Temperature-Driven flow through nano-channels

All discussions in this section will be performed in term of parameters like normalized flow rate and rarefaction factor, which have been described in the first paragraphs of section 4.2.

As we mentioned earlier, temperature driven flow (TDF) occurs when temperature gradient exists along the flow channel, i.e. when the flow becomes non isothermal. In contrast to the pressure driven flow, TDF is characterized by long transient time and lower mass flow rate.

4.3.1 Verification of the studied model and Diffuse BC

The model used for study on TDF is similar to the model described in section 4.2, but in this case T1≠T2. It should be noted, that TDF has limiting case - so called thermomolecular pressure difference (TmPD). This effect can be observed in a closed system (atoms can freely migrate between tanks, but they are not allowed to leave the system) of tanks connected by a channel or orifice. The value of pressure drop in case of TmPD can be computed using eq. (1.2-13). Parameter γ of eq. (1.2-13) significantly depends on rarefaction parameter δ, boundary conditions and shape of the channel’s cross-section[15, 36]. Equation (2.3-13) allows one to estimate the duration of transient period if TmPD effect is observed in case of ideal gas. These two equations are very important because they allow us to verify whether our model is able to reproduce such a tiny effect like TDF or not. Figure 4.3-1 represents the number of atoms in cold tank versus time. Initially atoms were distributed in order make pressures in both tanks equal. After the certain amount of time some atoms migrated from the left tank to the right one through orifice (Lch/hch=0) and pressure ration reached the value predicted by equation (1.2-13). One can see that that simulation result (black curve) agrees well with

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analytical prediction (green curve), it means that our model is able to reproduce TDF and TmPD, as well.

In contrast to the previous verification, Figure 4.3-2a shows the correlation of normalized temperature driven flow rate versus channel’s length-to-height ratio (diffuse BCs are applied). One can see that in case of long channel (Lch/hch>15) flow rate obtained in current study has good agreement with solution (2.3-15b) obtained by Sharipov [39]. Observation of Figure 4.3-2b allows us to conclude that flow rate approaches constant value while length-to-height ration tends to zero; in case of long and thin channel the flow rate becomes inversely proportional to the Lch/hch. However, nonlinear correlation between flow rate and channel’s length is observed if 0<<Lch/hch<15. It should be noted, that data points in Figure 4.3-2b corresponding to different rarefaction parameters are shifted with respect to each other, while the curvature is found to be unchanged within the range of studied parameters.

4.3.2. Realistic BC, New BC and studies on wall’s irregularities

Table 4.3-1 represents normalized flow rate through the channels with various BC. One can see that new statistical BC gives almost the same result like realistic wall.

It means that the novel boundary conditions are able to reproduce such a tiny effect like temperature driven flow through metal channel. On the other hand, diffuse BC gives value of flow rate that is very close to the value obtained in case of channel with irregularities on its wall. Taking in account that the model of diffuse BC supposes full accommodation of both energy and momentum of the impinged atom, one can conclude that momentum change upon collision influences on the temperature driven flow more than energy change (momentum and energy accommodation by a rough surface has

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been discussed in Section 4.1; it was found that surface’s irregularities don’t influence on the average energy of scattered atoms).

One can see that flow rate through the channel of Type D is very close to result given by diffuse BC, thus this channel’s type was selected for studies of correlation of flow rate with length-to-height ratio shown in Table 4.3-2. It should be noted that the gap between values of flow rate through the smooth channel and channel with diffuse BC decreases with respect to the channel’s length, but only up to the value of 14%. This observation is on contrast to the results found in case of pressure driven flow, where the deviation decreases steady with respect to the channel’s length. It should be noted that convergence of the gap to the constant value meant that flow rate through the smooth tungsten channel becomes inversely proportional to the first order of the channel’s length in case of extremely long channel (this behavior does not contradict with known experimental and theoretical results). The gap between flow rates corresponding to the case of diffuse BC and real channel Type D is found to be independent on the channel’s length. It means that use of diffuse BC may give significant underestimations of flow rate even in case rough surface. This error can be explained in terms of probability of momentum accommodation which depends on surface roughness, angle of incidence of the gas atom and ratio of gas atom’s incident kinetic energy to the surface temperature (see Figure 4.1-15 and discussions in Section 4.1). It means that the value of probability of momentum accommodation is nonconstant along the channel with atomic structure having longitudinal temperature gradient, while it is constant in case of diffuse BC.

Another result shown in Table 4.3-2 is that implementation of new BC gives the same flow rate like smooth tungsten channel. It means that this statistical model allows one to simulate nonisothermal rarefied gas flow through metallic channels with

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reasonable accuracy and without taking in account atomic of a solid structure, i.e this model allows one to save computation time.

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