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Viscosity Blending Equations Boris Zhmud, Ph.D., Assoc.Prof., MRSC Sveacon Consulting, Stockholm, Sweden

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Viscosity Blending Equations Boris Zhmud, Ph.D., Assoc.Prof., MRSC Sveacon Consulting, Stockholm, Sweden No.93 page 1 Lube-Tech No.93 page 1 LubePPUBLISHEDUBLISHED BBYY LLUBE:UBE: THETHE EUROPEANEUROPE-A-N LUBRICANTSLUTechBRICANTS INDUSTRYINDUSTRY MAGAZINE MAGAZINE Viscosity Blending Equations Boris Zhmud, Ph.D., Assoc.Prof., MRSC Sveacon Consulting, Stockholm, Sweden ABSTRACT: In lubricating and specialty oil industries, blending is routinely used to convert a finite number of distillation cuts produced by a refinery into an infinite number of final products matching given specifications regarding viscosity. To find the right component ratio for a blend, empirical or semi-empirical equations linking viscosity of the blend to viscosities of the individual components are used. Perhaps the best known among viscosity blending equations are the double- logarithmic equation of Refutas and the cubic-root equation of Kendall and Monroe. The kinetic theory led the way to a deeper understanding of viscosity blending principles for binary mixtures, culminating in Grunberg-Nissan, Oswal-Desai and Lederer-Roegiers equations. These equations have lifted viscosity blending calculations to a practically useful accuracy level. Ironically, despite being the most accurate one-parameter equation, the viscosity blending equation due to Lederer and Roegiers remained largely unknown to the oil research community until recently. INTRODUCTION calculates the viscosity, μ12, of the binary KINETIC THEORY APPROACH TO Viscosity blending is perhaps the most blend from viscosities and weight VISCOSITY BLENDING CALCULATIONS common operation in lubricant fractions of the components by At the moment there is no universal manufacture. All blenders rely upon their introducing the so-called viscosity theory which would allow exact viscosity-blending calculators, often used blending index (ASTM D7152), calculation of the viscosity of a complex as a magic black-box giving the right mixture from the viscosities of the blend composition [1,2]. Accurate (1) individual components [6,7]. The majority viscosity prediction for binary mixtures of of the existing theories are limited to the two components with a large difference where xi is the weight fraction, μi is the so-called ideal binary mixtures. in viscosity remains a challenging task kinematic viscosity of the ith component because viscosity blending curves may in the blend. Then, the blend viscosity is A binary mixture is said to be ideal if show a large degree of non-linearity. calculated as, mixing the components does not Strongly non-linear viscosity blending produce any change in volume. In other curves are often observed for binary (2) words, for an ideal mixture, the excess systems such as base oil / polymeric volume of mixing is zero. The dynamic thickener or viscosity index improver, where A12 is the average viscosity viscosity, η12, of an ideal binary mixture which are quite common in lubricant blending index, consisting of the components with formulation practice. viscosities η1 and η2 obeys the Arrhenius (3) equation, VISCOSITY BLENDING EQUATIONS Kendall-Monroe equation calculates the (5) COMMONLY USED IN THE blend viscosity as the cubic-root average PETROLEUM INDUSTRY of the component viscosities, where xi (i = 1,2) is the mole fraction the In the petroleum industry, empirical or ith component in the mixture. proprietary blending equations are (4) common. The best known are the The Arrhenius equation can be double-logarithmic equation of Refutas The above equations hardly afford any rationalised within the framework of the and the cubic-root equation of Kendall meaningful theoretical substantiation and absolute reaction rate theory [8]. and Monroe [3-5]. The Refutas equation have unsatisfactory accuracy. 2222L U B LUBEE M A GMAGAZINEA Z I N E N o . 1NO.1212 1 J U N EJUNE 2 0 1 42014 Lube 121.indd 22 13/05/2014 09:52 No.93 page 2 Lube-Tech No.93 page 2 LubePPUBLISHEDUBLISHED BBYY LUBE:LUBE: THETHE EUROPEANEUROPE-A-N LUBRICANTSLUTechBRICANTS INDUSTRY INDUSTRY MAGAZINE MAGAZINE According to this theory, the fluidity (the Another useful blending equation has Examples are polyalphaolefin / inverse of viscosity) of a liquid is related been proposed independently by Lederer olefin copolymer / fatty ester or to the flow activation energy, ΔE≠, which [11] and Roegiers, Sr. [12,13], polyalphaolefin / fatty ester / polyester is a measure of intermolecular cohesion, systems. In this case, changing the component ratio has a strong effect on (6) (11) the solvent power of the mixture, which, in its turn, has an effect on the random coil configuration of the polymeric thickener. For instance, the high activation energy corresponding to where α is an empirical parameter to mean radius of gyration of a polyester low fluidity and vice versa. This implies account for the difference in intermol- molecule in polyalphaolefin solution is that ecular cohesion energies between the smaller than it