Theory of ideal orifice pulse tube refrigerator

Theory of ideal orifice pulse tube refrigerator M. David,

J.-C.

Markhal,

Y. Simon

and C. Guilpin*

Ddpartement de Physique de I’Ecole Normale Superieure, 24 Rue Lhomond, 75005 Paris, France “Groupe de Physique des Solides, Universite Pierre et Marie Curie, Place Jussieu, 75005 Paris, France

Received 5 May 1992 The main purpose of this paper is to explain the operation of the orifice pulse tube refrigerator (OPTR). An analytical model of the ideal OPT has been developed. The mechanism of heat flow at the tube ends is clearly explained as the result of the hysteretic process of the elements of gas entering and leaving the tube. The motion of the buffer gas is deduced by numerical integration and the expected balance equation for the heat flows at the hot and cold exchangers is established. A numerical calculation of the velocity profile along the pulse tube is in good agreement with hot-wire anemometry data. In working conditions, we found, for the gross refrigeration power, (o), theory/experiment ratios as low as 1.2, whereas those previously reported by Starch and Radebaugh were about 3 - 5. The differences between the theory of Radebaugh et a/. and our model are following: (1) Radebaugh and co-workers assume small sinusoidal oscillations of the gas pressure in the tube (APIP 4 1) whereas we describe the gas flow in the tube for any time-dependence of the pressure oscillation P(t); (2) In our model, (cj), is expressed with a minimum number of independent and controlled parameters relative to the OPT. In a double inlet pulse tube configuration, our test apparatus was able to achieve a 32 K temperature limit. Keywords:

orifice pulse tube refrigerators; analytical modelling; refrigeration

Nomenclature A a C cr 7 K k L kl riz P P P ’

i r s T t

001 i 0

154

Dynamic pressure in the pulse tube Dynamic pressure in the reservoir Gas capacity of the reservoir Specific heat at constant pressure Specific heat at constant volume Pulse frequency Compression factor Instantaneous compression factor Length of the tube Amplitude of the mass flow rate Mas flow rate Pressure Average pressure P-P Heat flow rate Orifice resistance Ideal gas constant Cross-section area of the tube Temperature Time

- 2275/93/020154

1993

V

Butterworth

Cryogenics

-08

- Heinemann

Ltd

1993 Vol 33, No 2

X

Gas velocity Position along the tube

Greek letters i cp i P ; w

Thermal diffusivity Viscous penetration depth Angular displacement about t-axis Kinematic viscosity Time constant of the orifice and reservoir circuit Gas density Period of the pulse L-x Angular frequency

Subscripts a C

cp r t th

Hot end of tube Cold end of tube Compressor Reservoir Tube Thermal

Theory of idea/orifice pulse tube refrigerator: M. David et al. The orifice pulse tube refrigerator (OPTR) has been extensively studied in recent years 1-7. Its reliability and relative simplicity, as well as its high refrigeration capacity, make the pulse tube a suitable cooler for longlife space applications 4 (Figure la). The most reliable attempt to explain the working principle of the OPTR is due to Radebaugh 5 and then to Storch, Radebaugh and Zimmerman 6'7. As an introduction to the present work, some results of the Storch, Radebaugh and Zimmerman (RSZ) theory are worth mentioning and discussing briefly. Assuming small sinusoidal oscillations of the gas pressure in the tube, for instance in the form shown in Figure 3, P(t)= P A cos cot (with A ,~ P), RSZ derive several expressions for the gross refrigeration power (q) as a function of a set of measurable parameters. We may disregard those expressions that contain the swept volume of the compressor V~p, because they imply further arbitrary assumptions regarding the compressor-regenerator system, and thereby entail larger divergences between experimental and calculated magnitudes 7. Using the notation of the present paper, the RSZ equation 7 for (q), only expressed in terms of OPT parameters, reads

(q) = ~rlT(A21P)Vt f c o t ~o

(1)

where ~o is defined by cos ¢ =

(~t~/M~)ra/L

(2)

