- Email: [email protected]
Influence of the vacuum chamber shape on the non-uniformity of gas distribution
Vacuum 53 (1999) 193 — 196
Influence of the vacuum chamber shape on the non-uniformity of gas distribution S.B. Nesterov*, Yu.K. Vassiliev, A.P. Kryukov Low Temperature Department, Moscow Power Engineering Institute, Krasnokazarmennaya St., 14, Moscow, 111250, Russia
Abstract In this paper we describe the molecule concentration distribution obtained by calculation methods inside vacuum volumes of different shape (cube, cylinder, sphere and their combinations), assuming the isothermal conditions of all surfaces. It is assumed that the particles do not collide with each other (Kn<1). The reflection from solid boundary is described by cosine distribution. The results obtained by calculation show that clearly expressed anisotropy of concentration distribution takes place in the volume limited by curve surfaces (sphere, cylinder). The concentration distribution is uniform in the volume limited by flat surfaces (cube, polyhedron). Experiments were held to check this phenomenon. The analysis of experimental dependences shows that particle concentration value inside the spherical volume is constant by the radius in the range of Knudsen numbers 1(Kn(30. On investigating the inner surface of the experimental sphere with an electron microscope it was revealed that the real surface included a number of surfaces which were not smooth. Thus the real concentration distribution depends on the surface character forming the vacuum volume. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The term ‘‘non-uniformity of gas distribution’’ refers to the change of the flow, density and pressure values depending on the position in a vacuum system. Gas often does not obey the Maxwell distribution laws in real vacuum systems. Six reasons for the non-uniformity of gas distribution were formulated in [1]: E the presence of a pump; E the presence of a gas source; E the difference between the particle reflection law and the cosine law; E particle migration on the surface; E adsorption—desorption processes; E temperature differences inside the vacuum system. In this context the vacuum gauge position is also important [2]. Assuming the isothermal conditions of all surfaces the molecule concentration distribution is obtained inside vacuum volumes of different shape (a cube, a cylinder,
* Corresponding author. Tel.: 095 362 7610; fax: 095 362 8643; e-mail: [email protected]
and a sphere). It is supposed that particles do not collide with each other (Kn<1). The experimental apparatus was constructed and prepared for an experimental test of obtained results.
2. Method description Modelling of gas flow is made by the Monte-Carlo method. This is a numerical method to solve mathematical tasks by modelling of incidental values with a given distribution law, by the construction of probability models and statistical evaluation of the results. The results obtained are for a free-molecular mode of gas flow and for the diffuse law of particle reflection from the surface. The initial particle distribution in the volume is uniform. 2.1. Model testing To test the model operation Clausing coefficients calculated by means of the program realizing the given model were compared with the corresponding experimental data obtained from different sources. Comparison
0042-207X/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 3 6 1 - 3
194
S.B. Nesterov et al. / Vacuum 53 (1999) 193—196
has shown that the model operation error at least does not exceed 7% (and such cases are very few). Comparison of the calculated results of concentration ratio for the two volumes with different temperature connected by an orifice with theoretical conclusions of the Knudsen effect, has shown that the calculated error here does not exceed 2%. Another model test was made where Clausing coefficients of spherical channels were compared. Maximum diversion of results shown by the model and results of analytical calculations given in [3] did not exceed 1%. 2.2. Concentration distribution The aim of this analysis is to compare particle behavior obeying the diffuse law of reflection in structures consisting of linear and curvilinear surfaces. 2.2.1. Cube Concentration distribution inside a cube is shown in Fig. 1. The charts show that concentration distribution inside a cube is of uniform character. But in the cube angles a certain non-uniformity though small is observed which is probably due to the calculation error and not to the structure of the analyzed system. These results completely agree with general knowledge of particle behavior inside the vacuum chamber. 2.2.2. Cylinder A dual picture is observed inside a cylinder (Fig. 2). Along the longitudinal cylinder axis the distribution is of uniform character but in the diametrical direction (along the radius) distribution becomes obviously anisotropic, where the difference of concentrations is 70—100%.
Fig. 2. Particle distribution inside a cylinder along the radius in central part as well as along the longitudinal coordinate.
So one may suppose that the non-uniformity existence is explained by the presence of curvilinear surfaces. The point is that by reflecting from a flat surface a particle has a chance to fly in any direction (though not equal), and by leaving a curved surface there appear directions where a particle cannot go. Due to this the probability to fly in other directions grows (which is very small in a flat case).
Fig. 1. Particle distribution inside a cube.
2.2.3. Sphere Inside the sphere where effects of flat surfaces do not exist, the non-uniformity of distribution is expressed very distinctly (Fig. 3). This proves the previous hypothesis of how curvilinear surfaces influence the particle spatial distribution. Several calculations were made for a sphere to elucidate how various factors affect the distribution. A
S.B. Nesterov et al. / Vacuum 53 (1999) 193—196
195
Fig. 4. Vacuum introduction design.
