Dinc Ibrahim. Refrigeration systems and applications 2th edition

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Advanced Refrigeration Cycles and Systems 231

Table 5. 1 Various properties and performance parameters of the cycle in Figure 5.9 for T

= 25

◦

= 1 atm (0.101 MPa), and P

= 20 MPa. The ﬂuid is air.

= 298.4kJ/kg

= 263.5kJ/kg

= 61.9kJ/kg

= 78.8kJ/kg

=−126.1kJ/kg

= 6.86 kJ/kg · K

= 5.23 kJ/kg · K

= 2.98 kJ/kg · K

=−194.2

◦

= 0.9177

y = 0.0823

= 34.9 kJ/kg gas

= 424 kJ/kg liquid

actual

= 451 kJ/kg gas

actual

= 5481 kJ/kg liquid

rev

= 733 kJ/kg liquid

COP

actual

= 0.0775

COP

rev

= 0.578

= 0.134

Table 5. 2 Performance parameters of a simple Linde–Hampson cycle for various ﬂuids.

Item Air Nitrogen Oxygen Argon Methane Fluorine

Liquefaction temperature T

(

◦

C) −194.2 −195.8 −183.0 −185.8 −161.5 −188.1

Fraction liqueﬁed y 0.0823 0.0756 0.107 0.122 0.199 0.0765

Refrigeration effect q

(kJ/kg gas) 34.9 32.6 43.3 33.2 181 26.3

Refrigeration effect q

(kJ/kg liquid) 424 431 405 272 910 344

Work input w

(kJ/kg gas) 451 468 402 322 773 341

Work input w

(kJ/kg liquid) 5481 6193 3755 2650 3889 4459

Minimum work input w

rev

(kJ/kg liquid) 733 762 629 472 1080 565

COP

actual

0.0775 0.0697 0.108 0.103 0.234 0.0771

COP

rev

0.578 0.566 0.644 0.576 0.843 0.609

Exergy efﬁciency η

(%) 13.4 12.3 16.8 17.8 27.8 12.7

Here, T represents the temperature of the gas being liqueﬁed in Figure 5.10, which changes

between T

and T

during the liquefaction process. An average value of T may be obtained using

Equation 5.16 with COP

rev

= 0.578 and T

= 25

◦

C, yielding T =−156

◦

C. This is the temperature

a heat reservoir would have if a Carnot refrigerator with a COP of 0.578 operated between this

reservoir at −156

◦

C and another reservoir at 25

◦

C. Note that the same reservoir temperature T

could be obtained by writing Equation 5.15 in the form

rev

=−q



1 −



(5.17)

where q

= 424 kJ/kg, w

rev

= 733 kJ/kg, and T

= 25

◦

As part of the analysis, the effects of liquefaction temperature and gas inlet temperature on

various energy- and exergy-based performance parameters are investigated considering oxygen as

the gas being liqueﬁed. The results of these studies are given in Figures 5.11 through 5.18. The

results involving various gases are shown in Figures 5.14 and 5.18.

The data obtained for various ﬂuids in Table 5.2 show that different gases exhibit different

behaviors in terms of performance parameters. The differences are due to the thermophysical

properties of ﬂuids and the liquefaction temperatures. Figures 5.11 through 5.18 show that as

232 Refrigeration Systems and Applications

−180 −170 −160 −150 −140

0.106

0.108

0.11

0.112

0.114

0.116

0.118

0.12

0.122

1000

1500

2000

2500

3000

3500

4000

liq

(°C)

actual

(kJ/kg)

Figure 5.11 The liqueﬁed mass fraction y and the actual work input versus liquefaction temperature for

oxygen.

−180 −170 −160 −150 −140

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.6

0.7

0.8

0.9

1.1

1.2

COP

actual

COP

rev

liq

(°C)

Figure 5.12 The actual and reversible COPs versus liquefaction temperature for oxygen.

