Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 A NEW COMPUTERIZED APPROACH TO Z-FACTOR DETERMINATION Kingdom K. Dune, Engr. Oriji, Bright N, Engr. Dept. of Petroleum Engineering, Rivers State University of Science & Tech., Port Harcourt, Nigeria Abstract The compressibility factor of natural gases is a parameter that is used for various engineering purposes in the petroleum industry. The Standing and Katz correlation, among other methods, is a widely accepted method of determining z-factor manually from charts for natural gas of either known or unknown composition. A major setback is that, for computer-based applications, it is not convenient to obtain z by this means. This paper highlights the limitations of the other direct z-factor determination methods and then presents a new approach of computing z-factor based on the Standing and Katz correlation, which eliminates the limitations observed with the other methods. A set of z-factor equations were developed by regressing data (in different ranges of pseudo-reduced temperatures and pressures) obtained from the Standing-Katz and Brown et al correlations using a Visual Basic program. Results obtained from this approach were checked against those from the other correlations and were found to be more accurate than the other methods, with average absolute error in z less than 0.1%. A subroutine for calculating z-factor could easily be incorporated into any window-application program written in MS Excel, MATLAB or visual Basic, using equations developed in this approach when determining the properties of natural gas, estimating gas reserves, sizing oil and gas separators, designing gas transmission pipelines, and pressure traverses in pipes for multiphase flow conditions. A standard z-factor table may also be computed using the set of equations for all ranges of pseudo-reduced temperatures and pressures. Key Words: Compressibility, z-factor, correlation, pseudo-pressure, and pseudo-temperature 64 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 1. Introduction The z-factor comes into play for various engineering purposes, which include estimation of gas reserves, design of oil and gas separators, design of pipelines for the transmission of produced gas, among others. A number of these tasks or procedures have been developed in such a way that makes it necessary to employ the services of a computer in order to accomplish them in reasonable time. Carrying out such tasks by hand would make it lengthy, tedious and as such time consuming. An example of such tasks or procedures is the determination of pressure traverses in pipes for multiphase flow conditions. The Beggs and Brill method of calculating pressure traverses, for instance, is one that requires the gas compressibility factor. This method, involving about 21 steps, is an iterative one wherein a pressure drop is obtained at the end of each iteration using, among other data, an initial assumed pressure drop. If the difference between the initial and calculated pressure drops is substantial, the iteration is repeated with the calculated pressure drop in each iteration serving as the assumed pressure drop for the next iteration. This process is continued until the difference between the assumed and calculated pressure drops is small. Arriving at a value for the final pressure drop typically requires a number of iterations. What this means is that the working or operating pressure changes with each successive iteration making it a necessity to obtain, for each iteration, all data that are pressure dependent one of which is the gas z-factor. It is quite evident from the foregoing that manually obtaining z from the chart and entering the value into the computer, for each iteration, is rather inconvenient as it would undoubtedly slow down the