Kinematic Viscosity Prediction Guide: Reviewing and Evaluating Empirical Models for Diesel Fractions, and Biodiesel–Diesel Blends According to the Temperature and Feedstock

Experimental analysis of viscosity can be a straightforward and inexpensive analysis for few samples. However, in industrial processes that have high demands of properties measurements, the determination of viscosity and other properties involves time-consuming with sampling, analysis, and availability of results. Also in refineries, the sampling routines for experimental determination of the viscosity of streams are not enough to represent variations that occur in the process, such as the shift of an oil tank in distillation units. In addition, besides requiring cost of operating personnel and laboratory analyst, all of these steps can take up to one shift until the result is available. Therefore, as an alternative, the use of predictive methods of kinematic viscosity are essential. Empirical methods have been used in simulations and design calculations of streams and mixture at industries regarding kinematic viscosity (KV) of petroleum fractions and fuels at different temperatures. However, there are uncertainties about the most accurate method to use at specific condition (temperature, feedstock, volume fraction) which might affect the KV prediction of fuels with unknown composition. Therefore, we assembled and evaluated several methods to predict KV of different diesel systems. In addition, new methods for predicting KV of diesel fractions at several temperatures were also developed for improving the estimation accuracy. As a result, we developed a guide with suggestions of the most accurate models to be applied for diesel fraction from assays, diesel fractions S500 from blend system at several temperatures, and biodiesel–diesel blends at different temperatures, volume fractions, and feedstock.


