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Acoustic streaming in pulse tube refrigerators: tapered pulse tubes
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ELSEVIER
Acoustic streaming in pulse tube refrigerators: tapered pulse tubes J.R. Olson* and G.W. Swift Condensed Matter and Thermal Los Alamos, NM 87545, USA
Physics Group, Los Alamos
National
Laboratory,
Acoustic streaming is investigated in tapered tubes with axially varying temperature, in the boundary layer limit. By appropriately shaping the tube, the streaming can be eliminated. Experimental data demonstrate that an orifice pulse tube refrigerator with a conical pulse tube whose cone angle eliminates streaming has more cooling power than one with either a cylindrical pulse tube or a conical pulse tube with twice the optimum cone angle. 0 1998 Elsevier Science Ltd. Keywords: acoustic streaming;
pulse tube refrigerators;
In a simplified view of the operation of the orifice pulse tube refrigerator’, the gas in the pulse tube can be thought of as a long (and slightly compressible) piston, transmitting pressure and velocity oscillations from the cold heat exchanger to the orifice at higher temperature. In this view, the gas in the pulse tube must thermally insulate the cold heat exchanger from higher temperatures. Unfortunately, this simple picture can be spoiled by convective heat transfer within the pulse tube, which carries heat from the hot heat exchanger to the cold heat exchanger and thereby reduces the net cooling power. Such convection can be steady or oscillatory, and has causes as mundane as gravity or as subtle as jetting due to inadequate flow straightening at either end of the pulse tube. Here, we consider convection driven by streaming. In the acoustics literature, streaming2 denotes steady convection which is driven by oscillatory phenomena. In the context of the pulse tube, this driving can occur in the oscillatory boundary layer at the side wall of the pulse tube; in this layer, both viscous and thermal phenomena are important. For laminar oscillatory phenomena at angular frequency o, the relevant boundary-layer thicknesses are the viscous and thermal penetration depths 6, and S,, respectively, defined by -
J
2/J is,= ~ w
(1)
(2)
*Current address: Advanced Technology Center, Lockheed Martin Missiles and Space, Palo Alto, CA 94403, USA
shaped pulse tubes
where p is the dynamic viscosity of the gas, p is its density, cp is its isobaric specific heat per unit mass, and K is its thermal conductivity. In monatomic gases, the Prandtl numberal,so&6,. Much further from the wall than these penetration depths, the oscillatory temperature of the gas in the pulse tube is essentially adiabatic, and the axial oscillatory motion, parallel to the wall, is essentially independent of distance from the wall. Closer to the wall, the oscillatory temperature and motion are reduced by thermal and viscous contact with the wall; at the wall, the oscillatory temperature and motion are zero. One means by which streaming is generated within these penetration depths is easily visualized. Consider a small parcel of gas located approximately a penetration depth from the wall, oscillating up and down as shown in Figure la. On average, the gas between the parcel and the wall will have a different temperature during the parcel’s upward motion than during its downward motion, due to thermal contact with the wall and the phasing between oscillatory pressure and motion. Since the viscosity depends on temperature, the moving parcel will experience a different amount of viscous drag during its upward motion than during its downward motion, and hence will undergo a different displacement during its upward motion than during its downward motion. After a full cycle, the parcel does not return to its starting point; it experiences a small net drift. (In Appendix A, streaming is shown to be the sum of many processes; this paragraph simply provides an intuitively appealing explanation for one such process.) This drifting gas close to the wall has a profound effect on the entire pulse tube, because it drags gas further from the wall along with it. In the usual case, with pulse tube radius much larger than the penetration depths, an offset parabolic streaming velocity profile results, with the gas velocity near the wall equal to the drift velocity just outside the penetration depths, and the velocity in the centre of the
Cryogenics
1997 Volume
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769
Acoustic streaming in pulse tube refrigerators:
J.R. Olson and G. W. Swift
(4 Oscillatory motion
f Net drift
Wall of pulse tube ’
wave phasing between pressure and velocity and with constant tube cross-sectional area. Standing wave phasing is a poor assumption for the pulse tube in an orifice pulse tube refrigerator, because significant acoustic power flows along the pulse tube. The analysis presented here follows Rott’s, but incorporates variable cross-section and arbitrary phase between pressure and velocity. We restrict attention to ideal gases, so that p = pR,T and pa* = 3/p, where p is the pressure, R, is the gas constant, T is the temperature, a is the sound speed, and y is the ratio of specific heats. The geometry used in this analysis is shown in Figure lb, although the details of the geometry are largely unimportant (see Appendix A). The relevant variables are expanded to second order: u(x,yA
=
~e[h(w)e’“‘l
+ k,&,y)
v(x,y,t) = %e[v, (-w)e’“‘l +
v2,&,y
T(n,y,t) = T,(x) + ~dTAwk’“‘l p(x,y,t) pkr)
Central axis of pulse tube
/ Wall of pulse tube’
Figure 1 (a) Illustration of net drift caused by processes within a penetration depth 6, of the wall of a pulse tube. The box represents a parcel of gas; it is shown in three consecutive positions. Here, we imagine that the temperature dependence of the viscosity is the only important effect, and assume that the temperature is lower during upward motion than during downward motion. fb) The net drift near the wall, illustrated in (a), affects the entire pulse tube, causing an offset parabolic velocity profile which leads to a second-order mass flux density r&. For the signs chosen here for illustration, the gas moving downwards in the centre of the pulse tube is hotter than the gas moving upwards around it, so that heat is carried downwards. The coordinate y measures the distance from the wall, while r = R - y measures the distance from the centre of the pulse tube, of radius R. The calculations in Appendix A are for R>>S, but, for clarity, 6, is exaggerated in the figure
pulse tube determined by the requirement that the net mass flux along the tube must be zero (see Figure Ib). In turn, this parabolic streaming profile convects heat along the pulse tube. Hence, qualitatively, we imagine that oscillatory processes roughly a penetration depth from the wall cause a steady axial drift approximately a penetration depth away from the wall; this in turn establishes an offset parabolic mass-flux profile across the entire pulse tube, which convects heat. In this paper, we derive an expression for this streaming mass flux, and show that tapering the pulse tube slightly will suppress it. This possibility was first suggested by Lee et aL3. Our calculation of the optimum taper angle is confirmed by an experiment, in which the cooling power of an orifice pulse tube refrigerator is increased significantly by using a slightly tapered pulse tube.
