Pulse Tube Cryocoolers: A Cryogen-Free Path to 2K

J. Bert
December 10, 2007

(Submitted as coursework for Physics 210, Stanford University, Fall 2007)

Sharp rises in the in the price of liquid helium for terrestrial condensed matter and superconductivity experiments as well an increasing demand for compact, maintenance free cryocoolers for satellites have made pulse tube refrigerators an attractive and relatively simple way of reaching cryogenic temperatures. A pulse tube cooler does not require input of any cryogens, yet two stage pulse tube refrigerators have achieved temperatures below 2 K [1]. Additionally, pulse tube coolers have no moving parts at the cold end. This significantly increases their reliability and lifetime while decreasing the potential to couple vibrations into the sample mounted at the cold end.

This overview of pulse tube refrigerators traces the history of the pulse tube design from the first models capable of reaching 124K to today's state of the art cryogen free pulse tube systems which can go as low as 1.3K [1]. Improvement to the basic design including the addition of orifices, double inlets, and multiple stages are also discussed.

Fig. 1: Basic pulse tube refrigerator.

Basic Pulse Tube Refrigerator

The first pulse tube was built in 1963 by Gifford and Longsworth. [2] Its basic components include a pulse tube, a regenerator, a pressure wave generator, and two heat exchangers as shown in Fig. 1. The pulse tube is a simple tube with one open end and one closed end. The closed end is the hot end and is capped with a heat exchanger that cools it to the ambient temperature. The open end is the cold end. It is connected to the regenerator and a cold stage by a second heat exchanger. The regenerator is a periodic flow heat exchanger. It absorbs heat from gas pumped into the pulse tube precooling it, and stores the heat for half a cycle then transfers it back to outgoing cold gas in the second half of the cycle cooling the regenerator. The interior of the regenerator tube is filled with either stacked fine mesh screens or packed spheres to increase its heat capacity. A piston, compressor or similar pressure wave generator is attached to the warm end of the regenerator and provides the pressure oscillations that drive the refrigeration. Helium is used as the working gas due to its monotonic ideal gas properties and low condensation temperature. In systems with a base temperature below 2K the He3 isotope is used.

The pulse tube works by transporting heat against a temperature gradient in a process called surface heat pumping. Surface heat pumping is expected to occur in many systems subjected to pressure oscillations [3]. The piston compresses the working gas, and every gas particle moves towards the closed end of the pulse tube. At the same time the temperature of the gas particles rises as they undergo adiabatic compression. The pressure experienced by a gas molecule in the pulse tube is determined by when it entered the pulse tube as show in Fig. 2.

Fig. 2: Initial and compressed gas in the pulse tube.

All the gas that was initially in the tube will be compressed to the hot end. The extra gas that flows in from the regenerator has a pressure gradient. The pressure is highest closest to the hot end and decreases to zero at the bottom of the pulse tube. Gas at the bottom entered the tube just as equilibrium was established. The pressure gradient directly results in a temperature gradient. At the hot end of the pulse tube the gas conducts its heat to the heat exchanger and the gas temperature falls. The piston then retracts and the gas undergoes adiabatic expansion cooling it even more. As the expanding gas passes from the pulse tube into the regenerator it absorbs heat from the regenerator and the pulse tube walls cooling them. The next cycle starts by compressing the gas back through the pre-cooled regenerator. The gas begins at a lower temperature and it therefore reaches an even lower temperature after finishing its compression expansion cycle. The net result of the cycle is that each element of the gas transports heat against the temperature gradient toward the closed hot end of the pulse tube where it is absorbed by the heat exchanger.

Record low temperatures achieved with this basic pulse tube design are 124K with a single stage and 79K using two stages[4]. Two stages describes two pulse tube coolers in series powered by the same compressor. The hot end of the second pulse tube is thermally anchored to the cold end of the first pulse tube cooler. There is a valve at the cold end of the first regenerator that allows compressed gas to flow into both the second regenerator and the pulse tube of the first cooler.

Fig. 3: Orifice pulse tube refrigerator.

Orifice Pulse Tube Refrigerator

The basic pulse tube refrigerator and more generally the surface heat pumping technique is clearly of limited use for reaching very cold temperatures, and in 1984 Mikulin et. al improved the design by adding an orifice near the warm end of the pulse tube. An orifice is just a throttle valve or needle valve to regulate flow. Their new design had a base temperature of 105K [5]. Radebaugh et. al further improved the design, reaching a low temperature of 60K, by placing the orifice outside the heat exchanger and adding a reservoir closing the orifice [4]. The design of the Orifice Pulse Tube Refrigerator (OPTR) is shown in Fig. 3.

