Cooling capacity of Stirling cryocoolers — the split cycle and nonideal gas effects

The general expression for the cooling capacity of a Stirling cooler operating with a nonideal gas is derived. The result demonstrates that thermodynamic regions of negative Joule-Thomson coefficient should be avoided. It is also shown that heat transfer to the expansion space occurs during three of the four steps of the ideal Stirling cycle.

Cooling capacity of Stirling cryocoolers - - the split cycle and nonideal gas effects D.E. Daney Key words: cooling equipment, cryocoolers, nonideal gas effects, split cycle coolers, Stirling coolers, Stirling cycle

Nomenclature

X

position of displacer

Cp

specific heat at constant pressure

Y

position of piston

h

specific enthalpy

Z

compressibility

m

mass

O~

p

pressure

Q

heat transferred

r

(gcomp +

R

gas constant

s

specific entropy

Subscripts

T

tenrperature

C

u

specific internal energy

comp compressor

U

total internal energy

d

displacer

v

specific volume

h

warm (ambient) end

V

volume

max

maximum value during cycle

work

1,2~3,4

points on

W

Vd)/Va,

coefficient

compression ratio

A general discussion of the cooling capacity of Stifling coolers might seem out of date since it has been over 100 years since Schmidt published his analysis of Stifling enginesfl However, there has been a flurry of activity during the past few years in building and designing Stirling coolers for the range of 10 K and below - one of the principal applications being the cooling of superconducting electronics. At these temperatures the helium refrigerant behaves as a nonideal gas. However, analyses have generally either completely ignored nonideal gas effects or have considered only density effects. Nonideal thermal effects (eg, the Joule-Thomson effect) have generally been ignored. In a previous paper 2 we estimated nonideal gas effects on Stirling cooler performance by analyzing Brayton cycle performance over the temperature span of the low temperature stage, This analysis indicated that serious degradation of performance could result from operating at high pressures and low pressure ratios. In this paper we derive The author is at the Thermophysical Properties Division of the National Bureau of Standards, Boulder, Colorado 80303, USA. Paper received 25 May 1982.

(T/v) (~v/OT)p, dimensionless bulk expansivity Th/Tc, temperature ratio (p/T) (OT/OP)h, dimensionless Joule-Thomson

cold end, expansion space

T-s diagram, Figs

1 and 2.

the thermodynamically correct (and incidentally quite simple) expression for the cooling capacity of Stifling coolers operating with a nonideal, single-phase refrigerant. We arrive at the same conclusion regarding the operating pressure. In the process of doing the analysis, we also observed some interesting details generally ignored or forgotten in discussions of the Stifling cycle.

The Stirling Cycle - general discussion The Stifling cycle is generally represented on a temperatureentropy diagram as a combination of two isothermal and two constant volume processes as shown in Fig. 1. Isothermal compression occurs between points 1-2; constant volume transfer of the gas to the low temperature space between points 2-3; isothermal expansion (cooling) between points 3-4; and constant volume transfer of the gas to the ambient temperature space between points 4-1. The integral Stifling cooler (in which the piston and displacer operate in the same cylinder) may be operated to give the volume relationships shown in Fig. 1 for either the two piston cycle or the split cycle. However, the usual

0011-2275/82/010531-05 $03.00 © 1982 Butterworth & Co (Publishers) Ltd. CRYOG EN ICS . OCTOBE R 1982

531

Q1-2

'

/

represent a single point on the thermodynamic surface of the working fluid. The transfer of each differential element of gas takes place at nearly constant pressure, although there is a general decay in the pressure from P2 to P~ (P~ to P3) as the transfer proceeds. The gas in the warm end proceeds along path 2-2', and the gas in the cold end proceeds from 3'-3. The first element transferred follows path 2-3', and the last element follows path 2'-3. Because that portion of the gas present in the cold end undergoes a reduction in pressure (expands), refrigeration (heat adsorption at constant temperature) occurs during the transfer of gas from the warm to cold space. The magnitude of this cooling (process 3'-3) is of the same order as that occurring during process 3-4.

'

/, 3

4

/ 03_4

s Regenerotor

The isothermal expansion process 3-4 differs for the integral or two piston cycle and the split cycle. In the integral cycle all of the gas is expanded at temperature Tc. In the split cycle, however, some of the gas returns to the warm end as the piston is withdrawn. Thus the quantity of gas receiving heat decreases as the cycle preceeds from 3 to 4. (2)

(I)

(3)

C4)

Two piston cycle

iiiiliililill

iiIIifi;] I

H e a t t r a n s f e r r e d t o an i s o t h e r m a l e x p a n s i o n space general case

Split cycle

Fig. 1

Stirling cycle operating as a cooler

arrangement gives the volume relationships indicated for the two piston cycle. We consider this to be the case in this paper. The refrigeration capacity

The transfer of gas from the cold to warm end of the cooler, process 4-1 ; is essentially the reverse of process 2-3. The transfer of each differential element occurs at nearly constant pressure, and the system pressure gradually increases as the transfer proceeds. The gas remaining in the cold end, which follows path 4-4', gives up heat. The cold end heating that occurs during this process exactly cancels the cooling that occurs in process 2-3 for an integral cooler operating with an ideal gas. For a nonideal gas working fluid or a split cycle cooler, these two heat quantities are not equal and opposite. In fact they may differ by an order of magnitude.

