United States Department of Commerce

National Institute of Standards and Technology

A11103 475AMA

NISTIR 90-3936

EFFECT OF NOBLE GAS MIXTURES

ON THE PERFORMANCE OF

REGENERATIVE-TYPE CRYOCOOLERS

ANALYTICAL ESTIMATE

David E. Daney

-QC

100

.U56

90-3936

1990

C.2
NATIONAL INSTITUTE OF STANDARDS &

TECHNOLOGY

Research Information Center

Gaithersburg, MB 20899
NISTIR 90-3936

EFFECT OF NOBLE GAS MIXTURES

ON THE PERFORMANCE OF

REGENERATIVE-TYPE CRYOCOOLERS

ANALYTICAL ESTIMATE

David E. Daney

Chemical Engineering Science Division

Center for Chemical Technology

National Measurement Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

September 1990

Sponsored by

Air Force Space Technology Center

Kirtland Air Force Base, New Mexico 87117-6008

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director

CONTENTS

Page

Abstract 1

1. Introduction 1

2. Mixture Properties 3

3. Dependence of Heat Transfer on Prandtl Number 10

4 . Heat Transfer 12

4.1 Screens 14

4.2 Spheres 14

4.3 Gaps 14

4.4 Tubes 15

5. Pressure Drop 18

5.1 Screens 18

5.2 Randomly Packed Spheres 19

5.3 Gas with Laminar Flow 19

5.4 Tubes With Turbulent Flow 19

6. Discussion 22

Nomenclature 28

References 30

iii
List of Figures

Page

Figure 1. Transport properties of helium -krypton mixtures. 5

Figure 2. Thermal conductivity of helium-argon mixtures. 6

Figure 3. Dilute gas Prandtl number for He-Ar and He-Kr

calculated from mixing rules . 7

Figure 4. Dilute gas thermal conductivity for He-Ar and He-Kr

calculated from mixing rules. 8

Figure 5. Dilute gas viscosity for He-Ar and He-Kr calculated

from mixing rules . 9

Figure 6. Relative regenerator thermal loss for He-Ar mixtures. 16

Figure 7. Relative regenerator thermal loss for He-Kr mixtures. 17

Figure 8. Relative pressure drop through regenerator for He-Ar

mixtures. 20

Figure 9. Relative pressure drop through regenerator for He-Kr

mixtures. 21

Figure 10. Sum of thermal and pressure losses in regenerator as a

function of flow area. 24

Figure 11. Relative regenerator void volume as a function of heavy

component fraction. 26
List of Tables

Page

Table 1. Relative Enhancement of St (St/(St) Pr , 0 . 7 ) f° r St ~ 1/Ptn 12

v

Effect of Noble Gas Mixtures on the Performance of

Regenerative-Type Cryocoolers—Analytical Estimate

David E . Daney

National Institute of Standards and Technology

Boulder, Colorado 80303

We compare the performance of regenerators that use noble

gas mixtures to the performance of those that use pure helium gas.

Both helium-argon and helium-krypton mixtures are investigated.

For some heat transfer surfaces, a modest gain in heat transfer

can be achieved with these mixtures . The concomitant increase in

pressure drop, however, more than offsets the heat transfer gain

so the net regenerator loss increases for all cases we evaluated.

The dependence of heat transfer on Prandtl number has not

been measured for the range associated with noble gas mixtures,

0.2 < Pr <0.5, and we estimate that the uncertainty from this

source can exceed 20 percent. We give our own estimates for the

transport properties (Prandtl number, viscosity, and thermal conductivity) of helium-argon and helium-krypton mixtures because of

the absence of experimental data at low temperature.

Key words: cryocoolers; gas mixtures; heat exchangers; heat

transfer; noble gas mixtures; Prandtl number; regenerators;

thermal regenerators

1. Introduction

This report discusses the use of mixtures of noble gases with low Prandtl

number to enhance the performance of regenerative-type cryocoolers such as

Stirling cycle coolers. The potential for enhancement occurs because, all

other factors being equal, the Prandtl number determines how good the heat

transfer is, and, hence, how low the thermal losses are in the regeneration of

the cryocooler. Unfortunately, the changes in density, heat capacity,

viscosity, and thermal conductivity that give rise to low Prandtl numbers in

mixtures of noble gases generally degrade rather than enhance the performance.

