# A Capillary-Based Micro Gas Flow Measurement Method Utilizing Laminar Flow Regime
## 2 Materials and Methods  
### 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 \(R_{e}\leq 2300\)[19; 20]. The Reynolds number is calculated as shown in Equation (1)  
\[R_{e}=\frac{\rho vd}{\mu} \tag{1}\]  
where \(\rho\) denotes fluid density (kg/m\({}^{3}\)), \(v\) 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)  
\[q_{v}=\frac{\pi d^{4}\Delta p}{128\mu L} \tag{2}\]  
In this equation, \(q_{v}\) is the volumetric flow rate (cm\({}^{3}\)/min, CCM), \(\Delta p\) 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:  
1. Positions 1 to 2: Localized pressure differential due to sudden flow path contraction.  
Figure 1: Schematic diagram of fluid flow state.  
2. Positions 2 to 3: Frictional pressure differential during flow development toward fully developed laminar flow.
3. Positions 3 to 4: Frictional pressure differential under fully developed laminar flow.
4. 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.