a generalized thermodynamic correlation based on three-parameter corresponding states
DESCRIPTION
The volumetric and thermodynamic functions correlated by Pitzer andco-workers analytically represented with improved accuracy by a modifiedBWR equation of state. The representation provides a smooth transitionBYUNG IK LEEandMICHAEL G. KESLERbetween-the original tables of Pkzer et al. aAd more recent extensions tolower temperatures. It is in a form particularly convenient for computer use.TRANSCRIPT
-
feed to a typical coal processing fluidized bed (Skinner, 1970) ; fine particles in the distribution have a considerable effect on the calculation. Similarly, it is not clear at present exactly how dp should be specified in applying the correla- tion for jet penetration depth to a bed with a wide range of particle sizes.
ACKNOWLEDGMENT
This work was carried out at the Research Laboratories of Westinghouse Electric Corporation and was partially sponsored by the Office of Coal Research. The author wishes to thank Dr. D. L. Keairns for the encouragement and advice he has offered during many useful discussions.
NOTATION
& = jet nozzle diameter c& = mean particle diameter Db = initial bubble diameter
gC L Lo uo = jet nozzle velocity Vo = initial bubble volume y, yo Greek Letters 8 = jet half-angle p j = density of fluid pp = density of solids
= acceleration due to gavity = volumetric gas flow rate = jet penetration depth measured from nozzle = jet penetration depth measured from apex of cone
= length of conical section of jet (yJL = 0.55) = distance between apex of cone and jet nozzle
LITERATURE CITED
Basov, V. A., V. I. Markhevka, T. Kh. Melik-Akhnazarov, and D. I. Orochko, Investigation of the Structure of a Non-uni- form Fluidized Bed, Int. Chem. Eng., 9, 263 ( 1969).
Behie, L. A., and P. Kehoe, Grid Region in a Fluidized Bed Reactor, AIChE J . , 19, 1070 ( 1973).
Davidson, J. F. and D. Harrison, Fluidized Particles, Cam- bridge Univ. Press, England ( 1963 ).
Grace, J. R., Viscosity of Fluidized Beds, Can. J. Chem. Eng., 48,30 ( 1970).
Harrison, D., and L. S . Leung, Bubble Formation at an Orifice in a Fluidized Bed, Trans. Inst. Chem. Engrs., 39, 409 ( 1961 ).
Kunii, D., and 0. Levenspiel, Fluidization Engineering, Wiley, New York 1969).
Markhevka, V. I., V. A. Basov, T. Kh. Melik-Akhnazarov, and D. I. Orochko, The Flow of a Gas Jet into a Fluidized Bed, Theor. Found. Chem. Eng., 5, 80 (1971).
Merry, J. M. D., Penetration ot a Horizontal Gas Jet into a Fluidized Bed, Trans. Inst. Chem. Engrs., 49, 189 (1971).
Rowe, P. Iu., Experimental Properties of Bubbles, in J. F. Davidson and D. Harrison (Eds.), Fluidization, p. 139, Aca- demic Press, hew York (1971).
Shakhova, N. A., Outflow of Turbulent Jets into a Fluidized Bed, Inzh. Fiz. Zh., 14,61 ( 1968).
Skinner, D. G., The Fluidized Combustion of Coal, National Coal Board, London ( 1970).
Yang, W. C., and D. L. Keairns, Recirculating Fluidized-Bed Reactor Data Utilizing a Two-Dimensional Cold Model, AIChE Symp. Ser. No. 141, 70,27 (1970).
Pirie, (ed.), Fluidization, p. 136, Inst. Chem. Eng., London (1968). Manuscript received December 1 1 , 1974; recision received January
Zenz, F. A., Bubble Formation and Grid Design, in J. 11
28 and accepted Januaty 29, 1975.
A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States
The volumetric and thermodynamic functions correlated by Pitzer and co-workers analytically represented with improved accuracy by a modified BWR equation of state. The representation provides a smooth transition
BYUNG IK LEE and
MICHAEL G. KESLER between-the original tables of Pkzer et al. aAd more recent extensions to lower temperatures. I t is in a form particularly convenient for computer use.
Mobi, Research and Development Corporation Post Office Box 1026
Princeton, New Jersey 08540
SCOPE The 3-parameter corresponding states principle as pro-
posed by Pitzer and co-workers has been widely used to correlate the volumetric and thermodynamic properties needed for process design. The original correlations by Pitzer et al., based on that principle, were limited to reduced temperatures above 0.8. Several extensions to lower temperatures have appeared in the last five years. Most of these correlations are in tabular or graphical form, difficult to implement on the computer. Also, significant discrepancies appear at the interface (near T, = 0.8)
between the original and extended correlations. The objective of this work was to develop an analytical
correlation, based on the 3-parameter corresponding states principle and covering the whole range of T , and P, of practical interest in hydrocarbon processing. Another ob- jeclive was to improve the accuracy and consistency of the published correlations. This has been achieved by means of two equations of state, similar in form to that of Benedict, Webb, and Rubin, for the simple and refer- ence fluids.
Page 510 May, 1975 AlChE Journal (Vol. 21, No. 3)
-
CONCLUSIONS AND SIGNIFICANCE This paper describes a method of analytically represent-
ing volumetric and thermodynamic functions based on Pitzer's 3-parameter corresponding states principle. The functions include: densities, enthalpy departures, entropy departures, fugacity coefficients, isobaric and isochoric heat capacity departures, and the second virial coefficients.
