ADIABATIC METHOD FOR HELIUM LIQUEFACTION *

P. L. Kapitsa, Moscow

INTRODUCTION

The discovery by Kamerlingh Onnes of the method for liquefying helium has led, over the last thirty years, to a whole series of discoveries of extraordinary importance for the study of the solid state. Nevertheless, at the present time there are still very few laboratories equipped with the apparatus necessary for obtaining liquid helium. It would therefore be extremely desirable to simplify the complicated technique of obtaining liquid helium, which would make it possible to use it more widely. In the present work a more efficient method of liquefaction is described, based on the application of an adiabatic process, which considerably simplifies the production of liquid helium for research purposes.

There are two principal methods of cooling and liquefying gases. The first method is based on cooling the gas during its adiabatic expansion, this cooling being achieved through the performance of external work by the gas. This phenomenon was first observed as early as 1819 by Clément and Desormes¹, who discovered the cooling of a gas in a vessel when the pressure was lowered by releasing part of the gas through a tap. It can be shown that, in such an expansion, the gas remaining in the vessel imparts a certain kinetic energy to the escaping gas and is cooled adiabatically. In 1895 Olszewski² applied this method to the liquefaction of hydrogen; by compressing hydrogen to 190 atm and cooling it with liquid oxygen boiling under reduced pressure \((-211^\circ\mathrm{C})\), it was possible, upon subsequent rapid lowering of the pressure, to observe the appearance of a mist consisting of droplets of liquid hydrogen. By means of such experiments Olszewski succeeded in determining the critical constants of hydrogen. This method has recently been used by Simon³ for the liquefaction of helium. Simon made use of the fact that at very low temperatures the heat capacity of the vessel becomes so small that it practically does not take heat from the helium being liquefied. However, by means of this method only small quantities of liquid helium can be obtained, and for continuous liquefaction of helium it proves to be

* Proc. Roy. Soc. A 147, 188–211, 1934.

inapplicable; moreover, here a certain loss of cold is inevitable as a result of part of the gas leaving the vessel. This method is also complicated by the need to use high pressures and preliminary cooling by means of liquid hydrogen boiling at reduced pressure.

The fundamental advantages of a method in which the gas being cooled directly performs work in a suitable mechanical system have long been generally recognized, and these methods are now commonly used for the liquefaction of most gases. Unfortunately, however, in the case of helium or hydrogen its application is complicated, since it is very difficult to build an expansion machine operating at such low temperatures; for this reason the adiabatic method of liquefaction has hitherto not been applied to gases whose liquefaction temperature is lower than that of hydrogen.

The second method of cooling makes use of the internal work which the gas to be liquefied performs upon expansion. This phenomenon, known as the Joule–Thomson effect, was first used for the liquefaction of helium by Kamerlingh Onnes^4 and is still employed for obtaining liquid helium in considerable quantities. However, since helium in its properties is very close to an ideal gas, the magnitude of the Joule–Thomson effect for it proves to be very small, and only after preliminary cooling of helium to the temperature of liquid hydrogen boiling at reduced pressure does the magnitude of the effect become sufficient for liquefaction. The efficiency of this method is very small; for example, Meissner^5 considers that it does not exceed 1% of the efficiency that an adiabatic method could provide. In practice, in order to obtain 1 l of liquid helium, according to Meissner’s calculations it is necessary to expend 6 l of liquid air and 5 l of liquid hydrogen. These figures give some idea of the difficulties encountered in obtaining liquid helium by the method indicated.

POSSIBLE TYPES OF EXPANSION MACHINES

Taking into account the low efficiency of using the Joule–Thomson effect and the possibility of increasing it approximately 100-fold in an adiabatic process, we developed an expansion machine suitable for prolonged operation at low temperatures.

In order to cool a compressed gas adiabatically to a low temperature, a number of methods are possible a priori. The most natural would seem to be the use of a turbine, and at first glance this idea appears extremely attractive, chiefly because it eliminates the need to use a lubricant, which at low temperatures constitutes one of the principal difficulties. From the theory of turbine operation it is known that the velocity of motion of the blades must approach the critical velocity of the gas. The latter, under normal conditions, proves to be quite

high, which considerably complicates the construction of the machine. However, at temperatures approaching the liquefaction point, the critical velocity of helium decreases substantially and, for example, at \(15^\circ K\) is only \(153\ m/sec\); therefore in a helium machine the velocity of motion of the blades need not be excessively large. However, as a more detailed calculation shows, another difficulty arises here, connected with the dimensions of the turbine. Thus, for example, at \(15^\circ K\) and a pressure of \(30\ atm\), a turbine passing \(30\ m^3\) of helium per hour (which corresponds to the production of several liters of liquid helium per hour) would have to have a nozzle cross-section of no more than \(0.1\ mm^2\); and the other dimensions of the turbine parts would likewise turn out to be correspondingly extremely small. Hence it is clear that the use of a turbine could have practical meaning only in those cases where helium liquefaction is carried out on a very large scale. The installation described below, although different from a turbine, has so high an efficiency that it could hardly be further increased even by the use of a turbine.

