biosystems engineering 105 (2010) 495–506
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Research Paper
Development of a mathematical model to predict clean water head losses in hydrocyclone filters in drip irrigation systems using dimensional analysis H. Yurdem*, V. Demir, A. Degirmencioglu Department of Agricultural Machinery, Faculty of Agriculture, Ege University, 35100 Bornova-Izmir, Turkey
article info A model was developed using dimensional analysis, to predict head losses in hydrocyclone Article history:
filters. Different hydrocyclone filters with different specifications were used to measure
Received 19 August 2009
head losses at different flow rates in the laboratory. The parameters influencing head
Received in revised form
losses were considered to be the inside diameters of the inlet and outlet pipes, cylindrical
28 January 2010
section diameter of the filter, apex diameter of the conical part, cylindrical section length of
Accepted 5 February 2010
the filter, conical section length of the filter body, length of the outlet (vortex finder) pipe,
Published online 11 March 2010
water velocity in inlet pipe, acceleration of gravity, kinematic viscosity of water. A dimensional analysis was carried out, using Buckingham’s pi-theorem. To develop a model, experimental head loss data from 21 filters were considered in the study. The model accounted for 96.7% of the variation in the pressure coefficient. The predicted and the measured head losses were in close agreement with a correlation coefficient of 98.1%. The results showed that the model may be used to determine head losses in hydrocyclone filters with an acceptable accuracy if the variables are within the following ranges: inside diameter of inlet and outlet pipe 0.053–0.154 m; cylindrical section diameter of the filter 0.195–0.46 m; apex diameter of the conical part 0.04–0.06 m; cylindrical section length of the filter 0.16– 0.41 m; conical section length of the filter body 0.37–0.955 m; length of the vortex finder pipe 0.155–0.627 m; flow rate 3.7–98.48 m3 h1; and Reynolds number 18 860–421 065. The performance of the model was compared with models developed for industrial hydrocyclones and the necessary comparisons were established by using statistical test procedures. The model in this study provides better predictions as compared to some other models available in the literature. ª 2010 IAgrE. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
Energy consumption in drip irrigation systems is low since they operate at low pressures. However, their initial capital investment is high. As a result, the operating life of these systems must be extended. The operating life of the systems is directly
affected by the clogging of the emitters installed in the laterals. Emitters have very narrow water passages in order to reduce water energy and pressure and this increases the risk of clogging. The complete clogging of emitters can stop a system functioning and the clogging of some emitters in laterals results in non-uniform water distribution. Non-uniform water
* Corresponding author. E-mail address:
[email protected] (H. Yurdem). 1537-5110/$ – see front matter ª 2010 IAgrE. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.biosystemseng.2010.02.001
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Nomenclature B4 Da Dc Di Do EM ERMS F3 g KQO L Lc Lo LT Lv m N
system constant apex diameter of the conical part, m cylindrical section diameter of the filter (hydrocyclone diameter), m inside diameter of inlet pipe, m inside diameter of outlet (vortex finder) pipe, m modelling efficiency root mean square error calibration parameters acceleration of gravity, m s2 common material dependent constant in the generalised model for performance characteristic length in dimensional analysis matrix, m cylindrical section length of the filter, m conical section length of the filter, m total length of hydrocyclone length of the vortex finder pipe, m number of variables number of observations
