On the pressure losses through perforated plates

Flow Measurement and Instrumentation 28 (2012) 57–66 Contents lists available at SciVerse ScienceDirect Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst On the pressure losses through perforated plates Stefano Malavasi a, Gianandrea Messa a,n, Umberto Fratino b, Alessandro Pagano b a b Dip. IIAR, Politecnico di Milano, Piazza Leonardo da Vinci, 32-20133 Milano, Italy Dip. Ingegneria delle Acque e di Chimica, Politecnico di Bari, Via Orabona, 4-70125 Bari, Italy a r t i c l e i n f o abstract Article history: Received 22 February 2012 Received in revised form 18 July 2012 Accepted 22 July 2012 Available online 17 August 2012 Perforated plates are widely used in pipeline systems either to reduce flow nonuniformities or to attenuate the onset and the development of cavitation. This experimental work aims at investigating the dependence of the pressure losses through sharp-edged perforated plates with respect to the geometrical and flow key parameters. The data, collected in two large experimental campaigns carried out on different pilot plants, are reported and discussed. Several plates with different geometrical characteristics were tested. More precisely, perforated plates whose equivalent diameter ratio varies between 0.20 and 0.72; relative hole thickness between 0.20 and 1.44; and number of holes between 3 and 52. Experimental data from literature are also considered in order to ensure the reliability of the parametric investigation. The dependence of the pressure loss coefficient upon the Reynolds number, the equivalent diameter ratio, the relative thickness, and the number and disposition of the holes is studied. A comparison to different empirical equations, as available by the technical literature, and to the standard ISO 5167-2 single-hole orifice is also provided. & 2012 Elsevier Ltd. All rights reserved. Keywords: Perforated plates Pressure loss coefficient Parameters 1. Introduction Perforated plates are commonly used for the control and the maintenance of the efficiency of pressurized systems, being preferred over other hydraulic devices for their simple geometry and low cost. Generally, perforated plates are installed upstream to flowmeters to remove swirl and correct a distorted flow profile or, coupled with a control valve, used for preventing cavitation phenomena, assuring safe operating conditions (Tullis and Di Santo et al. [1,2]). The hydraulics of perforated plates was largely investigated in the technical literature, and most of the researches were aimed at investigating their functionality as flow conditioners. Laws and Ouazzane [3] focused their attention on the use of such devices for pre-conditioning a disturbed flow, whereas Schluter and Merzkirch [4] measured, by means of PIV techniques, the timeaveraged axial velocities downstream perforated plates for optimizing their geometry. A similar analysis was recently carried out by Xiong et al. [5]. Few investigations deal with the dissipation characteristics of perforated plates, being mostly focused on the occurrence and development of the cavitation phenomena (Govindarajan [6]; Tullis and Govindarajan [7]; Kim et al. [8]; and Testud et al. [9]). Tullis [1] investigated the pressure losses through different perforated plates and the pressure profile downstream them. n Corresponding author. Tel.: þ39 02 2399 6287. E-mail address: [email protected] (G. Messa). 0955-5986/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2012.07.006 Gan and Riffat [10] determined the pressure drop through a perforated plate in a square pipe by means of experimental tests and numerical simulations. Erdal [11] performed a numerical investigation of the parameters affecting the performance of a multi-hole plate used as flow conditioner, and discussed about its dissipation characteristics. Weber et al. [12] made a review of literature data about the pressure losses through perforated plates in circular and rectangular pipes, reporting data from different authors (Dannenberg; Kolodzie and Van Winkle; Wang et al.). In addition, some experimental tests on perforated plates and flat bar screens in a large rectangular pipe were discussed. Fratino [13] studied experimentally and numerically the flow through multihole orifices in circular pipes, and proposed a formula to estimate the pressure drop. Similar investigations are