Bureau oi Stana

354

The Single-Engine Claude Cycle

As A 4.2°K Refrigerator

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U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

SI

is

mi

;

SI
THE NATIONAL BUREAU OF STANDARDS

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UNITED STATES DEPARTMENT OF COMMERCE

Alexander B. Trowbridge, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

NBS TECHNICAL NOTE 354

ISSUED JUNE 1, 1967

THE SINGLE-ENGINE CLAUDE CYCLE AS A 4.2°K REFRIGERATOR

R. C. MUHLENHAUPT AND T. R. STROBRIDGE

Cryogenics Division

Institute for Materials Research

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402

Price 45 cents.

TABLE OF CONTENTS

Page

ABSTRACT 1

1. Introduction 1

2. Nomenclature List 2

3. Discussion of Process 3

4. Calculations „ „ 7

5. Results 13

6. Conclusions 19

7. References 21

8. Appendix 22

List of Figures

Figure 1. Schematic Flow Diagram of the Single-Engine Claude

Refrigerator 4

Figure 2. Temperature -Entropy Diagram of the Single-Engine

Claude Refrigerator 5

Figure 3. Cooling Curve for Heat Exchanger with SecondaryThrottle Valve 12

Figure 4a. Claude Refrigerator Performance -Compressor Power - 290°K Cycle Outlet Temperature - 100% Expander

Efficiency . 26

Figure 4b. Claude Refrigerator Performance - Expander Power - 290°K Cycle Outlet Temperature - 100% Expander

Efficiency 27

Figure 5a. Claude Refrigerator Performance - Compressor Power - 290°K Cycle Outlet Temperature - 80% Expander

Efficiency 28

in
TABLE OF CONTENTS (CONTINUED)

Page

Figure 5b. Claude Refrigerator Performance - Expander Power - 290°K Cycle Outlet Temperature - 80% Expander

Efficiency 29

Figure 6a. Claude Refrigerator Performance - Compressor Power - 290 °K Cycle Outlet Temperature - 70% Expander

Efficiency . 30

Figure 6b. Claude Refrigerator Performance - Expander Power - 290°K Cycle Outlet Temperature - 70% Expander

Efficiency 31

Figure 7a. Claude Refrigerator Performance - Compressor Power - 290°K Cycle Outlet Temperature - 50% Expander

Efficiency 32

Figure 7b. Claude Refrigerator Performance - Expander Power - 290°K Cycle Outlet Temperature - 50% Expander

Efficiency 33

Figure 8a. Claude Refrigerator Performance - Compressor Power - 295°K Cycle Outlet Temperature - 100% Expander

Efficiency 34

Figure 8b. Claude Refrigerator Performance - Expander Power - 295°K Cycle Outlet Temperature - 100% Expander

Efficiency 35

Figure 9a. Claude Refrigerator Performance - Compressor Power - 295°K Cycle Outlet Temperature - 80% Expander

Efficiency . . . . 36

Figure 9b. Claude Refrigerator Performance - Expander Power - 29 5° K Cycle Outlet Temperature - 80% Expander

Efficiency 37

Figure 10a. Claude Refrigerator Performance - Compressor Power - 295°K Cycle Outlet Temperature - 70% Expander

Efficiency 38

Figure 10b. Claude Refrigerator Performance - Expander Power - 295°K Cycle Outlet Temperature - 70% Expander

Efficiency 39

Figure 11a. Claude Refrigerator Performance - Compressor Power _ 295°K Cycle Outlet Temperature - 50% Expander

Efficiency 40

IV
TABLE OF CONTENTS (CONTINUED)

Page

Figure lib. Claude Refrigerator Performance - Expander Power - 295°K Cycle Outlet Temperature - 50% Expander

Efficiency 41

Figure 12a. Claude Refrigerator Performance - Compressor Power - 300°K Cycle Outlet Temperature - 100% Expander

Efficiency . 42

Figure 12b. Claude Refrigerator Performance - Expander Power - 300°K Cycle Outlet Temperature - 100% Expander

Efficiency 43

Figure 13a. Claude Refrigerator Performance - Compressor Power - 300°K Cycle Outlet Temperature - 80% Expander

Efficiency „ 44

Figure 13b. Claude Refrigerator Performance - Expander Power - 300°K Cycle Outlet Temperature - 80% Expander

Efficiency 45

Figure 14a. Claude Refrigerator Performance - Compressor Power - 300°K Cycle Outlet Temperature - 70% Expander

