2.1. Experimental set-up

The constant volume methodology (Arkilic et al. Reference Arkilic, Schmidt and Breuer1997; Colin et al. Reference Colin, Lalonde and Caen2004; Ewart et al. Reference Ewart, Perrier, Graur and Méolans2006; Rojas Cárdenas Reference Rojas Cárdenas2012; Yamaguchi et al. Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016) is applied to measure the pressure and temperature gradient driven flows through a microchannel. Each one of these two types of experiments was performed separately, i.e. either only a pressure gradient or only a temperature gradient was applied to generate a flow through the microchannel. A scheme representing the experimental set-up is shown in figure 1. It basically consists of two tanks, upstream and downstream tanks, which are connected by the microchannel. The upstream tank, represented in figure 1 by the blue colour (tank 1), is also called the high pressure tank, in the pressure gradient experiments, or the cold tank, in the temperature gradient experiments. The downstream tank, represented by the red colour (tank 2), is also called the low pressure tank in the pressure gradient experiments, or the hot tank, in the temperature gradient experiments. Both upstream and downstream tanks are made of stainless steel and their temperatures can be controlled by circulating water (tank 1, cold side) and by an electrical heater (tank 2, hot side), respectively. The positions of the cooling and heating systems entrances are indicated in figure 2 by the numbers 12 and 13, respectively.

Figure 1.

Experimental apparatus used in the pressure and temperature gradient experiments.

Figure 2.

(a) Exploded view of the flat plate, grooved plate and tanks. (b) Tanks and microchannel plates connected to the rest of the system.

Figure 2(a) presents an exploded view of the microchannel plates and tanks. The microchannel is grooved in the inner plate made in PEEK (polyetheretherketone) and covered by another flat plate of the same material, but thicker than the first one. The interface between the two plates is provided with a flat seal. These two plates are pressed up against each other by fourteen bolts and also pressed against the tanks by eight of these fourteen bolts. To improve the seal and reduce leaks, vacuum glue is added around each external interface and white paste at the top of the threaded rod, see figure 3(a).

Figure 3.

(a) Microchannel plates, blocks and part of the pipes and connections coated with a black paint. (b) Temperature field of the system measured by the IR camera for the highest temperature difference, $\Delta T=67.5\,^{\circ }\textrm {C}$.

The microchannel has a rectangular cross-section with the following dimensions: height $H=0.24\pm 0.01\ \textrm {mm}$, width $W=1.00\pm 0.01\ \textrm {mm}$ and length $L=72.00\pm 0.05\ \textrm {mm}$. The roughness of the microchannel, the Ra parameter, was measured with a three-dimensional digital microscope, following the criteria of ISO4287. This standard states that, to obtain a value of roughness from a topography of a sample, first a flatness correction is applied to it and then a set of roughness profiles is extracted. The Ra parameter is the arithmetic average value of the roughness profiles. The roughness was measured in the top and bottom faces of the channel. The roughness parameter for both faces is found to be equal to $113\pm 19\ \textrm {nm}$, which is of the order of $0.05\,\%$ of the channel height. The roughness of the lateral faces was not measured. Even if the material of the channel walls is the same, the milling process used to manufacture the channel cannot guarantee the same roughness for the horizontal and vertical walls. Since the areas of the lateral faces are much smaller (five times) in comparison with the top and bottom areas, we assume that the absence of this information cannot essentially impact further conclusions about the flow patterns.

Two additional volumes, referenced in the following as reservoirs 1 and 2, figure 1, are connected to tanks 1 and 2, respectively, to increase the total volume of the system. The additional reservoirs are connected to the upstream and downstream tanks by the valves $V_{R1}$ and $V_{R2}$, respectively. Therefore, depending on the position of the valves, these reservoirs can be included or not in the measuring system. For the pressure gradient experiments, the volumes of reservoirs 1 and 2 are equal ($208.9\pm 4.5\ \textrm {cm}^3$). For the temperature gradient experiments, we do not use any additional reservoirs in the cold side, while a small reservoir of volume equal to $57.6\pm 1.2\ \textrm {cm}^3$ is connected to the hot side. The effect of these additional volumes on the accuracy of mass flow rate measurements and duration of the experiments will be discussed in § 4.2.

The upstream and downstream tanks are connected not only by the microchannel, but also by a large diameter pipe system, called the secondary connection or secondary line, see figure 1. In order to allow or prevent a gas to flow between these two tanks by the secondary connection, a micro-valve was inserted in this circuit. During the pressure gradient experiments, this micro-valve was not used, remaining closed all the time, while in the temperature gradient experiments, this micro-valve was opened in the beginning of each experiment to insure the development of the stationary thermal transpiration flow.

The pressure variation in time in each tank is measured by a high-speed response (30 ms) capacitance diaphragm gauge (CDG), manufactured by Inficon. In the temperature gradient experiments, a single pair of CDGs is used in all the measurements, both of them with a full scale of 1.33 kPa, while in the pressure gradient experiments, three different pairs of CDGs are implemented, with full scales of 133–133 kPa, 133–13.3 kPa and 1.33–1.33 kPa, depending on the desired pressure measurement range.

During the pressure gradient experiments, the cooling and heating systems are switched off and both tanks are kept at the room temperature, while in the temperature gradient experiments both systems work to maintain the tanks at constant but different temperatures. The external temperature of each tank is measured by a K-type thermocouple. Obviously there should be a discrepancy between the measured temperatures (external walls of the tanks) and the temperatures at the microchannel entrances. To associate the measured temperature gradient driven flow with the applied temperature difference at the microchannel ends, the temperature at the microchannel surfaces was measured using an infra-red (IR) camera, and the details of these measurements are provided in § 3. The data measured from the pressure gauges (CDGs) and the thermocouples (TCs) are captured by a data acquisition system produced by the National Instruments Corporation.

The leakage rate in the system was evaluated using 13.3 kPa pressure sensors according to the following procedure: first, all the valves of the experimental set-up, except the valves $V_{G}$, see figure 1, were opened and the system was pumped down during 72 h using a vacuum pump, model Adixen Drytel 1025. Then, the valves $V_{A}$, $V_{R1}$ and $V_{R2}$ were closed and both CDGs captured the pressure evolution from initial measured pressure of 0.133 Pa for two hours. During this period, the pressure inside the system increased by 0.424 Pa, which is associated with the leakage rate into the system by the gaps of the microchannel plates, figure 2(a), and other connections, see figure 1, and also a possible outgassing. This increase in the pressure corresponds to a mass flow rate of the order of $2.18\times 10^{-14}\ \textrm {kg}\ \textrm {s}^{-1}$, which could be associated with a leakage rate into the system and an outgassing. However, the lowest value of mass flow rate measured through the microchannel driven by the temperature gradient was $2.88\times 10^{-12}\ \textrm {kg}\ \textrm {s}^{-1}$, which is more than two orders of magnitude larger than the leakage rate. Therefore, the leakages and outgassing were considered as a part of the uncertainties of the pressure driven and temperature driven experiments, see § 4.3.

