Influence of the impurity content on the density and viscosity of olive oily fluids

2.1. Test samples

In this work, 12 oily samples (approximately 10 l each) were collected between December 2019 and February 2020 from an olive oil mill (‘almazara’) located near Priego de Córdoba (Córdoba, Spain). Each of these samples had different characteristics and appearance (see Figure 1), mainly due to the date of collection and the point chosen for sampling. In summary, Table 1 shows some data of interest related to these samples. At this point, it should be noted that the production process carried out by the aforementioned ‘almazara’ was based on a ‘continuous conventional system’. In this system, each processing line uses (after cleaning, washing, milling and kneading) a two-way centrifugal decanter (‘horizontal centrifuge’) for solid-liquid separation, and a disc stack centrifuge (‘vertical centrifuge’) for liquid-liquid separation (Uceda et al., 2006Uceda M, Jiménez A, Beltrán G. 2006. Trends in olive oil production. Grasas Aceites57 (1), 25−31.). In addition, the oil must is not filtered, but it can be sieved after the decanter step and can be clarified (with dynamic settling tanks) after the vertical centrifuge.

2.2. Lab equipment

In order to conduct the necessary experiments, it was decided to build several tube viscometers (Steffe, 1996Steffe JF. 1996. Rheological methods in food process engineering. Freeman press, East Lansing, MI, USA.), whose geometry is described in Figure 2.

For manufacturing each viscometer, a calibrated plastic vessel was mainly required (to contain the fluid to be tested), as well as a transparent polyethylene tube (whose length and diameter were fixed by design to facilitate the development of laminar flow through the duct). The vessel utilized in each device had a capacity of one liter and it was marked with different reference lines to measure the fluid heads (h) reached inside it. The tube length (L) and the diameter (D), selected to construct the viscometers, followed these combinations to modulate different flow rates, shear rates or shear stresses throughout the experiments: L = 0.2 m, D = 4.07 mm; L = 0.4 m, D = 4.07 mm; L = 1 m, D = 7.63 mm; L = 1,20 m, D = 3.87 mm and L = 1.60 m, D = 4.10 mm. The tube insertion length (δ) inside the vessel (see Figure 2) was 0.01 m in all cases.

In addition to the above, other auxiliary materials were needed to run the tests: two glass thermometers (to determine the fluid temperature at the viscometer pipe ends) with a measuring range of -10 ºC to +60 ºC and with an accuracy of ±1 ºC; six laboratory graduated cylinders of 100 ml (with an accuracy of ±1 ml), and five more of 250 ml (with an accuracy of ± 2 ml), to collect the volumes that would be useful for determining densities and flow rates; two laboratory scales with accuracies of ±0.01 g and ±0.001 g, respectively; a digital chronometer (with a resolution of 0.01 s); A centrifuge with adjustable temperature and speed controls (suitable for 50 ml test tubes from -9 ºC to +40 ºC, and from 200 rpm to 14000 rpm, respectively); two refrigerating chambers (at 4.5 ºC and 12 ºC) and a freezing chamber (at -18 ºC) for storing samples during testing; ice blocks, hot water and 10-l buckets for rapid cooling (or heating) of oily samples by indirect contact; auxiliary bottles of 1.5 l to contain samples if required; calibrated 50-ml tubes for centrifugal separation; pipettes and syringes for handling or transferring samples; a measuring tape (calibrated in millimetres) and various office supplies (scissors, adhesive tape, markers, etc.).

2.3. Viscosity and density tests of oily samples

The experiments in this section were performed using a viscometer with a tube length (L) of 0.4 m and a diameter (D) of 4.07 mm. Having made the above clarification, it should be noted that the original samples (see section 2.1 and Table 1) were first homogenized by manual shaking and then divided into different 1.5-l auxiliary bottles to facilitate the tasks involved in the tests. The bottles were kept cold (at 4.5 ºC or -18 ºC, depending on the expected time before use) to avoid any microbiological activity (such as fermentation due to the presence of yeasts).

