Empirical correlation for calculating pressure drop in

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Cite This: Ind. Eng. Chem. Res. 2018, 57, 14202−14212

Empirical Correlation for Calculating the Pressure Drop in Microhydrocyclones Javier Izquierdo,† Roberto Aguado,*,† Ander Portillo,† Jorge Vicente,‡ Javier Bilbao,† and Martin Olazar† †

Department of Chemical Engineering, University of the Basque Country UPV/EHU, P.O. Box 644, E48080 Bilbao, Spain Novattia Desarrollos, Ltd., Scientific and Technologic Park of Bizkaia, Building 612, E48160 Astondo Bidea, Derio, Spain

Ind. Eng. Chem. Res. 2018.57:14202-14212. Downloaded from pubs.acs.org by UNIV OF SOUTH DAKOTA on 11/06/18. For personal use only.

‡

ABSTRACT: One of the most important parameters for the design, selection, and operation of a microhydrocyclone is pressure drop. Although there are certain correlations in the literature for determining microhydrocyclone capacity curves, their applicability range is rather limited because of the narrow range of operational and geometric parameters used in the runs for their proposal. Thus, they are not suitable for predicting the experimental results obtained in this study using 42 microhydrocyclones of different size, vortex, and spigot configuration. Therefore, a new empirical correlation containing operational and geometric parameters grouped into dimensionless numbers is proposed. The selection of the dimensionless moduli and determination of the corresponding exponential coefficients have been based on stepwise multivariate nonlinear regression and optimization. The new correlation proposed provides reasonable estimations of the pressure drop for all of the experimental systems analyzed.

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tillation for processing water with high nitrogen content.12 In view of the amount of sewage sludge generated in wastewater treatment plants and the increasingly more stringent legislation for its treatment, Mansour-Geoffrion et al.13 studied the use of microcyclones for the selective separation of sand from sewage sludge and obtained highly satisfactory results. Another application of these devices in the environmental sector deals with the treatment of water in crude oil wells by Hashmi et al.14 The relevant literature also deals with applications involving the recovery of lubricant oils, refrigerants, and drill lubricants.15,16 Catalyst recovery is another promising field for the use of these devices. Yang et al.8 use a cyclonic device for separating the catalyst in the methanol-to-olefins process with an efficiency of 88%. Likewise, Lv et al.17 studied the same process, with their recovery levels being in the range from 75 to 83%. Li et al.18 propose a hydrothermal treatment, followed by a separation treatment using hydrocyclones for regeneration and recovery of the catalyst. Microhydrocyclones have also been used successfully in the food industry,19 mainly for starch concentration.20,21 They have also been proposed for yeast separation in alcoholic fermentation processes.22−24 The diameter of these devices may change from 1000 to 5 mm. When the diameter is smaller than 100 mm, they are called minihydrocyclones17,25,26 or microhydrocyclones, which is the term commonly used in industry. Nowadays, efforts in the scientific community focus on the study of their hydrodynamic

INTRODUCTION Hydrocyclones have been extensively applied because of their versatility under different conditions in a variety of applications. These pieces of equipment are commonly found in many steps in processes related to the mining industry. Thus, they are used in closed-circuit grinding,1 desludging,2 liquid clarification,3 and thickening operations.4 Recently, they have been increasingly used in other sectors, mainly because of the better knowledge of flow patterns and effect of the design parameters on the separation efficiency. Thus, mention should be made to novel applications such as those involving separations in two immiscible liquid streams,5 or even in gas−liquid streams,6 in which hydrocyclones are used to remove CO2 from a gaseous stream by its injection in an alkaline solution. Microhydrocyclones are also applied in oil refining. Thus, Ma et al.7 propose a pilot plant for retaining the coke dust generated in the refining by 100-mm-diameter hydrocyclones, in which the separation efficiency is 99% for particles below 100 μm. Hydrocyclones are also being applied in processes related to the environmental sector. Thus, Yang et al.8 show in their review the impact that hydrocyclones may have in processes involving water treatment, such as clarification and thickening. Bayo et al.9 propose the use of these types of centrifugal separators for stabilizing sludge from urban wastewater, and Lee10 proposes their use for separating organic matter from these sludges. Rastogi et al.11 studied the use of coal ashes for adsorbing colorants from wastewater in textile industries. Hydrocyclones allow continuous operation by colorant adsorption and therefore their removal, which is a great advantage over conventional filters and centrifugation equipment. Strategies integrating microhydrocyclones in other operations have also been explored, i.e., microhydrocylones into membrane dis© 2018 American Chemical Society

Received: Revised: Accepted: Published: 14202

May 21, 2018 September 24, 2018 September 26, 2018 September 26, 2018 DOI: 10.1021/acs.iecr.8b02258 Ind. Eng. Chem. Res. 2018, 57, 14202−14212