Theory A general method for calculating streaming has been published previously by Rott4 for the case of axially varying wall temperature, in the boundary layer limit with standing
770
Cryogenics
1997 Volume 37, Number 12
= P,(X) = pm +
+
1
(4)
+ T2.0(x,~)
(5)
~d&-9y)ei”‘l + PZ&,Y)
~ebdxb+‘l
CL(X,Y,~ = CL,(X) +
(3)
+ Pi,&)
~d~dw)e’“‘l
(6)
(7)
+ CL~,&Y)
(8)
where u and v are the axial and lateral components of the velocity, x is the axial position, y is the lateral distance from the wall, t is the time, and %e[z] denotes the real part of z. In this notation, the variables with subscript ‘m’ are steady-state mean values, without time dependences; these represent the values the variables would have if there were no oscillating pressure or velocity. The mean temperature profile T,,,(x) is assumed to be known, and leads to the x dependences of pm and pm. We will omit the subscript ‘m’ on constant properties (such as c,, and y) and on variables for which terms of order higher than mean are unimportant in this work (such as a and K). The subscript ‘ 1’ indicates the first-order part of each variable, which accounts for oscillation at angular frequency w. The first-order variables are complex quantities, having both magnitude and phase to account for their amplitudes and time phasing. For the purposes of this paper, the oscillating pressure p, and the lateral spatial average (u,) of the oscillating axial velocity uI are assumed to be known, as they are experimentally accessible through measurements of oscillating pressure in the pulse tube and mass flow through the orifice. Expressions for the other oscillating variables (temperature, density, etc.) in terms of p, and (u,) are well known4*‘. To make the problem more tractable, the tube radius is assumed to be much larger than the viscous and thermal penetration depths, defined by Equations ( 1) and (2), respectively. It can be showrP that the dependence of pressure on y is negligible, either in the boundary layer limit, or when y dimensions are much smaller than the acoustic wavelength. Consequently, the pressure is given as a function of x only. Surprisingly, the oscillatory part of the viscosity cannot be neglected; we assume it to be independent of pressure and to depend on the oscillatory temperature via the temperature dependence of the viscosity, which takes the form
P(T) = /-M/To)’
(9)
Acoustic
streaming
Second-order, time-independent parts of variables are indicated by the subscript ‘2,O’. These include the axial streaming velocity u2,0 which is of great interest in this work. The complete expansion to second order would also include terms such as %e[u2,Jx,y)ei2”], with subscript ‘2,2’ indicating second order and 20 time dependence; these are ignored in this work as they have negligible influence on the terms above and on our results. Formally, when one expands variables as in Equations (3)-(g), the first-order terms are of order M x (mean value), and the second-order terms are of order Mz x (mean value), where M = lull/u - (PI//p,,,is the Mach number. The lowest-order energy fluxes are of order i@, and streaming might be expected to contribute an energy flux of order M4, due to terms such as ~,,,c~T~,~u~,~.In the case of the pulse tube, however, a streaming velocity u2,0 which is indeed of order M2 x a can lead to a large T2.0, of the same order as the temperature spanned by the pulse tube instead of Mz times the temperature spanned. Consequently, streaming can carry an appreciable amount of heat from the hot to the cold end of the refrigerator, reducing its coefficient of performance. The details of the calculation are outlined in Appendix A; here, we summarize the results in a form useful for designing tapered pulse tubes. The streaming mass flux density just outside the penetration depths is given by
in pulse
tube refrigerators:
J.R. Olson and G. W. Swift
neglect of the temperature dependence of viscosity, of thermodynamic effects (i.e. yf I), or of the phase between p, and (u,) leads to significant error. In particular, the conceptually subtle importance4 of the temperature dependence of viscosity has been missed in some previous work’ on streaming in pulse tubes. Third, Ij12,wdiverges as l/a, so gas mixtures with small values of cr will have higher streaming mass flux. Fourth, it is possible to make ’ = 0 by proper choice of the phase between p, and (u,). m2.w In the pulse tube of a traditional orifice pulse tube refrigerator, this is not possible, since 0 < 8 < 1r/2: p, and (u,) are in phase at the hot end of the pulse tube, and the sign of the phase between p, and (u,) in the rest of the pulse tube is determined by the positive compressibility of the gas. However, when inertance*~9 is used to provide more efficient phasing in an orifice pulse tube refrigerator, 8 can be negative, so that streaming might thereby be suppressed in an ordinary pulse tube. Fifth, as we show below, streaming can also be eliminated by using an appropriately shaped pulse tube, so that the dA/dx term in Equation ( 10) cancels the other terms at each value of x. Setting Equation (10) equal to zero, we find that streaming is eliminated in a tapered pulse tube if
1 dA
OlPll
1 +2(~-
YPml(4)I
AdX +1-
1)(1 -bo? 341 +a)
+ (1 + 2(y-3:i’:,p’:;)
cos8
sin 01
(12)
(10)
(1 - b)(l - +d P,lW w
3 dA/dx 4 A
(1 -b)(l + 40(1+sa)
-6)
d~,ldr Tm
1
-
where (u,) is the lateral spatial average of u, and 8 is the phase angle by which (u,) leads p,. The streaming profile far from the wall is then given by 23 1)12.o(r)= m2.w - - 1
( 1 R2
(11)
where r is the radial coordinate in the pulse tube, as illustrated in Figure lb. (To display the boundary-layer behaviour clearly, the figure does not have R>>&,, and hence the maximum of lj12,0falls short of ti2,+,,in the figure. However, our calculations are only valid for R>>S, when the maximum of &, = &_,.) For subsequent discussion, we adopt the convention that x increases towards the hot end of the pulse tube, so that 0 < 0 < 7r/2 in the pulse tube of a traditional orifice pulse tube refrigerator. Several things are worth noting about Equation (10). First, the coefficient of dT,,,/dx is rather small for helium, so the streaming depends only weakly on the temperature gradient. This is fortunate for two reasons: the optimally tapered pulse tube described below remains optimal over a wide range of operating conditions, and details of the actual axial temperature profile (generally deviating significantly from linear dependence on x) are of only minor significance. Second, the coefficients of the other terms in Equation ( 10) are of similar magnitude, so
- ~ =
&I
3(l+(T)(l+&)
dT,,,/&
TUll
(0.75 cos 8 + 0.64 sin 0) - 0.0058 F
PA(4>l
m(13)
In Equation ( 13), we have used numerical values for low-temperature helium gas: y = 5/3, cr = 0.69, and b = 0.68. For this value of dA/dx, the parabolic part of the velocity profile shown in Figure lb is eliminated; the only nonzero streaming occurs at distances from the wall comparable to the penetration depths. The derivation of these equations is based on the assumption that the flow is laminar, but for sufficiently high velocity, turbulence will probably invalidate the results. The transition to turbulence in oscillatory flo~‘O-‘~ is richer than that in steady flow, yielding several regimes as illustrated in Figure 2. We expect our results to be applicable in the weakly turbulent regime as well as in the laminar regime, because in the weakly turbulent regime the turbulence is generated outside” the viscous penetration depth, leaving the velocity close to the wall nearly the same as it would be for laminar flow. In the conditionally turbulent regime, the turbulence is generated within the penetration depth, and has a dramatic influence on the gas velocity within the penetration depth. Hence, it is likely that the quantitative details of streaming-driven convection in the conditionally turbulent regime will differ greatly from the results presented here. A brief survey of published data from seven
Cryogenics
1997 Volume
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771
Acoustic
streaming
in pulse tube refrigerators:
J.R. Olson and G. W. Swift Figure
‘03m’
102 -_
// //
Weakly 9
-
//
turbulent
// //
;
// // 10’ -_
// //
----
I/ ‘\
-
I/
Laminar
102
(2
103
i Turbulent Conditionally
1
:
turbulent
104
105
Re Figure2 Regimes of oscillatory flow in a circular pipe of radius 17.The Reynolds number Re = (u,)Zf?pl~ is based on the spatially-averaged velocity amplitude (u,) (adapted from references 10-12)
pulse tube refrigerators I3 showed that all seven operated in the weakly turbulent regime, as do our experiments described below. Although the energy flux density &@‘,,, convected by this streaming is formally of fourth order, it can in practice be as large as a typical second-order energy flux density, because T2,0 can be of the order of the temperature difference between the two ends of the pulse tube when RX=&. Accurate calculation of this heat flux density would probably require significant additional effort, because the weak turbulence in typical pulse tubes probably enhances the lateral heat transfer between the upward and downward streaming currents, reducing T2,,, and making accurate calculation of T2.0 difficult. (Note that this effect may also have driven the past evolution of pulse tubes toward long, slender aspect ratios.) In fact, with such large T2,0, the entire perturbation expansion presented here is essentially invalid, except when taper or phasing makes Gz?,~ (and thus T& very small. There are numerous other fourth-order energy flux terms in addition to the two (see Appendix A, Equation (A14)) contained in rit,,,~,T~,~, which would in principle have to be considered to obtain a formally correct fourth-order result. Fortunately, the only other large term, p2,0cp%e[T,(k,)]/2, is zero at the same taper angle that makes T2.0 zero, while the remaining terms, such as those involving products of first- and third-order quantities, are small for all angles.
