Miller Hadfield 1989

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J. agric. Engng Res. (1989) 42, 135-147

A Simulation Model o f the Spray Drift from Hydraulic Nozzles

P. C. H.

MILLER ; D. J. H ADFIELD?

A mod el to predict the trajectories of droplets from agricultural flat fan nozzles is

described. Droplet motion is considered in two phases: close to the nozzle where the

trajectory is dominated by the conditions associate d w ith droplet form ation, particularly the

initial velocity and entrained air conditions; and a second phas e wher e the effects of

atmo spheric turbulence are predicted using a

“random-walk” approach. Results from the

model are compared with laboratory measurements of droplet size/velocity profiles beneath a

nozzle and with field measurements of the downwind drift from boom mounted nozzles

oper ated conventionally. Pred icted total downwind drift deposits agre ed well with mea sured

values but the form of the vertical deposit distribution was less well predicted .

1 ntroduction

Previous work concerned with predicting spray droplet movem ent has used both

dispersion and random-w alk type models. Dispersion models have been used to study the

transport of sprays both towards and within crop canopies and have been applied

particularly to aerial spraying where droplet movem ents through the air can be relatively

large. Bathe and Sayer12 and Cramer (Dum bauld et a1. 3 derived exp ressions for the

deposit distribution downw ind of a line spray source and for the airborne flux just above

the crop canopy based on diffusion theory, and demonstrated some agreement with field

data. Schaefer and Allsop* discuss the application of gradient diffusion theory to spray

transport and demonstrate an improved prediction of deposit when compared with the

linearly expanding Gaussian plume used by Bathe and Sayer and by Cramer.

A number of authors have used Markov type simulation models to predict the

trajectories of droplets in turbulent air flows including Thom pson and Ley,’ Picot et

a l . 6

Wilson

et al . 7

and Legg and Raupauch.* Such models have been show n to provide an

acceptable description of spray deposit distributions downw ind from a defined source

given assum ed release conditions. These m odels have been used in conjunction with both

aerial and ground crop spraying systems and have examined the effects of operating

parameters on downw ind spray deposits. Both diffusion and “random -walk” type models

have been developed that account for droplet evaporation and sedimentation by

gravitational forces. However, droplet release conditions have been assum ed and not

related to sprayer characteristics and relatively little work has attempted to relate the

conditions of droplet generation to their subsequent transport particularly in the region

close to agricultu ral flat fan nozzles.

Such an analysis w ill provide a basis for the design of spraying systems to minimise drift

and is the subject of this paper.

Droplet trajectories are described in two distinct phases: close to the nozzle where

considerations of air drag and entrained air predominate, and further downw ind w here a

random-w alk approach is used.

*

Chemical Applications Group, AFRC Engineering, Silsoe, Bedford MK45 4HS, UK

7 Formerly of above address. Current address: Flat 3, 56 Ch aucer Road, B edford MK40 2A P, UK

Received 9 une 1988; accepted in revised form 6 November 1988.

Paper presented at Ag Eng 88, Paris, France, 2-6 March 1988.

135


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A MODEL OF SPRAY DRIFT

Notation

d droplet diameter, urn

z,

crop height, m

0, distance downw ind of nth sam- z, boom height, m

pling line, m

(Y factor indicating loss of velocity

h distance below nozzle, m correlation in successive time steps

K1 entrained air parameter,

y see Eqn (3)

dimensionless 6 constant in entrained air equation

L

Monin-O bukov length, m

rr_ L agrangian time scale, s

I, coherent length of liquid spray

At time step, s

sheet, m

Ap vapour pressure difference, P a

r see Eqn (3)

&

s

nozzle spacing on boom, m

rl

1

random variables from Gaussian

distribution with mean zero and

T, Stokesian response time, s

standard deviation of 1-O

u,

sheet velocity for fan nozzle, m/s p viscosity of air

u horizontal droplet velocity, m/s

p density of droplet, kg/m3

24, friction velocity, m/s

0”

rms horizontal velocity fluctua-

u,

entrained air velocity, m /s tions, m/s

v, droplet settling velocity, m/s

%I

rms vertical velocity fluctuations ,

w vertical droplet velocity, m/s

m s

z height above ground, m

2. Structure of the model

The model simulates a simplified spraying situation as shown in

Fig . 1

in which the

spray drift from a set of boom mounted nozzles operating above a cereal crop canopy is

measured

using ve al strings

mounted at various distances down wind. In structuring the

model the following simplifying assum ptions were made:

(1) Trajectories were considered in two dimensions only and effects due to the forward

9

W i nd d i r ec t i on

s

S a m p l i n g

l i nes

c-----D, -

.

