4.1. Main definitions

The mass flow rate, $\dot M$ [kg s $^{-1}$ ], through a tube connecting two tanks can be related to the pressure variation in both tanks. As shown in Sharipov & Graur (Reference Sharipov and Graur2014), the mass flow rates are the same in the capillary inlet and outlet when the capillary volume is significantly smaller than that of the tanks. The volume of the tube used in the set-up, is $4.7\times10^{-3}$ cm $^3$ , that is much smaller than the tank volume. Therefore, the mass flow rate can be related to the inlet pressure variation as

(4.1) \begin{align} \dot M=-\frac {V_{\textit{in}} {\mathcal M}}{T{\mathcal R}_g} \frac {\mbox{d} p_{\textit{in}}}{\mbox{d} t}, \end{align}

where $T$ is the gas temperature, ${\mathcal R}_g=8.314$ J K $^{-1}\,$ mol $^{-1}$ is the universal gas constant, $\mathcal M$ is the gas molar mass, $p_{\textit{in}}$ is the pressure in the inlet reservoir, $t$ is the time. This expression is derived under several assumptions: (i) the ideal gas law is valid; (ii) the volume $V_{\textit{in}}$ remains constant; (iii) the gas flow is isothermal. To quantify the deviation from ideal gas law conditions, the compressibility factor $Z=pV{\mathcal M}/(M{\mathcal R}_g T)$ is usually introduced (Reid, Prausnitz & Poling Reference Reid, Prausnitz and Poling1987). Using the CoolProp library (Bell et al. Reference Bell, Wronski, Quoilin and Lemort2014) with the specified pressure and temperature values, the compressibility factor $Z$ was calculated. In all cases, the relative deviation of $Z$ from unity was smaller than 0.0001 $\,\%$ , indicating that non-ideal gas effects are negligible, so the first assumption is valid. The second assumption is also fulfilled for the used set-up. The influence of the temperature variation is estimated via the quantity $\varepsilon =(\Delta T/T)/(\Delta p/p)$ . During the experiments, the maximum temperature variation is approximately 0.3 $^{\circ }$ C, leading to $\Delta T/T\sim 10^{-3}$ , while the pressure variation is large, i.e. $\Delta p/p \sim 1$ . Therefore, $\varepsilon$ is of the order of $10^{-3}$ , hence, the temperature variation can be neglected. Thus, the expression (4.1) is applicable to measuring the mass flow rate in the whole range of the pressure considered in the present work.

Theoretical results on the mass flow rate are, usually, reported in term of the Poiseuille coefficient $G_{\kern-1pt p}$ (Sharipov Reference Sharipov2016). For a long tube ( $a\ll L$ ), the mass flow rate is related to the Poiseuille coefficient $G_{\kern-1pt p}$ as described in Sharipov & Seleznev (Reference Sharipov and Seleznev1994) and Sharipov (Reference Sharipov1997),

(4.2) \begin{align} \dot M=-\frac {\pi a^3}{v{_{{m}}}}G_{\kern-1pt p}(\delta ) \frac {\mbox{d} p}{\mbox{d} x}, \end{align}

where $v{_{{m}}}$ is the most probable molecular speed, defined as

(4.3) \begin{align} v{_{{m}}}=\sqrt {\frac {2R_gT}{\mathcal M}}, \end{align}

where $p$ is the local gas pressure, $x$ is a coordinate along the tube, so $\mbox{d} p/\mbox{d} x$ is a local pressure gradient. The Poiseuille coefficient $G_{\kern-1pt p}(\delta )$ depends on the local rarefaction parameter $\delta$ defined as

(4.4) \begin{align} \delta =\frac {a}{\ell },\quad \ell =\frac {\mu v_m}{p}, \end{align}

where $\ell$ is the equivalent molecular free path. Considering the maximum pressure levels used in the experiments, the gas viscosity $\mu$ is assumed to depend solely on temperature (Ferziger & Kaper Reference Ferziger and Kaper1972; Shan et al. Reference Shan, Li, Tang, Torrilhon, Zhang and Guo2025) and is calculated from