reported in Malavasi et al. [14], Macchi [15], and Malavasi et al. [16], where the dependence of the pressure losses upon the most significant geometrical and flow parameters is considered. Zhao et al. [17] studied the dissipation characteristics of several multi-hole orifices of 2 mm thickness, and reported an empirical formula for estimating the pressure drop. Holt et al. [18] analyzed the dissipation and cavitation efficiency of baffle plates in circular pipes, introducing a method for evaluating the pressure losses in no cavitating conditions. The dissipation characteristics of perforated plates are usually quantified by means of the pressure loss coefficient, defined as Eu ¼ P U P D 1=2rV 2 ð1Þ 58 S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66 where PU and PD are, respectively, the pressure upstream and downstream of the device, V is the pipe bulk-mean velocity, and r is the fluid density (Fig. 1). As it will be discussed later, the reference sections U and D are defined in different ways by the existing standards. When no cavitation occurs, the pressure loss coefficient is influenced by the geometry of the plate, defined, for a squareedged plate with holes of uniform size, by the following characteristics: (1) the porosity of the screen, i.e. the ratio of the open area to the overall pipe section, usually expressed by means of its square root b (equivalent diameter ratio); (2) the plate thickness t, usually taken into account by the relative thickness t/dh, dh being the hole diameter; (3) the number of holes nh; (4) the distribution of the holes, usually quantified by the pitch P, i.e. the minimum distance between two adjacent holes. The dimensionless groups P/dh and P/D are considered in Weber et al. [12] and Zhao et al. [17] respectively, D being the diameter of the circular pipe. The losses are also influenced by the friction factor of the holes l, but such dependence, in the present work, was found to be absolutely negligible. An important role is played by the Reynolds number characteristic of the phenomenon, whose definition is still controversial. Some authors (Fratino [13]; Malavasi et al. [14]; Malavasi et al. [16]; Zhao et al. [17]) make reference to the pipe Reynolds number Rp ¼VpD/n, defined in terms of pipe diameter D and pipe bulk mean velocity Vp; other authors (Weber et al. [12]; Idelcick [19]) considered the hole Reynolds number Rh ¼Vhdh/n, defined in terms of hole diameter dh and hole bulk mean velocity Vh (therepffiffiffiffiffi fore, Rp ¼ Rh nh b); in Gan and Riffat [10] and Holt et al. [18] a Reynolds number R ¼DVh/n expressed in terms of pipe diameter D and hole bulk-mean velocity Vh is introduced. R is linked to Rp by the following relationship Rp ¼ b2R. Whatever Reynolds number is considered, the dependence between the pressure loss coefficient Eu and the Reynolds number is qualitatively sketched in Fig. 2. Under no cavitating conditions, as Reynolds increases two different regions can be identified: a low-Reynolds region (1), in which Eu is affected by the Reynolds number; and a selfsimilarity region (2), in which Eu is almost constant with respect to the Reynolds number. The occurrence of cavitation causes Eu to increase suddenly with the Reynolds number. The threshold value of Reynolds number at which a given device is subjected to cavitation depends on the plant pressure. Different formulas for evaluating the pressure loss coefficient in the self-similarity region with respect to RD (region (2) in Fig. 2) are available. All of them express Eu as a function of the equivalent diameter ratio b and, in some cases, the relative thickness t/dh and the friction factor l, therefore assuming that the effect of number and disposition of the holes is negligible. Some of them are derived for the single-hole orifice case and said to be applicable to the perforated plate case. Among them, more attention is given to that of Idelcick [19], valid for Rh 4105 and t/dh 40.015: 2 Eu ¼ 2 2 0:5ð1b Þ þ tð1b Þ1:5 þð1b Þ2 þ lt=dh ð2Þ 4 b in which is t is a tabular coefficient depending on t/dh, and to that of Miller [20]: 2 Eu ¼ C 0 ð1C C b Þ2 ð3Þ 4 C 2C b in which C0 is a coefficient depending on t/dh, while CC is the contraction coefficient of the jets. As reported in Fratino [13], C0 can be calculated by the following empirical expression, assumed valid for 0.1 ot/dh o3: C 0 ¼ 0:5 þ 0:178 ð4Þ 4ðt=dh Þ2 þ 