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Industrial & Engineering Chemistry Research behavior by means of computational fluid dynamics,27−32 optimization of the operation,33 and the development of new applications,17,34−38 as well as the effect that coarse particle dragging fines has on the underflow stream, which is known as fish-hook.25,39−41 Nevertheless, there is hardly any suitable correlation in the literature for determining the pressure drop for different operating conditions and geometric factors of these devices. These correlations are essential for the design of auxiliary equipment upstream and downstream of the separator, given that pressure drop conditions lead to not only the dimensioning of propulsion equipment but also the dimensioning of pipes and devices for controlling and monitoring flow rates. The few correlations in the literature for estimating the pressure drop in hydrocyclones are as follows:

Table 1. Experimental Conditions Used for Determining the Empirical Correlations Published in the Literature for Pressure Drop Estimation in Hydrocyclones eq 1

2 3

Rao et al.42 Q i = 0.00538Do 0.68Di 0.85Du 0.16Δp0.49

4

(1)

Plitt43 Δp =

1.316 × 105Q i1.78 exp(0.55c) D0.37Di 0.94 (L − l)0.28 (Du 2 + Do 2)0.87

5

(2)

Vallebuona et al.44 Q i = C1Δp + C2DoC3

6

Tavares et al.45

i Δp y Q i = K1D2jjjj zzzz k ρ {

(3)

0.5

ij Di yz jj zz kD{

0.45

ij Do yz jj zz kD{

0.68

ij L yz −0.10 jj zz θ kD{ 0.2

Neesse et al.

0.38

Q i = 4.896 × 10 Δp

D = 50 mm Do = 8 and 14 mm Du = 3.2 and 9.4 mm Δp = 50−250 Pa c = 0.13−0.32 D = 50 mm Do = 8 and 14 mm Du = 3.2 and 9.4 mm Δp = 50−250 Pa c = 0.13−0.32

D = 254 mm D = 152 mm Do = 76, 89, and 102 Do = 51, 64, 76, and 79 mm mm Du = 51 mm Du = 38 mm c = 0.17−0.29 D = 150 mm D = 25 mm Do = 5.5 and 7 mm Du = 2.2 and 3.2 mm Δp = 50−250 Pa c = 0.1−0.3 D = 25 mm Do = 5.5 and 7 mm Du = 2.2 and 3.2 mm Δp = 50−250 Pa c = 0.1−0.3 D = 10 mm Do = 2.5 mm Du = 1.6 mm Δp = 500−6000 Pa c = 0.03 D = 25 mm Do = 6 mm Du = 1.5 mm Δp = 500−6000 Pa

These authors propose a very simple correlation relating the flow rate with the pressure drop and vortex diameter. In view of the correlation, one may guess that the overall influence of both variables on the flow rate is the sum of individual effects, which is not so common in these types of systems. Although Vallebuona et al.44 carried out experimentation with microhydrocyclones, their correlation does not consider essential parameters, such as the hydrocyclone diameter or inlet diameter. The following correlation in Table 1, eq 4, is the one proposed by Tavares et al.45 and considers almost all of the parameters included in the aforementioned correlations, except the length of the vortex chimney, for which they use an alternative parameter such as the cone angle θ. Although this correlation is not entirely dimensionless, geometric and operational parameters are grouped into dimensionless moduli, except the flow rate, hydrocyclone diameter, and cone angle. Extrapolation of this correlation to other geometries is questionable because the experimental study was conducted using two hydrocyclone diameters, but neither the vortex diameter nor the spigot diameter was changed. Neesse et al.46 proposed one of the more simple correlations in the literature, eq 5, in which the flow rate is proportional to the pressure drop, and therefore only two fitting parameters are considered, i.e., the proportionality constant and exponential coefficient. Because it is such a simple correlation, its application under conditions different from those used in the experimental study (a rather narrow range) is questionable. Finally, eq 6 in Table 1, proposed by Yang et al.,8 is the only dimensionless correlation reported in the literature. The two dimensionless moduli considered are the Euler number, Eu, and Reynolds number, Re, which account for the fluid properties, operational parameters, and hydrocyclone diameter. This correlation is simple and easy to apply and has recently been

(5)

8

Yang et al.

