NBSIR 73-331

REFRIGERATION OF SUPERCONDUCTING ROTATING MACHINERY

Vincent D. Arp

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

Interim Report

June 1973

Prepared for

Defense Advanced Research Projects Agency

1400 Wilson Boulevard

Arlington, Virginia 22209

NBSIR 73-331

REFRIGERATION OF SUPERCONDUCTING ROTATING MACHINERY

Vincent D. Arp

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

Interim Report

June 1973

Prepared for

Defense Advanced Research Projects Agency

1400 Wilson Boulevard

Arlington, Virginia 22209

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS. Richard W Roberts. Director
FOREWORD

This work was performed under the sponsorship of the Defense

Advanced Research Projects Agency, under Order Number 2186,

beginning April 14, 1972. This report covers the 14-1/2 -month

period from that date to June 30, 1973, The project will terminate

on December 31, 1973, and a separate report will be issued covering

the last 6 -month period. The ARPA program code number is 3D10,

and the program monitor is Dr. E. C. Van Reuth.

The views and conclusions contained in this document are those

of the author and should not be interpreted as necessarily representing

the official policies, either expressed or implied, of the Advanced

Research Projects Agency or the U. S. Government.

ii
CONTENTS

Page

1. 0 Introduction 1

2. 0 High Pressure Helium Refrigeration Cycle For

Superconducting Components 2

2. 1 Basic Concept 2

2. 2 Expansion Engine Inefficiency 2

2. 3 Heat Exchanger Plus Expansion Engine , , , 11

2. 4 Mass Flow Rates 13

2. 5 Conclusion 13

3. 0 Flow of Helium in Cooling Passages 14

3. 1 Time -Dependent One -Dimensional Flow of a Real Fluid 14

3. 2 Steady State Equations 16

3. 3 Integration of the Steady State Equations ... 22

3. 4 Flow Through a Venturi 24

3. 5 The Ledinegg Instability 27

3. 6 High Velocity Flow 28

3. 7 The Polytropic Approximation 32

3. 8 Time Dependent Problems 35

4. 0 Helium State Equations For Refrigeration Analysis . . 36

4. 1 Structure of the Equations 39

4.2 Function T (P,H) 42

4.3 The Function H (P,S) 49

4.4 The Function S (P,H) 51

4. 5 The Two-Phase Region and Boundaries ... 53

4. 6 Accuracy. 55

5. 0 Refrigerator Performance Survey 56

6. 0 Program Relationships and Future Goals 57

iii
CONTENTS (Continued)

Page

7. 0 Nomenclature ajid Thermodynamic Relationship ... 60

7. 1 Nomenclature , 60

7. 2 Thermodynamic Definitions 62

7. 3 Typical Values in Liquid Helium Range ... 63

8. 0 Acknowledgements 66

9. 0 References 67

iv
LIST OF FIGURES

Page

Figure 1. "Engine inversion curves" for expansion

engines of stated efficiency 3

Figure 2. Low -temperature portion of the helium

refrigeration cycle 9

Figure 3. Minimum attainable temperature as a function

of expansion engine outlet pressure, for several

engine efficiencies 10

Figure 4. Specific heat as a function of temperature for

several pres sures 12

Figure 5. Calculation of pre s sure , temperature, and

irreversible entropy increase in helium flow

through a heated channel 23

Figure 6. Calculation of pressure and temperature

variations in an insulated channel of variable

diameter (approximate shape of a venturi). . . 26

Figure 7. Calculation of pressure and temperature

profiles as the helium flow approaches sonic

velocity 29

Figure 8. Critical channel lengths within which the helium

reaches sonic velocity, as a function of helium

velocity at the input. Q is a dimensionless

heating rate defined in the text 31

Figure 9. Difference between the true temperature and

the "ideal gas temperature" at the same enthalpy

and pressure, as defined in the text .... 43

V
LIST OF TABLES

Page

Table 1. dT/dP for expansion engine of 50% efficiency

as function of pressure and temperature .... 5

Table 2. dT/dP for expansion engine of 70% efficiency as function of pressure and temperature .... 6

Table 3, dT/dP for expansion engine of 85% efficiency as function of pressure and temperature .... 7

Table 4. The ratio au / C as a function of pressure and

temperature 18

Table 5. The specific heat ratio C /C as a function of ^ P V

pressure and temperature 19

Table 6. Numerical constcints for the equation T(P,H). . . 48

Table 7. Numerical constants for the equation H(P,S) ... 50

Table 8. Numerical constants for the equation S(P,H). . . 52

Table 9. Numerical constants for the equation H (P)

and H_^JP) 54

vi
REFRIGERATION OF

SUPERCONDUCTING ROTATING MACHINERY

Vincent D. Arp

1. 0 Introduction

This is a report to the Advanced Research Projects Agency of the

Department of Defense for work performed by the Cryogenics Division

of the National Bureau of Standards from April 1972 through June 1973.

Stated very briefly, the work is to assist on problems of refrigeration

and system design of superconducting motors and generators. In

programming this work, it has been necessary to be familiar with and

responsive to related problems within DoD Laboratories, particularly

the Naval Ship Research and Development Center at Annapolis, the

Aero-Propulsion Laboratory at Wright Patterson AFB, and the U. S. Army Mobile Equipment Research and Development Center at Ft. Belvoir.

Section 2, 0 summarizes a study of a conceivable all -high -pressure

refrigeration system which might offer reduced component size and

weight. Section 3. 0 summarizes analytical and computational work

which provides reliable basis for design of cooling channels, passages,

and fluid circuits. Section 4. 0 presents new forms of the equation of

state for helium, arranged so as to greatly speed the calculation of helium

refrigeration and flow problems. Section 5, 0 summarizes survey work

on refrigerator performance. Section 6. 0 summarizes the relationship

of this work to related work within NBS, and discusses future program

plans.

A continuation of this ARPA work is funded from April through

December 1973, and is oriented more than before to evaluation of liquid

helium pump design and testing. This work has begun, but is not

summarized here.
2. 0 High Pressure Helium Refrigeration Cycle For

Superconducting Components

2. 1 Basic Concept

A high pressure refrigerator would offer, in principle, the possibility of reduced size because of high fluid densities throughout all the

flow system. Thus, it may be useful where the refrigerator must be

fitted into a small space, as in mobile applications. Assuming that the

lowest pressure in the cycle is greater than the critical pressure (2. 2

2

atmospheres, or 0. 22 MN/m ), the flow system would not contain any

boiling liquid, and operating characteristics would not depend upon being

above or below the critical temperature (5. 2 Kelvin). This could be aji

advantage if overall system considerations should suggest optimum operation at some higher temperature, as may be feasible with Nb^Sn

superconductors.

We have briefly investigated high pressure helium refrigeration

cycles, since we can find no reference to previous studies. Analytical

and computer techniques are used to compute cycle efficiencies and

characteristics based upon realistic values of expansion engine efficiencies, heat exchanger efficiencies, and a recent NBS compilation of

the real properties of fluid helium [McCarty, 1972].

2. 2 Expansion Engine Inefficiency

The most important initial consideration for possible high pressure,

helium temperature refrigerators concerns the Joule - Thomson coefficient.

The curve labeled 0% in figure 1 is the inversion curve, along which

\j/ = 0. Above the curve, \^ is negative, meaning that an isenthalpic

(Joule - Thomson) expansion will cause the fluid temperature to increase.

It is evident from this figure that a refrigerator cycle using a final J-T

expansion from a high pressure down to about 3 atmospheres (an arbitrary

pressure, a little above the critical pressure) would not be capable of

2
TEMPERATURE, K

Figure 1. "Engine inversion curves" for expansion engines of

stated efficiency. _ ]
refrigerating much below 5 K even at zero thermal load; in fact, subsequent calculations indicate that the minimum temperature would be

closer to 10 K assuming realistic heat exchanger inefficiencies. However, if the final expansion is more nearly isentropic, as in an expansion engine, this limitation would be considerably relaxed. We consider

this alternative in more detail.