Here 7 is the ratio of heat capacities Cp/Cv, P and A as stated above are the average and dynamic pressures, Vt = sL is the pulse tube volume, f the pulse frequency and Tc and Ta are the temperature of the cold and hot ends respectively. A;/c and Ma stand for the amplitudes of the mass flow rates mc and m~ at the tube ends. It should be noted that Equation (1) was wrongly stated in Reference 6 (Equation (15)) and corrected in Reference 7 (Equations (2-50). When compared to experiment, Equation (1) suc-

cessfu,lly accounts for the observed linear dependence on Ma/Mc of (q). Yet the linear dependence on the involved quantities A, A/P, f and Ma/Mc, as displayed in Figure 6a of Reference 6, can hardly be consistent with Equation (1) except for small values of h)/,/h,:/c. On the other hand, predicted magnitudes are 3 to 5 times higher than experiment 6. Moreover, on applying different but equivalent equations for (q), which came out at different stages of the RSZ development 7, one finds a large dispersion of predicted values (by a factor of 2 to 3). Note also that some experimental values of the right-hand side of Equation (2) may exceed unity (see for example Figure 6a of Reference 6). We still believe that Equation (1), cumbersome though it is, has been derived correctly; we shall be able to retrieve the same result in another way. The discrepancies between theory and experiment rather indicate that the main underlying assumption of small and sinusoidal pressure oscillations is unrealistic in practical situations. Apart from the relevance of the harmonic approximation, there is another difficulty about Equation (1) (or similar expressions for (q) in References 6 and 7), because the parameters involved are not independent and cannot be controlled separately in a test apparatus. This makes such an expression unsuited to optimization of the OPTR. By rearranging Equation (1), one can obtain simpler equivalent forms (Equation (15) below). So one realizes that some variables have been introduced somewhat artificially in Equation (1), such as the pulse frequency or the tube volume Vt. In view of the foregoing remarks, this work has two objectives: While still assuming ideal working conditions, as recalled in the next section (in particular the adiabaticity along the tube), we wish to describe the gas flow in the tube for any time-dependence of the pressure oscillation P(t). By investigating the resulting velocity and temperature profiles, we may hope to understand the remaining divergences between theory and experiment, which now should

To the compressor

Reservoir1

)rifice

Regenerator

Solenoid Valve

Heat Exchanger

!

qa

Pulse

tube

qc

1

Heat Exchanger

a

b

c

Figure 1 Schematic of the different types of pulse tube: (a) OPT; (b) HPT; (c) DIPT

Cryogenics 1993 Vol 33, No 2

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Theory of ideal orifice pulse tube refrigerator: M. David et al. be attributed to deviations from the ideal behaviour, and thereby attempt to reduce them in designing an OPTR. 2 Given some form of P(t), (~1) should be otherwise expressed in terms of a minimum number of independent parameters relative to the OPT, such as the orifice opening, the volume of the reservoir or the tube dimensions. Actually the detailed waveform depends on the design of the compressorregenerator system. However, as the main features of the pressure oscillation are both easily measured and controlled, it may be regarded as a primary constraint. Of course we need a simple tractable expression for (t/) for easy numerical calculation, but the main point is to decide upon the relevant OPT parameters in optimizing the available refrigeration power of the ideal OPTR. The ideal refrigeration power, even though never achieved, represents a useful reference limiting value, somewhat like the Carnot efficiency of heat engines.

Describing the ideal process (the assumptions} Except for the harmonic approximation, which is just aimed at simplifying calculations, our analysis of the gas flow in the pulse tube relies essentially on the same assumptions as used by Radebaugh and co-workers. We will summarize them in this section, while stating our notations. It should be emphasized, however, that the ideal behaviour characterized by these assumptions only refers to the OPT part of the refrigerator such as sketched in Figure 2. Once given P(t), the kind of compressor or regenerator no longer has importance in our model. For definiteness helium will be used as the (best) working fluid. At / ~ - 1 0 bar, helium may be reasonably regarded as an ideal gas (precision _< 10%) down to 30 K. Its equation of state is P -- arT, where p is the fluid density and r the gas constant (r = 2.08 J g - l K-l). Heat capacities are cv = 3/2r (J g-l K-l) and 3' = 5/3. The first important assumption concerning the gas flow is that P is uniform along the pulse tube, including (as checked by experiment) upstream, just before the cold end exchanger, and downstream, behind the hot exchanger. This means that inertial and viscosity forces are unable to produce significant pressure gradients, so that, to a good approximation, mechanical equilibrium ( V P - 0) is achieved at any time in the tube. We can ascertain this fact by giving some orders of magnitude. Let us take a low pulse frequencyf = o~/2~r = 3 Hz, and

