A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime

Abstract

Accurate micro gas flow measurement is critical for medical ventilator calibration, environmental gas monitoring, and semiconductor manufacturing. Laminar flowmeters are widely employed in micro gas flow measurement applications owing to their inherent advantages of high linearity, the absence of moving components, and a broad measurement range. Nevertheless, due to the low measurement accuracy under micro gas flow caused by nonlinear errors and a relatively complex structure, traditional laminar flow measurement devices exhibit limitations in micro gas flow measurement scenarios. This study proposes a novel micro gas flow measurement method based on a single capillary laminar flow element, which simplifies the structure and enhances applicability in the field of micro gas flow. Through structural optimization with precise control of the capillary length–diameter ratios and theoretical error correction based on computational analysis, nonlinear errors were effectively reduced while improving the measurement accuracy in the field of micro gas flow. The proposed methodology was systematically validated through computational fluid dynamics simulations (ANSYS Fluent 2021 R1) and experimental investigations using a dedicated test platform. The experimental results show that the relative error of the measurement system within the full measurement range is less than ±0.6% (1–10 cm³/min; cm³/min means cubic centimeter per minute), and its accuracy is superior to 1% of reading (1% Rd) or 1.5% of reading (1.5% Rd) of conventional laminar flowmeters. The fitting curve of the flow rate versus the pressure difference derived from the measurement results maintains an excellent linear correlation (R² > 0.99), thus confirming that this method has practical application value in the field of micro gas flow measurement.

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 measurement accuracy. 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 Faraday’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 António Pedro 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 Poiseuille in 1838. Subsequent milestones include Ostwald’s pioneering application of fine-bore metal tubes for laminar flow measurement in 1908, and Richard et al.’s development of practical laminar flowmeters during the 1930s and 1940s internal combustion engine pulsation studies. Late-20th-century advancements by Professor Bonian Wang 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. Consequently, 21st-century research has prioritized minimizing nonlinear error impacts. In 2010, Lopez Pena et al. designed a laminar flowmeter with three pressure tapping points, selectively utilizing differential pressure from capillary tube ends at low flow rates and from downstream sections at higher rates, effectively reducing nonlinear error contributions. Haoqin Huang et al. enhanced linearity by implementing a dual-channel cross-symmetric laminar element configuration, mitigating inlet/outlet localized losses and nonlinear pressure drops in the capillary entrance region. Concurrently, Xiaolu Wang et al. developed a novel micro slit laminar flowmeter that circumvents nonlinear effects by strategically positioning pressure taps within fully developed flow regions [16,17,18].

Building upon these advancements, this study proposes a micro gas flow measurement method employing a single capillary laminar flow element. The objective of this method is to simplify the structural complexity of measurement devices in the low micro flow regime (1–10 cm³/min) and reduce nonlinear errors, thereby enhancing measurement accuracy.

This paper is mainly divided into five main chapters. The Section 1 is the Introduction, which introduces the origins and development history of laminar flow measurement devices, summarizes the limitations of existing laminar flowmeters, and elicits the research content of the paper. The Section 2 is Materials and Methods, which analyzes the principles underlying the limitations of laminar flowmeters, the structural parameters of the new laminar flow measurement device designed in this study, the establishment of error correction methods, and the construction of the experimental platform. The Section 3 is Results, which mainly presents the experimental results and data. The Section 4 is Discussion, which focuses on explaining the processing and analysis of the data. The Section 5 is Conclusions, which mainly summarizes and concludes the research content.

2. Materials and Methods

2.1. The Operating Principle of a Traditional Laminar Flowmeter

Laminar flowmeters operate based on the laminar flow state of fluids, which is determined by the Reynolds number (R_e). The flow state of the fluid is shown in Figure 1. A laminar flow state means R_e ≤ 2300 [19,20]. The Reynolds number is calculated as shown in Equation (1)

$$R_e={\displaystyle \frac{\rho vd}{\mu }}$$

(1)

where $\rho$ denotes fluid density (kg/m³), $v$ represents flow velocity (m/s), $d$ is the inner diameter of the circular flow channel (m), and $\mu$ is the dynamic viscosity of the fluid (Pa·s).

