Reddit Engineering Finite Element What is Continuity

I. INTRODUCTION

Section:

Pressure profile calculation is an important task when designing a large ultra-high vacuum (UHV) system, such as a particle accelerator. ^1–3 1. Y. Kalinin and V. G. Rogozinskij, Vacuum 46, 717 (1995). https://doi.org/10.1016/0042-207x(94)00091-3 2. M. N. El-Shazly, A. V. Tikhomirov, G. G. Gulbekian, and P. Kováě, Vacuum 52, 401 (1999). https://doi.org/10.1016/s0042-207x(98)00318-2 3. M. Michel, Nucl. Inst. Methods Phys. Res. A 322, 1 (1992). https://doi.org/10.1016/0168-9002(92)90350-D The calculated pressure profiles are used to determine the positions and pumping speeds of pumps in the vacuum system and to set parameters for the dimension and shape of tubes.

The early techniques of pressure calculation have mainly focused on developing analytical equations, but it is only effective for simple geometries. ^1,2 1. Y. Kalinin and V. G. Rogozinskij, Vacuum 46, 717 (1995). https://doi.org/10.1016/0042-207x(94)00091-3 2. M. N. El-Shazly, A. V. Tikhomirov, G. G. Gulbekian, and P. Kováě, Vacuum 52, 401 (1999). https://doi.org/10.1016/s0042-207x(98)00318-2 Numerical methods based on linear kinetic theory or Navier–Stokes equations are studied to simulate gas flow of more complex cases, such as a network geometry consisting of tubes, a cylindrical tube with end-effects, a system with temperature drops, and micro-channels with a sudden expansion or contraction. ^4–9 4. S. Pantazis, D. Valougeorgis, and F. Sharipov, Vacuum 101, 306 (2014). https://doi.org/10.1016/j.vacuum.2013.09.015 5. F. M. Sharipov and V. D. Seleznev, J. Vac. Sci. Technol. A 12, 2933 (1994). https://doi.org/10.1116/1.578969 6. S. Misdanitis and D. Valougeorgis, J. Vac. Sci. Technol. A 29, 061602 (2011). https://doi.org/10.1116/1.3645582 7. F. Sharipov, J. Vac. Sci. Technol. A 15, 2434 (1997). https://doi.org/10.1116/1.580904 8. C. Tantos, J. Vac. Sci. Technol. A 37, 051602 (2019). https://doi.org/10.1116/1.5111528 9. A. Agrawal, L. Djenidi, and R. A. Antonia, J. Fluid Mech. 530, 135 (2005). https://doi.org/10.1017/s0022112005003691 A direct simulation Monte Carlo (DSMC) is a more flexible numerical method to obtain pressure profiles over the wide range of gas rarefaction, even in micro- or nanochannels. ^10–13 10. G. A. Bird, The DSMC Method (CreateSpace Independent Publishing Platform, 2013). 11. V. Varade, V. S. Duryodhan, A. Agrawal, A. M. Pradeep, A. Ebrahimi, and E. Roohi, Comput. Fluids 111, 46 (2015). https://doi.org/10.1016/j.compfluid.2014.12.024 12. A. Ebrahimi and E. Roohi, Microfluid. Nanofluid. 21, 18 (2017). https://doi.org/10.1007/s10404-017-1855-1 13. A. Ebrahimi and E. Roohi, J. Therm. Eng. 2, 709 (2016). https://doi.org/10.18186/jte.53652