Fig. 3. Particle distribution inside a sphere (R"0.135 m).
calculation of concentration distribution inside the sphere was made at different start conditions. It is shown that distribution acquires uniform character in a short period of time irrespective of the start conditions (from a small sphere, in the center or from a volume). 5—10 collisions of particle with the surface are sufficient for stabilization. So it is easy to determine the time of distribution stabilization.
a sphere and back. A signal is sent to vacuum gauge by means of this ionization gauge helping to detect the change in particle concentration. Thus the dependence on the sphere radius can be obtained. Rough pumping of the sphere was made by the pumping system at room temperature. The installation allowed to obtain the dependence P"f (r), where r is a radius for Knudsen numbers in the range 1(Kn(3. To realize regimes corresponding to Kn'10 a moving gauge was installed with half-removed glass retort. To reduce background pressure the sphere was connected by means of cylindrical tubing with the other similar sphere immersed into liquid helium. This allowed to hold an experiment at conditions that corresponded to Kn"30 (a molecular mode). 3.2. Experimental results and their discussion
3. Experiment on particle distribution inside the spherical volume To test calculation results an experiment was held to determine the spatial distribution of concentration inside the spherical volume as this shape of vacuum system reflected the obtained results very clearly. 3.1. Experimental installation Experiments to determine concentration distribution inside a sphere are held in high vacuum installation which allows to make measurements in the low temperature range [4]. The experimental installation consists of four autonomous subsystems which can be used in other installations according to their purpose: (1) a cryoblock, (2) a system of rough pumping, (3) a system of helium vapor pumping, (4) an injection system (gas injection into the vacuum chamber). The installation is also equipped by the system of monitoring and measurement to register process parameters. A special vacuum introduction was made (Fig. 4) for this experiment which allowed to move the ionization gauge horizontally from the center to the inner surface of
The dependence obtained is shown in Fig. 5. The ionization gauge is connected to the side surface of the sphere through a flange connection. The analysis of dependence provides an opportunity to conclude that the particle concentration value inside a spherical volume is constant by radius in the Knudsen number range 1(Kn(30. Thus the experiment has shown clearly that the concentration distribution is uniform inside the spherical volume. Does the tested volume really have a spherical form? If it is viewed at the molecular level, i.e. at enlargement adequate to a molecule size, then it turns out that the sphere surface consists of a number of flat surfaces. It was observed by means of electron microscope with magnification order 10. So the hypothesis of the surface character effect on the concentration distribution is confirmed again. If a surface could be processed in such a way that at a dimension of about a molecule radius, it would have a spherical shape, then the result could be close to the calculated one. Hence the following conclusion can be made on the basis of the computation: spatial concentration distribution depends on the vacuum system configuration; if
196
S.B. Nesterov et al. / Vacuum 53 (1999) 193—196
Fig. 5. Experimental results.
curvilinear surfaces are considered then a non-uniform distribution occurs. An issue of particle behavior transformation when linear surfaces approach curvilinear ones remains open. Thus it is still not clear at what moment and how does the transition occur from a number of many flat surfaces to a curvilinear one. How will the transition take place from a polygonal polyhedron to a cylinder, for example. The following calculations were made: a polyhedron approached the infinite cylinder and a picture of particle distribution along the radial coordinate (like the cylinder radius) was found. The distribution is of uniform character, the same as the distribution inside the cube. By the growth of period number from 3 to 17 there are no serious changes in the distribution picture; that is why the issue of transition from a polyhedron to a cylinder remains open. 4. Conclusions 1. From calculations it is clear that the spatial concentration distribution depends on the vacuum system configuration — if curvilinear surfaces exist, then serious distortions of the concentration field are possible.
2. The experiment itself has shown that the real concentration distribution in a spherical volume is uniform. The analysis of the sphere surface has shown that in fact it consists of flat fragments; that is why we believe that the experimental results do not contradict the calculated ones. 3. The possible explanation of this difference is as follows: calculation of curvilinear structures has been made for ideal uniform surface. The experiment was made for real surfaces consisting of a number of planes.
References [1] Moore BC. Causes and consequences of non-uniform gas distributions in vacuum systems. J Vacuum Sci Technol 1968;6(1): 246—54. [2] Haefer RA. The application of ionization vacuum gauges to measurement in vacuum chambers provided with cryopumping surfaces. Vacuum 1980;30(415):193—5. [3] Koshmarov YuA, Rijov YuA. Applied rarefied gas dynamics, 1st ed. Moscow: Russia, 1977. [4] Kachalin GV, Kryukov AP, Nesterov SB. Adsorption of gaseous helium near ¹ at low pressures. Low Temp Phys 1998; H 24(2):97—9.