Although the isentropic expansion of a real fluid always causes a temperature decrease, the inefficiency in an expansion engine, i. e. , the deviation towards isenthalpic from isentropic performance, must

carefully be considered in determining the temperature decrease which

would be available to the refrigeration in a real system. In section 7, 3

we derive the equation relating the temperature change, dT, to the

pressure change, dP, through an expansion engine:

Expansion engine efficiencies, € , for helium liquefiers are typically

in the range 0. 60 to 0, 85 [Linhart, 1973, and Strobridge, 1973], Isentropic expansion corresponds to € =1,0 and isenthalpic expansion

corresponds to € =0, Tables 1, 2, and 3 give values of dT/dP for

engine efficiencies of 50, 70, and 85% respectively, calculated using

this equation. For any given engine efficiency, then, one can define

an "engine inversion curve, " which is the loci of points at which

dT/dP = 0, These are plotted in figure 1 for representative engine

efficiencies. Such curves have not been presented before to our

knowledge. Just as with J-T expansion, the maximum refrigeration is

obtained by an expansion process in which the initial fluid state (at the

engine input) lies on the engine inversion curve for the given efficiency

[Dean and Mann, 1965].

(I)

4


7
Actually there is a subtlety here which slightly clouds the accuracy

of applying these curves for practical systems. The engine efficiency

is defined implicitly as a function of the entropy at the engine input.

Equation 1, on the other hand implicitly assumes an entropy which is

defined from the pressure and temperature at each increment; in effect,

this means that the efficiency defined for a finite pressure drop through

the engine will not be identical to the incremental efficiency which is

used in defining the inversion curves. However, we have found by trial

that the differences are not large, e. g. , 70% vs 74%, in practical calculations. Hence, the engine inversion curves provide a good semi -quantitative limit to the desirable operating pressure of expajision engines. In

turn, this provides an absolute lower limit to the temperature attainable

with a high pressure helium cycle, independent of other parameters in

the cycle.

To explore this limit quantitatively, assume for the moment that

enthalpy balances in the heat exchangers and other components above the

expansion engine will permit a small value of AT = T^ - T^ to be obtained

at the low temperature end of the heat exchanger illustrated in figure 2.

For our purpose, we assume T - T = 0, 5 K, which is somewhere near

the minimum AT which might be expected in a good heat exchanger. Then

performing a thermodynamic balance on just the circuit between points 2

and 4, and assuming zero thermal load, the calculated minimum possible

T^ is shown in figure 3 as a function of the expansion engine efficiency

and exhaust pressure. The calculations were not carried below three

atmospheres since (1) the hoped-for advantages in size reduction disappear,

and (2) two-phase effects appear (below 2. 2 atmospheres) which would

have required a more elaborate computation. Considering that expansion

engine efficiencies above about 80% are difficult to achieve in practice,

the important conclusion from figure 2 is that the all-high-pressure cycle

is not feasible for 4 K refrigeration, though it could be for 6 or 8 K

8
P|.T| P5.T5

P4 •"'^4

P3.T3

t

Figure 2. Low-temperature portion of the helium refrigeration cycle .

9

It- 1038I5~|
10
as judged by this calculation. To explore this further, we must consider

the heat-exchanger and expansion engine combination as in the complete

figure 2,

2. 3 Heat Exchanger Plus Expansion Engine

Within the heat exchanger, the local temperature difference

between the high and low pressure streams varies with position, due to

the pressure and temperature dependence of the specific heat, as seen

in figure 4, Accordingly, there is a minimum temperature difference

(which must be greater than zero, from the second law of thermodynamics), somewhere within the heat exchanger. It is the minimum

temperature difference which typically may be 0. 5 K, but it usually does

not occur at the low temperature end of the heat exchanger, as was

assumed in the last section. As - increases from the require2 4

^ ments of enthalpy balance within this heat exchanger, the minimum

attainable refrigeration temperatures will rise above the values determined in the previous section.

From numerical calculations, assuming T^^ is more than just a few degrees above T , for most conditions the minimum difference in

^ 2'

T occurs as the top of the heat exchanger, and T - T may be as much

as 3 or 4 K larger than this minimum. However, it turns out that within

a restricted range of high pressures, which happen to be of interest here,

viz. , P,12% P_, % 30 atmos-oheres, "45P . P^ ^ 5 atmospheres, T 5 to 8 K,

4

and 30 K, the minimum AT does occur at or close to the low temperature end of the heat exchanger. For this parameter range the earlier

assumption of T - T = 0. 5 K would be realistic. Also it might be

possible to achieve this assumed value for a wider range of conditions by

adding a second expansion engine between points 1 and 4, thus unbalancing

the heat exchanger mass flow rates. We have chosen not to get into a

detailed mapping of these effects, since it would carry us beyond the

intended scope of inquiry.

11

2,4 Mass Flow Rates

With a non-zero thermal load, Q, the minimum attainable temperature will be higher than in figure 3 by an amount which depends

on further cycle details. The ratio of the mass flow rate to the

developed refrigeration is

m _ I rv. I g

Q 8(7^-13) W.s.

where this last equality assumes refrigeration temperatures in the

range 6 to 8 K and ^"^4 ^ atmospheres. The magnitude of

T^ - T^ will be limited by two factor (a) expansion engine inefficiencies,

limiting T^ - T^, as discussed earlier, and (b) the desirability of keeping

T^ - T^ small, since the load (superconductor) will experience temperatures up to T . 4

The corresponding value of m/Q for the conventional 4 K refrigerator with Joule - Thomson expansion to 1 atmosphere depends strongly

upon the precooling temperature T^ , but typically can be

«0.l to 0.5

Q W.s.

[Dean and Mann, 1965 ]. Without going into more detailed comparisons,

beyond the scope of this work, we can say that the required mass flow

rate for possible high pressure cycles would probably not be greatly

different than that for the conventional cycle provided that T - T ^ ^ ^ 4 3

0. 3 to 1. 0 K.

2. 5 Conclusion

We have not carried out the analysis in further detail, since the

cumulative factors indicate that the high pressure cycle would not be

13
practical at 4 K where most superconductors are designed to operate.

If expansion engines of 80% efficiency or more were used, this work

suggests that it could be engineered to produce useful refrigeration in a small package for temperatures above 7 or 8 K.

3. 0 Flow of Helium in Cooling Passages

Refrigeration of superconducting components generally involves

flow of helium through cooling passages, ducts, heat exchangers,

etc. One direction of our work has been to set up the mechanics so

that one can evaluate pressure and temperature profiles in such

ducts as a function of distributed thermal loads and variable channel

dimensions, etc. Because the temperature range of interest for

superconducting components, roughly 3 to 8 Kelvin, spans the transposed critical line, neither the incompressible liquid nor ideal gas

approximations to fluid behavior can be used in this type of analysis.

Unfortunately, almost all textbooks and papers on these topics are

based upon one or the other of these approximations, so that it has

been necessary for us to develop the appropriate general thermohydrodynamic equations from first principles. Some of this work

was reported in an earlier NBS report . For this report we have

extended the earlier work considerably.

The basic goal has been to develop general analytical and computer routines for design of helium cooling systems. Copies of the

computer programs have been delivered upon request to two National

Laboratories, Brookhaven and Los Alamos, and we invite direct

communication from other groups who may be interested in them.

3. 1 Time -Dependent One -Dimensional Flow of a Real Fluid

We present here the equations for fluid flow in a pipe or conduit

of cross section 'a' in which all radial fluid motions and radial

14
temperature gradients are neglected. This is a reasonable approximation for a large class of fluid flow problems in engineering design.

We do allow the conduit cross section to be a slow function of position, so that we can explore the fluid changes through Venturis and other

smooth restrictions, but we neglect changes in elevation. The equations presented here are based on the authoritative work of Bird,

Stewart, and Lightfoot (BSL) [1960].

Conservation of mass requires that

dm

dz

= - a

at ' (2)

and the momentum equation requires that

dP ^ DU

(3)

where F measures the volume force due to fluid friction, customarily

written for simple one -dimensional flow in terms of an empirical

friction factor f:

'-fa' u f (4)

Conservation of energy requires that

where T is the stress tensor defined by BSL, and the termT^VU

is the irreversible rate of conversion of mechanical energy into

15
internal energy of the fluid due to viscous effects. Using a dimensional

argument (which I find less than wholly convincing) BSL show, in effect,

that for our case of one dimensional flow in a conduit the average of

this rate of conversion is just equal to the average of the volume frictional force times the fluid velocity:

T :VU = FU (6)

M. C. Jones of this laboratory has derived this equality for the case

at hand using a more rigorous analytical technique (unpublished). Using

this equality, and eqs (2), (3), and (4), we can finally obtain the desired

expression

which does not explicitly contain frictional effects. The steady state

form of this expression was quoted in an earlier paper [Arp, 1972].

2 The quantity H + 1/2 U is known as the stagnation enthalpy.

In order to apply these equations to practical problems, one must

use them in conjunction with an equation of state for the fluid, giving

relationships between enthalpy, density, pressure and/or other thermodynamic properties.