Efficiency 46

Figure 14b. Claude Refrigerator Performance - Expander Power - 300°K Cycle Outlet Temperature - 7 0% Expander

Efficiency „ 47

Figure 15a. Claude Refrigerator Performance - Compressor Power - 300°K Cycle Outlet Temperature - 50% Expander

Efficiency 48

Figure 15b. Claude Refrigerator Performance - Expander Power - 300°K Cycle Outlet Temperature - 50% Expander

Efficiency 49

Figure 16a. Claude Refrigerator Performance - Compressor Power - 7 5. 36°K Cycle Outlet Temperature - 80% Expander

Efficiency 50

Figure 16b. Claude Refrigerator Performance - Expander Power - 7 5. 36 °K Cycle Outlet Temperature - 80% Expander

Efficiency 51

Figure 17a. Claude Refrigerator Performance - Compressor Power - 75. 36°K Cycle Outlet Temperature - 70% Expander

Efficiency 52
TABLE OF CONTENTS (CONTINUED)

Page

Figure 17b. Claude Refrigerator Performance - Expander Power - 75. 36°K Cycle Outlet Temperature - 70% Expander

Efficiency 53

Figure 18a. Claude Refrigerator Performance - Compressor Power - 76. 36°K Cycle Outlet Temperature - 80% Expander

Efficiency 54

Figure 18b. Claude Refrigerator Performance - Expander Power - 76. 36°K Cycle Outlet Temperature - 80% Expander

Efficiency 55

Figure 19a. Claude Refrigerator Performance - Compressor Power - 76. 36°K Cycle Outlet Temperature - 70% Expander

Efficiency 56

Figure 19b. Claude Refrigerator Performance - Expander Power - 76. 36°K Cycle Outlet Temperature - 70% Expander

Efficiency 57

Figure 20a. Claude Refrigerator Performance - Compressor Power - 77.36°K Cycle Outlet Temperature - 80% Expander

Efficiency 58

Figure 20b. Claude Refrigerator Performance - Expander Power - 77.36°K Cycle Outlet Temperature - 80% Expander

Efficiency 59

Figure 21a. Claude Refrigerator Performance - Compressor Power - 77.36°K Cycle Outlet Temperature - 7 0% Expander

Efficiency 60

Figure 21b. Claude Refrigerator Performance - Expander Power - 77. 36°K Cycle Outlet Temperature - 70% Expander

Efficiency 61

Figure 22. Expansion Engine Performance - 100% Efficiency . . 62

Figure 23. Expansion Engine Performance - 100% Efficiency . .- 63

Figure 24. Expansion Engine Performance - 80% Efficiency ... 64

Figure 25. Expansion Engine Performance - 80% Efficiency ... 65

Figure 26. Expansion Engine Performance - 7 0% Efficiency ... 66

Figure 27. Expansion Engine Performance - 70% Efficiency ... 67

VI
TABLE OF CONTENTS (CONTINUED)

Page

Figure 28. Expansion Engine Performance - 50% Efficiency ... fag

Figure 29. Expansion Engine Performance - 50% Efficiency ... 59

Figure 30. Helium Flow Rate ^ *. 70

Figure 31. Helium Flow Rate „ . j\

vi1

THE SINGLE-ENGINE CLAUDE CYCLE AS A 4. 2°K REFRIGERATOR

R. C. Muhlenhaupt and T. R. Strobridge

The performance of the 4. 2°K Claude-cycle refrigerator has been

computed taking into account the efficiencies of the various components.

The results are presented in graphical form. These charts give the

input power requirements, mass flow rates for both the compressor and

expander, pertinent temperatures, and allow selection of the optimum

high pressure for a given set of component characteristics.

Key Words: Claude, cryogenics, refrigeration

1. Introduction

The Claude cycle, as it has become known after the Frenchman

who constructed one of the first really successful low temperature expansion engines, has been used for many years in air liquefaction and

separation plants. This thermodynamic cycle has also found use in

other applications at lower temperatures and has been incorporated in

several different liquid helium temperature refrigerators. Virtually all

liquid helium temperature refrigerators have a heat exchanger and a Joule -Thorns on expansion valve as the lowest temperature stage of the

1 . cycle. Before the Joule -Thomson process becomes effective, the

helium gas entering the exchanger must be cooled below the inversion

temperature, about 55°K. The difference between the thermodynamic

cycles that are used for continuous cooling at about 4°K is the manner

in which the precooling is provided. Variations of the Cascade JouleThomson, the Gifford-Mc Mahon and the Stirling cycles have all been

incorporated in liquid helium temperature refrigerators. In the Claude

1. A complete analysis of the Joule-Thomson process is given by Dean

and Mann (1965).
cycle, the precooling is accomplished by a work producing expansion

machine. High pressure gas is cooled upon expanding, converting its

potential energy to work. This cold fluid, which must be brought below

the inversion temperature, is then used to refrigerate the gas which

circulates through the lowest temperature stage of the cycle.

