Simulation Analysis and Experimental Study of Pipeline Gas Resistance Modelling and Series Characteristics

Abstract

The principle of electro-analogy analysis treats a gas path structure as analogous to a circuit, offering significant potential for performance analysis in aerostatic systems. However, research on gas resistance remains in an early stage. This study investigates pipe gas resistance and its series characteristics using a slender circular pipe as the subject. First, gas resistance is redefined based on a derivation of the Bernoulli equation, resulting in formulas covering low and high speeds and a calculation model for series gas resistance. Simulations are conducted to model the pipe, focusing on the coefficient of frictional resistance at low speeds. The results provide insights into the gas resistance of pipes with varying inner diameters and related series connections. An experiment is conducted to validate predictions, indicating that, at low speeds, the defined and determined gas resistance values for pipelines with inner diameters ranging from 1 to 6 mm are largely consistent. Both gas and series gas resistances decrease as the pressure difference between the two pipe ends increases. Relative errors below 5% are typically regarded as very good, especially when dealing with complex systems. The maximum relative error between the experimentally measured single gas resistance, based on the defining formula and the simulation value, is 3.1%. Furthermore, the maximum relative errors for the measured single and series gas resistance values are 5% and 3.8%, respectively, according to the defining and determining formulas. The theoretical model is effective and reliable, providing valuable theoretical support for impedance analysis of aerostatic systems.

1. Introduction

Aerostatic systems, known for their high motion accuracy [1], rapid response [2], low friction [3], and pollution-free operation [4], have been widely applied in ultra-precision machining [5]. The core component determining system performance is the aerostatic restrictor [6], which is typically developed using engineering simplification algorithms, finite element and finite difference methods, computational fluid dynamics, multi-physics coupling techniques, and experimental approaches [7,8]. Each method, however, has its limitations: engineering simplification algorithms exhibit large errors and have limited applicability, experimental methods require sophisticated and precise testing equipment, and finite element and finite difference methods are hindered by the complexity of grid division and prolonged computation times.

In order to shorten the research process, researchers have drawn inspiration from the concept of electrical resistance, aiming to establish an impedance model for aerostatic throttle via quasi-electric analysis [9]. This research approach can be traced back to a century ago [10]. Recently, Xu et al. employed aerodynamic resistance networks to analyse the static characteristics of pneumatic components [11], and Sachan et al. developed a device to measure and analyse gas flow resistance in porous media [12]. Stanley et al. proposed a resistor–capacitor model of pneumatic loops and examined its response time, demonstrating that a specific pipeline inner diameter can minimise this response time [13]. Jing et al. investigated the resistance characteristics of flue tees under varying flow ratios [14], and Shen et al. explored the flow resistance characteristics of gas in slit flow channels [15]. Wu et al. applied quasi-electric analysis to study aerostatic bearings with closely spaced micro holes (ABMHs), established a resistance model, and evaluated the effects of series and parallel connections in restrictor orifice structures on the load-bearing capacity of the restrictor. Their findings indicated that the resistance of ABMHs is influenced by both the flow and differential pressure [16].

Calamas et al. analysed how bifurcation angles and apertures in flow channels affect pressure differentials [17], and Liao et al. studied porous materials in restrictors and assessed the effect of temperature on pressure drop versus flow rate characteristics [18,19]. Pant et al. examined the relationship between pressure drop and mass flow in automotive gas flow systems and developed a model consistent with empirical results, verifying its accuracy [20]. Liu et al. investigated flow–pressure drop curves for porous graphite using various equations [21].

Although the concept of gas resistance has been explored to some extent, most studies have merely drawn simplistic analogies with Ohm’s law when analysing aerostatic restrictors, as they have neither conducted in-depth investigations into the validity of establishing gas resistance models based on this relationship nor differentiated gas resistance models according to the compressibility of gases. Since the primary structure of the restrictor’s flow channel is a pipe, the restrictor can be regarded as a combination of pipe structures connected in series or parallel. Consequently, this study focuses on pipelines (i.e., thin and long pipes) as the research object to advance the theory of gas resistance, establish a gas resistance model for pipelines, and investigate the series characteristics of gas resistance. This study differentiates gas resistance models based on Bernoulli’s equation and the gas flow velocity, proposes new definitions of gas resistance corresponding to velocity ranges, and further reveals the nonlinear superposition characteristics of series-connected gas resistances through the established models. In summary, we find that our theoretical model is effective and reliable, providing theoretical support for impedance analysis of aerostatic systems. This study redefined the gas resistance model of pipelines under different states through the Bernoulli equation, established the theoretical calculation method of series gas resistance, and verified the model through simulation and experiments, with the error controlled within 5%. It provides high-precision theoretical support for the impedance analysis of gas static pressure systems.