3. 2 Steady State Equations

A major milestone of our work has been to formulate and program

the above equations for steady state studies without any approximations

on the fluid properties, valid for all velocities up to the velocity of

sound. They may also be valid for velocities greater than sound

16
velocity except for empirical friction factors and boundary layer heating

effects, which can be neglected anyway for some problems, though we

have not investigated this area. The work proceeds by performing the

indicated differentiation on eq (6) and relating derivatives to the independent thermodynamic variables as in the following example

dH _ dH

dz " dP

dP _^ aH

T dz dJ p dz

, etc.

In the course of performing this work, certain thermodynamic

identities turn out to be very useful. They are given as an addition to

the nomenclature section for convenience. Note in particular the

velocity

(X) = a

which might be called an expansion velocity, to distinguish it from

the sound velocity c. We have found for helium that the ratio of this

expansion velocity to the sound velocity is about equal to one and is

remarkably independent of pressure and temperature over the whole

range of fluid behavior, as can be seen from table 4. This fact can

be used to obtain order-of-magnitude "feel" for the importance of

various terms in the flow equations. Also for convenience, the value

of X = C /C is given in table 5. p V

An earlier NBS Report, (unpublished) presented equations for pres

sure and temperature gradients. However, they contain a small error

which becomes important as the velocity approaches sound velocity,

and they assume a pipe of constant diameter. Also, because their

formulation does not utilize (aJ or the dimensionless Mach number, the

17
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relative numerical magnitudes of the various forms are not as obvious

as in the formulation below. Leaving out many tedious steps, the results

of these corrected, more general calculations are:

(l-IVI^)^= -^/>u''ef..^(l-M^-N2) (10)

(i.M2)i£ = -^G^ef,-^ (II)

dz 2a ® mcj^

dz 2o ^ Qpu)^ a dz

where A s A + GU^-^. (13)

and f,^ iLf-^ aa (14)

® lUI p dz

20
We can find no previous publication of these exact expressions.

Especially worth noting are the effective friction factor, f , and the

effective heat input per unit length, Ag , defined by eqs (13) and (14).

Using these definitions, the correct thermodynamic and hydrodynamic

profiles of flow through Venturis, nozzles, etc. , can be obtained

by merely substituting these quantities for the usual f and A appearing

in conventional one -dimension equations presented here or elsewhere.

We must point out, however, that since the above equations neglect

energies and momenta perpendicular to the channel direction, such

substitution can be accurate only for slow channel variations, i. e„ only when

P <5a

4a dz

It is an easy illustrative exercise to derive Bernoulli's equation

Pg- p, + i^(u|-uf) = 0

from eq(8)by putting f = 0, assuming p = constant, approximating

2

1 -M ^ 1, and integrating between two fixed flow cross sections.

Parenthetically, eq(12)is correct as written, without the subscript e on f and A.

2 The factor 1 - M which appears in these expressions deserves

separate attention. It gives rise to infinite gradients at Mach 1, corresponding to a shock front. However, the Mach number

21
must be evaluated at the temperature of the moving stream, which maybe significantly different than the stagnation temperature which the fluid

would have if brought to rest adiabatically. From eq (9), neglecting

friction (f = 0) and/or when the term( + M^/PCy ) equals zero, it

can be seen that when M is greater than \ /*/y , adding thermal

energy to the moving stream causes a decrease in its stream temperature, and vice versa, whereas the stagnation temperature always

increases with added thermal energy. The divergence of the flow equations as the velocity approaches Mach 1 has also been explored by

Chive rs and Mitchell ^1971] using an ideal gas approximation. Our

result is in accord with theirs,

3. 3 Integration of the Steady State Equations

We have developed, tested, and used computer programs which

integrate these general steady- state equations along a flow channel

of specified geometry, using the properties of helium obtained from

McCarty's equations [1972, 1973] at every integration step. Using

predictor- corrector techniques, the integration steps are internally

adjusted to hold the integration accuracy within specified bounds. We

give here an example to show the general versatility of the program,

without detailed elaboration.

Figure 5 shows the profiles along a coolant passage of constant

diameter, but with varying heat input. In this particular example,

the pressure drops to the point that vapor would be developed within

the flow channel about 60 meters from the inlet. Since the equations

are valid only for single phase flow, the program stops at that point.

The temperature difference between the helium and the heated wall is

calculated using the correlation developed by Giarratano, et al, ri971].

22
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23
The irreversible entropy increases due to fluid friction and due to

the temperature difference at the wall, columns 12 and 13, are

important for analysis of the overall system refrigeration requirement. The stagnation enthalpy (next to last column) should be

constant over the last half of the pipe length where no heat is added,

even though the pressure drops due to friction and the temperature

drops due to yj; ; however, the printout shows it to be slowly

decreasing. Apparently this error, which is larger than most,

is due to residual inaccuracy in the equation of state as the phase

boundary near the critical point is approached, plus integration

inaccuracy when the specific heat changes rapidly. The last column

in the printout provides a thermodynamic consistency check, and

is the difference between the integral of the heat added per unit

length and the change in stagnation enthalpy over the stated length.

3. 4 Flow Through a Venturi

One question which we have been concerned with is the

measurement of helium flow in the near critical range through the

use of Venturis. Bernoulli's equation, as given above, is strictly

true only for incompressible fluid flow through a venturi. For

ideal gases, having a constant specific heat ratio y , charts and

tables prepared under ASME auspices are commonly used to

provide corrections to the simple Bernoulli expression. These

corrections are typically a few percent in magnitude. We were

concerned that the corrections due to real fluid properties in the

near critical range might be a factor of ten larger, since compressibilities and thermal expansion coefficients can be larger

than their ideal gas values by roughly this factor, as seen in

section 7. 3.

24
Figure 6 shows the profiles through an insulated venturi, with

the added artificial assumption that the friction factor equals zero. From basic thermodynamics, such flow should take place at constant

entropy and constant stagnation enthalpy. The program shows this

to be the case, aside from very small errors, even though the pressure and temperature change is very significant in this near critical

range. The residual errors are most probably due to slight inaccuracies

in the equation of state in this range; the integration error is probably

very small since the fluid velocity, at the exit, for example, is

almost identical to the velocity at the inlet, after integration through

all the venturi length. In this particular example, the calculated

pressure difference of 0. 991 atmospheres (between entrance and

center of the venturi) is only 4. 8% less than p(U

2

- U2.), using

3

p = 0. 08005 g/cm which is the density at the inlet. This is a comfortably small correction considering the close proximity to the

critical point (2. 25 atmospheres and 5. 2 Kelvin) and the fact that

the Mach number at the center reaches 0. 53. We have performed

such integrations for several fluid conditions and geometries, and

concluded that the errors due to using the incompressible fluid

Bernoulli equation even quite near the critical point will not be

greater than about 5% for any practical flow condition through Venturis.

Because errors of this magnitude are not immediately important

for the project at hand, we have decided not to take the time to quantify the corrections in some systematic way, e. g. , the ASME ideal

gas corrections. However, sometime in the future this could become

part of an interesting study in accurate, real-fluid metrology.

25
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3. 5 The Ledinegg Instability

A number of possible instabilities in helium flow systems were

outlined and discussed by Steward (unpublished report). Of these, the

Ledinegg instability has been studied directly through the use of our

program.

The Ledinegg instability refers to a situation which can occur when

the fluid enters the heated passage as a dense liquid, with density almost

independent of enthalpy, and picks up enough heat while flowing through

the channel to emerge from the downstream end in a gas -like state, with

a density approximately inversely proportional to enthalpy. For a given

mass flow, the gas -like fluid will be moving at relatively high velocity,

giving rise to a locally high pressure gradient due to fluid friction. As

a result, it is possible that the total pressure drop across the pipe, i. e. , the integral of the pressure gradient along its length, could increase if

the mass flow is decreased such that a longer fraction of the pipe length

contains fluid in the gas -like state [Ledinegg, 1954, and Zuber, 1966].

In effect, such a heated pipe would have a negative resistance characteristic over at least a portion of its flow range. It is well known that a component with negative resistance characteristic will promote unstable

system operation.

Most prior studies have been concerned with two-phase fluid flow,

wherein a significant quantity of vapor has been formed from boiling

liquid within the tube. For a supercritical fluid, the analogous situation

is for the thermodynamic state of the fluid to cross the transposed critical line between entrance and exit from the heated flow channel. Prior

analyses for this latter case have used crude approximations to the equation of state for the fluid, resulting in the qualitative prediction that

Ledinegg instability may likewise occur with a supercritical fluid. Zuber

27
further shows graphs from experimental work of Krasicikova and

Glusker ri965] and Semenkover [1964] with supercritical water supporting this contention, though experimental scatter in the data leaves some

room for question.

