a generalized thermodynamic correlation based on three-parameter corresponding states

18
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 pj = 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, h’ew 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 11, 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. It 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)

Upload: sebastian-montano-rodriguez

Post on 02-Oct-2015

22 views

Category:

Documents


1 download

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

    0.3114

    0.3122

    0.3138

    0.3164

    0.3182

    0.3204

    0.3229

    0.3260

    0.3297

    0 34

    52

    0.3953

    0.4760

    0 m5605

    3.00

    0

    0 .a648

    0.7775

    0 70

    95

    Urn6551

    0.6110

    0.5747

    0 m5446

    0.5197

    0.4991

    0.4823

    0.4690

    0.4591

    0.4527

    0.6507

    0.4501

    0.4504

    0.4508

    0.4514

    0 A522

    0.4533

    0.4547

    0.4604

    0.4770

    0.5

    04

    2

    0 54

    25

    0 63

    44

    0.7202

    0 78

    87

    0.8410

    0.8809

    5.000

    1 m4366

    1 .2902

    1 17

    58

    1.0841

    1 m0094

    0 mY475

    0 89

    59

    0 85

    26

    0.6161

    0 78

    54

    0 m7598

    0.7388

    0.7220

    0.7138

    0.7092

    0 70

    52

    0.7035

    0.7018

    0.7004

    0.6991

    0.6980

    0 m6956

    0.6950

    0.6987

    0.7069

    0 73

    58

    0.7761

    0.8200

    0.8617

    0.8984

    7.000

    2.0048

    1 m7987

    1 m6373

    1 e5077

    1.4017

    1.3137

    1 a2398

    1.1773

    1 1241

    1*07

    87

    1 m04

    00

    1 e0071

    0.9793

    0 96

    48

    0.9561

    0.9480

    0 m9442

    0.9406

    0 m9372

    0 m9339

    0.9307

    0 92

    22

    0.9110

    0.9033

    0.8990

    10.000

    2.8507

    2 55

    39

    2.3211

    2.1338

    1 e9801

    1 1.7440

    e8520

    1 m6519

    1 e5729

    1 m5047

    1 m4456

    1 e3943

    1 34

    96

    1 32

    57

    1.3108

    0 02

    90

    0.0261

    0 02

    39

    0.0221

    0.0207

    0.0195

    0.0186

    0.0178

    0.8958

    0.9165

    0.9319

    0 e9436

    0 95

    28

    0.9573

    0.9600

    0 e2892

    0.2604

    0 2379

    0.2200

    0.2056

    0.1939

    0.1842

    0.1765

    0.1703

    Om1656

    0 1626

    0.1614

    0 1630

    0 1664

    0.1705

    0.97

    0.98

    0.99

    1.00

    1.01

    1.02

    1 a05

    1.10

    1.15

    1 c20

    1 m30

    1 m40

    1 .so

    1 e60

    1.70

    1 .80

    1.90

    2.00

    2.20

    2.4

    0

    2.60

    2.80

    3.0

    0

    3.50

    4.00

    0 99

    63

    0 99

    65

    0 m9966

    0 e9967

    0 m9968

    0.9815

    0.9821

    0 m9826

    0.9832

    0.9837

    0 m9842

    0 e9855

    0 r9874

    0.9891

    0 a9

    904

    0 m9625

    0.9637

    0 m9648

    0 m9659

    0 m9669

    0 8338

    0.8398

    0 ma455

    0.8509

    0.8561

    0.5580

    0.5887

    0.6138

    0.6353

    0 a6542

    0.1779

    0 1844

    0 1959

    0.2901

    0.4648

    1 29

    68

    1 r2901

    1 28

    35

    1 m2772

    1.2710

    0 e9969

    0.9971

    0 a9975

    0 m9978

    0.9981

    0.9679

    0.9107

    0 a9747

    0.9780

    0.9808

    0.9343

    0.9401

    0 m9485

    0 m9554

    0 m9611

    0.8610

    0 m8743

    0 m8930

    0.9081

    0.9205

    0.6710

    0.7130

    0 e7649

    0.8032

    0.8330

    0.5146

    0.6026

    0.6880

    0.7443

    0.7858

    1 e2650

    1.2481

    1 22

    32

    1 .2021

    1 18

    44

    0 e9985

    0.9988

    0.9991

    0.9993

    0 99

    94

    0 a9995

    0 99

    96

    0 99

    97

    0 99

    90

    0 9999

    0 99

    26

    0.9942

    0 ~9954

    0.9964

    0.9971

    0.9977

    0.9982

    0 9

    986

    0 e9

    992

    0.9996

    0.9998

    1.0000

    1 moo

    02

    1 e00

    04

    1 m0005

    0 m9852

    0 9884

    0.9909

    0 e9928

    0.9943

    0 m9955

    0 9964

    0 e9972

    0 e9983

    0.9991

    0 99

    97

    1.0001

    1 m0004

    1.0008

    1.0010

    0.9792

    0.9768

    0.9818

    0.9856

    0.9886

    0.9910

    0 m9929

    0.9944

    0.9967

    0 a9983

    0 m9396

    0 953%

    0.9636

    0.9714

    0.9775

    0.9823

    0.9861

    0.9892

    0.9937

    0 99

    69

    0.9991

    1 e0007

    1.0018

    1 m0035

    1.0043

    0 m9083

    0 m9298

    0 m9456

    0.9575

    0.9667

    0.9739

