Api mpms 11 1 2004 (american petroleum institute)

Manual of Petroleum Measurement Standards Chapter II-Physical Properties Data Section I-Temperature and Pressure Volume Correction Factors for Generalized Crude Oils, Refined Products, and Lubricating Oils Adjunct to: ASTM D 1250-04 and IP 200/04 `,,`,,,-`-`,,`,,`,`,,` - MAY 2004 Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale `,,`,,,-`-`,,`,,`,`,,` - Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale Manual of Petroleum Measurement Standards Chapter II-Physical Properties Data Section I-Temperature and Pressure Volume Correction Factors for Generalized Crude Oils, Refined Products, and Lubricating Oils Adjunct to: ASTM D 1250-04 and IP 200/04 May 2004 `,,`,,,-`-`,,`,,`,`,,` - Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale `,,`,,,-`-`,,`,,`,`,,` - API publications necessarily address problems of a general nature With respect to particular circumstances, local, state, and federal laws and regulations should be reviewed API is not undertaking to meet the duties of employers, manufacturers, or suppliers to warn and properly train and equip their employees, and others exposed, concerning health and safety risks and precautions, nor undertaking their obligations under local, state, or federal laws Information concerning safety and health risks and proper precautions with respect to particular materials and conditions should be obtained from the employer, the manufacturer or supplier of that material, or the material safety data sheet Nothing contained in any API publication is to be construed as granting any right, by implication or otherwise, for the manufacture, sale, or use of any method, apparatus, or product covered by letters 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responsible for complying with all the applicable requirements of that standard API does not represent, warrant, or guarantee that such products in fact conform to the applicable API standard All rights reserved No part of this work may be reproduced, stored in a retrieval system, or transmitted by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from the publisher Contact the Publisher, API Publishing Services, 1220 L Street, N.W., Washington,D.C.20005 CopyrightO 2004 American Petroleum Institute Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale FOREWORD `,,`,,,-`-`,,`,,`,`,,` - M I publications may be used by anyone desiring to so Every effort has been made by the Institute to assure the accuracy and reliability of the data contained in them; however, the Institute makes no representation, warranty, or guarantee in connection with this publication and hereby expressly disclaims any liability or responsibility for loss or damage resulting from its use or for the violation of any federal, state, or municipal regulation with which this publication may conflict Suggested revisions are invited and should be submitted to AFT, Standards department, 1220 L Street, NW, Washington, DC 20005, standards@api.org Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale CONTENTS Temperature and Pressure Volume Correction Factors for Generalized Crude Oils, Refined Products, Section and Lubricating Oils 11.1.0 Implementation Guidelines 11.1.1 11.1.1.1 11.1.1.2 11.1.1.3 11.1.1.4 11.1.1.5 11.1.1.6 Introduction & History Early Temperature and Pressure Correction Tables 1952 Temperature Correction Tables 1980 Temperature Correction Tables 1981 Pressure Correction Tables Changes to Previous Standards Customary Temperature & Pressure Correction Tables 11.1.2 Purpose 11.1.2.1 Significance 11.1.2.2 Scope Temperature, Pressure, and Density Limits 11.1.2.3 11.1.2.4 Classification of Liquids 11.1.2.4.1 Crude Oil 5 11.1.2.4.3 Lubricating Oils 11.1.2.4.4 Special Applications 11.1.2.5 Application of Tables to S 11.1.2.5.1 Waxy Crudes 11.1.2.5.2 Natural and Drip Gasolines s 11.1.2.5.4 11.1.2.5.5 11.1.2.5.6 LNG Ethylene and Propylene Butadiene 11 11 11 11 11.1.2.5.9 11.1.2.5.10 11.1.2.5.11 Reformulated Fuels MTBE JP-4 12 12 11.1.2.5.13 12 13 Gasohol 11.1.3 11.1.3.1 11.1.3.2 11.1.3.3 11.1.3.4 11.1.3.5 11.1.3.6 11.1.3.7 11.1.3.8 11.1.3.9 11.1.3.10 11.1.3.11 Outline of Calculation Procedures Distinction Between “Standard,” “Base,” “Observed,” and “Alternate” Conditions Basic Equations Calculation of CTL and s Standard Base Pressure in This S Iteration Scheme to Determine Base Density from Observed Density Calculation of CTL and CPL Factors for Base Temperatures Other Than 60°F Calculation Types Calculating the Thermal Expansion Factor for Special Applications ections International Temperature Scale of 1990, ITS-90 13 13 14 15 16 16 17 18 18 19 19 19 11.1.4 Summary and Precision Statement 19 11.1.5 11.1.5.1 Implementation Procedures General 20 Method to Convert Units of Temperature, Pressure, Thermal Expansion Factor, and Density-Related Values 21 Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS ~ Not for Resale `,,`,,,-`-`,,`,,`,`,,` - ~ 11.1.5.2 11.1.5.3 11.1.5.4 11.1.5.5 11.1.6 11.1.6.1 11.1.6.2 11.1.6.3 11.1.7 11.1.7.1 11.1.7.2 `,,`,,,-`-`,,`,,`,`,,` - 11.1.7.3 11.1.8 11.1.8.1 11.1.8.2 11.1.8.3 11.1.8.4 11.1.8.5 11.1.8.6 11.1.8.7 11.1.8.8 11.1.8.9 11.1.8.10 11.1.8.11 11.1.8.12 11.1.8.13 11.1.8.14 11.1.8.15 11.1.8.16 11.1.8.17 11.1.8.18 11.1.8.19 Method to Calculate Thermal Expansion Factor from Density Measurements Method to Convert Temperature from ITS-90 to IPTS-68 Basis Rounding of Values Other Implementation Considerations 23 27 28 30 30 Implementation Procedures for Customary Units (60°F and O psig Base Conditions) Method to Correct a Measured Volume to Base Conditions and Density from Base Conditions to an 30 Alternate Temperature and Pressure Method to Correct Volume and Density from Observed Conditions to Customary Base Conditions46 Method to Correct Volume and Density from Observed Conditions to Alternate Conditions 63 Implementation Procedures for Metric Units (15°C or 20°C and O Wa Base Conditions) 80 Method to Correct a Measured Volume to Metric Base Conditions and Density from Metric Base 80 Conditions to an Alternate Temperature and Pressure Method to Correct Volume and Density from Metric Observed Conditions to Metric Base Conditions 92 Method to Correct Volume and Density from Observed Metric Conditions to Alternate Metric Conditions 108 Use of Implementation Procedures to Generate Correction Factors in Tabular Format 130 Instructions to Generate Table 5A API Gravity Correction to 60°F for Generalized Crude Oils 132 Instructions to Generate Table 5B API Gravity Correction to 60°F for Generalized Products 134 Instructions to Generate Table 5D API Gravity Correction to 60°F for Generalized Lubricating Oils 136 Instructions to Generate Tables 6A and 6B Correction of Volume to 60°F Against API Gravity at 138 60°F for Generalized Crude Oils and Products Instructions to Generate Tables 6D Correction of Volume to 60°F Against API Gravity at 60°F for for Generalized Lubricating Oils 141 Instructions to Generate Table 23A Correction of Observed Specific Gravity to Specific Gravity 60/60"F for Generalized Crude Oils 143 Instructions to Generate Table 23B Correction of Observed Specific Gravity to Specific Gravity 60/60"F for Generalized Products 145 Instructions to Generate Table 23D Correction of Observed Specific Gravity to Specific Gravity 60/60"F for Generalized Lubricating Oils 147 Instructions to Generate Tables 24A and 24B Correction of Volume to 60°F Against Specific Gravity 60/60"F for Generalized Crude Oils and Products 149 Correction of Volume to 60°F Against Specific Gravity Instructions to Generate Table 24D 60/60"F for Generalized Lubricating Oils 152 Instructions to Generate Table 53A Correction of Observed Density to Density at 15°C for Generalized Crude Oils 154 Instructions to Generate Table 53B Correction of Observed Density to Density at 15°C for Generalized Products 156 Instructions to Generate Table 53D Correction of Observed Density to Density at 15°C for Generalized Lubricating Oils 158 Instructions to Generate Tables 54A Correction of Volume to 15°C Against Density at 15°C for Generalized Crude Oils 160 Instructions to Generate Tables 54B Correction of Volume to 15°C Against Density at 15°C for Generalized Products 162 Correction of Volume to 15°C Against Density at 15°C for Instructions to Generate Tables 54D Generalized Lubricating Oils 164 Instructions to Generate Tables 59A Correction of Observed Density to Density at 20°C for Generalized Crude Oils 166 Instructions to Generate Tables 59B Correction of Observed Density to Density at 20°C for Generalized Products 168 Instructions to Generate Tables 59D Correction of Observed Density to Density at 20°C for Generalized Lubricating Oils 170 Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale 11.1.8.21 11.1.8.22 11.1.8.23 11.1.8.24 11.1.8.25 11.1.8.26 Instructions to Generate Table 60A Correction of Volume to 20°C Against Density at 20°C for Generalized Crude Oils 172 Instructions to Generate Table 60B Correction of Volume to 20°C Against Density at 20°C for Generalized Products 174 Instructions to Generate Table 60D Correction of Volume to 20°C Against Density at 20°C for Generalized Lubricating Oils 176 Instructions to Generate Tables 6C & 24C Volume Correction Factors for Individual and Special Applications Volume Correction to 60°F Against Thermal Expansion Coefficients at 60°F 178 Instructions to Generate Tables 54C & 60C Volume Correction Factors for Individual and Special Applications Volume Correction to 15°C or 20°C Against Thermal Expansion Coefficients 181 Instructions to Generate 1984 Chapter 11.2.1 Compressibility Factor Table Compressibility 184 Factors for Hydrocarbons Related to API Gravity and Metering Temperature Instructions to Generate 1984 Chapter 11.2.1M Compressibility Factor Table Compressibility Factors for Hydrocarbons Related to Density and Metering