agricultural water management 84 (2006) 130–136
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Study on hydraulic performance of drip emitters by computational fluid dynamics Qingsong Wei a,*, Yusheng Shi a,c, Wenchu Dong b, Gang Lu a, Shuhuai Huang a a
State Key Laboratory of Plastic Forming Simulation and Die and Mould Technology, Huazhong University of Science and Technology, Wuhan 430074, PR China b State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, PR China c State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, PR China
article info
abstract
Article history:
Because of small size and complex structure of labyrinth-channel used in most drip
Accepted 17 January 2006
emitters, it is not possible to observe the flow behavior of water passing through the
Published on line 23 March 2006
channel. In this study, the flow in labyrinth channels was simulated by using the method of computational fluid dynamics (CFD) to calculate the distributions of pressure and velocity
Keywords:
of the flow, and to calculate the relationship between pressure and rate of discharge for
Labyrinth channel
channels with three different shapes: triangular, rectangular, and trapezoidal. The trian-
Drip emitters
gular channel had the highest efficiency of hydraulic energy dissipation. Moreover, the
Hydraulic performance
results of simulation for the labyrinth channel well matched the measured data in labora-
Computational fluid dynamics (CFD)
tory. Computational fluid dynamics provides a promising tool to help in the design of labyrinth channels used in drip emitters. # 2006 Elsevier B.V. All rights reserved.
1.
Introduction
Drip emitters are the key component of drip irrigation systems. Many drip emitters use labyrinth channels to regulate water flow. Glaad (1974) reported results, obtained under laboratory conditions, which indicated the hydraulic performance of drippers was determined by the structural form, dimension and material of the emitter channel, and Ozekici et al. (1991) reported that the pressure loss during flow was the result of the tortuous corners of the emitter channel. However, one cannot observe how water flows through labyrinth channels because the channels are too small. Supposing the flow was a laminar, Palau-Salvador et al. (2004) simulated the relationship between pressure and
discharge rate for an in-line emitter labyrinth with the commercial CFD software, FLUENT, and pointed out that the mathematical model proposed had a good agreement with the experimental data. To study the development method for drip emitters, Wei et al. (2004). used the technique of CFD to simulate the flow in an eddy channel that was a novel geometry for drip emitters. The results indicated that the use of CFD could decrease the number of experimental channels and the times of laboratorial experiments. In this study, the distributions of pressure and flow velocities within in-line labyrinth channels with triangular, rectangular and trapezoidal shapes were simulated by using the method of computational fluid dynamics. The objective was to determine if CFD correctly predicted the effect of water pressure on discharge
* Corresponding author at: State Key Laboratory of Plastic Forming Simulation and Die and Mould Technology, School of Material Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, PR China. Tel.: +86 27 62412032; fax: +86 27 87548581. E-mail address:
[email protected] (Q. Wei). 0378-3774/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.agwat.2006.01.016
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agricultural water management 84 (2006) 130–136
Table 1 – The parameters provided by manufacturer for three types of drip emitters Emitter type
Manufacturer
Nominal discharge (L/h)
Rectangle
Xinjiang Tianye
3.0–4.0
Trapezoid
Beijing Lvyuan
2.0–3.0
Triangle
Shandong Laiwu
2.0–3.0
rate of the emitter. The paper concludes with a brief discussion of the potential application of CFD for design of labyrinth channels. Three types of emitters with in-line labyrinth channels that are manufactured in China by three different companies were used (Table 1). The nominal discharge ranged from 2 to 3 L/h for the trapezoidal and triangular shaped channels, and from 3 to 4 L/h for rectangular shaped channel. The structural parameters of three types of labyrinth channels are given in Table 2 and the shapes of individual structural cells within the channel are illustrated in Fig. 1.
1.1.