    0 m9796

    0.9

    84

    2

    0.9910

    0.9957

    0 &764

    0.9062

    0.9278

    0 m9439

    0 95

    63

    0 ~9659

    0.9735

    0 m9796

    0 m9886

    0 99

    48

    0.8438

    0 m8827

    0.9103

    0.9308

    0.9463

    0 e9583

    0.9754

    0 m9865

    0.9941

    0.9678

    0 m9993

    1 e0031

    1.0057

    1.0097

    1.0115

    0.8111

    0 85

    95

    0 me933

    0.9180

    0 m9367

    0.6908

    0 a7753

    0 ma328

    0 m8738

    0 e9043

    0 89

    98

    0.9112

    0 m9297

    0.9518

    0.9745

    1 15

    80

    1.1419

    1.1339

    1.1320

    1.1343

    1 1391

    1 14

    52

    1 15

    16

    1 16

    35

    1.1728

    z ;c P 0.9511

    0 m9624

    0.9715

    0.9847

    0 99

    36

    0.9413

    0 m9552

    0 m9664

    0 m9826

    0 e9935

    1.0010

    1 00

    63

    1.0101

    1.0156

    1.0179

    0.9275

    0 94

    56

    0 95

    99

    0.9806

    0 99

    45

    1.0040

    1.0106

    1.0153

    1 .0221

    1 m0249

    0.9118

    0 m9359

    0.9550

    0.9827

    1.0011

    1.0137

    1 m0223

    1 0

    28

    4

    1 m0368

    1.0401

    0 m9297

    0.9557

    0.9772

    1 m0094

    1.0313

    0.9961

    1.0157

    1 03

    28

    1 a0600

    1 07

    93

    1 09

    26

    1.1016

    1 10

    75

    1.1138

    1.1136

    1.0000

    1.0000

    1.0000

    1.0001

    1.0001

    0 99

    94

    1 moo02

    1.0008

    1.0017

    1.0021

    0.9990

    1.0013

    1 a0

    03

    0

    1 m0055

    1 m0066

    0.9990

    1.0021

    1 0043

    1.0075

    1 e0090

    0 9998

    1.0042

    1 m0074

    1.0120

    1.0140

    1.0463

    1 05

    65

    1 06

    35

    1 m0723

    1 m0747

    1.1792

    1 18

    30

    1.1848

    1 18

    34

    1 17

    73

    an

    an

    .I.

  • S L6 1 'ADW 91s a6*d (Q 'ON 'LZ

    00000

    999ma UIwoUlo ..... NmNrr ccccc c.h)o-DdD 4mulPw 00000 00000 .... ..... b-cccc ..... (urroo OUIOWN 4 n

    0

    0 I- 0

    0

    0 Ul 0

    . 0 . c

    Illlt CCOOO ..... 00000 OCrNGt *9lA,or PW4lnP

    IIIII 00000

    00000 WUI400 WO4PP 64N-W

    I1111 OOOGO

    OrOOO +roo0 Pm4mOD P-mNOI

    .....

    .....

    11l11 ooccc ..... 00000 GOO00 torcc 99099

    11111 00000 ..... 00000 00000 PPPPP w4amo IIIII 00000

    00000 00000 99990, OPUlOW

    ...-.

    111 oooco .... . 00000 00000 00000 PIUOUUI

    11 ooooc

    00000 00000 WrOrN SWOUlQ.

    II oocoo 00000 00000 WNONUl 9Wl-9-

    .....

    .....

    11111 oooco 00000 OCOOO OCCCr QL4QISO

    Il(11 00000

    00000 00000 WWWPUI OP9PO

    .....

    .....

    11111 00000 ..... 00000 ooooc rrrNf NUl99P

    11111 00000

    00000 OOOWN 649UlN NUl9NOJ

    11181 00000

    00000 WWNW+ NUlO-OD mPUl.04

    .....

    .....

    ccccc .... . 00000 00000 00000 WUI0CP-I

    00c.00

    00000 oooco ocooc IIIImmDoD

    ..... occco ..... 00000 00000 cooc0 wmm-4m

    00000

    00000 00000 NNNWW W*rg*W

    ..*.. 00000

    00000 00000 WWWPP Ul4900

    ..... 00000

    00000 00000 PPWWW 009mo

    ..... 11411 ooooc ..... 00000 00000 00000

    DUIUI6OI mIU9h)O'

    00000 ..... 00000 occcc SOrNW rW4Pr

    00000

    ..... ocooo ..... 00000 00000 4444m oPw.5-

    00000

    00000

    WPuIUIrn 94Ul9N

    00000

    ..... rrrrr

    00000 ..... 00000 00000 mOJ44m I-OODNUr

    00000 oooor OIOI460 09OOe- 0

    0 . N 0 0 0

    E 0 0

    . 0

    m 0 0

    . 0 . OD 0

    II 00000

    11111 00000

    81111 00000

    *I111 00000 ....*

    11111 00000 0.0..

    00000

    00000

    m4rUlD.N WNOD4UI

    00000

    ..... ccccc ..... 00000 rcrcN NPOImO OOePOD

    llill 00000

    ..... ooooc NWP4- OIlUOrOI N(rNUI0

    11881 00000

    008-00 Ul4rNN mmrOI4 9Wma3N

    88111 00000 ..... ~cooo D-QLWWP em960 0nIm-r

    .....

    ... 00000 ccucr PDUlOI4 WOJOIPN

    11111 00000

    .. . 00000

    mai9ODm -4oUlr

    11111 00000

    00000 WWWWW OI404N OPOOW

    I1111 00000

    00000 UIUIUllnP wm4Ulm 900fC

    rcccc

    .....

    .....

    00000 0000c mUloUl0 PN4PN

    11 00000 ..... ocooo -roar 9NW99 04TDNO)

    I1 00000

    00000 WNWON NWO-Drl m4m4-l

    I1 00000

    00000 PWNOW 99WWO 90.OINW

    I 00000

    00000 4mPNO cN4NOl 9UIOION

    00000

    .....

    .....

    ...a.

    ..... 00000 WNNNN ODOWEOI NPNUlO

    .... . oocco WWWWN NWNOOI 90WP-A

    ..... 00000 NNWWP PO)W.DlJI OUIUlOO

    ..... 00000 NNWWW OD90NP NPSrnW 6Womiii

    81811 00000

    00000 WPUICE-I UlWblP4 I-UIr-0 118l1 00000

    ooorr Pln4a-m NODOOP *me04

    .....