Experiment
v
Compliance
P
Orifice
valve
Hot heat exchanger
!c
Pulse tube
and results
To verify Equation (13), we tested an orifice pulse tube refrigerator with three pulse tubes: a right-circular cylinder, a truncated cone with the optimum angle determined by Equation (13), and a second cone with roughly twice the optimum angle. The double-angle cone should induce the same amount of streaming as the cylindrical pulse tube, but in the opposite direction, and so should exhibit roughly the same performance as the cylindrical pulse tube. The pulse tube refrigerator is shown schematically in
772
3. It was filled with 3.1 MPa helium and driven at 100 Hz by a thermoacoustic engine similar to one described previously’4. This system was not thoughtfully designed, so its performance was not impressive; it was simply assembled from parts on hand. The entire apparatus beyond the hot end of the regenerator was contained within a vacuum can where the pressure was 10m5torr. All cold parts were also wrapped in several layers of aluminized Mylar superinsulation. The regenerator was 4.6 cm in diameter, 3.0 cm long, and made of #325 stainless-steel screen with 28 pm wire diameter. The two hot temperature heat exchangers were stacked copper screens soldered into copper blocks through which 15°C cooling water flowed. The cold heat exchanger consisted of parallel copper plates. The flow straightener was simply four layers of #/40 stainless-steel screen. The compliance (i.e. reservoir volume) was a steel bulb with a volume of 150 cm3. The orifice was a needle valve which could be adjusted from outside the vacuum can. The three pulse tubes were stainless-steel with a wall thickness of 0.5 mm, a length of 4.65 cm, and a volume of 11 .O cm3. Hence, the cylinder had R = 8.7 mm, much larger than S, = 80 pm and S, = 97 pm at 200 K, which are in turn much larger than the 1 pm roughness of the inner surfaces of the pulse tubes. The optimum-angle cone pulse tube (see below) had a total included angle 4 of 0.049 rad = 2.8”, with the cold end larger than the hot end. This is a small enough angle that it is barely discernible by eye. The double-angle pulse tube had an angle of 0.103 rad. In both cases, the internal diameter at each end was calculated in order to make the volume of the pulse tube the same as in the cylindrical case. The oscillating pressure amplitude was measured with piezoresistive sensors in the compliance and near the two hot heat exchangers. The temperature of the cooling water in the hot heat exchangers was measured with thermocouples, and a thermocouple was inserted directly into the
Cryogenics
1997 Volume
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Figure3 Schematic representation of apparatus used in the measurements. ‘P’ denotes pressure sensors; ‘T’ denotes thermometers. The angle + of the conical pulse tube has been greatly exaggerated
Acoustic
streaming
gas space of the cold heat exchanger to measure the cold gas temperature T,. was optionally applied to the cold stage with a Heat Qcold resistive heater. The resistance of the heater and the applied voltage were measured to determine the applied power. The experimental operating conditions were determined as follows. The drive amplitude and orifice setting were selected experimentally by minimizing the cold gas temperature at zero applied heat load with the cylindrical pulse tube, without overloading the thermoacoustic driver (which operated near its high-temperature limit throughout these experiments). This determined the operating point that was reproduced for all the data shown in this work: b,l = 2.32 x lo5 Pa at the driven end of the hot heat exchanger adjacent to the regenerator and b,l = 0.59 x lo5 Pa in the compliance. Under these conditions, b,l at the heat exchanger at the hot end of the pulse tube was always within 0.5% of 1.95 x lo5 Pa. These operating conditions, along with the refrigerator geometry, were used to calculate* the magnitudes and relative phases of p, and (u,) at the ends and in the middle of the pulse tube. These calculated values were used in Equation ( 13) to determine the optimum dA/dx at the ends and in the middle of the pulse tube. The values were such that a simple cone was a reasonable approximation to the ideal shape, making more difficult fabrication unnecessary. The resulting cone angle was also shallow enough to eliminate concerns about flow separation at the wall. The results of our measurements are shown in Figure 4. The data corresponding to the cylindrical pulse tube are represented by the circles, while those of the optimumangle conical pulse tube are shown as triangles and those of the double-angle cone as squares. For all three pulse
in pulse
Acknowledgements The authors thank Ray Radebaugh, for numerous helpful discussions about pulse tube refrigerators and from whom we first heard the idea of shaping pulse tubes to reduce streaming. We are also indebted to Nikolaus Rott, for always doing things right, and to Mike Hayden and Nikolaus Rott for helping to improve the clarity of the manuscript. Support for this work was begun by the TTI program of the US DOE, and completed by the Division of Materials Sciences in the DOE’s Office of Basic Energy Sciences. Part of the hardware used for these measurements was supplied by Tektronix Corporation’6.
References I.
3.
4. 5. 6.
7. 8.
9. 10
20
30
IO.
Qcold (w)
Figure4 Experimental results, showing cold temperature as a function of heat load for the three pulse tubes. The pulse tube designed according to Equation (13) is the best
Il. 12.
13. *The pressure amplitude in the compliance, its volume, and the adiabatic compressibility of the gas yield the volumetric velocity of the gas entering the compliance, which is essentially equal to that at the hot end of the pulse tube. The volumetric velocity elsewhere in the pulse tube can then be estimated from its geometry and the adiabatic compressibility of the gas. We also incorporated small corrections due to, for example, the volume and pressure drop of the heat exchanger at the hot end of the pulse tube 15.
J.R. Olson and G. W. Swift
tubes, the temperature of the gas increases as more heat is applied to the cold stage, as expected. For all measured values of applied heat, the temperature corresponding to the optimum cone pulse tube is at least 5°C colder than the temperatures with either the cylindrical or the double-angle pulse tubes, indicating that the optimum cone performs significantly better than the other pulse tubes. From an altemative point of view, it appears that the streaming-driven convective heat load on the cold heat exchanger is 3 to 5 W greater with the cylindrical and double-angle pulse tubes than with the optimally tapered pulse tube. While it was expected that the cylinder and the double-angle cone would perform roughly the same, there is a slight difference between those two data sets which has not been explained.
2.
0
tube refrigerators:
14. 15.
Radebaugh, R., A review of pulse tube refrigeration. Adv. CQJO. Eng., 1990, 35, 1191-1205. Nyborg, W.L.M., Acoustic streaming. In Physical Acousfics, Vol. IIB, ed. W.P. Mason. Academic Press, New York, 1965, p. 265. Lee, J.M., Kittel, P., Timmerhaus, K.D. and Radebaugh, R., Flow patterns intrinsic to the pulse tube refrigerator. In Proc. 7th Inf. Cryocoofer Con$. Kirtland AFB, NM 87117.5776, Phillips Laboratory, 1993, p. 125. Rott, N., The influence of heat conduction on acoustic streaming. Z. Angew. Math. Phys., 1974, 25, 417. Rott, N., Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z. Angew. Math. Phys., 1969, 20, 230. Pierce, A.D., Acousfics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America, Woodbury, NY, 1989. Jeong, E.S., Secondary flow in basic pulse tube refrigerators. Ctyogenies, 1996, 36, 317-323. Zhu, S.W., Zhou, S.L., Yoshimura, N. and Matsubara, Y., Phase shift effect of the long neck tube for the pulse tube refrigerator. In Proc. Crycoolers 9, New Hampshire, 1996. Gardner, D.L. and Swift, G.W., Use of inertance in orifice pulse tube refrigerators. Cryogenics, 1997, 37, 117-I 2 I. Hino, S.M., Sawamoto, M. and Takasu, S., Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech., 1976, 75, 193-207. Ohmi, M. and Iguchi, M., Critical Reynolds number in an oscillating pipe flow. Bull. LYME, 1982, 25, 165. Ohmi, M., Iguchi, M., Kakehashi, K. and Tetsuya, M., Transition to turbulence and velocity distribution in an oscillating pipe flow. Bull. LYME, 1982. 25, 365. NASA Pulse Tube Database, on-line at available http://irtek.arc.nasa.gov/CryoGroup/PTDatabase~Data-~ilice.html (five cases); Proc. 7th Int. Cryocooler Con& (two cases). Swift, G.W., Analysis and performance of a large thermoacoustic engine. J. Acoust. Sot. Am., 1992, 92, 1563. Ward, W.C. and Swift, G.W., Design environment for low amplitude thermoacoustic engines. J. Acoust. Sot. Am., 1994, 95, 3671-3672. Fully tested software and a users guide are available from the Energy Science and Technology Center, US DOE, Oak Ridge, TN. For a beta-test version, contact [email protected] (Bill Ward) via the Internet.
Cryogenics
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Acoustic streaming
in pulse tube refrigerators:
J.R. Olson and G. W. Swift
16. Godshalk, K.M., Jin, C., Kwong, Y.K., Hershberg, E.L., Swift, G.W. and Radebaugh, R., Characterization of 350Hz thermoacoustic driven orifice pulse tube refrigerator with measurements of the phase of the mass flow and pressure. Adv. Cryo. Eng., 19%. 41, 14111418. 17. Landau, L.D. and Lifshitz, E.M., Fluid Mechanics. Pergamon, 1982, equation 15.5. 18. Mathematics, Wolfram Research, Champaign, IL (for example).