Te

o

Ta

M liiiiiiiiiiiiiiiiiiiiiiil

me_ w

i

i

i

~-

0

Xc

×a

L

Figure 2 Schematic of the orifice pulse tube. The gas (b) never leaves the tube

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Cryogenics 1993 Vol 33, No 2

a typical mass flow rate rh = OSV ~-1 g S - l , where s = 2 cm 2 is the cross-sectional area of the tube and v is the fluid velocity. Pressure gradients V P - DI~/L along the tube resulting from inertial terms in the equation of motion are the order of OOJVor pv2/L, giving DP - 100 Pa ,¢ P - 106 Pa. As for viscosity, consider the fluid labelled (a) and (c) in Figure 2 which moves in and out of the ends of the tube during a cycle. As the heat exchangers also act as flow straighteners, the velocity profile of the gas entering the tube is flat and the incident flow is laminar. Then the effect of viscosity comes in at the boundary layers and could both affect the profile and produce turbulence. But during the time At that gas (a) or (c) remains in the tube, viscous diffusion only takes place over a thin boundary layer. As At _< r = l/f, the pulse period, this does not exceed the viscous penetration depth ~ = (p/0o)i/2. Here v is the kinematic viscosity of helium; at P = 10 bar and T = 300 K (v maximum), we find 5 ~< 1 mm. The conclusion is more doubtful for the so-called buffer gas (labelled (b) in Figure 2), which never leaves the tube. Spreading of turbulence in the buffer gas is likely to happen at large Reynolds numbers. If such is the case, the buffer gas would behave as a well demarcated turbulent plug between laminar regions, such as is sometimes observed in pipe flow s . In fact turbulence growth in the pulse tube may be revealed as an important factor in OPTR performance and should be worth investigating. Nevertheless, for simplicity we shall neglect this complication. Thus we are led to the next assumption that no mixing of the gas occurs in the bulk of the tube and that we are dealing with one-dimensional laminar flow. Lastly we assume the absence of heat transfer either between elements of gas in the tube or between the gas and the tube wall. In other words, the flow in the tube is isentropic. The adiabatic assumption relies on the smallness of the thermal penetration depth dithcompared with the tube dimensions; ~th ~- (_X/o~)1/2 where X is the thermal diffusivity of helium. At P -- 10 bar, T = 300 K (X maximum), 8th -- I mm at a few Hz. The tube ends in two isothermal heat exchangers (Figure 2): at x = 0 (temperature To) and x = L (temperature Ta). We are concerned with steady-state conditions, after system cool-down, so that Tc and Ta are' constant. In our test apparatus a water-cooled coil at the hot end maintains Ta close to the ambient temperature, avoiding a temperature gradient between the reservoir and the hot heat exchanger. Since pressure variations, Pr = Pr - P, in the reservoir are very small, helium gas does not undergo significant temperature deviations T - Ta. Both exchangers are assumed to be perfect, in the sense that the temperature of the gas emerging from the exchanger is set to be Tc at the cold end and Ta at the hot end. As a result heat is delivered to or extracted from the gas at the tube ends, at rates depending on the gas temperature upstream of the exchangers. Let qc (qa) be the rate at which heat flows into the cold (hot) heat exchanger while withdrawn from the gas leaving the tube, i.e. during the half-period when Vo = v(0, 0 < 0 (rE = v(L, t) > 0). On the other hand, let t/" (t/:) be the heat flow rate during thecomplementary half-period when the gas enters the tube. It is useful to make a distinction between qc and t/" as they concern

