APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 ANALYSIS AND CORRELATIONS OF DIMENSIONLESS NUMBERS RELEVANT TO ORIFICES’ CAVITATING FLOW ABDUL-FATTAH MOHAMMED ALI1,2 AND MOHANAD JASIM MOHAMMED-RIDHA1* 1Department of Environmental Engineering, University of Baghdad, Baghdad, Iraq 2Department of Oil and Gas Refining Engineering, Al-Farabi University College, Baghdad, Iraq *Corresponding author: muhannadenviro@coeng.uobaghdad.edu.iq (Received: 5th December 2019; Accepted: 13th May 2020; Published on-line: 4th July 2020) ABSTRACT: The aim of this work was to establish a general design basis for pilot-scale units to treat textile dyeing wastewater containing recalcitrant organic chemicals by hydrodynamic cavitation (HC) using orifices of various geometries. Relevant tabulated data available in the literature were analyzed and correlated to obtain universal relationships to this end. In spite of extensive effort, most of the obtained correlations were system-specific, which still can be used for design using their respective orifice geometries as demonstrated. However, one salient general relationship links the pipe’s dimensionless loss coefficient (KLP) to the pipe’s Reynolds number (ReP), encompassing all data under consideration, which may serve as an additional design option to optimize such units. The implication of this relationship is a lower upstream pressure (P1) value with an increase in pipe diameter while using the same specified orifice and achieving the same desired cavitation number (Cv). The ratio of P1 value in the larger pipe to its value in the smaller pipe is a function of the smaller pipe diameter (DS) to the larger pipe diameter (DL) ratio: (P1 in DL) / (P1 in DS) = (DS /DL)2.33. A lower P1 value will increase the cavitation yield by decreasing the expended energy, especially if the required number of passes is large. Additionally, the variation of the orifices’ hole loss coefficient (KLh) with the ratio of the holes area to the pipe cross-sectional area (Ah/Ap) for cavitating flow is compared with that for non-cavitating/incipient cavitation flow reported in the literature. ABSTRAK: Tujuan kajian ini diadakan bagi mereka bentuk dasar umum unit skala- pandu bagi merawat pewarnaan air buangan tekstil yang mengandungi kimia organik rekalsitran daripada peronggaan hidrodinamik (HC) menggunakan orifis pelbagai geometri. Data berjadual berkaitan yang ada dalam kajian lepas dianalisa dan dikaitkan bagi mendapatkan kaitan universal hingga akhir. Walaupun pelbagai usaha telah dijalankan, banyak kaitan didapati mengguna pakai sistem-tertentu, di mana boleh digunakan bagi mereka cipta menggunakan geometri orifis yang ditunjukkan. Walau bagaimanapun, bagi menghubung kait pekali langsung tanpa dimensi (KLP) kepada paip nombor Reynolds (ReP), meliputi semua data di bawah pertimbangan, di mana membantu pilihan rekaan tambahan bagi mengoptimum unit tersebut. Implikasi hubungan ini adalah nilai tekanan hulu sungai bawah (P1) dengan penambahan diameter paip dengan menggunakan orifis sama yang sebenar dan mendapati nombor peronggaan yang sama diingini (Cv). Nisbah nilai P1 dalam paip besar kepada nilai paip kecil adalah berkadaran pada nisbah diameter paip kecil (DS) kepada diameter paip besar (DL): (P1 dalam DL) / (P1 in DS) = (DS /DL)2.33. 41 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 Nilai P1 yang lebih kecil akan menambah jumlah peronggaan dengan pengurangan tenaga pengembangan, terutama jika nombor laluan yang diperlukan adalah besar. Tambahan, variasi orifis pekali langsung lubang (KLh) dengan nisbah keluasan lubang kepada ruang keratan-rentas paip (Ah/Ap) bagi alur rongga dibandingkan dengan bukan rongga / permulaan rongga alur telah dilapor dalam kajian KEYWORDS: hydrodynamic cavitation; orifice geometry; cavitation numbers; upstream pressure; loss coefficients 1. INTRODUCTION Hydrodynamic cavitation (HC) is a well-established technique for a variety of applications including the treatment of wastewater containing pharmaceuticals [1], insecticides [2], phenolic compounds [3], and textile dyes [4-8]. HC is usually combined with the addition of an oxidizing agent (H2O2, NaOCl, etc.) or with acoustic cavitation to achieve an acceptable cavitation yield (CY) which is defined as the cavitation effect per unit energy supplied to the system. Over the last