Introduction
Production of petroleum fractions and transportation fuels requires monitoring and control of some properties, such as viscosity. Besides essential for flow and hydraulic calculations, this property is also required during the blending process of petroleum fractions to increase profits through giveaway decreasing, adding less noble fractions, without deviating from fuel specification [1,2].
The examination of kinematic viscosity (KV) at 40 °C is an important property, especially for fossil diesel, for the well-functioning of a given engine, influencing factors such as droplet size formation at the fuel atomization process for combustion quality [3]. Some concerns related to inadequate viscosity are leakage, in the case of low KV, and poor atomization, incomplete combustion, deposit formation, and malfunctioning during cold weather [4]. As such, depending on the type of diesel engines, the kinematic viscosity target of fuels varies from 1.3 mm 2 s −1 to 4.1 mm 2 s −1 at 40 °C for diesel fuel with light middle distillate, with specific sulfur concentration, according to ASTM D975-20a and ASTM 7467-20a [5,6].
Kinematic viscosity is also a crucial property for biodiesel, a biofuel derived from vegetable and animal oils applicable for Diesel engines. The biodiesel is a renewable source that can be an alternative of fossil diesel (non-renewable) in case of petroleum depletion and it is often blended with fossil diesel [7]. The application of biodiesel provides some benefits such as lower carbon dioxide emission and the increase of lubricity of the fuel in the engine, due to its higher KV [3,8]. However, the high KV of biodiesel, which varies according to feedstock, and volume fraction of biodiesel in a blended fuel, diminishes the quality of fuel atomization, and increases the formation of deposits in engines [9][10][11]. Thus, biodiesel KV must meet the specification, which ranges from 1.9 mm 2 s −1 to 6.0 mm 2 s −1 at 40 °C [12], according to the methods ASTM D445 and ASTM D7945 [5]. Those experimental tests depend on temperature and present satisfactory measurement for fluids with Newtonian flow behavior.
Experimental analysis of viscosity can be a straightforward and inexpensive analysis for few samples. However, in industrial processes, the determination of experimental viscosity or any other property involves sampling, analysis, and availability of results. Therefore, besides requiring cost of operating personnel and laboratory analyst, all of these steps can take up to one shift until the result is available. Also in refineries, the sampling routines for experimental determination of the viscosity of streams are not enough to represent the variations that occur in the process, such as the shift of an oil tank in the distillation unit. In addition, for minimizing giveaway in fuel blending systems, online optimization is necessary, with instantaneous values that is only possible through calculation methods. These methods are applicable for viscosity prediction of streams before and after blending (mixing calculations).
Viscosity prediction of fuels with high accuracy might not be a straightforward calculation, since it depends on its composition, chemical properties of each compound in the mixture [13], their interaction [14], temperature, and type of predictive model. For that, the prediction of kinematic viscosity can be carried out by two different approaches: rigorous calculations and empirical methods. Rigorous calculations for viscosity prediction are based on theoretical concepts, such as statistical mechanics [15][16][17][18][19][20][21]. There are also semi-theoretical models with theoretical basis but also has some adjusted parameters. Rigorous methods might require complex data at molecular level, such as repulsion diameter, intermolecular potential attractive energy, structure theory, molar mass, and so on [22].
Some theories for viscosity prediction were developed based on Brownian motion, and equations of state in function of density and complex parameters such as rigid sphere diameter of the molecules, and coefficients of shear [16]. There is also a theory relating the flow with activation energy so that the calculation includes several parameters such as the molar free energy of activation of flow, molar volume for Newtonian behavior [21,23]. For non-polar and polar pure liquid viscosity, the corresponding-states method has been applied for equilibrium and transport properties of pure materials [17] and requires critical properties, and size or shape parameter [24].
For hydrocarbon mixtures, rigorous methods also can be used. There are some studies for KV prediction of mixtures requiring composition content, pressure, temperature, thermal conductivity, some fitted parameters in the function of n-alkane and aromatics, molar mass, and critical properties [25][26][27].
In front of the complexity of theoretical models application, empirical methods are often used in simulations with interactive calculations and project optimization to predict KV based on a correlation between viscosities and other properties.
KV can also be obtained through conversion model instead of a predictive one applying the standard ASTM D341, which requires only KV at two different temperatures to obtain KV at specific temperature [28].
For feasibility, predictive empirical methods have been developed requiring as input only properties easily measured and frequently available, such as density and boiling temperature. These methods are highly useful when the composition and other difficult properties are unknown [1,2,[29][30][31][32]. However, as a drawback, empirical methods are developed for a specific system so that the range of validity is narrow and might not be applicable without restrictions [33].
Some individual studies have been conducted for kinematic viscosity prediction for fossil compounds, such as the development of new correlations for petroleum fraction [34,35], and crude oil blends [14,36]. There are also studies with assessment and development of models to predict kinematic viscosity of biodiesel-diesel blends [37][38][39][40][41][42] and specially a recent evaluation model at 40 °C [43].
Therefore, due to the abundance of empirical models and numerous individual assessments of KV prediction, we evaluated existing methods at several conditions (temperature, feedstock, volume fraction) to verify the applicability and accuracy of KV predictive models for various systems, such as diesel fractions, and several biodiesel-diesel blends at different temperatures and volume fraction. We also developed new methods to predict KV of diesel fractions with higher accuracy then presented existing models. After that, we also developed a guide with indication of the most suitable methods to be used, according to the obtained accuracy represented by the minor %AAD from each evaluation.

Predictive Models of Kinematic Viscosity
Viscosity can be predicted by different types of models, such as semi-theoretical, theoretical, and empirical. Semi-theoretical models are based on the principle of corresponding states, applying reduced properties and theoretical models, which are based on models of statistical mechanics [1].