The main components of a conventional laminar flowmeter include a laminar flow element, a differential pressure measurement device, etc. The most common laminar flow element consists of a bundle of parallel capillary tubes. When fluid passes through this element, it progressively develops into a fully laminar flow state. The pressure differential across the element is then measured by the differential pressure measurement device. Under laminar flow conditions, the volumetric flow rate can be derived from the Hagen–Poiseuille equation [21,22,23]. For incompressible fluids flowing through a horizontal circular pipe in the laminar regime, the volumetric flow rate exhibits a linear relationship with the pressure differential, pipe diameter, pipe length, and fluid dynamic viscosity, as expressed in Equation (2)

$$q_v={\displaystyle \frac{\pi d^4∆p}{128\mu L}}$$

(2)

In this equation, $q_v$ is the volumetric flow rate (cm³/min, CCM), $∆p$ is the pressure differential across the laminar element (Pa), and L is the distance between pressure taps (m). In conventional capillary-type laminar flowmeters, pressure taps are positioned at both ends of the laminar element (Figure 2). The total pressure differential between positions 1 and 5 comprises four distinct components:

• Positions 1 to 2: Localized pressure differential due to sudden flow path contraction.
• Positions 2 to 3: Frictional pressure differential during flow development toward fully developed laminar flow.
• Positions 3 to 4: Frictional pressure differential under fully developed laminar flow.
• Positions 4 to 5: Localized pressure differential due to sudden flow path expansion.

Among these components, only the frictional pressure differential between positions 3 and 4 satisfies the linear relationship described by the Hagen–Poiseuille equation. The remaining three nonlinear pressure differential components significantly degrade measurement accuracy and linearity. Minimizing the proportional contribution of these nonlinear components is critical for performance enhancement [24,25]. In addition, using capillary bundles composed of multiple capillaries as laminar flow elements will also lead to problems such as more complex structures difficult for maintenance and gaps between capillaries, which are not conducive to micro flow measurement. Therefore, it is necessary to optimize the structure of the laminar flow element.

2.2. Structural Design of the Novel Capillary Laminar Flow Component

For the purpose of simplifying the structure of the capillary laminar flow element, reducing the complex production processes caused by multiple capillaries, and facilitating maintenance while reducing measurement errors arising from the gap area between capillaries, making it more applicable to micro gas flow measurement, this study proposes a novel micro gas flow measurement device that replaces conventional capillary tube bundles with a single capillary element. The laminar flow element comprises four key structural components: external connection pipes, sealing chambers, pressure tap conduits, and a capillary element. The capillary element features a 0.15 mm inner diameter, with its ends interfacing with external pipes through specially designed sealing chambers. Hermetic sealing is achieved using compressible gaskets installed between the sealing chambers and capillary ends. Two pressure taps are precisely positioned 116 mm apart along the external pipes flanking the capillary element. Figure 3 illustrates the complete assembly configuration, while

Table 1. Dimensional diagram of capillary laminar flow device.

Structured Name	Structured Data
capillary length	100 mm
external connection pipes diameter	10.6 mm
diameter of sealing chambers	1 mm
sealing chambers length	2 mm
diameter of pressure tap conduits	6 mm

provides detailed dimensional specifications of the entire laminar flow measurement device.

2.3. 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 (Δp) under fully developed laminar flow is proportional to the ratio L/d⁴. 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.

$$R_e{\displaystyle \frac{d}{l}}\le 2$$

(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 cm³/min (the medium is air, 26.42 °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 (ΔP) is decomposed into four components as shown in Equation (4)

$$∆p={∆p}_{in}+{∆p}_{hp}+{∆p}_h+{∆p}_{out}$$

(4)

As delineated in Equation (4), the localized nonlinear pressure differentials induced by sudden flow path contraction and expansion are denoted as ${∆p}_{in}$ and ${∆p}_{out}$, respectively, while ${∆p}_h$ represents the nonlinear frictional pressure differential generated during the flow development phase prior to fully developed laminar conditions, and ${∆p}_{hp}$ 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 contraction/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 cm³/min and 100 cm³/min (the medium is air, 26.42 °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.

