Relationship between Carbon-Type Composition, Viscosity-Gravity Constant, and Refractivity Intercept of Viscous Fractions of Petroleum STEWART S. KURTZ, Jr., RICHARD W. KING, WILLIAM J. STOUT', DOROTHY G. PARTIKIAN, and E. A. SKRABEK2 Sun
Oil Co., Marcus Hook and Norwood, Pa.
A triangular graph has been developed which relates carbon-type composition in terms of per cent aromatic carbons, per cent naphthenic carbons, and per cent paraffinic carbons to viscosity-gravity constant and refractivity intercept for viscous petroleum fractions. If both viscosity-gravity constant and refractivity intercept are known, carbon-type composition can be determined from this graph with reasonable accuracy. Viscosity, gravity, and refractive index are the only experimental data required for the use of this graph. It is useful in following the effect of solvent extraction, acid treating, hydrogenation of the viscous fractions of petroleum, and for comparing the composition of the viscous fractions from various crude oils.
T
WO other laboratories have been working independently
on the correlation of density, refractive index, and viscosity to provide a simple method for the analysis of petroleum fractions (5, 41). The method presented here is believed to be as accurate as the other methods, and more simple to use. The information v hich has been accumulated in recent years concerning the lubricating oil fraction of petroleum indicates that the structures of the molecules in the lubricating oil fraction are fairly uniform ( I , 2, 5-11, 14, 16, 17, 20-22, 24-27, 29-33, 36-37, 39, 40, 43-46). It seems probable that only a very small portion of the theoretically possible structures actually exist in lubricating oil. Both viscosity-gravity constant (12) and refractivity intercept (16, 18, 19) were developed with the intent of obtaining constants which were indicative of the type of hydrocarbon composition and independent of molecular weight. Therefore, it seemed possible that lines of equal viscosity-gravity constant and equal refractivity intercept could be established on a three-component graph for carbon-type composition. Carbon type composition is used in the conventional sense (30, 31, 43). Specifically, per cent aromatic ring carbon means the per cent of the total carbon atoms in aromatic ring structures; per cent naphthenic ring carbon means the per cent of the total carbon atoms in naphthenic ring structures; and per cent paraffinic chain carbon means the per cent of the total carbon atoms in chain structures, free as \\ell as combined with naphthenic and aromatic rings. Van Xes and van Westen, in their book "Aspects of the Constitution of Mineral Oils" (31), have discussed both refractivity intercept (33) and viscosity-gravity constant (32) in relation to the carbon-type composition, and presented the following approximate equations: = 0.01065
ri
V.G.C.
=
( % Ca)
+ l l ( 5 6 C A ~ O - +~ )0.0103('% ~ CN) + 0.0105 ("/c C P ) (1)
O.OllO(G C a )
+ 0.00925(% CN)0.00743(% + CP)
2
address, University of Delaware, Newark, Del. Present address, University of Wisconsin, Madison 6, Wis.
V.G.C. T7.G.C.=
\-.G.C.
=
=
10G - 1.Oi52 log ( V , - 38) 10 - log ( J r i - 38)
G - 0.21 - 0.022 log (1'2 - 3 5 . 5 ) 0.755
d - 0.1384 log ( V , - 20) 0.1526[7.14 - log (V3 - 20)] ~~
+
0,579
(3) (-1)
(5)
where
G
= = VI = V 2= V 3=
d
specific gravity 60/60° F. density a t 20" C./4" C. Saybolt viscosity a t 100' F. Saybolt viscosity a t 210' F. kinematic viscosity in centistokes a t 20" C.