distribution can cause problems such as reduced yield and variable product quality. Therefore, filtration is necessary in order to prevent emitters from clogging. The main reasons for emitter clogging are suspended solids that have both organic and inorganic components. Problems encountered due to clogging can be prevented by proper filtration and chemical processing (Adin & Alon, 1986; Gilbert & Ford, 1986, chap. 3; Ravina et al., 1990). The filtration of the irrigation water is defined as the separation of solid materials by using the differences in their physical and mechanical properties from water. Different types of filters are used in drip irrigation systems. Gravel or sand filters, are used to withhold large particles of organic matter, and screen or disc type filters and hydrocyclones are used to filter out inorganic suspended (Benami & Ofen, 1993, chap. 3; Demir & Uz, 1994; Douglas & Bruce, 1985; Keller & Bliesner, 1990). Appropriate filters should be used by considering the suspended matter in the water. The last clean section of a drip irrigation system usually includes screen and/or disc filters. The time for the clogging of disc and screen filters can be extended if the separation of large size particles from water is achieved using hydrocyclone filters. Hydrocyclones can remove up to 98% of sand particles that could be contained by a 75 mm screen (Keller & Bliesner, 1990). Hydrocyclones depend on a centrifugal force, created by vortex flow inside the filter, to remove and eject high-density particles from the fluid stream. They are widely used as separation equipment in many branches of industry such as the mining, chemical, and water treatment industry (Martinez, Lavin, Mahamud, & Bueno, 2008). Various papers have reported on the performance of hydrocyclone filters used in these industries (Antunes & Medronho, 1994; Asomah & Napier-Munn, 1997; Kraipech, Chen, Dyakowski, & Nowakowski, 2006; Nageswararao, Wiseman, & Napier-Munn, 2004). However, studies on hydrocyclones used as water
n P Q r R2 Re T Vc
number of constants in the model probability value flow rate, m3 s1 rank of the dimensional analysis matrix coefficient of determination, % Reynolds number, dimensionless time in dimensional analysis matrix, s water velocity in cylindrical section of the filter, m s1 Vi water velocity in inlet pipe, m s1 q cyclone inclination from vertical, degree n kinematic viscosity of water, m2 s1 a cone angle of the hydrocyclone, degree chi-square c2 rf, and r density of feed slurry, liquid head loss (pressure drop), m DHf DHf exp,i experimental head loss, kPa DHf exp,mean mean value of experimental head loss, kPa DHf pre,i predicted head loss, kPa solid volume fraction in feed slurry 4sf
filtration devices in the field of micro irrigation are limited (Soccol & Botrel, 2004; Srivastava, Singh, Singh, & Singh, 1998). Chauhan (1998) suggested two main issues involving in testing hydraulic performance of hydrocyclone filters. These are the determination of the clean water head loss and the determination of filtration efficiency. Chauhan (1998) made an attempt to evaluate the performance of the hydrocyclone filter with clean water and with known concentration of impurities. The performance of the filter was studied by varying discharge, head loss, influent and effluent concentrations, and filtration efficiency with time of operation of the filter. Uz, Demir, and Eren (1994) investigated head losses from sand separators with 60.3 mm inlet and outlet diameter filters. The flow rate ranged from 2.4 to 12 m3 h1, head losses were in the range of 1.66–22 kPa for the hydrocyclone filter. Bulancak, Demir, Yurdem, and Uz (2006) studied the efficiency of nine different filters (discs, screens, hydrocyclone, sand separator and media filter) used in drip irrigation systems. From the study, it was found that the head losses for the hydrocyclone filters (200 mm diameter) were in the range of 20–60 kPa for the flow rate range of 17–32 m3 h1 when clean water was used. Mailapalli, Marques, and Thomas (2007) focused on a hydrocyclone filter used in drip irrigation. In their study a hydrocyclone filter with a cylindrical diameter of 200 mm was selected and its performance was evaluated by studying the variation of discharge, head loss, influent concentration, and filtration efficiency with elapsed time of operation. Martinez et al. (2008) studied the hydrocyclone filters with 50 and 100 mm cylindrical section diameter for the separation of chemicals with different densities and mainly focused on the optimum length of the vortex finder. The results obtained by Martinez et al. (2008), using these methodologies were in agreement, showing that the most efficient length for the vortex finder is 10% of the total length of the hydrocyclone.
