an analysis of the brayton cycle as a cryogenic ... - nist
TRANSCRIPT
jbrary, E-01 Admin. BJdg.
OCT 7 1968
NBS
366
An Analysis of the Brayton Cycle
As a Cryogenic Refrigerator ITATIONAL BURSAL'OF iTANDABD8
LIBRABT
MAR 6 J973
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v S. DEPARTMENT OF COMMERCENational Bureau of Standards
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NBS TECHNICAL NOTE 366
ISSUED AUGUST 1968
AN ANALYSIS OF THE BRAYTON CYCLE AS ACRYOGENIC REFRIGERATOR
R. C. MUHLENHAUPT AND T. R. STROBRIDGE
Cryogenics Division
Institute for Basic Standards
National Bureau of Standards
Boulder, Colorado 80302
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TABLE OF CONTENTS
PageABSTRACT 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 BraytonRefrigeration Cycle „ 5
Figure 2. Temperature-Entropy Diagram of the ModifiedBrayton Refrigeration Cycle 6
Figure 3. Temperature-Entropy Diagram of an Ideal BraytonRefrigerator 7
in
TABLE OF CONTENTS (CONTINUED)
PageFigure 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 WorkingFluid - Single and Double Minimum Curves . 25
Figure 8. Brayton Refrigerator Performance - HeliumRefrigerant - 300 K Cycle Inlet Temperature -
100% Expander Efficiency - No Pressure Drop 39
Figure 9« Brayton Refrigerator Performance - HeliumRefrigerant - 300 K Cycle Inlet Temperature -
90% Expander Efficiency - No Pressure Drop 40
Figure 10. Brayton Refrigerator Performance - HeliumRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - No Pressure Drop ...... 41
Figure 11. Brayton Refrigerator Performance - HeliumRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - No Pressure Drop ...... 42
Figure 12. Brayton Refrigerator Performance - HeliumRefrigerant - 77. 36 K Cycle Inlet Temperature -
100% Expander Efficiency - No Pressure Drop ..... 43
Figure 13. Brayton Refrigerator Performance - HeliumRefrigerant - 77. 36 K Cycle Inlet Temperature -
90% Expander Efficiency - No Pressure Drop 44
Figure 14. Brayton Refrigerator Performance - HeliumRefrigerant - 77. 36 K Cycle Inlet Temperature -
70% Expander Efficiency - No Pressure Drop 45
IV
TABLE OF CONTENTS (CONTINUED)
Page
Figure 15. Brayton Refrigerator Performance - HeliumRefrigerant - 77. 36 K Cycle Inlet Temperature -
50% Expander Efficiency - No Pressure Drop . . . . . 46
Figure 16. Brayton Refrigerator Performance - HeliumRefrigerant - 20. 27 K Cycle Inlet Temperature -
100% Expander Efficiency - No Pressure Drop .... 47
Figure 17. Brayton Refrigerator Performance - HeliumRefrigerant - 20. 27 K Cycle Inlet Temperature -
90% Expander Efficiency - No Pressure Drop ..... 48
Figure 18. Brayton Refrigerator Performance - HeliumRefrigerant - 20. 27 K Cycle Inlet Temperature -
70% Expander Efficiency - No Pressure Drop 49
Figure 19. Brayton Refrigerator Performance - HeliumRefrigerant - 20. 27 K Cycle Inlet Temperature -
50% Expander Efficiency - No Pressure Drop ..... 50
Figure 20. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 300 K Cycle Inlet Temperature -
100% Expander Efficiency - No Pressure Drop .... 51
Figure 21. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 300 K Cycle Inlet Temperature -
90% Expander Efficiency - No Pressure Drop 52
Figure 22. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - No Pressure Drop 53
Figure 23. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - No Pressure Drop 54
Figure 24. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 77. 36 K Cycle Inlet Temperature -
100% Expander Efficiency - No Pressure Drop .... 55
TABLE OF CONTENTS (CONTINUED)
Page
Figure 25. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 77. 36 K Cycle Inlet Temperature -
90% Expander Efficiency - No Pressure Drop 56
Figure 26. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 77. 36 K Cycle Inlet Temperature -
70% Expander Efficiency - No Pressure Drop 57
Figure 27. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 77. 36 K Cycle Inlet Temperature -
50% Expander Efficiency - No Pressure Drop 58
Figure 28. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