Theory of ideal orifice pulse tube refrigerator: M. David et al. unconnected features of the OPTR. The controlled temperature homogeneity of the hot part of the OPT should ensure that q" ~ 0. But g/~ > 0 results inevitably from the ineffectiveness of the regenerator and is to be counted in loss terms. Independently, the very purpose of an OPTR is to obtain g/~ < 0. The quantity of interest is ( q ) , averaged over a cycle, which is usually referred to as the gross refrigeration power 6'7. We may anticipate the following important result, pertaining to an ideal OPT 6 (q) = (-q~)

=
(4)

If the dead volume between the hot end of the tube and the orifice is neglected, rhr = dm~/dt is also the rate ma at which mass is flowing through the hot exchanger and the orifice. Assuming that the gas flow through the orifice and in the reservoir is adiabatic, a straightforward relation follows between the variation of m r and the small variations p~ = P r _ t 5 of the reservoir pressure m, = mr = 315(Vr/rT,)p~ = Cpr

(5)

Here Vr is the reservoir volume (Vr = 0.5 1) and C is defined as the reservoir capacity in an electrical analogue (current/mass flow, voltage/pressure). Then we assume that the mass flow through the orifice obeys a linear law in the form p - P~ = p - p~ = Rrh r

(6)

which defines the resistance R of the orifice. From Equation (5) and Equation (6), we have pr + OPt = p(t)

I

2a

y

~/2

/

.

t

(3)

Equation (3) will be rederived below as a simple consequence of our model, consistent with the enthalpy flow analysis by Radebaugh 6. In experiments, we only have access to the net refrigeration power (q~ + t / ' ) , such as resulting as a whole from the oscillating flow. Therefore, in so far as Equation (3) holds and ( q ' ) ~ 0, an estimation of the gross refrigeration power (g/) will be made more conveniently at the hot end. We thus begin by calculating (qa), especially as, for a given P(t), it turns out to be determined by tlae few parameters of the hot part of the OPT. Equation (1) does not exhibit this simple dependence. Let m(x, t) be the mass of gas downstream of a section x of the tube, including the mass mr(t) in the reservoir. The mass flow rate through section x is m = Om/Ot = psv

P

--t Figure 3

The case of sinusoidal pressure oscillations

as sinusoidal (Figure 3) or alternate exponentials (Figure 4 as used in our experiments), one may be content to measure the peak-to-peak amplitude 2a of pr; then, using Equation (7), pr(t) is calculated, while the only unknown parameter 0 or R is determined from 2a. In any case, the pressure oscillation p~(t) is observed to be quasi-sinusoidal. This is due to the fact that the o r i f i c e - r e s e r v o i r system acts as a low-pass R C filter, which essentially restores the first harmonic. Thus, if p~ = - a is minimum at t = tt, it is reasonably well approximated by (8)

p~(t) = - a cos w(t - tl)

According to Equation (7), at t = tl, P = Pr and /~ > 0. As a is small compared with the dynamic pressure A, tt is close to a zero of p(t), as shown in Figures 3 and 4. If p(t) = - A cos opt (Figure 3), Equation (8) follows rigorously from Equation (7), with a = A(1 + ~0202)-j/2 ~ A/o~O (coo ~, 1). In general, the amplitude of the first harmonic of p(t) is the order of A and again a ~ A/o~O ~ f - l .

Calculation of (q) at the hot end During the half-period [fi, t I -t-r/2], Pr > 0 and ma increases so that the gas is flowing out of the hot end. Consider the history of the element of gas that leaves the tube at some time t between tl and fi + r/2. It has entered the tube (x = L) at a time t' = g(t) (Figure 4) when the tube pressure was P ' = P(t') < P(t). The relationship between t and t' is obtained by simply writing that ma = mr(t) = mr(F), or equivalently

pr(/) = pr(t')

(9)

(7)

where O = R C - 1 s. In common working conditions o~0 _> 10, so that Pr "~ P and rhr ~- p / R . Introducing the orifice resistance or the time constant is useful for the discussion, but linear Equation (6) may be questionable at large dynamic pressures. So direct and precise measurements of p,(t) are desirable; ma is then deduced from the more reliable Equation (5). Nevertheless recording very small pressure oscillations may prove difficult. If p(t) is fitted by simple analytical forms, such

t'

A .................

tl + 1;/2

2aI~

Figure 4

The gas pressure in the tube (alternate exponential) and the pressure in the reservoir as functions of time when a valved compressor is used

Cryogenics 1993 Vol 33, No 2

157

Theory of ideal orifice pulse tube refrigerator: M. David et al.

