IRRIGATION AND DRAINAGE

Irrig. and Drain. 62: 407–413 (2013)

Published online 25 July 2013 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/ird.1759

SPRINKLER IRRIGATION SYSTEMS: STATE-OF-THE-ART OF KINEMATIC ANALYSISAND QUANTUM MECHANICS APPLIED TO WATER JETS†

DANIELE DE WRACHIEN1*, GIULIO LORENZINI2 AND MARCO MEDICI2

1Department of Agricultural and Environmental Sciences, University of Milan, Milan, Italy2Department of Industrial Engineering, University of Parma, Parma, Italy

ABSTRACT

The description of liquid jets crossing a gas is common to many scientific issues, and in particular sprinkler irrigation in theagricultural sector is definitely one of the most challenging issues of recent years. In the thematic literature, the classic approachhas recently been challenged by the quantum one, generally more suitable to describe interactions among several objects. Byusing a deeper approach, microscopic insight into the actual mechanics of single water droplets in sprinkler irrigation jets maybe carefully described and this represents the first move to find solutions for irrigation problems. Copyright © 2013 John Wiley& Sons, Ltd.

key words: sprinkler irrigation; droplet kinematics; classic and quantum mechanics; single- and multi-droplet systems

Received 10 March 2013; Accepted 10 March 2013

RÉSUMÉ

La description des jets de liquide traversant un gaz est commun à de nombreuses questions scientifiques; dans le secteur agricole, leproblème de l’irrigation par aspersion est certainement un des plus excitants à résoudre de ces dernières années. Dans la littératuresur le sujet, l’approche classique a été récemment remise en cause par la mécanique quantique, généralement plus appropriée pourdécrire les interactions entre plusieurs objets. En utilisant une approche plus approfondie, la vision microscopique de la mécaniquepeut décrire avec soin le comportement de simples gouttelettes d’eau dans les jets d’irrigation par aspersion; ceci représente lepremier pas pour trouver des solutions aux problèmes d’irrigation. Copyright © 2013 John Wiley & Sons, Ltd.

mots clés: irrigation par aspersion; cinématique des gouttelettes; mécanique classique et quantique; les systèmes mono et multi-goutte

INTRODUCTION

In order to design an optimal sprinkler irrigation system it isnecessary to fully understand how droplets mechanicallybehave during their flight from the nozzle of the sprinklerto the ground. In sprinkler irrigation jets the flight of a waterdroplet is related to several parameters and varies dependingalso on the mutual effects with other droplets; such interac-tions among particles and the big number of parametersinvolved greatly increase the complexity of the problem thathas to be faced. In these kinds of cases it is extremelydifficult to find analytical solutions. The first approaches to

* Correspondence to: Prof. Daniele de Wrachien, State University of Milan -Department of Agricultural and Environmental Sciences, Via GiovanniCeloria, 2 Milano 20133, Italy, Tel.: +390250316619 Fax: +390250316911.E-mail: daniele.dewrachien@unimi.it† Systèmes d’irrigation par aspersion: état de l’art sur l’analyse cinématiqueet la mécanique quantique appliquée à l’eau.

Copyright © 2013 John Wiley & Sons, Ltd.

treat this kind of problem were limited to practical experi-ences: in an initial moment each study was strongly basedon case- dependent descriptions. Examples of this ‘empiricistmovement’ are given in Kinzer and Gunn (1951), Edling(1985), Kincaid and Longley (1989) and Keller and Bliesner(1990). The strongly empirial emphasis which is the object ofthese works represents at the same time its pros and its cons:although one can benefit from the advantage of finding prob-ably the best description of the specific phenomenon, actuallycase-bound process descriptions almost fail to discoverreliable trends for real phenomena. To better describe thedroplet–-jets phenomena it has been necessary to partiallyabandon the case-dependent approach by using simplifying(but not empirical) hypotheses of the flight event: this is thecase of the classical viewpoints, generally based on relativelysimplified kinematic analysis of water droplets during theiraerial path (Edling, 1985; Keller and Bliesner, 1990;

408 D. DE WRACHIEN ET AL.

Thompson et al., 1993; Lorenzini, 2004, 2006; De Wrachienand Lorenzini, 2006). Another more recent approach is thatof adopting a quantum viewpoint: in this way a water dropletcan be treated as a quantum object characterized both bymaterial particle and by wave properties (Dirac, 1931).