Appendix A Much of this derivation follows the previous work of RotP. We assume that the tube radius R>>6, that the tube length is >>R, and that p, and (u,) (or equivalently p, and dp,/d.x) are given. Then the solution for the other first-order variables is well known4? The ideal-gas equation of state and the equations of conservation of mass, momentum and energy are solved to first order, yielding u,
=
~
(1
_
UPIn
_
e-Cl + i)Yh,) = (u,)< 1 _ e+’
In the gently tapered tubes under consideration here, these first-order solutions are identical to those found previously4 in straight tubes, with one exception: we find that v, has an additional term proportional to y, the final term in Equation (A5), which is zero in straight tubes by virtue of the spatially-averaged continuity equation. In tapered tubes, the spatially-averaged continuity equation adds additional complexity to the analysis, as will be discussed in more detail below. Although Rott adopted the boundary-layer approximation only in order to use exponential y dependences, we use a more restrictive approximation: that 6JR is algebraically <
+ W%)
dx (Al)
T
I
=y-l y
Ttn
_&I
pp,(l
(-46)
+WQ)
m
+;
%(u,,l--
~u +1-oe
1 e_’I + iMr 1-U
(A21
-(I +i)y/S,
with ( ) denoting the average over the cross-sectional area
A of the tube. As needed, p, can then easily be obtained
where we have neglected second viscosity. In the last terms, the viscosity must be kept inside the derivatives, since p varies significantly with temperature and thus position. The viscosity is assumed to vary as a power law of the temperature, as given by Equation (9), with b - 0.7 for monatomic gases. Equation (A6) can be written out to second order, keeping terms that are independent of time and dropping terms that are small:
from the first-order equation of state k!Yle -iwp,& (A3)
The velocity v, perpendicular to the wall can be obtained from the complete first-order continuity equation in and near the boundary layer
au,+ pmav,__ 0
pm-
ax
(A4)
ay
ax
+ pmv,
au ay
--J
(A7)
The tilde represents complex conjugate. For straight or gently tapered tubes, terms such as P,d2rZ,/axZare smaller than the terms remaining in Equation (A7) by factors of the order 6Jh < 1, where A is the acoustic wavelength, and can thus be neglected. The factors of l/2 arise because YIe[a]%e[b]
by integrating once with respect to y, and using the boundary condition v,(y = 0) = 0. This yields
1
1 a ati, = -- dp2.0 + cL azu2.0 i *-rRe[(p-)] dx m ay2 2 ’ ay 1 ay [
T, PlIl Pm T, PI PI -_=---
iw, +-dpm U, + dx
+ pm&
= t %e[ab] + 1 %e[a6];
the %e[ub]
terms
are those mentioned in the main text with time dependence of ei2&, which have been ignored. Equation (A7) can be regarded as a differential equation for u,,(y) in terms of dp,,,/dx and a complicated driving function consisting of the sum of products of known firstorder quantities. Upon integrating Equation (A7) once with respect to y, the left-hand side can be simplified using integration by parts on the third term within the brackets and using Equation (A4) to eliminate &Jay. The result is
(A-j)
i Se
=-zY +
774
Cryogenics
1997 Volume
37, Number
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$ (p,,,u,P,)dy
dpzo
/4nW)
+Prn
+ $ %e(u,P,)
1 auz,o +ZWe k,ati, ay
(
ay1
(A@
Acoustic streaming in pulse tube refrigerators: (cf. equation ( 11) in reference 4). Integrating Equation (A8) again with respect to y, expressing CL,in terms of T,, and solving for u~.~,we obtain
U*,okY)
L 2Pril
=
0
Y
u,V,dy-$%e
+&!Re
2Pm
(A9)
”
0
To arrive at this form of the result, we used dp,,,/dr = - (pJ’,,,)dZ’,,,/dx and dp,,,/dr = b(~,,,/JT,,,)d7’,&. Equation (All) reduces to Rott’s result upon substituting p, = (iyp,,,/Jo)d(u,)/dx (as appropriate for non-tapered tubes) and taking (u,) and d(u,)/d_x real. Rott’s final results were expressed in terms of u, and duJd.x, similar to our (u,) and d(u,)/dx, but we choose to eliminate d(u,)/dx so that our results are expressed in the more experimentally accessible variables (u,) and p,. To do so, we use the first-order y-averaged continuity equation
0
1 ho 2
+z
y* - C(x)y
dx
The first three terms on the right-hand side represent the (lengthy) driving terms referred to initially. They include exponential y dependences with characteristic lengths of 6, and S,, which are non-zero only close to the wall. The last two terms on the right-hand side contain no exponential y dependences; they contribute only to the parabolic velocity profile extending across the entire tube. Because of our choice of the lower limit of integration, Equation (A9) satisfies the boundary condition that u*,~= 0 at y = 0. Although we will not need them, the constant of integration C(x) could be determined if desired by the condition that &+,,&Iy = 0 at y = R, and the magnitude of d,u,,/dx could be determined if desired by the condition that the total second-order time-average mass flux through the tube is zero:
pm~2,0
1 !Re(p,E,)
+
dA = 0
(AlO)
1
Henceforth, our derivation differs importantly from that of Rott. Focusing solely on standing waves, Rott assumed that u, was real for all X, and hence that du,/& was real, implying that p, was purely imaginary (cf. equation ( 1) of reference 4). Here, to allow arbitrary phasing between p, and (u,), we must allow the phase of (u,) to vary with X, so that d(u,)/dx might have a different phase than (u,). We are interested in the velocity at the outer edge of the boundary layer, by which we mean y>S,, and y>>6,, yet y << R, so that the exponential parts of u~,~(Y)are essentially zero but the parabolic part of u*,~(Y)essentially has its value at the wall. We substitute all the first-order solutions into Equation (A9) and perform the y integrations. The evaluation of the integrals in Equation (A9) is tedious. After integration, to evaluate u*,~at the outer edge of the boundary layer, we let y-+03 within exponentials and set y = 0 in linear and quadratic terms. The result is
1 + dT, (I-@(I-&) 3
~~.~(m)= - - Se 4w
(1 + i)(q)
W,) ~ dx
Using Equations (A2) and (A3) and do,,& - (p,lT,,,)dT,,,/du, we can express this in the form
4wTzm CIJ 1
+-n?e 2YPln
(l+&)(l+a) (y-
Equations (A12) and (A13) are only valid for tubes with sufficiently slowly varying cross-sectional areas; the criteria for validity are summarized by Pierce6 in his derivation of Webster’s horn equation. Fortunately, the areas of tapered pulse tubes that we have considered vary so slowly that these criteria are extremely well satisfied. Straightforward substitution of Equation (A 13) into Equation (A 11) brings non-zero dAldx into the analysis. Finally, we are concerned not with uzTobut with the second-order, time-averaged mass density flux m2,okY) = Pm~*,o + 1 %e(p,ti, ) (the integrand of Equation (AlO)), shown schematically in Figure lb. In this paper, the quantity of greatest interest is the effective mass flux density at the wall r&, shown in the figure, which we write as
k2,, = AG~,~(~)+ 4 ~4pl(m)fil(~)l = P~u~.o(~) + f ~dhW1
dl+u)
-b)]
(A14)
The second form of the equation is due to our use once again of the approximation that 6,JR is algebraically <
=Prn
1
!!!$
g-g
[
m
Using Equations (All), (A14) quickly yields
(Al5)
(A13) and (A15) in Equation
3 (y- l)(l - bti) ~ + ~-~ 4 2a(l +a)
(u,)(h)
l)[l -bd+iid’*(l
=
(A13)
1
- 1 P&J 1
J. R. Olson and G. W. Swift
(Al 1)
z+(y-
1X1 -b)&
4
2(1 + a)
+& w (‘416 111
I Cryogenics
1997 Volume
37, Number
12
775
Acoustic streaming in pulse tube refrigerators:
JR
The result holds regardless of geometry (e.g. for rectang&r tapered tubes as well as circular tapered tubes), as long as a,,<< all lateral dimensions of the tube, although the last two terms in Equation (A9) will depend on the geometry.
