## 3 Results and Discussion
### Feasibility for Practical Use in Anaerobic Digestion Operation
#### 3.2.2 Quartz Crystal Microbalance Measurements of Anaerobic Digestate Samples  
Digestate samples were taken from the reactor during the semi-continuous operation at various intervals from day 30 to 79 (Table 1). To reduce LCFA concentration to below 100 mg/L, these samples were diluted 40 times with water. After the dilution and pH adjustment to 10.7, the samples were applied to the QCM sensor. Figure 5a indicates a comparison trend between LCFA concentration and absolute frequency-shift rate. All measurements were carried out in the same way as above, except the sample taken at day 79. The sample at day 79 shows very high increasing rates of resonance residence and reaches 1000 Ohm at about 100 s; then the frequency change becomes unstable. As such, the frequency-shift rate of that sample is calculated using a shorter linear range (50-100 s). The frequency-shift rates maintain a low level below 0.7 Hz/s during days 30-48 and then increase to a slightly higher level around 1.0 Hz/s at days 64 and 70. A sharp elevation of the frequency-shift rate occurs with a notable increase in LCFA during days 73-79. As a result, there is a significant correlation (\(R^{2}\) = 0.9751, \(p\)-value < 0.0001) between the frequency-shift rates and LCFA. The results shown in Figure 5b indicate that LCFA concentrations in unknown samples can be determined using an external calibration curve. Figure 5b plots the values of frequency-shift rate versus those of LCFA concentration. As discussed above, there are clear linear relationships between \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Day**} & \multicolumn{5}{c}{**Concentration (mg/L)**} \\ \cline{2-7}  & **C16:0** & **C16:1** & **C18:0** & **C18:1** & **C18:2** & **Total** \\ \hline
30 & 115 & 0 & 36 & 76 & 0 & 227 \\
40 & 147 & 0 & 48 & 116 & 18 & 328 \\
42 & 136 & 3 & 51 & 72 & 26 & 333 \\
44 & 154 & 11 & 49 & 174 & 42 & 473 \\
48 & 280 & 4 & 103 & 130 & 33 & 616 \\
64 & 229 & 12 & 87 & 173 & 21 & 543 \\
70 & 204 & 14 & 87 & 267 & 31 & 620 \\
73 & 924 & 14 & 122 & 79 & 0 & 1229 \\
77 & 1267 & 15 & 140 & 119 & 13 & 1686 \\
79 & 918 & 85 & 235 & 788 & 352 & 2602 \\ \hline \end{tabular}
\end{table} 
Table 1: The main composition of LCFAs in digestate samples (\(n=1\)). Figure 4: (a) Time courses of gas production rates and feeding rate of de-oiled grease trap waste (GTW); and (b) The change of pH and long-chain fatty acids (LCFA) concentration during the semi-continuous experiment. two types of values in both palmitate and digestion samples. Furthermore, the two lines are almost consistent with each other. This means that a regression line equation obtained from palmitate samples allows for an estimate of LCFA concentration in a digestate sample from the measured frequency-shift rate of the sample. From the view point of practical use, real anaerobic digestate has many organic contaminants. Above all, triglyceride is the major contaminant in de-oiled GTW-AD, and has the possibility of affecting the water-solubility of LCFA. According to [PERSON] et al. (2015), the solubility of an organic compound (i) between a liquid organic mixture (organic compound (i) and organic solvent) and an aqueous phase is defined as the following equation. \[\gamma_{w}^{-1}=\ C_{tot}^{sat}\ \cdot\ \exp\left(\frac{\Delta S_{m}}{R}\left[ \frac{T_{m}}{T}\ -\ 1\right]\right)=\ C_{tot}^{sat}\ \cdot\ \exp\left(\frac{\Delta H_{fus}}{RT_{m}}\left[ \frac{T_{m}}{T}\ -\ 1\right]\right) \tag{3}\]  
\[C_{i\ w}=\ \gamma_{i\ org}\ \cdot\ x_{i\ org}\ \cdot\ \gamma_{w}^{-1}=\ \gamma_{i\ org}\ \cdot\ x_{i\ org}\ \cdot\ C_{tot}^{sat}\ \cdot\ \exp\left(\frac{\Delta H_{fus}}{RT_{m}}\left[ \frac{T_{m}}{T}\ -\ 1\right]\right) \tag{4}\]  
where \(C_{tot}^{sat}\) is the total water-solubility of compound (i), \(\Delta H_{fus}\) is enthalpy of fusion (J/mol), \(T_{m}\) is melting temperature (K), \(\gamma_{i}\) org is the activity coefficient of compound (i) in the organic phase, and \(x_{i\ org}\) is the mole fraction in the organic solvent (mol/mol-solvent).