When it was convenient, one of the refrigerated bottles (at 4.5 ºC) was taken and tempered in the laboratory (at 23 ºC) for (1 - 2) h. Bearing in mind that the fluid temperature in the tests had to be similar to that present in the industrial production of olive oil (of the order of 20 ºC − 30 ºC), each viscosity experiment was started when the sample temperature was approaching 20 ºC. At that moment, the fluid was poured into the viscometer vessel (which was exposed to the atmosphere) and, after its stabilization and its flow through the tube, it was collected for a given time in a lab cylinder (placed in the lower part of the experimental setup, see Figure 2). Fluid volume (V), stored in the graduated cylinder, was also weighed with a laboratory scale to estimate its density ρ, (Eq. 1)

where m was the mass linked to V.

In addition, flow rate Q was estimated from V taking into account the time τ needed to collect such fluid volume (Eq. 2)

On the other hand, the fluid temperature was measured at two points (at the input and output ends of the viscometer pipe), and adjusted by indirect contact using hot or cold water before the fluid was poured into the viscometer vessel. The fluid height h inside the vessel (see Figure 2) was observed at the beginning and at the end of each test, and the mean of these measurements was taken as a representative value (usually in the range of 4 −6 cm).

From the procedure described above, which was applied 144 times (12 tests, in the range of 20 ºC−30 ºC, for each of the 12 types of fluid treated), and taking into account what it is described in section 2.2, the values for dynamic viscosity µ were estimated from the following expression (Eq. 3), based on the Hagen-Poiseuille equation and on the energy equation (Singh and Heldman, 2009Singh PR, Heldman DR. 2009. Introducción a la Ingeniería de los Alimentos. Acribia, Zaragoza, Spain.).

where ∆p = ρ g (h + L - δ) is the pressure loss between 1 and 2 (see Figure 2), g is the gravitational acceleration, and the other variables are those just indicated in this section and in paragraph 2.2. The use of Eq. 3, to estimate µ requires, in principle, the assumptions that the fluid tested is Newtonian, the kinetic heads are negligible and the flow is stationary and perfectly laminar throughout the entire length L. This last assumption implies that the effect of the inlet length and the local head losses at the pipe inlet can also be neglected. The accuracy of the previous approximations was analyzed for flows such as those treated here in Tirado (2020)Tirado, A. 2020. Estudio de la densidad y la viscosidad de fluidos oleosos procedentes de almazara. Trabajo Fin de Grado. ETSIAM - Universidad de Córdoba, Spain, where it was verified that for oily fluids at very low Reynolds numbers (of the order or less than 10) such premises could be accepted. To minimise possible errors, it should also be noted that the values of ρ employed in Eq. 3, were obtained, in each case, from polynomial interpolation taking 12 pair of values (t, ρ) available as reference for each type of fluid from the experiments.

2.4. Impurity content tests

Considering as impurities, as a whole, all those remains of vegetation water, pulp, olive stone (and even yeasts) present in the EVOO, in this work this matter content was determined according to an alternative method (Vallesquino and Tirado, 2023Vallesquino P, Tirado A. 2023. Un estudio sobre la densidad y la viscosidad de fluidos oleosos procedentes de almazara. Actas del XXI Simposio Científico Técnico - Expoliva 2023 IND - 8, 1−9.), different from the official one (IOC, 2019IOC. 2019. Trade standard applicable to olive oils and olive-pomace oils. COI/T.15/NC No 3/Rev.14/2019.), in order to obtain sufficiently accurate results while simplifying the experimental procedure. Therefore, the impurity content c, expressed as a percentage of the volume of impurities (V_imp) and the sample volume (V_sp) (Eq. 4):

could be estimated as follows: firstly, two samples (of around 40 ml) were taken from every type of fluid previously treated in the ‘viscosity and density tests of oily samples’ (see section 2.3), and they were poured into two centrifuge tubes like those shown in Figure 1. These tubes were allowed to decant naturally in the laboratory, between (24 - 48) h at 23 ºC, and then centrifuged at 3900 rpm for 12 min at 27 ºC. After that, the volume of impurities present in each sample was clearly visible (see the lower part of each tube in Figure 3, as an example), and this volume (in many cases very small) could be determined by applying the appropriate operations (Vallesquino and Tirado, 2023Vallesquino P, Tirado A. 2023. Un estudio sobre la densidad y la viscosidad de fluidos oleosos procedentes de almazara. Actas del XXI Simposio Científico Técnico - Expoliva 2023 IND - 8, 1−9.).

