Cryogenics 38 (1998) 1199–1206

 1999 Elsevier Science Ltd. All rights reserved

Printed in Great Britain
PII: S0011-2275(98)00110-6 0011-2275/99/$ - see front matter

Thermodynamic analysis of Collins helium

liquefaction cycle
M.D. Atrey

Cryogenics Section, Centre for Advanced Technology, Indore 452 013, India

Received 13 March 1998

The present paper gives a thermodynamic analysis of the Collins helium liquefaction
cycle with two reciprocating expanders. The results of the analysis make it clear
that, for a given efficiency of expanders and effectiveness of heat exchangers, there
exists an optimum mass flow fraction of total helium gas mass flow rate that should
be diverted through the expanders for which liquid yield is maximum and net power
input is minimum. The analysis quantitatively studies the effect of expander
efficiency and heat exchanger effectiveness on the performance of the liquefier. It
gives final steady state temperature distribution across the cycle, which is essential
data for carrying out the preliminary design of various components in the cycle. 
1999 Elsevier Science Ltd. All rights reserved
Keywords: helium liquefier; thermodynamic analysis

have
The helium liquefier based on the Collins cycle normally
consists of six heat exchangers and two reciprocating
expanders. The design of these would be possible only
when the design data in terms of nodal temperatures
across heat exchangers and expanders, effectiveness of
heat exchangers and efficiencies of expanders, mass
flow rate through compressor, expanders and J-T valve,
etc., are made available. The design is quite critical at
low tempera- tures due to changes in thermophysical
properties of helium gas. Different parameters like heat
exchanger effectiveness (c), expander efficiencies (μ1
and μ2), temperatures of gas before expansion, total mass
flow rate (m˙ ), mass flow frac- tion through expanders
(m˙ e1 + m˙ e2 ) etc., affect the per- formance of the
liquefier. Quite a bit of simulation work has been
presented in the earlier developmental period of these
machines. Hubbell and Toscano1 presented an entropy
generation concept for carrying out thermodynamic
optimisation of the helium liquefaction cycle. Minta and
Smith2 used a similar method of minimisation of the
gener- ated entropy in a cycle model with continuous
precooling. Khalil and McIntosh3 carried out an
exhaustive study to optimise inlet pressure, temperature
of first expander and number of expanders. Also, Hilal 4
analysed the effect of the number of expansion engines
in cascade form or in the independent form and pressure
on the COP of the refriger- ator and liquefier. He
showed that there is a significant increase in coefficient
of performance (COP) value in case of independent
expansion engines over the one obtained in case of
cascaded form. The required optimum pressure is also
lower. In the recent past, this topic of cycle simulation is
again gaining importance due to the increasing need of
the efficient helium liquefiers for cooling of supercon-
ducting magnets. Nobutoki et al.5 and Malaaen et al.6

Cryogenics 1998 Volume 38, Number 12 1

presented simulation programs for the Large Helical
Device (LHD) and the Large Hadron Collider (LHC)
projects, respectively, for helium liquefaction/refrigeration
plants in order to estimate, understand and analyse the
performance of cryogenic processes before investing in the
actual manu- facturing of these plants. However, none of
these analyses have referred to the optimum fraction of
total mass flow rate that has to be diverted through the
expanders, and have also not quantitatively analysed the
effect of expander efficiency and heat exchanger
effectiveness on this fraction and finally on the
performance of the liquefier. This may be due to the fact
that many of these simulation programs are classified in
nature. The cold produced in the expanders is directly
proportional to the mass flow rate diverted through them
and the liquefaction yield is proportional to the remaining
mass flow rate that passes through the J-T valve. If the
total mass flow rate that goes through the first and the
second expander, (m˙ e1 + m˙ e2 ), is less than a mini-
mum required quantity, there would not be any
liquefaction of helium gas. This is due to the fact that the
gas would never attain a low enough temperature for
liquefaction due to insufficient refrigeration effect, and
instead the machine would act as a refrigerator. Also, the
parameters like heat exchanger effectiveness and expander
efficiency affect the liquefaction yield considerably. The
inlet temperature of the gas at the expander depends on the
heat exchanger effec- tiveness at every stage and also on
the mass flow rates through different parts of the cycle.
The present paper aims to carry out an exhaustive
simulation study of the Collins helium liquefier with two
reciprocating engines. The analy- sis can also be extended
or interpreted for cycles with tur- boexpanders.