4

Fig. 1. Simulated spraying conditions


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P. C. H. MILLER; D. J. HADFIELD

137

movement of the sprayer and the turbulent wakes created by this forward

movement were neglected.

(2) The crop surface behaved as a perfect collector of spray with no movement or

reflection of droplets from within the canopy. Ground spray deposits were

calculated on this basis.

(3) Spray droplets from adjacent nozzles on a boom acted independently.

2.1.

Th e random-walk model

The random -walk model used in the simulation was based on that developed by

Thompson and Leg in which the velocity of a droplet at any time step was related to the

velocity in the previous time step but with an added random component due to

turbulence . The droplet velocity in the vertical direction, w, in the i + 1 th time step is

given by:5

wi+l

= r Wi + V si) + qj+lUw(l- 0 2)t - u s,+,

(1)

where Q = exp(-At/r,_) and 7 is a random variable. Eqn (1) is only valid if the time step

At is small compared with the Lagrangian time scale, rL. The effect of droplet settling

velocity, u,, is included as a direct term in the equation, and this was considered by

Thompson and Ley to be satisfactory for water based d roplets with diameters up to

450 urn based on data by Smith.’ Settling velocities, IJ,~,are therefore calculated using the

relationships given by Thompson and Ley’ as

v,, = 4.47 x 10-3d - 0.191

for d>lOOum

(2)

and

v,, = 3.2 x 10+d2 - 6.4 x 10+d3

for d


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A MODE L OF SPRAY DRIFT

2 .2 . Tra jecto ry ca lcu la t io n c lose to th e nozz l e

Trajectories were determined over a series of time steps using an integration routine

described by Marchan t” with drag coefficients equal to those for solid spheres. March ant

indicated that errors due to this assump tion would be less than 10% for the droplet

conditions simulated.

The effect of turbulence on initial droplet trajectories was simulated by includin g an

additional vertical air velocity component term which was sampled randomly from a

Gaussian distribution with a mean of zero and a standard deviation of a, and which was

assum ed to persist for a period equal to the Langrangian timescale, r,_.

The response time of a droplet in a moving air-stream is characterised by the Stokesian

response time, T where

Initial runs with the model indicated the droplets below a spray nozzle ha d reached

their settling velocities after a period of approximately 4T, and so this value w as used as

the basis for changing from the ballistic trajectory to random-w alk phases of the model.

2.2.1. Spray cha rac te r i s t i cs

Spray droplets were assum ed to form at a distance equal to the sheet coherent length

below the nozzle and have an initial velocity the same as the liquid sheet. Data relating to

coherent lengths and sheet velocities w as initially obtained from measurem ents made

using high-speed photography.”

Trajectory angles were sampled from a Gauss ian distribution with a mean of zero and a

standard deviation of O-4 times the nozzle angle. This value was determined experimen-

tally by comparing calculated volume distributions below a 110” flat fan nozzle with those

measured in patternator experiments.

The effect of the droplet size distribu tion wa s accounted for by simulating the

trajectories of between 500 and 5000 droplets in each of a number of size categories

(depending on the required accuracy and limitations of computing time) up to a maxim um

of 25 size categories. Size categories in increments of 20 urn have been found convenient

in applying the simulation to most agricultural flat fan nozzles. V olume distributions were

then determined by relating the proportion of droplets d eposited or remaining airborne in

a defined area, to the total volume of spray liquid output while travelling through the

sample volume and the measured droplet size/volume distribution for the spray produced

by the nozzle.

2.2.2.

En t r a i n ed a i r ve l o ci t i e s

Studies of air entrainment in liquid sprays have shown that the velocity of entrained air

along the axis of a fan jet nozzle, v,,

can be described by the equation:12

where u, is the sheet velocity and

h

the distance below the nozzle. 8 is a constant which

for sprays into a ir takes a value’*

of 0.4, w hile K1 is a constant defined by the width of the

spray fan at right angles to the spray sheet at a given distance from the nozzle. K1 was

determined from photographic measurem ents of a number of 80 and 110” nozzles, and a


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Start

139

Input nozzle

conditions

-8

Initiate droplet

flight

Fig. 2. Model

flow

chart


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140 A MODEL OF SPRAY DRIFT

mean value of 0.14 was used as input to the simulation. The entrained air velocity was

assum ed to be constant across the spray sheet.