100% Expander Efficiency - 2 K Heat ExchangerTemperature Difference - No Pressure Drop 59
Figure 29. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
90% Expander Efficiency - 2 K Heat ExchangerTemperature Difference - No Pressure Drop ..... 60
Figure 30. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature-70% Expander Efficiency - 2 K Heat ExchangerTemperature Difference - No Pressure Drop 61
Figure 31. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - 2 K Heat ExchangerTemperature Difference - No Pressure Drop ..... 62
Figure 32. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
100% Expander Efficiency - 4 K Heat ExchangerTemperature Difference - No Pressure Drop ..... 63
VI
TABLE OF CONTENTS (CONTINUED)
Page
Figure 33. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
90% Expander Efficiency - 4 K Heat ExchangerTemperature Difference - No Pressure Drop 64
Figure 34. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - 4 K Heat ExchangerTemperature Difference - No Pressure Drop 65
Figure 35. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - 4 K Heat ExchangerTemperature Difference - No Pressure Drop 66
Figure 36. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
100% Expander Efficiency - 6 K Heat ExchangerTemperature Difference - No Pressure Drop 67
Figure 37. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
90% Expander Efficiency - 6 K Heat ExchangerTemperature Difference - No Pressure Drop 68
Figure 38. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - 6 K Heat ExchangerTemperature Difference - No Pressure Drop 69
Figure 39. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - 6 K Heat ExchangerTemperature Difference - No Pressure Drop ...... 70
Figure 40. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
100% Expander Efficiency - 10 K Heat ExchangerTemperature Difference - No Pressure Drop 71
vn
TABLE OF CONTENTS (CONTINUED)
Page
Figure 41. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
90% Expander Efficiency - 10 K Heat ExchangerTemperature Difference - No Pressure Drop 72
Figure 42. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - 10 K Heat ExchangerTemperature Difference - No Pressure Drop 73
Figure 43. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - 10 K Heat ExchangerTemperature Difference - No Pressure Drop 74
Figure 44. Brayton Refrigerator Performance - HeliumRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - Pressure Drop. 75
Figure 45. Brayton Refrigerator Performance - HeliumRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - Pressure Drop. . „ 76
Figure 46. Brayton Refrigerator Performance - HeliumRefrigerant - 77. 36 K Cycle Inlet Temperature -
70% Expander Efficiency - Pressure Drop . 77
Figure 47. Brayton Refrigerator Performance - HeliumRefrigerant - 77. 36 K Cycle Inlet Temperature -
50% Expander Efficiency - Pressure Drop 78
Figure 48. Brayton Refrigerator Performance - HeliumRefrigerant - 20. 27 K Cycle Inlet Temperature -
70% Expander Efficiency - Pressure Drop 79
Figure 49. Brayton Refrigerator Performance - HeliumRefrigerant - 20.27 K Cycle Inlet Temperature -
50% Expander Efficiency - Pressure Drop 80
vm
TABLE OF CONTENTS (CONTINUED)
Page
Figure 50. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - Pressure Drop. ....... 81
Figure 51. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - Pressure Drop 82
Figure 52. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 77. 36 K Cycle Inlet Temperature -
70% Expander Efficiency - Pressure Drop 83
Figure 53. Brayton Refrigerator Performance - ParahydrogenRefrigerant - 77. 36 K Cycle Inlet Temperature -
50% Expander Efficiency - Pressure Drop 84
Figure 54. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - 2. K Heat ExchangerTemperature Difference - Pressure Drop 85
Figure 55. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - 2. K Heat ExchangerTemperature Difference - Pressure Drop 86
Figure 56. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - 4. K Heat ExchangerTemperature Difference - Pressure Drop 87
Figure 57. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - 4. K Heat ExchangerTemperature Difference - Pressure Drop 88