2 Cryogenics 1998 Volume 38, Number

12
 Thermodynamic analysis of Collins helium liquefaction cycle: M.D. Atrey

Figure 1 Schematic of the Collins helium liquefaction cycle

Thermodynamic analysis HX2, shown in Figure 1, can be integrated together to

Collins cycle reduce number of variables in the analysis. However, this
has been kept separate in the present analysis to study the
The Collins cycle or the modified Claude cycle is the one
option of LN2 precooling for the warm heat exchanger up
which is normally used for helium liquefaction. Figure 1 to a desired temperature level. This calls for special atten-
gives a schematic diagram of the Collins cycle and Figure
tion to attribute effectiveness to each division of the warm
2 gives its process representation on the T-S diagram. Six heat exchanger. It should be noted that if each of the two
heat exchangers, identified as HX1, HX2… HX6, respect-
warm heat exchangers has 96% effectiveness, the
ively, and two reciprocating expanders identified as EX1 integrated heat exchanger then would have a higher
and EX2 are shown in the schematic. m˙ is the total
resultant effec- tiveness than 96%.
mass flow rate of the helium gas through the compressor
while m˙ e1 and m˙ e2 are the mass flow rates diverted
through the expansion engine number 1 and 2, Assumptions
respectively. m˙ f is the liquefaction yield. The present
thermodynamic analysis is based on the steady state Following assumptions are made for carrying out the analy-
conditions at the time of liquefac- tion. c 1 to c6 represent sis.
the effectiveness of the heat exchangers from HX1 to HX6,
respectively, and μ1 and μ2 represent the isentropic 1. The maximum pressure (P h ) in the system is 15 bar
efficiencies of the expanders 1 and 2, respectively, Ph and and the minimum pressure (Pl) is 1 bar.
Pl represent discharge and suction pressure of the 2. The temperature of the gas after compression is 300 K
compressor. The heat exchangers, HX1 and and the return stream temperature of the helium gas
after liquefaction is at its boiling point, i.e. 4.21 K.
3. The pressure drop in the heat exchangers is negligible.
4. The J-T expansion is a perfect isenthalpic process.
5. Heat in-leak in the system is negligible.
6. Effectiveness of heat exchangers and efficiencies of
expanders are assumed to be constant; their depen-
dence on pressure, temperature and mass flow rate is
ignored.

Analysis
The thermophysical properties of the helium gas, at differ-
ent temperatures and pressures, are taken from Van Sciver7.
For any intermediate temperatures, the values for enthalpy,
entropy, etc. are linearly interpolated. Applying the first
law of thermodynamics to the system, excepting the com-
pressor, for the steady state condition, the ratio of liquid
yield to the total mass flow rate, y, is given as follows:

m˙ f h14 — h1 Ahe1
A
y h=e2 = + x1 + x2 (1)
m˙ h14 — hf h14 — h14 — hf
hf

where x1 = m˙ e1/m˙ and x2 = m˙ e2/m˙ and Ahe1 and

Ahe2 are the net enthalpy changes in helium occurring in
expander number 1 and 2, respectively. h represents
enthalpy at the respective points.
Figure 2 T-S diagram of the Collins helium liquefaction cycle A computer program is developed to analyse thermal
performance of the combined unit of six heat exchangers specific heat capacity of gas. Suffix c and h represent
and two expanders along with the J-T expansion valve. A cold and hot fluid respectively, Cmin indicates smaller
detailed flow chart for this analysis is given in Figure 3. quantity of Cc and Ch, suffix o and i represent outlet and
The crucial part of the analysis is that only two tempera- inlet, respectively.
tures are known initially, that is, the temperature of the gas
The efficiency of an expander, μ, is defined as:
after compression, T1, equal to 300 K, and the return
stream temperature of the gas after liquefaction, T8 g, equal
μ = actual enthalpy drop/maximum possible
to 4.21 K. All the intermediate temperatures are unknown
variables excepting the effectiveness of all the heat enthalpy drop = (h1 — h2)/(h1 — h2i) (4)
exchangers and the efficiencies of the expanders. The effec-
tiveness of heat exchangers, c, is defined as: where h1 is the enthalpy at the point from where expansion
takes place, h2 is the enthalpy at the actual point after
c = actual heat transfer/maximum possible heat transfer expansion, h2i is the enthalpy at the point if the expansion
is isentropic in nature.
c = Cc(Tco — Tci)/Cmin(Thi — Tci) (2) Based on the enthalpy balances in the system and
incorporating c and μ definitions at respective nodal points,
= Ch(Thi — Tho)/Cmin(Thi — Tci) (3) the temperatures at different nodes are calculated in an iter-
ative manner. Appendix A gives all the equations for
where, C is capacity rate, product of mass flow rate and
differ- ent important nodes in detail.

Figure 3 Flow chart for liquefaction cycle analysis

Cryogenics 1998 Volume 38, Number 12 1201

 Equation (1) assumes that the liquefaction of helium important parameters in order to get liquefaction and also
takes place in all the cases. Let us call the ‘y’ value
to get maximum yield. The parameters, x1, x2, (x1 + x2),
obtained from this equation y1. However, it is also possible
effectiveness of heat exchangers and efficiency of the
that due to changes in x1 and x2 or c and μ values, there
expanders together determine the liquefier performance. It
is no liquefaction of the gas. As a result of this, the isen-
is obvious that the effectiveness of the heat exchangers and
thalpic line indicating the J-T expansion may not fall in the
the efficiencies of the expanders should be as high as poss-
two-phase region and it may fall outside the dome of the
ible in order to get maximum yield from the liquefier and
two-phase region. This is taken into account by the bisec-
the higher values are fixed up mostly by fabrication or
tion equation in the two-phase dome of T-S diagram to
space limitations. However, x1 and x2, or the sum of x1 and
ensure liquefaction or no liquefaction cases. Let us call the
x2 are very important parameters in all the types of liquefi-
‘y’ value obtained from bisection equation as y2 which is
ers including the ones operated by using turboexpanders.
given below:
The above analysis is extended to understand the effect of
y2 = [(hg — h7)/hfg]*(1 — x1 — x2) (5) x1 and x2 on the output of the liquefier. The parameter OP,
to be optimised for a unit total mass flow rate, is given as:
where hfg is the latent heat of evaporation for He at 1 bar.
However, one has to be very careful to use the bisection OP = m˙ f /(Net Work) = y/(Wc — GxWe ) (7)
method alone to determine the value of y. This is due to
the fact that it may result in an oscillatory or diverging where Wc is the work done on the compressor and We is
solutions of the analysis due to very small values of y and the work done by the expander per unit mass. A routine is
therefore ‘y’ sensitivity of these calculations. As the isen- developed to calculate OP parameter for given c set for all
thalpic line may fall in the gas or two-phase region during the heat exchangers and μ of the expanders. The values of