Another possibility, which we have used, consists in the application of an ordinary piston machine. The major technical difficulty in designing such a machine for the case of very low temperature is that the piston must move in the cylinder with as little friction as possible, but at the same time must not allow leakage of the expanding gas. In ordinary machines of this type this is achieved by using a suitable lubricant; but at the temperatures of liquid hydrogen and helium it is impossible to find a suitable lubricating substance, since all known substances become solid. The first successful attempt to use an expansion machine at low temperatures belongs to Claude\(^{6}\). Claude at first used liquid air itself as the lubricant for the piston; subsequently, however, he came to the conclusion that liquid air is not a sufficiently good lubricant\(^{7}\), and replaced it with gasoline or ether, which limited the range of applicability of his expansion machine to \(-140^\circ C\). If liquid air proved to be an unsatisfactory lubricant, then it is obvious that liquid helium, possessing negligible surface tension, is entirely unsuitable as a lubricating substance and, consequently, the solution proposed by Claude is inapplicable in the case of helium liquefaction.

Seeking to overcome this difficulty, we undertook the development of a design for a piston expansion machine based on an entirely different principle, which does not require any lubrication for the piston and therefore can operate at any temperature\(^{8}\). The main feature of this machine was that the piston moved in the cylinder freely, with some clearance, so that when gas was admitted into the cylinder under high pressure a certain amount of it could escape from the cylinder through the clearance around the piston. However, the speed of motion of the piston was so great that during the negligible fraction of a second for which the expan-

gas, and this loss had practically no effect on the efficiency of the machine. Such an expansion machine has a very high efficiency (60%). With the apparatus described below, which uses this principle, it proved possible to liquefy helium without employing preliminary cooling by liquid hydrogen.

PRINCIPLE OF OPERATION

In designing a machine for liquefying helium, the aim was to achieve not only a high coefficient of performance, but also as short a cooling time as possible at start-up. A description of all the calculations connected with the design is beyond the scope of the present article, since they are of the usual technical character. We shall therefore confine ourselves to an analysis of the general principle of operation of the machine.

First of all it was necessary to choose a suitable distribution of the cooling stages in the liquefying apparatus. In principle, an expansion machine of the proposed type makes it possible to liquefy helium without preliminary cooling by liquid hydrogen or liquid air; however, this would require very large dimensions of the expansion machine and compressors, and the entire apparatus would become very bulky. Indeed, in order to liquefy 1 kg of helium, starting operation from room temperature, it is necessary to remove 380 cal. On the other hand, to liquefy the same amount of helium starting from the temperature of liquid nitrogen boiling under reduced pressure (65°K), it is sufficient to remove only 84 cal. Therefore an installation not using preliminary cooling by liquid nitrogen would have to be approximately four times larger than an installation operating with the aid of liquid nitrogen. The amount of nitrogen required in this case is very small; with ideal heat exchangers, no more than 0.5 l of liquid nitrogen would have to be expended for each liter of liquid helium obtained. Owing to the low cost of liquid nitrogen, its use for initial cooling proves, especially under laboratory conditions, to be very advantageous. Preliminary cooling by liquid hydrogen to 20°K would make it possible to reduce still further the amount of heat removed, down to 25 cal; however, this gain would be accompanied by the difficulties of obtaining liquid hydrogen and by the precautions necessary in working with it.

The heat-exchanger arrangement adopted by us is shown in Fig. 1. Compressed helium enters tube 1 and passes into heat exchanger A; then, passing around the annular vessel N, which contains liquid nitrogen boiling under reduced pressure, it is cooled to 65°K. The compressed helium then passes through heat exchanger B and enters the expansion machine E. On leaving the expansion machine, it passes through heat exchangers C, B, and A.

and returns to the compressor through tube 2. With the aid of these regenerative heat exchangers, the temperature of the gas leaving the expander could be gradually lowered until part of the helium left the machine in the liquid state. Then, separating the liquid helium from the gaseous helium, we obtain a process scheme similar to that used by Claude for the liquefaction of air. Such a process, however, is insufficiently effective for quite obvious reasons. Suppose that the gas enters the expansion machine at pressure \(p_1\) and fills the volume \(v_1\) at temperature \(T_1\), and then expands to the volume \(v_2\) at pressure \(p_2\) and temperature \(T_2\). The expansion process can be described sufficiently well by the familiar relation

\[ pv^\gamma=\mathrm{const}, \tag{1} \]

where \(\gamma\) is the ratio of the specific heats. The amount of cold obtained in one expansion will be

\[ \Delta i=\eta j p_1 v_1 \frac{\gamma}{\gamma-1} \left[ 1-\left(\frac{p_2}{p_1}\right)^{\frac{\gamma-1}{\gamma}} \right]. \tag{2} \]

Here \(\eta\) is the efficiency of the expansion machine, which in the ideal case would be equal to unity, and \(j\) is the mechanical equivalent of heat. Relation (2) shows that, for the case of an ideal gas, the amount of cold obtained in expansion does not depend on the temperature at which the expansion takes place. But since the density of the helium filling the expander is inversely proportional to the temperature \(T_1\), when \(T_1\) is lowered the quantity of gas delivered by the compressor will increase, while the amount of cold produced will remain unchanged. We arrive at this conclusion assuming that helium obeys almost exactly the equation of state of an ideal gas \((\gamma=\mathrm{const})\). It can be shown that deviations from the properties of an ideal gas only reinforce the conclusion reached, since the amount of cold produced \(\Delta i\) will decrease as the expansion temperature approaches the liquefaction point. A more detailed analysis of the phenomena shows that it is advantageous to keep the temperature \(T_1\) as high as possible and to cool the gas in the expansion machine to a temperature \(T_2\) appreciably above its boiling point; the last stages of the process,

Fig. 1.

up to liquefaction, must be attempted by some other methods.