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 ($∆p_{out}$)

$$∆p_{out}={k1(1-\beta )}^2{\displaystyle \frac{1}{2}}\rho {v1}^2$$

(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 cm³/min (the medium is air, 26.42 °C, 101.33 kPa), k1 is taken as 2.86. Similarly, the contraction loss (${∆p}_{in}$) is calculated via Equation (6)

$$∆p_{in}=k2(1-\beta ){\displaystyle \frac{1}{2}}\rho {v2}^2$$

(6)

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 cm³/min (the medium is air, 26.42 °C, 101.33 kPa), k2 is taken as 0.439 (β = 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 cm³/min (the medium is air, 26.42 °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-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)

$$L_e=Cd{R}_e$$

(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. L_e values of capillaries at different flow rates.

Flow (cm3/min)	Reynolds Number Re	Le (mm)
1	9.89	0.11
4	39.6	0.45
10	99.0	1.11
40	396.0	4.45
100	990.0	11.13

quantifies $L_e$ values for air under experimental conditions (26.42 °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 ${∆p}_h$ can be calculated based on $L_e$ using the following relationship:

$${∆p}_h=\alpha {\displaystyle \frac{L_e}{l}}{\displaystyle \frac{128\mu LQ}{\pi d^4}}$$

(8)

where α is a flow-dependent coefficient determined experimentally (under the flow rate of 10 cm³/min, α is taken as 0.0229). Now the derivation of computational formulas for the three nonlinear error components has been completed.

Furthermore, in practical applications, for a laminar element with fixed geometry, the term $\pi d^4/\left(128\mu L\right)$ in the Hagen–Poiseuille equation (Equation (2)) becomes a constant for a specific fluid. Incorporating this, the conventional Hagen–Poiseuille equation is modified as

$$q_v=C{\displaystyle \frac{\pi d^4}{128L}}\cdot {\displaystyle \frac{1}{\mu }}∆p=C{\displaystyle \frac{K}{\mu }}{∆p}_{hp}$$

(9)

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 ($\pi d^4/\left(128\mu L\right)$. Synthesizing all preceding equations, the flow rate calculation formula for the proposed laminar flow measurement device is derived as

$$q_v=C{\displaystyle \frac{K}{\mu }}(∆p-{∆p}_h-∆p_{in}-∆p_{out})$$

(10)

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 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 °C and 101.33 kPa, consistent with the experimental environment, where the density of the fluid (air) was 1.187 kg/m³ 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 cm³/min, 4 cm³/min, 10 cm³/min, 40 cm³/min, and 80 cm³/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. Comparison of nonlinear differential pressure proportion.

Identification Number	Flow (cm3/min)	Nonlinear Differential Pressure/ Total Differential Pressure (%)	Local Differential Pressure/ Total Nonlinear Differential Pressure (%)
1	1	0.4%	72.97%
2	4	4.59%	39.54%
3	10	7.62%	20.07%
4	40	11.03%	9.81%
5	100	13.21%	4.88%

.

Column 3 in Table 3 reveals that the percentage of nonlinear pressure differential increases progressively with flow rate, exceeding 10% at 40 cm³/min. And then, the data in the fourth column of the table indicate that with the increase in flow rate, the ratio of the local pressure difference to the total nonlinear pressure difference gradually becomes smaller than that of the nonlinear frictional pressure difference to the total nonlinear pressure difference. At flow rates of 4 cm³/min and higher, the nonlinear frictional pressure difference dominates the total nonlinear pressure difference. Simulation data analysis indicates an optimal operational flow range of 1–10 cm³/min for this laminar flow measurement device, where the nonlinear pressure differential remains below 8%.