Prediction of Kinematic Viscosity of Hydrocarbons
To predict the kinematic viscosity of hydrocarbons, there are several models that can be applied (Table 1). However, the calculation and the accuracy will mainly depend on the application range of each model and the required input parameters.
Abbot [2] has developed two methods to predict KV (cSt) at 37.8 °C and 98.9 °C, applying only API characterization factor ( K API and API gravity (°API), shown in Eqs. 1 and 2. Results from those models presented a deviation of around 10 % to 15 %, for petroleum fractions, with KV from 0.5 cSt to 200 cSt at 100 °F, 0.3 cSt to 40 cSt at 210 °F, and with °API from 0 to 25, and K API from 10 to 12.5 [2].
With only mid-boiling point (K), T b , it is also possible to predict KV (mm 2 s −1 ) of petroleum fractions and hydrocarbons at any temperature T (K) by applying Mehrotra et al. [1] model (Eq. 4).
Another model that requires only one input property is the one developed by Puttagunta et al. [44]. Their method (Eq. 5) requires KV (mm 2 s −1 ) at 37.8 °C and is applicable for fractions with API gravities in excess of 30 degrees and API at low pressure. It is also recommended by the API-TDB-American Petroleum Industry-Technical Data Book on Petroleum Refining [44].
With more and different input parameters, the KV of petroleum fraction can be predicted by Beg et al. [45] method (Eq. 6), applying parameters A and B, which are obtained from temperature at 50 % volume distilled ( T 50 ) and API gravity. This model can be applied only for data with boiling point from 200 °F to 850 °F [45]. Beg and Amin [30] also developed two more models to predict KV, Eq. 7 for heavy petroleum fraction and Eq. 8 for petroleum fraction with boiling point from 95 °C to 455 °C.
There is also another model to predict the KV of petroleum fractions at any temperature T (K) that requires three input properties, such as molecular weight, density at temperature T T , and temperature at 50 % volume distilled T 50 (K) (Eq. 9). In case of lack of molecular weight, density, and temperature T, authors Fang and Lei [31] have suggested equations to obtain them, using density at 15.6 °C ( 15,6 ) , and T 50 .
Also, there is a procedure to obtain KV from petroleum fractions by following the ASTM D341 [28] model. For that, one must apply two kinematic viscosities at two different temperatures (Eq. 10).
Santos and There are also two models of KV prediction at 98.8 °C, which are applicable for heavy oil with density from 929 kg/L to 1094 kg/L, and pour point of heavy oil from − 33 °C to 15 °C (Eqs. 11,12). Equation 11 requires kinematic viscosity at 60 °C and density at 20 °C, with Average Absolute Deviation, AAD = 1.54 mm 2 s −1 and %AAD = 4.46 Eq. 12 only requires kinematic viscosity at 60 °C, with AAD = 3.72 mm 2 s −1 and %AAD = 13.49 [32].

Prediction of Kinematic Viscosity of Blends
Kinematic viscosity of blends ( m ) can be predicted by two types of methods, such as mixture-viscosity equations and parameter mixing rules [1]. Their application depends on component similarity.
For KV prediction of a binary mixture based on mole fraction,x i , Kendal and Monroe model [54] (Eq. 17) can be applied. This model is recommended by API-TDB [44] and it is widely used due to its satisfactory results [1]. Arrhenius model and Kendal and Monroe model (Eqs. 14, 17, respectively) produce accurate results if the chemical structure of the two compounds is similar, such as non-polar and non-associated liquid, and if one of the compound is predominant [46].

Empirical Models with Additional Parameters
Equations 13-20 can predict the KV of blends ( m ) composed by components with similar properties. Then, when this similarity decreases, the prediction deviation increases. Therefore, equations with additional parameters (Table 3) are more suitable for complex mixtures to obtain higher accuracy [13,58].
Walther [36] model (Eq. 21) requires a parameter C, which is obtained from regression analysis. This parameter is related to type of mixture, such as C = 0.8 for petroleum mixtures [36]. Walther Lederer Power Twu and Bulls ln(ln(ν m + 0.7)) = m × (lnT) + b  [59]. According to the literature, Lederer [59] and Cragoe [48] apud [49] models provide a similar and satisfactory prediction for blend of petroleum fractions with low viscosity ratios [51]. In addition, Shu [60] developed a correlation to obtain alpha for heavy oil and petroleum fractions (Eq. 23) [60].
Power model, a modification of Kendal and Monroe (Shu, 1984) model (Eq. 24), and Miadonye et al. [35] model (Eq. 25) use mass fraction to predict KV. Miadonye et al. model has two parameters, related to viscosity interaction (a) and viscosity reduction (n). This model was developed to predict KF of bitumen B and diluent blend ( D [35]).
As a recommendation of API-TDB, there is Twu-Bulls [44] model to predict the KV of a binary mixture, requiring KV at two different temperatures (T, in degrees Rankine) to estimate m and b parameters (Eq. 26).