two decades, many HC research studies have been published, some with detailed tabulated data. Their size-scale ranged from 1 L lab units to 50 L pilot ones. Various devices were used to induce HC in these units such as single-/multi-hole orifices, venturis, stator-rotor equipment, etc. The operational mode was batch-wise based either on fixed time or on a specified number of passes (turnovers) where the wastewater volume was recirculated many times through the HC device to achieve an acceptable CY. This could be hundreds of times, lasting over two hours depending on the system’s flow rate (e.g. 260 passes lasting 130 min [7]). Single-/multi-hole orifices as HC inducing devices are simpler to manufacture and install than other alternatives which explain their wide use in HC studies. Their hole size generally ranges from 1 to 5 mm with multi-hole counts of 8, 16, 20, 25, or 33, although some studies used sizes and counts outside these values. The pipes in which the aforementioned orifices were placed had inside diameters ranging from 4 to 53 mm. The results obtained in all previously published HC studies were system-specific; a generalized or unifying approach were markedly lacking. Reviews stated only general trends based on the published works. Consequently, it is difficult to make practical use of available information, specifically a priori design of a functional wastewater-treating HC unit, save adopting a given system with all its particulars. The present work is an attempt at a unifying approach, relying entirely on tabulated data from a number of published research studies. The outcome of this endeavor was only partially productive as will be illustrated later by the obtained results. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. 2. GEOMETRICAL DETAILS AND REFERENCES OF ORIFICES WHOSE HC PUBLISHED DATA WERE USED IN THIS WORK Two categories of published data were excluded from the analysis and correlations performed in this study. The first concerned orifices with a hole size of one mm or less, whether single- or multi-hole ones. One mm holes or smaller are difficult/expensive to produce locally while ensuring their integrity (sharp edge, lack of burrs, etc.). The second category concerned a HC system in which the fully-recovered orifice’s downstream pressure (P2) was higher than atmospheric pressure (i.e. P2 > one atm. abs. or zero gage). 42 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 Data with P2 > 0 gage did not match those with P2 ≈ 0 gage. Table 1 gives the details of the orifices whose HC data were used in this work along with their references. The data of Testud et al. [10] and that of Mancuso et al. [7] were excluded due to P2 values being greater than one atm. abs. as pointed out earlier. It should be stressed that all the HC data of the references cited in Table 1 are relevant to pilot-scale units with P2 ≈ one atm. abs. Hence, all the obtained correlations of this work are only relevant to such systems. Table 1: Geometrical details and references of orifices whose HC data were employed in the analysis and correlations of this work. Hole dia. [mm] No. of holes Pipe I.D [mm] Ah/Ap Reference 2 1 25.4 6.2x10-3 [6] 2 1 25 6.4x10-3 [9] 2 1 19 0.011 [8] 2 1 19 0.011 [17] 2 8 38 0.02216 [11] 2 8 38 0.02216 [4] 2 33 38 0.0914 [11] 2 33 38 0.0914 [4] 3 1 25.4 0.01395 [6] 3 1 19 0.02493 [17] 3 16 38 0.099723 [11] 3 16 38 0.099723 [4] 3 20 38 0.12465 [11] 3 20 38 0.12465 [4] 5 8 38 0.1385 [11] 5 8 38 0.1385 [4] Notes: (i) Ah/Ap is the ratio of the total area of holes to the cross-sectional area of the pipe. (ii) Thicknesses of orifices in Table 1 references were not mentioned, apart from Pawar et al. [9] which was 4 mm leading to an aspect ratio of 2 (aspect ratio = orifice thickness/hole dia.). (iii) HC data listed in Table 1 references are extensive and can be referred to; they shall not be repeated in this article. (iv) A 4-mm single hole orifice datum from Madhu et al. [6] was excluded due to lack of 4-mm multi-hole data. 3. DIMENSIONLESS NUMBERS RELEVANT TO ORIFICE’S CAVITATING FLOW AND THEIR INTER-RELATIONSHIPS 3.1 Cavitation Number or Index There are several definitions for this parameter, the two most widely used ones are Cv and σ (notations used are similar to those in the cited references) defined as follows: Cv = (P2 – Pv) / (0.5 ρL uo2) and σ = (P2 – Pv) / (P1 – P2) where P1, P2 are the orifice’s upstream and fully-recovered downstream pressures, respectively. Pv is the liquid-vapor pressure and ρL is its density, both at the operating temperature. uo is the average liquid velocity at the orifice’s hole (single hole or one hole in a multi-hole geometry). Cavitating flow occurs when the value of Cv (or σ) is usually ˂ 1, with an increase in cavitation intensity as the value of the index is lowered. Very low values of the index (˂ 43 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 0.1) may induce a regime called supercavitation which is counterproductive and should be avoided. A modification of Cv (called Cvprime) was introduced and used by Vichare et al. [11] and Sivakumar and Pandit [4] as a unifying approach for comparison of HC data of orifices with various multi-hole geometries. Cvprime was defined as: Cvprime = Cv / [(total perimeter of holes/perimeter of pipe)] 3.2 Flow Head Loss Coefficients and Reynolds Numbers Two dimensionless flow head loss coefficients were used in this work. Their definitions are: KLh = (P1 – P2) / (0.5 ρL uo2) and KLP = (P1 – P2) / (0.5 ρL up2) KLh is based on the hole’s dynamic pressure, whereas KLP is based on the pipe’s dynamic pressure with up being the average flow velocity in the pipe. Additionally, two Reynolds numbers were used, one based on the hole average velocity and the other one on the pipe average velocity: Reo = ρL uo do / μL and Rep = ρL up D / μL where do and D are the hole diameter and the pipe inside diameter, respectively. μL is the liquid dynamic viscosity at the operating temperature. 3.3 Inter-relationships Among the Aforementioned Dimensionless Numbers KLh is related to KLP by (Maynes et al. [12]) Eq. (1): KLh= (Ah/Ap)2 KLP (1) and Rep is related to Reo by Eq. (2): ReP= Reo [n (Ah/Ap)]0.5 (2) where n is the number of holes in the orifice (hole count) and Ah/Ap is the ratio of the total area of holes to the pipe cross-sectional area as pointed out earlier. Additionally, σ is related to Cv by Eq. (3): σ= Cv / KLh (3) The relationships represented by Eqs. (1) to (3) can be easily obtained by considering the given definitions of the dimensionless numbers involved. The orifices’ HC tabulated data of Table 1 references constituted experimental values of P1, Cv, uo, Ah/Ap, flow rate, and temperature. P2 values were either given or calculated using the above relationships, where it was confirmed that its value was virtually one atm. abs. in all cases. Eq. (3) was used to calculate the corresponding σ values. 4. SPECIFIC CORRELATIONS OF ORIFICES’ CAVITATING FLOW DIMENSIONLESS NUMBERS AND THEIR UTILITY IN DESIGN Cavitating flow is a complex phenomenon that is still not-well understood and whose theoretical basis has not been fully established. Consequently, an empirical approach is usually adopted (Burzio et al. [18]). Accordingly, Eqs. (4) to (18) are empirical; only 44 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 characterized by values of the determination coefficient R2 ˃ 0.9. Furthermore, the data upon which they are based (Table 1) lacked standard deviation bounds which compromise their accuracy. Notwithstanding these limitations, the equations are of use in a priori design of HC pilot-scale units (as elucidated in section 4.5) where plus or minus 5% off specification is usually acceptable. Table 2 expounds the types of correlations employed and equations designation as related to five orifice geometries. Two other geometries pertaining to the 2mm and 3mm single hole orifices were excluded from the said correlations due to their trend-lacking scattered data. Additionally, Eqs. 13 to 15 were also excluded from Table 2 because they represent a test for Cvprime as a unifying parameter. Table 2: Types of correlation, equations designation and corresponding orifice geometries Correlation type 2mm-8h 2mm-33h 3mm-16h 3mm-20h 5mm-8h KLh vs. Reo Eq. (4) ---------- ---------- Eq. (5) --------- KLh vs. Cv Eq. (6) ---------- ---------- Eq. (7) --------- σ vs. Reo Eq. (8) Eq. (9) ---------- ---------- Eq. (10) σ vs. Cv Eq. (11) Eq. (12) Eq. (16) Eq. (17) Eq. (18) Notes: (i) In the design procedure, the value of Cv is initially assumed depending on the application (see sec. 4.5) which leads to the calculation of Reo. (ii) For a selected geometry, KLh or σ is then determined from an appropriate equation(s) leading to the value of P1 (a prime design parameter). (iii) There are 4 choices to accomplish note (ii) for 2mm-8holes, 3 choices for 3mm-20holes, 2 choices for either 2mm-33holes or 5mm-8holes, but only one choice for the 3mm-16holes geometry. 