Qref is given as 4

Qref = Qa-4 = TcASa-4 = W = t

pdV

(1)

!

$

However, (1), is correct only for the special case of an ideal gas working fluid and the integral or two piston (unsplit) cycle. The ideal gas restriction is apparent by referral to the first law of thermodynamics,

As a result of the above discussion, we observe the necessity of computing the cooling over the complete cycle, not just between points 3-4. Real coolers generally operate with sinusoidal motion of both the piston and displacer so that the separate processes are not distinct. Consequently, the cooling capacity is generally calculated as § pdV, ie, as the cyclic integral of the cold end work. Thus the conceptual error of associating the cooling only with process 3-4 is fortuitously avoided. We obtain the expression for the heat transferred to an isothermal space by two methods. First, we use the second law of thermodynamics to obtain the result in a very direct 2

AU = Q -

W + ham

Now consider the transfer of gas from the warm to cold ends. Although the end states 2 and 3 are at constant volume, the intermediate stages of the transfer do not

532

i

I'

..++'/////

The refrigeration Qref is equal to the work W only when the internal energy is independent of the pressure and when no mass enters or leaves the expansion space. A split cycle operating below 10 K satisfies neither condition. Let us consider the Stirling cycle in more detail, this time referring to Fig. 2. As before, isothermal compression occurs between points 1-2. No heating or cooling occurs at the low temperature space because there is no gas present there.

2'

/iv/ / j,?/ / / / S7/7

(2)

.,,L'J_/,f 3'

_ 3

j

4s

4

4

s Fig. 2 Stirling cycle showing details of gas transfer between working spaces

CRYOGENICS.

O C T O B E R 1982

Equation (10) is valid for any reversible isothermal process for both constant mass and variable mass systems. Displocer- regeneretor

For an ideal gas a = 1. With the additional restriction of an isothermal process, p d V = - V d p , (10) becomes

(Qnet)

~pdV

=

(1 1)

ideal gas [ I I I i

1 I I.,i - - Control volume I ._I

Te

!

the standard expression for computing the cooling capacity. Next let us verify (10) by applying the first law to process 3'-3 (Fig. 2) where cooling occurs as the displacer rises.

/-do Fig. 3

dU = dQ

Control volume for expansion space

-

(12)

dW + hdm

or

manner. Then we verify this result by the more familiar application of the first law to one of the processes in which

mdu

+ udm

= dQ

-

+ hdm

(13)

= u + pv

(14)

pdV

heat transfer occurs at the cold end of the cooler. We begin by considering the heat transferred to a control volume at the cold end of the cooler, Fig. 3. If the gas in the control volume is in equilibrium, and any gas entering or leaving is in equilibrium with the control volume gas (perfect regeneration), then the heat transferred to the control volume is dQ

(3)

mTds

=

Noting that d V = mdv

we obtain dQ = mdu

(15)

+ pmdv

or

for a differential process, and the net heat transferred over the cycle is the cyclic integral, Qnet = f

+ vdm, andh

dQ=V

--+p~-

(4)

mTds

For an isothermal process Note that (3) holds for both constant mass and variable mass reversible processes.

~v

dVT =

ds=

~-T P

(5)

T

dp, du T =

~

~

p

(6)

+ p

~v'

3u

= -

T

T

~T- p

(7)

SO that

~-f p

o~dp

(9)

substituting (9) into (4) we obtain the net heat transfer to the isothermal expansion space, Qnet =

~

&Vdp

CRYOG EN ICS. OCTOBE R 1982

(19)

eWdp

(20) Q = -

T

_ P

p

(8)

so that for an isothermal process the entropy change is given by dsT -

3v

2¢

or

v

T

(19) into (18) gives

dQ = -

The dimensionless bulk expensivity is defined as --- -

(18)

A thermodynamic identity gives

we obtain ds = Cp

] dp

and (16) becomes 1 ~u

~-T- p

(17)

T

and by using the thermodynamic identities CP = T

dp

3p ]

--T

For a real gas the specific entropy is given by

(10)

faVdp

for the displacer moving. First law consideration of the process with the displacer stationary (3-4) gives the identical expression, thus confirming the more direct second law approach. Computation of the cooling capacity of a Stirling cooler using (10) requires an indicator diagram (measured or calculated) and knowledge of thermal expansivity a as a function of pressure and temperature. Fig. 4 gives a for

533

helium a over much of the range of interest for low-temperature Stifling coolers. It is clear from Fig. 4 that relatively low pressures are favoured for Stirling cooler operation below 10 K.