THE CLASSIC APPROACH

Kinematic analyses are made adopting the Newtonianapproach. The following equations describe how a waterdroplet moves in its aerial path from the sprinkler nozzleto the ground. Using this method it is possible to obtaindiscrete results in terms of coordinates, velocities andaccelerations along the horizontal and vertical axes. Theforces applied to the system are weight, buoyancy andfriction. For a single droplet system it takes the form(Lorenzini, 2004, 2006; De Wrachien and Lorenzini, 2006):

x

y

Copy

mx ¼ �kx2

my ¼ �k y2 � ng

((1)

While in terms of parametric equation it can be written as

x tð Þ ¼ m

kln

voxk

mt þ 1

� �(2)

tð Þ ¼ mvoxmþ kvox t

(3)

y tð Þ ¼ h� m

kln

cos arctanffiffikm

pvoyffiffiffiffinmg

p� �

cos arctanffiffikm

pvoyffiffiffiffinmg

p � tffiffiffiffiffikngm

q� � (4)

tð Þ ¼ �ffiffiffiffiffing

k

rtan �

ffiffiffiffiffiffiffiffingk

pm

t þ arctan

ffiffiffiffiffik

ng

svoy

!" #

(5)

where x and y are respectively the horizontal coordinate andthe vertical coordinate (m), x and y are respectively thehorizontal velocity component and the vertical velocitycomponent (m s-1), v0x and v0y are respectively the horizontalvelocity and the vertical velocity component at t= 0 (m s-1);m is the particle mass (kg), n is the particle mass reducedbecause of buoyancy (kg); g (m s-2) is gravity acceleration;h is the y coordinate value at t = 0 (m) and k ¼ f ρA

2(kgm-1) is the friction coefficient, depending on the dimension-less friction factor according to Fanning f (Bird et al., 1960).

It is relevant in this approach that, being fully analytical,its results can be applied to any particular configurationwhich may occur, different from what happened with thepure empirical approach. However, more reliable results

right © 2013 John Wiley & Sons, Ltd.

were obtained in the case of describing a turbulent flow(i.e. for high Re).

Validation of the dynamic model

The validation of the procedure needs a quantitativeapproach to check how reliable the predictions are: thiscan be done introducing other Authors’ data in the model.The works chosen for these comparisons are Edling’s(1985) and Thompson et al.’s (1993). Among the casestudies by these authors, only those involving a no-windcondition were considered. Results are shown in Figures 1–7in terms of travel distance with respect to the jet inclination(regarding the horizontal direction) and in Table I in terms oftime of flight with respect to the droplet diameter. Both jetinclination and droplet diameter are studied parameters.

Facing a comparative approach, it can be stated that themodel defined here proves to be kinematically reliable inits predictions from a qualitative and quantitative points ofview, particularly when droplets having a ‘not too small’diameter are considered. This, being the model defined byneglecting most of the parameters typically introduced inthe others, can be considered as a first relevant result. Thecomparisons performed with the Thompson et al. (1993)data show that when the droplet gets close to a condition ofthe laminar flow law the model provides less accurate results.This is the limit to the model and it somehow defines the fieldof acceptability of the method. The model becomes weakerwhen it moves away from the turbulent flow law becauseof the approximation used to define the value of k in the othertwo flow patterns. The dependence of the results on the flowstate criterion can easily explain the different results obtainedfor the smallest droplets in the present work and inThompson et al. (1993).