Equation (9) yields t' versus t as illustrated in Figure 4. According to the approximate expression (8) for p,, t' - 2tl - t. In travelling back and forth along the tube, the element of gas has experienced an isentropic process. Consequently, its temperature has changed from Ta at t' to TL = T(L, t) in accordance with the isentropic law of an ideal monatomic gas, T = constant x p2/5 TL(t) = Ta (PIP') 2/5 > Ta

(10)

Thus the mass of gas mad/which flows out between t and t + dt rejects heat to the hot exchanger at a rate

qa = macp(TL - T~)

(11)

Integrating Equation (11) over the half-period, or alternatively over the whole period T, since TL = Ta between t~ + z/2 and t~ + r, we obtain t1 + r/2

(8)

~-- < q a )

:

Cp/T

I

#Ia(TL -- Ta)dt

tl

= Cplr I

Ta)dt

/fla(TL --

(12)

= Cplr f rhaTL dt

Integrals without specified limits are taken over a period r. The last integral in Equation (12) follows from the fact that (at any position) the integral of m, over a period vanishes. Using Equation (5) and Equation (10), the first integral in Equation (12) can be rewritten as l tl + r/2


dtj

prk(t)dt <<.3VrfaK

(13)

- cp(TJR)AK/r where k(t) is a bounded-above compression factor defined as

k(t) = (p/p,)2/5 _ 1 < K = [(P + A)I(P - A)] 2/5 _ 1 (14) Note that the upper bound in Equation (12), which gives a rough estimate of
= %/5 (Ta/R)A2/P

(15)

Experimental verification Equation (13) or Equation (15) clearly indicates that large (theoretical) refrigerationpowers will result from both a large pressure contrast (P + A)/(P - A) and a low

158

Cryogenics 1993 Vol 33, No 2

orifice impedance (meaning large mass flow rates ma =A/R). Now the pulse frequency or other parameters such as the tube volume, though not involved in Equations (13) to (15), may appear indirectly as important factors in the complex problem of optimization, depending on the specific constraints of an actual refrigerator. Thus the assumed adiabatic conditions should be better achieved by increasing frequency. But the amplitude A of the tube pressure (then K) decreases with increasing frequency because of the filtering action of the regenerator and OPT. In deriving Equations (10) to (13), we have implicitly assumed that any element of gas entering the tube at x = L could never reach the cold end. Therefore V, is required to be large enough to warrant the existence of the buffer gas. But large V, (i.e. small impedance) again contributes to make A lower. Furthermore, for the same reason, R and A become interdependent. We stress that the f and Vt dependence of (8) such as suggested by Equation (1) is not the point here, since, as shown below, Equation (1) reduces without further approximation to the much simpler expression Equation (15). Anyhow, it is preferable to start from the most simple expression for the quantity to be optimized. As k(t) and Pr are zero at both limits of the integral in Equation (13), the main contribution to (8) arises in a middle interval when pr is a maximum. This suggests that it would be desirable to open the orifice intermittently, for short periods near t = 0, r/2, r . . . . when P(t) = P 4- A, so that k(t) = K. This explains the better performances we obtained (Figure lb) by applying this principle in the so-called hybrid pulse tube (HPT) 2. Another way to take advantage of the maximum pressure contrast, such as produced at the input by the compressor, is the double inlet pulse tube (DIPT) (Figure lc). Recently (June 1991) using a DIPT with a square-wave input, we were able to obtain Tc = 32 K with the following parameters: L = 25 cm, s = 1 . 9 c m 2, Vr = 0 . 5 1 , Ta = 2 9 3 K a n d f = 2 . 7 H z . Independent of any optimization, Equation (13) is well-suited to calculate how much the OPT deviates from ideal behaviour. We have measured
Theory of idea/orifice pulse tube refrigerator: M. David et

al.

