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Analytical Study of Wind Influence on In
J. Eng. Technol. Sci., Vol. 47, No. 3, 2015, 296-305
296
Analytical Study of Wind Influence on In-Flight
Sprinkler Droplets
Frank Agyen Dwomoh1,2*, Yuan Shouqi1 & Li Hong1
1
Research Centre of Fluid Machinery Engineering and Technology,
Jiangsu University, 212013, Zhenjiang, Jiangsu, P. R. China
2
School of Engineering, Koforidua Polytechnic, P.O. Box 981 Ghana
*
Email: agyen2007@gmail.com
Abstract. An analytical model to describe the dynamics of in-flight droplets is
presented in this paper to augment information on wind influence on travel
distance of in-flight sprinkler droplets. The model is ballistic-theory based. It
employs a relatively simple, wide-range empirical relationship between drag
coefficient and Reynolds’ number to replace the several sets of relations for a
specified range of Reynolds numbers. The fourth-order Runge-Kutta numerical
integration techniques were used to solve the trajectory equations. A modified
exponential model for droplet size distribution was used during the simulation.
Comparative analysis showed that agreement exists between the predictions of
this model and that of earlier models. Droplets with a diameter smaller than
0.1 mm travelled farthest. Within the droplet range of 0.5 mm to 4.5 mm, as
droplet diameter increased, travelled distance increased with increasing wind
speed. The extent of drift increased sharply within the droplet range of 0.5 mm to
0.05 mm and increased mildly for droplet diameters greater than 0.5 mm. The
model also attempts to identify droplets that are likely to contribute to drift loss
and those that have a high probability of contributing only to distortion of the
distribution pattern.
Keywords: analytical model; droplet drift; distribution pattern; simulation; sprinkler
droplets; traveled distance; wind influence.
1
Introduction
0B
The influence of wind during sprinkler irrigation pose challenges that need
attention especially in this era of water conservation towards a sustainable use
of resources. Most sprinklers apply water to the ground by projecting water jets
into the air at high velocity, which later fall down as water droplets. Under
windy conditions, in-flight sprinkler water droplets may impact the ground or
plant canopy, experience droplet evaporation or be wind drifted [1-3].
Sprinkler droplet travel under no-wind condition is undisturbed and thus a
characteristic of the sprinkler nozzle for a given operation configuration. If
droplets travel beyond their characteristic distances for the same sprinkler
Received March 12th, 2014, 1st Revision October 16th, 2014, 2nd Revision November 27th, 2014, 3rd Revision
March 17th, 2015, Accepted for publication May 11th, 2015.
Copyright ©2015 Published by ITB Journal Publisher, ISSN: 2337-5779, DOI: 10.5614/j.eng.technol.sci.2015.47.3.5
Wind Influence on In-Flight Sprinkler Droplets
297
nozzle and pressure configuration, they are considered as drifted. These drifted
droplets contribute to wind distortion of the distribution pattern [3,4].
Several simulation studies have been carried out to model various aspects of
wind effect on sprinkler droplets over the years [3,5-9]. Several factors affect
the trajectories and losses of in-flight water droplets that complicate adequate
description and estimation of wind drift [10-12].
Studies that have simulated droplet drift loss are few. Notable among these are
Edling [13]; Seginer, et al. [8]; Martin and Newman [4]; Thompson, et al. [14];
McLean, et al. [15]; Teske, et al. [9]. Edling and Chowdhury [16] and Longley
[17] presented theoretical models for estimating spray evaporation and wind
drift from low-pressure spray sprinklers. Molle, et al. [18] also reported on
evaporation and wind drift loss during sprinkler irrigation. Lorenzini and Saro
