From Current Literature

THEORY OF HIGH-SPEED VACUUM PUMPS*)

Existing high-vacuum pumps—diffusion pumps—are based, according to generally recognized views, on Gaede’s diffusion principle. This principle, according to Gaede himself, briefly stated, is as follows: mercury vapor flowing along a gradient of decreasing pressure, at pressures of about \(1\) mm, is separated from the space being evacuated by a diaphragm with an aperture whose width does not exceed the mean free path of vapor molecules in the vicinity of the aperture. If the width of the aperture is greater than that indicated, then the density of the vapor passing through the aperture is so great that the molecules of the gas being pumped out will be thrown back into the evacuated space. With a normal aperture width, the gas molecules diffuse into the vapor jet and are carried away by it. The author of the article under review points out that, if the pumping action of pumps is described by this principle, then their pumping speed at permissible sizes is limited. He substantiates this as follows: the backing pressure according to Gaede is equal to \(0.1\) mm of mercury. This means that the gas pressure at any point between the aperture and the backing pipe is at least \(0.1\) mm. The vapor pressure in the jet must be considerably higher, in order to prevent gas from flowing back toward the aperture. It must be practically not less than \(1\) mm. At this vapor density the mean free path of vapor molecules must be of the order of \(0.1\) mm, and the width of the aperture therefore cannot exceed this value. The volume of air diffusing into the vacuum at normal temperatures is approximately \(12\) l/sec·cm\(^2\). In a pump, no more than half of this theoretical maximum can be attained. To obtain a pumping speed of even \(200\) l/sec, the aperture must be \(33\) cm\(^2\), and thus its length (even if the width is \(0.5\) mm) must be \(6.6\) m, i.e., the vapor would have to flow through a tube more than \(2\) m in diameter. This is an obvious absurdity!

Gaede asserted (Zeits. techn. Phys., 1923) that all mercury-vapor pumps capable of creating a high vacuum are based on his principle. If this is so, then high-vacuum pumps have no future, for their speed is fundamentally limited. However, Langmuir believes that his pump operates on a different principle. In Langmuir’s pump the vapor stream, flowing at high speed in the direction of pumping, carries gas molecules toward the walls and along the walls to the backing pipe. The walls of the pump must in this case be cooled; otherwise vapor molecules, reaching them, are adsorbed and then fly off uniformly in all directions, including in the direction opposite to the pumping direction. This reduces or even completely stops the pumping action. Langmuir attached special importance to the high velocity of the vapor stream. Crawford, in order to increase this velocity, used a divergent nozzle directed toward the evac—

) P. Alexander, Journal of Scientific Instruments 23*, No. 1, January 1946.

THEORY OF HIGH-SPEED VACUUM PUMPS

pumping, through which the vapor was released from the boiler. If one takes into account that Crawford’s pump was not cooled and nevertheless operated quite effectively, one may state that he clearly demonstrated the influence of the velocity of the vapor stream on the pumping action.

However, Gaede asserted that pumps of this type are also diffusion pumps, and that the aperture of the diaphragm here is the smallest distance between the edge of the nozzle and the wall of the pump—the throat of the pump. Applying Maxwell’s law of the distribution of molecular velocities, Gaede concluded that in this case a substantial fraction of the vapor molecules would always leave the vapor stream upward, opposite to the direction of pumping, and that the function of the diaphragm would be to reduce the number of these molecules. If the width of the throat is greater than the mean free path of the vapor molecules, then the pump cannot operate at low pressures. The author of the article under review himself counted the number of molecules tending to the side opposite to the direction of pumping, and arrived at a different conclusion.

If the number of molecules moving with velocities from \(c\) to \(c+dc\)

\[ N_{dc}=\frac{4N}{\sqrt{\pi}}x^2 e^{-x^2}\,dx, \]

where \(x=c/\alpha\), and \(\alpha\) is the most probable velocity, and if all these molecules are represented by vectors issuing from a single point, then all these vectors will terminate at points uniformly distributed over the surface of a sphere of radius \(c\). A motion of the vapor stream with velocity \(s\), directed straight downward, will be superposed on this molecular motion (Fig. 1).

