# A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime
## 3 Results  
The reference flowmeter in the experimental system records flow rates under standard conditions (0 \({}^{\circ}\)C, 101.35 kPa) in units of SCCM (standard cubic centimeters per minute), whereas the flow rates calculated via Equation (10) correspond to operational conditions in units of cm\({}^{3}\)/min (cubic centimeters per minute in the experimental environment, 26.42 \({}^{\circ}\)C, 101.33 kPa). To enable comparative analysis, the standard condition flow rate \(q1\) (SCCM) must be converted to the operational condition flow rate \(q_{\text{\tiny{ES}}}\) (cm\({}^{3}\)/min) using Equation (11)  
\[q_{\text{\tiny{ES}}}=q1\frac{p1+p2}{2 pa} \tag{11}\]  
where \(pa\) represents the atmospheric pressure in the standard condition. And \(p1\) and \(p2\), respectively, represent the absolute pressures (\(Pa\)) measured upstream and downstream of the capillary laminar element. During the experiments, \(p1\) was directly acquired from the upstream pressure gauge, while \(p2\) was approximated as \(p2=p1-\Delta p\), where \(\Delta p\) is the measured differential pressure.  
Under the experimental conditions, the dynamic viscosity of air was determined as 1.84 \(\times\) 10\({}^{-5}\) Pa*. The relative error \(\delta\) between the experimentally derived volumetric flow rate (\(q_{\text{\tiny{ES}}}\)) and the theoretically calculated flow rate (\(q_{\text{\tiny{E}}}\)) was quantified as  
\[\delta=\frac{q_{\text{\tiny{E}}}-q_{\text{\tiny{ES}}}}{q_{\text{\tiny{ES}}}} \times 100\% \tag{12}\]  
where now both \(q_{\text{\tiny{ES}}}\) (reference flow rate) and \(q_{\text{\tiny{E}}}\) (theoretical flow rate) are expressed in cm\({}^{3}\)/min (26.42 \({}^{\circ}\)C, 101.33 kPa).  
Table 4 consolides experimental data for the laminar flow element. Column 1 lists test identifiers; Columns 2 and 3, respectively, document the raw pressure differential measured across the laminar element under experimental conditions and the corrected linear frictional pressure differential obtained via nonlinear error compensation using Equations (5), (6), and (8). Columns 4 and 5 present the operational condition flow rates recorded by the reference meter and the theoretical flow rates calculated via Equation (10). To evaluate measurement consistency, Column 6 computes the repeatability of corrected pressure differentials across repeated trials. The final column reports the relative error between the theoretical flow rates and reference flow rates to validate the error compensation efficacy.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Identification**} & **Measured** & **Corrected** & & & & \\  & **Pressure** & **Pressure** & **Q\%s (cm\({}^{3}\)/min)** & **C\%s (cm\({}^{3}\)/min)** & **C\%s (cm\({}^{3}\)/min)** & **R\%s (cm\%)** & **Relative Error** \\
**Number** & **Difference \(\Delta p\)** & **Difference \(\Delta p^{\prime}\)** & **Q\%s (cm\({}^{3}\)/min)** & **Q\%s (cm\({}^{3}\)/min)** & **R\%s (cm\%)** & **Q\%s (\%)** \\ \hline \multirow{4}{*}{1} & 11.5 & 11.4 & 1.007 & 1.012 & \multirow{4}{*}{0.88\%} & 0.53\% \\  & 11.4 & 11.3 & 0.998 & 1.003 & & 0.55\% \\  & 11.3 & 11.2 & 0.989 & 0.994 & & 0.56\% \\ \hline \multirow{4}{*}{2} & 22.8 & 22.6 & 1.999 & 2.007 & \multirow{4}{*}{0.68\%} & 0.39\% \\  & 22.6 & 22.4 & 1.982 & 1.989 & & 0.36\% \\  & 22.5 & 22.3 & 1.973 & 1.980 & & 0.37\% \\ \hline \multirow{4}{*}{3} & 46.7 & 44.6 & 3.951 & 3.960 & \multirow{4}{*}{0.34\%} & 0.24\% \\  & 46.9 & 44.7 & 3.961 & 3.969 & & 0.21\% \\  & 47.01 & 44.9 & 3.978 & 3.987 & & 0.23\% \\ \hline \multirow{4}{*}{4} & 96.4 & 90.1 & 8.017 & 8.001 & \multirow{4}{*}{0.39\%} & \(-0.20\%\) \\  & 96.8 & 90.5 & 8.055 & 8.036 & & \(-0.23\%\) \\ \cline{1-1}  & 96.1 & 89.8 & 7.994 & 7.974 & & \(-0.25\%\) \\ \hline \multirow{4}{*}{5} & 121.6 & 112.5 & 9.976 & 9.990 & \multirow{4}{*}{0.14\%} \\  & 122.1 & 112.8 & 10.002 & 10.017 & & 0.35\% \\ \cline{1-1}  & 121.6 & 112.3 & 10.044 & 10.061 & & 0.17\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Micro tube laminar flow element experimental data.