2.4. Experimental System Construction

The experimental procedure was executed as follows: air was delivered into the capillary pipeline laminar flow element via an air compressor and a critical flow nozzle. Pressure signals were extracted through pressure tapping tubes connected to the laminar flow element, with the differential pressure across the element measured by a dedicated differential pressure gauge. To enable comparative analysis with theoretical data, a calibrated flowmeter was integrated into the experimental system to acquire standardized flow rate values. The fully assembled experimental platform is illustrated in Figure 9.

The experiments were conducted under controlled ambient conditions with a measured temperature of 26.42 °C and atmospheric pressure of 101.33 kPa, utilizing air as the working fluid. The system was equipped with a CONST221 (YIDU Intelligence, Xi’an, China) intelligent digital differential pressure gauge (range: −10 to 1000 Pa, accuracy class 0.02) to measure pressure differentials across the laminar flow element, alongside an S101 (Alicat Scientific, Tucson, AZ, USA) laminar volumetric flowmeter (range: −10 to 10 SCCM) serving as the calibrated reference for flow rate validation. All measurements strictly adhered to the standards outlined in JJG 736-2012 Verification Regulation for Gas Laminar Flow Transducers and BS ISO 11631:1999 Measurement of fluid flow—Methods of specifying flowmeter performance. Triplicate data acquisitions were performed at each of five predefined flow points (1, 2, 4, 8, and 10 cm³/min) under stabilized flow conditions, with both differential pressure and reference flow values manually recorded.

3. Results

The reference flowmeter in the experimental system records flow rates under standard conditions (0 °C, 101.35 kPa) in units of SCCM (standard cubic centimeters per minute), whereas the flow rates calculated via Equation (10) correspond to operational conditions in units of ${\mathrm{c}\mathrm{m}}^3/\mathrm{m}\mathrm{i}\mathrm{n}$ (cubic centimeters per minute in the experimental environment, 26.42 °C, 101.33 kPa). To enable comparative analysis, the standard condition flow rate $q1$ (SCCM) must be converted to the operational condition flow rate $q_{vs}$ (cm³/min) using Equation (11)

$$q_{vs}=q1{\displaystyle \frac{p1+p2}{2pa}}$$

(11)

where pa represents the atmospheric pressure in the standard condition. And p1 and p2, respectively, represent the absolute pressures (Pa) measured upstream and downstream of the capillary laminar element. During the experiments, p1 was directly acquired from the upstream pressure gauge, while p2 was approximated as p2 = p1 − $∆p$, where $∆p$ is the measured differential pressure.

Under the experimental conditions, the dynamic viscosity of air was determined as 1.84 × ${10}^{-5}$ Pa*s. The relative error δ between the experimentally derived volumetric flow rate ($q_{vs}$) and the theoretically calculated flow rate ($q_v$) was quantified as

$$\delta ={\displaystyle \frac{q_v-q_{vs}}{q_{vs}}}\times 100\%$$

(12)

where now both $q_{vs}$ (reference flow rate) and $q_v$ (theoretical flow rate) are expressed in cm³/min (26.42 °C, 101.33 kPa).

Table 4. Micro tube laminar flow element experimental data.

Identification Number	Measured Pressure Difference Δp (Pa)	Corrected Pressure Difference Δp′ (Pa)	Measured Flow qvs (cm3/min)	Theoretically Calculated Flow qv (cm3/min)	Repeatability RSD (%)	Relative Error After Correction δ (%)
1	11.5	11.4	1.007	1.012	0.88%	0.53%
	11.4	11.3	0.998	1.003		0.55%
	11.3	11.2	0.989	0.994		0.56%
2	22.8	22.6	1.999	2.007	0.68%	0.39%
	22.6	22.4	1.982	1.989		0.36%
	22.5	22.3	1.973	1.980		0.37%
3	46.7	44.6	3.951	3.960	0.34%	0.24%
	46.9	44.7	3.961	3.969		0.21%
	47.01	44.9	3.978	3.987		0.23%
4	96.4	90.1	8.017	8.001	0.39%	−0.20%
	96.8	90.5	8.055	8.036		−0.23%
	96.1	89.8	7.994	7.974		−0.25%
5	121.6	112.5	9.976	9.990	0.35%	0.14%
	122.1	112.8	10.002	10.017		0.15%
	121.6	112.3	10.044	10.061		0.17%