Temperature 186 History & Development of the 1980 Petroleum Measurement Tables Appendix A Background 188 Experimental Project Fluid Groups Separate Representation Needed for Crude and Product Classes Correlation Development Parameter Determination and Results Development of 1980 Tables Summary and Precision Statement Independent Test of the Correlation Comparison ofthe Pre-1980 and 1980 Tables Hydrometer Corrections Density & Relative Density References 195 188 History & Development of the 1981 Hydrocarbon Compressibility Factors Appendix B Basic Mathematical Model & Uncertainty Analysis References 204 203 203 Development of Modified CTLEquations for Base Temperatures Other Than 60" F 205 c.1 Introduction 205 c.2 Changing Temperature Bases 205 c.3 Consistency of Results 206 c.4 Original Equations 206 c.5 C.5.1 C.5.2 c.5.3 Mathematical Conversion fiom Customary to Metric Temperature Units Conversion of Temperatures Shift of Base Temperature Value Calculation of the 60°F Thermal Expansion Factor 206 206 207 208 C.6 Conversion to ITS-90 Temperature Scale 208 Appendix C 188 189 189 189 191 192 193 193 194 194 194 International Temperature Scale of 1990 ITS-90 Appendix D Changes to the International Temperature Scale Since 1980 Impact on the Petroleum Measurement Tables 60°F Water Density 210 210 211 211 Appendix E -Development of Thermal Expansion Regression Equations 213 Appendix F Development of Iteration Equations Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale 217 `,,`,,,-`-`,,`,,`,`,,` - 11.1.8.20 `,,`,,,-`-`,,`,,`,`,,` - Newton’s Method 17 Derivation of This Standard’s Newton’s Method Equations 217 Simplification of the Temperature Derivative Term 220 Special Applications - “C’Tables 221 Use of Iteration Equations to Shift 60°F Standard Density 221 Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS.REFINED PRODUCTS & LUBEOILS Chapter 11 - Physical Properties Data Section -Temperature and Pressure Volume Correction Factors for Generalized Crude Oils, Refined Products, and Lubricating Oils 11.1.0 Implementation Guidelines This Standard (Revised Standard) is effective upon the date of publication and supersedes the previous edition of the Standard(s) (Previous Standard(s)) referenced in Appendix A of this Revised Standard However, due to the nature of the changes in this Revised Standard, it is recognized that guidance concerning an implementationperiod may be needed in order to avoid disruptions within the industry and ensure proper application As a result, it is recommended that this Revised Standard be utilized on all new applications no later than TWO YEARS after the publication date An application for this purpose is defined as the point where the calculation is applied Once the Revised Standard is implemented in a particular application, the Previous Standard will no longer be used in that application If an existing application complies with the Previous Standard(s) then it shall be considered in compliance with this Revised Standard However, the use of API standards remains voluntary and the decision on when to utilize a standard is an issue that is subject to the negotiations between the parties involved in the transaction 11.1.1 Introduction 8, History The density and therefore the volume of hydrocarbons is sensitive to temperature and pressure Volume Correction Factors (VCFs) are used to correct observed volumes to equivalent volumes at a standard temperature and pressure These standard, or base, conditions serve as a way to use volumetric measures equitably in general commerce This Standard establishes a procedure for crude oils, liquid refined products, and lubricating oils by which density measurements taken at any temperature and pressure can be corrected to an equivalent density at the base conditions The Standard also provides a method for making a conversion to alternate base temperatures The volume correction factors, in their basic form, are the output of a set of equations derived from and based on empirical data relating to the volumetric change of hydrocarbons over a range of temperatures and pressures Traditionally, the factors have been listed in a tabular format called the Petroleum Measurement Tables In order to introduce this document and the work that serves as its foundation, a short history of these Tables is warranted 11.I.I.I Early Temperature and Pressure Correction Tables `,,`,,,-`-`,,`,,`,`,,` - Correction factors to account for the thermal expansion of liquid hydrocarbons were first formally developed in 1916 by the National Bureau of Standards (United States) under Circular No 57 These data were based on density and temperature pairs documented in the National Bureau of Standards ( N B S ) Technologic Paper No 77 Circular No 57 was superseded in 1924 by Circular No C154 which in turn was superseded by a more widely known Circular C410, in 1936 By 1945 The Institute of Petroleum (IP) was publishing