Eu = 21.9Re 0.486

D = 381 mm Do = 102, 133, and 149 mm Du = 51, 62, and 75 mm

(4)

46

−7

test conditions

(6)

The experimental conditions used for their proposal are detailed in Table 1. As observed, the range of geometric factors and operating conditions used for their proposal is narrow in most of the cases, which is a shortcoming for their use out of those ranges. Furthermore, most of them are based on parameters with given units, which also limits their application range and leads frequently to calculation errors. Equation 1, proposed by Rao et al.,42 is a dimensional correlation developed for hydrocyclones of diameter greater than 150 mm, and therefore out of microhydrocyclone range. This correlation relates the flow rate with the vortex diameter Do, inlet diameter Di, spigot diameter Du, and pressure drop Δp. A fact to be noted in this correlation is the absence of the hydrocyclone diameter, thus questioning its suitability for microhydrocyclones. Equation 2, proposed by Plitt,43 is also a dimensional correlation and contains more hydrocyclone geometric parameters than eq 1; i.e., apart from those in the latter correlation, it includes the hydrocyclone diameter D, hydrocyclone length L, and length of the vortex chimney l. In addition, it also considers the effect of the particle concentration defined by the volume fraction c. The original paper reporting this correlation was not available, but it was taken from another one.47 Therefore, we are not aware of the experimental conditions for its determination, except the hydrocyclone diameter, 150 mm, which is out of the range for microhydrocyclones. Equation 3, proposed by Vallebuona et al.,44 considers fewer parameters than those aforementioned. 14203

DOI: 10.1021/acs.iecr.8b02258 Ind. Eng. Chem. Res. 2018, 57, 14202−14212

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Industrial & Engineering Chemistry Research used in the literature.33 Nevertheless, the experimental range used for its determination is out of the one used in this study and, in addition, does not consider essential parameters influencing the capacity curve, such as the vortex and inlet diameters. On the basis of that aforementioned, the aim of this study is to check the validity of the correlations published in the literature for predicting the microhydrocyclone behavior in a wide range of configurations and experimental conditions. Thus, the factors changed are the diameters of the hydrocyclone, vortex, spigot, and inlet, the length of the hydrocyclone body, the pulp concentration, and the suspension material. Furthermore, a new empirical correlation of more general application than those reported in the literature has been proposed by grouping operational and geometric factors into dimensionless moduli. Determination of the significant moduli and their coefficients will be conducted based on statistical inference combined with multivariate and optimization techniques.

operation with microhydrocyclones of different size and is provided with a system for unit control and data acquisition. The stirred tank, TK-1, is of octagonal shape, and it is made of AISI304 stainless steel. It is 1200 mm high and 965 mm wide and has a maximum capacity of 700 L. The stirrer is located on the top of the tank, and the solid feeder on one side, together with two inserts for water feed lines. In the line from the stirred tank to the pump feeding vessel, there is a manual pinch DN-100 valve. At the lower section of the tank, there is a manual DN-50 valve for tank emptying. On another side of the tank, there is a screed for recirculation of the microhydrocyclone streams (overflow and underflow). Furthermore, the vessel is provided with a 800-mm-level meter made of transparent poly(vinyl chloride). The stirrer is a FluidMix VPP3-05 03 B 25/09.6 of 0.75 kW. It is provided with a 900 mm shaft coated with ebonite and three S-type helical axial blades of 250 mm diameter. The vessel for pump feeding, TK-2, is a square pyramid made of AISI 304 stainless steel, with a base of 990 mm, a height of 880 mm, and a slope angle of 53°. It has two outlet pipes at the lower end, one for feeding the P-1 pump and the other one for feeding the P-2 pump, both provided with DN-80 guillotine valves. P-1 is a positive displacement pump (BK 065-1L-G5-560-L14.1a-E/D from SYDEX). Its nominal power is 7.5 kW at 50 Hz and supplies a nominal flow rate of 20 m3 h−1 under 2 bar, but it may also operate up to 5 bar. It is actuated by means of a frequency variator operating in the 0−60 Hz range. At the outlet of the pump, there is a DN-80 guillotine valve. P-2 is also a positive displacement pump (BK 039-2S-G5-560-L1BQA-E/D 5H) from SYDEX. Its nominal power is 2.2 kW at 50 Hz and supplies a nominal flow rate of 2 m3 h−1 under 2 bar, but it may also operate up to 12 bar. This pump is also actuated by a frequency variator operating in the same range (0−60 Hz) and is also provided with a DN-80 guillotine valve. The flowmeter (Promag 55S1H-HC0B1AA0AAAA from Endress+Hauser) is an electromagnetic device, and its inside is coated with hard rubber in order to avoid abrasion of sensitive elements. Its nominal diameter is DN-80, and the measuring range is from 0 to 300 m3 h−1. Its calibration certificate is ISO/ IEC 17025. The pressure transmitter (Cerabar M PMP51AA21JA1PGJGRJA1+ADZ1 from Endress+Hauser) is a piezoresistive device and is provided with a welded metal membrane resistant to corrosion stress by oxidizing and reducing environments. Its measuring range covers from 0 to 16 bar. In addition to the electronic pressure indicator connected to the data acquisition system, the plant is provided with a manual manometer to verify the accuracy of the electronic device. The pilot plant is controlled by a LabView card, which allows changes of the rotational speeds of the pumps P-1 and P-2 and of the agitator. Furthermore, it also acquires and stores the data (every second) from the flowmeter and pressure meter and shows their plots in order to visualize their evolution with time. The plant described has been designed for the purpose of microhydrocyclone characterization, and therefore these devices may be easily replaced. Furthermore, in order to study different microhydrocyclone configurations, they have been manufactured by assembling three elements, namely, body, vortex, and spigot, all of them made of polyurethane and supplied by Novattia Desarrollos Ltd.. Three bodies of different length have been used in this study, one of 100 mm diameter (HCM-100) and two of 50 mm diameter (HCM-50A and HCM-50B). Figure 3 and Table 2 show the configurations of the three elements that make up the microhydrocyclone. Given that they are provided with a square inlet, an equivalent diameter, 4S/P, is calculated