Fig. 1.

All molecules whose resultant vectors end within the limits of the upper hemisphere (Fig. 1) can be regarded as moving backward. The vectors of such molecules terminate on the surface of a segment of height \(c-s\). Their number is

\[ N_- = N_{dc}\,\frac{2c(c-s)}{4c^2}=N_{dc}\,\frac{c-s}{2c}. \]

Expressing \(s\) through \(\alpha\), \(s=a\alpha\), we obtain:

\[ N_-=\frac{N_{dc}}{2}\left(1-\frac{a}{x}\right). \]

Substituting the Maxwellian value of \(N_{dc}\) into the right-hand side, we finally obtain

\[ N_-=\frac{2N}{\sqrt{\pi}}(x^2-ax)e^{-x^2}\,dx. \]

Integration from \(a\) to \(\infty\) gives the total number of molecules which go backward from a vapor stream having velocity \(a\). A rough estimate shows that

\[ \begin{aligned} N_- &= 80 \text{ out of } 1000 \ (N=1000), && \text{if } a=1,\\ N_- &= 17 \text{ out of } 1000 \ (N=1000), && \text{if } a=1.5,\\ N_- &= 2.3 \text{ out of } 1000 \ (N=1000), && \text{if } a=2. \end{aligned} \]

Thus, if the velocity of the vapor stream approaches a velocity equal to 1.5 times the most probable molecular velocity, then the jet of vapor, moving in the direction opposite to the direction of pumping, will become so small that it will not affect the pumping speed of the pump.

It remains to determine whether the vapor stream reaches such velocities and, if it does, under what circumstances. The answer to this question is given by experiment. To carry out the experiment, a pump was constructed which, in essence, was Langmuir’s pump with a divergent annular nozzle. This pump served as the prototype of a new high-speed pump. Naturally, the velocity of the vapor motion in it cannot be measured directly, but it can be calculated from the mass of mercury passing through the nozzle gap per unit time. In the experiment described, this measurement was made. In this case the vapor pressure in the nozzle gap was taken to be equal to \(3/4\) of the pressure in the boiler.

Examination of the experimental results (given in the article) shows that the value \(a\) varies from 1.2 to 3.4 and becomes less than 1.5 only in the case of a very large nozzle gap (2.33 mm).

The same result was also obtained in experiments on the evaporation of cadmium and the subsequent deposition of it on glass plates placed at different angles to the direction of motion of the stream: on the plates located on the side opposite to the direction of pumping, the deposit was vanishingly small.

All these data give a negative answer to the question of the validity of Gaede’s principle. At the same time, on the basis of Langmuir’s explanation of the pumping action, these results make it possible to create the following picture of the pumping process. Due to the possibilities discussed above, for example because of the velocities of the mercury molecules, the velocities of the gas molecules entering the vapor stream at the mouth of the pump attain higher values than those corresponding to the gas temperature. Therefore the velocity of the gas flow below the nozzle will be greater than above it, and the gas density in this region will be lower. Further, still lower than the nozzle, the vapor becomes less dense and its entraining action on the gas decreases and, finally, disappears (the decrease in vapor density is explained by its expansion in the nozzle region and by adsorption on the cold walls). Correspondingly, the velocity of the gas flow decreases, and its density increases, reaching a maximum below, closer to the fore-vacuum branch pipe. The mercury vapor must remain sufficiently dense throughout in order to prevent the vapor from flowing back from regions of high density to regions of low density. The vapor will be least dense at the periphery—along the pump walls. Widening of the throat leads to a greater decrease in density. At a certain width of it the vapor stream along the walls will cease to be sufficiently dense to lock the compressed gas, and it will break back, into the region of lower pressures.

These considerations are fully confirmed by the following experiment: pressures were measured at a number of points between the throat and the fore-vacuum branch pipe. Points at which the pressure proved to be the same were connected by lines forming a series of isobars (see Fig. 2).

