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Due to its unique properties, helium has wide application in different industries and scientific fields, which has turned it into a strategic material. Helium liquefaction plants include wide temperature range from 300 k to 4.2 k, so these plants have high energy consumption. A lot of studies have done to optimize the operation of these cycles. In this research, an exergy analysis is performed for a liquid helium production plant. The optimal performance of 3 and 4 stage cycles is extracted using parametric study and the results are compared with those of Collins dualexpander cycle. The results show that by increasing the number of cooling stages, not only the compressor optimum discharge pressure is reduced, but also the cycle efficiency dramatically increases and the power consumption of the cycle decreases. Further, a sensitivity analysis of the exergy efficiency of the 2, 3 and 4 expander cycles is compared to the heat exchangers effectiveness, expander’s efficiency and the input flow rate to the expanders.
cold box, cryogenic process, cycle efficiency, exergy analysis, helium liquefaction, parametric study
Due to its unique and various properties, e.g. low liquidstate viscosity, high conductivity coefficient, nonreactivity and being liquid at about 4.2 K, helium has wide application in different industries such as medicine, surgery, imaging, transportation and electronic equipment. Currently, natural gas is the only economic source of liquid helium production. In the last century, for the extraction and liquefaction of helium, different cycles have been designed and built based on regional conditions. The purpose of these researches is to enhance liquefaction and reliability, and reduce energy consumption. The helium liquefaction unit includes various equipment, e.g. compressors, heat exchangers and expanders.
Collins was the first researchers who designed an economical helium liquefier [1, 2]. In this cycle a JT valve and two expanders were used to liquefy helium. The analysis of Collins cycle as a basis cycle for today's liquefaction cycles can be a great help for the analysis of more complex cycles. Liquefaction cycles can be analyzed by two methods: The first one is to use the first law of thermodynamics for cycle optimization. This law is exclusively useful for the problems, in which energy conservation is examined [35]. The second method is to utilize exergy analysis, which is based on the second law of thermodynamics. Using this method provides adequate information on the quality of flow at any point in the cycle and so it’s more suitable for analyzing this type of problems. In fact, expressing the flow properties as exergy, integrates the expression of flow properties at different point in the cycle, thereby simplifying their comparison.
Considerable efforts have been made to study the effect of different thermodynamic parameters of liquefaction cycles, e.g. compressors discharge pressure, number of compression stages and intermediate pressures, total input flow rates to expanders and flow distribution between them.
In the study based on thermodynamic analysis (first law), Attery investigated the effect of distributing Collin cycle expander flow rate on cycle efficiency and showed that the energy required by this cycle for optimal performance depended on this parameter [6]. Using exergy analysis, Treep discussed the exergy losses in equipment of liquefier cycle [7], which formed the basis of liquefier optimization because it determined where the input energy and exergy of the cycle was destroyed. According to this study, approximately half of the exergy entering the cycle lost at the compression stage.
Following their previous research, Thomas et al. first determined the portion of exergy losses in each equipment for Collins dualexpander cycle and then investigate the effects of various performance parameters on cycle exergy efficiency [8]. They studied the effects of various parameters including compressor discharge pressure, expanders total input flow rate and flow distribution between them, expanders efficiency, effectiveness of heat exchangers, number of cooling stages used in the cycle and type of the cold end on the level of liquefaction and cycle efficiency [9]. By varying different parameters of the cycle such as expander efficiency, they attempted to show the trend of changes in cycle efficiency and the exergy losses of different equipment.
According to their studies on Collins dualexpander cycle, cycle efficiency is maximum at the pressure of 22 bar. By studying the effect of input flow rate to the expanders, they suggested that efficiency was maximum when assigning 80% of the compressors output flow rate equally to the expanders at any pressure. Moreover, as expected, cycle efficiency was maximized by assigning 80% flow rate equally to the expanders at the pressure of 22 bar [8].