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EXPERIMENTAL SECTION Materials. Microhydrocyclones are mostly used in mining because of their advantages related to compactness and capacity for particle classification in a sector in which large volumes have to be treated. Accordingly, the materials selected for this study have been taken from real streams in a plant for silica sand and kaolin production. Figure 1 shows the size distribution of the

Figure 1. Particle-size distribution of the materials used.

two materials used, which have been obtained by laser diffraction in a Mastersizer2000. Silica sand had an average particle size of 64 μm and kaolin of 8 μm (volume average dp). The particle densities of these materials were 2400 and 2600 kg m−3, respectively. The runs with silica sand were conducted using pulps with concentrations of 75 and 30 g L−1, whereas kaolin was only used in a concentration of 35 g L−1. These values correspond to the current values in real processes related to several industries in the mining sector. Equipment. In order to carry out the runs for determining the microhydrocyclone performance, an original pilot plant was designed, constructed, and fine-tuned. Figure 2a shows a 3D representation, whereas Figure 2b shows the corresponding flow sheet. As observed in Figure 2, the plant consists of a stirred tank, TK-1, a vessel to feed the pumps, TK-2, two positive displacement pumps, one for high flow rates and low pressures, P-1, and the other one for low flow rates and high pressures, P-2, an electromagnetic flowmeter, an electronic pressure indicator, and a tank for sample collection, TK-3. The pilot plant allows 14204


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Figure 2. Pilot plant for the experimental study: (a) 3D representation; (b) flow sheet.

vessel is filled with the required amount of water, the agitation is fixed at 350 rpm, and the amount of solid for the required concentration is added, e.g., 484 L of water and 37.5 kg of silica sand to prepare 500 L of pulp with a concentration of 75 g L−1. The pulp prepared in this way is stirred for at least 30 min in order to ensure mixture homogeneity and avoid material deposition zones. Meanwhile, the three elements that make up the microhydrocyclone are assembled, and the whole device is placed in the upper section of the plant. The valves are then opened, and the pump is started in order to attain the minimum pressure required for the microhydrocyclone. Prior to reading the values of pressure and flow rate, the unit is run for 5 min in order to attain steady state. The procedure described is repeated for every 50 kPa of pressure increment until the maximum corresponding to each experimental system (conditioned by the microhydrocyclone and pump) is attained. Once data have been acquired in the whole range of operating conditions, the flow rate is decreased until the pump is stopped, and the valves are closed. Each run has been repeated three times, and statistical criteria have been applied to validate the reproducibility of the results corresponding to each combination of the geometric parameters shown in Table 2. Once the required repetitions have been carried out, the separating device is modified by replacing the element corresponding to the new run. Following this procedure, the capacity curves (flow rate vs pressure) have been obtained for all of the combinations of body, vortex, and spigot set out in Table 2 for the two materials proposed. Prior to using a new pulp, the vessel has been emptied and the whole unit has been cleaned out in order to ensure the removal of any deposit corresponding to the previous pulp.

Figure 3. Scheme of the hydrocyclones used.

Table 2. Features of the Microhydrocyclones Used code D, mm Di, mm L, mm l, mm θ, deg lc, mm Do, mm Du, mm

HCM-100

HCM-50A

HCM-50B

100 28 1100 95 7.6 700 20, 25, 30, and 40 6, 10, 12, 14, 16 and 18

50 11 515 35 12.1 200 10, 14, and 18 3, 6 and 9

50 11 750 35 5.01 420 10, 14, and 18 3, 6 and 9

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for the inlet diameter, Di, with S being the inlet section and P the perimeter. Experimental Procedure. Prior to starting any run for microhydrocyclone characterization for a given material, the

RESULTS AND DISCUSSION

Figure 4 shows the capacity curves for the 42 microhydrocyclones mounted by assembling the body−vortex−spigot elements set out in Table 2. Figure 4a corresponds to the body HCM-100, Figure 4b to HCM-50A, and Figure 4c to HCM14205


