To build an ideal model for the viscosity of liquid vegetable oil, theoretical analysis is usually required. It is well known that liquid molecules are quite close together. However, the intermolecular forces are not strong enough to keep neighboring molecules in a fixed position and molecules are free to move past or slide over one another very easily.

In a system like this, there is a relationship between average molecular separation and density. If the volume occupied by each molecule is assumed as the sphere of radius r, and the liquid is closely packed with the spheres, then the average intermolecular separation (2r) is able to be estimated by the following equation (Yang and Li, 2015; Yang and Zhu, 2016):

where, m denotes molecular mass, N is the total number of molecules contained in volume V.

There is also a relationship between molecular separation and intermolecular potential energy in the freely sliding molecular system. This relationship can be presented by the potential model. A famous potential model is the Lennard-Jones model (1924) (Lennard-Jones, 1924), which takes the following form for a 3β-3α pattern:

where intermolecular separation is denoted by d,φ_σ and σ are the Lennard-Jones’ constants.

For the system closely packed with freely sliding spheres, the density can be adopted as the independent variable instead of the intermolecular separation through equation (15) (Yang and Li, 2015; Yang and Zhu, 2016). Then the Lennard-Jones potential at the average intermolecular separation, which is an estimation of the average molecular potential of the system, is written as the summation of repulsive and attractive terms:

here, ρσ=6m/(πσ³), stands for a characteristic density at which the potential equals zero.

Using the density dependent Lennard-Jones’ potential, symmetric molecular interaction approximation (Yang and Li, 2015; Yang and Zhu, 2016) and Maxwell-Boltzmann distribution function for the freely sliding molecules, liquid viscosity can be derived as:

where K, A, B are coefficients which are independent of density and temperature. This equation presents the mathematical model for viscosity in terms of both temperature and density without a restriction on pressure. Physically, the equation (18) involves two molecular mechanisms that govern the variations in viscosity with temperature and density (Yang and Zhu, 2016). The first is the momentum transfer due to the transportation of the freely sliding molecules, which is achieved through the term in the form of root square of temperature multiplying 2/3rd power of density. The second is the statistical effect of molecular potential realized in terms of the exponent of the ratio of average intermolecular potential to average kinetic energy. With respect to liquid vegetable oil at atmospheric pressure, the equation of state is found as a negatively linear relation between density and temperature (Esteban et al., 2012), which can be expressed as:

or

here, ξ and T₀ or ρ₀= ξ T₀ are coefficients.

Based on this equation of state, equation (19) is able to be simplified as a univariate function. When the independent variable is temperature, substituting equation (20) into equation (21) yields:

where, K, A, B, α, β, are adjusting coefficients, T the temperature in K, η the viscosity in mPa.s. This equation presents a new mathematical model for vegetable oil viscosity as the function of temperature at atmospheric pressure. When the independent variable is density, substituting equation (20) into equation (18) obtains:

here, ρ stands for the density in g/cm³, η the viscosity in mPa.s. This equation gives a new model for vegetable oil viscosity versus density at atmospheric pressure. Equations (21) and (22) are firstly derived upon the linear relation between density and temperature and an innovative approach that can deepen into the level of molecular motion and interaction. Therefore, it may be said that a considerable theoretical advancement has been made in modeling the viscosity of vegetable oil. Molecular mechanisms that govern the variations in vegetable oil viscosity with both the temperature and density have been found, and thus the aforementioned knowledge gap has been filled. For these reasons, the viscosity model’s quality and calculating accuracy are expected to be further improved with the two equations.

After a model has been constructed, a comparison against experimental data is usually required to assess the quality of the model. During the assessment, comprehensive statistical methods may be employed, for example, the curve fit with the least square method, the R squared determination, computer software utilization, and so on. In the present study, a non-linear robust fitting was used and carried out with the software named TableCurve 2D v5.01.