2.2. Volumes setting for pressure and temperature gradient experiments

The constant volume technique consists of measuring the pressure variation with time between two constant volumes connected by a microchannel (Ewart et al. Reference Ewart, Perrier, Graur and Méolans2007; Rojas Cardenas et al. Reference Rojas Cardenas, Graur, Perrier and Méolans2011; Rojas-Cardenas et al. Reference Rojas-Cardenas, Graur, Perrier and Méolans2013). The total volume $V_{i}$ connected to the side $i$ of the microchannel includes: volume of tank, $V_{tank,i}$, volume of additional reservoir, $V_{res,i}$, volume of connecting pipes, $V_{pipes,i}$, internal volumes of the valves, $V_{valves,i}$, and internal volume of pressure transducer, $V_{CDG,i}$

(2.1)\begin{equation} V_{i}=V_{res,i}+V_{tank,i}+V_{pipes,i}+V_{valves,i}+V_{CDG,i}, \end{equation}

where the subscript $i$ refers to the upstream ($i=1$) and downstream ($i=2$) sides of the microchannel, see figure 1. The total volumes of the upstream and downstream sides for all the settings used in the pressure and temperature gradient experiments are provided in table 1. It is clear that the volume of the microchannel, $V_{ch}=0.017\ \textrm {cm}^3$, is much smaller compared with the total upstream and downstream volumes.

Table 1.

Total volumes of both upstream and downstream sides, $V_{1}$ and $V_{2}$, respectively, used in the four different settings. The total volume of the system ($V_{T}=V_{1}+V_{2}$) and volume ratio ($V_{ratio}=V_{1}/V_{2}$) are also presented.

From table 1, it can be realized that the total volumes of the upstream, $V_{1}$, and downstream, $V_{2}$, sides are slightly larger for the two first settings, PG1 and PG2, in comparison with the third one, PG3, even if the same reservoirs are used in all the pressure gradient measurements. This happens because the CDGs with a full scale of 1.33 kPa have internal volumes slightly smaller than the other ones. This table also shows that only one volume setting is implemented in the temperature gradient experiments, since the same pair of CDGs is used in all the measurements.

4.1. Pressure gradient driven flow

A brief explanation of the steps followed during the pressure gradient experiments is given below using the sketch of the experimental set-up presented in figure 1. Firstly, before performing the measurements with a gas, the whole system is pumped down for 12 h with all the valves kept open, except the valves $V_G$. After this period, the valve $V_P$ is closed and one of the valves $V_G$ is opened, depending on the gas to be used, to fill the whole system with the chosen gas. Thereafter, the valves $V_A$ and $V_C$ are closed to stop the connections between the microchannel with the gas bottles and vacuum pump. Additionally, the micro-valve is closed too, then tanks 1 and 2 are connected only through the microchannel. After that, the initial pressure difference $\Delta p(t_0)=p_1(t_0)-p_2(t_0)$ (or with other notation: $\Delta p_0=p_{1,0}-p_{2,0}$) is imposed between the two tanks by quickly opening and closing the valve $V_A$, while the valve $V_P$ is kept open. Immediately after the opening and closing of the valve $V_A$, at $t=t_0$, the pressure inside the downstream tank, $p_2$, suffers a drastic reduction, while the pressure inside the upstream tank, $p_1$, does not change due to the high restriction imposed by the microchannel. Comments about the possible non-isothermal effects at this stage can be found in Appendix A. After that, for $t>t_0$, the pressure inside the upstream tank, $p_{1}$, starts to decrease while the pressure inside the downstream tank, $p_{2}$, starts to increase, until both pressures reach the same final value, $p_{f}$. The time when the pressures in both tanks become equal is denoted $t_f$ and its practical definition is given in the end of this section. The behaviour of the pressures inside the two tanks mentioned above for a generic pressure gradient experiment is shown in figure 6. The experiments for other different initial pressures are performed by opening and closing the valve $V_A$, keeping the valve $V_P$ opened, and waiting to the stabilization of pressures in both tanks.

Figure 6.

Behaviour of upstream, $p_{1}(t)$, downstream, $p_{2}(t)$, and mean, $p_{m}(t)=0.5(p_1(t)+p_2(t))$, pressures as function of time in a generic pressure gradient experiment. The exponential fittings of $p_{1}$ and $p_{2}$ are shown by dashed lines.

During the experiments, both tanks are kept at room temperature without any external heat sources. In addition, the amount of gas inside the microchannel is small and cannot change considerably the tank temperature because of the large thermal inertia of the tanks. During each experimental run, the external temperature of each tank was monitored, then its mean value over an experimental duration, $\bar {T}$, and corresponding standard deviation, $s$, are calculated. Both quantities appear as a pertinent evaluation of the probable temperature variations in the tanks as $\textrm {d}T_i/T_i\approx s_i/\overline {T_i}$, $i=1,2$. These values were found to be less than 0.0002 for both low and high pressure sides and they are associated with the corresponding temperature variations $dT_i/T_i$, $i=1,2$. The pressure variations in the high pressure tank were found to be in the range of $0.03< dp_1/p_1<0.24$, while the pressure variations in the lower tank are higher and were in the range of $0.03< dp_2/p_2<0.46$. Therefore, the values of $\epsilon _i$ parameters, present in (4.3), are estimated to be in the range of $0.0008<\epsilon _1<0.006$ and $0.0004<\epsilon _2<0.006$ for the higher and lower pressure tanks, respectively, being very low for all the pressure driven experiments, see table 4. Based on that, the mass flow rate for these experiments can be extracted from the measurements of pressure variation in any of the tanks, using one of (4.4).

Table 4.

Measurement of uncertainties of mass flow rate, when the pressure exponential evolution inside the tanks is used for the calculation. The results are presented for pressure (PGDF) and temperature (TGDF) gradient driven flows.