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Figure 4. Capacity curves for all of the microcyclones tested: (a) body HCM-100; (b) body HCM-50A; (c) body HCM-50B.

caused by a body of bigger diameter and the effect of a biggerdiameter vortex contribute to this increase. Therefore, although the Do/D ratios of the bodies studied are very similar (it is the optimum value for maximizing the separation capacity of the hydrocyclones), the higher opening area of the vortex in the hydrocyclone of higher diameter allows operation with higher flow rates. Finally, a comparison of the plots corresponding to the bodies of 50 mm (HCM-50A and HCM-50B; parts b and c of Figure 4, respectively) shows that operation at the same conditions (vortex, spigot, and pressure) allows treatment of a slightly higher flow rate when the body is longer; in other words, when a given flow rate is treated, the longer body generates a slightly lower pressure drop. Demir et al.48,49 observed the same trend in cyclones for gas separation, confirming that the pressure drop shows a decreasing trend when the conical height is increased. In order to ease the dimensional analysis of all of the experimental results obtained, Figure 5 shows the Euler (Eu) and Reynolds (Re) numbers calculated from the experimental data. As is well-known, the former relates pressure with inertial forces, i.e., the pressure drop in the hydrocyclone with the kinetic energy by volume unit of the flow rate crossing the device. Thus, because the Eu number is smaller, the friction generated by the fluid flow is lower. Furthermore, the Re number relates inertial and viscous forces in the fluid and provides an indication of the fluid pattern under different fluid flow regimes. In order to facilitate comprehension and data analysis and on the basis of the conclusions drawn from the data in Figure 4, the

50B, whose dimensions are shown in Table 2. In order to shorten the graphical representation of all of the results obtained in this detailed experimental study and bearing in mind that the vortex diameter has a greater influence on the capacity than the spigot diameter, lines have been traced to identify data series with the same vortex diameter and points to denote different spigot diameters, pulp concentrations, and types of material. Thus, the code used for the symbols in the three graphs in Figure 4 fits the format XX-YY-Z, with the first two characters corresponding to the spigot diameter, the second pair to the pulp concentration (30, 35, or 75 g L−1), and the third one to the material (S, silica; K, kaolin) . As observed, for a given pressure and the same percentage increase in the diameter of the vortex and spigot, the former leads to a highly significant change in the flow rate, which is of the same order as the percentage increase in the upper outlet orifice diameter. Nevertheless, the latter only leads to small changes in the flow rate, which in most cases lie within the experimental error. Therefore, for a given operating flow rate, the vortex diameter has a much greater influence on the operating pressure than the spigot diameter. Likewise, Figure 4a also shows that, in the range of concentrations studied with the two materials, these parameters hardly have any impact on the capacity curve. Furthermore, a comparison of the capacities for the body of 100 mm diameter (HCM-100; Figure 4a) with those of 50 mm diameter (HCM-50A and HCM-50B; parts b and c of Figure 4, respectively) reveals that, as expected, the operating flow rate is much higher with the bigger diameter under the same operating pressure. The lower pressure drop 14206


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experimental data. Table 3 shows the values of MSRR for eqs 1−6 analyzed in this study. 1 MSRR = n

ij Euexp − Eucalc yz zz ∑ jjjjj zz z Eu exp i=1 k { n

2

(7)

In view of Figure 6 and Table 3, the fit quality is poor or very poor in all cases, which must be attributed to several factors, which are (i) great differences between the geometric parameters of the hydrocyclones used in this study and those used to propose the correlations set out in Table 1, (ii) a narrow range of geometric and operating factors in the runs for proposing the literature correlations, and (iii) the absence of certain geometric factors of great influence on Eu in the literature correlations. The correlation with the lowest MSRR is eq 4. Although it includes all of the geometric parameters that may affect Eu, the authors did not conduct experiments with different vortex and spigot diameters for a given body, but they simultaneously changed D and Do. Therefore, a severe stratification is observed in the parity plot corresponding to eq 4 (Figure 6d) for the Eu values calculated using the correlation, which is evidence that certain geometric factors, operating conditions, or their combinations used in this study have not been contemplated in the correlation developed by Tavares et al.45 The MSRR value for eq 1 is slightly higher and also has certain stratification. It should be noted that this equation does not include the body diameter and was proposed using hydrocyclones of sizes different from those used in this study and therefore cannot be considered suitable for predicting the microhydrocyclone performance. Equations 2 and 3 provide similar MSRR values, but they are considerably higher than the previous ones. Equation 2 was obtained using hydrocyclones of higher diameter than the range corresponding to microhydrocyclones, which is reflected in the poor fit in Figure 6b. Although eq 3 was proposed using bodies corresponding to microhydrocyclones, Figure 6c shows that there is a set of points horizontally aligned along the x axis, which is evidence that the experimental value of Eu changes, but the correlation cannot predict it. Equation 6 is the only dimensionless one, but its fit is very poor, which is because no changes in Do and Du were considered in the experimentation for its proposal. Therefore, although this correlation has limitations, it is an interesting one as a starting point for the proposal of a more general dimensionless correlation for predicting the hydrocyclone performance. Finally, in view of the MSRR value and Figure 6e, the fit quality of eq 5 is very poor and has therefore not been considered further. Given that the fit quality of the correlations in the literature is not suitable, modifications in their proportionality and/or exponential coefficients should be tried to improve their predictions. To that end, a methodology has been used based on that established by Saldarriaga et al.50,51 for proposing correlations for the minimum operational velocity and heattransfer coefficients in conical spouted beds. This methodology is based on the combination of multivariate regression and optimization techniques for the stepwise minimization of the residual error according to significance tests applied to the coefficients. The procedure by these authors starts by fitting one by one the regressors in each literature correlation and determining the significance order and the need for modifying the corresponding coefficients. These decisions have been taken by applying Fisher F statistical tests in each step. Nevertheless, in view of Table 1 and the results of fit quality shown in Figure 6