0:355 while CC can be evaluated by C C ¼ 0:596 þ 0:0031eb=0:206 ð5Þ Empirical equations for estimating the pressure loss coefficient through perforated plates are reported in ESDU [21], Zhao et al. [17], and Holt et al. [18]. The first, said to be valid for Rh 4104, is ( 4 K 0 b la t=dh o 0:8 Eu ¼ ð6Þ 4 K 0:8 b lb t=dh 4 0:8 where K0 and K0.8 are given as function of b while la and lb depend on b and t/dh. All coefficients are provided in a graphical form. Zhao et al. [17] expressed Eu as a function of b using the following equation, valid for b ranging from 0.25 to 0.45: Eu ¼ Pm ðb 4:448 1Þ ð7Þ where 4 3 2 Pm ¼ 160:325ð71:467b 100:300b þ 52:021b 11:801b þ1Þ Fig. 1. Geometrical sketch of the system and identification of the reference sections U and D. ð8Þ At the end, according to Holt et al. [18], the pressure loss coefficient Eu can be evaluated as  8  2 0:4 0:8 0:4 > > K LA dt b o0:9 < 2:93:79 dth b þ 1:79 dth b h ð9Þ Eu ¼   > 0:4 t > : 0:876 þ 0:069 dt b0:4 K LA b 40:9 d h h where KLA is the pressure loss coefficient of a single-hole orifice as estimated by means of a theoretical model for reattached flow: ! 2 2 1 1 þ K LA ¼ 1 2 þ 4 1 ð10Þ C C 2C 2C b b Fig. 2. Qualitative trend of Eu as a function of Reynolds number. For a given device, the Reynolds number at which cavitation occurs depends on the plant pressure. The authors suggest setting the contraction coefficient of the jets CC equal to 0.72. The purpose of the present work is to investigate the dissipation characteristics of a multi-hole orifice under no cavitating conditions. The results of experimental campaigns performed in two different pilot plants are reported and discussed. Data from S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66 literature are made comparable and primarily checked for their consistency. Afterwards, they are added to our experimental data to create a large database for achieving a better awareness about the dependence of the pressure loss coefficient upon the most significant geometrical and flow parameters. A comparison with the above described formulas and to the standard ISO 5167-2 [25] orifice is also reported. 2. Experimental setup Tests were carried out by research groups of Polytechnic School of Milan and Polytechnic School of Bari. Experiments of the former group were conducted in a pilot plant located at Pibiviesse S.r.l, Nerviano, Italy. The rig, shown in Fig. 3, consists of 1000 and 1200 steel pipes, supplied by a pump able to guarantee pressures up to 10 bar at the reference section upstream the orifice. Control valves placed upstream and downstream the test area allow setting the proper fluid-dynamic conditions in each experimental test. Pressure drop was measured with a series of absolute and differential pressure transducers in reference sections located 2D upstream and 6D downstream the device, according to ISA-RP75.23 standard [22]. Other measurement points were placed at 1D upstream and 1D, 2D, 3D, 4D, 5D, 6D, and 7D downstream the plate. Flow rate was measured by a 1000 electromagnetic flow-meter, placed upstream the test section. During the tests, the water temperature was measured in order to monitor values of density, viscosity and vapor pressure of the fluid. The tests have been performed maintaining constant pressure at the upstream reference section PU and decreasing the downstream pressure PD in order to increase the discharge and consequently the Reynolds number. Complementary experimental tests have been performed by the research group of Bari in the Laboratory of the Department of Ingegneria delle Acque e Chimica at Polytechnic School of Bari. The laboratory setup, sketched in Fig. 4, is composed by 100 mm and 200 mm steel pipes, supplied by a pump able to guarantee pressures of about 9.0 bar and flow rates up to 100 l/s. 59 The pressure taps for evaluating the gross head drop were located 1D upstream and 10D downstream the device, but other measurement points were placed at 0.5D, 1D, 2D, 3D, 4D, 5D, 7D downstream the device. The pressure drop was measured by a mercury differential manometer and by Burdon tube pressure gauges, whereas the flow rate was evaluated by a flow measuring pipe orifice and by a volumetric tank. Even during this experimental campaign, the water temperature was measured in order to monitor the values of density, viscosity and vapor pressure of the fluid. Different pressure