An Analysis of the Brayton Cycle As a Cryogenic Refrigerator

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An Analysis of the Brayton Cycle

As a Cryogenic Refrigerator ITATIONAL BURSAL' OF iTANDABD8 LIBRABT

MAR 6 J973

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NBS TECHNICAL NOTE 366 ISSUED AUGUST 1968

AN ANALYSIS OF THE BRAYTON CYCLE AS A CRYOGENIC REFRIGERATOR

R. C. MUHLENHAUPT AND T. R. STROBRIDGE

Cryogenics Division Institute for Basic Standards National Bureau of Standards Boulder, Colorado 80302

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.

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TABLE OF CONTENTS

Page

ABSTRACT 1

1. Introduction 1

2. Refrigerants „ . 2

3. Nomenclature List . 2

4. Discussion of Process 4

5. Parameter Values 4

6. Ideal Gas Calculations 8

7. Real Gas Calculations 15

8. Pressure Drop „ 16

9. Results 18

10. Discussion of Results 19

11. Conclusions 30

12. References 33

13. Appendix 34

List of Figures

Figure 1. Schematic Flow Diagram of the Modified Brayton

Refrigeration Cycle „ 5

Figure 2. Temperature-Entropy Diagram of the Modified Brayton Refrigeration Cycle 6

Figure 3. Temperature-Entropy Diagram of an Ideal Brayton Refrigerator 7

in TABLE OF CONTENTS (CONTINUED)

Page Figure 4. Schematic Diagram of the Brayton Refrigerator

Performance with an Ideal Gas as the Working Fluid . . 12

Figure 5. Schematic Diagram of the Brayton Refrigerator Performance Showing a Double Minimum Curve ..... 22

Figure 6. Schematic Diagram of Enthalpy Difference as a Function of High Pressure For Parahydrogen 24

Figure 7. Schematic Diagram of Brayton Refrigerator Performance with an Ideal Gas as the Working

Fluid - Single and Double Minimum Curves . 25

Figure 8. Brayton Refrigerator Performance - Helium Refrigerant - 300 K Cycle Inlet Temperature - 100% Expander Efficiency - No Pressure Drop 39

Figure 9« Brayton Refrigerator Performance - Helium Refrigerant - 300 K Cycle Inlet Temperature - 90% Expander Efficiency - No Pressure Drop 40

Figure 10. Brayton Refrigerator Performance - Helium Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - No Pressure Drop ...... 41

Figure 11. Brayton Refrigerator Performance - Helium Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - No Pressure Drop ...... 42

Figure 12. Brayton Refrigerator Performance - Helium Refrigerant - 77. 36 K Cycle Inlet Temperature - 100% Expander Efficiency - No Pressure Drop ..... 43

Figure 13. Brayton Refrigerator Performance - Helium Refrigerant - 77. 36 K Cycle Inlet Temperature - 90% Expander Efficiency - No Pressure Drop 44

Figure 14. Brayton Refrigerator Performance - Helium Refrigerant - 77. 36 K Cycle Inlet Temperature - 70% Expander Efficiency - No Pressure Drop 45

IV TABLE OF CONTENTS (CONTINUED) Page

Figure 15. Brayton Refrigerator Performance - Helium Refrigerant - 77. 36 K Cycle Inlet Temperature -

50% Expander Efficiency - No Pressure Drop . . . . . 46

Figure 16. Brayton Refrigerator Performance - Helium Refrigerant - 20. 27 K Cycle Inlet Temperature - 100% Expander Efficiency - No Pressure Drop .... 47

Figure 17. Brayton Refrigerator Performance - Helium Refrigerant - 20. 27 K Cycle Inlet Temperature - 90% Expander Efficiency - No Pressure Drop ..... 48

Figure 18. Brayton Refrigerator Performance - Helium Refrigerant - 20. 27 K Cycle Inlet Temperature - 70% Expander Efficiency - No Pressure Drop 49

Figure 19. Brayton Refrigerator Performance - Helium Refrigerant - 20. 27 K Cycle Inlet Temperature - 50% Expander Efficiency - No Pressure Drop ..... 50

Figure 20. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 300 K Cycle Inlet Temperature - 100% Expander Efficiency - No Pressure Drop .... 51

Figure 21. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 300 K Cycle Inlet Temperature - 90% Expander Efficiency - No Pressure Drop 52

Figure 22. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - No Pressure Drop 53

Figure 23. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - No Pressure Drop 54

Figure 24. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 77. 36 K Cycle Inlet Temperature - 100% Expander Efficiency - No Pressure Drop .... 55 TABLE OF CONTENTS (CONTINUED)

Page

Figure 25. Brayton Refrigerator Performance - Para hydro gen Refrigerant - 77. 36 K Cycle Inlet Temperature - 90% Expander Efficiency - No Pressure Drop 56

Figure 26. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 77. 36 K Cycle Inlet Temperature - 70% Expander Efficiency - No Pressure Drop 57

Figure 27. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 77. 36 K Cycle Inlet Temperature - 50% Expander Efficiency - No Pressure Drop 58

Figure 28. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 100% Expander Efficiency - 2 K Heat Exchanger Temperature Difference - No Pressure Drop 59

Figure 29. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 90% Expander Efficiency - 2 K Heat Exchanger Temperature Difference - No Pressure Drop ..... 60

Figure 30. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature- 70% Expander Efficiency - 2 K Heat Exchanger Temperature Difference - No Pressure Drop 61

Figure 31. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 2 K Heat Exchanger Temperature Difference - No Pressure Drop ..... 62

Figure 32. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 100% Expander Efficiency - 4 K Heat Exchanger Temperature Difference - No Pressure Drop ..... 63

VI TABLE OF CONTENTS (CONTINUED)

Page

Figure 33. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 90% Expander Efficiency - 4 K Heat Exchanger Temperature Difference - No Pressure Drop 64

Figure 34. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - 4 K Heat Exchanger Temperature Difference - No Pressure Drop 65

Figure 35. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 4 K Heat Exchanger Temperature Difference - No Pressure Drop 66

Figure 36. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 100% Expander Efficiency - 6 K Heat Exchanger Temperature Difference - No Pressure Drop 67

Figure 37. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 90% Expander Efficiency - 6 K Heat Exchanger Temperature Difference - No Pressure Drop 68

Figure 38. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - 6 K Heat Exchanger Temperature Difference - No Pressure Drop 69