In this paper, Section 1 introduces the current research status and existing problems of the gas resistance theory. Section 2, through theoretical analysis, obtains the calculation models of gas resistance in pipelines under low-speed and high-speed conditions, as well as the series gas resistance. Section 3, based on the simulation results, derives the formula for calculating the friction factor along the pipeline and validating the calculation model proposed in Section 2 under low-speed conditions. Section 4 builds an experimental setup based on the calculation model proposed in Section 2 for low-speed conditions, and verifies the accuracy of the proposed gas resistance model by combining experimental and simulation results. Section 5 summarises the research content and looks towards future research directions.

2. Theory of Pipeline Resistance

2.1. Theory and Modelling of Pipeline Gas Resistance

Consider a horizontal straight pipe as an example, where steady and uniform gas flow is assumed within the pipe, implying that the gas flow rate and pressure are evenly distributed at each point along section 1-1, as shown in Figure 1.

Because the compressibility of gas is influenced by the gas flow rate, the theoretical analysis of pipeline gas resistance is conducted under two conditions.

2.1.1. Ma < 0.3

The Mach number (Ma) is defined as the ratio of the velocity at a point in the flow field to the speed of sound at that point. When the Mach number is less than 0.3, variations in the gas flow rate result in negligible changes in gas density, allowing the gas to be treated as an incompressible fluid. The Bernoulli equation, applied between sections 1-1 and 2-2 in Figure 1, is expressed as follows:

$${\displaystyle \frac{a_1{v}_1^2}{2g}}+{\displaystyle \frac{p_1}{{\rho }_{1}g}}+z_1={\displaystyle \frac{a_2{v}_2^2}{2g}}+{\displaystyle \frac{p_2}{{\rho }_{2}g}}+z_2+h_w$$

(1)

where $a_1$ and $a_2$ are kinetic energy correction coefficients, $v_1$ and $v_2$ are the gas flow rates at the centre of the two sections, and $g$ is the acceleration due to gravity. $p_1$ and $p_2$ are the static pressures at the centres of the two cross-sections, ${\rho }_1$ and ${\rho }_2$ are the gas densities at the centres of the two sections, $z_1$ and $z_2$ represent the potential energy due to the location at the centres of the two cross-sections, and $h_w$ is the energy loss between the two sections (head loss), which includes the frictional head loss $h_f$ and local head loss $h_j$.

The value of the kinetic energy correction coefficient depends on the flow state of the fluid, and the Reynolds number is the basis for judging the fluid state. It is generally believed that when the Reynolds number is below 2300, it is laminar flow, and when it is above 2300, it is turbulent flow. Based on calculation, for the pipe type studied in this paper, the flow inside the pipe is turbulent in most cases, so the kinetic energy correction coefficient is 1.

According to the law of mass conservation, the mass flow rate, $Q$, through the pipeline is expressed as $Q={\rho }_1{v}_1{A}_1={\rho }_2{v}_2{A}_2$, where the cross-sectional area of the pipeline remains constant (i.e., $A_1=A_2=A$), and the density of the incompressible fluid is constant (i.e., ${\rho }_1 = {\rho }_2 = \rho$). Therefore, we have the following:

$$v_1=v_2=v$$

(2)

For gases, the effect of changes in the positional potential energy is negligible; hence, the terms, $z_1$ and $z_2$, can be omitted from Equation (1). For the horizontal straight pipe depicted in Figure 1, the local head loss, $h_j$, is not considered. According to Darcy’s equation, the frictional head loss, $h_f$, can be expressed as follows:

$$h_f=\lambda {\displaystyle \frac{L}{D}}{\displaystyle \frac{v^2}{2g}}$$

(3)

where $\lambda$ is the friction factor, L is the length of the pipe, and D is the inner diameter of the pipe.