2.2.3.

St r u c t u r e o f t h e s imu l a t i o n mode l

A flow chart for the m odel of drift from a single nozzle is shown in Fig. 2. The drift

from boom mounted nozzles w as simulated by calculating the drift from a single nozzle

and overlapping that from successive nozzles off-set by the nozzle spacing distance s.

Inputs to the model in addition to those defining the spray nozzle included a description

of the samp ling system and parameters defining the meteorological conditions. Atmos-

pheric stability w as defined in terms of the Mo nin-Obukov length (L), and related to a,,

a, and tr by relationships derived in Thompson and Ley.5

Simulation outputs gave the vertical distribution of airborne spray in 10 cm increments

at each sampling line and the ground deposits between samp ling lines.

The predicted trajectories of ten 100~ urn droplets released in the spray from a flat fan

nozzle spraying at the rate of 0.6 l/min at a pressure of 3 bar are shown in Fig. 3.

1 Nozzle

; . ::

Crop

Fig. 3.

Simulated trajectories of l I pm droplets fr om a 110” nozzle operating 0.5m above a cereal

crop 50mm tall . . . . . . . -, from ballistic trajectory model;

-, fr om random walk model

3. xperimental verification of the model

3.1. M ea su r emen t s o f d ow nw i n d d r i f t f r om a mo v i n g b o om

Measurem ents of the drift from three boom m ounted nozzles were made using the

techniques reported by Sharp.

l3 The nozzles (Lurmark 110015 in Kemetal) were mounted


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P. C. H. MILLER; D. .I. HADFIELD

4

on a small boom and off-set from the axis of the tractor to minimise air disturbance and

wak e effects. The nozzles sprayed a 1% solution of a fluorescent tracer dye at a pressu re

of 3.0 bar in a total of 20 passes down a spray track aligned at right ang les to the mean

wind direction so as to accumulate measurable deposits. A spraying speed of 8 km/h was

used. Spray drift was captured using 2.5 mm diameter plastic tubing su spended in a

3 x 6 m framework. Meteorological conditions at the time of spraying were recorded

using a 10 m mast with four vane anemom eters, mean air temperature, relative humidity

and temperature difference sensors. The Mo nin-Obukov length was derived from the

Richardson number using the method described by Thompson and Ley5 for conditions

above a cereal crop with Richardson number determined directly from weather mast

measurem ents. The droplet size spectrum from the nozzles was determined from data

collected from a Particle Me asuring Systems size analyser and a laboratory x-y samp ling

arrangement. l4

To simulate the drift from the boom arrangemen t of nozzles, the velocity of the liquid

sheet (and hence the initial velocity of the droplets) was estimated from d ata obtained

from high speed photography of similar nozzles operating at the same pressure and with

the same spray liquid.”

Fig 4

shows measured and simulated total line deposits at distances up to 6 m

downw ind for two different weather conditions. Simulated values were calculated using

500 droplets in 20 size categories with a 20 urn increment between categories. The effects

360 -

320-

280 -

3

f 240-

5

“

200-

C

I=

0 160-

120-

EO-

, , --*-*-.

----.

- --.-_. _

o I

I\ _ _,

_ _

I

0

2 4

6

DMance d ownw md, rn

F ig. 4. Measured and predicted downwind dri f t prof il es fr om a moving boom for two atmospheric

conditions. -,

measured in mean wind speed of 6*6m/s at a height of 1Om; - - - - - , predicted

in mean wind speed of 6*6m/s at a height of 10m; - I - i - 1-, measured i n mean wind speed of

1_9m/s at a height of 10m; - * - ’ -, predicted in mean wind speed of 1_9m/s at a height of 10m


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142


of entrained air were not included in this initial simulation and model predictions

significantly over-estimated drift as expected.

3.2. D r oplet size/velocity prof i l es below a nozzle

Meas urements of droplet size and velocity were m ade 0 .5 m below the central axis of a

flat fan nozzle (Lurmark 110015 in Kemetal) spraying water plus 0.1% Agral (ICI plc)

using a Particle M easuring Systems size analyser,14 and the results are plotted on F ig. 5.

Data are for six replicated scans of the spray produced by the nozzle. A lso plotted on F ig .