Two equations of state, for the simple and reference fluids, accurately represent the volumetric and thermo- dynamic properties of vapor and liquid as a function of 3-parameters (reduced temperature, reduced pressure, and acentric factor) over the range of T, = 0.3 to 4 and P, = 0 to 10. This analytical form has led to improved repre- sentation of these properties near the critical region and at low temperatures. It has also provided a smooth transi- tion between the original correlations of Pitzer et al. and
more recent extensions by others to low temperatures. The method has been found to be reliable over a wide
range of conditions for nonpolar and slightly polar sub- stances and their mixtures. Its accuracy, like that of the original Pitzer correlations, is best in the subcooled and superheated regions. The accuracy is diminished, although to a lesser extent than in the Pitzer correlations, at saturated conditions and near the critical and retrograde regions when applied to widely boiling mixtures. Included in these developments is a new reduced vapor pressure equation and a means of estimating acentric factors from that equation, as well as a set of mixing rules for calculat- ing pseudo critical properties and acentric factors of mixtures.
In a series of papers, Pitzer and co-workers (Pitzer et al., 1955; Pitzer and Curl, 1957; Curl and Pitzer, 1958) demonstrated that the compressibility factor and other derived thermodynamic functions can be adequately rep- resented, at constant reduced temperature and pressure, by a linear function of the acentric factor. In particular, the compressibility factor of a fluid whose acentric factor is o, is given by the following equation:
where Z(O) is the compressibility factor of a simple fluid and Z ( l ) represents the deviation of the compressibility factor of the real fluid from 2'0). 2'0) and 2") are as- sumed functions of the reduced temperature and pressure.
Using this approach, Pitzer and co-workers correlated volumetric and thermodynamic properties in tabular form over the range of reduced temperatures, 0.8-4.0, and reduced pressures, 0-9.0. More recently a number of authors (Chao et a]., 1971; Chao and Greenkorn, 1971; Carruth and Kobayashi, 1972; Lu et al., 1973; Hsi and Lu, 1974) have extended the correlations to lower tem- peratures.
The correlations of Pitzer and co-workers have been used extensively to calculate compressibility factors and enthalpies of nonpolar substances and their mixtures. This approach, however, has proven inadequate when calcula- tions are made:
1. In the Critical region; 2. For liquids at low temperatures; 3. At the interface of the original correlations and the
4. For widely boiling mixtures, particularly those con- corresponding extensions;
taining high concentrations of very light and very heavy components. This difficulty is closely related to problems in defining the pseudo critical properties of the mixture.
One of the objectives of this work was to improve the correlations in the above areas. More generally, the ob- jective has been to provide a practical analytical frame- work for representing the volumetric and thermodynamic functions in terms of the three parameters of the corre- sponding states principle developed by Pitzer and co- workers.
To facilitate analytical representation, the compressi- bility factor of any fluid has been expressed in this work in terms of the compressibility factor of a simple fluid 2'0) and the compressibility factor of a reference fluid Z r ) , as follows:
When Z is expressed as in Equation (2), the deviation term 2") in Equation (1) is obviously equivalent to (2") - Z(O))/O(~). This expression is convenient since, as will be shown later, both 2(,) and Z'O) are given by the same equation with, however, different constants. A similar approach is used to represent analytically other derived thermodynamic functions, such as fugacity and the de- partures of enthalpy, entropy, and isobaric and isochoric heat capacities from the ideal gas state.
n-Octane has been chosen as the heavy reference fluid since it is the heaviest hydrocarbon for which there are accurate P-V-T and enthalpy data over a wide range of conditions. However, the final values of Zr) and of the corresponding equation of state constants were readjusted
TABLE 1. CONSTANTS FOR EQUATION ( 3 )
Constant Simple fluids
0.1181 193 0.265728 0.154790 0.030323 0.0236744 0.0186984
AIChE Journal (Vol. 21, No.
Reference fluids Constant
0.2026579 c3 0.331511 c4 0.027655 di x 104 0.203488 d2 x 104 0.0313385 B 0.0503618 Y
3)
Simple fluids
0.0 0.042724 0.155488 0.623689 0.65392 0.060167
May, 1975
Reference fluid
0.016901 0.041577 0.48736 0.0740336 1.226 0.03754
Page 511
-
to fit better the compressibility factors and the derived thermodynamic properties of other substances, in addition to those of octane.
Briefly, the work consisted of the following:
1. Modification of Benedict et al. equation of state ( 1940) as represented by Equation (3) .
2. Fitting the constants in Equation (3) using experi- mental data on enthalpy, P-V-T, and the second virial coefficient.
3. Derivation of a new vapor pressure equation and its use to derive an equation for estimating acentric factors. 4. Use of a new set of mixing rules to define critical
temperatures and pressures and acentric factors of mix- tures.
These steps are described below in further detail.
DESCRIPTION OF WORK The compressibility factors of both the simple fluid Z(O)
and the reference fluid Z(,) have been represented by the following reduced form of a modified BWR equation of state:
In determining the constants in these equations, the fol- lowing constraints, Equations (7) and ( 8 ) , were used along with the data shown in Tables 3 and 4.
fv = f L (at saturated condition) (7)
(>,, av, = (T) av, Tr = 0 (at critical point) (8 ) Additional care was exercised to obtain a smooth function of [Z() - Z(O)] around the loci of minimum compressibil- ity factors between P, = 1 and 2.