is in fatty ester solution. component 1 and 2. Vice versa, the mean radius of gyration (7) of an olefin copolymer molecule in Unfortunately, the authors published polyalphaolefin solution is greater than their original work in difficult-to-access it is in fatty ester solution. Generally Assuming that the flow activation energy sources, and as a result, the above speaking, such systems normally reveal for the mixture follows the additivity equation has remained largely unknown non-Newtonian rheology - their viscosity principle, to the lubricant research community until becomes a complex shear-rate- recently. dependent quantity. However, this is (8) outside the subject of the present The Lederer-Roegiers equation can be communication. As long as the mixture derived the same way as the Arrhenius behaves nearly like a normal liquid (and one immediately arrives at the Arrhenius equation but assuming an asymmetric not as a gel), the effect of random coil equation. mixing rule for the flow activation radius change on viscosity can be energy, described by an asymmetric correcting 2k+1 From a practical viewpoint, the Arrhenius (12) function of the type x1(1 - x1)(2x1 - 1) , equation lacks accuracy and has limited as has been done in eq.(10) by Oswal- value. Deviations from the Arrhenius Desai [10]. equation are normally linked to the fact the interaction energy between two where 0 < γ < 1. If γ < 0.5, the contri- It should also be commented that the unlike molecules is in general different bution of component 1 to the flow above blending equations refer to from the interaction energies of two like activation energy is greater than that of dynamic viscosity, η. The corresponding molecules. To account for non-ideality of component 2, and vice versa, if γ > 0.5, equations for kinematic viscosity, μ, are the system, an additional term has been the contribution of component 1 to the obtained using the relationship η = ρμ, included in the above equation by flow activation energy is less than that where ρ is the density. To link the Grunberg and Nissan [9], following the of component 2. Indeed, from a density of a binary mixture to the ideas of the regular solution theory, hydrodynamic viewpoint, viscosity densities of its components, the excess reflects the momentum flux from one volume of mixing should be determined. (9) liquid layer to another when two Luckily, most mineral oils behave as adjacent liquid layers slide against each normal fluids, for which the excess other. Since the molecules of the volume of mixing is close to zero. where is an empirical interaction components 1 and 2 differ in molecular Furthermore, since the densities of ε parameter. weight, size and electron density components in mineral oil blends are distribution, the energies of pair-wise usually close to each other, rarely falling An amendment to the Grunberg-Nissan interactions 1-1, 1-2, and 2-2 are also outside the range 0.8 to 1.0 g/cm3, the equation has been proposed by Oswal different. Putting α = γ / (1 - γ), the term lnρ12 - x1lnρ1 - x2lnρ2 is close to and Desai [10], who added two Lederer-Roegiers equation is arrived at. zero. Therefore, in practice, the above- additional terms, For γ = 0.5, α = 1, and the Lederer- mentioned viscosity-blending equations Roegiers equation (6) becomes identical (5), (9), (10) and (11) can well be (10) to the Arrhenius equation (1). applied for calculation of kinematic viscosities. For the same reason, mole The presence of highly polar compounds fractions are substituted by weight with strong orientational forces such fractions. However, the improvement in accuracy as hydrogen bonding may result in comes at a price of introduction of two S-shaped viscosity plots. Such behaviour additional fitting parameters with rather is often observed when blending obscure physical meaning. For that polymer-thickened oils having a reason, the Oswal-Desai equation will sufficiently large difference in solubility. not be considered here. LUBEL U MAGAZINEB E M A G A Z I NO.121N E N o . 1 JUNE2 1 J U N2014E 2 0 1 42323 Lube 121.indd 23 13/05/2014 09:52 No.93 page 3 Lube-Tech No.93 page 3 LubePPUBLISHEDUBLISHED BYBY LUBE:LUBE: THETHE EUROPEANEUROPE-A-N LUBRICANTS LUTechBRICANTS INDUSTRY INDUSTRY MAGAZINE MAGAZINE APPLICATIONS OF LEDERER-ROEGIERS EQUATION a. Determination of the intermol- ecular cohesion parameter The easiest way to determine the intermolecular cohesion parameter for a given blend is via measuring the blend viscosity for component weight ratio 50:50, in which case Table 1: Physical properties of base oils and petroleum distillates used in this study. (13) for a number of paraffinic, naphthenic From the theoretical viewpoint, the and synthetic oils. Some physicochemical Grunberg-Nissan equation and the and hence, properties of oils used in this study are Lederer-Roegiers equation are the most summarised in Table 1. Kinematic substantiated single-parameter viscosity (14) viscosity of the individual components blending equations. In the following, we and their binary mixtures was measured compare accuracy of those two at 40oC according to ASTM D445. equations.
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