Figure 39. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 6 K Heat Exchanger Temperature Difference - No Pressure Drop ...... 70

Figure 40. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 100% Expander Efficiency - 10 K Heat Exchanger Temperature Difference - No Pressure Drop 71

vn TABLE OF CONTENTS (CONTINUED)

Page

Figure 41. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 90% Expander Efficiency - 10 K Heat Exchanger Temperature Difference - No Pressure Drop 72

Figure 42. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - 10 K Heat Exchanger Temperature Difference - No Pressure Drop 73

Figure 43. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 10 K Heat Exchanger Temperature Difference - No Pressure Drop 74

Figure 44. Brayton Refrigerator Performance - Helium Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - Pressure Drop. 75

Figure 45. Brayton Refrigerator Performance - Helium Refrigerant - 300 K Cycle Inlet Temperature -

50% Expander Efficiency - Pressure Drop. . „ 76

Figure 46. Brayton Refrigerator Performance - Helium Refrigerant - 77. 36 K Cycle Inlet Temperature -

70% Expander Efficiency - Pressure Drop . 77

Figure 47. Brayton Refrigerator Performance - Helium Refrigerant - 77. 36 K Cycle Inlet Temperature - 50% Expander Efficiency - Pressure Drop 78

Figure 48. Brayton Refrigerator Performance - Helium Refrigerant - 20. 27 K Cycle Inlet Temperature - 70% Expander Efficiency - Pressure Drop 79

Figure 49. Brayton Refrigerator Performance - Helium Refrigerant - 20.27 K Cycle Inlet Temperature - 50% Expander Efficiency - Pressure Drop 80

vm TABLE OF CONTENTS (CONTINUED)

Page

Figure 50. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - Pressure Drop...... 81

Figure 51. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - Pressure Drop 82

Figure 52. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 77. 36 K Cycle Inlet Temperature - 70% Expander Efficiency - Pressure Drop 83

Figure 53. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 77. 36 K Cycle Inlet Temperature - 50% Expander Efficiency - Pressure Drop 84

Figure 54. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - 2. K Heat Exchanger Temperature Difference - Pressure Drop 85

Figure 55. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 2. K Heat Exchanger Temperature Difference - Pressure Drop 86

Figure 56. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - 4. K Heat Exchanger Temperature Difference - Pressure Drop 87

Figure 57. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 4. K Heat Exchanger Temperature Difference - Pressure Drop 88

Figure 58. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Oycle Inlet Temperature - 70% Expander Efficiency - 6. K Heat Exchanger Temperature Difference - Pressure Drop. 89

IX TABLE OF CONTENTS (CONTINUED) Page

Figure 59. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 6.0K Heat Exchanger Temperature Difference - Pressure Drop 90

Figure 60. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 70% Expander Efficiency - 10.0 K Heat Exchanger Temperature Difference - Pressure Drop 91

Figure 61. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 10. K Heat Exchanger Temperature Difference - Pressure Drop...... 92

Figure 62. Expander Performance - Helium Refrigerant - 100% Expander Efficiency - No Pressure Drop .... 93

Figure 63. Expander Performance - Helium Refrigerant - 90% Expander Efficiency - No Pressure Drop .... 94

Figure 64. Expander Performance - Helium Refrigerant - 70% Expander Efficiency - No Pressure Drop .... 95

Figure 65. Expander Performance - Helium Refrigerant - 50% Expander Efficiency - No Pressure Drop .... 96

Figure 66. Expander Performance - Parahydrogen Refrigerant - 100% Expander Efficiency - No

Pressure Drop . . , . 97

Figure 67. Expander Performance - Parahydrogen Refrigerant - 90% Expander Efficiency - No Pressure Drop 98

Figure 68. Expander Performance - Parahydrogen Refrigerant - 70% Expander Efficiency - No Pressure Drop 99 TABLE OF CONTENTS (CONTINUED)

Page

Figure 69. Expander Performance - Para hydro gen Refrigerant - 50% Expander Efficiency - No Pressure Drop 100

Figure 70. Expander Performance - Helium Refrigerant - 70% Expander Efficiency - Pressure Drop 101

Figure 71. Expander Performance - Helium Refrigerant - 50% Expander Efficiency - Pressure Drop. 102

Figure 72. Expander Performance - Parahydrogen Refrigerant - 70% Expander Efficiency - Pressure Drop 103

Figure 73. Expander Performance - Parahydrogen Refrigerant - 50% Expander Efficiency - Pressure Drop...... 104

Figure 74. Specific Flow Rate - Helium Refrigerant 105

Figure 75. Specific Flow Rate - Parahydrogen Refrigerant. . . . 106

AN ANALYSIS OF THE BRAYTON CYCLE AS A CRYOGENIC REFRIGERATOR

R. C. Muhlenhaupt and T. R. Strobridge

The performance of a Brayton-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 rate, pertinent temperatures, and allow selection of the optimum high pressure for a given set of component characteristics.

Key Words: Brayton, cryogenics, refrigeration

1. Introduction

In 1873 a Boston engineer, George Brayton, conceived an ideal heat engine which, when operating in a thermodynamic system, combines constant pressure and constant entropy processes. In actual practice, the real machine differs from this ideal because of fluid friction in the flow passages, heat transfer with the surroundings, and irreversibili- ties in the expansion machine, compressor, and heat exchanger.

In the same year Brayton, using the principles learned in the development of his heat engine, designed a refrigeration machine which operated on a reversed Brayton cycle. The working fluid in such a machine receives heat at some low pressure and temperature and then rejects heat at some higher pressure and temperature. Brayton' cycle is sometimes referred to as the Joule or the air- standard cycle.

This is because Joule, working independently of Brayton, proposed the same cycle, and air is commonly used as the working fluid.

During the past several years, the Brayton cycle, operating as an open cycle refrigerator, has been used extensively in aircraft cooling, and the cycle is considered by many to be the prime candidate for space power systems (Bernatowicz, [1963]; Harrach and Caldwell,

[1963]; Glassman and Stewart, [1964]; Schultz and Melber, [1964]; and Glassman et alv [ 1965]). A modified version of this refrigeration cycle has been analyzed and will be considered here.

Analytical expressions are derived to determine values for flow rate, high pressure, evaporator inlet temperature, and figure of merit

(the power input per unit of refrigeration) for selected values of expander efficiency, cycle inlet temperature, evaporator outlet temperature, and cycle temperature difference (the temperature differential across the warm end of the heat exchanger). The resulting data are presented in figures 8 thru 75.

2. Refrigerants

Fluids considered in this analysis are nitrogen, parahydrogen, and helium. The state equations of Strobridge, [1962], Roder and

Goodwin, [l96l], and Mann, [ 1962] were used to calculate the required thermodynamic properties. Data were obtained for refrigeration at, and above, the normal boiling points of each fluid which are 77. 364 K for nitrogen, 20. 269 K for parahydrogen and 4. 214 K for helium.

3. Nomenclature List

C specific heat at constant pressure (J/g mole»K)

C specific heat at constant volume (J/g mole»K)

F volumetric flow rate (scfm)

F/Q compressor flow per unit refrigeration (scfm/kW)

M molecular weight (He 4.0028 g/g mole) (H2 2.016 g/g mole) (N2 28.016 g/g mole) P pressure (atm)

R universal gas constant (8. 3143 J/g mole»K)

T temperature ( K)

T ambient temperature (300 K) a

AT temperature difference between 1-5 the fluid streams at warm end

of heat exchanger (T - T ) K) 15 ( W /Q figure of merit (compressor

power per unit of refrigeration) (W/W)

W /F expander power per unit flow (W/ scfm) h specific enthalpy (J/g)

k specific heat ratio (C /C ) (unities s) p v m mass flow rate (g/s) s specific entropy (J/g»K) scfm standard cubic feet per minute

measured at 273. 15 K and 3 1.0 atmosphere (ft /min)

3 v specific volume (cm /g)

v , specific volume of the gas at std p 5 273. 15 K and 1. atmosphere (cm /g)

T\ isothermal compressor

efficiency (unitless)

Q unit of refrigeration (W) s

T| isentropic expander efficiency (unitless)

p density (g mole/-t)

101.3278 a multiplication factor (J/^«atm)

3 0.002119 a multiplication factor (J«scfm/cm «W)

4. Discussion of Process

A schematic flow diagram of the modified Brayton refrigeration cycle is shown in figure 1, and the process paths are illustrated on temperature-entropy coordinates in figure 2. The thermodynamic processes involved are isobaric cooling between stations 1 and 2, adiabatic expansion between 2 and 3, isobaric heat absorption of the refrigeration load between 3 and 4, isobaric heating between 4 and 5, and isothermal com-

pression at ambient temperature between the low and high pressures .

For comparison, figure 3 shows the process paths of Brayton' ideal refrigeration cycle. The main difference between the two figures occurs in the ideal compression and expansion processes which are isentropic in figure 3 rather than isothermal and adiabatic as shown in figure 2.

5. Parameter Values

The performance of the refrigerator depends on the fluid used as the refrigerant, the pressure levels in the fluid circuit, the refrigeration temperature, the temperature at the warm inlet of the heat exchanger, and the efficiencies of the compressor, heat exchanger, and expansion

The process path shown in figure 2 between stations 5 and 1 is not isothermal, indicating that a finite temperature difference will exist in the heat exchanger of an actual refrigerator. It is assumed that the temperature of the fluid at the inlet of the compressor will be the same as the discharge due to heat transfer with the surroundings through the inter- connecting piping. w,

COMPRESSOR

EXPANDER W.-*/\'e WWWEVAPORATOR

Figure 1. Schematic Flow Diagram of the Modified Brayton Refrigeration Cycle. LjJ (Z HZ> < DC UJ

ENTROPY

Figure 2. Temperature-Entropy Diagram of the Modified Brayton Refrigeration Cycle. LU (T HZ) < (Z LU Q_

ENTROPY

Figure 3. Temperature -Entropy Diagram of an Ideal Brayton Refrigerator. engine. The power required for the compressor was in all instances calculated from the relation for the 100 percent efficient isothermal compression of an ideal gas at 300 K between the appropriate pressure levels.

The results presented here must therefore be corrected to take into account the isothermal efficiency of actual compressors. The efficiency of the heat exchanger is reflected by the temperature difference between the fluid streams at the warm end of the heat exchanger. Table I lists the values of the parameters used in the analysis.

TABLE I

Parameter Variations

T 300.0, 77. 36, & 20.27 K T 8-100 K (helium) 4 78-100 K (nitrogen) 22-100 K (parahydrogen)

AT 0.5, 1.0, 2.0, 3.0, 4.0, 6. & 10.0 K P 2-60 atm

P c 1 and 1. 2 atm

11 50, 70, 90 & 100%

6. Ideal Gas Calculations

Charts showing the performance of the modified Brayton refriger-

ator assuming that the refrigerants behave as ideal gases are useful in two ways. First, such charts, and the relations from which they are

derived, are helpful in explaining the characteristics of the various curves. Second, when the ideal gas performance charts are prepared

as transparent overlays to graphs calculated from real gas properties,

the direct visual comparison shows the necessity for using real gas

properties if accurate results are required.

8 The efficiency (or the figure of merit, as it is commonly called) of a refrigeration device is usually defined as the ratio of the net power input to the refrigeration effect. For the purpose of this analysis, the figure of merit is defined as the ratio of the isothermal compressor power input (rather than the net power input) to the refrigeration effect. Thus, P RT !n(--) W a \P_ / c 5 (1) Q Tl C (T - T.) c p 4 3

The compressor power W is negative because it represents power transferred to the system. However, for convenience, the absolute value is always used in this analysis.

Since the efficiencies of the expander and the heat exchanger are important considerations to the refrigerator designer, these properties must be included. An expression equivalent to (1) is

P k" X

w a VP / c 5 (2) Q k " 1 P - 1 T +AT iT ^cl^eL -^) j[ 4 l-5J- l-5} 5

For a given set of parameters (i.e., T , AT , T| , and the refrigerant), there is an optimum high pressure which will require a minimum amount of compressor power to produce a unit of refrigeration.