Our computer program allows a significant advance in the analysis

of the supercritical Ledinegg instability, specifically because it uses

the exact equation of state of the fluid. We have applied it to the pariicular case of the superconducting power line, using funding from Broolhaven

National Laboratory, and subsequently have been continuing the study in a more general sense under the ARPA funding. One would expect that the

occurrence of the instability would become more likely as the integration

path on the P - T plane crosses the transposed critical line closer to the

critical point, since the transition from liquid-like to gas -like behavior

becomes more sharp under these conditions. Surprisingly to us, we

have not yet been able to find any conditions of flow, heat flux, input

pressure and temperature, channel diameter, etc. , which leads to a negative resistance characteristic.

Our analysis will be completed within the next three months anci

published separately. At this time we can say that the Ledinegg inst ibility

should not occur for supercritical helium as easily as previously supposed,

and tentatively we feel that it would not be seen outside of the two-phase

flow regime.

3. 6 High Velocity Flow

While testing our computer-generated solutions to the flow equations, we were surprised at the number of times which seemingly re isonable input conditions led to sonic velocity not too far down a pipe of constant diameter. An illustrative output from one such computer run is

shown in figure 7; actually it is a fine-grained integration of the tail

28
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29
end of a previous run using a much more coarse grid. It is seen that the

velocity increases very rapidly with small increments in the axial coordinate as the position of the shock front is approached.

A basic assumption in the derivation of the equation is that the

fluid is in local thermodynamic equilibrium within each differential fluid

element of length dz and diameter D. This assumption is may be invalid

very near the shock front where the predicted fluid properties change significantly in a length comparable to the channel diameter. Nevertheless

the equations maintain mathematical consistency right up to Mach 1. To investigate the effects of friction and heat transfer on the development of a shock, consider just a straight pipe for which case

eq (12) can be written

P (l-4)fM^+" —aM 2a (J^ Qyooj^c

= (15)

It is further convenient to use a reduced coordinate z* = z/(4a/p),

where the denominator is the hydraulic diameter of the flow passage.

Separating variables, the equation becomes:

. * I I - dz = — TT- ; dM

2{\-^l) (fM^ + OM)

where

Q = :^—rz

Ga;c(l + K)

30
MACH No. at ENTRANCE to CHANNEL

Figure 8, Critical channel lengths within which the helium reaches

sonic velocity, as a function of helium velocity at the input. Q is a dimensionless heating rate defined in the text. 31
Conveniently, for a reasonable range of pressures, temperatures, ard

flows in helium-cooled systems, the velocities (J and c and the friction

factor are roughly constant (f has never really been measured as M—^1,

but the trend at lower flows is to become constant as the flow increases).

Thus, for an approximate estimate of the channel length over which sonic

velocity is reached, Q and f can be treated as constants, and the integration is straightforward, though tedious. Assuming M = M at z-i^ = 0 and

M = 1 at z* = (L/D) , the result is c

"j'^D^c 2f V Q+f / 2Q'"V^^2(Q+f)>'

The dimensionles s factor Q can vary over a wide range with different cooling systems and designs. A reasonable range would be per-

-4 -7

haps 10 as a maximum down to about 10 . With no heat transfer,

Q = 0„ Figure 8 shows (L/D) for a wide range of values of Q, assuming

f = 0. 003. At very low flow, for a given heat input, the fluid warms to

low density and sonic velocity in a relatively short length. At a very high

flow, fluid friction alone lends to sonic velocity. Obviously no system

would be designed close to the limit suggested by these approximate

curves, but this limit could become very important when it comes

to off-design, overload conditions in long channels, e.g. , superconducting power lines.

3. 7 The Polytropic Approximation

Some problems on fluid flow have been studied in the polytropic

approximation,

p dP

P dyO

32
where the index n is correlated with the type of process involved, i. e. ,

n = 0 for an isobaric process

n = 1 for an isothermal process

n ='^for an isentropic process

n = 00 for an isovolumetric process .

It is useful to evaluate the polytropic index n for the general case of fluid

flowing through a heated (or cooled) conduit. Simple substitution from the

eqs (8) and (11), and using thermodynamic definitions, yields the exact

result

We can now evaluate this for several approximations.

When there is no heat transfer between the fluid and the conduit,

and the cross section is constant, q =0, and the polytropic index becomes

e

.!4

For an ideal gas, this reduces to n = 1 in the limit of small velocities,

and n = in the limit U --c. Both of these are in accord with known

results [Chivers and Mitchell, 1971],

For flow in which heat is added (or taken from) the moving fluid,

and/or for flow through a conduit of changing cross section, three

33
regimes can be identified depending on the relative magnitude of the

terms in eq (16). When the flow velocity is much less than a lower crit cal

velocity, M ^

, given approximately by

U| 2qe

we obtain the result.

^

(!9)

When the flow velocity is much greater than an upper velocity,

>given approximately by

we obtain eq( 17), which applies when the heat transfer is negligible with

respect to kinetic effects.

When the flow velocity is between the two values and

M 2

, the index is approximately

n «

^ — (21)

Pk 2 qe

•

34
The following table gives approximate values of U and U for turbulent

1. 2

helium flow in the range 4-lOK, 1-5 atmospheres, using the approximation c ^ (jj 1 50 m/ s.

2 q[W/cm ] [m/s] [m/s]

3. 50 5

1.

' 35 1. 5

0. 1 15 0. 5 0.01 8 0.05

2

Heat fluxes encountered in design are typically under 1 W/cm , while

flow velocities required for reasonable heat transfer rates will be

^ 1 m/s. This suggests that typical designs will have ^ U < .

It is worthwhile to consider the polytropic equation in more detail for

this case.

2 2 Using the approximation U / CJ « 1, and considering just a straight tube for which p U = G = constant, we find

kdP « - xd (^m2)(I+ . (22)

This is a useful relationship between the pressure drop and the change

in kinetic energy for general, non-ideal flow in a conduit,

3. 8 Time Dependent Problems

We have given serious attention to integration of the timedependent equations 2, 3, and 6, for studies of transient effects in the

cooling of superconducting systems (flux jumps, for example). One

such problem which particularly needs study is the growth or decay

35
of a finite -length normal region of superconductor cooled by forced flow

FArp, 1972]. Two previous studies on this topic have both used a mathematical approach which rules a direct recovery process which we think

would be operative with forced single-phase flow ("Greene and Saibel,

1968, and Bald, 1970].

Basically, the mathematical procedure involves integration over

the spatial coordinate, sometimes iterating to match boundary conditions, once at each increment of time. This problem can be very time

consuming because we want to avoid approximations to the fluid equation of state.

After working with this problem for a while, we have concluded

that the integration could be performed more rapidly, at least when the

kinetic energy terms can be neglected, if enthalpy rather than temperature were one of the independent parameters of the equation of state.

This comes about because the enthalpy change is equal to the heat input

from the superconductor, and one iteration step could be eliminated.

Though this is not a critical factor in doing this particular calculation,

we came to realize from this work that iterative steps in a great body

of helium flow and refrigeration calculations can be eliminated by having

available different forms for the equation of state. Accordingly, we

have decided that the ARPA program would best be advanced by setting

aside, temporarily, this particular time -dependent flow calculation and

concentrating instead on the development of appropriate new forms of

the equations of state. This work is described in the following sec'sion.

4. 0 Helium State Equations For Refrigeration Analysis

The basic equations of state for the properties of helium has been

given by McCarty [1972, 1973]. This work will undoubtedly stand for

36
some time as the fundamentally accurate helium state equation, giving

both the PVT surface and properties such as specific heats, compressibilities, etc. , which are related to the derivatives of the PVT surface.

The refrigeration cycles analyses and helium flow analyses discussed

in this report have all been performed using McCarty's equations.

For practical analysis of refrigeration and flow problems, however,

the independent state parameters should be the enthalpy (related to heat

input), pressure (related to flow friction factors) and sometimes the entropy

(related to system reversibility), rather than the density and the temperature which are the independent parameters of McCarty's equations. This

is in no sense a criticism of McCarty's work since all fundamental

theories and equations of fluid properties are based on density and temperature expansions. But as a result, the equations are very cumbersome to use in engineering analysis. For example, a double iteration

must be done to find density and temperature and then other properties

of interest from a given pressure ajid enthalpy. Such procedure is time

consuming and very costly, and has been a significant limitation to the

refrigeration and fluid flow analysis to date.