Fig. 4: Sections of gas in the pulse tube. The regenerator is on the left side of region III the orifice is at point 0 which leads into a reservoir.

Enthalpy Model of Pulse Tube Cooling Power

OPTRs do not rely on surface heat pumping, but on enthalpy transfer into the pulse tube from the regenerator and out of the pulse tube through the orifice at the hot end of the pulse tube. Within the pulse tube are three regions of gas shown in Fig. 4.

The gas in region I is at the hot part of the pulse tube. During one cycle it flows into the pulse tube through the heat exchanger and orifice from the reservoir and then flows back out into the reservoir. The gas in region III is at the cold end of the pulse tube and flows in and out of the pulse tube through the regenerator. The gas in region II never leaves the pulse tube. It acts as a barrier between the cold and hot gasses in regions I and III. The energy for each section of gas is determined by the enthalpy flow through its region and the work done by the gas. This yields the following energy equation for sections I-III.

Section III Section II Section I

The work is the only energy transforming process in the pulse tube and in order to generate cooling enthalpy, H1, must flow out of the pulse tube, which means the gas in region III must do work outside the tube. Gas III does work on gas II which does work on gas I which causes the enthalpy flux H1 to flow out of the tube. The enthalpy flow and therefore the cooling power is maximized when gas III is optimized to do the most work on the other gases in the pulse tube.

This enthalpy transfer process is also described by simple harmonic model created by Radebaugh et al.[6]. We follow their derivation to calculate the net refrigeration power from the enthalpy transfer. Refrigeration in the OPTR is derived from the First and Second Law of Thermodynamics for an open system. Thermodynamic quantities are time averaged over one oscillation cycle of the refrigerator. According to the First Law the heat absorbed at the cold end is


<H1> is the time-averaged enthalpy flow in the pulse tube and <H4> is the time averaged enthalpy in the regenerator. <H1> will be zero for a perfect regenerator and an ideal working gas. Qc, the heat absorbed in the cold end at steady state, is also defined as the gross refrigeration power. Using the Second Law and the definition of enthalpy yields the time-averaged enthalpy flow at any point in the pulse tube.


Pd is the dynamic pressure, V is the volume flow rate, T0 is the average temperature of the gas and <S> is the time-averaged entropy flow. The first term on the right hand side of the equation, <PdV>, represents the amount of reversible work the gas can do at a pressure, Pd. It is also referred to as the hydrodynamic work or acoustic power.

If the pulse tube is assumed to be ideal then the expansion and compression processes are adiabatic and reversible. This means there is not net change in entropy throughout the cycle.


Assuming the pressure oscillations are sinusoidal, the average acoustic power in one cycle is


P1 is the sinusoidal pressure oscillation amplitude, V1 is the volume flow rate amplitude, θ is the phase angle between the flow and the pressure, R is the ideal gas constant per unit mass, and mi is the sinusoidal mass flow rate amplitude. Eqs. (1)-(3) yield the ideal or maximum refrigeration power for the pulse tube cooler.


This model for the maximum refrigeration power assumes that no heat is exchanged with the outside environment along the length of the pulse tube. To achieve this in practice pulse tubes are often operated in vacuum with heat exchangers just at the hot and cold ends. If the pulse tube remains insulated then the time averaged entropy flow through the pulse tube remains constant, and refrigeration is maximized.

Losses in Pulse Tube Cooler

A real pulse tube introduces nonideal conditions that reduce the cooling power stated in eq. (5). These include losses in both the regenerator and the pulse tube. The regenerator typically has some enthalpy, <H4>, which reduces the total acoustic power. Entropy is also generated in the pulse tube itself from heat exchanged with the walls, turbulent mixing of the hot and cold gas segments, circulation of the gas within the tube, and end effect losses from the adiabatic to isothermal transition in the cycle. All of these processes reduce the enthalpy flow in the pulse tube from the ideal acoustic power; however, these processes are difficult to calculate and no model can sufficiently simulate all these effects [6]. Instead the figure of merit for an pulse tube cooler is defined empirically by how much the actually enthalpy flow, <H1>, differs from the acoustic power, <PdV>.


Figures of merit between 0.55 and 0.85 have been reported for small orifice pulse tube refrigerators [7].