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Different prediction models for separation efficiencies and head losses have been developed by researchers focussing on hydrocyclones used in industry (Antunes & Medronho, 1994; Asomah & Napier-Munn, 1997; Kraipech et al., 2006; Nageswararao et al., 2004). Chen, Zydek, and Parma (2000) stated that under different operating conditions, different models could be required for the best fit for the experimental data. It can also be seen that no single model can be utilised if both pressure drop and separation performance are to be compared. A model may have good prediction for separation but perform poorly when predicting pressure drop. Although analytical models based on fundamental or basic principles can provide important insights into physical or biological process, some systems are too complex to be treated in this way (Upadhyaya, 2006). Filtration is such a complex process that it requires a method to predict head losses. Dimensional analysis develops general forms of equations that describe natural phenomena and is a useful tool for developing prediction equations for complex physical systems. Dimensional analysis applied to a particular phenomenon is based on the assumption that certain defined variables, are the independent variables of the problem, and that all other variables, other than the dependent variable, are redundant or irrelevant. This is considered to be the initial step for studying physical phenomena. The second step in the process is the formation of a complete set of dimensionless products of the variables. This reduces the physical quantities pertinent to a system to dimensionless groups called pi-terms (P terms) (Langhaar, 1987). Some studies have used dimensional analysis to predict head losses in filters. Puig-Bargue´s, Barragan, and Ramirez de Cartegena (2005) developed several equations for calculating
head loss in disc, screen and sand filters. They used dimensional analysis, but focused on filtering effluents. Predicted values were compared against to experimental data. Yurdem, Demir, and Degirmencioglu (2008) focused on disc filters and a model was developed using dimensional analysis to predict head losses in disc filters. The model accounted for 90.18% of the variation in the pressure coefficient. A comparison between the predicted and the measured head losses was in close agreement with a correlation coefficient of 99.5%. The objective of this study was to develop a prediction model using dimensional analysis for head losses in hydrocyclone filters at different geometry under different operating conditions.
2.
Materials and methods
2.1.
Experimental set-up
In laboratory experiments, 21 hydrocyclone filters with different specifications and designs were used. The technical properties were measured. Schematic representations of the hydrocyclone filters used are given in Table 1 and Fig. 1. Hydrocyclone filters mainly consist of two parts; a separator and a collector (Fig. 1). The separator is formed by cylindrical and conical sections that are attached together. The cylindrical section has inlet and outlet pipes welded onto these sections with different outside diameters. Inlet pipes are attached to the cylindrical section tangentially in order to create the vortex. The length of vortex finder pipe depends on the size of the hydrocyclone and it is located on the centre line of the
Table 1 – Technical properties of hydrocyclone filters used in experiments Filter type
E20 E25 H20 H25 H30 H40 K20 K25 K30 K40 M20 M30 M60 P20 P25 P30 P40 S20 S25 S30 S40
Inside diameter of inlet pipe (Di), mm
Cylindrical section diameter of the filter (Dc), mm
Apex diameter of the conical part (Da), mm
Cylindrical section length of the filter (Lc), mm
Conical section length of the filter body (Lo), mm
Length of the vortex finder pipe (Lv), mm
53 68 53 63 78 102 53 69 81 105 53 78 154 53 69 81 106 53 68 81 105
223 250 210 240 280 358 210 251 251 251 256 311 460 195 250 250 312 223 251 281 328
40 40 40 45 45 55 53 55 53 53 45 45 60 51 52 52 52 50 52 52 52
198 225 185 205 205 275 200 195 200 190 205 275 410 160 239 255 275 195 210 225 255
580 600 400 595 595 830 660 660 870 870 600 665 955 370 525 525 650 480 535 640 625
419 310 310 350 398 627 300 410 405 370 305 419 580 155 190 170 232 187 190 227 290
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forces. Particles separated from the water are collected in collection chamber and then removed by the use of a valve. The experiments in this study were conducted using set-up shown in Fig. 2. Clean water was used during the each experiment. Water was supplied from a tank via an electrically-driven centrifugal pump. Head losses at different flow rates were measured using two pressure transducers (PD21 type, Keller, Switzerland) with an accuracy of 0.2% when the flow was stabilised. The transducers were placed at the inlet and outlet of the filter. Flow rate during experiments was controlled by the valves and measured using an electromagnetic flowmeter (Emd-C100F type (DN100), Bass-Ela, Czech Republic and Optiflux 2000 type (DN65), Krohne, Germany) with an accuracy of 0.5% and in the range of 10–100% of maximum flow rate. Measurements were logged on a data-acquisition system (ADAM 4520 and ADAM 4017þ, Advantech Automation Corp., USA), using GeniDAQ version 4.25 data-acquisition software. Measurements were taken when fluctuations in flow rate ceased. During the experiments, water temperature was measured using a digital thermometer and it varied between 18 and 22 C. Details of the experiments carried out at different ranges of flow rates, Q, and Reynolds number at inlet pipe, Re¼ViDi/n, are given in Table 2 for each filter. In order to obtain head losses from different types of filters at different flow rates, a total of 1850 tests were carried out in the laboratory. The local head loss is directly proportional to the square of flow velocity and as a result, the selection of appropriate flow velocity in a system becomes important. The optimum flow velocity in pumped lines is usually considered from the point of construction and energy costs and is in the range 1–3 m s1 (Addink, Keller, Pair, Sneed, & Wolfe, 1983, chap. 15). Howell
Fig. 1 – Schematic representation of the hydrocyclone filters. Di, inside diameter of inlet pipe; Do, inside diameter of outlet pipe; Dc, cylindrical section diameter of the filter; Da, apex diameter of the conical part; Lc, cylindrical section length of the filter; Lo, conical section length of the filter body; Lv, length of the vortex finder (outlet) pipe.