776
Cryogenics
1997 Volume
37, Number
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Olson and G. W. Swift The mathematical challenge of going from Equation (A9) to Equation (Al 1) is considerable. Assistance from symbolic-mathematics softwareI is extremely useful.
in Great
37 ( 1997)
1998 Elsevier Britain.
PII: SOOll-2275@7)00037-4
All
169-776
Science
Ltd
rights reserved
001 I-2275/98/$19.00
ELSEVIER
Acoustic streaming in pulse tube refrigerators: tapered pulse tubes J.R. Olson* and G.W. Swift Condensed Matter and Thermal Los Alamos, NM 87545, USA
Physics Group, Los Alamos
National
Laboratory,
Acoustic streaming is investigated in tapered tubes with axially varying temperature, in the boundary layer limit. By appropriately shaping the tube, the streaming can be eliminated. Experimental data demonstrate that an orifice pulse tube refrigerator with a conical pulse tube whose cone angle eliminates streaming has more cooling power than one with either a cylindrical pulse tube or a conical pulse tube with twice the optimum cone angle. 0 1998 Elsevier Science Ltd. Keywords: acoustic streaming;
pulse tube refrigerators;
In a simplified view of the operation of the orifice pulse tube refrigerator’, the gas in the pulse tube can be thought of as a long (and slightly compressible) piston, transmitting pressure and velocity oscillations from the cold heat exchanger to the orifice at higher temperature. In this view, the gas in the pulse tube must thermally insulate the cold heat exchanger from higher temperatures. Unfortunately, this simple picture can be spoiled by convective heat transfer within the pulse tube, which carries heat from the hot heat exchanger to the cold heat exchanger and thereby reduces the net cooling power. Such convection can be steady or oscillatory, and has causes as mundane as gravity or as subtle as jetting due to inadequate flow straightening at either end of the pulse tube. Here, we consider convection driven by streaming. In the acoustics literature, streaming2 denotes steady convection which is driven by oscillatory phenomena. In the context of the pulse tube, this driving can occur in the oscillatory boundary layer at the side wall of the pulse tube; in this layer, both viscous and thermal phenomena are important. For laminar oscillatory phenomena at angular frequency o, the relevant boundary-layer thicknesses are the viscous and thermal penetration depths 6, and S,, respectively, defined by -
J
2/J is,= ~ w
(1)
(2)
*Current address: Advanced Technology Center, Lockheed Martin Missiles and Space, Palo Alto, CA 94403, USA
shaped pulse tubes
where p is the dynamic viscosity of the gas, p is its density, cp is its isobaric specific heat per unit mass, and K is its thermal conductivity. In monatomic gases, the Prandtl numberal,so&6,. Much further from the wall than these penetration depths, the oscillatory temperature of the gas in the pulse tube is essentially adiabatic, and the axial oscillatory motion, parallel to the wall, is essentially independent of distance from the wall. Closer to the wall, the oscillatory temperature and motion are reduced by thermal and viscous contact with the wall; at the wall, the oscillatory temperature and motion are zero. One means by which streaming is generated within these penetration depths is easily visualized. Consider a small parcel of gas located approximately a penetration depth from the wall, oscillating up and down as shown in Figure la. On average, the gas between the parcel and the wall will have a different temperature during the parcel’s upward motion than during its downward motion, due to thermal contact with the wall and the phasing between oscillatory pressure and motion. Since the viscosity depends on temperature, the moving parcel will experience a different amount of viscous drag during its upward motion than during its downward motion, and hence will undergo a different displacement during its upward motion than during its downward motion. After a full cycle, the parcel does not return to its starting point; it experiences a small net drift. (In Appendix A, streaming is shown to be the sum of many processes; this paragraph simply provides an intuitively appealing explanation for one such process.) This drifting gas close to the wall has a profound effect on the entire pulse tube, because it drags gas further from the wall along with it. In the usual case, with pulse tube radius much larger than the penetration depths, an offset parabolic streaming velocity profile results, with the gas velocity near the wall equal to the drift velocity just outside the penetration depths, and the velocity in the centre of the
Cryogenics
1997 Volume
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769
Acoustic streaming in pulse tube refrigerators:
J.R. Olson and G. W. Swift
(4 Oscillatory motion
f Net drift
Wall of pulse tube ’
wave phasing between pressure and velocity and with constant tube cross-sectional area. Standing wave phasing is a poor assumption for the pulse tube in an orifice pulse tube refrigerator, because significant acoustic power flows along the pulse tube. The analysis presented here follows Rott’s, but incorporates variable cross-section and arbitrary phase between pressure and velocity. We restrict attention to ideal gases, so that p = pR,T and pa* = 3/p, where p is the pressure, R, is the gas constant, T is the temperature, a is the sound speed, and y is the ratio of specific heats. The geometry used in this analysis is shown in Figure lb, although the details of the geometry are largely unimportant (see Appendix A). The relevant variables are expanded to second order: u(x,yA
=
~e[h(w)e’“‘l
+ k,&,y)
v(x,y,t) = %e[v, (-w)e’“‘l +
v2,&,y
T(n,y,t) = T,(x) + ~dTAwk’“‘l p(x,y,t) pkr)
Central axis of pulse tube
/ Wall of pulse tube’
Figure 1 (a) Illustration of net drift caused by processes within a penetration depth 6, of the wall of a pulse tube. The box represents a parcel of gas; it is shown in three consecutive positions. Here, we imagine that the temperature dependence of the viscosity is the only important effect, and assume that the temperature is lower during upward motion than during downward motion. fb) The net drift near the wall, illustrated in (a), affects the entire pulse tube, causing an offset parabolic velocity profile which leads to a second-order mass flux density r&. For the signs chosen here for illustration, the gas moving downwards in the centre of the pulse tube is hotter than the gas moving upwards around it, so that heat is carried downwards. The coordinate y measures the distance from the wall, while r = R - y measures the distance from the centre of the pulse tube, of radius R. The calculations in Appendix A are for R>>S, but, for clarity, 6, is exaggerated in the figure
pulse tube determined by the requirement that the net mass flux along the tube must be zero (see Figure Ib). In turn, this parabolic streaming profile convects heat along the pulse tube. Hence, qualitatively, we imagine that oscillatory processes roughly a penetration depth from the wall cause a steady axial drift approximately a penetration depth away from the wall; this in turn establishes an offset parabolic mass-flux profile across the entire pulse tube, which convects heat. In this paper, we derive an expression for this streaming mass flux, and show that tapering the pulse tube slightly will suppress it. This possibility was first suggested by Lee et aL3. Our calculation of the optimum taper angle is confirmed by an experiment, in which the cooling power of an orifice pulse tube refrigerator is increased significantly by using a slightly tapered pulse tube.