The bulk density of the impurities was determined separately, with the aim of estimating a more representative value than that which could be obtained from sub-samples like those shown in Figure 3. With this in mind, and depending on the content of non-oily remains, among 2 − 4 samples of each type of fluid were randomly selected (from those collected in 1.5-l bottles, as described in section 2.3), and the impurities present in their bottoms were extracted. Such sediments were formed by gravitational decantation until an effective separation between the substances could be visually verified. This task could take no less than (2-3) wk per sample, applying temperatures between 4.5 − 23 ºC to avoid unwanted fermentations. Next, the bottom content extracted from these samples was homogenized and 50 ml were poured into one or two centrifuge tubes (depending on the amount of impurities available) and centrifuged at 3900 rpm for 12 min at 27 ºC. Once this operation had been carried out, the appearance of the samples obtained was similar to that shown in Figure 3, but with a more abundant impurity phase (of the order of 30 ml), in which it was generally possible to observe vegetation water (about 55%) and insoluble solids (about 45%). The bulk density of each impurity sample could then be estimated by a simple fraction between the mass and the volume of the impurities (after carefully removing the residual oil that could be present in each tube).

2.5. Rheological tests of oily samples

These experiments were designed to verify whether the dynamic behavior of the fluids studied in this work was in accordance with Newton’s law of viscosity. Therefore, the relationship between σ_w (wall shear stress of the flow) and ${\left(\frac{du}{dr}\right)}_w$ (wall shear rate) was analyzed. According to Steffe (1996)Steffe JF. 1996. Rheological methods in food process engineering. Freeman press, East Lansing, MI, USA., σ_w can be expressed as follows (Eq. 5) for the laminar flow of a time-independent fluid through a cylindrical tube (Figure 2):

For its part, the shear rate ${\left(\frac{du}{dr}\right)}_w$ depends on the fluid treated. However, this variable could be expressed as a function ‘f [ ]’ of this elemental ratio (γ_e) (Eq. 6):

which can be used in further analysis.

Considering the above premises (together with the available resources), and trying to obtain an adequate set of pairs (γ_e , σ_w), it was considered appropriate to use four of the viscometers already mentioned in section 2.2 in these tests. The dimensions of these devices were: L = 0.2 m, D = 4.07 mm; L = 1 m, D = 7.63 mm; L= 1.20 m, D = 3.87 mm and L= 1.60 m, D = 4.10 mm. In addition, four samples were tested with each of these viscometers: M1, M4, M7 and M10, chosen because they showed a representative variability in impurity content. In order to avoid unwanted fermentations (due to the racking, aeration and duration of these tests), about 3 l of each of these samples were pasteurized (at 80 ºC for 45 min). They were then cooled (at a temperature close to 23 ºC) and analyzed 5 times with each of the viscometers cited above, following a procedure analogous to that described in section 2.3. Furthermore, it should be noted that the data previously obtained from the unpasteurized samples (according to section 2.3) were used as control data for the corresponding comparisons.