The performance of the Claude cycle has been computed taking

into account the efficiencies of the various components. The results

are presented in graphical form. These charts give the input power

requirements, mass flow rates for both the compressor and expander,

pertinent temperatures, and allow selection of the optimum high pressure for a given set of component characteristics.

2. Nomenclature List

m.

m.

total mass flow

expander flow

compressor flow rate per

unit of refrigeration

enthalpy

specific heat ratio

conversion factor

molecular weight

pressure

universal gas constant

temperature

specific volume at 1.0

atm and 273. 15°K

W /Q compressor power per

unit of refrigeration

available at 4. 2°K

h

k

K

M

P

R

T

V

std

(g/s)

(g/s)

(scfm/w)

(j/g)

(unities s)

(3. 5314455 x 10~ 5

ft /cm )

(4. 0028 g/g mole for helium)

(atm)

(8. 3143 J/g mole °K)

(°K)

3

(5603. 24 cm /g for helium)

(watts /watt)
W /Q expander power per unit of (watts/watt)

refrigeration available

at 4. 2°K

T) isothermal compressor (unitless)

efficiency

sentropic e efficiencyT| isentropic expander (unitless)

3 scfm standard cubic feet per (ft /min)

minute measured at 1 atm and 273. 15°K

Numerical subscripts refer to the state of the refrigerant as shown on

the schematic flow diagram in figure 1.

3. Discussion of Process

Basically the Claude refrigerator is a combination of the modified

Brayton and the Joule-Thomson cycles where the modified Brayton system precools the primary refrigerant stream for the Joule-Thomson process. The streams pass through the same compressor and heat exchangers

Depending upon the particular application, any number of expansion engines

or expansion valves (J-T) may be included in the system.

A schematic flow diagram of a double -valve Claude cycle is shown

in figure 1, and the corresponding process paths are shown on temperature-entropy coordinates in figure 2. A single-valve Claude cycle would

not have the lower heat exchanger, HX IV, or the first expansion valve between stations 4 and 5. The assumed processes for the single -valve cycle

are (primary refrigerant stream) isobaric cooling between stations 1 and

6, isenthalpic expansion between 6 and 7, isothermal heat addition constituting the refrigeration of the load between 7 and 8, isobaric heating

between 8 and 12, isothermal compression between the pressure levels

at T (heating from T to T is considered to take place in connecting

A 1 L. 1
COUNTERFLOW

HEAT

EXCHANGERS

(FOUR)

ITI2

w,

EXPANSION

ENGINE

EXPANSION

VALVE

EXPANSION

VALVE

COMPRESSOR

ol2

1 JW5wm

HEAT / t

ABSORBER Q

Figure 1. Schematic Flow Diagram of the Single -Engine Claude

Refrigerator
ENTROPY

Figure 2. Temperature-Entropy Diagram of the Single-Engine

Claude Refrigerator
lines by heat exchange with the ambient), and adiabatic expansion between

2 and 10 of the precoolant stream. The inclusion of a second valve produces essentially the same set of conditions with the addition of isenthalpic

expansion between 4 and 5 and isobaric cooling between 5 and 6 . Deviations from these assumed processes will occur in practice. Specifically, pressure drop in the piping and heat exchangers precludes isobaric

heat transfer and the compression will not be isothermal. A means of

estimating the effect of pressure drop is presented in the Appendix.

Further, it is possible to calculate the actual work of compression by

applying an efficiency factor to the energy required to isothermally compress an ideal gas.

The compressor circulates the helium at mass flow rate m

As the high pressure stream passes through the warmest of the counterflow heat exchangers, HX I, it is cooled by the low pressure stream to

temperature T At that point, some of the fluid is diverted through the

expansion engine at rate m . The remainder of the flow, m - m , is

further cooled in HX II by both the low pressure stream returning from

the load and by the cold exhaust of the expansion engine, and then passes

through the final heat exchanger before entering the J-T valve. A portion

of the stream is liquefied during the expansion process, and this liquid

is evaporated as the stream passes through the heat absorber. The vapor

then returns through the low pressure side of the heat exchangers.