To obtain the mass flow rate through the microchannel from (4.4), the derivative of the pressure in time must be calculated. According to Rojas-Cárdenas et al. (Reference Rojas-Cárdenas, Silva, Ho, Deschamps and Graur2017) and Johansson et al. (Reference Johansson, Testa, Zaier, Perrier, Bonnet, Moulin and Graur2018), we assume that the pressure variation in the upstream and downstream tanks can be represented as exponential functions in the following form:

(4.5a,b) \begin{align} p_1(t) &= p_f + (p_{1,0}-p_f) \exp (-(t-t_0)/\tau_1),\notag\\ p_2(t) &= p_f + (p_{2,0}-p_f) \exp (-(t-t_0)/\tau_2), \end{align}

where $\tau _1$ and $\tau _2$ are the pressure relaxation times for the upstream and downstream tanks, respectively. The exponential form of pressure variation in time, (4.5a,b), was obtained under the assumption of constancy of the pressure relaxation time during an experiment (Johansson et al. Reference Johansson, Testa, Zaier, Perrier, Bonnet, Moulin and Graur2018). The pressure difference between the tanks can be presented analogously as

(4.6)\begin{equation} \Delta p(t) = \Delta p(t_0)\exp (-(t-t_0)/\tau). \end{equation}

As this relaxation time depends on the mean pressure, the constancy of the mean pressure in time insures the constancy of $\tau$, and so the validity of (4.5a,b). In the following, we derive the conditions of the mean pressure constancy in the pressure driven experiments.

From the ideal gas law, (4.1), and considering the mass conservation along the microchannel at any time, it is possible to obtain an expression relating the pressure variation in both tanks, $\textrm {d}p_{1}$ and $\textrm {d}p_{2}$, with the volume ratio $V_{ratio}=V_1/V_2$, when the tanks are kept at the same temperature

(4.7)\begin{equation} \textrm{d}p_1 V_1={-}\textrm{d}p_2 V_2. \end{equation}

By integrating the previous relation from the initial stage of an experiment, with pressures $p_{1,0}=p_1(t_0)$ and $p_{2,0}=p_2(t_0)$, to the final stage, when the pressures are equal in both tanks, $p_1(t_f)=p_2(t_f)=p_f$, we obtain the following expression:

(4.8)\begin{equation} \frac{p_{1,0} - p_f}{p_f - p_{2,0}} = \frac{V_2}{V_1}, \end{equation}

which allows us to control the amplitude of the pressure variation in both tanks between its initial value, $p_{i,0}$, and its final value, $p_f$, by changing the volume ratio. From (4.8), it is still possible to obtain another expression to calculate the variation of the mean pressure from the beginning of the experiment, $p_{m,0}=0.5(p_{1,0}+p_{2,0})$, to its end, $p_{m,f}=p_f$, as

(4.9)\begin{equation} \frac{p_f}{p_{m,0}} = \frac{2 ( 1 + V_{ratio} p_{ratio}) }{(1 + V_{ratio})(1 + p_{ratio})}, \end{equation}

where $p_{ratio}=p_{1,0}/p_{2,0}$. When the volumes are equal, it follows from (4.9) that the mean pressure does not vary during an experiment and $p_f=p_{m,0}$. Considering that the total volumes of each side of the microchannel are different, that is $V_{ratio}$ is different from 1, the mean pressure may vary during an experiment. From a known volume ratio and using (4.9), we can calculate the maximum initial pressure ratio which ensures the mean pressure constancy with some given accuracy. The volume ratio, $V_{ratio}$, used in the pressure gradient experiments, is equal to 0.964, see table 1, thus ensuring a maximal mean pressure variation of the order of $1\,\%$, according to (4.9), we have to set the initial pressure ratio, $p_{ratio}$, less than 3.45. Therefore, in all the experiments this pressure ratio was fixed smaller than 3.

In practice, we have to define a time interval to fit the pressure (or pressure difference variation) in time for each experimental run. In pressure driven experiments, we fit the values of the pressure difference up to time moment $t_{\tau }$ defined as $\Delta p(t_{\tau })=\varepsilon _{\tau }\Delta p(t_0)$, where $\varepsilon _{\tau }$ is a low value parameter. As the pressure difference between the tanks follows an exponential decay, (4.6), we associate this time $t_{\tau }$ with the relaxation time as $t_{\tau }=t_0-\tau \ln {\varepsilon _{\tau }}$. In experiments, we fix the value of $\varepsilon _{\tau }$ equal to 0.02, so the fitting time becomes $t_{\tau }=t_0+3.9\tau$. The final experimental time is usually longer than the fitting time and it is defined as $t {_{{f}}}=t_0+7\tau$, which corresponds to 0.1 % of the deviation of the pressure difference from zero. This $t {_{{f}}}$ parameter is the time where the pressure equilibrium is asymptotically reached. Since the final time is proportional to the relaxation time, it depends on the gas species, pressure level and reservoir sizes.

Respecting the condition for the constancy of the pressure relaxation time and using the mass conservation property we can write that $\tau _1=\tau _2=\tau$. By calculating the derivative of pressure variation in time, (4.5a,b), and replacing it in (4.4), we obtain the expressions for the mass flow rate in each tank

(4.10a,b)\begin{align} \dot M_1(t) = \frac{V_1}{{\mathcal{R}} T} \frac{{p_f-p_{1,0}}}{\tau} \exp \left(-\frac{(t-t_0)}{\tau} \right), \quad \dot M_2(t) = \frac{V_2}{{\mathcal{R}} T} \frac{p_{f}-p_{2,0}}{\tau} \exp \left(-\frac{(t-t_0)}{\tau} \right). \end{align}

As was explained in Johansson et al. (Reference Johansson, Testa, Zaier, Perrier, Bonnet, Moulin and Graur2018), in the case of isothermal flow, the mass flow rate can be also calculated from the pressure difference variation in time, $\Delta p(t)=p_1(t)-p_2(t)$, as follows:

(4.11a,b)\begin{equation} \dot M(t) = \frac{V_0}{{\mathcal{R}} T} \frac{\Delta p_{0}}{\tau} \exp \left(-\frac{(t-t_0)}{\tau} \right), \quad V_0=\frac{V_1 V_2}{V_1+V_2}, \end{equation}

where $V_0$ is the reduced volume. From the mass conservation property we have ${-}\dot M_1(t)=\dot M_2(t)=\dot M(t)$. This equality is satisfied in the steady-state flow regime.

