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The effect of the liquid cone form on spray distribution of hollow cone nozzles
322
The Effect of the Liquid Cone Form on Spray Distribution of Hollow Cone Nozzles A. NORDBY*; J. HAMAN~ The form of droplet cone produced by the swirl nozzle and the spacing between nozzles have a strong influence on the liquid distribution over the field. The distribution characteristics obtained experimentally were compared with the theoretically calculated forms. It is suggested that a solid cone nozzle gives better results of spraying than hollow cone nozzles. Coverage deteriorates with decreasing cone skin thickness.
1.
Introduction
Field sprayers are very often equipped with swirl nozzles producing droplets in a stream forming a solid or hollow cone. In extreme cases, at small boom heights, there will be no overlapping of jets and hence unsprayed areas between adjacent nozzle patterns. The distance between nozzles on the boom is, in general, fixed. For economic reasons, the number of nozzles should be as small as possible. On the other hand all surfaces should be sprayed without excessive deposit variation. 2.
Theory
The analysis will refer only to swirl nozzles giving a cone of circular pattern (Fig. 1, left). There is no difference between solid and hollow cones because when the cone filling coeffi&nt Ezz-
R r
..
E = 1 when the thickness of the outer layer of the cone (i.e. R - r) is infinitely small E = cc when the cone is filled. The angle between the stream axis and sprayed area may not be a right angle so that the pattern of the sprayed area will have an elliptical form (Appendix I), but this case is very rare. On the majority of conventional booms the nozzles are mounted normal to the sprayed surface, therefore the theory and experiments given in this paper relate to the latter case. Assume that the delivery of liquid per unit area of the plane normal to the spray cone axis is
Fig. 1.
Pattern
of a single cone nozzle
constant and decreases with the square of boom height, and that there are no droplets in the hollow space within the cone. This last assumption is not quite justified because small droplets are falling very slowly inside the hollow section of the cone. The influence of these droplets on the pattern which is composed from bigger droplets descending with higher speed, is negligible. In view of the symmetry of the spray pattern only $ of the circle will be considered (Fig. I, right). By dividing the sprayed area into infinitely narrow strips dx, it can be shown that the amount of liquid falling on the strip area is proportional to Y = Y2 - Yl thus since v9.= 2/R2 - x2 and v, = z/r2 - x2 -;
=
j,2
_
x2
_
;e
. . . (2)
when x S r and when r < x < R because v, = 0 .I
Y = 2/R2 - x2
.
.
.
(3)
therefore the value of y increases when x increases from 0 to r, when x = r is a discrete point and when x > r the value of y decreases to 0 when x = R. Fig. 2 shows the values of Yfor several VdUeS Of
E.
I
0
2
I
--
R
x
3
5
4 E
Fi~q. 3.
Swelling
coefficient
plotted
~‘er.w.~ E
_
Y
I,
5R
’i,
324
EFFECT
OF
LIQUIII
(‘ONE
FORM
ON
SPKAI
I~1S’l‘KIHIJ-~IC)Ic
For estimating the pattern form we shall use the “swelling coefficient” .r
rmax
s=r,, where Y,,, and Ymb are maximum and minimum values of Yin the given pattern. Sometimes the difference between wanted chemicals deposit and scorch is small, therefore we have chosen this method of estimating. Fig. I shows that Y,,, corresponds to x = r. Assuming that positions of adjacent nozzles are so close that the value of Y for x < r is higher than for x > r, we can define Yminas corresponding to x = 0. Therefore Ymax= 1/R2 - r2 because y1 = 0 Ymin= R - r SE
d
R+r R-r
. . .(4)
and since from Eqn (1) r = -p -Efl .. E-l this coefficient decreases when E increases. Fig. 3 shows that the quality of work of a single cone s=
II
R
For thicker cones (E ;-. 2.5) the influence of boom height or nozzle spacing is not as great. The evenness of distribution can be sufficient at different boom heights.