This fact can be demonstrated by differentiating (2). Thus k-1 w k T T] -ill T +AT - AT c k {\b-(ir) ] 4 1-5J ,,} Q a d P I k-1 1 T 1 2 k T.+AT, r AT !-»}' \ ^{\[-(p:) ] _ 4 1 - 5_

P 2k- 1 P k-1 1 \ k IN k T 1] T] -^- T + AT ( A 1 a c e k \ 4 1-5/ V P ("p:Mt£ k-1

P, 1 - T + AT - AT H n A ! C c lie _ L 4 l-5_ ,-J which becomes W T T + AT {k ~ 1] Q a V 4 l-5 ) d P. Tl P. k c 1

AT k-1 k-1 1-5 1 k 1+ In \ T +AT kte) ]} V 4 l-5> (3) k-1 T + AT AT \ i\b-te) 1-5 J 1-5J

For simplification, two terms in the numerator are defined as follows: AT 1-5 A = 1 ,- and n (T ) e 4+ATl _ 5

k-1 k-1 k B = ' 1 (p7) [ — (p ) ]

10 For the combinations of component parameters, the term A is a constant greater than zero but less than one, while B varies from a maximum of one (as P approaches P ) to a minimum of zero (as P 15 1 approaches infinity).

At a pressure P equal to P , the term B equals 1. For this 1 5 condition only, (3) becomes W / \ d T (k - 1} r hf)iQ a (4) d P kAT P T \ l-5 l

The terms in (4) are positive; therefore, the slope is negative, and the

curve is decreasing with increasing pressure. Referring to (3), as the pressure ratio is increased, B decreases causing the slope to increase

is are until some pressure P w .» reached where A and B equal. Here the slope goes to zero establishing a minimum point on the curve.

Beyond this pressure the difference between A and B is always positive; therefore, the slope is also positive. As the pressure is increased further, the slope, still remaining positive, begins to decrease and continues to do so until the pressure becomes infinite. Here the slope goes to zero.

Equation (2) was evaluated holding AT. _, T| , 7] , and P_ con-

1-5 e . c 5 stant while varying P and T . The results were plotted as a function of P . The general shape of the curves is shown schematically in figure 4. For each T at constant AT the optimum high pressure can be found. For each AT at constant T again the optimum pressure can be calculated. Therefore, it is possible to plot the locus of optimum

pressures for both constant T „ and constant AT, ... 4 1-5

To determine these optimum pressures, (3) is set equal to zero, and after simplification, the expression becomes

11 11

u so +J n nJ •rH u ^ VM 14 •i-i £ <+H a; > rn (D Oi

.— a) nJ ^ X) t— <-H a 0) s 1 rt •i-i 0) Q

o oj

-t-> oi a a m

co Ph

0/ 3M

12 k" X AT. . ,P..-^± .P. 1 1 1+tn U °- 5 - < > I 11Tl (t'+'aT,(T +4T J) -Ip.....) [ (p:)- ] 5

The optimum pressure P was then calculated through an iteration

process for several sets of P , T , and AT for the ideal gas refrigerants, helium and hydrogen. For comparison to the corresponding real gas data, these charts are shown as transparent overlays to figures 8 thru 11 and 20 thru 23.

For additional information, a family of evaporator inlet (or expander outlet) temperature T curves is included on these ideal gas, optimum operating pressure plots. The efficiency of the expansion device is defined as

T T L2" *3 Tl = , (6) e 1 ^ - 1 „. and the temperature-pressure relationship for an isentropic expansion is

k-1 T P 2 , 2 M, T \P 3 1 3'

In addition

T =T +AT (8) 2 4 l-5

for an ideal gas. Substituting (8) and (7) into (6) and rear ranging> gives

13 '

p -i^

- 1 1 t = < t +at (9) 3 4 i- 5){\[(p:) ]" )- 5

where P equals P and P equals P . The evaporator inlet tempera- n 1 CD J tures were calculated from (9) using the optimum high pressures determined in evaluating (5).

As AT approaches zero for all refrigeration temperatures, the optimum high pressure becomes smaller and is equal to the low pressure in the limit. Although this is an impossible operating condition, it is informative to calculate the performance since it is a lower limit for the power required per unit of refrigeration. Equation (1) cannot be evaluated explicitly for a high pressure of one atmosphere; however, it can be changed to the equivalent form

T / 4 T In W a \ T c 3 10) W^ZL-T,,) '

where T equals T when P equals P . Although (10) also cannot be

evaluated explicitly for T equal to T , it can be solved using L'Hospital's rule. Thus, in the limit as T approaches T

W T c a ~Q" = (H) ' Tl Tl t c e 4

Equation (11) was used to determine W /Q at one atmosphere high pressure for all T values for both the ideal and the real gas cases.

: 7. Real Gas Calculations

The analysis that follows assumes no pressure loss in the system;

thus,P_ = and = = = one addition , T] P, P_ P P n atmosphere.r In T , , and 2 13 4 5 1 e T are constant. The compressor is 100 percent efficient with respect

to ideal gas isothermal compression. Steady state conditions are

assumed throughout and elevation and kinetic energy contributions to

the fluid energy are negligible. Application of the first law of thermo-

dynamics to the cycle gives

W RT a^(p; (12) . u n m c 4 3

Again the convention of using absolute values of W /Q is adopted. If (12)

is evaluated in the same manner as (2) and the results plotted, the curves

will again exhibit the general behavior shown in figure 4.

The development of (13), used to determine optimum pressures is

given in the Appendix. This expression is similar to (5) and equal to

zero at optimum conditions,

P 101 2779 '

, - i ;! )(p.u. 4 3'\ M / \ 1 P

T] T dh T] T i -i f / „ e 3 ' \ / 1 \ e 3 i - + = . (13) x ! 1) t /id^^ p t 2 2 2

Equation (13) with its logarithmic term cannot be solved explicitly for P ; therefore, an iteration process was employed to find the correct pressure

within 0.01 atmosphere.

15 As with the ideal gas plots, a family of evaporator inlet temperature curves is given to provide additional design information. Since the temperatures at stations 2 and 3' are determined in the course of evaluating

(13), the corresponding enthalpys are known. The enthalpy, and therefore the temperature, at station 3 is readily determined from the defini- tion of efficiency

h - h

e h h 2- 3-

As stated in the ideal gas calculations, the W /Q values at the one- atmosphere intercept are determined from (11).

Additional charts show the expander power per unit of volumetric flow as a function of expander inlet temperature, expander outlet temperature, and expander inlet pressure. These values were calculated directly from

W h - h (15) F 0.002119 v , std for each refrigerant.