THE QUANTUM MECHANICS APPROACH

Quantum mechanics for a single particle

The classic method, which uses simplifying hypotheses anddoes not employ case-dependent empiricism, is certainly aquite reliable approach, but is still not suitable to fullydescribe the phenomenon. Actually the trajectory of adroplet does not depend on the starting conditions just reported:in a sprinkler jet flow droplets are influenced by others in thevicinity. Thus, relying upon Newton’s second law of dynamics,the kinematic approach turns into the quantum one, also consid-ering a certain quantum potential. Firstly, for a single dropletsystem it takes the form (De Wrachien et al., 2012):

md2Q tð Þdt2

¼ F tð Þ (6)

where F (N) is the force,m (kg) is the droplet mass and Q (m)is the trajectory. Considering a more complete point of view, a

Irrig. and Drain. 62: 407–413 (2013)

Figure 2. Travel distance of sprinkler droplets: Edling’s (1985) data compared to Lorenzini’s (2004): flow rate = 1.4 × 10-4m3 s-1; nozzle diameter 3.96 × 10-3m;air temperature 29.4 °C; nozzle height = 2.44m; droplet diameter = 1.5 × 10-3m (R2 = 0.997)

Figure 3. Travel distance of sprinkler droplets: Edling’s (1985) data compared to Lorenzini’s (2004): flow rate = 1.4 × 10-4m3 s-1; nozzle diameter 3.96 × 10-3m;air temperature 29.4 °C; nozzle height = 3.66m; droplet diameter = 1.5 × 10-3m (R2 = 0.995)

Figure 1. Travel distance of sprinkler droplets: Edling’s (1985) data compared to Lorenzini’s (2004): flow rate = 1.4 × 10-4m3 s-1; nozzle diameter 3.96 × 10-3m;air temperature 29.4 °C; nozzle height = 1.22m; droplet diameter = 1.5 × 10-3m (R2 = 0.997)

409SPRINKLER IRRIGATION SYSTEMS

multi-droplet system composed of N droplets can be consid-ered. Thus Equation (6) can be written as

Copy

mkd2Qk tð Þdt2

¼ �∇kV �F tð Þ; 1≤k≤N (7)

where ∇ k is the 3-D gradient operator referred to the kthdroplet and V a potential function accounting for time

right © 2013 John Wiley & Sons, Ltd.

dependence (Lopreore and Wyatt, 1999). While in classicalmechanics the status of a particle can be described byspecific position and momentum as accurately as desired(see Equations (2)–(5)), in quantum mechanics the positionand momentum of a particle cannot be described asaccurately as required, according to Heisenberg’s UncertaintyPrinciple (Dirac, 1931). This problem could be sidesteppedby applying the Bohm interpretation to quantum mechanics,

Irrig. and Drain. 62: 407–413 (2013)

D2

Figure 4. Travel distance of sprinkler droplets: Edling’s (1985) data compared to Lorenzini’s (2004): flow rate = 1.4 × 10-4m3 s-1; nozzle diameter 3.96 × 10-3m;air temperature 29.4 °C; nozzle height = 1.22m; droplet diameter = 2.5 × 10-3m (R2 = 0.999)

Figure 5. Travel distance of sprinkler droplets: Edling’s (1985) data compared to Lorenzini’s (2004): flow rate = 1.4 × 10-4m3 s-1; nozzle diameter 3.96 × 10-3m;air temperature 29.4 °C; nozzle height = 2.44m; droplet diameter = 2.5 × 10-3m (R2 = 0.998)

410 D. DE WRACHIEN ET AL.

treating the system through the hidden variable theory: inthis way the trajectory of a particle could be expressed inthe form of a Bohomian. So Equation (7) takes the form

Copy

mkd2Qk tð Þdt2

¼ �∇kðV j VΨ tqu Þ Q tð Þ; 1≤k≤N (8)

where VΨ tqu is the quantum potential:

VΨ tqu ¼ �∑N

j¼1ℏ2

2mj

∇2j Ψj jΨj j 1≤j≤k≤N (9)

12∑

�¼ 2i

with ℏ = 1.055 × 10-34 J�s the Dirac constant andΨ →x; t

� �¼

R→x; t

� ��eS

→x;t

� �the wave function (R is the wave amplitude

and S is the wave phase). Comparing Equation (7) withEquation (6), if the quantum potential VΨ t

qu tends to zero,then the quantum and the classic kinematic trajectoriesbecome comparable.

The wave function Ψ is a probability amplitude describ-ing the quantum state of a particle and the pattern andbehaviour of such a wave function over time can bedescribed by Schroedinger’s equation. Thus a single particlequantum motion can be formulated as

right © 2013 John Wiley & Sons, Ltd.

∇2Ψ→x; t

� �� 12mV

→x; t

� �Ψ

→x; t

� �¼ �iD

∂Ψ→x; t

� �∂t

(10)

with D ¼ ℏ=2m (m2 s-1) the diffusion coefficient.

Quantum mechanics for many-particle systems

For a multi-droplet system Equation (10) needs to beadapted, not just considering at the same time several parti-cles (i.e. N droplets) but also taking into account the mag-netic fields due to polar covalent bonds between oxygenand hydrogen within the water molecule. For these reasons,Equation (10) takes the form (Ghosh, 2011)

Nj¼1 �2iD∇j �

→K

→xj; t

� �� �2þ 1mj

V→x N ; t

� ��Ψ

→x N ; t

� �D∂Ψ →x N ; t ∂t

(11)

in which appear→x N , which denotes all the N-particle

positions, the gradient operator ∇ j involving the coordinate→xj of the jth particle and the electrical potential due to

the particles V→x N ; t

� �, U

→x N ; t

� �being the mutual

Irrig. and Drain. 62: 407–413 (2013)

Figure 6. Travel distance of sprinkler droplets: Edling’s (1985) data compared to Lorenzini’s (2004): flow rate = 1.4 × 10-4m3 s-1; nozzle diameter 3.96 × 10-3m;air temperature 29.4 °C; nozzle height = 3.66m; droplet diameter = 2.5 × 10-3m (R2 = 0.998)

Figure 7. Travel distance of sprinkler droplets: Thompson et al. (1993) data compared to Lorenzini’s (2004): flow rate = 5.5 × 10-4m3 s-1; nozzle diameter4.76 × 10-3m; air temperature 38 °C; jet inclination = 25°; nozzle height = 4.5m (R2 = 0.994)

Table I. Time of flight of sprinkler droplets: Thompson et al. (1993) data compared to Lorenzini’s (2004): flow rate = 5.5 × 10-4m3 s-1; nozzleof diameter = 4.76 × 10-3m; air temperature 38 °C; jet inclination = 25°; nozzle height = 4.5m

Droplet diameter (m)

0.9 × 10-3 1.8 × 10-3 3.0× 10-3 5.1 × 10-3

Time of flight (s) Thompson et al. (1993) 1.54 1.63 1.75 1.84Lorenzini (2004) 1.35 1.73 2.00 2.26

411SPRINKLER IRRIGATION SYSTEMS

inter-electrical Coulomb repulsion and Φ→xj; t

� �the

external time-dependent scalar potential:

Copy

V→x

N

; t

� �¼ ∑N

j¼1 V0→rj � eΦ

→xj; t

� �h iþ U

→x N ; t

� �(12)

and the vector potential for the electromagnetic field→K

→xj; t

� �of a particle:

→K

→xj; t

� �¼ e

mjc

→A

→xj; t

� �(13)

where→A

→xj; t

� �is the vector potential of the classic

force, e is the charge of the particle and c is a coeffi-cient determined by initial conditions.

right © 2013 John Wiley & Sons, Ltd.