Advanced Refrigeration Cycles and Systems 233

−180 −170 −160 −150 −140

0.16

0.165

0.17

0.175

0.18

0.185

0.19

250

300

350

400

450

500

550

600

650

liq

(°C)

rev

(kJ/kg)

Figure 5.13 The exergy efﬁciency and reversible work versus liquefaction temperature for oxygen.

−200 −190 −180 −170 −160 −150 −140

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

liq

(°C)

Oxygen

Nitrogen

Fluorine

Argon

Air

Figure 5.14 The exergy efﬁciency versus liquefaction temperature for various gases.

234 Refrigeration Systems and Applications

0 5 10 15 20 25

0.105

0.11

0.115

0.12

0.125

0.13

0.135

0.14

2900

3000

3100

3200

3300

3400

3500

3600

3700

3800

actual

(kJ/kg)

gas

(°C)

Figure 5.15 The liqueﬁed mass fraction y and the actual work input versus gas inlet temperature for oxygen.

0 5 10 15 20 25

0.105

0.11

0.115

0.12

0.125

0.13

0.135

0.605

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0.645

COP

rev

gas

(°C)

Figure 5.16 The actual and reversible COPs versus gas inlet temperature for oxygen.

Advanced Refrigeration Cycles and Systems 235

0 5 10 15 20 25

0.16

0.17

0.18

0.19

0.2

0.21

0.22

628

628.2

628.4

628.6

628.8

629

629.2

rev

(kJ/kg)

gas

(°C)

Figure 5.17 The exergy efﬁciency and reversible work versus gas inlet temperature for oxygen.

0 5 10 15 20 25

0.12

0.14

0.16

0.18

0.2

0.22

gas

(°C)

Fluorine

Nitrogen

Air

Oxygen

Argon

Figure 5.18 The exergy efﬁciency versus gas inlet temperature for various gases.

236 Refrigeration Systems and Applications

the liquefaction temperature increases and the inlet gas temperature decreases the liqueﬁed mass

fraction, the actual COP, and the exergy efﬁciency increase, while actual and reversible work

consumptions decrease.

It is interesting to observe from Figure 5.16 that the reversible COP increases as the gas inlet

temperatures increase. This unexpected trend is due to the fact that the refrigeration effect increases

at a greater rate than the reversible work input when the inlet gas temperature increases. On the

other hand, the reversible COP increases as the liquefaction temperature increases as shown in

Figure 5.12 because of the fact that the reversible work input decreases at a greater rate than the

refrigeration effect.

The exergy efﬁciency increases with increasing liquefaction temperature and decreasing inlet

gas temperature for all gases considered as shown in Figures 5.14 and 5.18. In Figure 5.14, the

exergy efﬁciency reaches a maximum before decreasing at higher temperatures. The decreasing

trend at higher liquefaction temperatures is of no practical importance since liquefaction at these

high temperatures requires higher inlet pressures, which are not normally used.

Obtaining liqueﬁed oxygen at −183

◦

C requires exactly 2.1 times the minimum work required to

obtain oxygen at −145

◦

C (Figure 5.13). This ratio becomes 2.4 when actual work consumptions

at these temperatures are considered (Figure 5.11). Similarly, the reversible COP decreases almost

by half when the liquefaction temperature decreases from −140 to −190

◦

C (Figure 5.12). These

ﬁgures show that the maximum possible liquefaction temperature should be used to minimize the

work input. In another words, the gas should not be liqueﬁed to lower temperatures than needed.

As the inlet gas temperature decreases from 25 to 0

◦

C, the actual speciﬁc work input decreases

from 3755 to 2926 kJ/kg (Figure 5.15). The reversible work is not notably affected by the inlet gas

temperature (Figure 5.17).