    .....

    00000 ..... 00000 NWWWW 40*OIm OWUlUl4

    00000 ..... ooooc PPPP) cwm4m r404m

    00000 ..... oooco PWUPP bOb4N 4-449

    $ m 111ll 00000 .... . lllll 00000 ..... I1111 00000 ..... 00000 00000 UDPPU) UlOOIODW 4cOlWN

    ..... 00000 ..... 00000 UlUlmmm P4WWUI PS4UlN

    00000 ..... 00000 mmmmm m44mr 44IIWN

    00000 UIUIUIUIUI ProoN oPW9m

    00000 UlUlOIOIm UIIIJrPOD 09rOIN

    IIIII 00000 .....

    I1111 00000

    18111 00000

    88111 00000 00000

    00000 PPUIUI*

    W4UIrnP

    00000

    ..... 00000 e.... 00000 m444m 4rOI9r Uls4h)m

    00000

    00000 ..... 00000 mamma, WUIOIUlO- mu1449

    00000 .....

    1 . 0 0 0 c

    N 0 0

    c . Ul 0 0 h)

    0 0 0 . W

    0 0 0

    VI . 0 0 0 4 . 0 0 0 c 0 .

    .... 0 00000 mmOIm.0, OOONP 4NPNO)

    lllll 00000

    00000 mm44-4 -Imowm OD4cr4

    .....

    . .. 00000 moD0Imm N4mDN WSOrW

    IIIl4 00000

    ccccc 00000 owUIm9 mUlUlwOJ

    I 00000 ..... cI-cco PUImPUl IIPWUlN 4mO-P

    11111 00000 .....

    11014 00000 .....

    s1111 00000 .....

    00000

    coooo m4m0)9 UlNN4N rmOJOC

    ..... 00000 ..... ccccr NWWOP UlOPoJIU swvuo

    00000 4mODms mrPmP mooOD0

    11118 00000 .....

    11118 00000

    WllII 00000

    811#1 00000

    I1 00000 00000

    m4mm9 UlNO99 WS**W

    00000 ..... NNNNUN WWPWO OmO94 mew49

    ..... cccrc ..... occcc 900-N OIolJIWr 4499II 11111 00000 ... 0.

    111 00000

    coocc OWWODO SW4WN UINWm- IIIII 00000 ..a*. oorrr WOIOWUI PrN4UI VrrCOI

    I1118 ooooc

    .....

    .....

    11411 00000 00000

    NWUle4 rUINr0 9m9wm

    00000

    ....* c~ccc YUYYY I 0

    c 9 P W

    00000

    NNNNN m4mmm ONrPP N09'DCP

    ...*o

    I

    11841 00000 00000

    NNNNW WUlQ390 -4ODr-0 mP4uI9

    .... . 00000 B.... WWWWN or000: oUI44- awa-rl

    11111 00000 .....

    11811 00000 .....

    )(I81 00000 .....

    I~III 00000

    J1188 00000

    0 0

  • H" - H

    TA

    BLE

    ~.V

    AL

    UE

    SO

    E

    - _

    _

    [ R

    T,

    J

    PR

    1 *SO

    0

    5.95

    7 5.

    814

    5.6

    68

    5

    .51

    9

    5 36

    9

    5.22

    0 5.

    073

    4.92

    7 4.

    781

    4.6

    32

    4.47

    8 4.

    31Z

    4.

    127

    4.00

    0 3.

    904

    3 79

    6 3.

    736

    3.67

    0 3 . 598

    3.

    516

    3.42

    2 3.

    030

    2.2

    03

    1.

    719

    1 e44

    3

    3.00

    0

    5.86

    8 5.

    721

    5.57

    2 5.

    421

    5.27

    0

    5.12

    1 4.

    976

    4 . 833

    4.

    693

    4 . 554

    4.41

    3 4

    269

    4.11

    9 4

    9 0

    24

    3.95

    8

    3.89

    0 3.

    854

    3.81

    8 3

    782

    3.74

    4

    7.00

    0

    5.62

    8 5

    .46

    9

    5.31

    1 5

    .15

    4

    4 . 999

    4 . 849

    4.

    704

    4.56

    5 4

    43

    2

    4.3

    03

    4.17

    8 4.

    056

    3.93

    5 3

    86

    3

    3.81

    5

    3.76

    7 3 . 743

    3.

    719

    3.69

    5 3.

    671

    TR

    0.30

    0.

    35

    0.40

    0.

    45

    0.50

    0.01

    0 0.

    050

    6 04

    3 5

    904

    5.76

    1 5.

    612

    5.46

    3

    0.10

    0 0.

    200

    6.03

    4 5.

    895

    5.75

    1 5

    .60

    3

    5 . 453

    0.4

    00

    6.02

    2 5.

    882

    5.73

    8 5.

    590

    5.44

    0

    0.60

    0 0.

    800

    1.00

    0

    6.01

    1 5

    .99

    9

    5.98

    7 5.

    870

    5.85

    8 5

    ,04

    5

    5.72

    6 5.

    713

    5.70

    0 5.

    577

    5.56

    4 5.

    551

    5.42

    7 5.

    414

    5.40

    1

    5.27

    0 5

    .26

    5

    5.25

    2 5

    .12

    9

    5.11

    6 5.

    104

    4.98

    0 4.

    968

    4.95

    6 4.

    828

    4.81

    8 4.

    808

    4.67

    2 4.

    664

    4.65

    5

    4.50

    4 4.

    499

    4.49

    4 4

    .31

    3

    4.31

    6 4.

    316

    4.07

    4 4.

    094

    4.10

    8 0.

    960

    3.92

    0 3.

    953

    0.88

    5 3.