By substituting Equations (2) and (3) into Equation (1) [22], the pressure difference between the two sections can be expressed as follows:

$$\mathrm{\Delta }p=p_1-p_2={\displaystyle \frac{\lambda L\rho v^2}{2D}}={\displaystyle \frac{8\lambda L{Q}^2}{\rho {\pi }^2{D}^5}}$$

(4)

In electrical systems, resistors are energy-consuming elements that reduce the system energy. Similarly, in pipelines, energy loss occurs due to friction. Drawing on Ohm’s law, the ratio of the pressure difference between two sections to the square of the mass flow rate is defined as the gas resistance of the pipeline when the gas flows at a low speed [23]. This relationship, derived from Equation (4), is represented by $R_g$ and is expressed as follows:

$$R_g={\displaystyle \frac{p_1-p_2}{Q^2}}$$

(5)

$$R_g={\displaystyle \frac{8\lambda L}{\rho {\pi }^2{D}^5}}$$

(6)

Equation (5) serves as the definition formula for pipeline gas resistance at low speeds, whereas Equation (6) represents the determination formula for pipeline gas resistance under low-speed conditions.

2.1.2. Ma > 0.3

When the gas velocity in the pipe is high, the flow becomes compressible. Under these conditions, the flow can be approximated as adiabatic; however, due to friction, it is a non-isentropic adiabatic process. This results in certain flow coefficients (e.g., viscosity and friction) that vary with temperature. Consequently, in practical analyses, the approximate friction coefficient for incompressible flow is often used, and the isentropic equation is applied to account for density changes in the non-isentropic adiabatic flow.

At this stage, the Bernoulli equation for any fluid microelement between two sections of a horizontal pipeline is as follows:

$${\displaystyle \frac{\mathrm{d}p}{\rho }}+v\mathrm{d}v+{\displaystyle \frac{\lambda v^2}{2D}}\mathrm{d}l=0$$

(7)

The gas continuity equation and isentropic equation are expressed as Equations (8) and (9), respectively:

$$Q=\rho vA$$

(8)

$${\displaystyle \frac{p}{{\rho }^{\gamma }}}=C$$

(9)

where $\gamma$ is the specific heat volume ratio of the gas, and $C$ is a constant. By substituting Equations (8) and (9) into Equation (7), we eliminate $\rho$ and $v$ to obtain the following:

$${\displaystyle \frac{A^2}{Q^2}}{\left({\displaystyle \frac{p}{C}}\right)}^{{\displaystyle \frac{1}{\gamma }}}\mathrm{d}p-{\displaystyle \frac{1}{\gamma }}{\displaystyle \frac{\mathrm{d}p}{p}}+{\displaystyle \frac{\lambda }{2D}}\mathrm{d}l=0$$

(10)

By integrating Equation (10) along the pipeline between the two sections, we have the following:

$${\displaystyle \frac{A^2}{Q^2}}{C}^{-{\displaystyle \frac{1}{\gamma }}}{\displaystyle \frac{\gamma }{\gamma +1}}\left({p_1}^{{\displaystyle \frac{\gamma +1}{\gamma }}}-p_2^{{\displaystyle \frac{\gamma +1}{\gamma }}}\right)={\displaystyle \frac{\lambda }{2D}}L-{\displaystyle \frac{1}{\gamma }}\ln {\displaystyle \frac{p_2}{p_1}}$$

(11)

Because the logarithmic term, $\mathit{ln}\left(p_2/p_1\right)$, is very small, it is generally ignored. After rearrangement, the following is obtained:

$$p_1^{{\displaystyle \frac{\gamma +1}{\gamma }}}-p_2^{{\displaystyle \frac{\gamma +1}{\gamma }}}={\displaystyle \frac{p_1^{\frac{1}{\gamma }}}{{\rho }_1}}{\displaystyle \frac{\gamma +1}{\gamma }}{\displaystyle \frac{Q^2}{A^2}}{\displaystyle \frac{\lambda }{2D}}L$$

(12)

From Equation (12), the gas resistance of the pipeline under high-speed flow can be derived. This is represented by the symbol $R_{g^{\prime }}$:

$$R_{g\prime }={\displaystyle \frac{p_1-p_2^{\frac{\gamma +1}{\gamma }}/p_1^{\frac{1}{\gamma }}}{Q^2}}$$

(13)

$$R_{g\prime }={\displaystyle \frac{\gamma +1}{\gamma }}{\displaystyle \frac{8\lambda L}{{\rho }_1{\pi }^2{D}^5}}$$

(14)

Equation (13) is the definition formula for pipeline gas resistance under high-speed flow, whereas Equation (14) is the determination formula for pipeline gas resistance at high speeds.