cylindrical section. The vortex finder pipe is also part of the outlet pipe in the separator. Hydrocyclones used in irrigation have the same inlet and outlet diameters to make them easy to install. Air release valves are placed on top of the cylindrical part. Separation of sand and other solid matter is based upon the size and density with separation achieved by centrifugal
Fig. 2 – Schematic representation of the test apparatus: (1) electrically-driven centrifugal pump; (2) control valves; (3) electromagnetic flowmeter; (4) hydrocyclone filter; (5) transducers; (6) data-acquisition system; (7) computer.
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Table 2 – Flow characteristics of hydrocyclone filters used in experiments
Table 3 – Variables affecting head losses in hydrocyclone filters
Filter type
Type of variable
Flow characteristics
E20 E25 H20 H25 H30 H40 K20 K25 K30 K40 M20 M30 M60 P20 P25 P30 P40 S20 S25 S30 S40
Minimum– maximum flow rate (Q), m3 h1
Minimum– maximum Reynolds number (Re¼ViDi/n)
Minimum– maximum head loss (DHf), kPa
3.70–57.45 10.97–73.94 11.88–38.88 14.04–55.44 22.70–70.80 23.30–95.90 3.72–31.50 21.29–46.45 21.86–56.87 26.35–52.99 9.72–37.80 20.80–93.67 30.35–97.55 11.36–4.30 3.70–79.70 10.70–91.18 18.40–98.48 5.12–40.99 10.72–57.45 19.79–90.68 28.93–92.65
24 679–383 193 56 895–383 585 79 240–259 330 78 413–309 629 102 041–318 260 79 757–328 269 24 578–208 123 108 519–236 784 94 521–245 866 87 881–176 708 64 833–252 127 93 500–421 065 68 967–221 672 75 043–292 707 18 860–406 230 46 492–396 125 60 999–326 562 33 892–271 337 55 613–298 039 85 714–392 751 96 298–308 401
5.6–245.1 3.6–181.9 17.0–128.0 12.6–103.7 15.0–87.0 10.0–64.0 2.5–74.9 18.4–73.7 5.9–43.0 4.7–14.0 13.9–143.3 14.0–159.3 7.5–18.1 10.3–173.5 3.2–203.0 2.6–124.4 1.0–53.1 2.1–236.4 5.2–200.5 8.6–174.0 6.8–56.2
et al. (1983, chap. 16) indicated that drip irrigation pipes are usually made of plastic and many pipe manufacturers recommended an optimum flow velocity of 1.5 m s1. The recommended velocity ranges given above should be used for a better separation in a hydrocyclone filters design. However, the hydrocyclones were tested over a wider velocity range so that head losses can be predicted outside the recommended ranges.
2.2. Dimensional analysis of the head losses in hydrocyclone filters Buckingham’s pi-theorem requires the pertinent, nonredundant variables affecting the physical system. The variables, shown in Table 3, are considered to be relevant. Considering all the variables (m ¼ 10) and their dimensions of length, L, and time, T, the resulting dimensional matrix is given below.
L T
DHf
Di
Dc
Da
Lc
Lo
Lv
V
g
n
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 1
1 2
2 1
Variables in the above matrix are DHf, head loss; Di, inside diameter of inlet pipe (Di is considered to be equal to Do since the inlet and outlet pipe diameters of the hydrocyclones used in experiments were the same); Dc, cylindrical section diameter of the filter; Da, apex diameter of the conical part; Lc, cylindrical section length of the filter; Lo, conical section length of the filter body; Lv, length of the vortex finder pipe; V, water velocity in inlet pipe; g, acceleration of gravity; n, kinematic viscosity of water.