In another study, Thomas et al. investigated the effect of increasing the number of cooling stages and putting intermediate pressure in structure of the cycle [10]. Accordingly, increasing the number of cooling stages decreased exergy loss in the heat exchangers by reducing their performance temperature range and, thereby, decreasing the input temperature to the JT valve which, in turn, enhanced cycle efficiency. Also, by making more consistency between the heat capacities of the heat exchangers flow and reducing the expanders losses, intermediate pressure enhanced the cycle efficiency [10].
Further studies have been performed on the cycles with more expanders and cooling stages [1114].
In addition to the research on the helium liquefaction cycle, further studies have been conducted on the analysis and optimization of thermodynamic parameters in liquefaction cycles for other gases [1517]. Thomas et al. examined the effects of different parameters on cycle efficiency and attempted to identify the important parameters in designing an optimal cycle and propose a general trend for analyzing and optimizing largescale industrial cycles [8]. However, all their studies on different cycles were conducted at the compressor discharge pressure of 22 bar, while this pressure was optimal only for Collins dualexpander cycle. Consequently, they did not compare different cycles at their optimal pressures and, thus, their optimal performance states.
In the present research, by varying compressor discharge pressure in 3 and 4 expander cycles, first, the optimal pressure is calculated in each state. Then, through the parametric analysis of heat exchangers effectiveness, expanders efficiency, expanders total input flow rates, etc., the effects of these parameters on cycle optimal pressure are indicated and, finally, the optimal performance state of every cycle is determined.
Figure 1 shows the schematic representation of a liquefaction cycle with ncooling stages. Since the purpose of this research is to study 3 and 4 expander cycles, the value of n is equal to 3 and 4. Conventional helium liquefaction cycles include three units:
(1) Compression unit: This unit responsible for conducting a compression process near the constant temperature line, which includes several stages together with inter and after coolers;
(2) Precooling unit: This unit which comprises different cooling stages (each stage includes an expander and two heat exchangers) is responsible for reducing the temperature of highpressure helium flow below the maximum inversion temperature of helium.
(3) Liquefaction unit: in this unit, a fraction of gas flow is converted into liquid through constantenthalpy or constantentropy expansion. This unit is known as cold end.
In this research, the following assumptions are considered for analyzing and obtaining the equations:
(1) The system is operating in the steady state.
(2) The compressors and expanders efficiency do not change with pressure, temperature or flow rate.
(3) The heat transfer coefficient of heat exchangers does not change with pressure, temperature or flow rate.
(4) The effects of heat leakage in the heat exchangers and other irreversibilities, e.g. longitudinal heat conduction and flow maldistribution, are all considered in the effective UA of heat exchangers.
(5) The effects of heat leak into the pipelines have been neglected.
Figure 1. Schematic of the nexpander liquefaction cycle
The 32parameter modified BWR equation of state has been used for determining the thermodynamic properties of helium [8]. Also, the exergy balance equation was utilized for obtaining the cold box exergy efficiency, cycle efficiency and liquefaction:
$\sum_{\mathrm{j}} \mathrm{Q}^._{j}\left(1\frac{\mathrm{T}_{0}}{\mathrm{T}_{\mathrm{j}}}\right)\dot{\mathrm{W}}+\sum_{\mathrm{in}} \mathrm{E}^. \mathrm{x}_{\mathrm{flow}}$$\sum_{\mathrm{out}} \mathrm{{E}^.x}_{\mathrm{flow}}\mathrm{Ex}_{\mathrm{dest}}=0$ (1)