    763

    3.82

    5

    1.20

    0

    5.97

    5 5.

    833

    5.68

    7 5.

    538

    5 38

    8

    5 23

    9 5.

    091

    4.94

    5 4 . 797

    4

    646

    4 . 488

    4.

    316

    4.11

    8 3.

    976

    3 86

    5

    3.73

    2 3.

    652

    3.55

    8 3.

    441

    3 28

    3

    2.00

    0

    5.92

    7 5

    .78

    3

    5.63

    6 5.

    486

    5 33

    6

    5.18

    7 5.

    041

    4.89

    6 4.

    752

    4.60

    7

    5.00

    0

    5.74

    8 5 . 595

    5.

    442

    5.28

    8 5.

    135

    4 98

    6 4.

    842

    4.70

    2 4

    56

    6

    4.43

    4

    4.30

    3 4.

    173

    4 04

    3 3

    96

    3

    3.91

    0

    3 85

    6 3

    82

    9

    3.80

    1 3.

    774

    3.74

    6

    10.0

    00

    5.44

    6 5

    278

    5.1

    13

    4

    950

    4.79

    1

    6.04

    5 5.

    906

    5 76

    3 5.

    615

    5.46

    5

    6.04

    0 5.

    901

    5.75

    7 5

    609

    5.45

    9

    0.55

    0.

    60

    0.65

    0.

    70

    0.75

    0 03

    2 0

    027

    0 02

    3 0 . 020

    0.

    017

    0.01

    5 0.

    014

    0.01

    2 0.

    011

    0.01

    1

    5.31

    2 5.

    162

    0.11

    8 0.

    101

    0 . 088

    0.07

    0 0.

    069

    0 06

    2 0

    00

    58

    0

    056

    5.30

    9 5.

    159

    5.00

    8 0.

    213

    0.18

    3

    5.30

    3 5

    .15

    3

    5.00

    2 4

    .84

    8

    4.68

    7

    5.29

    0 5.

    141

    4.9

    91

    4.

    838

    4 67

    9

    4

    638

    4 49

    2 4

    .35

    3

    4.22

    1 4.

    095

    Z

    ? w

    Y

    0.80

    0.90

    0.

    93

    0.95

    0.85

    0.

    160

    0.14

    1 0.

    126

    0.11

    8 0.

    113

    0.34

    5 0

    .30

    0

    0.26

    4

    0.23

    5 0.

    246

    4.50

    7 4

    .30

    9

    0.59

    6 0

    545

    0.51

    6

    4 . 459

    4.

    302

    4.13

    2 4.

    020

    3.94

    0

    3.97

    4 3 . 857

    3.

    744

    3 6

    78

    3

    634

    0.97

    0.

    98

    0.99

    1.

    00

    1.01

    0.01

    1 0.

    010

    0.01

    0 0.

    010

    0.01

    0

    0 05

    4 0.

    053

    0.05

    2 0.

    051

    0.05

    0

    0 04

    9 0

    046

    0.04

    2 0

    039

    0 03

    6

    0.03

    1 0

    027

    0 02

    4 0.

    021

    0.01

    9

    0.01

    7 0.

    015

    0.01

    4 0.

    012

    0.01

    0

    0.0

    08

    0.

    007

    0.00

    6

    0 . 002

    0.

    004

    0.10

    9 0.

    107

    0.10

    5 0.

    103

    0.10

    1

    0.22

    5 0.

    221

    0.21

    6 0.

    212

    0.20

    8

    0.49

    0 0.

    478

    0 .4

    66

    0.45

    5 0.

    445

    0.8

    24

    1.

    356

    3.65

    8 0.

    797

    1.27

    3 3.

    544

    0.77

    3 1.

    206

    3.37

    6 0.

    750

    1.15

    1 2.

    584

    0.72

    0 1.

    102

    1.79

    6

    3.85

    3 3.

    806

    3.75

    8 3.

    706

    3 65

    2

    3.59

    1 3

    569

    3.54

    8 3.

    526

    3.50

    5

    1.02

    1 .

    05

    1.10

    1.

    15

    1.20

    0.01

    0 0.

    009

    0.0

    08

    0.

    008

    0.00

    7

    0 09

    9 0.

    094

    0 08

    6 0.

    079

    0.07

    3

    0.20

    3 0.

    192

    0.17

    5 0.

    160

    0.14

    8

    0.12

    7 0.

    110

    0.09

    7 0

    086

    0 07

    6

    0.06

    8 0.

    062

    0 05

    6 0.

    046

    0.03

    8

    0 03

    2

    0.02

    3 0.

    015

    0.00

    9

    0 . 027

    0.43

    4 0.

    407

    0.36

    7 0.

    334

    0.30

    5

    0.70

    8 1.

    060

    1.62

    7 0.

    654

    0.95

    5 1.

    359

    0.52

    3 0.

    732

    0.96

    8 0.

    474

    0.65

    7 0.

    857

    0.5

    ai

    0.82

    7 1.

    120

    3 03

    9 2

    034

    1 e48

    7 1 e

    239

    1.07

    6

    3.59

    5 3.

    398

    2 96

    5 2

    479

    2.07

    9

    3.70

    5 3.

    583

    3.35

    3 3.

    091

    2.80

    7

    3.71

    8 3

    63

    2

    3 . 484

    3.

    329

    3.16

    6

    3.64

    7 3 . 575

    3 . 453

    3

    329

    3.20

    2

    3 . 484

    3.

    420

    3.31

    5 3.

    211

    3.10

    7

    1.30

    1.

    40

    1 .50

    1 e

    60

    1.70

    0.00

    6 0.

    005

    0.00

    5 0.

    004

    0.00

    4

    0 06

    3 0

    055

    0.04

    8 0

    04

    3

    0.03

    8

    0.25

    9 0.

    224

    0.19

    6 0.