The data used to determine the constants in the equa- tion of state for the simple fluid were principally of Ar, Kr, and methane. Since the triple points of simple fluids are above T , = 0.5, additional data on light hydrocarbons were used to represent the low temperature region. This was done using Equation (2) along with the correspond- ing expression for other thermodynamic properties, and Equation (3 ) with the constants for ZC7) determined from n-octane data. However, as alluded to earlier, these con- stants were adjusted to obtain the best fit to all the data, as referenced in Tables 3 and 4. The constants are given in Table 1.
TABLE 2. COMPARISON OF REDUCED VAPOR PRESSURE
TI
1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.80 0.78 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30
-Log P,(O) Carruth-
This work Pitzer et al.
0.000 0.051 0.103 0.157 0.213 0.270 0.329 0.390 0.454 0.521 0.591 0.665 0.742 0.823 0.909 1.000 1.096 1.199 1.308 1.425 1.549 1.683 1.827 1.982 2.150 2.332 2.530 2.746 2.983 3.244 3.532 3.852 4.209 4.609 5.062 5.578
0.0 0.050 0.102 0.156 0.212 0.270 0.330 0.391 0.455 0.522 0.592 0.665 0.742 0.823 0.909 1.000 1.096 1.198 1.308 1.426 1.552 1.688 1.834 - - - - - - - - - - - - -
Kobayashi
- - - - - - - - - - - - - - -
1.000 1.096 1.198 1.308 1.424 1.544 1.680 1.818 1.965 2.130 2.315 2.515 2.730 2.970 3.240 3.540 3.870 4.220 4.600 5.005 5.450
Hsi-Lu
- - - - - - - - - -
0.591 0.664 0.740 0.823 0.908 1.000 1.099 1.199 1.307 1.420 1.543 1.679 1.823 1.982 2.149 2.333 - - - - - - - - - -
This work
0.000 0.042 0.086 0.132 0.179 0.229 0.282 0.339 0.400 0.466 0.537 0.615 0.699 0.790 0.890 1.000 1.120 1.252 1.397 1.557 1.734 1.929 2.144 2.383 2.648 2.944 3.273 3.642 4.056 4.521 5.047 5.644 8.323 7.102 8.000 9.042
-Log P r ( l ) Carruth-
Pitzer et al. Kobayashi Hsi-Lu
0.0 0.042 0.086 0.133 0.180 0.230 0.285 0.345 0.405 0.475 0.545 0.620 0.705 0.800 0.895 1.000 1.120 1.25 1.39 1.54 1.70 1.88 2.08 - - - - - - - - - - - - -
- - - - - - - - - - - - - - -
1.000 1.120 1.250 1.390 1.545 1.710 1.908 2.120 2.370 2.660 2.960 3.310 3.695 4.100 4.540 5.010 5.560 6.240 7.080 8.300 9.940
- - - - - - - - - -
0.542 0.620 0.709 0.789 0.880 1.000 1.078 1.206 1.343 1.518 1.688 1.867 2.061 2.273 2.526 2.826 - - - - - - - - - -
AlChE Journal (Vol. 21, No. 3)
-
To calculate 2 for the fluid of interest, given a T and P, first calculate the appropriate values of T,(= T / T , ) and P,(= P/P,) using critical properties of the fluid. From the simple fluid constants in Table 1 and Equation ( 3 ) , solve for V,. (This is not the correct reduced volume for the fluid of interest, but rather, it is dehed as P,V/RT, with V a simple fluid volume.) When V , is employed in the first equality of Equation (3), Z0) is calculated for the simple fluid. The next step is identical to the first except that the reference fluid constants of Table 1 are used, but with the same T , and P, values of the fluid of interest that were determined in the first step. The result of the second step is 27,). Finally, with Z0) from the first step and Z(+) from the second, the compressibility factor Z for the fluid of interest is determined from Equa- tion ( 2 ) where a(') = 0.3978.