iterations a very careful approach has to be taken. To over- x1 and x2 are varied during the execution. It is found that
come this problem, a weighted average method is adopted the solution of the program diverges for the cases in which
between the two y values, y1 value calculated by Equation no liquefaction occurs and these are considered as limiting
(1) and y2 value calculated by Equation (5). Optimum cases for the liquefaction.
weightage is worked out by various trials of iterations. The
optimum combination is determined for two reasons, first
to minimise the computer time and second to overcome the Results
oscillating or divergent solutions. The optimum weightage
Optimisation of the mass flow rate through the
for y1 and y2 are found to be 80 and 20%, respectively.
expanders
Considering this, the resultant y value is given as below:
It is obvious that the cold produced in the expanders and
y = (0.8*y1 + 0.2*y2) (6) in the J-T expansion valve is responsible for bringing down
the temperature of the helium gas from 300 K to below 7.5
For any liquefier, the y value calculated as y1 or y2 should K. The refrigeration effect produced in the expanders is
be the same and therefore as criteria for convergence, along proportional to the mass flow rate directed through them
with different temperatures, it is ensured that both y1 and and also to the inlet temperature of the gas of the engine.
y2 are practically the same. The refrigeration effect thus produced determines if the
To summarise the calculation procedure, the following machine would function as a liquefier or as a refrigerator
is the broad outline of different steps of the analysis. The depending upon the temperature levels of the expanders. In
flow chart for the same is given in Figure 3. a similar way, the liquefaction produced in the cycle is
1. Assume x1 and x2 and also the value of y. directly proportional to the mass flow rate directed to
2. Assume all the return line temperatures on 1 bar press- expand through the J-T valve. Considering this, it is really
ure line. a matter of conflict to decide what fraction of total mass
3. Based on the c definitions of the respective heat flow rate should be directed through the engines so that the
exchangers followed by enthalpy balances around the liquefier functions near an optimum value as given above.
heat exchangers, calculate unknown temperatures. As An optimisation routine is attached to the main program to
one advances from HX1 towards HX6, correct the earl- calculate the OP value for a different fraction of mass flow
ier assumed temperatures as given in Appendix A. rates that are directed through the first and the second
4. Calculate the temperature of the gas after expansion. expansion engines denoted by x1 and x2, respectively. The
Use gas enthalpy mixture formulae to find out resultant execution of this routine is quite a computer-intensive task.
temperatures after the mixing of gases from return line Figure 4 gives these results as a plot of OP versus x1. The
after liquefaction and from the expansion engine after curves are plotted for different values of x2.
expansion. Repeat calculations from HX1 to HX6 with The aim of this exercise is to find out a combination of x1
the new temperature values until the same and x2 for which OP is maximum. In the optimisation
temperatures are obtained. routine, x2 is kept constant and x1 is varied so as to deter-
5. Compute ‘y’ by Equation (6) and repeat from (3) until mine the local maximum OP value, termed (OP)max, for
y1 and y2 are found to be the same in the tolerance lim- this combination. It is seen that as x2 decreases from 0.5
its. to 0.45, the (OP)max value increases, indicating that (OP)max
obtained by the first combination is not an optimum one.
The (OP)max, thus obtained for each x2 curve, shows an
Optimisation of the mass fraction for expanders increase up to a certain point only and then starts
descending down. The OP value associated with this point
It has been found that the mass fractions x1 and x2 and also indicates an optimum combination of x1 and x2 for the
(x1 + x2 ), in case m˙ is assumed to be unity, are very present configuration and is termed (OP)opt, which is
Figure 4 Optimisation of helium mass flow rate fractions through expanders