As a result, we have spent some time developing three new state

equations for helium based on McCarty's work. We call these (1) S (P,H)

which calculates entropy from pressure and enthalpy, (2) H (P,S), which

calculates enthalpy from pressure and entropy, and (3) T(P,H), which

calculates temperature from pressure and enthalpy. In the future we

plan to add one more, V (P,H), which will calculate the specific volume.

In order to develop these, within just the last 3 to 4 months, we

have adopted guidelines which differ from McCarty's in several respects:

(1) The equations should be reliable for engineering calculations,

but need not retain the fundamental accuracy of McCarty's equations.

37
(2) Each relationship should be expressed by a single mathematical equation over the entire range of interest, McCarty, on the other

hand, uses three separate equations covering three different regions of

the pressure-temperature plane, with careful splicing techniques to

obtain smooth transitions from one equation to the next in the regions

of overlap. This system of equations is not easily programmed for the

computer.

(3) McCarty' s equations are structured so that it is reasonably

easy to take derivatives of the PVT surface. We have not retained this

requirement, thus making it much easier to find products of mathematical functions yielding reasonable fit of the stated variables, but whose

derivatives contain many factors and are algebraically complex,

(4) McCarty's equations cover the ranges 2 to 1500 K and 0 to 1000

atmospheres. For studies of helium refrigeration cycles, there is little

impetus to go to pressures above about 50 atmospheres. Our fitting is

0,25 done to 100 atmospheres, but weighted by the factor 2./ (1. 0 + P ), emphasizing the lower pressures. Our temperature range is from 3 to

400 K, though in fact they retain reasonable accuracy considerably

above 400 K,

The analysis was performed with a linear general least- squares

fitting program which determines the best coefficients for up to 36 independently-specified functions. Consistent with the above guidelines, we

have found it necessary in some cases to include additional numerical

constants within the specified functions in order to obtain desirable

accuracy in the result. These nonlinear constants were selected by

intermediate-stage studies, often of intuitive nature, with some repeated

testing to obtain approximate numerical values which optimize the fit. Accurate optimization of these nonlinear constants would be intolerably

expensive and probably would not gain much.

38
At this stage, we have not completed all the statistical analyses

and tests of the results, etc. , nor have we obtained even an approximate

equation V(P,H). This will be completed within the next few months and

submitted for separate publication. The equations are presented here

for those who may wish to use them in engineering calculations

without waiting for full documentation.

Throughout all the numerical equations, pressures are in atmospheres, enthalpies in J/g, and entropies in J/gK.

4. 1 Structure of the Equations

At temperatures above 300 K, helium behaves very nearly as an

ideal gas, and some rather simple expressions are found to satisfy the

equations at all pressures up to 100 atmospheres. We give these here

for further reference.

For an ideal gas, the enthalpy is just equal to the integral of

specific heat with respect to temperature, independent of pressure, with

one constant of integration. From McCarty ' s equations, we find, for

pressures up to 100 atmospheres, at T = 300 K,"^

T = - 2.82446 - 0.064533 P + 5.800 x lO'^P^,

C pi

and at T = 350 K,

T=— - 2.82419 - 0.0646I7P + 5.686 x 10® P^

Cpi

and at T = 400 K,

1=^-2.82394 - 0.064469 P + 5.615 x lO'® P^

^P' (23)

where Cpi = (5/2)R = ( 5/ 2 ) x 2.0772258 J / g • K . (24)

Throughout these calculations we retain more significant figures than

justified by the basic experimental data, in order to avoid roundoff error

in the calculated expressions.

^ 39
The differences between these equations are unimportant for this work,

indicating that one set of these parameters can reasonably fit the helium

data in the room temperature range. We have arbitrarily used the last

equation, evaluated at 400 K, in subsequent work. The standard deviation

between this expression and the 400 K data is 0. 000181 K,

The entropy of an ideal gas is given by

Si = So + Cpln(fJ-Rln(fJ

We find that McCarty's equations can be fitted at both 300 K and 400 K up

to 100 atmospheres by

Si = Cpi in T - R in P + 1.79056 + 0.0001077 P (25)

to an accuracy of about 0. 002% . Using eq (23) to eliminate T,

the "ideal" relationship between H and S is found to be

Hi = 3.6785757 exp (

-0000I077P N

+ 14.6649 + 0.33479 P - 0.00002913 P^ (26)

This expression can readily be inverted to give S^(H^,P).

One additional subsidiary equation turns out to be necessary for

this work.

Figure 4 shows the specific heat of helium as function of temperature for several pressures in the neighborhood of the critical

point (5. 2014 K, 2. 245 atmospheres). The loci of the peaks of the

40
specific heat curves, as a function of pressure and temperature, define

the transposed critical line, in the neighborhood of which all real properties show unusually strong dependencies on pressure and temperature

in a narrow range. In order to obtain equations free of significant error

in the vicinity of the transposed critical line, it is necessary to add

knowledge of its existence into the equations in some form. In this work

it is done by evaluating the enthalpy at the transposed critical line as a function of pressure. Calling this quantity , its equation is

Htc = 16.3713 + 2.21774 P - 0.002606 (27)

with a weighted standard deviation of about 0. 8 joules per gram. This

equation is probably much more precise than the accuracy with which

is really known from experimental work. However, it is also slightly

uncertain due to a slight lack of thermodynamic consistency which we

have found in McCarty's equations at higher pressures, as explained in

more detail in the discussion of equation T (P,H). In the equations to

follow, it is convenient to use a reduced enthalpy, h, given by

h = H/Htc . (28)

The subsequent equations for the properties of the real fluid are

developed from the differences between the real fluid property and that

given by the appropriate pressure-dependent "ideal gas" equation above.

In developing these equations, one cajinot avoid exploring a number of

mathematical approaches which turn out to be unprofitable. We give

here only our best results to date (June 15, 1973).

41
4. 2 Function T (P, H)

Figure 9 shows the difference between the temperature and that

predicted by the "ideal gas" eq (23), for several different pressures,

2

It looks somewhat like a Gaussian function, exp (-a«x ) or Lorentzian

2 -1

function (1-a.x ) , whose center position H . is a function of prescpi sure. Conveniently, its shape is not qualitatively changed as it bridges

the two-phase region.

While exploring these features, we uncovered a small error in

McCarty's work which needs some explanation. The center position of

the above Gaussian or Lorenzian curves can be obtained in two ways.

One way is to read the H (y ) directly from large and carefully plotted

graphs equivalent to figure 9. Secondly, one can use the fact that at the

center position the specific heat of the real fluid must equal that of the

ideal gas, as can be shown using just the thermodynamic definition

C = (dH/dT) and simple differentiation. However, we found that the

P P

center positions determined by the two methods differed by as much as 10% in enthalpy in the range of pressures of roughly 30 to 60 atmospheres.

After further study and discussion with McCarty, we deduce that

this thermodynamic inconsistency occurs only in the temperature range

from 10 to 15 K where two of his primary equations overlap and were

thought to be smoothly joined. Apparently there is a residual error in

the joining which shows up in this rather sensitive test.

The final data to which our equation for H . is fitted were selected

cpi

by excluding the region from 10 to 15 K, which in effect excludes the

region from about 30 to 60 atmospheres. The fitting was done to points

at 5, 10, 15, 20, 25, 70, 80, 90, and 100 atmospheres. At all of these

points except the first one the agreement between the two methods for

determining H (Y ) was about equal to the accuracy of the graphical

42
>1 '(2d9900000-clZl7+790H-t^28Z+ ^t^O- 1)

T3

• 1-1

m

d)

(1) a,

(U jH

a; U

-

Q

a0)

t-i

P4 bX)

43
method used, or about 1% or better. At 5 atrao spheres there was a small but distinct difference between the two, the zero-slope method

giving 14„ 3 J/g and the Cp method giving 14, 95 J/g, with the best

smooth curve predicting 15.2 J/g. We speculate that these differences

are due to small residual inaccuracies in McCarty's equations as the

critical point is approached; such inaccuracies are undoubtedly minor

in comparison with uncertainties in the experimental data to which the

equations are fitted.

At 30 to 50 atmospheres, the fitted curve turns out to agree to

within about 0. 6 J/g with the values deduced from C values at

P

these pressures, but differs by 3. 3 and 6. 7 J/g respectively from that

determined by the zero- slope method. The equation is

Hcpi = 8.6467 + 1.3220 P ~ 0.001823 (29)

with an rms error of 0. 34 J/g between fitted and calculated points.