In a free-molecular regime where the interactions between gases are neglected, test particle Monte Carlo (TPMC) simulation, ^14–17 14. R. Kersevan and J.-L. Pons, J. Vac. Sci. Technol. A 27, 1017 (2009). https://doi.org/10.1116/1.3153280 15. B. F. Macdonald, S. Bryan, H. S. Shiers, and M. P. Cox, Vacuum 84, 283 (2009). https://doi.org/10.1016/j.vacuum.2009.04.029 16. G. Y. Hsiung, C. K. Chan, and J. R. Chen, Vacuum 86, 1688 (2012). https://doi.org/10.1016/j.vacuum.2012.03.017 17. Y. Suetsugu, J. Vac. Sci. Technol. A 14, 245 (1996). https://doi.org/10.1116/1.579927 running on a personal computer, is more effective for pressure profile calculation and is widely used for evaluating the design of various vacuum systems recently. By tracking the test particles until they are evacuated, TPMC accurately determines the pressure profiles of a UHV system. However, if the system is complex, TPMC can take several hours or days to get a saturated result. Besides the calculation time, TPMC has the intrinsic difficulty that the 3D model of the vacuum system must be prepared several times before the design is finalized, which makes the optimization process slow and cumbersome.

The complexity of the pressure calculation increases with the size of the UHV system. Furthermore, the early stages of the design require frequent optimization iterations. Therefore, a simple and prompt calculation is desirable for this case. Pressure calculation that exploits the continuity of the gas flow or mass-balance equation ^3,18–25 3. M. Michel, Nucl. Inst. Methods Phys. Res. A 322, 1 (1992). https://doi.org/10.1016/0168-9002(92)90350-D 18. R. Reid, Vacuum 43, 31 (1992). https://doi.org/10.1016/0042-207x(92)90179-z 19. P. Li, L. Bozyk, Z. Dong, P. Spiller, M. Li, J. Yang, Y. Yuan, C. Luo, W. Xie, and W. Zhen, Vacuum 163, 15 (2019). https://doi.org/10.1016/j.vacuum.2019.02.004 20. Y. Miyahara, J. Vac. Sci. Technol. A 4, 111 (1986). https://doi.org/10.1116/1.573481 21. O. B. Malyshev and M. P. Cox, Vacuum 86, 1692 (2012). https://doi.org/10.1016/j.vacuum.2012.03.015 22. S. R. Wilson, J. Vac. Sci. Technol. A 5, 2479 (1987). https://doi.org/10.1116/1.574876 23. H. Hirano, Y. Kondo, and N. Yoshimura, J. Vac. Sci. Technol. A 6, 2865 (1988). https://doi.org/10.1116/1.575614 24. V. Ziemann, Vacuum 81, 866 (2007). https://doi.org/10.1016/j.vacuum.2006.10.003 25. J. P. J. van den Braand and A. P. Kaan, ESRF Report No. ESRF-IRM-61/84, ESRF, Grenoble, 1984. is one of the easiest ways to predict the pressure profiles. In this finite element analysis (FEA), a vacuum system is divided into many small subsections, and the mass balances between the subsections are considered to get the pressure of each element by matrix inversion. Once the input parameters for the elements are established, the result is obtained immediately, and the modification of the input parameters is undemanding.

Despite these advantages, the FEA method has several issues that need to be addressed and are not dealt with in the previous studies. ^{3,15,18,19,22} 3. M. Michel, Nucl. Inst. Methods Phys. Res. A 322, 1 (1992). https://doi.org/10.1016/0168-9002(92)90350-D 15. B. F. Macdonald, S. Bryan, H. S. Shiers, and M. P. Cox, Vacuum 84, 283 (2009). https://doi.org/10.1016/j.vacuum.2009.04.029 18. R. Reid, Vacuum 43, 31 (1992). https://doi.org/10.1016/0042-207x(92)90179-z 19. P. Li, L. Bozyk, Z. Dong, P. Spiller, M. Li, J. Yang, Y. Yuan, C. Luo, W. Xie, and W. Zhen, Vacuum 163, 15 (2019). https://doi.org/10.1016/j.vacuum.2019.02.004 22. S. R. Wilson, J. Vac. Sci. Technol. A 5, 2479 (1987). https://doi.org/10.1116/1.574876 First, the selection of the element's length L _e is arbitrary when divided into subsections from a large UHV system. Furthermore, the resulting element's conductance does not have strong theoretical legitimacy, and so the answer is doubtable. ¹⁸ 18. R. Reid, Vacuum 43, 31 (1992). https://doi.org/10.1016/0042-207x(92)90179-z Thus, L _e and element's conductance should be properly determined to ensure the validity of the result. Here, we introduce a modified finite element analysis (MFEA) to calculate the pressure profiles with the appropriate L _e and the element's conductance, considering the aperture conductance (Oatley method) ²⁶ 26. C. W. Oatley, Br. J. Appl. Phys. 8, 15 (1957). https://doi.org/10.1088/0508-3443/8/1/305 and the beaming effect. ^27,28 27. B. B. Dayton, J. Vac. Sci. Technol. 9, 243 (1972). https://doi.org/10.1116/1.1316568 28. S. In, Appl. Sci. Convergence Technol. 9, 1 (2000), see http://www.dbpia.co.kr/journal/articleDetail?nodeId=NODE09230554. Upon calculating the pressure profiles of a long vacuum tube, our MFEA result shows good agreement with the TPMC, both with and without wall sorption. Furthermore, the MFEA method can be extended for a two-dimensional vacuum system. We introduce the MFEA calculation for an accelerator vacuum system with a branch line as an example of practical application.