Theory A general method for calculating streaming has been published previously by Rott4 for the case of axially varying wall temperature, in the boundary layer limit with standing
770
Cryogenics
1997 Volume 37, Number 12
= P,(X) = pm +
+
1
(4)
+ T2.0(x,~)
(5)
~d&-9y)ei”‘l + PZ&,Y)
~ebdxb+‘l
CL(X,Y,~ = CL,(X) +
(3)
+ Pi,&)
~d~dw)e’“‘l
(6)
(7)
+ CL~,&Y)
(8)
where u and v are the axial and lateral components of the velocity, x is the axial position, y is the lateral distance from the wall, t is the time, and %e[z] denotes the real part of z. In this notation, the variables with subscript ‘m’ are steady-state mean values, without time dependences; these represent the values the variables would have if there were no oscillating pressure or velocity. The mean temperature profile T,,,(x) is assumed to be known, and leads to the x dependences of pm and pm. We will omit the subscript ‘m’ on constant properties (such as c,, and y) and on variables for which terms of order higher than mean are unimportant in this work (such as a and K). The subscript ‘ 1’ indicates the first-order part of each variable, which accounts for oscillation at angular frequency w. The first-order variables are complex quantities, having both magnitude and phase to account for their amplitudes and time phasing. For the purposes of this paper, the oscillating pressure p, and the lateral spatial average (u,) of the oscillating axial velocity uI are assumed to be known, as they are experimentally accessible through measurements of oscillating pressure in the pulse tube and mass flow through the orifice. Expressions for the other oscillating variables (temperature, density, etc.) in terms of p, and (u,) are well known4*‘. To make the problem more tractable, the tube radius is assumed to be much larger than the viscous and thermal penetration depths, defined by Equations ( 1) and (2), respectively. It can be showrP that the dependence of pressure on y is negligible, either in the boundary layer limit, or when y dimensions are much smaller than the acoustic wavelength. Consequently, the pressure is given as a function of x only. Surprisingly, the oscillatory part of the viscosity cannot be neglected; we assume it to be independent of pressure and to depend on the oscillatory temperature via the temperature dependence of the viscosity, which takes the form
P(T) = /-M/To)’
(9)
Acoustic
streaming
Second-order, time-independent parts of variables are indicated by the subscript ‘2,O’. These include the axial streaming velocity u2,0 which is of great interest in this work. The complete expansion to second order would also include terms such as %e[u2,Jx,y)ei2”], with subscript ‘2,2’ indicating second order and 20 time dependence; these are ignored in this work as they have negligible influence on the terms above and on our results. Formally, when one expands variables as in Equations (3)-(g), the first-order terms are of order M x (mean value), and the second-order terms are of order Mz x (mean value), where M = lull/u - (PI//p,,,is the Mach number. The lowest-order energy fluxes are of order i@, and streaming might be expected to contribute an energy flux of order M4, due to terms such as ~,,,c~T~,~u~,~.In the case of the pulse tube, however, a streaming velocity u2,0 which is indeed of order M2 x a can lead to a large T2.0, of the same order as the temperature spanned by the pulse tube instead of Mz times the temperature spanned. Consequently, streaming can carry an appreciable amount of heat from the hot to the cold end of the refrigerator, reducing its coefficient of performance. The details of the calculation are outlined in Appendix A; here, we summarize the results in a form useful for designing tapered pulse tubes. The streaming mass flux density just outside the penetration depths is given by
in pulse
tube refrigerators:
J.R. Olson and G. W. Swift
neglect of the temperature dependence of viscosity, of thermodynamic effects (i.e. yf I), or of the phase between p, and (u,) leads to significant error. In particular, the conceptually subtle importance4 of the temperature dependence of viscosity has been missed in some previous work’ on streaming in pulse tubes. Third, Ij12,wdiverges as l/a, so gas mixtures with small values of cr will have higher streaming mass flux. Fourth, it is possible to make ’ = 0 by proper choice of the phase between p, and (u,). m2.w In the pulse tube of a traditional orifice pulse tube refrigerator, this is not possible, since 0 < 8 < 1r/2: p, and (u,) are in phase at the hot end of the pulse tube, and the sign of the phase between p, and (u,) in the rest of the pulse tube is determined by the positive compressibility of the gas. However, when inertance*~9 is used to provide more efficient phasing in an orifice pulse tube refrigerator, 8 can be negative, so that streaming might thereby be suppressed in an ordinary pulse tube. Fifth, as we show below, streaming can also be eliminated by using an appropriately shaped pulse tube, so that the dA/dx term in Equation ( 10) cancels the other terms at each value of x. Setting Equation (10) equal to zero, we find that streaming is eliminated in a tapered pulse tube if
1 dA
OlPll
1 +2(~-
YPml(4)I
AdX +1-
1)(1 -bo? 341 +a)
+ (1 + 2(y-3:i’:,p’:;)
cos8
sin 01
(12)
(10)
(1 - b)(l - +d P,lW w
3 dA/dx 4 A
(1 -b)(l + 40(1+sa)
-6)
d~,ldr Tm
1
-
where (u,) is the lateral spatial average of u, and 8 is the phase angle by which (u,) leads p,. The streaming profile far from the wall is then given by 23 1)12.o(r)= m2.w - - 1
( 1 R2
(11)
where r is the radial coordinate in the pulse tube, as illustrated in Figure lb. (To display the boundary-layer behaviour clearly, the figure does not have R>>&,, and hence the maximum of lj12,0falls short of ti2,+,,in the figure. However, our calculations are only valid for R>>S, when the maximum of &, = &_,.) For subsequent discussion, we adopt the convention that x increases towards the hot end of the pulse tube, so that 0 < 0 < 7r/2 in the pulse tube of a traditional orifice pulse tube refrigerator. Several things are worth noting about Equation (10). First, the coefficient of dT,,,/dx is rather small for helium, so the streaming depends only weakly on the temperature gradient. This is fortunate for two reasons: the optimally tapered pulse tube described below remains optimal over a wide range of operating conditions, and details of the actual axial temperature profile (generally deviating significantly from linear dependence on x) are of only minor significance. Second, the coefficients of the other terms in Equation ( 10) are of similar magnitude, so
- ~ =
&I
3(l+(T)(l+&)
dT,,,/&
TUll
(0.75 cos 8 + 0.64 sin 0) - 0.0058 F
PA(4>l
m(13)
In Equation ( 13), we have used numerical values for low-temperature helium gas: y = 5/3, cr = 0.69, and b = 0.68. For this value of dA/dx, the parabolic part of the velocity profile shown in Figure lb is eliminated; the only nonzero streaming occurs at distances from the wall comparable to the penetration depths. The derivation of these equations is based on the assumption that the flow is laminar, but for sufficiently high velocity, turbulence will probably invalidate the results. The transition to turbulence in oscillatory flo~‘O-‘~ is richer than that in steady flow, yielding several regimes as illustrated in Figure 2. We expect our results to be applicable in the weakly turbulent regime as well as in the laminar regime, because in the weakly turbulent regime the turbulence is generated outside” the viscous penetration depth, leaving the velocity close to the wall nearly the same as it would be for laminar flow. In the conditionally turbulent regime, the turbulence is generated within the penetration depth, and has a dramatic influence on the gas velocity within the penetration depth. Hence, it is likely that the quantitative details of streaming-driven convection in the conditionally turbulent regime will differ greatly from the results presented here. A brief survey of published data from seven
Cryogenics
1997 Volume
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771
Acoustic
streaming
in pulse tube refrigerators:
J.R. Olson and G. W. Swift Figure
‘03m’
102 -_
// //
Weakly 9
-
//
turbulent
// //
;
// // 10’ -_
// //
----
I/ ‘\
-
I/
Laminar
102
(2
103
i Turbulent Conditionally
1
:
turbulent
104
105
Re Figure2 Regimes of oscillatory flow in a circular pipe of radius 17.The Reynolds number Re = (u,)Zf?pl~ is based on the spatially-averaged velocity amplitude (u,) (adapted from references 10-12)
pulse tube refrigerators I3 showed that all seven operated in the weakly turbulent regime, as do our experiments described below. Although the energy flux density &@‘,,, convected by this streaming is formally of fourth order, it can in practice be as large as a typical second-order energy flux density, because T2,0 can be of the order of the temperature difference between the two ends of the pulse tube when RX=&. Accurate calculation of this heat flux density would probably require significant additional effort, because the weak turbulence in typical pulse tubes probably enhances the lateral heat transfer between the upward and downward streaming currents, reducing T2,,, and making accurate calculation of T2.0 difficult. (Note that this effect may also have driven the past evolution of pulse tubes toward long, slender aspect ratios.) In fact, with such large T2,0, the entire perturbation expansion presented here is essentially invalid, except when taper or phasing makes Gz?,~ (and thus T& very small. There are numerous other fourth-order energy flux terms in addition to the two (see Appendix A, Equation (A14)) contained in rit,,,~,T~,~, which would in principle have to be considered to obtain a formally correct fourth-order result. Fortunately, the only other large term, p2,0cp%e[T,(k,)]/2, is zero at the same taper angle that makes T2.0 zero, while the remaining terms, such as those involving products of first- and third-order quantities, are small for all angles.