Piston

The various nonideal properties of gases are not unrelated, so that the adverse thermodynamic region for Stifling coolers is essentially the same as for Brayton and Claude coolers. That is, a bulk expansivity a less than one is associated with a negative Joule-Thomson coefficient ~. This is easily shown using thermodynamic identities to obtain

a = (Cp/R)~/Z + 1

I-~

7-.

v,

- - Displacer - regenerator

(21)

or conversely

d/

(ZR/Cp) (a

=

-

1)

(22)

Fig. 5

Clearly, the region of negative J - T coefficient ff is also the region where a < 1 and where the refrigeration capacity is adversely affected according to (10). Split

cycle cooling capacity -- ideal gas and ideal

Split

p =

cycle Stirling cooler geometry

Pmax 1 + ( / 3 - 1)x

(26)

The cooling during the process is calculated using (10),

cycle Finally, we quantitatively examine the consequences of the cooling and heating that occur during displacer travel in a split Stifling cooler. For simplicity, we consider an ideal gas.

Cooling during displacer movement (process 3'-3), piston in, displacer moving up. Referring to Fig. 5 we express

Q3'-3 = +

(3 "3 I

Vcdp = +

f' 0

XVdPmax (t3-- 1)

[1 + ( / 3 - 1)x] 2 dx (27)

or

the cold and warm volumes as

Vc

=

XVd

Vh = (1

-

Qa'- a = + Pmax 1 ° g Vd l 3 [ 3[- 1

(23)

x) Va

(fl--1)]__/3

(28)

(24)

Equating the warm plus cold masses to the total mass gives

Heating during displacer movement (process 4-4'), piston out, displacer moving down. The equation for the volume is

p(1 - x) Vd

RTh

+ PXVd - PmaxVd RTe RTh

(25) (29)

Vc + Vh = Vd + Vcomp

so that equating the warm plus cold masses to the total mass gives

Solving for the pressure p and defining the temperature ratio/3 = Th/Tc we obtain

[(1-x)

P

2.5

Va + Vcomp] p x V d + - RTh RTc

_

phVd RTh

(30)

Solving for p we obtain

2.0 IO bor

p _ 15

where r ~=

4O

(32)

I 4

8

12

r,K

534

(Vcomp -I- Va)lVa, the compression ratio.

t

1.5

4

=

(31)

As before, we use (10) to obtain the heat transferred:

I.O

rs

Fig.

Pmax r + (1-/3)x

Bulk

expansivity for helium

16

20

Note that the heating Q4-4' (32) is equal and opposite to the cooling Qa'-3 (28) for the special case of the compression ratio r = 1, ie, when the device does no cooling.

CRYOGENICS. OCTOBER 1982

Cooling with displacer stationary (process 3-4), displacer up, piston moving out. The mass balance for the system

Processes 3'-3 and 4-4' are evaluated using (28), and process 3-4 uses

gives

pVd + pyVcomp _ Pmax Vd R Tc R Th R Th

Pmax 13 + ( r - 1)y

(34)

Summary

The heat transferred using (10) is Q3-4 = + VdPmax [/3

(36)

The net cooling is greater for the two piston or integral cycle, but some of this gain may be offset by irreversibilities that will occur during 4-4' in a real machine.

and solving for the pressure we obtain p _

Onet - emax Vd log (r -- 1) /3

(33)

r+/3-1r--1 ]

(35)

Numerical example. In order to compare the heating and cooling that occur in the expansion space during the various parts of the cycle, let us consider a numerical example with the temperature ratio/3 = 4 and the compression ratio r = l 2. Equations (28), (32) and (35) give: Process

Q/pmax Vd

3'-3 3-4 4-4'

+.212 +. 183 -.023

Two very interesting results emerge from the example. First, the cooling as the displacer moves up is greater than that with the displacer stationary. Thus ignoring cooling during process 3'-3 neglects more than half the refrigeration. Second, the cooling as the displacer moves up can be an order of magnitude greater than the heating during the downward travel. By comparison the values for the two piston cycle are: Process

Q/PmaxVd

3'-3 3-4 4-4'

+.212 +.600 -.212

C R Y O G E N I C S . O C T O B E R 1982

Heat transfer to the expansion space of an ideal Stirling cooler occurs not just with the displacer stationary (process 3-4, Fig. 2) as is commonly thought, but also as the displacer transfers gas between the warm and {:old spaces (process 3'-3 and process 4-4'). In the idea[ split cycle the cooling during the transfer of gas to the expansion space is generally much greater than the heating that occurs during transfer of gas from the expansion space by the displacer. The thermodynamically correct expression for the cooling capacity of a Stifling cooler operating with a real gas is Qnet = - ¢; aVdp

(10)

where a is the dimensionless bulk thermal expansivity. For an ideal gas a = 1, and for a real gas a may be either greater or less than one. The inescapable conclusion of the analysis is that regions of low a (high pressures for T < 10 K) should be avoided.

References 1

Schmidt, G. Theorie der Lehmannschen calorischen Maschine,

Z Verb dtlng, 15 1 (1876) 2

3

Daney, D.E. Some thermodynamic considerations of helium temperature cryocoolers, NBS Special Publication 607, Refrigeration for Cryogenic Sensors and Electronic Systems, J.E. Zimmerman, D.B. Sullivan, S.E. McCarthy (eds) (1981) 48-56 McCarty, R.D. Thermophysical properties of helium-4 from 2 to 1500 K with pressures to 1000 atmospheres, NBS Technical Note 631 (1972)

535