Use of Distribution Functions: A Useful Tool To Calculate the

Ind. Eng. Chem. Res. 2000, 39, 5029-5036

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Use of Distribution Functions: A Useful Tool To Calculate the Properties of Condensate Gases Jean-Noe1 l Jaubert,* Roland Solimando, and Patrice Paricaud Laboratoire de Thermodynamique des Se´ parations, Institut National Polytechnique de Lorraine, Ecole Nationale Supe´ rieure des Industries Chimiques, 1 Rue Grandville, 54000 Nancy, France

Alain Barreau Institut Franç ais du Pe´ trole, 1 et 4 Avenue de Bois-Pre´ au, 92506 Rueil-Malmaison, France

In this paper, a compositional model is proposed to calculate the phase behavior and the physical properties of condensate gases. Such a model is based on a very classical and easy to handle cubic equation of state coupled with correlations devoted to the estimation of the hydrocarbon physical properties. Moreover, to completely define the composition of the C20+ distillation residue, three distribution functions associated with the families n-alkanes, n-alkylbenzenes, and polyaromatics are introduced. The method developed in this work shows clearly enhanced results in comparison to previously published papers in the same area. Introduction The phase behavior calculation of condensate gases is a key issue for the reservoir simulation, for the well calculations, and also for the process and transportation calculations. It is thus of high interest to have at one’s disposal a thermodynamic model able to calculate with high accuracy the properties of the phases in equilibrium (amount, density, enthalpy, etc.). However, it is well-known that thermodynamic petroleum fluid modeling suffers from some rather unfavorable conditions: on the one hand, the basic equilibrium computation tools have to be as little time consuming as possible, and only a one-fluid model using a cubic equation of state (EOS) can be used, although these are known to yield unsatisfactory results. On the other hand, the analysis of petroleum gases depends on the performance of the available chromatographic apparatus. Only the molar fractions of compounds lighter than undecane can be accurately measured if a standard chromatographic procedure is used. Heavier compounds cannot be measured as accurately because there exists a vast number of possible isomers. For this reason, a true boiling point (TBP) distillation is usually performed on the associated tank oil. In the classical procedure, the distillate is collected and divided into several cuts. Each distillated cut Cn is defined by a range of temperatures at the top of the column. By definition, the cut Cn includes all of the hydrocarbons, the boiling points of which are bounded by the boiling points of the two successive n-alkanes Cn-1 and Cn. Standard TBP distillation is generally never performed beyond the C19 cut so that the fraction C20+ collected at the bottom of the column is the heaviest component which can be dealt with. Because of the imperfect knowledge of the fluid composition and of the poor accuracy of cubic EOS, the common practice is to fit the EOS parameters of some components to the available experimental data. How* To whom the correspondence should be addressed. Fax: (+33) 3 83.35.08.11. Phone: (+33) 3 83.17.50.81. Email: [email protected].

ever, the tuning of parameters is always a difficult problem because it may lead to unreliable extrapolations and may yield errors in the prediction of properties which were not included in the fitting process.1 An alternative but less precise method is to work with purely predictive methods which only need standard analytical data.2-10 In this paper, to obtain a precise estimation of condensate gas properties and to avoid the problems linked to the tuning of the EOS parameters, it was decided first to propose a predictive way of modeling the intermediate cuts (from C12 to C19) and second to use three distribution functions to properly define the composition of the C20+ residue. In this work, EOS parameters are never tuned, but they are estimated with specific methods. Only the unknown internal composition of the heaviest fraction is adapted to experimental value restitution. In other words, the parameters of the distributions are fitted in order to reduce the deviations between calculated and experimental liquid dropout coming from a constant mass expansion at a given temperature. This paper is, in fact, an improvement of the previous work made by Sportisse et al.,11 who also used three distribution functions. However, such distributions were used to describe the composition of the C11+ fraction. Moreover, in this study, the mathematical forms of the distributions were revisited. Characterization Procedure Light Components. The light components until n-C11 need to be completely identified and analyzed. We advise the use of the following 23 pure compounds: hydrogen sulfide, nitrogen, carbon dioxide, methane, ethane, propane, isobutane, n-butane, isopentane, npentane, n-hexane, benzene, n-heptane, toluene, methylcyclohexane, n-octane, ethylbenzene, ethylcyclohexane, n-nonane, propylbenzene, propylcyclohexane, n-decane, and n-undecane. Intermediate Cuts from C12 to C19. In opposition to what was previously made by Sportisse et al.,11 the