II. CALCULATION METHOD

Section:

A. One-dimensional vacuum system

The gas flow Q in a vacuum chamber is expressed as

where P (mbar) is the pressure, V (L) is the volume, and S (L s⁻¹) is the pumping speed. For the FEA, we divide the system into small segments [Fig. 1(a) ] and determine P in each part by exploiting the continuity principle of the gas flow. In a steady-state with dP/dt = 0, the continuity equation for segment i is given by

$$Q_i+C_{i-1}\left(P_{i-1}-P_i\right)+C_i\left(P_{i+1}-P_i\right)-P_i{S}_i=0,$$

(2)

where C (L s⁻¹) represents the conductance. ^3,15,23 3. M. Michel, Nucl. Inst. Methods Phys. Res. A 322, 1 (1992). https://doi.org/10.1016/0168-9002(92)90350-D 15. B. F. Macdonald, S. Bryan, H. S. Shiers, and M. P. Cox, Vacuum 84, 283 (2009). https://doi.org/10.1016/j.vacuum.2009.04.029 23. H. Hirano, Y. Kondo, and N. Yoshimura, J. Vac. Sci. Technol. A 6, 2865 (1988). https://doi.org/10.1116/1.575614

FIG. 1. (a) Schematic view of the continuity principle of the gas flow in the one-dimensional vacuum system: Q _i is the outgassing rate, P _i is the pressure, C _i is the conductance, and S _i is the effective pumping speed of an element i. (b) A cylindrical tube of length L is divided into finite elements of length L _e for pressure calculation using the FEA: d is the diameter, C is the conductance of the entire tube, C _o is its aperture conductance, and C _i is the in-between conductance.

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When we consider two cases of net flow, (1) i → i − 1 and i → i + 1 and (2) i − 1 → i + 1 or i + 1 → i − 1, the corresponding continuity equations are

$$Q_i-C_{i-1}\left(P_i-P_{i-1}\right)-C_i\left(P_i-P_{i+1}\right)-P_i{S}_i=0,$$

(3)

$$Q_i+C_{i-1}\left(P_{i-1}-P_i\right)-C_i\left(P_i-P_{i+1}\right)-P_i{S}_i=0,$$

(4)

and

$$Q_i-C_{i-1}\left(P_i-P_{i-1}\right)+C_i\left(P_{i+1}-P_i\right)-P_i{S}_i=0.$$

(5)

Equations (2)–(5) are the same, which means we do not need to consider the direction of the gas flow. When we divide the vacuum tube into n elements, we get n linear equations by substituting numbers from 1 to n into the index i in Eq. (2). These equations are simplified into a matrix notation P = M ⁻¹ Q, where M includes parameters, such as conductance and pumping speed, and are expressed as follows:

$$\left(\begin{array}{c}{P}_1\\ P_2\\ ⋮\\ P_{n-1}\\ P_n\end{array}\right)={\left(\begin{array}{ccccc}{C}_1+S_1& -C_1& & & 0\\ -C_1& C_1+C_2+S_2& -C_2& & \\ & -C_2& \ddots & \ddots & \\ & & \ddots & C_{n-2}+C_{n-1}+S_{n-1}& -C_{n-1}\\ 0& & & -C_{n-1}& C_{n-1}+S_n\end{array}\right)}^{-1}\left(\begin{array}{c}{Q}_1\\ Q_2\\ ⋮\\ Q_{n-1}\\ Q_n\end{array}\right).$$

(6)

For example, as shown in Fig. 1(b) , a long cylindrical tube of length L and conductance C is divided into n elements of length L _e with the aperture conductance C _o for pressure calculation using the FEA (Algorithm 1).