8. Pressure Drop

The discussion thus- far has centered upon a simple, two-pressure refrigeration system. Since fluid friction occurs in the flow passages, pressure losses will result. Thus, the pressures P and P (see figure J. L* 1) will not be equal as assumed in the previous analysis. For the same reason, the pressures P and P will not equal P . Therefore it is desir- able to predict what, if any, effect this behavior has upon the parameters of interest.

16 '

The expression

k-1 W W '{^eL 1 r

T_ 1) 1 - { 2 e PD

derived from (2) for pressure drop and no pressure drop conditions,

predicts the change in the figure of merit W /Q with reasonable pre-

cision. The subscripts PD and NPD refer to pressure drop and no pressure drop conditions respectively. A similar expre ssion predict-

ing the corresponding shift in optimum pressure has not been developed.

It is impractical to perform a complete set of pressure drop

calculations because a large number of conditions can be conceived.

Therefore only a brief examination of this phase of the work was per-

formed. Equation (13) was modified slightly to obtain the necessary optimum operating expression W d f—E- 101. 32779 \ Q h h + p te = ( " ) L d P 4 3 M i p; 1 T 1

H T d h. T d P. e 3 \ 3< 1 - T\ + = (17) e T~ d P. T d P 2 2 1 T

Pressure drops of 10 and 20 percent in the high and low pressure streams

respectively, believed to be reasonable values were selected for the calcu-

lations. As before, an iteration process was employed to determine the

correct high pressure to within 0. 01 atmosphere.

17 9. Results

The data for optimum pressure plots for helium and parahydrogen shown in figures 8 thru 27 were calculated using real gas properties.

Several optimum pressure charts are also included for the ideal refrigerants helium and parahydrogen. For ease of comparison each ideal gas plot is a transparent overlay for the real gas refrigerant chart. Each graph is for a particular expander efficiency and a specific cycle inlet temperature. The isothermal compressor efficiency in all cases is 100 percent. The figure of merit W /Q is shown as a function of optimum pressure P for a series of curves representing the evaporator outlet

temperature T , the evaporator inlet temperature T , and the temperature difference between the fluid streams at the warm end of the heat exchanger AT 1-5

None of the optimum pressure plots show an evaporator outlet temperature equal to the liquid temperature of the refrigerant. This is because the optimum pressure is either beyond the range of available data or too high to be practicable.

Most optimum pressures were beyond the range of available data for nitrogen; therefore, high pressure--rather than optimum pressure-- plots were obtained for this refrigerant. This information is presented in figures 28 thru 43. Each chart is for a particular expander efficiency, a specific heat exchanger temperature difference between the fluid streams at the warm end of the heat exchanger, and an isothermal compressor efficiency of 100 percent. The figure of merit W /Q is shown as a func- c tion of high pressure P for a series of curves representing the evaporator inlet temperature T » the evaporator outlet temperature T , the 3 expander power per unit flow W /F, and the flow per unit of refrigeration F/Q.

18 Figures 44 thru 61 show performance data for pressure drops of 10 and 20 percent in the high and low pressure streams of the heat exchanger. These charts provide essentially the same information and cover the same general high pressure range as the corresponding ones for no pressure drop.

Additional performance information for helium and parahydrogen is given in figures 62 thru 73 which show expander power per unit flow

W /F as a function of expander inlet temperature T , cycle inlet e £ pressure P. , and expander outlet temperature T . Figures 62 thru 69 are for zero pressure drop in the heat exchanger, while figures 70 thru

73 are for pressure drops of 10 and 20 percent in the high and low pressure streams respectively of the heat exchanger.

The final two plots (figures 74 and 7 5) show the evaporator inlet temperature T as a function of evaporator outlet temperature T for a family of curves representing flow rate per unit refrigeration F/Q.

There are upper and lower limits on the value of T . It can equal (at an

infinite flow rate) but not exceed T . Likewise, it can equal but never be less than the liquid temperature of the refrigerant (i.e., for the conditions discussed in this work).

10. Discussion of Results

The optimum pressure charts for helium show reasonable agreement with the transparent overlays since the properties of helium, especially in the high temperature, low pressure regions, are very similar to those for an ideal gas. The corresponding plots for parahydrogen demonstrate the difference between the properties of parahydrogen and the ideal gas.

The discontinuities on the 300. K cycle inlet temperature parahydrogen curves are in disagreement with the smooth curves on the

19 transparent overlays; therefore, this anomalous behavior must be examined more closely. Accordingly, a generalized equation W Q RT A + B + C + D opt _ opt opt opt opt _ (18) d(P ) 2 2 1 opt M (P,) . A 1 opt opt representing the slope of the constant AT curves may be derived. The

terms in (18) for a variable evaporator outlet temperature T , a constant

cycle inlet temperature T , a constant cycle outlet temperature T , and 1- 5 a 100 percent expander efficiency (to simplify the equation) are defined as follows:

RT /M = a constant (approximately 1237), a

T d h. d h A + - v. opt LVdP d P opt 1 T 1 T opt

,2 2 d h d h d v. 3' = p + opt (—,») ( t d P opt dP, T, d P, T 1 T opt 1 1 1

d T d h. d h 3» 4 = C P + - v. , and opt 1 VT d P d P, 2J ' * ^ ' [y 2 opt l T 1 T IT opt

d T. d h. d h D = P - v opt 1 \T J Vd P d d 2 n P P J 2 opt IT 1 T 1 T opt

d h d 2 h It is difficult to obtain precise values for d P 1 T dP T d T 1 and because neither h nor T is an explicit function of P , IT 5 l

20 J

and no simple relationship between these properties exists. A brief examination will be made of the term C which, in fact, is responsible ° Pt dh dh —2 —4 for the discontinuities. The two derivatives {—r— - ) and ( ) are ^ d P, ^ d IT'rr, P^rrnIT

quite small relative to the specific volume v , thus these three terms combine to form a negative quantity. The ratio P /T is positive, while d T the derivative ( — is negative (the temperature T decreases 1 Tj with increasing pressure P,). Thus the C term provides a positive contribution to the slope equation.

Since T decreases as P is increased, there is some pressure

P, ... at which the temperature T„ , reaches the liquid temperature of the l(i) 3 ! refrigerant. When this occurs, the derivative must go to zero. Hence the positive contribution is no longer available, causing the slope to change abruptly to some lesser value. The remaining charts do not exhibit this behavior because the temperature T does not reach the liquid temperature of the refrigerant for the stated conditions. Had this happened, a discontinuity would be present.