Influence of the vacuum chamber shape on the non-uniformity of gas distribution S.B. Nesterov*, Yu.K. Vassiliev, A.P. Kryukov Low Temperature Department, Moscow Power Engineering Institute, Krasnokazarmennaya St., 14, Moscow, 111250, Russia
Abstract In this paper we describe the molecule concentration distribution obtained by calculation methods inside vacuum volumes of different shape (cube, cylinder, sphere and their combinations), assuming the isothermal conditions of all surfaces. It is assumed that the particles do not collide with each other (Kn<1). The reflection from solid boundary is described by cosine distribution. The results obtained by calculation show that clearly expressed anisotropy of concentration distribution takes place in the volume limited by curve surfaces (sphere, cylinder). The concentration distribution is uniform in the volume limited by flat surfaces (cube, polyhedron). Experiments were held to check this phenomenon. The analysis of experimental dependences shows that particle concentration value inside the spherical volume is constant by the radius in the range of Knudsen numbers 1(Kn(30. On investigating the inner surface of the experimental sphere with an electron microscope it was revealed that the real surface included a number of surfaces which were not smooth. Thus the real concentration distribution depends on the surface character forming the vacuum volume. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The term ‘‘non-uniformity of gas distribution’’ refers to the change of the flow, density and pressure values depending on the position in a vacuum system. Gas often does not obey the Maxwell distribution laws in real vacuum systems. Six reasons for the non-uniformity of gas distribution were formulated in [1]: E the presence of a pump; E the presence of a gas source; E the difference between the particle reflection law and the cosine law; E particle migration on the surface; E adsorption—desorption processes; E temperature differences inside the vacuum system. In this context the vacuum gauge position is also important [2]. Assuming the isothermal conditions of all surfaces the molecule concentration distribution is obtained inside vacuum volumes of different shape (a cube, a cylinder,
* Corresponding author. Tel.: 095 362 7610; fax: 095 362 8643; e-mail: [email protected]
and a sphere). It is supposed that particles do not collide with each other (Kn<1). The experimental apparatus was constructed and prepared for an experimental test of obtained results.
2. Method description Modelling of gas flow is made by the Monte-Carlo method. This is a numerical method to solve mathematical tasks by modelling of incidental values with a given distribution law, by the construction of probability models and statistical evaluation of the results. The results obtained are for a free-molecular mode of gas flow and for the diffuse law of particle reflection from the surface. The initial particle distribution in the volume is uniform. 2.1. Model testing To test the model operation Clausing coefficients calculated by means of the program realizing the given model were compared with the corresponding experimental data obtained from different sources. Comparison
0042-207X/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 3 6 1 - 3
194
S.B. Nesterov et al. / Vacuum 53 (1999) 193—196
has shown that the model operation error at least does not exceed 7% (and such cases are very few). Comparison of the calculated results of concentration ratio for the two volumes with different temperature connected by an orifice with theoretical conclusions of the Knudsen effect, has shown that the calculated error here does not exceed 2%. Another model test was made where Clausing coefficients of spherical channels were compared. Maximum diversion of results shown by the model and results of analytical calculations given in [3] did not exceed 1%. 2.2. Concentration distribution The aim of this analysis is to compare particle behavior obeying the diffuse law of reflection in structures consisting of linear and curvilinear surfaces. 2.2.1. Cube Concentration distribution inside a cube is shown in Fig. 1. The charts show that concentration distribution inside a cube is of uniform character. But in the cube angles a certain non-uniformity though small is observed which is probably due to the calculation error and not to the structure of the analyzed system. These results completely agree with general knowledge of particle behavior inside the vacuum chamber. 2.2.2. Cylinder A dual picture is observed inside a cylinder (Fig. 2). Along the longitudinal cylinder axis the distribution is of uniform character but in the diametrical direction (along the radius) distribution becomes obviously anisotropic, where the difference of concentrations is 70—100%.
Fig. 2. Particle distribution inside a cylinder along the radius in central part as well as along the longitudinal coordinate.
So one may suppose that the non-uniformity existence is explained by the presence of curvilinear surfaces. The point is that by reflecting from a flat surface a particle has a chance to fly in any direction (though not equal), and by leaving a curved surface there appear directions where a particle cannot go. Due to this the probability to fly in other directions grows (which is very small in a flat case).
Fig. 1. Particle distribution inside a cube.
2.2.3. Sphere Inside the sphere where effects of flat surfaces do not exist, the non-uniformity of distribution is expressed very distinctly (Fig. 3). This proves the previous hypothesis of how curvilinear surfaces influence the particle spatial distribution. Several calculations were made for a sphere to elucidate how various factors affect the distribution. A
S.B. Nesterov et al. / Vacuum 53 (1999) 193—196
195
Fig. 4. Vacuum introduction design.