computation process. Programming such tasks as the Beggs and Brill method (Brown and Beggs, 1977) for calculating pressure traverses in pipes for multiphase flow conditions cut down on the amount of time required for the calculation. Such reduction in computation time could be increased if a means was devised to incorporate the determination of gas compressibility factor into the program thus eliminating the need to manually obtain it from the chart for successive iterations. How can this be accomplished? This paper reviews existing literature and presents a new approach for determining z-factor for computer-based applications. Three other correlationsthat can be programmed for useconsidered in this paper are those of Hall and Yarborough, Beggs and Brill, and Drankchuk 65 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 and Abou-Kassem. The limitations that make them unfit for use for engineering purposes requiring precision are highlighted. 2. Background There are various correlations available for the calculation of gas compressibility factors. Using these correlations or equations of state (EOS), one can program the computer to solve directly for z. The correlations or equations of state considered for such purpose are Standing & Katz (1959), Hall and Yarborough, Beggs and Brill, and Dranchuk and Abou-Kassem. Standing and Katz Correlation Since z is a function of the gas pseudo-reduced temperature (Tpr) and pressure (Ppr), it is necessary to first determine the pseudo-critical temperature (Tpc) and pressure (Ppc) of the gas and subsequently use these to obtain the pseudo-reduced temperature (Tpr) and pressure (Ppr). For natural gas of known composition, the pseudo-critical pressure and temperature can be determined from Kay's mixing rule (Bradley, 1987) which gives these properties as: Ppc = yi Pci (1) Tpc = yi Tci (2) Where Ppc = pseudo-critical pressure of gas mixture, Tpc = pseudo-critical temperature of gas mixture, Pci = critical pressure of component i in the gas mixture, Tci = critical temperature of component i in the gas mixture, and yi = mole fraction of component i in the gas mixture For a gas whose complete analysis is not known, a correlation developed by Brown et al can be used. This correlation, presented in graphical form, relates the pseudo critical temperatures and pressures of naturally occurring systems with their specific gravities (Katz et al, 1959). Having determined the Tpr and Ppr, z may be obtained from either the Standing-Katz (Fig. 2) or the Brown et al chart (Fig. 3) Hall-Yarborough Equation The equation given by Hall and Yarborough (Ikoku, 1984) is given below: z 0.06125Pprte 1.21t y 2 (3) 66 Transnational Journal of Science and Technology where August 2012 edition vol. 2, No.7 t = Tc / T, and y = the reduced density which is obtained as the solution of the equation: y 0.06125Pprte 1.2 1 t 2 y y 2 y3 y 4 14.76t 9.76t 2 4.58t 3 y 2 3 1 y 90.7t 242.2t 2 42.4t 3 y 2.18 2.82t 0 This method is designed specifically to fit the Standing-Katz charts. Since the equation contains both z and M (which is a function of z) the solution is thus arrived at by iteration using the NewtonRaphson method. The Beggs and Brill Correlation The correlation by Beggs and Brill (Golan and Whitson, 1986) for the calculation of z is given below: z A 1 Ae B CPpr D (4) Where: A 1.39 Tpr 0.92 0.5 0.36Tpr 0.101 0.066 0.037 Ppr 2 0.32 Ppr 6 B 0.62 0.23T pr Ppr 9 T pr 0.86 10 T pr 1 C 0.132 0.32 log Tpr , D 10 and 0.3106 0.49T pr 0.1824T pr 2 The Dranchuk and Abou-Kassem Equation of State Dranchuk and Abou-Kassem (Lee and Wattenbarger, 1996) developed their equation of state primarily to estimate the z factor with computer routines. The form of the Dranchuk and AbouKassem EOS is: z 1 c1 Tpr pr c2 Tpr pr c3 Tpr pr c4 pr Tpr Where 2 5 2 (5) pr = 