B. Conductance correction

In order to solve Eq. (6), we need to know S _i and Q _i of the ith element and C _i between the ith and the (i + 1)th elements. Normally, S _i and Q _i are simply given in a problem, whereas C _i is not because it depends on the number of elements we divide into (or on the length of the element, L _e ). If we consider each element separately, C _i can be written in the form C _i = C _o α _i , where α _i is the transmission probability of the element i calculated using Berman's formula ²⁹ 29. A. S. Berman, J. Appl. Phys. 36, 3356 (1965). https://doi.org/10.1063/1.1702984 with the ratio L _e /d. However, since the elements are joined to each other in a series, the redundant count of the entrance aperture conductance C _o should be corrected. By using the Oatley method, ²⁶ 26. C. W. Oatley, Br. J. Appl. Phys. 8, 15 (1957). https://doi.org/10.1088/0508-3443/8/1/305 the corrected conductance C _i ′ or the corrected transmission probability α _i ′ of the element i is expressed as

$$\frac{1}{C_i^{\prime }}=\frac{1}{C_i}-\frac{1}{C_o}\mathrm{o}\mathrm{r}\frac{1}{{\alpha }_i^{\prime }}=\frac{1}{{\alpha }_i}-1.$$

(7)

Although the pressure profile calculated with this modified FEA (MFEA) shows better accuracy than that using the uncorrected FEA, there is still a discrepancy between the MFEA and the TPMC results (Fig. 3 ).

Particles in a tube generally tend to move in the axial direction as they proceed. This tendency increases as the length of a tube increases; this is the so-called beaming effect. ^27,28 27. B. B. Dayton, J. Vac. Sci. Technol. 9, 243 (1972). https://doi.org/10.1116/1.1316568 28. S. In, Appl. Sci. Convergence Technol. 9, 1 (2000), see http://www.dbpia.co.kr/journal/articleDetail?nodeId=NODE09230554. Because of the beaming effect, the reciprocal of the transmission probability of the entire tube 1/α _c is not the same as the reciprocally combined transmission probability Σ1/α _i ′, even if α _i ′ is already corrected with the Oatley method. Therefore, one should include correction (K) for the beaming effect in the FEA calculation, where K = (Σ1/α _i ′)/(1/α _c) = nα _c/α′, if α _i ′ = α′ for all i (Algorithm 1). The validity of the MFEA will be discussed intensively in Sec. III A.

C. Two-dimensional vacuum system

The one-dimensional continuity equation [Eq. (2)] can be extended into the two-dimensional system consisting of two channels, A and B, as described in Fig. 2 . The ith elements of both the channels are interconnected via conductance CT _i . In a steady-state with dP/dt = 0, the continuity equations for segment i are given by

$$\begin{array}{ll}\hfill & Q{A}_i+C{A}_{i-1}\left(P{A}_{i-1}-P{A}_i\right)+C{A}_i\left(P{A}_{i+1}-P{A}_i\right)\hfill \\ \hfill & +C{T}_i\left(P{A}_i-P{B}_i\right)-P{A}_{i}S{A}_i=0,\hfill \end{array}$$

(8)

$$\begin{array}{ll}\hfill & Q{B}_i+C{B}_{i-1}\left(P{B}_{i-1}-P{B}_i\right)+C{B}_i\left(P{B}_{i+1}-P{B}_i\right)\hfill \\ \hfill & +C{T}_i\left(P{B}_i-P{A}_i\right)-P{B}_{i}S{B}_i=0.\hfill \end{array}$$