Methodology
After reviewing predictive methods of KV (Tables 1, 2, 3), we calculated it applying, as input, some experimental properties provided by Brazilian oil industries and from the literature. Posteriorly, once R 2 might be inappropriate to validate the performance of non-linear models, we evaluated their performance by the average of absolute difference between the calculated property and the value of experimental property. We calculated average absolute deviation percentage, %AAD (Eq. 27), and average absolute deviation, AAD (Eq. 28). As a result, comparing the %AAD of each predictive method, the most appropriate and accurate one is the equation that provided predicted KV with the minor %AAD.

Approach for KV Prediction of Diesel Fraction from Petroleum Assays
To predict KV of diesel fraction, we applied data (56 samples) from petroleum assays. The characterization of the petroleum can be represented by the specific range of values for some properties, such as specific gravity at 15 °C from 0.82 to 0.91, °API from 24.15 to 41.83, KV at 20 °C from 3.82 mm 2 s −1 to 15.23 mm 2 s −1 , KV at 40 °C from 2.59 mm 2 s −1 to 7.83 mm 2 s −1 , KV at 50 °C from 2.20 mm 2 s −1 to 5.97 mm 2 s −1 , KV at 100 °C from 1.18 mm 2 s −1 to 2.26 mm 2 s −1 , and KV at 150 °C from 0.76 mm 2 s −1 to 1.24 mm 2 s −1 (Table 4) [61]. We also calculated KV at several temperatures for diesel fraction, applying methods developed to obtain KV through KV data at different temperatures (Eqs. 5, 10) ( Table 5). As mentioned previously, methods requiring KV at specific temperature to obtain KV in a different temperature are not considered predictive but a conversion one.
To estimate KV by Eq. 8, it was necessary to calculate molecular weight, since this is a difficult property to acquire, through mean average boiling point of petroleum fraction (°R), T b , and specific gravity at 60 °F/60 °F, s. The method for molar mass prediction (Eq. 29) is applicable for a boiling point from 90 °F to 1050 °F, API gravity from 14.4 to 93.1, and specific gravity from 0.63 to 0.97, and it is recommended by API-TDB [44].

Approach for KV Prediction of Diesel Fraction S500 from Blending Systems
To evaluate KV prediction methods, we applied 95 data of diesel S500 from blending system from Brazilian oil industries, which has specific properties ( Table 6). All 60F∕60F .  the models presented previously were evaluated, except Eqs. 11 and 12, once they have been developed for heavy oil, as well as Eq. 7, designated for heavy petroleum fraction.

Approach for Improving KV Prediction of Diesel Fractions
To obtain higher accuracy, we applied diesel data from petroleum assays to develop new models at temperatures from 20 °C to 150 °C and also a new method to predict KV at any temperature. In addition, since 40 °C is a crucial temperature to monitor kinematic viscosity for diesel specifications, we combined diesel fraction data from assays and diesel S500 from blending system to generate a model to amplify the restriction range for KV prediction at 40 °C.
Lastly, new methods were tested and validated with different data. Those new equations that predict KV at any temperature and KV at 40 °C were also evaluated through diesel S500 data to compare results from existing models.

Approach for KV Prediction of Diesel-Biodiesel Blends: Effect of Temperature, Volume Fraction, and Biodiesel Feedstock at KV Prediction
To evaluate the accuracy of predictive methods (models with additional parameters and models for mixing pure component viscosities) of KV and their application for mixtures of fossil fuel and renewable fuel, such as biodiesel, diesel and biodiesel, 431 data from literature were applied. Those data comprehend density and KV at several temperatures, such as at 20 °C, 30 °C, 40 °C, 60 °C, 80 °C, and 100 °C (Table 7). In addition, we applied data with varied volume fraction of diesel and data with several types of feedstock of biodiesel, such as soybean, yellow grass, castor, peanut, sunflower, and canola (Table 8). For this study, all the methods were evaluated, except Shu (Eq. 23), once it requires non-available data such as density of each compound at specific temperature.