Theoretical analysis
The Reynolds number, Re, is generally used to distinguish between laminar and turbulent flow, and can be described as follows (Ferziger and Peric, 1996): rVR ; (1) Re ¼ m where r (kg/m3) is the density of fluids, V (m/s) the average velocity of the flow, R (m) the hydraulic radius of flow, and m (kg/(m s)) is the viscosity coefficient of fluids. The relationship between V and discharge, q (L/h) is simply dependant on the cross sectional area of the emitter channel, S according to q (2) V¼ : S For most current labyrinth emitters, S is generally less than 1.0 mm2, and q generally ranges from 2 to 8 L/h. For a q of 2 to
Emitter photo
8 L/h and a S of 1.0 mm2, V ranges from 0.6 to 2.2 m/s. Since the cross-sectional shape of the three types of labyrinth emitters is rectangular, R can be determined as follow (Ferziger and Peric, 1996): R¼
wd ; 2ðw þ dÞ
(3)
where w and d are the width and depth of the emitter channels, which had been defined in Table 2. Based on these dimensions, R ranges from 0.125 to 0.25. Consequently, Re ranges from 75 to 550 assuming a water density of 103 kg/ m3, a value for R that ranges from 0.125 to 0.25 mm, and a viscosity of 103 kg/(m s). According to the traditional hydrodynamics, for this range of Reynolds number, the flow through labyrinth channels could be laminar. However, some studies (Pfahler et al., 1990; Harley et al., 1995; Kandilikar et al., 2003) indicated that the transition from laminar to turbulent in the channels with an area of about 1.0 mm2 occurs at a Re ranging from 100 to 700. Therefore, the flow in labyrinth channels could foster turbulence because the flow path in such small channels is not straight. Consequently, it is reasonable to adopt a model that describes the turbulent flow for the flow in labyrinthchannel drip emitters.
1.2.
Mathematical models
The standard k–e turbulent model is generally used for most engineering calculations, so it is chosen to describe the flow in
Table 2 – Structure parameters of three labyrinth channels Emitter type
Rectangle Trapezoid Triangle
Structural parameter w (mm)
u (8)
w1 (mm)
d (mm)
n
1.0 0.7 0.7
90 60 60
4 2.7 4
0.5 0.7 0.7
24 25 8
w is the width of flow passage, w1 the width of labyrinth channel, u the degree of corner, d the depth of flow passage, and n is the number of structural cell included in a single labyrinth channel.
Fig. 1 – The structurural parameters of structural cell for three labyrinth channels. w is the width of flow passage, w1 the width of channel, and u is the degree of corner.
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agricultural water management 84 (2006) 130–136
drip emitters in this study. The governing equations are given as follows (Meneveau and Katz, 2000): @ui ¼ 0; (3) continuity equation : @xi NavierStokes equation : " !# @ui u j @P @ @ui @u j ¼ þ m þ ; r @x j @xi e @x j @xi @xi
k equation :
@u j k @ r ¼ @xi @x j
me @k s k @x j
e equation :
@u j e @ ¼ r @x j @x j
me @e s e @x j
(4)
!
þ Gk re;
(5)
e e2 þ Ce1 Gk Ce2 r ; k k
(6)
(7)
me ¼ m þ mt ;
mt ¼
!
Cm rk2 e
(8)
and @u Gk ¼ m t i @x j
! @ui @u j þ : @x j @xi
(9)
In these equations, k is the turbulent kinetic energy, e the rate of dissipation, u the velocity of the flow, P the pressure, x the displacement vector, r the fluid density, m the fluid viscosity, and Cm, Ce1, Ce2, sk and se are the correcting coefficients. The authors chose the following values for these coefficients as proposed by Launder and Spalding (1974): Cm = 0.09, Ce1 = 1.44, Ce2 = 1.92, sk = 1.0 and se = 1.3. These constants are at the upper end of the ranges normally used in CFD, and were determined to be appropriate by Yakhot and Orszag (1986). Compared with the impact of water pressure, the effects of gravity is low, so the gravity of water is neglected here. The material of the drip emitters is plastic, with a low coefficient of surface roughness. Consequently, the effects of surface roughness on hydraulic performances was not considered. Additionally, because the pressures applied on the water passing through drip emitters was low, water was assumed to be incompressible and the rate of water flow was assumed to be steady, i.e., it did not change with time.
1.3.
Fig. 2 – Pressure (mH2O) distributions of the flow in partial channels in rectangular (A), trapezoidal (B) and triangular (C) shaped channels.