nozzle improves when the ratio 7 increases. All above considerations refer to work with a single nozzle only. With more nozzles on the boom, the most common case is when two adjacent streams overlap; this case will be the subject of the further considerations. The highest delivery at one point will be when Y,,, of adjacent streams coincide and the distance between nozzles is 2r. This concentration of liquid is often very critical for the crops, in particular when the thickness of the outside film of spray cone is small and the value of S very high. In Fig. 4 the value of S for superposition of two adjacent nozzles is plotted versus distance of nozzle axes as multiples of the cone pattern radius R. It can be seen that the lowest value of S appears when the streams overlap only at the edges. This position is very risky because a small modification of boom height causes either a rapidly increasing deposit or gives unsprayed areas.
3. 3.1.
Experiments Methods
The work was carried out at the Landbruksteknisk Institutt, Vollebekk, using a collector for spray distribution measurements (Fig. .5), developed by A. Nordby at that Institute. The edges on the top of the walls divide the collector into 50 mm wide channels. The knife edges and side Y
h CD
II
A
I
Fig. 6.
47
E
x,+1
Liquid quantity
I
x
R
deposited
into a channel
A.
hlORI)BY;
J.
HAMAN
ra c=l.6
c*l
t
c=3.5
t=i?
aTI
bll
c
Fig. 7.
. Yfor different E Relative distribution of Ii&c Arrows mark stream axes
walls are made from galvanised iron and painted with easily wetted acrylic paint. The end walls are Perspex and the frame construction of angle brass. The screen at the bottom of each measuring section lets some of the air through. In this way some of the drift over the table can be avoided. All the measuring cylinders can be tilted into a horizontal position for measuring and then emptied. This measuring method gives a discontinuous value of delivery in each 50 mm wide channel. Theoretical calculation of the liquid quantity deposited into a separate channel with walls at points xn and x~+~ (Fig. 6) for a solid cone can be given by equation . ..(6) and
Xn,1 -_ r 1
--
xn d(r2 - .Y,~) + r2 arc sin xn [
r
I)...
(7) The quantity Qn is proportional to the area ABCD in Fig. 6. For a hollow cone the quantity of liquid can be calculated by subtracting the
liquid adequate for a hollow cone of radius R r= -. Fii. 7 shows the relative quantity of liquid for hollow cone nozzles with constant radius R and variable coefficient E calculated for 14 channels. Only half of the pattern is illustrated because the other one is symmetrical. Comparing the diagrams obtained by measurements with Fig. 7 or Eqn (7), the approximate value of E and S can be defined from Fig. 4. 3.2.
Measurements
The measurements were made with the following nozzles : (1) Hardi 1553 No. 24, pressure 10 kg/cm2 (2) Hardi 1553 No. 24, pressure 5 kg/cm2 (3) Hardi 1553 No. 16, pressure 5 kg/cm2 (4) Birchmeier 1.5 mm dia core with 1 convolution, pressure 5 kg/cm2 (5) Birchmeier 1.O mm dia core with I convolution, pressure 5 kg/cm2. Fig. 8 shows the distribution of liquid in different channels for given nozzles. The measurements of stream superpositions were made by varying the distances between nozzles
326
l!F-FE<‘7
Nozzle
OF
LIQUID
CONI:
I-OKM
ON
Nozzle
I
I
SPRAY
2
r
-l
Nozzle
3
Fig. 8.
?OO
I)ISTRIHI;-II03
A Nozzle
4
Liquid distribution fbr different nozzles Arrows mark stream axes
mounted on the boom 50 cm above the collector. The results of the measurements in Fig. 9 agree with the theoretical values in Fig. 4. For example, Nozzle 5 with an almost solid cone gives the same evenness of distribution for different nozzle spacings. On the other hand, Nozzle 2 with a very thin outer film (Z z 1.4) is very sensitive to nozzle spacing. At increasing pressure (Nozzle l), the small droplets are filling out the cone, leading to a smaller S value. The effect of collisions between droplets from adjacent streams is negligible. Since the ratio 9 8 7
between liquid volume and cone volume is very small, (about 00IOl %), the collision of droplets is an improbable “accident”. 4.