Evaluating Empirical Models of KV
For KV calculation of diesel fraction from petroleum assays, we observed that the best accuracy was from Eq. 10, with minor deviation (%AAD = 0.002 %) for KV at temperatures from 20 °C to 50 °C. Differently, the deviation was higher from Eq. 10 regarding KV at 100 °C with %AAD = 0.45 and regarding KV at 150 °C with %AAD = 3.55. As a result from this evaluation, we observed that Eq. 10, independently of KV at a specific temperature, presented higher accuracy. However, we highlight that this is not a predictive model, since it requires two kinematic viscosities in different temperatures to obtain KV in another temperature, as well as Eq. 5. Therefore, if there are available KV in two different temperatures, we recommend applying Eq. 10 to obtain KV at the desired temperature. Otherwise, one must apply a predictive method. For that, evaluating several methods shown in Table 1, we obtained different behavior and results for each model, according to specific temperature (Table 9). In another words, even though all models can be applied for any temperature, we identified accuracy variation according to the temperature of kinematic viscosity.
Prediction of kinematic viscosity at 150 °C was more accurate if obtained through Eq. Therefore, we assumed that KV prediction can be performed by specific equations depending on the temperature range. For example, at 150 °C and 100 °C, KV presents the same performance. Then, KV can be predicted by the same model, which is Eq. 6. For lower temperatures, such as 50 °C, 40 °C, and 20 °C, Eq. 3 is the most appropriate model.  1.98 30.47 By this assessment of predictive methods, we realized that two input properties, such as mid-boiling point and specific gravity or API gravity, are recommended to predict flow liquid behavior at temperatures from 20 to 150 °C. Perhaps, because of that, Eq. 4 presented higher deviation, since it requires only mid-boiling point.
Assembling the results of KV predicted at several temperatures (20 °C to 150 °C), except results from Eq. 5, which did not predict KV at T = 37.8 °C once it is an input of the equation, we obtained that Eq. 6 was the most accurate method to predict KV at any temperature, since it presented minor deviation (AAD = 0.328 mm 2 s −1 and %AAD = 8.22) (Table 10).
Once diesel fractions data have presented K API from 12.56 to 13.86, Eqs. 1 and 2 were not applicable in this study. In addition, we did not evaluate Eqs. 11 and 12, since they have been developed for heavy oils. Equation 7 also was not evaluated because it is designated for heavy petroleum fraction.

Evaluating Empirical Models
Applying existing models to predict KV of diesel fraction S500 from blending systems at 40 °C, we observed that Eq. 3 presented a satisfactory accuracy (%AAD = 6.69) ( Table 11). Comparing this result with KV prediction at 40 °C from diesel fraction from petroleum assays, we observed that Eq. 3) is the most appropriate existing model to predict diesel fraction KV at 40 °C. Equations 7, 11, and 12 were not evaluated, since they are not applicable for KV data of diesel S500, but for heavier fractions. Also, Eq. 5 was not evaluated once it requires KV at 37.8 °C.
Equation 1 is originally developed to predict KV at 37.8 °C, not 40 °C. However, it presented the second minor deviation (%AAD = 7.52). Therefore, Eq. 1 can also be applied to predict KV at 40 °C with accuracy similar to Eq. 3.