The first term shows the exergy transferred by heat transfer; $\dot{\mathrm{Q}}_{\mathrm{j}}$ is the heat transfer rate from the control volume boundary (kW); $T_{j}$ shows the temperature of each equipment (K); and $T_0$ indicates the temperature of reference condition (K). The second terms show the exergy transferred as a work in control volume and is equal to the electrical and mechanical works that transfer to the system (kW). The third and fourth terms denote the input/output exergy to/from the control volume by mass flow rate (kW), is calculated by Eq. (2):
$\dot{\mathrm{E}} \mathrm{x}_{\mathrm{flow}}=\dot{\mathrm{m}} \times \psi$ (2)
where, $m ̇$ is the mass flow rate (kg s^{1}) and ψ indicates the exergy of flow per unit of mass (kJ.kg^{1}). Eq. (3). depicts the calculation of ψ, in which h and s respectively denote the specific enthalpy and entropy of the fluid:
$\psi=\left[\left(\mathrm{h}\mathrm{h}_{0}\right)\mathrm{T}_{0}\left(\mathrm{s}\mathrm{s}_{0}\right)\right]$ (3)
where, $h_0$ and $s_0$ are the specific enthalpy and entropy of the fluid at the temperature and pressure of reference condition, respectively. In this study, $T_{0}=300 \mathrm{K}$ and $P_{0}=1.013$ bar. The fifth term in Eq. (1). shows the thermodynamic irreversibility’s in the cycle that is calculated by Eq. (4).:
$\dot{\mathrm{E}} \mathrm{x}_{\mathrm{dest}}=\mathrm{T}_{0} \Delta^. \mathrm{S}_{\mathrm{g}}$ (4)
where, $\Delta^. S_{g}$ indicates the entropy generation rate (kW K^{1}) in the process. By applying the exergy balance equation to C.V. (Figure 1), the following equation is obtained:
$\mathrm{\dot{m}}\left(\psi_{2}\psi_{1^{\prime}}\right)=\dot{\mathrm{m}}_{l}\left(\psi_{l}\psi_{1^{\prime}}\right)+\sum_{\mathrm{i}=1}^{\mathrm{n}} \dot{\mathrm{m}}_{\mathrm{expn}} \Delta \psi_{\mathrm{expn}}$$+ {\dot{E}{x}_{destcoldbox}}$ (5)
The mass flow rate balance equation in Figure 1 will be as follow:
$\dot{\mathrm{m}}=\dot{\mathrm{m}}_{l}+\dot{\mathrm{m}}_{l p}$ (6)
where, $\dot{m}_{L}$ shows the output liquid flow rate of the cycle and $\dot{\mathrm{m}}_{\mathrm{LP}}$ indicates the flow rate of helium on the lowpressure line. The following equation is obtained by Eq. (5).:
$\frac{\dot{\mathrm{m}_{l}}}{\dot{\mathrm{m}}}\left(\psi_{l}\psi_{1^{\prime}}\right)=\left(\psi_{2}\psi_{1^{\prime}}\right)$$\sum_{i=1}^{n} \frac{\dot{m}_{e x p n}}{\dot{m}} \Delta \psi_{e x p n}\frac{ { {E^.x}_ {destcoldbox} }}{\dot{m}}=0$ (7)
The output liquid rate can be written in the following form based on exergy equations:
$\mathrm{y}=\frac{\dot{\mathrm{m}}_{1}}{\dot{\mathrm{m}}}=\frac{\left(\psi_{2}\psi_{1} \prime\right)}{\left(\psi_{1}\psi_{1} \prime\right)}\sum_{\mathrm{i}=1}^{\mathrm{n}} \frac{\dot{\mathrm{m}}_{\mathrm{expn}}}{\dot{\mathrm{m}}} \frac{\Delta \psi_{\mathrm{expn}}}{\left(\psi_{1}\psi_{1} \prime\right)}$ (8)
Since the expanders are rarely used in conventional smallscale cycles, Eq. (9)., and Eq. (10). demonstrate the input exergy rate to and the output exergy rate from the cold box, respectively:
Net exergy input $=\dot{\mathrm{m}}\left(\psi_{2}\psi_{1^{\prime}}\right)$ (9)
Net exergy Output $=\dot{\mathrm{m}}_{l}\left(\psi_{l}\psi_{1^{\prime}}\right)$ (10)
The cold box exergy efficiency, which indicates the ability of the cycle for liquefaction, can be defined based on Eq. (7). as:
$\eta_{\mathrm{ex}\mathrm{coldbox}}=\frac{\dot{\mathrm{m}}_{l}\left(\psi_{l}\psi_{1} \prime\right)}{\dot{\mathrm{m}}\left(\psi_{2}\psi_{1} \prime\right)} \times 100 \%$ (11)
Eq. (12). indicates the compressor exergy efficiency:
$\eta_{\mathrm{ex}\mathrm{comp}}=\frac{\dot{\mathrm{m}}\left(\psi_{2}\psi_{1} \prime\right)}{\dot{\mathrm{W}}_{\mathrm{comp}}} \times 100 \%$ (12)
The numerator of the above fraction is the input exergy to the cycle and its denominator is the total input electrical work to the compressors.