Model solutions
The finite element analysis software, ANSYS (Poole et al., 2002), was used to solve the k–e turbulent model described above. The 3D tetrahedron meshes of the emitter channels were generated by using the Meshing Tool of ANSYS. The boundary and initial conditions used were appropriate to the practical conditions of use: (1) at the channel inlet a pressure of 10 mH2O was assumed; (2) at the channel outlet the pressure was assumed zero; (3) the velocity vectors along the channel walls were set to zero. The results of simulation were the pressure and velocity distributions of flow in the three channels as shown in Figs. 2 and 3. The relationship between discharge, q, and pressure, P, is important for evaluating emitter performance. The model was solved using nine pressures: 5, 7.5, 10, 12.5, 15, 17.5, 20, 22.5, and 25 mH2O to calculate the discharge and flow exponent (Eq. (10)). The results of these simulations are given in Table 3 and Fig. 4.
agricultural water management 84 (2006) 130–136
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Fig. 4 – The curves of relationships of pressure and simulated and measured discharges.
Fig. 3 – Velocity (m/s) distributions of the flow in partial channels in rectangular (A), trapezoidal (B) and triangular (C) shaped channels.
2.
Results and discussion
The pressure distributions obtained from the CFD simulations (Fig. 2) show that an obvious pressure gradient exists in each of the three labyrinth channels. In each type of channel, the changes in pressure gradients occur mainly at the corner of the channels. When the flow arrives at the corner, the change in the direction of flow results in a large loss of the local pressure, which is the main cause of the hydraulic energy dissipation. Therefore, the pressure loss at the corner of the channel will finally determine the degree of hydraulic energy dissipation. The results of pressure distributions in the three emitter channels also show that: there are two pressure areas
(I and II) around the corner of rectangle channel (Fig. 2a); beside these two areas, a third (III) exists around the corner of trapezoidal channel (Fig. 2b); there are three equal pressure areas (I–III) in the triangular channel (Fig. 2c). The more the pressure varies, the greater the pressure loss will be. In terms of the number of pressure areas near the corner, the pressure loss of the three emitter channels can be described as the triangular > trapezoidal > rectangular. Consequently, a greater dissipation per unit length occurs in the triangular channel than in the rectangular and trapezoidal channels. The distributions of the velocities on the x–y plane shown in Fig. 3 clearly present the flow state of the water in the three channels. The shapes of the flow in the three channels are almost the same. An obvious turning appears at the corner of the channel, which makes the flow move along a S-shaped path. In addition, some obvious whirlpools form in the channels. These whirlpools will increase the dissipation of kinetic energy. The kinetic energy of the flow affects directly the efficiency of hydraulic energy dissipation, and the velocity of the flow along the length direction determines the discharge. Consequently, comparing the distributions of velocity along the flow path in the three channels, the rectangular channel has the highest discharge rate per unit of pressure. The relationship between the discharge and pressure can be described in the following equation (Keller and Bliesner, 1990): q ¼ kPx ;
(10)
where k is the discharge coefficient, and x is the flow exponent. These two coefficients can be calculated by using the following equations (Keller and Ron, 1990): P log k ¼
log qi
P P ðlog Pi Þ2 ðlog qi log Pi Þ log Pi P P m ðlog Pi Þ2 ð log Pi Þ2
P
(11)
and
x¼
m
P
P P log qi log Pi log qi log Pi ; P P 2 m ðlog Pi Þ ð log Pi Þ2
(12)
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Table 3 – The discharge, q (L/h), from simulation Emitter type
Rectangle Trapezoid Triangle
Pressure, P (mH2O) 5
7.5
10
12.5
15
17.5
20
22.5
25
2.15 1.62 2.03
2.76 2.01 2.45
3.21 2.24 2.80
3.66 2.52 3.11
4.03 2.78 3.38
4.41 2.92 3.63
4.89 3.14 3.87
5.26 3.34 4.08
5.51 3.54 4.28
where x is assumed to equal 2. In these equations, i the serial number of simulation, and m is the total number of data (m = 9 for this study). With the simulated results of discharge and pressure in Table 3, the discharge coefficient and the flow exponent for three types of channels are calculated by Eqs. (11) and (12). The calculated steps and results are given in Table 4. The values of x of the three channels from CFD simulation (Table 4) present a relationship of xrectangular > xtrapezoidal > xtriangular. The lower the value of x, the more effective the hydraulic energy dissipation of the emitter. Namely, the discharge variation with increasing pressure of an emitter with a lower value for x will be less than for an emitter with a higher value for x when the inlet pressure has been changed. Consequently, the triangular channel is more preferable than rectangular and trapezoidal channels.
2.1.