4.2. Temperature gradient driven flow

It is known that when a temperature gradient is applied along the axis of a microchannel connected to two reservoirs of infinite volume maintained at the same pressure but at different temperatures, the gas inside this channel flows continuously from the colder to the hotter side (Reynolds Reference Reynolds1879). This phenomenon is called thermal transpiration. When the volumes connected by a microchannel are finite, a counterflow from the hot to the cold tank is generated, which leads to a steady-state situation, when the temperature gradient driven flow is counterbalanced by the pressure gradient driven flow, and the total mass flow rate through a channel becomes zero.

A short description of the specificity of the temperature gradient driven flow experiments is presented below. Firstly, before performing the measurements with a certain gas, the water cooling of the upstream tank (tank 1 or cold tank) and the electrical heater inserted in the downstream tank (tank 2 or hot tank) are turned on. After that, the whole system is connected to the vacuum pump for 12 h, as was done for the pressure gradient experiments. After that, the temperatures of both tanks are stabilized and the whole system is under vacuum conditions. Then, the system is filled with the gas to be tested, by following the same steps as was done for the pressure gradient experiments. After the chosen working gas fills the system, the valve $V_A$ is closed, see figure 1. At this moment, even with the imposed temperature difference between the hot and cold tanks, the gas pressure inside both tanks is exactly the same, since the micro-valve is still open, and, consequently, the gas can flow from one tank to another not only through the microchannel but also through the secondary line (a large diameter pipe system), see figure 1. This fact of the pressure equality, $p_1(t)=p_2(t)=p_0$, can be clearly seen in figure 7, when $t< t_0$.

Figure 7.

Behaviour of cold-side, $p_{1}(t)$, hot-side, $p_{2}(t)$, and mean, $p_{m}(t)$, pressures as functions of time in a generic temperature gradient experiment for two different volume ratios: (a) $V_{ratio}=0.678$ and (b) $V_{ratio}=0.168$. The exponential fittings of $p_{1}(t)$ and $p_{2}(t)$ are also shown by the dashed lines.

At time $t_0$, the micro-valve is closed. From this moment, the two tanks are connected only by the microchannel, where the gas flows from the cold side to the hot one, so the thermal transpiration flow (Reynolds Reference Reynolds1879; Rojas Cardenas et al. Reference Rojas Cardenas, Graur, Perrier and Méolans2011) takes place in the microchannel. This thermal creep flow immediately generates an increase in the pressure inside the hot tank and a decrease in the pressure inside the cold tank. This pressure difference between the tanks leads to the appearance of a counterflow, so-called Poiseuille flow, from the high pressure side to the low pressure one, that is in the opposite direction to the thermal transpiration flow. To sum up, from time $t_0$, there is an increase in pressure $p_2(t)$ (hot tank) and a decrease in pressure $p_1(t)$ (cold tank), as can be seen in figure 7(a).

When the Poiseuille flow reaches the same magnitude as that of the thermal transpiration flow, the net mass flow rate through the microchannel becomes zero and the pressures inside both tanks achieve constant but different values, $p_{1,f}$ and $p_{2,f}$, in the cold and hot reservoirs, respectively, as can be seen in figure 7(a). As for the pressure driven flow, see § 4.1, we define the final experimental time in terms of the relaxation time as $t {_{{f}}}=7\tau +t_0$. Also, in the temperature driven experiments, the fitting of the pressure variation in the cold tank is done during the fitting time $t_{\tau }$, which is calculated for this tank as $p_1(t_{\tau })-p {_{{f}}}=\varepsilon _{\tau }(p_{1,0}-p {_{{f}}})$, with $\varepsilon _{\tau }=0.02$.

When this zero flow condition is achieved, an important characteristic of the temperature gradient driven flow can be obtained, the thermomolecular pressure difference (TPD), which is defined by the difference between the pressures inside the hot and cold tanks, respectively, according to the following expression:

(4.12)\begin{equation} \textrm{TPD} = p_{2,f} - p_{1,f} = \Delta p_1 + \Delta p_2, \end{equation}

where $\Delta p_1=p_0-p_{1,f}$ and $\Delta p_2=p_{2,f}-p_0$.

Since the temperatures of each tank are constant during an experiment, we can use the same reasoning as presented in § 4.1 and make use of expressions (4.4) to calculate the mass flow rate if the ratios of the thermal fluctuations to the pressure fluctuations, i.e. $\epsilon _i$ ($i=1,2$), (4.3), are small enough compared with unity. The estimations of $\epsilon _i$ are provided in § 4.2.1.

The pressure variation inside both cold and hot tanks can be approximated using the relations (Rojas-Cardenas et al. Reference Rojas-Cardenas, Graur, Perrier and Méolans2012, Reference Rojas-Cardenas, Graur, Perrier and Méolans2013; Yamaguchi et al. Reference Yamaguchi, Rojas-Cardenas, Perrier, Graur and Niimi2014b)

(4.13a,b)\begin{align} p_1(t) &= p_{1,f} + (p_{0}-p_{1,f}) \exp\left(-(t-t_0)/\tau_1\right),\notag\\ p_2(t) &= p_{2,f} + (p_{0}-p_{2,f}) \exp\left(-(t-t_0)/\tau_2\right), \end{align}

where $\tau _1$ and $\tau _2$ are the relaxation times for the cold and hot volumes, respectively. The behaviours of these relaxation parameters will be discussed in § 6. Taking the time derivative of the pressure variation in each tank, (4.13a,b), and replacing it in each of (4.4), we obtain the expressions for the temperature gradient driven mass flow rates

(4.14a,b)\begin{align} \dot M_1 = {\frac{V_1}{{\mathcal{R}} T_1}} \frac{p_{1,f}-p_0}{\tau_1} \exp \left(-\frac{(t-t_0)}{\tau_1} \right), \quad \dot M_2 = \frac{V_2}{{\mathcal{R}} T_2} \frac{p_{2,f}-p_0}{\tau_2} \exp \left(-\frac{(t-t_0)}{\tau_2} \right). \end{align}

As explained above, the maximum mass flow rate generated by the temperature difference in the present experiments is realized at $t=t_0$, since, at that moment, the Poiseuille counterflow is still negligible. If we evaluate mass flow rate, (4.14a,b), at the instant $t_0$, we obtain the expressions for the temperature gradient driven mass flow rates

(4.15a,b)\begin{equation} \dot M_1 = {\frac{V_1}{{\mathcal{R}} T_1}} \frac{p_{1,f}-p_0}{\tau_1}, \quad \dot M_2 = \frac{V_2}{{\mathcal{R}} T_2} \frac{p_{2,f}-p_0}{\tau_2}. \end{equation}