In the fitted equation we use a reduced enthalpy in some terms.

^ = (H/Hcpi) (30)

Various equations and modifications of equations were tested by

fitting to 932 data points generated from McCarty's equations. Because

of the thermodynamic consistency problems between 10 and 15 K, all

points below 20 atmospheres in this range were given weight 0. 5, and

all those above 20 atmospheres were given weight 0. 01 (essentially

zero).

The following equation for T (P,H) appears quite complex, since

we have in general chosen to expand in a series of terms characterized

44
by successive values of the nonlinear numerical constants, rather than

simpler power series in H and P. We do this because we found we can

obtain about three times the accuracy with three less terms using the

more complex functions rather than the simpler power series expansions

as used in functions S (P,H) and H (P, S). For final use on a computer,

the additional complexity is insignificant, whereas the increase in

accuracy may be important.

We start by defining functions

w = 0.6

F2(x) = (I + 0.1 371 1 501 loge(x/2)) / [l + U/zf)

F3(x,a,y,b) = x 6°^^ e'^^^

F^(x) = (logex)^/ (l + x^)

F5(x) = (logex)^/(l + x^)

and a pressure variable

p = loge (1.0 + P) . (31)

45
In terms of these functions, the deviations graphed in figure 12 between

the true temperature and the ideal temperature T. given by eq (23) is

(T-Tj) = F, (</),2.I0) x(a, p + a2P^+ a3P^ + a4p'^)

+ F, ((^, 1.85) xlogp + agp^+ayp^)

+ F| ((^, 1.60) X (Qgp + p^ + a,Q p^

)

+ Fi 1.35) X (a,,p + a,2P^+ aijp"*)

, -(P/3.) -(P/3.5) . -(P/4.0)n

+ Fzi<^) x(a,4e + 0,56 + ^\6^ ^

+ FjdogehJO., P,4.)x(a,7+a,8P + 0,9 p^+ OgoP^ )

+ Fgdoggh, 4., P,4.) x(a2i+a22P+ <^2Z^^ + ^24?^ ^

A + F^^{4>^ x(a25'^°26P °27P ^28 P ^29P ^

+ F^{(j>) X (a3oP+ 03, p^+ 032 p^ + 033 p'*)

(32)

where the constants a. are given in table 6 . Thus, the true temperature

is given by the sum of eq (32) for (T - T.) and eq (23) for T. , for a given P and H.
The standard deviation between this result and McCarty's defin'

equations is 0. 015 K. The accuracy in the vicinity of the two-phase

region is generally 0. 006 K or better, and the largest error on the

2. 3 atmosphere isobar, near the critical point, is 0. 022 K, At highe

pressures, the error increases.

47
Table 6. Numerical constants for the equation T(P,H)

10

11

12

13

14

15

16

17

13. 878902654

-9. 1294555014

1. 4753132952

-0. 064635371239

- 14. 085703051

-4. 3222209397

1. 4429523076

-1. 8712988904

18. 815813408

-3. 3611959286

3. 2417847568

-4. 1613166989

0. 1053291809

34. 867402925

-76. 900659027

47. 935651420

-4. 6708598724

18

19

20

21

22

23

24

25

^26

27

28

^29

30

31

32

33

6. 2461795885

-0. 4080674500

-1. 1548143734

-2. 3614901630

4. 6428182119

-2. 9101156545

0. 90838981816

-0. 23446642474

21. 976664669

•26. 313269947

5. 8053275951

-0. 29763970504

4. 5591407995

-6. 5729084580

2. 0788999721

-0. 17257458796

48
4. 3 The Function H (P,S)

This function takes pressure and entropy as the input variables

and calculates enthalpy^ The general procedure is to first calculate

an "ideal" enthalpy using eq (26). A correction term, A| H is

calculated from eq ( 33) shown below, and the final answer is obtained

by adding A| H to H. . In evaluating the constants for eq (33) , the

same low weight was given to the points between 10 and 1 5 K as was

described under eq T (P,H) .

j=l ^ (33)

where C^. is the ideal gas specific heat given by eq (24) and p is the

pressure variable given by eq (31). The coefficients bjj are given

in table 7 . Then,

H(RS) = A,H + Hj (eq.26) .

The standard deviation between the calculated enthalpy and that

given by McCarty's program is 0. 050 J/g . If we relate this to a temperature error at constajit pressure by

8T = ^

Cp

the equivalent temperature error is approximately 0. 010 K.

49
Table 7. Numerical constants for the equation H(P,S) .

Ni =: 6. 2095652653 X

_ 1 10

^ \4- 1. 1828344037 X

3 10^

S2 =: -2. 4559275204 X

1 10^ \5 -- -9. 3218939174 X

^3 = 2. 2297261674 X lo' hi - -4. 4998406724 X

N4 =: -6. 7492782747 \z -: -5. 4945892540 X

3

lo"^

^5 = 4. 1021573691 X

_ 1 10

^ 4. 9550341804 X lo"^

hl = -5. 2643461931 X 10 b

54

: -1. 3819212763 X lo"^

\z- 3. 6216988376 X

2

10 ^55 = 1. 1001849787 X

2 10^

^23

= -3. 2737113461 X

2 10^ -- 2. 9726956809 X2

10

^4 = 9. 9186574011 X

1 10 V = 3. 2678083338 X

3

10

^25 =: -7. 0819815035 ^3 =-- -2. 8933124992 X

3

10-^

^1 =: -5. 2181285811 X

1 10 V =: 7. 8724009925 X2

10

-1. 8942905449 X

3

10 •^65

=: -6. 2871925138 X

1 10

1. 7107602365 X

3 10^ Si =: -6. 6277471159 X1 10

S4- -5. 0693398133 X lO'^ "iz-: -7. 5478760174 X 10^

^35 = 3. 8921676371 X

1 10^ ^3 =: 6. 5427817221 X7 10"^

^1 = 2. 5428630050 X ^4 =: -1. 7447708487 X 10^

4. 5255682326 X ^5 = 1. 3939361536 X

^3 =^ -4. 1189023813 X 10^

50
4. 4 The Function S (P,H)

This function takes pressure and enthalpy as the input variables

and calculates entropy. The general procedure is to calculate a correction term A^H as a function of P, H, and (P), as shown in eq (34)

L tc

below. This A^H is added to the input H to give an "ideal" which in

turn is used to calculate S by inverting eq (26). The same low weight

was given to points between 10 and 15 K as was described under

T(P,H):

AzH = 2 1 Cijh(-) p(ri)

where h is the reduced enthalpy given by eq (28) and p is the pressure

variable given by eq (31), The coefficients Cjj are given in table 8.

Then

(34)

^/r...v r. . /H + AoH- 14.6649 - 0.33479 P + .00002913 P^\

S P,H =CDiIoge P 3.6785757 P^'^ '

+ 0.0001077 P.

The standard deviation between the calculated entropy and that

given by McCarty's program is 0.017 J/gK. If we relate the error at

each data point to a temperature error by

8S = ^ SS,

Cp

the standard deviation in temperature is 0. 014 K.

51
Table 8. Numerical constants for the equation S(P,H) .

11

96. 557792404 '44 90. 187334672

12

13

85. 342124216

93. 239184634

'45

'51

8. 3939686508

405. 81584516

14

32. 642437375 '52 123. 84837652

15

2. 8950104636 '53 321. 35063183

^21 ^ l^^O^^^l

54 89. 302684169

'22 337. 90821505

'55

-8. 2508383745

^23 ^ 1^24^^^^ c, , = -102. 61607953

6

'24

'25

31

32

'33

34

35

'41

'42

70. 146358374

6. 2193650992

884. 01 164347

359. 83067977

•24, 961141557

-6. 9236428132

0. 4327647102

821. 10597373

51. 518553781

'62

'63

'64

'65

'71

'72

'73

'74

'75

-72. 105848121

120. 99391576

-34. 006661384

3. 140489957

10. 469390204

11. 920614672

-16. 687143281

4. 7025518338

-0. 4348173542

'43 341. 82700671

52
4. 5 The Two-Phase Region and Boundaries

The equations presented in sections 4. 2 - 4. 4 are fitted to helium

properties data in both the liquid and the vapor phases, but not in the

two-phase region of liquid-vapor equilibrium. When the input data to

any one of these equations corresponds to a point in the liquid-vapor

equilibrium region, the equations continue to produce numerical results

of apparently correct order of magnitude, and a,n additional test must

be made to show that such calculation is in fact meaningless. To do

this, we have fitted the following two equations to respectively the

enthalpy of the saturated liquid H and the enthalpy of the saturated

OJ-J vapor H at equilibrium, as a function of pressure. If, for a given

o V

pressure, the enthalpy of the fluid is between the two values predicted

by these two equations, the point in question is within the two-phase

region and the eqs T (P,H), S (P,H), H(P,S), and V(P,H) will give

meaningless results. Because eq H (P,S), in particular, is single

valued, it can be used as input to these two equations to test whether

a given pressure and entropy lies within the two-phase region, even

though the calculated enthalpy will be otherwise meaningless within

the two -phase region.

