# A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime
## 2 Materials and Methods
### Processing Method for Nonlinear Errors  
To mitigate localized nonlinear pressure differentials at the junction between external connection pipes and the capillary element, as well as nonlinear frictional pressure losses within the capillary, this study implements two strategies. The first approach is increasing the capillary element's length-to-diameter ratio (\(L/d\)). As derived from the Hagen-Poiseuille equation, the linear frictional pressure differential (\(\Delta p\)) under fully developed laminar flow is proportional to the ratio \(L/d^{4}\). Thus, elevating this ratio amplifies the contribution of linear frictional pressure differentials within the total measured pressure differential. When the flow conditions satisfy Equation (3), nonlinear errors remain within acceptable bounds. \[R_{e}\frac{d}{l}\leq 2 \tag{3}\]  
where \(l\) is the capillary length (m). The design of this research employs a capillary element with a diameter of 0.15 mm and a length of 100 mm, for the specified flow range of 1-10 cm\({}^{3}\)/min (the medium is air, 26.42 \({}^{\circ}\)C, 101.33 kPa); calculations confirm that the threshold value remains below 2 for this capillary configuration, validating its efficacy in suppressing nonlinear effects. The second method analytically quantifies nonlinear pressure differentials and subtracts them from the total measured pressure differential. The total pressure differential (\(\Delta P\)) is decomposed into four components as shown in Equation (4)  
\[\Delta p=\Delta p_{in}+\Delta p_{hp}+\Delta p_{h}+\Delta p_{out} \tag{4}\]  
As delineated in Equation (4), the localized nonlinear pressure differentials induced by sudden flow path contraction and expansion are denoted as \(\Delta p_{in}\) and \(\Delta p_{out}\), respectively, while \(\Delta p_{h}\) represents the nonlinear frictional pressure differential generated during the flow development phase prior to fully developed laminar conditions, and \(\Delta p_{hp}\) corresponds to the linear frictional pressure differential under fully developed laminar flow. Structural analysis of the laminar element reveals that pressure variations caused by sudden con \begin{table}
\begin{tabular}{c c} \hline
**Structured Name** & **Structured Data** \\ \hline capillary length & 100 mm \\ \hline external connection pipes diameter & 10.6 mm \\ \hline diameter of sealing chambers & 1 mm \\ \hline sealing chambers length & 2 mm \\ \hline diameter of pressure tap conduits & 6 mm \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensional diagram of capillary laminar flow device. Figure 3: Structural diagram of a novel capillary laminar flow apparatus. traction/expansion predominantly occur at the interfaces between the external connection conduits and the capillary tube. The sealing chambers, due to their minimal length, are hypothesized to exert negligible influence on pressure dynamics. To validate this hypothesis, numerical simulations were conducted using ANSYS Fluent 2021 R1. A computational model was constructed and meshed, with experimental parameters replicated to simulate airflow at 10 cm\({}^{3}\)/min and 100 cm\({}^{3}\)/min (the medium is air, 26.42 \({}^{\circ}\)C, 101.33 kPa). Pressure distribution contour plots under these flow conditions are presented in Figure 4, illustrating localized pressure gradients at critical interfaces. From the simulation results and pressure reading results, it can be observed that when air flow enters the sealing chamber from the external connection pipe and exits from the sealing chamber into the external connection pipe at two flow rates, the pressure changes are almost negligible (both pressure variations are less than 0.3% of the total pressure difference). Therefore, it is proven that the presence of the sealing chambers has a negligible effect on the pressure change.