Conclusions
The best results of spraying with cone nozzles are obtained with a solid cone. When E is less than 2.5 it is difficult to get a good spray pattern from a boom with hollow cone nozzles. In order to get a sufficient distribution, the nozzles should be spaced very close on the boom, but from a practical point of view this is very inconvenient. The approximate value of E can be estimated from Fig. 7 and the value of S from Fig. 4. In this way the boom height and nozzle spacing can be worked out for a given coverage.
6 5 s4 3 2 I C
Fig. Y.
I 1.5R
SweNing coefficient plotted
2R
versus nozzle spacing
APPENDIX I Calculation of elliptical spray pattern When the spray cone axis is not perpendicular to the sprayed surface an elliptical spray pattern is obtained. Analytical calculation of the quantity of liquid on the sprayed surface is very intricate and may be solved only by using elliptical integrals. For simplification of that solution therefore a graphical method will be shown.
Fig. IO.
Graphic methodfor estimating the spray distribution from a nozzle with elliptical pattern
E.g. 10 shows a circular cone with vortex at 0. The cone is cut by an inclined (sprayed) plane PP.
The elliptical pattern is shown in the lower part of the illustration. Lines 1, 2. represent the walls of the channels of the measuring apparatus described. The method of calculation of delivery is demonstrated for Channel 9. Point A is projected on the plane PP and gives point B. A line is drawn through points 0 and B which gives point C on diameter LL. Projecting point C on the arc of circle with diameter LL we get point D. The same operation must be repeated for point F to get point G on the circle. The length of arc DC gives the quantity of liquid in channel 9 for E = I when the circumference of the circle is regarded as 100. When E ;c the same operation is carried out for points K and M. Thus we get points Q and R on the circle. The area DGOR gives the quantity of liquid in the channel in “,, when the full area of circle LL is taken as 100. When I <. s -: rc we can calculate the liquid cone for a hollow cone nozzle by subtracting the area of the hollow cone pattern from that of a solid cone pattern. This calculation can be made in the same way as mentioned before.
The Effect of the Liquid Cone Form on Spray Distribution of Hollow Cone Nozzles A. NORDBY*; J. HAMAN~ The form of droplet cone produced by the swirl nozzle and the spacing between nozzles have a strong influence on the liquid distribution over the field. The distribution characteristics obtained experimentally were compared with the theoretically calculated forms. It is suggested that a solid cone nozzle gives better results of spraying than hollow cone nozzles. Coverage deteriorates with decreasing cone skin thickness.
1.
Introduction
Field sprayers are very often equipped with swirl nozzles producing droplets in a stream forming a solid or hollow cone. In extreme cases, at small boom heights, there will be no overlapping of jets and hence unsprayed areas between adjacent nozzle patterns. The distance between nozzles on the boom is, in general, fixed. For economic reasons, the number of nozzles should be as small as possible. On the other hand all surfaces should be sprayed without excessive deposit variation. 2.
Theory
The analysis will refer only to swirl nozzles giving a cone of circular pattern (Fig. 1, left). There is no difference between solid and hollow cones because when the cone filling coeffi&nt Ezz-
R r
..
E = 1 when the thickness of the outer layer of the cone (i.e. R - r) is infinitely small E = cc when the cone is filled. The angle between the stream axis and sprayed area may not be a right angle so that the pattern of the sprayed area will have an elliptical form (Appendix I), but this case is very rare. On the majority of conventional booms the nozzles are mounted normal to the sprayed surface, therefore the theory and experiments given in this paper relate to the latter case. Assume that the delivery of liquid per unit area of the plane normal to the spray cone axis is
Fig. 1.
