J. ngrk.
Engng Res. ( 1977) 22, 93-96
RESEARCH
Calculation
of Spray
in a Moving
NOTES
Droplet Trajectory
Airstream
J. A. MARCHANT*:
1.
Introduction
Spray droplets can be deflected intentionally,
for example by a ducted fan designed to change
their direction, or by natural forces, i.e. wind, which is usually unintentional
and undesirable.
In both cases, some means of calculating the deflection is useful, to design a suitable deflection
system or to assess the extent of the undesirable
effects.
2.
Theory
2.1. Aerodynamics qf’spray droplets
Changes 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 and
is not rotating in relation to it, then there is no aerodynamic
lift force present and only the drag
force, along with any gravitational
force, need be considered.
Berry’ has summarized
the work of various authors and has shown that the drag coefficient of
a droplet is substantially
the same as that for a solid sphere for low Reynolds numbers.
The
discrepancy between the coefficients is zero for R,, ~=0 and about IO?/<)for R, 1 1000. increasing
rapidly after this figure.
For a droplet diameter of 500 urn, a Reynolds number of 1000 represents a velocity of about
30 m/s at the usual ambient temperatures.
Consequently
most crop spraying applications
are
well within this range, and drag coefficients for solid spheres can be used.
2.2. Equations of’motion
on the droplet. It is assumed that the forces are in a vertical
Fig. I (right) is a velocity diagram for
plane and hence the gravitational
force has been included.
the air and droplet velocities and the relative velocity.
Fig. I (left) shows the forces acting
Fig. I. Left, axis convention
‘National
Received
Institute
3 May
of Agricultural
1976;
accepted
Engineering,
in revised
form
and forces
actitrg on droplet.
Silsoe
7 July
1976
93
Right,
velocity diagram
94
SPRAY
DROPLET
TRAJECTORY
LIST OF SYMBOLS
area of droplet presented to airstream
acceleration of droplet
drag coefficient
diameter of droplet
aerodynamic drag force
acceleration due to gravity
mass of droplet
Reynolds number
time
velocity of droplet
velocity of air stream
V,, velocity of air stream relative to droplet
The velocity of the air stream relative
[ Vs, 1, is given by
1 VS,j =
and its inclination
to the positive
displacement in x-direction
displacement in y-direction
n
inclincation of V, to positive x-direction
inclination of V,,, to positive x-direction
/yl inclination of V to positive x-direction
” kinematic viscosity of air
p
density of air
X
Y
Subscripts
0
initial value
x,y component in x or y direction
to the droplet
[(V, cos a-
x-direction
cos y =
determines
VJ2+(
V, sin a-
the drag force.
V,v)2]+
Its magnitude,
. ..(I)
by
V, cos a-
V,
I K,I
’
V, sin a-V,
sin y =
IKPI
. ..(3)
’
where
The drag force i; in +he direction
i.e.
v, = vcose,
. ..(4)
V, = V sin 0.
. ..(5)
of Vs, and its magnitude
is a function
of the drag coefficient,
Fd = =$C,pAV2,,
where A is the area presented
to the air stream
. ..(6)
or
A = anD2.
The drag coefficient
tables,* where
Applying
Newton’s
is given as a function
Second
Law of Motion
of the Reynolds
. ..(7)
number
in standard
in the x- and y-directions
Fd cos y = ma,,
Fd sin y-mg
The velocities
in the x- and y-directions
aerodynamic
. ..(9)
= ma,.
can be obtained
. ..(lO)
by integrating
Eqns
(9) and (10).
95
.I.A. MARCHANT
The given initial conditions
are the components
of the initial velocity,
I’, = I’, cos 0, f
so
a,dt,
. ..(Il)
a,,dt
. ..(12)
0
‘f
V, sin B,,+
V, =
0
and the positions
can be obtained
by a further
integration,
f
X=
. ..(13)
VA,
0
i’ f
J
J=
V,dt.
..(14)
0
No initial conditions
from the origin.
need be included
in Eqns (13) and (14) as it is assumed
that the droplet starts
3. Examples
The following examples have been calculated using a computer program written in FORTRAN
for an I.C.L. 4-70 computer.
The integrations
were carried out using the Runge-Kutta
algorithm3
which integrates in a step-by-step fashion.
The time step for such a numerical integration
procedure must be chosen within the framework
of two conflicting
requirements.
It must be
small enough
to preserve
accuracy
and to prevent
numerical
instability
occurring
yet
not too small so as to give unacceptably
large solution times. Although rules are available3 giving
20 -
-~
0
--
F
E -2oE
E -4O-
L?
%
;5”
\
14
-80
-
-100
-
-,;o
??
/
/
+q
-100
0’
20
/
-80
-60
,
I
I
I
-40
-20
0
20
x
Fig. 2.
Droplet
trajectories.
the stability bounds
this non-linear case
and the calculations
time was reasonable
was acceptable.
Numbers
Dlsplocement
411
I
I
60
80
100
(mm)
at points shown thus: ??indicate rime in ms.
example numbers in Table I
Curve numbers correspond
for the numerical integration
of sets of linear equations, they do not
and so a trial and error method was used. A step size was chosen
made. This was then halved and the calculations
repeated. As the
and the two sets of results differed negligibly, it was decided that the
to
apply in
(0.002 s)
solution
step size
SPRAY
96
Table 1 summarizes
the initial
trajectories of the droplets.
conditions,
droplet
DROPLET
TRAJLCTORY
sizes and air speeds and Fig. 2 shows the
TABLE I
Summary of example run conditions
Example
no.
1
2
3
4
5
D, w
200
300
300
300
500
K, mls
(1,degrees
10
10
I.5
15
0
-~90
-90
180
180
0
V”, m/s
2.5
5
7.5
7.5
I
O,,degrees
0
0
45
-135
60
REFERENCES
Berry, E. X. Equations for calculating the terminal velocities of water drops. J. appl. Meteorol., 1974
13 (2) 108
* Streeter, V. L. Fluid Mechanics. New York: McGraw-Hill, 1962
3 Hamming, R. W. Numerical Methods for Scientists and Engineers. New York : McGraw-Hill, 1962
’