# Basics of Joule–Thomson Liquefaction and JT Cooling

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## Abstract

This paper describes the basic operation of Joule–Thomson liquefiers and Joule–Thomson coolers. The discussion is based on the first law of thermodynamics mainly using hT-diagrams. It is limited to single-component fluids. A nitrogen liquefier and a helium cooler are discussed as important examples.

## Keywords

Thermodynamics Joule–Thomson liquefiers and coolers Cryocoolers hT-diagrams## List of symbols

*A*Area (\(\hbox {m}^{2}\))

*C*Heat capacity (J/K)

*h*Specific enthalpy (J/kg)

*l*Heat exchanger length (m)

- \(\overset{*}{m}\)
Mass flow rate (kg/s)

*p*Pressure (Pa)

*P*Compressor input power (W)

*q*Specific heating power (J/kg)

- \(\dot{Q}_{\mathrm {L}}\)
Heating power (W)

*s*Specific entropy (J/kg K)

*v*Velocity (m/s)

*T*Temperature (K)

*x*Liquid mass fraction

- \(\rho \)
Density (\(\hbox {kg/m}^{3}\))

- \(\xi \)
Coefficient of performance

## List of lower indices

- a, b, c, d, e, f, g
Positions

- h, l
High- and low-pressure side

- co
Compressor

- ci
Compressor irreversible

- L
Low temperature

- Lim
Limiting

- min, max
Minimum, Maximum

## 1 Introduction

Joule–Thomson (JT) coolers are invented by Carl von Linde and William Hampson so these coolers are also called Linde–Hampson coolers [2]. Basically cooling, obtained by the JT process, is very simple and is widely applied in household refrigerators, air conditioners, cryocoolers, and as the final stage of liquefiers. They can be miniaturized [3, 4, 5], but they are also used on a very large scale in the liquefaction of natural gas [6, 7]. Even though this way of cooling is more than 160 years old published measurements of the cooling power of JT cryocoolers are rare. A critical comparison between theory and experiment is missing. For example, Ref. [5] does not contain one single plot of the measured cooling performance of a JT cooler. As a consequence, system properties discussed in this paper, such as in Fig. 16, are theoretical predictions.

The JT systems are often discussed using temperature–entropy (Ts) diagrams. However, entropies are needed mainly in loss analyses which will not be part of this paper. The basic principles can be understood much better with enthalpy–temperature (hT)-diagrams. Therefore, we will mainly deal with hT-diagrams. Irreversible processes and high pressures in JT coolers can be reduced by using gas mixtures instead of pure fluids [7, 8]. However, we will discuss pure fluids only. For the liquefier, we discuss nitrogen, and for the cooler we will treat helium four as the working fluid as important examples.

*T*of the fluid, the density \(\rho \), the velocity

*v*, and specific enthalpy

*h*vary over the cross section. The mass flow rate \(\overset{*}{m}\) is given by

*A*of a channel perpendicular to the flow. What we call “the” temperature \(T_{\mathrm {l}}\) of the gas in the low-pressure channel is defined by the relation

- 1.
Just like in Ref. [1], kinetic and potential energies of the fluid are neglected.

- 2.
The CHEX has no flow resistance. This condition is easier satisfied at the high-pressure side than at the low-pressure side. At the low-pressure side, the flow resistance immediately affects the value of the low temperature.

- 3.
We consider only the steady state.

- 4.
Heat conduction in the flow direction is neglected. This is justified since the heat flows in the flow direction usually are much smaller than the heat flows between the two sides of the CHEX.

## 2 JT Liquefier

### 2.1 System Description

*x*, which enters a reservoir. The reservoir can be used to collect liquid for some time with the exit valve at e closed. This valve can be opened to drain liquid from the reservoir when needed. Here we will discuss the steady state where all liquid, formed after the expansion in the JT valve, is removed from the reservoir instantaneously at e. The remaining gas flows into the low-pressure side of the CHEX at point f, leaving it at point g. In general, \( T_{\mathrm {g}}\ne T_{\mathrm {a}},\) so there will be some heat exchange with ambient in the line from g to a.

