# Effects of Velocity Profiles on Measuring Accuracy of Transit-Time Ultrasonic Flowmeter
## 4 Results  
### Verification  
Table 2 gives the results of flowrate measurement at the [PERSON] apparatus. The flowrate \(Q_{1}\) is measured by the ultrasonic flowmeter, while \(Q_{2}\) is measured by the weight discharged within a certain period of time. The flowrate ratio \(Q_{2}\):\(Q_{1}\) is the measured correction factor. The relative error (\(\delta_{1}\)) defined in Equation (31), is used to evaluate the difference between theoretical correction factor and measured data, where k indicates the theoretical correction factor.  
\[\delta_{1}=\left|\frac{k-\beta}{k}\right|\times 100\% \tag{31}\]  
From Equations (15) and (18), it is clearly indicated that the theoretical correction factor for laminar flow is 0.75. It can be seen from Table 2 that the flowrate ratios (\(\beta\)) are in range of 0.737 to 0.764, the average value of \(\beta\) is 0.747. The average relative error is 0.976%, and the maximum relative error is 1.931%, which is in good agreement with the experiment result. The last line in Table 2 is in the transition zone from laminar flow to turbulent flow, and the correction factor is a little bit larger than 0.75, which is caused by the gradual changing of velocity profile from laminar flow to turbulent flow.  
Table 3 gives the results of flowrate measured at the performance testing system for centrifugal pumps. The Reynolds numbers are from 4311 to 422102. The theoretical turbulent flow correction factor (\(k_{t}\)) is calculated by Equation (27). The average relative error is 0.25% and maximum error is 1.178%. Therefore, the theoretical correction factor is reliable.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Ultrasonic Flowmeter**} & \multicolumn{5}{c}{**Weighing Method**} & \multicolumn{3}{c}{**Flowrate**} \\ \cline{2-10}  & \(\Delta T\times 10^{9}\) & \begin{tabular}{c} **Flowrate** \\ **(s)** \\ \end{tabular} & \begin{tabular}{c} **Time** \\ **(s)** \\ \end{tabular} & \begin{tabular}{c} **Mass** \\ **(kg)** \\ \end{tabular} & \begin{tabular}{c} **Flowrate** \\ **(m/s)** \\ \end{tabular} & \begin{tabular}{c} **Velocity** \\ **(m/s)** \\ \end{tabular} & \
\begin{tabular}{c} **Re** \\ **(m/s)** \\ \end{tabular} &
\begin{tabular}{c} **Ratio** \\ **(m/s)** \\ \end{tabular} & \(\beta\) = Q2:Q1** & \(\delta_{1}\) **(\%)** \\ \hline
1 & 0.378 & 0.970 & 60 & 0.430 & 0.717 & 0.047 & 652 & 0.739 & 1.443 \\
2 & 0.462 & 1.185 & 60 & 0.524 & 0.873 & 0.057 & 795 & 0.737 & 1.772 \\
3 & 0.632 & 1.622 & 60 & 0.721 & 1.202 & 0.078 & 1094 & 0.741 & 1.192 \\
4 & 1.002 & 2.572 & 50 & 0.965 & 1.930 & 0.125 & 1757 & 0.751 & 0.052 \\
5 & 1.121 & 2.877 & 50 & 1.073 & 2.147 & 0.140 & 1954 & 0.746 & 0.498 \\
6 & 1.310 & 3.363 & 40 & 1.005 & 2.512 & 0.163 & 2286 & 0.747 & 0.406 \\
7 & 1.317 & 3.380 & 40 & 1.019 & 2.548 & 0.166 & 2318 & 0.754 & 0.513 \\
8 & 1.553 & 3.987 & 40 & 1.219 & 3.048 & 0.198 & 2774 & 0.764 & 1.931 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of flowrate measured at the [PERSON] apparatus.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Ultrasonic Flowmeter**} & \multicolumn{5}{c}{**Weighing Method**} & \multicolumn{3}{c}{**Flowrate**} \\ \cline{2-10}  & \(\Delta T\times 10^{9}\) & \begin{tabular}{c} **Flowrate** \\ **(s)** \\ \end{tabular} & \begin{tabular}{c} **Flowrate** \\ **(m/s)** \\ \end{tabular} & \begin{tabular}{c} **Mass** \\ **(s)** \\ \end{tabular} & \begin{tabular}{c} **Flowrate** \\ **(m/s)** \\ \end{tabular} & \begin{tabular}{c} **Velocity** \\ **(m/s)** \\ \end{tabular} & \
\begin{tabular}{c} **Re** \\ **(m/s)** \\ \end{tabular} &
\begin{tabular}{c} **Ratio** \\ **(m/s)** \\ \end{tabular} & \(\beta\) = Q2:Q1** & \(\delta_{1}\) **(\%)** \\ \hline
1 & 7.270 & 0.186 & 60 & 10.314 & 0.172 & 0.085 & 4311 & 6 & 0.925 & 0.923 & 0.216 \\
2 & 17.714 & 0.454 & 40 & 16.939 & 0.424 & 0.209 & 10619 & 6 & 0.934 & 0.923 & 1.178 \\
3 & 36.956 &