(9)

When we divide each channel into n elements, we get 2n linear equations by substituting numbers from 1 to n into the index i in Eqs. (8) and (9). Again, these equations are simplified into a matrix notation P = M ⁻¹ Q, where M consists of four sub-matrices of M ₁ , M ₂ , M ₃ , and M ₄ [Eqs. (10)–(13)]. The sectors M ₁ and M ₄ deal with the gas flow in channels A and B, respectively, whereas the sectors M ₂ and M ₃ deal with the gas flow between channel A and channel B,

(10)

where

$$M_1=\left(\begin{array}{ccccc}\hfill C{A}_1+S{A}_1+C{T}_1\hfill & \hfill -C{A}_1\hfill & \hfill \hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill -C{A}_1\hfill & \hfill C{A}_1+C{A}_2+S{A}_2+C{T}_2\hfill & \hfill -C{A}_2\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill -C{A}_2\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill C{A}_{n-2}+C{A}_{n-1}+S{A}_{n-1}+C{T}_{n-1}\hfill & \hfill -C{A}_{n-1}\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \hfill & \hfill -C{A}_{n-1}\hfill & \hfill C{A}_{n-1}+S{A}_n+C{T}_n\hfill \end{array}\right),$$

(11)

$$M_2=M_3=\left(\begin{array}{ccccc}\hfill -C{T}_1\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill -C{T}_2\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill -C{T}_{n-1}\hfill & \hfill \hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill -C{T}_n\hfill \end{array}\right),$$

(12)

and

$$M_4=\left(\begin{array}{ccccc}\hfill C{B}_1+S{B}_1+C{T}_1\hfill & \hfill -C{B}_1\hfill & \hfill \hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill -C{B}_1\hfill & \hfill C{B}_1+C{B}_2+S{B}_2+C{T}_2\hfill & \hfill -C{B}_2\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill -C{B}_2\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill C{B}_{n-2}+C{B}_{n-1}+S{B}_{n-1}+C{T}_{n-1}\hfill & \hfill -C{B}_{n-1}\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \hfill & \hfill -C{B}_{n-1}\hfill & \hfill C{B}_{n-1}+S{B}_n+C{T}_n.\hfill \end{array}\right)$$

(13)

FIG. 2. Schematic view of the continuity principle of the gas flow in the two-dimensional vacuum system consisting of two channels, A and B, which are interconnected via conductance CT _i ; the pressure, the outgassing rate, the pumping speed, and the conductance of the ith element of channel A are expressed as PA _i , QA _i , SA _i , and CA _i , and those of channel B are PB _i , QB _i , SB _i , and CB _i , respectively.

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FIG. 3. Pressure profiles of the cylindrical tube with L = 150d calculated with TPMC (black dashes), FEA (blue dots), and MFEA (red line). The MFEA shows a better agreement with TPMC, compared to the FEA.

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III. RESULTS AND DISCUSSION

Section:

A. One-dimensional vacuum system

For validation of the MFEA, we turn to the pressure calculation in a cylindrical vacuum tube with half-length L = 300 cm and diameter d = 2 cm. It has two UHV pumps, each with the pumping speed S = 0.6C _o , one at each end. The outgassing rate is 5 × 10⁻¹² mbar L s⁻¹ cm⁻², and L _e = 3d. We use the MFEA to calculate the pressure profile of N₂ gas with the parameters described above. With the same parameters, we show the results of the FEA and TPMC compared with that of the MFEA (Fig. 3 ). The MFEA agrees better than the FEA to the TPMC. This result suggests the importance of excluding the redundant aperture conductance of each segment in the FEA calculation.

To further investigate the pressure profile of the MFEA, we define a relative error ε = |1 − P _MFEA /P _MC | × 100%, where P _MFEA and P _MC are the average pressures obtained from MFEA and TPMC, respectively, and then plot ε as a function of L _e /d (d = 2 cm) [Fig. 4(a) ]. Several features are noteworthy. First, overall, the error of the MFEA is smaller than that of the FEA, as expected. Second, the error curve shows a minimum point L _min (dashed arrow), which signals that finding L _min is useful to the calculation. At element's lengths > L _min, the error grows as L _e /d of a segment increases. This tendency results from an intrinsic property of the FEA, in which the error increases as the mesh size increases .