    173

    0.15

    3

    0.39

    9 0.

    545

    0.69

    8 0.

    341

    0.46

    3 0.

    588

    0.29

    7 0

    .40

    0

    0.50

    5 0.

    261

    0.35

    0 0.

    440

    0.23

    1 0.

    309

    0.38

    7

    0.20

    6 0.

    275

    0.34

    4

    0.16

    7 0.

    222

    0.27

    6 0.

    137

    0.18

    2 0.

    226

    0.11

    4 0.

    150

    0.18

    7

    0.0

    95

    0.

    125

    0.15

    5 01

    080

    0.10

    5 0.

    130

    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.

    S" - s

    (0)

    VA

    LU

    ES

    OF [ ?

    ] PR

    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

    0.19

    4 0.

    162

    0.13

    7

    0.11

    7 0.

    102

    0 08

    9 0.

    070

    0.05

    6

    0.38

    5 0.

    303

    0 a2

    66

    0.20

    4 0.

    172

    0.14

    7 0.

    127

    0.11

    1 0.

    087

    0 07

    0

    0.47

    8 0.

    372

    0.29

    9 0.

    247

    0 a2

    08

    0.17

    7 0.

    153

    0.13

    4 0.

    105

    0 08

    4

    0.62

    8 0.

    478

    0.38

    1 0.

    312

    0.26

    1

    0.22

    2 0.

    191

    0.16

    7 0.

    130

    0.10

    4

    0 08

    6 0

    072

    0.06

    1 0

    042

    0.03

    1

    0.89

    1 0.

    663

    0.52

    0 0.

    421

    0.350

    0.29

    6 0

    255

    0.22

    1 0.

    172

    0.13

    8

    0.11

    3 0

    094

    0 . 080

    0 05

    6 0.

    041

    1 a29

    9 0.

    990

    0.77

    7 0.

    628

    0.51

    9

    1.55

    4 1 a303

    1.08

    8 0.

    913

    0.77

    3

    0.66

    1 0.

    570

    0.49

    7 0

    388

    0.31

    1

    0 25

    5 0.

    213

    0.18

    1 0.

    126

    0.09

    3

    1 a58

    1 1 a

    386

    1 a20

    8 1.

    050

    0.91

    5

    1 a55

    6 1 a

    402

    1 a26

    0 1.

    130

    1 a01

    3

    0,90

    8 0.

    815

    0 73

    3 0 a

    599

    0.49

    6

    0.438

    0.37

    5 0.

    325

    0.25

    1 0 a

    20 1

    0.16

    4 0.

    137

    0.11

    6 0.

    081

    0.05

    9

    0.79

    9 0.

    702

    0.62

    0 0.

    492

    0.39

    9

    Y

    (n

    W

    0

    rn

    (D

    0.04

    6 0.

    058

    0.06

    9 0.

    039

    0.04

    8 0.

    058

    0.033

    0.04

    1 0.

    049

    0.02

    3 0.

    029

    Om

    034

    0.01

    7 0.

    021

    0.02

    5

    0 32

    9 0.

    277

    0.23

    6 0.

    166

    0.12

    3

    0.41

    6 0.353

    0.303

    0.21

    6 0

    162

    2.60

    2.

    80

    3.00

    3.50

    1.

    00

  • 0.010

    16.782

    15.413

    13.990

    12.564

    11 e202

    0.115

    0 078

    0 055

    0.040

    0 029

    0.022

    0.017

    0.013

    0.011

    0.010

    0.010

    0.009

    0.009

    0 . 008

    0.00

    8

    0.008

    0.007

    0.005

    0 . 005

    0 00

    4

    0.003

    0.002

    0.001

    0.001

    0.001

    0.001

    0.001

    0.000

    0.000

    0.000

    0.00

    0 0.

    000

    0.00

    0 0.

    000

    0.00

    0

    0.05

    0

    16.774

    15.408

    13.986

    12.561

    11.200

    9.948

    8.828

    0.309

    0.216

    0.156

    0.116

    0 . 088

    0.068

    0 . 058

    0.