10.1021/ie0003786 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/11/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 12, 2000

Figure 1. Comparison between the experimental and calculated compositions of the intermediate cuts: +, experimental composition; -, calculated composition using Sportisse’s model obtained by summing the mole fractions of the n-alkane, n-alkylbenzene, and polyaromatic in each cut. Table 1. List and Physical Properties of the Pure Components Used To Model the Intermediate Cuts (from C12 to C19)

cut C12 C13 C14 C15 C16 C17 C18 C19

molecules used to model the cut Ci n-C12 pentylbenzene n-C13 hexylbenzene n-C14 heptylbenzene n-C15 octylbenzene n-C16 nonylbenzene n-C17 decylbenzene n-C18 undecylbenzene n-C19 dodecylbenzene

molar weight and standard (15 °C and 1 atm) density (g/cm3) of the selected molecules MW F 170.3 148.3 184.4 162.3 198.4 176.3 212.4 190.3 226.4 204.3 240.5 218.3 254.5 232.3 268.5 246.3

0.7541 0.8622 0.7612 0.8597 0.7674 0.8574 0.7726 0.8557 0.7773 0.8543 0.7814 0.8532 0.7851 0.8524 0.7885 0.8517

characterization procedure developed in this paper represents each cut from C12 to C19 by two pure components: a normal alkane (al) and an aromatic compound (ar), which is n-alkylbenzene. Such a choice was made after realizing that the global composition calculated for each intermediate cut using the three molar distributions developed by Sportisse et al.11 led to values very different from experimental ones. An example of such a behavior is shown in Figure 1 for a given gas condensate, the composition of which is given in Table 2 (gas 3). This figure shows that each cut from C11 to C15 is only modeled by the corresponding nalkane, whereas cuts from C16 to C19 are modeled by a mixture of two molecules: n-alkane and n-alkylbenzene. Indeed, in this example, the polyaromatic components only appeared in heavier cuts. Such a plot clearly evidences that the calculated mole fraction of each intermediate cut is much higher (3 times higher for the cut C11) than the corresponding experimental value. Sportisse et al.11 could not suspect such a shortcoming of their characterization procedure because the intermediate cuts of the condensate gases described in their paper were not analyzed. In our new approach, the molar proportions xal and xar of the n-alkane and of the n-alkylbenzene in a given

Table 2. Composition of the Five Gases Selected in This Study gas composition (mol %)

1

2

3

4

5

H2S N2 CO2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 n-C6 benzene n-C7 methylcyclohexane toluene n-C8 ethylcyclohexane ethylbenzene n-C9 propylcyclohexane propylbenzene n-C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+ MWC20+ FC20+/(g‚cm-3) Fstock tank oil/(g‚cm-3)

0.000 0.412 2.072 76.621 7.401 3.477 0.484 1.266 0.421 0.644 0.836 0.068 0.918 0.342 0.225 0.816 0.312 0.308 0.453 0.120 0.146 0.347 0.437 0.314 0.304 0.226 0.176 0.138 0.127 0.114 0.076 0.399 350.0 0.8715 0.7738

0.000 0.318 2.737 76.865 7.050 3.166 0.425 1.107 0.362 0.544 0.696 0.243 0.399 0.615 0.398 0.374 0.464 0.393 0.267 0.084 0.157 0.244 0.396 0.326 0.326 0.284 0.229 0.207 0.189 0.155 0.088 0.892 365.0 0.8606 0.7987

0.000 0.543 1.777 80.430 6.430 2.911 0.458 1.049 0.344 0.472 0.645 0.292 0.465 0.608 0.185 0.317 0.373 0.170 0.205 0.115 0.083 0.272 0.289 0.224 0.212 0.176 0.133 0.131 0.123 0.076 0.065 0.427 415.0 0.8694 0.7886

1.406 3.714 2.912 82.834 4.379 1.373 0.373 0.566 0.288 0.232 0.327 0.047 0.199 0.135 0.069 0.122 0.108 0.079 0.08 0.035 0.032 0.076 0.114 0.079 0.072 0.061 0.045 0.038 0.032 0.027 0.021 0.125 360.0 0.8675 0.7754

