Thermal creep flow by Thomas Hällqvist Veronica Eliasson

Transcription

1 Thermal creep flow by Thomas Hällqvist Veronica Eliasson Introduction Thermal creep It is possible to start rarefied gas flows with tangential temperature gradients along the channel walls, where the fluid starts creeping in the direction from cold towards hot. This is called the thermal creep phenomena. There is a growing interest for this phenomenon in micromachine engineering since pumping devices without any mechanical devices are attractive. (For example in space, where a good natural heat and sink source is available.) DSMC and DS2G DSMC (Direct Simulation Monte Carlo Method) is a technique for computer modeling of a real gas by some thousands or millions of simulated molecules. It is necessary when the NS equations do not provide a valid model for rarefied gases. The program used for simulations is called DS2G and is given by Bird et Al. It is applicable to a wide range of problems, from flows past aerodynamic bodies to internal flows in high vacuum equipment. DS2G is said to have a flexible system for the specification of geometries. The code uses plane 2D flow or axially symmetric flow. The flow field is divided into four sided regions. Typical examples of a flow field and an arrangement of regions are shown in Figures 3 and 4. Problem setup and cases The simulation is carried out on a micromachine pumping system featuring no moving parts. The goal is to find the optimal design in order to achieve most effective pumping system. Below in Fig. 3 is an example of a pumping system shown. In this example six modules are serially connected. The red dashed rectangle shows the domain for the simulation. This is possible as the geometry is both periodic and symmetric. simulated Reservoir Reservoir 2 P, T P2, T Y Flow Figure 3. Micromachinery pumping system. X b CELL 2 CELL L d CELL 3 Figure 4. Computational domain. D/2 Tw T Figure. A typical flow field. Figure 2. A typical arrangements of regions. T b L X Figure 5. Periodic temperature gradient on the walls.

2 Figure 4 shows an enlarged view of the computational domain and Fig. 5 the periodic temperature distribution on the walls. The present setup features no large temperature difference (doesn t depend on the length of the system, i.e. number of modules). This is so since there are ditches. If there where no ditches an infinite temperature difference would be needed for an infinite long system. Two reservoirs connected by a series of these systems have the same temperature (periodic temperature gradient, zero mean temperature gradient). The more modules the stronger pumping effect The cases performed and parameters studied is shown in Table, below Figure 6. Temperature contours for case, ranging from T =3K to T =9K. Figure 7. Velocity vectors for case. Velocity profiles superimposed (upper number is the local maximum velocity, lower number is the local averaged velocity). Mean velocity at X=b: U b =9 m/s Table. Performed cases Results Variation of d/d (case to 4). The first simulations were carried out with the default settings of the demonstration case, supplied by G.A.B. Consulting Pty Ltd. The only thing varied was d. Figure 6. shows the temperature distribution in the domain for case. The temperature distribution on the top-walls, according to Fig. 6 is clearly depicted, so also the streamwise periodic boundary conditions. According to the velocity profiles in Fig. 7 there is a dominant bulk flow in the main channel, whereas in the ditch, there is a large vortex. The number of sub-cells is 2 and 5 in the X- and Y-direction, respectively. The mean velocity in the channel is 9 m/s. Further downstream the mean velocity increases due to the increasing temperature. Setting d/d equal to.5, case 2, does not change the flow character. But when d/d is set equal to, case 3, the velocity in the main channel decreases to 7.3 m/s. The vortex in the ditch seems to extent out into cell. This can be even clearer seen for case 4 in Fig. 5. The mean velocity in this case is as low as 4.8 m/s Figure 8. Temperature contours for case 2. (d/d=.5, L/D=8) Figure 9. Velocity vectors for case 2. Mean velocity at X=b: U b =9. m/s.

3 Variation of L/D (case 5). The L/D ratio is changed by varying D. For case 5, below, D is half as large compared to the above cases, giving a L/D ratio of 6. The velocity profiles in Fig. 7, show also here, a distinct vortex within the ditch and a dominant bulk flow in the main channel. However, the shape of the velocity profiles is of different character, as the profiles are not as flat as in Fig Figure. Temperature contours for case 2. (d/d=.5, L/D=8) Figure. Velocity vectors for case 2. Mean velocity at X=b: U b =9. m/s Figure 6. Temperature contours for case 5. (d/d=4, L/D=6) Figure 7. Velocity vectors for case 5. Mean velocity at X=b: U b =7.2 m/s Variation of b/d (case 6 and 7) Figure2. Temperature contours for case 3. (d/d=, L/D=8) Figure3. Velocity vectors for case 3. Mean velocity at X=b: U b =7.3 m/s Figure 8. Temperature contours for case 6. (d/d=2, L/D=8, b/d=2) Figure 9. Velocity vectors for case 6. Mean velocity at X=b: U b =7. m/s Figure4. Temperature contours for case 4. (d/d=.5, L/D=8) Figure 5. Velocity vectors for case 4. Mean velocity at X=b: U b =4.8 m/s.