The Prandtl number (Pr = v/a = /iC

p

/k) is the ratio of the momentum

diffusivity (kinematic viscosity) to the thermal diffusivity. It is a measure

1
of the relative speed at which momentum and energy are propagated through a

convective system. For the simple case of laminar flow over a flat plate, the

heat transfer coefficient is related to the Prandtl number by

G • C

p

h _____ (1)

Re 1/2 Pr2/3

for Pr > 1 [1]. If other factors remain constant, then the lower the Prandtl

number, the greater the heat transfer coefficient. For mixtures the molar

heat capacity is constant, so the specific heat capacity changes with density

in such a way that G»C

p

is constant for constant velocity flow. The Reynolds

number (4pVrh

//i) , however, does not remain constant for constant velocity flow,

but varies with density and viscosity in such a way as to generally give a net

decrease in heat transfer for a constant velocity (constant machine speed)

,

constant molar density system. For other types of flow and 0.1 < Pr < l--the

range of interest here --the dependence of h on Pr may differ somewhat.

In evaluating the effect gas mixtures have on the performance of

regenerative cryocoolers, we avoid the general issue of optimization, because

the number of variables is so great that we know of no truly general procedure.

Thus it would not be clear whether the difference in performance between a

machine ''optimized” for mixtures and one "optimized" for helium would be due to

the thermophysical properties of the two gases, or just the result of the

optimization. We do consider one possible optimization where void volume in

the regenerator is minimized for fixed values of regenerator ineffectiveness,

pressure drop, and conduction loss. For that case, we calculate the effect of

mixed gases on the regenerator void volume. A larger void volume requires the

use of a larger compressor to maintain the same pressure ratio. Most of the

effort in this paper compares thermal and pressure losses in the regenerator of

a machine operating at the same speed and producing the same gross

refrigeration with the same regenerator geometry. Both criteria require that

the comparison be at constant pressure, because the refrigeration per mole of

an ideal, monatomic gas is constant for the same pressure ratio and

temperature

.

2
2. Mixture Properties

There are few data available on the transport properties of mixtures of

noble gases below ambient temperature, so we had to estimate them. We took two

different approaches. The first was to use kinetic theory of dilute gas

mixture and the second virial equation of state [2]. Because of the limited

time available, we did not "tune" parameters such as the potential well depth

to give the best agreement with experimental properties. The second approach

was to calculate the transport properties of the mixture from the pure

component values by using mixing rules [3, 4, 5]. This empirical approach

gives better agreement with the limited experimental results near ambient

temperature. Consequently, we adopted the mixing rule method to generate the

transport properties used in our heat transfer calculations.

Two different forms of mixing rules were used. The first,

xiYi (i-Xi )y2

y x1 +A(l-x1 ) +

(l-x1 )+Bx1 where x1 = mole fraction of component 1

y1 ,y2 = transport property of pure components

A,B = constants dependent on molecular species,

was used to estimate viscosity and thermal conductivity of helium- krypton

mixtures as well as the viscosity of helium-argon mixtures. For the thermal

conductivity of helium-argon mixtures, we used the expression

xiYi A + (l-x1 )y2 B ^ xx A + (1-X! )B ’ ^

which is recommended by Stephan and Heckenberger [5] for the thermal

conductivity of noble gas mixtures. We had already calculated the thermal

conductivity of helium -krypton using equation (2) before coming across the

Stephan-Heckenberger book, but the small difference in the results did not

justify the extra work required to redo all the calculations.

In calculating the specific heat capacity, we assume that the molar heat

capacity is constant. The heat capacity per unit mass is then

= 20.79

p MWj_ *x : + MW2 (1-xx )

( 4 )

where MW^^ and MW2 are the molecular weights of the pure components.

3
In figure 1 we compare our dilute kinetic theory calculations for gas

mixtures with equation (3) for a binary mixture of helium-krypton. The

viscosities for the two methods agree well, but the thermal conductivities

differ considerably. The difference in the Prandtl numbers results primarily

from the difference in the thermal conductivities calculated by the two

methods. Figure 2 compares the thermal conductivity calculated by the two

methods with smoothed experimental values for the helium-argon system. This

comparison is the basis for choosing mixing rule expressions 2 and 3 to

generate the properties for our heat transfer calculations.

Figures 3, 4, and 5 show the Prandtl number, thermal conductivity, and

viscosity used in the heat transfer calculations. The depression of Prandtl

number in these gas mixtures occurs because the specific heat initially

decreases faster than the thermal conductivity does. In contrast, liquid

metals enjoy a low Prandtl number in large part due to a high thermal

conductivity

.