Here three possibilities present themselves to us. The first corresponds to the method used by Claude for liquefying air, and consists in keeping the temperature \(T_2\) somewhat below the critical temperature \((5.2^\circ K)\) and then cooling the vessel in which the helium would be liquefied at a pressure exceeding the critical pressure \((2.26\ atm)\). However, this method proves unsuitable for helium, since the difference between the critical temperature and the liquefaction temperature is too small (only \(1^\circ\)).

The second method might also consist in cooling compressed helium in a vessel by the cooled gas entering the expansion machine, and in the subsequent liquefaction of the helium by reducing the pressure, as Simon did. Such a process would, obviously, have to be repeated periodically, each time emptying the vessel, which would of course cause complications.

The third method, which was the one used by us, coincides with the method of liquefying air used by Heylandt,* in which the last stage of the process is carried out by means of the Joule–Thomson effect. The scheme used is shown in Fig. 1. After passing through heat exchanger \(B\), part of the helium under high pressure is passed through heat exchanger \(C\), where it is cooled to the temperature \(T_2\) of the helium leaving the expansion machine. The gas then passes through heat exchanger \(D\) and throttling valve \(4\) into vessel \(5\), where liquefaction takes place. After part of the helium has passed into liquid, the remaining gas again passes through heat exchangers \(D\) and \(C\) and joins the gas leaving the expansion machine.

In order to calculate the output of the expander, it is necessary to know the equation of state of helium near the critical point. Its analytical expression, however, is not known with sufficient accuracy, and therefore it is more convenient here to use “entropy diagrams” or “Mollier diagrams.” In our work we used such diagrams, given for helium by Keesom and Gutkoff.^9 From these diagrams it was possible to obtain the following quantities: 1) the total heat content \(i_n\) and \(i'_n\), corresponding to the pressures \(p_1\) and \(p_2\) at the temperature \(T_n\), at which the compressed helium leaves the vessel containing liquid nitrogen; 2) the total heat content \(i_1\) of the gas entering the expansion machine at the temperature \(T_1\); 3) the total heat content \(i'_2\) after the gas has expanded adiabatically to the pressure \(p_2\) and assumed the temperature \(T'_2\). In reality, owing to various losses inevitable during expansion, the gas comes to a state \(p_2, T_2, i_2\), differing from the state \(p_2, T'_2, i'_2\) which would exist in the case of an ideal process, and the corresponding parameters of both states are connected by the expres—

* See Meissner, p. 314.

by the following expression for the efficiency coefficient of the expansion machine

\[ \eta=\frac{i_1-i_2}{i_1-i_2'} . \tag{3} \]

If \(D\) kg of helium passes through the apparatus, and a part \(K\) of it enters the expansion machine, the amount of cold produced will be \(DK\eta(i_1-i_2')\); however, owing to the imperfection of heat exchanger \(B\), not all this cold can be used for liquefaction. If \(\varphi\) is the efficiency coefficient of heat exchanger \(B\), the amount of cold carried away by the gas will be \(D(i_n'-i_1)(1-\varphi)\). In addition, there is inevitably a further certain loss of cold, due to the difference in heat content at temperature \(T_n\) in the cases of the real and ideal processes; however this loss, equal to \(D(i_n'-i_n)\), will be small in comparison with the other quantities.

The part of the gas \((1-K)\) is used for the Joule–Thomson effect between the pressures \(p_3\) and \(p_4\). The corresponding heat contents \(i_3\) and \(i_4\) at temperature \(T_2\) may be found from the entropy diagram, and in the liquid state at pressure \(p_4\) we shall have some value \(i_4'\). Since \(i-i_4'\) calories are required to liquefy 1 kg of helium, the amount of liquefied gas will be

\[ L=D\left[\frac{K\eta(i_1-i_2')-(1-\varphi)(i_n'-i_1)(i_n'-i_n)}{i_n-i_4'}\right]. \tag{4} \]

At the same time, the amount of helium liquefied by means of the Joule–Thomson effect will be (Keesom\(^{10}\))

\[ L=D(1-K)\frac{i_3-i_4}{i_3-i_4'} . \tag{5} \]

The quantities \(K\) and \(\dfrac{L}{D}\) can be determined from equations (4) and (5), and it turns out that there exists a certain temperature \(T_2\) at which \(\dfrac{L}{D}\) has a maximum. The arguments presented have meaning only when the temperature \(T_2\) is below \(40^\circ\)K and when the Joule–Thomson effect has a positive sign. When the temperature \(T_2\) is lowered below \(40^\circ\)K, the difference \(i_1-i_2'\) will decrease, while \(K\) will increase. At some temperature their product will reach a maximum, and the entire liquefaction plant will have the greatest efficiency coefficient. Experiment showed that this temperature is about \(10^\circ\)K; it proved impossible to calculate it from the state diagrams, since the available data are sufficiently accurate only down to \(15^\circ\)K and a pressure of 20 atm; for lower temperatures they are unreliable. We shall return again to this circumstance in one of the following chapters of the present article; for now we mention it because, in the experiment, we found an inversion of the Joule–Thomson effect that was not indicated on the given diagram, and this made it necessary to introduce a second throttle valve 6, which lowered

the pressure of the helium entering the regenerating coils \(C\) and \(D\), from 30 approximately to 17 atm.

Equation (4) shows that the productivity of the apparatus and its coefficient of performance are determined not only by the efficiency \(\eta\) of the expander, but to a considerable degree also by the efficiency \(\varphi\) of heat exchanger \(B\).

In practice we have \(i_n-i_1=61\) cal and the quantity of cold obtained in the expansion machine, \(K(i'_1-i'_2)=9.7\) cal. The efficiency of the heat exchanger of our apparatus \(\varphi\) reached 92%; but, despite this high value, it is evident that about half of the cold obtained in the expander will be carried away. This shows that it is necessary, as far as possible, to reduce losses in heat exchanger \(B\), which compelled us to make it very long and heavy (Fig. 3).