(35)

i = 2

8

+ (36)

i = 2

53
Table 9. Numerical constants for the equation H (P) and H (P)

oJ—i o V

11

31

41

11

3„ 0016449891 x 10

2. 6901274997

1. 5023185792 x 10

-2. 4216012980 x 10

-3

d = 3. 1430206769 x 10

51

• 2. 1707243329 x 10

7. 4933613037

1

1

1

1

d = -9. 9634989768 x 10

-1

Iv

5v

7v

8v

-1. 2529131754 x 10

-3

d^ = 2. 4289246713 x 10

2v

d, = 2. 8759510985 x 10

3v

d, = -6. 6130326207 x 10

4v

1

1

1

8, 5561182969 x 10

-6. 1554651598 x 10

2. 2519571060 x 10

-3. 2937094592

1

1

54
The constants for these equations are given in table 11. The standard

deviation is 0, 034 J /g at the liquid boundary (H^^ ), and 0, 0 50 J/g

at the vapor boundary (Hgy ).

4. 6 Accuracy

As stated earlier, we have not completed all appropriate tests

and measures of the accuracy of these three equations at this time.

We can state that the overall inaccuracy, expressed as a temperature

error, will generally be less than 0. 025 K. More explicit statements

can be made about specific regions of the pressure -temperature plane:

(1) The fittings were weighted to provide additional accuracy

near the liquid-vapor equilibrium line, where the errors are generally ^ 0. 008 K.

(2) The error rises rapidly in the compressed liquid region

below 3. 5 to 4 K as the pressure rises above a few atmospheres,

approaching a maximum of about 0. 10 K at 50 to 100 atmospheres.

(3) A difficult region exists at pressures above about 50 atmospheres and temperatures between about 200 to 400 K, where the first

deviations from the "ideal gas" eqs (23), (25), and (26) occur as a function of H or S. These deviations begin with increasing abruptness

as the pressure increases, and we have not yet found a really good

fit in this region. The error here can be several times the standard

devi ation.

(4) We have previously discussed a small residual error in

McCarty's equations in the region 10 to 15 K, which shows up in this

work as deviation from his predicted values by as much as 0. 10 K at

pressures from about 20 to about 70 atmospheres. For pressures

outside of this range, the deviation drops off noticeably. It is difficult

to say what the thermodynamically correct values are in this temperature region, so that an absolute error is hard to assess. It is conceivable that our equations could be accurate to within roughly their

standard deviation.

5 5
5. 0 Refrigerator Performance Survey

An important consideration in the operation of superconducting

systems is the long term refrigeration performance and reliability.

Statistically significant data in this area are scarce, especially on refrigerators for which long term performance data have been consistently

recorded, and on refrigerators serving in less-than-ideal conditions

which might typify field service.

At the inception of the ARPA contract, we began collecting

documentation on refrigerator reliability from a wide number of

manufacturer and user contacts throughout the USA and the world.

Shortly thereafter, our group received two other assignments which,

taken together, would explore this field in some depth, plus answering

more specialized questions from the Naval Ordnance Laboratory and

from the NBS Cryogenics for the Electrical Industry program (now

defunct). As a result, only a relatively small amount of ARPA funding

has gone into this type of survey, though the results remain important

to the ARPA program. Both of these projects have been completed,

and reports are currently in editorial process. They will be distributed

to the ARPA mailing list when they become available. Also they will

be presented at the Cryogenic Cooler Conference to be held at the

U. S. Air Force Academy in October 197 3. The first of these studies has been done with particular reference

to refrigerators for cooling infrared sensors. The approach has been

to consider all factors which contribute to refrigerator system performance under field conditions, with major weight being given to the

mean time between failures (MTBF), the maintainability, and the

maintenance interval. The project necessarily involves a degree of

subjective evaluation of the sparse data which exists. Our tentative

conclusion is that with careful selection of present technology one can

reasonably expect to obtain low capacity "mass purchased" (in numbers

of 50 to 100 at a time) refrigerators having a MTBF on the order of
6000 hours. As the system size increases, even less data are available,

but there is no reason to expect any dramatic decrease in achievable

MTBF from that of the small machines.

Though this particular survey has been directed towards operation

in the 60 to 90 K range, it is at the same time a significant measure of

the expected performance of helium temperature refrigerators. This is

because the room temperature compressor, more than auiy other single

component, is generally the controlling element in the system reliability,

although one must recognize additional problems in maintaining the helium

purity in the 4 K refrigerator. Also, many of the cycles considered

should work with essentially undiminished reliability down to their lower

limit of about 20 K, and could thus serve in conjunction with an additional

Joule-Thomson circuit to extend the lowest refrigeration temperature

down to the liquid helium range.

The second project of relevance to the ARPA program has been an

updating of the earlier work by Strobridge [1969] on refrigerator costs,

weights, volumes, and efficiencies as a function of capacity and temperature. Though much new data have been obtained in the intervening 4 years,

the earlier correlations remain essentially accurate, and no new trends

are evident using these basic coordinates.

6. 0 Program Relationships and Future Goals

The work reported here has been carried out in parallel with other

Cryogenics Division projects relating to helium-cooled superconducting

components. Our major activity in this particular field began in 1968

with surveys and analytical studies of various modes of helium heat transfer. Some of this heat transfer work continues to this day, but the program has steadily broadened to include system studies - the interrelationships of heat transfer mechanisms, fluid flow and stability problems,

refrigeration requirements, and auxiliary pumps and equipment [Smith,

1968; Sixsmith ajid Giarratano, 1970; Giarratano, Arp, and Smith, 1971;

Snyder, 1970; Arp, 1969, 1 97 1 ; Steward and Wallace , 1971; Arp, et al. , 1972; McConnell, 1973; McCarty, 1 972 ; and Mittag , 1973]. Of partiicular note at the present are:
(1) study of thermal stability of forced-flow helium cooling

systems, with application to superconducting power lines (funded by

Brookhaven National Laboratory).

(2) Studies of refrigeration requirements for superconducting

power lines (funding beginning momentarily by Stanford University).

(3) Study to evaluate transient heat transfer rates, and flow

system response to transient heat input (internal NBS funding),

(4) Study of helium pumps and pumping problems (funded by

WPAFB, and currently by ARPA) .

(5) Study of forced flow problems and heat transfer with

superfluid helium (funding by AEC and ONR).

From an overview of these programs, we feel that work has

progressed to the point where a major new experimental facility is

called for. Using modular construction, one component of the facility

would be a pumping unit, utilizing either one of our existing helium

pumps, or future pumps for test. The second component would be

a modular section of flow loop which could be changed from experiment to experiment to fit particular needs. The design would allow the

helium in the flow loop to be controlled at any pressure and temperature

interest, including subcooled liquid, normally boiling liquid, supercritical fluid, and even boiling and subcooled superfluid ranges. Flow

rate through the system would be controlled by a variable speed pump

and/or appropriate valving. We are presently working out the preliminary design for such a system. It is obviously complex, but no

more so than other cryogenic systems within our knowledge, and

entirely within the present state of the art within our laboratory. We

do not yet have funding for the project, and are restricting the work

to preliminary design at this time, using WPAFB and ARPA funds.

58
Between now and the end of the present contract period,

December 31, 1973, we plan to (1) coraplete the equation- of- state

studies described in section 4, 0 of this report, (2) complete a study

of helium pump performance which has begun but is not explicitlyreported here, (3) complete preliminary design on the pump-flow

facility described above, and begin assembly if additional funding

is obtained, and (4) complete and report upon possible Ledinegg

instabilities.

