## 2 Materials and Methods
### Experimental Methods
#### 2.2.2 Slug Frequency  
Different correlations can predict slug frequency. However, [PERSON] & [PERSON] [41] suggested a trustworthy velocity-dependent empirical equation where slug frequency was correlated with a form of Froude number, which is described as:  
\[N_{f_{s}}=\frac{v_{l}}{gd}\Bigg{[}\frac{\left(v_{m}^{0}\right)^{2}}{v_{m}}+v_ {m}\Bigg{]} \tag{4}\]  
Here, \(v_{m}^{0}\) was taken as 6 m/s and from the graphs of the slug frequency vs. slug Froude number by [PERSON] & [PERSON] [41], the following equation was derived:  
\[f_{s}=0.0157\Big{(}N_{f_{s}}\Big{)}_{slug}^{1.20}\sec^{-1}. \tag{5}\]  
From Equation (5), [PERSON] & [PERSON] [41] described a slug frequency correlation based on their liquid-gas two-phase flow experimental data where water and carbon dioxide are used in a 19-mm ID pipe:  
\[f_{s}=0.0226\Bigg{[}\frac{v_{l}^{s}}{gd}\Bigg{(}\frac{19.75}{v_{m}}+v_{m} \Bigg{)}\Bigg{]}^{1.2} \tag{6}\]  
where \(v_{m}\) and \(v_{l}^{s}\) represent the mixture velocity and superficial liquid velocity of liquid and gas, respectively. Therefore, this slug frequency can be combined with the Froude number established for superficial fluid velocity. [PERSON] & [PERSON] [43] reorganized the [PERSON] & [PERSON] [41] correlation as follows:  
\[f_{s}=\Bigg{[}0.0425\frac{v_{l}^{s}}{v_{m}}\Bigg{(}\frac{2.02}{d}+\frac{v_{m} ^{2}}{gd}\Bigg{)}\Bigg{]}^{\frac{6}{5}} \tag{7}\]  
Later, [PERSON] & others [44] described another correlation based on 399 data points with the smallest average absolute error and standard deviation for both horizontal and inclined pipe flow. This correlation is a modification of the [PERSON] & [PERSON] [41] correlation, using the imperial unit, which is shown in Equation (8), where \(\theta\) is the inclination angle. The experiment was conducted in a 2.54 cm and 10.16 cm ID pipe with air and water. \[f_{s}=\Bigg{[}0.0425\frac{v_{l}^{s}}{gd}\Bigg{(}\frac{1}{0.0506v_{m}}+v_{m} \Bigg{)}\Bigg{]}^{\frac{6}{5}}\Big{[}0.836+2.7 sin^{0.25}\theta\Big{]} \tag{8}\][PERSON] & [PERSON] (2009) determined the liquid volume fraction for air-water two-phase flow utilizing the gamma-ray technique in a 4.191 cm ID horizontal pipe. To determine the liquid volume fraction, they used a power spectral density function and probability density function. These features are also helpful to decide on different slug flow characteristics, such as the value of the average film and slug volume fraction, average slug frequency, and average slug length. The slug frequency correlation was determined by curve fitting the data, resulting in Equation (9), in which \(\lambda\) is the liquid volume fraction, where \(\lambda=v_{l}^{s}/\left(v_{l}^{s}+v_{g}^{s}\right)\), and \(d\) is the pipe diameter. \[f_{s}=\left[0.0462\ \lambda\left(\frac{1}{0.0126d}+\frac{v_{m}^{2}}{gd}\right) \right]^{1.02} \tag{9}\]  
[PERSON] et al. (2016) developed a correlation describing slug frequency as a function of pipe length. In Equation (10), \(v_{l}^{S}\) is the superficial liquid velocity, \(d\) is the pipe diameter, and \(l_{p}\) is the pipe length. This correlation is based on the curve fitting of field and laboratory data, and is not based on theoretical analysis. In this equation, it is also shown that the slug frequency is inversely dependent on the pipe length \(l_{p}\), which does not agree with the other theoretical analysis. According to [PERSON] (2016), the OLGA 2000 slug tracking model had some time delay problem between two slugs; to solve this issue, the [PERSON] et al. (2016) correlation was initially used. Moreover, pipe length can be questionable for a long-distance transmission system with hilly conditions:  
\[f_{s}=0.47\Bigg{[} \begin{array}{c}\left(\begin{array}{c}v_{l}^{s}\\ \end{array}\right)^{1.5}\\ \frac{l_{p}^{1.1}d^{2.4}}{l_{p}^{1.1}d^{2.4}}\end{array} \Bigg{]}^{0.5} \tag{10}\]  
[PERSON] et al. (2016) described a slug frequency equation that considers the rheology of the shear-thinning fluid. This equation is a modified version of the [PERSON] & [PERSON] (1998) correlation.