To avoid the use of such a cumbersome heat exchanger and to increase the coefficient of performance of the installation, it would have been possible, instead of one, to use a whole series of expansion machines, so that the gas leaving one machine would cool the gas entering the next. With three machines the exhaust gas of the last expander would attain a sufficiently low temperature to cool the compressed helium for liquefaction by means of the Joule–Thomson effect. An installation operating according to this scheme would not require heat exchanger \(B\), and its productivity would probably exceed that of our present installation by at least a factor of three. This scheme was not used by us in practice only because the construction of three expanders is technically rather complicated and was not justified at the initial stage.

EXPANSION MACHINE

We shall now dwell in more detail on the description of the general principle of operation and the design details of the expansion machine. In starting the design of the machine, we first had to choose its working pressure \(p_1\). As is seen from relation (2), the amount of cold obtained,

\[ \frac{\Delta i}{v_1 p_1}, \]

referred to unit mass of gas, changes comparatively little with a large increase in the working pressure \(p_1\). On the other hand, increasing the pressure of the incoming gas considerably complicates the design of the machine and makes it more bulky. The most convenient value of the pressure depends on various factors, which are very difficult to take into account exactly, so that it is most convenient to determine it experimentally. We settled on a pressure of 25–30 atm. The dimensions of the expansion machine depend on the amount of gas that we must pass through it and on the number of strokes of the piston per minute. The volume of the incoming gas and the magnitude of the expansion are determined by the specific volumes of helium on the state diagram.

Our expansion machine has from 100 to 120 strokes

of the piston per minute and passes about \(30\ \mathrm{m}^3\) of helium per hour at room temperature and normal pressure. The piston diameter is \(30\ \mathrm{mm}\), and the stroke length varied between \(3.5\) and \(5\ \mathrm{cm}\). In choosing this ratio of the piston diameter to the stroke length we had to take into account a whole series of factors: the thermal conductivity of the metal, the leakage of gas through the clearance between the piston and the cylinder walls, etc.

After the overall dimensions of the machine have been determined, one must choose a mechanism making it possible to produce a rapid motion of the piston during expansion and its slow return. It is obvious that the usual use of a connecting rod and crankshaft is unsuitable for our purpose. It is possible that the required character of the motion could be achieved with the aid of a suitable cam mechanism, but technically the simplest to us seemed to be the use of a mechanism operating on hydraulic principles. The arrangement of this mechanism is shown schematically in Fig. 2. The expansion machine \(E\) is connected with the hydraulic mechanism \(H\) by means of rod \(3\). The gas entering the expander sets piston \(13\) in motion; this piston in turn pushes piston \(8\); the space above piston \(8\) is filled with water, which is expelled in the form of a jet through orifice \(9\). In this way the work performed by the gas during expansion is transformed into the kinetic energy of the issuing jet of water. After the expansion stroke is completed and the exhaust valve of the expander is open, the cylinder of the hydraulic machine is gradually filled with water, and the expander piston slowly returns to its initial position. The cooled, expanded helium passes from the cylinder into tube \(10\) and through tube \(11\) enters the heat exchanger; packing of gland \(12\) prevents leakage of the gas. It should be noted that the diameter of connecting rod \(3\) is chosen equal to the diameter of the expander piston; because of this, during expansion the volume of gas in tube \(10\) does not change, and no sudden increase of pressure in tube \(10\) occurs. During the return stroke the gas in \(10\) is slowly compressed, and the cooled helium begins to circulate through the heat exchanger at a constant velocity.

Fig. 2.

As can be seen from the figure, the entire expansion machine is enclosed in a long thin-walled tube \(10\) (wall thickness \(0.5\ \mathrm{mm}\)), while the connecting rod \(3\) is also a thin-walled tube (wall thickness \(0.5\ \mathrm{mm}\)). Thanks to such thin tubes, good temperature insulation of the expansion machine from the surrounding medium is obtained.

The equation of motion of the entire system may be written in the form

\[ M\ddot{x}+C\dot{x}^{2}+F=P, \tag{6} \]

where \(x\) is the displacement of the expander piston, \(M\) is the mass of both pistons and the connecting rod, \(F\) is the frictional force, due mainly to the packing of gland 12; \(P\) represents the resultant force acting on the piston, equal to

\[ P=(p-p_{0})\pi R^{2}, \tag{7} \]

where \(R\) is the radius of the expander piston 13, and \(p_{0}\) is atmospheric pressure. The term \(C\dot{x}^{2}\) gives the resistance force caused by the hydraulic mechanism, where \(C\) is a constant having the following value:

\[ C=\frac{\pi r^{6}\sigma}{2\rho^{2}a^{4}} . \tag{8} \]

Here \(r\) is the radius of piston 8, \(a\) is the radius of the outlet opening for jet 9; \(\sigma\) is the density of water, and \(\rho\) is the contraction of the jet, which we have taken equal to 0.6. In the process of filling the expander with gas at constant pressure, \(P\) is constant, and we obtain

\[ \dot{x}^{2}=\dot{x}_{0}^{2}\left(1-e^{-\frac{2Cx}{M}}\right), \tag{9} \]

where

\[ \dot{x}_{0}^{2}=\frac{P-F}{C}. \tag{10} \]

If the factor \(\frac{2Cx}{M}\) is large, the influence of inertia will be small, and the piston will immediately acquire the velocity given by expression (10). Therefore, if the mass of the moving parts of the expander is small and \(C\) is large, one may consider that the motion of the piston will at all times be described by formula (10), even if \(P\) is variable. The change in \(P\) after the intake valve closes may be found approximately from expression (1), and the piston velocity for any of its positions may be calculated from (10). Although a more exact solution of equation (6) could be obtained by means of a series, the approximation indicated is quite sufficient for our practical purposes.