0.26 3.35 1.88 82.64 5.54 2.02 0.43 0.77 0.34 0.33 0.41 0.05 0.18 0.21 0.06 0.16 0.15 0.08 0.11 0.05 0.05 0.12 0.17 0.12 0.11 0.08 0.07 0.05 0.05 0.04 0.03 0.09 324.0 0.8695 0.7700

cut Ci are determined by solving the following system:

{

xal

Mal Mar Mexp(Ci) + xar ) Fal Far Fexp(Ci)

xal + xar ) 1

where Fal and Far are the densities of the two selected molecules. Mexp and Fexp are respectively the experimental molar weight and density of the cut Ci. If such experimental data are not available, it is advised to use

Ind. Eng. Chem. Res., Vol. 39, No. 12, 2000 5031

Figure 2. Constant-composition liquid dropout curve calculated with the model developed in this work (the fitted distribution parameters are given in Table 3): +, experimental values; -, curve calculated with the proposed model; - - -, curve calculated with Sportisse’s model. (a) Gas 1 at T ) 372.65 K. (b) Gas 2 at T ) 368.15 K. (c) Gas 3 at T ) 391.25 K. (d) Gas 4 at T ) 398.15 K. (e) Gas 5 at T ) 377.55 K.

the values recommended by Katz and Firoozabadi.12 Such a procedure allows us to ensure that the calculated density of the cut exactly matches the experimental one. If one of the two internal mole fractions (xal or xar) is negative, the cut is simply modeled by the pure component, the mole fraction of which is positive. The pure components used in this characterization procedure and their basic physical properties (molar weight and density) are given for each cut in Table 1. C20+ Fraction. As stated in the Introduction, three distribution functions associated with the families nalkanes, n-alkylbenzenes, and polyaromatics are introduced in order to have a complete description of the C20+ fraction composition. For the three families, a Γ function was selected. Indeed, there is very little experimental data concerning the composition of con-

densate gases beyond C20, meaning that the mathematical form of the distributions may be freely chosen. This lack of data comes from the very small amount of hydrocarbons (less than 1%) present in the C20+ residue. The general form of the n-alkanes distribution is given by

w j ′i ) r1wC20+

w′i n′

∑ w′i j)20

where w′i )

M′i M′19

-(i-19)

(i - 19)R-1e

In the previous formulas, i is the cut number, w j ′i is the weight fraction, in the whole fluid, of the n-alkane having i carbon atoms, r1 represents the weight propor-

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Figure 3. Molar distributions of the simulated heavy fraction versus cut number. The model developed in this work was used: (a) gas 1; (b) gas 2; (c) gas 3; (d) gas 4; (e) gas 5.

tion of the paraffinic distribution in the C20+ fraction (0 e r1 e 1), wC20+ is the weight fraction of the C20+ fraction in the whole fluid, M′i is the molar weight of the n-alkane having i carbon atoms, and R is the position of the concentration peak (R > 1). Knowledge of the two parameters r1 and R completely defines such a distribution. Similarly, the weight fractions w j i′′ of normal alkylbenzenes are given by the following expression:

w j ′′i ) r2wC20+

w′′i n′′

∑

carbon atoms, that is, dodecylbenzene. Such a hydrocarbon is present in the C19 cut because its normal boiling point lies between the normal boiling points of n-C18 and n-C19. M′′i is the molar weight of the nalkylbenzene present in the cut Ci, that is, having i - 1 carbon atoms. By the end, the weight fractions w j ′′′ i of polyaromatics are given by the following expression:

w j ′′′ i ) r3wC20+ where

w′′′ i n′′′

where

∑ w′′′i

j)14

w′′i

j)19

w′′i )

M′′i -1 -(i-18) (i - 18)β e M′′18

r2 and β are the two parameters of this distribution. r2 represents the weight proportion of the alkylbenzenes distribution in the C20+ fraction (0 e r2 e 1). M′′18 is the molar weight of the normal alkylbenzene having 18

w′′′ i )

M′′′ i -(i-13) (i - 13)γ-1e M′′′ 13

r3 and γ are the two parameters of this distribution. M′′′ i is the molar weight of the polyaromatic present in the cut Ci, that is, having i - 6 carbon atoms. It is indeed well-known that, for a given carbon atom number, the boiling points of polyaromatic compounds are much higher than those of n-alkanes.