0.27Ppr/(zTpr); c2 Tpr A6 A7Tpr A8Tpr ; 1 3 4 5 c1 Tpr A1 A2Tpr A3Tpr A4T4 A5Tpr ; 1 c3 Tpr A9 A7Tpr A8Tpr 2 ; and c 4 T pr pr A 1 A 2 10 11 pr 2 pr Tpr 2 The constants A1 through A11 are as follows: A1 = 0.3265; A2 = -1.07; A3 = -0.5339; 67 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 A4 = 0.01569; A5 = -0.05165; A6 = 0.5475; A7 = -0.7361; A8 = 0.1844; A9 = 0.1056; A10 = 0.6134; A11 = 0.721 The Dranchuk and Abou-Kassem EOS must be solved iteratively since the z factor appears on both sides of the equation. The solution of this equation can be obtained by employing a root solving technique such as the Newton's method or the secant method. Oriji (2003) while programming these methods made the following observations: The Beggs and Brill method, while being quite accurate for certain ranges, is not applicable when Tpr < 0.92. In determining the value of the temperature dependent term A, it is necessary to evaluate the square root of (Tpr – 0.92) which would mean an imaginary root when Tpr < 0.92. Also, for some values of Tpr and Ppr, the temperature and pressure dependent term B, gets so large that evaluating eB results in an overflow of values. Negative values for z were sometimes obtained from the method for some values of Tpr and Ppr The Dranchuk and Abou-Kasem method, for the most part, gave good results for z, but the EOS involves the use of an iterative method such as the Newton's method, necessitating an assumption before convergence would occur. Once convergence is obtained the final value is given as the calculated z factor. It was, however, observed that in some instances different initial or assumed values of z resulted in convergence to different values at the end of the iteration thus resulting in different final values for z for the same set of Tpr and Ppr values. There were even cases where using certain initial values for z resulted in a negative value for compressibility factor. So despite its accuracy, this method for obtaining z factor may not be incorporated into a design program since it is not possible to predict or determine when such erroneous values may result. Therefore, another method was sought that would give values for the gas compressibility factor without the limitations highlighted above. 3. Theoretical Development This approach is based on the Standing and Katz method which is generally accepted as the industry standard and were developed from data collected on methane and natural gases (Bradley, 1987). In addition to the Standing-Katz charts, the charts by Brown et al for low-pressure systems were also used. 68 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 The compressibility factor charts are essentially curves with the gas compressibility factor, z, being a function of the pseudo-reduced pressure Ppr. These curves appear on the charts for various values of Tpr the pseudo-reduced temperature. In this method, pseudo-reduced pressure values were selected and regressed with corresponding z values obtained from the charts to give equations that expressed z as a function of Ppr. This regression process had to be carried out for each pseudo-reduced temperature value on the chart. In a bid to ensure that the regression process gave rise to equations that were as accurate and reliable as possible, two regression exercises were carried out for each set of values. These two different exercises were carried out in such a way that they yielded two different equations – one linear and the other quadratic. Both equations expressed z in terms of Ppr. The equations were of the form: z = A(Ppr) + B (6) z = A(Ppr)2 + B(Ppr) + C (7) Where A, B, C are constants These two equations were subsequently tested over a range of Ppr values and the equation that gave z values that were more in agreement with values obtained from