(4.5) \begin{align} \mu (T)=\mu _{\textit{ref}}\left (\frac {T}{T_{\textit{ref}}}\right )^{\omega }\!, \end{align}

where $\mu _{\textit{ref}}$ is the gas viscosity at the reference temperature $T_{\textit{ref}}$ and $\omega$ is the viscosity index. According to Sharipov (Reference Sharipov2024), the viscosity index $\omega$ is variable. However, when the temperature variation is small compared with the reference temperature $T_{\textit{ref}}$ , which is the case of our experimental conditions, (4.5) holds. The dependence of the viscosity on the gas density is characterized by the viscosity virial coefficient. According to Berg & Burton (Reference Berg and Burton2013), the relative effect of the density on viscosity does not exceed $10^{-5}$ under the experiment conditions. The values of $\mu _{\textit{ref}}$ and $\omega$ based on ab initio calculations, (Sharipov & Benites Reference Sharipov and Benites2019, Reference Sharipov and Benites2021) and empirical representations (Cole & Wakeham Reference Cole and Wakeham1985; Kadoya, Matsunaga & Nagashima Reference Kadoya, Matsunaga and Nagashima1985), are given in table 1, together with the molar mass (Meija et al. Reference Meija2016).

Table 1.

Molar mass $\mathcal M$ (Meija et al. Reference Meija2016), viscosities $\mu _{\textit{ref}}$ at the reference temperature $T_{\textit{ref}}=300$ K and viscosity index $\omega$ based on the data reported in Sharipov & Benites (Reference Sharipov and Benites2019, Reference Sharipov and Benites2021), Cole & Wakeham (Reference Cole and Wakeham1985), and Kadoya et al. (Reference Kadoya, Matsunaga and Nagashima1985).

By integrating (4.2) along the tube and using the mass conservation one obtains

(4.6) \begin{align} \dot M=\frac {\pi a^3}{v{_{{m}}}L} \int _{p_{\textit{out}}}^{p_{\textit{in}}}G_{\kern-1pt p}(\delta ) \mbox{d} p. \end{align}

A combination of (4.1) and (4.6) leads to

(4.7) \begin{align} \frac {\mbox{d} p_{\textit{in}}}{\mbox{d} t}=-\frac {1}{\tau } \int _{p_{\textit{out}}}^{p_{\textit{in}}}G_{\kern-1pt p}(\delta ) \mbox{d} p, \end{align}

where $\tau$ is the characteristic time of the variation of $p_{\textit{in}}$ :

(4.8) \begin{align} \tau =\frac {2 L V_{\textit{in}}}{\pi a^3 v{_{{m}}}}. \end{align}

Therefore,(4.7) serves as the basic relation for extracting the Poiseuille coefficient from the inlet pressure variation.

4.2. Theoretical data on the Poiseuille coefficient

The Poiseuille coefficient for a long tube has been a subject of extensive study in recent decades (Sharipov & Seleznev Reference Sharipov and Seleznev1998). Here, an assess of the numerical uncertainty of $G_{\kern-1pt p}$ is given.

Figure 2.

Relative difference $\Delta G_{\kern-1pt p}$ versus rarefaction parameter $\delta$ : blue line, between S-model and Bhatnagar–Gross–Krook (BGK) model $\Delta G_{\kern-1pt p}=(G_{\kern-1pt p}^{S}/G_{\kern-1pt p}^{\textit{BGK}}-1)\boldsymbol{\cdot }100\,\%$ ; red symbols, between S-model and LBE $\Delta G_{\kern-1pt p}=(G_{\kern-1pt p}^{S}/G_{\kern-1pt p}^{\textit{LBE}}-1)\times100\,\%$ ; black line, relative error of approximation (5.1).

