## 3 Methodology
### Determination of the Properties, Parameters, Input Variables and Initial Conditions for the Model  
#### 3.3.1 Characterization of Glycerides and Esters  
Some properties of the system are defined by components in the reaction mixture. For example, the density and specific heat of the mixture depend on the densities and heat capacities of each of its elements. Moreover, the enthalpy of the reactions depends on the particular elements that take part as reactants and products in the reactions. Ideally, in order to characterize the components of the mixture, it is necessary to estimate the properties of each individual compound (e.g., by a group or fragment contribution method); however, because glyceride and ester molecules may be formed by different fatty acids, there is a large variety of these components in the reaction mixture. To simplify characterization, all glycerides and esters, regardless of whether they are formed by different fatty acids, are characterized with similar properties.  
In the the literature, there are three approaches for characterizing vegetable oil triglycerides: the mixed-TG method; the simple-TG method; and the pseudo-TG method [45]. The mixed-TG method determines the composition of each mixed triglyceride in the oil by means of either high-resolution gas chromatography (HRGC) or a probabilistic method [46]. The thermo-physical properties of the triglyceride mixture are then computed by applying a prediction method, such as a group contribution method or a fragment-based approach, using an appropriate mixing rule, although TG composition analysis is not commonly used. In both the literature and practice, the fatty acid (FA) composition is the most frequently used analytical criteria for characterizing feed oils [45]. For this reason, the use of methods based on FA composition is a more appropriate for oil triglyceride characterization. The simple-TG and pseudo-TG approaches use FA rather than TG composition. In the simple-TG method, the triglycerides are represented with an unique type of the fatty acids that appear in the FA composition and then the thermo-physical properties for each simple triglyceride are calculated via an appropriated method. Finally, the ideal mixing rule and the FA composition are used to estimate the properties of the triglyceride mixture. The pseudo-triglyceride approach represents the triglycerides as a simple molecule with the same weighted-average number of CH\({}_{2}\) groups in the FA chain and the same weighted-average number of CH=CH groups as the original feed oil mixture, based on the composition indicators and mole fractions for each FA component [47]. Then, the ideal mixing rule and the FA composition are used to estimate the properties of the simple-triglyceride mixture.  
The above described methods (mixed-TG, simple-TG, and pseudo-TG) are efficient and provide equally accurate estimations [47]. Additionally, due to the fact that, in practice, most biodiesel plants are only able to characterize the feed oil via the FA composition, the simple-TG and pseudo-TG methods are sufficient for estimating the thermo-physical properties of vegetable oils. The present study used the simple-molecule and pseudo-molecule methodologies to characterize triglycerides, diglycerides, monoglycerides and esters from palm, mustard, and sunflower oils.  
In the alkali-catalyzed transesterification of vegetable oils, the FA composition of methyl esters is identical to their corresponding oils [48]. For this reason, the FA composition of the oil was used to characterize the fatty acid methyl esters in the present study. Moreover, it is assumed that the FA composition of the diglycerides and monoglycerides is similar to their corresponding oils and thus the FA composition of the oil is used in the characterization of theses components.