L 0.5788 0.481 0.471 V0.9204 0.741 0.742 160 200 L 0.0480 0.0595 0.0595 V0.7568 0.935 0.940
2.6 5.7 3.6 5.0 7.9 7.9 0.8 3.0 14.3 0.0 13.9 1.4 1.8 3.1 4.6 1.2 4.1 2.8 7.1 2.4 6.3 1.0 7.9 4.6 2.3 7.4 12.7 4.6 0.4 4.1 9.8 0.7 8.0 0.3 9.5 1.5 0.7 9.6 16.2 1.1 5.5 3.2 13.1 10.7 11.4 4.4 10.4 2.7 0.6 3.9 10.9 4.6 19.8 5.5 19.5 3.2 1.8 2.0 6.3 3.7 7.7 8.9 1.7 2.8 7.4 1.1 0.6 1.5 11.6 0.9 1.1 0.9 10.3 1.0 0.5 1.8 7.5 0.4 1.8 0.6 9.3 0.4 1.3 2.2 5.2 0.5 1.3 2.3 0.3 0.9 5.1 1.6 6.2 1.9 4.0 1.9 1.9 0.5 3.3 2.5 1.3 1.0 5.5 4.3 0.8 2.9 6.5 1.6 0.3 2.3 2.8 2.0 2.1 3.1 6.4 ... 0.2 0.2 1.5 0.6 2.4 1 .o 4.4 1.3 1.5 7.1 4.5
- Error, % Molal Mole Press., Fraction ZCslod. av. Temp., P.S. CompoThis This criti.System F. I.A. nent 1 Z E ~ ~ ~Method I . method cal 3.0 160 2000 L 0.5460 0.494 0.479 1.6 ... V 0.8558 0.753 0.765 6.4 0.365 340 1000 L 0.2569 0.390 5.7 0.481 V 0.3364 0.510 2.8 Ci-nCio 100 2000 L 0.4469 0.706 0.686 v 0.9981 0.842 0.849 0.8 2.6 1.024 160 5000 L 0.8240 1.052 0.2 V 0.9364 0.989 0.991 Binary Paraffin Hydrocarbons Systems a t t h e Critical of the System ct-c, 160 1020 0.7665 0.392 0.382 2.5 ... Ci-nCa 100 1912 0.3602 0.528 0.520 3.3 . . . 1.5 ... 0.5211 0.531 0.523 160 1810 2.4 ... 0.7389 0.448 0.437 250 1264 1.5 ... 0.3228 0.658 0.648 Ci-nC5 160 2338 2.3 ... 0.7236 0.571 0.558 280 1610 1.019 2.2 . . . 0.8979 1.042 Cl-nCto 100 5310 1.007 2.1 0.8912 1.019 160 5180 0.833 4.3 . .. 0.6165 0.871 460 2911
... ... ... ... ...
... ...
...
Systems CcBntaining Polar Compounds and Components Other Than Paraffin Hydrocarbons" N2-nGb 175 2111 LO.1870 0.669 0.688 2.8 ... V 0.9882 1.009 1.030 2.0 ... 175 10025 L 0.7050 1.920 1.858 3.2 ... V0.9412 1.692 1.547 8.4 260 3022 L 0.2470 0.885 0.883 0.3 V0.98691.106 1.083 2.1 260 8005 L 0.7230 1.451 1.474 1.6 . .. V0.8340 1.463 1.420 2.9 CI-HzS 100 1000 LO.0828 0.130 0.125 3.8 ... V0.4707 0.694 0.675 2.7 160 1500 L 0.1245 0.234 0.209 10.2 ... V 0.2775 0.440 0.466 5.9 HzS-nCs 40 150 LO.8210 0.0249 0.0238 4.4 ... V 0.9930 0.867 0.884 1.9 ... 220 500 L 0.3838 0.121 0.113 6.6 ... V 0.7745 0.774 0.728 5.9 220 1000 L 0.7859 0.183 0.162 11.5 V0.9125 0.658 0.509 22.6 . . . HIS-nCiu 160 600 L 0.8107 0.116 0.108 6.9 . . . V0.9985 0.811 0.743 8.4 220 600 L 0.5681 0.163 0.151 7.4 V0.9938 0.822 0.818 0.5 220 1000 L0.8308 0.186 0.170 8.6 ... V0.9960 0.685 0.658 3.9 Propene- 100 200 LO.8560 0.0470 0.0462 1.7 I-buV 0.9380 0.778 0.776 0.3 tene 220 650 L0.7680 0.226 0.203 10.2 V0.8040 0.423 0.398 5.3 . Ca-ben220 500 L 0.7805 0.124 0.121 2.4 . .. zene V (No experimen- 0.619 .. tal value) 340 200 L0.0558 0.0416 0.0425 2.2 ... V (No experimen- 0.830 .. . tal value) co2-c3 40 200 L 0.1602 0.0479 0.0475 0.9 ... V0.6036 0.836 0.820 1.9 Crude Oil-Satural Gas System,e Critical Properties of C,+ Fraction Equated t o Critical Properties of nCla
... ... ... . .. ...
