ANALYSIS AND CORRELATIONS OF DIMENSIONLESS NUMBERS RELEVANT TO ORIFICES’ CAVITATING FLOW

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). 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 skalapandu 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).


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][5][6][7][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.

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).
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. Notes: (i) A h /A p 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.

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: 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 (˂ 0.1) may induce a regime called supercavitation which is counterproductive and should be avoided.
A modification of Cv (called Cv prime ) 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. Cv prime was defined as: Cv prime = Cv / [(total perimeter of holes/perimeter of pipe)]

Flow Head Loss Coefficients and Reynolds Numbers
Two dimensionless flow head loss coefficients were used in this work. Their 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: where do and D are the hole diameter and the pipe inside diameter, respectively. μL is the liquid dynamic viscosity at the operating temperature.

Inter-relationships Among the Aforementioned Dimensionless Numbers
KLh is related to KLP by (Maynes et al. [12]) Eq. (1): and Rep is related to Reo by Eq. (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): 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.

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 characterized by values of the determination coefficient R 2 ˃ 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 Cv prime as a unifying parameter. Table 2: Types of correlation, equations designation and corresponding orifice geometries σ vs. C v Eq. (11) Eq. (12) Eq. (16) Eq. (17) Eq. (18) Notes: (i) In the design procedure, the value of C v is initially assumed depending on the application (see sec. 4.5) which leads to the calculation of Re o . (ii) For a selected geometry, K Lh or σ is then determined from an appropriate equation(s) leading to the value of P 1 (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.

K Lh vs Re o Correlations
For this category, only two specific correlations were obtained as shown in Fig. 1. 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 > 16x10 4 . 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.

K Lh vs C v Correlations
The following specific correlations were obtained for this category.  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)].

σ vs Re o Correlations
For this category, the following specific relationships were obtained:

σ vs C v or C v prime Correlations
Specific correlations linking the two most widely-used definitions of cavitation number were as follows:  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. 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.  (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).

Outline of a Priori Design Procedure Utilising the Specific Correlations
(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. 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 (19) and noting that (P1 -P2) may be replaced by P1 as gage pressure, then Eq. (19) can take the following form: 0185x10 5 ρL -0.6707 μL 1.6707 up 0.3293 D -1.6707 (20)

GENERAL K LP vs Re P CORRELATION AND ITS IMPLICATIONS
where P1 has units of bar gage. 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.
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][15][16]  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.

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):    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 inbetween 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 noncavitating/incipient cavitation data and could very well be developed cavitation results.

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 R 2 = 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.