# A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime
## 5 Conclusions  
To address the issues of large nonlinear errors and complex structures in existing micro gas flow measurement methods, this paper investigates the accuracy of a novel gas small-flow measurement method based on capillary laminar flow elements. Using a single capillary as the laminar component, the capillary's length-to-diameter ratio is increased in structural design, and nonlinear errors are reduced through theoretical analysis. The correction equation and flow calculation formula are derived via ANSYS Fluent simulation analysis.  
After establishing the experimental platform, the working condition flow rate was inferred from the pressure difference across the laminar element when air flowed in a laminar regime. Multiple experiments yielded pressure difference data at 1-10 cm\({}^{3}\)/min. The repeatability RSD of measured pressure differences at all flow points was \(\leq\)1%, and the maximum expanded uncertainty (\(u_{C}\)) was less than 2%, indicating strong reliability of the experimental data. Theoretical flow rates calculated using the measured pressure differences were compared with data from the standard flowmeter in the experimental system. The results showed that the gas small-flow measurement method based on capillary laminar flow elements achieved good linearity, with the maximum residual ratio under full-scale pressure difference being less than 0.5% (R\({}^{2}\) > 0.99) and the relative error remaining within \(\pm\)0.6% in the micro-flow range (1-10 cm\({}^{3}\)/min).  
In practical measurement processes, compared with non-laminar flow measurement methods, the measurement method used in this study features a simple device structure, convenient operation, and excellent linearity of measurement results.  
Compared with other laminar flow measurement methods, the measurement device in this study requires shorter processing procedures. The detachable capillary elements provide different measurement ranges; using a single capillary with a diameter of 0.15 mm as the laminar flow element lowers the lower limit of the measurement range, avoids the problem of gaps caused by capillary bundles affecting measurement accuracy, and is more suitable for micro gas flow measurement. In terms of the measurement method, the application of a nonlinear error correction algorithm achieves a balance between structural simplification, maintenance of high linearity, and low measurement error.  
Therefore, the proposed micro gas flow measurement method is of great importance for the industrial application of laminar flowmeters in micro gas measurement.  
Conceptualization, Y.Z.; methodology, Y.Z. and D.X.; data curation, Y.Z.; formal analysis, Y.Z.; investigation, Y.Z.; resources, D.X., D.W. and H.Z.; writing--original draft  
Figure 10: Experimental data diagram. (**a**) The fitted curve correlating corrected pressure differentials with reference flow rates. (**b**) The post-correction relative error distribution.  
preparation, Y.Z.; writing--review and editing, D.X., Z.Q., Z.H. and Y.X.; visualization, Z.Q.; supervision, D.W. and H.Z.; funding acquisition, D.W. and H.Z.; project administration, H.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Science and Technology Plan Project (ZD2025014) of the Administration for Market Regulation of Zhejiang Province, China.  
Not applicable.  
The data in this paper can be obtained from the corresponding authors according to reasonable requirements.  
We sincerely thank the Administration for Market Regulation of Zhejiang Province for providing the necessary instrument resources and financial support for our research, and we also thank [PERSON] for his assistance with manuscript review.  
The authors declare no conflicts of interest.