These equations all give viscosity-gravity constants equivalent to the Hill and Coats equation for 100' F., which is Equation 3. Some of the carbon-type composition data were obtained by the Martin method (16, 26) of calculation and some by the n-d-M method (30, S1, 43). For aromatic extracts and other compounds rich in aromatic rings, which are outside the range recommended for the n-d-Lf method, the Martin analytical technique was used. VISCOSITY-GRAVITY CONSTANT CORRELATION
(2)
These equations were presented as somewhat tentative, and were not recommended for general use because of the uncertainty in regard to the effect of aromatic carbons on the physical properties. These authors quote Leendertse (20) in regard to the reliability of the refractivity intercept for determining the 1 Present
naphthene carbon content and the paraffin carbon content of aromatic-free lubricating oil fractions. They were impressed with its reliability for saturated mineral oil fractions. The intercept for paraffin carbons chosen by Leendertse (1.0502) and the value used here (1,0500) both correspond closely with the value for the limiting point of the paraffin homologous series (1.0498) (4). A lower value (1.0480) has also been investigated, because there is no reason to believe that the limiting CH, inO crement is necessarily the best increment to use in the C ~ to Cto molecular weight range. However, the higher value is better for data on lubricating oil fractions. This will be discussed in more detail below. The basic problem was to collect enough data for viscous oils and fractions of viscous oils so that the relation between carbontype composition, viscosity-gravity constant, and refractivity intercept could be established on an empirical basis. Carbontype composition can be obtained by the n-d-M procedure if data are available for density, refractive index, and molecular weight (31, 43). If molecular tveight is not available it can be estimated from published correlations of molecular weight with physical properties (28). D a t a were collected from the authors' own files (42) and from the literature (10, 25, S1) for approximately 258 lubricating oils and fractions of lubricating oils for which carbon-type composition, viscosity-gravity constant, and refractivity intercept were available. Viscosity-gravity constants were calculated with the following equations or the corresponding nomographs:
I n development of Equation 2 van Nes and van Westen observed ( 3 2 ) that for saturated oils a linear relationship existed between the viscosity-gravity constant (V.G.C.) and percentage carbon in paraffinic structures ('% Cp). There also seemed to be some relation between viscosity-gravity c o n s t p t and % CP for a number of oil fractions from widely differing crude sources. Their observations suggested that it might be possible to establish lines of constant viscosity-gravity constant on a triangular coordinate system by first establishing a relationship between viscosity-gravity constant and the percentage carbon in aro-
1928
V O L U M E 2 8 , NO. 1 2 , D E C E M B E R 1 9 5 6
1929
Table I.
Data Necessary for C o n s t r u c t i o n of ViscosityGravity Constant Lines" in F i g u r e s 5 and 7 Coordinate Point 1 Coordinate Point 2 V.G.C. % CA % CN % CP % CA % CN % CP 0.78 4.0 17.0 79.0 0 23.0 77.0 0.79 5.7 18.0 76.3 0 27.0 73.0 0.80 7.5 19.5 73.0 0 31.6 68.4 20 70.2 0 35.8 64.2 0.81 9.8 0.82 12.5 20 67.5 0 40.0 60.0 0.83 15.5 20 64.5 0 45.0 55.0 0.84 18.2 20 61.8 0 60.0 50.0 0.85 21.0 20 59.0 0 54.8 45.2 0.86 23.6 20 56.4 0 60.0 40.0 0.87 26.4 20 53.6 0 66.1 33.9 20 51.0 0 73.0 27.0 0.88 29.0 0.89 31.6 20 48.4 0.7 79.3 20 0.90 34.0 20 46.0 7.5 72.5 20 0.91 36.3 20 43.7 13.7 66.3 20 0.92 38.2 20 41.8 19.9 60.1 20 0.93 40.0 20 40.0 25.0 55.0 20 0.94 41.5 20 38.5 30.0 50.0 20 0.95 43.0 20 37.0 34.5 45.5 20 0.96 44.4 20 35.6 38.4 41.6 20 41.9 38.1 20 0.97 45.7 20 34.3 0.98 46.9 20 33.1 44.5 36.0 19.5 0.99 48.0 20 32.0 47.3 34.7 18.0 1.00 49.2 20 30.8 50.0 33.0 17.0 1.01 50.5 20 29.5 52.0 31.7 16.3
Table 11. Accuracy of Viscosity-Gravity C o n s t a n t Carbon-Type Correlation Source of D a t a
Lit. Ref.
No. of FracAv. tions V.G.C.
Std. Dev.
Dev. of Averagea
0 830 0 926 0.845 0,813 0.858
0.004 0 004 0.004 0,004 0.006
0.000
+O.OOQ
-0.002 - 0 002 -0,004 + O 006
-0,006 -0.007 -0,006 +0.010 +0.010 -0.007
Max. Dev.
