Measurements of the resistance of parallel-plate heat exchangers to

Measurements of the resistance of parallel-plate heat exchangers to oscillating flow at high amplitudes Ray Scott Wakeland and Robert M. Keoliana) The Pennsylvania State University, Graduate Program in Acoustics, P.O. Box 30, State College, Pennsylvania 16804-0030

共Received 29 July 2003; revised 30 January 2004; accepted 31 January 2004兲 Measurements of the acoustic resistance of parallel-plate heat exchangers are reported. The resistance is measured for two identical exchangers separated by a small gap, and also by a large gap. High amplitude deviations from linear theory are analyzed in terms of a minor loss coefficient. Results are compared to theoretical predictions made in a previous article. © 2004 Acoustical Society of America. 关DOI: 10.1121/1.1701901兴 PACS numbers: 43.35.Ud, 43.25.Ed, 47.60.⫹I 关RR兴

I. INTRODUCTION

The authors have recently completed a study of heat transfer between identical parallelplate heat exchangers in oscillating flow.1 Pressure data, collected during the same experimental runs in which the heat transfer data were collected, are reported here, with emphasis on the nonlinear, high-amplitude resistance to oscillating flow. In a previous article,2 we developed equations for calculating effective exit-flow minor loss coefficients for nonuniform, timevarying, oscillating flows, and applied the results to velocity profiles found between parallel plates in thermoacoustic devices. The present measurements provide some experimental support for that previous theoretical work. II. EXPERIMENTAL APPARATUS

The experimental apparatus is described in detail in Refs. 1 and 3. Essentially, a pair of identical parallel-plate heat exchangers is inserted into a high-amplitude, lowfrequency, oscillating flow in air at atmospheric pressure, and the pressure drop across the exchangers is measured. Reference 3 contains validation of the pressure measurement system used for the present measurements. The heat exchangers used in these measurements are made from flat extruded aluminum tubing of a type used in automotive air-conditioner condensers. The tubes are 2.0 mm ⫻22.0 mm in external cross section, having rounded ends with a nominal radius of 1.0 mm. Each 292 mm⫻292 mm exchanger is made up of 35 of these tubes separated by open spaces of 6.35 mm, for a center-to-center tube spacing of 8.35 mm and hydraulic radius of r h ⫽y 0 ⫽3.175 mm. 共The hydraulic radius for parallel plates is half of the plate separation. This half spacing is often referred to as y 0 in thermoacoustics, following the notation of Swift.4 Figure 1 shows a few of the tubes in cross sections, with dimensions labeled.兲 The total wetted perimeter of the tubes is ⌸⫽20.45 m, and the porosity ␴ 共exchanger void volume divided by total exchanger volume兲 is 0.76. A piece of foam is used to insulate the exchanger manifolds from each other, and to establish the separation 2x hx of adjacent ends of the exchanger tubes. Rea兲

Electronic mail: [email protected]

J. Acoust. Soc. Am. 115 (5), Pt. 1, May 2004

Pages: 2071–2074

sults are reported for two sizes of spacer: ‘‘small gap,’’ 2x hx⫽5.5 mm; and ‘‘large gap,’’ 2x hx⫽54.4 mm. The exchangers are aligned so that, if one could look axially down the duct, the tubes of the closer exchanger would lie directly in front of the tubes of the far exchanger, with a maximum amount of ‘‘free flow area.’’ Quantities used in the analysis include frequency f, angular frequency ␻ ⫽2 ␲ f , peak velocity amplitude u 1 of the gas in the duct, the higher amplitude u hx⫽u 1 / ␴ at locations within the exchangers, the kinematic viscosity ␯ of the air, viscous penetration depth ␦ ␯ ⫽ 冑2 ␯ / ␻ , total frontal area A fr of the exchangers 共which is also the cross-sectional area of the duct兲, and volume velocity U 1 ⫽A fru 1 . A mercury barometer is used to determine the density ␳ m of the air, with ␳ m ⫽1.145 kg/m3 for these experiments. Because the pressure drop across the exchangers is very small, the useful range of frequencies is much more limited for the pressure measurements than for the heat transfer measurements. A pair of bellows are used as part of the apparatus that drives the air back and forth between the exchangers, as described in detail in Ref. 3. In the direction of high frequency, as the first resonance of the bellows is approached, extraneous bellows motions affect the phase of the gas displacement, so that it can no longer be accurately inferred from the end-plate motion. Because the inertial component of the pressure increases with frequency squared 共for constant displacement amplitude兲, the phase shift due to the resistive component becomes very small, so that even a small error in phase produces a large error in the calculated resistance. In the low frequency direction, at some point the total pressure simply becomes too small to measure for the parallel plate exchangers, which present very little resistance to flow. The reliable range of frequencies is 1.5– 6 Hz, corresponding to 3.45⬎y 0 / ␦ ␯ ⬎1.72. A complete description of the measurements used to determine reliability is included in Wakeland’s dissertation.5 The oscillating flows within the exchangers are very likely laminar. The Reynolds numbers shown on the upper axes of Figs. 2–5 are peak Reynolds numbers, Rehx ⫽u hx4y 0 / ␯ , in terms of hydraulic diameter (⫽4y 0 ), with a maximum value of 2020. The acoustic, oscillating-flow Reynolds number, Reac⫽u hx␦ ␯ / ␯ , depends on frequency. Its