Verification of simulations
Verification of the results obtained from the CFD simulation was conducted according to the ISO standard ‘‘agricultural irrigation equipment-emitting specification and test methods’’ (ISO, 1991) using the drip-tapes described in Table 1. Some tests for the relationship between q and P were done in the State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University. Nine pressures (5, 7.5, 10, 12.5, 15, 17.5, 20, 22.5, and 25 mH2O) were used, which were the same as those used in the CFD simulations. In the experiments, the pressure of the water at the emitter inlet was regulated with a pressure regulator, and measured by using a needle pressure gauge. The outflow from the emitter within a given time, t (2 min), was collected in a beaker, and measured with an electronic scale. For each emitter, three measure-
Table 4 – The k and x parameters (Eq. (10)) calculated from the simulated effects of water pressure, P, on flow, q Emitter type Rectangle
P Trapezoid
P Triangle
P
P (mH2O) (I)
log P (II)
log q (III)
II2 (IV)
II III (V)
5 7.5 10 12.5 15 17.5 20 22.5 25
0.699 0.875 1 1.097 1.176 1.243 1.301 1.352 1.398
0.332 0.441 0.507 0.563 0.605 0.644 0.689 0.721 0.741
0.489 0.766 1 1.203 1.382 1.545 1.693 1.828 1.954
0.232 0.386 0.507 0.618 0.711 0.800 0.896 0.975 1.036
–
10.141
5.243
11.860
6.161
5 7.5 10 12.5 15 17.5 20 22.5 25
0.699 0.875 1 1.097 1.176 1.243 1.301 1.352 1.398
0.210 0.303 0.350 0.401 0.444 0.465 0.497 0.524 0.549
0.489 0.766 1 1.203 1.382 1.545 1.693 1.828 1.954
0.147 0.265 0.350 0.440 0.522 0.578 0.647 0.708 0.768
–
10.141
3.743
11.860
4.425
5 7.5 10 12.5 15 17.5 20 22.5 25
0.699 0.875 1 1.097 1.176 1.243 1.301 1.352 1.398
0.307 0.389 0.447 0.493 0.529 0.560 0.588 0.611 0.631
0.489 0.766 1 1.203 1.382 1.545 1.693 1.828 1.954
0.215 0.340 0.447 0.541 0.622 0.696 0.765 0.826 0.882
–
10.141
4.555
11.860
5.334
Eq. (11): P P P P P P log k ¼ ð III IV V IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:076;, k = 0.839; P P P P P Eq. (12): x ¼ ð9 V III IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:585
P P P P P P Eq. (11): log k ¼ ð III IV V IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:124;, k = 0.752; P P P P P Eq. (12): x ¼ ð9 V III IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:478
P P P P P P Eq. (11): log k ¼ ð III IV V IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:019, k = 0.957; P P P P P Eq. (12): x ¼ ð9 V III IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:465
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Table 5 – The discharge, q (L/h), measured from the tests in the laboratory Emitter type
Rectangle Trapezoid Triangle
Pressure, P (mH2O) 5
7.5
10
12.5
15
17.5
20
22.5
25
2.03 1.58 1.97
2.60 1.92 2.37
3.10 2.20 2.71
3.55 2.45 3.00
3.96 2.68 3.27
4.36 2.88 3.51
4.73 3.07 3.73
5.08 3.25 3.94
5.41 3.42 4.13
ments of the outflow were made. The average outflow per 2 min is given in Table 5. With the measured results in Table 5, k and x of the three types of emitters were calculated by the Eqs. (11) and (12), and the results are given in Table 6. These results also present the relationship of xrectangular >xtrapezoidal > xtriangular. The mea-
sured relationships between q and P in Fig. 4 are in rough agreement with the simulated results obtained from the CFD simulations. In this study, the deviation, d, of measured, qm, and simulated discharges (qs) was defined as jqm qs j d¼ 100% (13) qm
Table 6 – The k and x parameters (Eq. (10)) calculated from the measured effects of water pressure, P, on flow, q Emitter type Rectangle
P Trapezoid
P Triangle
P
P (mH2O) (I)
log P (II)
log q (III)
II2 (IV)
II III (V)
5 7.5 10 12.5 15 17.5 20 22.5 25
0.699 0.875 1 1.097 1.176 1.243 1.301 1.352 1.398
0.307 0.415 0.491 0.550 0.598 0.639 0.675 0.706 0.733
0.489 0.766 1 1.203 1.382 1.545 1.693 1.828 1.954
0.215 0.363 0.491 0.603 0.703 0.794 0.878 0.955 1.025
–
10.141
5.114
11.860
6.027
5 7.5 10 12.5 15 17.5 20 22.5 25