In a few instances two optimum pressures (i. e. , two minimum points) were found for parahydrogen precooled to 77. 36 K. The first minimum generally occurred in the low pressure region of two to ten atmospheres, while the second occurred in the high pressure region of

90 to 200 atmospheres.. Figure 5 shows a sketch of a typical double minimum curve which appears to be inconsistent when it is recalled that the ideal gas calculations predicted only one minimum.

Because the isothermal compressor power W is calculated in the same way for either the ideal or the real gas, it is evident that this

21 1

rt d i-t u

• H a mu Hi « H

>> (1) 0) 1— h 43 PQ

3 nJ n &0 fl

a "S «$ O h Men43 <9 Q o a

a o

a) co Ph

• I-l

0/ M

22 quantity is not responsible for the unusual behavior. The inconsistencies

therefore must be attributed to the refrigeration load Q which is defined as

Q = W + m (h_ - h) (19) e 5 1

for both cases. The expander power W changes with increasing pressure

for the ideal gas, while the enthalpy difference remains constant at some

fixed negative value. For the real gas, however, both terms change with increasing pressure. A typical plot of enthalpy difference as a function

of high pressure for parahydrogen is shown in figure 6. The diagram

shows that Ah is independent of pressure and temperature for the 1-5 ideal gas, while it is dependent upon both the high pressure and the cycle inlet temperature T, for the real gas. The temperature T. is less 1 1-a than T which is less than T ; therefore, the real gas curves begin to flatten out somewhat and the enthalpy difference tends to approach a

constant value at the higher cycle inlet temperature. Some of the iso-

a at the derivative 1 therms reach maximum value, which point VdP, _ l'T 1 must go to zero. For this special condition, and this condition only,

d T the Joule-Thomson coefficient f j will also be equal to zero.

1 h l Therefore, the locus of all such maxima represent the well-known

inversion curve.

If a refrigeration load Q for the ideal gas is recalculated using an

enthalpy difference that varies in a manner similar to that of the T

curve in figure 6, a revised figure of merit W /Q will be obtained. ° c rev line) Curves representing the original figure of merit W /Q . (solid r ° ° ° c orig and the revised figure of merit W /Q (dashed line) are shown in fig- c rev ure 7.

23 T|-Q

IT)

i <

deal

Figure 6. Schematic Diagram of Enthalpy Difference as a Function of High Pressure for Parahydrogen. 24 o «j Q

ID Pi U * .s ° CO

d i

bo -d h r1 « •wq o X U o « O CO

CO CO > u u

"i H 3 e a g

0/°M

25 At low pressures (point A) the new enthalpy difference is fairly- constant but slightly greater than the original value. Hence, the revised curve will lie slightly below the original curve. The minimum (point B) of the revised curve will occur in approximately the same general pressure region as that of the original, but it will be a lesser value.

As the pressure is increased beyond the minimum point, the new enthalpy difference becomes somewhat larger. Therefore, the magnitude

of the revised W /Q will be less than that of the original W / Q . , c rev c orig and the W /Q curve should begin to rise (point C). As the pressure c rev is increased further, the new enthalpy difference reaches its maximum

value causing W /Q to be considerably less than W /Q . . In this c rev c orig pressure region (point D) the revised W /Q curve can be expected to c rev have another minimum or, at the very least, an inflection point. As the pressure is increased further, the new enthalpy difference begins to decrease. In this region (point E) W /Q increases gradually at first, c rev and then as the magnitude of the new enthalpy difference approaches that of the original, it increases at a more rapid rate. Thus, for a cycle inlet temperature less than the maximum inversion temperature of the refrigerant, the J-T effect will force either a second minimum or an inflection point to occur on the figure of merit curve. The cycle inlet temperature of 300.0 K is above the maximum inversion temperature (about 208.0 K) for parahydrogen; therefore, no extreme deviation from the normal shape of the W / Q curve should be expected for this condition.

Cycle inlet temperatures of 300. and 77. 36 K are above the maximum inversion temperature (approximately 55. K) for helium.

Hence no unusual behavior is expected for the curves representing these conditions. On the other hand, the cycle inlet temperature of 20.27 K definitely is below the maximum inversion temperature, and although no double minima were observed for this condition, there was some

26 evidence of an inflection point. This appears to be reasonable because the maximum deviation of the enthalpy difference curves for helium precooled to 20. 27 K are considerably less than those for parahydrogen precooled to 77. 36 K.

For the few cases where double minima did occur, the first was always selected as the optimum operating condition because it was felt that the second occurred at a pressure too high to be practicable.

Some of the charts for parahydrogen and all of those for nitrogen show expansion engine exhaust temperatures that are the fluid saturation temperatures at the low pressure. This means that a portion of the refrigerant may change phase from gas to liquid in the expander. In many units, not designed for this condition, the presence of liquid will cause a drop in efficiency; however, the operation of "wet" expansion engines is becoming more common. While the problems encountered are not insurmountable, it is most important for the designer to be aware of the situation so that an allowance can be made for the liquid and for the possibility that a lower efficiency must be tolerated.

The flow data (i.e. , T as a function of T and F/Q) for 1. 2 atmospheres showed close agreement with that for 1.0 atmosphere, with differences in T of about 0. 01 K and 0. 5 K at the high and low ends of the T range respectively. Since these differences are negligible and unreadable on the charts, only the 1.0 atmosphere data are included.

The performance charts may be used in combination to determine the operating parameters, including the optimum pressure, for a given refrigeration requirement. Consider the following conditions:

27 Refrigerant helium Q 100 W 60% c isothermal

f] 70% e isentropic 300 K 1 30 K 296 K

0.9 P 1 1.2 P,

1.2 P, and

1. atm

Figure 44 shows that the optimum operating high pressure P is

11 atm for which T and W /Q are 21 K and 32 watts/watt respectively. 3 c The isothermal compressor efficiency must be considered, and this increases W /Q to 53. 3 watts/watt for the 60 percent efficient compressor.

Thus a 5. 33 kW compressor is needed to provide the required amount of refrigeration. As indicated, the chart accounts for pressure changes of 10 and 20 percent in the high and low pressure streams of the heat exchanger respectively. These losses result in pressures of 9.9 atm for P_ and 1. 2 atm for P„ and P „. 2 3 4

Flow rate information is given in figure 74. For an evaporator inlet temperature T of 21 K and an evaporator outlet temperature

T of 30 K the flow per unit of refrigeration F/ Q is 250 scfm/kW. Thus for a refrigeration load Q of 100 watts, the flow is 25 scfm.

