FLOW INSTABILITY IN VERTICAL CHANNELS

Transcription

1 FLOW INSTABILITY IN VERTICAL CHANNELS FLOW INSTABILITY IN VERTICAL CHANNELS Robert Stelling, and Edward V. McAssey, Jr. Department o Mechanical Engineering Villanova University Villanova, Pennsylvania Thomas Dougherty, and Bao Wen Yang Heat Transer Research Facility Columbia University New York, New York ABSTRACT A simple analytical model to predict the onset o Ledinegg instability in vertical channels under down-low conditions has been developed and evaluated. The model divides the low ield into our regions based upon the luid temperature. The pressure drop or each region is then ound by solving an appropriate set o equations or each region. The theoretical results are compared to an extensive set o experimental data covering a range o channel diameters and operating conditions. Agreement is excellent, and the prediction o the velocity at which the minimum point in demand curve occurs is well within 10% over the experimental results. A parameter, which is deined as the ratio between the lux level and the heat capacity lowrate is ound to be a very accurate indicator o the minimum point velocity. NOMENCLATURE C p = luid speciic heat D = tube inner diameter = riction actor h = enthalpy o vaporization k = conductivity Nu = Nusselt number Pe = Peclet number = RePr Pr = Prandtl number Re = Reynolds number St = Stanton number = φ/[ρ c p V (T sat - T,db )] T = temperature V = velocity v = speciic volume x = local quality Y = diameter z = distance rom entrance o channel DP = pressure drop DZ = length o region ε = channel roughness μ = viscosity ρ = density σ = surace tension τ = shear stress q = heat lux Subscripts b = bubble = luid db = ully developed sub-cooled boiling g = luid to gas ric = riction g = gas i = inlet onb = onset o nucleate boiling osv = onset o signiicant voiding sat = saturation sp = single phase tp = two-phase INTRODUCTION Flow instability is an important consideration in the design o nuclear reactors because o the possibility o low excursion during a postulated accident. For the purposes o this paper, low instability is deined as the occurrence o a minimum in the demand curve (i.e., the

2 channel s pressure drop versus velocity curve). This is commonly reerred to as a Ledinegg instability. In an operational system with alternate low paths, an increase in pressure drop in one channel can cause the low to be diverted to an alternate channel. This then results in a greater pressure drop and more low diversion. Eventually, burnout can occur in the unstable channel. One operational approach to solving this problem is to identiy the velocity at which the minimum point occurs and then to design the system so that during postulated accidents this condition is not achieved. An analytical and experimental program has been conducted to investigate this type o low instability in vertical channels under down-low conditions. Downlow was selected or this investigation because o the required reactor application. The analytical phase involved the development o a pressure drop model which could predict the single phase low regime through the ully developed nucleate boiling regime. Experimental data were obtained on circular tubes o varying diameter under a range o low conditions and heat luxes. BACKGROUND Some o the earliest data or low instability in downlow was obtained by Mirshak (1958). Since that time, a number o investigators have examined this problem but generally or up-low. Dormer and Bergles (1964) investigated the pressure drop in small diameter tubes under sub-cooled boiling conditions. These experiments were conducted in horizontal channels under controlled low so that data were obtained well beyond the minimum point. A large amount o data was correlated using the ratio o the channel pressure drop to the zero lux pressure drop versus the ratio o the surace heat lux to the heat lux required to raise the bulk luid temperature to saturation at the channel exit. Using these parameters, it was shown that the only remaining eect was geometry. Maulbetsch and Griith (1966) studied low instability in small diameter, high L/D ratio channels. They determined that during parallel low operation excursive low instability occurred when the pressure drop versus mass low reached a minimum. Whittle and Forgan (1967) examined the pressure loss in rectangular and circular channels under sub-cooled boiling conditions. These investigators determined that or a given L/D ratio the minimum in the demand curve occurred at ixed values o the ratio o the channel temperature rise to the inlet sub-cooling (T outlet -T inlet )/(T sat - T inlet ). Whittle and Forgan also determined that low direction at least or their low-rate regime did not aect the velocity at which the minimum point (OFI) occurred. Duey and Hughes (1990) developed an analytical model to describe static low instability (Ledinegg) in vertical channels. They showed that a linear relationship exists between the OFI point and the power density. In this paper, the authors developed a relationship or the low-rate at the minimum point based upon the rictional pressure drop and the buoyancy induced pressure drop. In this development, the assumption was made that the riction actor including two phase multiplier was independent o low-rate. Dougherty et al. (1989), (1990) presented experimental results or low instability studies in a vertical tube or L/D ratios rom 100 to 150. These authors showed that the results could be correlated using a Q ratio parameter deined by equation (1). This ratio is the same as that developed by Dormer and Bergles (1964). In addition, it is also in agreement with the Whittle and Forgan (1967) R coeicient. (1) q The Q ratio can be Qratio = mc appreciated i one p ( Tsat Tinlet ) considers that the pressure drop in the channel is composed o a single phase component and a two-phase component. As the channel velocity decreases, the ormer component decreases and the latter increases. These competing eects cause a minimum in the pressure drop to occur. Under sub-cooled boiling conditions, the wall temperature exceeds saturation and voids are ormed in the luid. As the luid temperature approaches saturation, the number o sub-cooled voids increases. The corresponding increase in Q ratio can be used as a measure o the void population increase. The ratio also contains all the test parameters, surace heat lux, inlet temperature, velocity, and exit pressure which determines the saturation temperature. Lee, Dorra, and Banko (1992) examined various models used to predict the onset o signiicant voids (OSV) under sub-cooled boiling conditions. Their results show that in vertical up-low the OFI point is almost coincident with OSV. These investigators concluded that the Saha-Zuber (1974) correlation was the best predictor o OSV. In addition, it was shown that the model o Levy (1967) was best among analytical approaches. Block et al. (1990) developed an analytical model to predict the pressure drop in a vertical channel under uplow conditions. This model considers our low regimes: single phase, nucleate boiling, ully developed sub-cooled boiling, and saturated boiling. For each regime, a pressure drop correlation and transition criteria are applied. Comparisons with limited test data showed