4.2.1. Influence of additional volume on the hot side

The initial idea for the temperature gradient experiments was to perform the measurements without considering the additional reservoirs 1 and 2, i.e. the valves $V_{R1}$ and $V_{R2}$ would remain closed during all the experiments. In this configuration, the total volumes of the cold and hot sides are equal to 12.6 cm$^3$ and 18.6 cm$^3$, respectively. For the lighter gases, such as helium and neon, this volume configuration could be perfectly used, since the amplitude of the pressure variation in the cold tank is sufficiently high for the whole covered pressure range. However, for heavier gases, such as argon and krypton, the amplitude of pressure variation inside the tanks is considerably lower, mainly for high pressures. In this case, it becomes very difficult to fit the pressure variation in the cold tank with an exponential function because the instantaneous pressure fluctuations start to be very important compared with the pressure variation in time. Figure 8 shows the pressure variation inside the hot and cold tanks for helium (figure 8a) and krypton (figure 8b) considering the same temperature difference and the same level of gas rarefaction. It is clear that the pressure variation inside the cold tank is much higher for helium ($\sim$3 Pa) than for krypton ($\sim$1 Pa). Consequently, in the case of krypton, the fluctuations of the pressure signal are more significant compared with the total pressure variation in a tank, so it is difficult to fit the pressure behaviour with the exponential function with a good accuracy.

Figure 8.

Behaviour of the pressure variation inside the hot and cold tanks in temperature gradient experiments using (a) helium and (b) krypton. The volume ratio, rarefaction parameter and temperature difference between the tanks used in both experiments are $V_{ratio}=0.678,\ \delta = 10$ and $\Delta T=58\,^{\circ }\textrm {C}$, respectively.

To overcome this problem of very small amplitude of the pressure variation in the cold tank, the total volume of the hot side was increased by including an additional reservoir connected to the system by the valve $V_{R2}$, see figure 1. For this new volume configuration, the increase in the amplitude of the pressure variation in the cold side due to the change of the volume ratio between hot and cold sides can be estimated. As was shown in § 4.1, the amplitude of the pressure variation in each tank is related to the volume of the tanks. However, in the case of the temperature gradient driven flow, (4.7) needs to be modified to take into account of the different temperatures of the tanks, so it becomes

(4.16)\begin{equation} \textrm{d} p_1 \frac{V_1}{T_1}={-}\textrm{d} p_2 \frac{V_2}{T_2}. \end{equation}

By integrating this equation between initial, $t_0$, and final, $t_f$, time instants, we have

(4.17)\begin{equation} \Delta p_1 =\Delta p_2 \frac{V_2}{V_1}\frac{T_1}{T_2}. \end{equation}

By using (4.17), the amplitude of the pressure variation in the cold tank can be estimated for both volume configurations.

By comparing figures 7(a) and 7(b) it can be observed that the amplitudes of the pressure variation inside the cold and hot tanks are changed when the volume ratio is modified. For the first scenario, figure 7(a), when $V_1=12.6\ \textrm {cm}^3$ and $V_2=18.6\ \textrm {cm}^3$, $V_2/V_1=1.476$ and $T_1/T_2=0.831$ the amplitude of the pressure variation inside the cold tank is approximately 1.23 times higher than the amplitude of the pressure variation inside the hot tank, i.e. $\Delta p_1 \approx 1.23 \Delta p_2$. However, for the second scenario, figure 7(b), where $V_1=12.6\ \textrm {cm}^3$ and $V_2=75.3\ \textrm {cm}^3$, $V_2/V_1=5.976$, the amplitude of the pressure variation inside the cold tank is much higher than the amplitude of pressure variation inside the hot tank, i.e. $\Delta p_1 \approx 4.87\Delta p_2$. Therefore, for the second set-up configuration with the highest difference between the volumes, the amplitude of the pressure variation on the cold side increases considerably, reducing consequently the importance of its fluctuations, then increasing the accuracy of exponential fit of the pressure curve. Besides, for the experiments carried out with additional volume at the hot side, the experiment duration increases, since the relaxation time is also increased, i.e. it takes more time to achieve the stabilization of the pressures in both tanks.

During each experiment, which always lasts no longer than 6 min, the temperature is monitored by an IR camera. The mean value of the temperature, $\overline {T_i}$, and its standard deviation, $s_i$, are calculated. As for the pressure driven flows, the ratio $s/\overline {T_i}$ is assumed to be a pertinent representation of the temperature variation. The temperature fluctuations in the cold tank are found to be $dT_1/T_1\approx 7\times 10^{-5}$, while in the hot tank they are $dT_2\approx 6\times 10^{-5}$. On the other hand, the pressure fluctuations on the cold side are higher than those at the hot side and they are in the ranges of $0.001< dp_1/p_1<0.03$ and $0.0002< dp_2/p_2<0.004$, respectively. Summing up, the temperature fluctuations are relatively small compared with the pressure fluctuations in the cold tank, i.e. $\epsilon _1$ is lower than 0.05. However, for the hot tank, the pressure fluctuations are very small, especially for heavier gases, and for these cases, $\epsilon _2$ is of the order of 0.30. Therefore, it is not possible to extract the mass flow rate from the pressure measurements in the hot tank (tank 2) using (4.3) (or derived from it under condition of smallness of $\epsilon _2$, right-hand side of (4.15a,b)), because $\epsilon _2$ is not small enough compared with unity. In the following, only the measurements made on the cold side are presented and analysed.

4.3. Uncertainty measurement of the mass flow rate

The classical uncertainty calculation technique is used to estimate the measurement uncertainty of the mass flow rate when it is calculated from the pressure evolution for both pressure and temperature gradient experiments, (4.10a,b) and (4.14a,b) respectively. The general expression of uncertainty on the mass flow rate measurements reads

(4.18)\begin{equation} {\frac{\delta \dot M_i}{\dot M_i} = \frac{\delta V_i}{V_i} + \frac{\delta p_i}{p_i} +\frac{\delta T_i}{T_i} + \frac{\delta \tau}{\tau},} \quad i=1,2. \end{equation}

All the terms presented on the right-hand side of (4.18) are described below and their estimations are given in table 4. As was explained in the previous section, for the temperature gradient experiments, the measurements are carried out on the cold side only.

To measure the uncertainty of the volume $V_i$, $\delta V_i/V_i$, a reference reservoir with known volume is used and the pressure is measured inside this volume (Johansson et al. Reference Johansson, Testa, Zaier, Perrier, Bonnet, Moulin and Graur2018). The volume of the reference reservoir implemented in this procedure has already an uncertainty of approximately 2 %.