Furthermore, the cycle exergy efficiency, which includes the exergy efficiencies of the cold box and compressor, is calculated based on Eq. (13).:
$\eta_{\text {excycle }}=\eta_{\text {excoldbox }} \times \eta_{\text {excomp }}=\frac{\dot{\mathrm{m}}_{l}\left(\psi_{l}\psi_{1}^{\prime}\right)}{\dot{\mathrm{W}}_{\text {comp }}} \times 100 \%$ (13)
The nondimensional heat transfer coefficient of heat exchangers is defined according to the equation below:
$(\mathrm{UA})^{*}=\frac{\mathrm{UA}}{\mathrm{\dot{m}c}_{\mathrm{p}}}$ (14)
Here, the specific heat capacity $c_p$ is measured at temperature and pressure of compressor suction flow.
The homemade code is used to simulate and obtain the results. In all the simulation states, the efficiency of heat exchangers and expanders is assumed 97% and 70%, respectively and the flow distribution between the expanders is assumed equal, unless note otherwise.
3.1 Effect of cooling stages
Generally, increasing the number of cooling stages reduces the heat exchangers temperature range and also input temperature to the JT valve, thereby enhancing cycle efficiency and liquefaction level. Assuming the high pressure of the cycle is 20 bar and low pressure of the cycle is 1.01 bar, Figure 2 indicates the effect of increasing cooling stages on the cold box exergy efficiency and input temperature to the JT valve. The following results are achieved from Figure 2.
Figure 2. Changes in cold box exergy efficiency based on the number of precooling stages in the liquefaction cycle at the compressor discharge pressure of 20 bar
3.2 Effect of compressor discharge pressure
Figure 3, Figure 4 and Figure 5 show the effect of variation in compressor discharge pressure on the cold box exergy efficiency, cycle exergy efficiency and level of liquefaction for 2, 3 and 4 expander cycles, respectively. In liquefier cycle, the optimal operating pressure of a liquefaction cycle is an important parameter since it shows the cycle power consumption. Results of this study show:
Figure 3. Changes in the cold box exergy efficiency for 2, 3 and 4 precooling stages in the liquefaction cycle based on the compressor discharge pressure
Figure 4. Changes in the cycle exergy efficiency for 2, 3 and 4 precooling stages in the liquefaction cycle based on the compressor discharge pressure
Figure 5. Changes in the liquefaction level for 2, 3 and 4 precooling stages of the liquefaction cycle based on the compressor discharge pressure
Furthermore, the exergy destruction in various equipment of 2, 3 and 4 expander liquefaction cycles are drawn based on total input exergy to the cold box in Figure 6 with the following results:
The ratio of exergy destruction to the input exergy of JT valve is significant due to the small amount of JT valve flow rate.
Figure 6. Exergy destruction in different equipment of the liquefaction cycle to the total input exergy to the cold box at the cycle optimal performance
3.3 Effect of total input flow rate to the expanders
According to Thomas, the highest exergy efficiency of the cold box in Collins dualexpander cycle occurred when the expanders total input flow rates equaled 80% of the input flow rate to the cold box [4, 8]. Figure 7 and Figure 8 demonstrate the results of investigating the effect of this parameter on the efficiency of the 3 and 4 expander cycles, respectively. Based on these two figures, it is found that:
Figure 7. Changes in the cold box exergy efficiency for various input flow rate to the expanders based on the compressor discharge pressure in the 3 expanders cycle
Figure 8. Changes in the cold box exergy efficiency for various input flow rate to the expanders based on the compressor discharge pressure in the 4 expanders cycle
3.4 Effect of specific flow rate distribution between expanders
In this step, different flow rate distribution between expanders is investigated in order to obtain maximum efficiency at various pressures. The analysis is as follows: at every step, 10% of the flow rate of an expander is increased from the state, in which the total flow rate is divided equally between the expanders, and the same amount is reduced in the next step. During each change, level of flow rate for other expanders is assumed constant in order to precisely determine the importance of the input flow rate to each expander in the cycle. Figure 9 shows the simulation results for the 3 expander cycle.