3 reasonable agreement. The present model is based upon the model proposed by Block et al. (1990). This paper presents experimental results over an L/D ratio range rom 90 to 270 or controlled low conditions under up-low and uniorm heating. Comparisons are made between these experimental data and a proposed analytical model. PRESSURE DROP MODEL In a heated tube o this nature, the luid has our possible regions o low: single-phase, nucleate boiling (partially developed sub-cooled boiling), ully developed sub-cooled boiling, and saturated boiling. The regions can be identiied by the vapor content and temperature o the luid at dierent points in the tube. The proposed theoretical model divides the tube into the appropriate regions by luid temperature only. The pressure drop or each region is then ound by solving an appropriate set o equations or each region. The regional pressure drops are then summed to determine a total pressure drop across the tube. The basic model used in this study was developed by Block et al. (1990). In this model, the total pressure drop is composed o two components, the rictional pressure loss and the gravitational pressure drop. The latter pressure drop is a unction o the luid temperature which under uniorm heating conditions is linear. The rictional pressure drop is a unction o the particular low conditions existing in the channel. The pressure drop or each low regime will be determined by the pressure gradient (dp/dz) and the length o the particular low regimes. The extent o each region is based upon the luid temperature at the onset o nucleate boiling, T onb, the luid temperature at OSV, T osv, and the saturation temperature, T sat. Equations (2) through (4) give the length o each low region. V C p D Δ Z sp = ( T onb Tinlet ) ( ρ / 4 ), φ In equation (4), the length o the ully developed, subcooled boiling region is simpliied because or V Cp D ΔZnb = ( T osv T onb ) ( ρ / 4 ),, φ the (2) (3) experimental data to be used or the comparisons OFI always occurred at ΔZ db = ( L ΔZsp ΔZnb ) luid temperatures below saturation. This model makes use o the luid energy balance equation, which is linear with temperature in the single phase and nucleate boiling regions under steady state conditions. It also uses wall heat transer equations by Dittus and Boelter (1930) and by Bowring and Thom (1965). Fluid temperatures at ONB and OSV points are ound by the Davis-Anderson (1966), and Saha-Zuber (1974) correlations respectively. Equations (5) and (6) present these correlations. T In equation (6), the high Peclet number orm (Pe φ T, osv = Tsat St ρ C p V 70,000) o the Saha-Zuber correlation is given because the experimental data in the present paper were obtained at Peclet number above this limit. Finally, the riction actors that are included in the pressure drop equations are ound using the Zigrang- Sylvester (1982) correlations, equation (7). Equation (7) is used in the single phase liquid region o the low. In the partially developed nucleate boiling ε / Dh 502 Dh 13 2 l = 025. ( 20. log( ( Re ) log( ε / 37. Re ))) region, a two-phase riction actor, tp, is obtained rom equation (7) by replacing the roughness actor, ε, by the Levy (1967) bubble size parameter, Yb. In the single phase and partially developed nucleate boiling regions, the rictional pressure gradient is then calculated rom equation (8). dp dz = 2 V D 2 ρ (4) 8 σφ T sat, = T + ( ) h k ρ onb sat g g φ D Nu (5) (6) (7) (8)