Taking into account the uncertainty on the internal volumes of the valves, open and closed during the procedure of the volume $V_i$ measurement, and the uncertainty of the pressure transducers, the total uncertainty of volume is 3 %.

The uncertainty on the pressure measurements is due to the uncertainty on the pressure sensors. This value is provided by the manufacturer and it varies from 0.2 % to 0.4 %, depending on the sensor type. The uncertainty on the temperature measurements is due to the uncertainty on the thermocouples, which is of the order of 0.3 %, and uncertainty of the IR camera, which is of the order of 0.7 %. As for the mass flow rate extraction, the values of the temperature obtained from the IR camera have been used, being the uncertainty of 0.7 % provided in table 4.

Finally, the uncertainty on the fitting parameter $\tau$ is obtained from the difference in magnitude of a 95 % confidence interval for $\tau$ to represent the experimental data. The value of this uncertainty is higher for temperature gradient experiments, due to the higher ratio between the fluctuation of the pressure signal and the pressure variation in time.

In table 4 we provide also two parameters which are not involved in (4.10a,b) and (4.14a,b) for the mass flow rate determination, but which impact the uncertainty of its calculation. The first quantity is the value of $\epsilon _i$, which represents the estimation of the non-isothermal effects, see also § 4, (4.3). These $\epsilon _i$ terms were neglected when deriving (4.10a,b) and (4.14a,b), therefore they do not appear in (4.18). However, the provision of these terms should give the idea on the accuracy of the model used for the mass flow rate extraction. This is why we included the values of $\epsilon _i$ in the total uncertainty on the mass flow rate. Finally, the last source of error in the mass flow rate measurement comes from the leakages, $\dot M_{leak}$, mainly through the gap between the microchannel plates. The values of $\dot M_{leak}$ are also provided in table 4 and they are included in $\delta \dot M_i/ \dot M_i$ calculations.

As can be seen in table 4, the uncertainties of the mass flow rate for pressure gradient experiments are very similar for high pressure and low pressure tanks. In the following, the data coming from the measurements in the low pressure tank are provided. For the temperature gradient experiments only uncertainties for the cold tank are given in table 4, since the pressure variations in the hot tank were very small compared with the initial pressure, especially for the high pressure experiments, which has led to the fact that the $\epsilon _2$ parameter becomes too high (of the order of 0.3) compared with unity and so cannot be neglected in the expression of mass flow rate ((4.3)), see also the comments in § 4.2.1. In the following, the data measured in the cold tank are used and analysed.

It should be also mentioned that the maximum values of uncertainties are presented in table 4. For example, the uncertainty on the fitting parameter $\tau$ is higher for heavier gases than for lighter ones, but the value obtained for krypton is provided in table 4. On the other hand, the leakage uncertainties are higher for light gases, thus the uncertainty of helium leakage is provided.

5.1. General definitions

The mass flow rate through the same rectangular microchannel was obtained from pressure gradient experiments for the Knudsen number range of $0.0016< K\!n<0.12$, which means for the hydrodynamic and slip flow regimes, and from temperature gradient flows for the Knudsen number range of $0.05< Kn<0.45$, which corresponds to the slip and beginning of transitional flow regimes. The Knudsen number is calculated by using the channel height, $H$, as the characteristic flow dimension

(5.1)\begin{equation} Kn = \frac{\ell}{H}, \end{equation}

where $\ell$ is the equivalent molecular free path, calculated as

(5.2)\begin{equation} \ell = \frac{\mu}{p} \sqrt{2 {\mathcal{R}} T}, \end{equation}

with $\mu$ the viscosity of the gas, calculated according to following expression (Bird Reference Bird1994):

(5.3)\begin{equation} \mu = \mu_{ref} {\left( \frac{T}{T_{ref}} \right)}^\omega, \end{equation}

where $\mu _{ref}$ is the gas viscosity at reference temperature $T_{ref}=273.15\ \textrm {K}$ and $\omega$ is the viscosity index. The rarefaction parameter is also used in the following and it is defined as the inverse of the Knudsen number

(5.4)\begin{equation} \delta = \frac{H}{\ell}=\frac{1}{Kn}. \end{equation}

5.2. Pressure gradient driven flow

The pressure gradient driven flow through a channel of a rectangular cross-section has been intensively studied in the last decades. Some numerical and analytical results can be found in Sharipov (Reference Sharipov1999a), Graur, Méolans & Zeitoun (Reference Graur, Méolans and Zeitoun2006) and Méolans et al. (Reference Méolans, Nacer, Rojas, Perrier and Graur2012). In our analysis we used the following expression for the mass flow rate obtained from the Stokes equation with the first-order velocity slip boundary condition (Méolans et al. Reference Méolans, Nacer, Rojas, Perrier and Graur2012):

(5.5)\begin{equation} \dot M = \dot M{_{{P}}}\left(1+6\sigma_p \frac{T_n}{S_n} Kn\right) , \end{equation}

where $\dot M {_{{P}}}$ is the Poiseuille mass flow rate, defined as

(5.6)\begin{equation} \dot M{_{{P}}} = \frac{H^3 W (1-{\mathcal{K}}) \Delta p p_m}{12 \mu {\mathcal{R}} T L}, \end{equation}

with $\sigma _p$ the velocity slip coefficient (Albertoni, Cercignani & Gotusso Reference Albertoni, Cercignani and Gotusso1963; Sharipov & Seleznev Reference Sharipov and Seleznev1998) and $p_m=(p_1+p_2)/2$ the mean pressure. The coefficient $\mathcal {K}$ allows us to take into account the influence of the lateral walls on the Poiseuille mass flow rate, (5.6), and it is obtained from (Sharipov Reference Sharipov1999a; Méolans et al. Reference Méolans, Nacer, Rojas, Perrier and Graur2012)

(5.7)\begin{equation} {\mathcal{K}} = 192 \frac{H}{W} \sum_{i=0}^{\infty} \frac{1}{n^5} \tanh \left( \frac{nW}{2H} \right) , \quad n={\rm \pi} (2i+1). \end{equation}

The coefficients $T_n$ and $S_n$ in (5.5) allow us to take into account the influence of the lateral walls on the mass flow rate in slip flow regime and they are calculated from the following expressions (Méolans et al. Reference Méolans, Nacer, Rojas, Perrier and Graur2012):

(5.8)\begin{gather} T_n = \frac{4}{3}S_n-\frac{1}{3} \left(1-\frac{H}{W}\right) \sum_{n=0}^{\infty} \frac{\tanh^2 \left(0.5 {\rm \pi}(2n+1)W/H\right)}{(2n+1)^4}, \end{gather}

(5.9)\begin{gather} S_n = \frac{{\rm \pi}^4}{96}-\frac{2H}{{\rm \pi} W} \sum_{n=0}^{\infty} \frac{\tanh \left(0.5 {\rm \pi}(2n+1)W/H\right)}{(2n+1)^5}. \end{gather}

It should be mentioned that the equivalent molecular free path, $\ell$, (5.2), and, consequently, Knudsen number, $K\!n$, (5.1), are calculated here using the mean pressure $p_m$.