5

are the droplet size/velocity profiles calculated using the ballistic trajectory routine in

the model and on the following assum ptions:

(1) a constant entrained air velocity between nozzle and samp ling point equal to the

mean m easured velocity of droplets in the 40-80 pm size range, and

(2) entrained air velocities defined by Eqn (7), and with values for the constants as

given in Section 2.2.2 (i.e d2/2K 1 = O-57).

Velocities of the larger droplet size are more v ariable because of the relatively sm all

number of droplets of this size in the spray sample.

The results show that the velocities predicted using the trajectory routine in the mod el

were some 2-5 m/s above those m easured and indicate that the initial (or liquid sheet)

velocity used in the simulation at 17.0 m/s was too high. It was also expected that the

theoretical plot in

F ig. 5,

based on a constant entrained air velocity between nozzle and

measuring point, w ould under-predict droplet velocities, and the fact that this was not the

case except above 450 pm also indicated that the release velocity was less than that

assumed.

Droplet size pm

0

F ig. 5. Measured and predicted droplet velociti es below 110” nozzle. * , measured values; -,

predicted using trajectory r outine in model with entrained air velocity determined from Eqn (7,);

-----

,

predicted assuming a constant entr ained air veloci ty


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P c

H MILLER; D J HADFIELD

143

14-

12-

0

I I

I

I

0 100 200

300 400 5

D r op l e t sue , pm

F ig. 6. Measured and predicted droplet veloci ties below a 110” nozzle with dif ferent entr ained air

conditions. -, 62/.2KI in Eqn (7) = O-57; * , measured values; -.

- . -,

Sz/2K, in Eqn

(7) = O-95; - - - - , 6 ‘ /2K, in Eqn (7) = 1.25

350-

300-

Ti

g 250-

b

D 200-

E

L

z 150-

t

IOO-

50-

:

I I

I I 1

0.5 I.5 2.5 3.5

4.5 5.5 c

D i st a n c e d o w n w m d , m

5

Fig. 7. Measured and predicted downwind dri ft profi les. * , measured;

-, predicted as Fig. 4;

----

, predicted with entrained ai r, sheet velocity = 17.0 S2/2K, = 0.57; - . - . - , predicted with

entr ained air , sheet velocity = 15.0 6 2/2K, = 0.95


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145

value of 6’/2K1 close to unity gave an improved agreement with both m easured

dow nwind drift and size/velocity profiles below the nozzle. Orig inal values of the

entrained air parameters were derived from photographic measurem ents of the spray

section and were probably inaccurate due to the difficulty in identifying spray boundaries

on such photographs. Further work is needed to define the effects of entrained air

conditions for a range of nozzle types and sixes and in different parts of the spray

structure. It can be seen from the results in

Figs 5

and 6 that with an entrained air

parameter of 0.085 (i.e. S2/2K1 = O-95) the model underestimates the vertical velocities of

droplets below 150 pm in diameter and yet it can be seen from Figs 7 and 8 that

dow nwin d drift values are not overestim ated. The droplet size/velocity profiles were

measured in the laboratory at a relatively low horizontal scanning speed of O-05 m/s, and

this velocity difference may affect the trajectories of drifting drop lets. Further work needs

to examine the interaction between the spray sheet and surrounding air movem ents and

to improve the definition of entrained air effects.

4.2. Change fr om trajectory to random walk rout ine

This was examined by examining the airborne drift volumes at distances up to 6 m from

the nozzle when simulating drift with different integer increments of the Stokesian

response time, T, in the trajectory routine of the model, and the results are shown in Fig.

9.

With values of less than 4, drift values were lower indicating that droplets entered the

random walk routine of the model w ith downw ard velocities greater than the settling

velocity.

4.3. General discussion

The agreement between measured and predicted drift profiles from 3 boom mou nted

nozzles was relatively good particularly in view of some of the assumptions made relating

to nozzle movement. With entrained air, predicted downw ind profiles, show n in Fig. 7,

4oo

t

,/._..__...........

.

01

’ ’ ’ ’ ’ ’ ’ ’ ’

I 2 3 4

5 6 7 8 9 IO

Time in trqectory routine nT

F ig. 9. Effect of increasing time in trajectov routi ne on the total airborne dri ft up to 6m from a

boom with three 110” nozzles. -, dri f t at 6m; - . - * -, drif t at 3 m; - - - - , drif t at 2m;

. . . . .

9

dri ft at 1 m


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146


decayed more rapidly than those measured, and this is probably related to the method

used for incorporating a settling velocity compon ent in the “random-w alk” model.