Among the results provided in Figures 5.11 through 5.18, the exergy efﬁciency values and trends

appear to provide the most valuable information by clearly showing that the system performance

increases with increasing liquefaction and decreasing inlet gas temperatures and that there is a

signiﬁcant potential for improving performance. Among the gases considered, argon performs best

while nitrogen performs worst (Figures 5.14 and 5.18). Noting that the cycle considered in this

example involves a reversible isothermal compressor and a 100% effective heat exchanger, the

exergy efﬁciency ﬁgures here are better than what they would be for an actual Linde–Hampson

cycle. In practice, an isothermal compression process may be approached by using a multistage

compressor. For higher effectiveness, a larger and thus more expensive heat exchanger would

be needed. The work consumption may be decreased by replacing the expansion valve with a

turbine. Expansion in a turbine usually results in a lower outlet temperature relative to that for an

expansion valve while producing work, thus decreasing the total work consumption in the cycle.

The complexity and added cost associated with using a turbine as an expansion device is only

justiﬁed in large liquefaction systems (Kanoglu, 2001). In some systems both a turbine and an

expansion valve are used to avoid problems associated with liquid formation in the turbine.

The system considered in this study involves an ideal isothermal compressor and a perfect heat

exchanger with an effectiveness of 100%. When a more realistic cycle for air liquefaction with an

isothermal efﬁciency of 70% and a heat exchanger effectiveness of 96.5% is analyzed, the liqueﬁed

mass fraction decreases by about 22% and the work consumption increases by 1.8 times compared

to ideal cycle. The actual exergy efﬁciencies of Linde–Hampson liquefaction cycle are usually

under 10% (Barron, 1985).

The difference between the actual and reversible work consumptions in liquefaction systems

are because of the exergy losses that occur during various processes in the cycle. Irreversible

compression in the compressor, heat transfer across a ﬁnite temperature difference in heat exchang-

ers (e.g., regenerator, evaporator, compressor), and friction are major sources of exergy losses

in these systems. In actual refrigeration systems, these irreversibilities are normally reduced by

applying modiﬁcations to the simple Linde–Hampson cycle, such as utilizing multistage compres-

sion and using a turbine in place of an expansion valve or in conjuction with an expansion valve

Advanced Refrigeration Cycles and Systems 237

(Claude cycle). Other modiﬁed cycles that have resulted in greater efﬁciency are known as the dual-

pressure Claude cycle and the Collins helium cycle. For natural gas liquefaction, mixed-refrigerant,

cascade, and gas-expansion cycles are used (Kanoglu, 2002). In most large natural gas liquefaction

plants, the mixed-refrigerant cycle is used in which the natural gas stream is cooled by the suc-

cessive vaporization of propane, ethylene, and methane. Each refrigerant may be vaporized at two

or three pressure levels to minimize the irreversibilities and thus increase the exergy efﬁciency of

the system. This requires a more complex and costly system but the advantages usually more than

offset the extra cost in large liquefaction plants.

5.4.2 Precooled Linde–Hampson Liquefaction Cycle

The precooled Linde–Hampson cycle is a well-known and relatively simple system used for the

liquefaction of gases including hydrogen (Figure 5.19). Makeup gas is mixed with the uncondensed

portion of the gas from the previous cycle and the mixture at state 1 is compressed to state 2.

Heat is rejected from the compressed gas to a coolant. The high-pressure gas is cooled to state 3

in a regenerative counter-ﬂow heat exchanger (I) by the uncondensed gas and is cooled further by

ﬂowing through two nitrogen baths (II and IV) and two regenerative heat exchangers (III and V)

before being throttled to state 8, where it is a saturated liquid–vapor mixture. The liquid is collected

as the desired product and the vapor is routed through the bottom half of the cycle. Finally, the

gas is mixed with fresh makeup gas and the cycle is repeated.

Using an energy balance of heat exchanger V and the throttling valve taken together, the fraction

of the liqueﬁed gas can be determined to be

liq

− h

(5.18)

m−m

III

Liquid

Makeup

gas

Figure 5.19 Precooled Linde–Hampson liquefaction cycle.