0001-4966/2004/115(5)/2071/4/$20.00

© 2004 Acoustical Society of America

2071

FIG. 1. Dimensions of the heat exchangers.

maximum value for any of these data points is 160. The flows within the exchanger, then, would be expected to be strictly laminar if the exchangers were long 共Ref. 4, Sec. 7.2兲. The shortness of the exchangers puts this conclusion in some doubt, but the low value of Rehx⬍2020 itself also suggests laminar conditions. The peak Reynolds number within the duct is 35 000, but the ratio 冑A fr/ ␦ ␯ ⬎26 puts these flows in the ‘‘conditionally turbulent’’ regime, so the boundarylayer calculation of resistance in the duct is appropriate.

FIG. 3. Measured acoustic resistance per exchanger at 1.5, 2, 2.25, and 2.5 Hz for small exchanger separation (2x hx⫽5.5 mm). There seems to be a knee in the data at around U 1 ⫽0.023 m3 /s.

Small-gap results are shown in Figs. 2 and 3, with largegap results in Figs. 4 and 5. Data are presented in terms of acoustic resistance (R⫽⌬p/U 1 ) versus volume-velocity am-

plitude U 1 , where ⌬p is the resistive component of the pressure difference across the test section, i.e., the component of the total pressure drop that is in phase with velocity. The total, measured resistance 共divided by two兲 is shown by the open circles for various values of y 0 / ␦ ␯ .

FIG. 2. Measured acoustic resistance per exchanger in the 2– 6 Hz range for small exchanger separation (2x hx⫽5.5 mm). Open circles are the measured resistance divided by two. Closed circles are resistance per exchanger minus the linear theory value and minus the boundary-layer value for the duct. The slope is related to the minor loss coefficient.

FIG. 4. Measured acoustic resistance per exchanger in the 2– 6 Hz range for large exchanger separation (2x hx⫽54.4 mm). Open circles are the measured resistance divided by two. Closed circles are resistance per exchanger minus the linear theory value and minus the boundary-layer value for the duct. The slope is noticeably higher than in Fig. 2.

III. RESULTS AND ANALYSIS

2072

J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004

R. S. Wakeland and R. M. Keolian: Acoustic resistance of parallel plates

for the rounded tube used here, the entrance minor loss is probably negligible. Looking at only the lowest reliable frequencies, in Fig. 3, we observe something of a ‘‘knee’’ at around U 1 ⫽0.023 m3 /s (Rehx⫽282), with a higher slope at amplitudes below this value. This slope corresponds to K ⫽0.21. In the plots of large-gap data 共Figs. 4 and 5兲, the slopes are significantly larger than those obtained from the small gap, corresponding to K⫽0.17 in the high amplitude region and K⫽0.26 at low amplitude for the lowest frequencies. The interaction between the exchangers in the small-gap case acts to lessen the overall effective minor loss coefficient. This is not surprising, since if the gap were reduced to zero, one entrance and one exit would be eliminated, cutting the per-exchanger effective minor loss in half. The wide gap results, therefore, are more representative of a single, isolated exchanger. The uniform-velocity exit minor loss coefficient K min for the parallel plate heat exchangers, which have porosity ␴⫽0.76, is2 K min⫽ 共 1⫺ ␴ 兲 2 ⫽0.057, FIG. 5. Measured acoustic resistance per exchanger at 1.5, 2, and 2.5 Hz for large exchanger separation (2x hx⫽54.4 mm). The knee in this data at U 1 ⫽0.029 m3 /s is more sharply defined than the knee in Fig. 3.