The Poiseuille coefficient was obtained in Cercignani & Sernagiotto (Reference Cercignani and Sernagiotto1966) applying the BGK model (Bhatnagar, Gross & Krook Reference Bhatnagar, Gross and Krook1954). The same data based on the S-model (Shakhov Reference Shakhov1968) are reported in Sharipov (Reference Sharipov1996). The linearized Boltzmann equation (LBE) with the hard sphere potential was solved to calculate $G_{\kern-1pt p}$ for a tube in Wu et al. (Reference Wu, Zhang, Liu, Zhang and Reese2017). The relative differences of the results based on the BGK model and LBE from those based on the S-model are plotted in figure 2, which shows that the maximum discrepancy of 2 % between these results is observed in the transitional regime ( $\delta =1$ ). In the limits $\delta \to 0$ and $\delta \to \infty$ , the discrepancy vanishes. Theoretical data on $G_{\kern-1pt p}$ for tube based on ab initio or Lennard–Jones potential are not available. In order to assess the influence of the potential, the data on the orifice flow by Sharipov & Strapasson (Reference Sharipov and Strapasson2014) at a small pressure drop can be used. These results point out that the flow rate based on ab initio potential is just slightly larger than that based on the hard sphere potential. The deviation does not exceed 1.4 %, i.e. the real Poiseuille coefficient should be between the values obtained from the S-model and those obtained from LBE for the hard sphere potential. Therefore, the red symbols in figure 2 indicate the theoretical uncertainty of the Poiseuille coefficient for the tube flow.

5.1. Main methods

The relation (4.7) allows us to express the rate of the pressure variation via the average Poiseuille coefficient in the pressure range from $p_{out}$ to $p_{in}$ . However, it is not trivial to express $G_{\kern-1pt p}$ at a specific value of the rarefaction $\delta$ via the pressure derivative.

The usual approach to this problem is to approximate the integral on the right-hand side of (4.7) as (Sharipov & Seleznev Reference Sharipov and Seleznev1994)

(5.1) \begin{align} \int _{p_{\textit{out}}}^{p_{\textit{in}}}G_{\kern-1pt p}(\delta ) \mbox{d} p\approx \left (p_{\textit{in}}-p_{\textit{out}}\right )G_{\kern-1pt p}\left (\delta _m\right ), \end{align}

where the rarefaction parameter $\delta _m$ is calculated from (4.4) using the mean pressure $p_m=(p_{in}+p_{out})/2$ . The error of this approximation is plotted in figure 2 by the black line, which shows that it reaches its maximum of 1.5 % at $\delta _m=2$ . Then it vanishes in both limits $\delta \to 0$ and $\delta \to \infty$ . Assuming that $p_{out}\ll p_{in}$ , a combination of (4.7) and (5.1) leads to

(5.2) \begin{align} G_{\kern-1pt p}(\delta _{in}/2)=-\tau \frac {\mbox{d} \ln p_{in}}{\mbox{d} t}. \end{align}

Another approach (Sharipov & Graur (Reference Sharipov and Graur2025) is based on the assumption that the outlet pressure $p_{\textit{out}}$ remains constant during the measurement period. This condition can be met either if the outlet reservoir volume is significantly larger than the inlet volume, or if the outlet reservoir is continuously pumped down to maintain a constant pressure. First, we calculate the derivative of both sides of (4.7) which leads to

(5.3) \begin{align} \tau \frac {\mbox{d}^2 p_{\textit{in}}}{\mbox{d} t^2}=-\frac {\mbox{d} p_{\textit{in}}}{\mbox{d} t}G_{\kern-1pt p}(\delta _{\textit{in}})+\frac {\mbox{d} p_{\textit{out}}}{\mbox{d} t}G_{\kern-1pt p}(\delta _{\textit{out}}). \end{align}

Then, the term with a derivative in time of the outlet pressure, $\mbox{d} p_{\textit{out}}/\mbox{d} t$ , is neglected under condition of constancy of the outlet pressure. Therefore, the expression of the Poiseuille coefficient is obtained as a function of the inlet pressure derivatives and the characteristic time

(5.4) \begin{align} G_{\kern-1pt p}(\delta _{\textit{in}})=-\tau \frac {\mbox{d}^2 p_{\textit{in}}}{\mbox{d} t^2}\left (\frac {\mbox{d} p_{\textit{in}}}{\mbox{d} t}\right )^{-1}\!. \end{align}

The advantage of this method is that it does not rely on any approximation. However, it requires to calculate the second derivative that is not easy, when the pressure $p_{in}$ is given as a table of measurement data.

