# A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime
## 2 Materials and Methods
### Processing Method for Nonlinear Errors
The velocity, pressure, and cross-sectional changes during sudden flow expansion are schematically illustrated in Figure 5. The figure reveals that abrupt boundary variations induce vortex regions between the mainstream flow and solid boundaries, causing flow separation and amplifying turbulent fluctuation intensity. This phenomenon generates localized energy losses. To quantify nonlinear pressure differentials, we can integrate the continuity equation, momentum conservation equation, and Bernoulli equation to yield Equation (5) for expansion loss (\(\Delta p_{out}\))  
\[\Delta p_{out}=k1(1-\beta)^{2}\frac{1}{2}\rho v1^{2} \tag{5}\]  
where \(\beta=A1/A2\) is the area ratio (pre-expansion to post-expansion) and \(v1\) represents upstream velocity (m/s). \(k1\) is an experimentally determined coefficient dependent on \(v1\). For example, under a flow rate of 10 cm\({}^{3}\)/min (the medium is air, 26.42 \({}^{\circ}\)C, 101.33 kPa), \(k1\) is taken as 2.86. Similarly, the contraction loss (\(\Delta p_{in}\)) is calculated via Equation (6)  
\[\Delta p_{in}=k2(1-\beta)\frac{1}{2}\rho v2^{2} \tag{6}\]  
Figure 4: Pressure distribution nephogram of laminar flow element. (**a**) Pressure nephogram at the inlet region and outlet region for airflow at 10 cm\({}^{3}\)/min. (**b**) Pressure nephogram at the inlet region and outlet region for airflow at 100 cm\({}^{3}\)/min. where \(v2\) is the downstream velocity after contraction (m/s) and \(k2\) is an experimentally determined coefficient dependent on \(v2\). For example, at the flow rate of 10 cm\({}^{3}\)/min (the medium is air, 26.42 \({}^{\circ}\)C, 101.33 kPa), \(k2\) is taken as 0.439 (\(\beta=A2/A1\), post-contraction to pre-contraction) (Figure 6). Since the flow velocity is the decisive factor for the magnitude of the local pressure difference, in order to compare the magnitude of localized nonlinear pressure differentials caused by flow path expansion and contraction, the fluid flow velocity before flow path expansion and after flow path contraction can be studied. Figure 7 presents ANSYS Fluent-derived velocity contours at 10 cm\({}^{3}\)/min (the medium is air, 26.42 \({}^{\circ}\)C, 101.33 kPa). As evident from the simulation, under the laminar flow element configuration proposed in this study, the post-contraction flow velocity (9.418 m/s) far exceeds the pre  
Figure 5: Schematic diagram of flow pipeline expansion. Figure 6: Schematic diagram of flow pipeline contraction. Figure 7: Velocity nephogram for air flow at 10 cm\({}^{3}\)/min. (**a**) Overall velocity nephogram. (**b**) Velocity nephogram at capillary element inlet and outlet. expansion velocity (1.878 m/s). This demonstrates that the nonlinear error induced by sudden flow path contraction surpasses that caused by sudden expansion. To calculate the nonlinear frictional pressure differential, we need the entrance length (\(L_{e}\)) for flow to achieve fully developed laminar conditions, which is calculated by Equation (7)  
\[L_{e}=CdR_{e} \tag{7}\]  
where \(d\) is the capillary diameter (\(m\)) and \(C\) is a constant determined by the development completeness (e.g., C = 0.075 for 99.9% development). Table 2 quantifies \(L_{e}\) values for air under experimental conditions (26.42 \({}^{\circ}\)C, 101.33 kPa) across varying flow rates. The data in the table reveal that the maximum \(L_{e}\) (11.13 mm) is substantially shorter than the capillary length (100 mm), ensuring fully developed flow throughout the capillary. This confirms the adequacy of the capillary element length employed in this study. The nonlinear frictional pressure differential \(\Delta p_{h}\) can be calculated based on \(L_{e}\) using the following relationship:  
Figure 3:0  
where \(\alpha\) is a flow-dependent coefficient determined experimentally (under the flow rate of 10 cm\({}^{3}\)/min, \(\alpha\) is taken as 0.0229). Now the derivation of computational formulas for the three nonlinear error components has been completed.