values, generally equal to 0.25, 0.5 and 1 bar, have been fixed downstream the plate to make the results independent from possible uncertainties due to the pressure scale effects in case of cavitation occurrence (Fratino [13]). It is worth mentioning that the different positions of the pressure taps in the two experimental pilot plants is related to laboratory constraints and arrangements and it can be verified that it has no influence on the reliability of the experimental data. In confirmation of it, the ISA-S39.2 standard [23] on testing procedures for estimating control valve capacity states that the location of the upstream pressure tap is between 0.5D and 2.5 D upstream the device. On the other hand, the difference in the downstream pressure tap locations is negligible, as in both cases the pressure recovery is completed and there are distributed friction losses without any significance over such a small pipe length. Considerations about the estimate of the uncertainty of measurements are reported in Appendix A. Several plates were tested in the two campaigns. Their geometrical characteristics, reported in Table 1, are different in terms of equivalent diameter ratio b (from 0.20 to 0.72); relative thickness t/dh (from 0.20 to 1.44); number of holes nh (from 3 to 52); and distribution of the holes. 3. Results and discussion Comments on the influence of geometrical and flow parameters on the dissipation characteristics of perforated plates are Fig. 3. Sketch of the test rig (Polytechnic School of Milan research group). 60 S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66 Fig. 4. Sketch of the test rig (Polytechnic School of Bari research group). Table 1 Geometrical characteristics and result obtained for all the plates tested (M-Series: tests of Polytechnic School of Milan research group; B-Series: tests of Polytechnic School of Bari research group). Plate label b [-] t/dh [-] nh [-] M1 0.40 0.24 13 53.6 7 1.0 M2 0.40 0.45 13 72.8 7 2.0 M3 0.40 0.73 13 38.8 7 0.8 M4 0.40 0.73 13 42.1 7 2.1 M5 0.40 1.00 13 50.7 7 2.7 M6 0.40 1.00 26 35.4 7 0.7 M7 0.40 1.40 13 37.8 7 1.2 M8 0.51 0.73 13 15.3 7 0.7 M9 0.51 1.00 26 15.8 7 0.8 M10 0.72 1.00 52 2.25 7 0.11 Distribution of the holes Eu [-] 61 S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66 Table 1 (continued ) Plate label b [-] t/dh [-] B1 0.20 0.69 3 787 7 88 B2 0.20 0.89 5 693 7 70 B3 0.20 1.06 7 741 7 68 B4 0.20 1.23 9 712 7 69 B5 0.20 1.32 11 588 7 40 B6 0.20 1.44 13 633 7 34 B7 0.40 0.35 3 69.4 7 4.3 B8 0.40 0.45 5 71.1 7 3.6 B9 0.40 0.53 7 71.3 7 1.8 B10 0.40 0.60 9 51.9 7 0.9 B11 0.40 0.72 13 35.5 7 1.6 nh [-] Distribution of the holes Eu [-] Fig. 5. Trend of the pressure loss coefficient Eu as a function of the pipe Reynolds number RD (lower horizontal axis) and the hole Reynolds number Rh (upper horizontal axis) for the following plates: (a) Milan: b ¼0.40, nh ¼ 26, t/dh ¼ 1.00; (b) Milan: b ¼ 0.40, nh ¼13, t/dh ¼ 1.00; (c) Bari: b ¼0.40, nh ¼ 13, t/dh ¼ 0.72. reported in the following. The effect of the Reynolds number is considered first. The dependence of the pressure loss coefficient Eu on the pipe Reynolds Rp (or the hole Reynolds number Rh) was found to be similar to the one qualitatively reported in Fig. 2. As an example, the experimental trend of the pressure loss coefficient Eu as a function of both Rp (lower horizontal axis) and Rh (upper horizontal axis) is depicted in Fig. 5 for three different plates. Since the present paper focuses on the pressure losses under noncavitating conditions, we first had to remove from the series all 62 S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66 the data subjected to cavitation, which, as remarked above, produces a sudden increase in Eu. As already noticed by Malavasi et al. [16], the dependence of Eu upon Rp (or Rh) in region (1) is not univocal; as the geometry of the plate changes, Eu can increase or decrease as Rp (or Rh) increases. Similar behavior results from the data of Zhang et al. [17]. The lower limit of the region of self-similarity with respect to the Reynolds number does not seem to depend significantly on the testing pressure, unlike the upper one, corresponding to the inception of cavitation. Hereafter, we will refer to Eu as the average among the values within the region of self-similarity with respect to the Reynolds number (region (2) in Fig. 2). Since the identification of that region