Measurements of shear stress in a square array rod bundle

Nuclear Engineering and Design 56 (1980) 385-391 © North-HoUandPublishing Company

MEASUREMENTS OF SHEAR STRESS IN A SQUARE ARRAY ROD BUNDLE R. EICHHORN, H.C. KAO and S. NETI *

Department of Mechanical Engineering, University of Kentucky, Lexington, KY40506, USA Received 15 February 1979

A flush-mounted hot film sensor was used to determine the shear stress distribution on the centrally located rod in a 1.4 pitch to diameter ratio square array, nine rod bundle with axial flow. The film sensor was calibrated in a concentric annulus flow geometry. Shear stress measurements were made at a position 65 hydraulic diameters from the flow entrance for Reynolds numbers from 12 000 to 32 000. The circumferential variation of the shear stress was nearly sinusoidal around the central rod and the maximum and minimum values occurred at the maximum and minimum subchannel spacing. The peak to peak variation of the sinusoidal shear stress distribution is about 4 to 6% of the mean value.

1. Introduction

the shear stress variations to a level comparable to the measured values. Flow field prediction methods in current use employ subchannel flow models which ignore many flow details [1]. They are quite successful in predicting overall pressure loss and heat transfer provided sufficiently general empirical formulas for such quantities as interchannel mixing are used along with them. More sophisticated calculation methods have been reported which attempt to account for both radial and transverse momentum transfer by an eddy viscosity model [2,3]. They require seemingly arbitrary assumptions to be made about the azimuthal eddy viscosity to achieve reasonable shear stress variations. Rehme [4] has shown that one such code [2] cannot be rationally modified to predict accurate details of the flow in a plane wall/rod region for a pitch/diameter ratio of 1.07. Still more sophisticated methods that use recent turbulence modeling techniques have also been developed [5]. They have achieved reasonable success in predicting secondary flows (although there are few measurements available) but such quantities as the local shear stress are still not satisfactorily predicted. In this work, we report the results of direct measurements of the shear stress on the central rod of

Axial flow along the outside of tubes occurs in a number of heat exchanger types, but the chief impetus for its recent study has been its importance to the design of nuclear reactor fuel rod bundles. In this application, fuel pellets are encased in rods that are assembled in an array. Since a large heat transfer rate occurs between the fuel rods and the coolant flow, it is important to know the detailed flow field both for pressure loss and for heat transfer design calculations. In fully developed flow along smooth rods, the pressure drop can be directly related to the skin friction. The skin friction varies around the periphery of the rods and the bounding walls. Its mean value determines the pressure gradient along the rod bundle. It takes on a minimum value at points of closest approach of neighboring rods and a maximum value at the largest spacing. Turbulent momentum transfer normal to the rod, acting alone, would give rise to much larger variations in shear stress around the periphery than are observed in practice. It appears that azimuthal turbulent momentum transfer and perhaps secondary flows are needed to reduce * Present address, Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA.

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R. Eichhorn et al. /Measurements o f shear stress in a rod bundle

a 3 × 3 square rod array of pitch to diameter ratio equal to 1.4. Hot film surface sensors were mounted on a rod and calibrated in an annulus. Measurements in the rod array were made with water flow over a Reynolds number range from 12 200 to 31 500. The local shear stress variation is clearly displayed, but the mean values obtained by integration of the local shear stress are of only modest accuracy.

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2. Experiment

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The experimental study was performed with the flow loop sketched in fig. 1. A centrifugal pump supplies water flow to a storage tank and thence to either the rod bundle test section or an annular calibration section. A heat exchanger is included in the loop to premit removal of the pump work and control the operating temperature of the system. The flow was adjusted by a valve at the pump outlet and metered by a calibrated square edge orifice. The rod bundle test section is shown in figs. 2 and 3. It consists of a three by three array of 1.9 cm diameter stainless steel rods 208 cm long, spaced on a pitch to diameter ratio of 1.4. The plane walls bounding the rod array are of plexiglass and aluminum and form an 8 cm × 8 cm square duct. The minimum distance between the rods and the plane walls is

equal to one-half the minimum distance between adjacent rods. The test section is fixed at the top and bottom to cylindrical plenum chambers about 35 cm diameter by 35 cm long. The flow inlet to the rod bundle is through holes drilled in the upper cover of the lower plenum chamber. The holes are located in the center of each subchannel and are sized to supply the same flow rate per unit area to each subchannel. The pressure drop across the lower plenum chamber cover is approximately 25 times the pressure drop in the rod bundles. The rods are screwed into the cover of the upper plenum chamber where a rLxture is used to place them in tension. The central rod, on which the shear stress measurements were made,