The expander performance may be determined from figure 70.

For a high pressure P of 1 1 atm and an evaporator inlet temperature

T of 21 K, the expander output power per unit flow W / F and the

28 expander inlet temperature are 6 watts/ scfm and ~ 35 K respectively.

Thus, for a flow of 25 scfm, the expander output power is 150 watts.

The various refrigeration parameters found from the graphs will then be

35 K

21 K

11 atm

9 . 9 atm

1 . 2 atm

1 . 2 atm W 5.33 kW c isothermal W 150 W e isentropic F 25 scfm and Q 100 W

In order to illustrate the effect of component efficiencies on the performance of the cycle, consider decreasing AT (i.e., increasing 1-5

T ) while holding T| and T. constant. The engine inlet temperature T , 5 e 1 d. and thus the evaporator inlet temperature T , will decrease. For the

same refrigeration capability, Q, the mass flow decreases therefore, the figure of merit W /Q will be smaller. The cycle, however, will c not be operating at the optimum high pressure which will have a new,

somewhat lower value.

Increasing the isentropic efficiency of the expander T} , while holding AT and T constant, produces an increase in the expander 1-5 1 power output. For the same refrigeration capability, Q, the mass flow

required decreases because of this additional power output and the value of W /Q becomes smaller. Again the high pressure P can be

reduced to achieve optimum operating conditions.

29 If component efficiencies could be slightly upgraded, considerable improvement in cycle performance would be obtained. For example, consider the following design requirements. The refrigerant is helium,

T] T\ T . is 20 K, T. is 300 K, AT, _ is 6 K, is 50 percent, is 60 percent, 4 1 1-5 e c and the pressure drop in the heat exchanger is 10 and 20 percent, respectively, in the high and low pressure streams. Figure 45 gives a value of 18 5 watts/ watt for W /Q and an optimum pressure of 28 atm c for a 100 percent efficient isothermal compressor. The 60 percent efficient compressor increases that value to about 310 watts/watt.

If the component efficiencies could be improved to the following values

AT = 1_5 4K

T] = 70% , and

T) = 70% c the figure of merit W / Q will be approximately 80 watts/ watt for the

70 percent efficient compressor which is nearly a 75 percent reduction.

In addition, the optimum high pressure will be about 15 atm which amounts to about a 50% reduction.

11. 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.

30 The modified Brayton refrigeration cycle has basic advantages which warrant its consideration when choosing a cryogenic refrigerator.

The system is relatively simple, requiring only a compressor, an expansion engine, and a heat exchanger as its main components. Com- pared to cycles using Joule- Thomson expansion valves which produce cooling isothermally only at the liquid temperatures of the refrigerant, the modified Brayton cycle permits a large selection of refrigeration temperatures. However, unless specific design allowances have been made, difficulty could be encountered in attempting to cool at liquid temperatures with the Brayton cycle because of the gas to liquid phase change which would occur in the expansion engine.

The optimum high pressures for helium and parahydrogen are not excessive so the problems of designing expansion engines for large pressure ratios may be eased. While computing the optimum pressures, it was evident that the minima in the performance curves were quite wide, so it would be possible to operate the refrigerator at pressures lower than the optimum without suffering a severe loss of efficiency.

The Brayton 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. However, calculations show that utilizing reheat expansion stages may increase the available refrigeration by 10 to 20 percent with no change in compressor capacity or input power requirements. Two-stage reheat involves expanding the high pressure gas not to the low pressure, but to some intermediate pressure. The cold gas passes through the first portion of the load exchanger which is now in two sections, where it is warmed to the expander inlet temperature as it absorbs energy from the refrigeration

31 load. After the final expansion to the low pressure, the fluid enters the second section of the load heat exchanger. Additional expansion stages and heat exchanger sections may be added if warranted. In addition to improved performance, reheat reduces the temperature rise in the load heat exchanger appreciably and tends to a more uniform heat sink temperature.

Precooling helium to 77. 36 K produces little if any advantage at

the same AT . Precooling to 20.27 K, however, results in a pro- 1-5 nounced reduction of the figure of merit at the expense of an increase in the optimum operating pressure. Precooling parahydrogen to 77. 36 K definitely decreases both the figure of merit and the optimum operating pressure. The benefit to be derived by precooling either refrigerant to 65. rather than 77. 36 K is negligible.

If the heat exchanger efficiency can be improved, both the optimum operating pressure and the figure of merit will be reduced. In addition, these values can also be decreased through the use of a more efficient expander. On the other hand, if a lower refrigeration temperature is required, the compressor power will be increased.

Any heat leak to the low temperature region of the refrigerator will decrease the amount of useful refrigeration. Thus 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.

32 12. References

Bernatowicz, D. T. (1963), NASA Solar Brayton-cycle Studies, Symposium on Solar Dynamic Systems, Interagency Advanced Power Group,

"Washington, D. C.

Glassman, A. J., R. P. Krebs, and T. A. Fox (1965), Brayton-cycle Nuclear Space Power Systems and their Heat Transfer Components,

American Institute of Chemical Engineers 61, No. 57, 306-314.

Glassman, A. J. and W. L. Stewart (1964), Thermodynamic Character-

istics of Brayton- cycles for Space Power, J. Spacecraft 1, No. 1, 25-64.

Harrach, W. G. and R. T. Caldwell ( 1963), System Optimization of Brayton-cycle Space Power Plants, ASME Winter Annual Meeting, Philadelphia, Pennsylvania.

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.

Roder, H. M. and R. D. Goodwin (1961), Provisional Thermodynamic

Functions for Parahydrogen, NBS Technical Note No. 130,

National Bureau of Standards, Boulder, Colorado.

Schultz, R. L. and W. E. Melber (1964), Brayton-cycle Radioisotope Space Power System, SAE National Aeronautical Space Engineer-

ing and Manufacturing Meeting, Los Angeles, California.

Strobridge, T. R. (1962), The Thermodynamic Properties of Nitrogen

from 64 to 300°K between 0. 1 and 200 Atmospheres, NBS Tech-

nical Note No. 129, National Bureau of Standards, Boulder, Colorado.