Figure 5. Plot of the experimental results grouped into dimensionless moduli.

same shape and color have been used in Figure 5 for the data points obtained with the same body and vortex because these two parameters are the more influential ones on the mentioned dimensionless moduli. Therefore, no distinction has been made concerning the points for different body lengths, spigot diameters, pulp concentrations, or particulate solids used because these parameters hardly have any effect on the Eu and Re moduli. The results in Figure 5 confirm that the fluid crosses the hydrocyclone in a well-developed turbulent regime (Re > 10000) under all of the experimental conditions assayed, with the Re values ranging from 18000 (50 mm body with Do/D = 0.2) to above 105 (the longest body and the largest Do/D ratio of 0.4). Furthermore, the Re values for the same Do/D ratio are lower with smaller body; i.e., they are lower for the smallest body of 50 mm than for the body of 100 mm. Furthermore, independent of the body size used, the minimum Eu value reached is close to 600. Therefore, the values shown in Figure 5 allow one to conclude that the effect of the less influential parameters (spigot diameter, body length, pulp concentration, and material) attenuates as the diameter of the upper outlet diameter is increased. Thus, for each body diameter, the experimental points corresponding to Do/D = 0.2 are scattered, whereas those corresponding to the highest Do/D ratio of ∼0.4 lie around almost horizontal lines. Under these conditions, Eu hardly depends on Re. On the basis of the experimental results shown in Figures 4 and 5, an analysis has been carried out on the validity of the equations published in the literature and gathered in the Introduction for predicting the capacity curves for the 42 microhydrocyclones used in this study. Although only one equation reported in the literature, eq 6, is a dimensionless one, the use of the Eu number is very convenient, given that it relates the two operating parameters involved this study, which are the pressure drop and flow rate (velocity). Therefore, from the values of the flow rate predicted by eqs 1−5 for the conditions used in each run, the corresponding Eu values have been calculated, Eucalc, and they have been compared with those experimentally determined, Euexp. Figure 6 shows the parity plots corresponding to the six equations gathered in the Introduction. In order to compare the fit quality of the literature correlations, the mean-square relative residual has been proposed as the objective function, MSRR. This parameter is the mean of the squared relative differences between the experimental values and those predicted by the correlations and is defined according to eq 7, with n being the number of 14207


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Figure 6. Comparison of the experimental Eu data and those predicted by the correlations in the literature, eqs 1−6.

included. In fact, certain correlations include all of the possible geometric factors in cyclones (although they have not been considered in the experimentation), and others are very short and specific, but no clear differences are observed in their prediction capacity. Accordingly, the procedure by Saldarriaga et al.50,51 has been tuned for this case. Thus, assuming that the proportionality coefficient cannot be zero, combinations of this coefficient with each one the remaining parameters or dimensionless moduli have been tested, and the one of highest significance (higher reduction in MSRR) has been taken. Once this first regressor has been considered in the correlation, the

Table 3. Mean of the Squared Relative Differences between the Experimental Values and Those Predicted by Equations 1−6 eq 1 2 3

MSRR 3.2 × 10 0.12 0.19

−2

eq

MSRR

4 5 6

2.2 × 10−2 7.3 × 107 16.2

and Table 3, the correlations proposed for pressure drop calculation in hydrocyclones are very different concerning both the type of equation and the geometric and operating factors 14208


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Figure 7. Algorithm for tuning the literature correlations.

diameter (Table 4). Furthermore, as observed in Table 4, the influence of the spigot diameter is not significant (a much lower

second one of highest significance is chosen among the remaining parameters. Parameter significance is determined based on an F test defined as