Figure 58. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Oycle Inlet Temperature -
70% Expander Efficiency - 6. K Heat ExchangerTemperature Difference - Pressure Drop. 89
IX
TABLE OF CONTENTS (CONTINUED)
Page
Figure 59. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - 6.0K Heat ExchangerTemperature Difference - Pressure Drop 90
Figure 60. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
70% Expander Efficiency - 10.0 K Heat ExchangerTemperature Difference - Pressure Drop 91
Figure 61. Brayton Refrigerator Performance - NitrogenRefrigerant - 300 K Cycle Inlet Temperature -
50% Expander Efficiency - 10. K Heat ExchangerTemperature 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 - ParahydrogenRefrigerant - 100% Expander Efficiency - NoPressure Drop . . , . 97
Figure 67. Expander Performance - ParahydrogenRefrigerant - 90% Expander Efficiency - NoPressure Drop 98
Figure 68. Expander Performance - ParahydrogenRefrigerant - 70% Expander Efficiency - NoPressure Drop 99
TABLE OF CONTENTS (CONTINUED)
Page
Figure 69. Expander Performance - ParahydrogenRefrigerant - 50% Expander Efficiency - NoPressure 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
XI
AN ANALYSIS OF THE BRAYTON CYCLE AS ACRYOGENIC REFRIGERATOR
R. C. Muhlenhaupt and T. R. Strobridge
The performance of a Brayton-cycle refrigerator has beencomputed 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'
s
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 ex-
pander efficiency, cycle inlet temperature, evaporator outlet temper-
ature, 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 between1-5
the fluid streams at warm end
of heat exchanger (T - T ) ( K)15W /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 and3
1.0 atmosphere (ft /min)
3v specific volume (cm /g)
v , specific volume of the gas atstd p 5
273. 15 K and 1. atmosphere (cm /g)
T\ isothermal compressor
efficiency (unitless)
Q unit of refrigeration (W)
T| isentropic expander efficiency (unitless)
p density (g mole/-t)
101.3278 a multiplication factor (J/^«atm)
30.002119 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 tem-
perature-entropy coordinates in figure 2. The thermodynamic processes
involved are isobaric cooling between stations 1 and 2, adiabatic expan-
sion 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'
s
ideal refrigeration cycle. The main difference between the two figures
occurs in the ideal compression and expansion processes which are isen-
tropic 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 refriger-
ation 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 notisothermal, indicating that a finite temperature difference will exist inthe heat exchanger of an actual refrigerator. It is assumed that thetemperature of the fluid at the inlet of the compressor will be the same asthe discharge due to heat transfer with the surroundings through the inter-connecting piping.
w,
COMPRESSOR
W.-*/\ EXPANDER'e
EVAPORATORWWWFigure 1. Schematic Flow Diagram of the Modified Brayton
Refrigeration Cycle.
LjJ
(ZZ>H<DCUJ
UJ
ENTROPYFigure 2. Temperature-Entropy Diagram of the Modified
Brayton Refrigeration Cycle.
LU(TZ)H<(ZLUQ_
LU
ENTROPY
Figure 3. Temperature -Entropy Diagram of an Ideal BraytonRefrigerator.
engine. The power required for the compressor was in all instances cal-
culated from the relation for the 100 percent efficient isothermal com-
pression 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
T4
8-100 K (helium)
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,
PRT !n(--)W a \P_ /
c 5
Q Tl C (T - T.)c p 4 3
(1)
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 proper-
ties must be included. An expression equivalent to (1) is
wc
Pk" X
a VP5
/
Q P -k " 1
^cl^eL 1 -^) j[ T4+AT
l-5J-iT
l-5}5
(2)
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
wQ
d P I
T T] -illa c k {\b-(ir)
k
]T4+AT
1-5J- AT
,,}
1 Tk-1
1
\2
^{\[-(p:)k
]T.+AT,
r_ 4 1 - 5_
AT !-»}'
T 1] T] -^- ( T A+ AT
1a c e k \ 4 1-5/ V P
2k- 1P P
1 \ k("p:Mt£
k-1
IN k
k-1
H P, nc lie 1 - T
A+ AT
! C_ L 4 l-5_
- AT ,-J
which becomes
WQ
d P.