2.2. Pipeline Gas Resistance Series Connection and Modelling

When two or more pipes are axially connected, they form a series gas resistance pipe. The following provides a brief analysis of two series cases:

2.2.1. Both Inner Pipe Diameters = $D$

It is assumed that there is no leakage at the connection between the two gas resistance pipes and no additional pressure loss at the connector. In this case, an air-resistant pipe of length $L_1$ combined with a gas resistance pipe of length $L_2$ constitutes a gas resistance pipe of a total length of $L_1+L_2$ in physical structure. Neglecting the influence of connectors, and following the analysis method for gas resistance pipelines, the series gas resistance is exactly the sum of the gas resistances of the two pipes.

The determination formulas for series gas resistance at low and high Mach numbers are provided in Equations (15) and (16), respectively. The order of the two pipelines in the series connection does not affect the total gas resistance of the system, which aligns with the series rule in electrical systems.

$$R_g={\displaystyle \frac{8\lambda (L_1+L_2)}{\rho {\pi }^2{D}^5}}$$

(15)

$$R_{g\prime }={\displaystyle \frac{\gamma +1}{\gamma }}{\displaystyle \frac{8\lambda (L_1+L_2)}{{\rho }_1{\pi }^2{D}^5}}$$

(16)

2.2.2. Inner Pipe Diameters Differ

It is assumed that the gas resistance pipe, $R_{g1}$, with an inner diameter of $D_1$, and the gas resistance pipe, $R_{g2}$, with an inner diameter of $D_2$, are directly connected, and that the axes of the two gas resistance pipes are aligned, as shown in Figure 2.

The series gas resistance must also be analysed in two cases based on the gas flow rate.

2.2.3. Ma < 0.3

In this case, the flow can be regarded as an incompressible fluid. The Bernoulli equation between sections 1-1 and 2-2 in Figure 2 is the same as in Equation (1). Owing to the difference in the inner diameters of the pipelines, in addition to the frictional head loss,$h_f$, the local head loss, $h_j$, must also be considered. Thus, the total head loss, $h_w$, is composed of the frictional head loss, $h_{f1}$, of the gas resistance pipeline, $R_{g1}$, the frictional head loss, $h_{f2}$, of the gas resistance pipeline, $R_{g2}$ and the local head loss, $h_j$, expressed as follows:

$$h_{f1}={\lambda }_1{\displaystyle \frac{L_1{v}_1^2}{2D_{1}g}}$$

(17)

$$h_{f2}={\lambda }_2{\displaystyle \frac{L_2{v}_2^2}{2D_{2}g}}$$

(18)

$$h_j=\zeta {\displaystyle \frac{v_1^2}{2g}}$$

(19)

where $L_1$ is the length of the gas resistance pipeline, $R_{g1}$, $L_2$ is the length of the gas resistance pipeline, $R_{g2}$ and $\zeta$ is the local resistance coefficient, determined by the inner diameters of the two pipelines.

For the structure shown in Figure 2, $\zeta$ can be expressed as follows:

$$\zeta ={\left(1-{\displaystyle \frac{D_1^2}{D_2^2}}\right)}^2$$

(20)

Note that the inner diameters of the two pipelines are not the same. For incompressible fluids, the density remains constant. Based on the mass conservation law, the gas velocity in the two pipelines will differ, resulting in varying frictional resistance coefficients for the two pipelines, which must be calculated separately.

Substituting Equations (17)–(19) into Equation (1), the pressure difference between the two sections can be expressed as follows:

$$\mathrm{\Delta }p={\displaystyle \frac{\rho }{1}}\left[{\displaystyle \frac{{\lambda }_1{L}_1{v}_1^2}{2D_1}}+{\displaystyle \frac{{\lambda }_2{L}_2{v}_2^2}{2D_2}}+{\displaystyle \frac{v_2^2+(\zeta -1)v_1^2}{2}}\right]={\displaystyle \frac{8Q^2}{\rho }}\left({\displaystyle \frac{{\lambda }_1{L}_1}{{\pi }^2{D}_1^5}}+{\displaystyle \frac{{\lambda }_2{L}_2}{{\pi }^2{D}_2^5}}\right.\left.+{\displaystyle \frac{\zeta }{{\pi }^2{D}_1^4}}+{\displaystyle \frac{1}{{\pi }^2{D}_2^4}}-{\displaystyle \frac{1}{{\pi }^2{D}_1^4}}\right)$$