Table 4. MSRRs Obtained in the Fitting of the Experimental Results to Equation 1

SSR j − SSR i

F=

i−j SSR j n−i

(8)

where SSR is the sum of squares of the residuals with any number of regressors and is calculated by summing the squared deviations between the experimental Euler values, Euexp, and those calculated with the regressors, Eucalc, j is the number of regressors prior to introducing a given parameter or modulus, and i is the number once the parameter or modulus has been introduced, i.e., the number of regressors in the correlation to be optimized. The value calculated for F statistic is compared with the critical one for the corresponding degrees of freedom and 95% confidence interval. If the value calculated for F is higher than the critical one, the reduction in MSRR is significant at the confidence interval considered, and therefore the regressor is included in the correlation. This procedure continues until all of the significant regressors are inserted into the correlation. Figure 7 shows the flow diagram of the algorithm written in Scilab, which has been adapted from that by Saldarriaga et al.50,51 The methodology described for regression analysis has been applied to eqs 1−6. Overall, this strategy improved the fit quality of the correlations for predicting the capacity curves of the microhydrocyclones studied in this paper, but acceptable fit levels were not attained. Nevertheless, this preliminary study allowed one to draw interesting conclusions. Thus, eq 1 indicates that the operational flow rate depends on the vortex diameter, inlet diameter, spigot diameter, and pressure drop, but the fitting methodology applied shows that the most influential is the vortex diameter, followed by the pressure drop and inlet

order

equation

MSRR

F0.05

F

1 2 3 4

Qi = kDoa Qi = kDoaΔpb Qi = kDoaΔpbDic Qi = kDoaΔpbDicDud

0.28 0.14 1.3 × 10−2 1.3 × 10−2

3.88 3.88 3.88

297.9 1440.7 0.02

value than the critical one). This conclusion is consistent with analysis of the experimental results detailed above. As observed in Table 4, the MSRR for the new version of eq 1 is almost one-sixth of the one corresponding to the original correlation proposed by Rao et al.42 The new version of eq 1, with the coefficients a, b, and c of best fit, is as follows: Q i = 0.0064Do 0.80Di 0.90Δp0.47

(9)

Figure 8 shows the parity plot for eq 9. As observed, the fit quality is much better than that of the original correlation, but there is still certain stratification in the calculated Eu values. Similarly, Figure 9 shows the parity plot corresponding to the fit of the experimental data to eq 6. Although the MSRR has been reduced to 0.16 (100 times lower than that of the original correlation) by changing the proportionality constant and exponential coefficient (5586 and −0.17, respectively, in the modified version), the prediction capability of the correlation is still very low. Nevertheless, the use of dimensionless moduli and inclusion of Re seem to be promising factors for improving the fit. In fact, Zhang et al.33 obtained a good fit between their experimental results obtained in a 50 mm mycrohydrocyclone by modyfing eq 6. Thus, they obtainted the values 281 and 0.23 14209


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that the proportionality constant cannot be zero, the first step in the multivariate analysis has involved the fitting of the proportionality constant with each one of the other moduli, which allowed identification of the regressor leading to the highest MSRR reduction. Subsequently, stepwise multivariate regression analysis has been applied to the remaining moduli. Table 5 shows the regressor introduction order and the values obtained for MSRR and F statistic. Table 5. MSRR and F Values Obtained in the Stepwise Procedure for Moduli Significance in Equation 10 order

equation

1

Figure 8. Parity plot for eq 9.

( ) Eu = kRe ( ) Eu = kRe ( ) ( ) Eu = kRe ( ) ( ) ( ) Eu = k

2 3 4

a

Di Do

a

Di Do

a

Di Do

b

b

L c D

b

L c D

F

1.29 × 10−2

3.88

116.8

1.24 × 10−2

3.88

15.6

Du lc

d

1.24 × 10−2

3.88

1.50 × 10−2

3.57

As observed in Table 5, the regressor leading to the highest MSRR reduction is the ratio between the inlet and overflow diameters, which is consistent with the experimental observation. Thus, the consideration of only this modulus leads to a great MSRR reduction (from 0.16 for modified eq 6 to 1.50 × 10−2), which is clear evidence that this modulus is much more influential than the Re modulus (the only one in eq 6). The following step has consisted of combining the proportionality constant and this modulus with each one of the remaining ones. In this case, the Re modulus led to the highest MSRR reduction and, in view of the F statistic, is highly significant and should be considered in the correlation. The next modulus leading to the highest reduction in MSRR (and also significant based on its F value) is the ratio between the length and diameter of the body, which is also consistent with the data shown in Figure 6b,c. Finally, as expected on the basis of the experimental results, the modulus relating the spigot diameter with the cone length is not significant. Table 6 shows the value of the proportionality

Figure 9. Parity plot for eq 6 retuned.