Whole oils and cuts thereof Water White Oils Borneo Crude OklahomaCrude Pennsylvania Crude Webster Crude Distillation fractions from Naphthenic Crudes P a . 180 S e u t r a l Midcontinent Neutral I California Neutral Gulf Coast Keutral Rodessa Pieutral Midcontinent Seutral 2
($6)
11
(91)
(91)
5 5
(91) (SI)
4 4
(Sf) (10)
21 20
0.883 0.819
0.004 0.002
0 000 0.000
17 17 15
0.860 0 892 0.879 0.830
0.003 0.002 0.003 0.002
+O 001 0.000 0.000 - 0 002
005 -0 005 $0.006 -0 004
+ O 006
(10) (10) (IO) (IO)
18
+O
19
0.849
0.004
+ O 001
156
0 858
0.003
0.000
(86) (48)
12 3
0.800 0.874
0.002 0.007
0,000 +O 006
+ O 004 + O 011
(42)
21
0.836
0.004
+0.003
+0.009
36
0.837
0.004
t 0 001
+0.011
(42)
3
0.802
0.008
-0
-0
(48) (48)
3
0.956
4
0.966
0.008 0.014
-0 007 - 0 014
-0,010 - 0 015
(48)
2
0.945
0.014
-0
014
- 0 015
(48)
6
0.9G9
0.005
-0
001
-0
010
1
32
0.948
0 009
- 0 006
-0
015
224
0.005
-0 001
-0
015
Coordinate points given are terminal points of the V.G.C. lines and define area of composition over which correlation has been tested. a
t
t
1 40
44
40
52
56
60
64
68
72
76
PER CENT Cp
F i g u r e 1.
Viscosity-gravity constant cs. saturated oils
7'
C p for
All whole oils Saturated oils Water White Oils Vebster Crude Distillation fractions froni Naphthenic Crudes All saturated oils Aromatic extracts Aromatic conc. froni solvent refined lube b Aromatic cone. from light lubes b Aromatic extracts Aromatic conc. from Webster Crudeb Aromatic conc. from Duosol and furfuralextractionsb Aromatic conc. of
(10)
-411 aromatic extracts Complete total
.92
z
z
4
L 0 ~
.90 .88
c
>
.86
a
i-
010
010
a Deviation from average of each d a t u m averaged, taking sign into account. b Concentrated on silica gel; a t least one aromatic ring per molecule.
.94
r
00;
+O
.84
v)
0
:-: . 8 2 >
.80
Viscosity-gravity constant plots for w h o l e o i l s
1930
ANALYTICAL CHEMISTRY
matic, naphthenic, or paraffinic structures on rectangular coordinates. For saturated oils, the linear relationship betvieen viscositygravity constant and % C p observed by van Xes and van Kesten (32) was confirmed, as shown in Figure 1. Because theae oils contained no carbon in aromatic structures, it was possible to transfer this line to the saturated base line of a triangular coThis gives a series of ordinate graph for % C P , % C.V, and lo CP-% CS side of the viscosity-gravity constant points on triangle. I n order to establish a second point for the viscosity-gravity constant lines, it was necessary to plot tn-o of the composition variables as a function of viscosity-gravity constant for the series of whole oils. Per cent C P and % C A gave the best approxiniations to a linear relationship when so plotted (Figures 2 and 3,.
I'ISk
B
.90
t
A plot of 70 C.V showed considerably greater scattering, and is presented in Figure 4 merely to show that a line constructed from % C A )fits the Figures 2 and 3 so that 70C.V = 100 -(70C p data reasonably well. It was possible t o fit the data for C P and % C A n-ith a straight line up to viscosity-gravity constant values of about 0.885. Above 0.885 the effect of the carbons in aromatic structures became large, and it was necessary to change the slope abruptly in order to fit the data in this region. These data mere transferred to triangular coordinates and points of equal viscosity-gravity constant connected to give a series of lines covering the viscosity-gravity constant range from 0.790 to 0.910 I n the range from 0.910 to 1.00 the lines were placed by inspection to give the best fit to the data. Adjustments to the correlation mere made until the average deviation was a minimum and the deviation of the average was close to zero. The viscosity-gravity constant lines on Figure 5 are the final adjusted lines. Table I gives the data necessary to lay out the lines for viscosity-gravity constant on a three-component graph. For the present, a t least, the lines are being restricted to that area of the graph for which supporting experimental data exist. Table I1 shows the degree of agreement obtained for 224 fractions between the viscosity-gravity constant determined experimentally and that derived from composition by using a chart corres onding to Figure 5 . The agreement between experimental and cafculated values, while not perfect, is satisfactory. It indicates that the reverse computation-that is, relating viscosity-gravity constant to composition-should be satisfactory, provided anothei function, also independent of molecular weight, could be plotted on the graph in order to obtain intersecting lines.