The thermodynamic functions derived from Equation
a. Fugacity coefficient ( 3 ) follow:
B C D l n ( + ) = z - l - l n ( z ) + - + - - - ; - + 7 v, 2 v , SV, + E
E = - { c4 B + l - ( B + I
(9) where
2 ~ ~ 3 ~
b. Enthalpy departure
b2 + 2b3/T, + 3b4/Tr2 H - H e = T , { Z - 1 - RTc T,V,
2T,VT2 5TrVT5 d2 + 3 E } (11) +- ~2 - 3c3/Tr2 - c. Entropy departure
TABLE 3. COMPARISON OF CALCULATED COMPRESSIBILITY FACTORS WITH LITERATURE DATA
Systems
Subcooled liquid and dense phase
Methane' ldutene Neopentane" 1-Pentene n-Ocehe' n-Octaneo n-Nonane n-Decane' n-Dodecane n-Heptadecane Cyclohexane Benzene H2S Ethane-n-pentane Propane-n-decane n-Decane-n-tetra-
n-Dodecane-n-hexa-
Hydrogen-n-hexane Methane-n-Butane-
decane
decane
n-Decane
Superheated-vapor
Methane' Neopentane" H2S
Saturated liquid 1-Butene n-Nonane Benzene H2S Ethane-n-pentane Hydrogen-n-hexane
Saturated vapor
1-Butene H2S Ethane-n-pentane Hy drogen-n-hexane
No. of points
21 16 18 17 12 9 13 9 9 15 14 17 12 13 16
10
9 15
20
16 24 u)
6 5 7 7 5 15
6 7 5 15
T,
0.79-1.44 0.74-1.06 1 .O-1.15 0.76-1.07 0.66-0.96 0.46-0.70 0.52-0.86 0.41-0.72 0.41-0.72 0.44-0.78 0.56-0.92 0.55-0.91 0.74-0.92 0.7-1.28 0.53-0.86
0.45-0.55
0.43-0.52 0.64-1.56
0.53-1.60
1.09-1.17 0.8-1.15 0.74-1.19
0.78-1.0 0.63-0.86 0.55-0.91 0.76-1.0 0.7-1.03 0.59-1.56
0.78-1.0 0.76-1 .O 0.7-1.03 2.8-8.7
P T
0.32-10 0.34-8.5 1.05-9.5 0.16-8.9 0.20-8.2 0.04
0.05 0.06
0.6-15
0.0-23 0.34-8.5 0.28-7.0 0.16-7.7 0.34-8.5 0.1-7.0
0.06-7.6
0.06-8.7 3.8-21
0.9-29
1.3-2.0 0.05-0.76 0.01-7.7
0.17-1.0 0.01-0.3 0-0.51 0.15-1 .O 0.3-1.57 4.4-21
0.17-1.0 0.15-1 .O 0.02-0.75 6.1-31.4
Avg. abs. dev., % References
1.06 1.06 1.17 2.22 1.87 1.26 1.09 0.99 1.14 2.82 0.43 3.15 1.12 0.86 1.03
2.84
3.46 0.95
1.84
0.3 0.44 0.41
1.55 1.36 3.15 1.96 0.97 1.68
1.51 4.12 2.52 1.33
Vennix et al. (1970) Sage and Lacey (1955) Dawson et al. (1974) Day and Felsing ( 1951 ) Felsing and Watson (1942) Chappelow et al. 1971 1 Sage and Lacey Rossini et al. Rossini et al. Doolittle Reamer and Sage Sage and Lacey Sage and Lacey Reamer et al. Reamer and Sage
Snyder et al.
Snyder et al. Reamer and Sage
Sage and Lacey
Vennix et al. Dawson et al. Sage and Lacey
Sage and Lacey Sage and Lacey Sage and Lacey Sage and Lacey Reamer et al. Reamer and Sage
Sage and Lacey Sage and Lacey Reamer et al. Reamer and Sage
(1955 j ( 1953) ( 1953) (1964) (1957) (1955) (1955) (1960) (19.66)
( 1974 1
(1974) (1957)
(19%)
(1970) (1974) (1955)
(1955) (1955) (1955) (1955) (1960) (1957)
(1955) (19%) (1960) (1957)
O Data used in correlational work.
AlChE Journal (Vol. 21, No. 3) May, 1975 Page 513
-
TABLE 4. COMPARISON OF CALCULATED ENTMLPIES WITH LITERATURE DATA
System
Methane' Ethane Propane' n-Pentane' n-Octane' n-Hexadecane ' cis-2-pentene C yclohexane Benzene Methane-propane
Ethane-propane
ra-Pentane-n-octane
n-pentane-cyclohexane
n-pentane-n-hexadecane
Benzene-n-hexadecane
49.4 % -50.6%
49.8 %-50.2%
59.7%-40.3%
61.2%-38.8%
58.7%-41.3%
58.1 %-41.9 %
* Data used in correlational work.
No. of points
40 70 42 66 50 30 25 36 37
44
28
13
19
26
18
Tr
0.61-1.49 0.4-1.35 0.32-1.07 0.63- 1.3 0.52-1.03 0.41-0.85 0.95-1.05 0.76-1.12 0.83-1.15
0.4-1.36
0.82-1.14
0.58-1.04
0.83-1.06
0.55-0.98
0.74-0.87
where P o = 1 atm. for API Research Project 44 S' data. d. Isochoric heat capacity departure
--- 3c3 6E (13) Cv - Cu* - 2 ( b ~ + 3b4/Tr) - R T2V, Tr3VrZ
e. Isobaric heat capacity departure
- 1 - T r ( ",',, (5) av, Tr c, - C,' c, - C,' - - R R aTr
(14) where
bi + b3/Tr2 + 2b4/Tr3 V,
c1 - 2c3/T2 dl 2 v 2 5vr5
+ +-
2B 3C 6D +-
TO calculate any of the quantities given in Equations (9) through (16), given a T and P for the fluid of interest, the following procedure, illustrated by the en- thalpy departure function, should be followed.
Page 514 May, 1975
p ,
0.37-3.0 0.28-2.83 0.81-3.25 0.41-2.87 0.55-3.88 0.19-6.8 0.38-2.64 0.17-2.37 0.28-1.97
0.37-3.0
0.38-3.02
0.46-3.25
0.38-2.66
0.08-4.22
0.31-4.4
Avg. abs. dev. J / K g
2600 2300 2100 3000 2600 4000 5600 2800 3300
6100
1600
2600
2600
7400
4600
References
Jones et al. (1963) Starling (1972) Yesavage (1968) Lenoir et al. (1970) Lenoir et al. ( 1970) Lenoir and Hipkin ( 1970) Lenoir et al. (1971) Lenoir et al. (1971) Lenoir et al. ( 1971 )
Yesavage (1968)
Starling ( 1972)
Lenoir et al. (1970)
Lenoir et al. (1971)
Lenoir and Hipkin ( 1970)
Hayworth et al. ( 1971 )
1. As described earlier, determine V, and Z(O) for the simple fluid at the T, and P, appropriate for the fluid of interest. Employ Equation (11) and, with the simple fluid constants in Table 1, calculate ( H - H')/RT,. Call this [ ( H - H*)/RT,](O). In this calculation, 2 in Equation (11) is 2'0).
