# Design and Application of New Aeration Device Based on Recirculating Aquaculture System
## 2 Materials and Methods
### Test Design and Methodology  
#### 2.4.1 Theoretical Considerations and Data Processing  
The tests in this study are based on the mass transfer theory of oxygen molecules proposed by [PERSON] in 1926, according to which the process of aeration equipment essentially consists of the penetration of oxygen molecules from the gas phase into the liquid phase through the gas-liquid interface; in a system containing two or more phases, due to the existence of a concentration gradient, there exists a tendency for substances to diffuse from the higher- to the lower-concentration phase, a process referred to as mass transfer [24]. The process of dissolving oxygen molecules into water is referred to as oxygen mass transfer and is represented in Equation (13):  
\[\frac{dC_{t}}{dt}=K_{L}a_{T}(C_{\infty}-C_{0}) \tag{13}\]  
where \(C_{t}\), \(C_{\infty}\), and \(C_{0}\) denote the current, saturated, and initial dissolved oxygen, respectively, at water temperature \(T\) (in mg/L), where the initial DO is generally 0. Integrating from Equation (13) yields Equation (14), with which the mass transfer coefficient (\(K_{L}a_{T}\)) for the total volume of the liquid phase in the aeration mass transfer system is calculated:  
\[-In\left(\frac{C_{\infty}-C_{t}}{C_{\infty}-C_{0}}\right)=K_{L}a_{T}\times t \tag{14}\]  
According to the Arrhenius temperature relation, Equation (15) is the transformation of the mass transfer coefficient (\(K_{L}a_{20}\)) in the standard state into that at temperature \(T\):  
\[K_{L}a_{20}=K_{L}a_{T}/\theta^{T-20} \tag{15}\]  
where \(\theta\) is the temperature correction factor (=1.024) for pure water. In order to evaluate the performance of an aerator, standard oxygen transfer rate (\(SOTR\)) and standard aeration efficiency (\(SAE\)) are generally used [25]:  
\[SOTR=K_{L}a_{20}\times(C_{\infty}-C_{0})\times V=K_{L}a_{20}\times 9.07\times V \times 10^{-3} \tag{16}\]  
Figure 3: Variations in natural conditions of test pond’s DO content during 24 h period. However, aerators used in pond applications, which are subject to outdoor conditions, present different \(SOTR\) and \(SAE\) from those measured under standard conditions. The oxygen transfer rate (\(OTR\)) calculation formula proposed by [PERSON] [26; 27] is shown in Equation (17):  
\[OTR=\left[SOR\times\alpha\times 1.024^{T-20}\left(\beta C_{\infty}-C_{0}\right) \right]/9.07 \tag{17}\]  
where \(OTR\) denotes the oxygen transfer rate under pond conditions (in kg/h); \(SOTR\) denotes the standard oxygen transfer rate (in kg/h); \(\alpha=K_{L}a_{20}\) pond water/\(K_{L}a_{20}\) tap water; and \(\beta\) = DO saturation concentration of pond water/DO saturation concentration of tap water. DO increase (\(R_{S}\)) refers to the increase in water DO induced by the aerator, expressed as shown in Equation (18):  
\[R_{s}=C_{2}-C_{1} \tag{18}\]  
where \(R_{s}\) denotes the average increase in pond DO level (in mg/L); \(C_{1}\) and \(C_{2}\) denote the dissolved oxygen concentrations recorded at times \(t_{1}\) and \(t_{2}\), respectively (in mg/L); and \(t_{1}\) and \(t_{2}\) denote the time elapsed since the device started the aeration process (in min). To determine the uniformity of the dissolved oxygen distribution in water, the DO data from each collection point are analyzed for standard deviation [28]. \[S_{1}=\sqrt{\frac{1}{x-1}{\sum\limits_{i=1}^{x}}\left(C_{i}-\overline{C} \right)^{2}} \tag{19}\]  
\[M_{1}=\left(1-\frac{S_{1}}{\overline{C}}\right)\times 100\% \tag{20}\]  
where \(S_{1}\) denotes the standard deviation of dissolved oxygen; \(x\) denotes the number of collection points; \(C_{i}\) are the dissolved oxygen data from collection points (in mg/L); \(\overline{C}\) denotes the average of dissolved oxygen at \(x\) collection points (in mg/L); and \(M_{1}\) denotes the uniformity of the dissolved oxygen distribution in water (in %).