0.699 0.875 1 1.097 1.176 1.243 1.301 1.352 1.398
0.199 0.283 0.342 0.389 0.428 0.459 0.487 0.512 0.534
0.489 0.766 1 1.203 1.382 1.545 1.693 1.828 1.954
0.139 0.248 0.342 0.427 0.503 0.571 0.634 0.692 0.747
–
10.141
3.633
11.860
4.303
5 7.5 10 12.5 15 17.5 20 22.5 25
0.699 0.875 1 1.097 1.176 1.243 1.301 1.352 1.398
0.294 0.375 0.433 0.477 0.515 0.545 0.572 0.595 0.616
0.489 0.766 1 1.203 1.382 1.545 1.693 1.828 1.954
0.206 0.328 0.433 0.523 0.606 0.677 0.744 0.804 0.861
–
10.141
4.422
11.860
5.182
P P P P P P Eq. (11): log k ¼ ð III IV V IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:120, k = 0.759; P P P P P Eq. (12): x ¼ ð9 V III IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:611
P P P P P P Eq. (11): log k ¼ ð III IV V IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:076, k = 0.723; P P P P P Eq. (12): x ¼ ð9 V III IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:483
P P P P P P Eq. (11): log k ¼ ð III IV V IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:076, k = 0.939; P P P P P Eq. (12): x ¼ ð9 V III IIÞ=ð9 IV ð IIÞ2 Þ ¼ 0:460
Table 7 – Discharge deviations, d(%), of the measured and simulated discharges Emitter type
Rectangle Trapezoid Triangle
Pressure, P (mH2O)
d
5
7.5
10
12.5
15
17.5
20
22.5
25
5.91 2.53 3.05
6.15 4.69 3.38
3.55 1.82 3.32
3.09 2.86 3.67
1.77 3.73 3.36
1.15 1.39 3.42
3.38 2.28 3.75
3.54 2.77 3.55
1.85 3.51 3.63
3.38 2.84 3.46
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agricultural water management 84 (2006) 130–136
and d is the average deviation: P di ; d¼ n1
(14)
where i is the serial number of measurement and n1 is the total number of the pressures used in the experiments (here n1 = 9). The calculated results in Table 7 show that the average discharge deviations for the there emitters are all less than 4%, which supports the suitability of the CFD simulations. However, qm is less than qs for all inlet pressures (Tables 3 and 5). The possible reasons are that the CFD simulations neglect the effects of the gravity, the surface tension of water and the surface roughness of the channel wall on water flow. All of these factors will increase the resistance of the labyrinth channel, so qm would be lower than qs.
3.
Conclusions
The flow simulation based on CFD method provides a mathematical method to generate figures that ‘visualize’ the flow within labyrinth channels of drip emitters. The distributions of pressure and velocity are helpful to understand the operating principles, and to analyze efficiency of hydraulic energy dissipation within the emitter channel. The pressure loss occurs mainly at the channel corners. Therefore, the corner structure of the channel is the main factor that affects the efficiency of hydraulic energy dissipation for the labyrinth emitters. Based on a comparison of predicted and measured flow rates as a function of pressure, the hydraulic energy dissipation within the triangle channel is better than that of the rectangular and trapezoidal channels. In addition, the results of the CFD simulation were verified by the measured data in laboratory. The average discharge deviations between the measured and simulated results were less than 4%, which indicates that the CFD method is a feasible and powerful technique to study the flow behavior of the water passing through labyrinth channels within drip emitters as well as to optimize the design of the labyrinth channel.
Acknowledgements This work is supported by the state ‘‘863’’ project of China (2002AA6Z3083), an open foundation of the State Key Laboratory of Water Resources and Hydropower Engineering Science (2004B014) and an open foundation of the State Key
Laboratory of Fluid Power Transmission and Control (GZKF2004003) and The State Key Laboratory of Plastic Forming Simulation and Die and Mould Technology, Huazhong University of Science and Technology is undertaking these three projects.
references
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