2. Repeat step 1, using the same T , and P,, but em- ploying the reference fluid constants from Table 1. A new V, and Z C r ) will be obtained. With these, Equation (11) allows the calculation of [ ( H - H')/RTc](r). In this calculation, 2 in Equation (11) is 22,).
3. Determine the enthalpy departure function for the fluid of interest from
[ ( H - H")/RTc] = [ ( H - H')/RT,]'O' + (ads")) { [ H - H')/RTc](r) - [ ( H - H')/RT,]''"}
where U J ( ~ ) = 0.3978. The tabulated values generated by this procedure are shown in Tables 5 through 14.
As with any other correlation based on the correspond- ing states principle, the success of this method depends on the accuracy of the values of the critical temperatures, critical pressures, and acentric factors. For mos't com- pounds the critical properties have been fairly well estab- lished, but values of the acentric factor have not been satisfactorily determined until recently. Passut and Danner (1973) have reported accurate values for a large number of hydrocarbons. However, some of their reported values (for example, 1-pentene, diolefins, and acetylenes) do not appear to be consistent. This is likely the result of the use of inaccurate data for the vapor pressure or critical properties for these compounds. In addition, Passut and Danner's tabulations do not include a number of compounds encountered in hydrocarbon processing. The above considerations have prompted the development of a new reduced vapor pressure equation and the use of the equation for estimating the acentric factor from the normal boiling point and critical properties of the fluid of interest. The vapor pressure equation is based on that of Riedel (1954) and on the experimental data of Prydz
AlChE Journal (vol. 211 No. 3)
-
b c T
AB
LE
5. V
ALUE
^ O
F Z
(0)
PR
- m
1 m500
0.4335
0.3901
0 35
63
0 32
94
0.3077
0 28
99
0.2753
0 26
34
0 m2538
0 m2464
0.2411
0.2382
0 23
83
0.2405
0 9 243
2
0 m2474
0 m2503
0 25
38
0 e2583
0.2640
0.2715
0.3131
0.4589
0.5798
0.6605
0 76
24
0 m8256
0 e8689
0.9000
0 92
34
TH
0.30
0.
35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.93
0.95
0.010
0 00
29
0.0026
0.0024
0 . 002
2 0.0021
0 m9804
0.9849
0.9881
0.9904
0 e9922
0 99
35
0 m9946
0 99
54
0 .9959
0.9961
0.05
0
0.0145
0.0130
0.0119
0.0110
010103
Om00
98
0.0093
0.9377
0.9504
0.9598
0.9669
0.9725
0.9768
0.9790
0.9803
0.100
0.200
0 0579
0 05
22
0 0477
0.0442
0.0413
0.0390
0.0371
0 0356
0.0344
0 0336
0 a8539
0.8810
0.9015
0.9115
0.9174
0.9227
0 a9253
0.9277
0.9300
0.9322
0.400
0.1158
0.1043
0 09
53
0 . 088
2 0 08
25
0.0778
0.0741
0.0710
0 0687
0 0670
0.0661
0.0661
0.7800
0.8059
0 a8206
0.600
0.1737
0 1564
0 1429
0.1322
0 1236
0.1166
0.1109
Om1063
0.1027
0.1001
0 0985
0.0983
0.1006
0 m6635
0.6967
0.7240
0.7360
0.7471
0.7574
0.7671
0.7761
0.8002
0 A323
0 m8S76
0 e8779
0.80
0
0.2315
0 e2084
0.1904
0.1762
0.1647
0.1553
0.1476
0.1415
0.1366
0.1330
0.1307
0.1301
0.1321
0.1359
0.1410
1 moo
0 1.200
0.3470
0.3123
0 m2853
0 m2638
0 24
65
0 a2323
0.2207
0.2113
0.2038
0.1981
0.1942
0.1924
0.1935
6.1963
0.1998
0.2055
0.2097
0.2154
0.2237
0 e2370
0 m2629
0 a4437
0 m5984
0.6803
0 73
63
2.000
0,5775
0.5195
0.4744
0.4384
0 m4092
0 m3853
0 36
57
0 34
95
0 m3364
0.3260
0.3182
0.3132
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0.06
7 0.
088
0.10
9 0
.04
3
0.05
6 0.
069
0.02
6 0.
033
0.04
1
0.18
5 0.
246
0.30
7
0.86
0 0.
716
0.61
1 0.
531
0.46
6
1.11
6 0.
915
0.77
4 0.
667
0.58
3
1 .56
0 1 a
253
1.04
6 0.
894
0177
7
2.27
4 1.
857
1.54
9 1.
318
1 13
9
2 9 8
25
2 48
6 2.
175
1 .90
4 1 m
672
2 94
2 2.
679
2.42
1 2.
177
1 .95
3
1.75
1 1.
571
1 .41
1 1
143
0.92
9
2 89
9 2.
692
2 48
6 2.
285
2.09
1
1 *9
08
1 .