The linear theory is well established.4 The purpose of the measurements is to try to learn something about highamplitude deviations from the acoustic theory. The closed circles show ‘‘excess resistance per heat exchanger,’’ in which the theoretical linear resistances of the plates and the duct have been subtracted from the measured resistances.6 The data collapse nearly to a single line, with a slope of 17 Pa s2/m6. 共In Figs. 2–5, all quantities are in MKS units, so the designation ‘‘17U 1 ⫹0.6’’ in the legend in Fig. 2 is shorthand notation for ‘‘R⫽(17 Pa s2 /m6 )U 1 ⫹0.6 Pa s/m3 ,’’ and similarly for the legends in Figs. 3–5.兲 Because the minor loss coefficient K relates pressure drop to the velocity in the heat exchanger u hx⫽u 1 / ␴ according to 2 , ⌬ p⫽K 21 ␳ m u hx

共1兲

and with R⬅

⌬p ⌬p ⌬p ⫽ ⫽ , U 1 A fru 1 ␴ A fru hx

共2兲

an excess resistance that increases linearly with velocity corresponds to a minor loss coefficient 2 ␴ 2 A fr2 dR . K⫽ ␳ m dU 1

共3兲

The slope of the fit line (dR/dU 1 ⫽17 Pa s /m ) in Fig. 2 corresponds to an effective minor loss coefficient K ⫽0.125. We say ‘‘effective’’ K because this is not a measurement of minor loss alone. Presumably there is also an ‘‘entrance effect,’’ since the hydrodynamic boundary layer is almost certainly thinner near the leading edge of the exchanger. There also may be some entrance minor loss, but, 2


6

共4兲

whereas the maximum coefficient K max , expected for a fully developed parabolic flow profile, is2 K max⫽ ␣ ⫺2 ␤ ␴ ⫹ ␴ 2 ⫽0.296,

共5兲

where ␣ and ␤ are the ‘‘kinetic energy coefficient’’ and ‘‘momentum coefficient,’’ which are 1.54 and 1.20 for parabolic flow between parallel plates. In our previous article on exit losses from nonuniform oscillating flow, we calculated effective values of ␣ and ␤ for laminar oscillating flow 共Fig. 3 of Ref. 2兲.7 Using these values we predict that the exit minor loss coefficient should range from a value of K⫽0.18 at y 0 / ␦ ␯ ⫽3.5 ( ␣ eff⫽1.31, ␤ eff⫽1.12) up to K⫽0.27 at y 0 / ␦ ␯ ⫽2.0 ( ␣ eff⫽1.48, ␤ eff⫽1.18). No trend in slope with frequency is discernable in Fig. 2. To the extent that the plates are short compared to the hydrodynamic entrance length, the flow is arguably not ‘‘fully developed’’ at the exit of the tubes, even in the oscillating flow sense that is assumed in Ref. 2. This would reduce the exit minor loss at higher amplitudes. At low amplitudes, we expect the exit velocity profile to be close to that of the linear theory used in Ref. 2. The low-frequency, small-amplitude, wide-gap value K⫽0.26 is quite close to the value 0.27 predicted by Ref. 2, and is 4.6 times larger than the value 0.057 obtained from (1⫺ ␴ ) 2 . This is evidence in support of the method used in Ref. 2, albeit for a single value of ␴. Shah and London8 give the steady-flow hydrodynamic entry length L hy for parallel plates as L hy ⫽0.031 25⫹0.011 Re. 4y 0

共6兲

The knees in Figs. 3 and 5 occur at Reynolds numbers 共for the gas within the exchangers兲 of around Rehx⫽282 and 355, for which Eq. 共6兲 puts the entry lengths at 43 and 53 mm. These entry lengths are 2–2.4 times the length of an exchanger. It might have been expected, then, that the knees would have occurred at lower values of U 1 . It is also curious