A final unified method for the computation of thekinematics of particles is provided from the union of thekinematic approach with the quantum one just discussed inthis work. For a single-particle system, taking into accountboth Newtonian and quantum features, the system of forcesassociated with the gradient of a potential representing themutual interactions can be described by consideringSchroedinger’s equation (Equation (10)) split into two‘quantum fluid dynamics equations’, namely the continuityequation and the Euler-type equation of motion respectively(Wyatt, 2005; Ghosh, 2011):

∂ρ→x; t

� �∂t

þ ∇ ρ→x; t

� �→x

→x; t

� �� �¼ 0 (14)

Irrig. and Drain. 62: 407–413 (2013)

d→

412 D. DE WRACHIEN ET AL.

Copy

d→v

→x; t

� �dt

¼ � 1m∇ V

→x; t

� �þ Q

→x; t

� �� �(15)

The same process can be followed in case of a multi-particlesystem, taking as reference Schroedinger’s equation forN par-ticles described in Equation (11) (Madelung, 1926; Bohm,1952a, 1952b; De Wrachien and Lorenzini, 2012):

∂ρN→x N ; t

� �∂t

þ∑Nk¼1∇k

→Jk

→x N ; t

� �¼ 0 (16)

vk→x N ; t

� �dt

þ∑Nj¼1

→vj

→x N ; t

� �∇k

� �→vj

→x N ; t

� �þ∑N

j¼1 1� δjk →

vj→x N ; t

� �∇k

→vj

→x N ; t

� �¼ � e

→E

→xk; t

� �þ e

c

→vk

→x N ; t

� �→B

→xk; t

� �� ��1m∇ V0

→x N ; t

� �þ U

→x N ; t

� �þ Q

→x N ; t

� �h i(17)

In Equation (16) ρN→x N ; t

� �¼ R2

→x N ; t

� �is theN-particle

density,while→Jk

→x N ; t

� �¼ ρN

→x N ; t

� �→v k

→x N ; t

� �is the

fluid current density. In Equation (17)→v k

→x N ; t

� �¼

ℏm∇kS

→x N ; t

� �� e

mc

→A

→x N ; t

� �is the velocity field of the

kth particle; δjk is the quantum equivalent of the electrical

potential V→x N ; t

� �;

→E

→xk; t

� �¼ �∇Φ

→xk; t

� �� 1

c

d→A

→x N ; t

� �dt

is the external electric field and→B

→xk; t

� �¼

curl→A

→x N ; t

� �is the corresponding magnetic field. It is

important to note in these last equations how quantumeffects, originating from the quantum potential, influencethe motion of both single and clusters of particles (Bohm,1952a, 1952b).

Numerical approximations of the quantumhydrodynamic equations

The hydrodynamic formulation of quantum mechanics is intui-tively appealing. The quantum potential and its associated forceappear on an equal footing with the classical potential and forcein the equations of motion, as previously shown. Thus, this ap-proach provides a unified computational method for includingboth classical and quantum mechanical effects in a moleculardynamics (trajectory-based) computational framework.

right © 2013 John Wiley & Sons, Ltd.

