## 4 Results  
### Flow Coefficients  
The \(CV\) regarding a change in the opening rate of the proposed cylinder-in-ball valve was calculated using Equation (1). The CFD analysis was carried out using boundary conditions such that the \(Q\) at the inlet and atmospheric pressure at the outlet met the requirement for a \(\Delta P\) of 74 kPa. The \(Q\) at the inlet regarding a change in the opening rate of the valve was found using a trial-and-error approach. A series of CFD analyses under pressure boundary conditions, such as 74 kPa at the inlet and atmospheric pressure at the outlet, were repeated until the desired values of the \(Q\) at the inlet was obtained. Then, the obtained \(Q\) at the inlet for each opening rate and atmospheric pressure at the outlet were designated as the boundary conditions for the CFD analysis. Subsequently, the final CFD analysis was observed. Table 1 summarizes the boundary conditions for the CFD analysis.  
Figure 4 shows the streamlines of the working fluid in the valve at an opening rate of 20%. As shown in Figure 1c, the opening started at the center of the cross-section in the flow path so that the streamlines approximated a simple straight line. Although several vortices were generated at the region of contact with the ball and cylinder, the vortices were small enough to not significantly affect the main flow path. These did not lead to a sudden change in the local pressure in the flow path.  
The obtained \(Q\) at the outlet and the \(CV\) calculated using Equation (1) with regard to changes in the opening rate of the valve are plotted in Figure 5. As the opening rate was changed to 20, 30, 50, 75, and 100%, the \(Q\) increased to 0.77, 2.20, 8.05, 21.82, and 80.78 m\({}^{3}\)/h, respectively. The \(CV\) values increased similarly to the rates for \(Q\), such as 1.05, 2.99, 10.95, 29.68, and 109.87. The changes of the \(CV\) with the opening rate followed in order  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Opening Rate (\%)**} \\ \cline{2-6}  & **20** & **30** & **50** & **75** & **100** \\ \hline Inlet flow rate & & & & & \\ (m\({}^{3}\)/h) & 0.71 & 2.10 & 7.92 & 21.6 & 81.09 \\ \hline Outlet pressure & & & & & \\ (atm) & & & & & \\ \hline \end{tabular}
\end{table}
Table 1: The boundary conditions of the CFD analysis according to the opening rates.  
Figure 4: Streamlines at an opening rate of 20% for the cylinder-in-ball valve.  
These flow characteristics were compared with those of a conventional ball valve and a commercial V-port ball valve with the same pipe diameter of 1-1/2\" and a V-port angle of 90\({}^{\circ}\). Figure 6 shows the \(CV\) values as the opening rate increases for the proposed cylinder-in-ball valve, a conventional ball valve, and the V-port ball valve. The increments of \(CV\) values showed very similar behaviors for three types of valves. The conventional ball valve showed higher \(CV\) values for the entire range of opening rates, but with irregular flow characteristics at some specific regions. The conventional ball valve was difficult to use as a valve for sophisticated flow control. Meanwhile, in the case of the V-port ball valve, no irregular flow characteristics were found for the entire region of opening rates, and the \(CV\) increased exponentially: 1.07, 4.00, 20.78, 40.05, and 90.50. For the region of opening rates between 40 and 80%, however, the proposed cylinder-in-ball valve provided an insensitive \(CV\) change compared with that of the V-port ball valve. This suggests that the proposed cylinder-in-ball valve could be considered a more sophisticated flow control mechanism. When fully opened, the proposed cylinder-in-ball valve has an even higher \(CV\) value of 109.87 compared with that of 90.50 for the V-port ball valve.