[19] studied thermal fluid dynamic modeling of a water droplet evaporating in
air by considering wind drift (but with uniform velocity field) by applying the
Runge-Kutta integration method.
This paper presents an analytical description of the dynamics of droplets from a
single operated irrigation sprinkler to augment the pioneering works of earlier
researchers on in-flight sprinkler droplets. Specifically, we seek to simulate the
dynamics of wind influence on the travel distance of sprinkler droplets.
2
Materials and Method
2.1
Model of Droplet Motion
Several models have already been developed by researchers that consider a
sprinkler as a device emitting numerous droplets with diameter as a function of
their travelled distances [5,6,8]. According to ballistic theory, droplets’ motions
are influenced by the initial velocity vector, the gravitational force, the wind
vector and the viscous drag force. Eqs. (1) to (3) were solved to compute the
droplet trajectories.
̈ =
2 
 2
= 4 � − �
̈ =
2 
 2
= − 4 � � − 
̈ =
2 
 2
�
3
�
3
= − 4 � − �
�
3
(1)
(2)
(3)
where ,  and z are the coordinates referring to the ground (with origin at the
sprinkler nozzle);  is the droplet diameter (mm); ̅ is the density ratio of air
298
Frank Agyen Dwomoh, et al.
and water respectively;  is time (s), and g is acceleration due to gravity. C is
the air drag coefficient of the droplet moving at the speed  .
2
2
 = �[� − � + � − � + 2 ]
(4)
 and  are the horizontal and vertical components of the droplet velocity,
respectively;  ,  and  are the , y and  components of the wind velocity
respectively. Since the logarithmic profile of wind speed is generally considered
to be a more reliable estimator of the actual field conditions, the average wind
speed ( ) at height r (cm) above the ground was calculated for all conditions
as:
ln[(−)/ ]
 =  ln[(−)/0 ]
(5)
0
U m = wind speed (m/s) measured at reference height m (cm) above the ground.
 and  are roughness height (cm) and roughness parameter (cm) respectively,
both are functions of crop height ℎ (cm), given by:
2.2
log  = 0.997ℎ − 0.1536
log  = 0.997 log ℎ − 0.883
(6)
Boundary and Initial Conditions
Height of sprinkler nozzle: 1.2 m (most sprinklers mounted on risers are within
the range of 0.8 to 1.5 m); droplet diameter range considered: 0 < droplet
diameter (mm) < 5; wind speeds: 0, 2.5, 3.5 and 4.5ms-1; operating pressure:
250,300,350 kPa.
The fourth-order Runge-Kutta numerical integration techniques were used to
solve equation (1), (2) and (3) for droplet movement with the specification of
initial conditions as follows:
( = 0) = 0;̇ ( = 0) = 0, ; ( = 0) = 1.2  (height from ground to
sprinkler nozzle = 1.2 m); ̇ ( = 0) = 0, . 0, = 0 cos  ; 0, = 0 sin  ;
 is the inclination of the sprinkler nozzle to the horizontal. The velocity of the
sprinkler jet exiting from the nozzle was calculated as:
0 = Cd (2)0.5
(7)
where () is the operating pressure head at the nozzle and Cd is the discharge
coefficient, equal to 0.98.
By setting  = 0 (soil surface) or catch can elevation, each trajectory solution is
constituted by the x and y coordinates. Two categories of simulations were
conducted: no-wind and in-wind conditions. Droplet travel distances were
simulated for both under-wind and no-wind conditions. The horizontal distance
Wind Influence on In-Flight Sprinkler Droplets
299
between the nozzle exit and the droplet landing point was simulated as the
droplet travel distance.
2.3
Empirical Model of the Drag Coefficient
To determine the trajectory of the droplet projectiles in the air, a relatively
simple, wide-range empirical relationship between the drag coefficient (C) and
the Reynolds’ number ( ), proposed by Holterman [20], was employed to
replace the several sets of relations for a specified range of Reynolds numbers,
as displayed in Eqs. (7) and (8).
=
 
