# ASTM D7152-11 (Reapproved 2016)

Designation: D7152 − 11 (Reapproved 2016)Standard Practice forCalculating Viscosity of a Blend of Petroleum Products1This standard is issued under the fixed designation D7152; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice covers the procedures for calculating theestimated kinematic viscosity of a blend of two or morepetroleum products, such as lubricating oil base stocks, fuelcomponents, residua with kerosine, crude oils, and relatedproducts, from their kinematic viscosities and blend fractions.1.2 This practice allows for the estimation of the fraction ofeach of two petroleum products needed to prepare a blendmeeting a specific viscosity.1.3 This practice may not be applicable to other types ofproducts, or to materials which exhibit strong non-Newtonianproperties, such as viscosity index improvers, additivepackages, and products containing particulates.1.4 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.5 Logarithms may be either common logarithms or naturallogarithms, as long as the same are used consistently. Thispractice uses common logarithms. If natural logarithms areused, the inverse function, exp(×), must be used in place of thebase 10 exponential function, 10×, used herein.1.6 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and to determine theapplicability of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2D341 Practice for Viscosity-Temperature Charts for LiquidPetroleum ProductsD445 Test Method for Kinematic Viscosity of Transparentand Opaque Liquids (and Calculation of Dynamic Viscos-ity)D7042 Test Method for Dynamic Viscosity and Density ofLiquids by Stabinger Viscometer (and the Calculation ofKinematic Viscosity)2.2 ASTM Adjuncts:Calculating the Viscosity of a Blend of Petroleum ProductsExcel Worksheet33. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1 ASTM Blending Method, n—a blending method atconstant temperature, using components in volume percent.3.1.2 blend fraction, n—the ratio of the amount of a com-ponent to the total amount of the blend. Blend fraction may beexpressed as mass percent or volume percent.3.1.3 blending method, n—an equation for calculating theviscosity of a blend of components from the known viscositiesof the components.3.1.4 dumbbell blend, n—a blend made from components ofwidely differing viscosity.3.1.4.1 Example—a blend of S100N and Bright Stock.3.1.5 inverse blending method, n—an equation for calculat-ing the predicted blending fractions of components to achievea blend of given viscosity.3.1.6 mass blend fraction, n—The ratio of the mass of acomponent to the total mass of the blend.3.1.7 McCoull-Walther-Wright Function, n—a mathematicaltransformation of viscosity, generally equal to the logarithm ofthe logarithm of kinematic viscosity plus a constant, lo-g[log(v+0.7)]. For viscosities below 2 mm2/s, additional termsare added to improve accuracy.3.1.8 modified ASTM Blending Method, n—a blendingmethod at constant temperature, using components in masspercent.3.1.9 modified Wright Blending Method, n—a blendingmethod at constant viscosity, using components in masspercent.1This practice is under the jurisdiction of ASTM Committee D02 on PetroleumProducts, Liquid Fuels, and Lubricants and is the direct responsibility of Subcom-mittee D02.07 on Flow Properties.Current edition approved Jan. 1, 2016. Published February 2016. Originallyapproved in 2005. Last previous edition approved in 2011 as D7152 – 11. DOI:10.1520/D7152-11R16.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.3Available from ASTM International Headquarters. Order Adjunct No.ADJD7152. Original adjunct produced in 2006.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.1.10 volume blend fraction, n—The ratio of the volume ofa component to the total volume of the blend.3.1.11 Wright Blending Method, n—a blending method atconstant viscosity, using components in volume percent.3.2 Symbols:fij= blending fraction of component i calculated at tem-perature tj. Blending fraction may be in mass percentor volume percent.mi= slope of the viscosity-temperature line,~Wi12 Wi0!~Ti12 Ti0!mi-1= reciprocal of the viscosity-temperature slope, mitB= temperature, in Celsius, at which the blend hasviscosity vBtij= temperature, in Celsius, at which component i hasviscosity vijTij= transformed temperatureTij5 log~273.151tij! (1)vB= predicted kinematic viscosity of the blend, in mm2/s,at temperature tBif component blend fractions areknown, or desired viscosity of the blend if componentblend fractions are being calculatedvij= viscosity of component i at temperature tjWij= MacCoull-Walther-Wright function, a transformationof viscosity:Wij5 log@log~vij10.71exp~21.47 2 1.84vij2 0.51vij2!!