TaV T
4+ AT
l-5 ) {k ~ 1]
Tl P. kc 1
ATk-1
1-5
V T4+AT
l-5>kte) 1+ In
k-1
1 \ k
]}(3)
k-1
i\b-te) T + AT AT \1-5 J 1-5J
For simplification, two terms in the numerator are defined as
follows:
ATA = 1 ,-
1-5
ne(T4+ATl _ 5
)
and
k-1 k-1
B =(p7) '
[— (p1
)
k]
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 P15 1
approaches infinity).
At a pressure P equal to P , the term B equals 1. For this1 5
condition only, (3) becomes
/W
\
rdhf)i T
a(k - 1}
Qd P
T \ kATl-5
Pl
(4)
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
until some pressure P w .» is reached where A and B are 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 fur-
ther, 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
uso
+J nnJ •rH
u ^V 14
M•i-i
£<+H
a; <u
«
8•M
TO
>> rn
(D Oi
.—
1
a) nJ
^ <v4-> X)<-H
t—
1
a
s0)
1rt
•i-i 0)
Q <J
ooj
-t->
oi a
a<i)m
43
co Ph
^'
<D
U3bfl
0/ 3M
12
AT. . ,P..-^± .P.k" X
I1
-11 (t'+'aT, J-Ip
1) [
1+tn(p:) ]
U °-< 5 >
Tl (T +4T ) ..... -5
The optimum pressure P was then calculated through an iteration
process for several sets of P , T , and AT for the ideal gas refriger-
ants, 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
T2" T3
Tl =L *
, (6)e 1 ^ - 1 „.
and the temperature-pressure relationship for an isentropic expansion
is
k-1T P2 , 2 M,
In addition
T \P3 1 3'
T2=T
4+AT
l-5(8)
for an ideal gas. Substituting (8) and (7) into (6) and rear ranging> gives
13
p -i^t3 = <t4
+ati- 5){\[(p:)
- 1 ]" 1 )- (9)
5
where Pnequals P and P equals P . The evaporator inlet tempera-
1 CD J
tures were calculated from (9) using the optimum high pressures deter-
mined 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 condi-
tion, 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
/T4
T InW a \ Tc 3
'
W^ZL-T,,) ' 10)
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 Tc a
~Q" =Tl Tl t '
(H)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.
14
:
7. Real Gas Calculations
The analysis that follows assumes no pressure loss in the system;
thus,P_ = P, and P_ = P = P n = one atmosphere. In addition T ,, T]
, and2 13 4 5 r
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 RTa^(p;(12)u n
cm <h
4. h
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,
,
-i
101;!
2779 )(p.u.P
'
4 3'\ M / \ 1 P
T] T dh T] Tf / „ e 3 ' \ / 1 \ e 3
x!
i - 1) +i -i
t2/id^^ p
2t2
= . (13)
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 tem-
peratures at stations 2 and 3' are determined in the course of evaluating
(13), the corresponding enthalpys are known. The enthalpy, and there-
fore the temperature, at station 3 is readily determined from the defini-
tion of efficiency
h - h
e h2- h3-
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 tempera-
ture, and expander inlet pressure. These values were calculated directly
from
W h - h
(15)
for each refrigerant.
F 0.002119 v ,
std
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 figureJ. 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
Wr <nL~o"JPD
W
Q JNPD
'{^eL 1
k-1
NPD
{T_ 1)
2 e1 -
k-1
PD
(16)
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
d
Wf—E-\ Q
L d P1 T
= (h4 " h
3 )+
101. 32779M p
i
tep;
1
1 - T\ +e
H Te 3
'
T~
d h.
d P.