+
REFRACTIVITY INTERCEPT CORREL4TION
I.80
I I I I I I I I I I I I I 0
IO
20
30
40
50
60
70
80
PER CENT C N
Figure 4.
Viscosity-gravity constant whole oils
TS.
YC C.v
for
It seemed reasonable to use refractivity intercept as the other plotted function, provided the effect of aromatic carbons could be evaluated on a reasonable basis. As the aromatic content of lubricating oil fractions increases, there is evidence that the proportion of condensed aromatic ring structures increases. Because the composition data for the 224 fractions that Rere available to the authors formed a fairly narrow band on the triangular diagram of carbon-type composition, it seemed reasonable to anticipate that a fairly smooth curve should be obtained if the equivalent refractivity intercept of the aromatic carbons were plotted against the observed refractivity intercept of the sample. Therefore, the equivalent refractivity interiept of the aromatic carbons was calculated using the follon-ing equation: r,A =
% AROMATIC RING CARBONS
% NAPHTHENE RING CARBONS
Figure 5 ,
r,sample(100) - 1.0300(7,C\) - l . O 5 O O ( % C p ) -
% PARAFFIN CHAIN CARBONS
Viscosity-gravity constant in relation to carbon-type composition
n
Figure 6 shows the calculated refractivity intercepts for aromatic ring carbons plotted against the refractivity intercept of the sample for those samples which had 5% or more of aromatic ring carbons. Although some scattering is observed, the anticipated trend is clear. Because the available data did not include points rich in naphthene ring carbons and having low intercept values, some assumption was needed to establish tentative intercept lines in the more naphthenic portion of the composition chart. -4s there cannot be less than one aromatic ring per molecule and as it was not believed necessary to extend the chart below 1.035 refractivity intercept, the intercept value of benzene (1.0616) was plotted opposite the value 1.035 for refractivity intercept to provide a tentative terminal point for this curve. This was a rather arbitrary step which xould only be justified if the chart ultimately developed was reliable. T h e refractivity intercept lines on Figure 7 were established by interpolating betxeen 1.0300 and 1,0500 on the naphthene and paraffin side of the triangle and between 1.0300 and r,A on the naphthene and aromatic side of the triangle, and 1.0500 and riA on the paraffin and aromatic side of the triangle. r,A is determined in each case using Figure 6 or Table 111. The data necessary for establishing the intercept lines in Figure i are given in Table IV. Study of the oints in Figure 6 suggests the possibility that &ere should be a sharp downward curvature of the line for the calculated aromatic carbon intercept betneen the values 1.045
1931
V O L U M E 28, NO. 12, D E C E M B E R 1 9 5 6
;tnd for API 42 compounds (2, 39, 40) would indicate that a paraffin intercept value of 1.0480 would be better in the range from 20 to 40 carbon atoms than a value of 1.0500. The corresponding naphthene value is 1.0320. A complete graph was constructed and evaluated. A statistical study showed little difference between this graph and Figure 7 . However, when compared by plotting data for a series of hydrogenated oils, the graph based on the paraffin intercept value of 1.0480 and a naphthene intercept value of. 1.0320 showed the completely hydrogenated naphthene fractions, falling about 2y0 below the base line representing 0% aromatic carbons. The graph based on 100% CP = 1.0500 and 1007, CN = 1.0300 showed the completely hydrogenated fractions falling on the base line (Figure 8 ) For this reason it was decided to adhere to values of 1.0500 and 1.0300 for the paraffin and naphthene apexes, respectively, In constructing this type of graph some thought needs to be given to what is meant by per cent composition. The data for per cent carbon type arp in terms of percentage of aromatic, naphthenic, and paraffinic carbons in the molecule (16, SI j. Refractivity intercept is, ingeneral, additive on a volume per cent basis for mixtures of liqui s. As shown by Leendertse (2Oj, there is good agreement between carbon-ty e composition and refractivity intercept for complex saturate8molecules. The graph in Figure 7 , as derived, represents carbon-type composition and not volume per cent composition or weight per cent compositiorl. This should be kept in mind if it is applied to blends.