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R. Eichhorn et al. /Measurements of shear stress in a rod bundle

was not so fixed; instead, provision was made to rotate it about its axis. The annular calibration section is shown in fig. 4. It is constructed of 3.8 cm I.D. copper tubing and is designed to permit the insertion of a 1.9 cm rod in the center of the tube. The annulus thus formed has a radius ratio of about 0.5. To ensure a uniform flow around the periphery of the annulus, water from the flow loop is introduced and removed through arrays of holes drilled in the outer tube. Centering screws are provided downstream of the measurement station. The central rod of the array was used for shear stress measurement. The rod is straight to within 0.15 cm per meter of length and has an eccentricity less than 5% of its diameter at the measurement station. It is fitted with two 1.6 mm diameter pressure taps located 9.9 hydraulic diameters upstream and downstream of the shear stress measurement station. A Thermosystems, Inc. Model 1472W subminature hot-film sensor is flush mounted in the surface of the tube wall 64 hydraulic diameters from the inlet plane of the rod array. The sensor

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consists of a quartz disc about 0.7 mm in diameter by 0.5 mm thick with a 0.5 by 0.1 mm platinum ribbon deposited on its surface. A coating of quartz covers the sensor surface to prevent decomposition of the platinum fdm in the water environment. The sensor was cast in a truncated epoxy cone, which was then cemented in a mating hole drilled in the surface of the rod. The resulting assembly is smooth to the touch and has the long dimension of the sensor film directed along the rod circumference. Electrical leads for the hot film sensor and tubing for the pressure taps are inside the rod and exit through the upper end. The instrumented rod can either be inserted in the annular calibration section or in the central location in the rod array. As discussed in detail below, calibration of the hot film sensor for shear stress measurements was done in the annular calibration section. When installed in the rod array, the instru. mented rod can be rotated 360 ° about its axis by means of a rotating table mounted on top of the upper plenum and attached by a collet arrangement to the end of the rod. Pressure drops were measured with U-tube manometers containing mercury, carbon tetrachloride on Merriam fluid No. 3, as appropriate for the pressure level. Fluid temperature was sensed with a chromelalumel thermocouple referenced to the ice point. Absolute temperature values were accurate to within -+0.I°C while relative temperatures were known to _+0.025°C. A Thermosystems, Inc. Model 1050 constant temperature anemometer with associated accessories was used to control the sensor operating temperature and indicate a voltage related to the shear stress level. Both during calibration and measurement, the anemometer output was averaged over 10 s with a Beckman Model 651 voltage to frequency converter (VFC) and events per unit time (EPUT) meter. The voltage measurement system was calibrated with a Fluke model 332B voltage reference source.

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2.2. Calibration and measurement procedure

In the usual mode of operation of hot film sensors, the sensor temperature is maintained at a constant value that is high enough so ambient temperature variations have only a slight effect on the system

388

R. Eichhorn et al. /Measurements o f shear stress in a rod bundle

calibration parameters. In that case, one needs only to determine the constants A and B in the calibration formula rl/3 =AV 2 + B W

(1)

and the hot film sensor can be operated at a fixed value of the overheat ratio. When making measurements in water, one must ensure that the film operating temperature is low enough so that boiling and degassing of the water is avoided. Since our loop operates at nearly atmospheric pressure, the sensor temperature is limited to about 70°C. Furthermore, the loop equilibrium temperature varies from 30 to 40°C depending on the ambient (outside) temperature. These considerations lead to a relatively insensitive maximum overheat ratio of about 1.08. Most importantly, they require the parametersA and B in eq. (1) to be expressed as functions of the ambient temperature. We were thus led to discard the usual anemometer practices based on a fixed overheat ratio and resort instead to a direct calibration of the sensor output voltage as a function of ambient temperature and shear stress level. Both during sensor calibration and rod bundle shear stress measurement, the sensor was maintained at a timed operating temperature and the overheat ratio allowed to vary with the ambient temperature. Each data point, whether collected during calibration in the annulus or measurement in the rod array, was taken as the average of 10 consecutive readings of the EPUT meter. From the calibration data, the average values of the voltage squared and the shear stress, determined as described below, were fitted by linear regression analysis to determine the parameters A and B in eq. (1). The coefficient of determination for the calibration curves was 0.98. The hot f'flm sensor was calibrated with fully developed turbulent flow in the annulus for shear stress levels that corresponded to those in the rod bundle. The flow loop temperature during both calibration and measurement varied from 31 to 35°C. A calibration run was made between each set of runs in the rod array. For fully developed flow in the annulus, the relationship between the inner surface shear stress, rw, and the pressure drop per unit length &p/A/is '