Steady state conditions are assumed throughout for the analysis.

Elevation and kinetic energy contributions to the fluid energy are negligible. With these considerations, application of the first law of thermodynamics to the various components of the system yields the following

equations. For the upper heat exchanger HX I Wh

12

- h

ii

• (1 '
For heat exchanger HX II

(mj - m

2

) (h

2

- h

3

) = m (h - h ) . (2)

For heat exchanger HX III

h3- h

4

=h10- h

9 • (3)

For the lower heat exchangers HX III and HX IV

h3-V h10- h

8 • < 4>

For the expansion valve

h

y

= h

6

. (5)

For the heat absorber evaporator

Q= (m

1 - m

2

)(h

g

- h

?

) -. (6)

4. Calculations

The performance of the Claude-cycle refrigerator is a function of

the efficiencies of the heat exchangers, the expander, and the compressor.

The amount of refrigeration that is lost in the counterflow heat exchangers

is reflected by the temperature difference, AT, between the streams at

the warm ends of the exchangers. All real expanders produce less work

than an ideal isentropic machine, reducing potential refrigeration.

The power required to produce a unit of refrigeration is a complicated function of the characteristics of the heat exchangers and the expansion machine, but is simply inversely proportional to the efficiency of the

compressor and drive motor. All the calculations were made assuming

100 percent efficient, isothermal compression of an ideal gas since the

results can be related to any real compressor through its actual efficiency.
Calculations were made for perfect heat exchangers and expander to

provide a reference for the degradation of performance caused by nonideal components. Further computations were performed for realistic

component efficiencies and for nitrogen precooling. A low pressure of

one atmosphere was used for all cases. The parameter values are

given in Table I.

Table I

Independent Parameters

AT AT 71 T, Pn (°K)

1 (°K)2

(percent)

e (°K)

11

(atm)

50 300 2 to 6o

1 0.25 70 77.36

2 0. 50

5 0. 75

10 1.

1. 5

The results give the compressor power requirement per unit of

refrigeration, the expander power output per unit of refrigeration, the

fraction of total mass flow through the expander, the total flow per unit

of refrigeration, the expander inlet temperature, and the expander outlet

temperature.

The thermodynamic properties of helium used for the computations

are from Mann (1962) and can be calculated directly from the equation of

state. However, this would be a time consuming effort since data at over

five million states were required for the present work. Four point

Tl e

(p<srcent)

50

70

80

100
interpolation in the tabular data was found to be sufficiently accurate

and a much more rapid way of obtaining the values. Comparisons of the

power required per unit of refrigeration, calculated from data obtained

by either method, show differences of less than 0. 01 percent. The differences in the other cycle characteristics are less than 0. 5 percent.

The power required to isothermally compress an ideal gas from

P to P at temperature T is

1 £> 1 1mjRT tn(P

12/P ) Wc = M '

< 7 >

or when an efficiency factor for a real machine compressing a real gas is

applied

ml RT

1 ln (P12/P1>

wc

= Tm • (8) c

Dividing (8) by (6) gives the compressor power requirement per unit of

refrigeration or the figure of merit

RT

1 ^n(Pi2/P

l

)

c' T] c M(m

1 - m

2

)(h

g

- h

?

)

The work of compression is negative; however, in the charts that follow,

the absolute values of W /Q are plotted.

An isentropic expansion machine would discharge the fluid at state

10', figure 2. The losses incurred in a real machine result in an entropy

increase and a discharge at a higher temperature at state 10. Since the

process is adiabatic, the work produced by the expander is equal to the

mass flow rate times the enthalpy change of the fluid from inlet to outlet.
The efficiency of such a device is defined as the ratio of the actual work

to the isentropic work or h

2

" h

10

T) = ./ . 10

(10)

e h2" h

10'

and the power produced per unit of refrigeration at 4. Z°K is

W m (h - h ) _SL = L_i LP. . ( n) Q (m

1 - m

2

)(h

g

- h

?

)

K

'

Rearranging (2) and substituting equalities gives the expression

for percent of total mass flow through the expander

m h - h + h - h

°1 A h2" h

3

'

The volumetric flow per unit of refrigeration is

W KMV , c std .' „. t " Q RT

1 ^(P^/Pj) ' (

'

For a given set of heat exchanger and expander efficiencies and

a high pressure, there is, in terms of the amount of power required per

unit of refrigeration, an optimum engine inlet temperature. Equation (12)

shows that the flow distribution is a function of the engine inlet temperature.