from the dataset collected is not always unequivocal, the values of Eu will be considered together with an estimation of their uncertainty. A summary of experimental pressure loss coefficient collected data is in the last column of Table 1. The dependence of the pressure loss coefficient in the Eu upon the equivalent diameter ratio b in the region of self-similarity with respect to the Reynolds number is discussed. Fig. 6 shows the trend of Eu as a function of b in which our experimental results are represented with other experimental data collected from technical literature. For all the experimental data, the shape of the pipe section (circular/rectangular) can be inferred from that of the marker points (the data about the rectangular pipe case are depicted using a square marker). In Fig. 6, the values of Eu obtained from the previously described literature formulas are reported too; since all the models, except that of Zhao et al. [17] (Eq. (6)), take the dependence upon the relative thickness into account, for clarity in Fig. 6 the curves with t/dh ¼0.5 only are drawn. The results confirm that the equivalent diameter ratio b is the dominant geometric characteristic affecting the losses (see Tullis [1]; Idelcick [19]; and Miller [20]), even if a significant dispersion can be detected, especially for low values of b. The dispersion of the experimental data could be related primarily to t/dh, nh and disposition of the holes, but also, as noticed by Weber et al. [12], by inaccuracies in the measurements of the pressure drop across the plate and of the geometrical characteristic of the device. Nevertheless, when examined at a large scale as in Fig. 6 with Eu plotted on a log axis, the behavior of the points seems fairly homogeneous; in particular, the shape of the pipe section (circular/rectangular) does not seem to have noticeable effect. At first sight, all the literature curves, even if referred to the arbitrary case of t/dh ¼0.5, appear able to catch the gross dependence of Eu upon b for the whole dataset, whatever the value of t/dh. However, a more detailed analysis reveals that the deviation between calculated and experimental data can be significant, reaching up to about 40%, even considering correct values of t/dh. The influence of the parameters other than the equivalent diameter ratio b upon the pressure loss coefficient Eu is investigated. To better highlight the effect of the relative thickness t/dh, we need to use only the data characterized by the constant values of equivalent diameter ratio b, number nh and disposition of the holes. The comparable data set came from our tests M1–M2–M4– M5–M7, Kolodzie and Van Winkle (in [12]), and Holt et al. [18]. Unfortunately, no information about the disposition of the holes in the plates is reported in the last two references. Fig. 7 shows the trend of Eu as a function of t/dh for the above discussed data, highlighting as, in the investigated range, Eu decreases as t/dh increases, whatever the values of b and nh, although not always monotonically. These results may be explained considering the effect of t/dh on the flow behavior; in fact, if t/dh is low, the jets remain separated from the inner wall of the holes; if t/dh is high, they reattach to the inner wall of the holes and then expand to the pipe (Malavasi et al. [16]). On the other hand, as reported by Miller [20], for t/dh between 0.1 and 0.8 flow instabilities can occur because of intermittent reattachment, and this may be the cause of the non-monotonous dependence of Eu upon t/dh. The effects of number and disposition of the holes should be considered together, as both parameters determine the curvature of the streamlines passing through the plate, and, as a consequence, influence the pressure losses. However, the role of these parameters is hard to investigate because of the difficulties in describing the distribution of the holes by means of few key parameters and because of the lack of detailed information about the disposition of the holes of the plates tested by the other authors. However, a preliminary approach aiming at analyzing the effect of nh on the pressure loss coefficient Eu is made. Among all the data considered, the only comparable ones, characterized by constant values of b and t/dh, are those reported in Table 2. Despite the uncertainties due to the fact that the data were collected by different authors and that the distribution of the holes were not considered because of the lack of information available, the results highlight that in most cases Eu decreases if nh is increased, and Fig. 6. Trend of the pressure loss coefficient Eu as a function of the equivalent diameter ratio b: comparison between our experimental data, experimental data from other authors, and literature formulas. Fig. 7. Dependence of the pressure loss coefficient Eu upon the relative thickness t/dh, the equivalent diameter ratio b and the number of holes nh being kept constant. 