Further work has now examined this feature of such models, e.g. Walklate,” and

incorporating an improved description of ‘heavy’ particle trajectories downw ind should

improve model predictions.

Work is being conducted to compare the model predictions with measurem ents of the

drift from single static and boom mounted nozzles to eliminate the effects of movement

and any nozzle interactions.

The predicted drift profiles differed from those measured directly above the

crop/ground surface indicating that the assum ptions made regarding droplet retention and

reflection were too simplified.

All the calculations assum ed that droplets behaved as spherical particles. This

assum ption was probably valid for droplets up to 350 urn in diameter, but it can be seen

from the results in

Figs 5

and 6 that the velocities of larger droplets are underestimated

because of the effects of shape distortion.

5. Conclusions and proposals for future work

The model provides a useful description of the spray drift from agricultural hydraulic

nozzles, with reasonable agreement between m easured and predicted downw ind drift

profiles.

Further work is required to extend the range of conditions for which the model h as

been validated. Work is also required to improve the description of entrained air effects,

to include the effects of forward speed including the generation of turbulent wakes from

the boom and spray vehicle,

and to improve the calculation of “heavy” particle

trajectories.

Acknowledgements

Thanks are d ue to Mr A. P. Chick and Mr J. E. Pottage for assistance with the field experiments

and to Mr C. R. Tuck for assistance with the laboratory measurements of droplet size/velocity

profiles.

References

’ Beche, D. H.; Sayer, W. J . D. Transp ort of aerial spray, I. A model of aerial dispersion.

Agricultural Meteorology 1975, 15: 257-271

Bathe, D. H.; Sayer, W. J . D.

Transpo rt of aerial spray. II. Transport within the crop canop y.

Agricultural Meteorology 1975, 15: 371-377

a

Dumbauld, R. K.; Rafferty, J . E.; Bjorkland, J . R.

Prediction of spray behaviour above and

within a forest canopy. U SDA Forest Service Report, 1977, TR-77-308-01

4

Schaefer, G. W.; Allsop, K.

Spray droplet behaviour above an d within a crop. Proceedings 10th

International Congress of Plant Protection 1983, 3: 1057-1065

5 Thompson, N.

J .; Ley, A J .

Estimating spray drift using a random -walk mod el of evaporating

drops. Journal of Agricultural Engineering Research 1983, 28: 418-435

’ Picot, J . J . C.; Kristmanson, D. D.; Basak-Brown, N. Canopy deposit and off-target drift in

forestry aerial spraying. The effects of operational parameters. Transactions of the ASAE

1986, 29: 80

’

Wilson, J . D.; llmtell, G. W.; Kidd, G. E.

Num erical sim ulation of particle trajectories in

inhomogeneous turbulence. IV: Com parison of predictions with experimental data for the

atmospheric surface layer. Boundary Layer Meteorology 1981, 21: 443-463.

Legg,

B. J .; Ranpach, M. R.

Markov chain simulation of particle dispersion in inhomogeneous


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47

flows. The mean drift velocity induced by a gradient in Eulerian velocity variance. Boundary

Layer Meteorology 1982,24: 3-13

g

Smith, F. B. The

turbulent sprea d of a falling cluster. In: Advances in Geoph ysics 6,

Atmospheric Diffusion and Air Pollution. London: Academic Press 1959, pp. 193-210

lo Mar ant, 1. A. Calculation of spray droplet trajectory in a moving airstream . Res earch Note .

Journal of Agricultural Engineering Rese arch 1977 , 22: 93-96

”

Lake, J . R.

Sprays, physical properties and deposition on cereals and weeks. P hD Thesis,

University o f London, 1982

l2

Briffa, F. E. J .; Dombrowski, N.

Entrainment of air into a liquid spray. Am erican Institute of

Chemical Engineers Journal 1966, l2(4): 708-717

” Sharp, R. B. Comparison of drift from charged and uncharged hydraulic nozzles. British Crop

Protection Council, Proceedings of Conference on Pests and Diseases, 1984, pp. 1027-1031

”

Lake, J . R.; Dix, A. J .

Measurements of droplet size with a PM S optical array probe using an

x-y nozzle transporter. Crop Protection 1985, 4(4): 464-472

”

Walklate, P. J .

A Mark ov-chain particle dispersion mode l based on airflow data: Extension to

large water droplets. Boundary Layer Meteorology 1986, 37: 313-318

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