Pump B is a multistage system, where each stage consists of one single pumping narrow microchannel with $d=50,20,10,5$ μm, followed by one wide channel with $D=100$ μm, targeting high pressure differences and small mass flow rates (as there is only one microchannel per stage).

The maximum pressure difference corresponding to zero net mass flow rate (closed system) at various inlet pressures

${P}_{in}=1,5,10,20,50,100$ kPa are provided in

Figure 6a,b versus the number of stages with

$N\le 1000$ and

$N\le 100$, respectively. The considered narrow and wide microchannels at each stage have diameters

$d=10$ μm and

$D=100$ μm, respectively. As seen in

Figure 6a, the pressure difference,

$\Delta P$, increases with the number of stages in a qualitatively similar manner for all inlet pressures, except for the lowest inlet pressure

${P}_{in}=1$ kPa and

$N\le 100$. It is also seen that for a small number of stages the pressure difference is rapidly increased and then it keeps increasing, but at a slower pace that gradually decreases as the number of stages increases. This is due to the fact that, for all the inlet pressures shown, except

${P}_{in}=1$ kPa, the pumps, independently of

$N$, always operate in the decreasing region of

$\Delta P$ in terms of

${P}_{in}$ (see

Figure 5a). In all these cases, each time a stage is added its contribution to the overall

$\Delta P$ is slightly reduced compared to the previous one, because the inlet pressure of the stage is increased. Therefore, the rate at which

$\Delta P$ is increased is slowly reduced with increasing number of stages. In the specific case of

${P}_{in}=1$ kPa the pump starts to operate in the increasing region of

$\Delta P$ in terms of

${P}_{in}$ (see

Figure 5a). Consequently, starting at

${P}_{in}=1$ kPa, every added stage, compared to the previous one, contributes with a larger

$\Delta P$ until the pressure of

${P}_{in}=5$ kPa, corresponding approximately to the peak value of the pressure difference

$\Delta {P}_{peak}$, is reached. Thus, a more detailed view for

$N\le 100$ is shown in

Figure 6b. It is clearly seen that the pressure difference with respect to the number of stages for

${P}_{in}=1$ kPa is qualitatively different compared to all other

${P}_{in}$ values and is increased more rapidly. Of course, once a sufficient number of stages has been added and the inlet pressure for the next stage becomes higher than

${P}_{in}=5$ kPa, then the pump operates in the decreasing region of

$\Delta P$ in terms of

${P}_{in}$ and it behaves qualitatively similar to all others. Overall, it is observed that for a multistage pump with

$N\ge 100$ the pressure difference generated is very significant and may be of the same order as the inlet pressure or even several times higher, when

${P}_{in}\le 20$ kPa. It is noted that the present results are in excellent qualitative agreement with the corresponding ones reported in [

10]. Furthermore, running the present code for some of the geometrical and operational parameters in [

10], it has been found that the deviation in the numerical results is small (about 10%) and is due to the corresponding deviation between the kinetic coefficients obtained by the ES and Shakhov models.

Although pump B targets high

$\Delta P$ and the mass flow rate is not of major importance, it is interesting to observe its variation versus the number

$N$ of stages. In

Figure 7, the maximum mass flow rate corresponding to zero pressure difference (i.e., a pump with the same pressure at both ends) is shown for the same parameters as in

Figure 6a. The mass flow rate is of the order of

${10}^{-12}$ or

${10}^{-13}$ kg/s and it is low since only one narrow channel is considered per stage. As expected,

$\dot{m}$ is decreased as

${P}_{in}$ is decreased, but more importantly,

$\dot{m}$ is kept constant as the number of stages is increased due to the fact that all stages have the same geometry and the same inlet and outlet pressures. It may be stated that for Knudsen pumps based on the architecture of pump B, it is desirable to add as many stages as possible and to operate the pump in low and moderate

${P}_{in}$, since

$\Delta P$ grows with

$N$, while the maximum

$\dot{m}$ remains constant. The corresponding results for other narrow microchannel diameters have a similar qualitative behavior.

The evolution of the pressure inside a system vacuumed with pump B connected at its inlet, while the outlet is at the atmospheric pressure (

${P}_{out}=100$ kPa), is analyzed in

Figure 8, showing the influence of the number of stages and of the narrow channels diameter

$d=5,10,20$ μm, with the diameter of the wide channel being always

$D=100$ μm. Three pressure drop regions can clearly be identified: at high pressures (green region), the pressure is reduced very slowly; at intermediate pressures (red region), the pressure is rapidly reduced; and finally, at low pressures (blue region), the pressure keeps reducing at a slower pace. This behavior can be understood by observing the corresponding results in

Figure 5, where the pressure difference is very small at high inlet pressures, then becomes quite large at intermediate inlet pressures, where the maximum pressure difference is reported (red symbols in

Figure 8) and, finally, becomes small again at low inlet pressures. These regions of

Figure 5 correspond to the green, red, and blue regions in

Figure 8. As expected, the required number of stages to reach the final low inlet pressure is increased by increasing the diameter of the narrow microchannels. These plots confirm that the optimal operating pressure range of the pumps B presented is, as pointed above, in the red zone (

Figure 8), i.e., at moderate and low inlet pressures corresponding to values of the gas rarefaction parameter close to

$\delta =3-4$ where

$\Delta {P}_{peak}$ is found (

Figure 5).