    053

    0 048

    0.046

    0 044

    0 042

    0.040

    0 039

    0 034

    0 . 028

    0.023

    0.019

    0.013

    0.010

    0.007

    0.005

    0.004

    0.003

    0.003

    0.002

    0.001

    0.001

    0.001

    0.001

    0.001

    0.000

    0.000

    0.100

    16.764

    15.401

    13.981

    12.558

    11.197

    9.946

    8.826

    7.832

    0.491

    0.34

    0

    0.246

    0.183

    0 140

    0.120

    0.109

    0 099

    0 094

    0 090

    0 086

    0.082

    0 078

    0 069

    0 055

    0 045

    0 037

    0.026

    0.019

    0.014

    0.011

    0.008

    0.006

    0.005

    0.004

    0.003

    0.002

    0 . 002

    0.001

    0.001

    0.001

    0.001

    0.20

    0

    16.744

    15.387

    13.972

    12.551

    11.192

    9.942

    8.823

    7.829

    6.951

    6.173

    0.578

    0.408

    0.301

    0 e254

    0.228

    0 e206

    0.196

    0 186

    0.177

    0.169

    0.161

    0.140

    0.112

    0.091

    0.075

    0 052

    0 037

    0 027

    0.021

    0.016

    0.013

    0.010

    U.0

    08

    0.006

    0.004

    0.00

    3 0.003

    0 . 002

    0.001

    0.001

    0.400

    16.705

    15.359

    13.953

    12.537

    11.182

    9.935

    8.817

    7.824

    6.945

    6.167

    5.475

    4 853

    0.744

    0.593

    0.517

    0.456

    0.429

    0.405

    0.302

    0.361

    0.342

    0 292

    0 *2

    29

    0.183

    0.149

    0.102

    0 072

    0.053

    0.04

    0 0.031

    0 024

    0.019

    0.016

    0.011

    0.008

    0.006

    0 005

    0.004

    0.003

    0.002

    TA

    BL

    E 10. V

    ALU

    ES OF

    - 1

    [

    0.600

    16.665

    15.333

    13.934

    12.523

    11.172

    9.928

    8.811

    7.819

    6.911

    6.162

    5.468

    4.841

    4.269

    1.219

    0.961

    0.797

    0.134

    0.680

    0.632

    0.590

    0 e552

    0.460

    0.3SO

    0.21

    5 0.220

    0.148

    0.104

    0.076

    0.05

    7 0 044

    0.035

    0 . 028

    0.023

    0.016

    0.012

    0.009

    0.008

    0.006

    0.004

    0.003

    0.80

    0

    16.626

    15.305

    13.915

    12.509

    11.162

    9.921

    8.806

    7.815

    6.937

    6.158

    5.462

    4.832

    4 249

    3.914

    3 69

    7

    1 .570

    1 e27O

    1.098

    0.977

    0.883

    0.807

    0.642

    0.470

    0.361

    0.286

    0.190

    0.133

    0.097

    0.073

    0.056

    0 044

    0 036

    0 029

    0.021

    0.015

    0.012

    0.010

    0 . 008

    01006

    0.005

    PR 1.000

    16.586

    15.278

    13.896

    12.496

    11.153

    9.914

    8.799

    7.810

    6.933

    6.155

    5.458

    4 826

    4.238

    3.894

    3.658

    3.406

    3 264

    3.093

    2.399

    1 e306

    1.113

    0.820

    0.577

    0.437

    0.343

    0.226

    0.158

    0.115

    0 086

    0 067

    0 053

    0.0

    43

    0 035

    0 025

    0.019

    0.015

    0.012

    0.010

    0.007

    0.006

    J 1 e200

    16.547

    150251

    13.877

    12.482

    11.143

    9.907

    8 . 794

    7.807

    6.930

    6.152

    5.455

    4.822

    4 232

    3.885

    3.647

    3.391

    3.247

    3.082

    2.868

    2.513

    1 .655

    0.831

    0 e640

    0.489

    0 385

    0.254

    0.178

    0.130

    0 098

    0.076

    0 060

    0 049

    0 040

    0.029

    0 . 022

    0.018

    0.014

    0.012

    0.009

    0.007

    1 e50

    0

    16.488

    15.211

    13.849

    12.462

    11.129

    9.897

    8 . 787

    7.801

    6 926

    6.149

    5 . 453

    4.820

    4.230

    3 884

    3.648

    3.401

    3 268

    3.126

    2.967

    2 784

    2.557

    1 a443

    0.618

    0.502

    0.412

    0.282

    0.200

    0.147

    0.112

    0 087

    0 070

    0.05

    7 0 048

    0 035

    0.027

    0.021

    0.018

    0.015

    0.011

    0.009

    2.000

    16.390

    15.144

    13.803

    12.430

    11.107

    9.882

    8.777

    7 . 794

    6.922

    6.147

    5.452

    4.822

    4 236

    3.896

    3.669

    3.437

    3.318

    3.195

    3.067

    2.933

    2.790

    2.283

    1.241

    0 e654

    0.447

    0.300

    0.220

    0.166

    0.129

    0.102

    0.0

    83

    0 069

    0 . 058

    0 043

    0 9 034

    0.028

    0.023

    0.020

    0.015

    0.012

    3.000

    16.195

    15.011

    13.714

    12.367

    11.063

    9.853

    8.760

    7 . 784

    6.919

    6.149

    5.461

    4 . 839

    4 a67

    3.941

    3.728

    3.517

    3.412

    3.306

    3.200

    3.094

    2 98

    6 2 65

    5 2.067

    1.471

    0.991

    0.481

    0 .206

    0.159

    0.127

    0.105

    0 089

    0.077

    0.060

    0 048

    0.041

    0 035

    0.031

    0.024

    0.020

    0.290

    5.00

    0

    15.837

    14.751

    13.541

    12.248

    10e9e5

    9.806

    8.736

    7.779

    6.929

    6.174

    5.501

    4 . 898

    4.351

    4.046

    31851

    3.661

    3.569

    3.477

    3 . 387

    3.297

    3.209

    2.949

    2 534

    2.138

    1 767

    1 147

    0 730

    0.479

    0.334

    0 248

    0.195

    0.160

    0.136

    0.105

    0.086

    0 074

    0.065

    0 05

    8 0 046

    0 03

    8

    7.000

    15.468

    14.496

    13.376

    12.145

    10.920

    9.769

    8.723

    7 . 785

    6.952

    6.213

    5 . 555

    4.969

    4.442

    4.151

    3 966

    3.788

    3.701

    3.616

    3.532

    3.450

    3.369

    3.134

    2.767

    2 428

    2.115

    1 e569

    1.138

    0.823

    0.604

    0.456

    0.355

    0.286

    0.238

    0.178

    0.143

    0.120

    0.104

    0 093

    0 073

    0 060

    10.000

    14.925

    14.153

    13.144

    11 .999

    10.836

    9 73

    2 8.720

    7.811

    7.002

    6 285

    5.648

    4.578

    4.300

    4.125

    3.957

    3 . 875

    3 796

    3.717

    3.640

    3.565

    3 . 348

    3.013

    2.708

    2.430

    1.944

    1 a544

    1.222

    0.969

    0.775

    0.628

    0.518

    0.434

    0 322

    0.254

    0.210

    5 . 082

    0.180

    0.158

    0.122

    0.100

  • PR 1.00

    0

    -5.5

    84

    -4

    359

    -3.4

    63

    -2.7

    85

    -2

    256

    -1 a8

    35

    -1.4

    94

    -1 e

    214

    -0.9

    81

    -0

    785

    00.6

    19

    -0.4

    79

    -0

    359

    -0.2

    96

    -0.2

    58

    00.2

    23

    -0.2

    06

    -0.1

    91

    -0.1

    76

    -0.1

    68

    -0.1

    61

    -0.1

    43

    -0 . 120

    -0

    .102

    -0

    .088

    -0.0

    66

    -0.0

    51

    -0

    039

    -0.0

    31

    -0 . 024

    -0.0

    19

    -0.0

    15

    -0.0

    12

    -0.0

    07

    -0.0

    03

    -0.0

    01

    0.00

    1 0.