the Tablesfor Measurement of Oil in British units The compressibility standard (API Standard 1101,Appendix ByTable 11) for hydrocarbons in the O to 90” API gravity ranges was developed in 1945 by Jacobson, et al It was based on limited data obtained mostly on pure compounds and lubricating oil type materials Standard 1101 was developed without the aid of a mathematical model 11.I.I.2 1952 Temperature Correction Tables In 1952 the British and the American temperature correction factor tables were joined together and made available in three units of measure: US units, British (Imperial) units, and metric units These tables were called The Petroleum Measurement Tables and were published jointly by the American Society for Testing and Materials (ASTM) and the IP These tables are commonly referred to as the 1952 Tables, or “Blue Book Tables.” Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS.REFINED PRODUCTS & LUBEOILS Because the temperature shift is so small, iterative equations can be applied in a very simple manner to get a pm that is consistent with IPTS-68, given a pm consistent with ITS-90 This means that the a60value can easily be calculated and the equations for the shifted base temperature used in this Standard For computations involving the customary base temperature of 60°F, the procedure that has been adopted is to shift the input temperature and the input p60,if given, to an IPTS-68 basis If pmis not given, a value is calculated using an iterative procedure and this is then shifted to an IPTS-68 basis The calculation of C , involves IPTS-68 values of pm,while At is the temperature differential expressed on the IPTS-68 scale For metric base temperatures, all relevant temperatures (base, observed and alternate) are first converted from "C to OF, where both units are expressed on the temperature scale ITS-90 If given, the p T value (where T is the metric base temperature) is used to calculate p60(ITS-90) The procedure then follows that described above for the 60°F system, with temperatures and base density shifted to IPTS-68 Since the procedures for the customary and metric base systems utilize the same routines, they deliver identical results when starting from equivalent input values `,,`,,,-`-`,,`,,`,`,,` - Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale 209 SECTION1 - VOLUME CORRECTION FACTORS FORCRUDEOILS REFINED PRODUCTS & LUBEOILS 210 Appendix D - International Temperature Scale of 1990, ITS-90 Changes to the International Temperature Scale Since 1980 The International Committee for Weights and Measures, CIPM, publishes the international temperature scale Its purpose is to define procedures by which specified practical thermometers of the required quality can be calibrated in such a way that the values of temperature obtained from them can be precise and reproducible while at the same time closely approximating the corresponding thermodynamic values Small amendments to the temperature scale are introduced from time to time by CIPM to improve precision and reproducibility and provide better continuity between sections of the scale Since the international temperature scale is used for the calibration of thermometers, the values of temperaturedependent physical parameters of materials will, in principle, depend on what scale is in force at the time the parameter is measured or referenced However, since changes between scales are relatively small, this effect will only become noticeable at high levels of precision When the 1980 Petroleum Measurement Tables were prepared, the temperature scale in effect was the International Practical Temperature Scale of 1968 (IPTS-68) The International Temperature Scale of 1990 (ITS-90) superseded this in 1990 The developers of ITS-90 fitted the differences between the ITS-90 and IPTS-68 temperature scales to an 8* order polynomial of the form: Note that this polynomial is useful over the range of -200°C to 630°C i -0.148759 -0.267408 1.O80760 1.269056 i -4.089591 -1.871251 7.438081 -3.536296 The following figure shows the magnitude of the differences between the ITS-90 and equivalent IPTS-68 temperature values in the range of this Standard Over this temperature range, the differences in the two temperature scales are no larger than 0.04"C (0.07"F) Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale `,,`,,,-`-`,,`,,`,`,,` - When the 1980 Petroleum Measurement Tables were prepared, the temperature scale in effect was the International Practical Temperature Scale of 1968 (IPTS-68) The International Temperature Scale of 1990 (ITS-90) superseded this in 1990 SECTION1 - VOLUME CORRECTION FACTORS FORCRUDEOILS REFINED PRODUCTS & LUBEOILS 21 0.02 Y 0.01 U 0.00 O B'1 -Omo1 -0.02 8S g! -0.03 -0.04 ' -0.05 -250 -200 -150 -100 -50 O 50 100 150 200 250 ITS-90 Temperature (b,90) (OC) Impact on the Petroleum Measurement Tables The principal physical parameter contained in the Petroleum Measurement