Table 1 Two sets of results obtained by varying orifice opening, for otherwise similar frequency and pressure conditions

/~ A f a 0 = RC

Tc Oa (exp) ~a (th) th/exp

(bar) (bar) (Hz) (bar) (s) (K) (W) (W)

Case A

Case B

7.9 2.03 2.70 0.106 1.2 100 7.3 8.71 1.2

7.7 2.5 2.64 0.187 0.91 135 9.1 15.65 1.8

0'6 tx\",,

0

.

4

'

~

, .. ~-er-ff-o--o---o-- o "7 E

"-"

0,2 -, / I

0

1/2

T

-0.2 -0.4

experiment = 1.8) should be explained by increasing deviations from ideality, as the gas (a) (Figure 2) penetrates more deeply into the tube. The lack of adiabaticity and/or the formation of turbulence in the buffer gas are the expected sources of trouble. An attempt to analyse unwanted heat transfer or gas mixing in the tube should require preliminary knowledge of the gas flow in the ideal OPT Velocity

profile in an ideal O P T

The velocity profile along the tube is easily deduced from the assumptions made of a one-dimensional isentropic flow in instantaneous mechanical equilibrium (P(t) uniform). The continuity equation can be written in the form

-(llp)Dp/Dt = div 7 = OvlOx

(16)

where the substantive derivative D/Dt denotes the rate of change following the fluid. Using the adiabatic connection between P and 0 for an ideal monatomic gas, P = constant x p 5/3, we then have

( 1/p)Dp/Dt) = 3/5( 1/P)DP/Dt = 3/5(P/P)

(17)

where P = dP/dt. From Equation (16) and (17), we see that v changes linearly along the tube v - vL = 3/5 (P/P) ( L - x )

(18)

where VL = v(L, t) at the hot end is simply related to Pr through ma~--Cpr = PLSVL

=PSVL/rTL

(19)

Then substituting TL from Equation (10) for halfperiods such as [fi, t~ + r/2], and TL = Ta elsewhere, vL can be expressed in terms of the same quantities, pr and k(t), as used in numerical calculations of (q,)

VL=3/5(Vr/s)p'Ill +k(t)

[tl - r/2, tl] [tl, tl + r/2]

(20)

Now that the gas dynamics is involved, note the important role of tube size, as v o¢ s-I. By using Equation (18) and Equation (20), v can be calculated for any point x along the tube. According to Equation (18), discontinuities in a0 at t = 0 and r/2 (Figure 5) have

-0.6

Figure 5 The gas velocity at the middle point of the pulse tube as function of time. Data are taken from the experiment described in Reference 9. The solid line represents the calculated velocity according to Equation (18) and Equation (20) and open circles are experimental. Unfortunately, turbulence generated along the anemometer holders, and convected with the gas, prevented negative velocities from being measured. From the occurrence of noisy signals between r/2 and r, we were just aware of the discontinuities in v as expected from numerical calculations. The dashed lines are obtained on substituting in Equation (18) arbitrary polytropic coefficients, viz 2/5 (lower line) and 4/5 (upper line). Experimental data near t = 0 confirm the 'adiabatic' coefficient 3/5

repercussions throughout the velocity field in the tube, except at x = L where VL is a continuous function of time. Velocity measurements have been reported in a previous work for a pulse tube 20 cm in length and 1.9 cm in diameter 9:°. A hot-wire anemometer was located at the middle point of the tube. Some data are shown in Figure 5. Allowing for the difficulties in obtaining precise and reliable velocity measurements, the fair agreement between theory and experiment supports the correctness of the present analytical model. Calculation

of

(qc)

at t h e cold e n d

The velocity at the cold end, v0 = v(0, t), is given by v0 = VL -- 3/5 (P/P)L

(21)

from which the mass flow rate rnc =psv0 can be calculated, as well as the heat flow qc = me(To- To). Following the same line of reasoning as above for (~¢a), we obtain an expression for (qc) similar to the last integral in Equation (12)

(qc) = - Cp/r i mc To dt

(22)

Multiplying Equation (21) by P/r = pT and integrating over a cycle, the last term vanishes, leaving

i p°T°v°dt = i pLTLVLdt

Cryogenics 1993 Vol 33, No 2

159

Theory of ideal orifice pulse tube refrigerator: M. David et al. Hence

tl+~..