4.1 KLh vs Reo Correlations For this category, only two specific correlations were obtained as shown in Fig. 1. Fig. 1: KLh vs Reo/104. KLh = 0.4964 – 0.019 (Reo/104) (4) (2mm-8holes, 5 ≤ Reo/104 ≤ 13, R2 = 0.9368) KLh = 0.2482 (Reo/104) – 0.1876 (5) (3mm-20holes, 4.5 ≤ Reo /104 ≤ 8, R2 = 0.9594) y = 0.2482x - 0.1867 R² = 0.9594 y = -0.019x + 0.4964 R² = 0.9368 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 12 14 16 18 KLh Re0/104 5mm- 8 holes 3mm- 16 holes 3mm- 20 holes 2mm- 8 holes 2mm- 33 holes 45 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 The trends of Eqs. (4) and (5) are opposite to one another. The 2mm-8holes KLh decreases with increasing Reo (Ah/Ap = 2.2%) whereas the 3mm-20holes KLh increases (Ah/Ap = 12%). Also shown in Fig. 1 are the data of the 2mm-33holes (Ah/Ap = 9.1%) and the 5mm-8holes (Ah/Ap = 13.85%). The former reflects an increasing-decreasing trend, whilst the latter depicts a clearly non-linear increasing trend for Reo > 16x104. Maynes et al. [12] had shown an increasing KLh trend with increasing uo for two multi-hole orifices with Ah/Ap values of 22% and 44% which they attributed to cavitating flow, since according to their results for non-cavitating flow KLh was constant and independent of uo. Any explanation for the above observations would be premature at this stage. 4.2 KLh vs Cv Correlations The following specific correlations were obtained for this category. KLh = 1.0192 Cv + 0.2095 (6) (2mm-8holes, 0.045 ≤ Cv ≤ 0.56, R2 = 0.9956, Fig. 2) KLh = 0.862 Cv-0.567 (7) (3mm-20holes, 0.3 ≤ Cv ≤ 0.75, R2 = 0.9681, Fig. 3) Fig. 2: KLh vs Cv 2mm-8holes. Fig. 3: KLh vs Cv 3mm-20holes. y = 1.0192x + 0.2095 R² = 0.9956 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 KLh Cv 2mm- 8 holes y = 0.862x-0.567 R² = 0.9681 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 KLh Cv 3mm- 20 holes 46 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 The trends represented by Eqs. (6) and (7) are opposite to one another. For each geometry, the above trend is the reverse of its KLh vs Reo trend [Eqs. (4) and (5)]. Hence, KLh increases with increasing Cv for the geometry of the 2mm-8 holes [Eq. (6)] whereas it decreases for the 3mm-20holes one [Eq. (7)]. 4.3 σ vs Reo Correlations For this category, the following specific relationships were obtained: σ = 0.8139 – 0.0159 (Reo/104) (8) (2mm-8holes, 3.7 ≤ Reo/104 ≤ 13, R2 = 0.9095, Fig. 4) σ = 1.0317 – 0.1757 (Reo/104) (9) (2mm-33holes, 3 ≤ Reo/104 ≤ 5, R2 = 0.9644, Fig. 4) σ = 65.504 (Reo/104)-2.027 (10) (5mm-8holes, 9 ≤ Reo/104 ≤ 17.5, R2 = 0.9559, Fig. 5) Equations (8) to (10) indicate a decreasing trend of σ with increasing Reo for different geometries. This was expected since higher Reo values imply higher P1 values making σ smaller since P2 and Pv are fixed, [σ = (P2 – Pv)/(P1- P2)]. Fig. 4: σ vs Reo/104 for 2 mm-8, 33holes. Fig. 5: σ vs Reo/104 for 5mm-8holes. y = -0.0519x + 0.8139 R² = 0.9095 y = -0.1757x + 1.0317 R² = 0.9644 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 σ Reo/104 2mm- 8 holes 2mm- 33 holes y = 65.504x-2.027 R² = 0.9559 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 9 10 11 12 13 14 15 16 17 18 σ Reo/104 5mm- 8 holes 47 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 4.4 σ vs Cv or Cvprime Correlations Specific correlations linking the two most widely-used definitions of cavitation number were as follows: σ= 1.016 Cv0.5509 (11) (2mm-8 oles, 0.045 ≤ Cv ≤ 0.56, R2 = 0.9647, Fig. 6) σ = 0.6848 Cv – 0.0262 (12) (2mm-33 holes, 0.35 ≤ Cv ≤ 0.75, R2 = 0.9799, Fig. 6) Fig. 6: σ vs Cv for 2mm-8, 33holes. The two geometries of Eqs. (11) and (12) were combined by a single σ vs Cvprime correlation: σ = 0.9478 Cvprime + 0.0593 (13) (2mm-8, 33 holes, 0.1 ≤ Cvprime ≤ 0.45, R2 = 0.9388, Fig. 7) Fig. 7: σ vs Cvprime for combined 2mm-8, 33holes. y = 1.016x0.5509 R² = 0.9647 y = 0.6848x - 0.0262 R² = 0.9799 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 σ Cv 2mm- 8 holes 2mm- 33 holes y = 0.9478x + 0.0593 R² = 0.9388 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 σ CVprime 