Figure 9. Changes in the cold box exergy efficiency for the distribution of specific input flow rate among the expanders based on the compressor discharge pressure in the 3 expanders cycle
Figure 10 shows the simulation results for the 4 expander cycles.
Figure 10. Changes in the cold box exergy efficiency for the distribution of specific input flow rate among the expanders based on the compressor discharge pressure in the 4 expanders cycle
To further explain for the result of Figure 10, the chart of exergy destruction in the cycle equipment per input exergy to the 4 expanders cold box is drawn at the optimal pressure of 14 bar in Figure 11. Also, in Figure 12, the chart of hot flow temperature change along the cycle for the equal distribution of specific flow rate, as well as the limit states of the unequal specific flow rate, i.e. EXP1 frac.=15% and EXP4 frac.=15%, is presented. The following results are obtained:
Figure 11. Exergy destruction ratio in different equipment of the liquefaction cycle to the total input exergy to the cold box in different states of the distribution of specific flow rate among the expanders of the 4 expanders cycle
Figure 12. Changes in the temperature of the hot flow along the cycle in different states of the distribution of specific flow rate among the expanders of the 4 expanders cycle
3.5 Effect of heat exchangers effectiveness
In liquefaction cycles, such as helium liquefier, the heat exchangers effectiveness is highly important for achieving appropriate liquefaction. Figure 13 depicts the minimum effectiveness of the heat exchangers (assuming the same efficiency for all) of different liquefaction cycles for achieving minimum liquefaction. Compressor discharge pressure in this simulation, are considered optimal pressure of each cycle.
Figure 13. Minimum effectiveness of the heat exchanger for generating liquid helium in different liquefaction cycles
Increasing the heat exchangers effectiveness must enhance the cold box exergy efficiency and liquefaction level because increasing the heat exchangers effectiveness leads to an increase in the level of heat transfer in it, thereby enhancing the cycle performance in precooling of the hot flow (based on the constant flow rate of the cycle). For the 3 and 4 expander cycles, the effects of increasing the heat exchangers effectiveness on the cold box exergy efficiency based on the compressor discharge pressure are obtained in Figure 14 and Figure 15, respectively, with the following results:
Figure 14. Changes in the cold box exergy efficiency for the heat exchanger effectiveness based on the compressor discharge pressure in the 3 expanders cycle
Figure 15. Changes in the cold box exergy efficiency for the heat exchanger effectiveness based on the compressor discharge pressure in the 4 expanders cycle
Table 1, Table 2 and Table 3, respectively compare the cycle various properties at 96% and 99% heat exchanger effectiveness for the 2, 3 and 4 expander cycles. These tables indicate that:
Table 1. Comparing the cycle various properties at 96% and 99% HX effectiveness for the dualexpander cycle

P_{opt} (bar) 
η_{coldbox} (%) 
y (%) 
y / Power [(kg/s) / (kW)] 
2 EXP (99%) 
16 
24.81 
6.266 
15.02×10^{6} 
2 EXP (96%) 
22 
19.42 
5.475 
11.47×10^{6} 
Table 2. Comparing the cycle various properties at 96% and 99% HX effectiveness for the 3 expanders cycle

P_{opt} (bar) 
η_{coldbox} (%) 
y (%) 
y / Power [(kg/s) / (kW)] 
3 EXP (99%) 
14 
33.81 
8.125 
20.75×10^{6} 
3 EXP (96%) 
17 
29.07 
7.503 
17.6×10^{6} 
Table 3. Comparing the cycle various properties at 96 % and 99% HX effectiveness for the 4 expanders cycle

P_{opt} (bar) 
η_{coldbox} (%) 
y (%) 
y / Power [(kg/s) / (kW)] 
4 EXP (99%) 
13 
39.5 
9.212 
24.39×10^{6} 
4 EXP (96%) 
15 
35.20 
8.682 
21.53×10^{6} 
To further clarify the effect of increasing the heat exchanger effectiveness in helium liquefaction cycle, the chart of $(U A)^{*}$ of heat exchangers, e.g. for the 3 expanders cycle, is shown in Figure 16.