5.2. Viscous and slip regime

In the case of viscous flow regime the Poiseuille coefficient is equal to

(5.5) \begin{align} G_{\kern-1pt p}(\delta )=\delta /4. \end{align}

In this case, (5.2) is easily integrated and the evolution of the inlet pressure with respect to the time is obtained as

(5.6) \begin{align} p_{\textit{in}}^{v\textit{is}}(t)=\frac {p_{\textit{in0}}}{At+1}, \end{align}

where

(5.7) \begin{align} A=\frac {\delta _{\textit{in0}}}{8 \tau }. \end{align}

Here $\delta _{\textit{in0}}$ is the rarefaction parameter, (4.4), calculated with initial inlet pressure $p_{\textit{in0}}$ at $t=0$ .

In the case of the slip flow regime ( $\delta _{\textit{in}}\gt 10$ ), the Poiseuille coefficient is obtained analytically as in Sharipov (Reference Sharipov2016),

(5.8) \begin{align} G_{\kern-1pt p}(\delta )=\delta /4+\sigma_{\kern-1pt p}, \end{align}

where $\sigma_{\kern-1pt p}$ is the slip coefficient (Sharipov Reference Sharipov2011). Then, (5.2) becomes

(5.9) \begin{align} \frac {1}{p_{\textit{in}}}\frac {\mbox{d} p_{\textit{in}}}{\mbox{d} t}=-\frac {1}{\tau }\left (\frac {\delta _{\textit{in}}}{8}+\sigma_{\kern-1pt p}\right )\!. \end{align}

By integrating (5.9) with respect to the time, the evolutions of the pressure $p_{\textit{in}}$ is obtained as

(5.10) \begin{align} p_{\textit{in}}^{\textit{slip}}(t)=\frac {p_{\textit{in0}}}{(B+1)e^{ct}-B}, \end{align}

where

(5.11) \begin{align} B=\frac {\delta _{\textit{in0}}}{8\sigma_{\kern-1pt p}}, \end{align}

and

(5.12) \begin{align} c=\frac {\sigma_{\kern-1pt p}}{\tau }. \end{align}

These expressions can be used to extract the tube radius $a$ and the velocity slip coefficient $\sigma_{\kern-1pt p}$ from the experimental data on $p_{\textit{in}}$ for large values of the rarefaction parameter. In this case, the quantities $B$ and $c$ are used as fitting parameters.

The experimental value of the relaxation time, $\tau ^{\textit{exp}}$ , is defined from the fit in the viscous regime as

(5.13) \begin{align} \tau ^{\textit{exp}}=\frac {\delta _{\textit{in0}}}{8A}. \end{align}

In § 6, the theoretical value of the relaxation time given by (4.8) is compared with the experimental one, (5.13). Additionally, when the relaxation time is normalized by the characteristic time defined as $t{^{{ch}}}=L/v{_{{m}}}$ it becomes to be independent from the gas nature.

Finally, the Poiseuille coefficient in the slip flow regime is extracted from (5.2). The pressure variation in the inlet tank is recorded and fitted using the form given in (5.10), with $B$ and $c$ as fitting parameters. The time derivative of the pressure is then calculated analytically with these fitted parameters and $G_{\kern-1pt p}$ is calculated from (5.2). The results are presented in § 6.