~m

--(qc)-----Cp/rf mcTodt=cp/T f rhaTLdt=(qa)

a (23)

which is the expected Equation (3). We emphasize that the presence of turbulence in the buffer gas, as discussed above, should invalidate Equation (18), and, therefore, Equation (23). Let us return to the case of small sinusoidal oscillations of the gas pressure, while still assuming the usual condition oJ0 ~, 1. Taking t~ ~ r/4 (Figure 3) and a = A/oJO, the flow through the orifice is proportional to the dynamic pressure rh~ = Cp, - - ( A / R ) cos o~t

b

t1+1]

(24)

C

Using Equation (21) and m~ = Psvo/rT o, and keeping terms only to first order in A in me, we obtain

t 1 + 31" 2

Ot¢ = -3/5(oJ/rT~)aVr cos(60t + ~o)/cos ~o = - A / R ( T J T c ) c o s ( o a t + ~p)/cos ~o

(25)

where tan p = A V t / a V ,. From the amplitude ratio K/~/M~, we find Equation (2). In agreement with the RSZ result, the mass flow through the orifice lags the mass flow at the cold end by the angle p (denoted as 0 in the RSZ paper). Finally, on substituting tan ¢ = Vto~O/Vr, Equation (1) reduces to Equation (15) as expected.

Motion of the buffer gas and the hysteresis effect The motion x(t) of any element of gas can be found by integration of Equation (18). Letting ~ = L - x , this equation reads - ~ + 3/5(P/P)~ = - VL(t)

(26)

d Figure 6 The motion of the buffer gas during a cycle in case A of Table I such as obtained by numerical calculation

It is instructive to plot the periodic motion of the 'hot' side of the buffer on a pressure-displacement diagram, P - x a . We obtain a hysteresis loop traversed in a clockwise direction as shown Figure 7. The system consisting of the gas (a) and the gas in the reservoir describes a cyclic process. According to the first law, the work done on this system during a cycle equals the amount of heat (q~) withdrawn from the system

W= ~ Ps d x a =

Consider an element of gas the position of which is ~0 at time to. Its subsequent motion is given by ~p3/5 = ~oP(to)3/5 _

it

(qa)

(28)

P (bars) 10

VLP3/5 dt

(27)

tO

Equation (27) can be used in particular to determine the 'cold' and 'hot' limits, xc and Xa, of the buffer gas (Figure 2) at any time of the cycle. Thus x, = L (~0 = 0) at t o = t~ + z/2 (Figure 6a). For the working pressure oscillation (Figure 3), Xa is a minimum at t = r (Figure 6b). In case A and B of Table 1, we find case A case B

Xami, = 20 cm Xamin= 15.3 cm

On the other hand xc = 0 (~0 = L) at to = 0 (or 7) and xc is a maximum at t = r/2 (or 3 r/2) case A case B

160

Xcm,x = 8.8 cm Xcmax= 14 cm

Cryogenics 1993 Vol 33, No 2

:S:.::U:""

5

m I

I

I

I

2O

I

I

25

x a (cm) Figure 7 Pressure-displacement hysteresis of the 'hot' side of the buffer gas. Full circles represent calculated results. The full line is a guide for the eye. The loop area represents the rejected heat per period to the hot exchanger which only depends on the pressure wave and the orifice opening