for the proportionality constant and exponential coefficient, respectively. The fit of the experimental data to the remaining correlations has not been improved by following the methodology described. Therefore, on the basis of the experimental evidence and those obtained by fitting the literature correlations, a different strategy has been followed for the proposal of a new correlation. Accordingly, eq 6 has been taken as a starting point based on the fact that it is dimensionless and relates the two key dimensionless moduli in the process. Subsequently, dimensionless moduli have been inserted in this equation to account for the diameters of the vortex, spigot, inlet, and microhydrocyclone and body and cone lengths. Although the experimental results seem to suggest that the spigot diameter is not influential, it has been included in the correlation in order to check its significance. On the basis of dimensional and statistical analysis, three dimensionless moduli have been defined, which are Di/Do, L/D, and Du/lc. The cone angle has not been considered because it is related to lc and D. Furthermore, neither the vortex chimney height nor the pulp concentration and type of solid suspended have been considered because the experimental trends and statistical analysis have proven they are not influential. Therefore, the new correlation with all of the possible moduli is as follows: b i Di yz i aj j zj

Di Do

F0.05

MSRR

b

Table 6. Parameters of Best Fit in Equation 10 regressor

value

regressor

value

k a

10045 −0.146

b c

1.47 −0.222

constant and those for the exponents of the moduli in eq 10, and Figure 10 shows the corresponding parity plot. The final MSRR is slightly better than the best one obtained in the previous strategies and, furthermore, there is no data stratification in Figure 10, which is clear evidence that all of the influential geometric and operational parameters are included in the correlation.

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c L y ij D yz Eu = kRe jj zz jj zzz jjj u zzz j Do z k D { j lc z (10) k { k { Following a similar methodology as the one described for the fitting of the correlations in the literature, i.e., a code written in Scilab, with the algorithm being similar to the one shown in Figure 7, the experimental data have been fitted to eq 10. Given d

CONCLUSIONS Microhydrocyclones are highly versatile equipment used in a wide range of processes. Accordingly, a hydrodynamic study has been carried out on the performance of microhydrocyclones based on 42 units differing in the inlet diameter and the geometry of the body, vortex, and spigot. Capacity curves covering a wide range of flow rates in the feed have been 14210


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Industrial & Engineering Chemistry Research c

solid concentration, volume per volume unit, dimensionless D hydrocyclone diameter, L Di equivalent inlet diameter, L Do vortex diameter, L dp particle diameter, L Du spigot diameter, L Eu Euler number, 2Δp/ρv2, dimensionless Eucalc calculated value of the Eu number, dimensionless Euexp experimental value of the Eu number, dimensionless F statistical F, dimensionless L length of the hydrocyclone, L l vortex chimney length, L lc cone length, L MSRR mean-square relative residual, dimensionless n number of experiments, dimensionless P inlet perimeter, L po atmospheric pressure, ML−1 t−2 Qi volume flow rate in the feed, L3 t−1 Re Reynolds number, DVρ/μ, dimensionless S inlet section, L2 SSR sum of square residuals, dimensionless v characteristic velocity of the hydrocyclone, 4Qi/πD2, L t−1

Figure 10. Parity plot for the new correlation proposed.

obtained. The vortex and body diameters have proven to have a great influence on the capacity curves. Few correlations have been proposed in the literature for predicting the evolution of the pressure drop with flow rate, and none of the them provides suitable results valid for the whole range of microhydrocyclones studied. This is mainly due to the narrow range of geometric and operational factors used in the runs for their proposal. Modifications in the coefficients of the literature correlations have been analyzed by means of a procedure developed based on stepwise nonlinear regression. Although great improvements have been attained in their fit, no satisfactory correlation has been obtained for the whole range of microhydrocyclones and operating conditions. Accordingly, a new dimensionless correlation, including all of the influential dimensionless moduli, has been developed. On the basis of the stepwise regression analysis, the more influential modulus is the ratio between the inlet and vortex diameters, followed by the Re number and the ratio between the body length and diameter. The correlation includes these moduli and faithfully predicts the microhydrocyclone performance in a wide range of experimental conditions.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +34 946 015 394. Fax: +34 946 013 500. ORCID

Javier Izquierdo: 0000-0002-7258-4995 Roberto Aguado: 0000-0001-8743-5696 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been carried out with financial support from the University of the Basque Country UPV/EHU (Projects US12/ 11 and US16/26) and the collaboration of Novattia Desarrollos Ltd. J.I. thanks the University of the Basque Country UPV/EHU for his Ph.D. grant.

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NOMENCLATURE d̅p average particle diameter, L Δp pressure drop, ML−1 t−2 μ liquid viscosity, ML−1 t−1 ρ fluid density, ML−3 θ cone angle, θ 14211


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Empirical correlation for calculating pressure drop in

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