3.1. Impurity density and content

Table 2 shows the data related to the impurity content c, as well as the bulk density values (ρ_ap) linked to them. From these data, it is worth noting that the minimum and maximum c values are of the order of 0.5% (sample M4) and 5.87% (sample M2), respectively. These data can be considered normal when compared to those obtained by other authors (Hermoso et al. 1996Hermoso M, González J, Uceda M, García-Ortiz A, Morales J, Frías L, Fernández A. 1996. Elaboración de aceite de calidad. Obtención por el sistema de dos fases. Consejería de Agricultura y Pesca - Junta de Andalucía. Sevilla, Spain.; Uceda et al. 2006Uceda M, Jiménez A, Beltrán G. 2006. Trends in olive oil production. Grasas Aceites57 (1), 25−31.; Bejaoui et al., 2013Bejaoui MA, Gila A, Allouche, Y, Aguilera MP, Jiménez A, Beltrán G. 2013. Eficiencia de la clarificación del aceite de oliva virgen mediante decantación y centrifugación. Actas del XVI Simposio Científico Técnico - Expoliva 2013IND - 36, 1- 6.; Dammak et al., 2015Dammak I, Neves M, Souilem S, Isoda H, Sayadi S, Nakajima M. 2015. Material balance of olive components in virgin olive oil extraction processing. Food Sci. Technol. Res.21 (2), 193−205. https://doi.org/10.3136/fstr.21.193 ; Gila, 2017Gila A. 2017. Clarificación de los aceites de oliva vírgenes y su efecto en las características físico-químicas y sensoriales. Doctoral dissertation. Universidad de Jaén, Spain.), who presented c results which varied from 1 to 15% for fluids taken from the decanter, and from 0.2 − 0.6% (Bejaoui et al., 2013Bejaoui MA, Gila A, Allouche, Y, Aguilera MP, Jiménez A, Beltrán G. 2013. Eficiencia de la clarificación del aceite de oliva virgen mediante decantación y centrifugación. Actas del XVI Simposio Científico Técnico - Expoliva 2013IND - 36, 1- 6.; Dammak et al., 2015Dammak I, Neves M, Souilem S, Isoda H, Sayadi S, Nakajima M. 2015. Material balance of olive components in virgin olive oil extraction processing. Food Sci. Technol. Res.21 (2), 193−205. https://doi.org/10.3136/fstr.21.193 ) for samples collected from the vertical centrifuge (note that samples M4, M6 and M12 came from the vertical centrifuge, and the other ones from the horizontal decanter; see Table 1). The factors causing this variability (in this study and in those cited) can be diverse and they could have their roots in agronomic and technological aspects (Hermoso et al., 1996Hermoso M, González J, Uceda M, García-Ortiz A, Morales J, Frías L, Fernández A. 1996. Elaboración de aceite de calidad. Obtención por el sistema de dos fases. Consejería de Agricultura y Pesca - Junta de Andalucía. Sevilla, Spain., Uceda et al., 2006Uceda M, Jiménez A, Beltrán G. 2006. Trends in olive oil production. Grasas Aceites57 (1), 25−31.; Vallesquino and Tirado, 2023Vallesquino P, Tirado A. 2023. Un estudio sobre la densidad y la viscosidad de fluidos oleosos procedentes de almazara. Actas del XXI Simposio Científico Técnico - Expoliva 2023 IND - 8, 1−9.). Most likely, in the absence of more data, the lower c results (for the samples taken from the decanter in this study) could be related to longer residence times, higher processing temperatures and smaller outlet radii for the oily phase. In this context, sample M11 could be a representative example of these operating conditions, as it appears to have been taken from a vertical centrifuge, but was actually collected from a decanter.

Regarding the bulk density values (ρ_ap) presented in Table 2, it is noteworthy that they all fell within a range of 1080 kg/m³ − 1240 kg/m³. Attempting to compare these data with those of other studies, it is notorious that Gila (2017)Gila A. 2017. Clarificación de los aceites de oliva vírgenes y su efecto en las características físico-químicas y sensoriales. Doctoral dissertation. Universidad de Jaén, Spain. points out that there are few references on this issue in the literature, placing ρ_ap (through simulations) in a range of 1025 kg/m³ to 1225 kg/m³. In Alba (2008)Alba J, 2008. Elaboración del aceite de oliva virgen, in Barranco D (Ed.) El cultivo del olivo. Madrid, Spain, 657-697., the density of oily wastewaters (alpechines) is set around 1050 kg/m³, and that of the pomace at 1200 kg/m³ (values compatible with those presented here).

It is also remarkable that ρ_ap seems to be smaller for the samples taken from the vertical centrifuge (M4, M6 and M12) compared to those obtained from the decanter. This could make sense if one considers that, according to Stokes’s law (Mafart and Béliard, 1994Mafart P, Béliard E. 1994. Ingeniería Industrial Alimentaria (Vol. II). Acribia, Zaragoza, Spain.), the densest and largest particles are easier to separate from the liquid phase in the first centrifugation (with the decanter). Similarly, although the number of data available is not large, the values for ρ_ap in Table 2 seem to show a decreasing trend as the production campaign progresses. This could be related to the changing nature of the raw material (maturity stage) and could be analyzed from a greater number of samples in future works.