marked in the figure. It could also be noted that no

increases the optimum point remains almost at the same
solutions were obtained for the (x1 + x2) combination less level, however, the minimum (x1 + x2) requirement shifts
than a particular value. This is attributed to the fact that the towards left or on the lesser side. The analysis highlights
tem- perature after the J-T expansion oscillates through the fact that for the case of all 98% efficient heat
liquefier to refrigerator region (inside or outside the two-
exchangers, the minimum requirement of (x1 + x2) is
phase dome) adding an imbalance in the program due to
around 75–76% as against 79% for the case of 95%
changes in the thermophysical properties of the liquid and
efficient heat exchangers. This explains why in the LN2
gas. To make sure that the machine functions as a liquefier precooled liquefiers, which is synonymous with more
it is safer to conclude that there is an unique value of (x1 + efficient heat exchangers case, the minimum (x1 + x2)
x2)min depending on the operating pressure, c and μ combi- values lie around 75% in practice.
nation, below which the machine will not function as a
Some industrial or actual machine data are available to
liquefier but as a refrigerator only. The designer should
therefore know the relationships of all these parameters substantiate the optimum mass flow rate arguments. How-
before he goes ahead with the design of the heat ever, due to the classified nature of the data, these can not
exchangers and expanders. be revealed.
It is noticed from the above figure that the combination
of x1 = 0.45 and x2 = 0.35 shows the maximum value of Effect of c on temperature distribution and
(OP)max as compared to any other combinations of x1 and performance of the cycle
x2 and this is the (OP)opt value for the given c and μ values It is clear from the T-S diagram that the most important
indicated in the figure. It states that, for this combination temperature which determines the amount of helium
of x1 and x2, the output in terms of liquefaction quantity is liquefaction is the one before J-T expansion, i.e., T7, and
maximum and the net power input is minimum. The also the mass flow rate through the J-T valve. The purpose
important point to be noted here is that for all the cases, of heat exchangers and expanders is mainly to reduce the
(OP)max value including (OP)opt lies at a combination where gas temperature from 300 K to a reasonable value of T7 in
x1 and x2 together constitute about 80–81% of the total order to get liquefaction after the J-T expansion. T7 should
mass flow rate while the remaining 19–20% of the total necessarily be around 7.5 K maximum for 15 bar pressure
mass flow rate goes through the J-T valve. It is also seen to get some liquid yield. If this temperature is lesser than
that as the (x1 + x2) value is below 79–79.5% there is no 7.5 K one can expect a higher quantity of liquefaction,
liquefaction indicated by the divergence of the program in how- ever this argument should be critically evaluated
the present case. This is due to the fact that in these cases, looking at the actual T-S diagram. It is quite difficult to
the point of the isenthalpic line after J-T expansion trans- bring down this temperature below 6.5 K without
lates into the gaseous region, i.e. outside the dome. As the increasing the com- plexity of the cycle, and this can be
values of (x1 + x2) exceed an optimum value there is a realised if one has an idea about the feasibility of design of
decrease in the OP value essentially due to the fact that the heat exchangers and the expanders.
effectively less mass flows through the J-T valve and this It is obvious that as the c of the heat exchanger increases,
decreases the values of y in these cases. The results of the the performance of the liquefier is better due to the
present analysis are valid for the liquefiers without LN2 decrease in the final value of T7 for a given mass flow rate
precooling. The c assumed for these cases is 95% for all through the compressor. However, this does not mean that
the heat exchangers and the μ assumed for both the the tem- peratures at all the points decrease by the same
expanders is 75%. amount. The fall in T7 could be achieved by various means,
To study the effect of increased c of the heat exchangers
i.e. merely by increasing the c of any of the heat
on this combination of x1 and x2, the routine was executed
exchangers or any two or all the heat exchangers, and
again. It shows that even if the c of the heat exchangers
could also be due to an increase in the μ of any or all
the expanders.
Table 1 Temperature distribution for different c of heat exchangers

Sr. c T2 T3 T4 T5 T6 T7 T9 T10 T11 T12 T13 T14 y

no. (%) (K) (K) (K) (K) (K) (K) (K) (K) (K) (K) (K) (K) (%)

1 95 232.47 90.18 48.43 20.77 10.46 6.35 9.98 14.54 46.27 73.17 224.5 296.22 5.82
2 96 239.92 90.57 48.11 20.59 10.26 6.27 9.88 14.5 46.37 73.85 233.29 297.33 6.18
3 97 249.27 91.23 47.93 20.44 10.08 6.2 9.79 14.47 46.61 74.72 244.03 298.32 6.52
4 c3 = 97
c1–6 = 95 233.52 91.92 47.49 20.61 10.37 6.33 9.9 14.44 46.14 74.8 225.59 296.27 5.93
5 c1 = 97
c2–6 = 95 222.18 87.45 46.92 20.51 10.32 6.31 9.85 14.38 44.81 71.01 214.62 297.43 6.01
6 c5 = 97
c1–6 = 95 233.11 90.99 18.88 21.12 10.39 6.33 10.10 14.85 46.7 73.85 225.15 296.25 5.91

μ1 = μ2 = 75%.
x1 = x2 = 0.4.