...
... ...
... ... ... ... ... ... ... ..
... . ...
...
... ... ...
I
0.0
0.4 2.1 0.1 0.0
0.5
1047
...
L 0.572
0.618 8.4 V 1.077 0.922 14.5 I1 200 8075 .. L 1.955 1.811 7.4 V 1.488 1.437 3.4 I11 160 5150 .,. L 1.383 1.543 10.4 V0.974 1.004 3.1 IV 160 7965 ... L 1.998 2.072 3.7 V 1.372 1.395 1.7 V 160 6678 L 1.919 1.882 1.9 V 1.178 1.198 1.7 VI 160 9694 ... L 2.322 2.413 3.9 V 1.862 1.779 4.4 VI1 160 3200 L 1.174 1.146 2.4 V0.795 0.806 1.4 VI11 160 5040 L 1.495 1.587 6.1 V 7.958 0.978 2.1 D a t a for binary systems (18,19). (1). C ( 1 7 ) .
.
... ... ... ...
... ... ... ... ... ... ... ...
200
...
... ...
a
VOL. 51, NO. 4
APRIL 1959
. .. ...
... ... ...
... ... ... ... ... . ,. ... .. , ... ...
. ..
599
there is no way to predict the proper pseudo value z,‘ for the mixture. The method of defining ze‘ is not important in calculating P,‘ by Equation 36 because numerical values of z, vary only slightly. If z,’ is used to select the reference substance, the method of defining z,’ is very important because slight numerical changes in z,’ may cause a wide variation in the reference substance chosen. The use of z, as a third parameter allows the generalized tabulation of Lydersen, Greenkorn, and Hougen (72) to be used for the reference properties. It is planned to investigate several possible third parameters for selecting the reference. The results of using z,’ as defined by Equation 31 are reported in this work. When z, is chosen as the third parameter, it cannot be equated directly to w in Equation 38. Equation 38 must be replaced by a series of functions each applying at a different constant z,. Equation 38 becomes [ V R = fl( TR,P R ) I r c l (39)
I VR
= f2( T R j PR)lr,z
Because no valid equation of state can exist in which VR is a function of TR?PR, and z,, where z, is a variable. Calculated Results
T h e pseudo-critical values defined here were tested for a series of saturated vapors and liquids. A reference substance was chosen as the normal paraffin hydrocarbon between CHI and nC,HI6 having a value of z, nearest to the value of zc’ for the mixture as defined by Equation 31. The P-V-T properties of each reference substance were expressed in terms of the Benedict-Webb-Rubin equation for the hydrocarbon. The calculations were carried out on the Bendix G-15 and Royal hlcBee LGP-30 computers in the following sequence:
and Po using the Benedict-Webb-Rubin equation. This value is the compressibility factor of the mixture a t T and P. The results of applying this method to a series of saturated liquids and vapors are shown in Table 11. The over-all average absolute per cent error is 2.3y0 for systems of hydrocarbons and nonpolar compounds. This is the same order of magnitude as the average errors (72) in applying corresponding states to determine compressibilities of pure components. I n Table I1 the method of this work is compared with Kay’s method of defining T,’ and P,’ as the molal average of the pure component criticals. In only 8 out of 58 different phases in Table I1 the method of molal average criticals gave slightly better results. In all other cases the method proposed here pioduced errors from one half to one tenth of the errors in using molal average criticals. For the hydrocarbon systems the method worked well in the retrograde region and even a t the actual critical of the system. The highest errors were for systems containing polar compounds. The chief difficulty in application to a natural gas-crude oil system is that in many cases greater than 50y0 of the liquid phase was described only as C7i and there is no accurate way to estimate its pseudo critical mithout further analysis of the fraction. However, if the C 7 + fraction can be characterized. surprisingly good results are obtained. Acknowledgment