From expression (10) it is clear that the use of the hydraulic device is advantageous in that the piston velocity is greatest at the beginning of expansion, when the leakage of helium through the gap between the piston and the cylinder is greatest, and decreases as the pressure in the cylinder drops. Another advantage is that the length of the piston stroke is not fixed and can be set by the temperature and pressure of the incoming gas, so that the final pressure will remain unchanged; this also increases the efficiency of the expander.

The velocity of the piston motion can be regulated by changing the size of opening 9. In this case, theoretically, it would seem that

the only condition in choosing the speed of the piston is that it must be less than the critical velocity of helium and that therefore it must be chosen as large as possible, since this thereby shortens the expansion time and reduces the leakage of gas through the clearance. In practice, however, it turns out that the speed of motion of the piston is limited by the difficulty of filling the cylinder with gas through the valve openings in a short interval of time. In choosing the dimensions of the valve the following conditions must be met: in order to avoid a large pressure loss, the area of the opening must be sufficiently large so that the velocity of motion of the gas through the valve is appreciably less than the critical velocity. On the other hand, the valve must not be very large, since in that case greater force would be required to actuate it. Taking these conditions into account, we settled on a round valve of diameter 1 cm, and chose an opening of 1 mm. Such a valve size proves sufficient if the total stroke time of the piston is about 0.1 sec; this is achieved with the following dimensions of the hydraulic mechanism: cylinder diameter 8–25 mm, opening diameter 9–2.7 mm.

If the speed of motion of the piston is known, then it is not difficult to calculate what the clearance between the piston and the cylinder wall should be so that the gas leakage is not too large. The maximum leakage can easily be calculated from the classical formulas, if one assumes that the gas behaves like a viscous liquid. The viscosity of helium at low temperatures is known from the measurements of Kamerlingh Onnes and Sophus Weber[^11]. The amount of leakage is proportional to the cube of the width of the clearance and is maximal when the piston is positioned most asymmetrically, touching the cylinder wall at some point. In reality, however, the gas flow will not be laminar but turbulent, and the amount of leakage will be smaller than that calculated in this way. Since exact measurements of the clearance are rather difficult, we also determined the leakage experimentally, observing the speed of motion of the loaded piston in the cylinder with the valves closed. Knowing the weight of the piston and the magnitude of the load, it was possible, with the aid of similarity theory, to extrapolate the leakage to low temperatures and high pressures. In this way we found that, for the clearance value we chose, 0.04–0.05 mm, the leakage would be of the order of several percent of the total amount of helium processed.

The details of the construction of the expansion machine are shown in Fig. 3. As in all technical constructions, many of the parts used could be replaced by other mechanisms without any substantial change in the operation of the machine; however, among them there are also some that are essentially necessary for the proper operation of the apparatus. We shall dwell on the description of the latter in somewhat more detail.

Since no lubrication is used in the cylinder of the expansion machine, it is very important to eliminate all lateral forces acting

Fig. 3.

on the piston; for this purpose two universal joints 26 and 28 were used, so that in practice the cylinder experiences no lateral pressures except those which may arise as a result of an inaccurate coincidence of the axes of the cylinder and the connecting rod.

Another essential circumstance is the necessity of cutting narrow grooves, \( \frac{1}{4} \) mm deep and \( \frac{1}{4} \) mm wide, around the circumference of the piston at intervals of approximately 5 mm from one another. This measure proves necessary because it is practically impossible to make the surface of the cylinder and piston perfectly smooth, and therefore there will always be certain irregularities of the surface. As was already mentioned above, the aerodynamic resistance of the gap is proportional to the cube of its width, and therefore even negligible changes of the gap will exert a significant influence on the magnitude of the pressure drop of the gas in the gap and will cause the appearance of lateral forces. The purpose of the described grooves is to equalize the pressure around the piston. Experience has shown that without such a precaution, almost immediately after the machine begins operating, such large lateral forces develop that the piston seizes in the cylinder.

In order to ensure uninterrupted operation of the piston in the cylinder, it is important that there should be no solid particles in the helium; therefore we had to filter carefully the gas entering the machine. Similarly, impurities of various kinds in the helium, for example air, which at low temperatures form solid crystals on the walls of the cylinder, scratch and wear down the piston, which can likewise cause it to seize and reduce the efficiency coefficient of the machine. It turned out that, in order to purify the helium, it is necessary to pass it before expansion through the charcoal absorber 30.

Finally, it proved essential to introduce a high-pressure reservoir 14 containing helium in an amount approximately ten times greater than is necessary for a single expansion. The presence of this reservoir eliminates pressure fluctuations during the admission of gas into the expander, which would undoubtedly cause a decrease in the efficiency coefficient of the installation.

The expansion machine is mounted at the center of the apparatus and is surrounded by the casing 16. To obtain the necessary thermal insulation, a very high vacuum is created inside the casing with the aid of activated charcoal.

In practice it is essential that the internal parts of the expansion machine be easily accessible for inspection. In the machine constructed, the cylinder is unscrewed from the support 34 by means of a special key; the inlet valve 37 can also be removed.