736
1.57
7 1 *
295
1 a0
58
1 .80
1.
90
2.00
2.
20
2.40
2.60
2.
80
3.00
3
.50
4
.00
0.00
3 0
.00
3
0.0
03
0 . 002
0.
002
0 03
4 0.
031
0.02
8 0
023
0.01
9
0.13
7 0.
123
0.11
1 0
092
0 07
6
0.41
3 0.
368
0.33
0 0
269
0.22
2
0.51
5 0.
451)
0.
411
0.33
4 0.
275
0.68
3 0.
606
0.54
1 0.
437
0 35
9
0.99
6 0
.88
0
0 78
2 0.
629
0.51
3
1 a47
6 1 .
309
1.16
7 0.
937
0.76
1
rg
Y
rn
W
to (0 m
0.00
2 0.
001
0.00
1 0.
001
0.0
00
0.01
6 0.
014
0.01
1 0.
007
0.00
5
0 06
4 0
.05
4
0 0
45
0.
029
0.01
7
00
18
5
0.15
4 0.
129
0.08
1 0
04
8
0.22
8 0.
190
0.15
9 0
099
0 05
8
0.29
7 0
246
0.20
5 0.
127
0.07
2
0.42
2 0
348
0.28
8 0.
174
0.09
5
0.62
1 0.
508
0.61
5 0.
239
0.11
6
0 75
6 0.
614
0.49
5 0.
270
0.11
0
0.8
58
0.
689
0.5
45
0.
264
0.06
1 2 Y
-
TR
0.3
0
0.35
0.40
0.45
0.50
0.55
0.60
0 065
0.70
0.75
0.80
0.85
0.90
0.93
0.95
0.97
0.98
0.99
1.00
1.01
1.02
1.05
1.10
1.15
1.20
1.30
1.40
1.50
1.60
1.70
i.eo
1.90
2.00
2.20
2.40
2.60
2.80
3.00
3.50
4.00
0.010
11.098
10.656
10.121
9.515
8086
8
0.080
0 05
9 0 04
5 0 03
4 0.027
0.021
0.017
0.014
0.012
0.011
0.010
0.010
0.009
0.009
0.009
0.008
0.007
0.006
0.005
0.004
0.0
03
0.002
0.001
0 . 000
0.
000
-0.000
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.002
0.05
0
11 00
96
10.655
10.121
9.515
8.869
8.211
7.568
0.247
O.le5
0.142
0.110
0.087
0.070
0.061
0 05
6
0 05
2 0.
050
0 048
0 . 046
0
044
0.042
0.037
0.03
0 0
025
0.020
0.013
0.00
8 0.005
0.002
0.000
-0.001
-0
003
-0.0
03
-0.0
05
-0.0
06
-0.006
-0.007
-0.007
-0.008
0.100
11.095
10.654
10.121
9.516
8.870
8.212
7.570
6.949
0.415
0.306
0.234
0.182
0 14
4 0.126
0.115
0.105
0.101
0.097
0 09
3 0 08
9
0 085
0.075
0.061
0.050
0.040
0 02
6 0.016
0.009
0.004
0.000
-0.0
03
-0
.005
-0.007
-0.010
-0.012
-0,013
-0.014
-0.014
-0.016
0.200
11.091
10.653
10.120
9.517
8.872
8.215
7.573
6.952
6.360
5 796
0.542
0.401
0.308
0.265
0.241
0.219
0.209
0.200
0.191
0.183
0.175
0.123
0 099
0.080
0 05
2 0 03
2 0.018
0.007
-0.0
00
-0 006
-0.011
-0.015
-0.020
-0 023
0.153
-0.026
-0.028
-0.029
0.400
11.083
10 065
0 10.121
9.519
8.876
8.221
7.579
6 959
6.367
S.802
5.266
4 753
0.751
0.612
0 05
42
0.483
0.457
0.433
0.410
0 389
0.370
0.318
0.251
0.199
0.158
0.100
0 060
0 032
0.012
-0.003
-0.015
-0 023
-0 030
-0 040
-0 047
-0 052
-0 055
-0 . 058
-0
062
TABL
E 8. V
ALU
ES 01 [ "'$I(')
0.600
11.076
10.646
10.121
3.521
8.880
8.226
7 0585
6.966
6.373
5.809
5.271
4 . 754
4.254
1.236
0.994
0.837
0.776
0.722
0.675
0.632
0.594
0.498
0.296
0.232
0.142
0.083
0 042
0.013
-0.0
09
-0.025
-0.037
-0.047
-0.062
-0.071
-0.078
-0.082
-0.086
-0.092
0.3ei
0.80
0
11.069
10.643
10.121
9.523
8.884
8.232
7.591
6.973
6.381
5.816
5.278
4 . 758
4
248
3.942
3.737
1.616
1 324
1 154
1.034
0.940
0.863
0.691
0.507
0.385
0 297
0.177
0.100
0.048
0.011
-0.017
-0.037
-0 053
-0 065
-0 083
-0 095
-0 104
-0.110
-0.114
-0.122
PR 1.000
11.062
10.640
10.121
9.525
8 . 888
8 238
7.596
6.388
5.824
5.285
4.763
4,249
3.934
3.712
6.980
3.470
3.332
3.164
2.471
1.375
1. 180
0.877
0.617
0.459
0 . 349
0.203
0.111
0 049
0.005
-0.027
-0.051
-0.070
-0 085
-0.106
-0.120
-0.130
*0.137
-0. 142
-0.152
1.200
11.055