3.2. Rheological tests results

Figure 4 shows the data of the shear stress σ_w (see Eq. 5), as a function of the elemental ratio ${\gamma }_e=\frac{32Q}{\pi D^3}$ , obtained from the rheological tests developed according to section 2.5 (at laboratory temperature of 23 ºC). The fluids treated in these experiments were: M1, M4, M7 and M10. It can be seen in Figure 4 that the relationship between σ_w and γ_e is linear, with zero ordinate at the origin. According to Steffe (1996)Steffe JF. 1996. Rheological methods in food process engineering. Freeman press, East Lansing, MI, USA., the data presented do not agree well with the Bingham or power law models, but they do with that expected for a Newtonian fluid. In such a case (Newtonian fluid), it could be stated directly (Eq. 7):

which implies that the shear rate ${\left(\frac{du}{dr}\right)}_w$ is directly equal to ${\gamma }_e=\frac{32Q}{\pi D^3}$ , and that the slope of the function ${\left(\frac{du}{dr}\right)}_w$ = f [γ_e] (see also Eq. (6)) corresponds to the dynamic viscosity (µ) of the fluids tested. In this scenario, it so happens that these fluids would have a µ value (at 23 ºC) that could be of the order of 0.07 Pa·s (sample M4) or 0.074 Pa·s (sample M10).

Other authors, like Fasina et al. (2006)Fasina OO, Hallman H, Craig-Schmidt M, Clements C. 2006. Predicting temperature-dependence viscosity of vegetable oils from fatty acid composition. J. Am. Oil Chem. Soc.83, 899−903. https://doi.org/10.1007/s11746-006-5044-8 , Bonnet et al. (2011)Bonnet JP, Devesvre L, Artaud J, Moulin, P. 2011. Dynamic viscosity of olive oil as a function of composition and temperature: A first approach. Eur. J. Lipid Sci. Technol.113 (8), 1019−1025. https://doi.org/10.1002/ejlt.201000363 or Gila (2017)Gila A. 2017. Clarificación de los aceites de oliva vírgenes y su efecto en las características físico-químicas y sensoriales. Doctoral dissertation. Universidad de Jaén, Spain., have also found that olive oil (without impurities) shows a Newtonian behavior, reporting values for µ which are similar to those indicated here. This circumstance confirms that the moderate presence of impurities does not seem to affect the Newtonian nature of olive oil. On the other hand, it is worth noting that the pasteurization treatment does not seem to have any effect on the rheological behavior of the fluids analyzed, since the control samples (without heat treatment, see section 2.3) have values for µ that are equivalent to those of the pasteurized samples. This aspect can be observed in Figure 4, where the values for M1, M4, M7 and M10 (marked with ‘O’ symbol and taken from the control samples) are perfectly in line with the rest of the values (obtained from pasteurized samples).

Besides the former, the data presented in Figure 4 are not evenly distributed. This effect is due to the design of the viscometers used, which conditions the values for γ_e that they can present. For instance, in those viscometers with D ≈ 4 mm, γ_e was in the range of 120 - 150 s^-1, and the remaining γ_e values were related to the device with D = 7.63 mm (including the singular point (0,0), explained by the fact that no fluid could be retained at the bottom end of the viscometer tube when it was finally emptied). Considering the results obtained in this study, it could be recommended for future works to follow an experimental design slightly different from the one treated here. In this regard, a better approach could be implemented by using a set of tube viscometers with a greater variety of tube diameters and fewer length options. Nevertheless, the results shown in Figure 4 could be useful references for the olive oil industry, since γ_e (see Eq. 7) is usually less than 250 s^-1 for oily flows of (1000 - 2000) l/h flowing through (3-5) cm diameter tubes (common operating values associated with decanters or vertical centrifuges).