The inlet valve 37 consists of a steel ball \( \frac{1}{2} \) inch in diameter, fitting tightly into a well-polished seat made of phosphor bronze. The careful manufacture of this valve is extremely important. At low temperatures, when the viscosity of the gas becomes very small, obtaining the necessary hermetic tightness—

of the valve is a very difficult task. The latter is achieved by the fact that the ball works at all times with only one of its parts; moreover, possible rotation of the ball is eliminated by a pin inserted into the hole inside the ball. The exhaust valve 38 is made in the wall of the cylinder; it is shown separately in Fig. 3. The inlet valve is opened by means of rod 39, which fits tightly into a cylindrical hole made in the cylinder. Gas leakage through the clearance at rod 39, as well as through the clearance of the cylinder, is small owing to the rapidity of expansion. In the cover of the expansion machine there are three stuffing boxes with packings, of which two (15) are intended for the valve rods, and the third (12) for the connecting rod. The dimensions of most parts of the expansion machine can be readily calculated by the standard methods of applied mechanics.

The expansion machine operates as follows: the cycle begins with energizing electromagnet 19, which opens water valve 18, thereby reducing the water pressure on the piston and releasing latch 20, which holds lever 21. The released lever is pulled back by spring 22 and, with the aid of connecting rod 23, opens the inlet valve. At the same time, by means of lever 24, the exhaust valve is closed. The compressed helium entering the cylinder moves the piston, and after the piston has traveled about 0.7 cm, filling of the expansion machine with gas is completed; cam 25 presses on the roller of lever 21, the lever rises upward, and the inlet valve closes. During the remainder of the piston stroke the gas expands adiabatically, and, evidently, the work performed is absorbed by the energy of the jet of water flowing out of orifice 9. The spent water is not wasted, but is used for cooling the compressor. When the piston stroke is completed, the current in the electromagnet is interrupted, its armature is pulled upward by spring 27, which opens the exhaust valve, returns latch 20 to its place, and finally closes water valve 18. Then water flows in through tube 17, and the piston slowly returns back to its initial position, pushing the cooled helium through exhaust tube 11 into the heat exchanger. The cycle thereby ends.

Electromagnet 19 is actuated automatically by means of a special contact device. The valve mechanism described works quite satisfactorily; however, it seems possible to us to simplify it considerably, which we hope to do in the next model.

In addition to purely design questions, the choice of materials is of great importance in building the expansion machine. As is known, the majority of substances that are sufficiently elastic at ordinary temperatures become brittle, like glass, at very low temperatures. This applies especially to ordinary mild steel. Although the tensile strength increases as the temperature is lowered, the loss of plasticity makes the use of such materials impossible. Therefore we had to choose materials,

preserving their plasticity at low temperatures. Secondly, the tubes connecting the expander with the hydraulic mechanism must have low thermal conductivity and considerable strength. It turned out that a very suitable material is Firth’s “Staybrite” steel. This stainless steel has very low thermal conductivity (0.033), retains its plasticity at low temperatures, and can be drawn into very thin-walled tubes. In our apparatus we used tubes whose wall thickness was 0.5—0.2 mm. One difficulty should be pointed out that may arise when using “Staybrite” steel, connected with the fact that this steel belongs to the austenitic class, in which the iron is in the nonmagnetic γ-modification. During cold working the steel acquires magnetic properties, which compels one to suppose the existence of an unstable equilibrium between the α- and γ-modifications and, at low temperatures, raises concern about the possibility of a transition to the α-state; if such a transition occurred, it could strongly affect the mechanical properties of the steel. To clarify the possibility of this transition, we kept samples of various grades of “Staybrite” for 8 days at the temperature of liquid hydrogen. Dr. Hatfield (research laboratory of the firm Thomas Firth and John Brown) was kind enough to subject these samples to careful investigation after exposure to low temperature. Most of the samples showed no changes in physical properties, and in the end we settled on the grade “FST.” Another feature of “Staybrite” steel is that it has a larger coefficient of linear thermal expansion than ordinary steel, approaching the coefficient of expansion of copper. This property makes the indicated steel a very suitable material for making the piston of the expansion machine from it. At first glance it seems that it would be most expedient to make both the cylinder and the piston from one and the same material; however, when in both cases we used phosphor bronze, it turned out that the piston quickly seized; investigation showed that the cause was the entry of foreign particles into the clearance, and, owing to the identical hardness of both surfaces, these particles could not cut into either of them and scratched both surfaces. If, however, the piston is made from the indicated steel and the cylinder from phosphor bronze, the foreign particles are pressed into the cylinder walls. It is also very important that all materials for parts in the machine which are to be subjected in the machine to the action of low temperatures should not have large internal stresses, and therefore must be annealed.

HEAT EXCHANGERS

For increasing the productivity of the installation, the choice of a rational design of the heat exchangers is of great importance. Generally speaking, the larger the dimensions of the heat exchangers, the higher

their productivity. On the other hand, under laboratory conditions one has to take into account not only productivity; large dimensions are associated with an increase in the bulkiness of the apparatus, which is a substantial drawback, since this increases the amount of cold expended on preliminary cooling, which lengthens the start-up time. In designing our apparatus we therefore deliberately accepted a certain reduction in productivity for the sake of greater rapidity of start-up and economy of liquid nitrogen for preliminary cooling, and tried to make the heat exchangers as small as possible.