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where rl and r= are the inner and outer annulus radii, respectively [6]. This equation assumes that the location of the radius of maximum velocity and the radius of zero shear are coincident. The resulting inner wall shear stress should be slightly high [7], but the uncertainty is believed to be less than 3%. Measured friction factors for the annulus flow are compared in fig. 5 with experimental results from the literature [8,9] for somewhat different radius ratios, with semi-empirical formulas due to Mal~k et al. [10] and Rehme [11] and with Prandtl's smooth circular tube equation [ 12]. The latter is given by the formula f-1/2 = 4 loglo 2 R e f 1/2 - 1.6,

(3)

where the friction factor f is defined by f = 2rw_ Dh Ap p~2 2p~2 A/ ,

(4)

and the Reynolds number is based on the hydraulic diameter. Our friction factors are somewhat low at high Reynolds numbers which could result from errors due to the fact that only two pressure taps were used to determine the shear stress. The analysis of Mal~k et al., is significantly higher than either our measurements or Rehme's theory. Ideally, the flow field about the central rod in the array has two planes of symmetry and two

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R. Eichhorn et al. / Measurements o f shear stress in a rod bundle

of antisymmetry. It should thus be necessary only to measure the shear stress in one of 8 subchannels or over an angular rotation of the sensor position through 45 ° . Actually, the rods are slightly bowed so we do not have perfect symmetry. Measurements were therefore made at 10° and occasionally 5° intervals over the full 360 ° angular span of the sensor location. Readings were taken at 5 Reynolds numbers from 12 200 to 31 500. The pressure drop along the test section was also measured to find the mean shear stress. Measurements in water with a hot film sensor are affected by gas bubbles that form on the sensor surface and by a contaminating film that builds up over a period of time. Both effects lead to a decrease in the anemometer voltage required to maintain a given sensor temperature. It is quite easy to detect and discard data which are affected by the formation of gas bubbles since the anemometer voltage decreases rapidly as a bubble is formed and returns abruptly to its previous level as the bubble is swept away by the flow. The effect of the contaminating film can only be eliminated by operating the sensor in a very clean system, or by repeatedly cleaning the sensor surface. In our experiments, the test rod was removed between runs at a given Reynolds number to clean the sensor and check its calibration. Even with this precaution, the anemometer bridge voltage was found tO decrease over the period of time ( 4 - 5 h) necessary to make measurements at one Reynolds number. In the worst case, the effect corresponded to a decrease of about 6% in the measured shear stress. Apart from the systematic uncertainties in the shear stress measurements discussed above, we estimate the random uncertainties (probable error) to be less then -+3%. The Reynolds numbers are known to about the same degree of uncertainty.

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with increasing angular coordinate. We believe this to be due to the buildup of a contaminating fdm on the surface of the hot film sensor and to slight bowing of the test section rods. The maxima in the shear stress at 0 °, 90 °, 180 ° and 270 ° are clearly apparent as are minima displaced 45 ° from the maxima. The maxima and minima occur at the maximum and minimum subchannel spacing, respectively. An enlargement of the 45 ° to 90 ° segment of the shear stress distributions is shown in fig. 7.

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3. Results and discussion Fig. 6 shows the wall shear stress measurements for the full 360 ° of rotation of the central rod. Measurements were made at five Reynolds numbers ranging from 12 200 to 31 500. The curves for each Reynolds number are roughly sinusoidal in shape but those for the three highest Reynolds numbers show a pronounced decline in the shear stress level

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R. Eichhorn et aL /Measurements o f shear stress in a rod bundle

These segments appear to indicate a decreasing relative magnitude for the peak to peak variation of the shear stress with the Reynolds number. However, this finding is most likely a result of changes in flow conditions during the course of the experiments and would be altered if we had selected other subchannels to display in fig. 7. Fakory and Todreas [13] have compiled results for triangular pitch arrays, from several investigators. They found the shear stress distributions to be essentially independent of Reynolds number. The mean relative magnitude of the peak to peak variation of the wall shear stress is between 0.04 and 0.06 for the conditions of our experiment. By integrating the measured shear stress around the periphery of the rod, we can find the average shear stress on the central rod. With this quantity, we can calculate a friction coefficient from eq. (4). From overall pressure drop measurements, the friction coefficient for the entire array can be found. Fig. 8 shows the results of these operations plotted as log f against log Re where the Reynolds number is based on the mean velocity and the hydraulic diameter. Also shown in fig. 8 are curves representing the semi-empirical analyses of Rehme [ 11 ] and Mal~ik et al. [10] and the experimental results of Darling [14]. The latter are for a 9 rod bundle of P / D = 1.44 and a somewhat larger wall to rod gap than in the present work. To apply the Rehme and Mal~ik analyses, one needs to know the laminar friction coefficient for the rod bundle. This was obtained from Rehme's [15] sub-

channel friction coefficients using the method outlined in [ 11]. The result of the analysis was f Re = 20.1 for the overall laminar value and f - l / 2 = 4.07 lOglo 2 R e f 1/2 - 2.12 ,