Pattern
of a single cone nozzle
constant and decreases with the square of boom height, and that there are no droplets in the hollow space within the cone. This last assumption is not quite justified because small droplets are falling very slowly inside the hollow section of the cone. The influence of these droplets on the pattern which is composed from bigger droplets descending with higher speed, is negligible. In view of the symmetry of the spray pattern only $ of the circle will be considered (Fig. I, right). By dividing the sprayed area into infinitely narrow strips dx, it can be shown that the amount of liquid falling on the strip area is proportional to Y = Y2 - Yl thus since v9.= 2/R2 - x2 and v, = z/r2 - x2 -;
=
j,2
_
x2
_
;e
. . . (2)
when x S r and when r < x < R because v, = 0 .I
Y = 2/R2 - x2
.
.
.
(3)
therefore the value of y increases when x increases from 0 to r, when x = r is a discrete point and when x > r the value of y decreases to 0 when x = R. Fig. 2 shows the values of Yfor several VdUeS Of
E.
I
0
2
I
--
R
x
3
5
4 E
Fi~q. 3.
Swelling
coefficient
plotted
~‘er.w.~ E
_
Y
I,
5R
’i,
324
EFFECT
OF
LIQUIII
(‘ONE
FORM
ON
SPKAI
I~1S’l‘KIHIJ-~IC)Ic
For estimating the pattern form we shall use the “swelling coefficient” .r
rmax
s=r,, where Y,,, and Ymb are maximum and minimum values of Yin the given pattern. Sometimes the difference between wanted chemicals deposit and scorch is small, therefore we have chosen this method of estimating. Fig. I shows that Y,,, corresponds to x = r. Assuming that positions of adjacent nozzles are so close that the value of Y for x < r is higher than for x > r, we can define Yminas corresponding to x = 0. Therefore Ymax= 1/R2 - r2 because y1 = 0 Ymin= R - r SE
d
R+r R-r
. . .(4)
and since from Eqn (1) r = -p -Efl .. E-l this coefficient decreases when E increases. Fig. 3 shows that the quality of work of a single cone s=
II
R
For thicker cones (E ;-. 2.5) the influence of boom height or nozzle spacing is not as great. The evenness of distribution can be sufficient at different boom heights.
nozzle improves when the ratio 7 increases. All above considerations refer to work with a single nozzle only. With more nozzles on the boom, the most common case is when two adjacent streams overlap; this case will be the subject of the further considerations. The highest delivery at one point will be when Y,,, of adjacent streams coincide and the distance between nozzles is 2r. This concentration of liquid is often very critical for the crops, in particular when the thickness of the outside film of spray cone is small and the value of S very high. In Fig. 4 the value of S for superposition of two adjacent nozzles is plotted versus distance of nozzle axes as multiples of the cone pattern radius R. It can be seen that the lowest value of S appears when the streams overlap only at the edges. This position is very risky because a small modification of boom height causes either a rapidly increasing deposit or gives unsprayed areas.
3. 3.1.
Experiments Methods
The work was carried out at the Landbruksteknisk Institutt, Vollebekk, using a collector for spray distribution measurements (Fig. .5), developed by A. Nordby at that Institute. The edges on the top of the walls divide the collector into 50 mm wide channels. The knife edges and side Y
h CD
II
A
I
Fig. 6.
47
E
x,+1
Liquid quantity
I
x
R
deposited
into a channel
A.
hlORI)BY;
J.
HAMAN
ra c=l.6
c*l
t
c=3.5
t=i?
aTI
bll
c
Fig. 7.
. Yfor different E Relative distribution of Ii&c Arrows mark stream axes
walls are made from galvanised iron and painted with easily wetted acrylic paint. The end walls are Perspex and the frame construction of angle brass. The screen at the bottom of each measuring section lets some of the air through. In this way some of the drift over the table can be avoided. All the measuring cylinders can be tilted into a horizontal position for measuring and then emptied. This measuring method gives a discontinuous value of delivery in each 50 mm wide channel. Theoretical calculation of the liquid quantity deposited into a separate channel with walls at points xn and x~+~ (Fig. 6) for a solid cone can be given by equation . ..(6) and
Xn,1 -_ r 1
--
xn d(r2 - .Y,~) + r2 arc sin xn [
r
I)...