\ T3<
2T2
d P.
d P1 T
= (17)
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 refrig-
erants 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 tempera-
ture difference between the fluid streams at the warm end of the heat
exchanger AT1-5
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 evapor-
ator inlet temperature T3
» the evaporator outlet temperature T , the
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 inlete £
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 pres-
sure 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 con-
ditions discussed in this work).
10. Discussion of Results
The optimum pressure charts for helium show reasonable agree-
ment 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 para-
hydrogen demonstrate the difference between the properties of para-
hydrogen and the ideal gas.
The discontinuities on the 300. K cycle inlet temperature para-
hydrogen 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
WRTQ
opt
d(P1
)
opt
A + B + C + D_ opt opt opt opt _
M (P,)2
.A 2
1 opt opt
(18)
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 , and1- 5
a 100 percent expander efficiency (to simplify the equation) are defined
as follows:
RT /M =a
a constant (approximately 1237),
Aopt
T
opt
d h.
LVdP +
1 T
d h
d P- v.
1 T opt
opt= p
3'
opt
,2 2d h d h
(—,»)+
( tdP, T, d P,
1 1 1
T
d v.
d P1 T opt
C = Popt 1 VT
d T3»
2 ' opt *
d Pl ' T
d h.
[y +
1 T
d h4
- v.d P, ^ 2JIT opt
, and
Dopt
= Pd T.
1 \T J Vd Pn2 opt IT
d h.
d P1 T
d h
d P- v
1 T2 J
opt
It is difficult to obtain precise values for
d h
d P1 T
andd T
d2 h
dP T1
because neither h nor T is an explicit function of P ,IT 5 l
20
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
2dh
4for the discontinuities. The two derivatives {—r——- ) and (
—) are
^ d P, 'rr, ^ d P^rrnIT IT
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 Tthe derivative ( —
J
is negative (the temperature T decreases1 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 thel(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
rt di-t u
•Hum
a
Hi H«
>>(1)
0) 1—
1
h 43
PQ
<u Q3
nJ
n &0fl
a "S«$ Oh 43Men<9
Q oa
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 the1-5
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 less1 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-
therms reach a maximum value, at which point the derivative 1 , _VdPl'T
1
must go to zero. For this special condition, and this condition only,
d Tthe Joule-Thomson coefficient f j will also be equal to zero.
1 hl
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
Curves representing the original figure of merit W /Q . (solid line)r ° ° ° c orig
and the revised figure of merit W /Q (dashed line) are shown in fig-c rev
ure 7.
23
IT)
i
T|-Q <T|-b<T
l-C
<
deal
rl
Figure 6. Schematic Diagram of Enthalpy Difference as aFunction of High Pressure for Parahydrogen.
24
0/°M
25
«joQ
<L> ID
Pi U* .s° CO
d i
bo -d
h r1
«
o
«O
q•wXUo
CO
CO
u
CO
>
u
"i H 3
e a g
U
fa
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 pres-
sure 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 pressurec rev
is increased further, the new enthalpy difference reaches its maximum
value causing W /Q to be considerably less than W /Q . . In thisc rev c orig
pressure region (point D) the revised W /Q curve can be expected toc rev
have another minimum or, at the very least, an inflection point. As the
pressure is increased further, the new enthalpy difference begins to de-
crease. 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 pre-
cooled 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 atmo-
spheres 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
Q
f]
c isothermal
e isentropic
1
helium
100 W60%
70%
300 K
30 K
296 K
0.9 P1
1.2 P,
1.2 P,
1. atm
and
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
Wc isothermal
WF
Q
e isentropic
35 K
21 K
11 atm
9 . 9 atm
1 . 2 atm
1 . 2 atm
5.33 kW
150 W25 scfm
100 Wand
In order to illustrate the effect of component efficiencies on the
performance of the cycle, consider decreasing AT (i.e., increasing1-5
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, willc
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 expander1-5 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 . is 20 K, T. is 300 K, AT, _ is 6 K, T] is 50 percent, T\ is 60 percent,4 1 1-5 e c
and the pressure drop in the heat exchanger is 10 and 20 percent, res-
pectively, 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 atmc
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
AT1_5
= 4KT] = 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 refrig-
eration 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 tem-
perature.