Improving KV Prediction of Diesel Fractions: Developing New Models
Several equations for KV prediction from the literature presented low deviation, mainly at 150 °C and 100 °C. However, to achieve higher accuracy, we developed new equations for several temperatures, such as 20 °C, 40 °C, 50 °C, 100 °C, and 150 °C, and also for any temperature T ( We analyzed the linear and non-linear correlation among kinematic viscosity and properties, such as T10, T30, T50 (temperature at 10, 30, and 50 vol% distilled, respectively) and specific gravity to choose efficient parameters to predict KV. After calculations, T30 and specific gravity presented the most satisfactory performance through a non-linear regression.
A new equation can predict KV at any temperature for diesel fraction (Eq. 35) and it also presented lower deviation than the one from the existing models. In addition, we developed empirical methods to predict KV at specific temperature to improve accuracy (Eqs. [30][31][32][33][34]. Those equations presented a deviation lower than 6 % (Eqs. [30][31][32][33][34][35]. For KV at 40 °C, we developed a new method including diesel fraction data from petroleum assays and diesel fraction S500 data from blending system. Therefore, Eq. 31) can be applied for a larger range of diesel fraction data, such as KV from 0.81 mm 2 s −1 to 7.98 mm 2 s −1 , T 30 % from 214.00 °C to 280.91 °C, and specific gravity from 0.83 to 0.90.
For KV prediction at any temperature, comparing results from the most accurate existing method (Eq. 6) with %AAD = 8.22 (Table 10), we observed that the developed method (Eq. 35) presented higher accuracy, since its deviation was lower, %AAD = 5.50 (Table 12).
For KV prediction at 40 °C, applying diesel S500 data from blending system, we also observed a satisfactory prediction. As a result, Eq. 35) presented deviation AAD = 0.11 mm 2 s −1 and %AAD = 4.00, showing higher accuracy than the existing models.
The same occurred when we applied diesel S500 data to evaluate Eq. 31), which presented AAD = 0.10 mm 2 s −1 and %AAD = 3.66. Therefore, with similar    4.77 (35) performance (Figs. 13, 14), to achieve higher accuracy from data similar to diesel S500, Eq. 35), and, mainly, Eq. 31), might be more efficient models to predict KV at 40 °C. We also observed that the test achieved minor deviation compared to the original data. This performance can be explained due to obedience to restriction ranges of these new models.

Diesel Blends
To evaluate the most applicable empirical methods to predict KV of diesel-biodiesel blend, we considered four categories of cases, such as (1°) KV prediction with no criteria of variables (temperature, volume fraction, and feedstock); (2°) KV prediction regarding volume fraction of biodiesel; (3°) KV prediction regarding temperature; and (4°) KV prediction regarding the biodiesel feedstock.

(2°) KV Prediction Regarding Volume Fraction of Biodiesel
To understand the effect of volume fraction for KV prediction, as a second case, we evaluated the predicted KV at any temperature in three categories, such as biodiesel with volume fraction below 50 %; with 50 %; and with volume fraction more than 50 %. In this case, mixture-viscosity equations presented higher accuracy with minor %AAD compared to models with additional parameter. We also observed that the lowest accuracy (highest %AAD) was performed with 50 % of volume fraction of biodiesel and, in general for all methods, the second lowest accuracy was performed with volume fraction of biodiesel more than 50 %. Arrhenius [50] apud [51] model (Eq. 14) was the most appropriate model for less than 50 % (%AAD = 1.49) and greater than 50 % (%AAD = 1.53) of volume  [46]. Also, Eq. 14 presented similar performance for volume fraction of biodiesel equal to 50 %, with %AAD = 2.59 (Fig. 17, Table 2.1 at Supplementary Material). For models with additional parameters, which presented minor deviation considering the specific volume fraction, we observed that the most appropriate    Eq.14 w/%v<50%B Eq.17 w/ %v=50%B Eq.14 w/ %v>50%B Eq.14 w/ %v=50%B model for volume fraction equal to 50 % of biodiesel was performed by Power model [35] (Eq. 24) (%AAD = 2.76). For volume fraction greater than 50 % of biodiesel, Lederer [59] (Eq. 22) and Power [35] methods were the most suitable, with %AAD = 2.30 and %AAD = 2.32, respectively. Also for volume fraction of biodiesel with less than 50 %, Lederer [59] and Power [35] methods were the most suitable with %AAD = 2.27 for both (Table 2.2 at Supplementary Material with appropriate  parameters).
Further studies with specific data relating volume fraction with temperature are needed to better evaluate the influence of both conditions. However, with this evaluation, we observed that kinematic viscosity prediction is influenced by composition, then we suggest to be aware of the possibility of greater deviation when biodiesel concentration increases.