(21)

By rearranging the terms in Equation (21), the determination formula for the gas resistance of the pipe in series at low speeds can be obtained.

$$R_g={\displaystyle \frac{8}{\rho }}\left({\displaystyle \frac{{\lambda }_1{L}_1}{{\pi }^2{D}_1^5}}+{\displaystyle \frac{{\lambda }_2{L}_2}{{\pi }^2{D}_2^5}}\right.\left.+{\displaystyle \frac{\zeta }{{\pi }^2{D}_1^4}}+{\displaystyle \frac{1}{{\pi }^2{D}_2^4}}-{\displaystyle \frac{1}{{\pi }^2{D}_1^4}}\right)$$

(22)

where the first and second terms in the right bracket represent the gas resistance values of the pipelines $R_{g1}$ and $R_{g2}$, respectively. The third term accounts for the local head loss caused by the difference in the inner diameters of the pipelines, and the fourth and fifth terms represent the kinetic energy changes caused by the different inner diameters. The last three terms depend solely on the structural parameters of the pipeline and are independent of the flow velocity.

Compared with the last three terms, the first two include an additional length-to-diameter ($L/D$) ratio. In practical pipeline structures, this ratio is typically greater than 10. Therefore, the series resistance value mainly depends on the first two terms in the determination formula, which correspond to the resistance values of the two series’ gas resistances.

When the inner diameters of the two pipelines are equal, the local resistance coefficient, $\zeta$, becomes 0, and Equation (22) simplifies to the sum of the gas resistances of the two pipelines. This result is consistent with Equation (15) in Case 1.

2.2.4. Ma > 0.3

Referring to the analysis process for pipeline gas resistance in incompressible fluids, the integral form of the series gas resistance Bernoulli equation can be expressed as follows:

$${\displaystyle \frac{A^2}{Q^2}}{C}^{-{\displaystyle \frac{1}{\gamma }}}{\displaystyle \frac{\gamma }{\gamma +1}}\left({p_1}^{{\displaystyle \frac{\gamma +1}{\gamma }}}-p_2^{{\displaystyle \frac{\gamma +1}{\gamma }}}\right)={\displaystyle \frac{{\lambda }_1{L}_1}{2D_1}}+{\displaystyle \frac{{\lambda }_2{L}_2}{2D_2}}+{\displaystyle \frac{\zeta }{2}}-{\displaystyle \frac{1}{\gamma }}\ln {\displaystyle \frac{p_2}{p_1}}$$

(23)

By rearranging the terms in Equation (23), the determination formula for the series gas resistance during high-speed flow can be derived.

$$R_{g\prime }={\displaystyle \frac{8(\gamma +1)}{\gamma {\rho }_1}}\left({\displaystyle \frac{{\lambda }_1{L}_1}{{\pi }^2{D}_1^5}}+{\displaystyle \frac{{\lambda }_2{L}_2}{{\pi }^2{D}_2^5}}\right.\left.+{\displaystyle \frac{\zeta }{{\pi }^2{D}_1^4}}-{\displaystyle \frac{1}{\gamma }}\ln {\displaystyle \frac{p_2}{p_1}}\right)$$

(24)

When the inner diameters of the two series pipelines are equal, the local resistance coefficient, $\zeta$, becomes 0. In this case, the logarithmic term, $\ln \left(p_2/p_1\right)$, in Equation (24), is negligible and can be ignored. Consequently, Equation (24) reduces to the sum of the gas resistances of the two pipelines, which is consistent with Equation (16) in Case 2.

3. Simulation Analysis of Pipeline Gas Resistance Value

3.1. Simulation Model and Boundary Conditions

To verify the accuracy of pipeline gas resistance modelling, COMSOL software was used to model and simulate a single gas resistance pipeline under low-Mach-number conditions. Owing to the simple structure of the model, a free tetrahedral mesh was used for partitioning. For instance, a gas resistance pipeline model with an inner diameter of 5 mm and a length of 500 mm was divided into an average unit size of 0.70 mm, resulting in 571,281 units, as shown in Figure 3.

To minimise the impact of grid size on the simulation results, the model was validated for grid size independence. Five different grid sizes were used for calculations, and the results are presented in

Table 1. Grid size independence test results.