Experiment
v
Compliance
P
Orifice
valve
Hot heat exchanger
!c
Pulse tube
and results
To verify Equation (13), we tested an orifice pulse tube refrigerator with three pulse tubes: a right-circular cylinder, a truncated cone with the optimum angle determined by Equation (13), and a second cone with roughly twice the optimum angle. The double-angle cone should induce the same amount of streaming as the cylindrical pulse tube, but in the opposite direction, and so should exhibit roughly the same performance as the cylindrical pulse tube. The pulse tube refrigerator is shown schematically in
772
3. It was filled with 3.1 MPa helium and driven at 100 Hz by a thermoacoustic engine similar to one described previously’4. This system was not thoughtfully designed, so its performance was not impressive; it was simply assembled from parts on hand. The entire apparatus beyond the hot end of the regenerator was contained within a vacuum can where the pressure was 10m5torr. All cold parts were also wrapped in several layers of aluminized Mylar superinsulation. The regenerator was 4.6 cm in diameter, 3.0 cm long, and made of #325 stainless-steel screen with 28 pm wire diameter. The two hot temperature heat exchangers were stacked copper screens soldered into copper blocks through which 15°C cooling water flowed. The cold heat exchanger consisted of parallel copper plates. The flow straightener was simply four layers of #/40 stainless-steel screen. The compliance (i.e. reservoir volume) was a steel bulb with a volume of 150 cm3. The orifice was a needle valve which could be adjusted from outside the vacuum can. The three pulse tubes were stainless-steel with a wall thickness of 0.5 mm, a length of 4.65 cm, and a volume of 11 .O cm3. Hence, the cylinder had R = 8.7 mm, much larger than S, = 80 pm and S, = 97 pm at 200 K, which are in turn much larger than the 1 pm roughness of the inner surfaces of the pulse tubes. The optimum-angle cone pulse tube (see below) had a total included angle 4 of 0.049 rad = 2.8”, with the cold end larger than the hot end. This is a small enough angle that it is barely discernible by eye. The double-angle pulse tube had an angle of 0.103 rad. In both cases, the internal diameter at each end was calculated in order to make the volume of the pulse tube the same as in the cylindrical case. The oscillating pressure amplitude was measured with piezoresistive sensors in the compliance and near the two hot heat exchangers. The temperature of the cooling water in the hot heat exchangers was measured with thermocouples, and a thermocouple was inserted directly into the
Cryogenics
1997 Volume
37, Number
12
Figure3 Schematic representation of apparatus used in the measurements. ‘P’ denotes pressure sensors; ‘T’ denotes thermometers. The angle + of the conical pulse tube has been greatly exaggerated
Acoustic
streaming
gas space of the cold heat exchanger to measure the cold gas temperature T,. was optionally applied to the cold stage with a Heat Qcold resistive heater. The resistance of the heater and the applied voltage were measured to determine the applied power. The experimental operating conditions were determined as follows. The drive amplitude and orifice setting were selected experimentally by minimizing the cold gas temperature at zero applied heat load with the cylindrical pulse tube, without overloading the thermoacoustic driver (which operated near its high-temperature limit throughout these experiments). This determined the operating point that was reproduced for all the data shown in this work: b,l = 2.32 x lo5 Pa at the driven end of the hot heat exchanger adjacent to the regenerator and b,l = 0.59 x lo5 Pa in the compliance. Under these conditions, b,l at the heat exchanger at the hot end of the pulse tube was always within 0.5% of 1.95 x lo5 Pa. These operating conditions, along with the refrigerator geometry, were used to calculate* the magnitudes and relative phases of p, and (u,) at the ends and in the middle of the pulse tube. These calculated values were used in Equation ( 13) to determine the optimum dA/dx at the ends and in the middle of the pulse tube. The values were such that a simple cone was a reasonable approximation to the ideal shape, making more difficult fabrication unnecessary. The resulting cone angle was also shallow enough to eliminate concerns about flow separation at the wall. The results of our measurements are shown in Figure 4. The data corresponding to the cylindrical pulse tube are represented by the circles, while those of the optimumangle conical pulse tube are shown as triangles and those of the double-angle cone as squares. For all three pulse
in pulse
Acknowledgements The authors thank Ray Radebaugh, for numerous helpful discussions about pulse tube refrigerators and from whom we first heard the idea of shaping pulse tubes to reduce streaming. We are also indebted to Nikolaus Rott, for always doing things right, and to Mike Hayden and Nikolaus Rott for helping to improve the clarity of the manuscript. Support for this work was begun by the TTI program of the US DOE, and completed by the Division of Materials Sciences in the DOE’s Office of Basic Energy Sciences. Part of the hardware used for these measurements was supplied by Tektronix Corporation’6.
References I.
3.
4. 5. 6.
7. 8.
9. 10
20
30
IO.
Qcold (w)
Figure4 Experimental results, showing cold temperature as a function of heat load for the three pulse tubes. The pulse tube designed according to Equation (13) is the best
Il. 12.
13. *The pressure amplitude in the compliance, its volume, and the adiabatic compressibility of the gas yield the volumetric velocity of the gas entering the compliance, which is essentially equal to that at the hot end of the pulse tube. The volumetric velocity elsewhere in the pulse tube can then be estimated from its geometry and the adiabatic compressibility of the gas. We also incorporated small corrections due to, for example, the volume and pressure drop of the heat exchanger at the hot end of the pulse tube 15.
J.R. Olson and G. W. Swift
tubes, the temperature of the gas increases as more heat is applied to the cold stage, as expected. For all measured values of applied heat, the temperature corresponding to the optimum cone pulse tube is at least 5°C colder than the temperatures with either the cylindrical or the double-angle pulse tubes, indicating that the optimum cone performs significantly better than the other pulse tubes. From an altemative point of view, it appears that the streaming-driven convective heat load on the cold heat exchanger is 3 to 5 W greater with the cylindrical and double-angle pulse tubes than with the optimally tapered pulse tube. While it was expected that the cylinder and the double-angle cone would perform roughly the same, there is a slight difference between those two data sets which has not been explained.
2.
0
tube refrigerators:
14. 15.
Radebaugh, R., A review of pulse tube refrigeration. Adv. CQJO. Eng., 1990, 35, 1191-1205. Nyborg, W.L.M., Acoustic streaming. In Physical Acousfics, Vol. IIB, ed. W.P. Mason. Academic Press, New York, 1965, p. 265. Lee, J.M., Kittel, P., Timmerhaus, K.D. and Radebaugh, R., Flow patterns intrinsic to the pulse tube refrigerator. In Proc. 7th Inf. Cryocoofer Con$. Kirtland AFB, NM 87117.5776, Phillips Laboratory, 1993, p. 125. Rott, N., The influence of heat conduction on acoustic streaming. Z. Angew. Math. Phys., 1974, 25, 417. Rott, N., Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z. Angew. Math. Phys., 1969, 20, 230. Pierce, A.D., Acousfics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America, Woodbury, NY, 1989. Jeong, E.S., Secondary flow in basic pulse tube refrigerators. Ctyogenies, 1996, 36, 317-323. Zhu, S.W., Zhou, S.L., Yoshimura, N. and Matsubara, Y., Phase shift effect of the long neck tube for the pulse tube refrigerator. In Proc. Crycoolers 9, New Hampshire, 1996. Gardner, D.L. and Swift, G.W., Use of inertance in orifice pulse tube refrigerators. Cryogenics, 1997, 37, 117-I 2 I. Hino, S.M., Sawamoto, M. and Takasu, S., Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech., 1976, 75, 193-207. Ohmi, M. and Iguchi, M., Critical Reynolds number in an oscillating pipe flow. Bull. LYME, 1982, 25, 165. Ohmi, M., Iguchi, M., Kakehashi, K. and Tetsuya, M., Transition to turbulence and velocity distribution in an oscillating pipe flow. Bull. LYME, 1982. 25, 365. NASA Pulse Tube Database, on-line at available http://irtek.arc.nasa.gov/CryoGroup/PTDatabase~Data-~ilice.html (five cases); Proc. 7th Int. Cryocooler Con& (two cases). Swift, G.W., Analysis and performance of a large thermoacoustic engine. J. Acoust. Sot. Am., 1992, 92, 1563. Ward, W.C. and Swift, G.W., Design environment for low amplitude thermoacoustic engines. J. Acoust. Sot. Am., 1994, 95, 3671-3672. Fully tested software and a users guide are available from the Energy Science and Technology Center, US DOE, Oak Ridge, TN. For a beta-test version, contact [email protected] (Bill Ward) via the Internet.
Cryogenics
1997 Volume
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Acoustic streaming
in pulse tube refrigerators:
J.R. Olson and G. W. Swift
16. Godshalk, K.M., Jin, C., Kwong, Y.K., Hershberg, E.L., Swift, G.W. and Radebaugh, R., Characterization of 350Hz thermoacoustic driven orifice pulse tube refrigerator with measurements of the phase of the mass flow and pressure. Adv. Cryo. Eng., 19%. 41, 14111418. 17. Landau, L.D. and Lifshitz, E.M., Fluid Mechanics. Pergamon, 1982, equation 15.5. 18. Mathematics, Wolfram Research, Champaign, IL (for example).