Fig. 5: Double inlet pulse tube refrigerator. Gas flow through the regenerator into the pulse tube and through the secondary orifice directly to the warm end.

Double Inlet Pulse Tube Refrigerators

Orifice pulse tube refrigerators have achieve significantly lower base temperatures and higher cooling power because the addition of the orifice and the external reservoir cause an enthalpic heat flow rather than surface heat pumping. However, even higher efficiencies can be induced by introducing a phase shift between the oscillations of the pressure and mass flow within the pulse tube system.

Only a small amount of the gas that flows through the regenerator does external work. The efficiency of the pulse tube refrigerator can be increased by maximizing the refrigeration power per unit mass flow. The extra gas, even though it provides no refrigeration power, still must be cooled by the regenerator which increases the heat transfer load but does no work and therefore limits refrigeration.

Zhu, Wu and Chen addressed this problem by adding a direct connection, or secondary orifice, between the warm end of the regenerator and the warm end of the pulse tube [8]. Their design called a double inlet pulse tube refrigerator is show in Fig. 5.

The secondary orifice is designed to allow about 10% of the gas, specifically the gas that does not contribute to refrigeration, to travel directly from the pressure oscillator to the warm end of the pulse tube, bypassing the regenerator pulse tube circuit. This direct flow assists in compressing and expanding the warm working gas in the pulse tube, and reduces the amount of gas that needs to be pre-cooled by the regenerator. The extra gas flow also adjusts the phase angle between the pressure and mass flow in the system.

Obtaining the correct phase between the gas flow and the pressure is crucial to increasing refrigeration efficiency. Eq. (5) shows that the maximum acoustic power is achieved when the mass flow and pressure and in phase with each other. However, the phase angle between the mass flow and pressure is different at different points in the pulse tube system due to the large volumes contained by the pulse tube and regenerator. To achieve maximum efficiency pressure and mass flow should be in phase at the cold end of the pulse tube where the gas begins to do work. In an ordinary orifice pulse tube the point where the mass flow and pressure are in phase is at the entrance to the orifice at the hot end of the tube not the cold end.

Fig. 6: Series arrangement of a multistage pulse tube.

If mass flow and pressure are in phase at the orifice, then the mass flow will lead the pressure by about 30° at the cold end of the pulse tube and will lead the pressure by as much as 60° at the warm end of the regenerator. Such a large pressure drop in the regenerator results in poor heat exchange between the regenerator and the working gas, which lowers the overall efficiency of the cooler. Heat exchange in the regenerator is maximized when the pressure amplitude averaged throughout the regenerator is minimized. This means it is optimal to have mass flow at the warm end lead the pressure and mass flow at the cold end lag the pressure. Radebaugh reports optimal working conditions when the mass flow lags the pressure by 30° at the cold end of the regenerator and pulse tube [6]. To achieve this phase difference the mass flow must lag the pressure by 60° at the warm end of the pulse tube. Flow through the secondary orifice adjusts the phase difference to this optimal point as well as diverting some of the flow out of the regenerator.

Fig. 7: Parallel arrangement of a multistage pulse tube.

Multistage Pulse Tube Refrigerators

Introduction of the orifice and double inlet pulse tube pushed the limit of the system's cooling power. Today's modern pulse tube coolers incorporate these improvements; however, to reach liquid helium temperatures with a pulse tube cooler requires multiple cooling stages [9]. The series pulse tube configuration described earlier in this paper was the first multistage assembly built to improve the base temperature of the pulse tube refrigerator. However, the series pulse tube is not an efficient design. In a series configuration the heat generated by the hot end of the lower stage pulse tube must be removed by the upper stage cooling power which decreases the cooling capacity of the upper stage. Matsubara and Gao redesigned the multistage pulse tube cooler by putting the regenerators in series but the pulse tubes in parallel [9]. This revolutionary design allowed them to reach a minimum temperature of 3.6K and a cooling power of 100 mW at 4.8K [1]. Their design is now the current model for today's pulse tube cryocoolers.

In this paper, we have briefly covered the operation mechanisms and advances associated with the pulse tube refrigerator. It is a compact, vibration and cryogen free refrigerator capable of reaching liquid helium temperatures and maintaining cooling power. By attaching a dilution refrigerator or adiabatic demagnetization refrigerator to the cold end the whole system can provide cooling power at milikelvin temperatures creating a revolutionary and reliable way of running low temperature terrestrial and space based experiments without expensive cryogens.

© 2007 J. Bert. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


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