Theory of Meal orifice pulse tube refrigerator: M. David et al. The loop area thus represents the heat rejected per period to the hot exchanger (and/or to the tube wall in a real OPT). Figure 8 shows the consequent hysteresis effect at the cold end. In cases A and B the temperature distributions in gases (a) and (c) during a cycle are quite similar. The only difference lies in that, in case B, this distribution spreads over a larger depth ~,, since velocities are roughly twice as large as in case A. Therefore, we do not expect that heat transfer between elements of gases (a) and (c) should be enhanced in case B as compared with case A. In order to explain the definitely lower performance in case B, we must suspect other sources of thermal losses: The growth of turbulence at higher velocities and heat transfer through the buffer gas. Heat transfer between the tube wall and the gas, as gases (a) and (c) penetrate more deeply into the tube. If no heat flux exists from the gas to the wall, the d.c. heat current conveyed by the tube itself from the hot end to the cold end is negligible ( q - 0.1 W for Tc - 100 K) and the temperature gradient along the tube wall is nearly constant (the thermal conductivity of stainless steel is weakly T-dependent between 100 and 300 K). Now, under ideal conditions, the temperature T(xa) of the gas at t = r, when Xa is a minimum, is nearly the same for both cases A and B;

from the isentropic relation PT -2/~= constant, we find 255 K (case A) and 250 K (case B). On the other hand, assuming a linear T-profile, the wall temperature at Xaminwould be 254 K in case A and 231 K in case B. Therefore, in case B, heat is more likely to flow from the gas (a) to the tube wall and symmetrically from the tube wall to the gas (c) in the cold part. This in turn might affect the temperature profile along the tube wall, so that a significant part of the heat extracted from the gas would be bypassed through the tube wall. As easily seen from Equation (20) and Equation (27), ~a oc r/s can be reduced by increasing either the pulse frequency or the tube size. As pointed out above, however, the dynamic pressure and the ideal power (qa) will decrease for a given compressor-regenerator system; again a compromise is to be sought. Further measurements are clearly required to understand how an actual OPT deviates from ideal behaviour. We have just outlined the above discussion as an illustration of the model. Experiments are planned to investigate the turbulence of the gas flow and the role of the tube wall. Also we shall attempt to measure both (qa) and (qc) in the same apparatus. The question is to what extent Equation (3) holds, regardless of whether the OPT working is close to ideal or not. References 1 Liang, J., Zhou, Y. and Zhu, W. Development of a single-stage pulse tube refrigerator capable of reaching 49 K Cryogenics (1990) 30 49

110

100,

90

I

I

I

I

2

4

x Figure 8

I

I

,

6

(cm)

The temperature of the element of gas entering the tube at a time near t 1 (Figure 2) in case A of Table 1. After travelling back and forth into the tube, its temperature has decreased significantly. This is the mechanism for the refrigeration power

2 David, M. and Mar~chal, J.C. How to achieve the efficiency of a Gifford-MacMahon cryocooler with a pulse tube refrigerator Cryogenics (1990) 30 262 3 Baks, M.J.A., Hirschberg, A., Van der Ceelen, B.J. and Gijsman, H.M. Experimental verification of an analytical model for orifice pulse tube refrigeration Cryogenics (1990) 30 947 4 Tward, E., Chan, C.K. and Butt, W. Pulse tube refrigerator performance Adv Cryog Eng (1990) 35 1207 5 Radebaugh, R. Pulse tube refrigeration: a new type of cryocooler Proceedings of the 18th International Conference on Low Temperature Physics (Kyoto, Japan 1987) J Appl Phys (1987) 26 2016 6 Storch, P.J. and Radebaugh, R. Development and experimental test of an analytical model of the orifice pulse tube refrigerator Adv Cryog Eng (1988) 33 851 7 Storch, P.J., Radebangh, R. and Zimmerman, J. Analytical model of refrigeration power of the orifice pulse tube refrigerator Technical Note 1343 National Bureau of Standards 8 Tritton, D.J. Physical Fluid Dynamics, van Nostrand Reinhold (1977) 15 9 David, M., MarShal, J.-C. and Encrenaz, P. Measurements ot" instantaneous gas velocity and temperature in a pulse tube refrigerator Adv Cryog Eng (1992) 37 939 10 David, M. Refrig6ration par tube h gaz puls6: 6tude th~orique et exp6rimentale, Thesis (1992)

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