#(2)where log is the common logarithm (base 10) andexp(x) is e (2.716.) raised to the power x.WH= arbitrary high reference viscosity, transformed usingEq 2WL= arbitrary low reference viscosity, transformed usingEq 24. Summary of Practice4.1 The Wright Blending Method calculates the viscosity ofa blend of components at a given temperature from the knownviscosities, temperatures, and blending fractions of the com-ponents. The viscosities and temperatures of the componentsand the blend are mathematically transformed into MacCoull-Walther-Wright functions. The temperatures at which eachcomponent has two reference viscosities are calculated. Thetransformed reference temperatures are summed over all com-ponents as a weighted average, with the blend fractions as theweighting factors. The two temperatures at which the blend hasthe reference viscosities are used to calculate the blendviscosity at any other temperature.4.2 The Inverse Wright Blending Method calculates theblend fractions of components required to meet a target blendviscosity from the known viscosities and temperatures of thecomponents. The viscosities and temperatures of the compo-nents and the blend are mathematically transformed intoMacCoull-Walther-Wright functions. The temperatures atwhich each component has the target blend viscosity arecalculated. The component transformed temperatures aresummed over all components, as a weighted average, to meetthe target blend transformed temperature. The weighting fac-tors are the desired blend fractions, which are obtained byinverting the weighted summation equation.4.3 The ASTM Blending Method calculates the viscosity ofa blend of components at a given temperature from the knownviscosities of the components at the same temperature and theirblending fractions. The viscosities of the components and theblend are mathematically transformed into MacCoull-Walther-Wright functions. The transformed viscosities are summedover all components as a weighted average, with the blendfractions as the weighting factors. The transformed viscosity isuntransformed into viscosity units.4.4 The Inverse ASTM Blending Method calculates theblend fractions of components required to meet a target blendviscosity at a given temperature from the known viscosities ofthe components at the same temperature. The viscosities of thecomponents and the blend are mathematically transformed intoMacCoull-Walther-Wright functions. The component trans-formed viscosities are summed over all components, as aweighted average, to equal the target blend transformed vis-cosity. The weighting factors are the desired blend fractions,which are obtained by inverting the weighted summationequation.5. Significance and Use5.1 Predicting the viscosity of a blend of components is acommon problem. Both the Wright Blending Method and theASTM Blending Method, described in this practice, may beused to solve this problem.5.2 The inverse problem, predicating the required blendfractions of components to meet a specified viscosity at a giventemperature may also be solved using either the Inverse WrightBlending Method or the Inverse ASTM Blending Method.5.3 The Wright Blending Methods are generally preferredsince they have a firmer basis in theory, and are more accurate.The Wright Blending Methods require component viscositiesto be known at two temperatures. The ASTM BlendingMethods are mathematically simpler and may be used whenviscosities are known at a single temperature.5.4 Although this practice was developed using kinematicviscosity and volume fraction of each component, the dynamicviscosity or mass fraction, or both, may be used instead withminimal error if the densities of the components do not differgreatly. For fuel blends, it was found that viscosity blendingusing mass fractions gave more accurate results. For base stockblends, there was no significant difference between massfraction and volume fraction calculations.5.5 The calculations described in this practice have beencomputerized as a spreadsheet and are available as an adjunct.36. ProcedureProcedure A6.1 Calculating the Viscosity of a Blend of Components WithKnown Blending Fractions by the Wright Blending Method:D7152 − 11 (2016)26.1.1 This section describes the general procedure to predictthe viscosity of a blend, given the viscosity-temperatureproperties of the components and their blend fractions. Anynumber of components may be included. If the blend fractionsare in volume percent, this is known as the Wright BlendingMethod. If the blend fractions are in mass percent, this isknown as the Modified Wright Blending Method.6.1.2 Compile, for each component, its blend fraction, andviscosities at two temperatures. The viscosity of component i attemperature tijis designated vij, and its blend fraction is fi.Iftheviscosities are not known, measure them using a suitable testmethod. The two temperatures may be the same or different foreach component.NOTE 1—Test Methods D445 and D7042 have been found suitable forthis purpose.6.1.3 Transform the viscosities and temperatures of thecomponents as follows:Zij5 vij10.71exp~21.47 2 1.84vij2 0.51vij2! (3)Wij5 log@log~Zij!