For a gas mixture to be a useful refrigerant in a regenerative type

cryocooler it is important that the mixture not condense in the cold part of

the machine. Thus the dew point of the heavier component should be below the

minimum operating temperature of the cooler. For mixtures of ideal gases, the

dew point is equal to the saturation temperature of the pure heavier component.

For mixtures of non- ideal gases, the dew point may be depressed below the

saturation temperature of the pure components. The enhancement factor

Pyi

<f> = (5)

ro 1

where P = total pressure

y x = heavier component mole fraction

Pol = vapor pressure of saturation heavier component

is a measure of how much the partial pressure of the heavier component may

exceed the vapor pressure of the pure component. For example, if <f> = 1.2 the

partial pressure of the heavier component may be up to 20 percent greater than

the saturated vapor pressure of the pure component before condensation occurs.

For He-Ar, the measured enhancement factor is 1.14 at 20 bar and 92 K [6].

For He-Kr the measured value is 1.09 at 20 bar and 110 K [7]. The enhancement

factor is only a weak function of temperature, so the value at the dew point

will differ only slightly.

4
1.4

HELIUM FRACTION

Figure 1. Transport properties of helium -krypton mixtures.

5
HELIUM FRACTION

Figure 2. Thermal conductivity of helium-argon mixtures.

6
PRANDTL

NUMBER

HELIUM FRACTION

Figure 3. Dilute gas Prandtl number for He-Ar and He-Kr calculated from mixing

rules

7
HELIUM FRACTION

Figure 4. Dilute gas thermal conductivity for He-Ar and He-Kr calculated from

mixing rules.

8
HELIUM FRACTION

Figure 5. Dilute gas viscosity for He-Ar and He-Kr calculated from mixing

rules

9
Thus we expect no major depression of the dew point of these mixtures at

pressures up to 20 bar or so. An exact value for the lower useful temperature

of these gas mixtures depends on pressure and mole fraction. A rough

approximation, however, is given by the normal boiling point of the heavier

component, which is 87 K for argon and 120 K for krypton.

3. Dependence of Heat Transfer on Prandtl Number

Results for heat transfer from compact heat exchanger surfaces are often

presented by giving the Colburn " j" factor (j = StPr2/3

) as a function of the

Reynolds number. For example, Compact Heat Exchangers by Kays and London [8],

the definitive work on the subject, uses this method of presentation. As Kays

and London are careful to point out, however, their measurements were all made

with air (Pr = 0.7), so the Prandtl number was not a test parameter.

The dependence of the Prandtl number on the two- thirds power comes from

analytical solutions for laminar flow [1,9] which assume Pr > 1 . For this case

the thickness of the thermal boundary layer is less than or equal to that of

the momentum boundary layer, and the energy equation can be integrated in a

single step from the wall to 6 T. An integral method approximation to the

heated flat plate problem gives the ratio of the thermal to momentum boundary

thickness as f = S T /S = 1/1.026 Pr 1/3

, and the local j factor as

Stx Pr2/3 = . (6)

Re 1 / 2 X

For Pr down to 0.6 the error in using Pr2/3 is quite small.

For Pr < 1 , the thickness of the thermal boundary layer is greater than

that of the momentum boundary layer and it becomes necessary to integrate the

energy equation in two steps: wall to 8 and 8 to 6 T --the second step being

with uniform velocity. Using the integral technique, Eckert [10] obtained

/RePr

Nil* = (7

)

1.55/Pr + 3 . 09Jo .372-0. 15 Pr

for a heated flat plate. For Pr « 1 such as is the case for liquid metals,

St - (8)

Pr 1/2 Re 1/2

10
is a good approximation to expression 7.

For 0.5 > Pr > 0.1--the region of interest for noble gas mixtures- -the

denominator in equation (7) varies such that

Nu, ~ Pr°

39 Re 1/2

, (9)

or,

St (10)

Pr°

61 Re 1 ' 2

gives a good approximation.

For turbulent and separated flow, Zukauskas [11] shows that the exponent

of the Prandtl number should be about 0.4 in the Nusselt number relation

Nu, ~ Pr° • 4

,

or -0.6 for the Stanton number relation

Pj-0 . 6

Zukauskas ' s experiments were for fluids with Pr between 0.7 and 109, and the

exponent varied from 0.34 for a flat plate to 0.37 for a cylinder in cross

flow. The moderately low range of Prandtl number was not investigated.