Average Unit Size (mm)	Q (kg/s)	$\overline{\mathit{v}}$ (m/s)	${\mathit{v}}_{\mathit{max}}$ (m/s)
1.7374	2.593 × 10−3	114.96	116.03
1.3353	2.419 × 10−3	107.23	111.48
0.8955	2.194 × 10−3	94.42	96.96
0.7000	2.085 × 10−3	89.28	90.66
0.4888	2.045 × 10−3	86.98	92.71

. When the average unit size was less than 0.7 mm, there were minimal differences in the calculation results for the average flow velocity, $\overline{v}$, maximum flow velocity, $v_{max}$, and mass flow rate, $Q$.

The standard $k-\omega$ model for turbulence in fluid flow was selected, and steady-state research was conducted for the physical field. Gaseous air was chosen as the material with default material properties. The two sections of the pipeline were configured with inlet pressure and outlet pressure conditions (atmospheric pressure), while all other surfaces were set as non-slip wall surfaces. Surface roughness was not considered, and the fluid temperature was set to 293.15 K.

3.2. Validation of Friction Factor Fitting and Gas Resistance Model

The friction factor, $\lambda$, in the determination formula for gas resistance, must either be obtained experimentally or calculated using empirical formulas. However, as Wang et al. [24] noted, when derived from empirical formulas, $\lambda$ is not universally applicable to all fluid media and can deviate significantly, with errors sometimes exceeding 50%, compared with experimentally obtained friction factors.

To address this issue, the $\lambda$ for air in a circular tube under a low Mach number was determined via simulation and curve fitting. By substituting the definition Formula (6) into the determination Formula (7), the value of $\lambda$ was calculated. A pipeline with an inner diameter ranging from 1 to 6 mm and a length of 0.5 m was simulated, with the results illustrated in Figure 4.

Owing to the small dynamic viscosity coefficient of gas, most of its flow within the pipeline occurs in the hydraulically smooth pipe region. Consequently, the friction factor, $\lambda$, is primarily related to the Reynolds number, $Re$. As shown in Figure 4, although $\lambda$ varies with different gas resistance pipeline inner diameters, the overall trend of its variation with $Re$ exhibits good consistency. Referring to the classical empirical formula for $\lambda$, the calculation formula for the $\lambda$ of air in a circular tube is given as $\lambda =a\cdot R{e}^{-b}$. The best estimated values of coefficients $a$ and $b$ were obtained by fitting $\lambda$ to $Re$ using the least-squares method. Therefore, the formula for calculating $\lambda$ of air in a circular tube is expressed as follows:

$$\lambda =6.426R{e}^{-0.5912}$$

(25)

To verify the accuracy of the proposed gas resistance model, simulations were conducted on pipelines with inner diameters of 1 and 6 mm at a length of 1 m. Equation (25) was substituted into Equation (6) to calculate their gas resistance, which was then compared with the gas resistance values calculated using Equation (5). The results, shown in Figure 5, indicate that the relative error between the gas resistance values calculated using Equations (5) and (6) remained within 10%, confirming that the proposed gas resistance model at low Mach numbers is accurate.

3.3. Simulation Results of Series Gas Resistance

Using the proposed series gas resistance model, simulations were conducted to examine two cases of series gas resistance under low-Mach-number conditions, employing the same grid division method. Case 1 involved a series gas resistance model composed of pipes with inner diameters of 2 and 3 mm, both with lengths of 0.5 m. Case 2 consisted of pipes with inner diameters of 1 and 6 mm, also with lengths of 0.5 m. The deterministic gas resistance values were calculated using Equation (22).

Taking Case 1 as an example, the determination formula for series gas resistance includes five terms, and their values are shown in

Table 2. Numerical values of the terms in the series gas resistance determination formula for Case 1.