2mm- 8 holes 2mm- 33 holes 48 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 Equation (13) is the only case where Cvprime acted as a unifying parameter. Consequently, two additional σ vs Cv specific relationships were deduced for imaginary geometries of 2mm-16holes and 2mm-20holes: σ = 1.1255 Cv + 0.0593 (14) (2mm-16holes, 0.0842 ≤ Cv ≤ 0.379) σ = 0.9004 Cv + 0.0593 (15) (2mm-20holes, 0.1053 ≤ Cv ≤ 0.477) Additionally, the following specific σ vs Cv correlations were obtained: σ = 0.5995 Cv – 0.0081 (16) (3mm-16holes, 0.37 ≤ Cv ≤ 1.15, R2 = 0.9602, Fig. 8) σ = 1.2227 Cv – 0.206 (17) (3mm-20holes, 0.3 ≤ Cv ≤ 0.75, R2 = 0.9936, Fig. 8) σ = 1.2676 Cv (18) (5mm-8holes, 0.15 ≤ Cv ≤ 0.55, R2 = 0.9732, Fig. 9) Equations (11) to (18) reflect a trend of increasing σ with increasing Cv for the various orifice geometries. This trend was also expected since an increase in Cv value means a lower uo value which entails a lower P1 value leading to an increase in the value of σ. Fig. 8: σ vs Cv for 3mm-16, 20holes. Yan and Thorpe [13] derived a theoretical equation linking σ to Cv for a single hole orifice. Their relationship included the orifice’s discharge coefficient Cd , its contraction coefficient Cc , and Ah/Ap . It was not possible to compare Eqs. (11) to (18) with their equation because Cd and Cc were not given for the orifices under consideration in this work. Estimating the values of Cd and Cc from correlations available in the literature would be uncertain since they are related to the aspect ratio, which is also missing for all the multi-hole orifices of Table 1. y = 0.5995x - 0.0081 R² = 0.9602 y = 1.2227x - 0.206 R² = 0.9936 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 σ Cv 3mm- 16 holes 3mm- 20 holes 49 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 Fig. 9: σ vs Cv for 5 mm-8holes. 4.5 Outline of a Priori Design Procedure Utilising the Specific Correlations (a) Initially an appropriate Cv value for the intended application is chosen, e.g. for wastewater containing bio-refractory organic dyes, a suitable range is 0.2 ≤ Cv ≤ 0.4 (Rajoriya et al. [19]) (b) Since Cv = (P2 – Pv)/(0.5 ρL uo2), uo can be determined after an operating temperature is set, fixing the values of Pv, ρL, and μL. Hence the value of Reo can be established. (c) After selecting an orifice geometry, one or more correlation may be used to calculate either KLh or σ from which the value of P1 can be determined. This P1 value is based on 38 mm (1.5 inch) pipe diameter (see Table 1). (d) The 38mm-based P1 value can be lowered by specifying a larger pipe diameter, e.g. 53mm (2 inch) according to the findings of sec. 5 below. (P1 – P2) represents the main pressure drop of the system whose volumetric flow rate (Q) can be calculated from the value of uo and the selected geometry leading to the estimation of the system’s total pressure drop (∆P). (e) A centrifugal pump whose best efficieny point (BEP) matches the system’s Q-∆P point should be employed. Conversely, if a pump is already available, an appropriate Cv value, the ensuing P1 and reduced P1 values, and selected geometry should be manipulated to achieve the pump’s BEP-system’s Q-∆P match. 5. GENERAL KLP vs ReP CORRELATION AND ITS IMPLICATIONS Figure 10 shows a plot of KLP vs. Rep for all the HC data of the orifices listed in Table 1, including the single hole data which were excluded from the specific correlations of sec. 4 due to their trend-lacking scatter. It is noteworthy that four pipe sizes are included in Fig. 10 with a range of 19 ≤ I.D. ≤ 38 mm, which is in contrast with the specific correlations where only a single pipe size of I.D = 38 mm was involved. The unifying characteristic of KLP vs Rep is clear, resulting in the following general correlation Eq. (19): KLP = 4228.5 (Rep/104)-1.6707 (19) {(Rep/104) ≤ 18, R2 = 0.9353} y = 1.2676x R² = 0.9732 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 σ Cv 5mm- 8 holes 50 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 Retrieving the definitions of KLP = (P1- P2)/(0.5 ρL up2) and Rep = (ρL up D)/μL, Fig. 10 indicates that increasing Rep by increasing up, which can only come about by increasing P1 reduces KLP in a nearly exponential decay manner. If the definitions of KLP and Rep are substituted into Eq. (19) and noting that (P1 – P2) may be replaced by P1 as gage pressure, then