Pickett, et al. [12] measured the turbulent heat transfer of helium-argon

mixtures in tubes and found the expression recommended by Kays [13]

Nu = 0.022 Re0 • 8 Pr°

6

correlated their data well for Prandtl number down to 0.42. The DittusBoelter expression [14]

Nu = 0.023 Re0 * 8 Pr°

A

overestimated the Nusslet number by 8 to 20 percent.

What we are left with is some uncertainty in the dependence of the heat

transfer on the Prandtl number. An estimate of this uncertainty can be made by

assuming that

11
( 11 )

and evaluating the enhancement in St for various values of the exponent n and

Pr, using Pr=0.7 as the base- -the value which was used in most heat transfer

measurement for compact surfaces. Table 1 gives the results of such a

comparison.

Table 1, Relative Enhancement of St (St/(St) Pr » 0 . 7 ) f°r St ~ 1/Prn

n

Pr 2/3 0.6 0.5

0.6 1.11 1.10 1.08

0.5 1.25 1.22 1.18

0.4 1.45 1.40 1.32

0.3 1.76 1.66 1.53

0.2 2.31 2.12 1.87

0.1 3.66 3.21 2.65

For modestly low Pr (say Pr = 0.4) we expect an error of 10 percent in St if

the correct value of n is 0.5 rather than 2/3. For a mixture with Pr as low as

0.2, the possible error jumps to 24 percent.

4. Heat Transfer

The influence of heat transfer on the performance of a regenerative

a refrigerator can be expressed in terms of the regenerator's thermal loss Qreg . Radebaugh, Louie, and Arp [15,16] show that the dimensionless thermal loss is

given by

Qreg K-Hl]

T— = U-O

Qr >. - where e = regenerator effectiveness.

The regenerator ineffectiveness is a complex function of several

variables

,

( 12 )

1 - € I (NTU, Cr/Cf , Cmin /Cmax’ '"'v o i

d

/Cf ) (13)

12
For the special case of Cvoid /Cf = 0, Cmin /Cmax = 1, and Cr /Cf > > 1 the

relationship is given exactly by [8]

l-c

2

2 + NTU

( 14 )

which is identical with the expression for a counterflow heat exchanger. For

regenerators with large NTU which are typical of cryocooler regenerators, cr /cf is generally large for a cryocooler operating at 80 K and above, because the

large surface area required also results in high matrix heat capacity. For our

analysis it is more convenient to approximate equation (14) by

!_ e = I: 89

. , (15)

NTU0 • 9 9 2

which agrees with (14) within 1 percent for NTU greater than 50.

The NTU is related to the Stanton number (St = h/G*C

p

) by

NTU = St (A/A

g

)(16)

Substituting (15) and (16) into (12) gives

Qr e g 1.89

Qr (A/A_

Hu-Hl

l qr St0.992

(17)

as the relationship between the Stanton number and the thermal loss in either a

regenerator or counterflow heat exchanger with balanced flow.

The relationship between the Stanton number and the Reynolds number

depends on the geometry of the flow passage, so no general conclusion is

possible in relating the refrigerant properties to thermal losses. Instead, we

consider four particular geometries: screens, randomly packed spheres, gap

passages with laminar flow, and circular tubes with turbulent flow.

13
4.1

Screens

We use the expression from Kays and London [8]

St*Pr2/3 = K (p)Re' 0 - 43 (18)

S c

for the heat transfer in screen regenerators. Substituting (18) into (17)

gives

Qr eg 1.89

Qr [a/a8

-k (p)]

& c 0.992

Hu-Hl

0 . 6 6 d 0 . A 3 Pr° • e

l qr

For a fixed geometry and pressure, equation (19) gives

•Qr eg2 Pr2 0.66 mw2

' 0 . A 3 f \

Ml

•Qr e g 1 P*l. MW,, CM

(19)

( 20 )

4.2

Spheres

We again turn to Kays and London [8] and use

to obtain

St • Pr2 1

3

= K (p)Re~ 0 • 3 Sp

•Qr eg2 Pr2 0.66 ^w2

" 0 . 3 fMi

•Qr e g 1 .MWi. J*2.