The Joule -Thomson portion of the cycle becomes more efficient as the

inlet temperature of the exchanger (governed by the inlet temperature of

the expansion machine and its efficiency) is lowered. However, at lower

temperatures the precooling refrigeration from the expander decreases

so the expander flow rate must increase and more compressor power

for precooling is required. The optimum engine inlet temperature occurs

at a balance between these two opposing effects. For each set of

10
conditions, the best value of T was found to within 0. 05°K and the curves

in figures 4 through 21 are all for optimum T .

The data for the constant T and T curves were found by iteration to within 0. 05° K, and the data for the constant expander flow curves

were also found by iteration to within 0. 1 percent.

It is possible to select numerous sets of independent parameters

from Table I which are invalid for a single expansion-valve refrigerator

due to implied violations of the second law of thermodynamics in the

Joule-Thomson heat exchanger. The fluid entering the low pressure side

of that heat exchanger is assumed to be saturated vapor at one atmosphere,

station 8, figure 1 and state 8, figure 2. Optimization of T determines

state 10, selection of AT gives state 3, and solution of (4) determines

state 6. All these manipulations are straightforward applications of the

first law of thermodynamics. However, because the specific heats of the

high and low pressure streams are different, it may be that under the

assumed conditions, the calculated temperature at state 6 is lower than

at state 8. This impossible situation is illustrated schematically in figure

3 where the temperatures of the high and low pressure streams are plotted as a function of the heat transferred between them. This cooling curve

shows, with the dashed high pressure line, the invalid temperature profile

in the heat exchanger calculated from the first law of thermodynamics for

a single-valve arrangement. The effect of a secondary valve between 4

and 5 is to lower the pressure to some intermediate level where the

specific heat of the fluid will give a positive temperature difference

throughout the exchanger. The heat balance below the expansion machine

is not affected by the second valve. During the computations, if a negative temperature difference was found between 6 and 8, a second valve

was included in an attempt to correct the situation. The region where

such valves are required is indicated on the performance charts. Under

11
UJ

(T

I-

<

(Z

LU

Q_

UJ

SECONDARY

4 THROTTLE VALVE \\

5 %N ^

C

.0/Vn

hx m hx nr \

\

\

\

HEAT TRANSFERRED

Figure 3„ Cooling Curve for Heat Exchanger with Secondary

Throttle Valve

12
certain circumstances, the thermal properties of the fluid made it impossible to have a positive temperature difference even with two valves,

and the curves for AT = are discontinued at higher pressures for this

reason. In practice, additional valves could be used.

5. Results

The calculated performance characteristics of the single-engine

Claude-cycle refrigerator at 4. 2°K are shown in figures 4 through 31.

Figures 4a through 15a show the compression power per unit of refrigeration W /Q as a function of high pressure P for various values of

expander inlet temperature T and upper heat exchanger inlet temperature

(low pressure side) T . The values on the charts represent the conditions for which the expander inlet temperature T is at an optimum

level and the cycle inlet temperature T is at a temperature of 300°K.

The solid line marked "two valves" indicates the limit to the range of

possible operating pressures for the single-valve refrigerator. Hence,

an additional valve must be included in the system before higher pressures

are valid. In general, W /Q reaches a minimum value; thus, the optimum

high pressure P (i.e., the pressure at which the minimum occurs) can

be determined. Therefore, the cycle can be optimized for high pressure

as well as for expander inlet temperature. Figures 4b through 15b show

expander power per unit of refrigeration at 4. 2°K W /Q as a function of

high pressure P for various values of expander flow m .

J. ^ The same information (W /Q versus P and W /Q versus P ) is

presented for nitrogen precooling (i. e. , T equals 77. 36°K) in figures

16 through 21.

Figures 22 through 29 show the expansion engine outlet temperature T as a function of the expander inlet temperature T. for several

out in

13
values of helium inlet pressure P. . For an isentropic expansion of an

in

ideal 5gas. the relationship^ of T to T is represented by

out in r y

k-1

T = T

out in

out

m

(14)

where k is the specific heat ratio of the gas and P is some constant

out

value. Therefore, a plot of T versus T. would be a set of straight

out in

lines with slopes

dT

out

dT.m

> k-1

_out\ k

in

(15)

and the slope of a high pressure line will always be less than that of a low pressure line. A plot of T versus T for helium should be very

out in

similar, and this is true except for the few cases where the high pressure

curves intersect. It can be shown that this behavior results from the nonideal properties of the gas and the efficiency of the expander. The more

general variable subscripting in (14) and (15) and figures 22 through 29

was used because the equations and data apply to expansion machines in

any cycle.