Eq. (19) can take the following form: P1 = 1.0185x105 ρL-0.6707 μL1.6707 up0.3293 D-1.6707 (20) where P1 has units of bar gage. Fig. 10: KLpvs Rep/104 for all single and multi-hole orifices. If for a specific orifice geometry and a desired Cv value, two pipes are considered (in which the specified orifice is to be installed in one of them) one pipe with a smaller inside diameter DS and the other with a larger inside diameter DL , then the application of Eq. (20) and the equation of continuity (conservation of mass flow rate) for the two pipes will result in the following ratio Eq. (21): (see Appendix A), (P1 in DL pipe / P1 in DS pipe) = (DS / DL)2.33 (21) Hence, a lower value of P1 is required to obtain the desired Cv value by installing the specified orifice in a larger I.D pipe compared to a smaller I.D. pipe. It is emphasized that this result is limited to the applicability range of Eq. (19) and the data upon which it is based. An additional advantage of using a relatively larger inside diameter pipe is the lower pressure drop in the system, apart from the orifice’s pressure drop (P1 – P2), especially when the required number of passes is large. This will have a positive effect on the expended energy and consequently on the cavitation yield. A possible drawback of lowering P1 value by using a larger inside diameter pipe is the ensuing decrease in the value of the cavity collapse pressure given by the following empirical correlation [14-16] Eq. (22). Pcollapse = 7527 [100 (Ah/Ap)]-2.55 P12.46 Ro-0.8 do2.37 (22) which is related to the cavitation yield by Eq. (23): y = 4228.5x-1.671 R² = 0.9353 1 10 100 1000 10000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 KLp Rep/104 3mm-16 holes 3mm- 20 holes 3mm single hole 2mm- 8 holes 2mm- 33 holes 2mm-single hole 5mm- 8 holes 51 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 CY = K ( Pcollapse )W (23) where Ro is the initial cavity size, K and w are constants whose values depend on the system’s particulars (device geometry, operating parameters, type of reaction). However, the possible negative effect of lower P1 value may be partially or totally offset by the decrease in the value of Ah/Ap with a larger inside diameter pipe and/or the value of w being positive or negative. Applying Eq. (22) with DS or DL leads to: (Pcollapse in DL pipe)/(Pcollapse in DS pipe) = (DS/ DL)0.632 (24) 6. COMPARISON WITH FINDINGS OF MAYNES et al. [12] Maynes et al. [12] showed a graphical representation of KLh vs Ah/Ap in which data points pertaining to 16 multi-hole orifices of their work plus data points from three references were all bounded by two theoretical models, namely Eqs. (25) and (26): (KLh)Detached = [Cc-1 – (Ah/Ap)]2 (25) (KLh)Attached = 2 [1- (Ah/Ap) – Cc-1] + Cc-2 + (Ah/Ap)2 (26) These models were derived by Testud et al. [10] for single hole orifice using Bernoulli’s principle and conservation of momentum for non-cavitating flow. Eq. (25) is relevant to a thin orifice where the flow does not reattach within the hole, whereas Eq. (26) is for a thick orifice where the flow is reattached. Cc value for Eq. (25) was determined from Busemann theoretical relation cited by Maynes et al. [12] (a function of Ah/Ap ) whilst for Eq. (26) it was given a fixed value of 0.75. Maynes et al. emphasized that all the shown data were non-cavitating/incipient-cavitation points. However, the two included points from Testud et al. are developed cavitation results (one for a single hole and the other for multi-hole orifices with aspect ratios of 0.64 and 4.67, respectively). Fig. 11: KLh vs Ah/Ap for all single and multi-hole orifices. Figure 11 is a reproduction of Maynes et al. [12] presentation where the upper line represents Eq. (25) and the lower line Eq. (26). The plotted points are all the data of Table 0 0.5 1 1.5 2 2.5 3 3.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 KLh Ah/Ap 2mm - single hole 2mm- 8 holes 2mm- 33 holes 3mm- 16 holes 3mm- 20 holes 3mm- single hole 5mm- 8 holes Testud et al.