Fig. 3. Particle distribution inside a sphere (R"0.135 m).
calculation of concentration distribution inside the sphere was made at different start conditions. It is shown that distribution acquires uniform character in a short period of time irrespective of the start conditions (from a small sphere, in the center or from a volume). 5—10 collisions of particle with the surface are sufficient for stabilization. So it is easy to determine the time of distribution stabilization.
a sphere and back. A signal is sent to vacuum gauge by means of this ionization gauge helping to detect the change in particle concentration. Thus the dependence on the sphere radius can be obtained. Rough pumping of the sphere was made by the pumping system at room temperature. The installation allowed to obtain the dependence P"f (r), where r is a radius for Knudsen numbers in the range 1(Kn(3. To realize regimes corresponding to Kn'10 a moving gauge was installed with half-removed glass retort. To reduce background pressure the sphere was connected by means of cylindrical tubing with the other similar sphere immersed into liquid helium. This allowed to hold an experiment at conditions that corresponded to Kn"30 (a molecular mode). 3.2. Experimental results and their discussion
3. Experiment on particle distribution inside the spherical volume To test calculation results an experiment was held to determine the spatial distribution of concentration inside the spherical volume as this shape of vacuum system reflected the obtained results very clearly. 3.1. Experimental installation Experiments to determine concentration distribution inside a sphere are held in high vacuum installation which allows to make measurements in the low temperature range [4]. The experimental installation consists of four autonomous subsystems which can be used in other installations according to their purpose: (1) a cryoblock, (2) a system of rough pumping, (3) a system of helium vapor pumping, (4) an injection system (gas injection into the vacuum chamber). The installation is also equipped by the system of monitoring and measurement to register process parameters. A special vacuum introduction was made (Fig. 4) for this experiment which allowed to move the ionization gauge horizontally from the center to the inner surface of
The dependence obtained is shown in Fig. 5. The ionization gauge is connected to the side surface of the sphere through a flange connection. The analysis of dependence provides an opportunity to conclude that the particle concentration value inside a spherical volume is constant by radius in the Knudsen number range 1(Kn(30. Thus the experiment has shown clearly that the concentration distribution is uniform inside the spherical volume. Does the tested volume really have a spherical form? If it is viewed at the molecular level, i.e. at enlargement adequate to a molecule size, then it turns out that the sphere surface consists of a number of flat surfaces. It was observed by means of electron microscope with magnification order 10. So the hypothesis of the surface character effect on the concentration distribution is confirmed again. If a surface could be processed in such a way that at a dimension of about a molecule radius, it would have a spherical shape, then the result could be close to the calculated one. Hence the following conclusion can be made on the basis of the computation: spatial concentration distribution depends on the vacuum system configuration; if
196
S.B. Nesterov et al. / Vacuum 53 (1999) 193—196
Fig. 5. Experimental results.
curvilinear surfaces are considered then a non-uniform distribution occurs. An issue of particle behavior transformation when linear surfaces approach curvilinear ones remains open. Thus it is still not clear at what moment and how does the transition occur from a number of many flat surfaces to a curvilinear one. How will the transition take place from a polygonal polyhedron to a cylinder, for example. The following calculations were made: a polyhedron approached the infinite cylinder and a picture of particle distribution along the radial coordinate (like the cylinder radius) was found. The distribution is of uniform character, the same as the distribution inside the cube. By the growth of period number from 3 to 17 there are no serious changes in the distribution picture; that is why the issue of transition from a polyhedron to a cylinder remains open. 4. Conclusions 1. From calculations it is clear that the spatial concentration distribution depends on the vacuum system configuration — if curvilinear surfaces exist, then serious distortions of the concentration field are possible.
2. The experiment itself has shown that the real concentration distribution in a spherical volume is uniform. The analysis of the sphere surface has shown that in fact it consists of flat fragments; that is why we believe that the experimental results do not contradict the calculated ones. 3. The possible explanation of this difference is as follows: calculation of curvilinear structures has been made for ideal uniform surface. The experiment was made for real surfaces consisting of a number of planes.
References [1] Moore BC. Causes and consequences of non-uniform gas distributions in vacuum systems. J Vacuum Sci Technol 1968;6(1): 246—54. [2] Haefer RA. The application of ionization vacuum gauges to measurement in vacuum chambers provided with cryopumping surfaces. Vacuum 1980;30(415):193—5. [3] Koshmarov YuA, Rijov YuA. Applied rarefied gas dynamics, 1st ed. Moscow: Russia, 1977. [4] Kachalin GV, Kryukov AP, Nesterov SB. Adsorption of gaseous helium near ¹ at low pressures. Low Temp Phys 1998; H 24(2):97—9.