Figures 30 and 31 show total volumetric flow F as a straight

line function of the compression work per unit of refrigeration W /Q

for several values of high pressure P . The slope of each curve is

represented by

dF KMV

std

Ld(W

c /Q)J

p

= RT

2 In (P^/Pj)

(16)

14
The logarithmic term in (16) is negative; consequently, the slope is also

negative. For convenience, however, the curves are plotted with a positive slope. The values of W /Q used to find the flow rates from figures

30 and 31 must be for 100 percent compressor efficiency with no pressure

drop in the system .

Since the calculations were performed assuming 100 percent isothermal compression, the plotted values of W /Q in figures 4a through

21a must be divided by the actual isothermal compressor efficiency to

obtain the power requirement for a lower efficiency.

The charts do not contain any information regarding the effect of

pressure drop on the performance characteristics. This is because many

conceivable sets of conditions could be assumed, thereby making the work

approach undesirable proportions. In addition, the effect of pressure drop

can be readily estimated. As will be shown later, the power requirement

increases, the amount depending upon the pressure loss in the system.

The shapes of the curves in figures 4 through 21 reflect the characteristics of the Brayton cycle and the Joule-Thomson process, which in

turn depend upon the thermodynamic properties of the working fluid. The

charts are in pairs (a and b) with each pair giving data for constant AT

and T| . For illustration, refer to figures 4a and 4b which are for a 10°K

temperature difference at the warm end of HX I and a 100 percent efficient

expander. In figure 4a, for a AT of 1. 0°K, the minimum W /Q occurs

at about 25 atmospheres high pressure with the engine inlet temperature

of about 39 °K. The outlet temperature of a 100 percent efficient expander

with an inlet at 39° K will be ~ 11 °K (see fig. 22). Since AT is 1 . 0°K,

the inlet to the Joule -Thomson heat exchanger will be at 12°K. From

Dean and Mann (1965), figure 12, the optimum high pressure for the

Joule -Thomson process, with the inlet at 12°K and a 1 °K temperature

15
difference, is ~ 22 atmospheres with the performance decreasing as the

high pressure is decreased. The optimum high pressure for a Brayton

cycle with the discharge of the expander at 11°K is greater than 25

atmospheres for an ambient temperature difference in the heat exchanger

of 10°K. Therefore, the best high pressure for the combination of the

cycles falls somewhere in between - in this case ~ 25 atmospheres.

Consider now the effect of changing the value of AT while holding

the high pressure constant at ~ 25 atmospheres. The performance of the

cycle increases as AT becomes smaller because the Joule-Thomson

process efficiency increases. The optimum engine inlet temperature

rises slightly, since the most favorable inlet temperature for the Joule - Thomson process varies in the same way. Helium behaves almost as an ideal gas at moderate pressures and down to fairly low temperatures,

This means that the temperature difference at the cold end of HX I will

be about the same as at the warm end, and the curves for constant values

of T will roughly parallel, or at least have the same general characteristics as, those for constant T . c* At lower pressures, the performance of both cycles becomes

progressively unfavorable and the power requirement per unit of refrigeration increases rapidly. There is still a general trend of increasing

optimum engine inlet temperature with better Joule -Thomson heat exchanger performance at constant pressure (see the curve for T = 30°K.).

In figure 4b, the expansion engine power output, the amount of

precooling performed by the expander, is given as a function of pressure

for several values of AT At low pressures, two effects are evident.

The relative amount of flow through the expander increases as does the

amount of energy removed from the cycle per unit of refrigeration. The

total flow rate is proportional to W /Q,so the amount of precooling required is also proportional to W /Q as reflected by the shape of the

16

J
curves in figure 4b. The relative flow through the expander decreases

at higher pressures where a greater amount of work is produced per unit

flow.

Figures 4 through 21 for the single-engine Claude cycle show that

reasonable thermal efficiency can be expected for this type of refrigerator - much better than a cascade Joule -Thomson cycle, for example,

but less efficient than a two-expander cycle. For discussion purposes,

consider a set of realistic component efficiencies (i. e. , realistic in the

sense that no advance in technology would be required to design and

fabricate such components today). Depending on the mass flow rate, expander efficiencies between 50 and 80 percent can be achieved. The temperature difference at the warm end of HX I could be between 5 and 10°K,

while AT across HX III could be less than l°K,with 0. 5°K considered to

be a reasonable value. Choosing AT as 5°K, AT as 0. 5°K, and T) as 70 percent, figure 10a shows a power requirement of about 345 watts/watt

