# A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime
## 4 Discussion  
The data in Table 4 show that the repeatability RSD of the measured pressure differences at all flow points is \(\leq\)1%, demonstrating the good repeatability and high consistency of the experimental data. To improve the reliability of the measurement results, the uncertainty of the corrected pressure difference needs to be calculated. Through analysis, the standard deviation of the pressure difference after nonlinear error correction is taken as the Type A uncertainty \(u_{A}\), while the Type B uncertainty \(u_{B}\) mainly originates from the resolution of the pressure difference measuring device. The minimum scale value of the pressure difference measuring device in the experimental system is 0.1 Pa, with the half-width of the interval being 0.05 Pa, and the pressure difference follows a uniform distribution. The calculation results of the uncertainty are shown in the following table (the coverage factor k is taken as 2, corresponding to a confidence probability of 95%).  
Table 5 indicates that the maximum ratio of the expanded uncertainty \(u_{C}\) to the average pressure difference is 1.77%, which is less than the 2-3% required for general industrial measurements. Therefore, the reliability of the measurement results can be considered high.  
Observing the data in Table 4, it is found that within the flow range of 1-10 cm\({}^{3}\)/min, the relative errors between the theoretical flow rates obtained after nonlinear error correction and the flow rates measured by the standard meter are all less than \(\pm\)0.6%. This represents a significant improvement compared to conventional laminar flowmeters, whose measurements can only meet the requirement of a relative error less than or equal to \(\pm\)1% to \(\pm\)1.5%. This indicates that both the optimization of the laminar element structure in this study and the correction using theoretical equations can effectively reduce the proportion of nonlinear errors and improve measurement accuracy.  
Figure 10a shows the curve fitted by the least square method for the pressure difference after nonlinear error correction and the flow rate measured by the standard flowmeter. To verify the linearity of the fitted curve, the determination coefficient R\({}^{2}\) is calculated to be \(>\)0.99, and the maximum residual ratio under full-scale pressure difference is 0.34%, which is lower than the general industrial requirement of \(\pm\)1%. This demonstrates that the flow-pressure difference fitted curve after correction has good linearity, and the experimental data can well illustrate the characteristic that the laminar flow measurement method has high linearity. Figure 10b shows the distribution diagram of the corrected relative errors. It can be observed from the relative error distribution diagram that the relative error is relatively large during the measurement at the 1 cm\({}^{3}\)/min flow point. Through analysis, it is considered that the main reason is that the gas source was in the initial start-up state, resulting in unstable output gas.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Flow Rate/cm\({}^{3}\)/min**} & \multirow{2}{*}{\begin{tabular}{c} **Type A Uncertainty** \\ \(u_{A}\)/Pa \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Type B Uncertainty** \\ \(u_{B}\)/Pa \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **The Expanded** \\ **Uncertainty \(u_{C}\)/Pa** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **The Maximum Ratio of the** \\ **Expanded Uncertainty \(u_{C}\) to the** \\ **Average Pressure Difference (\%)** \\ \end{tabular} } \\ \hline
1 & 0.10 & 0.03 & 0.20 & 1.77\% \\ \hline
2 & 0.15 & 0.03 & 0.30 & 1.34\% \\ \hline
4 & 0.15 & 0.03 & 0.30 & 0.67\% \\ \hline
8 & 0.35 & 0.03 & 0.70 & 0.78\% \\ \hline
10 & 0.40 & 0.03 & 0.80 & 0.71\% \\ \hline \end{tabular}
\end{table}
Table 5: Standard uncertainty data.