## 4 Discussion  
Many studies have been carried out to investigate the thermal inactivation of bacteria and biofilm inhibition [33; 34; 35; 36; 44; 45]. Some of these studies, especially those in the fields of food engineering, health care and water supply system, focused on the number of live bacteria cells. In such studies, bacterial cell counts or the dry biomass were measured. However, in the TSSHPS, the efficiency decrease is mainly caused by the thermal resistance of the biofilms. Previous studies [11; 41] have shown that the thermal resistance of the biofilms were linearly correlated with the biofilm thickness and the wet weight. In this study, the thermal resistance is of the main interest and the wet weight of the biofilms, instead of the cell count or the dry biomass, was measured. In the experiments, the coupons were suspended in air for 5 min to drain off the surface water-droplets before being weighed. A previous study [13] has reported that 5 min was long enough for all of the extraneous water in the biofilms to be eliminated. The remaining water is stabilized water in the biofilms, and such water acts as a static layer in the biofilms and causes a thermal resistance increase in the heat exchangers. Such water should not be eliminated and the biofilms were weighed with the stabilized water. Many studies have investigated the data fitting and modelling of the thermal inactivation of planktonic bacteria cells [33; 34; 35; 36]. These studies have employed the log linear bacterial survival curves, as shown by Equation (6):  
\[\log N_{\rm t}=\log N_{0}-\frac{t}{D} \tag{6}\]  
where \(N_{\rm t}\) is the number of live cells after the thermal shock, \(N_{0}\) is the number of the live cells without being thermally shocked, \(t\) is the exposure time, and \(D\) is a constant correlated with temperature. However, it has been reported that thermal shock efficient to eliminate planktonic bacteria cells barely killed any cells in biofilms [46], suggesting that thermal shock has different bactericidal effects on planktonic cells and on cells in biofilms. Similarly, Equation (6) could not be used for data fitting of the 120 h biomass (when used, \(R^{2}\) in all temperature groups would be less than 0.5). Thus, a new formula was needed to correlate the 120 h biomass \(m_{\rm f}\) to the exposure time \(t\). In this study, a hypothesis was raised that the biomass decrease was correlated with the amount of heat transferred to the biofilms, and the hypothesis could be written as:  
\[m_{\rm f}=m_{0}-c\times Q_{t} \tag{7}\]  
where \(m_{\rm f}\) is the 120 h biomass, \(m_{0}\) is the biomass of biofilms without heat treatment, \(c\) is a constant and \(Q_{t}\) is the heat transferred to the biofilms within the exposure time. The biofilms, which were 20 mm in length and width and about 100 \(\upmu\)m in thickness, were considered as infinite plates, and the Biot number of the biofilms was less than 0.05. The lumped parameter method could be used for heat transfer analysis of the biofilms, and \(Q_{t}\) could be written as:  
\[Q_{t}=(T-T_{0})\rho c_{\text{P}}V[1-\exp(-\frac{hA}{\rho c_{\text{P}}V})t] \tag{8}\]  
where \(T\) is the thermal shock temperature, \(T_{0}\) is the initial temperature of the biofilms (25 \({}^{\circ}\)C), \(\rho\), \(c_{\text{P}}\), \(V\), \(h\) and \(A\) are the density, special heat, volume, convective heat transfer coefficient, and surface area of the biofilms, and \(t\) is the exposure time [47]. For a certain thermal shock temperature and a certain biofilm, Equation (8) could be written as:  
\[Q_{t}=a_{0}[1-\exp(-bt)] \tag{9}\]  
where \(a_{0}\) and \(b\) are coefficients correlated with temperature. Equation (9) was substituted into Equation (7), and we got Equation (4) as the fitting equation. The results showed that Equation (4) yielded tight correlations (Figure 5, average \(R^{2}\) = 0.9211), indicating it was appropriate to correlate the biomass decrease with the total heat and use Equation (4) as the fitting equation.