63 S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66 Table 2 Dependence of the Euler number Eu upon the number of holes nh, the equivalent diameter ratio b and the relative thickness t/dh being the same. Reference b [-] t/dh [-] Zhao et al. [17] Weber et al. [12] 0.30 0.33 6 13 423 273 Weber et al. [12] Weber et al. [12] 0.30 0.50 0.52 13 33 237 227 Zhao et al. [17] Weber et al. [12] 0.33 0.30 0.32 6 1119 312 124 Zhao et al. [17] (form 1) Zhao et al. [17] (form 2) M1 0.40 0.24 6 6 13 113 117 53.6 7 1.0 0.40 0.45 5 13 71.1 7 3.6 72.8 7 2.0 M5 M6 0.40 1.00 13 26 50.9 72.7 35.4 7 0.7 Weber et al. [12] Weber et al. [12] 0.48 0.48 3052 7103 21.0 18.8 Weber et al. [12] Weber et al. [12] 0.57 0.32 1948 4534 10.8 9.9 Weber et al. [12] Weber et al. [12] 0.71 0.32 3048 7093 2.4 3.2 B8 M2 such behavior seems to be more evident for lower values of b and t/dh. A possible explanation of such phenomenon, shared by Erdal [11], is that an increase in the number of holes reduces the size of the recirculation zones between the holes and between the outer holes and the pipe wall, resulting in a lower pressure drop. However, much information about the position of the holes is required to confirm this hypothesis. A comparison between the experimental data and the existing above mentioned literature models, at a more detailed scale with respect to Fig. 6, is reported in Fig. 8, which depicts, for different values of equivalent diameter ratio b, the pressure loss coefficient Eu versus the relative thickness t/dh. The overall trend of Eu as a function of t/dh is generally well represented by the curves of Idelcick (Eq. (2)), Miller (Eq. (3)), ESDU (Eq. (6)), and Holt et al. (Eq. (9)), even if none of them is able to catch the nonmonotonous behavior observed in some cases (Fig. 7). On the other hand, the correlation of Zhao et al. [17] (Eq. (7)), which takes into account only b, shows disagreement with the experimental evidence. Probably, some geometrical peculiarities that characterize the experiments of Zhao et al. [17] may contribute to explain why their Eu values – and as consequence those derived by the application of Eq. (7) – are considerably higher if compared to all other. However, the dispersion of the data, especially for low values of b and t/dh, indicates that the number and the disposition of the holes have some influence on the pressure losses, so reducing the validity of all the existing literature formulas which, as remarked in Section 1, neglect the effect of these parameters. The results reported in Fig. 7 and Table 2 show that the pressure loss coefficient decreases as both the relative thickness t/dh and the number of holes nh increase. This suggests that a very thin single-hole orifice would have the maximum loss. Such behavior is investigated by making a comparison between the Euler number of the multi-hole orifices reported in Fig. 8 and those of the standard single-hole orifices introduced in the ISO 5167-2 [25] normative, characterized by j ¼451þ151 and 0.005D rt r0.02D (Fig. 9). Specific constrains are then imposed on dh, D, and the dimensionless parameters b and Rp; in particular, dh 412.5 mm, 50 mmoD o1000 mm, and 0.10o b o0.75. According to the normative, the pressure loss coefficient of the single-hole orifice nh [-] Eu [-] can be estimated from the following formula: 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32 4 1b ð1C 2 Þ 4 Eu ¼ 15 2 Cb ð11Þ where C is the discharge coefficient, i.e. the ratio of the actual flow rate to the maximum theoretical flow rate. ISO 5167-2 [25] prescribes to make use of the Reader-Harris/Gallagher correlation for evaluating C: !0:7 106 b 2 8 C ¼ 0:5961þ 0:0261b 0:0216b þ 0:000521 ReD " !0:3  0:8 # 19000b 106 þ 0:0188 þ 0:0063 b3:5 Rp Rp 1:3 0:031ðM d 0:8M 1:1 þ ð0:043þ 0:08e10L1 d Þb "  0:8 # 19000b b4 0:123e7L1 Þ 10:11 