10.637
10.121
9.527
8 892
8 243
7.603
6.987
6 39
5 5.832
5 293
4.771
4 255
3.937
3.713
3.467
3.327
3. 164
2 952
2 59
5
1.723
0.878
0.673
0.503
0.381
0.218
0.115
0 046
-0.004
-0.0
40
-0 067
-0 . 088
-0.105
-0 . 128
-0
144
-0. 156
-0.164
-0.170
-0.181
1 050
0
11.044
10.632
10.121
9.531
8.899
8.252
7.614
6.997
6.407
5 . 845
5.306
4 . 784
4
266
3.951
3.730
3 49
2 3.363
3 22
3 3
065
2.880
2.650
1.496
0.617
0.487
0.381
0.218
01108
0.032
-0 02
3 -0
06
3
-0 09
4 -0.117
-0.136
-0.163
-0e101
-0 19
4 -0
20
4 -0.211
-0 22
4
2.000
11.027
10.624
10.122
9.537
8.909
8.267
7.632
7.017
6.429
5.868
5.330
4.810
4.298
3.987
3.773
3.551
3 . 434
3.313
3.186
3.051
2.906
2.381
1.261
0.604
0.361
0.178
0.070
-0.008
-0 065
-0.109
-0.143
-0.169
-0.190
-0.221
-0.242
-0.257
-0 26
9 -0
27
8 -0.294
3.000
10.992
10.609
10.123
9.549
8.932
8 298
7 66
9 7
059
6.475
5.918
5.385
4.371
4.073
3.873
3.670
3.568
3.464
3 35
8 3.251
3.142
2.800
2.167
1.497
0.934
0.3
00
0
044
-0.078
-001
51
-0.202
-0.241
-0.271
-0 0295
-0.331
-0 e
356
-0.376
-0.391
-0.403
-0.425
4.a~
5.000
10.935
10.581
10.128
9.576
8.978
8.360
7.745
7.147
6 574
6.027
5.506
5.008
4.530
4.251
4 06
8
3.885
3 . 795
3.705
3.615
3 52
5
3.435
3.167
2.720
2.275
1 r8
40
1.066
0.504
0 142
-0 . 082
-0.223
-0.317
-0.381
-0.428
-0.493
-0.535
-0.567
-0.591
-0.611
-0.650
7.000
10.872
10.554
10.135
9.611
9.030
8.425
7.824
7 23
9 6.677
6. 142
5.632
5.149
4.688
4.422
4 24
8
4.077
3.992
3.909
3.825
3 74
2
3.661
3.418
3.023
2.641
2.273
1.592
1.012
0.556
0.217
-0 . 028
-0.203
-0.330
-0.424
-0.551
-0.631
-0 068
7 -0.729
-0 76
3 -0 .a27
10.000
10.781
10.529
10.150
9.663
9.111
8.531
7.950
7.381
6 83
7 6.318
5.824
5 . 358
4.916
4.662
4.497
4 33
6 4.257
4.178
4.100
4.023
3.947
3 72
2 3 36
2 3.019
2.692
2.086
1.547
1.080
0.689
0 36
9
0.112
-0 . 092
-0
25
5 -0
0489
-0 064
5
-0 . 754
-0.836
-0.899
-1.015
-0.031
.~
.
-0i0
02
-0
0008
-0
0016
-0.032
-0.064
-0.096
-0.127
-0.158
-0.188
-0.233
-0.306
-0.442
-0.680
-0.874
-1.097
-
b
TA
BL
E
9.
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(0)
VA
LU
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OF [ ?
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0.800
1.00
0 1.
200
7.30
4 7.
099
6.93
5 6.
869
6.66
3 6.
497
6.48
3 6.
275
6.10
9 6.
132
5.92
4 5.
757
5.81
6 5.
608
5.44
1
5.53
1 5.
324
5.15
7 5.
273
5.06
6 4.
900
5.03
6 4.
830
4.66
5 4.
814
4.61
0 4.
446
4.60
0 4.
399
4.23
8
4.38
8 4.
191
4.03
4 4.
166
3.97
6 3.
825
3.91
2 3.
738
3.59
9 3.
723
3.56
9 3.
444
3.55
6 3.
433
3.32
6
1.05
6 3.
259
3.18
8 0.
971
3.14
2 3.
106
0.90
3 2.
972
3.01
0 0.
847
2.17
8 2.
893
0.79
9 1.
391
2.73
6
0.75
7 1.
225
2.49
5 0.
656
0.96
5 1.
523
0.53
7 0,
742
1.01
2 0.
452
0.60
7 0.
790
0.38
9 0.
512
0.65
1
F m L 0 3 c TR
0.
010
11.6
14
11.1
85
10.802
10.4
53
10.1
37
0.10
0
9.31
9 8 8
90
8.50
6 8.
157
7.84
1
0.20
0
8.63
5 8.
205
7 a8
2 1
7.47
2 7.
156
6.87
0 6.
610
6.36
8 6.
140
5.91
7
0 a29
4 0.
239
0.19
9 0.
179
0.16
8
0.15
7 0.
153
0.14
8 0.
144
0.13
9
0.13
5 0
124
0.10
8 0.
096
0.085
0 06
8 0.
056
0 04
6 0
039
0 03
3
0 02
9 0.
025
0.02
2 0.
018
0.01
4
0.01
2 0.