# (4)Tij5 log@tij1273.15# (5)where vijis the kinematic viscosity, in mm2/s, of componenti at temperature tijin degrees Celsius, exp() is e (2.716) raisedto the power x, and log is the common logarithm (base 10).6.1.3.1 If the kinematic viscosity is greater than 2 mm2/s,the exponential term in Eq 3 is insignificant and may beomitted.6.1.3.2 Transform the temperature at which the blend vis-cosity is desired using Eq 5. This transformed temperature isdesignated TB.6.1.4 Calculate the inverse slope for each component, asfollows:mi215~Ti12 Ti0!~Wi12 Wi0!(6)6.1.5 Calculate the predicted transformed viscosity, WB,ofthe blend at temperature TB, as follows:WB5TB1(fi~mi21Wi02 Ti0!(~fimi21!(7)where the sum is over all components.6.1.6 Calculate the untransformed viscosity of the blend, νB,at the given temperature:Z B5 10WB(8)ZB5 10Z B2 0.7 (9)vB5 ZB2 exp@20.7487 2 3.295ZB10.6119ZB22 0.3193ZB3# (10)where Z Band ZBare the results of intermediate calculationsteps with no physical meaning.NOTE 2—For viscosities between 0.12 and 1000 mm2/s, the transform-ing Eq 3 and Eq 4 and the untransforming equations Eq 9 and Eq 10 havea discrepancy less than 0.0004 mm2/s.NOTE 3—See the worked example in Appendix X3.Procedure B6.2 Calculating the Blend Fractions of Components to Givea Target Viscosity Using the Inverse Wright Blending Method:6.2.1 This section describes the general procedure to predictthe required blending fractions of two components to meet atarget blend viscosity at a given temperature, given theviscosity-temperature properties of the components. This isknown as the Inverse Wright Blending Method.6.2.1.1 In principle, the blend fractions may be calculatedfor more than two blending components, if additional con-straints are specified for the final blend. Such calculations arebeyond the scope of this practice.6.2.2 Compile the viscosities of the components at twotemperatures each. The viscosity of component i at temperaturetijis designated vij. If the viscosities are not known, measurethem using a suitable test method. The two temperatures do nothave to be the same for both components, nor do they have tobe the same as the temperature at which the target viscosity isspecified.NOTE 4—Test Methods D445 and D7042 have been found suitable forthis purpose.6.2.3 Transform the viscosities and temperatures of thecomponents using Eq 3, Eq 4, and Eq 5.6.2.4 Use the target blend viscosity, vB, as a referenceviscosity. Transform vBto WBusing equations Eq 3 and Eq 4.6.2.5 Calculate the transformed temperatures at which eachcomponent has that viscosity:TiL5~Ti12 Ti0!~Wi12 Wi0!~WL2 Wi0!1Ti0(11)6.2.6 Calculate the predicted blend fraction of the firstcomponent:f15~TB2 T0L!~T1A2 T0L!(12)and the fraction of the second component is f2=(1–f1)because the total of the two components is 100 %.NOTE 5—See the worked example in Appendix X4.Procedure C6.3 Calculating the Viscosity of a Blend of Components WithKnown Blending Fractions Using the ASTM Blending Method:6.3.1 This section describes the general procedure to predictthe viscosity of a blend at a given temperature, given theviscosities of the components at the same temperature and theirblend fractions. Any number of components may be included.If the blend fractions are in volume percent, this is known asthe ASTM Blending Method. If the blend fractions are in masspercent, this is known as the Modified ASTM BlendingMethod.6.3.2 Compile the viscosities of the components at a singletemperature (the reference temperature). The viscosity ofcomponent i at that temperature is designated vi.Iftheviscosities are not known, measure them using a suitable testmethod.NOTE 6—Test Methods D445 and D7042 have been found suitable forthis purpose.6.3.2.1 If the viscosity of a component is not known at thereference temperature, but is known at two other temperatures,D7152 − 11 (2016)3use Viscosity-Temperature Charts D341 or Eq 10 to calculateits viscosity at the reference temperature.6.3.3 Transform the viscosities of the components using Eq2.6.3.4 Calculate the transformed viscosity of the blend as aweighted average of the component transformed viscosities,using the blend fractions as the weighting factors:WB5(@fiWi#(@fi#(13)where WBis the transformed viscosity of the blend, fiis theblend fraction of component i, and Wiis the transformedviscosity of componenti.6.3.4.1 Normally, the sum of blend