4 Figure 1 Schematic o the Test Loop In the ully developed sub-cooled boiling region, equation (8) has to be modiied to include the eect o void ormation. Equation (9) presents the rictional pressure gradient relationship used or this region. In equation (9), the quality gradient was based upon a relationship presented in Collier (1972). This orm o the 2 dp 2 tpv ρ 2 2 l dx = + V ρ dz D ( ρ ρ ρ ρ ) l dz quality gradient (see equation (10)) was used instead o the one used by Block et al. (1990) because it provided a more accurate estimate o the pressure drop beyond the minimum point. (9) (10) As noted earlier, the data that was used to evaluate the theoretical predictions achieved the minimum in the dx 4 φ h o ( Tsat T ( z)) = ( ) dz DGh g 23. demand curve beore the luid temperature reached saturation. For this reason, the above model does not include a saturated boiling regime. EXPERIMENTAL APPARATUS The test loop was designed to provide vertical downlow into test sections o various diameters with ixed length. Figure 1 presents a schematic o the test loop. Flow was supplied by either a centriugal pump or a screw type pump. The latter pump was added to the loop in order to provide the capacity to test small diameter tubes. With the centriugal pump, the loop can provide a maximum low o m 3 /min. at a head o 480 kpa. The screw pump provides a head o 5000 kpa at lows down to 0.05 m 3 /min. All components in the low loop were stainless steel or nickel plated. The water in the loop is maintained at a ph o slightly less than 7. As shown in igure 1, a 100 gallon accumulator supplied deionized water to the pump inlet. System pressure was maintained by adjusting the liquid level and the helium blanket pressure in the accumulator. The test section inlet temperature was controlled by adjusting the loop external heat rejection rate. A number o dierent diameter test sections were used; however, each test section had the same heated length (2.44 m). The test sections were constructed rom either 300 series stainless steel or 600 series inconel. Upstream and downstream o the heated portion o the test section, a 0.38 m calming length was provided. These sections were constructed rom copper and had the same inside diameter as the heated length. The connection to the power system was accomplished through these copper sections. To insure that power dissipation was minimal in the calming regions, the wall thickness was approximately ten times the test section wall thickness. Table 1 provides a description o the various test sections used in the program.