-mulit- holes Testud et al. -single hole Eq. 26 Eq. 25 52 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 1 plus all the developed cavitation results of Testud et al. [10]. Hence, all the points in Fig. 11 are cavitating flow results. The 2 mm and 3 mm single hole points having very small Ah/Ap values lie mainly below or in-between the bounds representing Eqs. (25) and (26). As for the multi-hole orifices data and that of Testud et al. [10], there seems to be a trend with increasing value of Ah/Ap starting below the lower bound (2mm-8holes), then in- between the two bounds and finally back to the lower bound (5mm-8holes). It is therefore concluded that results falling in-between these two bounds are not necessarily non- cavitating/incipient cavitation data and could very well be developed cavitation results. 7. CONCLUSIONS Correlations represented by Eqs. (4) to (12) and Eqs. (16) to (18), although specific to their respective orifice geometries, can be used to design and/or predict the performance of pilot-scale HC units for wastewater treatment. This includes a priori determination of P1 and (P1 – P2) values for the desired Cv value using a specific orifice geometry. Furthermore, they can be used to design systems to suit available pumps in order to obtain operating points at or very close to the pump’s best efficiency points. This will increase the cavitation yield by lowering the value of the expended energy. The pipe’s loss coefficient KLP and the pipe’s Reynolds number Rep were the only dimensionless parameters that unified all the HC data of orifices of various geometries and pipe sizes in a single general correlation with R2 = 0.9353. This resultant correlation, Eq. (19), implies that for the same specified orifice geometry and desired Cv value, using a relatively larger pipe inside diameter entails a reduction in the value of the required upstream pressure P1. The ratio of P1 value in the larger pipe to its value in the smaller pipe is a function of the ratio of the smaller pipe diameter (DS) to the larger pipe diameter (DL): (P1 in DL) / (P1 in DS) = (DS / DL)2.33. A lower P1 value will increase the cavitation yield by decreasing the expended energy especially if the number of passes is large. However, a lower P1 value will also decrease the value of the cavity collapse pressure which may affect the cavitation yield negatively, positively, or not at all depending on Ah/Ap variation and on constants whose values are system-specific. The findings of Maynes et al. [12] that non-cavitating/incipient cavitation orifices’ KLh vs Ah/Ap data fall in-between two theoretical models, Eqs. (25) and (26), are not exclusive. Orifices’ cavitating flow results may also behave likewise, but with a discernible trend. REFERENCES [1] Braeutigam P, Franke M, Schneider RJ, Lehmann A, Stolle A, Ondruschka B. (2012) Degradation of carbamazepine in environmentally relevant concentrations in water by hydrodynamic-acoustic cavitation (HAC). Water Res., 46:2469-2477. [2] Patil PN, Bote SD, Gogate PR. (2014) Degradation of imidacloprid using combined advanced oxidation processes based on hydrodynamic cavitation. Ultrason. Sonochem., 21:1770-1777. [3] Pradham AA, Gogate PR. (2010) Removal of p-nitrophenol using hydrodynamic cavitation and Fenton chemistry at pilot scale operation, Chem. Eng. J., 156:77-82. [4] Sivakumar M, Pandit AB. (2002) Wastewater treatment: a novel energy efficient hydrodynamic cavitational technique. Ultrason. Sonochem., 9:123-131. [5] Saharan VK, Pandit AB, Kumar PSS, Anandan S. (2012) Hydrodynamic Cavitation as an Advanced Oxidation Technique for the Degradation of Acid Red 88 Dye. Ind. Eng. Chem. Res., 51:1981-1989. 53 IIUM Engineering Journal, Vol. 21, No. 2, 2020 Ali and Ridha https://doi.org/10.31436/iiumej.v21i2.1306 [6] Madhu GM, Thomas A, Deepak S, Preetham HS, Rajanandam KS. (2015) Escalation of degradation of malachite green and methyl violet using hydrodynamic cavitation using different orifice geometry. Int. J. Env. Sci., 5(4):880. [7] Mancuso G, Langone M, Laezza M, Andreottola G. (2016) Decolourization of Rhodamine B: A swirling jet-induced cavitation combined with NaOCl. Ulatrason. Sonochem., 32:18- 30. [8] Rajoriya S, Bargole S, Saharan VK. (2017) Degradation of reactive blue 13 using hydrodynamic cavitation: Effect of geometrical parameters and different oxidizing additives. Ultrason. Sonochem., 37:192-202, DOI:http://doi.org/10.1016/j.ultsonch.2017.01.005 [9] Pawar SK, Mahulkar AV, Pandit AB, Roy K, Moholkar VS. (2017) Sonochemical Effect Induced by Hydrodynamic Cavitation: Comparison of Venturi/Orifice Flow Geometries. AIChE Journal, 63(10):4705-4716. [10] Testud P, Moussou