4 Rp 1b ð12Þ in which the parameters L1 and Md are defined differently according to the pressure tap arrangement (corner, D D/2, flange) considered for the determination of the discharge coefficient. ISO 5167 [25] leaves open the question of whether the discharge coefficient in Eq. (11) is Ccorner, CD D/2 or Cflange. As in Urner [26], we will show the calculation for the first two tap arrangements. The values of L1 and Md with flange tap arrangements depend on the pipe diameter; however, Cflange was found to lie between Ccorner and CD D/2 in the range of D and b specified by the standard. The estimated values of pressure loss coefficient for the values of diameter ratio b considered in Fig. 8 are reported in Table 4, and indicate that, except for the case of b ¼0.72, the tap arrangement has a negligible effect on Eu. Moreover, although Eq. (12) expresses the discharge coefficient C as a function of the pipe Reynolds number Rp, the trend of the pressure loss coefficient Eu evaluated by Eq. (11) as a function of Rp is qualitatively similar to that depicted in Fig. 2 for non-cavitating flows, with Eu almost independent of Rp for Rp sufficiently high. The values of Eu reported in Table 4 belong to the self-similarity region with respect to Rp. Table 4 and Fig. 8 indicate that the pressure loss 64 S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66 Fig. 8. Trend of the pressure loss coefficient Eu as a function of the relative thickness t/dh for different values of equivalent diameter ratio b; comparison between experimental data and several proposed formulas. Fig. 8 except for those of Zhao et al. [17], which, as already noticed, are considerably higher than all others. At last, it is worth noticing that the different positions of the pressure taps for the evaluation of the gross pressure drop indicated by the ISO 5167-2 [25] standard, i.e. 1D upstream and 6D downstream the plate, do not affect the reliability of the comparison, since the pressure recovery is completed and the distributed friction losses are absolutely negligible. 4. Conclusion Fig. 9. The single-hole orifice defined in the ISO 5167 [25] normative. coefficient of the sharp single-hole ISO 5167-2 [25] orifice is an upper limit to those of multi-hole orifices with the same equivalent diameter ratio b. This appears true for all the data reported in In this work the dissipation characteristics of perforated plates under no cavitating conditions have been investigated on the basis of the data collected in the experimental campaigns performed by two research groups of Polytechnic School of Milan and Polytechnic School of Bari. Data from literature, made comparable and primarily checked for their consistency, were also considered. The dependence of the pressure loss coefficient upon the most significant parameters involved in the process, like the Reynolds number, the equivalent diameter ratio, the relative thickness, and the number and disposition of the holes was studied. Based on our investigations, the following major conclusions can be done:  The pressure loss coefficient is independent of the Reynolds number as long as this parameter stays in the self-similarity 65 S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66    range (Fig. 5). For lower values of the Reynolds number, the pressure loss coefficient can either increase or decrease with the Reynolds number. The lower limit of the self-similarity range depends on the geometry of the plate but not on the testing pressure, unlike the upper one. A reduction of the equivalent diameter ratio – this being, as well known, the dominant parameter affecting the pressure losses – causes the pressure loss coefficient to increase, and the effect of the other parameters to get more relevant (Fig. 6). However, the general behavior of all collected data seems quite homogeneous; in particular, the shape of the pipe section (circular/ rectangular) does not have significant influence on the value of pressure loss. The relative thickness has noticeable effect on the pressure loss coefficient. The modification of the behavior of the jets causes the pressure loss coefficient to globally decrease as the relative thickness increases, if all other significant parameters are kept constant (Fig. 7). The dependence of the pressure loss coefficient upon the relative thickness is often non-monotonic, probably due to flow instabilities. Number and disposition of the holes influence the pressure losses. The analysis of comparable data revealed that in most cases the pressure loss coefficient decreases if the number of holes increases, due to a reduction of the size of the recirculation zones between the holes. Such behavior is however dependent upon the disposition of the holes (Table 2).  