( 21 )

( 22 )

4.3

Gaps

For laminar flow in gaps the Stanton number is given by [8]

St*Pr2/3 = t^Re-1

, (23)

which yields

•Qr e g 2

r _ ^ Pr2 0.66 rMW2

' 0.99 r \

Mi

•Qr e g 1 P

^

J*2,

( 24 )

14
4.4 Tubes

For turbulent flow through tubes, we use the Colburn expression [1,14]

St*Pr2/3 = 0.023 Re-0 - 2

, (25)

which gives

•Qr eg2

0.66 mw2

' 0 . 2 f \

Ml

•Qr e g 1 JPn. MW,.

(26)

An alternative approach, which is equivalent and yields the same numerical

results, is to formulate the problem in terms of the heat transfer coefficient.

For tubes with turbulent flow, the Nusselt number equivalent of (25) is

Nu = 0.023 Re°

8 pr 1/3

, (27)

or

and

0.023

h =

pVDh

l 4

0 . 8 (Pr) 1 / 3 (28)

V f > Pi

0 . 8 f > Ml0 . 8 ?*2

UJ LplJ ^2> [Prj (29)

The relative thermal loss, using equations 12 and 14, is related to the heat

transfer coefficient by

Qr e g 2

1

Qregl O^/h,) 0 - 992

' (30)

Figures 6 and 7 show the relative thermal loss, Qreg 2 /Qregi> f° r he-Ar and HeKr mixtures. Only tubes and packed spheres show improved heat transfer

(reduced thermal loss)

and only He-Kr mixtures in tubes show enough

improvement to be interesting.

15
RELATIVE

THERMAL

LOSS

MOLE FRACTION, Ar

Figure 6. Relative regenerator thermal loss for He-Ar mixtures.

16
RELATIVE

THERMAL

LOSS

MOLE FRACTION, Kr

Figure 7. Relative regenerator thermal loss for He-Kr mixtures.

17
( 31 )

5. Pressure Drop

The pressure drop in the case of the regenerator is given by

To simplify the calculations, we adopt a modified Colburn analogy [1] and

assume that the gas friction is related to the heat transfer by the expression

[15]

f = StPr2/3

(32)

where the parameter a is taken as a constant for a particular geometry. In

fact, a is sometimes a weak function of the Reynolds number, but for our

purposes assuming constant a is adequate. Combining equations (31) and (32)

gives

AP ' L

1 V

2

'

P l 2rh°J [p

pStPr2 /

3

. (33)

The first grouping of terms on the RHS is fixed by the geometry of the

regenerator and the second grouping depends on the operating conditions -- machine speed and fill pressure. As with the heat transfer, we consider four

geometries: screens, randomly packed spheres, gap passages with laminar flow,

and circular tubes with turbulent flow.

5 . 1 Screens

Substituting (18) into (33) we obtain

AP Ksc (P)Ll V

2

P l 2rh a J Ip J [Re

same geometry, operating speed, and

then

41*2 ^Pz

0.57 r 'k P’2

4^i IPi, UJ

p

T.TT (34)

( 35 )

18
5.2

Randomly Packed Spheres

Substituting (21) into (33) gives

AP 'K(p).

p

L' v

2

' f P

P k. 2 rh o , Ip J Re0 • 3

and

ap2 apT

% Pi

0 . 7 f \

P’2 UiJ Mu

for the same machine operating at the same speed and pressure.

( 36 )

(37)

5 .

Gas with Laminar Flow

Substituting (23) into (33) gives

AP X;L

' V

2

' r \

p

P k 2rh aJ Ip J

and

APj

ap:

Pi

Mu

which is the expected result for pure laminar flow.

(38)

(39)

5.4

Tubes with Turbulent Flow

Substituting (25) into (33) gives

and

AP kt l

' V

2

' f \

P

P - 2rh aj Ip J Re 0 • 2

r

ap2 apT

f y pi

o CO f \

Pi

UiJ

(40)

(41)

Figures 8 and 9 give the ratio of the pressure drop of the mixture to

that of helium for He-Ar and He-Kr mixtures. In all cases, mixtures give

increased pressure drop, and for screens, spheres, and tubes the effect of

density makes these increases large.

19
RELATIVE

PRESSURE

DROP

(AP

mi

MOLE FRACTION, Ar

Figure 8. Relative pressure drop through regenerator for He-Ar mixtures.

