Hydraulic resistance at sudden pipe expansion-the influence of cavitation I Department of Hydraulic Structures and Water Resources Management, Technical University Graz 2 Department of Hydraulic Machinery, Technical University Graz 3 Faculty of Mechanical Engineering, University of Maribor Abstract On a cavitation test stand the hydraulic resistance-head loss coefficient was measured at a sudden pipe expansion in a closed circuit. The results were compared using a well-tried calculating method (impulse loss after Borda- Carnot). Then, the head loss coefficients were measured for certain cavitation numbers. This showed that only very severe cavitation influences the head loss coefficients. Over a large range of cavitation numbers, the loss coefficient remained unaffected. The same behaviour was revealed by velocity profile measurements. Surprisingly, the fluctuating components of the velocities (RMS values) corresponded for cavitating and non-cavitating flows. 1 Introduction Head loss coefficients, hydraulic grade lines along the pipe wall, and velocity profiles had to be measured for a sudden pipe enlargement. In order to determine the influence of cavitation, measurements had to include a range from cavitationfree to severe cavitation. The measurements had to determine configuration and extension of the cavitation zone that develops downstream of the sudden expansion of discharge section. The results were to serve as a basis for comparisons with the results of a numerical flow analysis [2] and provided information on two-phase flow behaviour in the cavitation zone downstream of the sudden expansion.
198 Computational Methods in Multiphase Flow 2 Test Set-up 2.1 Test Stand The tests were carried out on a cavitation test stand (see Fig. 1). The closed circuit mainly consisted of a 10m3-capacity underwater tank a 250kW centrihgal pump, a Venturi tube and a choke control for measuring and controlling flow through the model. Provisions were made for varying the air pressure above the water surface in the underwater tank between 0.9 bar subatmospheric and 5 bar above atmospheric. This allowed control of the pressure level within the circuit and, thus, the cavitation condition in the sudden pipe expansion. 2.2 The Hydraulic Model The model was constructed with a maximum possible length of 6.5m. It consisted of long straight plexiglass pipes extending upstream and downstream of the sudden expansion as shown in Fig. 1. The straight undisturbed upstream length is approx. 180, and the downstream length approx. 23D2. This design was meant to insure developed turbulent pipe flow both upstream of the expansion and in the outlet, towards the end of the test length. 1 - Entrance pipe from pump 2 - Flow rectifier 3 - Sudden expansion 4 - water reservoir All measures are in mm Figure 1 : Test stand In order to provide favourable conditions for the velocity measurements, large pipe diameters 0.140 m for the entrance pipe, 0.172 m for the exit pipe - were selected. This selection was a compromise between two contradicting requirements, i.e. large diameter for the velocity measurements, small diameter to ensure a large lengthldiameter ratio in the upstream and downstream pipe sections for a given model length.
Computational Methods in Multiphase Flow 199 Figure 2 is a drawing showing and explaining the locations of the measuring cross sections with the taps. Figure 2: Locations of Measuring Cross Sections for Pressure Difference 3 Measurements and Results Pipe friction, head loss coefficient and hydraulic grade lines on the wall were measured by differential pressure measurements. For this purpose, measuring cross sections were provided at regular intervals along the model. These were 14D1. 8DI. and 2D1 (DI = internal diameter of the entrance pipe) upstream and 602, 12D2, 18D2, and 21 4 (D2 = internal diameter of outlet pipe) downstream of the sudden expansion. Each measuring cross section was provided with four tap boreholes located at the quarter points of the pipe circumference. The taps were connected by a closed circular pipe, which served to compensate potential minor pressure differences over the pipe circumference so as to form a mean wall pressure over the measuring cross section. This wall pressure was transmitted to the pressure transducer via a measuring line. 3.1 Pipe Friction The head loss coefficient for the sudden enlargement is determined fiom the total loss reduced by the pipe fi-iction losses in the upstream and downstream pipes. Pipe ffiction losses were determined within the section of uniform turbulent flow by measuring the head losses between two cross sections. The zone of disturbance at the sudden expansion extended fiom a point immediately above the expansion, at a distance smaller than 1 D, to a point about 84 to 104 below. The list below shows that the measuring locations were situated outside the disturbance zone and within the zones of uniform turbulent flow. Data concerning the pipe fkiction measuring lengths in the upstream and downstream pipes: Upstream Downstream Material: plexiglass, extruded plexiglass extruded Internal pipe diameter: D, = 0.14 m D2 = 0.172 m Measuring length: ALI = 0.843 m AL2 = 1.552 m Measuring cross sections: MP2 and MP2a MP4 and MP6 The locations of measuring cross sections are 8DI and 2D1 upstream and 12D2 and 21D2 downstream fiom the sudden enlargement.