3.3. Density and viscosity of the oily samples

Figure 5 presents the density values (ρ) for all the samples tested. On the left side of this Figure, ρ is plotted as a function of the impurity content c (see Eq. 4), taken as reference, as a first approximation, only three temperature values (t = 20 ºC, t = 25 ºC and t = 30 ºC) to carry out a simpler analysis. On the right side of Figure 5, ρ is shown according to the model results which will be presented later.

Observing the relationship between ρ, c, and t in the cited Figure, it is clear (despite the scatter) that the density showed a linear increasing trend as the impurity content increased. However, ρ tended to decrease as temperature t increased (at 30 ºC the values for ρ were, in general, smaller than at 25 or 20 ºC). Likewise, the gap (separation) between the functions ρ = f(c) maintained a certain linearity (the ordinate at the origin and the slope of such functions grew linearly with t), which suggests that the general relationship between ρ, c and t could be of the form:

where a₁, a₂, a₃ and a₄ are fitting parameters. In the scientific literature, the density of different types of oils, including olive oil, is usually adjusted by means of linear functions that depend on temperature (Esteban et al., 2012Esteban B, Riba JR, Baquero G, Rius A, Puig R. 2012. Temperature dependence of density and viscosity of vegetable oils. Biomass Bioener.42, 164−171. https://doi.org/10.1016/j.biombioe.2012.03.007 ; Gila, 2017Gila A. 2017. Clarificación de los aceites de oliva vírgenes y su efecto en las características físico-químicas y sensoriales. Doctoral dissertation. Universidad de Jaén, Spain.). In addition, their resulting values are similar to those presented here, taking into account that the oils treated in this study contained impurities. Considering the above, ρ data were fitted to Eq. (9) as follows:

where the determination coefficient (R²) was equivalent to 0.402 and the residual mean square error (RMS) was of the order of 11.070 (kg/m³)².

Figure 5 (right side) presents the fluid density values ρ_est , estimated from Eq. (10), in comparison to those measured in the experiments (ρ_ms). Except for the scale factor of this Figure, which in this case tends to exaggerate the magnitude of the deviations, the main trend established by Eq. (10) is in agreement with the experimental data. Moreover, Figure 5 shows that, in order to accurately take into account the effect of the impurity content on ρ, it is necessary to measure the fluid density with a lower tolerance (of the order of 0.1%), since the density variations are not very sensitive to changes in c values. In this respect, it should be noted that the data scatter shown in this Figure is the result of possible errors associated with the measurement of volumes using the materials mentioned in section 2.2. These errors are unlikely to be greater than ± 1.5%, but they are sufficient to produce a maximum deviation of ± 14 kg/m³, which is compatible with the RMS value just indicated.

Regarding the dynamic viscosity (µ), Figure 6 presents the values of µ in a similar way to that already performed with ρ. On the left side of such a Figure, three reference t values have been taken to analyze the variation in µ with respect to the impurity content c. It is observed, as in the case of ρ, that µ follows a linearly increasing trend with c, and a decreasing one (but not linear) with respect to t. On this occasion, the intercept and the slope of the fitting functions µ = f(c) do not vary linearly with t, which leads to the assumption that the possible relationship between µ, c and t could be of the type:

where b₁, b₂, b₃, b₄, and b₅ are fitting parameters.

According to several authors (Fasina et al. 2006Fasina OO, Hallman H, Craig-Schmidt M, Clements C. 2006. Predicting temperature-dependence viscosity of vegetable oils from fatty acid composition. J. Am. Oil Chem. Soc.83, 899−903. https://doi.org/10.1007/s11746-006-5044-8 ; Esteban et al., 2012Esteban B, Riba JR, Baquero G, Rius A, Puig R. 2012. Temperature dependence of density and viscosity of vegetable oils. Biomass Bioener.42, 164−171. https://doi.org/10.1016/j.biombioe.2012.03.007 ; Gila, 2017Gila A. 2017. Clarificación de los aceites de oliva vírgenes y su efecto en las características físico-químicas y sensoriales. Doctoral dissertation. Universidad de Jaén, Spain.; Sahasrabudhe et al., 2017Sahasrabudhe SN, Rodriguez-Martinez V, O’Meara M, Farkas BE. 2017. Density, viscosity, and surface tension of five vegetable oils at elevated temperatures: Measurement and modeling. Int. J. Food Prop.20 (sup2), 1965-1981. https://doi.org/10.1080/10942912.2017.1360905 ), the viscosity of diverse oils (including olive oil) could be estimated from different models which, essentially, are based on the Arrhenius equation:

where A is an empirical parameter, E_a is the activation energy, R is the universal gas constant, and T is the absolute temperature. Taking into account the exponential nature of the previous model, and the development of Taylor’s series for any exponential function (Spiegel and Ribero, 1970Spiegel MR, Ribero OG. 1970. Manual de fórmulas y tablas matemáticas. McGraw-Hill, Mexico. ):

it is obvious that Eq. (13) could be reformulated as a function such as:

which is directly related to the Slotte’s equation (Herschel, 1922Herschel, WH. 1922. The change in viscosity of oils with the temperature. Ind. Eng. Chem.14 (8), 715−722.). Note that, again, d₁, d₂, d₃ … are fitting parameters. For short temperature ranges (as occur in this study), where the slope of the function µ = f (T) does not change excessively with temperature, it may be feasible to apply an expression like Eq. (15) by only using, as an approximation, two or three equation terms. Furthermore, if T is well above 273 K, Eq. (15) could even be adapted for Celsius degrees, achieving quite reasonable results (Vallesquino and Sánchez, 2023Vallesquino P, Sánchez M. 2023. Un método para la determinación del contenido de agua de emulsiones de agua en aceite de oliva. Actas del XXI Simposio Científico Técnico - Expoliva 2023 IND - 1, 1−9.). Bearing in mind the above, µ data were adjusted for two models, such as those shown in Eqs. (11) and (12), in which the Celsius scale was used in the first one (Eq. 16) and the Kelvin scale was applied in the second one (Eq. 17):

On the right side of Figure 6, the values for dynamic viscosity estimated by means of Eq. (17) (µ_est) are plotted against those obtained from the experiments (µ_ms). It should be noted that, for the sake of clarity, the data related to Eq. 16 are not displayed in the Figure because they would overlap with those estimated from Eq. 17, (as can be inferred from R² and RMS values associated to these equations).

On the other hand, it is also remarkable that the fitting accuracy of Eqs. 16 and 17 is higher than that of Eq. 10, (see also the right side of Figures 5 and 6). This fact reveals that the dynamic viscosity of olive oil is a variable that is sensitive to variations in temperature (this is well-known), but it is sensitive, as well, to variations in the impurity content. This aspect is relevant because, similar to the approach presented by Vallesquino and Sánchez (2023)Vallesquino P, Sánchez M. 2023. Un método para la determinación del contenido de agua de emulsiones de agua en aceite de oliva. Actas del XXI Simposio Científico Técnico - Expoliva 2023 IND - 1, 1−9., by combining expressions such as those established in Eqs. 3 and 16 (or in Eqs. 3 and 17) it could be possible to calibrate empirical functions (ψ) of the form:

or

These ψ functions could, in the end, make it possible to estimate the impurity content c in real time (present in any oily must) if some simple sensors were installed on the production line. As an example, if Eqs. 3 and 16 were considered in this case, Eq. 18.a could be reformulated as:

where c is obtained, as a percentage, from two plant design parameters (L and D, in m) and three process line variables like Q (flow rate, in m³/s), t (temperature, in ºC) and ∆p (pressure drop, in Pa). Although Eq. (19) is only valid, as a rough approximation, for certain operating conditions (related to some samples taken from a determinate conventional olive oil mill), this approach opens the possibility of studying similar models that could be applied in any ‘almazara’, which could be of interest for the improvement of processes and products in the olive oil sector. In this regard, it should be pointed out that, in real manufacturing processes, knowledge of the impurity content in oily fluids could help to make appropriate decisions on factors such as the residence time of the processed fluids in decanters (or in vertical centrifuges) and the choice of the most suitable type of clarification to be applied after the second centrifugation.