20
RELATIVE

PRESSURE

DROP

(A

Pmbc

/

A

PHe

)

MOLE FRACTION, Kr

Figure 9. Relative pressure drop through regenerator for He-Kr mixtures.

21
6. Discussion

The use of noble gas mixtures in regenerative cryocoolers does not appear

to give an overall enhancement in performance. For example, although a

reduction of 10 percent in the thermal loss of a sphere regenerator can be

achieved with a 0.1 mole fraction Kr , He-Kr mixture, the resulting 227-percent

increase in pressure drop would give a net loss of performance in a welldesigned regenerator.

The loss associated with the high pressure drop of the mixture can be

mitigated by increasing the flow area of the regenerator- -although at the

expense of increasing the thermal loss. We analyze the effect of flow area on

the thermal and pressure losses by combining equations 17 and 21 to get, for

spheres

,

If we hold constant the regenerator surface area, A, and the hydraulic radius,

rh , then the volume of the regenerator is constant

1.89

pj.0 . 6 6 . 4 3 (42)

Qr [A/Ag-K (p) ]°

992

l qr , Sp

Volume ~ LAg = constant:

or

and the velocity V is related to the flow area by

(43)

Substituting (43) into (42) gives

Qr

In a similar fashion, we obtain the pressure loss as

22
-2.7 (44)

QP AP

The sum of the pressure and thermal losses as a function of flow area is

then

Qr e g

QP 'Qre 6

' \' 0.69

Qp r\

| •Qr

H

O*

1 1

. Qr - o

T

.Qr- o ^go>

where the subscript o refers to some given reference geometry.

In figure 10, we use equation 45 to compare the combined pressure and

• • • • thermal losses for a regenerator having Qreg /Qr = Qp/Qr = 0-1 as a reference

condition for two gases--pure helium and the helium -krypton mixture of 0.1 mole

fraction krypton cited above. Although increasing the flow area Ag does much

to mitigate the increased pressure loss for the mixture, the losses for the

mixture are never quite as low as the minimum for pure helium. The heat

conduction loss is not included in figure 10, so the optimum regenerator does

not occur at the minimum in the curves. Instead the optimum will be at a lower

value of A

g

/Ag

0 because the conduction loss goes as (A

g /A0 ) 2

. Regenerator void volume has a significant effect on the performance of

regenerative cryocoolers, and in general, the smaller the void volume, the

greater the efficiency. For ideal, loss-free Stirling coolers, increased void

volume reduces the cooling power, but not the efficiency, which is always 100

percent of Carnot efficiency. For non-ideal Stirling coolers, however,

increased void volume reduces the efficiency by compounding the effect of

thermal and pressure losses.

We estimate the effect of mixtures on the void volume of the regenerator

by using the expression

V /V r g

/

v

«

*2 NTU(Cr /Cf )Pr2/3 wC

p

Jkdf

. «P A qr (Qc/Qr )(Pm Cm)

(46)

of Radebaugh and Louie [15] which relates the regenerator void volume to

regenerator losses and thermal properties of the working fluid. If we note

that pqr in the denominator can be expressed as (p/p e )p e qr and that p e qr , the

23
0.40 w

co

sson aanssaud smd iviAiaam

24

Figure

10.

Sum

of

thermal

and

pressure

losses

in

regenerator

as

a

function

of

flow

area.
refrigeration per unit volume of the expansion space, is independent of the

mixture mole fraction, then

( 47 )

gives the relationship between the void volume and mixture properties when the

regenerator ineffectiveness, pressure drop, and conduction loss are held

constant. Noting that the molar heat capacity and density are constant, we use

as the relationship between the regenerator void volume, mixture Prandtl

number, and molecular weight.

Figure 11 shows that both He-Ar and He-Kr mixtures increase the

regenerator void volume over that for pure helium.

The heat transfer strongly depends on the geometry of the heat transfer

surface as illustrated by figures 6 and 7. In general, the weaker the

dependence of the Stanton number on the Reynolds number, the greater the

enhancement of heat transfer. If we express the relationship between St and Re

in the general form

(48)

and

P =

to obtain

(49)

St«Pr2/3 = K Re” n

, (50)

then the thermal loss in the regenerator is

1.89 f

p^.0 . 66pg0 . 99n (51)

Qr [ (A/Ag)K ] 0 • 9 9 qr

25
•h

Ca/

Bj

a)/

x, “Ca/

b,

a)

26

Figure

11.