    002

    0.00

    4 01

    005

    TR

    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

    0.97

    0.

    98

    0.99

    1.

    00

    1.01

    1.02

    1.

    05

    1.10

    1.

    15

    1.20

    1.30

    1 e

    40

    1.50

    1.

    60

    1 e70

    1 .a0

    1 e

    90

    2.00

    2.

    20

    2.40

    2.60

    2.

    80

    3.00

    3.

    50

    - 4.

    00

    0.01

    0

    -3.7

    08

    -2.4

    71

    -1 e

    566

    -0.8

    79

    -0

    344

    -0.0

    00

    -0.0

    07

    -0.0

    05

    -0.0

    04

    -0

    .003

    -0.0

    03

    -0.0

    02

    -0.0

    02

    -0.0

    02

    -0.0

    02

    -0.0

    02

    -0.0

    02

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    00

    -0.0

    00

    -0.0

    00

    -0.0

    00

    -0.0

    00

    -0.0

    00

    -0.0

    00

    -0.0

    00

    -0.0

    00

    0.00

    0 0

    000

    0.00

    0 0.

    000

    0 05

    0

    -4.4

    02

    -3.1

    66

    -2.2

    61

    -1 e

    575

    -1 e

    04

    0

    -0.6

    14

    -0 e

    269

    -0

    026

    -0.0

    21

    -0.0

    17

    -0.0

    14

    -0.0

    12

    -0.0

    10

    -0.0

    09

    -0.0

    00

    -0.0

    08

    -0.0

    08

    -0.0

    07

    -0.0

    07

    -0.0

    07

    -0.0

    07

    -0.0

    06

    -0.0

    05

    -0.0

    05

    -0.0

    04

    -0.0

    03

    -0.0

    03

    -0.0

    02

    -0.0

    oZ

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    01

    -0.0

    00

    -0.0

    00

    -0.0

    00

    0.00

    0 0.

    000

    0.00

    0 0

    .00

    ~

    0.10

    0

    -4.6

    96

    -3.4

    61

    -2

    557

    -1 e

    871

    91.3

    36

    -0.9

    11

    -0

    566

    -0

    283

    -0

    043

    -0.0

    35

    -0

    029

    -0

    024

    -0

    020

    -0.0

    18

    -0.0

    17

    -0.0

    16

    -0.0

    16

    -0.0

    15

    -0.0

    15

    -0.0

    14

    -0.0

    14

    -0.0

    13

    -0.0

    11

    -0.0

    09

    -0.0

    08

    -0.0

    06

    -0.0

    05

    -0.0

    04

    -9.0

    03

    -0

    002

    -0

    002

    -0 m

    oo2

    -0.0

    01

    -0.0

    01

    -0.0

    00

    -0.0

    00

    0.00

    0 u.

    000

    0.00

    0 0.

    000

    0.20

    0

    -4.9

    85

    -3.7

    51

    -2

    848

    02.1

    62

    -1.6

    28

    -1 e

    204

    -0 a

    859

    -0 -

    576

    -0.3

    41

    -0.1

    44

    -0.0

    59

    -0

    049

    -0

    04 1

    -0

    .037

    -0

    03

    5

    -0

    033

    -0.0

    32

    -0.0

    31

    -0.0

    30

    -0

    029

    -0.0

    28

    -0

    025

    -0.0

    22

    -0.0

    19

    -0.0

    17

    -0.0

    13

    -0.0

    10

    -0.0

    08

    -0.0

    06

    -0.0

    05

    -0.9

    04

    -0.0

    03

    -0.0

    02

    -0.0

    01

    -0.0

    01

    -0.0

    00

    0.00

    0 0.

    000

    0.00

    1 0.

    001

    0.40

    0

    -5.2

    61

    -4.0

    29

    -3.

    128

    -2.4

    44

    -1 e

    912

    -1 e

    488

    -1

    144

    -0 e

    862

    -0 r

    627

    -0.4

    30

    -0.2

    64

    -0.1

    23

    -0.0

    86

    -0.0

    77

    -0.0

    72

    -0.0

    67

    -0 e

    065

    -0 e

    063

    -0.0

    61

    -0

    059

    -0

    057

    -0

    052

    -0

    045

    -0.0

    39

    -0

    034

    -0.0

    26

    -0.0

    20

    -0.0

    16

    -0.0

    12

    -0.0

    10

    -0.0

    08

    -0.0

    06

    -0.0

    05

    -0.0

    03

    -0.0

    01

    -0.0

    00

    0.00

    0 0.

    001

    0.00

    1 0.

    002

    0.60

    0

    -5.4

    12

    -4.1

    83

    -3.2

    83

    -2.6

    01

    -2.0

    70

    -1 e

    647

    -1 e3

    04

    -1.0

    23

    -0.7

    89

    -0.5

    92

    -0.4

    126

    -0.2

    85

    -0.

    166

    -0.

    122

    -0.1

    13

    -0.1

    05

    -0.1

    01

    -0.0

    98

    -0.0

    95

    -0.0

    91

    -0

    088

    -0.0

    80

    -0.0

    69

    -0.0

    59

    -0

    .051

    -0.0

    39

    -0.0

    30

    -0e0

    24

    -0.0

    19

    -0.0

    15

    -0.0

    12

    -0.0

    09

    m

    O.0

    07

    -0.0

    04

    -0.0

    02

    -0.0

    01

    O.OO

    U 0.