(7) The quantity Qn is proportional to the area ABCD in Fig. 6. For a hollow cone the quantity of liquid can be calculated by subtracting the
liquid adequate for a hollow cone of radius R r= -. Fii. 7 shows the relative quantity of liquid for hollow cone nozzles with constant radius R and variable coefficient E calculated for 14 channels. Only half of the pattern is illustrated because the other one is symmetrical. Comparing the diagrams obtained by measurements with Fig. 7 or Eqn (7), the approximate value of E and S can be defined from Fig. 4. 3.2.
Measurements
The measurements were made with the following nozzles : (1) Hardi 1553 No. 24, pressure 10 kg/cm2 (2) Hardi 1553 No. 24, pressure 5 kg/cm2 (3) Hardi 1553 No. 16, pressure 5 kg/cm2 (4) Birchmeier 1.5 mm dia core with 1 convolution, pressure 5 kg/cm2 (5) Birchmeier 1.O mm dia core with I convolution, pressure 5 kg/cm2. Fig. 8 shows the distribution of liquid in different channels for given nozzles. The measurements of stream superpositions were made by varying the distances between nozzles
326
l!F-FE<‘7
Nozzle
OF
LIQUID
CONI:
I-OKM
ON
Nozzle
I
I
SPRAY
2
r
-l
Nozzle
3
Fig. 8.
?OO
I)ISTRIHI;-II03
A Nozzle
4
Liquid distribution fbr different nozzles Arrows mark stream axes
mounted on the boom 50 cm above the collector. The results of the measurements in Fig. 9 agree with the theoretical values in Fig. 4. For example, Nozzle 5 with an almost solid cone gives the same evenness of distribution for different nozzle spacings. On the other hand, Nozzle 2 with a very thin outer film (Z z 1.4) is very sensitive to nozzle spacing. At increasing pressure (Nozzle l), the small droplets are filling out the cone, leading to a smaller S value. The effect of collisions between droplets from adjacent streams is negligible. Since the ratio 9 8 7
between liquid volume and cone volume is very small, (about 00IOl %), the collision of droplets is an improbable “accident”. 4.
Conclusions
The best results of spraying with cone nozzles are obtained with a solid cone. When E is less than 2.5 it is difficult to get a good spray pattern from a boom with hollow cone nozzles. In order to get a sufficient distribution, the nozzles should be spaced very close on the boom, but from a practical point of view this is very inconvenient. The approximate value of E can be estimated from Fig. 7 and the value of S from Fig. 4. In this way the boom height and nozzle spacing can be worked out for a given coverage.
6 5 s4 3 2 I C
Fig. Y.
I 1.5R
SweNing coefficient plotted
2R
versus nozzle spacing
APPENDIX I Calculation of elliptical spray pattern When the spray cone axis is not perpendicular to the sprayed surface an elliptical spray pattern is obtained. Analytical calculation of the quantity of liquid on the sprayed surface is very intricate and may be solved only by using elliptical integrals. For simplification of that solution therefore a graphical method will be shown.
Fig. IO.
Graphic methodfor estimating the spray distribution from a nozzle with elliptical pattern
E.g. 10 shows a circular cone with vortex at 0. The cone is cut by an inclined (sprayed) plane PP.
The elliptical pattern is shown in the lower part of the illustration. Lines 1, 2. represent the walls of the channels of the measuring apparatus described. The method of calculation of delivery is demonstrated for Channel 9. Point A is projected on the plane PP and gives point B. A line is drawn through points 0 and B which gives point C on diameter LL. Projecting point C on the arc of circle with diameter LL we get point D. The same operation must be repeated for point F to get point G on the circle. The length of arc DC gives the quantity of liquid in channel 9 for E = I when the circumference of the circle is regarded as 100. When E ;c the same operation is carried out for points K and M. Thus we get points Q and R on the circle. The area DGOR gives the quantity of liquid in the channel in “,, when the full area of circle LL is taken as 100. When I <. s -: rc we can calculate the liquid cone for a hollow cone nozzle by subtracting the area of the hollow cone pattern from that of a solid cone pattern. This calculation can be made in the same way as mentioned before.