(3°) KV Prediction Regarding Temperature
Analyzing the performance according to the temperature, as the third case, we observed that the most appropriate method and its accuracy vary according to the temperature (Tables 1, 15).
KV prediction presented the highest deviation at 20 °C (almost the double of average deviation, considering all temperatures with %AAD = 1.62). Since the difference between KV diesel and KV biodiesel is almost always the double, regardless of the temperature, we observed that it is not the value of KV at 20 °C that affects the prediction result, but it might be the type of model. Hence, KV prediction at 20 °C may not be appropriate to apply mixture index models, or KV at 20 °C might not follow an additive property performance. Nevertheless, more studies need to be carried out with the same biodiesel source and with the same volume fraction to verify the influence of those parameters. In this study, we could not follow this procedure due to scarcity of data.
After evaluating methods with additional parameter and comparing results (Tables 14, 15), we concluded that methods with additional parameters performed higher accuracy than mixture-viscosity methods. In addition, we observed that Lederer [59] method performed satisfactory results for almost all temperatures but for each temperature its parameter has a different value. Depending on the temperature, others also can be applied. We observed that methods with additional parameter performed with minor AAD among specific temperatures such as 30 °C to 60 °C. Therefore, to obtain the most satisfactory performance, we suggest applying determined methods for a specific temperature (Table 16). For example, for coconut, jatropha, and waste oil at 30 °C, the minor and similar deviation can be obtained applying Eq. 24 (n = 0.61) or Eq. 21 (C = 11.57) or Eq. 22 (α = 1.19) ( Table 16, Table 4 in Supplementary material). We just highlight that due to lack of data, for each temperature, a different type of data (different biodiesel feedstock) was applied. High varieties of biodiesel feedstock were used at 40 °C.

(4°) KV Prediction Regarding the Biodiesel Feedstock
For the fourth case of applying empirical methods of KV prediction regarding biodiesel feedstock, we observed that there was not only one the most suitable model to be applicable as well. Actually, the most appropriate model varied according to the biodiesel type, as well as temperature (Tables 3.1, 3.2 at Supplementary Material). However, in general, Lederer [59] method was the method with the minor deviation and for each type of feedstock there is a specific optimum parameter to be applied (Table 17).
On the other hand, the system with the lowest accuracy was that of soybean FAEE biodiesel-diesel blend, with AAD = 1.04 mm 2 s −1 and %AAD = 22.17, applying Linear [55] model at 40 °C (Table 5 at Supplementary Material). This might imply that the type of alcohol used during transesterification reaction affects the flow property, presenting higher KV, and consequently the accuracy of specific predictive method of KV. More studies might be necessary to verify the prediction of FAEE biodiesel. One of the most feedstock used for biodiesel production is soybean and we observed different models for each temperature to better predict KV. For any temperature, the most appropriate method was Linear [55] method with minor deviation, %AAD = 2.07 (Fig. 18 (Table 5 at Supplementary Material). This might imply that the type of alcohol used during transesterification reaction affects the flow property, presenting higher KV, and consequently the accuracy of specific predictive method of KV. More studies might be necessary to verify the prediction of FAEE biodiesel.
Finally, we highlight that Refutas [55] apud [36], Chirinos [47] apud [36], and Kendal and Monroe [54] models are not based on volumetric fraction to predict KV for blends. However, precise and similar results were obtained compared to models based on volumetric fraction.
In general, Miadonye et al. [35] and Power [35] models presented the highest deviation, with %AAD = 9.55 and %AAD = 8.45, respectively. However, Miadonye et al. model might have lower accuracy, since it was developed for bitumen. We also highlight that due to lack of density of compounds and/or of mixtures at specific temperatures, Shu [60] method was not evaluated and Walther [36] and Lederer [59] methods were evaluated through volume fraction instead of mass fraction.