the charts was adopted. In some cases the linear equation proved more accurate while in others the quadratic was more accurate. Table 1 shows a sample data and the resultant regression equation. In many cases, however, it was not possible to obtain an equation that gave accurate results for the entire range of Ppr values. Thus to ensure that the computer generates accurate values for z, it was necessary in such cases to divide the range of Ppr values into several sub-ranges and then obtain equations for these sub-ranges. In these instances also, two equations – one linear and the other quadratic – were obtained and tested for each of these sub-ranges. The more accurate and reliable equations were adopted (See Table 2). There were instances where it was not possible to obtain satisfactory equations that give accurate z-factor for certain ranges of Ppr values. In these cases, a number of values of Ppr along with the corresponding z-factor values were selected for interpolation. Using these predetermined values 69 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 one can obtain accurate z values for this range of Ppr via interpolation. A summary of the equations derived from the regression processes and by means of which z values may be obtained is found in Table 3 As was the case for the z-factor, various values of specific gravity were regressed with the corresponding values of pseudo-critical temperature and pressure respectively setting the specific gravity as the independent variable. This process yielded two (2) equations of the form: Xpc = A (g) 2 + B (g) + C (8) Where Xpc = pseudo-critical constant (temperature or pressure), g = specific gravity of natural gas and A, B, C are constants Dune and Oriji (2007) regressed data obtained from the Standing and Katz method and obtained the following equations: Tpc = 158.01 + 342.12(g) – 16.04(g)2 (9) Ppc = 688.634 – 21.983(g) – 13.886(g)2 (10) 4. The Computer Program Using the equations in Table 3 in conjunction with equations (9) and (10), a Visual Basic program (Siler and Spotts, 1998), to compute compressibility factor (z), was developed. For a gas of unknown composition, the program accepts the gas specific gravity as an input parameter and with this computes the gas pseudo-critical temperature and pressure from equations (9) and (10). If, however the gas composition is known, equations (1) and (2) are utilised to compute Tpc and Ppc after the gas composition has been entered as required. With the Tpr and Ppr values, z can be got using the appropriate equation from Table 3. For those range of Ppr values for which an adequate equation could not be found, a routine that interpolates between predetermined values of Ppr and z, to give the required z value, was incorporated into the program. The interpolation routine was extended to cover the entire process of determining z so that even if the pseudo-reduced temperature value is not explicitly incorporated into the program, it is still able to give a value for z. For example, if the input data results in a Tpr value of 1.45 and a Ppr 70 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 value of 1.2, the program obtains a value for z by determining values for z at Tpr = 1.4 and Tpr = 1.5 when Ppr =1.2 and then interpolating between these values to give the appropriate value for z. 5. Results and Discussion Results obtained from the computer program were analyzed to ascertain their level of accuracy. In order to determine the accuracy of the compressibility factor values obtained from this method, z values obtained from it were compared in Table 4 with those obtained from the correlations by Hall and Yarborough, Beggs and Brill, Dranchuk and Abou-Kassem and manual reading of the appropriate compressibility factor charts (Standing and Katz) for