for a 100 percent efficient compressor and no pressure drop. Typical

isothermal compressor efficiencies are about 60 percent. Estimates can

be made of the effect of fluid pressure losses. For example, if the pressure drop were 10 percent of total pressure in both the high and low pressure streams, the intake and discharge at the compressor would be 0.9

and 27. 5 atmospheres, respectively, for optimum conditions. Ideally, this

would raise W /Q to about 365. If the compressor efficiency is included,

W /Q would be about 610. In addition,there are other losses, such as heat

c leak to the low temperature space,which becomes critical in smaller

capacity refrigerators and which should be considered when translating

the information given here to real situations.

Inspection of the performance charts shows the possible improvement in thermal efficiency if the components could be slightly upgraded.

Consider making the following changes in efficiencies given in the example,

17
1. Decrease AT from 0. 5°K to 0. 25°K,

2„ Increase T| from 70 to 80 percent, and

3. Increase 7| from 60 to 70 percent while holding AT constant

c 1 at 5°K.

From figure 9a, W /Q is about 245. The assumed pressure drop increases that value to about 260, and the compressor efficiency increases

it further to the final value of 371 which is a considerable improvement.

About the same improvement in performance as in the previous

example is shown in figure 17a which illustrates the effect of precooling

the cycle with liquid nitrogen. Precooling involves depressing T to the

temperature of an evaporating liquid-nitrogen bath and providing an additional heat exchanger to communicate with the ambient temperature compressor. The liquid nitrogen consumption is determined by the efficiency

of the warmest heat exchanger. The cooling provided by the liquid can

be converted into an equivalent power requirement to be added to the compressor power. In most instances it is estimated that the increase in

W /Q would not exceed ten percent and is considered to be negligible.

Inclusion of the equivalent power required for precooling would involve

additional parametric studies beyond the scope of this work. Typical

conditions could be

T = 77.36°K,

AT = 2.0°K,

AT = 1.0° K,

T| =70 percent,

e

T] =100 percent, and

W /Q = 243 at a high pressure of 19 atmospheres with no pres

sure drop.

18
Many existing Claude-cycle refrigerators are liquid-nitrogen precooled,

but only the largest of these approach the power requirement of 468 watts

per watt of refrigeration which comes from adjusting the above conditions

for 10 percent pressure drop and 60 percent compressor efficiency (see

Strobridge and Chelton, 1966).

6. Conclusions

The proper choice of a low temperature refrigeration cycle to

meet a given set of duty requirements rests on a balance between

economic and technical requirements. Present applications for such

refrigerators are so varied that it is impossible to make definite statements about the "best" cycle for general use. It is, however, possible to

discuss the advantages and disadvantages of a particular cycle.

The single-engine Claude-cycle refrigerator is capable of reaching potentially high thermal efficiencies with reasonable improvements in

present heat exchangers, expanders, and compressors, and competitive

performance can be obtained with present components. This cycle selected for analysis is an elementary configuration. The single expansion

engine allows for simple control. Refrigerators operating in the liquid

region provide refrigeration at constant temperatures which is imperative for some applications. The refrigeration temperatures are limited,

but control of those temperatures is easily affected by regulating the

pressure over the liquid in the evaporator.

The minima in the performance curves are reasonably flat; thus,

for the sample set of conditions, where W /Q is a minimum of 345 at 25

c atmospheres, the high pressure could vary between 17 and 36 atmospheres

without increasing W /Q by more than 10 percent. This behavior permits

good flexibility in component selection. The optimum pressures are

generally between 20 and 35 atmospheres.

19
The Carnot efficiency, a useful tool of the refrigerator designer,

indicates the extent to which the real refrigerator deviates from ideal

performance. In general^the Carnot efficiencies of the single-engine

Claude refrigerator, operating at a cycle inlet temperature of 300. 0°K»

are somewhat low. For example, the following reasonable conditions

AT = 5. 0°K,

AT

2

= 0. 5°K,

T] =80 percent,

e r

T] =60 percent, and

Pressure drop = 10 percent,

yield a Carnot efficiency of only ~ 15 percent at the optimum high pressure of 24 atmospheres.

A simple hand calculation of the two-engine Claude-cycle refrigerator was performed for comparison purposes. In addition to the same

set of conditions, the isentropic efficiency of the second engine was

assumed to be 80 percent, and the temperature difference at the top of the

additional heat exchanger was assumed to be 0. 25 °K. This configuration

yielded a Carnot efficiency of approximately 23 percent. Liquid nitrogen

precooling of the single-engine cycle would produce about the same effect

without the added complexity of the second expander.