Tables that is affected by the change from IPTS-68 to ITS-90 is density and the related properties of relative density and API gravity Values of the density of a substance are a function of its temperature For petroleum hydrocarbons, the Petroleum Measurement Tables are based on average relationships between density and temperature according to broad classifications of products The precision of the VCF values, which represent the change of density with temperature, is to decimal places The changes due to ITS-90 give differences at high temperatures of no more than kO.00003 Even though this is well within the inherent accuracy of the correlations, because this would be noticeable, it was decided to fully incorporate procedures to account for differences between ITS-90and IPTS-68 Converting the input ITS-90 temperatures to equivalent IPTS-68 values before making any calculations in 11.1.6.1 did this One other subtle effect of the temperature scale correction is that the customary standard temperature, 60°F, has undergone a slight shift What is 60°F in ITS-90 is an equivalent 60.00687490"F in IPTS-68 Because of this, any input 60°F standard density values must be corrected to an equivalent IPTS-68 60°F value, before the value can be usedin the amand Fp correlations 60°F Water Density Another parameter in the Tables that is affected by the change of temperature scale is the density of water at 60°F, which affects the relative density and API gravity While density is a simple physical property (the mass of a substances divided by its volume at a specified temperature), calculations of relative density and API gravity relate to the density of water as well as the density of the hydrocarbon Since pure water is a defined substance, it has different density values on IPTS-68 and ITS-90 dependent on the temperature of the measurement In addition, a new, more accurate equation of state has been adopted for giving water densities The value at 60°F (ITS-90) has been re-defined as 999.016 kg/m3 for air-free water, based on new laboratory work by Patterson & Morris `,,`,,,-`-`,,`,,`,`,,` - Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS.REFINED PRODUCTS & LUBEOILS 212 (Metrologia, 1994,31,277-288) The value used in the 1980 Tables was 999.012 kg/m3 (0.999012 g / d ) based on earlier laboratory work and applying IPTS-68 `,,`,,,-`-`,,`,,`,`,,` - The water density value of 999.016 kg/m3 at 60°F has been adopted for use in any conversion between density and relative density or API gravity Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION1 - VOLUME CORRECTION FACTORS FORCRUDEOILS REFINED PRODUCTS & LUBEOILS 213 Appendix E - Development of Thermal Expansion Regression Equations The correlations for the 60°F thermal expansion factor give results for an “average” liquid of a specific commodity type However, there may be occasions when one wants to make density measurements on a particular liquid in order to determine its actual a60value The following is a development of the equations given in 11.1.5.2 as the procedure to determine the thermal expansion factor from a set of measured density data The p60and a60values must be determined from a set of density measurements using non-linear regression In general, this would involve doing a two-variable minimization process However, because of the form of the equations used to relate density to the thermal expansion factor, the process can be simplified to the solution of a single 3d order polynomial The goal of this regression is to determine p60and a60values so that some measure of the difference between the measured density values and values estimated from the CTL equation is minimized Mathematically, we can express the function that represents this difference using the least squares criteria: where there are N measured density values, pm,iis the i-th measured density value, and pi is the estimate of the density using the corresponding temperature of the i-th measurement The function y is the “objective” function, the sum of the squared residuals (All subsequent summations will not show the limits on the summation, but they remain to N ) This objective function has the following features: y Since all terms are squared, each term in positive errors is positive Negative errors will add to and not cancel out If this objective function is at an unconstrained minimum, then the lStderivatives with respect to each variable will be zero and all of the 2ndpartial derivatives will be positive The equation relating density and temperature is: is the density at the base temperature 60°F, a is where p is the density at O psig and any valid temperature, the 60°F thermal expansion coefficient , At is the difference between the alternate temperature and the 60°F base temperature, and 6, is the base temperature shift factor (0.01374979547’F) The temperatures used to calculate At must be adjusted using the procedure in 11.1.5.3 and the value for the base temperature should also be shifted consistent with 11.1.5.3,60.0068749”F Using Equation (E.2) the objective function can be expressed as: y=-C(lnp,,, -lnp, -E( ln p,,, =1 - ln p, +amAt, [1+0.8a,(Ati +6,)])2 + a,At + ~ ~(At,) ; (Ati + T ) ) This expression can be fully expanded to: `,,`,,,-`-`,,`,,`,`,,` - Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS.REFINED PRODUCTS & LUBEOILS 214 The various summations in Equation (E.4) need only be calculated once and then be treated as constants when adjusting the p60and a60values Equation (E.4) can be restated as: ?