010
0.00
8 0
006
0.00
4
0.40
0
7.96
1 7.
529
7.14
4 6.
794
6.47
9
0.60
0
7.57
4 7.
140
6 75
5 6.
404
6.98
9
1 a50
0 2.
000
3.00
0
6.18
2 5.
728
5.33
0 4.
974
4.65
6
4 . 373
4. 1
20
3.89
2 3.
684
3.49
1
5.00
0 7.
000
5.68
3 5.
194
4.77
2 4.
401
4.07
4
3.78
8 3.
537
3.31
5 3.
117
2.93
9
2.77
7 2.
629
2.49
1 2.
412
2 36
2
2.31
2 2.
287
2 26
3 2.
239
2.21
5
2.19
1 2.
121
2.00
7 1 a
897
1 a78
9
10.0
00
5.57
8 5.
060
4.61
9 4.
234
3 . 899 3.607
3.
353
3.13
1 2.
935
2.76
1
0.05
0
10.0
08
9.57
9 9.
196
8.84
7 8.531
8.24
5 7.
983
0.12
2 0.
096
0.07
8
0.06
4 0.
054
0 04
6 0
042
0 03
9
0.03
7 0.
036
0 03
5 0.
034
0.03
3
0.03
2 0
030
0.02
6 0.
023
0.02
1
0.01
7 0.
014
0.01
1 0.
010
0.008
0.00
7 0.
006
0.00b
0.00
6
0.00
3 0.
002
0.00
2 0.
001
0.00
1
0 . 004
0.30
0.
35
0.40
0.45
0.50
6.74
0 6
299
5.90
9 5.
557
5.24
0
6 49
7 6.
052
5.66
0 5.
306
4.98
9
4.70
6 4.
451
4.22
0 4.
007
3.80
7
5.84
7 5
376
4 96
7 4.
603
4 20
2
3.99
8 3.
747
3 52
3 3
322
3.13
8
< > 2 0.
55
0.60
0.
65
0170
0.
75
0.80
0.85
0.90
0.
93
0.95
0.97
0 a
98
0 a9
9 1.
00
1.01
1 a02
1.
05
1.10
1.
15
1.20
1 a30
1 a
40
1 a50
1 a
60
1.70
1.80
1.
90
2.00
2.
20
2.40
0.03
8 0.
029
0.02
3 0.
018
0.01
5
0101
3 0.
011
0.00
9 0
008
0.00
8
0 00
7 0.
007
0.00
7 0
007
0.00
7
0.00
6 0
006
0.00
5 0.
005
0.00
4
0.00
3 0.
003
0.00
2 0 . 002
0.
002
0.00
1 0.
001
0.00
1 0.
001
0.00
1
0.00
1 0.
000
0.00
0 0 . 000
0 . 000
7.55
5 7.
294
7 05
2 0.
206
0.16
4
0.13
4 0.
111
0 09
4 0
085
0.08
0
0.07
5 0
073
0.07
1 0
069
0 06
7
0.06
5 0.
060
0.053
0 04
7 0.
042
0.03
3 0
027
0 02
3 0.
019
0.01
7
0.01
4 0.
013
0.01
1 0.
009
0.00
7
0.00
6 0.
005
0.00
4 0.
003
0.00
2
6.19
3 5.
933
5.69
4 5
467
5 24
8
5 02
6 4.
785
0.46
3 0.
408
0.37
7
0.35
0 0.
337
0.32
6 0.
315
0.30
4
0.29
4 0
a267
0.
230
0.20
1 0.
177
0.14
0 0.
114
0 09
4 0.
079
0 06
7
0 . 058
0.05
1 0
044
0.035
0 . 028
0.02
3 0
020
0.01
7 0.
012
0.00
9
5.80
3 5.
544
5.30
6 5.
U82
4
866
4.95
6 4.
700
4.46
7 4.
250
4.04
5
Z
0 k?
4.64
9 4.
418
4.14
5 0.
750
0.67
1
0.60
7 0.580
0.55
5 0.
532
0.51
0
0 a4
9 1
0.43
9 0.
371
0.31
9 0.
277
0.21
7 0.
174
0.14
3 0.
120
0.10
2
0 . 088
0.
976
0 06
7 0.
053
0 04
2
0 03
5 0
029
0.02
5 0.
017
0.01
3
3 84
6 3.
646
3.43
4 3
295
3.19
3
3.08
1 3.
019
2 95
3 2.
879
2.79
8
3.61
5 3.
425
3.23
1
3.02
3
2.93
2 2
884
2.83
5 2
784
2.73
0
3.io
e
3.31
0 3.
135
2.96
4 2.
860
2.79
0
2.97
0 2.
812
2.66
3 2.
577
2.52
0
2.60
5 2.
463
2 33
4 2.
262
2.21
5
2.71
9 2 be
2 2.
646
2.60
9 2.
571
2.46
3 2.
436
2.40
8 2.
380
2.35
2
2.17
0 2.
148
2.12
6 2.
105
2.08
3
2.70
6 2
328
1 a55
7 1.
126
0.89
0
2.67
3 2.
483
2.08
1 1 a
649
1 a308
2.53
3 2.
415
2.20
2 1 a
968
1.72
7
2 32
5 2.
242
2.10
4 1 a
966
1 a82
7
2.06
2 2.
001
1 a90
3 1.
810
1 a72
2
0.29
8 0.
237
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