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J. ngrk. Engng Res. 1977) 22, 93-96

RESEARCH NOTES

Calculation of Spray Droplet Trajectoryin a Moving irstream

J. A. MARCHANT*:

1. Introduction

Spray dro plets can be deflected intentionally, for exam ple by a ducted fan designed to changetheir direction, or by natural forces , i.e. wind, which is usually unintentional and undesirable.In both case s, som e means of calculating the deflection is useful, to design a suitable deflectionsystem or to assess the extent of the undesirable effects.

2. Theory

2.1. Aerodynamics qfspray dropletsChanges in the speed or direction of a spray droplet are brought about by aerodynamic and

gravitational forces acting on it. If the droplet presents a symmetrical aspect to the air stream andis not rotating in relation to it, then there is no aerodynam ic lift force presen t and only the dragforce, along with any gravitational force , need be considered .

Berry has summarized the work of various authors and has shown that the drag coefficient ofa droplet is substantially the sam e as that for a solid sphere for low Reynolds numbers. Thediscrepancy between the coefficients is zero for R,, ~= 0 and about IO?/

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94 SPR Y DROPLET TR JECTORY

LIST OF SYMBOLS

area of droplet presented to airstream displacement in x-directionacceleration of droplet Y displacement in y-directiondrag coefficient n inclincation of V, to positive x-directiondiameter of droplet inclination of V ,,, to positive x-directionaerodynamic drag force /yl inclination of V to positive x-directionacceleration due to gravity kinematic viscosity of airmass of droplet p density of airReynolds numbertime Subscriptsvelocity of droplet 0 initial valuevelocity of air stream x,y compon ent in x or y direction

V,, velocity of air stream relative to droplet

The velocity of the air stream relative to the droplet determines the drag force. Its magnitude,[ Vs, 1, is given by

1 VS,j = [ V, cos a- VJ2+ V, sin a- V,v)2]+

and its inclination to the positive x-direction by

. .. I)

cos y =V, cos a- V,

I K I

sin y =V, sin a-V,

IKPI . .. 3)

where

v = vcose, . .. 4)

V, = V sin 0. . .. 5)

The dra g fo rce i; in +he direction of Vs, and its magnitude is a function of the drag c oefficient,i.e.

Fd = = C,pAV2,, . ..(6)

where A is the area presented to the air stream or

A = anD 2. . ..(7)

The drag coefficient is given as a function of the Reynolds number in standard aerodynam ictables,* where

Applying New tons Second Law of Motion in the x- and y-directions

Fd cos y = ma,, . ..(9)

Fd sin y-mg = ma,. . .. lO)

The velocities in the x- and y-directions can be obtained by integrating Eqns 9) and 10).

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.I. . MARCHANT 95

The given initial conditions are the compon ents of the initial velocity, so

I, = I, cos 0, f a,dt,0

. .. Il)

f

V, = V, sin B,,+ a,,dt0

and the positions can be obtained by a further integration,

. .. 12)

f

X= VA, . .. 13)0

i f

J= dt.0

.. 14)

No initial conditions need be included in Eqns 13) and 14) as it is assum ed that the droplet starts

from the origin.3. Examples

The following exam ples have been calculated using a com puter progr am written inFORTRANfor an I.C.L. 4-70 com puter. The integrations wer e carried o ut using the Runge-Ku tta algorithm3which integrates in a step-by-step fashion. The time step for such a numerical integration pro-cedure must be chosen within the framew ork of two conflicting requirements. It must besmall enough to preser ve accurac y and to prevent n umerical instability occurring yetnot too small so as to give unacceptably large solution times. Although rules are available3 giving

20 -

0 -~ --

FE 2o

E

EL? -4O-

;5\

14

-80 -/

020

-100 - /

-,;o +q , I I I I I-100 -80 -60 -40 -20 0 20 411 60 80 100

x Dlsplocem ent (mm)

F ig. 2. D ropl et trajectori es. Nu mbers at poin ts shown thu s: ndi cate ri me in ms. Cur ve nu mbers corr espond t o

example numbers i n Table I

the stability bounds for the numerical integration of sets of linear equations, they do not apply inthis non-linear case and so a trial and error method was used. A step size was chosen 0.002 s)and the calculations ma de. This was then halved and the calculations repea ted. As the solutiontime was reasonable and the two sets of results differed negligibly, it was decided that the step sizewas acceptable.

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96 SPR Y DROPLET TR JLCTORY

Table 1 summ arizes the initial conditions, droplet sizes and air speed s andFig. 2 shows thetrajectories of the droplets.

T BLE I

ummary of example run conditions

Exampleno.

D, w K, mls 1, egr ees V, m/s O,, degr ees

1 200 10 -~ 0 2.5 02 300 10 -90 5 0

3 300 I .5 180 7.5 454 300 15 180 7.5 -135

5 500 0 0 I 60

REFERENCES

Berry, E. X. Equations for calculating the termi nal velociti es of water dr ops. J. appl. Meteorol., 197413 2) 108

* Streeter, V. L. F lui d M echanics. New York: McGraw-Hill, 19623 Hamming, R. W. Nu meri cal M ethods for Scienti sts and En gin eer s. New York : McGraw-Hill, 1962