# A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime
## 2 Materials and Methods
### Processing Method for Nonlinear Errors
Furthermore, in practical applications, for a laminar element with fixed geometry, the term \(\pi d^{4}/(128\mu L)\) in the Hagen-Poiseuille equation (Equation (2)) becomes a constant for a specific fluid. Incorporating this, the conventional Hagen-Poiseuille equation is modified as  
Figure 3:1  
where \(C\) represents a correction coefficient accounting for manufacturing uncertainties in the capillary element (through experimental analysis, it is believed that \(C\) in this article is set as 1.03), and \(K\) consolidates the geometric constant from the Hagen-Poiseuille equation (\(\pi d^{4}/(128\mu L)\). Synthesizing all preceding equations, the flow rate calculation formula for the proposed laminar flow measurement device is derived as  
Figure 3:2  
To compare the proportions of nonlinear and linear pressure differences, simulations were performed using ANSYS (Figure 8). In setting the simulation meshing parameters, face size adjustment was adopted to refine the mesh at the capillary and its connection. The mesh element size was set to 0.015 mm, which is one-tenth of the capillary diameter (0.15 mm), while coarser meshes were used for other parts to simplify pressure calculation. Meanwhile, a boundary layer was configured in the mesh of the capillary section to \begin{table}
\begin{tabular}{c c c} \hline
**Flow (cm3/min)** & **Reynolds Number \(R_{e}\)** & \(Le\) (mm)** \\ \hline \(1\) & 9.89 & 0.11 \\ \hline \(4\) & 39.6 & 0.45 \\ \hline \(10\) & 99.0 & 1.11 \\ \hline \(40\) & 396.0 & 4.45 \\ \hline \(100\) & 990.0 & 11.13 \\ \hline \end{tabular}
\end{table} 
Table 2: \(L_{e}\) values of capillaries at different flow rates. accurately capture the flow characteristics near the wall. The boundary layer was designed with 5 layers and a growth rate of 1.2 times, and the final number of meshes exceeded 250,000, ensuring the accuracy of the simulation process. For the setting of simulation operation parameters, to accurately simulate the flow under laminar flow conditions, the laminar model was selected as the viscosity model, with steady-state simulation and double-precision calculation employed. The physical environment was set to 26.42 \({}^{\circ}\)C and 101.33 kPa, consistent with the experimental environment, where the density of the fluid (air) was 1.187 kg/m\({}^{3}\) and the mass flow inlet was set according to the flow rate magnitude. Post-modeling, meshing, parameter configuration, and simulation execution, the total pressure differential across the capillary's pressure taps and the linear frictional pressure differential within the capillary element were extracted. This enabled quantification of the nonlinear pressure differential's contribution. A 0.15 mm capillary laminar element was modeled, with simulations performed at three flow rates: 1 cm\({}^{3}\)/min, 4 cm\({}^{3}\)/min, 10 cm\({}^{3}\)/min, 40 cm\({}^{3}\)/min, and 80 cm\({}^{3}\)/min. To minimize extraneous variables, all parameters except flow rate remained consistent across simulations. The comparative results between simulation and theoretical calculations are summarized in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Identification Number**} & \multirow{2}{*}{**Flow (cm\({}^{3}\)/min)**} & \multicolumn{2}{c}{**Local Differential Pressure/**} \\  & & **Total Differential Pressure (\%)** & **Total Nonlinear** \\ \hline
1 & 1 & 0.4\% & 72.97\% \\ \hline
2 & 4 & 4.59\% & 39.54\% \\ \hline
3 & 10 & 7.62\% & 20.07\% \\ \hline
4 & 40 & 11.03\% & 9.81\% \\ \hline
5 & 100 & 13.21\% & 4.88\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of nonlinear differential pressure proportion. Figure 8: Simulation grid division diagram. Column 3 in Table 3 reveals that the percentage of nonlinear pressure differential increases progressively with flow rate, exceeding 10% at 40 cm\({}^{3}\)/min.