Appendix A Much of this derivation follows the previous work of RotP. We assume that the tube radius R>>6, that the tube length is >>R, and that p, and (u,) (or equivalently p, and dp,/d.x) are given. Then the solution for the other first-order variables is well known4? The ideal-gas equation of state and the equations of conservation of mass, momentum and energy are solved to first order, yielding u,
=
~
(1
_
UPIn
_
e-Cl + i)Yh,) = (u,)< 1 _ e+’
In the gently tapered tubes under consideration here, these first-order solutions are identical to those found previously4 in straight tubes, with one exception: we find that v, has an additional term proportional to y, the final term in Equation (A5), which is zero in straight tubes by virtue of the spatially-averaged continuity equation. In tapered tubes, the spatially-averaged continuity equation adds additional complexity to the analysis, as will be discussed in more detail below. Although Rott adopted the boundary-layer approximation only in order to use exponential y dependences, we use a more restrictive approximation: that 6JR is algebraically <
+ W%)
dx (Al)
T
I
=y-l y
Ttn
_&I
pp,(l
(-46)
+WQ)
m
+;
%(u,,l--
~u +1-oe
1 e_’I + iMr 1-U
(A21
-(I +i)y/S,
with ( ) denoting the average over the cross-sectional area
A of the tube. As needed, p, can then easily be obtained
where we have neglected second viscosity. In the last terms, the viscosity must be kept inside the derivatives, since p varies significantly with temperature and thus position. The viscosity is assumed to vary as a power law of the temperature, as given by Equation (9), with b - 0.7 for monatomic gases. Equation (A6) can be written out to second order, keeping terms that are independent of time and dropping terms that are small:
from the first-order equation of state k!Yle -iwp,& (A3)
The velocity v, perpendicular to the wall can be obtained from the complete first-order continuity equation in and near the boundary layer
au,+ pmav,__ 0
pm-
ax
(A4)
ay
ax
+ pmv,
au ay
--J
(A7)
The tilde represents complex conjugate. For straight or gently tapered tubes, terms such as P,d2rZ,/axZare smaller than the terms remaining in Equation (A7) by factors of the order 6Jh < 1, where A is the acoustic wavelength, and can thus be neglected. The factors of l/2 arise because YIe[a]%e[b]
by integrating once with respect to y, and using the boundary condition v,(y = 0) = 0. This yields
1
1 a ati, = -- dp2.0 + cL azu2.0 i *-rRe[(p-)] dx m ay2 2 ’ ay 1 ay [
T, PlIl Pm T, PI PI -_=---
iw, +-dpm U, + dx
+ pm&
= t %e[ab] + 1 %e[a6];
the %e[ub]
terms
are those mentioned in the main text with time dependence of ei2&, which have been ignored. Equation (A7) can be regarded as a differential equation for u,,(y) in terms of dp,,,/dx and a complicated driving function consisting of the sum of products of known firstorder quantities. Upon integrating Equation (A7) once with respect to y, the left-hand side can be simplified using integration by parts on the third term within the brackets and using Equation (A4) to eliminate &Jay. The result is
(A-j)
i Se
=-zY +
774
Cryogenics
1997 Volume
37, Number
12
$ (p,,,u,P,)dy
dpzo
/4nW)
+Prn
+ $ %e(u,P,)
1 auz,o +ZWe k,ati, ay
(
ay1
(A@
Acoustic streaming in pulse tube refrigerators: (cf. equation ( 11) in reference 4). Integrating Equation (A8) again with respect to y, expressing CL,in terms of T,, and solving for u~.~,we obtain
U*,okY)
L 2Pril
=
0
Y
u,V,dy-$%e
+&!Re
2Pm
(A9)
”
0
To arrive at this form of the result, we used dp,,,/dr = - (pJ’,,,)dZ’,,,/dx and dp,,,/dr = b(~,,,/JT,,,)d7’,&. Equation (All) reduces to Rott’s result upon substituting p, = (iyp,,,/Jo)d(u,)/dx (as appropriate for non-tapered tubes) and taking (u,) and d(u,)/d_x real. Rott’s final results were expressed in terms of u, and duJd.x, similar to our (u,) and d(u,)/dx, but we choose to eliminate d(u,)/dx so that our results are expressed in the more experimentally accessible variables (u,) and p,. To do so, we use the first-order y-averaged continuity equation
0
1 ho 2
+z
y* - C(x)y
dx
The first three terms on the right-hand side represent the (lengthy) driving terms referred to initially. They include exponential y dependences with characteristic lengths of 6, and S,, which are non-zero only close to the wall. The last two terms on the right-hand side contain no exponential y dependences; they contribute only to the parabolic velocity profile extending across the entire tube. Because of our choice of the lower limit of integration, Equation (A9) satisfies the boundary condition that u*,~= 0 at y = 0. Although we will not need them, the constant of integration C(x) could be determined if desired by the condition that &+,,&Iy = 0 at y = R, and the magnitude of d,u,,/dx could be determined if desired by the condition that the total second-order time-average mass flux through the tube is zero:
pm~2,0
1 !Re(p,E,)
+
dA = 0
(AlO)
1
Henceforth, our derivation differs importantly from that of Rott. Focusing solely on standing waves, Rott assumed that u, was real for all X, and hence that du,/& was real, implying that p, was purely imaginary (cf. equation ( 1) of reference 4). Here, to allow arbitrary phasing between p, and (u,), we must allow the phase of (u,) to vary with X, so that d(u,)/dx might have a different phase than (u,). We are interested in the velocity at the outer edge of the boundary layer, by which we mean y>S,, and y>>6,, yet y << R, so that the exponential parts of u~,~(Y)are essentially zero but the parabolic part of u*,~(Y)essentially has its value at the wall. We substitute all the first-order solutions into Equation (A9) and perform the y integrations. The evaluation of the integrals in Equation (A9) is tedious. After integration, to evaluate u*,~at the outer edge of the boundary layer, we let y-+03 within exponentials and set y = 0 in linear and quadratic terms. The result is
1 + dT, (I-@(I-&) 3
~~.~(m)= - - Se 4w
(1 + i)(q)
W,) ~ dx
Using Equations (A2) and (A3) and do,,& - (p,lT,,,)dT,,,/du, we can express this in the form
4wTzm CIJ 1
+-n?e 2YPln
(l+&)(l+a) (y-
Equations (A12) and (A13) are only valid for tubes with sufficiently slowly varying cross-sectional areas; the criteria for validity are summarized by Pierce6 in his derivation of Webster’s horn equation. Fortunately, the areas of tapered pulse tubes that we have considered vary so slowly that these criteria are extremely well satisfied. Straightforward substitution of Equation (A 13) into Equation (A 11) brings non-zero dAldx into the analysis. Finally, we are concerned not with uzTobut with the second-order, time-averaged mass density flux m2,okY) = Pm~*,o + 1 %e(p,ti, ) (the integrand of Equation (AlO)), shown schematically in Figure lb. In this paper, the quantity of greatest interest is the effective mass flux density at the wall r&, shown in the figure, which we write as
k2,, = AG~,~(~)+ 4 ~4pl(m)fil(~)l = P~u~.o(~) + f ~dhW1
dl+u)
-b)]
(A14)
The second form of the equation is due to our use once again of the approximation that 6,JR is algebraically <
=Prn
1
!!!$
g-g
[
m
Using Equations (All), (A14) quickly yields
(Al5)
(A13) and (A15) in Equation
3 (y- l)(l - bti) ~ + ~-~ 4 2a(l +a)
(u,)(h)
l)[l -bd+iid’*(l
=
(A13)
1
- 1 P&J 1
J. R. Olson and G. W. Swift
(Al 1)
z+(y-
1X1 -b)&
4
2(1 + a)
+& w (‘416 111
I Cryogenics
1997 Volume
37, Number
12
775
Acoustic streaming in pulse tube refrigerators:
JR
The result holds regardless of geometry (e.g. for rectang&r tapered tubes as well as circular tapered tubes), as long as a,,<< all lateral dimensions of the tube, although the last two terms in Equation (A9) will depend on the geometry.
776
Cryogenics
1997 Volume
37, Number
12
Olson and G. W. Swift The mathematical challenge of going from Equation (A9) to Equation (Al 1) is considerable. Assistance from symbolic-mathematics softwareI is extremely useful.