However, despite their deceptively simple form, thequantum hydrodynamic equations of motion are verychallenging to solve numerically. Notwithstanding therefinements provided by recent progress in describing thephenomenon, analytical solutions of the ‘quantum fluiddynamics equations’ are impossible, for both approaches(single-particle and multi-particle). Thus, once again someapproximation, numerical or dynamic, needs to be made,even if their effectiveness is still to be fully improved(Kendrick, 2011; De Wrachien and Lorenzini, 2012). Inparticular, the numerical approximation relies upon threeapproaches based on the following reference frames:Lagrangian, Eulerian and arbitrary Lagrangian–Eulerian(ALE). Although each such approach is characterized byits own pros and cons, the ALE procedure combines the bestproperties of both the Lagrangian and Eulerian frames(De Wrachien and Lorenzini, 2012). Within each of thethree procedures, a given numerical approach can be furthersubdivided into different algorithms for evaluating deriva-tives and propagating in time such as meshless moving leastsquares (MLS) (Kendrick, 2011). The MLS tends to averageout any numerical error which may be accumulating in thesolution, helping by this means to stabilize the computationalprocess. The dynamicl approximation procedures, instead,consist in imposing some particular condition on the originalfluid dynamic equations: for example, the fluid can be consid-ered incompressible, or some fluid feature may be ignored.These approximations are based on the idea that some termsare smaller or less important than others. Thus, the goal isto identify these terms and include them in those taken intoaccount. Unfortunately, due to the nonlinear nature of thequantum hydrodynamic equations, this goal is difficult toachieve in practice. Several approximate methods have beendeveloped in recent years, such as the linearized quantumforce (LQF) , the derivative propagation method (DPM) andthe vibrational decoupling scheme (VDS) (Wyatt, 2005).Finally, a combination of both numerical and dynamic ap-proximations may be considered to make possible a practicalcomputational solution (De Wrachien and Lorenzini, 2012).

CONCLUSIONS

A correct analytical and physical description of the phenom-enon of a water droplet travelling from the exit of a sprinklenozzle down to the ground is definitely hard to make.

In this work a parallel classic–quantum description both forsingle-droplet and multi-droplet systems is shown. The aim isto achieve a full description of the phenomenon, starting fromearlier approaches that made the single droplet an analysed ob-ject and classical mechanics as a method of analysis of the firstpivotal points. As demonstrated in depth, if single-dropletmodelling is an extremely important part of the problem, it is

Irrig. and Drain. 62: 407–413 (2013)

413SPRINKLER IRRIGATION SYSTEMS

not the whole problem. In fact in a sprinkler jet the kth drop-let tends to follow its own potential aerial path determinedby initial conditions (as well described by the classicalapproach), but in addition the influence derived from allthe other droplets in the vicinity needs to be taken into ac-count. In this way the inter-droplet electrical interactionsdue to the difference in electric charges between oxygenand hydrogen, give birth to external magnetic fields whichcan be fully described using a quantum approach. Thusthe ‘quantum fluid-dynamic equations’ derived fromSchroedinger’s equation extended to an N-particle systemmay represent the highest point ever reached in the state-of-the-art in the description of liquid jets crossing a gas.However, since analytical solutions of the ‘quantum fluiddynamics equations’ are impossible, future studies shouldfocus on how both numerical and dynamic approximationshould simplify the problem in order to find criticizablesolutions.

NOTATION

List of symbols and definitions for classical and quantummechanics:

→A

Copyrig

Vector potential of the classic force

→B Magnetic field c Numerical coefficient D Diffusion coefficient (m2 s-1) δ Quantum equivalent of the electrical potential V e→ Elementary charge (C) E External electric field f Fanning friction factor F Force (N) g Gravity acceleration (m s-2) h Vertical coordinate value at t= 0 (m) ℏ Dirac constant (J s) →j Fluid current density k→ Friction coefficient (kgm-1) K Vector potential for the electromagnetic field m Particle mass (kg) n Particle mass reduced because of buoyancy (kg) Q Particle trajectory (m) R Wave amplitude Re Reynolds number ρ Particle density S Wave phase U Mutual inter-electrical Coulomb repulsion V Potential function Vqu Quantum potential v0x Horizontal velocity component at t= 0 (m s-1) v0y Vertical velocity component at t= 0 (m s-1) →v Velocity vector

ht © 2013 John Wiley & Sons, Ltd.

x

Horizontal coordinate (m) x Horizontal velocity component (m s-1) →x Position vector y Vertical coordinate (m) y Vertical velocity component (m s-1) φ External time-dependent scalar potential Ψ Wave function ∇ 3-D gradient operator

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