In Fig. 2, the pressures refer to typical values for a nitrogen liquefier. At the inlet of the compressor, the gas is at room temperature (taken here at 300 K) and a pressure of 1 bar (point a).^{1} After compression, it is at 300 K and 200 bar (point b). In point d, it has a temperature \(T_{\mathrm {d}}=\) 77.244 K and a pressure of 1 bar.

### 2.2 The Maximum Liquefaction Rate

First we assume that the gas at the low-pressure side leaves the CHEX at point g with a temperature equal to room temperature so \(T_{\mathrm {g}}=T_{ \mathrm {a}}\). Later on, we will see that this means that we calculate the maximum liquid fraction \(x_{{\mathrm {\max }}}\). The thermodynamic properties of nitrogen are obtained from Ref. [9] from which the hT-diagram (Fig. 3) is derived.

Thermodynamic values of the JT nitrogen liquefier [9]

| | | |
---|---|---|---|

a | 300 | 1 | 311.20 |

b | 300 | 200 | 279.11 |

c | 165 | 200 | 62.316 |

d | 77.244 | 1 | 62.316 |

e | 77.244 | 1 | \(-122.25\) |

f | 77.244 | 1 | 77.073 |

g | 300 | 1 | 311.20 |

### 2.3 More Detailed Analysis

So far we assumed that \(T_{\mathrm {g}}=T_{\mathrm {a}}\), but this is not generally true, e.g., due to the finite length of the CHEX. In this subsection, we will drop this assumption and also derive relations for the temperatures inside the CHEX.

*x*. An example is given for \(x=0.04\) in Fig. 5. In this case, the temperature at the exit \(T_{ \mathrm {g}}\approx 286\) K. Improving the CHEX will increase

*x*and “lower” the low-pressure curve through the factor \(1-x\) in Eq. (12) until it touches the high-pressure curve at room temperature if \(x=x_{{\mathrm {\max }}}\) . This is the maximum production rate of the liquefier as given in Eq. (6).

## 3 JT Coolers

Now we turn our attention to JT coolers. In this case, there is no liquid production, but the JT cooling effect is used in a cryocooler to cool a particular application.

### 3.1 System Description

The right-hand side in Figs. 1 and 6 represents a JT cooler. In some applications, there is no compressor, but the gas is supplied by a high-pressure container and the gas leaves the system at 1 atm (open cycle), but for the thermodynamics this makes no difference. A heating power \(\dot{Q} _{\mathrm {L}}\) is applied at the low temperature \(T_{\mathrm {L}}=T_{\mathrm {f }}\) at the coldest point which usually is called the evaporator.^{2} As mentioned before, we will take helium as an important example of a cooling medium. It is necessary to precool the helium with a cryocooler or a (pumped) liquid hydrogen bath. Figure 7 shows the hT-diagram of helium with several isobars.

### 3.2 Exit Temperature

### 3.3 Temperature Profiles

### 3.4 Zero Heat Load

*h*(horizontal line in the diagram) can be found on the 1 bar isobar at about \(T_{\mathrm {l}}=\) 12.3 K. A \(T_{\mathrm {h}}\) of 14 K gives a \(T_{\mathrm {l}}\) of about 11 K, etc. In this way, the \(T_{\mathrm {l}}-T_{\mathrm {h}}\) dependence can be constructed and is shown in Fig. 10. It is important to note that we know the \(T_{\mathrm {l}}-T_{\mathrm {h}}\) relationship, but we do not know at which position in the CHEX we have these particular values of \(T_{\mathrm {l}}\) and \(T_{\mathrm {h}}.\) This is determined by the design of the CHEX (shape and size of the channels, wall material, \(\ldots \)) and the mass flow rate.