consolidates experimental data for the laminar flow element. Column 1 lists test identifiers; Columns 2 and 3, respectively, document the raw pressure differential measured across the laminar element under experimental conditions and the corrected linear frictional pressure differential obtained via nonlinear error compensation using Equations (5), (6), and (8). Columns 4 and 5 present the operational condition flow rates recorded by the reference meter and the theoretical flow rates calculated via Equation (10). To evaluate measurement consistency, Column 6 computes the repeatability of corrected pressure differentials across repeated trials. The final column reports the relative error between the theoretical flow rates and reference flow rates to validate the error compensation efficacy.

4. Discussion

The data in Table 4 show that the repeatability RSD of the measured pressure differences at all flow points is ≤1%, demonstrating the good repeatability and high consistency of the experimental data. To improve the reliability of the measurement results, the uncertainty of the corrected pressure difference needs to be calculated. Through analysis, the standard deviation of the pressure difference after nonlinear error correction is taken as the Type A uncertainty $u_A$, while the Type B uncertainty $u_B$ mainly originates from the resolution of the pressure difference measuring device. The minimum scale value of the pressure difference measuring device in the experimental system is 0.1 Pa, with the half-width of the interval being 0.05 Pa, and the pressure difference follows a uniform distribution. The calculation results of the uncertainty are shown in the following table (the coverage factor k is taken as 2, corresponding to a confidence probability of 95%).

Table 5. Standard uncertainty data.

Flow Rate/cm3/min	Type A Uncertainty uA/Pa	Type B Uncertainty uB/Pa	The Expanded Uncertainty uC/Pa	The Maximum Ratio of the Expanded Uncertainty uC to the Average Pressure Difference (%)
1	0.10	0.03	0.20	1.77%
2	0.15	0.03	0.30	1.34%
4	0.15	0.03	0.30	0.67%
8	0.35	0.03	0.70	0.78%
10	0.40	0.03	0.80	0.71%

indicates that the maximum ratio of the expanded uncertainty $u_C$ to the average pressure difference is 1.77%, which is less than the 2–3% required for general industrial measurements. Therefore, the reliability of the measurement results can be considered high.

Observing the data in Table 4, it is found that within the flow range of 1–10 cm³/min, the relative errors between the theoretical flow rates obtained after nonlinear error correction and the flow rates measured by the standard meter are all less than ±0.6%. This represents a significant improvement compared to conventional laminar flowmeters, whose measurements can only meet the requirement of a relative error less than or equal to ±1% to ±1.5%. This indicates that both the optimization of the laminar element structure in this study and the correction using theoretical equations can effectively reduce the proportion of nonlinear errors and improve measurement accuracy.

Figure 10a shows the curve fitted by the least square method for the pressure difference after nonlinear error correction and the flow rate measured by the standard flowmeter. To verify the linearity of the fitted curve, the determination coefficient R² is calculated to be >0.99, and the maximum residual ratio under full-scale pressure difference is 0.34%, which is lower than the general industrial requirement of ±1%. This demonstrates that the flow-pressure difference fitted curve after correction has good linearity, and the experimental data can well illustrate the characteristic that the laminar flow measurement method has high linearity. Figure 10b shows the distribution diagram of the corrected relative errors. It can be observed from the relative error distribution diagram that the relative error is relatively large during the measurement at the 1 ${\mathrm{c}\mathrm{m}}^3/\mathrm{m}\mathrm{i}\mathrm{n}$ flow point. Through analysis, it is considered that the main reason is that the gas source was in the initial start-up state, resulting in unstable output gas.

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 cm³/min. The repeatability RSD of measured pressure differences at all flow points was ≤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% (R² > 0.99) and the relative error remaining within ±0.6% in the micro-flow range (1–10 cm³/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.