The authors acknowledge the helpful suggestions of Z. W. Salsburg of T h e Rice Institute and thank C. R . Hocott of the Houston Research Center, Humble Oil and Refining Co., for making available the Bendix G-15 computer. Nomenclature
--
a, b = constants in the equation of state
P
xd ( i- = 1-
Tc)i from a given phase comJ
position and selection of the value of a nearest to one of the values in Table I. 2. Calculation of T,’ and P,’ from Equations 35 and 36. 3. Calculation of TR’ = TIT,’ and P,’ = P/P,’ from the given temperature and pressure, T and P. 4. Selection of a pure reference substance with a zc value most nearly equal to zc’* 5 . Calculation of the equivalent temperature To = T R I P c and the equivalent pressure Po = PR’Pco, where T,o and Pco are the criticals of the pure reference substance. 6 . Calculation of the compressibility for the pure reference substance a t TO
600
for a pure substance which has the same properties as the mixture when each are a t the same P-V- T conditions a, b = constants in a two-constant equation of state B = second virial coefficient C = third virial coefficient f = function k = Boltzmann’s constant = number of components in a phase n N = Avogadro’s number P = pressure r = intermolecular distance T = temperature U = intermolecular potential function V = volume x = mole fraction z = compressibility factor a , p, 7 , = exponents in virial expansion e, = parameters in the intermolecular potential function E a, = parameters in the intermolecular
INDUSTRIAL AND ENGINEERING CHEMISTRY
Z:
6
= = =
w
=
?r
potential function for a hypothetical pure substance which has the same properties as the mixture when each are a t the same P-V-T conditions summation 3.1416 residual function measuring the deviation of a term in the actual virial expansion for a mixture from the term predicted by pseudo properties third parameter in the law of corresponding states function
SUBSCRIPTS = critical property i , j , k = components i, j , k M = mixture property R = reduced property C
SUPERSCRIPTS
’
0
of a mixture, a property of a hypothetical pure substance which has the same properties as the mixture when each are a t the same P-V- T conditions = property of a pure reference substance = pseudo property
literature Cited (1) Akers, W. W., .4ttwell, L. L., Robinson, J. A,, IND.ENG.CHEM.46, 2536 (1954). (2) Benson, S. W., Golding, R. A., J.Chem. Phys. 19, 1413 (1951). (3) Bloomer, 0. T., Peck, R. E., A.1.Ch.E. Meeting, Louisville (1955). (4) Corner, J., Proc. Roy. SOG. (London) A192, 275 (1948). (5) Guggenheim, E. A., “Mixtures,” pp. 154-65, Clarendon Press, Oxford, 1952. (6) Hirschfelder, J. O., Curtis, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” pp. 162-3, 131--3, Wiley, New York, 1954. (7) Joffe, J.: IND. ENG. CHEM.39, 837 (1947). (8) Kay, W. B., Zbid., 1014 (1936); 33, 590 (1941). (9) Kihara, T., J . Phys. Soc. Japan 6 , 289 (1951). ( I d ) Kilpatrick, J. E., J . Chem. Phys. 21, 274 (1953). (11) Lennard-Jones, J. E., Proc. Roy. Soc. (London) A106, 463 (1924). (12) Lydersen, A. L., Greenkorn, R. A.,
Hougen, 0. A., University of Wisconsin Engineering Experiment Station, Madison, Wis., Rept. 4, 1955. (13) Meissner, H. P., Seferian, R., Chem. Eng. Progr. 47, 579 (1954). (14) Pitzer, K. S., IND.ENG. CHEM.50, 265 (1958). (15) Pitzer. K. S., J . Chem. Phys. 7, 583 . (1939). ’ (16) Riedel, L., Chem. Ingr. Tech. 26, 83, 259, 679 (1954). (17) Roland. C. H.. IND.ENG. CHEM. ‘ 37, 930 (1945). ‘ (18) Sage, B. H., Lacey, W. N., “Thermodynamic Properties of the Lighter Par-
affin Hydrocarbons and Nitrogen,” American Petroleum Institute, New York, API Research Project 37, 1950. (19) Sage, B. H., Lacey, W. N., API Research Project 37, American Petroleum Institute, New York, 1955. (20) Salsburg, Z. W., Kirkwood, 3. G., J . Chem. Phys. 21, 2169 (1953). (21) Scott, R. L., Zbid.,25, 193 (1956).
RECEIVED for review July 7, 1958 ACCEPTED November 10, 1958