The flow was varied between 0.09 and 0.24 cubic meters per second so as to be sure to cover the entire range of flows of between 0.1 m3/s and 0.21 m3/s occurring during the tests that followed. The instrument readings head loss h,,, and flow rate Q were used to calculate the dimensionless pipe friction coefficient A, which,was then entered in the graph shown in Figure 3. 0.008 0.007,-U--- -I-?, X, - entrance plpe - 11,=0.14 m -, meas. positions MP2-MP2a.- Xz - exit pipe - D2 =0,172 m, meas. positions MP4 -- - MP6.- W Figure 3: Pipe friction coefficient /? The pipe pressure fi-iction coefficient is defined as where AL is the length of pipe, c is the flow velocity, g is the acceleration, and h,/ is the pipe head loss as expressed by where Ap is the total pressure difference, p is flow density. The Reynolds number is defined by CD -& = ----, I/ where D is pipe diameter and U is kinematic viscosity of flow. The flow velocity c is determined by the continuity equation 4Q c=- 71- D2 ' where Q is discharge. The calculated values of pipe friction coefficient are shown plotted in Figure 3 for the entrance pipe (ill) and the exit pipe(&) for measuring cross sections MP2 to MP2a in the entrance pipe and MP4 to MP6 in the exit pipe.
4 Estimation the Coefficients 4.1 Loss Coefficients The zone where the head loss is determined extends between MP2 and MP5. It is composed of a length L, = 1.122 m in the entrance pipe (D,) and a length L2 = 3.095 m in the exit pipe (D2). The energy balance for these two flow sections can be expressed by the equation Rearranging equation (S), the total head loss is given as The first and third terms on the right-hand side of equation (6) above can be used to calculate the total pressure difference between measuring cross sections MP2 P, - P, and MP5 as being --- + 2, - z, = -dh. Where a measured discharge Q is PS available, the second term (velocity) can be written as The total loss is determined between the measuring cross sections MP2 and MP5. The total loss is h7.,p/ = hp/l + hsi; + hpf2 ' (8) where hm is the fiiction loss in the entrance pipe, h,ye is the loss at the sudden pipe expansion, and hfl is the head loss in the exit pipe. Using equation (l), the friction losses of entrance and exit pipes can be expressed as The head loss at the sudden expansion of the discharge section can be determined by calculating the friction losses using equations (9) above: hs,=h -h -h PJC P J ~ p12 ' (lo) which means that the difference between the measured and the calculated components of friction loss in the entrance and exit pipes can be attributed to the head loss at the sudden pipe expansion. The loss at the sudden expansion of the discharge section can express in dimensionless form as a function of the flow velocity in the entrance pipe: where ts, is the dimensionless coefficient for head loss at the sudden expansion. This coefficient considers the total head loss due to the sudden expansion of the discharge section and can be determined experimentally.