'Table 111. Data Needed for Drawing Curve of Equivalent Intercept of Aromatic Carbons os. Intercept of Lube Oil a s Shown in Figure 6 Intercept Intercept of of Aromatic Oil Carbons 1 0616 1.0350 1 ,0695 1.0400 1 0775 1.0450 1 0855 1.0500 1 0926 1,0550 1.0986 1.0600 1.1042 1 ,0650 1.1094 1 ,0700 1.1140 1.0750 1.1186 1.0800
Data for Establishing Refractivity Intercept Lines" NaphtheneNaplitheneParaffinItefracParaffin Aromatic* Aromatic Base Line Side of Triangle Side of Triangle tivity % CP % CS 5% CA 7%C.V yo CA 7% C P Intercept 25 75 15 8 84 2 1 035 74.7 50 25.3 i 040 50 68.4 25 31.6 75 1.045 63.9 36.1 0 1.050 100 60 1 88.3 39 9 11.7 1.055 56 3 79.5 20 5 43 7 1 060 72.3 52 8 27 7 47 2 1 065 po 4 49 6 33 7 66 3 1 070 39 1 46 4 93 6 BO 9 1 n7.5 i &I 56.4 43 4 43.7 56 3 In constructing Figure i values for intermediate lines can he 011tained by plotting columns 2 to 7 us. column 1 and interpolating. h working graph should have lines for each 2% of carbon-type (TJYIposition, for each unit in the third decimal place of the refractirit?. intercept, and for each unit in the second decimal of viscoeit!.-gravit!. constant.
Table IV.
~~
APPLICATIO\
The carbon-type composition of the lubricating oils and frwtions of lubricating oils used in developing Figure 7 , as well as :L number of other oils on n hich data were available, was determined by plotting the viscosity-gravity constant and refractivity intercept for all these points on a large graph and reading off the corresponding composition. This study showed that the aromatic. carbon contents are usually in agreement with the authors' best analyses to within 1 or 2%, although occasionally deviation3 as large as 4 or 5% may be observed. On the paraffin and naphthene compositions, the agreement is good for fractions containing 30% or less of aromatic carbons. Between 30% and 50% aromatic carbons, there is an increase of uncertainty in the values for the paraffin and naphthene carbons, because the angle at 11-hich the correlation lines intersect decreases rapidly. Thp chart should not be used for samples having viscosity-gravity constant values between 0.95 and 1.01 which fall below 20% C,vor 20% CP. This is practical because very few samples having less than 20% CNor 2O%Cp have been found in this range. Table V presents data on a number of samples for which rather complete analytical data were obtained in connection with studies
and 1.050 for the intercept of the oliginal oil. This was tried, but led to an irregular spacing of the intercept lines on the triangular diagram. The lines so placed did not agree well with the limited available data for this region. Therefore, as explained above, the coordinatee 1.0616 and 1.0350 were tried as the terminal point of the curve. This led to more reasonable spacing of the intercept lines and better agreement with the data. Considerable thought has been given to the significance of pure compound data in relation to locating the intercept lines on Figure 7 . Consideration of data from the A4PI44 tables ( 3 , 35)
I
I In
I
I
I
I
I
I
1.12
m
a 4:
g gI
1.11
1.10
a LL
..-
0
1.09
n w
tj
1.08
i8
1.07
1.030
1.040
1.050
1.060
1.070
1.080
r i OF SAMPLE
Figure 6.
Refractivity intercept of aromatic carbons us. refractivity intercept of sample
ANALYTICAL CHEMISTRY
1932 of rubber processing oils (16, 48). I n general, the agreement between composition obtained with Figure i and the composition obtained by more complete analysis is satisfactory-that is, within 1 or 2%. -4few deviations larger than 4% are shonn. Table V also contains data obtained by the n-d-bI procedure of Tadema, van Xes, and associates (31, 43). I n the range for which the n-d-11 method is recommended-that is, for samples in which the ratio of % C A to yo Cy is 1.5 or less-the agreement is quite good. I n the higher aromatic range the data obtained by Figure 7 are in better agreement with detailed analyses b j the Martin (16, 26) method than are the data by the n-d-11 method . Table VI compares carbon-type analyses obtained x ith Figuie i and with the n-d-?