The general theory of heat exchangers and the data for their calculation are given in Nusselt’s works[^12]. We used Nusselt’s formulas, however, only for an approximate determination of the size of the heat exchanger, and then applied the formulas of the theory of similarity. It follows from Nusselt’s formulas that, for a given quantity of gas, heat transfer in a heat exchanger consisting of parallel tubes is proportional to \(n^{0.21}\), \(l^{0.95}\), and \(d^{-0.79}\), where \(n\) is the number of tubes, \(d\) their diameter, and \(l\) the length of the heat exchanger. Therefore, to increase heat transfer one may either lengthen the tubes or reduce their diameter; changing the number of tubes does not have a substantial influence on the magnitude of heat exchange.

In choosing the dimensions of the heat-exchanger tubes it is extremely important that the pressure drop in them not be too large; this applies especially to the return tubes. It can be shown that, for a constant quantity of gas passed through, the pressure drop is proportional to \(n^{-1.75}\), \(l\), and \(d^{-4.75}\), whence it is clear that even very small changes in diameter lead to significant changes in the pressure drop. Thus it turns out that the most convenient way of reducing the pressure drop without a substantial change in heat exchange is to increase the number of tubes.

In our apparatus the heat exchangers consisted of one high-pressure tube surrounded by six low-pressure tubes for the reverse flow of gas. The tubes were made of a copper–nickel alloy (composition 80 : 20); they were soldered together and bent into a coil. Heat exchanger \(A\) (Figs. 1 and 3) is rather complex and could be considerably improved. Its upper part has a length of 4 m and consists of seven tubes of internal diameter 4 mm; the lower part has a length of 3 m and consists of seven tubes with internal diameter 5 mm. Three tubes of the lower part are intended for the return of the gases of evaporated liquid nitrogen to vessel \(N\).

The heat exchanger connected with vessel \(N\) has a length of 8 m and consists of 5-millimeter tubes.

Heat exchanger \(B\) is the most important in the whole system; it has a length of 16 m and is divided into three equal sections, made of tubes with diameters 4, 3.5, and 2.3 mm.

Heat exchangers \(C\) and \(D\) have lengths of 2 and 6 m and are made of tubes of diameter 2.3 mm.

If special precautions are not taken, the expansion machine will feed gas in spurts, which will considerably impair the operation of the heat exchangers, especially heat exchanger B. To obtain a more uniform flow of the incoming gas into vessel 14, there was a nozzle 29, introducing resistance into the gas stream. The nozzle had a diameter of 0.4 mm and a length of 1 cm, and could be unscrewed when the cylinder was dismantled.

Let us note that, when soldering the tubes and other parts of the installation, special care is required in the work in order to obtain the necessary tightness for maintaining a high vacuum.

Among the other parts of the installation one should mention the centrifugal separator for separating droplets of liquid helium from the cooled gas. After valve 4 (Fig. 3) the helium passes through three nozzles 31, which impart rotational motion to it, and enters the narrow gap between the walls of the vessel and hemisphere 32, where deposition occurs.

Vessel N with liquid nitrogen is connected with the copper hemisphere 40, which, owing to thermal conductivity, maintains a low temperature and protects the helium heat exchanger located in it and the expansion machine from heat losses. The outer tube of the expansion machine is cooled to the temperature of liquid nitrogen in part 33 by the return flow of helium, thereby reducing the heat leakage from the expansion machine into the external space due to thermal conductivity.

The installation is provided with a number of manometers and two simplified helium thermometers, by means of which the temperature of the gas before and after the expansion machine is measured. In addition, there are two level indicators: one for helium, the other for liquid nitrogen. Liquid helium is discharged through outlet tap 7, mounted at the bottom of the apparatus; the latter consists of two thin-walled tubes (0.2 mm), of which the inner one is provided with a nozzle that closes when pressure is applied to the outer tube. Sealing of the entire device is achieved by means of flexible bellows 35. A similar tap 36 is used for discharging liquid nitrogen into an external vessel. Taps of this type, as experience has shown, operate quite satisfactorily.

For proper operation of the apparatus it is absolutely necessary that the helium be sufficiently pure, and therefore we shall devote special attention to the development of a helium-storage system in which the possibility of the entry of air and other impurities is excluded. Helium is kept in a cylinder at a pressure of up to 10 atm and, as needed, is automatically fed into a gas-holder of capacity 0.5 m³, filled with heavy compressor oil (it turned out that only very heavy oil is suitable for this purpose, since otherwise air diffuses into the helium). When the gas-holder is filled with helium, it is automatically pumped back into the cylinder by means of a special hermetically sealed pump. The system is designed to store from 7 to 8 m³ of helium. The general view of the installation is shown in Figs. 4 and 5.

Operation of the apparatus

It turned out that, from the thermodynamic point of view, our expansion machine is a very efficient mechanism. With an initial gas pressure of 30 atm and a final pressure of 2.2 atm, the gas temperature decreases from 19 to 10°K; the state diagram makes it possible to conclude that about 60% of the free energy of the gas is thereby converted into mechanical work. This high efficiency in such a small machine is evidently a consequence of the small friction losses between the piston and the cylinder. Apparently, the small quantity of gas that leaks through the gap acts as an “ideal” lubricant. This is confirmed by the fact that even after many hours of operation of the machine no wear of the cylinder and piston is observed, and those scratches and irregularities of their rubbing surface which were noticed during assembly of the machine remain unchanged.

Fig. 4.

From the data given one may conclude that, feeding 32 m³ of helium per hour by the compressor and passing 28 m³ of it through the expansion machine, we obtain 49 cal. This cold should have been sufficient to liquefy 4.7 liters of helium; however, in practice we were unable to obtain more than 1.9–2 l/hour; the difference must be attributed to losses in the heat exchanger \(B\), mentioned above.

Fig. 5.

