## 2 Raw Material, Experimental Device and Procedure  
The schematic diagram of batch-type reactor manufactured by Parr Instrument Co. (Moline, IL, USA) with volume of 25 mL is demonstrated in Figure 1. Heating jacket with temperature controller (Hanwoul Engineering, Gunpo, Korea) was used to maintain the desired transesterification temperature. The admissible pressure is 55 MPa. Methanol (99.5% purity) and ethanol (99.5% purity) were used as solvents manufactured by Ducksan Chemical Co. (Ansan, Korea) and OCI company Ltd. (Seoul, Korea), respectively. The frying temperature and length of use of the oil along with the material used for frying was responsible for the composition of the waste oil. After repetitive use of the soybean oil for a minimal of 4 h per day for four consecutive days, the content of free fatty acid (FFA) present in oil sample increased. The fatty acid (FA) and FFA composition of WVO was established by titration against 0.1 M KOH solution using phenolphthalein as the indicator.  
Table 1 illustrates the FA and FFA contents of WVO and virgin oil indicating the elevation in FFA content in WVO from 0.28% to 3.15% during cooking. Further, it can be observed that the FA content of 18:2 is notably above than virgin oil, though the FA content of 18:3 is lower for WVO, as a result of the oxidation of the oil during cooking WVO [24]. The experiments were carried out by varying the transesterification temperature (250, 270 and 290 \({}^{\circ}\)C), WVO to alcohol molar ratios (1:6, 1:12 and 1:18), alcohols (SCM and SCE), and reaction retention time (0, 15, 30, 45 and 60 min). At room temperature, the WVO with alcohol at definite molar ratio was loaded into the autoclave reactor. The time required to reach the set point is dependent on heating rates. The average heating rate was 6.2 \({}^{\circ}\)C/min. The reaction pressure was controlled by varying the volume of alcohol and WVO fed into the reactor. The total amount of WVO and alcohol fed into the reactor was determined to ensure that supercritical pressure was reached. The critical temperatures and pressures of ethanol and methanol are \(T_{\rm c}\) = 243.0, 239.4 \({}^{\circ}\)C and \(P_{\rm c}\) = 6.4, 8.1 MPa, respectively. The pressure ranged 7.9-12.8 and  
Figure 1: Schematic diagram of the experimental apparatus.  
9.2-14.7 MPa for SCM and SCE, respectively, making them fall in proper supercritical zone. To mix the WVO and alcohol solution homogeneously, a magnetic stirrer was used for rigorous stirring at a rate of 500 rpm during the reaction. The stirring speed used in this study was obtained from a study by [PERSON] et al. [30]. Maintaining a constant heating rate was difficult. Accordingly, set point of the temperature controller was set to a higher temperature than the needed temperature for this experiment. On reaching the desired condition, the heating of the vessel was stopped promptly and the vessel was cooled in an ice bath to halt the reaction instantaneously.  
Gas Chromatography Mass Spectrometry (GC-MS) (Agilent GC-6890 with MSD-5975 detector, Santa Clara, CA, USA) was used for the characteristic analysis of liquid products on differing transesterification temperature, retention time, and WVO to alcohol ratios. DB-WAX column (\(30\;\mathrm{m}\times 250\;\mathrm{\SIUnitSymbolMicro m}\), \(0.25\;\mathrm{\SIUnitSymbolMicro m}\) thickness) was used for all samples. The area percentage method was used to measure the amount of alkyl esters in the product.