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As an example, the cut C30 is modeled by n-C30, an n-alkylbenzene having 29 carbon atoms and a polyaromatic component having 24 carbon atoms. To complete our description, it is necessary to strictly state that the distributions are truncated if the weight fractions are below 10-6 or if the cut number reaches 80. As a conclusion, this representation of the C20+ fraction requires six distribution parameters: r1, r2, r3, R, β, γ. However, because r1 + r2 + r3 ) 1, only five parameters are independent. These five parameters are determined in order to minimize the deviations between the calculated and experimental liquid dropout coming from a constant mass expansion. The fitting procedure is similar to the one described by Sportisse et al.11 By the end, let us recall that the liquid dropout is defined by

liquid dropout ) 100 ×

Table 3. Deviations (%) on the Dew-Point Pressure and on the Stock Tank Oil Density and Values of the Corresponding Fitted Parameters % deviation

gas

on the dew-point pressure

on the stock tank oil density

1

0.0

1.1

2

-0.1

4.6

3

0.2

1.5

4

0.1

5.9

5

0.6

2.7

average

0.2

3.2

Vliq(P) Vsat(Psat)

where Vliq(P) is the liquid volume at the pressure P and Vsat(Psat) is the gas volume at the dew pressure Psat. By using such a characterization procedure, it is not scarce that a given gas condensate was modeled by more than 80 pure components. This is not a problem any more for petroleum companies because personal computers are nowadays very powerful. The Thermodynamic Model EOS. To calculate the PVT properties of the condensate gases, the Peng-Robinson13 (PR) EOS with classical mixing rules (linear on b and quadratic on a) was used. Critical Properties and Acentric Factor. For n-alkane and n-alkylbenzenes, the critical parameters (Tc and Pc) and the acentric factor were determined using the correlations developed by Rogalski and Neau14 and by Neau et al.3 For polyaromatics, the correlations developed by Sportisse et al.11 were used. Volume Translation. To improve the calculation of the liquid volumes, a volume correction was introduced. For light components, for n-alkanes, and for n-alkylbenzenes, the volume correction was determined so that the EOS perfectly matches the liquid standard density (15 °C and 1 atm) of each pure component. For heavy hydrocarbons, the liquid standard density is not experimentally available and was thus estimated by the correlations developed by Rogalski and Neau.14 For polyaromatics, the correlation developed by Sportisse et al.11 which directly gives the volume translation was used. Binary Interaction Parameters. To enhance the predictive capabilities of the PR EOS, binary interaction parameters were introduced for binary systems involving H2S, N2, CO2, methane, ethane, or propane as lighter components. The parameters proposed by Robinson and Peng15 were used. However, when available, the values recommended by Sportisse et al.11 were preferably used. Results The compositional model developed in this study was used in order to calculate the liquid dropout curve obtained during a constant mass expansion. Many condensate gases coming from different countries were

values of the five fitted parameters

{ { { { {

r1 ) 0.251 r2 ) 0.420 R ) 69.99 β ) 14.08 γ ) 54.72 r1 ) 0.217 r2 ) 0.529 R ) 18.26 β ) 5.279 γ ) 13.11 r1 ) 0.158 r2 ) 0.578 R ) 66.83 β ) 17.41 γ ) 8.918 r1 ) 0.157 r2 ) 0.507 R ) 66.97 β ) 14.22 γ ) 6.569 r1 ) 0.378 r2 ) 0.343 R ) 6.119 β ) 15.391 γ ) 6.0725