several pseudoreduced temperature and pressure values. Table 4: Compressibility Factor, Z, for Various Methods. Tpr Ppr This Hall- Beggs & Dranchuk & Standing & Approach Yarborough Brill Abou-Kassem Katzs 0.75 0.048 0.9512 1.0069 N/A 0.9546 0.951* 1.30 0.020 0.9950 0.8840 0.9977 0.9969 0.9967* 1.62 0.065 0.9970 1.0005 0.9959 0.9950 0.9956* 1.20 1.500 0.6740 0.1605 0.6759 0.6532 0.6730* 1.80 0.900 0.9562 1.0036 0.9599 0.9550 0.9700 2.00 1.200 0.9718 1.0041 0.9690 0.9621 0.9700 2.60 13.000 1.2693 1.3987 0.8222 1.2732 1.0500 2.88 6.000 1.0615 1.1436 -0.4410 1.0607 1.0620 1.76 3.500 0.8776 0.9739 0.8727 0.8818 0.8768 1.25 10.200 1.1824 1.0038 N/A 1.1818 1.1825 *Values were obtained from z-factor charts developed by Brown et al (Bradley, 1987; Fig 3) The extent to which results from this approach and the other methods deviated from those obtained from the Standing-Katz method for a given Tpr of 1.20 can be seen from Table 5. A plot of absolute error versus pseudo reduced pressure for the different methods were made (Fig. 1). A look at Table 5 as well as the plot (Fig. 1) reveals that the results from this approach are actually more accurate and in harmony with those obtained from the Standing-Katz and Brown et al methods than those from the other correlations considered. A measure of the degree of accuracy of the various methods is seen when one considers the error in computing z-factor from the various methods. These errors are computed with the results obtained from the Standing-Katz and Brown et al charts serving as the reference. Whereas errors greater than 1% were obtainable (in some cases) with the 3 correlations considered, the error 71 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 associated with this approach do not exceed 0.2%. Indeed, the average error associated with this approach, for the Tpr and Ppr values indicated, is less than 0.1%. This is much better than the averages of the other correlations which are in excess of 0.5%. Thus utilizing z values from this approach yields results that are accurate, reliable and acceptable. Table 5: Absolute Error in z, % Deviation from Standing & Katz for Tpr = 1.20 z-factor Absolute Error in z, % Ppr HallYarboro ugh Beg Dranchuk gs & & Brill AbouKasem 0.0 0.9934 48 0.5 0.99 0.9904` 23 0.8934 0.90 This Appro ach Standi ng & HallYarboro ugh Beggs Dranchu & k- Brill Abou- Katz This Appro ach Kasem 0.9897 0.9899 0.3536 0.2424 0.0505 0.0202 * 0.8951 0.9000 0.900* 0.7333 0.2889 0.5444 0.000 0.8027 0.8160 0.815* 1.8528 0.2822 1.5092 0.1227 0.6532 0.6740 0.673* 2.2883 0.4309 2.9421 0.1486 0.5181 0.5199 0.520 0.3462 4.2692 0.3654 0.0192 0.5632 0.5661 0.567 0.9171 1.1464 0.6702 0.1587 0.7937 0.7897 0.790 0.0759 1.9367 0.4684 0.0380 0.9870 0.9902 0.989 0.4247 0.9707 0.2022 0.1213 26 0.9 0.7999 0.81 27 1.5 0.6576 0.67 59 2.5 0.5218 0.49 78 3.5 0.5618 0.56 05 6.0 0.7906 0.80 53 8.0 0.9848 0.99 86 72 Transnational Journal of Science and Technology 10. 1.1963 1.20 2 13. August 2012 edition vol. 2, No.7 1.1959 1.1952 1.195 0.1088 1.1967 0.0753 0.0167 93 1.4602 N/A 1.4550 1.4565 1.458 0.1509 N/A 0.2058 0.1029 1.6452 N/A 1.6358 1.6432 1.643 0.1339 N/A 0.4382 0.0122 0 15. 0 Graph of Error in z-Factor Vs Ppr 4.5 3.75 Absolute Error (%) 3 Hall-Yarborough Beggs & Brill 2.25 Dranchuk & Abou-Kassem This Approach 1.5 0.75 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Ppr Average Absolute Error, % 0.6714 1.196 0.6792 0.0691 *Values were obtained from z-factor charts developed by Brown et al4 (Fig 3) Figure 1: Absolute Error in z, % Deviation from Standing & Katz for Tpr = 1.20 73 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 6. Conclusion A comparison of results from this approach with those obtained from other methods shows that the results from this approach are more reliable and accurate. The method is