�� �

 =



1�

+ �
(8)
(9)
where  = 24;  = 0.32;  = 0.52; =droplet diameter (), v=velocity (ms-1)
and =the kinematic viscosity of the air (2  −1). The adopted relationship
compares very well with the well-known set of relations by Fukui, et al. [5].
The model is applicable not only to the turbulent-flow regime, but also to the
Stokes regime. However, it shows some deviation from the experimental data
for  > 104 .
2.4
Estimation of Droplet Size Distribution
Several mathematical models and data have been published for drop size
distribution for distinct types of sprinkler devices using different methods and
operated at varying pressures, nozzle sizes and heights [21-25]. In this study,
the simple exponential model used by Li, et al. [25], which was later modified
by Kincaid, et al. [24], was used. The exponential model is given by Eq. (10).
 
) ��
50
 = 100 �1 −  �−0.693(
(10)
50 =  +   and  =  +  
(11)
Where  is the percentage (%) of the total drops that are smaller than d;  is
the drop diameter (mm); 50 is the volume mean drop diameter (mm);  is the
dimensionless exponent. Kincaid, et al. [24] found out that Eq. (10) together
with the following suggested adjustment factors gave reasonable predictions
that cater for smaller diameter droplets.
The regression coefficients used for estimating the drop size distribution
parameters for the impact sprinkler with small round nozzle (3 mm) are:  =
0.31;  = 11, 900;  = 2.04;  = -1,500;  is the ratio of the nozzle
diameter to the pressure at the base of the sprinkler device. Sprinkler droplets
were assumed to be spherical in shape (this is consistent with the photographic
300
Frank Agyen Dwomoh, et al.
studies by Okaruma and Nakanishi [26]). It was also assumed that the volume
of the droplet is invariant during its flight from the nozzle to the ground. The
droplet sizes distributions derived from Eqs. (10) and (11) that were used in the
analysis are shown in Figure 1.
Volume cumulative
frequency (%)
100
80
250kPa
300kPa
350kPa
60
40
20
0
0
2
4
Droplet diameter (mm)
Figure 1 Droplet size distribution derived and used for model analysis.
3
Results and Discussion
3.1
Comparative Analysis of Models
Travel Distance [m]
A comparative analysis of droplet travel distance by the models of Molle, et al.
[18], Fukui, et al. [5], von Bernuth and Gilley [26] and our model are presented
in Figure 2. Agreement exists between our model and that of Molle, et al. [18]
for droplets with diameters greater than 2.5 mm, while some differences exist
for droplets with diameters smaller than 2.5 mm.
18
16
14
12
10
8
6
4
2
0
Molle, et al. [18]
Fukui, et al. [5]
Bernuth & Gilley [26]
Our Model
0
2
3
Droplet Size [mm]
5
Figure 2 Comparison between other simulated travel distances.
Wind Influence on In-Flight Sprinkler Droplets
301
Even though similarity exists in terms of the shape of the trajectory, the travel
distances simulated by our model are longer than those of Fukui et al. [5] and
von Bernuth & Gilley [27] but shorter than those of Molle et al. [18]. The
disparities can be attributed mainly to differences in the operating parameters
and assumptions used in the simulation.
3.2
Effect of Wind Speed on Droplet Travel Distance
Simulated droplet travel distances from the sprinkler for three wind speeds with
downwind direction, and zero-wind condition at constant pressure (300 kPa) are
compared in Figure 3. Droplets with diameters smaller than 0.1 mm travelled
farthest, travelling beyond 24 m from the nozzle exit. This is in agreement with
the work of Molle, et al. [18]. Wind increased droplet travel distance
downwind. Within the range of 0.5 mm to 4.5 mm, as droplet size and wind
speed increased, travelled distance also increased. The extent of drift is defined
here as the difference between the droplet travel distance under no-wind and inwind situations for the same sprinkler nozzle-pressure configuration. The extent
of drift increased sharply within the droplet range of 0.5 mm to 0.05 mm and
then increased mildly for droplets diameters greater than 0.5 mm (Figure 4).
From Figure 1 (for 300 kPa), droplets with a mean diameter larger than 4.0 mm,
representing a frequency of 0.92% of the total number of droplets, and droplets
with a mean diameter smaller than 0.2 mm, representing a frequency of less
than 3%, traveled beyond the wetted radius (Figure 3). Such categories of
droplets, apart from contributing to distortion of the distribution pattern, also
have a higher probability of contributing to wind drift for sprinklers that are
located at the periphery of the irrigated field.
0[m/s]
3.5 [m/s]
Travel distance (m)
30
2.5[m/s]
4.5 [m/s]
24
18
12
6
0
0
1
2
3
Droplet diameter (mm)
4
Figure 3 Comparison of droplet travel distance between no-wind and in-wind
conditions at constant pressure (300 kPa).
Frank Agyen Dwomoh, et al.
Extent of Drift (m)
302
30
2.5m/s
3.5m/s
4.5m/s
25
20
15
10
5
0
0
2
4
Droplet diameter (mm)
Figure 4 Comparison of extent of drift (m) as a function of droplet size (mm) at
three wind speeds at constant pressure (300 kPa).
The remaining droplets are more likely to contribute to distortion of the
distribution pattern. Even though larger droplets (with diameter > 4 mm)
represent a small percentage of the number of droplets in the droplet
distributions considered (Figure 1) due to their high volume per droplet; if they
are wind-drifted they will constitute a high percentage loss.
For example, at a constant operating pressure of 300 kPa, at wind speeds of
3.5 m/s and 4.5 m/s, 20% and 32% of the total volume travelled beyond 17 m
(the wetted radius), respectively. Of these percentages, 70-90% were larger
drops (> 3.9 mm) representing 3.6% and 6.3% of the total number of drops in
the distribution, respectively. Hence the percentage of large droplets in the
distribution spectrum should not only be of interest for predicting water droplet
impact [24], but can also be critical for estimating wind drift losses as well. The
above observation is particularly important as it partially identifies droplets that
are likely to contribute to drift losses and those that have a high probability of
contributing only to distortion of the distribution pattern.
4
Conclusion
The paper presented an analytical model to describe the dynamics of wind effect
on in-flight sprinkler droplets. A comparative analysis showed that agreement
exists between the predictions of this model and those of earlier models.
Droplets with a diameter smaller than 0.1 mm travelled farthest. Within the
droplet range of 0.5 mm to 4.5 mm, as the droplet diameter increased travelled
distance increased with increasing wind speed. The extent of drift increased
sharply within the droplet range of 0.2 mm to 0.05 mm and then increased
gently for droplet diameters greater than 0.5 mm. The model also identified
Wind Influence on In-Flight Sprinkler Droplets
303
droplets within the mean diameter ranges of 0.05 to 0.1 mm and greater than 3.9
mm as likely to contribute to both distortion of the distribution pattern as well
as wind drift, especially for sprinklers located at the periphery of the irrigated
area when wind speeds are greater than or equal to 3.5m/s.
Acknowledgements
The financial support provided by the program for National Hi-tech Research
and Development (863 Program No. 2011AA100506 and 2011GB2C100015) of
China is gratefully acknowledged.
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