# A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime
## 1 Introduction  
Flow measurement refers to the process of quantitatively measuring the volume or mass of fluid (gas or liquid) passing through a specific cross-section per unit time via certain technical means. Its technological development has a significant impact on fields such as industrial process control, energy metering, and environmental monitoring. Currently, there are numerous flow measurement methods on the market, mainly distinguished by different physical principles: Turbine flowmeters operate by a fluid driving an impeller to rotate, featuring fast response speed and wide rangeability, but they are susceptible to fluid viscosity and particulates; impeller wear can lead to degraded long-term measurementaccuracy. Ultrasonic flowmeters calculate flow velocity using the time difference in sound wave propagation in forward and reverse flows, offering the advantage of non-contact measurement to avoid pipe wear, yet they have higher costs. Electromagnetic flowmeters, based on [PERSON]'s law of electromagnetic induction, are suitable for measuring conductive fluids, with characteristics of low pressure loss and a wide range, but they cannot be used for gases or non-conductive fluids and must be installed away from strong magnetic field interference. The aforementioned flowmeters are mostly applicable to medium and large flow measurement, while micro flow measurement in practical applications is also an important aspect of scientific research and production [1; 2; 3; 4; 5]. Laminar flowmeters, functioning as differential pressure-based flow measurement devices, have evolved various structural configurations of their laminar flow elements, including dual-cone, slit-type, and capillary-type designs. These devices operate by measuring the pressure differential generated across the flow element during fluid passage. Compared to alternative measurement technologies such as thermal or rotameter-type flowmeters, laminar flowmeters demonstrate superior linearity in pressure-flow rate relationships, feature no moving components, and achieve enhanced measurement accuracy. These characteristics make them particularly advantageous for low-flow measurement applications, leading to their widespread adoption across industrial production, chemical and pharmaceutical manufacturing, and aerospace systems [6; 7; 8]. For instance, in the semiconductor manufacturing field, thin-film deposition processes (CVD/PVD) require capillary laminar flowmeters to monitor the minute flow rates of reactive gases in real time, ensuring that the deposition rate meets the standard and avoiding film defects caused by flow fluctuations. In the medical field, where various ventilators used in hospitals rely on capillary laminar flowmeters to measure and feed back the minute flow rates of mixed gases with different ratios, thereby forming a closed-loop control system with pressure sensors. In the aerospace field, scholars such as [PERSON] have used laminar flowmeters as flow measurement devices for aircraft oxygen regulators, ensuring the accuracy of gas flow measurement through data acquisition systems and sensors [9; 10; 11]. The theoretical foundation of laminar flow measurement originates from the Hagen-Poiseuille law, empirically established through extensive experimentation by French physician [PERSON] in 1838. Subsequent milestones include [PERSON]'s pioneering application of fine-bore metal tubes for laminar flow measurement in 1908, and [PERSON] et al. 's development of practical laminar flowmeters during the 1930s and 1940s internal combustion engine pulsation studies. Late-20 th-century advancements by Professor [PERSON] established critical operational parameters, including maximum mean velocity and optimal pipe length under constrained volumetric flow conditions [12; 13; 14; 15]. Nevertheless, most laminar flowmeters inherently suffer from nonlinear errors during flow measurement due to structural and operational principles, compromising measurement accuracy.