The mass flow rate, (5.5), can be presented also in the dimensionless form

(5.10)\begin{equation} S{^{{T}}} = \dot M / \dot M{_{{P}}} = 1+6\sigma_p \frac{S_n}{T_n} Kn=C{^{{T}}}_0+C{^{{T}}}_1Kn, \end{equation}

where

(5.11a,b)\begin{equation} C{^{{T}}}_0=1, \quad C{^{{T}}}_1=6\sigma_p \frac{S_n}{T_n}. \end{equation}

We can fit the measured dimensionless mass flow rate analogously to the (5.10) form as a function of Knudsen number

(5.12)\begin{equation} S{^{{F}}} =C{^{{F}}}_0+C{^{{F}}}_1Kn, \end{equation}

and then extract the velocity slip from the relation $C {^{{T}}}_1=C {^{{F}}}_1$, as follows:

(5.13)\begin{equation} \sigma_p=\frac{C{^{{F}}}_1}{6} \frac{T_n}{S_n}. \end{equation}

When the velocity slip coefficient for each pair gas–surface is extracted from (5.13), then the accommodation coefficient can be also obtained. The authors of Loyalka, Petrellis & Stvorick (Reference Loyalka, Petrellis and Stvorick1975) calculated the velocity slip coefficient for a given accommodation coefficient using kinetic modelling, namely the Bhatnagar–Gross–Krook (BGK) model, and the Maxwell specular-diffuse scattering kernel. Then, a simple expression associating the slip and accommodation coefficients was proposed

(5.14)\begin{equation} \sigma_p (\alpha) = \frac{2-\alpha}{\alpha}(\sigma_p(1) - 0.1211(1-\alpha)) , \end{equation}

where $\alpha$ is the accommodation coefficient and $\sigma _p(1)$ is the slip coefficient for $\alpha =1$, with $\sigma _p(1)=1.016$ (Albertoni et al. Reference Albertoni, Cercignani and Gotusso1963).

The second-order (in Knudsen number) polynomial fit was used recently by several authors to extract the accommodation coefficient from pressure driven flows (Maurer et al. Reference Maurer, Tabeling, Joseph and Willaime2003; Colin et al. Reference Colin, Lalonde and Caen2004; Graur et al. Reference Graur, Perrier, Ghozlani and Méolans2009; Perrier et al. Reference Perrier, Hadj-Nacer, Méolans and Graur2019). The comparison between the fit of the experimental data of helium to polynomials of first and second orders was carried out in the Knudsen number ranges of $[0.0016;0.12]$ and $[0.0016;0.67]$, respectively. It was found that the difference between the accommodation coefficients derived from the respective fits is of the order of $0.1\,\%$, see table 13, so the first-order polynomial fit is used for the coefficient extraction. The values of the velocity slip and accommodation coefficients for each gas–surface pair, obtained for the five analysed gases, are provided in § 6.

5.3. Temperature gradient driven flow

As was underlined in § 4.2, the thermal creep flow is evaluated at instant $t_0$, where the pressures in both tanks are equal between them and the counterflow from the hot side to the cold has not yet started. At this stage of established thermal creep between two tanks with different temperatures and equal pressures, a very small pressure gradient is generated inside the microchannel (Sharipov Reference Sharipov1999b; Méolans & Graur Reference Méolans and Graur2008; Graur & Ho Reference Graur and Ho2014; Rojas Cárdenas et al. Reference Rojas Cárdenas, Graur, Perrier and Méolans2015; Yamaguchi et al. Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016). The total mass flow rate $\dot M$ through the channel can be presented as

(5.15)\begin{equation} \dot M ={-}\dot M{_{{P}}}+\dot M{_{{T}}}, \end{equation}

where $\dot M {_{{P}}}$ is the pressure driven flow rate induced by the thermal creep flow and $\dot M {_{{T}}}$ is the thermal creep mass flow rate. The pressure profile along the channel has a parabolic shape, with maximum value near to its central point and it was numerically established in Sharipov (Reference Sharipov1999b) and Méolans & Graur (Reference Méolans and Graur2008). Using the previously developed kinetic modelling, the authors of Yamaguchi et al. (Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016) estimated the ratio $\dot M {_{{P}}}/\dot M$ in the case of very similar temperature gradients and in the case of the flow through a rectangular microchannel to be smaller than 0.2 %. Therefore, the total measured mass flow rate could be identified with a temperature driven mass flow rate as

(5.16)\begin{equation} \dot M \approx \dot M{_{{T}}}. \end{equation}

To obtain an explicit expression of the mass flow rate driven by thermal transpiration $\dot M {_{{T}}}$ in the slip flow regime, the Stokes equation subjected to the thermal slip boundary conditions in the following form

(5.17)\begin{equation} u{_{{slip}}}=\sigma{_{{T}}} \frac{\mu}{\rho T}\frac{\textrm{d}T}{\textrm{d}x} \end{equation}

was integrated over the channel cross-section. Then, the mass flow rate reads (Sharipov Reference Sharipov2011; Yamaguchi et al. Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016)

(5.18)\begin{equation} \dot M{_{{T}}}=\sigma{_{{T}}}HW\frac{\mu}{T}\frac{\textrm{d}T}{\textrm{d}x}. \end{equation}

In previous expressions, $\sigma {_{{T}}}$ is the thermal slip coefficient (Sharipov Reference Sharipov2011), $\rho$ is the gas density and $\textrm {d}T/{\textrm {d}x}$ is the temperature gradient along the channel walls in the $x$-direction. The thermal slip coefficient could be extracted from (5.18) if the temperature gradient along the channel is known (Yamaguchi et al. Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016). However, this is only a first-order solution according to the Knudsen number and it is not accurate enough (Rojas Cárdenas et al. Reference Rojas Cárdenas, Graur, Perrier and Méolans2015; Yamaguchi et al. Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016). The higher-order solution for the dimensionless temperature driven mass flow rate between two infinite parallel plates was obtained from the kinetic theory in Loyalka & Hickey (Reference Loyalka and Hickey1989)