Any heat leak to the low temperature region of the refrigerator

will decrease the amount of useful refrigeration,so the calculated performance is better than can be realized in practice. The magnitude of

the heat leak depends upon the size and geometry of the low temperature

enclosure and is especially critical in small capacity refrigerators.

20
7. References

Dean, J. W. , and D. B. Mann (1965), The Joule-Thomson Process in

Cryogenic Refrigeration Systems, NBS Technical Note No. 227,

National Bureau of Standards, Boulder, Colorado.

Mann, D. B. (1962), The Thermodynamic Properties of Helium from 3

to 300°K between 0. 5 and 100 Atmospheres, NBS Technical Note

No. 154, National Bureau of Standards, Boulder, Colorado.

Strobridge, T. R. , and D. B. Chelton (1967), Size and Power Requirements of 4. 2°K Refrigerators, Advances in Cryogenic Engineering 12, 576, K. D. Timmerhaus, Ed. (Plenum Press, New York,

N. Y.)

21
8. Appendix

The performance charts may be used in combination to determine

the design parameters of a single-engine Claude refrigerator. An illustration will best describe their use. Consider the conditions

T

1 — 300°K,

e lsentr opic

= 80 percent,

c is othe rmal

= 60 percent,

AT = 5. 0°K,

AT

2

= 0. 5°K,

Q

P

= 1 watts, and

1 . atm,

which are practical and readily attainable in the current state-of-the-art.

In addition, the heat exchangers experience some pressure losses with a

10 percent drop considered to be an average value. The greatest pressure

losses occur in HX I, because the fluid passing through it is at a lower

density than in the other exchangers and HX I is the largest (in terms of

heat transferred) of the exchangers. Hence, it is reasonable to assume

that the total drop occurs here, and the example that follows will be based

on this assumption.

The appropriate performance chart (i.e., figure 9a) shows that

the optimum operating high pressure is 23 atm for which T , T , and

W /Q are 28. 5°K, 21 °K, and 265 watts /watt, respectively. In addition,

it shows that a second expansion valve is required to prevent second law

violations from occurring in the lower heat exchanger.

The W /Q value must be modified to account for pressure loss,

and a close approximation may be obtained through the use of

22
i n i —L\

wc wc \p, y _^ 12 PD _ p (17)

PD NPD , ( 1

v

12 NPD

where the subscripts PD and NPD refer to pressure drop and no pressure

drop, respectively. The pressure drop terms may be found from

P

l NPD P

l PD

= —079— and (18)

P12PD =

°' 9 P

12NPD- (19)

Thus,

W

-— = 282 watts/watt. (20) Q PD

The isothermal compressor efficiency must also be considered, and

this results in

Wc W Q

c PD

isothermal c isothermal

282

(21)

0. 6

= 470 watts/watt,

23
Thus, a 4. 7 kw compressor is needed to provide the required amount of

refrigeration.

The expander performance may be determined from figure 9b.

Using a high pressure of 23 atm, slightly less than half the total flow

(i. e. , about 49. 5%) will pass through the expander, and this device will

deliver about 5. 5 watts of power per watt of refrigeration. The expander outlet temperature is found in figure 24, and its value is about

12. 0°K.

Figures 24 and 25 give flow rate information for an isothermal

compressor efficiency of 100 percent. Thus, for a W /Q of 265 watts/watt

and a high pressure of 23 atm, the flow will be about 1. 7 scfm/watt or

17 scfm for the required amount of refrigeration.

The various refrigerator parameters will then be

P = 25. 5 atm,

P = p = p = 23. atm,

c* D rr P

ll = P10 = P

8 = K ° atm ' P

12

= °* 9 atm ' F = 17 scfm,

M = 50 percent,

W =47 00 watts,

c W = 55 watts, and

e Q = 10 watts.

1

1 3UU iS.

T

2

= 28. 5°K

T

3

= 12. 5°K

T

8 = 4. 2°K

T

io

= 12°K

T

ll

= 21. 5°K

T

12

= 295°K

\

= 80 percent

c = 60 percent

24
NOTE

Figures 4a through 31

follow.

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300.00 °K CYCLE INLET TEMP.

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49
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53
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54
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56
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REFRIGERANT : HELIUM /

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57
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60
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64
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66

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68
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69
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2200 2600

71

GPO 836-483


U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE

OFFICIAL BUSINESS