Term 1	Term 2	Term 3	Term 4	Term 5
9.61 × 1011	1.63 × 1011	1.30 × 1010	8.33 × 109	−4.21 × 1010
6.38 × 1011	1.08 × 1011	1.30 × 1010	8.33 × 109	−4.21 × 1010
5.02 × 1011	8.50 × 1010	1.30 × 1010	8.33 × 109	−4.21 × 1010
4.23 × 1011	7.17 × 1010	1.30 × 1010	8.33 × 109	−4.21 × 1010
3.71 × 1011	6.28 × 1010	1.30 × 1010	8.33 × 109	−4.21 × 1010
3.33 × 1011	5.64 × 1010	1.30 × 1010	8.33 × 109	−4.21 × 1010
3.04 × 1011	5.15 × 1010	1.30 × 1010	8.33 × 109	−4.21 × 1010
2.81 × 1011	4.76 × 1010	1.30 × 1010	8.33 × 109	−4.21 × 1010
2.62 × 1011	4.44 × 1010	1.30 × 1010	8.33 × 109	−4.21 × 1010
2.46 × 1011	4.17 × 1010	1.30 × 1010	8.33 × 109	−4.21 × 1010

. The results indicate that the pressure difference increases from top to bottom under varying pressure conditions. The sum of the five terms in each row corresponds to the series gas resistance value at the respective pressure difference.

From Table 2, it can be seen that because the last three terms depend on the structural parameters of the pipeline and are unaffected by the flow state, their influence on the series gas resistance value increases with the pressure difference across the series pipeline. For example, in Case 1, when the pressure difference is 1.5 kPa, the last three terms account for only 3% of the series gas resistance. However, when the pressure difference increases to 34 kPa, the contribution of these terms rises to 13%.

When the inner diameters of the two series gas resistance pipelines differ significantly, the impact of the last three terms on the series gas resistance value is minimal. Using the same method, the terms in the determination formula for Case 2 were calculated. At a pressure difference of 92 kPa, the last three terms had the most significant influence on the series gas resistance value, yet they still accounted for only 0.3% of the total series gas resistance value.

The gas resistance values were calculated using both the definition Formula (5) and the determination Formula (22). The gas resistance value–pressure difference relationship curves were plotted based on the calculation results, as shown in Figure 6. The results indicate that the gas resistance value of the series pipeline decreases as the pressure difference at its ends increases. Overall, the gas resistance values calculated using the definition Formula (5) were consistent with those calculated using the determination Formula (22), with discrepancies appearing only at small pressure differences.

In Case 1, the maximum relative error was 6.8% at a pressure difference of 1.5 kPa, whereas the relative error at other points was less than 3.6%. In Case 2, the maximum relative error was 13.4% at a pressure difference of 4.4 kPa, with errors at other points below 7.8%. These discrepancies can be attributed to the low Reynolds number at lower pressures, which introduces larger errors in the friction factor, $\lambda$, calculated from $Re$. These results demonstrate that the proposed low-Mach-number series gas resistance model is suitable for pipe combinations with inner diameters ranging from 1 to 6 mm.

4. Experiment and Result Analysis

4.1. Gas Resistance Experimental Device and Process

The experimental setups shown in Figure 7 and Figure 8 were designed for the gas resistance experiment and the series gas resistance experiment, respectively. The pipeline to be measured and the sensor installation module were connected via threaded joints. The sensor installation module was flexibly connected to the electrical proportional valve and flow metre using quick connectors, and the measuring device was placed on a vibration isolation platform.

The air source consisted of a two-stage series closed-loop stabilised pressure system, capable of continuously and stably providing compressed air with a maximum pressure of 0.5 MPa. The air supply pressure was controlled by the electric proportional valve to adjust the pressure difference at both ends of the gas resistance. The pressure at both ends of the gas resistance was measured using a micro pressure sensor, and the mass flow rate through the gas resistance was measured using a flow metre, ensuring that the flow rate remained within the low-Mach-number range. The micro pressure sensor model utilised was HM91, with a range of 0–0.5 MPa and an accuracy of ±0.1%. The flow metre model employed was HK25, with a range of 0–8 m³/h and an accuracy of 0.75%. The electrical proportional valve model utilised was ITV2050, with a pressure regulation range of (0.005 to 0.9) MPa and a sensitivity better than 0.2% F.S. The oil mist separator model employed was AFM30, with a precision of 0.5 μm. The ambient temperature during the experiments was 18.3 °C.

4.2. Analysis of Experimental Results of Gas Resistance

In Figure 7, the measured pipe was a cold-rolled seamless 304 stainless steel round pipe with an inner diameter of 6 mm and a length of 0.5 m. The mass flow through the measured pipe was obtained under different pressure differences. The experimental gas resistance value was calculated using Equations (5) and (6), and the gas resistance value–pressure difference relationship curve was plotted, as shown in Figure 9a.