=-s,, -inp,s, +0.8a;6,SfP +-(lnp,)’ N+a,S, -amln pmSf- 0.8ab6, lnp,S, +-a;Sn + 0.8ai06,Sn + 0.32ak6;Sn + 0.8a;S& -0.8a; ln pmSn+ 0.8a&Sm+ 0.64ak6,Sm + 0.32a&Sm with the following definitions of the variables to represent the summations: `,,`,,,-`-`,,`,,`,`,,` - sp Clnpm,i S, = C A t i Sip= CAti lnpm,i S, = C(Ati)’ (E-10) S,,, = C(Ati)’ -lnpm,i (E-11) S, = C(Ati)3 (E.12) S, = C(Ati)4 (E 13) In the data fitting process, the two adjustable variables are p60and a60.An unconstrained minimum will be at an extremum point, Le., a point where each lStderivative with respect to a variable is zero The lStderivatives for these variables are: (E-14) = -(-Sp +lnp,N-a& -0.8a;6,Sf -0.8a&Sn) P60 and: Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS.REFINED PRODUCTS & LUBEOILS = S,, + 1.6a,6,Sf, +2.4a;6,Sn - ln p,S, - 1.6a,6, 215 ln p,S, + a,& +1.28a&6&S,+1.6a,S,, -1.6a, lnp,S, (E 15) + 2.56a&6,Sm + 1.28a&Sm +2.4a;S, Denote the values of the two adjustable variables at the extremum point as p* and a*.These values are solutions to the two simultaneous equations resulting when setting the lStderivatives equal to zero: O=-S,+lnp*N-a*S, -0.8(Sn +660Sf)a*2 (E 16) and: O = S,, + 1.6a*6,Sfp - ln p*S, - 1.6a*6, ln p*S, + a*S, + 1.28a*36&S,+ 1.6a*S,, - 1.6a*ln p*Sn + 2.56a*3660S,+ 1.28a*3Sm +2.4aY6,S, +2.4aYS, (E 17) Both equations are non-linear with respect to a*but the first equation gives a direct relationship between the optimal values of p* and a*: * S, N S, * O.8(S,+6,Sf) a*2 + N N lnp =-+-a This can be substituted into Equation (E.17) to give a single non-linear equation in a:, : [ -%]+ O = S,, [S, + 1.6(S,,, + 6,Sfp) - S: +1.6(Sn +6,S,)S, N Notice that (E.19) is just a 3d order polynomial equation in a:, : O = a, + ala:, + a,azz + a3az$ (E.20) with the coefficients defined from: (E.21) a, =S,+1.6(Sn,+6,Sfp)- S: +1.6(S, +6,S,)S, N (E.22) (E.23) N Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale `,,`,,,-`-`,,`,,`,`,,` - (E 19) SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS.REFINED PRODUCTS & LUBEOILS 216 (E.24) `,,`,,,-`-`,,`,,`,`,,` - The equation leading to a* is a cubic and there is a procedure to determine its roots analytically However, this procedure is fairly complex Sometimes it is simpler to use a numerical method (such as Newton’s method) to determine the numerical values of the roots A Newton’s iteration method with the following steps is used in 11.1.5.2: Initialize the value for a* The result from a linear regression of lnp, vs At with 6, = O is used as the initial guess: Determine the “residual” for this guess: f = ao +ala*+ a2a*2+ a3a*3 Determine the derivative of the residual for this guess: f ’ = a, + 2a2a*+ ~ , a * ~ Determine a correction to this guess: Update the value for a*: a*t a *+Aa* Return to step until some convergence criteria is achieved Some convergence criteria are small values of the residual, small values of the update value, andíor a set number of iterations Since the equation leading to a* is a cubic, it is guaranteed to always have at least one real root, but may have two or three Because the coefficients are generated from expressions that involve sums and differences of (mostly) positive summations, we cannot tell apriori the signs of the coefficients and how many positive real roots we will have From geometric considerations, we will have the following: The objective function surface should be a quartic function in the plane of the ln p* value(s) and a quadratic in the plane of the a* values Both of these should be concave down If there are multiple roots to the cubic equation giving the a* value, two of the roots should be associated with local minima & the third with a local maximum The root associated with the local maximum should lie between the two roots associated with the local minima Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION1 - VOLUME CORRECTION FACTORS FORCRUDEOILS REFINED PRODUCTS & LUBEOILS Appendix F 217 - Development of Iteration Equations `,,`,,,-`-`,,`,,`,`,,` - The basic iteration scheme to determine the 60°F base density p60from the observed density po is outlined in 11.1.3.5 It is pointed out that a Newton's method defines a specific way to calculate a new p60 estimate from the previous estimate In this appendix the actual equations used for the Newton's methods will be developed In the following