In statistics, the R² of a fit for a model to measured data is a number that indicates the explanatory power for the model to explain the data. It is thus an ideal parameter to quantitatively demonstrate the performance of the author’s new model. The higher the degree of the amount of R² approaching 1 may suggest more internal laws of vegetable oil involved in the model.

Calculated viscosity versus temperature. In order to visualize the variation in viscosity with temperature by the author’s model, equation (19) was numerically calculated when lnK= − 4.3, A=−0.1, B=−8×10⁻⁵, β=3, T₀=500, and α=1, 1.4, 1.6.

Consequently, three viscosity curves were obtained and plotted in figure 1 during the temperature increase from 250 to 500 K. They show that the viscosity quickly decreases with temperature in a low temperature region, in contrast to the small decreasing rate in the high temperature region. This trend qualitatively agrees with the commonly accepted knowledge about the viscosity of vegetable oil. The lower α value leads to higher viscosity at a given temperature, indicating a high sensitivity for the molecular interaction parameter to alter the viscosity (Gupta et al., 2007; Noureddini et al., 1992).

In order to compare the new model with existing models, the 61 viscosity data points of soybean oil and 58 data points of corn oil were used again. The comparing samples included the 7 models presented in the introduction section. The fitted results from all the models are listed in Table 3, which shows that the new model gives the fits of R²=0.988 and 0.997 for the two oils. These two numbers not only mean the comprehensively best performance among the models in question, but also the considerable improvement in the model’s accuracy and quality.

Calculated curves of density dependent viscosity. As mentioned in the introduction section, the viscosity as function of density is very useful in the vegetable oil industry. The equation (20) from the author’s model can give a theoretical prediction to the variation viscosity with density. When the coefficients of equation (20) took the amounts: lnK=-17.3, A=8.45, B=9.50, β=3, ρ₀=0.953, and α=0.40, 0.43, 0.48, three curves of viscosity versus density were calculated as Figure 3 in the density range from 0.84 to 0.93 g/cm³. According to this figure, it is easily found that viscosity increases with density. This nature has been commonly accepted for vegetable oils. The increasing rate raises in the high density region. At a fixed density, the higher α value corresponds to the lower viscosity value.

Fits to the density dependent model. As above, soybean oil was chosen as the first sample to be fitted. The fitted results are presented in both Figure 4 and Table 5. The results show that the curve by equation (20) meets most of the 34 data points with the R² value of 0.994, suggesting a good fit and high explanatory ability for the author’s model to the experimental viscosity data of soybean oil. Similar good fits were also found with the other four vegetable oils listed in Table 5. The minimum R squared value is 0.917, which shows how the author’s new model can successfully describe the experimental viscosity data versus density, and then can include the major internal laws of the vegetable oil.

Comparison to existing density dependent models. The density dependent models reported for vegetable oil are much fewer than the temperature dependent ones. For the mentioned density dependent models in the introduction section plus a power law model, comparisons were made against the model of the present work using soybean and corn oils.

The results are illustrated in Table 6, which shows that equation (20) of the author’s new model considerably improves the accuracy and quality, and comprehensively performs the best among these models.

It has been demonstrated that the new model developed upon the freely sliding molecules with Lennard-Jones’ potential and linear density-temperature relation can fit both the temperature and density dependent experimental data of vegetable oils in good agreement and comprehensively performs the best among the models in question. The good fits and out-performances propose an inclusion of the major internal laws of the vegetable oil into the model, and therefore the above-presented molecular mechanism that governs the variations of vegetable oil viscosity has been certified from a point of view. The goodness and out-performance also suggest considerable progress in the improvements to the accuracy and quality of the viscosity model for vegetable oil over broad temperature, density and viscosity ranges. The parameter ranges cover the common industrial working circumstances so as to encompass considerable application potentials of the new model in industry associated with vegetable oils. It may also be stated that the new model suits the usual vegetable oils and their mixture, biodiesel, and other liquid oils. However, it cannot be applied to crystalline oil, or other systems composed of either large molecular polymers or inorganic molecules.