(5.19)\begin{equation} \dot M{_{{T}}} =HW \frac{ \mu}{T}\frac{\textrm{d}T}{\textrm{d}x} \left({\mathcal{C}}_0{^{{T}}} + \frac{{\mathcal{C}}_1{^{{T}}}}{\delta} + \frac{{\mathcal{C}}_2{^{{T}}}}{\delta^2}+ O \left(\frac{1}{\delta^3}\right)\right), \end{equation}

where the values of the two coefficients ${\mathcal {C}}_0 {^{{T}}}$ and ${\mathcal {C}}_1 {^{{T}}}$ were calculated in Loyalka & Hickey (Reference Loyalka and Hickey1989), being equal to ${\mathcal {C}}_0 {^{{T}}}=0.9924$ and ${\mathcal {C}}_1 {^{{T}}}=-1.3284$. Following Yamaguchi et al. (Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016) and Rojas Cárdenas et al. (Reference Rojas Cárdenas, Graur, Perrier and Méolans2015), we integrated this asymptotic solution, but up to the term of the order of $O(1/\delta ^3)$, so the expression for the mass flow rate becomes

(5.20)\begin{equation} \dot M{_{{T}}} = \dot M{_{{ref}}} \left({\mathcal{C}}_0{^{{T}}} + \frac{{\mathcal{C}}_1{^{{T}}} {\mathcal{C}}_1'}{\delta{_{{m}}}} + \frac{{\mathcal{C}}_2{^{{T}}} {\mathcal{C}}_2'}{\delta{_{{m}}}^2}+ O \left(\frac{1}{\delta{_{{m}}}^3}\right)\right), \end{equation}

where

(5.21)\begin{equation} \dot M{_{{ref}}} = \frac{HW \mu{_{{ref}}}}{T{_{{ref}}}^{\omega}} \frac{T_2^{\omega}-T_1^{\omega}}{\omega L}. \end{equation}

The mean value of the rarefaction parameter, $\delta {_{{m}}}$, is calculated according to

(5.22)\begin{equation} \delta{_{{m}}} = \frac{p_0 H}{\mu(T{_{{m}}}) \sqrt{2 {\mathcal{R}} T{_{{m}}}}}, \end{equation}

where $T_m=0.5(T_1+T_2)$ is the mean temperature; the viscosity coefficient is calculated also using this mean temperature. The term ${\mathcal {C}}_0 {^{{T}}}$ in (5.20) can be identified as the thermal slip coefficient, $\sigma {_{{T}}}$. Two additional coefficients come from the integration of (5.19) along the channel and they are equal to

(5.23a,b)\begin{equation} {\mathcal{C}}_1' = \frac{T_2^{2\omega+0.5} - T_1^{2\omega+0.5}}{(2\omega+0.5)T_m^{\omega+0.5}}\frac{\omega}{T_2^{\omega}-T_1^{\omega}}, \quad {\mathcal{C}}_2' = \frac{T_2^{3\omega+1} - T_1^{3\omega+1}}{(3\omega+1)T_m^{2\omega+1}}\frac{\omega}{T_2^{\omega}-T_1^{\omega}}. \end{equation}

The values of both coefficients are very close to one with a deviation less than 1 %, so they are assumed to be equal to 1 in the following. Therefore, (5.20) can be rewritten in more convenient form by dividing it by the reference mass flow rate, $\dot M {_{{ref}}}$, and by neglecting the terms of the order of O($1/\delta {_{{m}}}^3)$

(5.24)\begin{equation} {\mathcal{G}}{^{{T}}}= \frac{\dot M{_{{T}}}}{\dot M{_{{ref}}}} = {\mathcal{C}}_0{^{{T}}} + \frac{{\mathcal{C}}_1{^{{T}}}}{\delta{_{{m}}}}+\frac{{\mathcal{C}}_2{^{{T}}}}{\delta{_{{m}}}^2}={\mathcal{C}}_0{^{{T}}} + {\mathcal{C}}_1{^{{T}}}Kn{_{{m}}}+{\mathcal{C}}_2{^{{T}}}Kn{_{{m}}}^2. \end{equation}

As the fit in the polynomial form is more convenient compared with the previous expression, we use the following expression to fit the experimental data

(5.25)\begin{equation} {\mathcal{G}}{^{{F}}}(Kn{_{{m}}}) = \frac{\dot M{_{{T}}}}{\dot M{_{{ref}}}} = {\mathcal{C}}_0{^{{F}}} + {\mathcal{C}}_1{^{{F}}}Kn{_{{m}}}+{\mathcal{C}}_2{^{{T}}}Kn{_{{m}}}^2, \end{equation}

in the slip and beginning of transitional flow regimes, $0.05< Kn {_{{m}}}<0.45$. The fitting coefficient ${\mathcal {C}}_0 {^{{F}}}$ is associated with the coefficient $\sigma {_{{T}}}$.

Finally, the accommodation coefficient, $\alpha$, was calculated from the thermal slip coefficient by using the expression proposed in Sharipov (Reference Sharipov2011) and Suetin & Chernyak (Reference Suetin and Chernyak1977) and obtained using the S-model kinetic equation with Maxwellian specular-diffuse boundary condition

(5.26)\begin{equation} \sigma_T = 0.75 (1+0.5\alpha). \end{equation}

For the case of a polyatomic gas such as nitrogen, it is necessary to take into account the effects of the internal degrees of freedom of the molecule, as formulated in the expression proposed in Loyalka & Storvick (Reference Loyalka and Storvick1979) and Loyalka, Storvick & Lo (Reference Loyalka, Storvick and Lo1982), where the model of Hanson and Morse was used. Thus the expression of the thermal slip coefficient for a polyatomic gas reads

(5.27)\begin{equation} \sigma_T = \tfrac{3}{10}\,f_{tr} (1+0.5\alpha), \end{equation}

where $\,f_{tr}$ is the translational Eucken factor, which is equal to 2.25 for nitrogen (Porodnov et al. Reference Porodnov, Kulev and Tuchvetov1978).

The thermal slip and accommodation coefficients obtained from temperature gradient driven flow, as well as the fitting coefficients for all the five gases used in the present work (He, Ne, N$_2$, Ar and Kr), are presented in the next section.