238 Refrigeration Systems and Applications

Energy balances for the heat exchangers can be written as

− h

= (1 − f

liq

)(h

− h

) (5.19)

− h

= (1 − f

liq

)(h

− h

) (5.20)

− h

= (1 − f

liq

)(h

− h

) (5.21)

Since the gas behaves ideally during compression, the speciﬁc compression work may be deter-

mined from

ln(P

)

comp

(per unit mass of gas in the cycle) (5.22)

where η

comp

is the isothermal efﬁciency of the compressor, R is the gas constant, and P is the

pressure. The numerator of the right side represents the work input for a corresponding isothermal

process. The speciﬁc work input to the liquefaction cycle per unit mass of liquefaction is

in,liq

liq

(per unit mass of liquefaction) (5.23)

Example 5.3

Hydrogen gas at 25

◦

C and 1 atm (101.325 kPa) is to be liqueﬁed in a precooled Linde–Hampson

cycle. Hydrogen gas is compressed to a pressure of 10 MPa in the compressor which has an

isothermal efﬁciency of 65%. The effectiveness of heat exchangers is 90%. Determine (a) the

heat removed from hydrogen and the minimum work input, (b) the fraction of the gas liqueﬁed,

efﬁciency of the cycle if the work required for nitrogen liquefaction is 15,200 kJ/kg of hydrogen

gas in the cycle. Properties of hydrogen in the cycle at various states are as follows:

= 271.1kJ/kg

= 4200 kJ/kg

= 965.4kJ/kg

= 1147.7kJ/kg

= 17.09 kJ/kg · K

= 70.42 kJ/kg · K

Solution

(a) The heat rejection from hydrogen gas is

= h

− h

= (4200 − 271.1) kJ/kg = 3929 kJ/kg

Taking the dead state temperature to be T

= T

= 25

◦

C = 298.15 K, the minimum work

input is determined from

min

= h

− h

− T

− s

)

= (4200 − 271.1) kJ/kg − (298.15 K)(70.42 − 17.09) kJ/kg · K

= 11,963 kJ/kg

Advanced Refrigeration Cycles and Systems 239

(b) The fraction of the gas liqueﬁed is

liq

− h

1147.7 − 965.4

1147.7 − 271.1

= 0.208

ln(P

)

comp

(4.124)(298.15) ln(10,000/101.325)

0.65

= 8682 kJ/kg

Per unit mass of liquefaction,

in,liq

liq

8682

0.208

= 41,740 kJ/kg

(d) The total work input for the cycle per unit mass of liqueﬁed hydrogen is

in,total

+ w

in,nitrogen

liq

8682 + 15,200

0.208

= 114,800 kJ/kg

(e) The second-law efﬁciency is determined from

min

in,total

11,963

114,800

= 0.104 = 10.4%

5.4.3 Claude Cycle

Claude cycle may be used to liquefy various gases including hydrogen (Figure 5.20). In this cycle,

an expander (turbine) makes work production during the expansion process possible. Feed gas

is compressed to approximately 40 bar pressure. About 75% of the gas after the primary heat

exchanger is expanded in a turbine before mixing with the cold returning gas. An expansion valve

is used to obtain liquid. An energy balance on the entire cycle with no heat leak into the cycle

gives

( ˙m −˙m

+˙m

−˙mh

−˙m

= 0 (5.24)

The fraction of mass ﬂowing through the expander is deﬁned as

x =˙m

/ ˙m

(5.25)

Then the fraction of mass liqueﬁed becomes

y =

˙m

− h

+ x

− h

(5.26)

The last term is greater than zero and thus Claude cycle is a deﬁnite improvement over the

Linde–Hampson cycle. The work produced in the expander is given by

=˙m

− h

) (5.27)

The total work input is the difference between the work consumed in the compressor and the

work produced in the expander:

w = w

comp

− w

[

− s

) − (h

− h

)

]

− x(h

− h

) (5.28)

240 Refrigeration Systems and Applications

Liquid

12 3

987

Expander

Evaporator

Temperature T

Entropy s

h = const

s = const

p = const

T = const

(b)

(a)

Figure 5.20 A Claude low-pressure process cycle using an (a) expansion machine and (b) its T − s diagram

(Adapted from Barron, 1985).