equations variables that may change from one iterative step to the next will be denoted with a superscript '(m)' where the m denotes the m-th iterative step For example, the value of p60used on the first iterative ~ so on the value used on the second iterative step will be ~ ( ) and step will be Newton's Method Newton's method gives a procedure for finding a numerical value for the zero of a function, Le., for a function f(x) finding a specific value of X such that f ( X ) = O The function f ( x ) is approximated as a straight-line about a point xo;this approximate equation can then be solved for X Using a little bit of calculus, the function f ( x ) can be exactly expanded as a Taylor series polynomial: n! (x- x0)"f'"'(xo) + f - where the derivatives are evaluated at the point x = xo The approximate straight-line equation is obtained by dropping all terms after the first derivative: f ( 4= f ( x o ) + ( x - x o ) f ' ( x o ) F-2) - Remembering that X is defined as the point where the function's value is zero, we can solve this approximate equation for X : f(X)=o=f(xo)+(x-xo)f'(xo) X=x, f(xo1 fl x F-3) If we expand the function about each iterative value then x ( m ) plays the role of xo and the next iteration's value x(m+l) plays the role of X The recursion formula becomes: Two notation changes can be made: the function and derivative values may be denoted with the iteration's superscript ( f ( m ) and f ' ( m ) instead of f ( x ( m ) ) and f ' ( x ( m ) ) )and the ratio of the function to its derivative can be referred to as the step change, h ( m ) So, this recursion formula can also be expressed as: x(m+l) = x ( m ) +h ( m ) where f(m) h ( m ) E F-5) Y(") Derivation of This Standard's Newton's Method Equations The Newton's method iteration procedure to find &o given an observed p is: Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS,REFINED PRODUCTS, & LUBEOILS 218 where the step change Apg) is calculated from: DF' = Dr'a',"'At [1 + 1.6a',"' (At + 6, )] (F-9) = - 2Cg)PF9)(7.93920+ 0.02326t) PP2 (F.1 O) These recursion equations can be developed by starting with the residual function: f(P60) (where f = cTPL ( ~ )= O 'P6O -P= 'TL cPL 'P60 (F 1) -P corresponds to the solution) The Newton's method recursion equation will come from: (F.12) where: (F.13) Using this definition for the residual function and its derivative, the iteration scheme can be expressed as: (F.14) We can change the appearance by dividing the numerator and denominator by @)CE) and factoring out a -1 from the numerator (changing the order of the subtraction): (F.15) We can simplie the looks of equation (F.15) by defining two derivative terms: (F.16) (F.17) SO: `,,`,,,-`-`,,`,,`,`,,` - Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS.REFINED PRODUCTS & LUBEOILS 219 (F 18) The various derivatives can be determined fiom the basic equations Neglecting any differences between the ITS-90 and IPTS-68 temperature scale densities (i.e y dp,/dp* = 1), the CTL expressions: C , =exp(-a,At[1+0.8a,(At+6,)]) (F 19) K,+K,P*+K,P*~ KO KI =?+*+K2 P*, P P (F.20) am= give the derivatives: = -C, (At [i +1.6a, (At +6,)]) ab (F.21) dP60 (F.22) (F.23) The CPL expressions: CPL = 1- 10-5 F,P (F.24) -1.9947+0.000134270t+ 793920+2326t P* (F.25) give the derivatives: (F.26) FI - dFp P- SO dFP y = -2F ( 79392;; 2326t 793920 + 2326t (F.27) dP6Q d P DY)is: (F.28) Note that the derivative terms not have to be extremely accurate since they only establish the magnitude of the change fiom one iteration to the next If they values are offyusually one extra iteration is needed to achieve convergence That is why we can interchange the pm and pio values in these derivative terms `,,`,,,-`-`,,`,,`,`,,` - Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION1 - VOLUME CORRECTION FACTORS FORCRUDEOILS REFINED PRODUCTS & LUBEOILS 220 Simplification of the Temperature Derivative Term Let's go back to the term in Equation (F.23) The a60derivative could be replaced with Equation (F.24), but a manipulation can be made simplie the calculation Equation (F.23) can be modified to: (m) DT = -p,At[l+l.6a',"'(At+6,)][~~&] = ~~',")At[1+1.6~~',")(At+6~)] [ &] (m) DT =a',"'At[1+1.6a',"'(At+6,)] - (F.29) = a',")At(1+1.6a',"'At)[Dp)] where the term in the square brackets is the dimensionless a derivative and is designated as 0,.The expression for Da is: Da= 2K0 + Kip* KO+ Kip* + K2py 2Ko +KIP, KO+ Klp, + K2pk (F.30) This definition is convenient because of the nature of D, for the Equation (F.20) for a Depending upon the values of KO, KI ,and K2,the 0,values are constant or are nearly constant For example, for the Generalized Crude Oils, KI = K2 = O , so Da= for all p, values Similarly, for the Generalized Lubricating Oils, KO= K2 = O , so Da= for all p, values For nearly all of the commodity groups the 0,values are or are nearly constant, but different, values The only exception is the Transition Zone (770.3554 < p60 < 787.5224) where there is a significant change in the value (see the following figure) -however, a constant value of 8.5 does not significantly harm the convergence properties of the iteration equation Refined Product CL Correlations 0.0009 10.0 !