59
0 Nomenclature and Thermodynamic Relationship

7. 1 Nomenclature

cros s -sectional area of flow channel

velocity of sound

hydraulic diameter = p/4-Q

efficiency

mass flow rate/unit area = yO U

heat transfer coefficient

enthalpy /unit mass

heat transferred into the fluid per unit

length of flow channel

thermal conductivity

total mass flow rate = p\Jo

, , , , fluid velocity Mach number - :; ; r

sound velocity

fluid velocity (see thermodynamics

"expansion velocity"' definitions)

Nusselt number = hD/X

perimeter of flow channel

pres sure

heat flux = Ap

density

Reynold number = GD /jJ,

SI Units

m2

m/s

m

kg/(m^s

W/m^K

J /kg

W/m

W/mK

kg/s

m

N/m

W/m

kg/m"

60
SI Units

entropy /unit mass J /kg K

temperature K

flow velocity (average) = GV m/s

specific volume - \ / p m /kg

viscosity m. s /kg

dimension in direction of flow m

61
7. 2 Thermodynamic Definitions

X = Cp /Cv = I + aT (^2)

SI Units

a 1 ^

V dT

K

1 ^

ap

m^/N

V aH

kg/J

dT

ap H

= (aT-l) / (yoCp) m^'k/ N

1 ^yo ap H

= k - a i/r m^/ N

oj = ( Cp / a "expansion velocity" m/s

1//Q e = (///5k) , velocity of sound m /s

pQ

cu'^ c^

52
7. 3 Typical Values in Liquid Helium Range

Subcooled Vapor Near-Critical

Liquid (ideal gas) Fluid

T dV I ^^'^^varlp- ^-^

P dp I Pk = ^ 0. 03 1. 0 10.

P dP Ij

0) « C ^ 150 meters per second

7. 4 The Temperature Change of a Real Fluid Through a Pump or Engine

The input mechanical work per unit mass flow, dW, through a pump or engine is equal to the increase in the stagnation enthalpy

2

(H + 1/2 U ) from inlet to outlet. Neglecting the kinetic energy term,

this can be written

dW = dH = TdS + VdP .

In this convention dW, dH, and dP are positive for pumps and negative

for engines. On the other hand, the entropy change dS from inlet to

outlet will always be > 0, by the Second Law. The input work to an ideal

(isentropic) device operating over the same pressure range dP will be

dW. = dH. = VdP

1 1

For a pump, dW >dW., and the efficiency £p is defined

dWj

" dw"'

63
By simple substitution the entropy increase as a function of the pump

efficiency is then found to be

dS = VdP (--0 ^

For an engine, |dW^| > |dW| , and the efficiency is defined

IdWI

^« " IdWil

so that the entropy increase is

dS = (I - £e) f(-dP)

In either case, the temperature increase through the device, from inlet

to outlet, is given by

dp dS

where the first term can be related to real fluid properties in an ideal

(isentropic) process, and the second term can be related to irreversibilities in the process itself. Using straightforward thermodynamic

identities, this last equation can be written

dT V \ T

64
Substituting dS (€p) for a pump, we find

dT

dP PUMP

= i// +

€yOCp

and for an engine

dT

dP ENGINE pCp

A real pump or engine may operate over a large AP such that

these differential equations are inadequate as they stand. They can be

integrated between required inlet and outlet pressures (or temperatures),

but to our initial surprise we found that the integrated results correspond

to overall efficiencies which differ from the given efficiency by up to

10 or 20%, for typical refrigeration calculations. After some thought,

we deduced that the discrepancy occurs because the entropy corresponding to the isentropic compression dH^ changes implicitly at each integration step, whereas the entropy used to check the integrated results always

equals that at the inlet to the device. Nevertheless, the equations provide

a useful semi -quantitative estimate of AT/AP in a real device.

65
8. 0 Acknowledgements

The research output of the Cryogenics Division depends

strongly upon informal and cooperative input from the many employees

whose expertise overlaps the project at hand. Michael Jones and

co-workers have been a valuable source of ideas and information on

the hydrodynamics of helium flow. Robert McCarty has given generously of advice and review on the development of the helium state

equations. Jesse Hord has provided valuable input to the systems

concepts within the program. Roland Voth and Richard Strobridge

have done the survey work summarized in section 5. 0. Neil Houbolt

has been a most patient and persistent programmer for many of the

computer analyses.

66
9. 0 References

Arp, V. , Heat transport through He 11, Cryogenics J_0, 96 (1970).

Arp, V. , Forced-flow, single phase helium cooling systems. Book,

Advances in Cryogenic Engineering _1_7, p. 342 (1972),

Bald, W. B. , Supercritical helium cooling of hollow superconductors;

Part II, Thermal instabilities, Brookhaven National Laboratory

Note 15805 (1970).

Bird, R. B. , Stewart, W. E. , and Lightfoot, E. N. , Transport

Phenomena (John Wiley and Sons, New York, N. Y. , I960).

Chivers, T. C. , and Mitchell, L. A. , On the limiting velocity through

parallel bore tubes, J. Phys. D: Appl. Phys. 4, 1069(1971).

Dean, J. W. , and Mann, D. B. , The Joule-Thomson Process in Cryogenic

Engineering, Nat. Bur. Stand. (U.S.), Tech. Note 227 ( 1965).

Giarratano, P. J. , Arp, V. , and Smith, R. V. , Forced convection

heat transfer to supercritical helium, Cryogenics J_l_, 385 (1971).

Greene, W. J. , and Saibel, E. , Stability of internally -cooled superconductors, Book, Advances in Cryogenic Engineering 14, p. 138

(1968).

Krasiakova, I. I. , and Glusker, B. N. , Energomashinostroenie 9 , 18 (1965).

Ledinegg, M. , Instability of flow during natural and forced circulation,

Die Warme 61 , No. 8 (1938); AEC translation 1861 (1954).

Linhardt, H. D. , Cryogenic turboexpanders , LNG/Cryogenics 1^, No. 7 (1973).

McCarty, R. D. , Thermophysical properties of helium-4 from

2 to 1500 K with pressures to 1000 atmospheres, Nat. Bur.

Stand. (U.S.) Tech. Note 631 (1972).

McCarty, R. D. , Thermophysical properties of helium from 2 to

g 1500 K at pressures to 1 x 10 pascals, To be published (1973).

67
McConnell, P. M. , Liquid helium pumps, Nat. Bur. Stand. (U.S.)

Interagency Report 73-316 (1973).

Mittag, K. , Kapitza conductance and thermal conductivity of copper,

niobium, and aluminum in the range 1. 3 to 2. 1 K, Cryogenics 13,

94 (1973).

Semenkover, I. E. , Energomashinostroenie ^ , 16 (1964).

Sixsmith, H. , and Giarratano, P. J. , A miniature centrifugal pump.

Rev. Sci. Inst. 41_, 1570 (1970).

Smith, R. V. , Review of heat transfer to helium I, Cryogenics 9 , No. 11 (1968).

Snyder, N. S. , Thermal conductance at the interface of

and helium II, Nat. Bur. Stand. (U.S.) Tech. Note 385 (1969).

Steward, W. G. , and Wallace, G. H. , Helium viscosity measurements

2

4 to 20 K, 0 to 10 MN/m , Nat. Bur. Stand. Report (unpublished)

(1971).

Strobridge, T. R. , Private communication (1973).

Strobridge, T. R. , Refrigeration for superconducting and cryogenic

systems, IEEE Trans, on Nuclear Science NS- 16 (1969),

Zuber, N. , An analysis of thermally induced flow oscillations in the

near- critical and super -critical thermodynamic region, NASA

Report NAS8-11422, DDC accession number N67-13534 (1966).

68
FORM KlBS.114A (1-71) U.S. DEPT. OF COMM. 1 . PUBLICATION OR REPORT NO. 2. dov't Accession

BIBLIOGRAPHIC DATA NBSIR 73-331

5n c t T

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

REFRIGERATION OF SUPERCONDUCTING ROTATING

MACHINERY

5. Publication Date June 1973

6. Performing Organization Code

7. AUTHOR(S)

Vincent D. Arp

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

10. Project/Task/ Work Unit No. 2750452

11. Contract/Grant No. 2186

12. Sponsoring Organization Name and Address Defense Advanced Research Projects Agency

1400 Wilson Boulevard

Arlington, Virginia 22209

13. 1 ype of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) Recent work at the NBS Cryogenics Division in three areas related to

helium refrigeration is summarized: (1) Analysis is given of a possible high

pressure refrigeration cycle which offers in principle a reduced component

size, but which turns out to be impractical because of expansion engine

inefficiencies. (2) Exact equations for flow of a real fluid are derived and

applied to problems of fluid flow near the critical point, as may occur with

some helium-cooled superconducting systems, and (3) Three new equations

of state for heliami are given, each using different state variables, to

eliminate the need for iterative techniques in helium refrigeration cycle

and fluid flow analysis.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Equation of state; helium; hydrodynamics; near-critical flow; refrigeration;

superconductors

;

thermodynamics.

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