FIG. 4. (a) The error curves between FEA and TPMC (black squares) and between MFEA and TPMC (red circles) as a function of L _e /d or the corresponding transmission probability α calculated using Berman's formula. ²⁹ 29. A. S. Berman, J. Appl. Phys. 36, 3356 (1965). https://doi.org/10.1063/1.1702984 (b) The error curve after the beaming effect correction (blue triangles) and that of the MFEA (red circles).

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However, below L _min, the error also increases as the L _e /d decreases. This is different from the general tendency of the FEA in which the smaller element size produces the more accurate result. This result is a consequence of the beaming effect. The error that comes from the beaming effect is corrected by multiplying the transmission probability of each segment by the correction factor K. This correction reduces the error evolution to as low as 5% at L _e /d $\le$ 5 [Fig. 4(b) ].

So far, we have investigated the validity of the MFEA in a simple cylindrical tube and found that the calculated pressure profile is as accurate as the result from TPMC. For practical applications, we have also checked the validity of the MFEA for vacuum tubes with non-cylindrical cross-sections. In this case, an effective diameter d _eff , which is defined by 2 × (cross-sectional area/π)^1/2, is used instead of d in the calculation of the MFEA. With the element length L _e = 3d _eff , where the relative error is minimum as shown in Fig. 4(b) , we calculated the pressure profiles of the vacuum tubes with various cross-sectional aspect ratios. Figure 5 shows relative errors of the MFEA from TPMC concerning the aspect ratios. The relative error is less than 5% for the aspect ratio <2, and the accuracy is similar to that for the cylindrical tubes. Even though relatively large errors appear when the aspect ratio is larger than 2, these are still in the range of confidence of practical ionization gauges (e.g., 20% of the reading error). ²⁸ 28. S. In, Appl. Sci. Convergence Technol. 9, 1 (2000), see http://www.dbpia.co.kr/journal/articleDetail?nodeId=NODE09230554. The aspect ratios of the vacuum tubes in a particle accelerator are usually between 1 and 10. Therefore, the MFEA is also applicable to predict the pressure distribution of various cross-sectional tubes of an accelerator vacuum system using the effective diameter.

FIG. 5. The error curve between MFEA and TPMC concerning the aspect ratio of the non-cylindrical cross-sectional tubes.

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We now consider the effect of the wall sorption of a tube for H₂ gas [e.g., the wall coated by non-evaporable getter (NEG) materials]. At L _e < L _min [Fig. 4(a) ], we calculate the relative error with a wide span of sticking coefficients 10⁻⁷ ≤ s ≤ 10⁻² for the FEA [Fig. 6(a) ] and MFEA [Fig. 6(b) ]. Both results show that the error starts to increase at s ≈ 10⁻⁴ and tends to saturate at s ≈ 10⁻⁶ where the wall pumping is negligible. At s $\ge$ 10⁻⁴, the errors become as less as 20% for the FEA and 6% for the MFEA, regardless of the element's length. This result indicates that the wall-pumping effect dominates the effect of conductance between the elements in the pressure profile calculation. The line profiles obtained along transects through color maps (broken white lines) show that the errors reduce at any element length with increasing s because of the effect of the wall pumping.

FIG. 6. Color maps of ε with L _e /d and s in (a) FEA and (b) MFEA. The line profiles taken along the dotted lines in (a) and (b) are shown as (c) and (d) to indicate the trend of decreasing errors with increasing sticking coefficient.