considered. All of these data were nicely given by the French petroleum company TOTALFINA. For all of the gases studied, it was possible to reproduce with a good accuracy the liquid dropout curve. Moreover, our model was able to perfectly calculate the dew-point pressure and to predict satisfactorily the stock tank oil density. In comparison with the model previously published by Sportisse et al.,11 the results were improved. An example of such an improvement is given in Figure 2c. To illustrate our results, five gases for which the maximum liquid dropout varies between 2.0 and 24.0 were selected in our database. The composition of these five gases is given in Table 2. In Figure 2 is represented the liquid dropout curve, and in Figure 3 is represented the corresponding distributions obtained. By the end, Table 3 gives the values of the distribution parameters and the deviations on the dew-point pressure and on the stock tank oil density. For a possible comparison with the previous work by Sportisse et al.,11 Figure 4 gives the C11+ fraction composition obtained with such a model. Figure 4 concerns gas 3 and has to be compared with Figure 3c. In Figure 4, we can notice a huge amount of polyaromatic compounds in the cuts C22-C24. Such an amount is highly improbable, all the more because such components were never found during the analysis of these cuts. On the other hand, the compositional model developed in this work (see Figure 3c) makes appear such polyaromatics in heavier cuts and in much less amount, which is much more realistic. To be honest, it is important to outline that the calculated end cut composition with the proposed model is not completely satisfactory. This is because the tuned parameters are those of the distribution functions. As an example, even if there is no data to confirm or infirm the tuned composition, the hole in the C20 cut composition (see


Figure 4. Molar distributions of the simulated heavy fraction versus cut number using Sportisse’s model. The gas studied is gas 3.

Figure 5. Constant-composition liquid dropout curve calculated with different models: +, experimental values; -, curve predicted with the proposed model after fitting the distribution parameters at T ) 368.15 K; - - -, curve predicted with Avaullee et al.’s model; ‚‚‚, curve predicted with Pedersen et al.’s model. (a) Gas 2 at T ) 353.15 K. (b) Gas 2 at T ) 378.15 K.

Figure 3) is meaningless. Such a phenomenon appears because the distribution functions start at the cut C20. As a consequence, there is a discontinuity between the C19 and C20 compositions (such a problem also existed with Sportisse’s model). We, unfortunately, were unable to satisfactorily correct such a behavior. Moreover, the small but increasing amount of n-alkane which appears above the cut C65 for gases 1, 3, and 4 is probably senseless (who knows?). As a conclusion, it is probable that the calculated composition has nothing to do with the real gas condensate composition. However, the key question is rather whether the calculated composition is able to predict the gas condensate properties in other conditions. For this reason, in Figure 5 are plotted the calculated dropout curves of gas 2 at two new temperatures. In this figure, the C20+ composition has been determined by fitting the distribution parameters at T ) 368.15 K. Using such a composition, the liquid dropout curves were predicted at T ) 353.15 and 378.15 K. In Figure 5 are also plotted the curves obtained with two purely predictive models. The first one was developed by Avaulleé et al.9 and the second one by Pedersen et al.2 From this figure, it is clear that the best results are obtained with the proposed method. It is thus possible to conclude that the use of distribution

functions is a useful tool to calculate the properties of condensate gases. Conclusion The compositional model developed in this work lies on the following key points: (i) The light components are each considered as a pure compound. (ii) Each intermediate cut is modeled by two pure components: n-alkane and n-alkylbenzene. (iii) The C20+ fraction composition is represented by three distribution functions. By fitting the five parameters of the distribution functions on experimental liquid dropout data, it becomes possible first to reproduce with a good accuracy such data but also to perfectly calculate the dew-point pressure and to estimate with a small error the stock tank oil density. Moreover, even if not completely satisfactory, the composition of the C20+ fraction thus obtained looks more realistic than the one obtained with Sportisse’s model. Another key feature of the proposed method is also to be able to predict the behavior of condensate gases at temperatures different from those at which the parameters were fitted.

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Nomenclature Mexp(Ci) ) experimental molecular weight of an intermediate cut M′i ) molar weight of the n-alkane present in the cut Ci M′′i ) molar weight of the n-alkylbenzene present in the cut Ci M′′′ i ) molar weight of the polyaromatic present in the cut Ci r1 ) weight proportion of the paraffinic distribution in the C20+ fraction r2 ) weight proportion of the monoaromatic distribution in the C20+ fraction r3 ) weight proportion of the polyaromatic distribution in the C20+ fraction w j ′i ) weight fraction, in the whole fluid, of the n-alkane present in the cut Ci w j ′′i ) weight fraction, in the whole fluid, of the n-alkylbenzene present in the cut Ci w j ′′′ ) weight fraction, in the whole fluid, of the polyi aromatic present in the cut Ci xal ) internal mole fraction of the n-alkane used to model an intermediate cut xar ) internal mole fraction of the n-alkylbenzene used to model an intermediate cut Greek Letters R ) parameter of the distribution function of n-alkanes β ) parameter of the distribution function of n-alkylbenzenes γ ) parameter of the distribution function of polyaromatics Fal ) density of the n-alkane used to model an intermediate cut Far ) density of the n-alkylbenzene used to model an intermediate cut Fexp(Ci) ) experimental density of an intermediate cut