therefore fit for use for all computer-based applications that require the gas compressibility factor. A look at the summary of equations obtained and utilized by this method reveals that the equationsfor the various ranges of Tpr and Ppr are numerous. Programming these equations for use is therefore recommended as the most practical way of using this method. References: Bradley, H. B.: Petroleum Engineering Handbook, SPE, Richardson, TX, chap. 12, chap 20, 1987. Brown, K. E. and Beggs, H. D.: The Technology of Artificial Lift Methods Vol. 1, PennWell Books, Tulsa, OK, 1977, p 85. Dune, K. K. & Oriji, B. N.: “Alternative correlation for the computation of critical temperature & pressure.” Global Journal of Engineering Research, Calabar, vol. 5, no. 1, 2007, pp 69-74, Golan, M. and Whitson, C. H.: Well Performance, D. Reidel Publishing Co., Dordrecht, Holland, 1986, pp. 17 – 21. Ikoku, C. U.: Natural Gas Production Engineering, John Wesley and Sons, N.Y., 1984. Katz, D. L. et al: Handbook of Natural Gas Engineering, McGraw-Hill Books Co., N.Y., 1959. Lee, J. and Wattenbarger, R. A.: Gas Reservoir Engineering, SPE, Richardson, TX, 1996, pp. 6 – 7. Oriji, B. N.: Separator Design: A Computerized Approach, B.Tech Thesis, Rivers State University of Science and Technology, 2003 pp 14 – 20. Siler, B. and Spotts, J.: Special Edition Using Visual Basic 6, Que, 1998. Nomenclature e -Euler's constant 2.7182 74 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 -gas specific gravity M -Molar mass of gas P -pressure, psia Pc -critical pressure, psia Pci -critical pressure of ith component in gas mixture, psia Ppc -pseudo-critical pressure, psia Ppr -pseudo-reduced pressure g -density of gas, lbm/ft3 pr -pseudo-reduced density T -temperature, oR Tc -critical temperature, oR Tci -critical temperature of ith component in gas mixture, oR Tpc -pseud0-critical temperature, oR Tpr -pseudo-reduced temperature yI -mole fraction of ith component in the gas mixture z -gas compressibility factor Table 1: Regression Data Obtained From Brown et al z Factor Chart Tpr = 0.9 0 Ppr 0.07 Selected Ppr Corresponding z 0.010 0.9947 0.020 0.9893 0.035 0.9814 0.050 0.9732 0.060 0.9678 Resultant Equation: z = - 0.537647 (Ppr) + 1.000098 75 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 Table 2: Regression Data Obtained From Standing-Katz z Factor Chart for Tpr = 2.2 10 Ppr 15 4 Ppr 10 Ppr z Ppr Z 10.5 1.17 5 0.988 12.0 1.238 7 1.045 13.0 1.280 8 1.077 14.0 1.322 9 1.113 15.0 1.361 10 1.156 Resultant equation: z = 0.04109 Ppr + 0.745537 Resultant equation: z 0.001988Ppr2 0.003488Ppr 0.921786 76 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 Figure 2: Compressibility Factor Chart of Natural Gases After Standing and Katz 2 Fig 3: Gas Compressibility at Low Reduced Pressures After Brown et al 4 Table 3: Summary of Equations used to evaluate compressibility factor, z Tpr Range of Ppr Equation for z 0.60 0 ≤ Ppr ≤ 0.016 z 3.527273Ppr 1.000075 0.65 0 ≤ Ppr ≤ 0.036 z 1.9Ppr 1 0.70 0 ≤ Ppr ≤ 0.07 1.3610345Ppr 0.9998172 0.75 0 Ppr 0.07 z 1.017619Ppr 1.000017 0.8 0 Ppr 0.07 z 0.815Ppr 0.99994 0.8 0.07 < Ppr 0.27 0.85 0 Ppr 0.07 z 0.65923Ppr 0.99995 0.90 0 Ppr 0.07 z 0.537647 Ppr 1.000098 0.90 0.07 < Ppr 0.54 0.95 0 Ppr 0.07 z 0.641854Ppr 0.611208Ppr 0.929185 2 Interpolate Using Predetermined z 0.459742Ppr 1.000105 Values 77 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 0.95 0.07 < Ppr 0.72 1.0 0 Ppr 0.07 1.0 0.07 < Ppr 0.7 1.0 0.7 < Ppr 0.9 1.0 0.9 < Ppr 1.0 1.0 1.0 < Ppr 1.2 1.0 1.2 < Ppr 1.6 1.05 0 Ppr 0.8 1.05 0.8 < Ppr 1.6 Interpolate Using Predetermined 1.05 1.05 1.6 < Ppr 2.0 2.0 < Ppr < 4.0 Values 1.05 4.0 Ppr < 7.0 z 0.002286Ppr 0.149143Ppr 0.026571 1.05 7 Ppr 15 z 0.000515Ppr 0.116867 Ppr 0.115012 1.10 0 Ppr 0.07 1.10 0.07 < Ppr 1.6 1.6 < Ppr < 2.3 Interpolate Using