An analogue simulation model of the experimental device was constructed in COMSOL, and the simulated gas resistance values were calculated using Equation (6). The relationship curve between the gas resistance value and the pressure difference is shown in Figure 9b.

Figure 9 shows that the gas resistance values calculated using Equation (6) closely matched those calculated using Equation (5), with a maximum relative error of 4.8%. The good consistency between the two curves demonstrates that the friction factor calculation formula is suitable for circular pipes with an inner diameter of 6 mm. As the pressure difference between the two ends of the gas resistance pipeline increased, both the experimental and simulated gas resistance values showed a gradual decreasing trend, with a maximum relative error of 3.5% between the two.

The error between the experimental and simulated gas resistance values arises mostly from practical issues, such as the gap between the measured gas resistance pipeline and the sensor installation module. Although the experimental device was designed to avoid damaging the pipeline flow channel structure, inevitable gaps create additional cavities that form a series gas resistance. These gaps are not accounted for in the simulation model or in Equation (6).

Because the assembly gap length is smaller and the inner diameter is larger than those of the gas resistance pipe, the gas resistance value of the assembly gap is smaller than that of the pipe. As a result, the gas resistance value calculated using Equation (6) is slightly lower than that calculated using Equation (5). Nevertheless, the good agreement between the experimental and simulated gas resistance values in the pressure difference curve demonstrates that the proposed low-Mach-number gas resistance model and formula for the coefficient of friction fitted are suitable for pipelines with an inner diameter of 6 mm.

4.3. Analysis of Experimental Results of Series Gas Resistance

The device shown in Figure 8 was used to conduct the series gas resistance measurement experiment. The lengths of the two series pipes were 0.5 m each, with inner diameters of 4 and 6 mm. From the gas resistance experiment analysis, it can be observed that there is a noticeable difference between the experimental flow channel and the simulation model of the experimental device.

In the series gas resistance experiment, the increased number of connectors made it challenging to build a simulation model that accurately represented the experimental pipeline flow channel. Additionally, it was difficult to strike a balance between the simulation model’s accuracy and computational efficiency. The experimental gas resistance was calculated using Equations (5) and (22), and the gas resistance–pressure difference relationship curve was plotted, as shown in Figure 10.

From Figure 10, it can be observed that the gas resistance values calculated using Equations (5) and (22) decrease as the pressure difference at both ends of the gas resistance increases. The maximum relative error between the gas resistance values calculated using the two methods is 3.8%, primarily due to the connection between the two gas resistance pipes.

The analysis of simulated series gas resistance shows that, for series gas resistance, the influence of frictional resistance loss on gas resistance is significantly greater than that of local resistance loss and kinetic energy variation. Owing to the complexity of the flow channel at the connection points, accurately calculating the local resistance loss is challenging. When using the determination formula to calculate gas resistance, only the frictional resistance loss at the connection points is considered.

The relative error between the gas resistance values calculated using both methods remained within 4% during the experiment, demonstrating that the proposed series gas resistance model for low Mach numbers is applicable to series pipelines composed of pipes with inner diameters of 4 and 6 mm. Due to the limitation of the experimental conditions, this paper does not conduct experiments on single and series gas resistance under high-Mach conditions.

5. Conclusions

Inspired by Ohm’s law on electricity, a concept of pipeline gas resistance was proposed. Using the Bernoulli equation, gas resistance and series gas resistance models for pipelines at both low and high Mach numbers were developed and verified through simulations and experiments. Based on the simulation results, the friction factor, $\lambda$, of gas resistance pipelines with inner diameters ranging from 1 to 6 mm at low Mach numbers was fitted, and the accuracy of the $\lambda$ expression for this diameter range was confirmed. Unlike electrical resistance, the gas resistance value decreases as the pressure difference at both ends of the pipeline increases.

The experimental results indicate that the maximum relative error between the theoretical analysis and simulation results is 3.1%, whereas the maximum relative error between the definition formula and determination formula is 3.8%. These findings confirm that the theoretical model is effective and reliable, providing strong theoretical support for the impedance analysis of aerostatic systems.

There are limitations in this study. The inner diameter of the air resistance pipeline studied in this paper is in millimetres, but there are structures with smaller inner diameters in the aerostatic pressure system, and whether the flow state of the gas in these structures will change is not discussed in depth in this paper. It is believed that after the gas resistance theory is perfected in the future, it will become an important method for the analysis and design of gas static pressure systems.