\I procedure on the cuts of five tlpical oils studied by Hill and Ferris (13). The data on these oils xere used by Hill and Coats (12) in deriving the viscosity-gravity constant, The t a o methods agree xell for these typical oils. The range of carbon-type composition is about 20% for % C.4 and % C Vand about 40% for % C p . To prepare Table VI data weie read from photostatic enlargements of the curves published by Hill and Ferris. Data were corrected to 20' C. when necessary. A statistical study of all the data (Table YII) shows that, for oils having a viscosity-gravity constant of 0.900 or less, Figure i nil1 give the value of yo C A n ith a standard deviation of ahout
1.0%. For % C.V and % C p the standard deviation is roughly 1.5% for samples m-ith a viscosity-gravity constant below 0.900. Most naturally occurring crude oils have viscosity-gravity constant values below 0.900. For samples with viscosity-gravity constants between 0.900 and 1.000, the standard deviation for % C A is approximately 2.0% and for % C,y and % C p , approximately 4.5%. The samples having viscosity-gravity constant above 0.900 are, for the most part, aromatic concentrates obtained either by solvent extraction or gel separation. These samples are outside the range of materials for x hich the viscosity-gravity constant was originally developed. Table 1-111presents corresponding data arranged b j groups based on per cent carbon in aromatic rings. For samples having less than 30% carbon in aromatic rings the standard deviation is as follons: per cent aromatic carbons, 1.2%; per cent naphthenic carbons, 2 . 0 % ; per cent paraffinic carbons, 1.7%. The data used in developing this chart represent a Fide variety of samples. Consideration of the data indicates that Figure 7 is sufficiently reliable for obtaining many of the carbon-type composition data ahich are needed in practical petroleum refining. The relation betn-een refractivity intercept and composition, and beta een viscosity-gravity constant and composition is more accurately presented by Figure 7 than hy Equations 1 and 2
% AROMATIC RING CARBONS
.os00 LRBONS
V O L U M E 2 8 , NO. 1 2 , D E C E M B E R 1 9 5 6
1933
Table V. Comparison of Carbon-Type Analysis by Viscosity-Gravity Constant-Refractivity Intercept and Other Methods 011
No. 1
Mol. wt. V.G.C
.
T%
n
d 2
Mol. nt. V.G.C. 1L
n d
3
A I O l , n.t. T.G.C. ri n
d 4
1101. a t . T.G.C. ri ri
d 5
698 0.798 1.0448
1.4865 0,8835
T.G.C. rt
d 7
>roi.art. V.G.C. T l
n d
8
3101. wi.
1.. G.C. rt 12
d
9
JIol. rvt. T.G.C. ri n
d 10
IIOl. wt.
V.G.C. rt
n d 11
Mol. ai. V.G.C. 1%
d 12
\r01.nt. V.G.C. rt 71
d 13
Mol. wt. 1'. G .C. rl n d
14
Ll01.nt. T.G.C. IL
n
d 1.5
16
x-z
352 0.818 1.0440 1.4748 0.8613
Fig. 7 x n-d-XI u Martin z
464 0,842 1.0419 1.4971 0.9105
Fig. 7 z n-d-XI y Martin z
x-2
2-y 2-2
FIE. 7
2-2/
z-z
442 0 892 1 ,0332 1.5352 0.9600
Fig. 7 x n-d-XI y Martin z 2-Y
365 0,915 1.0501 1.5291 0,9580
Fig. 7 s n-d-11 y Martin z
427 0 927 1.0631 1 5550 0 9837
Fig. 7 x n-d-ll y Martin z
369 0.93fj1 ,066.2
Fig, 7 n-d-115 hIartin
x-z
x-2
x-y 2-2
z-y 2-2
x y z x-y
0.9783
2'-2
373 0.936 1 0039 1.5537 0.9797
Fig. 7 x n-d-Ma y Martin z
373 0,943 1.0654 1.5649 0,9990
Fig. 7 x n-d-MQ y LIortin z
361 0.970 1.0735
Fig. 7 x n-d-Ma y Martin z
x-y x-z
s-y x-z
1,5804
z-y
1.0138
5-2
Fig. 7 x n-d-Ma u Nartin z 2-y 2-2
Fig. 7 z n-d-lIe y Martin z x-y
l f o l . wt. V.G.C.
Fig. 7 x n-d-3Ia y
n
d
310 0 997 1.0734 1.5872 1.0236
x-z
Martin
z x-y s-2
0
70 69 70 + I 0
3 3 3
33 33 35
64 64
0 0
0 0
3 3 3
34 34 34
0 0
0 0
4
45 43 45 0 0
50 0 + I
4
64 0 0
63 63 63 0 0 51
-
2 3 3 1 1
49 47 47 + 7
49 50 50 - 1
+ 2
- 1
20 19 16 1 4
38 39 42 - 1
42 42 42 0 0
25 24 25 1 0
34 35 30
i - 4
41 41 45 0 - 4
45 43 45 + ? 0
3