-9.0 0.0008 -: Function Value @ Dimensionless Derivative 0.0007 -: LL !-7.0 0.0006 -! !-6.0 O u al 0.0005 -! ->m 0.0004 e -I O a" 0.0003 -I !-5.0 Oe :-4.0 II !-2.0 0.0001 -: !-1.0 " " " " ' ~ " " " " ' ~ " ' " " " ~ ' " " " " ~ " " " " ' ~ " " " ' " ~ ' " " " " ~ " " " " ' ~ " " " " ' ~ " " ' " " ~ " ' " " "- Using the approximate constant 0,values for the different commodity groups, the DT term becomes: Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS P ;-3.0 0.0002 -I 0.0000 O a" Not for Resale 0.0 `,,`,,,-`-`,,`,,`,`,,` - d n !-8.0 SECTION1 - VOLUME CORRECTION FACTORS FORCRUDEOILS REFINED PRODUCTS & LUBEOILS DF' = D?'a',"'At [1 + 1.6a',"' (At + 6, )] 221 (F.30) This equation shows Da changing with the iterations; however, as long as the commodity group does not change during the iterations,this would actually be a constant The following table shows the appropriate Da to use for the various commodity groups Crude Oil Fuel Oils Jet Fuels Transition Zone Gasolines Density Range(kg/m3) 610.6 < p60 < 1163.5 838.3154 < p60 Da 2.0 < 1163.5 1.3 < 838.3 154 2.0 770.3554 < p60< 787.5224 8.5 787.5224 < 610.6 < < 770.3554 1.5 Lubricating Oil 800.9 < p60 < 1163.5 o Special Applications All p60values 0.0 Special Applications - "C" Tables What happens to the iteration equations if the a60value is not calculated from the correlationsbut instead is precalculated and specified? The need for iteration is reduced In fact, if no pressure correction is applied, the need for iteration could be eliminated entirely Equation (F.29) gives the DT factor in the recursion formula If the a value is specified then it does not change with the p60values and ak0= O ,leading to Da = O and D P ) = O This simplifies the recursion formula Equation (F.18) to: (F.31) `,,`,,,-`-`,,`,,`,`,,` - When a pressure correction is applied iterations will be needed since the D Y ) term is not necessarily zero However, if no pressure correction is applied, then the p60value could be directly calculated, or, keeping the iteration procedure, only one iteration need be done When no pressure correction is applied, C,, = ,and D Y ) = O ,leading to: (F.32) When a60is specified, CTLis not a function of p60and will not change from one iteration to the next For any estimate, only one Newton step is necessary to find the correct value of p60 Use of Iteration Equations to Shift 60°F Standard Density The Newton's Method equations developed here have been used to develop the equation in 11.1.6.1 to shift pm to p* for subsequent use in the equations to calculate amand Fp This was easy to accomplish since the temperature difference between p* is very small When done as an iterative procedure, only one iteration is ever needed Since only a single iteration is needed, the specific starting values can be put into the iteration equations and it can be recast as a single equation to shift the pm value to p* Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale SECTION - VOLUME CORRECTION FACTORS FORCRUDE OILS.REFINED PRODUCTS & LUBEOILS 222 The shift equation can be most easily developed using the equations based upon the IPTS-68 60°F value being the standard temperature and the ITS-90 60°F value (to be denoted as t,) as the alternate temperature The initial estimate to the IPTS-68 60°F density, p* ,will be the given ITS-90 60°F density, p60 Starting with (F.18): (F.33) Since the IPTS-68 60°F value is being used as the standard temperature (F 19) becomes: Cg) = exp (-aE)At [1+ O.da"At]) [ = exp (-a:) (t, - 60) 1+ 0.8az)(t, - 6O)]) (F.34) where a:) is calculated using p60in (F.20): (F.35) Or) is calculated using p60in (F.9): `,,`,,,-`-`,,`,,`,`,,` - 0;) = @')a:) (T, [+ - 60) 1.6~~:) (T, - 6O)] (F.36) and Dao) is calculated using p, in (F.30): (F.37) The temperature difference (t, - 60) is based upon the IPTS-68 values for temperature and is used to define the value of 6, ; specifically, it is %6, Combining this with (F.34), (F.36), and (F.37) changes (F.33) to: 1+ [ exp 0.5ag6, (i + O.4ag6,)I - i (F.38) 1+ 0.5ag6, (1+ 0.8az)6, Finally, making the following symbolic substitutions: A = 0.5ag6, = $[(:+KI)-& + K2] (F.39) (F.40) the form for the final shift equation has been developed: exp [A( 1+ 0.8A)I - (F.41) 1+ A( 1+ 1.6A) B Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale `,,`,,,-`-`,,`,,`,`,,` - Additional copies are available through Global Engineering Documents at (800) 854-7179 or (303) 397-7956 Information about API Publications, Programs and Services is available on the World Wide Web at http://www.api.org Copyright American Petroleum Institute Reproduced by IHS under license with API No reproduction or networking permitted without license from IHS Not for Resale