Predetermined z 0.026257 Ppr 0.050591Ppr 0.355425 1.10 2.3 Ppr 3.5 3.5 < Ppr < 7.0 1.10 7 Ppr 15 z 0.102012Ppr 0.189738 1.2 0 Ppr 0.07 z 0.216207 Ppr 1.000147 1.2 1.2 0.07 < Ppr 1.6 1.6 < Ppr < 3.4 1.2 3.4 Ppr < 5.5 1.2 5.5 Ppr 8.0 1.2 8 Ppr 15 1.3 1.3 0 Ppr 1.6 1.6 < Ppr < 3.8 1.3 3.8 Ppr < 6.0 1.3 6 Ppr 15 1.40 0 Ppr 0.07 1.40 0.009738Ppr 0.121092Ppr 1.000687 1.40 0.07 < Ppr 1.6 1.6 < Ppr < 4.0 1.40 4.0 Ppr < 6.0 z 0.011371Ppr 0.056954Ppr 0.76088 1.40 6 Ppr 15 1.50 0.0 Ppr 1.6 0.013426Ppr 0.10381Ppr 1.004459 1.50 1.6 < Ppr 3.0 z 0.015808Ppr 0.127019Ppr 1.013774 1.10 1.10 Interpolate Using Predetermined z 0.3818375Ppr 1.000112 Values Interpolate Using Predetermined Interpolate Using Predetermined Values Interpolate Using Predetermined Values 2 z 2.817158Ppr 6.451697 Ppr 3.892978 Values z 0.131016Ppr 0.215878Ppr 0.27348 2 z 0.083333Ppr 0.291905Ppr 0.997714 2 z 0.15Ppr 0.465Ppr 0.61 2 z 0.001027 Ppr 0.121228Ppr 0.033131 2 2 2 z 0.275172Ppr 1.000131 z 0.230864Ppr 0.884788Ppr 1.21503 2 Values 2 z 0.0007 Ppr 0.120452 pr 0.086866 2 Interpolate Using Predetermined Interpolate Using Predetermined Values 2 z 0.002686Ppr 0.062909Ppr 0.313023 Values z 0.000464Ppr 0.093761Ppr 0.210439 2 z 0.09333Ppr 0.243254 Interpolate Using Predetermined Interpolate Using Predetermined Values 2 z 0.010114Ppr 0.026463Ppr 0.59884 Values z 0.086983Ppr 0.28228 z 0.12459Ppr 1.000008 2 z 0.03299Ppr 0.221064Ppr 1.0735735 2 2 z 0.077786Ppr 0.361643 2 2 78 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 1.50 3.0 < Ppr < 5.0 z 0.018409Ppr 0.128492Ppr 0.994725 1.50 5.0 Ppr < 8.0 z 0.003429Ppr 0.015229Ppr 0.647786 1.50 8.0 Ppr 11.5 z 0.071736Ppr 0.404538 1.50 11.5 < Ppr 15 z 0.0711Ppr 0.4254 1.60 0 Ppr 0.07 1.60 0.07 < Ppr 1.6 z 0.008579Ppr 0.074544Ppr 0.999739 1.60 z 0.012337 Ppr 0.0975Ppr 1.006528 1.60 1.6 < Ppr 3.6 3.6 < Ppr <4.9 1.60 4.9 Ppr <8.0 z 0.002429Ppr 0.020486Ppr 0.682321 1.60 8.0 Ppr 15 z 0.000345Ppr 0.059171Ppr 0.505621 1.70 z 0013197 Ppr 0.06913Ppr 1.006746 1.70 0 Ppr 1.6 1.6 < Ppr < 4.5 1.70 4.5 Ppr < 7.0 z 0.004964Ppr 0.016089Ppr 0.835179 1.70 7.0 Ppr 10 z 0.001427 Ppr 0.030492Ppr 0.682271 1.70 10 < Ppr 15 z 0.000282Ppr 0.069231Ppr 0.46612 1.80 z 0.006354Ppr 0.053396Ppr 0.999106 1.80 0 Ppr 1.6 1.60 < Ppr < 4.0 z 0.008273Ppr 0.062679Ppr 1.009321 1.80 4.0 Ppr < 7.0 z 0.005333Ppr 0.026152Ppr 0.910257 1.80 7 Ppr 11.2 z 0.00205Ppr 0.014764Ppr 0.783503 1.80 11.2 < Ppr 15 z 0.000571Ppr 0.073114Ppr 0.454186 1.90 0.0 Ppr < 3.0 z 0.005859Ppr 0.045613Ppr 1.0009 1.90 3.0 Ppr < 6.0 z 0.008275Ppr 0.057658Ppr 1.015644 1.90 6.0 Ppr 10 z 0.001357 Ppr 0.02163Ppr 0.787448 1.90 10 < Ppr 15 z 0.0536Ppr 0.6024 2.0 0 Ppr 0.07 z 0.031Ppr 1.000065 2.0 2.0 0.07 < Ppr 1.6 1.6 < Ppr < 3.5 z 0.006051Ppr 0.03969Ppr 1.002002 2.0 3.5 Ppr < 5.5 z 0.0086Ppr 0.06098Ppr 1.04517 2.0 5.5 Ppr < 10.5 z 0.001968Ppr 0.008278Ppr 0.864989 2.0 10.5 Ppr 15 z 0.0004999Ppr 0.061723Ppr 0.575096 2.2 0.0 Ppr 1.5 z 0.003754Ppr 0.023475Ppr 0.999756 2.2 1.5 < Ppr 4.0 z 0.005729Ppr 0.033753Ppr 1.011104 2.2 4.0 < Ppr 10.0 z 0.001988Ppr 0.003488Ppr 0.921786 2.2 10.0 < Ppr 15 z 0.04109Ppr 0.745537 2.4 0.0 Ppr 1.5 1.5 < Ppr < 4.0 z 0.000906Ppr 0.007386Ppr 0.998156 2.4 2 2 z 0.067933Ppr 1.000132 2 2 z 0.011365Ppr 0.075853Ppr 0.939908 2 2 2 2 z 0.011669Ppr 0.087546Ppr 1.02058 2 2 2 2 2 2 2 2 2 2 2 2 z 0.00524Ppr 0.030058Ppr 1.000274 2 2 2 2 2 2 2 2 2 z 0.005246Ppr 0.026683Ppr 1.013352 2 79 Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7 2.4 4.0 Ppr < 9.5 z 0.0013045Ppr 0.009525Ppr 0.931105 2.4 9.5 Ppr 15 z 0.000458Ppr 0.048873Ppr 0.7169 2.6 0.0 Ppr < 4.0 z 0.002286Ppr 0.007958Ppr 1.001055 2.6 4.0 Ppr < 10.0 z 0.001098Ppr 0.010063Ppr 0.949436 3.0 z 0.000699Ppr 0.006209Ppr 0.99927 3.0 0.0 Ppr 3.0 3.0 < Ppr < 10.0 z 0.001576Ppr 0.0000815Ppr 1.01047 3.0 10.0 Ppr 15 z 0.000071Ppr 0.031414Ppr 0.845286 2 2 2 2 2 2 2 80