(5)

and f - 1 / 2 = 3.83 loglo 2 R e f 1/2 - 1.75 ,

(6)

respectively, for the Rehme and Mal~k et al. turbulent friction coefficients. One can also use Rehme's analysis to determine the shear stress on a single rod. The result is (f,)-~/2 = 4.07 loglo 2 R e f 1/2 - 1.17,

(7)

where f ' = 2~w/pW 2, is the friction coefficient for the single rod and f i s given by eq. (5). The constants 4.07 and 1.17 were determined from the laminar friction coefficient for the central subchannel, f ' Re' = 116, and curves presented by Rehme. A curve for f ' is also presented in fig. 8. The overall pressure drop measurements agree well with the earlier measurements of Darling and the semiempirical theory of Rehme. Malfik's theory is high by about 8%. The results from integration of the local shear stress around the rod are somewhat higher than Rehme's analysis applied to the central rod for the data at the three lowest Reynolds numbers. The agreement of theory and analysis for the two highest Reynolds numbers is encouraging, but it may be fortuitous. More experiments in more carefully controlled circumstances are needed.

4. Conclusions I0

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Measurements of the local shear stress on the central rod of a 9 rod square array rod bundle clearly show a variation in the shear stress with circumferential position. The variation is similar in shape to a sine wave with the maximum value occurring at the maximum subchannel spacing. The peak to peak variation of the shear stress is between 4 and 6% of the mean value. Overall pressure drop results agree well with previous experimental studies and with the semi-empirical analysis of Rehme. The mean local shear stress on the central rod is not accurately determined in this study but the results support Rehme's analysis, which

R. Eichhorn et al. / Measurements o f shear stress in a rod bundle

predicts a lower shear stress on the central rod than for the bundle as a whole.

of the investigation. Mr. E.B. Yates constructed much o f the apparatus.

Nomenclature

References

A,B

D; Dh f;f' P Re

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temperature dependent functions in eq. (1) rod diameter; hydraulic diameter overall friction factor; friction factor for the central rod pitch of rod array Reynolds number based on mean velocity and hydraulic diameter annulus inner and outer radii anemometer bridge voltage mean velocity pressure gradient fluid density local shear stress; mean shear stress

Acknowledgement We are pleased to recognise financial support provided by the Department of Mechanical Engineering and the Graduate School of the University of Kentucky. Professor O.J. Hahn offered helpful comments throughout the conduct

391

[ 1] D.S. Rowe, BNWL-1695, Battelle-Northwest, Laboratories, Richland, WA (1973). [2] W. Eifler and R. Nijsing, Report EUR-4950e (1973). [3] W. Slagter, Nucl. Sei. Eng. 66 (1978) 84-92. [4] K. Rehme, Nucl. Eng. Des. 45 (1978) 311-323. [5] P. Carajilescov and N.E. Todreas, Trans. ASME, J. Heat Transfer 98 (1976) 262-268. [6] M.R. Doshi and W.N. Gill, Trans. ASME J. Appl. Mech. 38 (1971) 1090-1091. [7] K. Rehme, J. Fluid Mech. 64(2) (1974) 263-287. [8] C.J. Lawn and C.J. Elliott, J. Mech. Eng. Sci. 14 (1972) 195-204. [9] A.P. Kratz, H.J. Maclntire and R.E. Gould, University of Illinois, Eng. Exp. Sta. Bull. 222 (1931). [10] J. Mal~k, J. Hejna and S. Schmid, Int. J. Heat and Mass Transfer 18 (1975) 139-149. [11] K. Rehme, Int. J. Heat Mass Transfer 16 (1973) 933-950. [12] H. Schlichting, Boundary Layer Theory, 6th ed. (McGrawHill Book Co., New York, 1968). [13] M.R. Fakory and N.E. Todreas, private communication (1979). [14] C.W.W. Darling, MS. Thesis, Department of Chem. Eng., Queen's University, Hamilton, Ontario (1961). [15] K. Rehme, Chemie Ing. Technik 43 (1971) 962-966.