Figure 16. Value of the nondimensional heat transfer coefficient of heat exchangers based on their efficiency in the 3 expanders cycle
3.6 Effect of expanders efficiency
Similar to the effect of increasing the heat exchangers effectiveness on the cold box exergy efficiency, the precooling of highpressure helium flow is improved by enhancing the expanders efficiency, thereby increasing the cold box exergy efficiency and liquefaction level. For the 3 and 4 expander cycles, the effects of increasing the expanders efficiency based on the compressor discharge pressure are calculated in Figure 17 and Figure 18, respectively, with the following results:
Figure 17. Changes in the cold box exergy efficiency in various expander efficiency based on the compressor discharge pressure in the 3 expanders cycle
Figure 18. Changes in the cold box exergy efficiency in various expander efficiency based on the compressor discharge pressure in the 4 expanders cycle
Same to the previous section, the effect of increasing the expanders efficiency in each cycle on the cold box exergy efficiency is studied by keeping the cycle high pressure constant. Results are presented in Figure 19.
Figure 19. Changes in the cold box exergy efficiency for various stages in the liquefaction cycle based on the expander efficiency
The effect of increasing the expander efficiency by 5% is higher on the cold box exergy efficiency and liquefaction level compare with the effect of increasing the heat exchanger efficiency by 2%. For example, for the 4 expanders cycle, increasing the expanders efficiency from 70% to 75% (i.e. 5% increase) results in the 3.5% increase in the cold box exergy efficiency and 0.95% increase in the liquefaction level. Nevertheless, increasing the heat exchanger effectiveness from 97% to 99% (i.e. by 2%) results in 2.44% increase in the cold box exergy efficiency and 0.66% increase in the liquefaction level.
Due to the unique properties of helium, it has been widely used in different applications which makes it a strategic material in the world. Most of the current largescale helium Liquefaction industries are based on the Collins Liquefaction cycle. This cycle includes 3 parts namely compression, cooling, and Liquefaction. Each cooling stage has two heat exchangers and one expander. In this work, the effects of different parameters such as the number of cooling stage, discharge pressure of compressor unit, expander flow rate distribution, etc. on the performance of the Liquefaction cycle were investigated according to the exergy analyses.
The most important results of the present study are:
${EX}^.$ 
Exergy rate, kW 
$Q^.$ 
heat transfer rate, kW 
$W^.$ 
work rate, kW 
$c_p$ 
specific heat, J. kg^{1}. K^{1} 
$m^.$ 
Mass flow rate, kg .s^{1} 
C.V 
control volume 
COMP 
compressor 
eff 
efficiency 
EXP 
Expander 
h 
Specific enthalpy, kJ. $\mathrm{kg}^{1}$ 
HX 
Heat exchanger 
JT 
JouleThomson valve 
s 
Specific entropy, kJ. $\mathrm{K}^{1}$ 
SEP 
separator 
UA 
Overall heat transfer coefficient W/(m^{2}K). 
(UA)* 
Nondimensional heat transfer coefficient. 
A 
Heat transfer area (m^{2}). 
Greek symbols 

Ψ 
exergy of flow per unit of mass, kJ kg^{1} 
Δ 
difference 
$\eta_{e x}$ 
exergy efficiency 
Subscripts 

l 
liquid flow 
lp 
lowpressure 
comp 
compressor 
exp 
expander 
JT 
JouleThomson valve 
flow 
flow 
dest 
destruction 
j 
each component 
0 
reference condition 
s 
suction 
g 
generation 
ex 
exergy 
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