Symbol
Variable
Dimension
Dependent
DHf
Head loss (m)
L
Independent
Di
Inside diameter of inlet pipe (m) Cylindrical section diameter of the filter (m) Apex diameter of the conical part (m) Cylindrical section length of the filter (m) Conical section length of the filter (m) Length of the vortex finder pipe (m) Water velocity in inlet pipe (m s1) Acceleration of gravity (m s2) Kinematic viscosity of water (m2 s1)
L
Dc Da
Lc Lo Lv Vi
g n
L L
L L L LT1
LT2 L2T1
The number of P terms resulting from the dimensional analysis is found by subtracting the number of the pertinent variables from the rank of the matrix or the difference between the number of variables and number of dimensions. In both cases, the result would be eight dimensionless P terms that could be used to explain the head losses. The resulting dimensionless groups are given in Table 4. The dependent P term, the head loss in filter (DHf), can be expressed as a function of dimensionless groups. It can be written as a function of dimensionless groups as in the following theoretical form: DHf Vi Di Lc Vi Di Dc Lc Da ; ; ; ¼f ; ; ; Di gDi Dc Lv n Da Lo Lo
(1)
The above dimensionless groups from 21 different filters were tabulated in a spreadsheet and transformed by taking the log of each dimensionless group. The values were then transferred to a statistical package program (Minitab release 13.20, Minitab Ltd., United Kingdom) for fitting a multiple linear regression model using a stepwise procedure.
2.3. Statistical test methods to evaluate the goodness of fit of the models The suitability of the developed model was evaluated using the reduced chi-square, c2, root mean square error, ERMS and modelling efficiency, EM with other design equations. These statistical parameters, which have been used in other studies (Demir, Gunhan, Yagcioglu, & Degirmencioglu, 2004; Karayel, Barut, & Ozmerzi, 2004; Willmott, 1982), were calculated as follows:
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Table 4 – Dimensionless groups for predicting the head losses in disc filters and their significance
Table 5 – Results of multiple regressions analysis for predicting the head losses
Dimensionless terms
P terms
Function
P1
DHf /Di
P2 P3
V2i /gDi Di /Dc
P4
Lc /Lv
P5 P6
ViDi /n Dc /Da
P7
Lc /Lo
P8
Da /Lo
Ratio of head loss to inside diameter of inlet pipe (pressure coefficient) Froude number Ratio of inside diameter of inlet pipe to cylindrical section diameter of the filter Ratio of cylindrical section length of the filter body to length of the vortex finder pipe Reynolds number (Re) Ratio of cylindrical section diameter of the filter to apex diameter of the conical part Ratio of cylindrical section length of the filter body to conical section length of the filter body Ratio of apex diameter of the conical part to conical section length of the filter body
DHf, head loss; Vi, water velocity in inlet pipe; g, acceleration of gravity; n, kinematic viscosity of water; Di, inside diameter of inlet pipe; Dc, cylindrical section diameter of the filter; Da, apex diameter of the conical part; Lc, cylindrical section length of the filter ; Lo, conical section length of the filter body; Lv, length of the vortex finder pipe.
ERMS
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 PN i¼1 DHf pre;i DHf exp;i ¼ N PN
EM ¼
c2 ¼
i¼1
2 P 2 DHf exp;i DHf exp;mean Ni¼1 DHf pre;i DHf exp;i 2 PN i¼1 DHf exp;i DHf exp;mean
PN i¼1
DHf pre;i DHf exp;i Nn
(2)
Log (constant) V2i /gDi Di /Dc Lc /Lv ViDi /n Dc /Da Lc /Lo Da /Lo
Exponent
Standard error
R2, %
P-value
5.1467 0.4233 3.3584 0.4608 0.9522 0.3332 0.9341 0.4368
0.37883 0.03685 0.15900 0.03769 0.07445 0.16805 0.09712 0.09970
– 94.47 95.17 95.48 96.08 96.50 96.67 96.70