2073

that the knee occurs at a higher velocity for the wide-gap data, though the location of the knee on the small-gap graph is somewhat vague, and the difference may not be significant. It should also be pointed out that, at these low frequencies, the peak gas displacement is roughly the same as the 22 mm length of the exchangers at the knee, but again, the poor definition of the knee limits our ability to comment further. None of the plots of excess resistance 共Figs. 2–5兲 has an intercept of zero. That is, there is some real or apparent linear resistance not accounted for by the linear theory for the plates and the ducts. Unfortunately, we do not consider the data at the very lowest amplitudes to be sufficiently reliable to judge whether this is a real or apparent offset in resistance at U 1 ⫽0. To understand what we mean by an ‘‘apparent’’ linear resistance, consider Fig. 3, and imagine that the lowamplitude data extrapolate along the straight line R⫽28U 1 ⫹0.35 to a value of R⫽0.35 Pa/共m3 /s) at U 1 ⫽0. This 0.35 would then be a ‘‘real’’ linear resistance of R 0 ⫽0.35 Pa/共m3 /s) because this would be the actual value obtained in the limit as U 1 →0. The high-amplitude fit line, R ⫽17U 1 ⫹0.60, on the other hand, has an intercept of 0.60 Pa/共m3/s兲. The high-amplitude resistance, therefore, is not given by R⫽17U 1 ⫹R 0 , but rather by R⫽17U 1 ⫹R 0 ⫹R app , where R app⫽0.25 Pa/共m3 /s) the additional ‘‘apparent’’ linear resistance caused by the downward curvature of the excess resistance. Petculescu and Wilen9 made very careful, accurate measurements of minor losses in jet pumps. Their data also exhibit an apparent linear resistance 关see their Figs. 4共a兲 and 5共a兲兴. This means the two quantities linear resistance and effective minor loss coefficient are not sufficient to calculate the high-amplitude resistive pressure drop. A full description would have to include the apparent excess linear resistance as well. In conclusion, these measurements support the general approach of calculating pressure drops in thermoacoustic devices by starting with the results of linear theory and adding

2074


a minor loss correction, with some caveats. The exit minor loss coefficient used in the correction may be larger than that predicted by K⫽(1⫺ ␴ ) 2 , due to the development of a nonuniform velocity profile. On the other hand, for high amplitudes in very short ducts, such as these heat exchangers, the minor loss coefficient may be less than that predicted by Ref. 2 because full development of the oscillating-flow velocity profile does not occur. Other influences of ‘‘entry effects’’ have yet to be worked out. ACKNOWLEDGMENTS

This work was supported by the Office of Naval Research, the Penn State Applied Research Laboratory, and the Pennsylvania Space Grant Consortium. 1

R. S. Wakeland and R. M. Keolian, ‘‘Effectiveness of parallel-plate heat exchangers in thermoacoustic devices,’’ J. Acoust. Soc. Am. 共in press兲. 2 R. S. Wakeland and R. M. Keolian, ‘‘Influence of velocity profile nonuniformity on minor losses for flow exiting thermoacoustic heat exchangers,’’ J. Acoust. Soc. Am. 112, 1249–1252 共2002兲. 3 R. S. Wakeland and R. M. Keolian, ‘‘Measurements of resistance of individual square-mesh screens to oscillating flow at low and intermediate Reynolds numbers,’’ J. Fluids Eng. 125, 851– 862 共2003兲. 4 G. Swift, Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators 共Acoustical Society of America, Mellville, NY, 2002兲. 5 R. S. Wakeland, ‘‘Heat exchangers in oscillating flow, with application to thermoacoustic devices that have neither stack nor regenerator,’’ Ph.D. thesis, The Pennsylvania State University, 2003. 6 The linear resistances are given by R⫽( ␻␳ m L/ ␴ A fr兲Imag(⫺1/1⫺ f ␯ ), where L is the length of the element. The ‘‘Rott functions’’ f ␯ are given by Eqs. 共4.56兲 and 共4.58兲 of Ref. 4 for the duct and heat exchangers, respectively. For the exchangers, ␴⫽0.76 and L⫽22 mm; for the duct, ␴⫽1 and L is the total length of open duct between the pressure sensors, which depends on the gap size. 7 There is a typo in the labeling of Fig. 3 of Ref. 2. The tick marker ‘‘1.000’’ on the ␤ eff scale should be ‘‘1.100.’’ 8 R. K. Shah and A. L. London, Laminar Flow Forced Convection in Ducts, Advances in Heat Transfer, supplement 1, edited by T. F. Irvine, Jr. and J. P. Harnett, Eq. 共291兲共Academic, New York, 1978兲. 9 A. Petculescu and L. A. Wilen, ‘‘Oscillatory flow in jet pumps: Nonlinear effects and minor losses,’’ J. Acoust. Soc. Am. 113, 1282–1292 共2003兲.