P, Hirschberg A, Auregan Y. (2007) Noise generated by cavitating single-hole and multi-hole orifices in a water pipe. J. Fluids and Structures, 23:163-189. [11] Vichare N P, Gogate PR, Pandit AB. (2013) Optimization of hydrodynamic cavitation using a model reaction. Chemical Engineering Technology, 23:683-690. [12] Maynes D, Holt GJ, Blotter J. (2013) Cavitation Inception and Head loss Due to Liquid Flow Through Perforated Plates of Varying Thickness. J. Fluids Eng., 135: 031302-1— 031302-11. [13] Yan Y, Thorpe RB. (1990) Flow Regime Transitions Due to Cavitation in the Flow Through an Orifice. Int. J. Multiphase Flow, 16 (6):1023-1045. [14] Gogate PR, Pandit AB. (2000) Engineering Design Methods for Cavitation Reactors II: Hydrodynamic Cavitation. AIChE J., 46 (8):1641-1649. [15] Gogate PR, Shirganokar IZ, Sivakumar M, Senthilkumar P, Vichare NP, Pandit AB. (2001) Cavitation Reactors: Efficiency Assessment Using a Model Reaction. AIChE J., 47 (11):2526-2538. [16] Gogate PR, Pandit AB. (2005) A review and assessment on hydrodynamic cavitation as a technology for the future. Ultrason. Sonochem., 12:21-27. [17] Carpenter J, Badve M, Rajoriya S, George S, Saharan VK, Pandit AB. (2017) Hydrodynamic cavitation: An emerging technology for the intensification of various chemical and physical processes in a chemical process industry. Reviews in Chemical Engineering, 33(5):433-468. https://doi.org/10.1515/revce-2016-0032. [18] Burzio E, Bersani F, Caridi GCA, Vesipa R, Ridolfi L, Manes C. (2020) Water disinfection by orifice-induced hydrodynamic cavitation. Ultrasonics-Sonochemistry, 60, 104740: 1-13. [19] Rajoriya S, Carpenter J, Saharan VK, Pandit AB. (2016) Hydrodynamic cavitation: an advanced oxidation process for the degradation of bio-refractory pollutants. Rev. Chem. Eng., 32: 379-411. APPENDIX A The mass flow rate and volumetric flow rate (Q) will be the same in both DL and DS pipes, therefore, P1 in DL pipe = 1.0185x105 ρL-0.6707 μL1.6707 (4Q/πDL2)0.3293 DL-1.6707 P1 in DS pipe = 1.0185x105 ρL-0.6707 μL1.6707 (4Q/πDS2)0.3293 DS-1.6707 Dividing: (P1 in DL pipe / P1 in DS pipe) = ( DL-0.6586 DL-1.6707 ) / ( DS-0.6586 DS-1.6707 ) = DL-2.33 / DS-2.33 = (DS / DL)2.33 54 << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles false /AutoRotatePages /None /Binding /Left /CalGrayProfile (Gray Gamma 2.2) /CalRGBProfile (None) /CalCMYKProfile (None) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Warning /CompatibilityLevel 1.7 /CompressObjects /Off /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams true /MaxSubsetPct 100 /Optimize false /OPM 0 /ParseDSCComments false /ParseDSCCommentsForDocInfo false /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo false /PreserveFlatness true /PreserveHalftoneInfo true /PreserveOPIComments false /PreserveOverprintSettings true /StartPage 1 /SubsetFonts false /TransferFunctionInfo /Remove /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 200 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages false /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.76 /HSamples [2 1 1 2] /VSamples [2 1 1 2] >> /ColorImageDict << /QFactor 0.76 /HSamples [2 1 1 2] /VSamples [2 1 1 2] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 200 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages false /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.76 /HSamples [2 1 1 2] /VSamples [2 1 1 2] >> /GrayImageDict << /QFactor 0.76 /HSamples [2 1 1 2] /VSamples [2 1 1 2] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV /HUN /ITA (Utilizzare queste impostazioni per creare documenti Adobe PDF adatti per visualizzare e stampare documenti aziendali in modo affidabile. I documenti PDF creati possono essere aperti con Acrobat e Adobe Reader 6.0 e versioni successive.) /JPN /KOR /LTH /LVI /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken waarmee zakelijke documenten betrouwbaar kunnen worden weergegeven en afgedrukt. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 6.0 en hoger.) /NOR /POL /PTB /RUM /RUS /SKY /SLV /SUO /SVE /TUR /UKR /ENU (Use these settings to create Adobe PDF documents suitable for reliable viewing and printing of business documents. Created PDF documents can be opened with Acrobat and Adobe Reader 6.0 and later.) >> >> setdistillerparams << /HWResolution [600 600] /PageSize [595.440 841.680] >> setpagedevice