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 tempera-
ture 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
1RT £nW a VPc
Q ^ M(h4
- h3
)
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
(h4 " h
3> ¥1+
dh. P3 1
d P1 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
h2
= hl
"h5+ h
4 '(21)
34
Differentiating (21) with respect to pressure P yields
d h.
dpiJ
d h.
Ld P.(22)
or
d h2 1
d h.= 1
4 5
The isentropic efficiency of an expander is given by (14) as
(23)
*n=K - K2 3
6 h2" h3-
Solving for h and differentiating with respect to P yields
d h.
LdV = *n
d h
Ld P JIT+ (1 - T1J
d h.
e' Ld P J(24)
Substituting (22) into (24) and (24) into (20) gives
(h4 - h
3>IT+
d h3'
L e Vd P+ (1 - Tl )
1 T
d h.
e' \d P1 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
hl
= f (pl'
Tl
) {26)
where P is an independent variable, and T is a constant;
h2
= g (hr *V
h5K {27)
where h is a dependent variable, and h „ and h_ are constants:1 .
r 4 5
s2 = j (h
2'P2>- (28)
where h is a dependent variable, and P equals P , an independent
variable; and
h3
,= k(s
2, P
3), (29)
where s is a dependent variable, and P is a constant.
Differentiating the above four equations, yields
dhi= (rp7)„ dp
i• <
30 >
1
d hdh
z= Gnr) dh
i<31 >
1 h4h5
dS2
=(^hT)^ dh
2+ (^) u
dP2 •
and (32)
2 P2
2 h2
d h3»
dh3'
= (^^) pds
2 <33
>
36
respectively. Substituting (30), (31), and (32) into (33) produces the
expression
d h3
=KS s
2 P
d' h. a h1
d h J Vd h J va p2 P. 1 hh 1 T.
2 4 5 1
d P +i \a p
d P2 h.
2J• (34)
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
d P.
a h.
a s2 P
[(£)_(£).. Q) <~ir)2 P_ 1 h^h. 1 T.
2 4 5 12 h.
(35)
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 \ v3~py,
~h
(37)
37
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62
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63
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65
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66
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67
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68
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71
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Figure 46. Brayton Refrigerator Performance - Helium Refrigerant - 77. 36 KCycle Inlet Temperature - 7 0% Expander Efficiency - Pressure Drop.
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83
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Figure 56.
10 12 14 16
HIGH PRESSURE, atmBrayton Refrigerator Performance - Nitrogen Refrigerant - 300 KCycle Inlet Temperature - 7 0% Expander Efficiency -4.0K Heat Ex-changer Temperature Difference - Pressure Drop.
87
II.RFFRIfiFRANIT- NITRDGFN
300.00 K CYCLE INLET TEMP50 % EXPANDER EFF
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20 60 6828 36 44 52HIGH PRESSURE, atm,
Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 KCycle Inlet Temperature - 50% Expander Efficiency - 4. K Heat Ex-changer Temperature Difference - Pressure Drop.
88
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Brayton Refrigerator Performance - Nitrogen Refrigerant - 3 00 K.
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Figure 59.
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changer Temperature Difference - Pressure Drop.
68
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HIGH PRESSURE, atm.Figure 60. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 K
Cycle Inlet Temperature - 7 0% Expander Efficiency - 10. K Heat Ex-changer Temperature Difference - Pressure Drop.
91
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Figure 61. Brayton Refrigerator Performance - Nitrogen Refrigerant - 300 KCycle Inlet Temperature - 50% Expander Efficiency - 10. K Heat Ex-changer Temperature Difference - Pressure Drop.
92
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120
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98
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