^{3}For example, for \(p_{\mathrm {h}}=40\) bar and \(T=T_{{\mathrm {Lim}}}=4.2\) K, \(h=19.5\) J/g. In the pT-diagram in Fig. 13, the \(T-p\) relationship for this isenthalp is given. Usually only the beginning and the end of the curve are relevant since, inside the flow constriction, the velocities usually are very high (speed of sound). If the diameters of the supply and drain lines are large enough (which is usually the case), the velocities at the ends are small and the kinetic energy can be neglected. If the flow resistance would consist of, e.g., a long porous plug, where the gas velocity would remain far below the speed of sound, one would actually see that the temperature goes up to 7.5 K in the middle of the plug. In Fig. 13, it can be seen that the expansion follows the saturated vapor line in the pT-diagram after it “hits” this line at 1.6 bar. At the low-pressure side, the curve ends just inside the two-phase region as shown in Fig. 12.

As noted before, we know the \(T_{\mathrm {l}}{-}T_{\mathrm {h}}\) relationship inside the CHEX, but we do not know at which point in the CHEX we have a particular \(T_{\mathrm {l}}\)–\(T_{\mathrm {h}}\) combination. Figure 14 shows the temperatures \(T_{\mathrm {l}}\) and \(T_{\mathrm {h}}\) in a CHEX with \(p_{ \mathrm {h}}=\) 40 bar and \(p_{\mathrm {l}}=\) 1 bar as function of the position with the length *l* in arbitrary units. With zero heat load, as discussed in this section, the fluid leaves the evaporator with the same composition as after the JT expansion. For \(p_{\mathrm {h}}<42\) bar and if the heat exchanger is long enough, this means that two-phase flow enters the low-pressure side of the CHEX at the cold end. When this gas–liquid mixture flows further inside the CHEX, the liquid evaporates until, at some point, only gas remains. In the region of two-phase flow, \(p_{\mathrm {l}}\) and \(T_{ \mathrm {l}}\) are constant. In this part of the CHEX, \(T_{\mathrm {h}}\) approaches \(T_{\mathrm {l}}\) exponentially. If the CHEX is not as long as assumed in Fig. 14, the temperature profile is simply a part of the curves shown. So the figure can also be regarded as giving the temperatures at the cold end as functions of the CHEX length.

### 3.5 Finite Heat Load

*the hot end*of the CHEX are the same (\(T_{ \mathrm {g}}=T_{\mathrm {b}},\) pinch point) so there is no temperature difference any more to drive the cooler to temperatures below the entrance temperature. Hence, Eq. (21) is the relation for the maximum specific heat load. If \(q_{\mathrm {L}}\) is increased from below \(q_{{\mathrm {\max }}}\) to above \(q_{{\mathrm {\max }}}\), the cooler “suddenly” stops working in the sense that the temperature starts to drift away indefinitely. This is unlike other cryocoolers, such as pulse tube or GM coolers, where the cooling power smoothly increases with \(T_{\mathrm {L}}\) until room temperature is reached. The strange behavior of JT coolers can only be observed in its pure form in an infinitely long CHEX. For a CHEX of finite length, \(T_{\mathrm {c}}\) increases with \(q_{\mathrm {L}}\) especially if \(T_{\mathrm {g}}\) approaches \( T_{\mathrm {b}}\). The specific enthalpy of the saturated vapor at 1 bar is 20.648 J/g. The isenthalp in the pT-diagram (compare Fig. 13) has a maximum at 7.674 K at a pressure of 13.8 bar. So the highest temperature with which 4.2 K can be reached is 7.7 K and is realized if \(p_{\mathrm {h}}=13.8\) bar. For all other \(p_{\mathrm {h}}\), the \(T_{\mathrm {c}}\) with which 4.2 K can be reached is lower. So, in any case, \(T_{\mathrm {L}}>4.2\) K if \(T_{\mathrm {c} }>7.7\) K (Fig. 15).

Figure 16 shows calculated plots of \(q_{\mathrm {L}}\) as functions of \(T_{ \mathrm {L}}\) for four values of \(p_{\mathrm {h}}\). The \(q_{{\mathrm {\max }}}\) values are indicated with the little circles. For \(p_{\mathrm {h}}\) below about 19 bar, \(T_{\mathrm {L}}\) remains constant and equal to the boiling point value of 4.2 K until \(q_{{\mathrm {\max }}}\) is reached. If \(p_{\mathrm {h} }>19\) bar, the \(q_{\mathrm {L}}\) curve contains a part of constant \(T_{\mathrm { L}}\) and one where \(T_{\mathrm {L}}\) increases above 4.2 K (see also Fig. 15). It is often suggested that the fluid that leaves the cold heat exchanger (at point f) is saturated vapor (see, e.g., Ref. [5] Fig.3.3, [11] Fig. 1, [12] Fig. 1). This is correct for the liquefier (almost by definition) but incorrect for the cooler. In a cooler, the composition of the fluid that leaves the “evaporator” is either a liquid-gas mixture or superheated gas. The case that *saturated* gas leaves the cold CHEX is accidental.