5 Test sections 1 through 9 were constructed rom commercial grade tubing with wall thickness o 1.65 mm (+/- 10%). Test section number 10 was a specially bored and ground tube with the same wall thickness but a tolerance approximately one hal o commercial grade tubing. Table 1 Description o Test Sections Test section Material Inside Diameter ST-1 stainless steel 25.4 mm 304 ST-2 stainless steel 19 mm 304 ST-3 stainless steel 25.4 mm 304 ST-4 inconel mm ST-5 inconel mm ST-6 inconel mm ST-7 stainless steel 15.2 mm 304 ST-8 inconel mm ST-9 inconel mm ST-10 inconel mm Table 1 contains several duplicate test sections which were required to replace burned out units. Heating was accomplished by passing a DC electric current directly along the test section wall. Each test section was designed to be capable o producing a maximum average surace heat lux o 3.16 Mw/m 2. The entire exterior o the test section was insulated with approximately 0.3 m thickness o iberglass insulation. To minimize any circumerential lux variation, there were no penetrations in the heated length. The instrumentation or this program consisted o three categories, primary, secondary, and loop housekeeping. The primary instrumentation included parameters necessary to establish OFI. These were power, test section exit pressure, test section inlet temperature, lowrate, and test section pressure drop. Test section wall temperatures were measured at 9 or 10 axial locations; however, these were considered to be secondary measurements. Other secondary measurements included test section inlet pressure and outlet temperature. To insure proper loop operation, there were a number o low-rate, pressure, and temperature measurements also made. For each test point, there were a total o 60 to 70 items recorded. An uncertainty analysis was conducted or all instrumentation. This analysis included instrument error, line error, and digitizing error. Table 2 presents the results o this analysis or the primary instrumentation. Table 2 Uncertainty Analysis Measurement Uncertainty Exit pressure +/ % o ull scale Dierential pressure +/ % o ull scale Inlet temperature +/ o C Flow-rate +/ % o ull scale Power +/ % o ull scale TEST PROCEDURE Under controlled low, suicient pressure head was provided to assure that low excursion did not occur. In this mode o operation, the test section inlet temperature, exit pressure, and surace heat lux were set to meet the test matrix requirements. Test conditions were established with high enough low to prevent local boiling. The test section low-rate was then gradually reduced in steps to insure that each data point was recorded at steady state. Whenever possible, several points were measured beyond the minimum point in test section pressure drop. The low-rate was then increased gradually. Data points were also measured as the lowrate increased. The number o points taken during a test varied depending upon test conditions. The physical limitations o the luid temperature sensors restricted the maximum test section low-rate to approximately 0.19 m 3 /min. For the largest diameter test sections, this limit was close to the minimum point. The data system scanned all the sensors in approximately 100 milliseconds. A total o 40 scans were averaged to provide a single data point. Allowing or recording and unit conversions, the resulting test point represents approximately a ive second average. The 40 scans were also used to compute standard deviations or the primary data. With the exception o the pressure drop data, the standard deviations were small over the entire test range. The standard deviation in test section pressure drop grew rapidly as the OFI point was reached. At this minimum point, the test section pressure drop oscillated violently due to the ormation and collapse o bubbles. However, these oscillations did not aect the test section low-rate. DISCUSSION A total o 88 demand curves were generated during the experimental phase o this program. Table 3 presents the OFI points on the basis o L/D (test section). For each test section, the average, maximum, and minimum Q ratio is given. In addition, the predicted Q ratio is also presented.

6 Figure 2 presents a comparison between the measured demand and the theoretical demand curves or the 9.14 mm tube. Figures 3 and 4 present similar results or the 15.6 mm and 25.3 mm tubes respectively. All three igures show very good agreement with respect to the prediction o the minimum point. The theoretical model tends to predict a larger pressure drop ater the minimum point. Analysis shows that the pressure gradient given by equation (9) is predominated by the quality gradient term. Equation (10) presents the model used or this gradient which was selected over the model used by Block et al (1990). The latter model produced an even larger deviation rom the experimental results. Considering the simplicity o the model, the results are very encouraging, particularly with respect to the accuracy o the minimum point prediction. As noted earlier, Dougherty et al., (1989), (1990) correlated the experimental results using the Q ratio given by equation (1). Figures 5 and 6 present comparisons between measured pressure drop and calculated pressure drop as a unction o Q ratio. Again, the agreement is very good. From these igures, it can be seen that the minimum point in the pressure is correlated by the Q ratio

7 parameter. In the theoretical model this parameter does not appear explicitly; however, it can be calculated or each point in the demand curve. Table 3 presents a comparison o all the measured Q ratio and the corresponding calculated Q ratio. Figure 7 shows the eect o L/D ratio on the minimum point Q ratio. The open symbols represent the experimental results. The solid line presents the theoretical predictions. In addition, igure 7 presents the Q ratio expressed in terms o the L/D ratio by means o the Saha-Zuber correlation (equation (6)). Equation (11) presents the unction used in igure 7. Q ratio = (1+0.25t(L/D)) -1 (11) Equation (11) represents a very good tool or predicting the minimum point (OFI) at least or these up-low data. Other experimental work in annular geometry and in down-low shows similar accuracy although in downlow at low velocities, the accuracy o equation (11) deteriorates. CONCLUSIONS The results rom this investigation have shown that the theoretical model presented in equations (2) through (10) can be used to obtain the pressure drop in a heated tube under down-low conditions. The model predicts low conditions rom single phase low through ully developed sub-cooled boiling.