Relative

regenerator

void

volume

as

a

function

of

heavy

component

fraction.
and

•Qr e g 2 ?r2 0.66 MW2

' 0 . 9 9 n

rPi

•Qr e g 1 N>

Because the molecular weight increases more rapidly than the Prandtl number

decreases, geometries with a low value of n show better heat transfer

performance with gas mixtures

Our conclusion that mixtures of noble gases do not give an overall

enhancement in the performance of Stirling type machines contrasts with that

drawn by Vanco [18] for Brayton power cycles. Vanco compares heat transfer and

pressure drop at constant molecular weight (for example, performance with pure

argon versus performance with a helium-xenon mixture of the same molecular

weight) and finds enhanced heat transfer with no change in pressure drop. We,

in contrast, compare systems at constant molar density and constant pressure.

Because the pressure rise per stage in turbo machinery is proportional to

the density, constant molecular weight maintains a constant gross power output

in a Brayton cycle with turbines. With reciprocating compressors and expanders

such as in Stirling machines, however, the pressure rise is independent of

density- -depending only on the geometry of the machine- -and constant molar

density, not constant molecular weight, maintains constant gross refrigeration.

Of all the noble gases, helium has both the lowest density, which gives

the lowest pressure drop, and the highest thermal conductivity, which gives the

best heat transfer. Adding heavier species always increases pressure drop with

little or no improvement in the heat transfer.

Cryocoolers with turbo compressors might benefit from the use of mixtures

for the working fluid because of the resulting reduction in the number of

stages in the compressor. Of course, the dew point of the mixture needs to lie

below the refrigeration temperature.

27
Nomenclature

A

As

cm cm a x Cmin » C f Cr Cvo i d

C

P

f

G

h

Hu

HL k

K

L

•m

MW

NTU

Nu

P

P

Ho 1 Pr

Qc

<lr

Qr e g Qr

rn Re

St

regenerator total surface area

regenerator mean gas flow area, pA,,,

matrix cross-sectional area

matrix specific heat capacity

maximum flow stream heat capacity rate

minimum flow stream heat capacity rate

regenerator matrix heat capacity

heat capacity of gas in regenerator void volume

specific heat capacity at constant pressure of gas

friction factor defined by equation 31

flow stream mass velocity, m/Ag

heat transfer coefficient

enthalpy per unit mass of high pressure gas at warm end of

regenerator

enthalpy per unit mass of low pressure gas at cold end of regenerator

thermal conductivity of gas

proportionality constant between Colburn j factor and Reynolds number

defined by equation 50. Subscripts refer to particular geometry,

length of regenerator

mass flow rate of flow stream

molecular weight of either pure component or mixture.

number of heat transfer units, hA/Cp

Nusselt number, 4hrn /k

porosity

pressure

vapor pressure of saturated heavier component

Prandtl number = u/a = /xC p /k

regenerator conduction loss

refrigeration per unit mass

regenerator thermal loss

gross refrigeration power

hydraulic radius, Ag L/A

Reynolds number, 4rn G//i

Stanton number, h/G»C

p

28
T temperature

V flow stream velocity

V# maximum expansion space volume

Vrg regenerator void volume

x1 mole fraction of lighter component of mixture

Yi ,y2 transport properties of pure lighter and heavier components

Greek

a thermal diffusivity; proportionality constant for modified Colburn

analogy, equation 31

6 momentum boundary layer thickness

$ T thermal boundary layer thickness

e regenerator effectiveness

/i viscosity

v kinematic viscosity

p density

pn matrix density

pM molar density

<f> enhancement factor, defined by equation 5

<acycle frequency

29
References [1]

> Rohsenow, W.M. and Choi, H. , Heat. Mass, and Momentum Transfer . Englewood

Cliffs, N.J., Prentice-Hall, Inc., 1961.

[2] Ely, J.F., Private communication.

[3] Ibele, W. , Thermophysical properties, Handbook of Heat Transfer . W.M.

Rohsenow and J.P. Hartnett, eds .

,

New York, McGraw Hill, 1973.

[4] Brokaw, R.S., Approximate Formulas for Viscosity and Thermal Conductivity

of Gas Mixtures . NASA TN D-2502, 1964.

[5] Stephan, K. and Heckenberger , T., Thermal Conductivity and Viscosity Data

of Fluid Mixtures . DECHEMA Deutsche Gesellschaft fur Chemisches

Apparatewesen, Chemische Technik und Biotechnologie e.V., Frankfurt/Main,

1988.