Table 1 gives the values of the temperatures at various case 1. Case 6 shows the effect for increased c of HX5,
locations in the cycle, to understand changes in the tem-
wherein only T6 and T7 show a decrease in the tempera-
perature distribution across the cycle obtained by the ture. Thus, the table shows that any increase in c of any
present cycle analysis, when the c of all the heat
heat exchanger results ultimately in the decrease in the
exchangers or any of the heat exchangers is increased. The value of T7, which finally affects the liquefaction process
expander efficiency is assumed to be 75% for these cases
directly. Figure 5 shows the effect of the c of the heat
and also that 40% of the total mass flow rate is diverted exchangers on the performance of the liquefier
through each of the expanders. Cases 1, 2 and 3 in the table
graphically. It shows the effect of variation of the c of a
give the temperature distribution in the liquefier in which particular heat exchanger on the performance of the
the c of all the heat exchangers is changed simultaneously
liquefier. The figure shows the relative importance of the c
by the same amount. It is seen from the table that as c of all of each heat exchanger. It can be seen that the c of heat
the heat exchangers increases from 95 to 97%, the
exchangers 3, 4 and 5 should necessarily be higher in
liquefaction, y, increases by 12%, which is quite order to ensure liquefaction, while for other heat
substantial. Also, the temperatures T2 and T3 show an
exchangers, c can have little less values as shown in the
increase with the increase in the effectiveness. However, curves. The figure also shows that there is a significant
after the first expansion point the temperature drops down
change in the performance of the liquefier if the c of all
from T4 up to T10 while T11 to T14 shows an increase the heat exchangers are increased simultaneously as
again. Cases 4, 5 and 6 show the implications of increased
compared to an increase in individual c of any of the
c of an individual heat exchanger, keeping c of the rest of heat exchangers. The curves are significant data to under-
the heat exchangers as 95%. The results of these cases
stand the implications of changes in c of any heat
could be compared with case 1 where all the heat
exchangers.
exchangers have a c of 95%. Cases 4, 5 and 6 show the
effect of increased c of 97% for heat exchangers numbers
3, 1 and 5, respectively. It is seen in case 4 that the increase Effect of μ on the performance of the liquefier
in c decreases the temperatures after HX3 onwards, from
Figure 6 shows the effect of μ on the performance of the
T4 to T11. Similarly, case 5 is for the increased c of the
liquefier. It is quite clear from the curves that as the μ
first heat exchanger, in which tempera- tures T2 to T13
increases the performance of the liquefier increases
show a decreasing trend as compared to
linearly. Also, it shows that if the μ2 is 75%, the minimum
μ1 should

Figure 5 Effect of heat exchanger effectiveness (c) on the performance of the liquefier
Figure 6 Effect of expander efficiency (μ) on the performance of the liquifier