5.3. Transitional regime

To extract the Poiseuille coefficient from (5.2) and (5.4) we need to calculate the first and second derivatives of the measured pressure with respect to time. To calculate them, the measured pressure is locally represented around a point $t_i$ , within the interval $[t_i-\Delta t, t_i+\Delta t]$ , by means of a nonlinear regression of the form

(5.14) \begin{align} p_{\textit{in}}(t)=C_0\exp \left (\sum _{n=1}^4 C_n(t-t_i)^n\right )\!, \end{align}

where $C_n$ ( ${1}\le n\le 4$ ) are coefficients to estimate, obtained by the least square method. Then, the first and second derivatives read

(5.15) \begin{align} \frac {\mbox{d} p_{\textit{in}}}{\mbox{d} t}\Big |_{t=t_i}=C_0 C_1, \\[-28pt] \nonumber \end{align}

(5.16) \begin{align} \frac {\mbox{d}^2 p_{\textit{in}}}{\mbox{d} t^2}\Big |_{t=t_i}=C_0 \big(C_1^2+2C_2 \big). \\[2pt] \nonumber \end{align}

When, the approximation (5.1) is used, (5.15) is substituted into (5.2) and the Poiseuille coefficient is calculated by the first method as

(5.17) \begin{align} G_{\kern-1pt p}(\delta {_{{i}}}/2)=-\tau C_1. \end{align}

Substituting (5.15) and (5.16) into (5.4), the coefficient $G_{\kern-1pt p}$ is calculated by the second method as

(5.18) \begin{align} G_{\kern-1pt p}(\delta {_{{i}}})=-\tau \left (C_1+2\frac {C_2}{C_1}\right )\!, \end{align}

where the rarefaction parameter $\delta {_{{i}}}$ is calculated via (4.4) using $C_0$ for pressure $p_{\textit{in}}(t{_{{i}}})$ .

Finally, expression (5.18) is more correct from theoretical point of view because allows to obtain the expression for the Poiseuille coefficient directly from (4.7) without any additional approximation, required for derivation of (5.17). In § 6, two expressions will be used to extract the Poiseuille coefficient values from the pressure measurements.

5.4. Experimental uncertainty

The experimental uncertainties associated with the measurements performed using the set-up arise from the inlet volume ( $u_V$ ), tube radius ( $u_a$ ), tube length ( $u_L$ ), time ( $u_t$ ), gas temperature ( $u_T$ ) and gas pressure ( $u_p$ ). All these uncertainties are summarized in table 2. The uncertainty of the pressure sensors ( $u_p$ ) depends on their measurement range. For the 10 Torr sensor,

(5.19) \begin{align} u_p=\left \{ \begin{array}{ccc} 0.50\,\% & \mbox{at} & 0.1\le p \le 0.25,\\ 0.25\,\% & \mbox{at} & 0.25\le p \le 1,\\ 0.20\,\% & \mbox{at} & 1\le p \le 10,\\ \end{array}\right . \end{align}

where $p$ is in Torr. The 100 and 1000 Torr sensors have the maximum error of 0.15 % of the reading, according to the manufacturer. There is no information on the minimum detectable pressure for the sensors. However, for 10 Torr sensors the oscillations occur for reading pressure below 30 Pa for lighter gases and below 10 Pa, for the heavier ones. When the amplitude of the oscillations becomes a half of the measured values, the measurement is discarded. Table 2 contains the highest uncertainty $u_p$ , which is 0.50 %.

Table 2.

Uncertainties (%) of the main measured parameters.

The mass flow rate and Poiseuille coefficient are determined using the fitting of the measured pressure data. To evaluate the fit quality, standard statistical parameters are used, including the coefficient of determination, the sum of squared residuals and the standard error ( $E{_{{s}}}$ ). Only the standard error is taken into account for the uncertainty estimation, and its highest value is reported in table 2.

The combined standard uncertainties $u$ were calculated following the standard procedure (GUM 2008) as $u=\sqrt {\sum _i u_i^2}$ . The uncertainty of the mass flow rate $\dot M$ contains $u_V$ , $u_t$ , $u_T$ and $u_p$ , so that $u=4.3\,\%$ . The Poiseuille coefficient uncertainty includes all those given in table 2 and equal to $u=5.2\,\%$ .

The uncertainty of $\tau$ is

(5.20) \begin{align} u_\tau =\sqrt {u_L^2+u_V^2+(3u_a)^2+(0.5u_T)^2}=5.2\,\%. \end{align}

According to (5.17) and (5.18), this is the main uncertainty of $G_{\kern-1pt p}$ besides that of $C_1$ and $C_2$ .