[6] Mullins, J.C. and Ziegler, W.T.

,

Phase Equilibria in the Argon-Helium and

Argon-Hydrogen systems from 68° to 108° K and Pressures to 120

Atmospheres, International Advances in Cryogenic Engineering . K.D.

Timmerhaus, Editor, New York, 171-181, 1965.

[7] Kate, F.H.

,

A Study of Solid-Vapor Equilibrium in the Helium- Krypton and

Helium-Xenon Systems . Ph.D. Thesis, Stillwater, Oklahoma, Oklahoma State

University, 1972.

[8] Kays, W.M. and London, A.L., Compact Heat Exchangers . 3rd ed.

,

New York,

McGraw-Hill, 1984.

[9] Pohlhausen, E., Z. Angew, Math. u. Mech. , 1 , 115, 1921.

[10]

Eckert, E.R.G. and Drake, R.M. , Jr., Heat and Mass Transfer . 2nd ed.

,

New

York, McGraw-Hill, 1959.

[ill Zukauskas , A., Heat Transfer in Turbulent Flows of Various Fluids,

Turbulent Forced Convection in Channels and Bundles . S. Kakai and D.B.

Spalding, eds., New York, Hemisphere Publishing Corp., 547-561, 1979.

[12] Pickett, P.E., Taylor, M.F., and McEligot, D.M., Heated Turbulent Flow of

Helium-Argon Mixtures in Tubes, Int . J. Heat Mass Transfer, 22., 705

(1979).

[13] Kays, W.M.

,

Convective Heat and Mass Transfer . New York, McGraw-Hill

(1966)

.

[14] McAdams, W.H.

,

Heat Transmission . 3rd ed.

,

New York, McGraw-Hill (1954).

[15] Radebaugh, R. and Louie, B., A simple, first step to optimization of

regenerator geometry, Proc

.

Third Cryocooler Conf .

,

NBS Special

Publication 698, 177-196, 1983.

30
[16] Arp, V. and Radebaugh, Interactive Program for Microcomputers to

Calculate the Optimum Regenerator Geometry for Cryocoolers (REGEN and

REGEN2) , AFWAL-TR-87-3040 , 1987.

[17] Colburn, A.P., Trans. Am. Inst. Chem. Engrs . , 29, 174, 1933.

[18] Vanco, M.R.

Analytical Comparison of Relative Heat-Transfer Coefficients

and Pressure Drops of Inert Gases and Their Binary Mixtures, NASA TN D2677, 1964.

31

u.S. GOVERNMENT PRINTING OFFICE: 1990 - 573-695/45022

NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

4. TITLE AND SUBTITLE

PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

NISTIR 90-3936 September 1990

Effect of Noble Gas Mixtures on the Performance of Regenerative Type

Cryocoolers—Analytical Estimate

5. AUTHOR(S)

David E. Daney

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions) 7. Contract/Grant No.

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY 8. Type of Report & Period Covered

BOULDER, COLORADO 80303-3328

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City . State. ZIP)

Air Force Space Technology Center

Kirtland Air Force Base, New Mexico 87117-6008

10.

SUPPLEMENTARY NOTES

: | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

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

bi bl iography or literature survey, mention it here)

We compare the performance of regenerators that use noble gas mixtures

to the performance of those that use pure helium gas. Both helium-argon and

helium-krypton mixtures are investigated. For some heat transfer surfaces,

a modest gain in heat transfer can be achieved with these mixtures. The concomitant

increase in pressure drop, however, more than offsets the heat transfer gain

so the net regenerator loss increases for all cases we evaluated.

The dependence of heat transfer on Prandtl number has not been measured

for the range associated with noble gas mixtures, 0.2 < Pr < 0.5, and we estimate

that the uncertainty from this source can exceed 20 percent. We give our own

estimates for the transport properties (Prandtl number, viscosity, and thermal

conductivity) of helium-argon and helium-krypton mixtures because of the absence

of experimental data at low temperature.

12.

KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semi colon s) cryocoolers; gas

Prandtl number;

l3. AVAILABILITY

”

mixtures; heat exchangers; heat transfer; noble gas mixtures;

regenerators; thermal regenerators

14. NO. OF

PRINTED PAGES

f^~] Uni imited

Q7] For Official Distribution. Do Not Release to NTIS 40

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