8 Table 3 Comparison o Measured and Theoretical Results L/D average Q ratio measured Qratio maximum Qratio minimum Qratio theoretical The model also provides very accurate prediction o the velocity at which the minimum point in the demand curve occurs. This is an important reactor saety parameter. The experimental data shows that the occurrence o Ledinegg low instability depends upon the L/D ratio o the channel, the inlet temperature, exit pressure, and surace heat lux. These parameters are combined in the Q ratio deined in equation (1). The experimental results show that Q ratio is an excellent predictor o the minimum point velocity (OFI). In addition, the Q ratio was calculated rom the theoretical results, and the agreement was also very good. Finally, equation (11), which is based upon the Saha- Zuber correlation was shown to be an excellent predictor o the minimum point velocity. This equation contains all the signiicant parameters aecting the onset o low instability. REFERENCES Barry, J. J., Crowley, C. J., Qureshi, Z., 1989, "Modeling the Onset o Flow Instability or Subcooled Boiling in Down-low," Creare, Inc. and Westinghouse Savannah River Co. Block, J. A., Crowley, C. J., Dolan, F. X., Sam, R. G., Stoedealke, B. H., 1990, "Nucleate Boiling Pressure Drop in an Annulus," Creare, Inc. Bowring, R. W., 1962, "Physical Model Based on Bubble Detachment and Calculation o Steam Voidage in the Subcooled Region o a Heated Channel," OECD Halden Reactor Project Report HPR-10. Cheh, H. Y., Dougherty, T., Fighetti, C. F., Maciuca, C., McAssey, E. V., Reddy, D. G., Yang, B. W., 1990, Flow Excursion Experimental Program Single Tube Uniormly Heated Tests, Westinghouse Savannah River Co. Collier, J. G., 1972, Convective Boiling and Condensation, McGraw-Hill International Book Co. Davis, E. J. and Anderson, G. H., 1966, "The Incipience o Nucleate Boiling in Forced Convection Flow," AIChE J., V.12, pp Dittus, F. W. and Boelter, L. M. K., 1930, "Heat Transer in Automobile Radiators o Tubular Type," Publ. in Engineering, U. o CA, Berkeley, p Dormer, J and Bergles, A. E., 1964, "Pressure Drop with Surace Boiling in Small Diameter Tubes," Report no , Dept. o Mech. Eng., MIT. Dougherty, T., Fighetti, C., McAssey, E., Qureshi, Z., C., Reddy, G., and Yang, B., 1990, "Boiling Channel Flow Instability," Columbia University, Villanova University, and Westinghouse Savannah River Co. Dougherty, T., Fighetti, C., McAssey, E., Qureshi, Z., C., Reddy, G., and Yang, B., 1990, "Flow Boiling in Vertical Downlow," Columbia University, Villanova University, and Westinghouse Savannah River Co. Duey, R. B., Hughes, E. D., 1989, "Downlow in Annuli Part 2: Flow Instability At High Flowrates," EG&G Idaho, Inc. Griith, P. and Maulbetsch, J. S., 1966, "System Induced Instabilities in Forced-Convective Flows with Sub-cooled Boiling," Proc. o the Third Int. Heat Transer Con., pp Lee, S.C., Dorra, H., and Banko, S. G., 1992, A Critical Review o Predictive Models or the Onset o Signiicant Void in Forced-Convection Subcooled

9 Boiling, Fundamentals o Subcooled Flow Boiling, ASME HTD-Vol. 217, pp Levy, S., 1967, "Forced Convection Subcooled Boiling Prediction o Vapor Volumetric Fraction," Journal o Heat and Mass Transer, Vol 10, pp Saha, P. and Zuber, N., 1974, "Point o Net Vapor Generation and Vapor Void Fraction Subcooled Boiling," Proceedings o the 5th International Heat Transer Conerence, B4.7, Tokyo, Japan. Thom, J. R. S. et al., 1965, "Boiling in Subcooled Water During Flow Up Heated Tube Annuli," Paper 6 presented at the Symposium on Boiling Heat Transer in Steam Generating Units and Heat Exchangers, Inst. Mech. Engrs., Manchester, England. Zigrange, D. J., and Sylvester, N. D., 1982, "Explicit Approximations to the Solution o Colebrook's Friction Factor Equation," AIChE Journal, Vol. 28, No. 3, pp