33 13. Appendix

The development of (13) proceeds in the following manner. A

series of initial cycle calculations were performed using the basic

equation ( 12)

P RT £n , 1 W a VP c - Q M(h h ) ^ 4 3

Again the output data were plotted as a function of high pressure for

several values of evaporator outlet temperature, and as expected, these

curves also contained minimum points. Therefore, it was again neces-

sary to derive a suitable means of determining their value.

Differentiating (12) with respect to pressure P , and setting the

derivative equal to zero, yields

dh. P 3 1 (h h + 4 " 3> ¥1 d P 1 5

where P now represents the optimum operating pressure.

Because the enthalpy h is not an explicit function of the pressure

P , the derivative (dh /dP ) can not be evaluated directly; hence, an

equivalent relationship must be found.

Taking an energy balance around the heat exchanger (see figure 1) gives

h = + h " h h (21) 2 l 5 4 '

34 Differentiating (21) with respect to pressure P yields

d h. d h. (22) dp J Ld P. i or

d h 2 1 = 1 (23) d h. 4 5

The isentropic efficiency of an expander is given by (14) as

- K2 K3 *n = 6 h h 2" 3-

Solving for h and differentiating with respect to P yields

d h. d h d h.

= *n + (1 - T1J (24) Ld V Ld P ITJ e' Ld P J

Substituting (22) into (24) and (24) into (20) gives

d h d h. 3' (h h + + - Tl - IT (1 ) 4 3> L e Vd P e' \d P 1 T 1 T

x tn = . (25)

Because h is not an explicit function of P , the derivative

(dh /dP ) can not be readily obtained; therefore, some relationship between these two properties remains to be determined. The following functions are known:

35 = h f (p T ) {26) l l' l where P is an independent variable, and T is a constant;

h = g (h h K {27) 2 r *V 5

where h is a dependentr variable, and h „ and h_ are constants: 1 . 4 5

s (h P (28) 2 = j 2' 2>-

where h is a dependent variable, and P equals P , an independent variable; and

h = k(s , , P ), (29) 3 2 3 where s is a dependent variable, and P is a constant.

Differentiating the above four equations, yields

dh dp 30 = • < > i (rp7)„ i

1 d h dh = dh 31 < > z Gnr) i 1 h h 4 5

dS = dh + dP and (32) 2 (^hT)^ 2 2 • 2 P (^)2 hu 2 2

d h 3» dh = ds 33 3' < > (^^) p 2

36 respectively. Substituting (30), (31), and (32) into (33) produces the expression

d h = 3 KS s 2 P

d' h. a h 1 d P + d P • (34) d h J Vd h J va p i \a p 2J 2 P. 1 hh 1 T. 2 h. 2 4 5 1

Since the pressure at station 2 of figure 1 is the same as that at

station 1, d P must equal d P „ Thus (34) becomes

d h a h. d P. a s 2 P

[(£)_(£).. Q) <~ir) (35) 2 P_ 1 h^h. 1 T. 2 h. 2 4 5 1

From basic thermodynamics it is known that

dh = Tds + vdP,

When the pressure is held constant,

' 9 ^ .T (36) and when the enthalpy is held constant,

a s \ ~v 3~py, (37) h

37 If the specific volume-density relationship is considered, (37) becomes

5 s 1 (38) 3 P PT '

Inserting the proper subindices and substituting (23), (36), and (38) into

(35) yields

d h. d h 3'\ 3' 1 1 (39) d L\dP VTj 1 T 1

Finally, substituting (39) into (25), rearranging, and inserting the proper conversion factors, produces

"' (h,-h, )+ ";"%?,{,r n 4 3' f M A r R

11 T d h Tl T_, e 3 ' \ / 1 \ e 3' = , (40) [('-.v-%^) d P P T 1 T 2 2

where the enthalpy h is an explicit function of pressure P , and therefore the derivative (d h /d P,) T can be readily evaluated. 1

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57 2 3 4 6 HIGH PRESSURE, atm.

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60 —

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~~75 3 4 6 8 10 HIGH PRESSURE, atm. Figure 45. Brayton Refrigerator Performance - Helium Refrigerant - 3 00 K Cycle Inlet Temperature - 50% Expander Efficiency - Pressure Drop.~~

~~76 200~~

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~~77 300~~

~~200~~

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~~81 REFRIGERANT: PARAHYDROGEN 300.00 K CYCLE INLET TEMP 100 50 % EXPANDER EFE~~

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3 4 6 8 10 HIGH PRESSURE, atm. Figure 51. Brayton Refrigerator Performance - Parahydrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - Pressure Drop. 22 REFRIGERANT: PARAHYDROGEN 77.36 K CYCLE INLET TEMP. 70 % EXPANDER EFF.

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~~I 2 3 4 6 8 10 HIGH PRESSURE, atm. Figure 52. Brayton Refrigerator Performance - Parahydrogen Refrigerant 77. 36 K Cycle Inlet Temperature - 70% Expander Efficiency - Pressure Drop.~~

~~83 —~~

~~22 REFRIGERANT: PARAHYDROGEN 77.36 K CYCLE INLET TEMR 50 % EXPANDER EFF~~

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~~84 <~~

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5 7 9 II 13 15 17 19 HIGH PRESSURE, atm. Figure 54. Brayton Refrigerator Performance - Nitrogen Refrigerant - 3 00 K Cycle Inlet Temperature - 7 0% Expander Efficiency - 2. K Heat Ex- changer Temperature Difference - Pressure Drop.

~~85 RFFRIGFRANT NITRDfiFN~~

~~1 1.0 300.00 K CYCLE INLET TEMR 50 % EXPANDER EFF~~

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~~86 refrigerant; NITROGEN 300.00 K CYCLE INLET TEMR 70% EXPANDER EFE~~

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12 20 28 36 44 52 60 68 HIGH PRESSURE, atm, Figure 57. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K Cycle Inlet Temperature - 50% Expander Efficiency - 4. K Heat Ex- changer Temperature Difference - Pressure Drop.

~~88 1~~

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~~89 DCFBIftCDAMT i uiTRnriFM 300.00 K CYCLE INLET TEMP 50 % EXPANDER EFF.~~

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~~90 RFFRIfiFRANT NITRDftFN 300.00 K CYCLE INLET TEMR 70 % EXPANDER EFE~~

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~~92 45~~

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~~97 ti> REFRIGERANT! PARAHYDR0GEN 90% EXPANDER EFF.~~

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