4 Figure 2. Temperature contours for case 7. (d/d=2, L/D=8, b/d=6) Figure 2. Velocity vectors for case 7. Mean velocity at X=b: U b =2.6 m/s Figure 24. Temperature contours for case Figure 25. Velocity vectors for case 9. Mean velocity at X=b: U b =8.5 m/s. Refined mesh (case 8 and 9). When increasing the number of cells, the time-step, dt, needs to be shortened. The movement of one molecule during dt should be about one third of the cell-size. The number of sub-cells is 6 and 4 in the X- and Y-direction, respectively, for case 8 and 2 and 8, respectively, for case 9. No significant differences are seen between case 8 and case. However, case 9 show larger differences. For both these refined cases the flow does not seem to be fully converged, particularly regarding case 9. This can be seen from the contour lines and the velocity profiles. The total sampling time for case 9 is, due to the small dt, short compared to the rest of cases. Also, the possibility for collision becomes smaller in each sub-cell, if the total number of molecules is held constant. Thus, the total sampling time needs to be larger in order to achieve adequate statistics. Rescaled domain (case to ). By rescaled domain is meant that the d/d ratio is constant, by changing d and D with same factors Figure 26. Temperature contours for case. (d/d=2, L/D=2.) Figure 27. Velocity vectors for case. Mean velocity at X=b: U b =7.2 m/s Figure 22. Temperature contours for case 8. Figure 23. Velocity vectors for case 8. Mean velocity at X=b: U b =8.8 m/s.

5 Influence from number of simulated molecules (case 3). Here the influence from the number of simulated molecules is studied. In this case 55 molecules have been simulated, compared to for case. No direct qualitative differences can be seen compared to the default case. The mean velocity in Fig. 33 is, though, slightly higher. However, this might be an effect from the slightly higher Knudsen number as will be studied later Figure 28. Temperature contours for case. (d/d=2, L/D=6) Figure 29. Velocity vectors for case. Mean velocity at X=b: U b =5.8 m/s Variation of T (case 2). When increasing the temperature ratio there is an inherent increase in massflow. As seen in Fig. 3 the velocity is 2.8 m/s, compared to 9 m/s for the default case. The temperature ratio is 5% higher and the mean velocity 42% higher Figure 32. Temperature contours for case Figure 33. Velocity vectors for case 3. Mean velocity at X=b: U b =.7 m/s Pumping effect (case 4). The pumping effect is studied by setting the periodic boundaries to solid surfaces, making the domain a closed one. The pumping effect is a measure of the maximum allowed pressure difference between outlet and inlet, dp. As can be seen in Fig. 37, there is a 7% pressure build up along the symmetry-axis of the channel. As a reference also the pump effect for higher T was studied. As seen in Fig. 37 the pump efficiency increases Figure 3. Temperature contours for case 2. (d/d=2, L/D=8, T /T =4.5) Figure 3. Velocity vectors for case 2. Mean velocity at X=b: U b =2.8 m/s Figure 34. Temperature contours for case 4. Figure 35. Velocity vectors for case 4.

6 P/P.5 Micromachine pumping system featuring no moving parts has been studied. The influence from the geometrical setup has been studied. The system features zero temperature gradient on the average. However, steady one dimensional flow is thermally induced in the system. Even a small temperature gradient is enough to achieve a steady flow. The design of the ditch is critical. A Long ditch (3/4L) high massflow, a shallow ditch decreased massflow and no ditch one-way flow absent. Good results have been achieved even with few sub-cells and few simulated particles. The maximum possible pressure difference (pumping effect) verified for case. Massflow features a maximum at Kn.4. One possible application is in space as there are good natural heat source and sink present Figure 36. Pressure contours for case X Figure 37. Distribution of pressure along the symmetry-axis for case 4 (dashed line: T /T =6). Influence from Knudsen number (case 5 to 9). The Knudsen number is varied by redefine the average number density over the whole domain. Study the influence from Kn on the total massflow. Five different Kn was studied;.36,.47,.63,.8 and 2.9. The massflow experience a maximum at Kn.4, as seen in Fig. 38. Thereafter follows a continuous decrease to zero velocity (Kn= ). The results are in good agreement with data by Sone et. al. Massflow as function of Kn,2 massflow,8,6,4,2,5,5 2 2,5 3 3,5 Kn Figure 38. Massflow as function of Kn. Figure 39. Massflow as function of Kn (Sone et al). Conclusions