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B. Two-dimensional vacuum system

The MFEA method can be applied to the practical vacuum system, such as the particle accelerator beam chamber, near the crotch area where the beam chamber is split into two beam channels (e.g., an electron storage ring and a photon beam line), as shown in Fig. 7(a) . In this example, the length of the electron beam chamber (L _A ) and photon beam line (L _B ) is 294 and 114 cm, respectively. The diameters of both chambers are the same (6 cm). The inner wall of the electron beam chamber is assumed to be NEG coated, which has a sticking coefficient s = 0.005. There are additional four identical lumped pumps of pumping speed 100 L/s in the system. The outgassing rate from the inner chamber is 5 × 10⁻¹² mbar L s⁻¹ cm⁻² and that from the crotch area of 82.53 cm² is 1.2 × 10⁻⁸ mbar L s⁻¹ cm⁻², as indicated in Fig. 7(a) .

FIG. 7. (a) The particle accelerator beam chamber near the crotch area where the beam chamber is split into two beam channels. (b) Schematic view of the continuity principle of gas flow of the split vacuum system for the MFEA calculation.

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Using the matrix equation expressed in Eqs. (10)–(13), the pressure profiles of this two-dimensional system can be obtained by the MFEA. First, we divide each channel into 49 elements, as shown in Fig. 7(b) . Channel A and channel B are assigned to the electron beam chamber and the photon beam chamber, respectively. Every inter-channel conductance CT _i is zero except for the 30th element where the chamber is split. The conductance of the 29th element of channel B is also zero because there is no photon beam line before the beam chamber is split. The outgassing rate of all the elements, QA_i and QB_i , is 5.65 × 10⁻¹⁰ mbar L s⁻¹ except for the 31st element, which includes the crotch area (QA₃₁ = 1 × 10⁻⁶ mbar L s⁻¹). The pumping speed of all the elements, SA_i and SB_i , is 6.6 L/s except for the four elements, including the lumped pumps (SA ₈ = SA ₂₁ = SA ₄₆ = SB ₁₆ = 100 L/s). Solving Eq. (10) with these parameters, we get the pressure profiles of channels A and B, as shown in Fig. 8 . The MFEA results show good agreement with those of TPMC.

FIG. 8. Comparison of the pressure profiles of the split chamber described in Fig. 7 calculated by the MFEA and the TPMC. Solid lines indicate the pressure profiles of the electron beam chamber by the TPMC (black line) and the MFEA (blue triangles), and the dashed lines indicate those of the photon beam chamber by the TPMC (red dots) and the MFEA (green squares).

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IV. CONCLUSION

Section:

The MFEA method for designing a large UHV system has been proposed in which the element's conductance is corrected not only by using the Oatley method but also by compensating the beaming effect. To validate the MFEA, we have calculated the pressure profiles of a one-dimensional cylindrical tube and compared the relative errors between the results from the MFEA and the TPMC. The MFEA has also been tested in several practical cases, such as vacuum chambers with wall sorption, non-cylindrical tubes with various aspect ratios, and the split chamber of a particle accelerator. The results are summarized as follows:

(1)	The average pressure of a 1D cylindrical tube calculated by the MFEA is as accurate as that by the TPMC, and the relative error is only 5% or less compared to the TPMC for L e /d ≤ 5, which means the length of the element for the FEA calculation is desirable to be selected in the range L e /d ≤ 5.
(2)	For the non-cylindrical tubes, the relative error is less than 5% for the aspect ratio < 2, and even for the aspect ratio close to 10, it is in the similar range of error as the practical ionization gauge (20%).
(3)	The validity of the MFEA has been checked for a vacuum chamber with wall sorption, such as the NEG coated chamber. When the sticking coefficient s is larger than 10−4, the relative error is only 6% or less, regardless of the element's length because the gas flow by the wall-pumping dominates the gas flow through the elements of the tube in the highly sticky condition.
(4)	The MFEA can be extended to the calculation of a practical 2D system, such as the accelerator vacuum system, with a split chamber around a crotch area with a high degree of accuracy.

The simple and efficient MFEA calculation provides a significant reduction in workload for pressure calculation in a UHV system while retaining remarkable accuracy. This method will be of great help when applied to large UHV systems that require frequent design iterations.

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Source: https://aip.scitation.org/doi/10.1063/6.0001583

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