Literature Cited (1) Pedersen, K. S.; Thomassen, P.; Fredenslund, Aa. On the Dangers of Tuning Equation of State Parameters. Chem. Eng. Sci. 1988, 43, 269-278. (2) Pedersen, K. S.; Thomassen, P.; Fredenslund, Aa. Properties of oils and natural gases; Gulf Publishing Company: Houston, 1989. (3) Neau, E.; Jaubert, J. N.; Rogalski, M. Characterization of Heavy Oils. Ind. Eng. Chem. Res. 1993, 32, 1196-1203.

(4) Jaubert, J. N.; Neau, E.; Peńeloux, A.; Fressigne´, C.; Fuchs, A. Phase Equilibrium Calculations on an Indonesian Crude Oil Using Detailed NMR Analysis or a Predictive Method To Assess the Properties of the Heavy Fractions. Ind. Eng. Chem. Res. 1995, 34, 640-655. (5) Jaubert, J. N.; Neau, E. Characterization of Heavy Oils. 2. Definition of a Significant Characterizing Parameter to Ensure the Reliability of Predictive Methods for PVT Calculations. Ind. Eng. Chem. Res. 1995, 34, 1873-1881. (6) Jaubert, J. N.; Neau, E.; Avaulleé, L. Characterization of Heavy Oils. 3. Prediction of Gas Injection Behavior: Swelling Test, Multicontact Test, Multiple Contact Minimum Miscibility Pressure and Multiple Contact Minimum Miscibility Enrichment. Ind. Eng. Chem. Res. 1995, 34, 4016-4032. (7) Avaulleé, L.; Trassy, L.; Neau, E.; Jaubert, J. N. Thermodynamic Modeling for Petroleum Fluids. I. Equation of State and Group Contribution for the Estimation of Thermodynamic Parameters of Heavy Hydrocarbons. Fluid Phase Equilib. 1997, 139, 155-170. (8) Avaulleé, L.; Neau, E.; Jaubert, J. N. Thermodynamic Modeling for Petroleum Fluids. II. Prediction of PVT Properties of Oils and Gases by Fitting One or Two Parameters to the Saturation Pressures of Reservoir Fluids. Fluid Phase Equilib. 1997, 139, 171-203. (9) Avaulleé, L.; Neau, E.; Jaubert, J. N. Thermodynamic Modeling for Petroleum Fluids. III. Reservoir Fluid Saturation Pressures. A Complete PVT Property Estimation. Application to Swelling Test. Fluid Phase Equilib. 1997, 141, 87-104. (10) Avaulleé, L.; Duchet-Suchaux, P.; Durandeau, M.; Jaubert, J. N. A New Approach in Correlating the Thermodynamic Oil Properties. J. Pet. Sci. Eng. 2000, submitted for publication. (11) Sportisse, M.; Barreau, A.; Ungerer, P. Modeling of Gas Condensates Properties Using Continuous Distribution Functions for the Characterization of the Heavy Fraction. Fluid Phase Equilib. 1997, 139, 255-276. (12) Katz, D. L.; Firoozabadi, A. Predicting Phase Behavior of Condensate/Crude-Oil Systems Using Methane Interaction Coefficients. J. Pet. Technol. 1978, 20, 1649-1656. (13) Peng, D. Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59-64. (14) Rogalski, M.; Neau, E. A group contribution method for prediction of hydrocarbon saturated liquid volumes. Fluid Phase Equilib. 1990, 56, 59-69. (15) Robinson, D. B.; Peng, D. Y. The Characterization of the heptanes and heavier fractions for the GPA Peng-Robinson programs. GPA Research Report 28, Mar 1978.

Received for review April 3, 2000 Accepted September 8, 2000 IE0003786

Use of Distribution Functions: A Useful Tool To Calculate the

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