In diagrams I and II the distribution of gas densities can be compared with standing waves having a node in the point lying directly under the nozzle and an antinode lower down, toward the fore-vacuum branch pipe. The vapor must be sufficiently dense that, by impacts of its molecules, it drives the gas molecules into the region of the antinode from the region of the node and thereby produces the locking action. If along some vertical line the vapor is not sufficiently dense, there is no wave; that is, there is no constant gradient of decreasing density of the gas molecules downward—there is no locking, and the gas breaks through. Such a situation is shown in diagrams III and IV. The pumping speed falls from 130 l/sec in cases I and II to 83 and 34 l/sec in cases III and IV. It is noted that at low pressures the decisive influence on the entraining action of the vapor and on the locking of the gas is exerted by the velocity of the vapor stream. At high pressures

... in applications, the amount of circulating mercury proves to be more important (the amount of heat consumed, the width of the throat).

Thus it has been established that the condition for effective operation of the pump is the presence of a maximum and a minimum of the gas density along some vertical line. When this condition is fulfilled, the pumping speed of the pump is directly proportional to the area of the throat.

Fig. 2.

On the basis of this theory a new high-speed pump was created with a large operating range—from the very lowest pressures to 0.1 mm of mercury. Its design is based on the following conclusions of the theory: 1) the area of the throat must be as large as possible; 2) the nozzle must be directed straight downward; 3) the vapor density along the outer wall must be as great as possible; 4) both walls of the vessel bounding the pumping space must be cooled. Conditions 1) and 2), it would seem, contradict condition 3), since a large throat area and a downward-directed nozzle predetermine a lower vapor density along the wall of the pump. This difficulty is eliminated by making the outer wall not parallel to the direction of pumping—it follows the broken dashed line shown in Fig. 3 (\(N\) — nozzle, \(h\) — throat, \(W\) — wall).

Fig. 3.

The new pump has diameter \(D = 27.5\) cm. The width of the throat varies from 1.5 to 5 cm, its area from 86 to 333 cm\(^2\), and the nozzle slit from 1 to 3 mm. The mercury is heated by an electric furnace. The highest pumping speed at an operating pressure of \(5 \cdot 10^{-2}\) mm Hg is equal to 1000 l/sec. In this case 12 kW of heat is consumed, and the throat must be the smallest. At the lowest pressures the throat must be at its maximum, and the heat consumption is only 2.7 kW. Figure 4 shows comparative curves of the dependence of pumping speed on operating pressure both for this new pump and for the original prototype pump.

Fig. 4.

As can be seen, the new pump has advantages both in the size of the operating range and in the magnitude of the pumping speed. At \(10^{-2}\) mm Hg, for each square centimeter of throat area, the pumping speed of the prototype pump is 1.2 l/sec, and that of the new pump is 2.8 l/sec. The backing pressures are the same in this case. In general, it was found that the pumping speed over wide limits does not depend on the backing pressure, especially at low operating pressures. Thus, at pressures below \(1.5 \cdot 10^{-2}\) mm, the pumping speed is constant at a backing pressure of 2.5 mm.

Following this, a second pump was created in which, thanks to certain changes in the internal dimensions, its characteristics were significantly improved, namely: the maximum speed was brought up to 1500 l/sec; the operating range with a speed of 1400 l/sec extends from \(10^{-2}\) to \(10^{-4}\) mm Hg; the maximum pumping speed per 1 cm\(^2\) of throat width is 4 l/sec; the maximum pumping speed per 1 kW of expended energy is 470 l/sec, whereas for the first pump it was 280, and for the prototype pump only 85 l/sec.

The construction presented is all the more interesting because existing pumps have a low pumping speed at pressures of the order of \(10^{-2}\) mm: high-vacuum pumps give a high pumping speed at pressures below \(10^{-3}\) mm, while oil rotary pumps do so at pressures above \(10^{-1}\) mm.

V. V. Fedorov