    001

    0.00

    2 0.

    003

    0.80

    0

    -5.5

    12

    -4

    285

    -3.3

    87

    -2.7

    07

    -2.1

    77

    - 1 75

    5 -1

    .413

    -1

    .132

    -0

    .899

    -0

    .703

    -0.5

    37

    -0.3

    96

    -0.2

    76

    -0.2

    14

    -0.1

    76

    -0.1

    48

    -0.1

    42

    -0.1

    37

    -0.1

    32

    -0.1

    27

    -0.1

    22

    -0.1

    10

    -0.0

    93

    -0

    080

    -0.0

    69

    -0.0

    52

    -0 e

    O4O

    -0

    03

    2 -0

    02

    5 -0

    .020

    10.0

    15

    -0.0

    12

    -0.0

    09

    -0

    005

    -0.0

    03

    -0.0

    01

    0.00

    1 0.

    002

    0.00

    3 0

    00

    4

    1 e20

    0

    -5.6

    38

    -4.4

    16

    -3.5

    22

    -2.8

    45

    -2.3

    17

    - 1 . 897 - 1 557

    -1

    a27

    8 - 1 0

    45

    -0.8

    50

    -0 a

    685

    -0

    544

    -0 a

    424

    -0.3

    61

    -0.3

    22

    -0.2

    87

    -0.2

    70

    -0.2

    54

    -0

    238

    -0 a

    224

    -0.2

    10

    -0 . 180

    -0

    .148

    -0

    .125

    -0

    10

    6

    -0.0

    80

    -0

    06 1

    -0

    .047

    -0

    03

    7 -0

    .029

    -0

    023

    -0.0

    18

    -0.0

    14

    -0.0

    00

    *0.0

    04

    -0.0

    01

    0.00

    1 0

    003

    0.00

    5 0.

    006

    1 e50

    0

    -5.6

    97

    -4.4

    79

    -3

    588

    92.9

    13

    -2.3

    87

    -1.9

    69

    -1 e

    630

    -1 r3

    52

    -1 e

    l20

    -0

    925

    -0.7

    60

    -0e6

    2U

    -0.5

    00

    -0.4

    37

    -0

    398

    -0.3

    62

    -0

    344

    -0 e

    328

    -0.3

    12

    -0

    297

    -0.2

    82

    -0

    242

    -0.1

    93

    -0.1

    60

    -0.1

    35

    -0.1

    00

    -0

    076

    -0.0

    59

    -0

    046

    -0

    036

    -0.0

    28

    -0.0

    22

    -0.0

    17

    -0.0

    09

    -0.0

    04

    w0.

    001

    0.00

    2 0.

    003

    0.00

    6 0.

    007

    2.00

    0

    -5 . 759

    -4

    .547

    -3

    .661

    -2

    .990

    -2

    .468

    -2.0

    52

    11.7

    15

    -1.4

    39

    -1 e

    208

    -1.0

    15

    -0.8

    51

    -0.7

    11

    -0.5

    91

    -0.5

    27

    -0.4

    88

    00.4

    52

    -0.4

    34

    -0.4

    17

    ~0

    .40

    1

    -0.3

    85

    -0.3

    70

    -0.3

    27

    -0

    267

    -0.2

    20

    -0.1

    84

    -0.1

    34

    -0.1

    01

    -0

    077

    -0

    060

    -0

    046

    -0

    036

    -0

    028

    -0.0

    21

    -0.0

    12

    -0.0

    05

    -0.0

    01

    0.00

    3 0.

    005

    0.00

    8 0.

    010

    3.00

    0

    -5.8

    10

    -4.6

    11

    -3.7

    35

    -3.0

    71

    -2.5

    55

    -2.1

    45

    -1.8

    12

    -1.5

    39

    -1 e

    312

    -1.1

    21

    -0 e

    958

    -0.8

    19

    -0.7

    00

    -0.6

    37

    -0.5

    98

    -0.5

    61

    -0.5

    43

    -0

    e52

    6 -0

    SO

    9

    -0.4

    93

    -0.4

    77

    -0.4

    33

    -0

    .368

    -0

    .312

    -0

    .266

    -0.1

    95

    -0

    146

    -0.1

    11

    -0.0

    85

    -0

    065

    -0.0

    50

    -0.0

    38

    -0

    029

    -0.0

    15

    -0.0

    06

    0.00

    1 0.

    005

    0.00

    9 0.

    013

    0.01

    6

    5.00

    0

    -5.7

    82

    -4.6

    08

    -3.7

    52

    -3.1

    04

    -2.6

    01

    -2.2

    01

    -1 e

    878

    -1.6

    12

    -1.3

    91

    -1.2

    04

    - 1 046

    -0

    .911

    -0

    79

    4 -0

    .732

    -0

    .693

    -0.6

    57

    -0.6

    39

    -0.6

    22

    -0.6

    05

    -0.5

    89

    -0.5

    73

    -0.5

    29

    -0 A

    62

    -0

    .403

    -0

    .352

    -0.2

    69

    -0.2

    05

    -0.1

    57

    -0.1

    20

    -0.0

    92

    -0

    069

    -0

    052

    -0

    037

    -0.0

    17

    -0.0

    03

    0.00

    7 0.

    014

    0.01

    8 0.

    025

    0.02

    8

    7.00

    0

    -5.6

    79

    -4.5

    30

    -3.6

    94

    -3.0

    63

    -2

    572

    92.1

    83

    -1 e

    86

    9

    -1.6

    11

    -1 a

    396

    -1 e

    215

    -1 0

    62

    -0.9

    30

    -0.8

    17

    -0

    756

    -0.7

    19

    -0 e

    683

    -0 e

    666

    -0.6

    49

    -0.6

    33

    -0.6

    17

    -0.6

    01

    -0 e