be 70% in order to have liquefaction. Similarly, if μ 1 is

75% the minimum μ2 should be 74% in order to have Appendix A
liquefaction. This highlights the importance of minimum μ
of the expanders and also their interdependence. The thermal analysis of the liquefier involves solution of
following equations in an iterative manner. Figures 1 and
2 should be referred to to understand the nodal nomencla-
ture used in these equations. Excepting temperatures at
Conclusions points 2 and 8g from these figures, no other temperatures
are known. Temperatures T13d, T12d, T11d, T10d and T9d
The paper presents a cycle simulation for the Collins indicate assumed values for T13, T12, T11, T10 and T9,
helium liquefaction cycle with six heat exchangers and two respectively. The temperature values which could be
reciprocating expanders. It highlights the concept of an obtained from these equations are indicated at the right side
optimum mass flow rate through expanders for the of the arrow. The suffix i in the following equations indi-
liquefier. At the same time, the paper analytically puts cates the temperature of the gas under ideal conditions of
forward the importance of heat exchanger effectiveness (c) heat exchange.
and expander efficiency (μ) on the performance of the
liquefier. The optimum mass flow rate concept holds good
for the liquefiers also with the turboexpanders. The Heat exchangers 1, 2 and 4
simulation can be adapted to bring about any changes in the
configuration of the liquefaction cycle and to make a Heat exchangers 1, 2 and 4 are the cases where the capacity
quantitative com- parison of different cycles based on their rate on the warm side is higher than that on the cold side.
merits and demerits. The analysis is very important to get a Therefore, the equations for computing heat balance are
prelimi- nary design data for the heat exchangers and the similar in nature. In the case of heat exchanger 1, T14i
expanders for a required helium liquefaction rate. could be equal to T1. So the enthalpy of gas at pressure Pl
and temperature T14i can be given as:
References h14i = h(T1,Pl) (A.1)
1. Hubbell, R.H. and Toscano, W.M., Thermodynamic optimisation of
helium liquefaction cycles. Adv. Cryo. Engng., 1980, 25, 551.
2. Minta, M. and Smith, J.L., An entropy flow optimisation technique
Applying heat exchanger effectiveness definition:
for helium liquefaction cycles. Adv. Cryo. Engng., 1984, 29, 469.
3. Khalil, A. and McIntosh, G.E., Thermodynamic optimisation study of h14 = c1(h14i — h13d) + h13d = > T14 (A.2)
the helium multiengine Claude refrigeration cycle. Adv. Cryo.
Engng., 1978, 23, 431.
4. Hilal, M.A., Optimisation of helium refrigerators and liquefiers for Applying enthalpy balance:
large superconducting systems. Cryogenics, 1979, 19, 415.
5. Nobutoki, M., Iwamoto, K. and Matsuda, H., Simulation of the large h2 = h1 — (1 — y)(h14 — h13d) = > T2 (A.3)
helium refrigeration plant for LHD. Proceedings of the 16th ICEC
Cryogenics (Suppl), Vol. 36. 1996, 71.
6. Malaaen, E., Owren, G., Wadahl A. and Wagner, U., Simulation pro- The equations for heat exchangers 2 and 4 should be
gram for cryogenic plants at CERN. Proceedings of the 16th ICEC similar to the ones given above. However, it is always a
Cryogenics (Suppl), Vol. 36. 1996, 99. good practice to verify the capacity rates of each stream in
7. Van Sciver, S.W., Helium cryogenics. Plenum Press, New York,
USA, 1986.
each case due to the fact that Cp of helium goes on increas-
ing at lower temperatures.
Heat exchangers 3 and 5 gas at low pressure after the heat exchanger 4 before
mixing with expansion stream at temperature Te1. From
For heat exchangers 3 and 5, the capacity rate on the cold the defi- nition of μ1 the following equation could be
side will be more than the warm side. This changes the obtained:
enthalpy balance relationship as compared to heat
exchangers 1, 2 and 4. For heat exchanger 3, T4 i could be he1 = h3 — μ1(h3 — he1d) = > Te1 (A.7)
equal to T11. So, the enthalpy of gas at pressure Ph and
temperature T11d can be given as: Mixer equations for three gas streams:
h4i = h(T11d,Ph) (A.4) h11id = [(1 — y)(h11d) — (x1*he1)]/(1 — x1 — y) (A.8)

h4 = h3 — c3(h3 — h4i) = > T4 (A.5) = > T11id

h12 = h11d + (1 — x1)(h3 — h4)/(1 — y1) (A.6) Mixer equation for three gas streams at 11:

= > T12 h11 = [(1 — x1 — y)(h11i) + (x1*he1)]/(1 — y) (A.9)

= > T11
Mixer 1 and 2 In a similar way, the equations for mixer 2 are estab-
lished.
After expansion, the expanded gas mixes with the return
stream coming back after the liquefaction. The resultant
temperature of the stream after mixing depends on
tempera- ture and respective mass flow rates of the two
Heat exchanger 6
streams. In the case of expander 1, he1d is the enthalpy of The analysis of this heat exchanger has to be correctly car-
the gas at a point just after isentropic expansion from ried out as the inlet temperature on the cold side is very
temperature T3 and he1 is the enthalpy of the gas after near to boiling point of He and Cp of the gas at this tem-
expansion taking into consideration the isentropic perature is quite high. So, one has to verify in what cate-
efficiency, μ1, defined in Equation (4). T11id is the gory this heat exchanger falls, and accordingly it has to be
temperature of the return stream evaluated as given above.