ADIABATIC METHOD OF LIQUEFYING HELIUM

The apparatus is started up in the following way. Reservoir \(D\) (Fig. 1) is filled with liquid nitrogen and, at the same time, the expansion engine is put into operation. To cool the heat exchangers rapidly, a large quantity of helium is passed through tube \(1\), reservoir \(5\), valve \(7\), and returned to the gas holder, bypassing the return tube \(2\). In 45 min the whole apparatus is brought to the temperature of liquid nitrogen, after which the expansion engine lowers the temperature to \(10^\circ\mathrm{K}\) in approximately 25–30 min. In order to cool reservoir \(5\) to this temperature as quickly as possible, a weaker steady flow of helium through valve \(7\) is maintained. Then valve \(7\) is closed, and almost immediately liquid helium begins to deposit on the bottom of the reservoir. To start the apparatus, \(7\ \mathrm{kg}\) of liquid nitrogen is required.

Experience has shown that the amount of liquefied helium depends strongly on the pressure of the gas leaving through the throttle. We observed that, at a pressure equal to \(30\ \mathrm{atm}\), the quantity of liquid helium formed after the moment at which liquefaction begins continuously decreases; at the same time the temperature of the gas leaving the expansion engine also decreases, and in the end liquefaction ceases altogether. This phenomenon at first puzzled us, and it seems to us that its only explanation is that, at these temperatures and pressures, an inversion of the Joule–Thomson effect takes place. When the quantity of gas used in the Joule–Thomson effect does not remove a sufficient amount of heat for liquefaction, the temperature of the expansion engine falls and, in accordance with this lowering of the temperature, owing to inversion, the magnitude of the Joule–Thomson effect continues to decrease, which entails a reduction in liquefaction and further cooling of the engine. At a temperature of the gas leaving the expansion engine of about \(6^\circ\mathrm{K}\), the amount of cold obtained is so small that it is sufficient only to compensate the losses in heat exchanger \(B\), and liquefaction stops. At present there are no data whatever on the existence of such an inversion point at low temperatures. Meissner\(^5\), discussing this question, likewise arrives at no definite results. The entropy diagrams of Keesom and Guttof\(^9\) also do not indicate the existence of an inversion near \(14^\circ\mathrm{K}\) and \(20\)–\(25\ \mathrm{atm}\), where, according to our data, it should be located. Incidentally, Kamerlingh Onnes had long ago found that the optimum pressure for liquefying helium is \(20\ \mathrm{atm}\), and suggested that at higher pressures the reduction in yield is explained by the fact that there is considerable entrainment of liquid-helium droplets by the return flow of gas. At first we supposed that this explanation was applicable in our case; however, by improving the separation of droplets with the aid of the centrifugal separator \(31\) (Fig. 3), we found no improvement in the operation of the apparatus. By lowering the pressure by means of throttle \(4\) (Fig. 1) from \(30\) to \(17\ \mathrm{atm}\), while at the same time keeping the gas flow through the throttle constant, so that the entrainment of droplets did not change, we were able to increase the yield of liquid helium from \(0.6\) to \(2\ \mathrm{l}\). It is very probable that the optimum conditions

liquefaction observed by Kamerlingh-Onnes are in fact also connected with the existence of an inversion point. Our apparatus is unsuitable for an exact study of the equation of state of helium at low temperatures; however, it is evidently difficult to give any other satisfactory explanation of the observed phenomena. At a pressure of 17–18 atm we found the output of the machine to be maximal, equal to about 2 l/hour, with a consumption of approximately 2 kg of liquid nitrogen per hour. Thus, for 1 l of liquid helium the apparatus described requires less than 1.5 l of liquid nitrogen.

The liquefier was built in the workshop of the Mond Laboratory of the Royal Society in Cambridge by the mechanic H. Pearson.

Additional Remarks

The type of expansion machine described may find application not only in the liquefaction of helium, but also in other cases where the temperature of the apparatus must be 7–8°K. We used this machine for liquefying hydrogen; however, it seems to us that in this case it can have practical importance only when liquid hydrogen is produced in large quantities.

Our apparatus operated for several months and, after a number of minor defects had been eliminated, proved entirely reliable. In the future we intend to introduce a number of improvements, such as, for example, simplifying the valve mechanism and reducing all the refrigerated parts of the apparatus. It is also proposed to use several expansion machines operating in series, as was already mentioned above; with such a system the yield of liquid helium should increase at least twofold and the starting time should be reduced by approximately one half.

Literature

1. Clement et Desormes, J. phys. Chim. Hist. nat., 89, 321, 428, 1819.
2. Olszewski, Wiedemann’s Ann. d. Phys., 56, 133, 1895; Phil. Mag., 39, 188, 1895.
3. Simon, Z. Physik, 81, 816, 1933.
4. Kammerlingh-Onnes, C. R., 147, 321, 1908.
5. Meissner, Hdb. d. Phys., B II.
6. Claude, C. R. 134, 1568, 1902.
7. Claude. Ibid., 141, 762, 1905.
8. Kapitza, Nature, 133, 208, 1934.
9. Keesom a. Houthoff, Comm. phys. Lab. Univ. Leiden, 17, 1924–1928; supplement 65e, 1928.
10. Keesom, Comm. phys. Lab. Univ. Leiden, 17, 1924–1928, supplement 65 f, 1928.
11. Kammerlingh-Onnes a. Sophus Weber, Comm. phys. Lab. Univ. Leiden, 13, No. 1346, 1913–14.
12. Gröber, Wärmeübertragung, Springer, p. 84, 1926.