202 Computational Methods in Multiphase Flow 4.2 Definition of the cavitation Coefficient The cavitation coefficient can be defined as h, - h, g=- 29 where h. is the absolute static pressure head at the end of the entrance pipe, directly above the sudden expansion, h, is the water evaporation pressure head, and c. is the velocity in the entrance pipe, determined from operating discharge 4Q c =c =- (13) I, l,d;? The static pressure head h,,cannot be measured directly, but can be determined by taking the measured total pressure head at measuring cross section MP2 and allowing for the ambient pressure (air pressure) and the entrance pipe head loss: g (12) 4.3 Experimentally Determined Coefficients The tests were performed for a discharge Q = 0.21 m3/s with a Reynolds number Re, = 2 106 in the entrance pipe. The results are shown plotted in Figure 4. 0.04- Cut-of of -Cavitation cloud No-cavitation - cavitation I 0.02-shock 01 1 I I 0 0.1 0.2 0.3 0.4 0.5 0.600.7 O.$ Figure 4: Loss coefficient as a function of the cavitation coefficient a The cavitation coefficient decreases as the cavitation process increases. Loss coefficient stay in wide range of cavitation constant, but at very strong cavitation it increases rapidly. The development of cavitation, based on visual observation (fig. 5)
Conzpututiorlal Methods ill Multipl~ase Flow 203 ---.,.----C-.. -,-. - ---., _._ -_ Strong cavitation - u6= 0.13 1 d) Very strong cavitation - U F 0.08 1 +on mm L e) Super cavitation - U 9= 0.076 - fully developed cavitation Figure 5: Cavitation development by changed acoefficient. Shock losses For comparing the experimental results with known theoretical analyses (Borda Carnot) it is necessary to consider the reduced pipe friction downstream of a sudden expansion. Idelchik [l] has introduced a reduction factor (~n*) for pipe l?iction downstream of a sudden expansion. Using the Idelcik factor, the friction loss in the exit pipe between the sudden expansion and measuring cross section MP5 can be determined as where giving resulting for the shock losses or in dimensionless form 'I."?k = hsti + (h,,2 - " J2)
Compared results are given in diagram (Fig. 6). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0- Figure 6: Measured losses (Shock loss ~ SE, Jshock with reduced friction) and calculated shock losses (&,ock.after Borda-Carnot, after Idelchik) At any rate, as can be seen from the comparative diagram in Figure 6 above, the shock loss can be determined: a) for the rectangular flow patterns in the entrance and exit pipes by using the well-known Borda-Carnot equation [6]: and b) for the hlly developed velocity profiles in turbulent pipe flow, which can be expressed by the equation: where for Reynolds number Re, = 2 106, following the Idelchik recommendation, for determining shock loss: J3aock 4, = Experimentally determined shock loss values show good agreement with the analytically determined values.
5 Flow velocity profiles Conlputntio~~al Methods irl M~rltiplrase F~OM. 205 Velocity profiles are determined by use of the Laser Doppler Anemometer method [3]. We measured seven profiles at seven measuring cross sections under three different cavitation regimes (c = 0,456, 0.3 15,O. 13) and two different flow rates (Q = 0.1 and 0.21 m3/s). The measurements [S] concerned axial velocity profiles (Figure 7.a) and their corresponding RMS (root means square) values (Figure 7.b). The characteristic location was 6 (Figure 2) right below the sudden expansion of the discharge section. Under the regime of filly developed extreme cavitation intensity, it was not possible to measure the flow velocity profiles as the presence of many gas and vapour pockets made the water non- transparent. Figure 7: Axial flow velocity profile at measuring cross section 6 directly below the sudden expansion of the discharge section (a) and RMS values @). It is interesting to note that there is no significant difference between the results from non cavitating flow up to strong cavitation. This allows the conclusion that for small cavitation bubble concentrations flow behaviour is that of single-phase flow. The percentage of gas and vapour has no influence on flow velocity. The reasons for this phenomenon may be the fact that the inertia of flow is large enough to avoid the influence of small concentrations of gas or water vapour filled bubbles on two phase flow [4]. Under very strong and super cavitation flow regimes, the flow velocity profiles are probably different because the gas and vapour percentage is larger. However, for lack of transparence under those
206 Con~putatiorlal Methods in Multiphase Flow conditions, it is not possible to use the LDA measuring method, so that this statement cannot be supported by experimental results. 6 Conclusions and Comments The experiments have shown that the hydraulic resistance measured at a sudden pipe expansion remains unaffected over a large range of cavitation numbers. Very severe cavitation - especially super cavitation - involves substantial increasing head loss coefficient. The results obtained have been used in the development of a numerical model for cavitating flow. For calculating head loss in the case of cavitation, except for super cavitation, it is possible to use a singlephase numerical model. References Idelchik, I. E.: Handbook of Hydraulic Resistance. Springer; Berlin, Heidelberg, New York, 1986. R. Klasinc, H. Knoblauch, T. Durn, M. Seiner : Determination of hydraulic losses - Two selected examples. Proceeding: Flow Modeling and Turbulence Measurement Thalahasse, USA, 1996 BURSTware User's Guide, DANTEC, Denmark, 1991. Hammit, F. G.: Cavitation and Multiphase Flow Phenomena. McGraw- Hill Inc., USA, 1980. Durn, T.: Verifikation eines numerischen Stromungsmodells anhand physikalischer Modelle. Dr. Thesis, TU Graz, 1996. Miller, D. S.: Internal Flow Systems. BHRA, Fluid engineering, Series 4, 1978.