Results Overview: A KV Prediction Guide
As an overview of the results obtained in this assessment, we developed a guide with suggestions of the most appropriate models, regarding minor deviation, to predict kinematic viscosities according to diesel system, its properties, and temperature (Fig. 19).
For diesel fraction at 40 °C, temperature to attend specifications, the most appropriate model was the new Eq. 31 presenting %AAD = 5.00, or the existing model Eq. 3, with %AAD = 10.21. For any other temperature, we suggest applying the new Eq. 35, with %AAD = 5.50, or the existing model Eq. 6, with %AAD = 8.22, or verifying a specific temperature and system at Table 12.
For diesel fraction blend system at 40.0 °C (temperature to meet specifications), the most appropriate model was Eq. 31, presenting %AAD = 5.00, and the existing model Eq. 3, with %AAD = 6.69.
To achieve higher accuracy of KV prediction of diesel-biodiesel blend with a different biodiesel, and also at another temperature different from 40 °C, others models are more suitable (Tables 16, 17). In addition, in the case of lack of biodiesel KV to be applied on calculations, we suggest consulting a list (Table 7) with average KV of several biodiesel types at 40 °C, obtained according to the literature.
In this guide, no model was suggested regarding volume fraction, since more studies are needed considering at specific temperature and feedstock biodiesel. However, we highlight that increasing biodiesel volume fraction might decrease the accuracy of these prediction models.

Conclusion
In this study, several models of KV prediction were analyzed to different conditions of diesel fraction and diesel blends.
For KV prediction of diesel fraction from petroleum assays, we obtained different behavior and results for each model, according to specific temperature. In another words, even though all models can be applied for any temperature, we identified accuracy variation according to the temperature of kinematic viscosity.
For KV prediction of diesel fraction S500 from blending systems at 40 °C, we observed that Eq. 3 presented a satisfactory accuracy (% AAD = 6.69).
To achieve higher accuracy, we developed new equations for KV at several temperatures, such as 20 °C, 40 °C, 50 °C, 100 °C, and 150 °C, and also for any temperature T (Table 12, Figs. 1, 2, 3 , 4, 5, 6, 7, 8, 9, 10, 11, 12) for diesel fractions. As input for new methods, we studied T10, T30, T50, and specific gravity. We verified that the most satisfactory performance through a non-linear regression to predict KV are applying T30 and specific gravity. Therefore, only methods with T30 as input were presented in this study. Those methods were tested with different data and provided satisfactory results.
For KV prediction of diesel blends, we considered four categories of cases, such as (1°) KV prediction with no criteria of variables (temperature, volume fraction, and feedstock); (2°) KV prediction regarding volume fraction of biodiesel; (3°) KV prediction regarding temperature; and (4°) KV prediction regarding the biodiesel feedstock.
As result for the first case, Eq. 14 was the most suitable method for KV prediction, with the minor deviation, AAD = 0.06 mm 2 s −1 and %AAD = 1.62.
For considering the study of the impact of volume fraction on KV prediction as the second case, we evaluated the KV prediction of blend with 50 % of the biodiesel, greater than 50 % of biodiesel and less than 50 % of biodiesel. As results, Eq. 14 was the most accurate method for diesel blend with less and greater than 50 % of biodiesel. For volume fraction of biodiesel equal to 50 % the accuracy is poorer than the other two fractions and presented Eq. 17 as the most accurate method. For models with additional parameters the higher accuracy was obtained through Eqs. 22 and 24. For the third and fourth cases, we observed that the most appropriate method and its accuracy vary according to the temperature and biodiesel feedstock. Therefore, to achieve higher accuracy a specific method must be applied for a specific temperature and feedstock.
We concluded in general that, to obtain the highest accuracy for KV prediction, it is adequate to apply an appropriate model for specific condition and fuel properties. Except in the case of soybean biodiesel-diesel blend, any type of KV prediction model (if mixture-viscosity equations or model with additional parameter) can provide similar performance. Also, some models might have their accuracy decreased when the volume fraction of biodiesel increases but more studies must be carried out to verify the influence of the volume fraction on the predictive methods.
Overall, interested researchers can follow the developed guide to choose the most appropriate model for achieving higher accuracy to predict KV of diesel fraction and its derivatives, according to the object of study, KV range, and temperature.