Instead of a discrete expansion by a JT valve, one might consider a continuous expansion by having a high flow resistance at the low-temperature side of the high-pressure channel. The continuous JT expansion is shown in the right figure in Fig. 17. In this case, there is heat exchange between the expanding fluid and its surroundings, so, strictly speaking, this is not a JT expansion. The flow resistance inside the CHEX is of the same order as the flow resistance of the final expansion. That means that one can make the high-pressure channel very narrow, e.g., as a narrow gap between a capillary and a wire which closely fits into the capillary. As a result, the thermal resistance in the fluid at the high-pressure side will be practically zero which enhances the heat exchange at this side of the CHEX. Also the tube diameters can be very small which may decrease the thermal mass at the low-temperature side of the cooler and speed up all processes. Care should be taken that the heat exchange at the low-pressure side remains adequate. It may be interesting to analyze the case where the whole expansion takes place inside the high-pressure channel so that there is no final expansion.

## 4 *COP*

*P*the electrical power supplied to the compressor. The second law reads

*P*, where \(\dot{S}_{{\mathrm {ci}}}=0\), satisfies

*COP*is defined by

*COP*of the entire system, the energy consumption by the precooler has to be taken into account.

## 5 Measured Cooling Powers

Another source of a measured JT cooling power is the CryoTiger manual (Ref. [16]). Figure 19 shows its cooling power characteristic. The CryoTiger is a JT cooler which operates not with a pure fluid but with a gas mixture. There is no part in the \(T_{\mathrm {L}}-\dot{Q}_{\mathrm {L}}\) where \(T_{ \mathrm {L}}\) is constant as shown in Fig. 16. The \(\dot{Q}_{\mathrm {L}}-T_{\mathrm {L }}\) plot has a maximum of 24.2 W at about 140 K. If the heating power would be increased above 24.2 W, the temperature would rise unlimited in accordance with the behavior described above.

During cool down, one has to be aware of the fact that the flows in the CHEX may differ from point to point due to the changing density distribution. If there is a reservoir in the system there may be a phase in the cool down during which it is filling with liquid. In that case, the flows at the warm and cold sides differ significantly. Also the finite heat capacity of the CHEX comes into play. If there is a big thermal mass at the cold end, the cool down is very slow and we are dealing with a quasi-steady situation.

## 6 Conclusion

The first law of thermodynamics is a very powerful tool for analyzing JT liquefiers and JT coolers. Important properties such as liquefaction rates, cooling powers, temperature distributions, and system limitations can be easily visualized using hT-diagrams.

## Footnotes

- 1.
For convenience, we take the pressure at the inlet of the compressor equal to 1 bar while, in reality, it will be around 1 atm which is 1.01325 bar. The boiling point of a substance is defined as the saturated temperature at 1 atm. For nitrogen, it is 77.355 K. This differs slightly from the saturation temperature at 1 bar which is 77.244 K.

- 2.
In practice, the evaporator is a reservoir which may contain some liquid that is just sitting there. The position of the liquid level is determined by the position of the entrance of the exit tube. Evaporation of the liquid can take place in the evaporator but also in the counterflow CHEX, and in other situations, superheated gas can leave the evaporator. So, what is called the evaporator could better be called the cold heat exchanger.

- 3.
During the expansion, the fluid temperature changes (see, e.g., Fig. 13) With an

*isothermal*expansion, we mean that the end points of the expansion are at the same temperature.

## Notes

### Acknowledgements

I thank the members of the cryogenic group (Cryoboat) of the Zhejiang University in Hangzhou, China, for their kind hospitality and their stimulating interest in my work.

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