The effect of the distribution of the holes, the number of holes being the same, seems to be instead minor (Table 3). The gross dependence of the pressure loss coefficient upon the equivalent diameter ratio is well caught by all the considered formulas (Fig. 6). However, at a more detailed scale, they appear to be inadequate to describe all the characteristics of the phenomenon. Only the overall trend of the pressure loss coefficient as a function of the relative thickness is generally quite well represented by the equations proposed by Idelcick (Eq. (2)), Miller (Eq. (3)), ESDU (Eq. (6)), and Holt et al. (Eq. (9)), which takes into account only the effect of equivalent diameter ratio and relative thickness. It is worth mentioning as all Table 4 Pressure loss coefficient Eu for the ISO-5167 [25] orifice, obtained by Eq. (11). The discharge coefficient is evaluated by Eq. (12) for different tap arrangements. The values of Eu in the self-similarity region with respect to the pipe Reynolds number are reported. b [-] Eucorner [-] EuD  D/2 [-] 0.20 0.30 0.40 0.72 1668 307 87.0 4.33 1672 308 87.4 4.60 Table 3 Effect of the distribution of the holes upon the pressure loss coefficient Eu. Reference b [-] t/dh [-] nh [-] Distribution of the holes Eu [-] Zhao et al. [17] 0.40 0.24 6 113 Zhao et al. [17] 0.40 0.24 6 117 Zhao et al. [17] 0.40 0.30 9 95 Zhao et al. [17] 0.40 0.30 9 97 Zhao et al. [17] 0.40 0.30 9 98 Zhao et al. [17] 0.40 0.36 13 102 Zhao et al. [17] 0.40 0.36 13 103 M3 0.40 0.73 13 38.8 7 0.8 M4 0.40 0.73 13 42.1 7 2.1 B11 0.40 0.72 13 35.5 7 1.6 66  S. Malavasi et al. / Flow Measurement and Instrumentation 28 (2012) 57–66 these equations appear to partially fail their prediction skill, especially at low b and relative thickness values (Fig. 8), if the number and the disposition of the holes became more significant. The pressure losses of perforated plates appear to be lower than those of a standard single-hole ISO 5167-2 [25] orifice (Fig. 9) with the same equivalent diameter ratio. Therefore, an upper limit to the pressure loss coefficient of multi-hole orifices with a certain equivalent diameter ratio may be estimated from Eqs. (11) and (12). Appendix A. Pressure loss coefficient uncertainty The estimate of the uncertainty of the pressure loss coefficient Eu was provided in respect to the International Organization of Standardization-GUM [24]. Application of the error combination law on Eu yields ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2  2  2ffi @Eu @Eu @Eu uðEuÞ ¼ uðDpÞ þ uðrÞ þ uðV p Þ @Dp @r @V p ðA:1Þ where u(Dp), u(r), and u(Vp) are the absolute errors on pressure drop, density and pipe bulk-mean velocity respectively. For the experiments carried out by the research group of Polytechnic School of Milan, u(Dp) was equal to 44 Pa, 155 Pa or 260 Pa depending on the used transducer. The density r was not directly measured, but inferred from the fluid temperature T by means of an empirical curve obtained by a fitting of experimental data (Macchi [15]) and u(r) was found always lower than 2.2 kg/m3. The uncertainty on the pipe bulk-mean velocity was assumed equal to 0.002Vp, as indicated by the manufacturer of the used flowmeter. The maximum relative error of the pressure loss coefficient u(Eu)/Eu was found to be about 1.5% for the worst-case condition. As far as the results of Polytechnic School of Bari are concerned, u(Dp) was considered equal to 250 Pa, due to the accuracy of the reading on the differential manometer. The uncertainty on the density r was computed referring to the influence of both fluid temperature (a potential variation between 7 1C and 25 1C was considered) and compressibility, giving a result of about 3.5 kg/m3. At last, the uncertainty on the pipe bulk-mean velocity was assumed equal to 0.25% of the full scale (10 m/s) of the measuring device as indicated by the manufacturer. The relative error of the pressure loss coefficient u(Eu)/Eu was found to be 2.5% at maximum. 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